Tectonoph~vstcs, 130 (1986) 431-458

Elsevier Science Publishers B.V.. Amsterdam - Printed in The Netherlands

437

SOME ASPECTS OF THE ANALYSIS OF GEODETIC STRAIN

OBSERVATIONS IN KINEMATIC MODELS

W.M. WELSCH

Insfirnre of Gende.v, Hochschuie der Bundeswehr, D-8014 Neubiberg {F.R. of German?)

(Accepted March 18. 1986)

ABSTRACT

Welsch. W.M.. 1986. Some aspects of the analysis of geodetic strain observations in kinematic models.

In: H.G. Henneberg (Editor). Recent Crustal Movements, 1985. Tectonophysic.7. 130: 437-458.

Frequently. deformation processes are analyzed in static models. In many cases, this procedure is

justified. in particular if the deformation occurring is a singular event. If. however, the deformation is a

continuous process, as is the case. for instance, with recent crustal movements, the analysis in kinematic

models is more commensurate with the problem because the factor “ time” is considered an essential part

of the model.

Some specialities have to be considered when analyzing geodetic strain observations in kinematic

models. They are dealt with in this paper. After a brief derivation of the basic kinematic model and the

kinematic strain model, the following subjects are treated: the adjustment of the pointwise velocity field

and the derivation of strain-rate parameters; the fixing of the kinematic reference system as part of the

geodetic datum; statistical tests of models by testing linear hypotheses; the invariance of kinematic

strain-rate parameters with respect to transformations of the coordinate-system and the geodetic datum:

the interpolation of strain rates by finite-element methods.

After the representation of some advanced models for the description of secular and episodic

kinematic processes. the data analysis in dynamic models ts regarded as a further generalization of

deformation analysis.

1. STATIC AND KINEMATIC MODELS

The det~r~nation of recent crustal movements results as an inverse problem,

usually from repeated observations of geodetic monitoring networks if geodetic

methods are applied. The geodetic data can be processed in static or kinematic

models.

In static models time does not appear as a parameter. With respect to the

description of deformations, static models are especially suited to movements which

are singular events. In this case, the only aim of a static analysis is the knowledge of

the solitary change of the geometry of the network or the object under investigation.

This procedure is justified considering various problems.

~-1951/86/~03.50 ‘c> 1986 Elsevier Science Publishers B.V.

43x

If, however, the movement or distortion monitored is a continuous process, as is, for instance, the case with recent crustal movements, one is frequently interested in more general statements. The generalization consists of quantities describing the deformations and distortions by time-independent, constant parameters. Such quan- tities are, for instance, velocities or accelerations or, regarding strain analyses, strain rates. One can achieve those parameters by considering the factor “time” as a parameter of the inverse problem. By this means, the model becomes a kinematic one.

2. THE BASIC KINEMATIC MODEL

A system, i.e. a geodetic network, subject to movements or distortions, is at any time well defined with respect to a spatial and temporal reference frame. In this way, the points of a geodetic horizontal network may at a time t, be fixed by the space-time-coordinates P,(x,, y,, tl), at another time by PZ(x,, y2, t2). Within a kinematic model the movements of the points between tl and c, are expressed by the relation:

=P,+i’At+$BAt2+...

or in more detail:

(2.1)

x,=x,+fAt+;ZAt’f...

y,=y1+j,At+:jjAt2+ . . . (2.2)

The principal model (2.1) contains a reIative statement, as only spatial and temporal changes of the system are described. The absolute reference has to be given by a spatial and a temporal datum. A given or an arbitrary fixing of the temporal reference as a temporal datum eliminates the number of uncertainties (datum defects) being increased compared to a static model of a geodetic network.

In model (2.2) k, j and X, ji are the coordinate-related velocities and accelera- tions, respectively. As shown below, also different types of movements can be modeled and be tested with regard to their statistical significance. If a model is suitable for describing the deformations and distortions of a system, also any number of additional observations dues not increase the number of kinema,tic parameters. This is considered an advantage of the kinematic approach.

3. THE KINEMATIC STRAIN MODEL

The kinematic analysis of a geetic network will be treased in the following. The network was repeatedly observed in order to monitor horizontal recent crustal movements. We proceed from the assumption of infinitesimal strain.

439

According to model (2.2) the linear relation:

(3.1)

(3.2)

x?=x,+i At

y;?=y1+jAt

exists. Therein x:, j, are linear, position-dependent polynomials . . . x = a, + azx, + ir,Y,

j = b, + b,x, + b3y,

The physico-mechanical interpretation of the kinematic coefficients f, _$ and cii, bi

results from the theory of infinitesimal strain. According to it, { f, Y} is the velocity

field of the point movements:

x At=x?-x, =u

$At=y,-yi=v (3.3)

Those movements are rigid body movements plus distortions of the object (crust of

the earth) being monitored by the network (e.g. Welsch, 1982):

u=(dR+E)x+t (3.4)

with:

U T= JU VI coordinate changes

antisymmetric rotation matrix

e E= -‘-’ e,, I I e e strain tensor

X .” ?‘.V

tT = I t.x t,. I translations

x T=Ix Yl coordinates of points of comparison.

After rearranging, (3.4) can be dented:

u=H.p

with:

(3.5)

HzX ’ 0 y 10

0 x y -x 0 1

PT = I e,., e,,. e w t, t, I ?>

If the parameters p are related to the unit of time, (3.5) becomes:

u=H.i, At

from which, in comparison with (3.3):

i

I I P =H.j,

(3.6)

(3.7)

440

results. Therefore. in (3.2) the polynomial coefficients are:

ir, = i,. 12, = 6,, . ir, = C,, + L3

A, = i,., b, = P,, - cj, &, = CT, (3.8)

For a network analysis according to (3.1) and (3.2), the separation of the

unknown kinematic parameters is desirable so that the final kinematic strain model

is:

x,=x,tH.j, At (3.9)

4. THE SOLUTION OF THE INVERSE PROBLEM USING KINEMATIC MODELS

The solution of the inverse problem using kinematic models is favourably carried

out in two steps.

4.1. Determination of the velocity field

In the first step, the velocity field {x, _$} of model (3.1) is determined. Consider-

ing that the coordinates of the network points are usually not observed immediately

but mediately by the traditional geodetic instruments, model (3.1) has to be

transformed in order to obtain a set of observation equations.

In the static model the linearized observation equation of a distance s between

the points P and Q is:

l+c~=a;dx (4.1)

with I = s - S. S is the distance calculated from approximate coordinates:

a,=[-coscp-sincpcosvsinp,I, ‘p bearingcpPQ

dxT = Idx, dy, dx, dy, 1

The corresponding equation in the kinematic model is:

I+u= la, a;AtI. y I I

(4.2)

Combining the equations of all (direction-, distance- etc.) observations of the

various measurement epochs i = 1,. . . , k, the following model is obtained:

I, ui Ai A, . At,

I, 02 A2

.+.= . A2- “2 . dx

I I J? (4.3)

1; ok A, . A,. At,

In this case not time t, is chosen as the reference time but an arbitrary time to so

that the matrices A, of all epochs can be established. However, this is a formal point

of view only.

441

Following the usual procedure of solving the inverse problem on the basis of the

Gauss-Markov model, the unknowns dx and i can be derived from (4.3), including

the other quantities of interest such as the unit variance ~0’ and the cofactor

matrices Q,, and Q,, of the estimated parameters. Whether the established

kinematic model corresponds to the movements and distortions occurring in reality

has to be verified by statistical tests.

4.2. Determination of strain rates

In the second step, the results -i-, Q,,? ~0’ of the first step are utilized to determine

the polynomial coefficients in (3.2) or the kinematic parameters p in (3.7). Regard-

ing the velocities i as correlated observations, the new setup of equations is:

f+&=H-b, ey (4.4)

For the derivation of the weight matrix:

e, = (Q,, + HHT) - (4.5)

from the usually singular cofactor matrix Q,, (see next paragraph), reference is

made to Caspary (1984).

The solution is

@ = &HT&i (4.6)

with:

QP = (HEATH)-’ (4.7)

If H, too. is singular, a general or the pseudo-inverse has to be used in (4.7).

The parameters:

t:= IL e,, e,.,. I (4.8)

are the elements of the strain-rate tensor.

Its covariance matrix

c, = S,’ .& (4.9)

is the corresponding submatrix of 8,.

From the strain rate-tensor. the further strain rate parameters:

j$ = P,.,. - e,, pure shear rate

=rz = 2P ‘il engineering shear rate

d = e,, + e,., rate of dilatation

j/ = (j/f + 9;)“2 rate of total shear

*,=:(A+;/) (4.10)

rate of maximum principal strain

6,=$(A-+) rate of minimum principal strain

B = +arctan( - +2/+,) bearing of 2,

$=8+45” bearing of 9

can be derived, including their covariance matrices (Welsch, 1983).

442

5. ESTABLISHMENT OF THE KINEMATIC REFERENCE SYSTEM

It is generally known that the velocity field of the network points derived from traditional geodetic observations has to be considered a relative quantity. The kinematic parameters f derived from model (4.3), however, pretend to be absolute. Of course, they are not. Those fictitious absolute quantities, which are not justified by observations, cause the well-known problems of the geodetic datum.

The rank defects occurring in static models are increased in kinematic models. Using a velocity model means doubling the defects of the static model, adding components of acceleration means even tripling. Therefore, it is necessary to establish reference systems for velocities, accelerations, etc. This is. as usual, performed by introducing condition equations for the unknowns.

For a two-dimensional strain-rate model of rank defect d = 4, and the special case of the “inner” velocity solution, the condition equations are as follows (Pap0 and Perelmuter. 1983):

(1) The componentwise vector sum of the velocity-unknowns is to be zero:

xm,=z_C,=o

(2) The sum of the pointwise vector products of position and velocity-unknowns is to be zero:

z(xxi),=o~z(x,j:-y,~~)=o

(3) The sum of the pointwise scalar products of position and velocity-unknowns

is to be zero:

c(x-i),=o~):(xi~j+y,j:)=o

These four condition equations correspond to the condition equations of a Helmert-transformation:

Gri=O (5.1)

with:

Y, --Xl Y2 --x2 ...

GT= x1 *o’ “1’ ; ‘*.

1 (5.2)

0 1 0 1 :::

which disposes of the rigid body velocities iT = ) i riz px &, 1 of the network establishing the kinematic reference frame:

kTk = mm. (5.3)

Within the frame defined by the type of the observations, the choice of the reference system is optional and arbitrary. By intro&c@ a “selective identity matrix” I, one is in a position to choose those kinematic parameters which should pa&&ate in the definition of the reference system. Using this matrix I,, (5.1) becomes:

GTIsf = 0 (5.4)

In Is the diagonal elements to be filled in with a one can be chosen freely, as long as

the number of the rank defect is considered. According to the arrangement and

number of elements in Z,, the network can be transformed to the kinematic

reference system without discrepancies or with a minimal sum of the squares of the

discrepancies in the case of a datum-redundancy defining the reference system:

jzTZb_+ = min. (5.5)

If Z, = Z the “inner” solution (5.3) is obtained.

With some kinematic systems, e.g., recent crustal movements, landslides. glacier

movements, etc., one is frequently in a position to define a pre-given reference

frame outside of the kinematic system. In this case, the stationary points are those

which represent the reference system.

For some reasons, it is sometimes desirable to change from one geodetic datum

to another. This can easily be achieved by S-transformations (e.g., Van Mierto,

1978; Prescott, 1981). If, in general, notation -i-, is a solution on the basis of a

certain geodetic datum and &, is the corresponding cofactor matrix, then the same

unknowns related to another reference frame _i-, and their cofactor matrix & are

given by:

i, = i, - G( GTZsG) -‘GTZ$, (5.6’)

and:

&=(I- G(GTZsG)-‘GTZs)~,,(Z- G(GTfI,G)-‘GrZ,)T (5.7)

The transformation parameters i which transfo~ the solution -i-r to the solution -i-,

are:

i= - (GT~s~)p'~Tzs_kI (5.8)

For more details refer to Perelmuter (1984).

6. MODEL TESTS BY LINEAR HYPOTHESES

Apart from tests investigating the accuracy and reliability of individual observa-

tions or groups of observations. two questions arise which concern tests of the

kinematic model:

(1) There is the principle of establishing the most simple model which is able to

describe the reality of the kinematic event. Unnecessary model parameters entail

some numerical difficulties and problems of interpretation. Therefore, it has to be

tested if the model set-up meets this basic principle, or if one can do without certain

non-significant parameters. Included here is the problem of detecting incoherent

movements of individual points, a problem which occurs frequently within geodetic

networks;

(2) It has to be tested whether new observations being introduced are in

conformity with the model. If necessary, the model has to be modified.

444

In answer to the first question, several strategies have been developed. One is to calculate all combinations of suitable parameters in order to find the optimal one by certain criteria. This method, however, takes much computer time. Another usual procedure is to expand stepwise a model containing primarily only the most essential parameters. In contrast to this method, the following starts from a model with a set of parameters which is as complete as possible. After elimination of non-significant parameters. the model will be reduced to an adequate size (Schwintzer, 1984a).

Within the Gauss-Markov model:

l+v=Ax, P

the well-known relation:

vrPv = !?I- ITPA

exists. The parameters x may also be of kinematic nature f (the dot for kinematic characters is neglected for the rest of the paragraph).

(6.1)

(6.2)

Using the estimation:

f = Q,_ATPl

the last term of (6.2) becomes:

(6.3)

ITPA = l=PA(QXXQ;x)QxxATPt = 2Q,_; (4.4)

and:

VrPv = l=Pi - FQx;f (6.5)

The quadratic forrn 3TQ;X3 is independent of the type of the inverse of the form

matrix Q,,, so that also the pseudo-inverse Q,’ or the regular inverse Q_&‘. together with their respective solution vectors f, can be used.

The quadratic forms (6.5) also show the independence of the parameters .? from the residuals v. This fact can be utilized to establish two variances (see Section 4):

vTPv .s;l = -

f

and:

( f, number of independent eIer%nts of f)..The ratio:

E;,= < so”

(6.8)

must not exceed the critical v&e I”‘/,,r,a of the Fisher-distribution ( IY error probabil- ity):

KY 2 +.a j6.9)

if the null-hypothesis:

H,: E(2) =o

holds.

(6.10)

This test is also known as the so-called global test (Pelzer, 1971) which investi-

gates the significance of deformations of a monitoring network in general.

With a kinematic model, a specific group of parameters x, is ordinarily consid-

ered essential and significant while the significance of a second group x2 is to be

tested. For instance, the significance of acceleration parameters against velocity

parameters, or some strain rates against others has to be determined. For this

purpose, one splits up the parameters x and their estimates P into the respective

components:

9,=JfT _;;I (6.11)

Using (6.11). the quadratic form q_y becomes (the regular inverse Qr,’ for conveni-

ence) :

(6.12)

(N coefficient matrix of the normal equations), or after some conversions:

q,= (a, - + N,;,N,232)TN,,( P, + N,-i’N&) + i;( N,, - N,,N,;,N,,)i, (6.13)

Substituting:

2, = f, + N,-,‘N,,P, (6.14)

and, as known from the inversion of block matrices:

Q;; = N,z - N,&‘N,, (6.15)

( QzZ submatrix corresponding to iZ), (6.13) becomes:

q, = i;N,,i, + i;Q,‘P, = qr, + q_,L (6.16)

For illustration of this subdivision of q,, one can imagine a model reduced by x2;

x, = 0 no longer appears. As an effect of this elimination P, will change to 2, and

N,, to N,, = 0,‘.

The quadratic form qx of the reduced model will be:

qy = 22,‘2, = qx,

The square sum of the new residuals 0 will be:

CTPC = rTpr - iz,;,;,

and will be increased by the term:

q,? = 2;Q-‘.&

(6.17)

(6.18)

(6.19)

The increase is not significant, if the null-hypothesis:

H,: E{.S*) =o

holds.

(6.20)

This is tested with the help of the variance:

(f,, number of independent elements of 3,) and the test statistic:

(6.21)

(6.22)

The null-h~othesis holds in the case:

FX, 1 F;,J.a

If so, the parameters x2 are not significant,

(6.23)

If one cannot split the parameters x according to (6.11) from the beginning, one could allot systematically parameter by parameter to group x2, calculate (6.15) and (6.19), test the smallest quadratic form qX, according to (6.21)-(6.23), and if the test fails, reject the respective parameter as non-significant. Using (6.14), (6.17) and (6.18), the unit variance (6.6) valid for the reduced model could be calculated, and the process be repeated until only significant parameters were left in the model. The specialities of multiple test procedures are referred to.

Schwintzer (1984a) reports on an aIgorithm systemat~ng the process. In eliminating non-significant parameters, attention must be paid to the datum prob- lem or the possibly necessary change of the datum.

The question of whether new observations are in conformity with the existing model is answered with the help of the increase qAO of the square sum of the residuals (Linkwitz, 1968). This increase serves for the caknlation of another variance:

2 _ %u sag - -

fA

(6.24)

( fA increase of degrees of freedom), which is again tested against the unit variance $0” (6.6). If the test statistic:

F,, = d! s(f

(6.25)

does not exceed the critical value:

FA, 5 Fr, . f . a

the null-hypothesis:

H,: E{Au} =0

(6.26)

(6.27)

447

cannot be rejected: the new observations are in conformity with the model. If they

are not, the reasons have to be traced. If necessary, the model has to be expanded or

modified.

7. TRANSFORMATION OF STRAIN RATES AND DATUM INVARIANCE

7.1. Transformation of strain rates

Frequently, the coordinate system of a geodetic network is arbitrarily oriented

with respect to the object to be monitored. In some cases, it may be desirable to

change the orientation of the coordinate system, e.g., parallel to a tectonic fault etc.

The following investigates how the strain rates are to be transformed according to

the change of the orientation of the coordinate system (Fig. 1).

The reflexions will be based on the relation (Welsch, 1982):

p=p .,_~ cos2q + 6,) sin 2~ + gJY sin2q

(6 rate of linear extension). The change

extension rate which is an object-related

tensor elements. Thus, the new relation is:

e=; _~_~ cos’@ + Z_,,. sin 2@ + Z_ sin’+

After replacing the new bearing @ by:

+=(P---E

(7.1)

of the orientation does not affect the

quantity, but does influence the strain

(7.2)

(7.3)

and after some trigonometrical conversions (7.2) becomes:

6 = cos’~(S~~~ cos’~ - k,, sin 2r + ZYY sin2e)

+ sin 2~ (+Z,, sin 2~ + d,,. cos 2~ - fZ,, sin 2~)

+ sin2q ( Z,, sin’r + e’, ~ sin 2~ + S.PJ cos4 )

Fig. 1. Change of orientation of the coordinate system.

448

The comparison of the coefficients with (7.1) leads to:

21-v cos2c -sin 26 sin’6 L

p,,. = isin 2~ cos 2C _ - isin 2~ . I,,./ (7.4)

P,., sin% sin 2~ COS% &Vi I

or with (4.8):

t,= T-i, (7.5)

The inverse relation:

d, = T-‘6, (7.6)

is due to the reverse rotation (e ---) -e):

L COS2C sin 2r sini 6X.X Z,, = - _tsin 2e cos 2e isin 26 . & (7.7)

& sin2e -sin 2r cos2< b,.,.

If one replaces further on the shear component 6,,. and d,, by the engineering shear T2 and 9, (4.10), (7.7) becomes:

k = R&R; (7.8)

after some trigonometric conversion and re-arrangement (Dermanis, 1981). The rotation matrix is:

R, = cos c sin c

- sin e cos e

From (7.4) the transformation of the pure shear rate +, and the engineering shear rate T2 can immediately be derived as:

Yl I-l I cos 2e sin 26 5, =

j12 -sin 2~ cos 2C . 9, /I I

or:

(7.9)

(7.9a)

Due to the orthogonality of the rotation matrix R2Cr the inverse relation is:

(7.10)

Analogously follows:

h = e,, + e,.? = e’,, + z.,I

~=(~f+~i)‘“=(t:+~2’)“’

P, = z, (7.11’)

c’z = 82

The new bearing 8 of the maximum principal strain rate ii can be calculated

from:

- R tan 28 = L

91

sin 2e” -& tan28=-=y

cos 28 +,

5, = cos 28. &= -sin 28

tan 28 = sin 2r $, - cos 2~ $, = sin(2c + 28)

cos 2~ $, + sin 2~ $, COS(2C + 28)

and finally:

8=0-C (7.12)

With formulae (7.8)-(7.12) all strain-rate parameters can be transformed due to a

re-orientation of the coordinate system. It turns out that the strain rate tensor ,?? is

to be transformed according to the congruence transformation of matrices. The rate

components +, and +2 experience a rotation by R2,, while the rest of the elements,

the dilatation rate A, the total shear rate ‘, and the principal strain rates 6, and &,.

as semi-axes of the strain rate ellipse remain invariant. The orientation 8 of the

strain-rate ellipse, however, follows the re-orientation of the coordinate system so as

to keep the position of the ellipse with respect to the object being distorted.

7.2. Datum invariance of strain rate parameters

The kinematic datum is not defined by observations but arbitrarily fixed.

Reasons of expediency or clarity lead to the choice of the one or the other

establishment. This can be done without influence on the relative information

contained in the observations.

In Fig. 2 the original shape of an equilateral triangle has been distorted. The

appearance of the deformation velocities ic, and ri, is different and depends only on

the choice of the kinematic datum. When investigating the distorted figure and

describing the distortions by strain rate parameters, the question arises whether

these quantities depend on the datum. This is examined in the following,

Fig. 2. The effect of datum chmges on the deformation velocity.

Ei, may be the vector of deformation velocities related to a particular datum. By S-transformation according to (5.6) and (5.8) another vector of deformation veloci- ties:

ir,=&+Gi (7.13)

can be produced. According to (3.5) the deformation velocities ic, can be analyzed by:

12, = H-Q (7.14)

resulting in:

fizH-‘& (7.15)

(7.13) solved for hi, and put in (7.15) leads to:

P=H-‘ir,-H-‘Gt=B*_H-‘Gi

and:

b* =B +H-IGi

The final result (Welsch, 1984) is 8.

e* 6: 0 1 0 0

e* 0 0 0 0 0 1 0 0 p*=iY+9+, * o o‘

i* x 0 0 1 0 i* 0 0 0 1 Y

or:

p* i= xx P,, -t- ti

P* =E,, XY &3* = - YY eYY+ ri?

&*=&++i

i,* =i;+px

i_; = i, +pY

(7.16)

i tit A PY

(7.17)

(7.17a)

451

Applied to the other strain-rate parameters, the following relations are obtained:

Yl ** =j/,

‘*_’ Y2 - Y2

A*=i+2ri? +*=jl

8: = ;(i + 2ri? + jJ) = ;(A* + ;/)

eZ .*+(~+2++)+:(~*--;I)

e* =I3

+*=\i,

(7.18)

The conclusion can be drawn that the change of the rigid body velocities caused

by the datum transformation is immediately brought forward to the respective

parameters of the deformation analysis. A scale of velocity influences the rate of the

normal strain components, the dilatation and the principal strain rates as well.

Apart from the scale, however, the strain rates are independent of S-transformations

of the point velocities ti. In this respect they are invariants.

8. INTERPOLATION OF KINEMATIC PARAMETERS BY FINITE ELEMENTS

If one is not only interested in a single set of strain-rate parameters representing

the distortions of the whole network but also in the strain behaviour of individual

parts of network, the dissection of the network into finite elements is an effective

tool. The finite element method is a field with many applications. From a geometri-

cal point of view it is a method for interpolation and approximation (Moritz, 1978).

The simplest decomposition of the surface of the earth’s crust is by means of

triangles constructed from the observations or from functions of the same type

(observables), e.g., distances derived from adjusted network coordinates.

Within a horizontal network of n points, f, = 2n - d linear independent distor-

tions, e.g., linear extension rates P or other strain-describing elements. can be

calculated. The number of triangles the network has been decomposed into, may be

m. Therefore, 3 . m strain-rate elements are necessary to describe the distortions of

all the triangles. Since always m >f,, the mutual correlations between the strain

parameters of adjacent triangles have to be utilized for condition equations in order

to obtain the necessary number of elements (Fig. 3).

First two triangles are investigated. Besides five observation equations (7.1)

using, for instance, the extension rates &, (i = 1,. . ,5). a condition equation can be

derived for the extension rate common to the two triangles:

-6 YX 2 cos*‘p3.1 - ery.2 sin 2~~5, - @Y.v.2 sin’q,,, = 0 (8.1)

Fig. 3. The correlation of strain rates between adjacent triangles.

Using (8.1) as an additional equation (“side condition equation”), it becomes possible to calculate six strain parameters rigorously and definitely from five linear extension rates. Because there is no redundancy, a weight matrix is not needed. The resulting strain-rate elements are correlated.

Within a central system the following situation occurs: If the kinematic datum defect is d = 3, nine linear independent strain rates are necessary and sufficient to describe the (relative) distortion of the figure. They can be calculated from the distortions of lines, angles or coordinates by observation equations. However, in order to describe the individual strain patterns of the five triangles, fifteen strain-rate parameters have to be calculated. In addition to the nine observation equations, five side equations can be set up (Fig. 4). The last equation still missing is taken from the “closure condition equation”:

gs+(gi+gz+&+g~)=o

(g, distortion of the angles ar,; Welsch, 1982).

(8.2)

Fig. 4. The correlation of strain rates within a central system.

453

Fig. 5. The correlation of strain rates of adjacent triangles from point-change velocities.

At full length (8.2) is:

*, K5tcos 2% - cos 2q6.,) + +(PVr.s - 6x.X.5 )(sin 2q6., - sin 2v6., )

+P,,.I(cos 2% - cos 2q6.,) + +(.6,.,.., - e,,,, )(sin 2v6,, - sin 2q6,, )

+ . . .

+ 6, ,..‘l(cos W6.5 - cos 2Q76.4) + :t 2!,..4 - 6, r.4 )(sin 2q6., - sin 29~~.~) = 0 (X.3)

As side equations are also contained in the system of observation and condition

equations. the terms (& - e,,,,) can be separated. The full system of fifteen

equations is consistent. The observation equations do not need weighting.

If one uses the velocities of the coordinate changes ti instead of the distortion

rates of distances and angles, the following picture is relevant: According to (3.7).

six deformation rates have to be calculated for each triangle, i.e., three strain rates

6T= le.,, e,,.

triangle there-are

P,.,. 1 and three rigid-body movement rates 3, i,, i,. Within one

six coordinate velocities available for this purpose. As long as the

kinematic datum is kept, the task is definitively solvable. If. however. the datum is

changed, only rigid-body movement parameters (b, i,. i,.) change, while the strain

rates P, are not influenced, as shown in section 7.

If there are two adjacent triangles (see Fig. 5) twelve deformation rate parame-

ters are needed. Eight out of the twelve equations necessary are set up as observa-

tion equations according to (3.7) while the remaining four are “coordinate condition

equations”. For instance, the coordinate velocity ti.,,, of point 1 influences both

triangles in the same way. This fact leads to the condition equation:

%x.1X1 + ~,,..,Y, + ~1.3 + ix., - (~x,.,.~, + e,,.2Yl + 4Yl + L> = 0 (8.4)

Each additional triangle adds two observation and four condition equations. In this

way, the thirty strain-rate elements needed for the strain-rate pattern of the central

system (Fig. 4) can be obtained from twelve observation and eighteen condition

equations. The equation system is consistent. there is no need for weighting.

454

Even if the parameters describing rigid-body movements depend on the datum, the information about relative rigid-body movements between the individual trian- gles, such as differences of translations and rotations, is not dependent.

The advantage of the finite-element analysis is that an instructive insight into the local deformation distribution is made possible. Disadvantageous is the big expendi- ture of computer storage and time as already small networks create big systems, especially if the strain-rate pattern is derived from point velocities.

9. ADVANCED MODELS FOR KINEMATIC EVENTS

Kinematic models are functions describing point displacements in their depen- dence on time. With simple models, this dependence is the only one; with advanced models the point displacements can additionally be made dependent on the point positions themselves. Also, particular events interrupting smooth movements such as earthquakes, etc., can be considered. In the following, some typical models are presented.

A simple function of time is given by the model:

&f, :=

i

x,=x,,, + h, At + ti, At2 + .

y, =yz, + b, At + A, At’ + . . . (9.1)

Compare model (2.2) where ci, = 1, 6, = $3, b, =j, b, = $. This is concentrated to model 1:

( n,,

x, = xt, + c ci, Ath

&f, := k=l

n,, (9.2)

Y, =Y,~ + c A, Atk k=l

(n IX) nty degree of the polynomial in time). According to the degree of the polynomial expansion, the point displacement

rates depend linearly, quadraticalIy, etc., on the time At = t - t, elapsed since the reference time point t,. Each coordinate may have its own velocity, acceleration, etc. The model avoids “ temporal centering” to common epochs (Welsch, 1981; Van Mierlo, 1983) in the case of long observation campaigns (e.g., the time-consuming observation of leveling networks), as For each observation the observation time point is introduced into the model as an individual coefficient. Attention should be paid to some particular problems of polynomials: the difficulty of a meaningful physical interpretation grows with the increasing degree of the polynomial, and so does the tendency towards oscillating between the nodal points so that an interpolation can become meaningless. Nevertheless, eapeeia@ with then one-dimensional processes of height changes, many problems have been solved with the help of the simple, solely time-dependent polynomial of model 1.

455

Modeling movements as a function of time and position becomes possible by

model M,:

M, := X,=X,,, + Li, At + ti2 At' + . . .

y,=y,,+b, At+b, At2+...

where:

Summarized. model 2 is:

x, 1=).x,,, + c c c cii,~x’_rJ At” !=O J=o t=l

I=0 J=o k=l

(9.3)

(9.4)

(n ~ degree of the polynomial in x, n,. degree in J ).

In model MZ the position of a point can also be seen as its relative position with

respect to a particular reference point or to a phenomenon characteristic for the

kinematic process, e.g., to a fault. For the one-dimensional case of height changes,

MZ describes a vertical movement-rate surface. This is well suited for the demonstra-

tion of some properties of a continuous position-time model. It contains a “built-in”

mechanism for the interpolation of coordinates at any positional and temporal spot

within the model. The model generalizes the information of the observations. This

can be judged from different view points; it is advantageous if one is interested in

regional trends rather than in local details: it is, however, disadvantageous, too, if

essential variations of the movement-rate pattern do not appear because they have

not been modeled and therefore smoothed off.

The position- and time-dependent model M2 was applied, e.g., by Snay and

Gergen (1978) to a kinematic model of recent crustal movements in California. A

general review on various approaches to the analysis of vertical networks is given by

Holdahl(l978). The disadvantages of the smoothing mentioned above can partly be

avoided if the models M, or M2, respectively, are used as trends in collocation

models. In this case, M, and M, represent, as trend functions, the regional

movements while the local structures are added as signals. This approach was

proposed by Pelzer (1980) and applied to the analysis of vertical movements of the

North Sea coast (Pelzer. 1981). Schwintzer (1984b) has worked out further details of

adjustment and hypothesis test theory.

The most comprehensive models are those which model not only continuous.

secular movements but also unique, episodic kinematic events. The set-up of those

models is partly purely kinematic, partly under consideration of dynamic quantities.

456

Van&k et al. (1979) describe an expansion of model Mz considering linear episodic movements by which a continuous process is brought to an end. There are IQ= i . . . . . np, np+ _,_‘... nl,in,, movement epochs considered. Out of 12~ = 11~ + tzC

epochs n,, are continuous, II,. episodic. All points of time t are related to the reference t,, the beginning and the end of each episodic epoch is marked by t,,i and t rl respectively; n, and ?z,. are as in model Mz. The model of VaniEek reads

(slightly modified, for one coordinate component only):

My := x,=x,<, + 2 s 2 ci,,kx'yJT; r=O j=o k=l

(9.5)

where:

T,=tk k=l,...,n,

T, =

i

0 r < fbl

(t - 16, )/( t,k - tb,) f/,, I t I II,* k=np+l,...,np+n,

1 t > t+

The model can be modified for other than linear movements. Snay et al. (1983) modify model MS for the special case of recent crustal

movements being superimposed by dismcations d, which were episodically caused by earthquakes at the time t,. The conttrtuous secular movements are regarded as inf~ites~~ strains. This model iM4 reads ~s~~tly modified; compare (3.1)):

x, = x,~ + i i ciijx’yJ At + r( t, t,) .A,d,

&f4 := i-0 j=O

(9.6)

y,=y,,+ i i bijxzy’bt+rft, tm).B,,,d, i=(-j ,j=$j

(i +j f 2), with the stepfunction r(t, t,):

r Ct. t,) := i

tm < to i

-1 t<t,

1 t>t,

\ f ?n > ta

i

0 t<t,

1 t>t,

The dislocations d, OCGUX inside of the earth’s crust, are determined from seismic activities by focal solutions, and are transformed according to the functions A, and B,,, to the surface of the crust.

Model M4 goes beyond the purely kinematic models in that dynamic aspects are considered by the geophysical transformation functions A, and B,,,.

451

9. FURTHER GENERALIZATION OF KINEMATIC MODELS

Kinematic models include the factor time and describe the temporal process of a

movement or a deformation of a system not by spatial (positions of points) but by

temporal parameters (point velocity rates, strain rates). This means a completion

and extension of the static viewpoint. However, the analysis of kinematic processes

in kinematic models is descriptive only by investigating the appearance but not the

causative forces of the movement. Exterior parameters, i.e.. those which are not

inherent in the geodetic network observations, are considered nuisance quantities.

They are to be eliminated in order to be able to work in the purely kinematic realm.

This is worth the effort and helps to solve many tasks and problems.

However, if one inspects a deformation process more thoroughly, at once the

question arises about the causes and forces which may have created that deforma-

tion. From this point of view, the exterior parameters may exactly be those the

quality and quantity of which are of great interest. Mathematic-physical models

representing the relations between causative forces and deformations caused lead to

the establishment of dynamic models. They are the most extensive generalization of

deformation models.

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