research article some aspects of sensitivity analysis in...

Research ArticleSome Aspects of Sensitivity Analysis in Variational DataAssimilation for Coupled Dynamical Systems

Sergei Soldatenko1 Peter Steinle1 Chris Tingwell1 and Denis Chichkine2

1Centre for Australian Climate and Weather Research 700 Collins Street Melbourne VIC 3008 Australia2University of Waterloo 200 University Avenue West Waterloo ON Canada N2L 3G1

Correspondence should be addressed to Sergei Soldatenko ssoldatenkobomgovau

Received 18 December 2014 Revised 24 February 2015 Accepted 11 March 2015

Academic Editor Guijun Han

Copyright copy 2015 Sergei Soldatenko et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Variational data assimilation (VDA) remains one of the key issues arising in many fields of geosciences including the numericalweather prediction While the theory of VDA is well established there are a number of issues with practical implementationthat require additional consideration and study However the exploration of VDA requires considerable computational resourcesFor simple enough low-order models the computational cost is minor and therefore models of this class are used as simple testinstruments to emulate more complex systems In this paper the sensitivity with respect to variations in the parameters of one ofthe main components of VDA the nonlinear forecasting model is considered For chaotic atmospheric dynamics conventionalmethods of sensitivity analysis provide uninformative results since the envelopes of sensitivity functions grow with time andsensitivity functions themselves demonstrate the oscillating behaviour The use of sensitivity analysis method developed on thebasis of the theory of shadowing pseudoorbits in dynamical systems allows us to calculate sensitivity functions correctly Sensitivityestimates for a simple coupled dynamical system are calculated and presented in the paper To estimate the influence of modelparameter uncertainties on the forecast the relative error in the energy norm is applied

1 Introduction

The earth system consists of several interactive dynamicalsubsystems and each of them covers a broad temporal andspatial spectrum of motions and physical processes Thecomponents of the earth system have many differences intheir physical properties structure and behavior but arelinked together by fluxes of momentum and mass as well assensible and latent heat All of these subsystems interact witheach other in different ways and can be strongly or weaklycoupled Prediction of future state of the earth system and itscomponents is one of the most important problems of mod-ern science The most significant progress has been achievedin the forecasting of the atmosphere via numerical modelswhich describe the dynamical and physical processes in theearthrsquos gaseous envelope It is clear that further improvementof forecasts can be pursued via the development of coupledmodeling systems that primarily combine the atmosphere

and the ocean and describe the interactions between thesetwo systems Since numerical weather prediction systemscalculate a future state of the atmosphere and ocean byintegrating a set of partial differential equations that describethe fluid dynamics and thermodynamics initial conditionsthat accurately represent the state of the atmosphere andocean at a certain initial time must be formulated Numericalweather prediction systems use data assimilation proceduresto estimate initial conditions for forecasting models fromobservations Data assimilation remains one of the key issuesnot only in the numerical weather prediction (NWP) but alsoin other geophysical sciences

One of the most advanced and effective data assimilationtechniques is four-dimensional variational data assimilation(4D-Var) In particular the weather forecasts produced by theACCESS (Australian Community Climate and Earth SystemSimulator) at the Bureau of Meteorology use 4D-Var in theincremental formulation developed at the Met Office [1 2]

Hindawi Publishing CorporationAdvances in MeteorologyVolume 2015 Article ID 753031 22 pageshttpdxdoiorg1011552015753031

2 Advances in Meteorology

A comprehensive historical review and current status of 4D-Var are presented for example in [3ndash7] In our view pointsome papers and books such as [8ndash17] have contributedsignificantly to the development of mathematical foundationtheory and practice of 4D-Var

In general the main objective of 4D-Var is to defineas perfectly as possible the state of a dynamical system bycombining in statistically optimal manner the observationsof state variables of a real physical system together withcertain prior information In NWP this prior information isusually referred to as the background Mathematically 4D-Var procedures are formulated as an optimization problemin which the initial condition plays the role of control vectorand model equations are considered as constraints Whilethe theory of variational data assimilation is well establishedthere are a number of issues with practical implementationthat require additional consideration and study The perfor-mance of 4D-Var schemes depends on their key informationcomponents such as the available observations estimatesof the observation and background error covariances thatare quantified by the corresponding matrices as well asthe background state All of those information componentsstrongly impact the accuracy of calculated initial conditionsthus influencing the forecast quality Thus it is importantto estimate the sensitivity of a certain forecast aspect withrespect to variations in the observational data backgroundinformation and error statistics This problem is usuallyformulated within the framework of the adjoint-based sen-sitivity analysis (eg [18ndash29])

However an adjoint model used to calculate sensitivityfunctions is derived from a linearized forward propagationmodel and contains numerous input parameters Conse-quently linearization of strongly nonlinear NWPmodels andalso uncertainties in their numerous parameters generateerrors in the initial conditions obtained by data assimilationsystems The influence of linearization and parameter uncer-tainties on the results of data assimilation can be studied ide-ally for each particular NWPmodel using sensitivity analysis[30 31] Evenmore problems arise when considering coupled4D-Var data assimilation schemes since the atmosphere andocean have very different physical properties and time-spacespectrum of motions generating initialization shock Severalcoupling strategies are being developed for use in NWPsystems however all of them introduce issues that requireadditional detailed consideration For example the influenceof coupling strength on the initial conditions obtained by 4D-VAR procedures is one such issue that is important to studyand analyze

Good practice in the development of NWP models anddata assimilation systems requires evaluating the confidencein the model In this context it is important to estimate theinfluence of parameter variations on system dynamics and tofind those parameters that have the largest impact on systembehaviour Sensitivity analysis which is an essential elementof model building and quality assurance addresses this veryimportant issue

The exploration of coupled 4D-Var systems parameterestimation and sensitivity analysis require considerable com-putational resources For simple enough low-order coupled

models the computational cost is minor and for that reasonmodels of this class are widely used as simple test instrumentsto emulate more complex systems In this paper we describea coupled nonlinear dynamical system which is composedof fast (the ldquoatmosphererdquo) and slow (the ldquooceanrdquo) versions ofthe well-known Lorenz [32] model This low-order coupledsystem allows us to mimic the atmosphere-ocean systemand therefore serves as a key element of a theoretical andcomputational framework for the study of various aspectsof coupled 4D-Var procedures [33 34] Under certain con-ditions the Lorenz model exhibits a chaotic behaviour andusing conventional methods of sensitivity analysis can bequestionable in terms of interpretation of the obtained results[35ndash37] The ldquoshadowingrdquo method [36 37] for estimatingthe system sensitivity to variations in its parameters allowsus to calculate the average along the trajectory sensitivitiesand therefore to make a clear conclusion with respect to thesystem sensitivity to its parameters This method is based onthe pseudoorbit shadowing in dynamical systems [38 39]Calculated sensitivity coefficients obtained via conventionalmethods and the ldquoshadowingrdquo approach are presented in thepaper

We also succinctly consider commonly used techniquesfor sensitivity analysis and parameter estimations of dynam-ical systems and study the influence of coupling strengthparameter on the dynamical behaviour of the coupled systemusing Lyapunov characteristic exponent analysis It was foundthat the coupling strength parameter strongly affects thesystem dynamics both quantitatively and qualitatively Thisfact should be taken into consideration when choosing thecoupling strategy in data assimilation systems

2 Low-Order Coupled Dynamical System

In this section we consider a low-order coupled nonlineardynamical system obtained by coupling of two versions ofthe original Lorenzmodel (L63) [32] with distinct time scaleswhich differ by a factor 120576 (eg [33 34])

= 120590 (119910 minus 119909) minus 119888 (119886119883 + 119896)

119910 = 119903119909 minus 119910 minus 119909119911 + 119888 (119886119884 + 119896)

= 119909119910 minus 119887119911 + 119888119911119885

(1a)

= 120576120590 (119884 minus 119883) minus 119888 (119909 + 119896)

= 120576 (119903119883 minus 119884 minus 119886119883119885) + 119888 (119910 + 119896)

119885 = 120576 (119886119883119884 minus 119887119885) minus 119888

119911119911

(1b)

where lower case letters represent the fast subsystem andcapital letters the slow subsystem 120590 119903 and 119887 are theparameters of L63 model 119888 is a coupling strength parameterfor the 119909 and 119910 variables 119888

119911is a coupling strength parameter

for 119911 119896 is an ldquouncenteringrdquo parameter and 119886 is a parameterrepresenting the amplitude scale factor The value 119886 = 1

indicates that two systems have the same amplitude scaleThus the state vector of the coupled model (1a) and (1b) isx = (119909 119910 119911 119883 119884 119885)

119879 and the model parameter vector is

Advances in Meteorology 3

120572 = (120590 119903 119887 119886 119888 119888119911 119896 120576)119879 Without loss of generality we can

assume that 119886 = 1 119896 = 0 and 119888 = 119888119911 then 120572 = (120590 119903 119887 119888 120576)

119879

and the system (1a) and (1b) can be rewritten in the operatorform

119889x119889119905

= (L +Q) x (2)

where the nonlinear uncoupled operator L and linearcoupled operatorQ are represented by the followingmatrices

L =

[

[

[

[

[

[

[

[

[

[

[

[

minus120590 120590 0

119903 minus1 minus119909 0

0 119909 minus119887

minus120576120590 120576120590 0

0 120576119903 minus120576 minus120576119883

0 120576119883 minus120576119887

]

]

]

]

]

]

]

]

]

]

]

]

Q =

[

[

[

[

[

[

[

[

[

[

[

[

minus119888 0 0

0 0 119888 0

0 0 119888

minus119888 0 0

0 119888 0 0

0 0 minus119888

]

]

]

]

]

]

]

]

]

]

]

]

(3)

The unperturbed parameter values are taken as

1205900

= 10 1199030

= 28 1198870

=

8

3

1205760

= 01 1198880

isin [01 12]

(4)

Chosen values of 120590 119903 and 119887 correspond to chaotic behaviourof the L63 model For 120590 = 10 and 119887 = 83 the critical value ofparameter 119903 is 2474 which means that any value of 119903 largerthan 2474 induces chaotic behaviour [32]The parameter 120576 =01 indicates that the slow system is 10 times slower than thefast system

The coupling strength parameter 119888 plays a very importantrole in qualitative changes in the system dynamics since thisparameter controls the interactions between fast and slowsubsystems Qualitative changes in the dynamical propertiesof a system can be detected by determining and analyz-ing the systemrsquos spectrum of Lyapunov exponents whichcharacterize the average rate of exponential divergence (orconvergence) of nearby trajectories in the phase space Inthe analysis of coupled dynamical systems we are dealingwith conditional Lyapunov exponents that are normally usedto characterize the synchronization with coupled systemsSystem (2) has six distinct exponents If the parameter 119888

tends to zero then system (2) has two positive two zero andtwo negative Lyapunov exponents The influence of couplingstrength parameter 119888 on the two largest conditional Lyapunovexponents is illustrated in Figure 1 The numerical experi-ments demonstrated that those initially positive exponentsdecrease monotonically with an increase in the parameterc At about 119888 asymp 08 they approach the 119909-axis and at about

Coupling strength parameter00 04 08 12

Lyap

unov

expo

nent

s

08

06

04

02

00

12058211205822

Figure 1 Two largest conditional Lyapunov exponents as functionsof coupling strength parameter

119888 asymp 095 negative values Thus for 119888 gt 095 the dynamicsof both fast and slow subsystems become phase-synchronous[40] For 119888 gt 10 a limit circle dynamical regime is observedsince all six exponents become negative

Apart from Lyapunov exponents autocorrelation func-tions (ACFs) enable one to distinguish between regular andchaotic processes and to detect transition fromorder to chaosIn particular for chaotic motions ACF decreases in time inmany cases exponentially while for regular motions ACF isunchanged or oscillating In general however the behaviourof ACFs of chaotic oscillations is frequently very complicatedand depends on many factors (eg [41 42]) Autocorrelationfunctions can also be used to define the so-called typical timememory (typical timescale) of a process [43] If it is positiveACF is considered to have some degree of persistence atendency for a system to remain in the same state from onemoment in time to the next The ACF for a given discretedynamic variable 119909

119894119873minus1

119894=0is defined as

119862 (119904) = ⟨119909119894119909119894+119904⟩ minus ⟨119909

119894⟩ ⟨119909119894+119904⟩ (5)

where the angular brackets denote ensemble averagingAssuming time series originates from a stationary andergodic process ensemble averaging can be replaced by timeaveraging over a single normal realization

119862 (119904) = ⟨119906119898119906119898+119904

⟩ minus ⟨119906⟩2

(6)

Signal analysis commonly uses the normalized ACF definedas 119877(119904) = 119862(119904)119862(0) ACF plots for realizations of dynamicvariables 119909 and119883 and 119911 and 119885 calculated for different valuesof the coupling strength parameter 119888 are presented in Figures2 and 3 respectively For relatively small parameter 119888 (119888 lt

04) the ACFs for both 119909 and 119883 variables decrease fairlyrapidly to zero consistently with the chaotic behaviour of thecoupled system However as expected the rate of decay of


the ACF of the slow variable 119883 is less than that of the fastvariable 119909 The ACF envelopes for variables 119911 and 119885 alsodecay almost exponentially from the maximum to zero Forcoupling strength parameter on the interval 04 lt 119888 lt 06 theACF of the fast variable 119909 becomes smooth and converges tozero At the same time the envelopes of the ACFs of variables119883 119911 and 119885 demonstrate a fairly rapid fall indicating thechaotic behaviour As the parameter 119888 increases the ACFsbecome periodic and their envelopes decay slowly with timeindicating transition to regularity For 119888 gt 08 calculatedACFs show periodic signal components

3 The Basics of Four-Dimensional VariationalData Assimilation

Atmospheric models used for operational NWP are mainlydeterministic and derived from a set of multidimensionalnonlinear differential equations in partial derivatives whichare the equations of fluid dynamics and thermodynamics thatdescribe atmospheric processes and atmosphere-underlyingsurface interactions A generic atmospheric model canbe represented by the following continuous autonomousdynamical system

119889x (119905)119889119905

= 119891 (x (119905) 120572 (119905)) 119905 isin [0 120591] = T x (0) = x0 (7)

Here x(119905) is the state vector belonging to a Hilbert space X120572 isin P is a parameter vector where P is a Hilbert space(the space of the model parameters) 119891 is a nonlinear vector-valued function such that 119891 X times P times T rarr X and x

0

is a given vector function This infinite-dimensional modelhas to be truncated by some means to finite-dimensionalapproximate model for which a solution can be soughtnumerically Applying either a projection onto a finite set ofbasic functions or a discretization in time and space onecan derive the discrete atmospheric model which can berepresented as 119904 discrete nonlinear dynamical system givenby the equation

x119894+1

= M119894119894+1

(x119894) + 120576119898

119894 (8)

where x119894isin R119899 is the 119899-dimensional state vector representing

the complete set of the model variables that determine theinternal state of the atmosphericmodel at time 119905

119894M119894119894+1

is thenonlinear operator that indirectly containsmodel parametersand propagates the state vector from time 119905

119894to time 119905

119894+1for 119894 =

0 119873minus 1 and 120576119898119894is the model errors It is usually assumed

that model (8) is ldquoperfectrdquo (120576119898119894= 0) that is the forecast has

no errors if the initial condition is perfect In this case giventhe model operator and the initial condition x

0 (8) uniquely

specifies the orbit of the dynamical system Let y0119894isin R119898 be the

119898-dimensional vector of observations measured at a discretetime 119905

119894 119894 = 0 119873 that are linked to the system state via the

following equation

y0119894= H119894(x119894) + 1205760

119894 (9)

where H119894

R119899 rarr R119898 is the nonlinear observationoperator that maps the state vector to observation space It is

usually assumed that the observation errors 1205760119894are unbiased

serially uncorrelated and normally distributed with knowncovariance matrices R

119894isin R119898times119898

Suppose that at the initial time the prior (background)model state x119887

0is known and represents the ldquobestrdquo estimate

of the ldquotruerdquo state x1199050before any observations are taken This

background state is provided by a previous forecast It isassumed that x119887

0has unbiased and normally distributed errors

120576119887 with known covariance matrix B

0isin R119899times119899

x1198870= x1199050+ 120576119887

(10)

Given the observations y119900119894at time 119905

119894 the corresponding

observation error covariance matrices R119894(119894 = 0 119873) the

background initial state x1198870 and the error covariance matrix

B0 the 4D-Var data assimilation seeks to minimize with

respect to x0 a certain cost function 119869(x) expressing the

ldquodistancerdquo between observations and corresponding modelstate using the model equations as constraints

x1198860= arg min 119869 (x) (11)

subject to x satisfying the set of the ldquoidealrdquo (120576119898119894= 0) model

equations with initial state x0

x119894+1

= M0119894+1

(x0) 119894 = 1 119873 (12)

The 4D-Var cost function is usually written as (eg [17 18])

119869 (x0) =

1

2

(x0minus x1198870)

119879

Bminus10(x0minus x1198870)

+

1

2

119873

sum

119894=0

(H119894(x119894) minus y0119894)

119879

Rminus1119894(H119894(x119894) minus y0119894)

(13)

The optimization problem (11) is nonlinear with strongconstraints and an iterative minimization algorithm (eggradient-based technique) is required to obtain the solutionThe gradient of the cost function (13) is as follows

nablax0119869 (x0) = Bminus10(x0minus x1198870) +

119873

sum

119894=0

M1198790119894H119879119894Rminus1119894(H119894(x119894) minus y0119894)

(14)

where M1198790119894

is the adjoint of the linearized model operatorM0119894

= M10158400119894(x119894) and H119879

119894is the adjoint of the linearized

observation operator H119894

= H1015840119894(x119894) If the model is not

ldquoperfectrdquo then we need to take into account the model errors120576119898 which are sometimes taken as Gaussian noise

120576119898

isin N (0Q) (15)

whereQ is a model error covariance matrix Thus we obtainthe weakly constrained 4D-Var data assimilation and thefollowing term

119869119898=

1

2

119873

sum

119894=1

(x119894minusM119894minus1119894

(x119894minus1))119879Qminus1119894(x119894minusM119894minus1119894

(x119894minus1)) (16)


Fast variable x Slow variable X

10

05

00

10

05

00

10

05

00

minus05

10

05

00

minus05

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus100 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

c = 015c = 015

c = 05c = 05

c = 08c = 08

c = 10c = 10

Figure 2 Autocorrelation functions for dynamic variables 119909 and119883 for different parameter 119888


Fast variable z Slow variable Z

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

c = 015c = 015

c = 05c = 05

c = 08c = 08

c = 10c = 10

Figure 3 Autocorrelation functions for dynamic variables 119911 and 119885 for different parameter 119888


should be added to the right-hand side of the cost function(13) [6 19]

It is important to make the following comments WhileNWPhas significantly improved over the last several decadesweather forecasts are still intrinsically uncertain This isbecause mathematical models used in NWP have severalsources of uncertainty and therefore a number of sourcesof errors The first one is an intrinsic uncertainty due tochaotic nature of the system The second one is a structuraluncertainty which is how the model itself represents thephysical processes incorporated into it It is important tounderline that our knowledge of the earth system is alwaysimperfect and therefore we can only theoretically design theldquoidealrdquo model However improving model physics demon-strated that even theoretically well-posed models failed toaccurately simulate and predict the dynamics of real atmo-spheric processes This is because numerical models have aparametric uncertainty (how accurate model parameters are)initial and boundary uncertainty (are initial and boundaryconditions known precisely) and in addition numericalerrors All of those uncertainties (errors) impair the weatherforecast accuracy and limit the time horizon of accurateNWP Current time horizon of synoptic-scale NWP is severaldays

The climate study has significantly longer time horizonseveral decadesThe sensitivity analysis of the climate systemis associated with stability of characteristics of climatic modelattractors with respect to perturbations in model parametersA priori estimation of the behaviour of state vector whenthe perturbations in the climate model parameters tend tozero is generally an unresolved problem since it is not knownwhether or not the invariant measure of climate modelingsystem is continuous with respect to the small perturba-tions in the differential matrix operator of the numericalmodel Indeed at certain model parameter values differentbifurcation can occur in the system phase space Thereforedynamics on the attractor generated by themodelmay changeconsiderably even for small parameter perturbations

4 Sensitivity Analysis Essentials ofConventional Approaches

One of the commonly used measures for estimating theinfluence ofmodel parameter variations on the state variablesis the sensitivity coefficient which is the derivative of a certaincomponent of a model state vector 119909

119894with respect to some

model parameter 120572119895[30 31 44]

119878119894119895equiv

120597119909119894

120597120572119895

1003816100381610038161003816100381610038161003816100381610038161003816120572119895=120572

0

119895

= lim120575120572rarr0

[

[

119909119894(1205720

119895+ 120575120572119895) minus 119909119894(1205720

119895)

120575120572119895

]

]

(17)

where 120575120572119895is the infinitesimal perturbation of parameter 120572

119895

around some fixed point 1205720119895 Differentiating (8) with respect

to 120572 we obtain the set of nonhomogeneous ODEs the so-called sensitivity equations which can be written as

119889S119895

119889119905

= M sdot S119895+D119895

119895 = 1 119898 (18)

where S119895= (120597x120597120572

119895) = (119878

1119895 1198782119895 119878

119899119895)119879 is the sensitivity

vector with respect to parameter 120572119895 D119895= (1205971198911120597120572119895 1205971198912120597120572119895

120597119891119899120597120572119895)119879 and M is a Jacobian matrix Thus to analyze

the sensitivity of system (7) with respect to parameter 120572119895one

can solve the following set of differential equations with giveninitial conditions

119889x119889119905

= 119891 (x120572) x (0) = x0

119889S119895

119889119905

= M sdot S119895+D119895 S119895(0) = S

1198950

(19)

Sensitivity equations describe the evolution of sensitivitycoefficients along a given trajectory and therefore allowtracing the sensitivity dynamics in time The procedurefor computing sensitivity coefficients includes the followingsteps

(1) Obtain initial conditions on the system attractor attime 119905 = 119905

0by integrating the nonlinear model

equations (7) for a long enough time range [1199050 120591]

starting from random initial conditions(2) Solve the nonlinear model equations (7) to calculate a

trajectory x(119905) 119905 isin [1199050 120591]

(3) Calculate a model Jacobian matrix and a parametricJacobian matrix

(4) Solve the sensitivity equations (18) with given initialconditions to obtain the desired sensitivity coeffi-cients

Sensitivity analysis allows us also to explore the sensitivityof a generic objective function (performancemeasure) whichcharacterizes the dynamical system (7)

J (x120572) = int

120591

0

Φ (119905 x120572) 119889119905 (20)

where Φ is a nonlinear function of the state variables x andmodel parameters 120572 The gradient of the functional J withrespect to the parameters 120572 around the unperturbed statevector x0

nabla120572J (x01205720) = (

119889J

1198891205721

119889J

119889120572119898

)

119879100381610038161003816100381610038161003816100381610038161003816x0 1205720

(21)

quantifies the influence of parameters on the model outputresults In particular the effect of the 119895th parameter can beestimated as follows

119889J

119889120572119895

10038161003816100381610038161003816100381610038161003816100381610038161205720

119895

asymp

J (x0 + 120575x0 12057201 120572

0

119895+ 120575120572119895 120572

0

119898) minusJ (x01205720)

120575120572119895

(22)

where 120575120572119894is the variation in parameter 1205720

119894 Note that

119889J

119889120572119895

=

119899

sum

119894=1

120597J

120597119909119894

120597119909119894

120597120572119895

+

120597J

120597120572119895

=

119899

sum

119894=1

119878119894119895

120597J

120597119909119894

+

120597J

120597120572119895

(23)


This approach is acceptable for low-order models Howeverthe accuracy of sensitivity estimates strongly depends onchoice of the perturbation 120575120572

119894 By introducing the Gateaux

differential the sensitivity analysis problem can be consid-ered in the differential formulation eliminating the need toset the value of 120575120572

119894[30 31] The Gateaux differential for the

objective function (20) has the following form

120575J (x01205720 120575x 120575120572) = int

120591

0

(

120597Φ

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575x + 120597Φ

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575120572)119889119905

(24)

Here 120575x is the state vector variation due to the variationin the parameter vector in the direction 120575120572 Linearizing thenonlinear model (7) around an unperturbed trajectory x0(119905)we obtain the following system of variational equations theso-called tangent linear model for calculating 120575x

120597120575x120597119905

=

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575x + 120597119891

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575120572

119905 isin [0 120591] 120575x (0) = 120575x0

(25)

Then using (24) we can calculate the variation 120575J Since120575J(x01205720 120575x 120575120572) = ⟨nabla

120572J 120575120572⟩ where ⟨sdot sdot⟩ is a scalar

product the model sensitivity with respect to parametervariations can be estimated by calculating the components ofthe gradient nabla

120572J However this method is computationally

ineffective if the number of model parameters119898 is large Theuse of adjoint equations allows obtaining the required sen-sitivity estimates within a single computational experiment(eg [6 30 31]) since the gradient nabla

120572J can be calculated by

the following equation

nabla120572J (x01205720) = int

120591

0

[

120597Φ

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

minus (

120597119891

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

)

119879

sdot xlowast]119889119905 (26)

where the vector function xlowast is the solution of adjoint model

minus

120597xlowast

120597119905

minus (

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

)

119879

xlowast = minus

120597Φ

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

119905 isin [0 120591] xlowast (120591) = 0

(27)

This equation is numerically integrated in the inverse timedirection Thus the algorithm for computing sensitivityfunctions is as follows

(1) Obtain initial conditions on the system attractor attime 119905

0by integrating the nonlinear model equations

(7) for a long enough time range [1199050 120591] starting from

random initial conditions(2) Solve the nonlinear model equations (7) to calculate a


(3) Calculate the right-hand side of (27) and then inte-grate numerically this equation in the inverse timedirection with the initial conditions xlowast(120591) = 0

(4) Calculate the gradient (26)

5 Testing the Variational Data Assimilation

Wewill consider the dynamics of system (2) on its attractor Inorder to obtain themodel attractor the numerical integrationof (2) is started at 119905

119863= minus20 with the initial conditions

x (119905119863) = (001 001 001 002 002 002)

119879 (28)

and finished at 1199050= 0 to guarantee that the calculated model

state vector x0= x(0) is on the model attractor The forecast

obtained by NWPmodels is substantially determined by ini-tial conditions which calculated via 4D-Var The accuracy ofinitial conditions strongly depends on numerous parametersof 4D-Var systems Some of these parameters are uncertainTo estimate the influence of parameter variations on theforecast obtained by model (2) the ldquotruerdquo and ldquoforecastrdquotrajectories were calculated The ldquotruerdquo trajectory x119905(119905

119894) and

the ldquotruerdquo state x119905(119905119891) at the verification time 119905

119891are obtained

by integrating the model equations over the time interval[1199050 119905119891] with unperturbed parameter vector 1205720 and initial

conditions x0 Then the forecast trajectory x119891(119905

119896) and the

forecast state x119891(119905119891) at 119905119891are obtained by integration of the

ldquoforecastrdquo model (2) with initial conditions x0and a certain

perturbed model parameter Thus the ldquoforecastrdquo model hasthe same set of equations as the ldquotruerdquo model however somemodel parameter is slightly changed (ie this parameter isknown with uncertainty) In order to measure forecast errorsthe relative error in energy norm is used

119890 (x119891) = [

[

(x119891 minus x119905)119879

(x119891 minus x119905)

(x119905)119879 x119905]

]

12

(29)

As an example variations in parameters 119903 and 119888 areconsidered and forecast is made for two time periods 119905

119891

25 and 50 of nondimensional time units The forecast error(29) is estimated at time 119905

119891 Table 1 shows the results of

forecast verifications It is obvious that the less the forecasterror measure 119890(x119891) the higher the forecast skill Qualita-tively calculated results are consistent with real numericalweather forecasts obtained with complex state-of-the-artNWP models longer 119905

119891leads to lower forecast accuracy and

smaller parameter variations (difference between the ldquotruerdquoparameter value and the value used in the ldquorealrdquo model)lead to better forecast accuracy It can also be observed thatparameter 119903 influences the forecast accuracy almost twice asmuch as parameter 119888

Synthetic data assimilation requires the ldquotruerdquo state x119905(119905119894)

the background (first guess) state x119887(119905119894) and observations

y0(119905119894) inside the assimilation window [119905

0 119905119873] as well as error

covariance matrices of the prior guess B0and observations

R119894 The length of data assimilation window should be defined

as well The ldquotruerdquo and background trajectories on the dataassimilation interval [119905

0 119905119873] represent some portions of x119905(119905

119894)

and x119887(119905119894) respectively Observations should be provided

every 5ndash10 time steps inside the assimilation window andcan be generated by adding the Gaussian random noise (withzero mean and variance 1205902

0= 02) to the ldquotruerdquo state Since

observation grid and model grid are the same observation


Table 1 Relative errors in energy norm (subscript at parameters denotes the nondimensional time unit)

Parameter variations in percent of unperturbed value1 minus1 5 minus5 10 minus10

11990325

01273 01266 02871 02796 04081 0389011988825

00584 00583 01312 01300 01866 0183311990350

01663 01689 03617 03202 05004 0574211988850

00705 00704 01583 01569 02250 02210

operator H is simply an identity mapping To take intoconsideration the background covariances for simplicity theassumption B

0= 1205902

119887I can be used where 1205902

119887= 02 is the

variance of background error and I is the identity matrixUnder assumption that the observation quality is the samefor all variables the observation covariance matrix can bedefined as R

119894= R = 120590

2

0I

Testing the TL model and its adjoint is required toensure the convergence of theminimization algorithm in dataassimilation procedures If 120577120575x is a small perturbation of themodel state then

M (x + 120577120575x) minusM (x) asymp M (x) 120577120575x (30)

To verify the applicability of theTLmodel on the time interval[1199050 119905119873] the relative error

119890119877=

M (x + 120577120575x) minusM (x)M (x) 120577120575x

(31)

should be calculated The TL model is valid if 119890119877

rarr 0

when 120577 rarr 0 The results of numerical experiments showedthat the TL model passed this test with 119890

119877tending towards

zero (Table 2) The TL adjoint correctness can be tested byverification of the inner product identity

⟨M120575xM120575x⟩ = ⟨120575xM119879M120575x⟩ (32)

It was found that this equality is essentially correct thedifference was observed only in the 7th digit which isconsistent with a round-off errorThe second test to verify theadjoint model is the so-called gradient test [45] which aimsto compare a finite difference representation of the gradientof 4D-Var cost function (13) with the gradient obtained viaadjoint model nabla119869(x

0) A linear Taylor approximation of the

cost function can be written as

119869 (x0+ 120577120575x) asymp 119869 (x

0) + 120577 (120575x)119879 nabla119869 (x

0) (33)

Let us introduce the following function

Ψ (120577) =

119869 (x0+ 120577120575x) minus 119869 (x

0)

120577 (120575x)119879 nabla119869 (x0)

(34)

If the gradient is estimated correctly then the functionΨ(120577) rarr 1 as 120577 rarr 0 The perturbation vector 120575x is takento be [45]

120575x =nabla119869 (x0)

1003817100381710038171003817nabla119869 (x0)1003817100381710038171003817

(35)

Table 2 Results of verification of tangent linear model for 119888 = 08

and 120575x = 10minus2

times x0

120577 119890119877

1 0918206602754424910minus1 0999727996578274310minus2 0999992546815546310minus3 0999999192961153110minus4 0999999953496501210minus5 0999999991121788310minus6 0999999991442708710minus7 09999997435447022

Table 3 Results of verification of 4D-Var cost function gradient for119888 = 08 and 120575x = 10

minus2

times x0

120577 Ψ (120577) log10(1003816100381610038161003816Ψ (120577) minus 1

1003816100381610038161003816)

10minus4 08727731981461396 minus0895421389700905610minus5 09975483775420343 minus2610546408968684010minus6 09998765512756632 minus3908513393511378710minus7 09999884562844123 minus4937654381865784510minus8 09999979865432855 minus5696057702470860010minus9 09999998912431426 minus6963543350089321010minus10 09999999244103234 minus71215375125562080

where sdot is the 1198712norm Table 3 manifests the success of the

gradient test

6 Sensitivity of the System withrespect to Parameters

According to the sensitivity theory [44] general solutionsof sensitivity equations for oscillatory nonlinear dynamicalsystems grow unbounded as time tends to infinity thereforesensitivity functions calculated by conventional approacheshave a high degree of uncertaintyThe reason is that nonlineardynamical systems that exhibit chaotic behavior are verysensitive to its initial conditionsThus the solutions to the lin-earized Cauchy problem (7) grow exponentially as 120575x(119905) asymp120575x(0)119890120582119905 where 120582 gt 0 is the leading Lyapunov exponentAs a result calculated sensitivity coefficients contain a fairlylarge error [35ndash37] To illustrate this point let us explorethe sensitivity of model output to changes in the coupling


strength parameter Let us introduce the following sensitivitycoefficients

1198781119888=

120597119909

120597119888

1198782119888=

120597119910

120597119888

1198783119888=

120597119911

120597119888

1198784119888=

120597119883

120597119888

1198785119888=

120597119884

120597119888

1198786119888=

120597119885

120597119888

(36)

The corresponding sensitivity equations can be written as

1198781119888= 120590 (119878

2119888minus 1198781119888) minus 119888119878

4119888minus 119883

1198782119888= 1199031198781119888minus 1198782119888minus 1199091198783119888minus 1199111198781119888+ 1198881198785119888+ 119884

1198783119888= 1199091198782119888+ 1199101198781119888minus 1198871198783119888+ 1198881198786119888+ 119885

1198784119888= 120576120590 (119878

5119888minus 1198784119888) minus 119888119878

1119888minus 119909

1198785119888= 120576 (119903119878

4119888minus 1198785119888minus 1198831198786119888minus 1198851198784119888) + 119888119878

2119888+ 119910

1198786119888= 120576 (119883119878

5119888+ 1198841198784119888minus 1198871198786119888) minus 119888119878

3119888minus 119911

(37)

Sensitivity coefficients can be introduced for any particularmodel parameter Since the parameter vector120572 consists of fivecomponents (120590 119903 119887 119888 and 120576) five sets of sensitivity equationscan be derived from the model equations (2) The dynamicsof sensitivity coefficients (36) can be traced by solving thesensitivity equations (37) along with the nonlinear model (2)

Sensitivity coefficients (36) calculated on the time inter-val [0 20] are shown in Figure 4 Envelopes of these coeffi-cients grow over timewhile sensitivity coefficients themselvesexhibit oscillating behavior Figure 5 provides more detailedinformation on the evolution of sensitivity coefficients (36)calculated on the time interval [0 5] It is known that sensitiv-ity coefficient is a measure of the change in state variable dueto the variation of the estimated parameter Unfortunatelyobtained sensitivity coefficients are inherently uninformativeand misleading We cannot make a clear conclusion fromthem about system sensitivity to variations in the parameter119888 In this regard the average values of sensitivity functionsnabla120572⟨J(120572)⟩ over a certain period of time can be considered as

one of the most important measures of sensitivity where Jis a generic objective function (20) However the gradientnabla120572⟨J(120572)⟩ cannot be correctly estimated within the frame-

work of conventional methods of sensitivity analysis since forchaotic systems it is observed that [35 36]

nabla120572⟨J (120572)⟩ = ⟨nabla

120572J (120572)⟩ (38)

This is because the integral

I = lim119879rarrinfin

int

119879

0

lim120575120572rarr0

[

J (120572 + 120575120572) minusJ (120572)

120575120572

] 119889119905 (39)

does not possess uniform convergence and two limits (119879 rarr

infin K 120575120572 rarr 0) would not commuteSimilar results were obtained when we considered the

influence of variations in the parameter 119903 on the systemdynamics This parameter plays an important role in theformation of systemrsquos dynamical structure and transition tochaotic behavior [32] Figure 6 shows the differences between

components of state vector x(1199030) obtained with 119903 = 1199030

= 28

and x(1199030 +120575119903) obtained with 119903 = 1199030

+120575119903 where 120575119903 = 0011199030

=

028 Even a small perturbation in the parameter 119903 generatesa tangible difference between corresponding state variablesLet us define the following sensitivity coefficients

1198781119903=

120597119909

120597119903

1198782119903=

120597119910

120597119903

1198783119903=

120597119911

120597119903

1198784119903=

120597119883

120597119903

1198785119903=

120597119884

120597119903

1198786119903=

120597119885

120597119903

(40)

The associated system of sensitivity equations can be writtenas

1198781119903= 120590 (119878

2119903minus 1198781119903) minus 119888119878

4119903

1198782119903= 119909 + 119903119878

1119903minus 1198782119903minus (1199091198783119903+ 1199111198781119903) + 119888119878

5119903

1198783119903= (119909119878

2119903+ 1199101198781119903) minus 119887119878

3119903+ 1198881198786119903

1198784119903= 120576120590 (119878

5119903minus 1198784119903) minus 119888119878

1119903

1198785119903= 120576 [119883 + 119903119878

4119903minus 1198785119903minus (119883119878

6119903+ 1198851198784119903)] + 119888119878

2119903

1198786119903= 120576 [(119883119878

5119903+ 1198841198784119903) minus 119887119878

6119903] minus 119888119878

3119903

(41)

Envelopes of calculated sensitivity coefficients (40) growover time and sensitivity coefficients demonstrate the oscil-lating behavior (Figures 7 and 8) Obtained sensitivitycoefficients are also uninformative and inconclusive Theldquoshadowingrdquo approach for estimating the system sensitivityto variations in its parameters [36 37] allows us to calculatethe average sensitivities ⟨nabla

120572119869(120572)⟩ and therefore to make a

clear conclusion with respect to the system sensitivity to itsparameters A detailed description of two variants of theldquoshadowingrdquo approach is provided in an appendix and someresults of numerical experiments are presented below

The main problem arising in the ldquoshadowingrdquo methodis to calculate the pseudotrajectory We consider two setsof numerical experiments weak coupling (119888 = 001) andstrong coupling (119888 = 08) between fast and slow systemsFast and slow variables that correspond to the original andpseudoorbits are shown in Figures 9 and 10when the couplingstrength parameter 119888 = 001 The Least Square Shadowingvariant of the ldquoshadowingrdquo approach was used to calculatepseudotrajectories The differences between state variablescorresponding to the original and pseudotrajectories ofthe fast and slow systems are plotted in Figure 11 Thesefigures show that the calculated pseudoorbits are close tocorresponding true trajectories over a specified time intervaldemonstrating the shadowability The strong coupling doesnot introduce significant qualitative and quantitative changesin the behavior of pseudotrajectories with respect to thetrue orbits The original and pseudo fast and slow variablesfor 119888 = 08 are shown in Figures 12 and 13 and thedifferences between these state variables are presented inFigure 14 Sensitivity estimateswith respect to the parameter 119903calculated over the time interval [0 20] for different values ofcoupling strength parameter are shown in Table 4 The mostsensitive variables are 119911 and 119885 The sensitivity of variables 119909


0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

Fast subsystem Slow subsystem

Am

plitu

de

2000

0

minus2000

Am

plitu

de

3000

0

minus3000

Am

plitu

deA

mpl

itude

2000

0

minus2000

Am

plitu

de

3000

0

minus3000

Am

plitu

de

2000

0

minus2000

1000

0

minus1000

S1

S2

S3

S4

S5

S6

Figure 4 Time dynamics of sensitivity functions with respect to parameter 119888 on the time interval [0 20] for 1198880 = 09

119910119883 and119884with respect to 119903 is significantly less than variables119911 and 119885

7 Concluding Remarks

We considered a coupled nonlinear dynamical system whichis composed of fast (the ldquoatmosphererdquo) and slow (the ldquooceanrdquo)versions of the well-known Lorenz model This low-ordermathematical tool allows us to mimic the atmosphere-oceansystem and therefore serves as a key part of a theoretical andcomputational framework for the study of various aspects

of coupled 4D-Var procedures Numerical models used topredict the weather are highly nonlinear but tangent linearapproximations and their adjoints are used in VDA algo-rithms Linear approximation of strongly nonlinear NWPmodels and also uncertainties in their numerous parametersgenerate errors in the initial conditions obtained via dataassimilation systems The influence of parameter uncertain-ties on the results of data assimilation can be studied usingsensitivity analysis

We discussed conventional methods of sensitivity analy-sis and their inefficiencywith respect to calculating sensitivity



Am

plitu

de

S1

S2

S3

S4

S5

S6

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

200

100

0

minus100

Am

plitu

de

100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de

50

0

minus50minus200


Table 4 Sensitivity estimates of fast and slow variables with respect to parameter 119903

119888 120597119885120597119903 120597119884120597119903 120597119883120597119903 120597119911120597119903 120597119910120597119903 120597119909120597119903

10 110 005 001 108 004 00308 069 008 003 102 007 00704 095 003 minus001 103 009 009015 091 minus008 minus009 101 minus001 minus00110minus4 104 minus002 minus003 102 minus001 minus001


Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

3

0

minus3

Am

plitu

de

3

0

minus3

minus6

Δx

Δy

Δz

ΔXΔY

ΔZ

Figure 6 Difference between x(1199030) and x(1199030 + 0011199030

) for fast andslow subsystems

coefficients for chaotic dynamics To calculate sensitivitycoefficients with acceptable accuracy the sensitivity analy-sis method [36 37] developed on the basis of theory ofshadowing of pseudoorbits in dynamical systems [38 39]was applied Previously this method was used to analyze thesensitivity of the periodic van der Pol oscillator the originalLorenz system and simplified aeroelastic model that exhibitboth periodic and chaotic regimes

Calculated sensitivity coefficients obtained via conven-tional methods and the ldquoshadowingrdquo approach are presentedand discussed It was shown that envelopes of sensitivitycoefficients obtained by conventional methods grow overtime and the coefficients themselves exhibit the oscillatingbehaviour Using the ldquoshadowingrdquo method allows us tocalculate the average sensitivity functions (coefficients) thatcan be easily interpreted

In conclusion two comments should be highlighted(1) The shadowing property of dynamical systems is a

fundamental feature of hyperbolic systems that was firstdiscovered by Anosov [46] and Bowen [47] However mostphysical systems are nonhyperbolic Despite the fact thatmuch of shadowing theory has been developed for hyperbolicsystems there is evidence that nonhyperbolic attractors alsohave the shadowing property (eg [48ndash51]) In theory thisproperty should be verified for each particular dynamicalsystem but this is more easily said than done

(2) The applicability of the shadowing method for sen-sitivity analysis of modern atmospheric and climate modelsis a rather complicated problem since these models arequite complex and they contain numerous input parametersThus further research and computational experiments arerequired However we are confident that by using the basicideas of the shadowing method it is possible to better

understand the sensitivity analysis of atmospheric models ofvarious levels of complexity

Appendix

The novel sensitivity analysis method for chaotic dynamicalsystems developed in [36 37] is based on the theory ofpseudoorbit shadowing in dynamical systems [38 39] whichis one of the most rapidly developing components of theglobal theory of dynamical systems and classical theory ofstructural stability [52] Naturally pseudo- (or approximate-)trajectories arise due to the presence of round-off errorsmethod errors and other errors in computer simulation ofdynamical systems Consequently we will not get an exacttrajectory of a system but we can come very close to anexact solution and the resulting approximate solution will bea pseudotrajectory The shadowing property (or pseudoorbittracing property) means that near an approximate trajectorythere exists the exact trajectory of the system considered suchthat it lies uniformly close to a pseudotrajectory The shad-owing theory is well developed for the hyperbolic dynamicswhich is characterized by the presence of expanding andcontracting directions for derivativesThe study of shadowingproblem was originated by Anosov [46] and Bowen [47]

Let (119872 dist) be a compact metric space and let 119891 119872 rarr

119872 be a homeomorphism (a discrete dynamical systemon119872)A set of points 119883 = 119909

119896 119896 isin Z is a 119889-pseudotrajectory

(119889 gt 0) of 119891 if

dist (119909119896+1

119891 (119909119896)) lt 119889 119896 isin Z (A1)

Here the notation dist(sdot sdot) denotes the distance in the phasespace between two geometric objects within the brackets

We say that 119891 has the shadowing property if given 120576 gt 0

there is 119889 gt 0 such that for any 119889-pseudotrajectory119883 = 119909119896

119896 isin Z there exists a corresponding trajectory 119884 = 119910119896 119896 isin

Z which 120576-traces119883 that is

dist (119909119896 119910119896) lt 120576 119896 isin Z (A2)

The shadowing lemma for discrete dynamical systems[53] states that for each 120576 gt 0 there exists 119889 gt 0 such thateach 119889-pseudotrajectory can be 120576-shadowed

The definition of pseudotrajectory and shadowing lemmafor flows (continuous dynamical systems) [38] are morecomplicated than for discrete dynamical systems Let Φ119905 R times119872 rarr 119872 be a flow of a vector field 119883 on119872 A function119892 R rarr 119872 is a 119889-pseudotrajectory of the dynamical systemΦ119905 if the inequalities

dist (Φ119905 (119905 119892 (120591)) 119892 (120591 + 119905)) lt 119889 (A3)

hold for any 119905 isin [minus1 1] and 120591 isin R The ldquocontinuousrdquoshadowing lemma ensures that for the vector field 119883 gener-ating the flow Φ

119905 the shadowing property holds in a smallneighborhood of a compact hyperbolic set for dynamicalsystem Φ

119905It is very important to note that the shadowing problem

for continuous dynamical systems requires reparameteriza-tion of shadowing trajectories This is the case because for



S1

S2

S3

S4

S5

S6

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

Am

plitu

de

500

0

minus500

Am

plitu

de

500

0

minus500

Am

plitu

de

6000

4000

2000

0

minus2000

Am

plitu

de

1000

0

minus2000

minus1000 Am

plitu

de

8000

4000

0

minus4000

Am

plitu

de

1500

0

minus1500

Figure 7 Time dynamics of sensitivity functions with respect to parameter 119903 on the interval [0 25] for 1198880 = 09

continuous dynamical systems close points of pseudotra-jectory and true trajectory do not correspond to the samemoments of time A monotonically increasing homeomor-phism ℎ R rarr R such that ℎ(0) = 0 is called areparameterization and denoted by Rep For 120576 gt 0 Rep(120576)is defined as follows [38]

Rep (120576) = ℎ isin Rep

100381610038161003816100381610038161003816100381610038161003816

ℎ (1199051) minus ℎ (119905

2)

1199051minus 1199052

minus 1

100381610038161003816100381610038161003816100381610038161003816

le 120576

for any different 1199051 1199051isin R

(A4)

For simplicity we will consider a generic autonomousdynamical system with one parameter 120572

119889119909

119889119905

= 119891 (119909 120572) 119909 isin R119899

(A5)

The new sensitivity analysis method [36 37] is based on theldquocontinuousrdquo shadowing lemma and the following two basicassumptions

(a) Model state variables are considered over long timeinterval 119905 isin [0 119879] where 119879 rarr infin and an averaged


S1

S2

S3

S4

S5

S6


Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de3

6

0

minus3

Am

plitu

de

3

6

0

minus3

Am

plitu

de

2

4

0

minus2


performancemeasure ⟨119869(120572)⟩ is of themost interest forus

⟨119869 (120572)⟩ = lim119879rarrinfin

1

119879

int

119879

0

119869 (119909 (119905 120572) 120572) 119889119905 (A6)

(b) The dynamical systemunder consideration is ergodic

With these assumptions we can use the arbitrarily chosentrajectory of the system to trace the state variables alongthe orbit and calculate the performance measure 119869(120572) Forexample the arbitrary trajectory 119909(119905) can be chosen as a

solution of the model equation such that it is located nearbya certain reference trajectory 119909

119903(119905) Taking into account the

shadowing lemma the closest orbit 119909(119905) to 119909119903(119905) satisfies the

following constrained minimization problem [37]

min119909120591

1

119879

int

119879

0

[1003817100381710038171003817119909 (120591 (119905)) minus 119909

119903(119905)1003817100381710038171003817

2

+ 1205782

(

119889120591

119889119905

minus 1)

2

]119889119905

such that 119889119909

119889119905

= 119891 (119909 120572)

(A7)


Am

plitu

de

45

30

15

Am

plitu

de

40

20

00 5 10 15 20

Time0 5 10 15 20

Time

zrzs

Zr

Zs

Figure 9 Original trajectory (in red) and pseudoorbit (in blue) for the fast 119911 and slow 119885 variables for 1198880 = 001

0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

minus6

minus8

minus10

minus12

Am

plitu

de

0

20

10

minus10

minus20

Figure 10 Original trajectory (in red) and pseudoorbit (in blue) for fast 119909 and slow119883 variables for 1198880 = 001

where 120578 is the parameter that provides the same order ofmagnitude of the two terms in the integrand and 120591(119905) isa time transformation The second term in the integranddescribes reparameterization Problem (A7) is called thenonlinear Least Square Shadowing (LSS) problem and itssolution denoted by 119909(119879)

119904(119905 120572) and 120591

(119879)

119904(119905 120572) is a solution of

the model equation and time transformation that provides

the trajectory 119909(119879)119904(119905 120572) to be close to 119909

119903(119905) The performance

measure (A6) averaged over the time interval 119905 isin [0 119879] canbe then approximated as

⟨119869 (120572)⟩ asymp ⟨119869(119879)

119904(120572)⟩ =

1

120591 (119879) minus 120591 (0)

int

120591(119879)

120591(0)

119869 (119909(119879)

119904(119905 120572) 120572)

(A8)


since 119909(119879)119904(119905 120572) satisfies the model equation at a different 120572

The desired sensitivity estimate nabla120572⟨119869(119879)

119904(120572)⟩ can be computed

by solving the following linearized LLS problem [37]

min119878120583

1

119879

int

119879

0

[1198782

+ 1205782

1205832

] 119889119905

such that 119889119878

119889119905

=

120597119891

120597119909

119878 +

120597119891

120597120572

+ 120583119891 (119909119903 120572)

(A9)

The solutions of this problem 119878(119905) and 120583(119905) relate to thesolutions of the nonlinear LSS problem (A7) as follows

119878 (119905) =

119889

119889120572

(119909(119879)

119904(120591(119879)

119904(119905 120572) 120572))

120583 (119905) =

119889

119889120572

119889120591(119879)

119904(119905 120572)

119889119905

(A10)

The time-dependent parameter 120583 is called a time-dilationvariable and corresponds to the time transformation from the

shadowing lemma Using 119878 and 120583 we can estimate the desiredsensitivity (the derivative of the linearized performancemeasure (A8) with respect to the parameter 120572)

nabla120572⟨119869(119879)

119904(120572)⟩ asymp

1

119879

int

119879

0

(

120597119869

120597119909

119878 +

120597119869

120597120572

+ 120583 (119869 minus 119869)) 119889119905

where 119869 = 1

119879

int

119879

0

119869 119889119905

(A11)

Several numerical algorithms can be used to solve thelinearized LSS problem (A9) One such method is basedon variational calculus which is used to derive optimalityconditions representing a system of linear differential equa-tions that are then discretized and solved numerically tocalculate variables 119878 and 120583 [37] Let Δ119905 = 119879119898 be a uniformdiscretization time step then denoting 119909

119894+12= 119909119903((119894 +

12)Δ119905) 119909119894minus12

= 119909119903((119894 minus 12)Δ119905) 119894 = 0 119898 minus 1 and using

the trapezoidal rule to approximate the time derivative of 119878and 120583 we can obtain the following discrete Karush-Kuhn-Tucker (KKT) system [37 54]

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119868 119860119879

1

1205782

119861119879

1

119868 119862119879

1119860119879

2

1205782

119861119879

2

119868 119862119879

2

d d 119860119879

119898minus1

1205782

119861119879

119898minus1

119868 119862119879

119898minus1

119860111986111198621

119860211986121198622

d d

119860119898minus1

119861119898minus1

119862119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11987812

1205831

1198781+12

1205832

1198782+12

120583119898minus1

119878119898minus12

1198861

1198862

119886119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0

0

0

0

0

0

minus1198891

minus1198892

minus119889119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(A12)

where

119860119894= minus

119868

Δ119905

minus

120597119891

120597119909

(119909119894minus12

120572) 119861119894=

119909119894+12

minus 119909119894minus12

Δ119905

119862119894=

119868

Δ119905

minus

120597119891

120597119909

(119909119894+12

120572)

119889119894=

1

2

[

120597119891

120597120572

(119909119894minus12

120572) +

120597119891

120597120572

(119909119894+12

120572)]

(A13)

The KKT system can be solved using for example iterativemethods Gauss elimination approach for solving the KKTsystem was considered in [37] The convergence of the LSSmethod was proved in [55] Calculated variables 119878 and 120583 are

then used to determine the sensitivity estimate (A11)TheLSSalgorithm is summarized as follows

(1) Define a temporal grid 119905119894= 119894Δ119905 119894 = 0 119898 and then

discretize the model equation (A5) on this grid(2) Calculate a solution of (A5) 119909

119903on the time interval

[0 119879](3) Compute the vectors 119860

119894 119861119894 119862119894 and 119889

119894

(4) Solve the KKT system to obtain variables 119878119894and 120583

119894

(5) Compute the gradient components (sensitivity func-tions) (A11)

Anothermethod that also uses the concept of pseudoorbitshadowing in dynamical systems for sensitivity analysis ofchaotic oscillations was developed in [36] This method


0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

120575x

04

02

00

minus02

minus04

08

04

00

minus04

minus08

04

00

minus04

minus08

02

00

12

08

04

00

10

15

05

00

minus02

minus04

120575y

120575z

120575X

120575Y

120575Z

Figure 11 Difference between variables that correspond to the original trajectory and pseudoorbit for 1198880 = 001

is based on inverting the so-called ldquoshadowing operatorrdquothat requires calculating the Lyapunov characteristic (covari-ant) vectors However the computational cost needed forthe Lyapunov eigenvector decomposition is high when thedynamical system hasmany positive Lyapunov exponents Toillustrate this approach suppose that the sensitivity analysisof system (A5) aims to estimate the following sensitivitycoefficient 119878

120572= 120597119909120597120572 Let us introduce the following

transform 1199091015840(119909) = 119909 + 120575119909(119909) where 119909 and 1199091015840 are true

trajectory and pseudoorbit respectively The orbit 1199091015840 is

generated due to the variation in parameter120572 It can be shown[36] that 120575119891(119909) = 119860120575119909(119909) where

119860 = [minus(

120597119891

120597119909

) + (

119889

119889119905

)] (A14)

is a ldquoshadowrdquo operator Thus to find a pseudotrajectory weneed to solve the equation 120575119909 = 119860

minus1

120575119891 that is we mustnumerically invert the operator 119860 for a given 120575119891 To solvethis problem functions 120575119909 and 120575119891 are decomposed into theirconstituent Lyapunov covariant vectors 120592

1(119909) 120592

119899(119909)

120575119909 (119909) =

119899

sum

119894=1

120595119894(119909) 120592119894(119909) (A15a)

120575119891 (119909) =

119899

sum

119894=1

120593119894(119909) 120592119894(119909) (A15b)

Note that each 120592119894(119909) satisfies the following equation

119889120592119894(119909 (119905))

119889119905

=

120597119891

120597119909

120592119894(119909 (119905)) minus 120582

119894120592119894(119909 (119905)) (A16)

where 1205821 120582

119899are the Lyapunov exponents From (A14)

one can obtain

119860 (120595119894120592119894) = [minus120595

119894(119909)

120597119891

120597119909

+

119889120595119894(119909)

119889119905

] 120592119894(119909) + 120595

119894(119909)

119889120592119894(119909)

119889119905

(A17)

Substitution of (A16) into the last term of (A17) gives

119860 (120595119894120592119894) = [

119889120595119894(119909)

119889119905

minus 120582119894120595119894(119909)] 120592

119894(119909) (A18)

From (A15a) and (A15b) and (A18) and the relation 120575119891(119909) =119860120575119909(119909) we get

120575119891 (119909) =

119899

sum

119894=1

119860 (120595119894120592119894) =

119899

sum

119894=1

(

119889120595119894

119889119905

minus 120582119894120595119894)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120593119894

120592119894 (A19)

Equation (A19) gives the following relationship between120595119894(119909) and 120593

119894(119909) along the orbit

119889120595119894(119909)

119889119905

= 120593119894(119909) + 120582

119894120595119894(119909) (A20)

Thus we can calculate 120595119894(119909) using (A20) by first decom-

posing 120575119891 as a sum (A15b) and then the desired 120575119909 canbe obtained from (A15a) However if dynamical system hasa zero Lyapunov exponent 1205820

119894= 0 then the algorithm

described above fails to compute 120575119909 [36] The problem can


0 5 10 15 20

Time0 5 10 15 20

Time

Am

plitu

de

80

60

40

20Am

plitu

de

45

30

15

0

minus20

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

40

30

20

10

0

minus20

minus30

minus10

Am

plitu

de

20

10

0

minus20

minus10


be resolved by introducing a time-dilation variable 120583 thatsatisfies the following equation

120583 + ⟨1205930

119894⟩ = 0 (A21)

where

⟨1205930

119894⟩ = lim119879rarrinfin

1

119879

[1205950

119894(119909 (119879)) minus 120595

0

119894(119909 (0))] (A22)

In the presence of the variable 120583 the expression forcalculating 120575119909 takes the following form 120575119909 = 119860

minus1

(120575119891 + 120583119891)The supplement 120583119891 affects (A20) only for 1205820

119894= 0

1198891205950

119894(119909)

119889119905

= 1205930

119894(119909) + 120583 (A23)


0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

120575x

16

08

00

minus08

16

06

00

minus08

05

00

minus05

minus05

2

1

20

15

10

05

00

08

16

00

minus08

0

minus1

120575y

120575z

120575X

120575Y

120575Z


In general the procedure for solving a sensitivity analysisproblem represents the following set of steps

(1) Obtain initial conditions of the system attractor byintegrating the model equations (A5) from 119905

119863= minus20

to 1199050= 0 starting from random initial conditions

(2) Solve (A5) to obtain a trajectory 119909(119905) 119905 isin [0 20] onthe attractor

(3) Compute the Lyapunov exponents 120582119894and the Lya-

punov covariant vectors 120592119894(119909(119905)) 119894 = 1 119899

(4) Define 120575119891 = (120597119891120597120572)120575120572 and execute the Lyapunovspectrum decomposition of 120575119891 along the trajectory119909(119905) to obtain 120593

119894(119909) 119894 = 1 119899

(5) Calculate the time-dilation variable 120583 using (A21)(6) Compute 120575119909 along the trajectory 119909(119905)(7) Estimate the sensitivity 119878

120572= 120597119909120597120572 by averaging over

the time interval 119905 isin [0 20]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] F Rawlins S P Ballard K J Bovis et al ldquoThe met officeglobal four-dimensional variational data assimilation schemerdquo

Quarterly Journal of the Royal Meteorological Society vol 133no 623 pp 347ndash362 2007

[2] K Puri G Dietachmayer P Steinle et al ldquoImplementationof the initial ACCESS numerical weather prediction systemrdquoAustralian Meteorological and Oceanographic Journal vol 63no 2 pp 265ndash284 2013

[3] R Daley Atmospheric Data Analysis Cambridge UniversityPress New York NY USA 1991

[4] E Kalnay Atmospheric Modeling Data Assimilation and Pre-dictability Cambridge University Press New York NY USA2003

[5] F Rabier ldquoOverview of global data assimilation developmentsin numerical weather-prediction centresrdquo Quarterly Journal ofthe Royal Meteorological Society vol 131 no 613 pp 3215ndash32332005

[6] I M Navon ldquoData assimilation for numerical weather predic-tion a reviewrdquo in Data Assimilation for Atmospheric OceanicandHydrologic Applications S K Park and L Xu Eds SpringerNew York NY USA 2009

[7] D G Cacuci I M Navon and M Ionescu-Bujor Compu-tational Methods for Data Evaluation and Assimilation CRCPress Boca Raton Fla USA 2013

[8] I Sasaki ldquoAn objective analysis based on variational methodrdquoJournal of the Meteorological Society of Japan vol 36 no 3 pp29ndash30 1958

[9] J L Lions Optimal Control of Systems Governed by PartialDifferential Equations Springer Berlin Germany 1971

[10] V V Penenko and N N Obraztsov ldquoVariational method forassimilation of meteorological fieldsrdquoMeteorology and Hydrol-ogy no 11 pp 3ndash16 1976


[11] F-X Le Dimet ldquoUne application des methodes de controleoptimal rsquoa lrsquoanalyse varationellerdquo Rapport Scientifique LAMP63170 Universite Blaise Pascal Clermont-Ferrand France 1981

[12] O Talagrand ldquoA study of the dynamics of fourmdashdimensionaldata assimilationrdquo Tellus vol 33 no 1 pp 43ndash60 1981

[13] D G Cacuci ldquoSensitivity theory for nonlinear systems I Non-linear functional analysis approachrdquo Journal of MathematicalPhysics vol 22 no 12 pp 2794ndash2802 1981

[14] D G Cacuci ldquoSensitivity theory for nonlinear systems IIExtensions to additional classes of responsesrdquo Journal of Math-ematical Physics vol 22 no 12 pp 2803ndash2812 1981

[15] J M Lewis and J C Derber ldquoThe use of adjoint equationsto solve a variational adjustment problem with advective con-straintsrdquo Tellus A vol 37 no 4 pp 309ndash322 1985

[16] F-X Le Dimet and O Talagrand ldquoVariational algorithmsfor analysis and assimilation of meteorological observationstheoretical aspectsrdquo Tellus Series A vol 38 no 2 pp 97ndash1101986

[17] O Talagrand and P Courtier ldquoVariational assimilation ofmeteorological observations with the adjoint vorticity equationPart 1 Theoryrdquo Quarterly Journal of the Royal MeteorologicalSociety vol 113 pp 1311ndash1328 1987

[18] P Courtier andO Talagrand ldquoVariational assimilation of mete-orlogical observations with the adjoint vorticity equation IInumerical resultsrdquoQuarterly Journal of the RoyalMeteorologicalSociety vol 113 no 478 pp 1329ndash1347 1987

[19] F-X le Dimet H-E Ngodock B Luong and J VerronldquoSensitivity analysis in variational data assimilationrdquo Journal ofthe Meteorological Society of Japan vol 75 no 1 pp 245ndash2551997

[20] N L Baker and R Daley ldquoObservation and background adjointsensitivity in the adaptive observation-targeting problemrdquoQuarterly Journal of the Royal Meteorological Society vol 126no 565 pp 1431ndash1454 2000

[21] G Hello and F Bouttier ldquoUsing adjoint sensitivity as a localstructure function in variational data assimilationrdquo NonlinearProcesses in Geophysics vol 8 no 6 pp 347ndash355 2001

[22] R H Langland and N L Baker ldquoEstimation of observationimpact using theNRL atmospheric variational data assimilationadjoint systemrdquo Tellus Series A vol 56 no 3 pp 189ndash203 2004

[23] R M Errico ldquoInterpretations of an adjoint-derived observa-tional impact measurerdquo Tellus Series A Dynamic Meteorologyand Oceanography vol 59 no 2 pp 273ndash276 2007

[24] R Gelaro Y Zhu and R M Errico ldquoExamination of various-order adjoint-based approximations of observation impactrdquoMeteorologische Zeitschrift vol 16 no 6 pp 685ndash692 2007

[25] D N Daescu ldquoOn the sensitivity equations of four-dimensionalvariational (4D-Var) data assimilationrdquo Monthly WeatherReview vol 136 no 8 pp 3050ndash3065 2008

[26] Y Tremolet ldquoComputation of observation sensitivity and obser-vation impact in incremental variational data assimilationrdquoTellus Series A Dynamic Meteorology and Oceanography vol60 no 5 pp 964ndash978 2008

[27] D N Daescu and R Todling ldquoAdjoint estimation of thevariation in model functional output due to the assimilation ofdatardquo Monthly Weather Review vol 137 no 5 pp 1705ndash17162009

[28] R Gelaro and Y Zhu ldquoExamination of observation impactsderived from observing system experiments (OSEs) and adjointmodelsrdquo Tellus Series A Dynamic Meteorology and Oceanogra-phy vol 61 no 2 pp 179ndash193 2009

[29] D N Daescu and I M Navon ldquoSensitivity analysis in non-linear data assimilation theoretical aspects and applicationsrdquoin Advances Numerical Methods for Complex EnvironmentalModels Needs and Availability I Farago and Z Zlatev Eds pp276ndash300 Benham Science Publishers 2013

[30] D G Cacuci Sensitivity and Uncertainty Analysis Volume ITheory CRC Boca Raton Fla USA 2003

[31] DGCacuciM Ionescu-Bujor and IMNavon Sensitivity andUncertainty Analysis CRC Boca Raton Fla USA 2005

[32] E N Lorenz ldquoDeterministic nonperiodic flowrdquo Journal of theAtmospheric Sciences vol 20 no 2 pp 130ndash141 1963

[33] M Pena and E Kalnay ldquoSeparating fast and slow modes incoupled chaotic systemsrdquoNonlinear Processes in Geophysics vol11 no 3 pp 319ndash327 2004

[34] L Siqueira and B Kirtman ldquoPredictability of a low-orderinteractive ensemblerdquoNonlinear Processes in Geophysics vol 19no 2 pp 273ndash282 2012

[35] D J Lea M R Allen and T W N Haine ldquoSensitivity analysisof the climate of a chaotic systemrdquo Tellus vol 52 no 5 pp 523ndash532 2000

[36] Q Wang ldquoForward and adjoint sensitivity computation ofchaotic dynamical systemsrdquo Journal of Computational Physicsvol 235 no 15 pp 1ndash13 2013

[37] Q Wang R Hu and P Blonigan ldquoLeast squares shadowingsensitivity analysis of chaotic limit cycle oscillationsrdquo Journal ofComputational Physics vol 267 pp 210ndash224 2014

[38] S Y Pilyugin Shadowing in Dynamical Systems vol 1706 ofLecture Notes in Mathematics Springer Berlin Germany 1999

[39] K J Palmer Shadowing in Dynamical Systems Theory andApplications Kluwer Academic Dordrecht The Netherlands2000

[40] M G Rosenblum A S Pikovsky and J Kurths ldquoPhasesynchronization of chaotic oscillatorsrdquo Physical Review Lettersvol 76 no 11 pp 1804ndash1807 1996

[41] F Christiansen G Paladin and H H Rugh ldquoDetermination ofcorrelation spectra in chaotic systemsrdquo Physical Review Lettersvol 65 no 17 pp 2087ndash2090 1990

[42] C Liverani ldquoDecay of correlationsrdquoAnnals of Mathematics vol142 no 2 pp 239ndash301 1995

[43] S Panchev and T Spassova ldquoSimple general atmosphericcirculation and climate models with memoryrdquo Advances inAtmospheric Sciences vol 22 no 5 pp 765ndash769 2005

[44] E Rosenwasser andR Yusupov Sensitivity of Automatic ControlSystems CRC Press Boca Raton Fla USA 2000

[45] I M Navon X Zou J Derber and J Sela ldquoVariational dataassimilation with an adiabatic version of the NMC spectralmodelrdquoMonthly Weather Review vol 120 no 7 pp 1433ndash14461992

[46] D V Anosov ldquoOn class of invariant sets of smooth dynamicalsystemsrdquo in Proceedings of the 5th International Conference onNonlinear Oscillations vol 2 pp 39ndash45 Kiev Ukraine 1975

[47] R Bowen Equilibrium States and the Ergodic Theory of AnosovDiffeomorphisms vol 470 of Lecture Notes in MathematicsSpringer Berlin Germany 1975

[48] A Koropecki and E R Pujals ldquoSome consequences of theshadowing property in low dimensionsrdquo Ergodic Theory andDynamical Systems vol 34 no 4 pp 1273ndash1309 2014

[49] D Richeson and J Wiseman ldquoSymbolic dynamics for nonhy-perbolic systemsrdquo Proceedings of the American MathematicalSociety vol 138 no 12 pp 4373ndash4385 2010


[50] A A Petrov and S Y Pilyugin ldquoShadowing near nonhyperbolicfixed pointsrdquo Discrete and Continuous Dynamical Systems vol34 no 9 pp 3761ndash3772 2013

[51] S Y Pilyugin Spaces of Dynamical Systems De Gruyter BerlinGermany 2012

[52] A Katok and B Hasselblatt Introduction to the Modern Theoryof Dynamical Systems vol 54 of Encyclopedia of Mathematicsand its Applications Cambridge University Press New YorkNY USA 1995

[53] D V AnosovGeodesic Flows on Closed RiemannManifolds withNegative Curvature vol 90 of Proceedings of the Steklov Instituteof Mathematics 1967

[54] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press Cambridge UK 2004

[55] QWang ldquoConvergence of the least squares shadowing methodfor computing derivative of ergodic averagesrdquo SIAM Journal onNumerical Analysis vol 52 no 1 pp 156ndash170 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Applied ampEnvironmentalSoil Science

Volume 2014

Mining


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Journal of

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ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Geological ResearchJournal of


Geology Advances in


A comprehensive historical review and current status of 4D-Var are presented for example in [3ndash7] In our view pointsome papers and books such as [8ndash17] have contributedsignificantly to the development of mathematical foundationtheory and practice of 4D-Var

In general the main objective of 4D-Var is to defineas perfectly as possible the state of a dynamical system bycombining in statistically optimal manner the observationsof state variables of a real physical system together withcertain prior information In NWP this prior information isusually referred to as the background Mathematically 4D-Var procedures are formulated as an optimization problemin which the initial condition plays the role of control vectorand model equations are considered as constraints Whilethe theory of variational data assimilation is well establishedthere are a number of issues with practical implementationthat require additional consideration and study The perfor-mance of 4D-Var schemes depends on their key informationcomponents such as the available observations estimatesof the observation and background error covariances thatare quantified by the corresponding matrices as well asthe background state All of those information componentsstrongly impact the accuracy of calculated initial conditionsthus influencing the forecast quality Thus it is importantto estimate the sensitivity of a certain forecast aspect withrespect to variations in the observational data backgroundinformation and error statistics This problem is usuallyformulated within the framework of the adjoint-based sen-sitivity analysis (eg [18ndash29])

However an adjoint model used to calculate sensitivityfunctions is derived from a linearized forward propagationmodel and contains numerous input parameters Conse-quently linearization of strongly nonlinear NWPmodels andalso uncertainties in their numerous parameters generateerrors in the initial conditions obtained by data assimilationsystems The influence of linearization and parameter uncer-tainties on the results of data assimilation can be studied ide-ally for each particular NWPmodel using sensitivity analysis[30 31] Evenmore problems arise when considering coupled4D-Var data assimilation schemes since the atmosphere andocean have very different physical properties and time-spacespectrum of motions generating initialization shock Severalcoupling strategies are being developed for use in NWPsystems however all of them introduce issues that requireadditional detailed consideration For example the influenceof coupling strength on the initial conditions obtained by 4D-VAR procedures is one such issue that is important to studyand analyze

Good practice in the development of NWP models anddata assimilation systems requires evaluating the confidencein the model In this context it is important to estimate theinfluence of parameter variations on system dynamics and tofind those parameters that have the largest impact on systembehaviour Sensitivity analysis which is an essential elementof model building and quality assurance addresses this veryimportant issue

The exploration of coupled 4D-Var systems parameterestimation and sensitivity analysis require considerable com-putational resources For simple enough low-order coupled

models the computational cost is minor and for that reasonmodels of this class are widely used as simple test instrumentsto emulate more complex systems In this paper we describea coupled nonlinear dynamical system which is composedof fast (the ldquoatmosphererdquo) and slow (the ldquooceanrdquo) versions ofthe well-known Lorenz [32] model This low-order coupledsystem allows us to mimic the atmosphere-ocean systemand therefore serves as a key element of a theoretical andcomputational framework for the study of various aspectsof coupled 4D-Var procedures [33 34] Under certain con-ditions the Lorenz model exhibits a chaotic behaviour andusing conventional methods of sensitivity analysis can bequestionable in terms of interpretation of the obtained results[35ndash37] The ldquoshadowingrdquo method [36 37] for estimatingthe system sensitivity to variations in its parameters allowsus to calculate the average along the trajectory sensitivitiesand therefore to make a clear conclusion with respect to thesystem sensitivity to its parameters This method is based onthe pseudoorbit shadowing in dynamical systems [38 39]Calculated sensitivity coefficients obtained via conventionalmethods and the ldquoshadowingrdquo approach are presented in thepaper

We also succinctly consider commonly used techniquesfor sensitivity analysis and parameter estimations of dynam-ical systems and study the influence of coupling strengthparameter on the dynamical behaviour of the coupled systemusing Lyapunov characteristic exponent analysis It was foundthat the coupling strength parameter strongly affects thesystem dynamics both quantitatively and qualitatively Thisfact should be taken into consideration when choosing thecoupling strategy in data assimilation systems

2 Low-Order Coupled Dynamical System

In this section we consider a low-order coupled nonlineardynamical system obtained by coupling of two versions ofthe original Lorenzmodel (L63) [32] with distinct time scaleswhich differ by a factor 120576 (eg [33 34])

= 120590 (119910 minus 119909) minus 119888 (119886119883 + 119896)

119910 = 119903119909 minus 119910 minus 119909119911 + 119888 (119886119884 + 119896)

= 119909119910 minus 119887119911 + 119888119911119885

(1a)

= 120576120590 (119884 minus 119883) minus 119888 (119909 + 119896)

= 120576 (119903119883 minus 119884 minus 119886119883119885) + 119888 (119910 + 119896)

119885 = 120576 (119886119883119884 minus 119887119885) minus 119888

119911119911

(1b)

where lower case letters represent the fast subsystem andcapital letters the slow subsystem 120590 119903 and 119887 are theparameters of L63 model 119888 is a coupling strength parameterfor the 119909 and 119910 variables 119888

119911is a coupling strength parameter

for 119911 119896 is an ldquouncenteringrdquo parameter and 119886 is a parameterrepresenting the amplitude scale factor The value 119886 = 1

indicates that two systems have the same amplitude scaleThus the state vector of the coupled model (1a) and (1b) isx = (119909 119910 119911 119883 119884 119885)

119879 and the model parameter vector is




119879


119889x119889119905

= (L +Q) x (2)


L =

[

[

[

[

[

[

[

[

[

[

[

[

minus120590 120590 0

119903 minus1 minus119909 0

0 119909 minus119887

minus120576120590 120576120590 0

0 120576119903 minus120576 minus120576119883

0 120576119883 minus120576119887

]

]

]

]

]

]

]

]

]

]

]

]

Q =

[

[

[

[

[

[

[

[

[

[

[

[

minus119888 0 0

0 0 119888 0

0 0 119888

minus119888 0 0

0 119888 0 0

0 0 minus119888

]

]

]

]

]

]

]

]

]

]

]

]

(3)


1205900

= 10 1199030

= 28 1198870

=

8

3

1205760

= 01 1198880

isin [01 12]

(4)





Lyap

unov

expo

nent

s

08

06

04

02

00

12058211205822




119894119873minus1


119862 (119904) = ⟨119909119894119909119894+119904⟩ minus ⟨119909

119894⟩ ⟨119909119894+119904⟩ (5)


119862 (119904) = ⟨119906119898119906119898+119904

⟩ minus ⟨119906⟩2

(6)







119889x (119905)119889119905

= 119891 (x (119905) 120572 (119905)) 119905 isin [0 120591] = T x (0) = x0 (7)


0


x119894+1

= M119894119894+1

(x119894) + 120576119898

119894 (8)



119894M119894119894+1


119894to time 119905

119894+1for 119894 =




0 (8) uniquely




following equation

y0119894= H119894(x119894) + 1205760

119894 (9)

where H119894




119894isin R119898times119898








x1198870= x1199050+ 120576119887

(10)








x1198860= arg min 119869 (x) (11)



x119894+1

= M0119894+1

(x0) 119894 = 1 119873 (12)


119869 (x0) =

1

2

(x0minus x1198870)

119879


+

1

2

119873

sum

119894=0

(H119894(x119894) minus y0119894)

119879

Rminus1119894(H119894(x119894) minus y0119894)

(13)



119873

sum

119894=0


(14)

where M1198790119894


= M10158400119894(x119894) and H119879





120576119898

isin N (0Q) (15)


119869119898=

1

2

119873

sum

119894=1

(x119894minusM119894minus1119894


(x119894minus1)) (16)



10

05

00

10

05

00

10

05

00

minus05

10

05

00

minus05

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus100 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

c = 015c = 015

c = 05c = 05

c = 08c = 08

c = 10c = 10




10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

c = 015c = 015

c = 05c = 05

c = 08c = 08

c = 10c = 10









model parameter 120572119895[30 31 44]

119878119894119895equiv

120597119909119894

120597120572119895

1003816100381610038161003816100381610038161003816100381610038161003816120572119895=120572

0

119895

= lim120575120572rarr0

[

[

119909119894(1205720

119895+ 120575120572119895) minus 119909119894(1205720

119895)

120575120572119895

]

]

(17)


119895



119889S119895

119889119905

= M sdot S119895+D119895

119895 = 1 119898 (18)

where S119895= (120597x120597120572

119895) = (119878

1119895 1198782119895 119878






119889x119889119905

= 119891 (x120572) x (0) = x0

119889S119895

119889119905

= M sdot S119895+D119895 S119895(0) = S

1198950

(19)










J (x120572) = int

120591

0

Φ (119905 x120572) 119889119905 (20)


nabla120572J (x01205720) = (

119889J

1198891205721

119889J

119889120572119898

)

119879100381610038161003816100381610038161003816100381610038161003816x0 1205720

(21)


119889J

119889120572119895

10038161003816100381610038161003816100381610038161003816100381610038161205720

119895

asymp

J (x0 + 120575x0 12057201 120572

0

119895+ 120575120572119895 120572

0

119898) minusJ (x01205720)

120575120572119895

(22)


119894 Note that

119889J

119889120572119895

=

119899

sum

119894=1

120597J

120597119909119894

120597119909119894

120597120572119895

+

120597J

120597120572119895

=

119899

sum

119894=1

119878119894119895

120597J

120597119909119894

+

120597J

120597120572119895

(23)







120575J (x01205720 120575x 120575120572) = int

120591

0

(

120597Φ

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575x + 120597Φ

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575120572)119889119905

(24)


120597120575x120597119905

=

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575x + 120597119891

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575120572

119905 isin [0 120591] 120575x (0) = 120575x0

(25)








nabla120572J (x01205720) = int

120591

0

[

120597Φ

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

minus (

120597119891

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

)

119879

sdot xlowast]119889119905 (26)


minus

120597xlowast

120597119905

minus (

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

)

119879

xlowast = minus

120597Φ

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

119905 isin [0 120591] xlowast (120591) = 0

(27)












x (119905119863) = (001 001 001 002 002 002)

119879 (28)




119894) and


119891are obtained



119896) and the




119890 (x119891) = [

[

(x119891 minus x119905)119879

(x119891 minus x119905)

(x119905)119879 x119905]

]

12

(29)


119891














119894)








11990325

01273 01266 02871 02796 04081 0389011988825

00584 00583 01312 01300 01866 0183311990350

01663 01689 03617 03202 05004 0574211988850

00705 00704 01583 01569 02250 02210


0= 1205902


119887= 02 is the


119894= R = 120590

2

0I




119890119877=

M (x + 120577120575x) minusM (x)M (x) 120577120575x

(31)


rarr 0




⟨M120575xM120575x⟩ = ⟨120575xM119879M120575x⟩ (32)




119869 (x0+ 120577120575x) asymp 119869 (x

0) + 120577 (120575x)119879 nabla119869 (x

0) (33)


Ψ (120577) =

119869 (x0+ 120577120575x) minus 119869 (x

0)

120577 (120575x)119879 nabla119869 (x0)

(34)


120575x =nabla119869 (x0)

1003817100381710038171003817nabla119869 (x0)1003817100381710038171003817

(35)



times x0

120577 119890119877



minus2

times x0

120577 Ψ (120577) log10(1003816100381610038161003816Ψ (120577) minus 1

1003816100381610038161003816)



gradient test





1198781119888=

120597119909

120597119888

1198782119888=

120597119910

120597119888

1198783119888=

120597119911

120597119888

1198784119888=

120597119883

120597119888

1198785119888=

120597119884

120597119888

1198786119888=

120597119885

120597119888

(36)


1198781119888= 120590 (119878

2119888minus 1198781119888) minus 119888119878

4119888minus 119883


1198783119888= 1199091198782119888+ 1199101198781119888minus 1198871198783119888+ 1198881198786119888+ 119885

1198784119888= 120576120590 (119878

5119888minus 1198784119888) minus 119888119878

1119888minus 119909

1198785119888= 120576 (119903119878


2119888+ 119910

1198786119888= 120576 (119883119878

5119888+ 1198841198784119888minus 1198871198786119888) minus 119888119878

3119888minus 119911

(37)





nabla120572⟨J (120572)⟩ = ⟨nabla

120572J (120572)⟩ (38)



int

119879

0

lim120575120572rarr0

[

J (120572 + 120575120572) minusJ (120572)

120575120572

] 119889119905 (39)





= 28


+120575119903 where 120575119903 = 0011199030

=


1198781119903=

120597119909

120597119903

1198782119903=

120597119910

120597119903

1198783119903=

120597119911

120597119903

1198784119903=

120597119883

120597119903

1198785119903=

120597119884

120597119903

1198786119903=

120597119885

120597119903

(40)


1198781119903= 120590 (119878

2119903minus 1198781119903) minus 119888119878

4119903

1198782119903= 119909 + 119903119878

1119903minus 1198782119903minus (1199091198783119903+ 1199111198781119903) + 119888119878

5119903

1198783119903= (119909119878

2119903+ 1199101198781119903) minus 119887119878

3119903+ 1198881198786119903

1198784119903= 120576120590 (119878

5119903minus 1198784119903) minus 119888119878

1119903

1198785119903= 120576 [119883 + 119903119878

4119903minus 1198785119903minus (119883119878

6119903+ 1198851198784119903)] + 119888119878

2119903

1198786119903= 120576 [(119883119878

5119903+ 1198841198784119903) minus 119887119878

6119903] minus 119888119878

3119903

(41)






0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time


Am

plitu

de

2000

0

minus2000

Am

plitu

de

3000

0

minus3000

Am

plitu

deA

mpl

itude

2000

0

minus2000

Am

plitu

de

3000

0

minus3000

Am

plitu

de

2000

0

minus2000

1000

0

minus1000

S1

S2

S3

S4

S5

S6









Am

plitu

de

S1

S2

S3

S4

S5

S6

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

200

100

0

minus100

Am

plitu

de

100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de

50

0

minus50minus200



119888 120597119885120597119903 120597119884120597119903 120597119883120597119903 120597119911120597119903 120597119910120597119903 120597119909120597119903



Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

3

0

minus3

Am

plitu

de

3

0

minus3

minus6

Δx

Δy

Δz

ΔXΔY

ΔZ









Appendix





(119889 gt 0) of 119891 if

dist (119909119896+1

119891 (119909119896)) lt 119889 119896 isin Z (A1)






dist (119909119896 119910119896) lt 120576 119896 isin Z (A2)



dist (Φ119905 (119905 119892 (120591)) 119892 (120591 + 119905)) lt 119889 (A3)







S1

S2

S3

S4

S5

S6

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

Am

plitu

de

500

0

minus500

Am

plitu

de

500

0

minus500

Am

plitu

de

6000

4000

2000

0

minus2000

Am

plitu

de

1000

0

minus2000

minus1000 Am

plitu

de

8000

4000

0

minus4000

Am

plitu

de

1500

0

minus1500




100381610038161003816100381610038161003816100381610038161003816

ℎ (1199051) minus ℎ (119905

2)

1199051minus 1199052

minus 1

100381610038161003816100381610038161003816100381610038161003816

le 120576


(A4)


119889119909

119889119905

= 119891 (119909 120572) 119909 isin R119899

(A5)




S1

S2

S3

S4

S5

S6


Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de3

6

0

minus3

Am

plitu

de

3

6

0

minus3

Am

plitu

de

2

4

0

minus2



⟨119869 (120572)⟩ = lim119879rarrinfin

1

119879

int

119879

0

119869 (119909 (119905 120572) 120572) 119889119905 (A6)







min119909120591

1

119879

int

119879

0

[1003817100381710038171003817119909 (120591 (119905)) minus 119909

119903(119905)1003817100381710038171003817

2

+ 1205782

(

119889120591

119889119905

minus 1)

2

]119889119905

such that 119889119909

119889119905

= 119891 (119909 120572)

(A7)


Am

plitu

de

45

30

15

Am

plitu

de

40

20

00 5 10 15 20

Time0 5 10 15 20

Time

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

minus6

minus8

minus10

minus12

Am

plitu

de

0

20

10

minus10

minus20



119904(119905 120572) and 120591

(119879)

119904(119905 120572) is a solution of





⟨119869 (120572)⟩ asymp ⟨119869(119879)

119904(120572)⟩ =

1

120591 (119879) minus 120591 (0)

int

120591(119879)

120591(0)

119869 (119909(119879)

119904(119905 120572) 120572)

(A8)






min119878120583

1

119879

int

119879

0

[1198782

+ 1205782

1205832

] 119889119905

such that 119889119878

119889119905

=

120597119891

120597119909

119878 +

120597119891

120597120572

+ 120583119891 (119909119903 120572)

(A9)


119878 (119905) =

119889

119889120572

(119909(119879)

119904(120591(119879)

119904(119905 120572) 120572))

120583 (119905) =

119889

119889120572

119889120591(119879)

119904(119905 120572)

119889119905

(A10)



nabla120572⟨119869(119879)

119904(120572)⟩ asymp

1

119879

int

119879

0

(

120597119869

120597119909

119878 +

120597119869

120597120572

+ 120583 (119869 minus 119869)) 119889119905

where 119869 = 1

119879

int

119879

0

119869 119889119905

(A11)


119894+12= 119909119903((119894 +

12)Δ119905) 119909119894minus12



[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119868 119860119879

1

1205782

119861119879

1

119868 119862119879

1119860119879

2

1205782

119861119879

2

119868 119862119879

2

d d 119860119879

119898minus1

1205782

119861119879

119898minus1

119868 119862119879

119898minus1

119860111986111198621

119860211986121198622

d d

119860119898minus1

119861119898minus1

119862119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11987812

1205831

1198781+12

1205832

1198782+12

120583119898minus1

119878119898minus12

1198861

1198862

119886119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0

0

0

0

0

0

minus1198891

minus1198892


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(A12)

where

119860119894= minus

119868

Δ119905

minus

120597119891

120597119909

(119909119894minus12

120572) 119861119894=

119909119894+12


Δ119905

119862119894=

119868

Δ119905

minus

120597119891

120597119909

(119909119894+12

120572)

119889119894=

1

2

[

120597119891

120597120572

(119909119894minus12

120572) +

120597119891

120597120572

(119909119894+12

120572)]

(A13)







119894 119861119894 119862119894 and 119889

119894


119894




0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

120575x

04

02

00

minus02

minus04

08

04

00

minus04

minus08

04

00

minus04

minus08

02

00

12

08

04

00

10

15

05

00

minus02

minus04

120575y

120575z

120575X

120575Y

120575Z







119860 = [minus(

120597119891

120597119909

) + (

119889

119889119905

)] (A14)


minus1


1(119909) 120592

119899(119909)

120575119909 (119909) =

119899

sum

119894=1

120595119894(119909) 120592119894(119909) (A15a)

120575119891 (119909) =

119899

sum

119894=1

120593119894(119909) 120592119894(119909) (A15b)


119889120592119894(119909 (119905))

119889119905

=

120597119891

120597119909

120592119894(119909 (119905)) minus 120582

119894120592119894(119909 (119905)) (A16)

where 1205821 120582


one can obtain

119860 (120595119894120592119894) = [minus120595

119894(119909)

120597119891

120597119909

+

119889120595119894(119909)

119889119905

] 120592119894(119909) + 120595

119894(119909)

119889120592119894(119909)

119889119905

(A17)


119860 (120595119894120592119894) = [

119889120595119894(119909)

119889119905

minus 120582119894120595119894(119909)] 120592

119894(119909) (A18)


120575119891 (119909) =

119899

sum

119894=1

119860 (120595119894120592119894) =

119899

sum

119894=1

(

119889120595119894

119889119905

minus 120582119894120595119894)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120593119894

120592119894 (A19)



119889120595119894(119909)

119889119905

= 120593119894(119909) + 120582

119894120595119894(119909) (A20)






0 5 10 15 20

Time0 5 10 15 20

Time

Am

plitu

de

80

60

40

20Am

plitu

de

45

30

15

0

minus20

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

40

30

20

10

0

minus20

minus30

minus10

Am

plitu

de

20

10

0

minus20

minus10



120583 + ⟨1205930

119894⟩ = 0 (A21)

where

⟨1205930


1

119879

[1205950

119894(119909 (119879)) minus 120595

0

119894(119909 (0))] (A22)


minus1


119894= 0

1198891205950

119894(119909)

119889119905

= 1205930

119894(119909) + 120583 (A23)


0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

120575x

16

08

00

minus08

16

06

00

minus08

05

00

minus05

minus05

2

1

20

15

10

05

00

08

16

00

minus08

0

minus1

120575y

120575z

120575X

120575Y

120575Z




119863= minus20






119894(119909) 119894 = 1 119899






References




































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in




119879


119889x119889119905

= (L +Q) x (2)


L =

[

[

[

[

[

[

[

[

[

[

[

[

minus120590 120590 0

119903 minus1 minus119909 0

0 119909 minus119887

minus120576120590 120576120590 0

0 120576119903 minus120576 minus120576119883

0 120576119883 minus120576119887

]

]

]

]

]

]

]

]

]

]

]

]

Q =

[

[

[

[

[

[

[

[

[

[

[

[

minus119888 0 0

0 0 119888 0

0 0 119888

minus119888 0 0

0 119888 0 0

0 0 minus119888

]

]

]

]

]

]

]

]

]

]

]

]

(3)


1205900

= 10 1199030

= 28 1198870

=

8

3

1205760

= 01 1198880

isin [01 12]

(4)





Lyap

unov

expo

nent

s

08

06

04

02

00

12058211205822




119894119873minus1


119862 (119904) = ⟨119909119894119909119894+119904⟩ minus ⟨119909

119894⟩ ⟨119909119894+119904⟩ (5)


119862 (119904) = ⟨119906119898119906119898+119904

⟩ minus ⟨119906⟩2

(6)







119889x (119905)119889119905

= 119891 (x (119905) 120572 (119905)) 119905 isin [0 120591] = T x (0) = x0 (7)


0


x119894+1

= M119894119894+1

(x119894) + 120576119898

119894 (8)



119894M119894119894+1


119894to time 119905

119894+1for 119894 =




0 (8) uniquely




following equation

y0119894= H119894(x119894) + 1205760

119894 (9)

where H119894




119894isin R119898times119898








x1198870= x1199050+ 120576119887

(10)








x1198860= arg min 119869 (x) (11)



x119894+1

= M0119894+1

(x0) 119894 = 1 119873 (12)


119869 (x0) =

1

2

(x0minus x1198870)

119879


+

1

2

119873

sum

119894=0

(H119894(x119894) minus y0119894)

119879

Rminus1119894(H119894(x119894) minus y0119894)

(13)



119873

sum

119894=0


(14)

where M1198790119894


= M10158400119894(x119894) and H119879





120576119898

isin N (0Q) (15)


119869119898=

1

2

119873

sum

119894=1

(x119894minusM119894minus1119894


(x119894minus1)) (16)



10

05

00

10

05

00

10

05

00

minus05

10

05

00

minus05

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus100 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

c = 015c = 015

c = 05c = 05

c = 08c = 08

c = 10c = 10




10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

c = 015c = 015

c = 05c = 05

c = 08c = 08

c = 10c = 10









model parameter 120572119895[30 31 44]

119878119894119895equiv

120597119909119894

120597120572119895

1003816100381610038161003816100381610038161003816100381610038161003816120572119895=120572

0

119895

= lim120575120572rarr0

[

[

119909119894(1205720

119895+ 120575120572119895) minus 119909119894(1205720

119895)

120575120572119895

]

]

(17)


119895



119889S119895

119889119905

= M sdot S119895+D119895

119895 = 1 119898 (18)

where S119895= (120597x120597120572

119895) = (119878

1119895 1198782119895 119878






119889x119889119905

= 119891 (x120572) x (0) = x0

119889S119895

119889119905

= M sdot S119895+D119895 S119895(0) = S

1198950

(19)










J (x120572) = int

120591

0

Φ (119905 x120572) 119889119905 (20)


nabla120572J (x01205720) = (

119889J

1198891205721

119889J

119889120572119898

)

119879100381610038161003816100381610038161003816100381610038161003816x0 1205720

(21)


119889J

119889120572119895

10038161003816100381610038161003816100381610038161003816100381610038161205720

119895

asymp

J (x0 + 120575x0 12057201 120572

0

119895+ 120575120572119895 120572

0

119898) minusJ (x01205720)

120575120572119895

(22)


119894 Note that

119889J

119889120572119895

=

119899

sum

119894=1

120597J

120597119909119894

120597119909119894

120597120572119895

+

120597J

120597120572119895

=

119899

sum

119894=1

119878119894119895

120597J

120597119909119894

+

120597J

120597120572119895

(23)







120575J (x01205720 120575x 120575120572) = int

120591

0

(

120597Φ

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575x + 120597Φ

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575120572)119889119905

(24)


120597120575x120597119905

=

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575x + 120597119891

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575120572

119905 isin [0 120591] 120575x (0) = 120575x0

(25)








nabla120572J (x01205720) = int

120591

0

[

120597Φ

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

minus (

120597119891

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

)

119879

sdot xlowast]119889119905 (26)


minus

120597xlowast

120597119905

minus (

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

)

119879

xlowast = minus

120597Φ

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

119905 isin [0 120591] xlowast (120591) = 0

(27)












x (119905119863) = (001 001 001 002 002 002)

119879 (28)




119894) and


119891are obtained



119896) and the




119890 (x119891) = [

[

(x119891 minus x119905)119879

(x119891 minus x119905)

(x119905)119879 x119905]

]

12

(29)


119891














119894)








11990325

01273 01266 02871 02796 04081 0389011988825

00584 00583 01312 01300 01866 0183311990350

01663 01689 03617 03202 05004 0574211988850

00705 00704 01583 01569 02250 02210


0= 1205902


119887= 02 is the


119894= R = 120590

2

0I




119890119877=

M (x + 120577120575x) minusM (x)M (x) 120577120575x

(31)


rarr 0




⟨M120575xM120575x⟩ = ⟨120575xM119879M120575x⟩ (32)




119869 (x0+ 120577120575x) asymp 119869 (x

0) + 120577 (120575x)119879 nabla119869 (x

0) (33)


Ψ (120577) =

119869 (x0+ 120577120575x) minus 119869 (x

0)

120577 (120575x)119879 nabla119869 (x0)

(34)


120575x =nabla119869 (x0)

1003817100381710038171003817nabla119869 (x0)1003817100381710038171003817

(35)



times x0

120577 119890119877



minus2

times x0

120577 Ψ (120577) log10(1003816100381610038161003816Ψ (120577) minus 1

1003816100381610038161003816)



gradient test





1198781119888=

120597119909

120597119888

1198782119888=

120597119910

120597119888

1198783119888=

120597119911

120597119888

1198784119888=

120597119883

120597119888

1198785119888=

120597119884

120597119888

1198786119888=

120597119885

120597119888

(36)


1198781119888= 120590 (119878

2119888minus 1198781119888) minus 119888119878

4119888minus 119883


1198783119888= 1199091198782119888+ 1199101198781119888minus 1198871198783119888+ 1198881198786119888+ 119885

1198784119888= 120576120590 (119878

5119888minus 1198784119888) minus 119888119878

1119888minus 119909

1198785119888= 120576 (119903119878


2119888+ 119910

1198786119888= 120576 (119883119878

5119888+ 1198841198784119888minus 1198871198786119888) minus 119888119878

3119888minus 119911

(37)





nabla120572⟨J (120572)⟩ = ⟨nabla

120572J (120572)⟩ (38)



int

119879

0

lim120575120572rarr0

[

J (120572 + 120575120572) minusJ (120572)

120575120572

] 119889119905 (39)





= 28


+120575119903 where 120575119903 = 0011199030

=


1198781119903=

120597119909

120597119903

1198782119903=

120597119910

120597119903

1198783119903=

120597119911

120597119903

1198784119903=

120597119883

120597119903

1198785119903=

120597119884

120597119903

1198786119903=

120597119885

120597119903

(40)


1198781119903= 120590 (119878

2119903minus 1198781119903) minus 119888119878

4119903

1198782119903= 119909 + 119903119878

1119903minus 1198782119903minus (1199091198783119903+ 1199111198781119903) + 119888119878

5119903

1198783119903= (119909119878

2119903+ 1199101198781119903) minus 119887119878

3119903+ 1198881198786119903

1198784119903= 120576120590 (119878

5119903minus 1198784119903) minus 119888119878

1119903

1198785119903= 120576 [119883 + 119903119878

4119903minus 1198785119903minus (119883119878

6119903+ 1198851198784119903)] + 119888119878

2119903

1198786119903= 120576 [(119883119878

5119903+ 1198841198784119903) minus 119887119878

6119903] minus 119888119878

3119903

(41)






0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time


Am

plitu

de

2000

0

minus2000

Am

plitu

de

3000

0

minus3000

Am

plitu

deA

mpl

itude

2000

0

minus2000

Am

plitu

de

3000

0

minus3000

Am

plitu

de

2000

0

minus2000

1000

0

minus1000

S1

S2

S3

S4

S5

S6









Am

plitu

de

S1

S2

S3

S4

S5

S6

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

200

100

0

minus100

Am

plitu

de

100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de

50

0

minus50minus200



119888 120597119885120597119903 120597119884120597119903 120597119883120597119903 120597119911120597119903 120597119910120597119903 120597119909120597119903



Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

3

0

minus3

Am

plitu

de

3

0

minus3

minus6

Δx

Δy

Δz

ΔXΔY

ΔZ









Appendix





(119889 gt 0) of 119891 if

dist (119909119896+1

119891 (119909119896)) lt 119889 119896 isin Z (A1)






dist (119909119896 119910119896) lt 120576 119896 isin Z (A2)



dist (Φ119905 (119905 119892 (120591)) 119892 (120591 + 119905)) lt 119889 (A3)







S1

S2

S3

S4

S5

S6

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

Am

plitu

de

500

0

minus500

Am

plitu

de

500

0

minus500

Am

plitu

de

6000

4000

2000

0

minus2000

Am

plitu

de

1000

0

minus2000

minus1000 Am

plitu

de

8000

4000

0

minus4000

Am

plitu

de

1500

0

minus1500




100381610038161003816100381610038161003816100381610038161003816

ℎ (1199051) minus ℎ (119905

2)

1199051minus 1199052

minus 1

100381610038161003816100381610038161003816100381610038161003816

le 120576


(A4)


119889119909

119889119905

= 119891 (119909 120572) 119909 isin R119899

(A5)




S1

S2

S3

S4

S5

S6


Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de3

6

0

minus3

Am

plitu

de

3

6

0

minus3

Am

plitu

de

2

4

0

minus2



⟨119869 (120572)⟩ = lim119879rarrinfin

1

119879

int

119879

0

119869 (119909 (119905 120572) 120572) 119889119905 (A6)







min119909120591

1

119879

int

119879

0

[1003817100381710038171003817119909 (120591 (119905)) minus 119909

119903(119905)1003817100381710038171003817

2

+ 1205782

(

119889120591

119889119905

minus 1)

2

]119889119905

such that 119889119909

119889119905

= 119891 (119909 120572)

(A7)


Am

plitu

de

45

30

15

Am

plitu

de

40

20

00 5 10 15 20

Time0 5 10 15 20

Time

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

minus6

minus8

minus10

minus12

Am

plitu

de

0

20

10

minus10

minus20



119904(119905 120572) and 120591

(119879)

119904(119905 120572) is a solution of





⟨119869 (120572)⟩ asymp ⟨119869(119879)

119904(120572)⟩ =

1

120591 (119879) minus 120591 (0)

int

120591(119879)

120591(0)

119869 (119909(119879)

119904(119905 120572) 120572)

(A8)






min119878120583

1

119879

int

119879

0

[1198782

+ 1205782

1205832

] 119889119905

such that 119889119878

119889119905

=

120597119891

120597119909

119878 +

120597119891

120597120572

+ 120583119891 (119909119903 120572)

(A9)


119878 (119905) =

119889

119889120572

(119909(119879)

119904(120591(119879)

119904(119905 120572) 120572))

120583 (119905) =

119889

119889120572

119889120591(119879)

119904(119905 120572)

119889119905

(A10)



nabla120572⟨119869(119879)

119904(120572)⟩ asymp

1

119879

int

119879

0

(

120597119869

120597119909

119878 +

120597119869

120597120572

+ 120583 (119869 minus 119869)) 119889119905

where 119869 = 1

119879

int

119879

0

119869 119889119905

(A11)


119894+12= 119909119903((119894 +

12)Δ119905) 119909119894minus12



[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119868 119860119879

1

1205782

119861119879

1

119868 119862119879

1119860119879

2

1205782

119861119879

2

119868 119862119879

2

d d 119860119879

119898minus1

1205782

119861119879

119898minus1

119868 119862119879

119898minus1

119860111986111198621

119860211986121198622

d d

119860119898minus1

119861119898minus1

119862119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11987812

1205831

1198781+12

1205832

1198782+12

120583119898minus1

119878119898minus12

1198861

1198862

119886119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0

0

0

0

0

0

minus1198891

minus1198892


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(A12)

where

119860119894= minus

119868

Δ119905

minus

120597119891

120597119909

(119909119894minus12

120572) 119861119894=

119909119894+12


Δ119905

119862119894=

119868

Δ119905

minus

120597119891

120597119909

(119909119894+12

120572)

119889119894=

1

2

[

120597119891

120597120572

(119909119894minus12

120572) +

120597119891

120597120572

(119909119894+12

120572)]

(A13)







119894 119861119894 119862119894 and 119889

119894


119894




0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

120575x

04

02

00

minus02

minus04

08

04

00

minus04

minus08

04

00

minus04

minus08

02

00

12

08

04

00

10

15

05

00

minus02

minus04

120575y

120575z

120575X

120575Y

120575Z







119860 = [minus(

120597119891

120597119909

) + (

119889

119889119905

)] (A14)


minus1


1(119909) 120592

119899(119909)

120575119909 (119909) =

119899

sum

119894=1

120595119894(119909) 120592119894(119909) (A15a)

120575119891 (119909) =

119899

sum

119894=1

120593119894(119909) 120592119894(119909) (A15b)


119889120592119894(119909 (119905))

119889119905

=

120597119891

120597119909

120592119894(119909 (119905)) minus 120582

119894120592119894(119909 (119905)) (A16)

where 1205821 120582


one can obtain

119860 (120595119894120592119894) = [minus120595

119894(119909)

120597119891

120597119909

+

119889120595119894(119909)

119889119905

] 120592119894(119909) + 120595

119894(119909)

119889120592119894(119909)

119889119905

(A17)


119860 (120595119894120592119894) = [

119889120595119894(119909)

119889119905

minus 120582119894120595119894(119909)] 120592

119894(119909) (A18)


120575119891 (119909) =

119899

sum

119894=1

119860 (120595119894120592119894) =

119899

sum

119894=1

(

119889120595119894

119889119905

minus 120582119894120595119894)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120593119894

120592119894 (A19)



119889120595119894(119909)

119889119905

= 120593119894(119909) + 120582

119894120595119894(119909) (A20)






0 5 10 15 20

Time0 5 10 15 20

Time

Am

plitu

de

80

60

40

20Am

plitu

de

45

30

15

0

minus20

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

40

30

20

10

0

minus20

minus30

minus10

Am

plitu

de

20

10

0

minus20

minus10



120583 + ⟨1205930

119894⟩ = 0 (A21)

where

⟨1205930


1

119879

[1205950

119894(119909 (119879)) minus 120595

0

119894(119909 (0))] (A22)


minus1


119894= 0

1198891205950

119894(119909)

119889119905

= 1205930

119894(119909) + 120583 (A23)


0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

120575x

16

08

00

minus08

16

06

00

minus08

05

00

minus05

minus05

2

1

20

15

10

05

00

08

16

00

minus08

0

minus1

120575y

120575z

120575X

120575Y

120575Z




119863= minus20






119894(119909) 119894 = 1 119899






References




































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in





119889x (119905)119889119905

= 119891 (x (119905) 120572 (119905)) 119905 isin [0 120591] = T x (0) = x0 (7)


0


x119894+1

= M119894119894+1

(x119894) + 120576119898

119894 (8)



119894M119894119894+1


119894to time 119905

119894+1for 119894 =




0 (8) uniquely




following equation

y0119894= H119894(x119894) + 1205760

119894 (9)

where H119894




119894isin R119898times119898








x1198870= x1199050+ 120576119887

(10)








x1198860= arg min 119869 (x) (11)



x119894+1

= M0119894+1

(x0) 119894 = 1 119873 (12)


119869 (x0) =

1

2

(x0minus x1198870)

119879


+

1

2

119873

sum

119894=0

(H119894(x119894) minus y0119894)

119879

Rminus1119894(H119894(x119894) minus y0119894)

(13)



119873

sum

119894=0


(14)

where M1198790119894


= M10158400119894(x119894) and H119879





120576119898

isin N (0Q) (15)


119869119898=

1

2

119873

sum

119894=1

(x119894minusM119894minus1119894


(x119894minus1)) (16)



10

05

00

10

05

00

10

05

00

minus05

10

05

00

minus05

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus100 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

c = 015c = 015

c = 05c = 05

c = 08c = 08

c = 10c = 10




10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

c = 015c = 015

c = 05c = 05

c = 08c = 08

c = 10c = 10









model parameter 120572119895[30 31 44]

119878119894119895equiv

120597119909119894

120597120572119895

1003816100381610038161003816100381610038161003816100381610038161003816120572119895=120572

0

119895

= lim120575120572rarr0

[

[

119909119894(1205720

119895+ 120575120572119895) minus 119909119894(1205720

119895)

120575120572119895

]

]

(17)


119895



119889S119895

119889119905

= M sdot S119895+D119895

119895 = 1 119898 (18)

where S119895= (120597x120597120572

119895) = (119878

1119895 1198782119895 119878






119889x119889119905

= 119891 (x120572) x (0) = x0

119889S119895

119889119905

= M sdot S119895+D119895 S119895(0) = S

1198950

(19)










J (x120572) = int

120591

0

Φ (119905 x120572) 119889119905 (20)


nabla120572J (x01205720) = (

119889J

1198891205721

119889J

119889120572119898

)

119879100381610038161003816100381610038161003816100381610038161003816x0 1205720

(21)


119889J

119889120572119895

10038161003816100381610038161003816100381610038161003816100381610038161205720

119895

asymp

J (x0 + 120575x0 12057201 120572

0

119895+ 120575120572119895 120572

0

119898) minusJ (x01205720)

120575120572119895

(22)


119894 Note that

119889J

119889120572119895

=

119899

sum

119894=1

120597J

120597119909119894

120597119909119894

120597120572119895

+

120597J

120597120572119895

=

119899

sum

119894=1

119878119894119895

120597J

120597119909119894

+

120597J

120597120572119895

(23)







120575J (x01205720 120575x 120575120572) = int

120591

0

(

120597Φ

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575x + 120597Φ

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575120572)119889119905

(24)


120597120575x120597119905

=

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575x + 120597119891

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575120572

119905 isin [0 120591] 120575x (0) = 120575x0

(25)








nabla120572J (x01205720) = int

120591

0

[

120597Φ

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

minus (

120597119891

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

)

119879

sdot xlowast]119889119905 (26)


minus

120597xlowast

120597119905

minus (

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

)

119879

xlowast = minus

120597Φ

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

119905 isin [0 120591] xlowast (120591) = 0

(27)












x (119905119863) = (001 001 001 002 002 002)

119879 (28)




119894) and


119891are obtained



119896) and the




119890 (x119891) = [

[

(x119891 minus x119905)119879

(x119891 minus x119905)

(x119905)119879 x119905]

]

12

(29)


119891














119894)








11990325

01273 01266 02871 02796 04081 0389011988825

00584 00583 01312 01300 01866 0183311990350

01663 01689 03617 03202 05004 0574211988850

00705 00704 01583 01569 02250 02210


0= 1205902


119887= 02 is the


119894= R = 120590

2

0I




119890119877=

M (x + 120577120575x) minusM (x)M (x) 120577120575x

(31)


rarr 0




⟨M120575xM120575x⟩ = ⟨120575xM119879M120575x⟩ (32)




119869 (x0+ 120577120575x) asymp 119869 (x

0) + 120577 (120575x)119879 nabla119869 (x

0) (33)


Ψ (120577) =

119869 (x0+ 120577120575x) minus 119869 (x

0)

120577 (120575x)119879 nabla119869 (x0)

(34)


120575x =nabla119869 (x0)

1003817100381710038171003817nabla119869 (x0)1003817100381710038171003817

(35)



times x0

120577 119890119877



minus2

times x0

120577 Ψ (120577) log10(1003816100381610038161003816Ψ (120577) minus 1

1003816100381610038161003816)



gradient test





1198781119888=

120597119909

120597119888

1198782119888=

120597119910

120597119888

1198783119888=

120597119911

120597119888

1198784119888=

120597119883

120597119888

1198785119888=

120597119884

120597119888

1198786119888=

120597119885

120597119888

(36)


1198781119888= 120590 (119878

2119888minus 1198781119888) minus 119888119878

4119888minus 119883


1198783119888= 1199091198782119888+ 1199101198781119888minus 1198871198783119888+ 1198881198786119888+ 119885

1198784119888= 120576120590 (119878

5119888minus 1198784119888) minus 119888119878

1119888minus 119909

1198785119888= 120576 (119903119878


2119888+ 119910

1198786119888= 120576 (119883119878

5119888+ 1198841198784119888minus 1198871198786119888) minus 119888119878

3119888minus 119911

(37)





nabla120572⟨J (120572)⟩ = ⟨nabla

120572J (120572)⟩ (38)



int

119879

0

lim120575120572rarr0

[

J (120572 + 120575120572) minusJ (120572)

120575120572

] 119889119905 (39)





= 28


+120575119903 where 120575119903 = 0011199030

=


1198781119903=

120597119909

120597119903

1198782119903=

120597119910

120597119903

1198783119903=

120597119911

120597119903

1198784119903=

120597119883

120597119903

1198785119903=

120597119884

120597119903

1198786119903=

120597119885

120597119903

(40)


1198781119903= 120590 (119878

2119903minus 1198781119903) minus 119888119878

4119903

1198782119903= 119909 + 119903119878

1119903minus 1198782119903minus (1199091198783119903+ 1199111198781119903) + 119888119878

5119903

1198783119903= (119909119878

2119903+ 1199101198781119903) minus 119887119878

3119903+ 1198881198786119903

1198784119903= 120576120590 (119878

5119903minus 1198784119903) minus 119888119878

1119903

1198785119903= 120576 [119883 + 119903119878

4119903minus 1198785119903minus (119883119878

6119903+ 1198851198784119903)] + 119888119878

2119903

1198786119903= 120576 [(119883119878

5119903+ 1198841198784119903) minus 119887119878

6119903] minus 119888119878

3119903

(41)






0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time


Am

plitu

de

2000

0

minus2000

Am

plitu

de

3000

0

minus3000

Am

plitu

deA

mpl

itude

2000

0

minus2000

Am

plitu

de

3000

0

minus3000

Am

plitu

de

2000

0

minus2000

1000

0

minus1000

S1

S2

S3

S4

S5

S6









Am

plitu

de

S1

S2

S3

S4

S5

S6

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

200

100

0

minus100

Am

plitu

de

100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de

50

0

minus50minus200



119888 120597119885120597119903 120597119884120597119903 120597119883120597119903 120597119911120597119903 120597119910120597119903 120597119909120597119903



Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

3

0

minus3

Am

plitu

de

3

0

minus3

minus6

Δx

Δy

Δz

ΔXΔY

ΔZ









Appendix





(119889 gt 0) of 119891 if

dist (119909119896+1

119891 (119909119896)) lt 119889 119896 isin Z (A1)






dist (119909119896 119910119896) lt 120576 119896 isin Z (A2)



dist (Φ119905 (119905 119892 (120591)) 119892 (120591 + 119905)) lt 119889 (A3)







S1

S2

S3

S4

S5

S6

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

Am

plitu

de

500

0

minus500

Am

plitu

de

500

0

minus500

Am

plitu

de

6000

4000

2000

0

minus2000

Am

plitu

de

1000

0

minus2000

minus1000 Am

plitu

de

8000

4000

0

minus4000

Am

plitu

de

1500

0

minus1500




100381610038161003816100381610038161003816100381610038161003816

ℎ (1199051) minus ℎ (119905

2)

1199051minus 1199052

minus 1

100381610038161003816100381610038161003816100381610038161003816

le 120576


(A4)


119889119909

119889119905

= 119891 (119909 120572) 119909 isin R119899

(A5)




S1

S2

S3

S4

S5

S6


Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de3

6

0

minus3

Am

plitu

de

3

6

0

minus3

Am

plitu

de

2

4

0

minus2



⟨119869 (120572)⟩ = lim119879rarrinfin

1

119879

int

119879

0

119869 (119909 (119905 120572) 120572) 119889119905 (A6)







min119909120591

1

119879

int

119879

0

[1003817100381710038171003817119909 (120591 (119905)) minus 119909

119903(119905)1003817100381710038171003817

2

+ 1205782

(

119889120591

119889119905

minus 1)

2

]119889119905

such that 119889119909

119889119905

= 119891 (119909 120572)

(A7)


Am

plitu

de

45

30

15

Am

plitu

de

40

20

00 5 10 15 20

Time0 5 10 15 20

Time

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

minus6

minus8

minus10

minus12

Am

plitu

de

0

20

10

minus10

minus20



119904(119905 120572) and 120591

(119879)

119904(119905 120572) is a solution of





⟨119869 (120572)⟩ asymp ⟨119869(119879)

119904(120572)⟩ =

1

120591 (119879) minus 120591 (0)

int

120591(119879)

120591(0)

119869 (119909(119879)

119904(119905 120572) 120572)

(A8)






min119878120583

1

119879

int

119879

0

[1198782

+ 1205782

1205832

] 119889119905

such that 119889119878

119889119905

=

120597119891

120597119909

119878 +

120597119891

120597120572

+ 120583119891 (119909119903 120572)

(A9)


119878 (119905) =

119889

119889120572

(119909(119879)

119904(120591(119879)

119904(119905 120572) 120572))

120583 (119905) =

119889

119889120572

119889120591(119879)

119904(119905 120572)

119889119905

(A10)



nabla120572⟨119869(119879)

119904(120572)⟩ asymp

1

119879

int

119879

0

(

120597119869

120597119909

119878 +

120597119869

120597120572

+ 120583 (119869 minus 119869)) 119889119905

where 119869 = 1

119879

int

119879

0

119869 119889119905

(A11)


119894+12= 119909119903((119894 +

12)Δ119905) 119909119894minus12



[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119868 119860119879

1

1205782

119861119879

1

119868 119862119879

1119860119879

2

1205782

119861119879

2

119868 119862119879

2

d d 119860119879

119898minus1

1205782

119861119879

119898minus1

119868 119862119879

119898minus1

119860111986111198621

119860211986121198622

d d

119860119898minus1

119861119898minus1

119862119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11987812

1205831

1198781+12

1205832

1198782+12

120583119898minus1

119878119898minus12

1198861

1198862

119886119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0

0

0

0

0

0

minus1198891

minus1198892


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(A12)

where

119860119894= minus

119868

Δ119905

minus

120597119891

120597119909

(119909119894minus12

120572) 119861119894=

119909119894+12


Δ119905

119862119894=

119868

Δ119905

minus

120597119891

120597119909

(119909119894+12

120572)

119889119894=

1

2

[

120597119891

120597120572

(119909119894minus12

120572) +

120597119891

120597120572

(119909119894+12

120572)]

(A13)







119894 119861119894 119862119894 and 119889

119894


119894




0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

120575x

04

02

00

minus02

minus04

08

04

00

minus04

minus08

04

00

minus04

minus08

02

00

12

08

04

00

10

15

05

00

minus02

minus04

120575y

120575z

120575X

120575Y

120575Z







119860 = [minus(

120597119891

120597119909

) + (

119889

119889119905

)] (A14)


minus1


1(119909) 120592

119899(119909)

120575119909 (119909) =

119899

sum

119894=1

120595119894(119909) 120592119894(119909) (A15a)

120575119891 (119909) =

119899

sum

119894=1

120593119894(119909) 120592119894(119909) (A15b)


119889120592119894(119909 (119905))

119889119905

=

120597119891

120597119909

120592119894(119909 (119905)) minus 120582

119894120592119894(119909 (119905)) (A16)

where 1205821 120582


one can obtain

119860 (120595119894120592119894) = [minus120595

119894(119909)

120597119891

120597119909

+

119889120595119894(119909)

119889119905

] 120592119894(119909) + 120595

119894(119909)

119889120592119894(119909)

119889119905

(A17)


119860 (120595119894120592119894) = [

119889120595119894(119909)

119889119905

minus 120582119894120595119894(119909)] 120592

119894(119909) (A18)


120575119891 (119909) =

119899

sum

119894=1

119860 (120595119894120592119894) =

119899

sum

119894=1

(

119889120595119894

119889119905

minus 120582119894120595119894)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120593119894

120592119894 (A19)



119889120595119894(119909)

119889119905

= 120593119894(119909) + 120582

119894120595119894(119909) (A20)






0 5 10 15 20

Time0 5 10 15 20

Time

Am

plitu

de

80

60

40

20Am

plitu

de

45

30

15

0

minus20

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

40

30

20

10

0

minus20

minus30

minus10

Am

plitu

de

20

10

0

minus20

minus10



120583 + ⟨1205930

119894⟩ = 0 (A21)

where

⟨1205930


1

119879

[1205950

119894(119909 (119879)) minus 120595

0

119894(119909 (0))] (A22)


minus1


119894= 0

1198891205950

119894(119909)

119889119905

= 1205930

119894(119909) + 120583 (A23)


0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

120575x

16

08

00

minus08

16

06

00

minus08

05

00

minus05

minus05

2

1

20

15

10

05

00

08

16

00

minus08

0

minus1

120575y

120575z

120575X

120575Y

120575Z




119863= minus20






119894(119909) 119894 = 1 119899






References




































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in



10

05

00

10

05

00

10

05

00

minus05

10

05

00

minus05

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus100 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

c = 015c = 015

c = 05c = 05

c = 08c = 08

c = 10c = 10




10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

c = 015c = 015

c = 05c = 05

c = 08c = 08

c = 10c = 10









model parameter 120572119895[30 31 44]

119878119894119895equiv

120597119909119894

120597120572119895

1003816100381610038161003816100381610038161003816100381610038161003816120572119895=120572

0

119895

= lim120575120572rarr0

[

[

119909119894(1205720

119895+ 120575120572119895) minus 119909119894(1205720

119895)

120575120572119895

]

]

(17)


119895



119889S119895

119889119905

= M sdot S119895+D119895

119895 = 1 119898 (18)

where S119895= (120597x120597120572

119895) = (119878

1119895 1198782119895 119878






119889x119889119905

= 119891 (x120572) x (0) = x0

119889S119895

119889119905

= M sdot S119895+D119895 S119895(0) = S

1198950

(19)










J (x120572) = int

120591

0

Φ (119905 x120572) 119889119905 (20)


nabla120572J (x01205720) = (

119889J

1198891205721

119889J

119889120572119898

)

119879100381610038161003816100381610038161003816100381610038161003816x0 1205720

(21)


119889J

119889120572119895

10038161003816100381610038161003816100381610038161003816100381610038161205720

119895

asymp

J (x0 + 120575x0 12057201 120572

0

119895+ 120575120572119895 120572

0

119898) minusJ (x01205720)

120575120572119895

(22)


119894 Note that

119889J

119889120572119895

=

119899

sum

119894=1

120597J

120597119909119894

120597119909119894

120597120572119895

+

120597J

120597120572119895

=

119899

sum

119894=1

119878119894119895

120597J

120597119909119894

+

120597J

120597120572119895

(23)







120575J (x01205720 120575x 120575120572) = int

120591

0

(

120597Φ

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575x + 120597Φ

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575120572)119889119905

(24)


120597120575x120597119905

=

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575x + 120597119891

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575120572

119905 isin [0 120591] 120575x (0) = 120575x0

(25)








nabla120572J (x01205720) = int

120591

0

[

120597Φ

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

minus (

120597119891

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

)

119879

sdot xlowast]119889119905 (26)


minus

120597xlowast

120597119905

minus (

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

)

119879

xlowast = minus

120597Φ

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

119905 isin [0 120591] xlowast (120591) = 0

(27)












x (119905119863) = (001 001 001 002 002 002)

119879 (28)




119894) and


119891are obtained



119896) and the




119890 (x119891) = [

[

(x119891 minus x119905)119879

(x119891 minus x119905)

(x119905)119879 x119905]

]

12

(29)


119891














119894)








11990325

01273 01266 02871 02796 04081 0389011988825

00584 00583 01312 01300 01866 0183311990350

01663 01689 03617 03202 05004 0574211988850

00705 00704 01583 01569 02250 02210


0= 1205902


119887= 02 is the


119894= R = 120590

2

0I




119890119877=

M (x + 120577120575x) minusM (x)M (x) 120577120575x

(31)


rarr 0




⟨M120575xM120575x⟩ = ⟨120575xM119879M120575x⟩ (32)




119869 (x0+ 120577120575x) asymp 119869 (x

0) + 120577 (120575x)119879 nabla119869 (x

0) (33)


Ψ (120577) =

119869 (x0+ 120577120575x) minus 119869 (x

0)

120577 (120575x)119879 nabla119869 (x0)

(34)


120575x =nabla119869 (x0)

1003817100381710038171003817nabla119869 (x0)1003817100381710038171003817

(35)



times x0

120577 119890119877



minus2

times x0

120577 Ψ (120577) log10(1003816100381610038161003816Ψ (120577) minus 1

1003816100381610038161003816)



gradient test





1198781119888=

120597119909

120597119888

1198782119888=

120597119910

120597119888

1198783119888=

120597119911

120597119888

1198784119888=

120597119883

120597119888

1198785119888=

120597119884

120597119888

1198786119888=

120597119885

120597119888

(36)


1198781119888= 120590 (119878

2119888minus 1198781119888) minus 119888119878

4119888minus 119883


1198783119888= 1199091198782119888+ 1199101198781119888minus 1198871198783119888+ 1198881198786119888+ 119885

1198784119888= 120576120590 (119878

5119888minus 1198784119888) minus 119888119878

1119888minus 119909

1198785119888= 120576 (119903119878


2119888+ 119910

1198786119888= 120576 (119883119878

5119888+ 1198841198784119888minus 1198871198786119888) minus 119888119878

3119888minus 119911

(37)





nabla120572⟨J (120572)⟩ = ⟨nabla

120572J (120572)⟩ (38)



int

119879

0

lim120575120572rarr0

[

J (120572 + 120575120572) minusJ (120572)

120575120572

] 119889119905 (39)





= 28


+120575119903 where 120575119903 = 0011199030

=


1198781119903=

120597119909

120597119903

1198782119903=

120597119910

120597119903

1198783119903=

120597119911

120597119903

1198784119903=

120597119883

120597119903

1198785119903=

120597119884

120597119903

1198786119903=

120597119885

120597119903

(40)


1198781119903= 120590 (119878

2119903minus 1198781119903) minus 119888119878

4119903

1198782119903= 119909 + 119903119878

1119903minus 1198782119903minus (1199091198783119903+ 1199111198781119903) + 119888119878

5119903

1198783119903= (119909119878

2119903+ 1199101198781119903) minus 119887119878

3119903+ 1198881198786119903

1198784119903= 120576120590 (119878

5119903minus 1198784119903) minus 119888119878

1119903

1198785119903= 120576 [119883 + 119903119878

4119903minus 1198785119903minus (119883119878

6119903+ 1198851198784119903)] + 119888119878

2119903

1198786119903= 120576 [(119883119878

5119903+ 1198841198784119903) minus 119887119878

6119903] minus 119888119878

3119903

(41)






0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time


Am

plitu

de

2000

0

minus2000

Am

plitu

de

3000

0

minus3000

Am

plitu

deA

mpl

itude

2000

0

minus2000

Am

plitu

de

3000

0

minus3000

Am

plitu

de

2000

0

minus2000

1000

0

minus1000

S1

S2

S3

S4

S5

S6









Am

plitu

de

S1

S2

S3

S4

S5

S6

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

200

100

0

minus100

Am

plitu

de

100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de

50

0

minus50minus200



119888 120597119885120597119903 120597119884120597119903 120597119883120597119903 120597119911120597119903 120597119910120597119903 120597119909120597119903



Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

3

0

minus3

Am

plitu

de

3

0

minus3

minus6

Δx

Δy

Δz

ΔXΔY

ΔZ









Appendix





(119889 gt 0) of 119891 if

dist (119909119896+1

119891 (119909119896)) lt 119889 119896 isin Z (A1)






dist (119909119896 119910119896) lt 120576 119896 isin Z (A2)



dist (Φ119905 (119905 119892 (120591)) 119892 (120591 + 119905)) lt 119889 (A3)







S1

S2

S3

S4

S5

S6

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

Am

plitu

de

500

0

minus500

Am

plitu

de

500

0

minus500

Am

plitu

de

6000

4000

2000

0

minus2000

Am

plitu

de

1000

0

minus2000

minus1000 Am

plitu

de

8000

4000

0

minus4000

Am

plitu

de

1500

0

minus1500




100381610038161003816100381610038161003816100381610038161003816

ℎ (1199051) minus ℎ (119905

2)

1199051minus 1199052

minus 1

100381610038161003816100381610038161003816100381610038161003816

le 120576


(A4)


119889119909

119889119905

= 119891 (119909 120572) 119909 isin R119899

(A5)




S1

S2

S3

S4

S5

S6


Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de3

6

0

minus3

Am

plitu

de

3

6

0

minus3

Am

plitu

de

2

4

0

minus2



⟨119869 (120572)⟩ = lim119879rarrinfin

1

119879

int

119879

0

119869 (119909 (119905 120572) 120572) 119889119905 (A6)







min119909120591

1

119879

int

119879

0

[1003817100381710038171003817119909 (120591 (119905)) minus 119909

119903(119905)1003817100381710038171003817

2

+ 1205782

(

119889120591

119889119905

minus 1)

2

]119889119905

such that 119889119909

119889119905

= 119891 (119909 120572)

(A7)


Am

plitu

de

45

30

15

Am

plitu

de

40

20

00 5 10 15 20

Time0 5 10 15 20

Time

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

minus6

minus8

minus10

minus12

Am

plitu

de

0

20

10

minus10

minus20



119904(119905 120572) and 120591

(119879)

119904(119905 120572) is a solution of





⟨119869 (120572)⟩ asymp ⟨119869(119879)

119904(120572)⟩ =

1

120591 (119879) minus 120591 (0)

int

120591(119879)

120591(0)

119869 (119909(119879)

119904(119905 120572) 120572)

(A8)






min119878120583

1

119879

int

119879

0

[1198782

+ 1205782

1205832

] 119889119905

such that 119889119878

119889119905

=

120597119891

120597119909

119878 +

120597119891

120597120572

+ 120583119891 (119909119903 120572)

(A9)


119878 (119905) =

119889

119889120572

(119909(119879)

119904(120591(119879)

119904(119905 120572) 120572))

120583 (119905) =

119889

119889120572

119889120591(119879)

119904(119905 120572)

119889119905

(A10)



nabla120572⟨119869(119879)

119904(120572)⟩ asymp

1

119879

int

119879

0

(

120597119869

120597119909

119878 +

120597119869

120597120572

+ 120583 (119869 minus 119869)) 119889119905

where 119869 = 1

119879

int

119879

0

119869 119889119905

(A11)


119894+12= 119909119903((119894 +

12)Δ119905) 119909119894minus12



[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119868 119860119879

1

1205782

119861119879

1

119868 119862119879

1119860119879

2

1205782

119861119879

2

119868 119862119879

2

d d 119860119879

119898minus1

1205782

119861119879

119898minus1

119868 119862119879

119898minus1

119860111986111198621

119860211986121198622

d d

119860119898minus1

119861119898minus1

119862119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11987812

1205831

1198781+12

1205832

1198782+12

120583119898minus1

119878119898minus12

1198861

1198862

119886119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0

0

0

0

0

0

minus1198891

minus1198892


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(A12)

where

119860119894= minus

119868

Δ119905

minus

120597119891

120597119909

(119909119894minus12

120572) 119861119894=

119909119894+12


Δ119905

119862119894=

119868

Δ119905

minus

120597119891

120597119909

(119909119894+12

120572)

119889119894=

1

2

[

120597119891

120597120572

(119909119894minus12

120572) +

120597119891

120597120572

(119909119894+12

120572)]

(A13)







119894 119861119894 119862119894 and 119889

119894


119894




0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

120575x

04

02

00

minus02

minus04

08

04

00

minus04

minus08

04

00

minus04

minus08

02

00

12

08

04

00

10

15

05

00

minus02

minus04

120575y

120575z

120575X

120575Y

120575Z







119860 = [minus(

120597119891

120597119909

) + (

119889

119889119905

)] (A14)


minus1


1(119909) 120592

119899(119909)

120575119909 (119909) =

119899

sum

119894=1

120595119894(119909) 120592119894(119909) (A15a)

120575119891 (119909) =

119899

sum

119894=1

120593119894(119909) 120592119894(119909) (A15b)


119889120592119894(119909 (119905))

119889119905

=

120597119891

120597119909

120592119894(119909 (119905)) minus 120582

119894120592119894(119909 (119905)) (A16)

where 1205821 120582


one can obtain

119860 (120595119894120592119894) = [minus120595

119894(119909)

120597119891

120597119909

+

119889120595119894(119909)

119889119905

] 120592119894(119909) + 120595

119894(119909)

119889120592119894(119909)

119889119905

(A17)


119860 (120595119894120592119894) = [

119889120595119894(119909)

119889119905

minus 120582119894120595119894(119909)] 120592

119894(119909) (A18)


120575119891 (119909) =

119899

sum

119894=1

119860 (120595119894120592119894) =

119899

sum

119894=1

(

119889120595119894

119889119905

minus 120582119894120595119894)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120593119894

120592119894 (A19)



119889120595119894(119909)

119889119905

= 120593119894(119909) + 120582

119894120595119894(119909) (A20)






0 5 10 15 20

Time0 5 10 15 20

Time

Am

plitu

de

80

60

40

20Am

plitu

de

45

30

15

0

minus20

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

40

30

20

10

0

minus20

minus30

minus10

Am

plitu

de

20

10

0

minus20

minus10



120583 + ⟨1205930

119894⟩ = 0 (A21)

where

⟨1205930


1

119879

[1205950

119894(119909 (119879)) minus 120595

0

119894(119909 (0))] (A22)


minus1


119894= 0

1198891205950

119894(119909)

119889119905

= 1205930

119894(119909) + 120583 (A23)


0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

120575x

16

08

00

minus08

16

06

00

minus08

05

00

minus05

minus05

2

1

20

15

10

05

00

08

16

00

minus08

0

minus1

120575y

120575z

120575X

120575Y

120575Z




119863= minus20






119894(119909) 119894 = 1 119899






References




































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in



10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

10

05

00

minus05

minus10

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

0 20 40 60 80

Time0 20 40 60 80

Time

c = 015c = 015

c = 05c = 05

c = 08c = 08

c = 10c = 10









model parameter 120572119895[30 31 44]

119878119894119895equiv

120597119909119894

120597120572119895

1003816100381610038161003816100381610038161003816100381610038161003816120572119895=120572

0

119895

= lim120575120572rarr0

[

[

119909119894(1205720

119895+ 120575120572119895) minus 119909119894(1205720

119895)

120575120572119895

]

]

(17)


119895



119889S119895

119889119905

= M sdot S119895+D119895

119895 = 1 119898 (18)

where S119895= (120597x120597120572

119895) = (119878

1119895 1198782119895 119878






119889x119889119905

= 119891 (x120572) x (0) = x0

119889S119895

119889119905

= M sdot S119895+D119895 S119895(0) = S

1198950

(19)










J (x120572) = int

120591

0

Φ (119905 x120572) 119889119905 (20)


nabla120572J (x01205720) = (

119889J

1198891205721

119889J

119889120572119898

)

119879100381610038161003816100381610038161003816100381610038161003816x0 1205720

(21)


119889J

119889120572119895

10038161003816100381610038161003816100381610038161003816100381610038161205720

119895

asymp

J (x0 + 120575x0 12057201 120572

0

119895+ 120575120572119895 120572

0

119898) minusJ (x01205720)

120575120572119895

(22)


119894 Note that

119889J

119889120572119895

=

119899

sum

119894=1

120597J

120597119909119894

120597119909119894

120597120572119895

+

120597J

120597120572119895

=

119899

sum

119894=1

119878119894119895

120597J

120597119909119894

+

120597J

120597120572119895

(23)







120575J (x01205720 120575x 120575120572) = int

120591

0

(

120597Φ

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575x + 120597Φ

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575120572)119889119905

(24)


120597120575x120597119905

=

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575x + 120597119891

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575120572

119905 isin [0 120591] 120575x (0) = 120575x0

(25)








nabla120572J (x01205720) = int

120591

0

[

120597Φ

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

minus (

120597119891

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

)

119879

sdot xlowast]119889119905 (26)


minus

120597xlowast

120597119905

minus (

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

)

119879

xlowast = minus

120597Φ

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

119905 isin [0 120591] xlowast (120591) = 0

(27)












x (119905119863) = (001 001 001 002 002 002)

119879 (28)




119894) and


119891are obtained



119896) and the




119890 (x119891) = [

[

(x119891 minus x119905)119879

(x119891 minus x119905)

(x119905)119879 x119905]

]

12

(29)


119891














119894)








11990325

01273 01266 02871 02796 04081 0389011988825

00584 00583 01312 01300 01866 0183311990350

01663 01689 03617 03202 05004 0574211988850

00705 00704 01583 01569 02250 02210


0= 1205902


119887= 02 is the


119894= R = 120590

2

0I




119890119877=

M (x + 120577120575x) minusM (x)M (x) 120577120575x

(31)


rarr 0




⟨M120575xM120575x⟩ = ⟨120575xM119879M120575x⟩ (32)




119869 (x0+ 120577120575x) asymp 119869 (x

0) + 120577 (120575x)119879 nabla119869 (x

0) (33)


Ψ (120577) =

119869 (x0+ 120577120575x) minus 119869 (x

0)

120577 (120575x)119879 nabla119869 (x0)

(34)


120575x =nabla119869 (x0)

1003817100381710038171003817nabla119869 (x0)1003817100381710038171003817

(35)



times x0

120577 119890119877



minus2

times x0

120577 Ψ (120577) log10(1003816100381610038161003816Ψ (120577) minus 1

1003816100381610038161003816)



gradient test





1198781119888=

120597119909

120597119888

1198782119888=

120597119910

120597119888

1198783119888=

120597119911

120597119888

1198784119888=

120597119883

120597119888

1198785119888=

120597119884

120597119888

1198786119888=

120597119885

120597119888

(36)


1198781119888= 120590 (119878

2119888minus 1198781119888) minus 119888119878

4119888minus 119883


1198783119888= 1199091198782119888+ 1199101198781119888minus 1198871198783119888+ 1198881198786119888+ 119885

1198784119888= 120576120590 (119878

5119888minus 1198784119888) minus 119888119878

1119888minus 119909

1198785119888= 120576 (119903119878


2119888+ 119910

1198786119888= 120576 (119883119878

5119888+ 1198841198784119888minus 1198871198786119888) minus 119888119878

3119888minus 119911

(37)





nabla120572⟨J (120572)⟩ = ⟨nabla

120572J (120572)⟩ (38)



int

119879

0

lim120575120572rarr0

[

J (120572 + 120575120572) minusJ (120572)

120575120572

] 119889119905 (39)





= 28


+120575119903 where 120575119903 = 0011199030

=


1198781119903=

120597119909

120597119903

1198782119903=

120597119910

120597119903

1198783119903=

120597119911

120597119903

1198784119903=

120597119883

120597119903

1198785119903=

120597119884

120597119903

1198786119903=

120597119885

120597119903

(40)


1198781119903= 120590 (119878

2119903minus 1198781119903) minus 119888119878

4119903

1198782119903= 119909 + 119903119878

1119903minus 1198782119903minus (1199091198783119903+ 1199111198781119903) + 119888119878

5119903

1198783119903= (119909119878

2119903+ 1199101198781119903) minus 119887119878

3119903+ 1198881198786119903

1198784119903= 120576120590 (119878

5119903minus 1198784119903) minus 119888119878

1119903

1198785119903= 120576 [119883 + 119903119878

4119903minus 1198785119903minus (119883119878

6119903+ 1198851198784119903)] + 119888119878

2119903

1198786119903= 120576 [(119883119878

5119903+ 1198841198784119903) minus 119887119878

6119903] minus 119888119878

3119903

(41)






0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time


Am

plitu

de

2000

0

minus2000

Am

plitu

de

3000

0

minus3000

Am

plitu

deA

mpl

itude

2000

0

minus2000

Am

plitu

de

3000

0

minus3000

Am

plitu

de

2000

0

minus2000

1000

0

minus1000

S1

S2

S3

S4

S5

S6









Am

plitu

de

S1

S2

S3

S4

S5

S6

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

200

100

0

minus100

Am

plitu

de

100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de

50

0

minus50minus200



119888 120597119885120597119903 120597119884120597119903 120597119883120597119903 120597119911120597119903 120597119910120597119903 120597119909120597119903



Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

3

0

minus3

Am

plitu

de

3

0

minus3

minus6

Δx

Δy

Δz

ΔXΔY

ΔZ









Appendix





(119889 gt 0) of 119891 if

dist (119909119896+1

119891 (119909119896)) lt 119889 119896 isin Z (A1)






dist (119909119896 119910119896) lt 120576 119896 isin Z (A2)



dist (Φ119905 (119905 119892 (120591)) 119892 (120591 + 119905)) lt 119889 (A3)







S1

S2

S3

S4

S5

S6

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

Am

plitu

de

500

0

minus500

Am

plitu

de

500

0

minus500

Am

plitu

de

6000

4000

2000

0

minus2000

Am

plitu

de

1000

0

minus2000

minus1000 Am

plitu

de

8000

4000

0

minus4000

Am

plitu

de

1500

0

minus1500




100381610038161003816100381610038161003816100381610038161003816

ℎ (1199051) minus ℎ (119905

2)

1199051minus 1199052

minus 1

100381610038161003816100381610038161003816100381610038161003816

le 120576


(A4)


119889119909

119889119905

= 119891 (119909 120572) 119909 isin R119899

(A5)




S1

S2

S3

S4

S5

S6


Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de3

6

0

minus3

Am

plitu

de

3

6

0

minus3

Am

plitu

de

2

4

0

minus2



⟨119869 (120572)⟩ = lim119879rarrinfin

1

119879

int

119879

0

119869 (119909 (119905 120572) 120572) 119889119905 (A6)







min119909120591

1

119879

int

119879

0

[1003817100381710038171003817119909 (120591 (119905)) minus 119909

119903(119905)1003817100381710038171003817

2

+ 1205782

(

119889120591

119889119905

minus 1)

2

]119889119905

such that 119889119909

119889119905

= 119891 (119909 120572)

(A7)


Am

plitu

de

45

30

15

Am

plitu

de

40

20

00 5 10 15 20

Time0 5 10 15 20

Time

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

minus6

minus8

minus10

minus12

Am

plitu

de

0

20

10

minus10

minus20



119904(119905 120572) and 120591

(119879)

119904(119905 120572) is a solution of





⟨119869 (120572)⟩ asymp ⟨119869(119879)

119904(120572)⟩ =

1

120591 (119879) minus 120591 (0)

int

120591(119879)

120591(0)

119869 (119909(119879)

119904(119905 120572) 120572)

(A8)






min119878120583

1

119879

int

119879

0

[1198782

+ 1205782

1205832

] 119889119905

such that 119889119878

119889119905

=

120597119891

120597119909

119878 +

120597119891

120597120572

+ 120583119891 (119909119903 120572)

(A9)


119878 (119905) =

119889

119889120572

(119909(119879)

119904(120591(119879)

119904(119905 120572) 120572))

120583 (119905) =

119889

119889120572

119889120591(119879)

119904(119905 120572)

119889119905

(A10)



nabla120572⟨119869(119879)

119904(120572)⟩ asymp

1

119879

int

119879

0

(

120597119869

120597119909

119878 +

120597119869

120597120572

+ 120583 (119869 minus 119869)) 119889119905

where 119869 = 1

119879

int

119879

0

119869 119889119905

(A11)


119894+12= 119909119903((119894 +

12)Δ119905) 119909119894minus12



[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119868 119860119879

1

1205782

119861119879

1

119868 119862119879

1119860119879

2

1205782

119861119879

2

119868 119862119879

2

d d 119860119879

119898minus1

1205782

119861119879

119898minus1

119868 119862119879

119898minus1

119860111986111198621

119860211986121198622

d d

119860119898minus1

119861119898minus1

119862119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11987812

1205831

1198781+12

1205832

1198782+12

120583119898minus1

119878119898minus12

1198861

1198862

119886119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0

0

0

0

0

0

minus1198891

minus1198892


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(A12)

where

119860119894= minus

119868

Δ119905

minus

120597119891

120597119909

(119909119894minus12

120572) 119861119894=

119909119894+12


Δ119905

119862119894=

119868

Δ119905

minus

120597119891

120597119909

(119909119894+12

120572)

119889119894=

1

2

[

120597119891

120597120572

(119909119894minus12

120572) +

120597119891

120597120572

(119909119894+12

120572)]

(A13)







119894 119861119894 119862119894 and 119889

119894


119894




0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

120575x

04

02

00

minus02

minus04

08

04

00

minus04

minus08

04

00

minus04

minus08

02

00

12

08

04

00

10

15

05

00

minus02

minus04

120575y

120575z

120575X

120575Y

120575Z







119860 = [minus(

120597119891

120597119909

) + (

119889

119889119905

)] (A14)


minus1


1(119909) 120592

119899(119909)

120575119909 (119909) =

119899

sum

119894=1

120595119894(119909) 120592119894(119909) (A15a)

120575119891 (119909) =

119899

sum

119894=1

120593119894(119909) 120592119894(119909) (A15b)


119889120592119894(119909 (119905))

119889119905

=

120597119891

120597119909

120592119894(119909 (119905)) minus 120582

119894120592119894(119909 (119905)) (A16)

where 1205821 120582


one can obtain

119860 (120595119894120592119894) = [minus120595

119894(119909)

120597119891

120597119909

+

119889120595119894(119909)

119889119905

] 120592119894(119909) + 120595

119894(119909)

119889120592119894(119909)

119889119905

(A17)


119860 (120595119894120592119894) = [

119889120595119894(119909)

119889119905

minus 120582119894120595119894(119909)] 120592

119894(119909) (A18)


120575119891 (119909) =

119899

sum

119894=1

119860 (120595119894120592119894) =

119899

sum

119894=1

(

119889120595119894

119889119905

minus 120582119894120595119894)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120593119894

120592119894 (A19)



119889120595119894(119909)

119889119905

= 120593119894(119909) + 120582

119894120595119894(119909) (A20)






0 5 10 15 20

Time0 5 10 15 20

Time

Am

plitu

de

80

60

40

20Am

plitu

de

45

30

15

0

minus20

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

40

30

20

10

0

minus20

minus30

minus10

Am

plitu

de

20

10

0

minus20

minus10



120583 + ⟨1205930

119894⟩ = 0 (A21)

where

⟨1205930


1

119879

[1205950

119894(119909 (119879)) minus 120595

0

119894(119909 (0))] (A22)


minus1


119894= 0

1198891205950

119894(119909)

119889119905

= 1205930

119894(119909) + 120583 (A23)


0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

120575x

16

08

00

minus08

16

06

00

minus08

05

00

minus05

minus05

2

1

20

15

10

05

00

08

16

00

minus08

0

minus1

120575y

120575z

120575X

120575Y

120575Z




119863= minus20






119894(119909) 119894 = 1 119899






References




































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in








model parameter 120572119895[30 31 44]

119878119894119895equiv

120597119909119894

120597120572119895

1003816100381610038161003816100381610038161003816100381610038161003816120572119895=120572

0

119895

= lim120575120572rarr0

[

[

119909119894(1205720

119895+ 120575120572119895) minus 119909119894(1205720

119895)

120575120572119895

]

]

(17)


119895



119889S119895

119889119905

= M sdot S119895+D119895

119895 = 1 119898 (18)

where S119895= (120597x120597120572

119895) = (119878

1119895 1198782119895 119878






119889x119889119905

= 119891 (x120572) x (0) = x0

119889S119895

119889119905

= M sdot S119895+D119895 S119895(0) = S

1198950

(19)










J (x120572) = int

120591

0

Φ (119905 x120572) 119889119905 (20)


nabla120572J (x01205720) = (

119889J

1198891205721

119889J

119889120572119898

)

119879100381610038161003816100381610038161003816100381610038161003816x0 1205720

(21)


119889J

119889120572119895

10038161003816100381610038161003816100381610038161003816100381610038161205720

119895

asymp

J (x0 + 120575x0 12057201 120572

0

119895+ 120575120572119895 120572

0

119898) minusJ (x01205720)

120575120572119895

(22)


119894 Note that

119889J

119889120572119895

=

119899

sum

119894=1

120597J

120597119909119894

120597119909119894

120597120572119895

+

120597J

120597120572119895

=

119899

sum

119894=1

119878119894119895

120597J

120597119909119894

+

120597J

120597120572119895

(23)







120575J (x01205720 120575x 120575120572) = int

120591

0

(

120597Φ

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575x + 120597Φ

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575120572)119889119905

(24)


120597120575x120597119905

=

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575x + 120597119891

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575120572

119905 isin [0 120591] 120575x (0) = 120575x0

(25)








nabla120572J (x01205720) = int

120591

0

[

120597Φ

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

minus (

120597119891

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

)

119879

sdot xlowast]119889119905 (26)


minus

120597xlowast

120597119905

minus (

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

)

119879

xlowast = minus

120597Φ

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

119905 isin [0 120591] xlowast (120591) = 0

(27)












x (119905119863) = (001 001 001 002 002 002)

119879 (28)




119894) and


119891are obtained



119896) and the




119890 (x119891) = [

[

(x119891 minus x119905)119879

(x119891 minus x119905)

(x119905)119879 x119905]

]

12

(29)


119891














119894)








11990325

01273 01266 02871 02796 04081 0389011988825

00584 00583 01312 01300 01866 0183311990350

01663 01689 03617 03202 05004 0574211988850

00705 00704 01583 01569 02250 02210


0= 1205902


119887= 02 is the


119894= R = 120590

2

0I




119890119877=

M (x + 120577120575x) minusM (x)M (x) 120577120575x

(31)


rarr 0




⟨M120575xM120575x⟩ = ⟨120575xM119879M120575x⟩ (32)




119869 (x0+ 120577120575x) asymp 119869 (x

0) + 120577 (120575x)119879 nabla119869 (x

0) (33)


Ψ (120577) =

119869 (x0+ 120577120575x) minus 119869 (x

0)

120577 (120575x)119879 nabla119869 (x0)

(34)


120575x =nabla119869 (x0)

1003817100381710038171003817nabla119869 (x0)1003817100381710038171003817

(35)



times x0

120577 119890119877



minus2

times x0

120577 Ψ (120577) log10(1003816100381610038161003816Ψ (120577) minus 1

1003816100381610038161003816)



gradient test





1198781119888=

120597119909

120597119888

1198782119888=

120597119910

120597119888

1198783119888=

120597119911

120597119888

1198784119888=

120597119883

120597119888

1198785119888=

120597119884

120597119888

1198786119888=

120597119885

120597119888

(36)


1198781119888= 120590 (119878

2119888minus 1198781119888) minus 119888119878

4119888minus 119883


1198783119888= 1199091198782119888+ 1199101198781119888minus 1198871198783119888+ 1198881198786119888+ 119885

1198784119888= 120576120590 (119878

5119888minus 1198784119888) minus 119888119878

1119888minus 119909

1198785119888= 120576 (119903119878


2119888+ 119910

1198786119888= 120576 (119883119878

5119888+ 1198841198784119888minus 1198871198786119888) minus 119888119878

3119888minus 119911

(37)





nabla120572⟨J (120572)⟩ = ⟨nabla

120572J (120572)⟩ (38)



int

119879

0

lim120575120572rarr0

[

J (120572 + 120575120572) minusJ (120572)

120575120572

] 119889119905 (39)





= 28


+120575119903 where 120575119903 = 0011199030

=


1198781119903=

120597119909

120597119903

1198782119903=

120597119910

120597119903

1198783119903=

120597119911

120597119903

1198784119903=

120597119883

120597119903

1198785119903=

120597119884

120597119903

1198786119903=

120597119885

120597119903

(40)


1198781119903= 120590 (119878

2119903minus 1198781119903) minus 119888119878

4119903

1198782119903= 119909 + 119903119878

1119903minus 1198782119903minus (1199091198783119903+ 1199111198781119903) + 119888119878

5119903

1198783119903= (119909119878

2119903+ 1199101198781119903) minus 119887119878

3119903+ 1198881198786119903

1198784119903= 120576120590 (119878

5119903minus 1198784119903) minus 119888119878

1119903

1198785119903= 120576 [119883 + 119903119878

4119903minus 1198785119903minus (119883119878

6119903+ 1198851198784119903)] + 119888119878

2119903

1198786119903= 120576 [(119883119878

5119903+ 1198841198784119903) minus 119887119878

6119903] minus 119888119878

3119903

(41)






0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time


Am

plitu

de

2000

0

minus2000

Am

plitu

de

3000

0

minus3000

Am

plitu

deA

mpl

itude

2000

0

minus2000

Am

plitu

de

3000

0

minus3000

Am

plitu

de

2000

0

minus2000

1000

0

minus1000

S1

S2

S3

S4

S5

S6









Am

plitu

de

S1

S2

S3

S4

S5

S6

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

200

100

0

minus100

Am

plitu

de

100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de

50

0

minus50minus200



119888 120597119885120597119903 120597119884120597119903 120597119883120597119903 120597119911120597119903 120597119910120597119903 120597119909120597119903



Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

3

0

minus3

Am

plitu

de

3

0

minus3

minus6

Δx

Δy

Δz

ΔXΔY

ΔZ









Appendix





(119889 gt 0) of 119891 if

dist (119909119896+1

119891 (119909119896)) lt 119889 119896 isin Z (A1)






dist (119909119896 119910119896) lt 120576 119896 isin Z (A2)



dist (Φ119905 (119905 119892 (120591)) 119892 (120591 + 119905)) lt 119889 (A3)







S1

S2

S3

S4

S5

S6

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

Am

plitu

de

500

0

minus500

Am

plitu

de

500

0

minus500

Am

plitu

de

6000

4000

2000

0

minus2000

Am

plitu

de

1000

0

minus2000

minus1000 Am

plitu

de

8000

4000

0

minus4000

Am

plitu

de

1500

0

minus1500




100381610038161003816100381610038161003816100381610038161003816

ℎ (1199051) minus ℎ (119905

2)

1199051minus 1199052

minus 1

100381610038161003816100381610038161003816100381610038161003816

le 120576


(A4)


119889119909

119889119905

= 119891 (119909 120572) 119909 isin R119899

(A5)




S1

S2

S3

S4

S5

S6


Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de3

6

0

minus3

Am

plitu

de

3

6

0

minus3

Am

plitu

de

2

4

0

minus2



⟨119869 (120572)⟩ = lim119879rarrinfin

1

119879

int

119879

0

119869 (119909 (119905 120572) 120572) 119889119905 (A6)







min119909120591

1

119879

int

119879

0

[1003817100381710038171003817119909 (120591 (119905)) minus 119909

119903(119905)1003817100381710038171003817

2

+ 1205782

(

119889120591

119889119905

minus 1)

2

]119889119905

such that 119889119909

119889119905

= 119891 (119909 120572)

(A7)


Am

plitu

de

45

30

15

Am

plitu

de

40

20

00 5 10 15 20

Time0 5 10 15 20

Time

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

minus6

minus8

minus10

minus12

Am

plitu

de

0

20

10

minus10

minus20



119904(119905 120572) and 120591

(119879)

119904(119905 120572) is a solution of





⟨119869 (120572)⟩ asymp ⟨119869(119879)

119904(120572)⟩ =

1

120591 (119879) minus 120591 (0)

int

120591(119879)

120591(0)

119869 (119909(119879)

119904(119905 120572) 120572)

(A8)






min119878120583

1

119879

int

119879

0

[1198782

+ 1205782

1205832

] 119889119905

such that 119889119878

119889119905

=

120597119891

120597119909

119878 +

120597119891

120597120572

+ 120583119891 (119909119903 120572)

(A9)


119878 (119905) =

119889

119889120572

(119909(119879)

119904(120591(119879)

119904(119905 120572) 120572))

120583 (119905) =

119889

119889120572

119889120591(119879)

119904(119905 120572)

119889119905

(A10)



nabla120572⟨119869(119879)

119904(120572)⟩ asymp

1

119879

int

119879

0

(

120597119869

120597119909

119878 +

120597119869

120597120572

+ 120583 (119869 minus 119869)) 119889119905

where 119869 = 1

119879

int

119879

0

119869 119889119905

(A11)


119894+12= 119909119903((119894 +

12)Δ119905) 119909119894minus12



[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119868 119860119879

1

1205782

119861119879

1

119868 119862119879

1119860119879

2

1205782

119861119879

2

119868 119862119879

2

d d 119860119879

119898minus1

1205782

119861119879

119898minus1

119868 119862119879

119898minus1

119860111986111198621

119860211986121198622

d d

119860119898minus1

119861119898minus1

119862119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11987812

1205831

1198781+12

1205832

1198782+12

120583119898minus1

119878119898minus12

1198861

1198862

119886119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0

0

0

0

0

0

minus1198891

minus1198892


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(A12)

where

119860119894= minus

119868

Δ119905

minus

120597119891

120597119909

(119909119894minus12

120572) 119861119894=

119909119894+12


Δ119905

119862119894=

119868

Δ119905

minus

120597119891

120597119909

(119909119894+12

120572)

119889119894=

1

2

[

120597119891

120597120572

(119909119894minus12

120572) +

120597119891

120597120572

(119909119894+12

120572)]

(A13)







119894 119861119894 119862119894 and 119889

119894


119894




0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

120575x

04

02

00

minus02

minus04

08

04

00

minus04

minus08

04

00

minus04

minus08

02

00

12

08

04

00

10

15

05

00

minus02

minus04

120575y

120575z

120575X

120575Y

120575Z







119860 = [minus(

120597119891

120597119909

) + (

119889

119889119905

)] (A14)


minus1


1(119909) 120592

119899(119909)

120575119909 (119909) =

119899

sum

119894=1

120595119894(119909) 120592119894(119909) (A15a)

120575119891 (119909) =

119899

sum

119894=1

120593119894(119909) 120592119894(119909) (A15b)


119889120592119894(119909 (119905))

119889119905

=

120597119891

120597119909

120592119894(119909 (119905)) minus 120582

119894120592119894(119909 (119905)) (A16)

where 1205821 120582


one can obtain

119860 (120595119894120592119894) = [minus120595

119894(119909)

120597119891

120597119909

+

119889120595119894(119909)

119889119905

] 120592119894(119909) + 120595

119894(119909)

119889120592119894(119909)

119889119905

(A17)


119860 (120595119894120592119894) = [

119889120595119894(119909)

119889119905

minus 120582119894120595119894(119909)] 120592

119894(119909) (A18)


120575119891 (119909) =

119899

sum

119894=1

119860 (120595119894120592119894) =

119899

sum

119894=1

(

119889120595119894

119889119905

minus 120582119894120595119894)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120593119894

120592119894 (A19)



119889120595119894(119909)

119889119905

= 120593119894(119909) + 120582

119894120595119894(119909) (A20)






0 5 10 15 20

Time0 5 10 15 20

Time

Am

plitu

de

80

60

40

20Am

plitu

de

45

30

15

0

minus20

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

40

30

20

10

0

minus20

minus30

minus10

Am

plitu

de

20

10

0

minus20

minus10



120583 + ⟨1205930

119894⟩ = 0 (A21)

where

⟨1205930


1

119879

[1205950

119894(119909 (119879)) minus 120595

0

119894(119909 (0))] (A22)


minus1


119894= 0

1198891205950

119894(119909)

119889119905

= 1205930

119894(119909) + 120583 (A23)


0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

120575x

16

08

00

minus08

16

06

00

minus08

05

00

minus05

minus05

2

1

20

15

10

05

00

08

16

00

minus08

0

minus1

120575y

120575z

120575X

120575Y

120575Z




119863= minus20






119894(119909) 119894 = 1 119899






References




































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in







120575J (x01205720 120575x 120575120572) = int

120591

0

(

120597Φ

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575x + 120597Φ

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575120572)119889119905

(24)


120597120575x120597119905

=

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575x + 120597119891

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

sdot 120575120572

119905 isin [0 120591] 120575x (0) = 120575x0

(25)








nabla120572J (x01205720) = int

120591

0

[

120597Φ

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

minus (

120597119891

120597120572

10038161003816100381610038161003816100381610038161003816x0 1205720

)

119879

sdot xlowast]119889119905 (26)


minus

120597xlowast

120597119905

minus (

120597119891

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

)

119879

xlowast = minus

120597Φ

120597x

10038161003816100381610038161003816100381610038161003816x0 1205720

119905 isin [0 120591] xlowast (120591) = 0

(27)












x (119905119863) = (001 001 001 002 002 002)

119879 (28)




119894) and


119891are obtained



119896) and the




119890 (x119891) = [

[

(x119891 minus x119905)119879

(x119891 minus x119905)

(x119905)119879 x119905]

]

12

(29)


119891














119894)








11990325

01273 01266 02871 02796 04081 0389011988825

00584 00583 01312 01300 01866 0183311990350

01663 01689 03617 03202 05004 0574211988850

00705 00704 01583 01569 02250 02210


0= 1205902


119887= 02 is the


119894= R = 120590

2

0I




119890119877=

M (x + 120577120575x) minusM (x)M (x) 120577120575x

(31)


rarr 0




⟨M120575xM120575x⟩ = ⟨120575xM119879M120575x⟩ (32)




119869 (x0+ 120577120575x) asymp 119869 (x

0) + 120577 (120575x)119879 nabla119869 (x

0) (33)


Ψ (120577) =

119869 (x0+ 120577120575x) minus 119869 (x

0)

120577 (120575x)119879 nabla119869 (x0)

(34)


120575x =nabla119869 (x0)

1003817100381710038171003817nabla119869 (x0)1003817100381710038171003817

(35)



times x0

120577 119890119877



minus2

times x0

120577 Ψ (120577) log10(1003816100381610038161003816Ψ (120577) minus 1

1003816100381610038161003816)



gradient test





1198781119888=

120597119909

120597119888

1198782119888=

120597119910

120597119888

1198783119888=

120597119911

120597119888

1198784119888=

120597119883

120597119888

1198785119888=

120597119884

120597119888

1198786119888=

120597119885

120597119888

(36)


1198781119888= 120590 (119878

2119888minus 1198781119888) minus 119888119878

4119888minus 119883


1198783119888= 1199091198782119888+ 1199101198781119888minus 1198871198783119888+ 1198881198786119888+ 119885

1198784119888= 120576120590 (119878

5119888minus 1198784119888) minus 119888119878

1119888minus 119909

1198785119888= 120576 (119903119878


2119888+ 119910

1198786119888= 120576 (119883119878

5119888+ 1198841198784119888minus 1198871198786119888) minus 119888119878

3119888minus 119911

(37)





nabla120572⟨J (120572)⟩ = ⟨nabla

120572J (120572)⟩ (38)



int

119879

0

lim120575120572rarr0

[

J (120572 + 120575120572) minusJ (120572)

120575120572

] 119889119905 (39)





= 28


+120575119903 where 120575119903 = 0011199030

=


1198781119903=

120597119909

120597119903

1198782119903=

120597119910

120597119903

1198783119903=

120597119911

120597119903

1198784119903=

120597119883

120597119903

1198785119903=

120597119884

120597119903

1198786119903=

120597119885

120597119903

(40)


1198781119903= 120590 (119878

2119903minus 1198781119903) minus 119888119878

4119903

1198782119903= 119909 + 119903119878

1119903minus 1198782119903minus (1199091198783119903+ 1199111198781119903) + 119888119878

5119903

1198783119903= (119909119878

2119903+ 1199101198781119903) minus 119887119878

3119903+ 1198881198786119903

1198784119903= 120576120590 (119878

5119903minus 1198784119903) minus 119888119878

1119903

1198785119903= 120576 [119883 + 119903119878

4119903minus 1198785119903minus (119883119878

6119903+ 1198851198784119903)] + 119888119878

2119903

1198786119903= 120576 [(119883119878

5119903+ 1198841198784119903) minus 119887119878

6119903] minus 119888119878

3119903

(41)






0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time


Am

plitu

de

2000

0

minus2000

Am

plitu

de

3000

0

minus3000

Am

plitu

deA

mpl

itude

2000

0

minus2000

Am

plitu

de

3000

0

minus3000

Am

plitu

de

2000

0

minus2000

1000

0

minus1000

S1

S2

S3

S4

S5

S6









Am

plitu

de

S1

S2

S3

S4

S5

S6

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

200

100

0

minus100

Am

plitu

de

100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de

50

0

minus50minus200



119888 120597119885120597119903 120597119884120597119903 120597119883120597119903 120597119911120597119903 120597119910120597119903 120597119909120597119903



Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

3

0

minus3

Am

plitu

de

3

0

minus3

minus6

Δx

Δy

Δz

ΔXΔY

ΔZ









Appendix





(119889 gt 0) of 119891 if

dist (119909119896+1

119891 (119909119896)) lt 119889 119896 isin Z (A1)






dist (119909119896 119910119896) lt 120576 119896 isin Z (A2)



dist (Φ119905 (119905 119892 (120591)) 119892 (120591 + 119905)) lt 119889 (A3)







S1

S2

S3

S4

S5

S6

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

Am

plitu

de

500

0

minus500

Am

plitu

de

500

0

minus500

Am

plitu

de

6000

4000

2000

0

minus2000

Am

plitu

de

1000

0

minus2000

minus1000 Am

plitu

de

8000

4000

0

minus4000

Am

plitu

de

1500

0

minus1500




100381610038161003816100381610038161003816100381610038161003816

ℎ (1199051) minus ℎ (119905

2)

1199051minus 1199052

minus 1

100381610038161003816100381610038161003816100381610038161003816

le 120576


(A4)


119889119909

119889119905

= 119891 (119909 120572) 119909 isin R119899

(A5)




S1

S2

S3

S4

S5

S6


Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de3

6

0

minus3

Am

plitu

de

3

6

0

minus3

Am

plitu

de

2

4

0

minus2



⟨119869 (120572)⟩ = lim119879rarrinfin

1

119879

int

119879

0

119869 (119909 (119905 120572) 120572) 119889119905 (A6)







min119909120591

1

119879

int

119879

0

[1003817100381710038171003817119909 (120591 (119905)) minus 119909

119903(119905)1003817100381710038171003817

2

+ 1205782

(

119889120591

119889119905

minus 1)

2

]119889119905

such that 119889119909

119889119905

= 119891 (119909 120572)

(A7)


Am

plitu

de

45

30

15

Am

plitu

de

40

20

00 5 10 15 20

Time0 5 10 15 20

Time

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

minus6

minus8

minus10

minus12

Am

plitu

de

0

20

10

minus10

minus20



119904(119905 120572) and 120591

(119879)

119904(119905 120572) is a solution of





⟨119869 (120572)⟩ asymp ⟨119869(119879)

119904(120572)⟩ =

1

120591 (119879) minus 120591 (0)

int

120591(119879)

120591(0)

119869 (119909(119879)

119904(119905 120572) 120572)

(A8)






min119878120583

1

119879

int

119879

0

[1198782

+ 1205782

1205832

] 119889119905

such that 119889119878

119889119905

=

120597119891

120597119909

119878 +

120597119891

120597120572

+ 120583119891 (119909119903 120572)

(A9)


119878 (119905) =

119889

119889120572

(119909(119879)

119904(120591(119879)

119904(119905 120572) 120572))

120583 (119905) =

119889

119889120572

119889120591(119879)

119904(119905 120572)

119889119905

(A10)



nabla120572⟨119869(119879)

119904(120572)⟩ asymp

1

119879

int

119879

0

(

120597119869

120597119909

119878 +

120597119869

120597120572

+ 120583 (119869 minus 119869)) 119889119905

where 119869 = 1

119879

int

119879

0

119869 119889119905

(A11)


119894+12= 119909119903((119894 +

12)Δ119905) 119909119894minus12



[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119868 119860119879

1

1205782

119861119879

1

119868 119862119879

1119860119879

2

1205782

119861119879

2

119868 119862119879

2

d d 119860119879

119898minus1

1205782

119861119879

119898minus1

119868 119862119879

119898minus1

119860111986111198621

119860211986121198622

d d

119860119898minus1

119861119898minus1

119862119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11987812

1205831

1198781+12

1205832

1198782+12

120583119898minus1

119878119898minus12

1198861

1198862

119886119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0

0

0

0

0

0

minus1198891

minus1198892


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(A12)

where

119860119894= minus

119868

Δ119905

minus

120597119891

120597119909

(119909119894minus12

120572) 119861119894=

119909119894+12


Δ119905

119862119894=

119868

Δ119905

minus

120597119891

120597119909

(119909119894+12

120572)

119889119894=

1

2

[

120597119891

120597120572

(119909119894minus12

120572) +

120597119891

120597120572

(119909119894+12

120572)]

(A13)







119894 119861119894 119862119894 and 119889

119894


119894




0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

120575x

04

02

00

minus02

minus04

08

04

00

minus04

minus08

04

00

minus04

minus08

02

00

12

08

04

00

10

15

05

00

minus02

minus04

120575y

120575z

120575X

120575Y

120575Z







119860 = [minus(

120597119891

120597119909

) + (

119889

119889119905

)] (A14)


minus1


1(119909) 120592

119899(119909)

120575119909 (119909) =

119899

sum

119894=1

120595119894(119909) 120592119894(119909) (A15a)

120575119891 (119909) =

119899

sum

119894=1

120593119894(119909) 120592119894(119909) (A15b)


119889120592119894(119909 (119905))

119889119905

=

120597119891

120597119909

120592119894(119909 (119905)) minus 120582

119894120592119894(119909 (119905)) (A16)

where 1205821 120582


one can obtain

119860 (120595119894120592119894) = [minus120595

119894(119909)

120597119891

120597119909

+

119889120595119894(119909)

119889119905

] 120592119894(119909) + 120595

119894(119909)

119889120592119894(119909)

119889119905

(A17)


119860 (120595119894120592119894) = [

119889120595119894(119909)

119889119905

minus 120582119894120595119894(119909)] 120592

119894(119909) (A18)


120575119891 (119909) =

119899

sum

119894=1

119860 (120595119894120592119894) =

119899

sum

119894=1

(

119889120595119894

119889119905

minus 120582119894120595119894)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120593119894

120592119894 (A19)



119889120595119894(119909)

119889119905

= 120593119894(119909) + 120582

119894120595119894(119909) (A20)






0 5 10 15 20

Time0 5 10 15 20

Time

Am

plitu

de

80

60

40

20Am

plitu

de

45

30

15

0

minus20

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

40

30

20

10

0

minus20

minus30

minus10

Am

plitu

de

20

10

0

minus20

minus10



120583 + ⟨1205930

119894⟩ = 0 (A21)

where

⟨1205930


1

119879

[1205950

119894(119909 (119879)) minus 120595

0

119894(119909 (0))] (A22)


minus1


119894= 0

1198891205950

119894(119909)

119889119905

= 1205930

119894(119909) + 120583 (A23)


0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

120575x

16

08

00

minus08

16

06

00

minus08

05

00

minus05

minus05

2

1

20

15

10

05

00

08

16

00

minus08

0

minus1

120575y

120575z

120575X

120575Y

120575Z




119863= minus20






119894(119909) 119894 = 1 119899






References




































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in




11990325

01273 01266 02871 02796 04081 0389011988825

00584 00583 01312 01300 01866 0183311990350

01663 01689 03617 03202 05004 0574211988850

00705 00704 01583 01569 02250 02210


0= 1205902


119887= 02 is the


119894= R = 120590

2

0I




119890119877=

M (x + 120577120575x) minusM (x)M (x) 120577120575x

(31)


rarr 0




⟨M120575xM120575x⟩ = ⟨120575xM119879M120575x⟩ (32)




119869 (x0+ 120577120575x) asymp 119869 (x

0) + 120577 (120575x)119879 nabla119869 (x

0) (33)


Ψ (120577) =

119869 (x0+ 120577120575x) minus 119869 (x

0)

120577 (120575x)119879 nabla119869 (x0)

(34)


120575x =nabla119869 (x0)

1003817100381710038171003817nabla119869 (x0)1003817100381710038171003817

(35)



times x0

120577 119890119877



minus2

times x0

120577 Ψ (120577) log10(1003816100381610038161003816Ψ (120577) minus 1

1003816100381610038161003816)



gradient test





1198781119888=

120597119909

120597119888

1198782119888=

120597119910

120597119888

1198783119888=

120597119911

120597119888

1198784119888=

120597119883

120597119888

1198785119888=

120597119884

120597119888

1198786119888=

120597119885

120597119888

(36)


1198781119888= 120590 (119878

2119888minus 1198781119888) minus 119888119878

4119888minus 119883


1198783119888= 1199091198782119888+ 1199101198781119888minus 1198871198783119888+ 1198881198786119888+ 119885

1198784119888= 120576120590 (119878

5119888minus 1198784119888) minus 119888119878

1119888minus 119909

1198785119888= 120576 (119903119878


2119888+ 119910

1198786119888= 120576 (119883119878

5119888+ 1198841198784119888minus 1198871198786119888) minus 119888119878

3119888minus 119911

(37)





nabla120572⟨J (120572)⟩ = ⟨nabla

120572J (120572)⟩ (38)



int

119879

0

lim120575120572rarr0

[

J (120572 + 120575120572) minusJ (120572)

120575120572

] 119889119905 (39)





= 28


+120575119903 where 120575119903 = 0011199030

=


1198781119903=

120597119909

120597119903

1198782119903=

120597119910

120597119903

1198783119903=

120597119911

120597119903

1198784119903=

120597119883

120597119903

1198785119903=

120597119884

120597119903

1198786119903=

120597119885

120597119903

(40)


1198781119903= 120590 (119878

2119903minus 1198781119903) minus 119888119878

4119903

1198782119903= 119909 + 119903119878

1119903minus 1198782119903minus (1199091198783119903+ 1199111198781119903) + 119888119878

5119903

1198783119903= (119909119878

2119903+ 1199101198781119903) minus 119887119878

3119903+ 1198881198786119903

1198784119903= 120576120590 (119878

5119903minus 1198784119903) minus 119888119878

1119903

1198785119903= 120576 [119883 + 119903119878

4119903minus 1198785119903minus (119883119878

6119903+ 1198851198784119903)] + 119888119878

2119903

1198786119903= 120576 [(119883119878

5119903+ 1198841198784119903) minus 119887119878

6119903] minus 119888119878

3119903

(41)






0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time


Am

plitu

de

2000

0

minus2000

Am

plitu

de

3000

0

minus3000

Am

plitu

deA

mpl

itude

2000

0

minus2000

Am

plitu

de

3000

0

minus3000

Am

plitu

de

2000

0

minus2000

1000

0

minus1000

S1

S2

S3

S4

S5

S6









Am

plitu

de

S1

S2

S3

S4

S5

S6

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

200

100

0

minus100

Am

plitu

de

100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de

50

0

minus50minus200



119888 120597119885120597119903 120597119884120597119903 120597119883120597119903 120597119911120597119903 120597119910120597119903 120597119909120597119903



Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

3

0

minus3

Am

plitu

de

3

0

minus3

minus6

Δx

Δy

Δz

ΔXΔY

ΔZ









Appendix





(119889 gt 0) of 119891 if

dist (119909119896+1

119891 (119909119896)) lt 119889 119896 isin Z (A1)






dist (119909119896 119910119896) lt 120576 119896 isin Z (A2)



dist (Φ119905 (119905 119892 (120591)) 119892 (120591 + 119905)) lt 119889 (A3)







S1

S2

S3

S4

S5

S6

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

Am

plitu

de

500

0

minus500

Am

plitu

de

500

0

minus500

Am

plitu

de

6000

4000

2000

0

minus2000

Am

plitu

de

1000

0

minus2000

minus1000 Am

plitu

de

8000

4000

0

minus4000

Am

plitu

de

1500

0

minus1500




100381610038161003816100381610038161003816100381610038161003816

ℎ (1199051) minus ℎ (119905

2)

1199051minus 1199052

minus 1

100381610038161003816100381610038161003816100381610038161003816

le 120576


(A4)


119889119909

119889119905

= 119891 (119909 120572) 119909 isin R119899

(A5)




S1

S2

S3

S4

S5

S6


Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de3

6

0

minus3

Am

plitu

de

3

6

0

minus3

Am

plitu

de

2

4

0

minus2



⟨119869 (120572)⟩ = lim119879rarrinfin

1

119879

int

119879

0

119869 (119909 (119905 120572) 120572) 119889119905 (A6)







min119909120591

1

119879

int

119879

0

[1003817100381710038171003817119909 (120591 (119905)) minus 119909

119903(119905)1003817100381710038171003817

2

+ 1205782

(

119889120591

119889119905

minus 1)

2

]119889119905

such that 119889119909

119889119905

= 119891 (119909 120572)

(A7)


Am

plitu

de

45

30

15

Am

plitu

de

40

20

00 5 10 15 20

Time0 5 10 15 20

Time

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

minus6

minus8

minus10

minus12

Am

plitu

de

0

20

10

minus10

minus20



119904(119905 120572) and 120591

(119879)

119904(119905 120572) is a solution of





⟨119869 (120572)⟩ asymp ⟨119869(119879)

119904(120572)⟩ =

1

120591 (119879) minus 120591 (0)

int

120591(119879)

120591(0)

119869 (119909(119879)

119904(119905 120572) 120572)

(A8)






min119878120583

1

119879

int

119879

0

[1198782

+ 1205782

1205832

] 119889119905

such that 119889119878

119889119905

=

120597119891

120597119909

119878 +

120597119891

120597120572

+ 120583119891 (119909119903 120572)

(A9)


119878 (119905) =

119889

119889120572

(119909(119879)

119904(120591(119879)

119904(119905 120572) 120572))

120583 (119905) =

119889

119889120572

119889120591(119879)

119904(119905 120572)

119889119905

(A10)



nabla120572⟨119869(119879)

119904(120572)⟩ asymp

1

119879

int

119879

0

(

120597119869

120597119909

119878 +

120597119869

120597120572

+ 120583 (119869 minus 119869)) 119889119905

where 119869 = 1

119879

int

119879

0

119869 119889119905

(A11)


119894+12= 119909119903((119894 +

12)Δ119905) 119909119894minus12



[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119868 119860119879

1

1205782

119861119879

1

119868 119862119879

1119860119879

2

1205782

119861119879

2

119868 119862119879

2

d d 119860119879

119898minus1

1205782

119861119879

119898minus1

119868 119862119879

119898minus1

119860111986111198621

119860211986121198622

d d

119860119898minus1

119861119898minus1

119862119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11987812

1205831

1198781+12

1205832

1198782+12

120583119898minus1

119878119898minus12

1198861

1198862

119886119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0

0

0

0

0

0

minus1198891

minus1198892


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(A12)

where

119860119894= minus

119868

Δ119905

minus

120597119891

120597119909

(119909119894minus12

120572) 119861119894=

119909119894+12


Δ119905

119862119894=

119868

Δ119905

minus

120597119891

120597119909

(119909119894+12

120572)

119889119894=

1

2

[

120597119891

120597120572

(119909119894minus12

120572) +

120597119891

120597120572

(119909119894+12

120572)]

(A13)







119894 119861119894 119862119894 and 119889

119894


119894




0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

120575x

04

02

00

minus02

minus04

08

04

00

minus04

minus08

04

00

minus04

minus08

02

00

12

08

04

00

10

15

05

00

minus02

minus04

120575y

120575z

120575X

120575Y

120575Z







119860 = [minus(

120597119891

120597119909

) + (

119889

119889119905

)] (A14)


minus1


1(119909) 120592

119899(119909)

120575119909 (119909) =

119899

sum

119894=1

120595119894(119909) 120592119894(119909) (A15a)

120575119891 (119909) =

119899

sum

119894=1

120593119894(119909) 120592119894(119909) (A15b)


119889120592119894(119909 (119905))

119889119905

=

120597119891

120597119909

120592119894(119909 (119905)) minus 120582

119894120592119894(119909 (119905)) (A16)

where 1205821 120582


one can obtain

119860 (120595119894120592119894) = [minus120595

119894(119909)

120597119891

120597119909

+

119889120595119894(119909)

119889119905

] 120592119894(119909) + 120595

119894(119909)

119889120592119894(119909)

119889119905

(A17)


119860 (120595119894120592119894) = [

119889120595119894(119909)

119889119905

minus 120582119894120595119894(119909)] 120592

119894(119909) (A18)


120575119891 (119909) =

119899

sum

119894=1

119860 (120595119894120592119894) =

119899

sum

119894=1

(

119889120595119894

119889119905

minus 120582119894120595119894)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120593119894

120592119894 (A19)



119889120595119894(119909)

119889119905

= 120593119894(119909) + 120582

119894120595119894(119909) (A20)






0 5 10 15 20

Time0 5 10 15 20

Time

Am

plitu

de

80

60

40

20Am

plitu

de

45

30

15

0

minus20

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

40

30

20

10

0

minus20

minus30

minus10

Am

plitu

de

20

10

0

minus20

minus10



120583 + ⟨1205930

119894⟩ = 0 (A21)

where

⟨1205930


1

119879

[1205950

119894(119909 (119879)) minus 120595

0

119894(119909 (0))] (A22)


minus1


119894= 0

1198891205950

119894(119909)

119889119905

= 1205930

119894(119909) + 120583 (A23)


0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

120575x

16

08

00

minus08

16

06

00

minus08

05

00

minus05

minus05

2

1

20

15

10

05

00

08

16

00

minus08

0

minus1

120575y

120575z

120575X

120575Y

120575Z




119863= minus20






119894(119909) 119894 = 1 119899






References




































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in



1198781119888=

120597119909

120597119888

1198782119888=

120597119910

120597119888

1198783119888=

120597119911

120597119888

1198784119888=

120597119883

120597119888

1198785119888=

120597119884

120597119888

1198786119888=

120597119885

120597119888

(36)


1198781119888= 120590 (119878

2119888minus 1198781119888) minus 119888119878

4119888minus 119883


1198783119888= 1199091198782119888+ 1199101198781119888minus 1198871198783119888+ 1198881198786119888+ 119885

1198784119888= 120576120590 (119878

5119888minus 1198784119888) minus 119888119878

1119888minus 119909

1198785119888= 120576 (119903119878


2119888+ 119910

1198786119888= 120576 (119883119878

5119888+ 1198841198784119888minus 1198871198786119888) minus 119888119878

3119888minus 119911

(37)





nabla120572⟨J (120572)⟩ = ⟨nabla

120572J (120572)⟩ (38)



int

119879

0

lim120575120572rarr0

[

J (120572 + 120575120572) minusJ (120572)

120575120572

] 119889119905 (39)





= 28


+120575119903 where 120575119903 = 0011199030

=


1198781119903=

120597119909

120597119903

1198782119903=

120597119910

120597119903

1198783119903=

120597119911

120597119903

1198784119903=

120597119883

120597119903

1198785119903=

120597119884

120597119903

1198786119903=

120597119885

120597119903

(40)


1198781119903= 120590 (119878

2119903minus 1198781119903) minus 119888119878

4119903

1198782119903= 119909 + 119903119878

1119903minus 1198782119903minus (1199091198783119903+ 1199111198781119903) + 119888119878

5119903

1198783119903= (119909119878

2119903+ 1199101198781119903) minus 119887119878

3119903+ 1198881198786119903

1198784119903= 120576120590 (119878

5119903minus 1198784119903) minus 119888119878

1119903

1198785119903= 120576 [119883 + 119903119878

4119903minus 1198785119903minus (119883119878

6119903+ 1198851198784119903)] + 119888119878

2119903

1198786119903= 120576 [(119883119878

5119903+ 1198841198784119903) minus 119887119878

6119903] minus 119888119878

3119903

(41)






0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time


Am

plitu

de

2000

0

minus2000

Am

plitu

de

3000

0

minus3000

Am

plitu

deA

mpl

itude

2000

0

minus2000

Am

plitu

de

3000

0

minus3000

Am

plitu

de

2000

0

minus2000

1000

0

minus1000

S1

S2

S3

S4

S5

S6









Am

plitu

de

S1

S2

S3

S4

S5

S6

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

200

100

0

minus100

Am

plitu

de

100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de

50

0

minus50minus200



119888 120597119885120597119903 120597119884120597119903 120597119883120597119903 120597119911120597119903 120597119910120597119903 120597119909120597119903



Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

3

0

minus3

Am

plitu

de

3

0

minus3

minus6

Δx

Δy

Δz

ΔXΔY

ΔZ









Appendix





(119889 gt 0) of 119891 if

dist (119909119896+1

119891 (119909119896)) lt 119889 119896 isin Z (A1)






dist (119909119896 119910119896) lt 120576 119896 isin Z (A2)



dist (Φ119905 (119905 119892 (120591)) 119892 (120591 + 119905)) lt 119889 (A3)







S1

S2

S3

S4

S5

S6

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

Am

plitu

de

500

0

minus500

Am

plitu

de

500

0

minus500

Am

plitu

de

6000

4000

2000

0

minus2000

Am

plitu

de

1000

0

minus2000

minus1000 Am

plitu

de

8000

4000

0

minus4000

Am

plitu

de

1500

0

minus1500




100381610038161003816100381610038161003816100381610038161003816

ℎ (1199051) minus ℎ (119905

2)

1199051minus 1199052

minus 1

100381610038161003816100381610038161003816100381610038161003816

le 120576


(A4)


119889119909

119889119905

= 119891 (119909 120572) 119909 isin R119899

(A5)




S1

S2

S3

S4

S5

S6


Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de3

6

0

minus3

Am

plitu

de

3

6

0

minus3

Am

plitu

de

2

4

0

minus2



⟨119869 (120572)⟩ = lim119879rarrinfin

1

119879

int

119879

0

119869 (119909 (119905 120572) 120572) 119889119905 (A6)







min119909120591

1

119879

int

119879

0

[1003817100381710038171003817119909 (120591 (119905)) minus 119909

119903(119905)1003817100381710038171003817

2

+ 1205782

(

119889120591

119889119905

minus 1)

2

]119889119905

such that 119889119909

119889119905

= 119891 (119909 120572)

(A7)


Am

plitu

de

45

30

15

Am

plitu

de

40

20

00 5 10 15 20

Time0 5 10 15 20

Time

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

minus6

minus8

minus10

minus12

Am

plitu

de

0

20

10

minus10

minus20



119904(119905 120572) and 120591

(119879)

119904(119905 120572) is a solution of





⟨119869 (120572)⟩ asymp ⟨119869(119879)

119904(120572)⟩ =

1

120591 (119879) minus 120591 (0)

int

120591(119879)

120591(0)

119869 (119909(119879)

119904(119905 120572) 120572)

(A8)






min119878120583

1

119879

int

119879

0

[1198782

+ 1205782

1205832

] 119889119905

such that 119889119878

119889119905

=

120597119891

120597119909

119878 +

120597119891

120597120572

+ 120583119891 (119909119903 120572)

(A9)


119878 (119905) =

119889

119889120572

(119909(119879)

119904(120591(119879)

119904(119905 120572) 120572))

120583 (119905) =

119889

119889120572

119889120591(119879)

119904(119905 120572)

119889119905

(A10)



nabla120572⟨119869(119879)

119904(120572)⟩ asymp

1

119879

int

119879

0

(

120597119869

120597119909

119878 +

120597119869

120597120572

+ 120583 (119869 minus 119869)) 119889119905

where 119869 = 1

119879

int

119879

0

119869 119889119905

(A11)


119894+12= 119909119903((119894 +

12)Δ119905) 119909119894minus12



[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119868 119860119879

1

1205782

119861119879

1

119868 119862119879

1119860119879

2

1205782

119861119879

2

119868 119862119879

2

d d 119860119879

119898minus1

1205782

119861119879

119898minus1

119868 119862119879

119898minus1

119860111986111198621

119860211986121198622

d d

119860119898minus1

119861119898minus1

119862119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11987812

1205831

1198781+12

1205832

1198782+12

120583119898minus1

119878119898minus12

1198861

1198862

119886119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0

0

0

0

0

0

minus1198891

minus1198892


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(A12)

where

119860119894= minus

119868

Δ119905

minus

120597119891

120597119909

(119909119894minus12

120572) 119861119894=

119909119894+12


Δ119905

119862119894=

119868

Δ119905

minus

120597119891

120597119909

(119909119894+12

120572)

119889119894=

1

2

[

120597119891

120597120572

(119909119894minus12

120572) +

120597119891

120597120572

(119909119894+12

120572)]

(A13)







119894 119861119894 119862119894 and 119889

119894


119894




0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

120575x

04

02

00

minus02

minus04

08

04

00

minus04

minus08

04

00

minus04

minus08

02

00

12

08

04

00

10

15

05

00

minus02

minus04

120575y

120575z

120575X

120575Y

120575Z







119860 = [minus(

120597119891

120597119909

) + (

119889

119889119905

)] (A14)


minus1


1(119909) 120592

119899(119909)

120575119909 (119909) =

119899

sum

119894=1

120595119894(119909) 120592119894(119909) (A15a)

120575119891 (119909) =

119899

sum

119894=1

120593119894(119909) 120592119894(119909) (A15b)


119889120592119894(119909 (119905))

119889119905

=

120597119891

120597119909

120592119894(119909 (119905)) minus 120582

119894120592119894(119909 (119905)) (A16)

where 1205821 120582


one can obtain

119860 (120595119894120592119894) = [minus120595

119894(119909)

120597119891

120597119909

+

119889120595119894(119909)

119889119905

] 120592119894(119909) + 120595

119894(119909)

119889120592119894(119909)

119889119905

(A17)


119860 (120595119894120592119894) = [

119889120595119894(119909)

119889119905

minus 120582119894120595119894(119909)] 120592

119894(119909) (A18)


120575119891 (119909) =

119899

sum

119894=1

119860 (120595119894120592119894) =

119899

sum

119894=1

(

119889120595119894

119889119905

minus 120582119894120595119894)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120593119894

120592119894 (A19)



119889120595119894(119909)

119889119905

= 120593119894(119909) + 120582

119894120595119894(119909) (A20)






0 5 10 15 20

Time0 5 10 15 20

Time

Am

plitu

de

80

60

40

20Am

plitu

de

45

30

15

0

minus20

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

40

30

20

10

0

minus20

minus30

minus10

Am

plitu

de

20

10

0

minus20

minus10



120583 + ⟨1205930

119894⟩ = 0 (A21)

where

⟨1205930


1

119879

[1205950

119894(119909 (119879)) minus 120595

0

119894(119909 (0))] (A22)


minus1


119894= 0

1198891205950

119894(119909)

119889119905

= 1205930

119894(119909) + 120583 (A23)


0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

120575x

16

08

00

minus08

16

06

00

minus08

05

00

minus05

minus05

2

1

20

15

10

05

00

08

16

00

minus08

0

minus1

120575y

120575z

120575X

120575Y

120575Z




119863= minus20






119894(119909) 119894 = 1 119899






References




































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time


Am

plitu

de

2000

0

minus2000

Am

plitu

de

3000

0

minus3000

Am

plitu

deA

mpl

itude

2000

0

minus2000

Am

plitu

de

3000

0

minus3000

Am

plitu

de

2000

0

minus2000

1000

0

minus1000

S1

S2

S3

S4

S5

S6









Am

plitu

de

S1

S2

S3

S4

S5

S6

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

200

100

0

minus100

Am

plitu

de

100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de

50

0

minus50minus200



119888 120597119885120597119903 120597119884120597119903 120597119883120597119903 120597119911120597119903 120597119910120597119903 120597119909120597119903



Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

3

0

minus3

Am

plitu

de

3

0

minus3

minus6

Δx

Δy

Δz

ΔXΔY

ΔZ









Appendix





(119889 gt 0) of 119891 if

dist (119909119896+1

119891 (119909119896)) lt 119889 119896 isin Z (A1)






dist (119909119896 119910119896) lt 120576 119896 isin Z (A2)



dist (Φ119905 (119905 119892 (120591)) 119892 (120591 + 119905)) lt 119889 (A3)







S1

S2

S3

S4

S5

S6

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

Am

plitu

de

500

0

minus500

Am

plitu

de

500

0

minus500

Am

plitu

de

6000

4000

2000

0

minus2000

Am

plitu

de

1000

0

minus2000

minus1000 Am

plitu

de

8000

4000

0

minus4000

Am

plitu

de

1500

0

minus1500




100381610038161003816100381610038161003816100381610038161003816

ℎ (1199051) minus ℎ (119905

2)

1199051minus 1199052

minus 1

100381610038161003816100381610038161003816100381610038161003816

le 120576


(A4)


119889119909

119889119905

= 119891 (119909 120572) 119909 isin R119899

(A5)




S1

S2

S3

S4

S5

S6


Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de3

6

0

minus3

Am

plitu

de

3

6

0

minus3

Am

plitu

de

2

4

0

minus2



⟨119869 (120572)⟩ = lim119879rarrinfin

1

119879

int

119879

0

119869 (119909 (119905 120572) 120572) 119889119905 (A6)







min119909120591

1

119879

int

119879

0

[1003817100381710038171003817119909 (120591 (119905)) minus 119909

119903(119905)1003817100381710038171003817

2

+ 1205782

(

119889120591

119889119905

minus 1)

2

]119889119905

such that 119889119909

119889119905

= 119891 (119909 120572)

(A7)


Am

plitu

de

45

30

15

Am

plitu

de

40

20

00 5 10 15 20

Time0 5 10 15 20

Time

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

minus6

minus8

minus10

minus12

Am

plitu

de

0

20

10

minus10

minus20



119904(119905 120572) and 120591

(119879)

119904(119905 120572) is a solution of





⟨119869 (120572)⟩ asymp ⟨119869(119879)

119904(120572)⟩ =

1

120591 (119879) minus 120591 (0)

int

120591(119879)

120591(0)

119869 (119909(119879)

119904(119905 120572) 120572)

(A8)






min119878120583

1

119879

int

119879

0

[1198782

+ 1205782

1205832

] 119889119905

such that 119889119878

119889119905

=

120597119891

120597119909

119878 +

120597119891

120597120572

+ 120583119891 (119909119903 120572)

(A9)


119878 (119905) =

119889

119889120572

(119909(119879)

119904(120591(119879)

119904(119905 120572) 120572))

120583 (119905) =

119889

119889120572

119889120591(119879)

119904(119905 120572)

119889119905

(A10)



nabla120572⟨119869(119879)

119904(120572)⟩ asymp

1

119879

int

119879

0

(

120597119869

120597119909

119878 +

120597119869

120597120572

+ 120583 (119869 minus 119869)) 119889119905

where 119869 = 1

119879

int

119879

0

119869 119889119905

(A11)


119894+12= 119909119903((119894 +

12)Δ119905) 119909119894minus12



[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119868 119860119879

1

1205782

119861119879

1

119868 119862119879

1119860119879

2

1205782

119861119879

2

119868 119862119879

2

d d 119860119879

119898minus1

1205782

119861119879

119898minus1

119868 119862119879

119898minus1

119860111986111198621

119860211986121198622

d d

119860119898minus1

119861119898minus1

119862119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11987812

1205831

1198781+12

1205832

1198782+12

120583119898minus1

119878119898minus12

1198861

1198862

119886119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0

0

0

0

0

0

minus1198891

minus1198892


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(A12)

where

119860119894= minus

119868

Δ119905

minus

120597119891

120597119909

(119909119894minus12

120572) 119861119894=

119909119894+12


Δ119905

119862119894=

119868

Δ119905

minus

120597119891

120597119909

(119909119894+12

120572)

119889119894=

1

2

[

120597119891

120597120572

(119909119894minus12

120572) +

120597119891

120597120572

(119909119894+12

120572)]

(A13)







119894 119861119894 119862119894 and 119889

119894


119894




0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

120575x

04

02

00

minus02

minus04

08

04

00

minus04

minus08

04

00

minus04

minus08

02

00

12

08

04

00

10

15

05

00

minus02

minus04

120575y

120575z

120575X

120575Y

120575Z







119860 = [minus(

120597119891

120597119909

) + (

119889

119889119905

)] (A14)


minus1


1(119909) 120592

119899(119909)

120575119909 (119909) =

119899

sum

119894=1

120595119894(119909) 120592119894(119909) (A15a)

120575119891 (119909) =

119899

sum

119894=1

120593119894(119909) 120592119894(119909) (A15b)


119889120592119894(119909 (119905))

119889119905

=

120597119891

120597119909

120592119894(119909 (119905)) minus 120582

119894120592119894(119909 (119905)) (A16)

where 1205821 120582


one can obtain

119860 (120595119894120592119894) = [minus120595

119894(119909)

120597119891

120597119909

+

119889120595119894(119909)

119889119905

] 120592119894(119909) + 120595

119894(119909)

119889120592119894(119909)

119889119905

(A17)


119860 (120595119894120592119894) = [

119889120595119894(119909)

119889119905

minus 120582119894120595119894(119909)] 120592

119894(119909) (A18)


120575119891 (119909) =

119899

sum

119894=1

119860 (120595119894120592119894) =

119899

sum

119894=1

(

119889120595119894

119889119905

minus 120582119894120595119894)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120593119894

120592119894 (A19)



119889120595119894(119909)

119889119905

= 120593119894(119909) + 120582

119894120595119894(119909) (A20)






0 5 10 15 20

Time0 5 10 15 20

Time

Am

plitu

de

80

60

40

20Am

plitu

de

45

30

15

0

minus20

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

40

30

20

10

0

minus20

minus30

minus10

Am

plitu

de

20

10

0

minus20

minus10



120583 + ⟨1205930

119894⟩ = 0 (A21)

where

⟨1205930


1

119879

[1205950

119894(119909 (119879)) minus 120595

0

119894(119909 (0))] (A22)


minus1


119894= 0

1198891205950

119894(119909)

119889119905

= 1205930

119894(119909) + 120583 (A23)


0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

120575x

16

08

00

minus08

16

06

00

minus08

05

00

minus05

minus05

2

1

20

15

10

05

00

08

16

00

minus08

0

minus1

120575y

120575z

120575X

120575Y

120575Z




119863= minus20






119894(119909) 119894 = 1 119899






References




































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in



Am

plitu

de

S1

S2

S3

S4

S5

S6

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

200

100

0

minus100

Am

plitu

de

100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de100

0

minus100minus200

Am

plitu

de

200

100

0

minus100

Am

plitu

de

50

0

minus50minus200



119888 120597119885120597119903 120597119884120597119903 120597119883120597119903 120597119911120597119903 120597119910120597119903 120597119909120597119903



Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

3

0

minus3

Am

plitu

de

3

0

minus3

minus6

Δx

Δy

Δz

ΔXΔY

ΔZ









Appendix





(119889 gt 0) of 119891 if

dist (119909119896+1

119891 (119909119896)) lt 119889 119896 isin Z (A1)






dist (119909119896 119910119896) lt 120576 119896 isin Z (A2)



dist (Φ119905 (119905 119892 (120591)) 119892 (120591 + 119905)) lt 119889 (A3)







S1

S2

S3

S4

S5

S6

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

Am

plitu

de

500

0

minus500

Am

plitu

de

500

0

minus500

Am

plitu

de

6000

4000

2000

0

minus2000

Am

plitu

de

1000

0

minus2000

minus1000 Am

plitu

de

8000

4000

0

minus4000

Am

plitu

de

1500

0

minus1500




100381610038161003816100381610038161003816100381610038161003816

ℎ (1199051) minus ℎ (119905

2)

1199051minus 1199052

minus 1

100381610038161003816100381610038161003816100381610038161003816

le 120576


(A4)


119889119909

119889119905

= 119891 (119909 120572) 119909 isin R119899

(A5)




S1

S2

S3

S4

S5

S6


Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de3

6

0

minus3

Am

plitu

de

3

6

0

minus3

Am

plitu

de

2

4

0

minus2



⟨119869 (120572)⟩ = lim119879rarrinfin

1

119879

int

119879

0

119869 (119909 (119905 120572) 120572) 119889119905 (A6)







min119909120591

1

119879

int

119879

0

[1003817100381710038171003817119909 (120591 (119905)) minus 119909

119903(119905)1003817100381710038171003817

2

+ 1205782

(

119889120591

119889119905

minus 1)

2

]119889119905

such that 119889119909

119889119905

= 119891 (119909 120572)

(A7)


Am

plitu

de

45

30

15

Am

plitu

de

40

20

00 5 10 15 20

Time0 5 10 15 20

Time

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

minus6

minus8

minus10

minus12

Am

plitu

de

0

20

10

minus10

minus20



119904(119905 120572) and 120591

(119879)

119904(119905 120572) is a solution of





⟨119869 (120572)⟩ asymp ⟨119869(119879)

119904(120572)⟩ =

1

120591 (119879) minus 120591 (0)

int

120591(119879)

120591(0)

119869 (119909(119879)

119904(119905 120572) 120572)

(A8)






min119878120583

1

119879

int

119879

0

[1198782

+ 1205782

1205832

] 119889119905

such that 119889119878

119889119905

=

120597119891

120597119909

119878 +

120597119891

120597120572

+ 120583119891 (119909119903 120572)

(A9)


119878 (119905) =

119889

119889120572

(119909(119879)

119904(120591(119879)

119904(119905 120572) 120572))

120583 (119905) =

119889

119889120572

119889120591(119879)

119904(119905 120572)

119889119905

(A10)



nabla120572⟨119869(119879)

119904(120572)⟩ asymp

1

119879

int

119879

0

(

120597119869

120597119909

119878 +

120597119869

120597120572

+ 120583 (119869 minus 119869)) 119889119905

where 119869 = 1

119879

int

119879

0

119869 119889119905

(A11)


119894+12= 119909119903((119894 +

12)Δ119905) 119909119894minus12



[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119868 119860119879

1

1205782

119861119879

1

119868 119862119879

1119860119879

2

1205782

119861119879

2

119868 119862119879

2

d d 119860119879

119898minus1

1205782

119861119879

119898minus1

119868 119862119879

119898minus1

119860111986111198621

119860211986121198622

d d

119860119898minus1

119861119898minus1

119862119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11987812

1205831

1198781+12

1205832

1198782+12

120583119898minus1

119878119898minus12

1198861

1198862

119886119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0

0

0

0

0

0

minus1198891

minus1198892


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(A12)

where

119860119894= minus

119868

Δ119905

minus

120597119891

120597119909

(119909119894minus12

120572) 119861119894=

119909119894+12


Δ119905

119862119894=

119868

Δ119905

minus

120597119891

120597119909

(119909119894+12

120572)

119889119894=

1

2

[

120597119891

120597120572

(119909119894minus12

120572) +

120597119891

120597120572

(119909119894+12

120572)]

(A13)







119894 119861119894 119862119894 and 119889

119894


119894




0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

120575x

04

02

00

minus02

minus04

08

04

00

minus04

minus08

04

00

minus04

minus08

02

00

12

08

04

00

10

15

05

00

minus02

minus04

120575y

120575z

120575X

120575Y

120575Z







119860 = [minus(

120597119891

120597119909

) + (

119889

119889119905

)] (A14)


minus1


1(119909) 120592

119899(119909)

120575119909 (119909) =

119899

sum

119894=1

120595119894(119909) 120592119894(119909) (A15a)

120575119891 (119909) =

119899

sum

119894=1

120593119894(119909) 120592119894(119909) (A15b)


119889120592119894(119909 (119905))

119889119905

=

120597119891

120597119909

120592119894(119909 (119905)) minus 120582

119894120592119894(119909 (119905)) (A16)

where 1205821 120582


one can obtain

119860 (120595119894120592119894) = [minus120595

119894(119909)

120597119891

120597119909

+

119889120595119894(119909)

119889119905

] 120592119894(119909) + 120595

119894(119909)

119889120592119894(119909)

119889119905

(A17)


119860 (120595119894120592119894) = [

119889120595119894(119909)

119889119905

minus 120582119894120595119894(119909)] 120592

119894(119909) (A18)


120575119891 (119909) =

119899

sum

119894=1

119860 (120595119894120592119894) =

119899

sum

119894=1

(

119889120595119894

119889119905

minus 120582119894120595119894)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120593119894

120592119894 (A19)



119889120595119894(119909)

119889119905

= 120593119894(119909) + 120582

119894120595119894(119909) (A20)






0 5 10 15 20

Time0 5 10 15 20

Time

Am

plitu

de

80

60

40

20Am

plitu

de

45

30

15

0

minus20

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

40

30

20

10

0

minus20

minus30

minus10

Am

plitu

de

20

10

0

minus20

minus10



120583 + ⟨1205930

119894⟩ = 0 (A21)

where

⟨1205930


1

119879

[1205950

119894(119909 (119879)) minus 120595

0

119894(119909 (0))] (A22)


minus1


119894= 0

1198891205950

119894(119909)

119889119905

= 1205930

119894(119909) + 120583 (A23)


0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

120575x

16

08

00

minus08

16

06

00

minus08

05

00

minus05

minus05

2

1

20

15

10

05

00

08

16

00

minus08

0

minus1

120575y

120575z

120575X

120575Y

120575Z




119863= minus20






119894(119909) 119894 = 1 119899






References




































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

3

0

minus3

Am

plitu

de

3

0

minus3

minus6

Δx

Δy

Δz

ΔXΔY

ΔZ









Appendix





(119889 gt 0) of 119891 if

dist (119909119896+1

119891 (119909119896)) lt 119889 119896 isin Z (A1)






dist (119909119896 119910119896) lt 120576 119896 isin Z (A2)



dist (Φ119905 (119905 119892 (120591)) 119892 (120591 + 119905)) lt 119889 (A3)







S1

S2

S3

S4

S5

S6

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

Am

plitu

de

500

0

minus500

Am

plitu

de

500

0

minus500

Am

plitu

de

6000

4000

2000

0

minus2000

Am

plitu

de

1000

0

minus2000

minus1000 Am

plitu

de

8000

4000

0

minus4000

Am

plitu

de

1500

0

minus1500




100381610038161003816100381610038161003816100381610038161003816

ℎ (1199051) minus ℎ (119905

2)

1199051minus 1199052

minus 1

100381610038161003816100381610038161003816100381610038161003816

le 120576


(A4)


119889119909

119889119905

= 119891 (119909 120572) 119909 isin R119899

(A5)




S1

S2

S3

S4

S5

S6


Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de3

6

0

minus3

Am

plitu

de

3

6

0

minus3

Am

plitu

de

2

4

0

minus2



⟨119869 (120572)⟩ = lim119879rarrinfin

1

119879

int

119879

0

119869 (119909 (119905 120572) 120572) 119889119905 (A6)







min119909120591

1

119879

int

119879

0

[1003817100381710038171003817119909 (120591 (119905)) minus 119909

119903(119905)1003817100381710038171003817

2

+ 1205782

(

119889120591

119889119905

minus 1)

2

]119889119905

such that 119889119909

119889119905

= 119891 (119909 120572)

(A7)


Am

plitu

de

45

30

15

Am

plitu

de

40

20

00 5 10 15 20

Time0 5 10 15 20

Time

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

minus6

minus8

minus10

minus12

Am

plitu

de

0

20

10

minus10

minus20



119904(119905 120572) and 120591

(119879)

119904(119905 120572) is a solution of





⟨119869 (120572)⟩ asymp ⟨119869(119879)

119904(120572)⟩ =

1

120591 (119879) minus 120591 (0)

int

120591(119879)

120591(0)

119869 (119909(119879)

119904(119905 120572) 120572)

(A8)






min119878120583

1

119879

int

119879

0

[1198782

+ 1205782

1205832

] 119889119905

such that 119889119878

119889119905

=

120597119891

120597119909

119878 +

120597119891

120597120572

+ 120583119891 (119909119903 120572)

(A9)


119878 (119905) =

119889

119889120572

(119909(119879)

119904(120591(119879)

119904(119905 120572) 120572))

120583 (119905) =

119889

119889120572

119889120591(119879)

119904(119905 120572)

119889119905

(A10)



nabla120572⟨119869(119879)

119904(120572)⟩ asymp

1

119879

int

119879

0

(

120597119869

120597119909

119878 +

120597119869

120597120572

+ 120583 (119869 minus 119869)) 119889119905

where 119869 = 1

119879

int

119879

0

119869 119889119905

(A11)


119894+12= 119909119903((119894 +

12)Δ119905) 119909119894minus12



[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119868 119860119879

1

1205782

119861119879

1

119868 119862119879

1119860119879

2

1205782

119861119879

2

119868 119862119879

2

d d 119860119879

119898minus1

1205782

119861119879

119898minus1

119868 119862119879

119898minus1

119860111986111198621

119860211986121198622

d d

119860119898minus1

119861119898minus1

119862119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11987812

1205831

1198781+12

1205832

1198782+12

120583119898minus1

119878119898minus12

1198861

1198862

119886119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0

0

0

0

0

0

minus1198891

minus1198892


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(A12)

where

119860119894= minus

119868

Δ119905

minus

120597119891

120597119909

(119909119894minus12

120572) 119861119894=

119909119894+12


Δ119905

119862119894=

119868

Δ119905

minus

120597119891

120597119909

(119909119894+12

120572)

119889119894=

1

2

[

120597119891

120597120572

(119909119894minus12

120572) +

120597119891

120597120572

(119909119894+12

120572)]

(A13)







119894 119861119894 119862119894 and 119889

119894


119894




0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

120575x

04

02

00

minus02

minus04

08

04

00

minus04

minus08

04

00

minus04

minus08

02

00

12

08

04

00

10

15

05

00

minus02

minus04

120575y

120575z

120575X

120575Y

120575Z







119860 = [minus(

120597119891

120597119909

) + (

119889

119889119905

)] (A14)


minus1


1(119909) 120592

119899(119909)

120575119909 (119909) =

119899

sum

119894=1

120595119894(119909) 120592119894(119909) (A15a)

120575119891 (119909) =

119899

sum

119894=1

120593119894(119909) 120592119894(119909) (A15b)


119889120592119894(119909 (119905))

119889119905

=

120597119891

120597119909

120592119894(119909 (119905)) minus 120582

119894120592119894(119909 (119905)) (A16)

where 1205821 120582


one can obtain

119860 (120595119894120592119894) = [minus120595

119894(119909)

120597119891

120597119909

+

119889120595119894(119909)

119889119905

] 120592119894(119909) + 120595

119894(119909)

119889120592119894(119909)

119889119905

(A17)


119860 (120595119894120592119894) = [

119889120595119894(119909)

119889119905

minus 120582119894120595119894(119909)] 120592

119894(119909) (A18)


120575119891 (119909) =

119899

sum

119894=1

119860 (120595119894120592119894) =

119899

sum

119894=1

(

119889120595119894

119889119905

minus 120582119894120595119894)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120593119894

120592119894 (A19)



119889120595119894(119909)

119889119905

= 120593119894(119909) + 120582

119894120595119894(119909) (A20)






0 5 10 15 20

Time0 5 10 15 20

Time

Am

plitu

de

80

60

40

20Am

plitu

de

45

30

15

0

minus20

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

40

30

20

10

0

minus20

minus30

minus10

Am

plitu

de

20

10

0

minus20

minus10



120583 + ⟨1205930

119894⟩ = 0 (A21)

where

⟨1205930


1

119879

[1205950

119894(119909 (119879)) minus 120595

0

119894(119909 (0))] (A22)


minus1


119894= 0

1198891205950

119894(119909)

119889119905

= 1205930

119894(119909) + 120583 (A23)


0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

120575x

16

08

00

minus08

16

06

00

minus08

05

00

minus05

minus05

2

1

20

15

10

05

00

08

16

00

minus08

0

minus1

120575y

120575z

120575X

120575Y

120575Z




119863= minus20






119894(119909) 119894 = 1 119899






References




































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in



S1

S2

S3

S4

S5

S6

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

0 2010

Time0 2010

Time

Am

plitu

de

500

0

minus500

Am

plitu

de

500

0

minus500

Am

plitu

de

6000

4000

2000

0

minus2000

Am

plitu

de

1000

0

minus2000

minus1000 Am

plitu

de

8000

4000

0

minus4000

Am

plitu

de

1500

0

minus1500




100381610038161003816100381610038161003816100381610038161003816

ℎ (1199051) minus ℎ (119905

2)

1199051minus 1199052

minus 1

100381610038161003816100381610038161003816100381610038161003816

le 120576


(A4)


119889119909

119889119905

= 119891 (119909 120572) 119909 isin R119899

(A5)




S1

S2

S3

S4

S5

S6


Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de3

6

0

minus3

Am

plitu

de

3

6

0

minus3

Am

plitu

de

2

4

0

minus2



⟨119869 (120572)⟩ = lim119879rarrinfin

1

119879

int

119879

0

119869 (119909 (119905 120572) 120572) 119889119905 (A6)







min119909120591

1

119879

int

119879

0

[1003817100381710038171003817119909 (120591 (119905)) minus 119909

119903(119905)1003817100381710038171003817

2

+ 1205782

(

119889120591

119889119905

minus 1)

2

]119889119905

such that 119889119909

119889119905

= 119891 (119909 120572)

(A7)


Am

plitu

de

45

30

15

Am

plitu

de

40

20

00 5 10 15 20

Time0 5 10 15 20

Time

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

minus6

minus8

minus10

minus12

Am

plitu

de

0

20

10

minus10

minus20



119904(119905 120572) and 120591

(119879)

119904(119905 120572) is a solution of





⟨119869 (120572)⟩ asymp ⟨119869(119879)

119904(120572)⟩ =

1

120591 (119879) minus 120591 (0)

int

120591(119879)

120591(0)

119869 (119909(119879)

119904(119905 120572) 120572)

(A8)






min119878120583

1

119879

int

119879

0

[1198782

+ 1205782

1205832

] 119889119905

such that 119889119878

119889119905

=

120597119891

120597119909

119878 +

120597119891

120597120572

+ 120583119891 (119909119903 120572)

(A9)


119878 (119905) =

119889

119889120572

(119909(119879)

119904(120591(119879)

119904(119905 120572) 120572))

120583 (119905) =

119889

119889120572

119889120591(119879)

119904(119905 120572)

119889119905

(A10)



nabla120572⟨119869(119879)

119904(120572)⟩ asymp

1

119879

int

119879

0

(

120597119869

120597119909

119878 +

120597119869

120597120572

+ 120583 (119869 minus 119869)) 119889119905

where 119869 = 1

119879

int

119879

0

119869 119889119905

(A11)


119894+12= 119909119903((119894 +

12)Δ119905) 119909119894minus12



[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119868 119860119879

1

1205782

119861119879

1

119868 119862119879

1119860119879

2

1205782

119861119879

2

119868 119862119879

2

d d 119860119879

119898minus1

1205782

119861119879

119898minus1

119868 119862119879

119898minus1

119860111986111198621

119860211986121198622

d d

119860119898minus1

119861119898minus1

119862119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11987812

1205831

1198781+12

1205832

1198782+12

120583119898minus1

119878119898minus12

1198861

1198862

119886119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0

0

0

0

0

0

minus1198891

minus1198892


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(A12)

where

119860119894= minus

119868

Δ119905

minus

120597119891

120597119909

(119909119894minus12

120572) 119861119894=

119909119894+12


Δ119905

119862119894=

119868

Δ119905

minus

120597119891

120597119909

(119909119894+12

120572)

119889119894=

1

2

[

120597119891

120597120572

(119909119894minus12

120572) +

120597119891

120597120572

(119909119894+12

120572)]

(A13)







119894 119861119894 119862119894 and 119889

119894


119894




0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

120575x

04

02

00

minus02

minus04

08

04

00

minus04

minus08

04

00

minus04

minus08

02

00

12

08

04

00

10

15

05

00

minus02

minus04

120575y

120575z

120575X

120575Y

120575Z







119860 = [minus(

120597119891

120597119909

) + (

119889

119889119905

)] (A14)


minus1


1(119909) 120592

119899(119909)

120575119909 (119909) =

119899

sum

119894=1

120595119894(119909) 120592119894(119909) (A15a)

120575119891 (119909) =

119899

sum

119894=1

120593119894(119909) 120592119894(119909) (A15b)


119889120592119894(119909 (119905))

119889119905

=

120597119891

120597119909

120592119894(119909 (119905)) minus 120582

119894120592119894(119909 (119905)) (A16)

where 1205821 120582


one can obtain

119860 (120595119894120592119894) = [minus120595

119894(119909)

120597119891

120597119909

+

119889120595119894(119909)

119889119905

] 120592119894(119909) + 120595

119894(119909)

119889120592119894(119909)

119889119905

(A17)


119860 (120595119894120592119894) = [

119889120595119894(119909)

119889119905

minus 120582119894120595119894(119909)] 120592

119894(119909) (A18)


120575119891 (119909) =

119899

sum

119894=1

119860 (120595119894120592119894) =

119899

sum

119894=1

(

119889120595119894

119889119905

minus 120582119894120595119894)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120593119894

120592119894 (A19)



119889120595119894(119909)

119889119905

= 120593119894(119909) + 120582

119894120595119894(119909) (A20)






0 5 10 15 20

Time0 5 10 15 20

Time

Am

plitu

de

80

60

40

20Am

plitu

de

45

30

15

0

minus20

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

40

30

20

10

0

minus20

minus30

minus10

Am

plitu

de

20

10

0

minus20

minus10



120583 + ⟨1205930

119894⟩ = 0 (A21)

where

⟨1205930


1

119879

[1205950

119894(119909 (119879)) minus 120595

0

119894(119909 (0))] (A22)


minus1


119894= 0

1198891205950

119894(119909)

119889119905

= 1205930

119894(119909) + 120583 (A23)


0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

120575x

16

08

00

minus08

16

06

00

minus08

05

00

minus05

minus05

2

1

20

15

10

05

00

08

16

00

minus08

0

minus1

120575y

120575z

120575X

120575Y

120575Z




119863= minus20






119894(119909) 119894 = 1 119899






References




































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


S1

S2

S3

S4

S5

S6


Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Time0 1 2 3 4 5

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de

1

2

0

minus1

minus2

Am

plitu

de3

6

0

minus3

Am

plitu

de

3

6

0

minus3

Am

plitu

de

2

4

0

minus2



⟨119869 (120572)⟩ = lim119879rarrinfin

1

119879

int

119879

0

119869 (119909 (119905 120572) 120572) 119889119905 (A6)







min119909120591

1

119879

int

119879

0

[1003817100381710038171003817119909 (120591 (119905)) minus 119909

119903(119905)1003817100381710038171003817

2

+ 1205782

(

119889120591

119889119905

minus 1)

2

]119889119905

such that 119889119909

119889119905

= 119891 (119909 120572)

(A7)


Am

plitu

de

45

30

15

Am

plitu

de

40

20

00 5 10 15 20

Time0 5 10 15 20

Time

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

minus6

minus8

minus10

minus12

Am

plitu

de

0

20

10

minus10

minus20



119904(119905 120572) and 120591

(119879)

119904(119905 120572) is a solution of





⟨119869 (120572)⟩ asymp ⟨119869(119879)

119904(120572)⟩ =

1

120591 (119879) minus 120591 (0)

int

120591(119879)

120591(0)

119869 (119909(119879)

119904(119905 120572) 120572)

(A8)






min119878120583

1

119879

int

119879

0

[1198782

+ 1205782

1205832

] 119889119905

such that 119889119878

119889119905

=

120597119891

120597119909

119878 +

120597119891

120597120572

+ 120583119891 (119909119903 120572)

(A9)


119878 (119905) =

119889

119889120572

(119909(119879)

119904(120591(119879)

119904(119905 120572) 120572))

120583 (119905) =

119889

119889120572

119889120591(119879)

119904(119905 120572)

119889119905

(A10)



nabla120572⟨119869(119879)

119904(120572)⟩ asymp

1

119879

int

119879

0

(

120597119869

120597119909

119878 +

120597119869

120597120572

+ 120583 (119869 minus 119869)) 119889119905

where 119869 = 1

119879

int

119879

0

119869 119889119905

(A11)


119894+12= 119909119903((119894 +

12)Δ119905) 119909119894minus12



[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119868 119860119879

1

1205782

119861119879

1

119868 119862119879

1119860119879

2

1205782

119861119879

2

119868 119862119879

2

d d 119860119879

119898minus1

1205782

119861119879

119898minus1

119868 119862119879

119898minus1

119860111986111198621

119860211986121198622

d d

119860119898minus1

119861119898minus1

119862119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11987812

1205831

1198781+12

1205832

1198782+12

120583119898minus1

119878119898minus12

1198861

1198862

119886119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0

0

0

0

0

0

minus1198891

minus1198892


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(A12)

where

119860119894= minus

119868

Δ119905

minus

120597119891

120597119909

(119909119894minus12

120572) 119861119894=

119909119894+12


Δ119905

119862119894=

119868

Δ119905

minus

120597119891

120597119909

(119909119894+12

120572)

119889119894=

1

2

[

120597119891

120597120572

(119909119894minus12

120572) +

120597119891

120597120572

(119909119894+12

120572)]

(A13)







119894 119861119894 119862119894 and 119889

119894


119894




0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

120575x

04

02

00

minus02

minus04

08

04

00

minus04

minus08

04

00

minus04

minus08

02

00

12

08

04

00

10

15

05

00

minus02

minus04

120575y

120575z

120575X

120575Y

120575Z







119860 = [minus(

120597119891

120597119909

) + (

119889

119889119905

)] (A14)


minus1


1(119909) 120592

119899(119909)

120575119909 (119909) =

119899

sum

119894=1

120595119894(119909) 120592119894(119909) (A15a)

120575119891 (119909) =

119899

sum

119894=1

120593119894(119909) 120592119894(119909) (A15b)


119889120592119894(119909 (119905))

119889119905

=

120597119891

120597119909

120592119894(119909 (119905)) minus 120582

119894120592119894(119909 (119905)) (A16)

where 1205821 120582


one can obtain

119860 (120595119894120592119894) = [minus120595

119894(119909)

120597119891

120597119909

+

119889120595119894(119909)

119889119905

] 120592119894(119909) + 120595

119894(119909)

119889120592119894(119909)

119889119905

(A17)


119860 (120595119894120592119894) = [

119889120595119894(119909)

119889119905

minus 120582119894120595119894(119909)] 120592

119894(119909) (A18)


120575119891 (119909) =

119899

sum

119894=1

119860 (120595119894120592119894) =

119899

sum

119894=1

(

119889120595119894

119889119905

minus 120582119894120595119894)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120593119894

120592119894 (A19)



119889120595119894(119909)

119889119905

= 120593119894(119909) + 120582

119894120595119894(119909) (A20)






0 5 10 15 20

Time0 5 10 15 20

Time

Am

plitu

de

80

60

40

20Am

plitu

de

45

30

15

0

minus20

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

40

30

20

10

0

minus20

minus30

minus10

Am

plitu

de

20

10

0

minus20

minus10



120583 + ⟨1205930

119894⟩ = 0 (A21)

where

⟨1205930


1

119879

[1205950

119894(119909 (119879)) minus 120595

0

119894(119909 (0))] (A22)


minus1


119894= 0

1198891205950

119894(119909)

119889119905

= 1205930

119894(119909) + 120583 (A23)


0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

120575x

16

08

00

minus08

16

06

00

minus08

05

00

minus05

minus05

2

1

20

15

10

05

00

08

16

00

minus08

0

minus1

120575y

120575z

120575X

120575Y

120575Z




119863= minus20






119894(119909) 119894 = 1 119899






References




































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


Am

plitu

de

45

30

15

Am

plitu

de

40

20

00 5 10 15 20

Time0 5 10 15 20

Time

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

minus6

minus8

minus10

minus12

Am

plitu

de

0

20

10

minus10

minus20



119904(119905 120572) and 120591

(119879)

119904(119905 120572) is a solution of





⟨119869 (120572)⟩ asymp ⟨119869(119879)

119904(120572)⟩ =

1

120591 (119879) minus 120591 (0)

int

120591(119879)

120591(0)

119869 (119909(119879)

119904(119905 120572) 120572)

(A8)






min119878120583

1

119879

int

119879

0

[1198782

+ 1205782

1205832

] 119889119905

such that 119889119878

119889119905

=

120597119891

120597119909

119878 +

120597119891

120597120572

+ 120583119891 (119909119903 120572)

(A9)


119878 (119905) =

119889

119889120572

(119909(119879)

119904(120591(119879)

119904(119905 120572) 120572))

120583 (119905) =

119889

119889120572

119889120591(119879)

119904(119905 120572)

119889119905

(A10)



nabla120572⟨119869(119879)

119904(120572)⟩ asymp

1

119879

int

119879

0

(

120597119869

120597119909

119878 +

120597119869

120597120572

+ 120583 (119869 minus 119869)) 119889119905

where 119869 = 1

119879

int

119879

0

119869 119889119905

(A11)


119894+12= 119909119903((119894 +

12)Δ119905) 119909119894minus12



[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119868 119860119879

1

1205782

119861119879

1

119868 119862119879

1119860119879

2

1205782

119861119879

2

119868 119862119879

2

d d 119860119879

119898minus1

1205782

119861119879

119898minus1

119868 119862119879

119898minus1

119860111986111198621

119860211986121198622

d d

119860119898minus1

119861119898minus1

119862119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11987812

1205831

1198781+12

1205832

1198782+12

120583119898minus1

119878119898minus12

1198861

1198862

119886119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0

0

0

0

0

0

minus1198891

minus1198892


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(A12)

where

119860119894= minus

119868

Δ119905

minus

120597119891

120597119909

(119909119894minus12

120572) 119861119894=

119909119894+12


Δ119905

119862119894=

119868

Δ119905

minus

120597119891

120597119909

(119909119894+12

120572)

119889119894=

1

2

[

120597119891

120597120572

(119909119894minus12

120572) +

120597119891

120597120572

(119909119894+12

120572)]

(A13)







119894 119861119894 119862119894 and 119889

119894


119894




0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

120575x

04

02

00

minus02

minus04

08

04

00

minus04

minus08

04

00

minus04

minus08

02

00

12

08

04

00

10

15

05

00

minus02

minus04

120575y

120575z

120575X

120575Y

120575Z







119860 = [minus(

120597119891

120597119909

) + (

119889

119889119905

)] (A14)


minus1


1(119909) 120592

119899(119909)

120575119909 (119909) =

119899

sum

119894=1

120595119894(119909) 120592119894(119909) (A15a)

120575119891 (119909) =

119899

sum

119894=1

120593119894(119909) 120592119894(119909) (A15b)


119889120592119894(119909 (119905))

119889119905

=

120597119891

120597119909

120592119894(119909 (119905)) minus 120582

119894120592119894(119909 (119905)) (A16)

where 1205821 120582


one can obtain

119860 (120595119894120592119894) = [minus120595

119894(119909)

120597119891

120597119909

+

119889120595119894(119909)

119889119905

] 120592119894(119909) + 120595

119894(119909)

119889120592119894(119909)

119889119905

(A17)


119860 (120595119894120592119894) = [

119889120595119894(119909)

119889119905

minus 120582119894120595119894(119909)] 120592

119894(119909) (A18)


120575119891 (119909) =

119899

sum

119894=1

119860 (120595119894120592119894) =

119899

sum

119894=1

(

119889120595119894

119889119905

minus 120582119894120595119894)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120593119894

120592119894 (A19)



119889120595119894(119909)

119889119905

= 120593119894(119909) + 120582

119894120595119894(119909) (A20)






0 5 10 15 20

Time0 5 10 15 20

Time

Am

plitu

de

80

60

40

20Am

plitu

de

45

30

15

0

minus20

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

40

30

20

10

0

minus20

minus30

minus10

Am

plitu

de

20

10

0

minus20

minus10



120583 + ⟨1205930

119894⟩ = 0 (A21)

where

⟨1205930


1

119879

[1205950

119894(119909 (119879)) minus 120595

0

119894(119909 (0))] (A22)


minus1


119894= 0

1198891205950

119894(119909)

119889119905

= 1205930

119894(119909) + 120583 (A23)


0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

120575x

16

08

00

minus08

16

06

00

minus08

05

00

minus05

minus05

2

1

20

15

10

05

00

08

16

00

minus08

0

minus1

120575y

120575z

120575X

120575Y

120575Z




119863= minus20






119894(119909) 119894 = 1 119899






References




































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in






min119878120583

1

119879

int

119879

0

[1198782

+ 1205782

1205832

] 119889119905

such that 119889119878

119889119905

=

120597119891

120597119909

119878 +

120597119891

120597120572

+ 120583119891 (119909119903 120572)

(A9)


119878 (119905) =

119889

119889120572

(119909(119879)

119904(120591(119879)

119904(119905 120572) 120572))

120583 (119905) =

119889

119889120572

119889120591(119879)

119904(119905 120572)

119889119905

(A10)



nabla120572⟨119869(119879)

119904(120572)⟩ asymp

1

119879

int

119879

0

(

120597119869

120597119909

119878 +

120597119869

120597120572

+ 120583 (119869 minus 119869)) 119889119905

where 119869 = 1

119879

int

119879

0

119869 119889119905

(A11)


119894+12= 119909119903((119894 +

12)Δ119905) 119909119894minus12



[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119868 119860119879

1

1205782

119861119879

1

119868 119862119879

1119860119879

2

1205782

119861119879

2

119868 119862119879

2

d d 119860119879

119898minus1

1205782

119861119879

119898minus1

119868 119862119879

119898minus1

119860111986111198621

119860211986121198622

d d

119860119898minus1

119861119898minus1

119862119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

11987812

1205831

1198781+12

1205832

1198782+12

120583119898minus1

119878119898minus12

1198861

1198862

119886119898minus1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0

0

0

0

0

0

0

minus1198891

minus1198892


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(A12)

where

119860119894= minus

119868

Δ119905

minus

120597119891

120597119909

(119909119894minus12

120572) 119861119894=

119909119894+12


Δ119905

119862119894=

119868

Δ119905

minus

120597119891

120597119909

(119909119894+12

120572)

119889119894=

1

2

[

120597119891

120597120572

(119909119894minus12

120572) +

120597119891

120597120572

(119909119894+12

120572)]

(A13)







119894 119861119894 119862119894 and 119889

119894


119894




0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

120575x

04

02

00

minus02

minus04

08

04

00

minus04

minus08

04

00

minus04

minus08

02

00

12

08

04

00

10

15

05

00

minus02

minus04

120575y

120575z

120575X

120575Y

120575Z







119860 = [minus(

120597119891

120597119909

) + (

119889

119889119905

)] (A14)


minus1


1(119909) 120592

119899(119909)

120575119909 (119909) =

119899

sum

119894=1

120595119894(119909) 120592119894(119909) (A15a)

120575119891 (119909) =

119899

sum

119894=1

120593119894(119909) 120592119894(119909) (A15b)


119889120592119894(119909 (119905))

119889119905

=

120597119891

120597119909

120592119894(119909 (119905)) minus 120582

119894120592119894(119909 (119905)) (A16)

where 1205821 120582


one can obtain

119860 (120595119894120592119894) = [minus120595

119894(119909)

120597119891

120597119909

+

119889120595119894(119909)

119889119905

] 120592119894(119909) + 120595

119894(119909)

119889120592119894(119909)

119889119905

(A17)


119860 (120595119894120592119894) = [

119889120595119894(119909)

119889119905

minus 120582119894120595119894(119909)] 120592

119894(119909) (A18)


120575119891 (119909) =

119899

sum

119894=1

119860 (120595119894120592119894) =

119899

sum

119894=1

(

119889120595119894

119889119905

minus 120582119894120595119894)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120593119894

120592119894 (A19)



119889120595119894(119909)

119889119905

= 120593119894(119909) + 120582

119894120595119894(119909) (A20)






0 5 10 15 20

Time0 5 10 15 20

Time

Am

plitu

de

80

60

40

20Am

plitu

de

45

30

15

0

minus20

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

40

30

20

10

0

minus20

minus30

minus10

Am

plitu

de

20

10

0

minus20

minus10



120583 + ⟨1205930

119894⟩ = 0 (A21)

where

⟨1205930


1

119879

[1205950

119894(119909 (119879)) minus 120595

0

119894(119909 (0))] (A22)


minus1


119894= 0

1198891205950

119894(119909)

119889119905

= 1205930

119894(119909) + 120583 (A23)


0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

120575x

16

08

00

minus08

16

06

00

minus08

05

00

minus05

minus05

2

1

20

15

10

05

00

08

16

00

minus08

0

minus1

120575y

120575z

120575X

120575Y

120575Z




119863= minus20






119894(119909) 119894 = 1 119899






References




































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

0 5 10 15 20

Time0 5 10 15 20

Time

120575x

04

02

00

minus02

minus04

08

04

00

minus04

minus08

04

00

minus04

minus08

02

00

12

08

04

00

10

15

05

00

minus02

minus04

120575y

120575z

120575X

120575Y

120575Z







119860 = [minus(

120597119891

120597119909

) + (

119889

119889119905

)] (A14)


minus1


1(119909) 120592

119899(119909)

120575119909 (119909) =

119899

sum

119894=1

120595119894(119909) 120592119894(119909) (A15a)

120575119891 (119909) =

119899

sum

119894=1

120593119894(119909) 120592119894(119909) (A15b)


119889120592119894(119909 (119905))

119889119905

=

120597119891

120597119909

120592119894(119909 (119905)) minus 120582

119894120592119894(119909 (119905)) (A16)

where 1205821 120582


one can obtain

119860 (120595119894120592119894) = [minus120595

119894(119909)

120597119891

120597119909

+

119889120595119894(119909)

119889119905

] 120592119894(119909) + 120595

119894(119909)

119889120592119894(119909)

119889119905

(A17)


119860 (120595119894120592119894) = [

119889120595119894(119909)

119889119905

minus 120582119894120595119894(119909)] 120592

119894(119909) (A18)


120575119891 (119909) =

119899

sum

119894=1

119860 (120595119894120592119894) =

119899

sum

119894=1

(

119889120595119894

119889119905

minus 120582119894120595119894)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

120593119894

120592119894 (A19)



119889120595119894(119909)

119889119905

= 120593119894(119909) + 120582

119894120595119894(119909) (A20)






0 5 10 15 20

Time0 5 10 15 20

Time

Am

plitu

de

80

60

40

20Am

plitu

de

45

30

15

0

minus20

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

40

30

20

10

0

minus20

minus30

minus10

Am

plitu

de

20

10

0

minus20

minus10



120583 + ⟨1205930

119894⟩ = 0 (A21)

where

⟨1205930


1

119879

[1205950

119894(119909 (119879)) minus 120595

0

119894(119909 (0))] (A22)


minus1


119894= 0

1198891205950

119894(119909)

119889119905

= 1205930

119894(119909) + 120583 (A23)


0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

120575x

16

08

00

minus08

16

06

00

minus08

05

00

minus05

minus05

2

1

20

15

10

05

00

08

16

00

minus08

0

minus1

120575y

120575z

120575X

120575Y

120575Z




119863= minus20






119894(119909) 119894 = 1 119899






References




































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


0 5 10 15 20

Time0 5 10 15 20

Time

Am

plitu

de

80

60

40

20Am

plitu

de

45

30

15

0

minus20

zrzs

Zr

Zs


0 5 10 15 20

Time0 5 10 15 20

Time

xrxs

Xr

Xs

Am

plitu

de

40

30

20

10

0

minus20

minus30

minus10

Am

plitu

de

20

10

0

minus20

minus10



120583 + ⟨1205930

119894⟩ = 0 (A21)

where

⟨1205930


1

119879

[1205950

119894(119909 (119879)) minus 120595

0

119894(119909 (0))] (A22)


minus1


119894= 0

1198891205950

119894(119909)

119889119905

= 1205930

119894(119909) + 120583 (A23)


0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

120575x

16

08

00

minus08

16

06

00

minus08

05

00

minus05

minus05

2

1

20

15

10

05

00

08

16

00

minus08

0

minus1

120575y

120575z

120575X

120575Y

120575Z




119863= minus20






119894(119909) 119894 = 1 119899






References




































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

0 5 10 15 20

Time

120575x

16

08

00

minus08

16

06

00

minus08

05

00

minus05

minus05

2

1

20

15

10

05

00

08

16

00

minus08

0

minus1

120575y

120575z

120575X

120575Y

120575Z




119863= minus20






119894(119909) 119894 = 1 119899






References




































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in

























































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in

research article some aspects of sensitivity analysis in...

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