a surface distortion decomposition for vector csamt … ja 1991.pdf · single source csamt data,...

A SURFACE DISTORTION DECOMPOSITION

FOR VECTOR CSAMT DATA OVER 1-D EARTH

by

Jorge A. Arzate

A Thesis submitted in conformity with the requirements for the Degree of Master of Science in the

University of Toronto

© Copyright by Jorge A. Arzate 1991

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

1 Introduction 5

2 Principles of CSAM T M ethods 9

2.1 Introduction............................................................................. 9

2.2 The CSAMT and the Plane W ave.......................................... 11

2.3 Plane Wave Source G eo m etry .................................................14

2.4 Receiver-Transmitter R eciprocity.......................................... 16

2.5 Single and multiple sources.......................................................21

2.6 Scale Modeling considerations................................................ 24

2.7 EM Fields in a Layered E a r th ................................................ 26

3 The D istortion Problem 37

3.1 Channeling and D i.' o rtio n ....................................................... 37

3.2 The Impedance T ensor............................................................. 41

3.3 Impedance Tensor Decomposition.......................................... 46

4 D ecom positions for vector CSAM T 53

4.1 Introduction.................................................................................53

4.2 Zero-Splitting 1-D Factorization............................................. 57

4.3 Zero-Shear 1-D Factorization....................................................61

4.4 S u m m ary ....................................................................................63

5 A Case History of CSAM T D istortion 65

5.1 Introduction.................................................................................65

5.2 1-D Expected Electrical B ehavior.......................................... 66

5.3 2-D Earth without E -d is to rtio n .............................................69


5.4 Test for Zero-Splitting Model ............................................ 71

5.5 Test for Zero-Shear M o d e l...................................................... "u

5.6 Conclusions................................................................................76

A Programs for 1-D Galvanic D istortion 50

B Rotation Angles of Polarization Ellipses 55

References 58


3

Acknowledgements

I would like to gratefully acknowledge to Dr. Richard Bailey for the sug

gestion of this work, his supervision, the corrections he has made to it, and

for his support throughout its completion. I am indebted to Dr. Gordon

West, not only for finding the tim e to read this thesis, but for the opportu

nity of being his assistant in the Geophysics Lab course which I have found

a very useful experience. I also wish to thank Dr. M arianne Mareschall

for her observations and support in the last stage of this work a t Ecole

Polytechnique.

I enjoyed the courses given by Drs. Derek York, David Dunlop

and Nigel Edwards from which I have benefited very much. My friends,

Claire Sam son and Vincenzo Costanzo encouraged me always I needed en

couragement, and Guadalupe my wife, offered me perm anently her unvalu

able support.

Finally, I wish to thank to Phoenix Geophysics LTD for providing

the d a ta used in this work, and to the Universidad National A u tonom a de

Mexico who has provided the funds for my studies abroad.


A bstract

Single source CSAMT data, also known as vector CSAMT

data, has been routinely collected in recent years and processed us

ing the conventional 2-D induction model which does not incorpo

rate galvanic distortion effects, although the importance of these has

been widely recognized. The small number of data available in vector

CSAMT though, does not allow models that simultaneously incorpo

rate both 2-D induction and local galvanic distortion. Either the 2-D

inductive model with no distortion or alternatively a 1-D model with up

to two distortion parameters can be used. Two 1-D distortion models

were tested that are specialized cases of the general Groom and Bailey’s

(1989) M T decomposition method, (A) one which assumes the distor

tion parameter splitting (s) can be made zero choosing an appropriate

coordinate system, and (B) other in terms of polarization ellipses that

assumes the distortion parameter shear (e) can be zeroed by rotating

the principal axes to the measurement axes. The former produces the

distortion parameters twist and shear while the later generates twist

and splitting. Computer algorithms were implemented and a data set


was used to test the models and later compared with the classical 2-

D model. Except for the magnitudes which differ by a scaling factor,

the apparent resitivities computed with the three different approaches

show very similar electrical distributions but the phases show discrep

ancies. The distortion parameters distribution of the distortion model

(i4) for the used data set has a complicated pattern and are frequency

dependent. Distortion parameters obtained using model (B ) define

better the conductive structure and are approximately constant in a

reasonable frequency interval. The results of the test suggest that the

zero-split assumption is likely not valid while the zero-shear approach

is a more realistic one, although further testing is necessary to confirm

this.


C hapter 1

Introduction

The objective of the present work is to adapt as fax as possible the

theory of surface distortion decomposition (Groom and Bailey, 1989),

which has been recently introduced to the interpretation of magnetotel-

luric (M T) data, to electromagnetic measurements of controlled source

in the frequency domain. The Controlled Source Audio-frequency Mag-

netotelluric (CSAMT) method was introduced as an alternative geo

physical system for shallow prospecting by Goldstein (1971) and by

Goldstein and Strahgway (1975) to overcome some of the problems en

countered in the conventional A M T surveys, such as a poor signal to

noise ratio. It can be argued on the basis of the availability of reference

papers that the advances in instrumentation and data acquisition since

the introduction of the CSAMT method, lead by far those achieved in

the interpretation techniques. The advantages that a controlled source

survey is supposed to provide are overshadowed because the traditional


7

interpretation approach (Cagniard, 1953) which is often used, is often

inadequate. Although there have been efforts to improve the process

ing techniques for single source data using Swift’s (1967) method to

retrieve electric impedances and strike angle (i.e. Yamashita, 19S9),

they do not normally take into account distortion effects due to local

electrical structures.

For magnetotelluric methods (M T) however, there have been

attempts on how to extract the earth’s electromagnetic response from

field measurements in the presence of electrical distortion. Among the

latest are the works of Eggers (19S2), Yee and Paulson (1984), LaTor-

raca et al. (19S6), and Groom and Bailey (1989). These axe impedance

decomposition methods developed to cope with the general three di

mensional case of conductivity structure using a variety of different ap

proaches. The controlled source analogue to the M T situation for which

these methods are applied is that of having more than one generator

of electromagnetic waves. This means that any one of these methods,

which often involve up to 8 distortion parameters, are restricted to be

used only with CSAMT data collected with at least two different not

collinear sources. Although there exist tensor CSAMT data, most of

the survey data collected is usually done using a single source. Thus,

vector data is abundant and waiting to be interpreted.


8

The decomposition used by Groom and Bailey (19S9) is used

in M T to separate the effects of local and regional induction, but has

too many parameters to be used with single source vector CSAMT

data. The simpler case of such decomposition, where the underlying

inductive response is one dimensional, is appropriate for use with vector

CSAMT data. Using real data, we compare the results obtained with

this method to the conventional case of 2-D inductive response without

distortion assuming a 1-D inductive response with known strike and

unknown surface distortion. An alternative 1-D impedance tensor de

composition that allows a clearer visualized factorization is also tested

and compared with these cases.

In the following two chapters of this thesis, a review is done

of basic principles and concepts of the controlled source EM meth

ods in the frequency domain. The justification of the M T concepts

to the CSAMT case is also discussed as well as the restrictions that

apply. Also discussed here are the forms of the impedance tensor for

1, 2, and 3-D local and regional structures. The difference between

vector and tensor data will be addressed as will be the concept of

electrical anisotropy in the context of the distortion of EM fields. Also

discussed here will be the effect of the source setting on the EM mea

surements, and consequently the applicability of the reciprocity prin


9

ciple to CSAMT surveys. In the Chapter 3, a review of Groom and

Bailey’s (1989) distortion problem will be done, as well as a discussion

of the two 1-D distortion approaches proposed here. The last part of

this work will be used to test the decomposition method on a data set

of a vector CSAMT survey. Discussion of results and conclusions follow

an analysis of the data.


C hapter 2

P rincip les o f C SA M T M ethod s

2.1 In trod u ction

The basic concepts behind the magnetotelluric (M T) exploration method

were presented in 1953 by Cagniard, who described the scalar relation

ship between the electric and magnetic tangential fields at the surface

of the earth. The relation among these naturally varying orthogonal

fields for a uniform half space define the apparent resistivity as

pa = 0 .2T (E i/H j)2 (2.1)

T being their oscillating period and E{ and H j the induced electric and

magnetic orthogonal field components. The apparent resistivity has

been since a widely used concept to assess the electrical structure of

the Earth at depth. Cagniard (1953) showed that the natural magnetic


fluctuations in the frequency range of 0.0001 to few thousands of Hz

can be approximately uniform over areas as large as 100 km? or more,

and therefore could be approximated by a plane wave so as to greatly

simplify the interpretation process. As we will point out in the fol

lowing section, the validity of this assumption, which will be adopted

by CSAMT, depends strongly upon the scale of the EM survey. Al

though there were several papers after the publication of his results

(e.g. Wait, 1954; Price, 1962; Wait, 1962) objecting to the validity of

the approximation to a plane wave, all these dealt with errors intro

duced when the horizontal plane wave is not infinite in length. It was

pointed out then, that the harmonic components for the electric and

magnetic fields tangential to the ground are only proportional to one

another if the fields are sufficiently slowly varying (Wait, 1954). It was

also observed (Price, 1962) that at great probing depths the approxi-■*

mation to a plane wave is no longer valid because the dimensions and

distribution of the ionospheric inducing field become significant. These

and others observations were useful to fix bounds on the applicability

of the M T exploration method more than to discourage its use. The

problem was clearly one of scale.

With the use of frequencies in the audio range (i.e. between 10 Hz to

10 kHz), the M T method has been used for shallow exploration by the


mining industry. The source of such frequencies is known to be light

ning discharges from remote thunderstorms. The generated energy that

propagates, trapped in the waveguide formed between the Earth’s sur

face and the ionosphere, contains a wide spectrum of frequencies but

many of these tend to be attenuated as it propagates. Although the

audio-frequency magnetotelluric (A M T) method has been applied with

success to the location of massive layered sulfides and other type of

mineral deposits associated with a sharp lateral resistivity contrast, it

relies heavily on the stability of the signals and distance from their

source location which in general are of low amplitude and highly vari

able.

2.2 T h e C S A M T and th e P la n e W ave

The CSAMT technique is one of several electromagnetic exploration

methods where the source is either a grounded electric bipole or a cur

rent loop on the surface of the ground. The operating frequencies of

the transmitter are usually in the range of 1 to 10 000 Hz, but it may

vary depending on the objective of a survey. The horizontal orthogonal

components of the electric and magnetic fields E and H respectively,

are measured over the survey area. The apparent resistivity is com

puted from the ratio of two of the horizontal orthogonal components of

E and H , using equation (2.1). When the transmitter is too close to


the area under investigation, the fields do not correspond with plane

wave geometry and therefore the resistivity of deeper strata cannot be

determined in this way. In order for the electrical structure of deep

layers to be determined, the transmitter is located a distance of the

order of 3 to 5 skin depths from the area of interest (Goldstein and

Strangway, 1975; Zonge and Hughes, 19SS). Skin depth for a homo

geneous medium is the depth at which the amplitude of the EM field

decreases to 1/e (37%) of its value at the surface. In an inhomogeneous

medium, the skin depth (or apparent skin depth) which is indicative of

the depth of penetration of EM fields when 2-D and 3-D structures are

absent, is given in terms of the apparent resistivity pa as

where u> = 2x / is the angular frequency, and po = 4jt x 10~7H /m the

magnetic permeability of free space.

The use of a controlled source has benefits as well as prob

lems. Among the advantages are that the source characteristics are

well known and can be located so as to configure the excitation fields

in the most advantageous geometrical form. Knowledge of the primary

field polarization will enhance the ability to interpret the data. Also,


signals are stronger and in consequence the signal to noise ratio is highly

improved. Another no less important achievement is that because of

the coherence of the signal the processing and enhancement techniques

are far more effective. There is however an important limitation of

this method, and that is to model CSAMT fields is significantly more

difficult than modelling plane wave fields. This is probably one of the

reasons why multidimensional modeling for controlled source methods

is in a very elementary state.

As noted before, it is often the case to approximate controlled

source fields to the plane wave case by locating the dipole source at

least three skin depths away from the observation point and to avoid

experimental configurations and geological environments in which the

EM response is not interpretable. Sandberg and Hohmann (19S2) have

shown that when this condition is met, the apparent resistivities calcu

lated from CSAMT measurements using equation (2.1) axe within 10

percent of the plane wave natural field A M T apparent resistivities.

It is well known that electrical inhomogeneities distort signif

icantly the horizontal electric fields which are controlled by the bulk

material properties. Whereas .electrical inhomogeneities are consid

ered a source of noise for M T fields (often reffered as ‘near-surface’

conductivity structures because of the M T deeper exploration depths)


15

for CSAMT, the interpretation of the generated secondary fields from

these structures is in general the ultimate goal. However, due to its

finite character, the sources used in the CSAMT method are normally

located far away form the receivers in order to comply with the plane

wave constriction. This configuration, often called far field setting, is

compatible with the standard M T theory and it is more or less routinely

used in CSAMT data. On the other hand, near field measurements are

seldom carried out because the complexity of the interpretation pro

cess. Most of the data available is taken in the far field, thus when we

refer to CSAMT data further in this text unless otherwise specified, it

means far field data.

2 .3 P la n e W ave Source G eom etry

I t was mentioned previously that beyond a range of about three skin

depths, a dipole source can be considered as a plane wave to a good

approximation. But because a single dipole source is not a symmetri

cal source in the horizontal plane, there are sectors in which the field

measurements are more accurate than in others. Figure 1 shows the

distribution of far field zones generated by a single dipole in a homoge

neous earth. The areal limitations of a CSAMT field survey for scalar

measurements is a consequence of the single orthogonal pair of electric

and magnetic field components measured. Using an additional non-


16

collinear electric dipole will fill the gaps between the shaded sectors

although it will not extent the domain of validity further out or in.

The dotted areas in the figure are the far field zones where the

fields behave approximately as plane wave fields and large enough to be

measured accurately. Within this regions Cagniard’s apparent resistiv

ity relationship (eq. 2.1) can be reliably applied. The phase difference

<t>E — < j>H can also be calculated in these areas where it will be equivalent

to the plane wave phase. Sandberg and Hohmann (19S2) determined

that the fax field of a dipole transmitter in a homogeneous earth began

at a distance of three skin depths for the broad side configuration and

five skin depths for the collinear configuration (see Fig. 1).

The regions between the far field zones and the transmitter is

usually known as the near field zone or the transition zone, depending

how far from the source the measurements are made. In the near field

area the fields are frequency dependent and the impedance is known to

be proportional to 1 f r (Zonge and Hughes, 19SS), r being the distance

to the center of the dipole. Application of the plane-wave apparent

resistivity to data collected in this region yields apparent resistivity

values which increase linearly as frequency decreases in a log-log scale.

The reason of this ” near-field rise” occurs because Cagniard’s apparent

resistivity is proportional to 1 / / in the near field zone where the electric


and magnetic field components are constant in a 1-D earth. Addition

ally, the phase difference tends to zero in this zone. The behavior of the

fields within the transition zone are in general gradational from near to

far field response.

In the following paragraph, the source and receiver reciprocity is stated,

thus although only receiver site distortion is discussed in Chapter 3 this

will show that the distortion arguments can be applied to transmitter

site distortion as well.

2.4 R ece iv er-T ra n sm itter R ec ip ro city

In general terms, the reciprocity theorem in electrodynamics states that

if the role of transmitter and receiver are interchanged the signal in the

new receiver remains the same as in the previous one. A transmitter

of an electromagnetic field is a kind of electric circuit that generates

and drives a current into the earth. The terminals used in frequency

domain CSAMT are a pair of electrodes (or electric dipole).

The basic measurement performed at the receiver, which de

tects a secondary electromagnetic field, is in the form of a voltage. In

electromagnetic work this voltage is a complex magnitude with real

and imaginary components and is often expressed in terms of apparent

resistivity and phase. The reciprocity of a given receiver-transmitter


18

array is attained when the current generator at the transmitter termi

nals is replaced by the receiver, and viseversa. There is not reciprocity

if the transmitter and receiver interchange their respective orientations

or positions in space.

In a homogeneous (or linear) medium the reciprocity theorem

for a time-varying electric and magnetic fields takes the form (Parasnis,

1988)

J j \E i{ l)d v \ = J j 2E 1(2)dv2 (2.3)

where j \ is the current density occupying the volume t>i, and Ei(2) is

the electric field of the transmitter at a point of the receiver occupying

a volume v2, while j 2 and £ 2 ( 1 ) axe the corresponding vectors when

the receiver and transmitter position are interchanged. The theorem

is valid for any arbitrary distribution in the medium of the magnetic

permeability fi, the dielectric permitivity e and the electric conductivity

< 7 if these physical properties do not depend on the magnetic or electric

field intensities H and E.

The derivation of this equation is outlined here following the

work of Parasnis (19SS). The current density at any point in the

medium with position vector r is given by


19

J{r) = a (f )E {f ) + j { r ) (2.4)

where the first term of the right hand side is the density due to the

electromagnetic field E (r ) and the second term due to the current fed

by the generator in the case that the point under consideration is part

of the transmitter unit. If and D i(r ) denotes the fields

at r when the transmitter currents are distributed in volume 1, then

Maxwell’s equations at the point r are written as

V x E \(r ) = iuB i(r) (2.5)

V x H \( f ) = - iwDi(r) + <rEi(r) + j i ( r ) (2.6)

B1{ r ) = p H i {r) (2.7)

and

D 1(r) = e £ ( f ) (2.8)

Similarly, for some other distribution 2 of the transmitter currents at

the same point r

V x Ez{f) = iuBz{r) (2.9)

V x Hzir) = —iu D iir ) + aE i{r) + .7 2 (f) (2.10)


20

&(#D = t*H * (?) (2.11)

and

A ( r ) = eE2(f). (2.12)

Combining these two sets of equations by taking the scalar products of

equation (2.5) and (2.6) by H 2 and E 2 respectively and equations (2.9)

and (2.10) by H \ and E\ then combining them and rearranging we get

( tf2V x i a - i i V x H 2) + ( i 2V x H i - t f xV x E 2) +

— H 2E \) -f* iw(E2D i — E \D 2) -|-

j 2E i — j i E 2 = 0 (2.13)

Substituing equations (2.7), (2.8), (2.11) and (2.12) into this expression

we have

( # 2V x J? i—.E jV x H 2) + ( E 2V x H i —H i V x E 2) = j i E 2- j 2E t , (2.14)

which in turn can be written using the equation

V ( i x B) = B V x A - A V x B, (2.15)


21

as

V(i?i x H 2 - E 2 x H i ) = j iE j — j 2E\. (2.16)

Using the divergence theorem and integrating throughout a sphere of

radius R which include either transmitter current distribution within

it, the previous equation can be expressed as:

/ [(Jx x H 2)n - (E 2 x H i)n}dS = / Z & d u i - f n E 2dv2. (2.17)J R JR JR

The left hand side term approaches to zero as R tends to infinity (Paras

nis, 19S8) for a sufficiently homogeneous medium. Thus the reciprocity

theorem (eq. 2.3) holds. Thus, although only receiver site distortion

is discussed in Chapter 3, because of the reciprocity principle it can as

well be applied to transmitter site distortion

2.5 S in g le and m u ltip le sources

Both single and multiple electrical dipolar sources have been used in

CSAMT methods. Of these, the multiple source experiments are analo

gous to M T surveys in that the estimation of impedance tensor elements

can be done in a similar way to that used for M T fields (see Sims et


22

al., 1971). The procedure of using more than one source eliminates

the influence of the individual current source orientation via generat

ing data redundancy. This is a way to overcome the impedance tensor

dependence on the orientation of a single source which is one of the

major limitations of the CSAMT technique. An equally important

form of data redundancy is by repeating spatial sampling. For a ID

earth, data acquired with the same source-receiver configuration but at

different lateral positions are purely redundant.

When conductive inhomogeneities are present, they distort the

horizontal electrical field in their surroundings. The magnitude and di

rection of these distortions depend greatly on the bulk electrical prop

erties of the inhomogeneity. The electromagnetic response to these

variations involves the interaction between inductive and frequency in

dependent or galvanic effects, the electric field being the most affected.

Groom and Bailey (19S9) introduced the distortion model based upon

the paxametrization of the impedance tensor to deal with local galvanic

distortion in the form of a product factorization. Their approach was

designed to take into account the distortion effects of the M T electric

currents due to 3-D structures induced on a 1-D or 2-D regional scale.

The key assumptions are that the regional structure is at most 2-D

and the local structure causes only galvanic scattering of the electric


23

fields. Conventional magnetotelluric measurements can be represented

in terms of an impedance tensor which at the same time defines the

distortion parameters. The character of M T data is similar to that col

lected in multiple controlled source EM measurements in the far field,

not only due to the data redundancy but because the field components

in the magnetotelluric relation for the frequency domain

E = Z H (2.1S)

axe complete and can be solved for Z, the magnetotelluric impedance

tensor, if E and H are known. In multiple source CSAMT experi

ments one relies on having an ensemble of events (e, h) with which

the four complex elements of Z can be evaluated. Although for single

source measurements we have a set of field components as well, these

measurements are not enough to evaluate all the impedance tensor

components via a least-squares or other fitting method. Therefore, in

subsequent sections the impedance tensor computed from single source

data will be referred to as vector impedance. Unless specified other

wise, CSAMT will be synonymous with ‘vector controlled source audio

magnetotellurics’ or alternatively of single controlled source audio mag-

netotellurics. Here we will make use of the decomposition of the magne

totelluric impedance tensor approach based on local galvanic distortion

models, for controlled single source audio-magnelotelluric data. The


24

models will be applied to experimental data to test and subsequently

discuss receiver site distortion and their relation with transmitter site

distortion.

2.6 Scale M od elin g con sid eration s

Physical scale modeling has been used for many years to study the

electromagnetic response of a large variety of geologic settings. Frisch

knecht (19S9) has done a general review of the theory, types, design, etc.

of scale modeling in electromagnetic methods. Physical scale models

are only approximations to actual earth structures. They are mostly

used to reproduce targets related to mineral exploration which often

consist of an overburden of varying conductivity and thickness over-

lying a host rock (air or liquid) containing the ore deposits simulated

by more conductive materials. Modeling has been necessary, and still

is, because the development of analytical solutions and computational

techniques for determining the response of 2-D and 3-D structures have

not kept up to date with the development of instrumentation, which

have had a much more faster evolution. Thus, physical modeling in the

laboratory has created its own methodology to study the efFects that

two and three dimensional structures have upon EM fields in well con

trolled environments. Although the results are seldom used directly to

interpret quantitatively field profiling and sounding curves, they usually


provide indirect ways of interpreting field results by comparing them

with type curves for different geological settings. Usually, it is possible

to extract physical parameters from this comparison and make assump

tions on the nature of the structure that produces the field anomaly.

An additional advantage is that physical scale modeling provides a rel

atively easy and low-cost geophysical tool to study systematically the

EM responce under a variety of simplified electrical environments. A

chief disadvantage though, specially in the case of CSAMT lab mea

surements, is to reproduce a plane wave that mimics the far field sit

uation. The design of transmitters for modeling controlled source EM

in the near field does not present problems. It is the generation of a

plane wave required for far field measurements the one which present

difficulties to reproduce. Rigorous CSAMT measurements could be

possible by placing a grounded dipole at the properly scaled distance

(i.e. between 3 and 5 skin depths) but this may require extremely

large tanks and, at the same time, a large source strength. There are

several techniques to model a plane wave, each of which present par

ticular problems. Helmholtz coils for example, can be used to generate

a uniform field within a tank, however, its use is limited to extremely

low frequencies. Their use is not warranted when the frequency is high

enough to establish proper impedance relations due to induction effects

(Frischknecht, 1989). An alternative way to generate a uniform electric


26

and magnetic fields within a tank is to place an electrode at either end

of it to drive a current. At very low frequencies the current distribution

will be uniform, whereas, at higher frequencies the current will tend to

concentrate in the outer regions of the tank unless additional current

electrodes are used. Edwards (19S0) on the other hand, suggested the

use of a vertical wire grounded at the surface of the tank. Such a source

will generate a tangential magnetic field but the current will flow in ver

tical planes whereas in a uniform earth, real induced currents flow in

horizontal planes. He notes though, that at the M T limit (i.e. in the

far field), these two kinds of current flows are not distinguishable and

that Cagniard’s approximations are obeyed by both.

2.7 E M F ie ld s in a L ayered E a rth

All electromagnetic phenomena are governed by Maxwell’s equations.

There are a large number of places where the reader can find detailed

derivation of the field expressions for a homogeneous layered earth (i.e.

Wait, 19S2, Kaufman and Keller, 1983, Ward and Hohmann, 1987).

What follows here is a synthesis of the formulation to obtain the field

equations due to a dipolar horizontal source in a 1-D environment.

A general solution of a boundary-value problem is obtained by

the combination of the solution of the homogeneous Maxwell equations


27

V x E + iftcuiH = 0 (2.19)

V x H - (a + ieu)E = 0 (2.20)

and a particular solution of the inhomogeneous differential equations

V x E + in<xuH = - Jm‘ (2.21)

V x H - (a + iew)E = Jea (2.22)

in which Jm4 and Je‘ are a ’magnetic’ and an electric source respec

tively. /i, e and a are the magnetic permeability, dielectric permittivity

and electric conductivity in that order. The homogeneous solution is

worked out in terms of the Schelkunoff potentials A and F (Ward and

Hohmann, 1989) related to the fields by

E = - V x F

and

f f = V x A

with the additional condition that they can be expressed as

V F = - i f iu U (2.25)

(2.23)

(2.24)


28

and

V A = -(<r + i«w)V, (2.26)

where U and V are two arbitrary scalar functions. Thus the homoge

neous equation can be rewritten in the form

V 2A + & 2A = 0 (2.27)

and

V 2F + k2F = 0. (2.28)

If the vector functions A and F have only a single component in the Z

direction, then these become the ordinary differential equations given

by

(2.29)

(2.30)

where A and F are the Fourier transforms of A and F and u2 =

kx 2 + kv 2 — k2, with kx and kv coefficients in the Fourier space to be

determined and k is the wave number. The solution to these equations

is

and

(PA 2 I ns r - “ ,A = °


29

A(kx, ky, z) = A + (kx, ky)e-us + A - (k x, ky)eU! (2.31)

and

kv, z) = F +(kx , ky)e~uz + F~(kx, ky) e (2.32)

Here the symbols ” + ” and ” - ” refer to the downward and

upward decaying solutions respectively. A particular solution of the

inhomogeneous field equation considering a point source located above

the earth (z = —h) can be written as

Ay(kx,ky)e-W 1**1

and

FP(kx,ky)e~'‘° ^

where Ap and Fp are the amplitudes of the incident field which depend

on the type and location of the used source. If the source is located at

the earth’s surface, i.e. at z — 0, then the coefficients A ,=o- and F , - 0~

can be expressed as

A_.=0- = r TMApeu‘ h (2.35)

(2.33)

(2.34)


30

and

F „ o~ = rTEFpe',°h (2.36)

where ttm and tte are the transverse magnetic and electric mode re

flection coefficients given by

- Z\vtm = — — f (2.37)

and

Y0 - Yxtte — --------~ (2.38)

Y0 + Yi K 1

where Y0 and Y\ are the free space and surface admittance respectively

and Za and Zi are the free space and surface impedances. In the trans

form space, the general solutions which apply to any source type is given

by the linear combination of the particular and homogeneous solutions,

i.e.

A = j4pe_“oA(e -Uo* + rTMeUo1) (2.39)

and

F = Fpe~Uoh(e~'la‘ + rrBeu*3),- (2.40)

whose inverse Fourier transforms are given by


31

A = — y ° °y 00/lpe-,,-’',(e -,,<’, + r T M e ' ^ e ^ + ^ d k t d k y (2.41)

and

^ = A / °° f °°FPe~'loK{e~u°1 + rr i!e ^ , )e(* '*+fc»1',d M V (2.42)47r */—oo •/—oo

For electric sources E and H are given by

E = - z A + i v ( V • A ) (2.43)

and

H =5 V x A (2.44)

from where the fields generated by an electric dipole along the X axis

are

£ = (2.45)a + iwe axoz

and

H , = (2.46)

whose Fourier transforms are respectively


32

& = (~z~-— ^ w ir (2,47)cr + iu e o x o z

and

6 , = . (2.48)

For a horizontal electric dipole oriented along the X direction

and located on the z axis at z = — h above the earth surface, Banos

(1966) has demonstrated that the vector potential for this source in a

volume v is given by

^ ( 0 = / . (2-49)Ju 4jr|r — r | J ( r )

where J ( f ) is the current density. Ward and Hohmann (1987) have

found using this result that the explicit expression for the vector po

tential A due to a dipolar source is

A = ^ e - u«<I+ ',>uI (2.50)

Equation (2.50) can be substituted into equation (2.47) and (2.4S) re

spectively to give

E = il: ro r-i \


33

and

H , = l^»e-u0(*+fc) (2.52)2 u„

Thus, the coefficient Ap in equation (2.41) is obtained by equating

equation (2.51) with the field expression for the transform of E, given

by

and then applying the inverse Fourier transform, which results in

-* Ids ik*t f t s - <2-54>

In a similar way, it can be shown that Fp is given by

Substituing these expressions into equations (2.41) and (2.42) then the

T M and T E potentials for an electric dipole are obtained, these are

kx 2 + k

and

%kx e ^ + ^ d k r d k , , (2.56)


34

F (x ,y ,z ) = J _ J [ e- ^ K ) +rTEeM ^ ) ]

'& . . e ^ + ^ d k j k y (2.57)KX + ky

Finally, substituting these equations into the expressions for

each component of the electric and magnetic field given by the sum of

the T M and T E modes by

1 82A Z dFz ydxdy dy 1 82AS dFz ydydz + dx

(2-58)dAz 1 d2 Fz dy zdxdz dAz 1 d2 Fz

8 x z dydz1 / 5 2 , 2 x n

+ ) F ‘

(2.59)

we have the complete description of the EM fields in a homogeneous

earth due to an electric dipolar source. The explicit expressions axe

(Ward and Hohmann, 1987)

Ex =

E y =

E z =

H x =

H y =

H z =


35

Z0Ids roo + _A Jg^ p)dX (2 6Q)JO Un4x Jo

Ids d y /■00r. .u 0 Z0.E y = J— Q / l ( l — r T M ) f 7 — (1 + » T b ) ]

47T dxpJo Yo U»

J1 {Xp)dX (2.61)

Id s d v r°°Hx = 4V d ip Jo (r™ + r™ V M X p ) d X (2.62)

Hy = (7'tjw + r i ’s)eUo*Ji(A/>)</A777© roo

~ J 0 ( l - r TE)e ^ X J 0 (Xp)dX (2.63)

f t = - ^ j f ^ + r r s J e ^ ^ A r t d A . (2.64)

These are valid expressions for any layered earth if vtm and

r?E are suitable functions of the frequency u and wavelength A. Similar

expressions for the fields were used by Goldstein and Strangway (1975)

and more recently by Zonge and Hughes (19SS) to determine zones

where the plane wave approximation can be applied to CSMT data

(Fig. 1).


C hapter 3

T he D istortion P rob lem

3.1 C hanneling and D isto r tio n

As in the case of M T, CSAMT sounding curves have been interpreted

mostly in terms of horizontally homogeneous strata which gives good

results when conductivity inhomogeneities do not significantly distort

the EM field. However, these favorable conditions are rare, specially

when surveys are focused on mineral targets which axe seldom asso

ciated with sedimentary basin-like geological environment. Field dis

tortions and scattering in the vicinity of a 3D electrical inhomogeneity

are either due to accumulation of charges or to excess of currents or

to a combination of both. The excess of charges alone produce purely

galvanic effects though the excess of currents leads to the generation

of induction effects. At low frequencies the effects of currents tend to

zero while the excess of charges remain important. For high frequen

cies though, the mutual induction between the anomalous body and


37

the host as well as self-induction become important.

Distortion of electromagnetic fields by conducting structures

has been of concern at least since the early 70’s (e.g. Ward, 1971).

Initial attempts to take into account effects of 3D electrical inhoino-

geneities where focused in solving numerically the integral equation

representation of the distorted fields produced by regular bodies. The

equation is given in general form by

E ,(p ,s ,6 ,u ) = J J K (p ,z , 0 ,w) E p dp dz dO (3.1)

where K(p, z, 0 ,u>) is the kernel that describe a subsurface distribution

of the anomaly parameters, p, z, and 0 describes the coordinate sys

tem, while u>, Ep, and E , are the probing angular frequency, and the

primary and secondary electric fields. Examples of the evaluation of

expressions of this type, which includes anomalous inductive response,

can be found in the literature (e.g. Schmucker, 1971). Expressions of

this type involve complicated integrals which are, in general, difficult to

evaluate. To simplify the interpretation of distorted EM field measure

ments a different approach to account for 2D and 3D anomalies was

introduced some 10 years ago (Eggers, 1981). To address the effects

of distortion due to electrical inhomogeneities this approach assumes


38

only galvanic effects, i.e. it is assumed a very weak inductive response.

This, as observed, may not be true for high frequencies but is a good

first approximation to account for some of the electric distortion which

causes EM data to be misinterpreted.

The buildup of free charges producing galvanic distortion oc

curs at conductivity gradients in the surroundings of the electrical inho

mogeneity. The array of charges in its vicinity tends to channel currents

along the conductivity structure and consequently distort the horizon

tal components of the electric field. In order for the anomalous body to

distort significantly this electric field, two skin depth criteria have been

suggested in the past (Berdichevski and Dmitriev, 1976a; Wannamaker

et a1., 1984b) namely: a) the skin depth in the host medium should

be long compared with the distance from the observation point to the

inhomogeneity, and b) the skin depth within the inhomogeneity should

be long enough compared with its size, which assumes a negligible self

induction inside the anomaly.

One of the most commonly encountered type of distortions is

what is known as ‘Static shift’, which is reflected as a parallel vertical

displacement in the M T log apparent resistivity curves. Static shift

occurs because the magnetic fields that are measured at any point are

affected primarily by the hemisphere of earth whose radius is of the


39

order of a skin depth and will in a sense, integrate all the minor resis

tivity variations within this volume. Similarly, currents will be induced

throughout this hemisphere. However, the measurement of the electric

fields is a local measurement which depends on the resistivity in the

vicinity of the electrode array. This resistivity is the local surface re

sistivity which can vary very rapidly from one location to other. An

electrode array set up at one location often sees a different surface re

sistivity than an array set up a short distance away. Static shift is then

a frequency independent effect due to near surface small-scale inhomo

geneities. There axe however, frequency dependent small and large-scale

EM responses. Large-scale inhomogeneities, i.e. on the order or greater

than a skin depth, will cause frequency-dependent shift in the apparent

resistivity curve rather than a parallel shift. These 2 and 3D effects

may be interpreted from the M T curve using numerical modeling only

when small-scale effects are separated. Small-scale frequency indepen

dent effects other than static shift are often assumed to be independent

of their counterpart large-scale effects (Bahr, 1985). Based upon this

hypothesis and the premise that the electric distortion is of galvanic

origin, several decomposition methods have been suggested to sepa

rate local from regional fields. In a subsequent section, this problem

will be addressed, but before it is necessary to introduce the concept


40

of Impedance Tensor subject to galvanic distortion, discussed in the

following section.

3.2 T h e Im pedan ce Tensor

For a horizontally stratified earth only a single component of the hori

zontal magnetic field vector and the electric field in the perpendicular

direction are measured. The linear relationship between the natural

fields is through the impedance Z(w), a complex scalar containing in

formation about the amplitude and phase between the two orthogonal

fields. It has been observed experimentally that the electric field re

sponse of the earth depends upon the direction in which E and H

are measured. This fact can be attributed to the earth’s apparent

anisotropy due to lateral changes in physical properties. In its more

general case, the horizontal components of E and H are connected

through a tensor quantity rather than a scalar. As for a homogeneous

earth, for laterally inhomogeneous structures a linear relationship be

tween the field components is often assumed and given by

Ex = ZxxHx + ZxyHy (3 .2)

and

Ey = ZyXH x + ZyyHy (3.3)


41

In matrix notation this reduces to

E = Z H (3.4)

where

z = ( 5 * z7xy) (3.5)\ 4yx &yy /

is termed the impedance tensor with each of its elements complex. The

vectors E = (Ex, E y) and H = (H x, H y) jure field measured quantities

taken in the far-field when artificial sources are employed.

Because of the complex nature of the matrix Z, the physical sig

nificance of its coefficients is not readily apparent. This has motivated

different approaches to extract scalar parameters from it with more

physical meaning with respect to the subsurface conductivity structure.

I t has been a common practice, before more elaborated decomposition

methods were available, to assume that the subsurface is uniform along

one axis. In this case, the impedance tensor can be rotated so that the

coordinate axes correspond to the principal axes of the 2D structure. If

the fields are linearly polarized parallel to the symmetry or strike axis

of the structure, Cagniard’s scalar relationship (eq. 2.1) still holds. If

R is a matrix which rotates the components of Z through an angle 0,


42

then the impedance tensor Z$ along the regional inductive prin

cipal axis system is given by

Z , = R Z R t (3.6)

where the rotation operator is given by

R = ( “ ! * *“ ! ) . (3.7)\ —sind cosO) ' '

By solving equation (3-6) it can be shown that under this rotation

(Zxx + Zvy) and (Zxy — Zyx) are both invariant, and Zxx -I- Zyy = 0 and

in general Zxy ^ —Zyx (e.g. Cevallos, 1986). Thus for a 2D structure

with the coordinate axis parallel to the principal axis of the structure,

the impedance tensor Z has zero as elements on the diagonal, i.e.

= ( - 6 ;)■• m

The standard procedure to arrive at this form is to fit the elements

of Z in the least square sense (e.g. Sims et al., 1971). .More often

than in M T, experimental CSAMT data does not conform an ideal 2D

impedance tensor even after a rotation is performed. This is mainly

because of local galvanic distortion is more likely present in geological

environments associated with ore bodies, thereof their significance.


43

When relatively small near surface inhomogeneities are present,

one can neglect any induced secondary fields produced by them. Thus

a simple electrostatic distortion model will be appropriate when the

earth is excited by a uniform primary electric field Ep. For this case

Groom (1988) has shown that a first approximation model of electric

distortion (static shift only) can be represented by

E = C Z 2H (3.9)

where C is a distortion operator with purely real, frequency indepen

dent elements given by

This relation assumes that the measurement axes are coincident with

the principal axes of the two-dimensional structure. In general, the

orientation of the 2D structure is unknown and therefore the measure

ment axes are others but the principal axes of the structure. In this

case, the true impedance tensor could be obtained performing the same

operation as in equation (3.6),


44

Z = R C Z aR r . (3.11)

this assumes that the principal impedances Zxy and Zyx (or Z\\ and

Z±. respectively) can be obtained by rotating the impedance tensor

to an off-diagonal form and that the scaling factors si and sa (which

represent the static shift elements) can be estimated in some other way.

The implementation of this method is done by minimizing the sum of

the square of the magnitudes of the diagonal elements of

Z ’ = R Z mR r (3.12)

where Z m is the measured impedance tensor. For a more general dis

tortion problem, i.e. for one which does not make any geometrical

simplification, the distortion vector C may have off-diagonal elements

different from zero even after a rotation to the principal axes is per

formed. In this case C may be factorized to separate possible distortion

effects and provide physical insight on the nature of electrical inhome-

geneities. Several alternative surface impedance tensor representations

that account differently for two and three-dimensional galvanic and

weak induction effect have been recently suggested. In the following

section Groom and Bailey’s (19S9) impedance tensor decomposition

for 2-D is summarized.


45

3.3 Im p ed an ce Tensor D eco m p o sitio n

To obtain a representative picture of the earth’s conductivity structure

when the electric held is distorted by charge accumulation on conduc

tivity boundaries or gradients, it is necessary to unmix the inhomegene-

ity effects from the homogeneous electrical information. Under these

circumstances there is no reason for the measured impedance tensor

to be close to a true 2D impedance tensor. Assuming that galvanic

distortion does not destroy the information about an underlying 2D

induction process (Bahr, 1985), Groom and Bailey (19S9) proposed an

alternative product factorization of the impedance tensor. Their de

composition allows the explicit parametrization of the tensor in such a

way that the separation of local and regional effects is possible when the

regional structure is at most two dimensional. Although the decompo

sition given by (3.11) expresses the underlying conductivity structure,

the system of eight real equations that it defines has a greater num

ber of unknowns. Nine real parameters are present: the rotation an

gle 0 (in R ), four distortion tensor elements (in C ), and two complex

impedances (in Z ). To overcome this problem they proposed a different

factorization for the distortion tensor, namely

C = s T S A (3.13)


46

where g is a scaling factor and T , S, and A are tensors which have

physical meaning by themselves defined by

S = J\fi(I + eE0 (3.14)

T = JV2( I + fE2) (3.15)

A = -/V3( I + sE3) (3.16)

where N{ are normalization factors to ensure that the elements of the

tensors remain bounded during any computation. E,- are a set of ma

trices given by

-(! !)■ (3.17)

= (? o1)-and

(3.18)

* - ( ! i)- (3.19)


47

and I is the identity 2x2 range matrix. The tensor A is known as the

anisotropy or splitting tensor because it stretches the Held components

by different factors through

A = Ws( ‘ r ! - . ) • <3'20>

which generates anisotropy and is added to the anisotropy existing in

the regional induction impedance tensor Z. Thus, this type of distortion

is static shift and is indistinguishable experimentally from the inductive

anisotropy. The tensor T on the other hand, called also the twist tensor,

produces the effect of rotating the regional electric field vectors through

a clockwise angle <j>t = tan~l t and is represented by

T = N i ( - t !)• (3-2i)

The effect of the tensor S on the electric field components is to deflect

each component by an angle <j>e = tan~l e towards each other having

their maximum angular changes when they are aligned along the prin

cipal axis. The tensor S is called shear tensor because of the analogy

it bears with the same concept in theory of deformation. Similarly to

the previous expressions, S is given by


48

S = M ( * [ ) . (3.22)

As observed before, g is a constant defined as the ‘site gain’ which per

forms an overall scaling of the electric fields. If the new decomposition

of C (eq. 3.13) is substituted into (3.11) then the impedance tensor

takes the form

Z = $ R T S A Z 2R t , (3.23)

and because neither g nor A can be determined separately from Z 2

they are absorbed into Z 2, i.e.

Z ' = g A Z 2 (3.24)

which is an equally valid (but static shifted) ideal 2D impedance ten

sor. In the case that telluric distortion is frequency independent this

absorption will not change the shape of the M T apparent resistivity

curve or the phases. This implies that the recovered apparent resistiv

ity is correct except for a static shift. Because Z2 and Z '2 cannot be

distinguished from field measurements, the prime (') is dropped leading

to


49

Z2 = R TS Z2R r . (3.25)

This equation is Groom and Bailey’s decomposition for the

impedance tensor which has seven real distortion parameters to be de

termined, these are : the scaled real and imaginary part of the principal

impedance, the real and imaginary parts of the minor impedance, the

a resistivity, the shear e, and the twist t. An important characteristic

of this representation is that if the physical model is correct for the

impedance tensor, then there is a unique decomposition (3.25). There

is also a unique solution to the system (3.4) with Z given by equation

(3.25) if the shear and twist angles are restricted to magnitudes less

than 45° although, this is not true for the 1-D model as we will see in

the following chapters.


C hapter 4

D ecom p osition s for vector C SA M T

4.1 In tro d u ctio n

Although the decomposition of the impedance tensor method was de

veloped originally for M T, its use can be extended to frequency domain

CSAMT. An important difference though is the target nature and size

of the electric scatterers. In M T the objective of applying distortion

analysis is to reduce as far as possible the unwanted distortion effects

on the measured electric fields due to electrical inhomogencities in the

vicinity of the measuring station. In CSAMT, one would like to know

the distortion parameters associated with shallow conductivity bodies

not to filter them out but to assess their approximate geometry and con

ductivity distribution because of the economic interest that they may

represent, i.e., they represent additional information to the commonly


computed pa(u) and 4>(u).

The CSAMT field analogous to the M T plane wave survey is

the case where the electric and magnetic field components are measured

in the far field zone using at least two sources used to excite the earth.

Under this situation the electromagnetic fields are related by equation

(3.4) and tensor decomposition methods can be still applied. As in

M T, one may have an ensemble of events (E , H ) with which all the

elements of Z can be evaluated. When this is the case the technique

is known as Tensor CSAMT. For example, for two independent source

measurements of the horizontal electric field components, the elements

of Z at a given frequency are given by

(4.1)

(4.2)

(4.3)A

and

(4.4)A


52

where

a = ( £ &) ■ <4-5>

provided that (HxiH y 2 — H xiH y\) r 0- 1° general (i.e. for N indepen

dent measurements), a least-square approach is used to evaluate Zy. In

addition a 2-D distortion model (6571.(3.25)) can be applied to solve for

the three Groom and Bailey’s distortion parameters, t, e, and s because

enough data is available. In practice, multisource CSAMT surveys have

not been very popular in part because they are expensive to carry out.

Instead single source surveys have been used routinely in mineral ex

ploration projects, with the disadvantage that because only a single

coherent excitation mode of H is available at a given site, there are

not enough equations to solve (3.4) for all the elements of the complex

impedance tensor Z. Therefore some simplifying assumptions have to

be done in order to apply it to vector CSAMT data. The small number

of data parameters available in vector CSAMT (four in total) do not

allow models for the data which simultaneously incorporate both 2D

induction and local galvanic distortion. The alternative simplification

is to assume a one-dimensional inductive structure whose impedance

tensor Z i is of the form


53

with a a complex. In the general distortion model given by equation

(3.25) applied to the 1-D case, there are 6 parameters to solve for,

namely, Re a , Im a, t, e, s, and 9. Thus we are unable to solve because

the equation (3.4) defines only four equations. Because for a 1-D earth

model the regional strike is undefined, it can be chosen arbitrarily. So

a reasonable question to ask is whether or not would be possible to find

a 6 for which one of the distortion parameters t, e or s can be made

zero. A yes answer is plausible because the same distortion applied

to regional models with a different strike angle yields to different local

distortion parameters.

The next question to answer is if any (but one at a time) of the distor

tion parameters can be zeroed by choosing an appropriate coordinate

system. Here, the answer seems to be no. The case of a pure twist for

example, appears as the same twist even when the coordinate system

is rotated (see Fig. 6.3.1 of Groom and Bailey, 19S9). The amount of

twisting will remain the same in any rotated coordinate system. So we

assume that twist cannot be made zero no matter what regional strike

is used, therefore, any 1-D distortion factorization considered has to

include it. There are two possibilities left, either splitting or shear have


54

to be made zero using an appropriate azimuth angle 0. That the split

ting can be zero is not clear apriori, so a factorization that excludes it

has to be constructed and tested in order to evaluate this possibility.

A zero shear factorization though, can easily be constructed which has

a particularly simple visualization in terms of polarization ellipses. In

the following two section these two factorizations are discussed.

4.2 Z ero-S p littin g 1-D F actoriza tion

Assume that there is a rotation of the axes such that the splitting can

be made zero. In such a case, the Groom and Bailey’s impedance tensor

decomposition is given by

Z = R T S Z i R t (4.7)

or more explicitly by

Z = I { ( cos® / 1 t \ / I\ —sin0 cosOJ V— t 1 / \ e l )

( 0 a \ ( cosQ — sin9\ . .I - a OJ VsinO cos6 J ^

with K = IVjjVj the normalization coefficient given by

K VTTe5 vT+t1' ^


55

Substituting this form of impedance tensor in (3.4) we have the five

real parameters (e ,t,R ea ,Im a ,0 ), and four equations defined by the

real and imaginary parts of the complex fields. As we choose 0, we are

left with a set of four (non linear) equations with four unknowns. Thus,

assigning a?i,a?2 , X3 and X4 to a,-, ar, e and t respectively, the functional

relations to be zeroed are represented by

fi(^l)®2)®3)®4) = 0, (4.10)

with i = 1 . . . 4. The fs are a set of nonlinear equations obtained

by explicit expansion of equation (4.8). To linearize this system, each

function /,• are expanded in Taylor series (Press et al., 1988) resulting

in

f i ( x + SX) = f t{X ) + + Oi(SX2), (4.11)

where X denote the entire vector solution. Neglecting quadratic or

der terms SX2 and higher, we obtain a set of linear equations for the

correction SX that moves each function closer to zero simultaneously.

This system is expressed as

Pi - EatijSxj, (4.12)


56

where /?,• = - / ; and ay = f £ .

The matrix equation (4.12) was solved using LU decomposition (Press

et al., 1988). The corrections were then added to the solution and tested

for convergence using the damped Newton-Raphson method (Conte and

de Boor, 19S0). Thus for the ith attempt

X ,neu' = X i 0 ,d + S X i ^ , (4.13)

with n = 1,2,3, — The X , n e w solution is accepted only if it leads to a

reduction in the residual error, i.e., only if

MA'n+OI < l« (* » ) l, (4.14)

where

|e,(A')| = £ ( 7 ^ - & ( * ) ) * . (4.15)

The process is iterated to convergence if possible. An initial

guess is needed to start the iteration process. A total of 34 different

initial guesses (xi°, x?0, X3 0, X4 0) was attempted at each frequency. Nev

ertheless, the solutions that constrain the values of the parameters e

and t to lie within the intervals


57

e, < < |45°| (4.16)

are very often sparse. Groom’s results that such solutions are always

possible for the M T factorization is not necessarily true for the zero-split

CSAMT factorization. A more detailed discussion on this will be done

in the next chapter. The algorithm and subroutines of the program

implemented to find the vector impedances and distortion parameters

using this approach can be found in Appendix A.

4 .3 Zero-Shear 1-D F actorization

If in the impedance decomposition given in equation (3.23) we choose

9 such that the shear e is zero, then it takes the form

Z = R T A Z xRt (4.17)

where the site gain factor g has been absorbed in A . Then the distortion

model can be written according to equation (3.4) as

E = R T A Z xRt H . (4. IS)

This is a useful factorization that has a direct geometrical visualization

in terms of polarization ellipses as follows: First rotate the equation to


58

the principal axis system of ft, i.e.

R T£ = T A Z xRt H . (4.19)

so that the electric field in the new reference axes is

E' = R TE = T A Z l H ' , (4.20)

where H 1 = TLt H The rotation angle in terms of the field components

is give by the expression (see Appendix B):

fl — 2 {HxrHyr + H sjHyi)Oh 2 (Hyr2 + Hyi2 - H xr2 - H J ) ( )

Thus 0 =1 O h 'is the required azimuth. The next step is to rotate E so

that the major axis of its polarization ellipse is perpendicular to the

major axis of the //-polarization ellipse as one would expect from the

Cagniard relation. This rotation,

1 2(E'xrE lyr + E'xjEyj)' • 2 ( E ^ + E ' / - E ^ - E ' J ) (4-22)

is the twist angle, and may be computed also as

0t = \ - \ 0 ' - 6 h\, (4.23)

where 0 e is the angle required to rotate the electric field polarization

axis to the original measurement system (see Appendix B). Once 0 and

0 t are known we are left with a decomposition that includes the splitting


59

tensor A which stretches or elongates the field components adjusting to

the same ratio the E and H polarization ellipses. This decomposition

has the form

or, explicitly

Ei = %Z i{ \ + s)HiVl + sJ

E 2 = — = L = Z i ( l - s ) H 2, (4.25)V l + s 2

which can easily be solved for the s and a.

Since there is by construction, a shear-free decomposition, it is clear

that shear is not needed to model a 1-D earth that includes distortion

galvanic effects. This model generates the two distortion parameters

splitting s and twist t and a single impedance a calculated as the ratio

of the major axes of the E and H polarization ellipses. It is important

to note that the distortion parameters obtained using the factorization

(4.17) depend on the source field and would be different if different

source location is used. This method of evaluating an apparent resistiv

ity from CSAMT data was originally suggested by Yamashita (personal

communication) as an empirical way of dealing with distortion effects.


60

Here it is shown that it is based on a valid distortion model. The result

of applying this factorization on the sample field data is discussed in

the next chapter. An algorithm of the program implemented to find

the vector impedance and distortion parameters using this approach

can be found in Appendix A.

4.4 Sum m ary

The general Groom and Bailey (19S9) 1-D distortion model has a total

of 6 real parameters, three related to distortion (t,e, and s), the real

and imaginary parts of the 1-D impedance and the strike angle 9. In or

der to be able to apply this model to CSAMT data, where a maximum

of 4 reril parameters is allowed because of the limited amount of data

available, we have considered the possibility of finding a 9 such that one

of the distortion parameters is set to zero. I f this can be done, then the

resulting factorization will have the desired 4 real parameters to solve

for, two of which are distortion related parameters. There are three

possible factorizations resulting from making zero one of the distor

tion parameters. The zero-twist distortion factorization was discarded

because t is not likely affected by the choice of our reference system.

Thus a zero-splitting and zero-twist decompositions were suggested as

potentially useful distortion models usable with vector CSAMT data


61

for 1-D earth. A test of these methods using experimental field data is

discussed in the following chapter.


C hapter 5

A Case H istory o f C S A M T D istortion

5.1 In trod u ction

The vector CSAMT data used in this thesis is part of a contract work

done by Phoenix Geophysics for PNC Exploration Co. The objective

of the survey, which was carried out over the Midwest uranium ore

zone in Saskatchewan, was to study the effectiveness of the method for

prospecting uranium deposits in the Athabaska basin. A total of four

lines, with between 8 and 12 stations each, were accomplished in such

survey. All lines are parallel to each other and cross an elongated con

ductor at approximately right angles (Fig. 2). Due to the similarities

encountered during the processing, only line 8 is used here to illustrate

the results obtained with the decomposition factorizations described in

the previous chapter.


63

Figure 3 is a simplified geological cross section of the ore zone.

Details have been intentionally excluded by request of the owner of

the data (PNC Exploration Co.). Basically, a Uranium ore deposit

is unconformably overlying a Precambrian basement and underlying

a sandstone formation at a depth of approximately 200 m from the

surface. The uranium deposit is surrounded by a conductive clay al

teration zone, which makes it the actual geoelectrical target. A 4 km

long transmitter bipolt was used at approximately approximately 6.3

km South-West of the survey area and roughly parallel to the survey

lines (Fig. 2). A set of three H field components (Hx, H y and H c) were

collected for each two sets of perpendicular E field components (Ex and

E y). This configuration assumes a negligible variation of the magnetic

field over two consecutive electric dipole locations which collected data

confirmed to be correct (Phoenix Geoph., 1989).

5.2 1-D E x p ec ted E lectr ica l B eh av ior

For a 1-D layered eaxth the electrical and magnetic horizontal com

ponents are perpendicular to each other. In the far field, the E and

H polarization ellipses are expected to be degenerated to a straight

line if the earth is truly 1-D. In the presence of a 2-D and 3-D electrical

structure though, the fields are distorted due to the preferential current

directions along more conductive zones. Depending on the orientation


64

of the transmitter and the electrical character of the anomalous body

(if it is conductive or resistive), the amplitude of some field components

will be enhanced more than others, but more important than this, their

direction and phases could be drastically modified. This results in po

larized fields of varying ellipticity and orientation and consequently

non-orthogonal E and H horizontal components.

The data used in this work was collected using the survey lay

out of figure 2. Here the dipole source Tx is parallel to the x-coordinate

axis of the reference system, it is almost perpendiculax to the strike of

the surveyed ore conductive zone. If the regional structure of the area

were nearly 1-D, the horizontal components of the magnetic field would

show very little variation from one station to the other even across a

moderate conductive structure. The H z component of the magnetic

field is expected to be very close to zero in the far field zone and have

a significant value as we approach to the dipole source. For a survey

done mainly in the far field wave zone, most of the H : component must

be small compared with H x or H y, and H y will have comparatively a

greater amplitude than H x if the transmitter dipole is parallel to the

X axis. The x —component of the electric field however, should be

of relatively high amplitude and in principle, strongly distorted in the

neighborhood of a conductive body. The y component as well, although


of lesser magnitude, might be modified in magnitude and direction in

the presence of a conductive structure. In consequence, the polarization

ellipse of the horizontal E field components would be highly distorted

both in ellipticity and orientation across a conductive body embedded

in a homogeneous earth. This expected change in pattern provides an

a priori diagnostic to define approximate boundaries of a conductive

ore zone.

For a one dimensional earth and in the far-field, the phase angle

between E x and H y must be 45° at all frequencies. This is an indication

of the homogeneity of the electrical structure of the medium. If the

phase angle is greater than 45° as probing frequency decreases, it could

mean, that the resistivity of the surveyed area is increasing. If the phase

falls below the 45° value as probing frequency decreases it is likely that

the resistivity of the ground is increasing. In the near-field however, the

phase has values close to zero, while the apparent resistivity increases

linearly with frequency showing a slope of 45° in a log-log plot. In the

case of a 2-D earth containing electrical scatterers, phases are no longer

easy to predict and resistivities may vary differently depending on the

scale of the measurements and physical distribution of the conductors.

Thus, the diagnostic parameters provided by the decomposition of the

impedance vector are expected to provide additional information to


66

assess the presence of electrical conductors, in the first place, and to

help interpret the conductivity structure i" a later stage.

5.3 2-D E arth w ith o u t E -d isto rtio n

Assuming a 2-D earth with no surface distortion, the impedance tensor

would conform an ideal tensor in which the diagonal elements are both

zero. That is

One of the non-zero elements of the tei'.sor is associated with current

parallel to the strike (Electric polarization) and the other with the mag

netic field also parallel to the strike (Magnetic polarization), and can

be calculated from the measured fields. As in M T , in controlled audio

frequency M T methods, this has been the usual approach to compute

apparent resistivity and phase from survey data. An example of it for

line 8 is shown in Fig. 4. Here, the assumption was done that the mea

surement axes were aligned with the principal axes of the structure.

The figure shows the magnetic polarization mode (also known as the

transverse electric or T E mode), i.e., the electric field is perpendicular

to the strike of the conductor. The apparent resistivity and phase in

the figure, were normalized using their peak value of the respective data


sets in order to compare the different approaches considered here. The

normalized apparent resistivity pseudo section (magnetic polarization)

shows a well defined low resistivity zone coincident with the extension

of the conductive halo surrounding the uranium ore. Over the conduc

tor where the resistivity of the clay halo can be as low as 1 ohm-m the

skin depth at 2 Hz is only about 250 m. This is probably why very little

or no detail is observed below this zone. The phase angle between the

field components Ex and H y is also shown in this figure. At interme

diate and low frequencies there is an increase in the phase magnitude

across the conductor. A rotation of 90° of the principal axis does not

alter significantly the observed resistivity distribution but produces a

phase shift such that the maxima and minima are interchanged (see

Fig. 5). This represents the electric polarization mode (also known

as the transverse magnetic or T M mode), where the electric field is

parallel to the strike. Both the T E and T M modes show very similar

behavior. I f the principal axes of the conductor were exactly aligned

with the measurement axes there would be a noticeable difference be

tween these polarization modes. Their similarity is a result of having a

large E-polarization ellipse which does not coincide with the principal

axes such that their projection on their X and Y coordinate axis results

in similar E-field magnitudes and consequently apparent resistivities.

Thus an error in choosing 9 in the presence of 2-D highly conductive


bodies may lead to undistinguishable T E and T M polarization modes

which appears to be the case here.

5.4 T est for Z ero-S p littin g M o d el

Applying the zero splitting model (4.7) to experimental data, often

failed to find any good fitting solution for the non-linear algorithm de

scribed earlier. In fact, two almost equally bad solutions were obtained

consistently at each frequency and the fit is worst within the anoma

lous conductivity zone. A noticeable characteristic of them is that the

absolute value of the distortion parameters twist t and shear e are al

ways greater than 45°. Groom and Bailey have concluded that there

exist two types of solutions for the distortion parameters, a solution

of magnitude greater than one (|e|, |t| > 1 or large) and other whose

magnitude is less than one (|e|, |t| < 1 or small).

Because of the impossibility of selecting the ’’better” solution on the

basis of Groom and Bailey’s criteria, which states that a meaningful

solution will be such that |f| and verte| are less than 45°, then we can

anticipate that the zero splitting approach is not an appropriate model.

These can be observed for the particular data set used. One of the so

lutions showed consistently large impedance magnitude than the other,

thus, they were separated using this criterion. Figure 6 shows the solu


69

tion with smaller apparent resistivity, the corresponding phase is also

plotted here. As noted before, these quantities have been normalized

using their respective peak value in the profile. In Figure 7, the solu

tion whose apparent resistivity is consistently larger is plotted as well

as the computed phase. Although the general shape of the apparent

resistivity is preserved, the range of amplitudes widely varies. For ex

ample, while in the first case the resistivity profile was normalized using

a value of 16,596 ft — m in the former case a norm of 496,909 ft — m

was used. Differences in phase can also be observed between them,

but more noticeable is their relative complexity compared with the 2-D

case, even though there occur phase changes across the electric discon

tinuity, particularly in Fig. 7b. The distortion parameters on the other

hand (Figs. 10 and 11) show a complicated pattern and little or no

correlation with the assumed 1-D conductor. None of the two encoun

tered solutions seems to fit the model properly. The complexity of the

twist and shear distribution show that these parameters have a highly

frequency dependent behavior. Even in the case that the assumed one

dimensionality may not be valid everywhere, it was expected that at

relatively high frequencies, the distortion parameters would be constant

within the conductivity structure. The results however, suggests that

the anisotropy or splitting effects cannot be disregarded.


70

5.5 T est for Zero-Shear M o d el

Consider now the factorization give by equation (4.17). The analytical

nature of the polarization approach or zero-shear model, prevents the

ambiguity of multiple solutions and provides an exact fit to the data.

The electrical resistivity computed following the procedure described

in Section 5.3 and normalized using the peak value afterwards, bears

a general resemblance to the 2-D classical approach as well as to the

previous 1-D distortion model (see Figs. 5 to 8). The conductive zone

is well defined and, as for the previous cases, no detail is observed at

depth. The resistive zone though, has been shifted downwards and a

relatively conductive overburden is also observed at most of the stations

of the profile. Larger contrast in the resistivity profile are observed as

can be noted from the maximum and minimum normalized values in the

scale bar of the same figure. The phase plot, on the other hand, show

significant changes across the anomaly at relatively high frequencies,

but after all a more simpler distribution is observed (Fig. Sb).

Recall that the distortion parameters obtained using this method are

the twist t and the split s. In Figure 11a it is shown that the twist bears

a good correlation with the conductive zone showing high values where

the fields are expected to be strongly rotated, as it may be expected.

Within the conductor and at high frequencies, t is approximately inde


71

pendent of frequency which suggests that a 1-D interpretation might be

approximately valid at least for a given frequency interval. A param

eter close in meaning to this twist was obtained earlier by Yamashita

(1989) although not specifically as a distortion parameter. The split

distribution in Fig. l ib also shows relatively frequency independent

high values of anisotropy coincident with the conductor location, par

ticularly at high frequencies. This supports the validity of this model

in a similar frequency interval as for t. Thus, both twist and split are

approximately constant for frequencies greater than about 256Hz. For

frequencies lower than this, and outside the conductor, 2-D distortion

effects are likely to be important and cannot be modeled with this fac

torization. Figure 13 shows the frequency dependence of the distortion

parameters at stations 4525 (*), 4425 (o), 4275 (*), and 4175 (+ ). Sta

tion 4425 is on top of the conductor, where we expect our 1-D model

to be valid. At this station, the zero-splitting distortion parameters t

and e are frequency dependent particularly at high frequencies, while

the zero-shear parameters t and s show a much less pronounced vari

ation for a similar frequency interval suggesting that a 1-D distortion

model which does not include the distortion parameter shear is a better

factorization of the impedance tensor.


72

5.6 C onclusions

The main task of this thesis work was to study the possibilities of ap

plying Groom and Bailey’s decomposition of the magnetotelluric tensor

approach to far-field vector CSAMT data in the presence of electrical

inhomogeneities. Two factor decompositions of 1-D distortion models

were proposed and tested with an experimental data set to assess their

utility and limitations.

The conventional 2-D model, that does not consider distortion effects,

may be useful for qualitatively locating the conductive anomalies be

cause small misalignments of the electric axes in the presence of high re

sistivity ratios leads to practically undistinguishable polarization modes.

Even if the undistorted model were appropriated, under this circum

stances it can only resolve 1-D worth of information. Thus, unless the

measurement direction coincides with the principal induction directions

and there is no distortion present, we will not get results which are

quantitatively useful.

The zero-split distortion approach gives large fitting errors particularly

on top of the conductor (.Fijr.12). There were systematically two so

lutions for the resistivity, phase and distortion parameters using this

method. Both gave values for t and e consistently greater than 45°

which discard them as physically meaningful solutions. The assump


73

tion that it is possible to find a rotation angle such that the split can

be made zero is likely not correct, although more theoretical work has

to be done in this respect.

The zero-shear approach was constructed in such a way that warran

tee the existence of a rotation 6 such that the distortion parameter e

can be made zero. Although this is a source dependent approach, it is

shown to be a valid 1-D distortion model within certain limits even for

the relatively complex data set used to test it. However, the parame

ters obtained depend on the location of the source and thus, they do

not have absolute meaning. Future work on this method might take

advantage of physical scale modeling to study with more detail their

dependence with source location. Furtheremore, additional testing us

ing data sets covering a wider range of geological settings have to be

done to asses the utility of the distortion model presented here.


A p p e n d ix A

Programs for 1-D Galvanic Distortion


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PROGRAM CSAMT

6 T h i s p r o g r a n c a l l s t h e s u b r r o u t l n a s o a s t s , o a s a b t a d c a a e c w h ic h C o o a p u ta t h e a l a a a o t s o f t h a la p a d a o o a t a a s o r f o r a A) 2 0 e a r t h s t r u c t u r e G ( w i t h o u t E l e c t r i c { l a i d d i s t o r t i o n ) . B) 10 s t r u e t u r o ( w i t h t d i s t o r t i o n ) ,C a n d C ) ID a a r t h s t r u o t u r a ( w i t h o u t C d i s t o r t i o n ) r o a p e c t l v a l y . T ha f i r s t o f G t h a s a e a s a s (A ) i s t h a o l a s s i o a l w ay t o c o a p u t a t h a l a p a d a o o a o o a p o o a o ta C w h l l a (B ) i s t h a O ro o a a n d B a i l e y ' s d a e o a p o s i t l o a a d a p t a d f o r 1 - 0 a a r t h .C (C ) o o n a l d a r s t h a o a s a o f 1 -D a a r t h w i t h o u t d i s t o r t i o n , l . a . . t h i s l a t h a G s c a l a r o l a s s i o a l a p p r o a c h .C ...................... .................................................... .................................................. ..

PARAMETER (N F -2 0 .N S -2 3 )DIMENSION APPRESM AJ(NF).APPRESM IN(ND,

/PH A SEM AJ(NF).PBASEM INCNT),PBASE(NF), A PPRES(N F).SH EA R (N T),/TW 1ST( NF) , ERRORMIN(NF)

INTEGER Q,CHOOSE.N GOMMON /T E S T S / FREQ(NF),NRF COMMON /W tT Q C / MCOMMON /T E S T D / TET __________ ______

WRXTE(6. * ) 'B ow a a n y s t a t i o n s I n t h i s p r o f l l a ? ! 1-5E&Si5i:it!____________________0 0 LQO 3 " l ,MHRITEC6, .................w r i t e ( 6 > • ) 'Y o u h a v a t h a f o l l o w l n 9 a a n u t o o h o o s a f r o n t 1W RITE(S, ............................................................ * .....WRITE( 6 ,1 5 0 )FO R M A T (///,

> .> ..> ' 1 ) 2 0 a a r t h s t r u c t u r e w i t h o u t E f t a l d d i s t o r t i o n * ,> * (ENTER 1 ) ' / / .> '2 ) 1 0 a a r t h s t r u o t u r a w i t h E f l a l d d i s t o r t i o n * ,> * (ENTER 2 ) * / / ,> '3 ) ID a a r t h s t r u o t u r a w i t h o u t E f l a l d d i s t o r t i o n * ,> * (ENTER 3 ) ' / / ,> *0 ) EXITS t h e p r o g r a a * ,> * (ENTER 0 ) * / / ,> .................................> '«.............. ......

CBOOSE-OWRITE ( 6 , *) * W h a t i s y o u r c h o i c e ? ■ >*R E A O (S ,* )CBOOSE

II_W R ITE(61; ) * s t a t l o n _ N u a b e r t j j J __________________ ______________

IF (C B 00S E .E Q .1 )T 8E NCALL CASEA( APPRE5MAJ, APPRESMIN, P8ASEMAJ, P BAS EMIN,

/T E T ,N R F ,F R E Q )OPEN( U N IT "S 0 , STATUS*1N E W '.F I L E - 'c a s a a .x y x * ,

/A C C E S S " ' APPEND1 .E R R -96)W R X T E (S0,325)DO 5 I-L .N R FWRITE( 5 0 ,3 0 0 ) FREQ( I ) , APPRESMAJ( I ) , APPRESMIN( X) ,

/PHASEMAJ ( I ) ,PBASEMXN(I)CONTINUE

E LSE1F( C B 005E . EQ. 2 ) TBD) CALL CAS£B(APPRES,PHASE.SHEAR,TWIST,EBRORMAX)

ELSEXF(CBOOSE.EQ.3 )THENCALL CASEC( APPRES, PBASE«NRF,FREQ)

OPEN( U N IT -70 ,S T A T U S "1 NEW• , F IL E - ' e s i a o . x y z * , /A C C E SS"'A PPEN D *, ERR-9B)

W R IT E (7 0 ,4 7 S )DO 7 I - l .N R FW R X T E (70,450)F R E Q (I),A P P R E S (X ),P B A S E (X )CONTINUE

E L S E !F ( CBOOSE. EQ. 0 ) TBEN GO TO 20

ENDIF

1 0 0 CONTINUE

98

CLOSE(SO)CLO SE(70)Q -0W R IT E (S ,* ) 'O o y o u w ia b t o c o n t i n u e ( l / 0 ) ? i * R E A D (5 ,* )Q

X F (Q .E Q .l) THEN GO TO 10

ELSEGO TO 20

ENDIFDO 9 5 R -l.N R F

ERRORMIN(K)"0 S 8£A R (R )«0 TW IST(K )>0 P B A SE(K )"0 A PPR E S(K )-0 APPRESMAJ( X)" 0 APPRESMIN( E )" 0 P8A5EMAJ( K )-0 PBASEMZN(K)"0

CONTINUE WRITE( 6 , * ) 'PROGRAM CSAMT STOPEO*

GO TO 500WRZTE(• , • ) ' FIL E c a s e s . x y * ALREADY E X IS T S )' W RITE(» Q ' FIL E c a s a c . x y t ALREADY E0AXIST3 ) '

3 0 0 FORMATC ' . 1 F 1 0 . 1 , ' ' . 4 F 1 0 . 2 )3 2 5 FORMATC' FREQ(Hz) A R M A J(ohn) A R M IN (O hn)

/ 'P B M A j( d e g ) PB M IN (dag) ' )4 5 0 FORMATJ* ' , 1 F 1 0 . 1 , ' ' . 2 F 1 0 . 2 )4 7 5 .FORMATC FREQ(Hz ) APRES(ohsW ) PBAS E (d a g ) ' )

5 0 0 END

SUBROUTINE CASEA(APPRESMAJ, APPRESMIN, PBASEMAJ, PBASEMZN, TET, /N R F,FR E Q )

c T h i s s u b r o u t i n e a n a l i z e t h a c a s e o f t cc a ) 2D l n p a d a n c a c b ) No l o c a l E d i s t o r t i o n c c ) S t r i k e tn o w n


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PARAMETER ( P I * 1 . 1 4 1 3 * 2 1 5 4 )PARAMETER ( PKRMFREESPACEM. 0 *PI * 0 .0 0 0 0 0 0 1 )PARAMETER (N * 4 ,N P > 4 ,T IN Y > L .0 E -2 0 ,N F " 2 0 ,N S "2 3 )DIMENSION A ( N P . N P ) , B ( H P ) . r R E Q ( N F ) , C V R R < N F ) , 8 X < N r ) , H Y < N r ) , B E ( N F ) ,

/ s i o 8 X ( N r ) , * i c r f ( N r ) , i i o u z ( N r ) , E X ( N r ) . r r ( N r ) , s x o E x < N f ) , s i a c Y ( N r ) , / P H K N D , P H 2 ( N r ) .P R ] ( N P ) . P B 4 ( N P ) , P B S < N r ) . P f l l ( N r ) , S P f l l ( N T ) , S P H 2 < N F ) , / S P B J ( N F ) » S P H 4 ( N T ) « S P R 9 ( N F ) «J P H I ( N F ) ( R E H X ( N P ) <A Z K R X (N P ) </RERY(NP) ( AIMBY(NP) «RCX(NP) ( AZEX(NP)«REY(NP)<AZEY(NP)«/PB E X (N P)<PR E Y (N P), ANOFRCQ(Nf), APPRESMAJ(NT)»APPRESMIN(NF) , /PHAJCMAJ<NP)»PHASEMtN(NF)>IN0X(N)

REAL SQMODMAJ.SOMODMIN. D INTEGER N .K.NRF

0a E n t e r t h e n r . o f f r e q u e n c i e s a t s o u n d in g • • • • * .

N R IT E (4 ,* ) ' I n p u t n r . o r f r o q u a n o lo a a t s o u n d ln g i*R E A D (5 »* ) N R F0 .

oa N o to > A c c o r d in g t o t h e P h o e n ix r e p o r t , t h e a m p l i t u d e s o f t h e a l e o t r l o o a n d m a g n e t i c f i e l d s a r e n o r m a l i s e d t o t h e o u r r e n t a e m p lo y e dc I n t h e a u r v e y . T h e r e f o r e , t h e v a l u e s p r o v id e d b y t h e a o a oo b e u s e d d i r e c t l y i n t h e c o m p u ta t io n o f t h e Im p e d a n c e t e n s o r s ,oO .... .c T h e p h a s e o f R t w as c o n s i d e r e d t o b e t e r o . t h u soc P 8 i< J ) - p R t- P U x — PHx— ZmRx/ReHx o r la B x /-R e R xo P U 2(j> «P E y-P H x-> P E y - p a a ( J ) - P H i ( j )o P 0 1 < J )« P U t-P H y » -p a y a - In a y /R e H y o r XmHyZ-ReBy0 P R 4 (J |-PU X -PB Y *PB 3(J ) - P K l ( J \O P R 5(J)»PE X -PH y»> P E X *P H 3 (J)~ P 8 3 ( J )0

CALL REAOA(NP.NRP. FREQ. CURR. OX. SZGBX. BY. SXGBY. BE»SIG B Z./E X .S IG E X .E Y . S Z G E Y .P H l.S P B l.P R 2 .S P B 2 >PB3.SPB3.PB4<SPH 4 «PHS /.S P H S .P U C .S P B 6 )

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c o e p u t a c l o n o f m a t r i x c o e t f l e n t s a n d v e c t o r B t o r a l l t h e f r e q u e n c i e s

W R iT E < 4 ,M 'I« p u t s t r i k e a n g l e ( i n d e g r e e s ) i ' READ(5 . * )TET

OO 200 J-l.NRFREBX (J)>SQ R T(( (H X ( J ) ) * * 2 ) / ( ( (T A N D (P 8 1 (J )) ) • • 2 ) + l ) ) A 1M R X (J)*R E 8X (J)*T A N 0(P B 1(J))R E R Y (J )* S Q R T ( ( (R Y (J ) )* * 2 ) /( ( (T A N D (P 8 3 (J )) ) * * 2 ) + l ) ) A IM B Y (J)*R C 8Y (J) *TAND(PB3{J) )R E X (J ) -S 0 R T ( (E X ( J ) » « 2 ) / ( ( ( T A N D { P H 5 < J ) - P B 3 ( J ) ) ) « « 2 ) f l ) ) AIEX( J)» (T A N O (P B S ( J)**PB3( J ) ) ) *R EX (J)R E Y (J )-S Q R T ((E Y (J )• • ! ) / ( ( (TAND(PR2( J ) - P 8 1 ( J ) ) ) ” 2 ) + l ) )

■ _ A IE Y (J)* tT A N 0 (P H 2 < J)» P B 1 (J)) ) *REY (J)

IT ((PRl(J).GE.O .ANO. PB1<J).LT.90) .OR./ <PBl<J).QE.-3*0 .AND. PB1(J).LT.-270)} TBENZr (REIIX(J).LT.0) TBEN REBX(J)—REHX(J)ENDZF< Zr (AZKBX(J).GT.O) TBENAIKBX(J)—AIMHX(J)ENDIFEUEIF ( (PB14J) .GE.90 .AND. PB1(J) .LT.180) .OR./ (PBl(J).CE.-270 .AND.PBl(J).LT.-lBO)) TBENZr (REBX(J).GT»0) TBEN REBX(J)»*REHX(J)ENDIFtr <AZMHX(J).GT.O) TBEN AZMBX(J)—AIKHX(J)ENDIFELSEZF ((PBL(J).GE.100 .ANO. PBl(J).LT.270) .OR./ (PB1<J).CE.-180 .AND. PB1(J).LT.-90)) TBEN.90 IP (REHX(J).GT.O) TBEN199 REBX(J)—REHFi *)200 ENOir251 tp (AIMHX(J).LT.0) TBEN202. AIMBX(J)—AIKHX(J).203 ENO ir204 ELSEZF ((PRl(J).LT.O .ANO. PB1(J).CE.-90) .OR.205 / (PR1(J).GE.270 AND. PB1(J).LT.3S0)) TBEN206 tr (REItX(J).LT.O) TBEN207 REBX<J)—REHX(J)208 ENDZF209 tr <AIMHX<J).LT.Q) TBEN210 AIMHX («I) ••AIMHX(J)2M ENDIF212 ENOir2 1 3 o — --------------------------------------------------------------------------------------------------------------------214 c ---------------------------------------------------------. 215 tr ((PB3(J) .CE. 0 .AND. PB3(J) .LT.90) .OR..216 / <PR3(J).CE.-3<0 .AND. PB3(J).LT.-270)) TBEN.217 tf (RCUY<J).LT.0) TBEN210 REBY(J)—REHY(J).219 ENOtr220 tr (AINBY(J),CT.0) TBEN-221. AIMBY(J) —AIMHY(J).222 ENOir223 ELSEZF <<PB3(J).GE.90 .ANO. PB3(J).LT.180) .OR.224 / (PB3(J).CE.~270 .AND.PB3(J).LT.-100)) TBEN2« XT (REHY{3) .GT.O) TBEN226 RERY(J)«-REItY(J)227 ENOtr220 tr (AIMHY(J) ,GT. 0) TBEN229 AZMHY(J)*-AIMHY(J). 230 ENOir231 CLSEir ((PR3(J).GE.180 .AND. PB3(J).LT.270) .OR.2?2 / (PlfJ(J).GE.-1S0 .AND. PR3(J).LT.-90)) TBEN233 XT (REHY(J).GT.O) TBEN234 RERY(J)"»RCHY(J)235 ENOir236 ir (AIMBY(J).LT.O) TBEN237 AtM8Y(J)a*AIM8Y(J)230 ENOir239 ELSEtr ((PRim LT.0 .AND. PR3(J).GE.-90) .OR.240 1 / (PMJ(J).GE.270 .AND. PK3(J).LT.3*0)) THEN


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XT (REHYtJ) . L T .0 ) TBEN JUEtrnJ) — l»EtfT(J)

END I ft r (A X h X Y tJ ).L T .O ) TBEN

AIMRY<J)«-AXMMY(J)ENDZF

ENDtr

PBEX(J)«PH5< J)-P B 3 <J)

t r (<PBEX(J).Q E.O .AND. P H EX(J). L T .90 ) .OR./ (P BEX(J) .C E .-3 I 0 .AND. PBEX(J) . L T . - 2 70 ) ) TBEN

t r (R C X (J ).L T .O ) TBEN REX (J)«-R E X (J)

ENDIFz r (A Z E X (J ).L T .O ) TBEN

AXEX(J)«-AXEX(J)ENOir

C L S E ir ((P B E X (J ).G E .9 0 .ANO. P H E X (J ).L T .180) .OR./ (P U EX(J).G E.-2 7 0 .A N D .P B E X (J ).L T .- IS O )) TBEN

t r (R E X (J).O T .O ) TBEN REX(J)— REX(J)

ENDZFt r (A X E X (J ).L T .O ) TBEN

AX E X (J)»A X E X <J)ENDIF

ELSEXr ( (P B E X (J ). OC. 180 .ANO. P B E X (J ).L T .270) .OR./ (P H EX(J).G E.-1 8 0 .AND. P B E X (J). L T . - 9 0 ) ) TBEN

ZF (R E X (J).G T .O ) TBEN R E X (J)— REX(J)

ENDZFt r (A X E X (J).O T .O ) TBEN

AXEX(J)«~AXEX(J)ENDIF

ELSEXr ( (P R E X (J). LT.O .AND. P H E X (J ).G E .-90 ) .OR./ (P B E X (J).G E .2 7 0 .AND. P B E X (J). I T . 360 ) ) TBEN

X r (R E X (J ) . IT .O ) THEN R E X (J)— REX(J)

ENDIFX r (A X E X (J).G T .O ) TBEN

A IE X (J )— A IE X (J )ENDXF

ENO ir

X r ( (P B E Y (J ) .G E .Q .AND. P B E T (J ) ,L T .9 0 ) .OR ./ < P B E Y (J ) .G e .-3 6 0 .AND. P B E T ( J ) .L T . - 2 7 0 ) ) TBEN

X r (R E Y (J ) .L T .O ) TBEN R E Y (J)« -R E Y (J)

E N D iri r (A X E T (J) .L T .O ) TBEN

A IE Y (J )— A IE T (J )ENDIF

ELSEXF ( (P H E Y (J ) .G E .9 0 .AND. P 8 E Y ( J ) .L T .1 8 0 ) .OR./ (P B E T (J ) .G E .-2 7 0 .A N D .P B E T (J ) .L T .-1 8 0 ) ) TBEN

XT (R E T (J ) .G T .O ) TBEN R E T (J )— R E T (J)

ENDXFIT (A IE T (J ) .L T .O ) TBEN

A IE Y (J )— A IE Y (J)ENDXF

ELSEXF ( (P B E T (J ) .G E .1 8 0 .ANO. P B E Y (J ) .L T .2 7 0 ) .OR ./ (P B E Y (J ) .G E .-1 8 0 .AND. P B E T ( J ) . I T . - 9 0 ) ) TBEN

X r (R E Y (J ) .G T .O ) TBEN R E Y (J)— R E Y (J)

ENDXFXF (A IE V (J ) .G T .O ) TBEN

AX EY (J)— AXET(J)ENDXF

ELSEXF ( (P B E T (J ) .L T .O .AND. P B E Y (J ) .G E .-9 0 ) .O R ./ (P B E Y (J ) .G E .2 7 0 .AND. P B E T ( J ) .L T .3 S 0 ) ) TBEN

XF (R E T (J ) .L T .O ) TBEN R E V (J)— R E Y (J)

ENDXFX r (A X E T (J).G T .O ) TBEN

A IE Y (J)*—A IE Y (J)ENDXF

p f o x r

M l , l ) « ( tS lN D tT E T ) ) * tC 0 S D ( T E T ) ) * R E B X ( J ) ) 4 / ( ( (C O SD (TET)) * * 2 ) *R E B Y (J))

A < 1 ,2 )» -(((S X N D (T E T ))* (C 0 S D (T E T ))« A IM B X (J))+/ ( < (C O SD (TET)) *«2)•A X M B Y (J)) )

A ( l # 3 ) « - ( ( (S X N D (T E T ))* (C O S D (T E T ))*R E B X (J)) •/ ( ( (SXND(TET)) * * 2 ) * R E B T (J)) )

A < 1 ,4 ) • ( ( (SX N D (TET)) *(C O SD (TET)) *AXKBX(J) ) - / ( ( (SX N D (TET))**2)*A X M H Y (J)) )

B ( j)» R E X (J ) _______________________ .

A ( 2 .1 ) — A ( l , 2 )A ( 2 , 2 ) - A ( l . l )A ( 2 , 3 ) * - A ( l , 4 )A ( 2 .4 ) - A ( 1 . 3 )

5i 2il£ £ 5* i i lA (3 . ! ) • - ( { (SZN D (TET)) * (CO SD (TET)) • R E B T (J ) )*

/ ( ( (S Z N D (T E T ))* * 2 )* R E B X (J)) )A (3 » 2 )* ((S X N D (T E T )) * (CO SD (TET)) *AXMVY(3))♦

/ ( ( (SZ N D (T E T ))**2)*A X M B X (J))A<3>! ) ■ ( (S IN D (T B T )) * (C O S D (T E T ))* R E B T (J)) "

/ ( { (C O SD (TET)) * * 2 ) *R E B X (J))A < 3 ,4 ) — (((S Z N D (T E T ))*(C O S D (T E T ))«A X M B Y (J))>

/ ( ( (C O SD (TET)) * * 2 ) *A IM B X (J)) ) 2i 2J £ S £ ii i___

A < 4 ,1 )— A ( 3 .2 )A( 4 . 2 ) " A ( 3 . 1 )A < 4 .3 )— A ( 3 .4 )A (4 « 4 )« A (3 . 3 )B (4 )» A IE Y (J )

O pto t i l t Cor t h t l o l u t l o o a t t h i s CSAMT s t a t io n

O P E N (u n tt" l5 . s t a t u s * ' u n k n o w n * , f i l o « . l i t * #/ s c c o s s * 1 sppond1)_________________________ ____________


U N I X ™Text .File Listing

FILE date 9 /2 5 /9 1 7Sgbnospiit ■nwe 8:17:05 pm

LINE $

„ 161 . 362

36) * 364

36S >"366 . .367

166 _ 389 _ 370

371 372>.373 _ J74

375 76

. 77 _ 78

79 J>80 : si " 82 > 63

8485

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36889

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1392^ 9 3

394J 9 S...396

397__398-J99 400 401— 4024033 0 43 0 53 0 6 3 0 7 3 0 6 '40931'?-3 U- M3 t s'J l i3*31*3 2 2 :3 2 3 .“ 4241425

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440"441"442

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446447

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450451

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457.4 5 8

459460

“ 461 3 6 2

463464

' 465 "466467468

“ 469470

'471*472_47)474

“ 475478477478479 ,480 !

Q P C N < u a i t* l t ,a t4 tu a * 'u a k o o w a ' » { l l e * 'e a a e a . d a t ' , / a c o e a a - 'a p p e a d ' )

HR I T E ( 1 9 , ’ ) ' S t r i k e A n g le * ',T E TWRITC(15, *}' Frequency <lfi H k ) - ‘ ,niEQ(J)WR r TB ( 1 9 , • | • ----------------- — -------------------------------------------------------------------------- -WR1TE< 1 9 . • ) ' t a p e d a o c e T e a a o r C o a p o o e a ta ( R e a » la a ,R e b ,e o d l a b ) '

CALL LUDCMP(A«N,NP,INDX.O) CALL LUBK5B(A,N,NP,INOX,0)

A N O FR EQ tJ)-2. 0 * P 1 * rR E Q (J)3QM0DMAJ*(B( 1 ) * * 2 )+ (B (2 )* * 2 ) SQ H O O K lN -tB (3 )** l)-M B < 4)««2)APPRESMAJ < J) - < SQMODMAJ) / (ANGrREQ( J) • PERMFREESPACE) APPRE9MIN(J)*fSQMOOMIN)/(ANGrRCQ(J)*PERMrR£E3PACE) PHASCMAJ<J)-ATAN20(D(2),D(1))PHASEHIN( J)*ATAN3D( B( 1 ) , B( 4 ) )

W r i te t h a r e a u l t a t o t b o 'e a a e a . d a t * f i l e (« a w a l l a s o a t h a s e r a a a ( t a a p l )

WRITE < 6 . • ) * — — — — — — — — — — — — — — — — •DO 100 K *l,N WRITE ( 1 3 . ' )D (K )WRITE ( t ,« ) D ( K )

I CONTINUE

WRITE ( « , » ) ' — — ----- >WRITE ( I f ,3 0 0 ) F R E Q ( J ) , APPRESM AJ(J) ,A PPR ESM IN (J) , PHASEMAJ(J) ,

/PU A SEM IN (J)I CONTINUEI FORMAT(‘ ' , 3 r l 2 . 2 , 2 r i 0 . 2 )

CLOSE (1 9 )CLOSE (1 6 )

SUBROUTINE CASES ( APPRES«PEASE, SHEAR, TWIST.ERRORMAX)

a ) 10 l a d u o t l o a s t r u o t u r ab ) l o c a l B d l s t o r t l o a

PARAMETER ( P I* 3 .1 4 1 5 9 2 S 5 4 )PARAMETER <PERM rREESPA C E*4.0*PZ*0.0000001)

. PARAMETER tH P*lS ,N r*20,N *4,N S-23 ,N T R X A L -100,M A X X X *lO O ) DIMENSION X 0(N )<A N G FR EQ (N F),A PPR ES(N F),PB A SE(N r),

/SHEAR( N F ), TWIST( NF) , ERROR( NP)REAL A ,D ,C .D .SQ M 0D ,aTTP2,A PPL.A FP2,ER R l.ER R 2,C X X TER X 02,

/PHAl»PUA2,3IBl«SBE2,TW ll,TNX2,XM »CRXTERXOl<AVAPL,AVAP2 INTEGER NRF,Q«5TN.INDEX,XN0XCAL,1N0XCA2 CHARACTER*50 STR COMMON /T E S T 1 / X(NP)COMMON /T E S T 2 / FREQ(NF),NRF COMMON /T E S T ) / JCOMMON /T E S T 4 / R X (N F ),8 Y (N F ),E X (N F ),E Y (N F ),P B 1 (N F ) ,P H 2 ( N r ) ,

/P H 3 (N F ),P H 4 (N F ),P H 5 (N F )COMMON /T E S T 1 0 / ITS.ERRF.ERRX COMMON /F IE L D / ETYP2 COMMON /A L L / STN

c E n ta r t h a a r . o f f r e q u e n c i e s a t s o u a d lo g

K R IT E (6 , * ) ' I n p u t t h a n r . o f C r a q u a n c la a a t a t a t i o m ' READ(S.*)NRF

R aad l a t h a d a t a

CALL READS

OPEN(UNIT*23,STATUS*'UNKNOWN1 , F X L E - 'c a s e b .d a t ‘ ,/ACCESS*'APPEND*)

OPEN(UNIT-24,STATUS-'UNKNOWN*, F I L E - 'b i g a o l . X T * • ,/A C C E SS-'A PPE N D ')

OPEN(UNIT-13,STATUS*'UNKNOWN', FX LE*' s a a a o l . x r t ' ,/A C C E SS-'A PPE N D ')

WRITE(2 4 , * ) .............................. B O .1 ' ,STNW R I T E ( 2 3 , ................. .. O O .1 • ,STN

c C o a p u ta t h a 10 t a p e d a a c e t a a a o r f o r a a c h f r e q u e n c y

DO 6 0 0 J - l .N R F 0 -0

T ry a l l p o a a t b l e I n i t i a l g u e a a e s ( x l , x 2 , x J , x 4 ) a u c h t h a t x l l a l a t h a i n t e r v a l ( - 1 , 1 ) a o d t a k a t h a d l a c r a t a v a l u a a x i - x l + 0 . 3

INDEX-0 INOICKI-O IN D ICA 2-0

E R R l-0 A P P I*0 PHA1-0 T W ll-0 S S E l-0 A V A Pl-0

ERR2-0 A PP2-0 PHA2-0 T N I2 -0 S R E 2-0 AVAP2-0

DO 100 A— 1 ,1 ___ X0( t )-A


- 1 • '■ -- - *■ . ' 4U\V«Text File. Listing* ‘ gbnosplit

O A Tt 9 /25 /91

8:17:05 pm79

LINE «481

“ 482483

"484‘■’ASS “ "486 “ "'487

488 3 8 9 —*90

'4913 9 2 :3 9 3

494“ “495“ 496“4973 « .499

-3 0 0 " 501.30215031504 “505.5061507:508509 “510.:sn:1512 “5| 3. 1514*3 is;15165171 5 1 83 * 9 '520“521152215231 5 2 41525 32652732813 2 9 -530t 33i;

3 5 3 2 ?3 5 , 3 3 .

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=?&3545*5461547;35483549

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r i n3S753763775 7 8579 580.

3 8 1582583584

3 5 8 5__S863S 873 8 83 6 9590591392,3 9 3 . -5 9 4595"596

597598599600

DO 110 0 — 1 .1 X 0(2 )*B

DO 120 C— l . l X 0 (3 ) -C

DO U O D— 1 .1 X 0 < 4 )-0

INDEX-INDEXU

WRITE ( 6 , * ) 'PROCESSING FREQUENCY)1 .F R E Q (J) WRITE ( 6 , « ) ♦ ATTEMPT N O .1 '.IN D EX

DO 3 5 I - l . N X < X )-X 0<I)

CONTINUE

CALL DAMPED(N.TOLX.TOir.NTRIAL)

XF ( IT S , EQ. 3 0 ) TBENIF ( INDEX. EQ .D T8EN

INDEX-0 ENDIF

GO TO 130 ELSE

N o n a l l a a t l o o i K * s q r t( 1+X 3-X 3) • • q r c ( l+ X 4 * X 4 )

KN *<SQ RT(1+X (3)*X <1)-K N *X (1)X (2)*K N *X (2)X(3)«EN*X(3)X (4)*K N *X (4)

'2 ) )* (3 0 R T (1 + X (4 ) - - 2 ) )

C o a p u ts d i s t o r t i o n p a r a a a t a r s

A N G FR C Q (J)"2.0*PX *FR EQ (J)5QM0Da ( X ( l ) * * 2 ) + ( X ( 2 ) **2)APPRES( J ) a (SQMOD)/(ANGFREQ(J)*PERMFREESPACE) P B A S E (J)a A T A N 2 D (X (2 ),X (l))SBEAR( J ) "ATANO( X ( 3 ) )TN X ST (J)-A TA N D (X (4))ERRO R(J) • < ERRF*1 0 0 )/ETYP2 HTYP2a ( B X ( J ) " 2 ) + ( S Y (J ) " * 2 )

ENDXF

XF (P B A S E (J).G T .9 Q .A N D .P B A S E (J).L E .1 8 0 ) TBEN P B A S E (J)-1 0 0 -P B A S E (J)ENDXF

i r (P B A S E (J)< L T .~ 9 0 .A N D .P 8 A S E (J).G E .“ 180)THEN P 8 A S E (J )a 180+PB A SE(J)ENDXF

W r i t* a l l s o l u t i o n s

WRITE (25 ,800)X N O E X .F R E Q < J)aA P P R E S (J),E R R O R < J),P B A S E (J)a /S B E A R (J ) .T W IS T (J )

! D i v i d e s o l u t i o n s i n i t s ( o o r a a l l y ) tw o t y p o s l o o p i n g ttao o n o s ! v i t b a i n i a u a o r r o r .

XF(INDEX.EQ.1)TBEN ERRl-ER R O R (J)A P P l-A P P R E S (J)SBE1*S8EAR<J)T W ll-TW X ST(J)PBA1>PHA5E(J)A V A Pl-A PPRES(J)XNDICAl* INDEX

GO TO 130ENOXF

IF<A V A P1.LT.A PPRES(J))TBEN CRXTERX01a AVAPl/APPRES( J )

ELSECRITERX01a APPRE5(J)/A V A Pl

ENOXF

XF(CRXTERXOl.GT.0.8S)TBENAVAPla <AVAPL+APPRES<J))/2

IF tE R R O R (J) .LT.ERR1)TBEN A P P la A PPRES(J)ERRl-ERRO R(J)SB E l-SB E A R (J)TWI1-TWXST<J)PBA1*PHASE(J)IN 0IC A 1-IN 0EX

GO TO 130ENDIF

ELSEX F(A PP2.E Q .0)T B D f

XF<( ( S B E A R (J)/S B E l) .L T .O ).O R ./ ( (T W IS T (J)/T W X 1). LT.O))TBEN

A PP2"A PPRES(J)ERR2*ERR0R(J)S8E2"SB EA R (J)TW T2-TW IST(J)PHA2<*PBASE( J )AVAP2"APPRE5(J)XNOICA2*INDEX

ENDXFGO TO 130

ENOXFENDXF

X F(A PP2.EQ .0)TB EN GO TO 130

ENDXF

XF(A V A P2.LT .A PPRES(J))T8ENC R IT E R I02a AVAP2/APPRES(J)

ELSECRXTERX02-APPRES<J)/AVAP2

ENOXF

XF(CRXTERX02 .G T . 0 . SS)TBENI F ( ( ( SHEAR( J)/S B E 2 ) .G T .0 ) . ANO.

/ < (T N IS T (J ) /T M I2 ) .G T .0 ) |T B D fKWNP1MKVKP1+NP?1)/1

X F(E R R O R !J)• L T .ERR2)TREN ________ A PP2"A PPRES(J)


6' N IX™: ? x t Fi le Listing. ) gbnosplit

9 /2 5 /9 1

8:17:05 pm soa**.!-.1w ftsw jm m y

LINE #. 001 . 002 00)

004 -005 000 ■ 007

003 _eo9 010 Oil -.012 — 01) -014 -.0 15 __0I0

017 _"0 I8 -0 1 9 020 .02! -022

-0 2 3 -0 2 4 -0 2 5 020 1027 .020 1020 -0 3 0

031 -0 3 2

—.033 034zoos— 030— 037_03B

039_040

— 041— 042— 043

044— 045—040—047— 048— 049—050—051*—052—053— 054—055.HOS 8_ 6S7“ fisa3 5 9nooo_ 66l—002—003304—005 660— 007rooo-009 _670 -071 —072 —073 —074 —075 —070 377 —078 379 080 —081 —082 —083 084 3 8 5 —088 —087 —088 —089 _ 890 391 —092 -093 ~094 -095 090 697

098099 —700

704 t **705 706 -1707 308709710711712713714 .715710717718719 ;720

ENOtr

CRRl-CRRORJJ) SHE2*S8KAR(J) TWI2-TWZST<J) PHA2«PHASE(J) INU1CA2*IN0EX

ENOIF eNoir130 CONTINUE 130 CONTINUE 110 CONTINUE too CONTINUE

I r< A m .GT. A PP2) TIIENMRITE!2 4 « • ) INOICA1. !K E Q <J) , A PP1.PB A 1, CRR1*

/ SHE1, T H U , AVAP1W R IT E (2 3 ,•)lN D IC A 2 .F R E Q tJ),A P P 2 ,P H A 2 ,E R R 2 .

/ s b e 2 . t w i2 , a v a p2ELSE

WRITE( 2 3 , • ) INOICA1. FREQ(J 1«APP1, PBA1, ERRL« / SOE1, TNI1 , AVAP1

WRITE!2 4 , • ) INOICA2, F R E Q (J) , APP2, PBA2, ERR2, / SHE2.TWI2.AVAP2

EN O tr < 00 CONTINUE

5 ............ • • • • • • • • • • • • • • • • • • • • • ............ ..9 B aap t o a n n o u n c a i t f l o l s h a d w i t h t h i s a t a t l o a9

a w - * END OF STATION1 //C H A R ( 7 )

PRINT^ « \ STRWRITE! < , * > • ------ 1WRITE!< • • J W R lT E (< ,»)W R ITE(<,*)

J • • * • • • ........... .. ...............................• 0 0 FORMAT( * ' , 1 4 . * ■ ,3 1 1 2 .1 * ' ' * i r 7 . 4 . ' ‘ , 3 r i 0 . 3 )

CLOSE(23)CLOSE!24}CLOSC(25}RETURNEND

SUBROUTINE CASCC!APPRES, PHASE,NRf, FREQ)

e T h la s u b r o u t i n e a a a l l s a s t h a e a a a o f t

a ) ID l s p e d a a e eb j No l o o a l E d i s t o r t i o n

PARAMETER ( P I - 1 . 1 4 1 5 9 2 6 5 4 )PARAMETER ! PERMFREESPACE-4. 0 * P I • 0 .0 0 0 0 0 0 1 )PARAMETER !N P -4 .N F -2 0 ,N S -2 S )DIMENSION FREQtNF) *CURR!NF) .BX(NT) *SY(NT) ,B Z (N F ) ,

/S IG H S (N F ) .SIG H Y (N F) ,S IC H Z (N F ) ,E X (N D ,EY (N F) ,SZG EX|N F) ,S IG E Y (N F ), /P B 1 (N F )* P H 2 (N ff)«P H 3 !N F ),P B 4 (N F )f P H S (N F )«P H 6(N F )«S P H 1(N F )*3P B 2!N F )* /S P H 3 !N F ) , S P H 4 (N F ),S P H 5{N F ),S P B 6(N F ),R L B X (N P ), AIM H X (N PJ,RESY (N P), /A IM H Y (N P ),R E X (N P ),A IE X (N P ),R E Y (N P ),A IE T (N P ),P H E X (H P ),P H E Y (N P ), /ANGFREQ(NF) .A P PR E S(N F), PHASE(NF)

REAL X .Y «A ,8.C .D .C*F«SQ M 0D INTEGER NRF COMMON /H IT H C / M

o E a t a r t h a o r . o f f r a q u a a c l a a a t s o u n d in g

H R X T E ( 6 f * ) 't a p u t o r . o f t x e q u a n e i a i a t s o u n d i n g s •REAO!5 , * )NRFe

o T h a p h a s * o f H i v a a c o n s i d e r e d t o b a z e r o , t h u s ee P B l( J ) - P H t- P B x — PHx— IaH x/R *H x o r I* H x /-R a H xO P U 2 (J )» P F ‘/-PHX»> P E y -P H 2 ( J ) - P H l( J )e PH 3<J)«PH z-PH y— PHy— IaH y/R aH y o r X a 8 y /-R e B ye P C 4 ! J ) - P B x - F B y P B 3 ! J ) - P B l ( J )c P R 5 (J)« P E X - 'P h y .> P E X -P H 5 (J ) -P H 3 (J )c .

CALL RCADC!NF.NRF.FREO.CURR,HX.SXGHX,BY.SXCBy,HZ»SIGaZ» /E X 'S X C C X ,£ Y ,S IC E Y .P H l,S P B i.P R 2 .5 P H 2 ,P 8 3 ,5 P H 3 ,P H 4 ,S P B 4 ,P B 5 / .S P H 5 ,P H 6 ,S P H 6 )c

c C o a p u ta t h a r e a l a a d i a a g l n a r ; p a r t s o f t h a a l a c t r l c a a d a a u o e t l c c f l a l d a f o r NRF f r a q u a a c l a a .

DO 200 J - l .N R F

R E B X !J ) -S Q R T !! !H X (J ) )* * 2 ) /! ( !T A N D !P B Z !J ) ) ) • * 2 ) * 1 ) )• • taX(J)-REnX(J)«TAND<PHl(J)). <fY(J)-SQRT(! (HY(J))«*2)/(((TAND(PB3(J)))**2)+!}| AIKHY!J)>RCBY!J)*TAND(PB3(J))R E X !J)BS Q R T ((E X (J)* * 2 ) / { ( ( T A N D ( P B S (J ) -P B 3 (J ) ) )* * 2 )+ l ) ) A IE X (J ) - (T A N D (P H 5 (J ) -P H 3 (J ) ) )» R E X (J )REV{J )* S Q R T ((E Y (J ) " * 2 ) / ( ( (T A N D (P B 2 !J ) -P B 1 !J ) ) ) • • 2 ) + l ) )

_____________ T o » t_ f o r _ t h a _ a l q o .o t _ t h e _ B a a o d l a c o e p o o a o t s _

I F ! ! P B l ( J ) . e C . O .AND. P B t(J ) . L T .9 0 ) .OR./ ( P H l ( J ) .C E . - ) 6 0 .AND. P H l ( J ) . L T .- 2 7 0 ) ) TBEN

IT (R E B X (J) . LT.O ) TBEN R 6M X IJ)— REHXIJ)

ENDIFIF (AXKBXtS) .G T .Q ) TBEN

AIMHX(4) — AIMBX(J)ENDIF

CLSEIF < (P B 1 (J ) .G E .9 0 .ANO. P H 1 ( J ) .L T .1 8 0 ) .OR ./ ( P B 1 ( J ) .C E .- 2 7 0 .A N D .P B 1 (J ) .L T .- 1 8 0 ) ) THEN

IF (R E H X !J ) .C T .0 ) TBEN R C R X !J)— REHX(J)

EN O tr______________ IT |A IM H X tJ) .G T .Q ) THEN_____________


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A IH H X (J)— AIMHX(J)ENDIF

ELSEXF ( (P R 1 < J ) .G E .1 8 0 .ANO. P B l ( J ) .L T .2 7 0 ) .OR./ ( P H l ( J ) .C E . - U O .AND. P B l ( J ) . L T . - 9 0 ) ) TBEN

XT (R E H X (J).G T .O ) TBEN REHX(J) — REHX(J)

ENOXFX r (A IM B X (J). LT .O ) TBEN

AXK8X( J ) •-AIM HX(J )ENOXF

ELSEXF ( ( P H 1 ( J ) . L T .0 .AND. P B l ( J ) . G B .- 9 0 ) .OR ./ (P B L (J ) .G E .2 7 0 .AND. P B L (2 > .L T .3 6 0 ) ) THEN

I F (R E H X (J).L T .O ) TBEN .R B H X (J)«-R EH X (J)

ENDXFXF (AXMHX(J). L T .0 ) TBEN

A 1M BX(J)>-AIM BX(J)ENDXF

ENDXF__________________________________________________________

XF ( ( P B 3 (J ) .G E .O .AND. P B 3 < J ) .L T .9 0 ) .OR./ ( P B 3 ( J ) .G E . - 3 6 0 .AND. P H 3 < J ) .L T . - 2 7 0 ) ) TBEN

I F (R C H Y (J).L T .O ) TBEN R E B Y (J)— REKY(J)

ENDXFir (A IM H Y (J).G T .O ) TBEN A IM H Y (J)--M M H Y (J)

ENDXFELSEXF ( ( P B 3 ( J ) .G E .9 0 .AND. P B 3 ( J ) .L T .1 8 0 ) .OR.

/ ( P B 3 ( J ) .G E . - 2 7 0 .A N D .P H 3 (J ) .L T .-1 8 0 ] ) TBENXF (R E I(Y (J).C T .O ) TBEN

R E B Y (J)— REtlY (J)ENDIFXF (A X M FY (J).GT .O ) THEN

A IM H Y (J)— AIMHY(J)ENDXF

ELSEXF ( < P B 3 (J ) .G E .X 8 0 .AND. P B 3 < J ) .L T .2 7 0 ) .OR./ ( P B 3 ( J ) .G E . - 1 8 0 .AND. P B 3 < J ) .L T . - 9 0 ) ) TBEN

XT (R E tfY (J).G T .O ) TBENr eiiy( J ) — rehv w i

ENDXFXF (A X M H Y (J).LT.O ) TBEN

AXMBY (J ) — AXMBY < J )ENOIF

ELSEXF ( ( P B 3 ( J ) .L T .O .AND. P B 3 ( J ) .G E .- 9 0 ) .O R ./ ( P B 3 ( J ) .G E .2 7 0 .AND. P B 3 ( J ) .L T .3 6 0 ) ) TBEN

XF (R E H Y (J).L T .O ) TBEN R EH Y (J)— REKY(J)

ENDXFXF <A X K S¥<3).LY .O ) TBEN

A IK B Y (J)— AIMHY(J)ENOXF

ENDXF

P B E X < J)> P B 5 < J)-P B 3 (J)P B E Y (J ) -P B 2 (J )> P B 1 (J )

XF t( P H E X ( J ) .G £ .0 .AND. P B E B t? ) .L T .9 0 ) .OR./ (P B E X (J ) .G E .-3 6 0 .AND. P B E X ( J ) .L T .- 2 7 0 ) ) TBEN

XF (R E X (J ) .L T .O ) TBEN R E X (J)— R E X (J)

ENDXFXF (X X E X (J).L T .O ) THEN

A IE X (J) — AXEX(J)ENDXF

ELSEXF ( (P B E X (J ) .G E .9 0 .AND. P B E X (J ) .L T .l t 'T ) .OR./ (P H E X (J ) .G E .-2 7 0 .A N D .P B E X (J ) .L T .-1 8 0 )) TBEN

I F (R E X (J ) .G T .O ) TBEN R E X (J)— R E X (J)

ENDXFXF (A X E X (J).L T .O ) TBEN

A IE X (J) — AXEX(J)ENDXF

ELSEXF ( (P B E X (J ) .G E .IS O .AND. P B E X (J ) .L T .2 7 0 ) .OR./ (P H E X (J ) .G E .-1 8 0 .AND. P B E X ( J ) .L T .- 9 0 ) ) TBEN

XF (R E X (J ) .G T .0 ) TBEN R E X (J)— R E X (J)

ENDXFXF (A X E X (J).G T .O ) TBEN

A IE X (J)» -A X E X (J)ENDIF

ELSEXF ( (P B E X (J ) .L T .0 .AND. P B E X (J ) .G E .-9 0 ) .OR./ ( P 8 E X (J ,.G E .2 7 0 .AND. P B E X < J ) .L T .3 6 0 ) ) TBEN

i r 'R E X (J) .L T T O ) TBEN R E X (J)— R EX (J)

ENDIFI F (A IE X (J ) .G T .O ) TBEN

A IE X (J )— AXEX(J)ENDXF

ENDIF ________ _________________

X r ( (P B C Y (J ) .G E .O .AND. P B E Y (J ) .L T .9 0 ) .OR./ (P B E Y (J ) .G E .-3 6 0 .AND. P B E Y ( J ) .L T .- 2 7 0 ) ) TBEN

I f (R E Y (J ) .L T .O ) TBEN R E Y (J )" -R E Y (J )

ENDXFI F (A IE Y (J ) .L T .O ) TBEN

AX EY (J)— A IE T (J )ENOIF

ELSEXF ( (P H E Y (J ) .G E .9 0 .AND. P B E Y (J ) .L T .1 8 0 ) .OR./ (P B E Y (J ) .G E .-2 7 0 .A N D .P B E Y (J ) .L T .-1 B 0 )) TBEN

XF C R E Y (J).G T .O ) TBEN R C Y (J) — R EY (J)

ENDIFXF (A X E Y (J).L T .O ) TBEN

M E Y ( J ) — A IE Y (J)ENDXT

ELSEXF { (P H E Y (J ) . C E .1 8 0 .AND. P B E Y (J ) .L T .2 7 0 ) .OR./ (P H E Y (J ) .G E .-1 8 0 .AND. P B E Y ( J ) .L T .- 9 0 ) ) TBEN

i r (R E Y (J ) .C T .O ) TBEN R E Y (J)— REY<J»

ENDXFI F (A X E Y (J) .G T .O ) TBEN

A IE Y (J ) " -A IE Y (J )ENDIF

E L SE IF t tPHEVf T > . L T .0 .AND. P H E Y tJ ) .0 E .- 9 0 ) .OR.


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LINE * TEXT/ ' ( P U E Y < J ) . G E . 2 7 0 .ANO. PMBY (<3 J . L t 7 j « 0 ) ) ^THEN

I F ( R E Y ( J ) . L T . O } TBEN R E Y ( J ) — R E Y ( J )

EN DI FI T ( A I E Y ( J ) . G T . O ) TBCN

a x ey ( J ) — a i e y ( J ) e n d it

E N O t r

C o a p U t« tlO B o t X I SOd X2 trom

a - ( i / 2 ) ( ( C x / H y M E y / a x ) l

XI s o d X2 a r e

A - ( A I E X ( J ) » R E H Y < J ) ) - ( * E X ( J ) « A I M H Y ( J ) )B - ( R E B X ( J ) * A I E Y ( J ) ) - ( R E Y ( J ) * A 1 M B X ( J ) ) C - ( ( X C B Y ( J ) ) « 0 ) + ( { A X W n r ( J ) ) * ' 2 J 0 s ( ( R E U X ( J ) ) • • 2 } + ( ( A I M B X ( J ) ) * * 2 )E - < R E X { J ) * H E H T ( J ) ) * ( A t E X ( J ) » A I M H Y < J ) )

_ . r ? ( R C Y ( j i « B E H X ( J ) i * ( A I E Y ( j i ? R I M H X ( J ) ) _ in - - ______ -

X " 0 . s * ( ( ( c * 0 ) - t r * c j j / ( c * o j >____________________A N G F R E Q (J)-2 .0 * P I« rR £ Q (J)S Q M 0 D - (X * * 2 ) + (Y* *2 )AFPR£S(J)*(SQNOD)/(ANGFREQ(J)*PERM rREE5PACE)P U A S E t J )» A T A N 2 D ( Y , X )

O pen f i l e t o r th o s o l u t i o n a t t h i s CSAMT s t a t i o n

OPEN( un i t * 1 5 , s t a t u s * 1unk n o w n ' , f i l s - ' e a s o o . l i s ' , a c c e s s - 1a p p e n d * ) O P E N < u o l t* l£ ,s t a tu s - 'u n k n o w n * , f i l e * ' c a s e o . d a t * , a c c e s s - ‘ a p p e n d * ) W R iT E (1 5 ,* )* ta p e d a n c e T o o s o r a t f r e q u e n c y ( i n B t)* * ,F R E Q (J )

W r i t* t h e r e s u l t s t o t h o ' c a s e o . d a t * f i l e ( a s v a i l a s o n t h o s o r t o n ( t o a p l )

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WRITE ( 1 3 , • ) * — ------- --------------------------------------------WRITE ( € , « ) ' --------- ---------------------------------------------WRITE ( 1 5 , • )X ,Y WRITE ( 6 , * )X»YWRITE {I S , " ) 1----------------------------------------- --------------WRITE ( 6 , « ) 1------------------------------------------------------- --WRITE ( 1 6 , 3 0 0 )F R E Q (J ) ,A P P R £ 5 (J ) ,P B A S E { J ) CONTINUErORMATC ' ,2 F 1 2 .2 , 1 F 1 0 .2 )CLOSE ( I S )CLOSE (1C )

SUBROUTINE OAMPED(N,TOLX,TOLF,NTRIAL)

• I t r e f u s e t o a c c e p t t h o n o x t N ew ton i t e r a t e

X( NTRI AL+1 ) -X (NTRIAL)4 c o r z o x t l o n o f X

i f i t l o a d s t o a n i n c r o a s o i n t h o r e s i d u a l e r r o r , i . e . i f

| r < x a * u l > | r ( i n | | . . . o

I n s u c h a c a s e , i t l o o k s a t t h e r e c t o r s X n + ( h /2 * * l ) f o r 1 * 1 , 2 , . . . , a n d t a V e s X o+ l t o b e t h e f i r s t s u c h r e c t o r f o r w h ic h t h e r e s i d u a l e r r o r i s l e s s t h a n | r ( X o ) | .

PARAMETER <N P-15 ,M P -15 ,JM A X -30)DIMENSION X (N P ),A (N P ,N P ),B (N P ),D E L T A X (N P ), FX(JMAX),F(JMAX)

/ ,B ( N P ) ,X C ( N P ) , U (N P ,N P ), W (N P),V (N P,N P)REAL HX.ETYP2 INTEGER Q .COMMON /T E S T 1 / X(NP)COMMON /T E S T 1 0 / IT S , ERRr.ERRXCOMMON /H E L D / ETYP2T O L F-.0 1T O L X -.00001N -4M-4

S t a r t t h e I t e r a t i o n s

DO 12 X -l.N T R IA L

c a l c u l a t i o n o f Xo+1

CALL USRFU N (X ,A ,0,ETY P2)

ERRF-0 DO 11 I - l . N

E R R F-E R R F+A B 5(B (I))CONTINUE

I F (E R R F.L E .(T O L F*£T Y P2)) RETURN

C c o e p u te I p (X ts) i u s l a q Xn ( f r o a l a s t a t t e a p t w i t h USRFUN.f)C .................

F ( K ) - S Q R T ( ( ( B ( l ) ) * * 2 ) + ( ( B ( 2 ) ) * * 2 ) + ( ( 8 ( 2 ) ) * * 2 ) - ( ( 8 ( 4 ) ) * * 2 ) )

C s o l r e a l i n e a r s y s t e a o f e q u a t i o n s f o r X s u c h t h a t ( * ) h o ld sC

DO 25 t ' l . N DO 24 J * l , N

U ( I , J ) - A ( I , J )24 CONTINUE25 CONTINUE

CALL SVDCM P(U,N.N,NP,NP,W ,V)IF ( IT S .C Q .2 0 )G O TO 26 NMAX-0

O O IO T .J .N


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s t o p ' r x i i m E : . n o x x r c a r x k x s p o o c d r a x ? x s s o c s s r s s e r w m i *

13 COS XXSCEK l I T t ( » ,« r « X * N * lN C ! '.X l l TEE ITERATIONS K O tl BSED1 * * R IT E (* ,-> * _________________________________________________ •nxar.M'_______________________ *KRX7X(• . • j ’ Do t o u v » n t t o t r y s o r * i t « r « t i o a * ( 1 / 0 ) v R E A D (S .*)C 1 7 ( 0 .6 7 . 3 ) SEEN

CO 7 0 9 c tD x r

3* RETURNBfD

SU BRO U TIN E L U B R S B (A ,N .N P ,X K D X .B )

DIMENSION A (N P.N P),X N D X (N ),B (N ) I t - 9DO 13 X -l.M

LL-XNDX(X)SUM-B(LL)B ( L L ) - 8 ( I )x r ( i i .n e .O )then

DO 11 J - I I . I - 1SDM-5UN-A(X. J ) * B ( J )

CONTINUE ELSE XF (SU M .N E .O .) TBEN

X X -I ENDXF B (I)-S U M

CONTINUE DO 14 I - N . l . - l

S U M -B (I)I F ( t .W .N )T B E N

DO 13 J - I + l . N5UM*SUM-A(X. J ) * B ( J )

CONTINUE ENDXFB(X)-SUM/A(X,X)

CONTINUERETURNEND

11

SUBROUTINE LUDCMP(A.N.NP,INOX,D)

PARAMETER (N M A X -4.T 1N Y -1.0E -20)DIMENSION A (N P .N P ), IN D X (N ). W(NMAX)D - l.DO 12 I - l . N

AAMAX-0.DO 1 1 J - l . N

i r ( A B S ( A ( I .J ) ) .CT.AAMAX) AAM AX-ABS(A(Z,J)) CONTINUEXF (AAMAX.EQ.O.) PAUSE 'S l n g u l i r « « t r t x . •W ( I)-I ./A A M A X

CONTINUE DO 19 J - L .N

I T ( J . C T . l ) THEN DO 14 I - L . J - l

S U M -A d .J )______ IF (X.GT.l)THEN


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00 13 K-l.1-1SU H "SU M -A (I,K )*A <*..7)

CONTINUEA (I ,J )* S U H

E N O trCONTINUEENOir

AAMAX-Q.0 0 16 I - J . N

SU H "A {I, J ) ir (J .G T .l)T B E N

0 0 L5 E - l . J - l 'SUM"SUM-A(I, K )• A (K ,J )

CONTINUE A ( I ,J ) -S U H

E N O trDUM"VV(X) *AD3(3UM)Z f (OUM.CE.AAMAX) TOEN

IMAX-I AAMAX"0UM

ENOIF CONTINUEIF (J.NE.IM AX)TBEN

DO 17 K"1»N DUM"A(IMAX,X) A (IM A X ,X )"A (J 'X )A (J,K)"DUM

CONTINUE D*-DW (IM A X )" W (J )ENOir

INDX(J)"IM AXX F(J.N E.N )TBEN

( F (A (J ,J 1 .E Q .0 . ) A ( J ,3 ) " T X N Y D U M " l./A (J ,J )0 0 18 I " J + 1 ,N

A (1 ,J )" A (X ,J )* 0 U MCONTINUE

ENOXFCONTINUEX ftA (N »N ).E Q .O .)A (N «N )"T X N YRETURNEND

SUBROUTINE READA<NF,NRr,FREQ,CURR,HX,SIGHX,BY,SIGHY#HZ,SIGHZ,EX, /SXGEX, BY »SIGEY, P H I,3 P B 1 , PB S, S P S S . PB S, SPH 3, P 8 4 , 3P B 4 ,P B S ,S P H 5 , PB 6, /S P Q fi)

S u b r o u t i n e t o r a i d l a t h e H o l d d a t a C or a c s a a t s t a t i o n a t t b a d iC C a r a o t a o a a u r a d f r e q u e n c i e s ( u s u a l l y 14 )

EXPLANATION

N R r*ouB b«r o f f r e q u e o o l e s F R E Q « a o tu a l f r e q u e n c i e s ( l a Hz)C U R R -a p p lle d o u r r a n t t o t b a g r o u n d ( A s p ) . O sa d t o n o r a a l l z a t b a

a m p l i t u d e o t t h a E l a o d H I t l a l d a ( P h o e n ix d a t a l a a l r a a d y n o r m a l i z e d )

H i - m a g n i tu d e o f t b a 1 t h e o a p o o a n t o f t h a m a g n e t i c f l a l dE l* a a g n l t u d a o f t h a 1 t h o o a p o n a o t o f t h o a l a e t r l c f l a l d

3XGB1" s t a n d a r d d e v i a t i o n o f t h o a a g n a t l c f l a l d ‘ I 1 e o a p o o a n t 3XGE1" ■ “ * a l a e t r l c • • •

P H I" p b a a a 8 z -B x ( I n d a g r a a a )PH 2-PB3"PH4"PBS"PBS"

E y-B x H z -a y B x -a y E x -a y B y-E x

(PB3 l a a y o o t a s )

(PBS l a a y n o t a s ) (P H I l a a y n o t a s )

S PB o" c o r r e s p o n d i n g a t a r d a r d d e v i a t i o n o f t h o p h a s e a n g l e s

D e f i n i t i o n s

CHARACTER*20 FXLENAMREAL FREQ(NF),CURR(NT), H X (N T)«SIG B X (N F),B Y (N F),SX G H Y (N F),B Z(N F ),

+ S X G 8Z (N F).E X (N F)«SX G E X (N F)»E Y (N F),SX G E Y (N F),PH 1(N F).SPH l(N F), + P B 3(N F )»5P B 3(N F )«P B J(N F )«3P H S (N F )«P B 4(N F )*S P H 4(N P ) ,♦ P H 3 (N F ),S P B 3 (N F ),P B 6 (N F ),S P B 6 (N F |

N R IT E (6«• ) * ENTER NAME OF TBE DATA F I L E i1 R E A D (S ,3) FXLENAM FORMAT( AS0 )N"XTRMU*{ FXLENAM)OPEN(UNXT*7, FILE"FX LEN A M (liN )> STATUS-'OLD*) c h a r a c t e r * 20 f l l a a a a

lo o p t o r e a d d a t a - f o r NRF f r e q u e n c i e s

READ(?,M rREQ<J),CURR(J),aX{-7),SXGaX(J),HY(J),5XG&y(J), +BZ(J),3ICB2(J),EX(J)«5ICEX(J)«EY(J)»31GEY(J)«PR1(J)»3PB1(J)f +PHS(J),3PH3(J),PB3(J)f3PH3(J)«PH4(J)«3PB4(J)«PB3(J)t +5PRS(J)rPRS(J)f3PQ6(J)w r i t e d a t a o n t h a s c r e e n

N R Z T E (6 ,«) rR E Q (J ) |C U R R (J ) .H X (J ) ,S X G B X (J ) , & Y (J )* S IG S Y (J )> * R Z ( J ) .S I G B Z < J ) ,C X ( J ) ,S I G E X ( J ) ,E Y ( J ) ,S I G E Y ( J ) ,P B 1 ( J ) ,S P B 1 ( J ) , + P B 2 ( J ) ,5 P H 2 ( J ) ,P H 3 ( J ) ,3 P B 3 ( J ) ,P B 4 ( J ) ,S P H 4 ( J ) ,P H 5 ( J ) , * S P U 5 ( J ) ,P H 4 ( J ) .S P a 4 ( J )

CONTINUECLOSE(7)END

FUNCTION ZTRMLN (STRING)

T h i s f u n c t i o n r e t u r n s t h a l e n g t h o f a c b a r a c t o r s t r i n g w i t h a l l t r a i l i n g b l a n k s r e e o v e d . I t r e a d s t h a c h a r a c t e r s t r i n g b a c k w a rd s u n t i l a n o n - b l a n k c h a r a c t e r La e n c o u n te r e d *

0»TE 9 / 2 5 / 9 1

TIME 8 :1 7 :0 S p m


gbnosplit9 / 2 5 / 9 1

t|mc 8 :17 :05 pmS5

LINE # TEXT1201120212031204 J20S1206 1207 1206

Z\ 209 J2I0'i2 i r '1212 :I2I3 1214 1215' J'216 ri2t? 1216 1219 11220 1221 1222 1223 J.224, 3225 I I226 1227 J.226 322912301231 ;i232 1233'_J2B4.11235J236237_236239240 241“ .242 .243244245246247 248" 249. 250253254 .255 256 257. .258 .259 -260_ 261 .262263264265266267268 269 '270.. 271 272' 273. .274275276277278279280 281 282'„283. 3 284 285 1286 1287. 3-288. 3289 1290 J29i: 3 29212931294129512961297 3 2981299.

t o o o .3 301. 3.302.33033304 “1305’ 3*306"1307 “1308 “13091310 ,1311 1312* 1313J314;D3I51316'"*13171318'“1319"1320

CHARACTER*)*) STRING INTEGER L i I

CBECR TBS LENGTB OT THE CHARACTER STRING

. - LEN(STRING)

I f ( L .L E .O ) TBEN 1T T M L N -0 RETURN

ENO I F

X- IIT (ST R X N G (X iI) .NE. 1*1-1.GO TO 10

CON TIN U E x r ( X .G T .O ) TBEN

ITRMLN - I E L SE

ITRMLN • 0 END r r RETURN

1 ) GO TO 20

SUBROUTINE SV B K SB {U ,W ,V ,M ,N ,M P,N P,S ,X )

PARAMETER (NMAX-100)DIMENSION U (M P ,N P ),H (N P ),V (N P 'N P ),8 (M P ),X (H P ),T M P (N M A X ) 0 0 12 J - l . N

S - 0 .X F (K (J) .N E .O .)T B E N

DO 1 1 I - l . NS -S + U (X ,J )* B (X )

CONTINUES - S /W (J )

ENDIFW P ( J ) - S

CONTINUEDO 14 J - l . N •

S -O .DO 13 J J - l . N

S « S + V (J ,M )*T M P (J J )CONTINUEX ( J ) - S

CONTINUERETURNEND

SUBROUTINE SVDCMP ( A,H ,N ,M P,N P,W ,V)

PARAMETER (NMAX-100)DIMENSION A (M P,N P),M (N P)/V (N P,N P),R V 1(N M A X ) COMMON /T E S T 1 0 / IT S G - 0 .0 S C A L E -0 .0 ANORM-O.O DO 25 I - l . N

L - I + lRYL(I)«SCALE*G G - 0 .0 S -O .0 S C A L E -0 .0 i r (X .L E .M ) TBEN

DO 1 1 E-X »MSCALE-SCALE+ABS( A (X , I ) )

CONTINUEXF (S C A L E .N E .0 . 0 ) TBEN

DO 12 R - I .MA ( K , I ) - A ( X , X)/SCALE S -S+A (K »X )*A (K »X )

CONTINUE F - A ( I , 1 )G --S X G N (S Q R T (S ),F )S -F * G -5 A ( X » I ) -F —G I F ( I .N E .N ) TBEN

DO 1 5 J - L .N 5 - 0 . 0DO 13 K -t.M

S -S + A (K ,X )* A (K .J )CONTINUEr - s /aDO 14 K-X,M

A ( K ,J ) - A ( R ,J ) + F * A ( E ,I )CONTINUE

CONTINUEENDIFDO 16 K - I,M

A (X.X)-SCALE*A(K<X)CONTINUE

ENDIFENDXFH (t)-S C A L E *G G - 0 .0 S - 0 . 0 S C A L E -0 .0x r ( ( X .L E .M ) .A N D .( I .N E .N ) ) TBEN

DO 1 7 K -L .NSCALE-SCALE+A8S(A( I , X) )

CONTINUEXF (SCA LE.NE.O ? ) TBEN

DO 18 X -L .NA (X ,E )-A (X .X )/S C A L E S -S + A ( I .X ) * A ( I ,X )

CONTINUEr - A ( I . L )

_________ G— S IC N (S Q R T (S ).F )_________________________ _


> Text File. UstiniaFILE

gbnosplitBATE 9 /2 5 /91

t im e 8:17:05 pm8 6

LINE 0 TEXT

1121 1322 1)2313241325 1320 1322 1 32013291330 |33»332

133313341335 1330 1332 13301339134013411342 1)4313441345 1340 1342 13401349135013511352 353 J54355356 ;357 350 359300301 362363.304305 366 307 300 309370371372373

m&378379380381382383384 3 0 5 ' 380' 307 380389390 391'392393394 395' 396 39?398399

1400"1401140214031404

*14051400

V407T4Q8'1409*1410*141114121413

*141414151416

*141714101419142014211422 M23142414251426

*14271428U2914J01431143214331434143514361417 1438 14)9 1440

a - r « c - s

00 19 K -L .NM V i( K ) - A ( I ,K ) / l l

19 CONTINUEI T ( I .N E .N ) THEN

0 0 2 ) J -L .M S - 0 . 000 21 K -L.N

S -S « A (J .X )» A ( I .K )21 CONTINUE

00 22 K -L .NA (J .K )-A (J .X )* S * R V 1 (K )

22 CONTINUE22 CONTINUE

CNDir00 24 K -L.N

A (I ,X t-S C A L E » A (I ,K )24 CONTINUE

ENoir CNDirANORM-KAX( ANOHH, ( ABS{W (I) )*A D S(R V t( I ) ) ) )

23 CONTINUEDO 32 I - N .1 , - 1

IT ( I .L T .N ) THEN IT (G .NE.O .O ) THEN

00 26 J -L .NV ( J .X ) - ( A ( X . J ) /A (X < L) )/G

26 CONTINUEDO 29 J -L .N

5 - 0 .000 27 K -L .N

S » S V A (X .K )*V {X .J)27 CONTINUE

00 26 K -L .NV (K ,J ) -V (K ,J )+ S * V (K .X )

28 CONTINUE29 CONTINUE

ENOtrDO 31 J - L .N

V ( I . J ) - 0 . 0 V ( J . 2 ) - 0 . 0

31 CONTINUE EN D ir v < i , n - i . oC -R V l( I )L - r

32 CONTINUEDO 39 Z - N .1 , - 1

L - I + l G -H (IJx r ( I .L T .N ) TUEN

DO 33 J -L .N A ( I . J ) - 0 . 0

33 CONTINUE E N D irx r (G .NE.O .O ) THEN

C -1 .0 /0x r (X .N E .N ) TBB4

DO 36 J - L .N S - 0 . 0DO 34 K-L.M

S -S * A (K ,X )« A < K ,J)34 CONTINUE

r - ( S /A (X ,X ) ) * G DO 35 K -X .N

A ( K .J ) - A ( K .J ) + F * A ( K .I )39 CONTINUE36 CONTINUE

ENDIFDO 37 J -X .N

A ( J iX ) -A (J .X )* G37 CONTINUE

ELSEDO 30 J - Z .H

A ( J . X ) - 0 . 0 36 CONTINUE

E N D irA ( I , I ) - A ( I «I )+ 1 .0

39 CONTINUEDO 49 K - N . l . - l

DO 46 IT S - 1 .3 0 DO 41 L - K . l . - l

N M -L - lI F ( (AB5(RV1(L)J+ANORM).EO.ANORN) GO TO 2 XT ((AB5(W(NM))+ANOftH).EQ.ANORM) GO TO 1

41 CONTINUE1 C -0 .0

S -L .ODO 43 I - J . .K

F -S » R V l( I )IT ( (AB5(F)+ANORM).NE.ANORM) THEN

G -H ( I )H-SQRT(F*F+G«G)N <X )-«H -1 .0 /H C - (C *« )s— ( r * H )DO 42 J - l . N

Y -A (J .N H )E -A (J .X )A (J ,N N )- (Y * C )+ (£ * S )A ( J . I )— (Y »S) + ( J» C )

42 CONTINUE ENOir

43 CONTINUE2 E-N(R)i r (L .E Q .K ) TEEN

IT (2 .L T .0 .0 ) TBEN »(*) — 2 DO 44 J - l . N

V ( J .K ) — V (J .K )44 CONTINUE

ENOtrGO TO 3

ENDirI f ( I T S .E Q .3 0 ) GO TO SO X-W (L)N K -K -lY-N(NM) ____________________


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1510□ 5 IT □*5T2" 4:SI3:□ SI4□STS J 5 I6 " 4517; □ .S I 8 □ 5 19 □>20 □.S2I 4 52 2□ 523□ 524 □.525 4-526□ 527 j;S 2 8 □.529 21530 □.531' □.532.□ 533 1534□ S3s;□ 5 3 64 -537

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G-RVl(NM)B -R V l(K )F " ( ( Y - Z ) • < Y * Z )* (G -B )* < G + B ))/< 2 .0 * B * Y ) G -S Q R T (F » r* l.O )r-{<x-z)«(X-»*)*K'<(y/(P+s:cN(c,r)))-aii/xC - l .O3 - 1 .00 0 47 J-L .N M

1 - j + iG -R V l( I )Y -M (t)a -s *cG>G*GZ*SQRT<r*F+B*B)R V 1< J)-ZC T / ls - a / 2 r - ( X » o * ( G « s )G— <X *S)+(C*C)8 -V SY-Y*CDO 45 NM-L.N

X -V (N M .J)Z -V (N M .l)V (H M ,JJ- <X*C)+<Z«S)V (N M ,t) — (X «S)+<Z*C )

CONTINUE2 -s Q R T (r* r+ a « a j

XF < Z .N E .O .0 ) TBEN Z - 1 .0 /Z C»F«Z S-B *Z

ENOXFF - <C»G)*<S«Y)X— (S*G ) + (C*Y)DO 46 N M -l.M

Y-A<NM*J)Z -A (N N .I)A (N N .J ) - <Y*C)+<Z*S)A (N M .I) — <Y*S)+<Z«C)

CONTINUE CONTINUE R V M M -0 .0 R V 1(R )-F H(R)-X

CONTINUE CONTINUE

CONTINUE GO TO S IN RX TE(6,• ) 'N o c o n v e r g e n c e a f t e r 3 0 i t e r a t i o n s 'RETURNEND

SUBROUTINE U S R F U N (X ,A L P B A ,8 E T A .& T Y P 2 )

PARAMETER (N P -1 5 .N F -2 0 )INTEGER JREAL F 1 ,F 2 ,F 3 ,F 4 ,Z 1 .Z 2 ( Z 3 ,Z 4 ,8 ,0 ,E ,F ,£ T Y P 2DIMENSION X (N P) ,ALP8A <N P,N P),SETA <N P) .REBX(NP) ,A IK B X (N P ),

/R E B Y (N P ), AXMBY(NP)«REX<NP) ,AXEX<NP) ,REY<NP) ,A IE Y (N P ),/PBEX (N P),PBEY <N P)

COMMON /T E S T 3 / JCOMMON /T E S T 4 / HX(NF) ,HY (N F) ,EX<NF) ,CY<NF) ,P8L<N F) ,P B 2 (N F ) ,

/P B 3(N F )*P 84< N F )> P B 5(N F )N -4

S u p p ly m a t r i x c o e f f i c i e n t s f o r t h e e a a e l a v h l e h t h e n o r m a l i z a t i o n f a c t o r K l a *NOT* t a k e n I n t o a c c o u n t l a t h e s o l u t i o n o f t h e s y s t e m , b u t l a t e r . T h e c o m p u te d m a t r i x c o e f f i c i e n t s h e r e a r e

ALPHA<1 , j J-d < f i ) / d X j w i t h f i - f i < X l , x 2 , x 3 , x 4 )

i n s t e a d o f f l " E ( x 3 , x 3 ) f ' l ( x l #x 3 ,x 3 » x 4 )

T h e p h a s e o f a t w a s c o n s i d e r e d t o b e z e r o , t h u s

PH l (J ) “ PH z-PH x— PHx— Im H x/R eH x o r IoB X /-R «B X P B 2 (J )-P E y -P H x -> P E y > P 8 2 ( J ) -P B l< J ) P B 3 (J )-P B £ -P B y — P B y - I a a y /R e B y o r XBHy/~ReBy PB 4<J )-P H x -P B y -P B 3 < J ) - P B l ( J )P B 5 (J)-P E X “ PBy*> P E x -P B 5 (J )~ P B 3 (J )

R E B X < J)-S Q R T (< (U X < J))* * 2 )/< < (T A N D (P B 1 (J)) ) * * 2 ) + l ) ) AIM HX<J)-REBX(J) *TAND<PB1(J) )R E B Y < J)-S Q R T < (< H Y (J))* * 2 )/< < <TAND<PB3<J ) ) ) » * 2 ) + l ) ) A IM BY (J)-REBY <J) *TA N D (PB 3<J))REX<J)-SQRT< < E X < J )* * 2 )/< <(T A N D <PB 5<J)~PB 3<J)) ) • • 2 ) + l ) ) A I£ X < J )- (T A N 0 (P H 5 (J ) -P B 3 { J ) ) ) *R£X<J)REY<J)-SQRT( < E Y < J)* * 2 )/< < <TAND<PB2<J ) * - P 8 1 ( J ) ) ) « « 2 ) + l ) )

-M_gYiJl;(TAND<PH2(Jl-PHl(Jin;«CYIJJ____________XF < (P B 1 (J ) .G E .0 .AND. P B 1 ( J ) .L T .9 0 ) .OR .

/ < P B 1 (J ) .G E .-3 6 0 .AND. P B 1 < J ) .L T . - 2 7 0 ) ) TBENI F (R E H X < J).L T .O ) TBEN

REBX(J ) - -R E H X (J )ENDIFI F <AXM BX<J).LT.O) TBEN

AIMBX<J)»-AIMBX<J)ENDIF

ELSEZF < < P R 1 < J).G E .9 0 .AND. P B L < J) .L T .1 B 0 ) .OR./ < P B 1 < J ) .G E .-2 7 0 . A N D .P B 1 < J ) .IT .- 1 8 0 ) ) TBEN

I F (R E H X (J).G T .O ) TBEN R EH X (J)— REHX(J)

ENDIFt r (A IM B X (J).L T .O ) TBEN

AIM HX(J)— AIMBX(J)E N D ir

ELSEZF ( (P B 1 < J ) .G E .180 .AND. P B 1 < J ) . L T .2 7 0 ) .OR./ ( P B 1 ( J ) . G E . - 1 8 0 .A N O . P B 1 < J ) . I T . - 9 0 ) ) TBEN

I P ( R E H X ( J ) .G T .O ) TBEN R E R X ( J ) — R E H X U l

E N D ir


U N i x;™: 'T ex t*Hi I etfiils tl rig'...A... • -v S ' ■

file DATE 9 /2 5 /9 1 ss•&3 -M gbnosplit time 8:17:05 pm

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638 . 639

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IT (A IM liX (J).C T .O ) THEN A IM U X (Jl--M K M X (J J

ENDirC U B IT ( ( P U l ( J ) .L T .O .ANO. P H l< J ) .C E .- > 0 ) .OR.

/ ( P H l( J ) .G E .2 7 0 .AND. P S 1 ( J ) . L T .3 6 0 ) ) T8SNi r (R EIIX (J) .L T .O ) TBEN

R tU J ( J ) — REHX(J)E N O tr

IT (A IM H X (J).G T .O ) TBEN AZK H X (J)*-AIKHX(J)

ENDirEN D ir

xr (< ? H 3 U ).C B .0 -AND. P K 3 1 3 ).L T .9 0 ) . o r ./ (P H )< J ).C C .-3 1 0 .AND. P B 3 (J ). L T .- 2 7 0 ) | TBEN

I f (REIfY(J) .L T .O ) TBEN REBY<J)— REItY<J)

ENDiri r (A IM B Y (J ).L T .O ) TBEN

A IH B Y (J)— AIMUY(J)ENDir

E L S E ir ( (P H J(J ) .O E .90 .AND. P H 3(J) . L T .180) .OR./ (P U 3 (J ).C E .-2 7 0 .A N D .P U 3(J). L T . -1 8 0 ) ) TBEN

I T (REH Y(J).O T.O ) TBEN REIIY<J)— REIIY(J)

ENDiri r (A Z K U Y (J).LT .O ) TBEN

AIM BY(J)«-AIM M Y(J)E N D ir

C U B IT U P a 3 {0 )-C E .1 8 0 .AND. P 8 3 (J ) .L T .2 7 0 ) .OR./ (P U 3(J ) .C E .-1 0 0 .AND. P H 3(J) .L T . - 9 0 ) ) TBEN

Z r (REH Y(J).G T.O ) TBEN REUYtJ)— REHY(J)

ENDirt r (A IM H Y (J).G T .O ) TBEN

M M U Y <J)« -U K U Y t3 )ENDZF

ELSEXF < (P B 3 (J ).L T .C .AND. P H 3< J).G C .-90 ) .OR./ (P 8 3 (J ).G E .2 7 0 .ANO. P B 3(J) . L T .3 6 0 ) ) TBEN

t r (R E IIY (J ).L T .O ) TBEN REBY(J )"-R E t!Y (J )

ENDITt r (AZM HY(J).GT.O) TBEN

AZMffY(J)— A IM ffY (J)ENOZr

E N D ir

PBCX<J) - P 3 5 t J )-P H 3 (0 )gSSYijlTggllJlTPgllJl-Z r ((P B E X (J ) .G E .O .AND. P B E X (J ) .L T .9 0 ) .OR .

/ (P O E X (J ) .G E .-3 6 0 .AND. P B E X ( J ) .L T .- 2 7 0 ) ) TBEN Z r < R E X < J).L T .Q ) TBEN

REX<J) — R EX (J)EN O irzr (A Z E X (J) .L T .O ) TBEN

A IE X (J )— A IE X (J)ENDIT

ELSEXr ( (P B E X (J ) .G E .90 .AND. P B E X (J ) .L T .1 8 0 ) .OR ./ (P R E X (J ) .G E .-2 7 0 .A N D .P B E X (J). L T .- lG w j j T2SN

i r (R E X (J ) .G T .O ) TBEN R E X (J)— R E X (J)

ENDXFi r (A X E X (J).L T .O ) TSOI

A 1E X (J)— AZEX(J)ENDZF

E L S E ir { (P B E X (J ) .G E .1 8 0 .ANO. P 8 E X < J ) .L T .2 7 0 ) .OR ./ (P ltE X (J ) .G E .- lB O .AND. PB E X (J) . L T . - 9 0 ) ) TBEN

i r < R E X (3).G T .O ) TBEN R E X (J)— REX (J)

ENDZFI F { A IE X (J) .G T .Q ) TBEN

A IE X (J )— AZEX(J)ENDIF

ELSEZF ( 'P B E X (J ) .L T .O .AND. P B E X (J ) .G E .-9 0 ) .OR./ (P R E X (J ) .G E .2 7 0 .AND. P B E X ( J ) .L T .3 6 0 ) ) TBEN

/ IF (R E X (J ) .L T .O ) THENREX <J)— REX (J)

E N o rrI F (A Z E X (J).C T .O ) THEN

A IE X fJ)» * A IE X (J)EN D ir

ENOIF _____

z r ( (P H E Y < J).G E .0 .AND. P B E Y (J ) .L T .9 0 ) .OR./ (P B E Y (J) .G E .-3 6 0 .AND. P U E Y ( J ) .L T .- 2 7 0 ) ) TBEN

XF (A E Y (J ) .L T .O ) TBEN R E Y (J)— REY(J)

ENOIFXT <A X CY {3).LT.O ) TBEN

A Z E Y (J)— AZEY(J)ENDZF

E L S E i r ( (P K E Y (J) .G E .9 0 .AND. P H E Y (J ) .L T .1 8 0 ) .OR ./ (F B E Y (J ) .G E .-2 7 0 .A N O .P B E Y (J). L T . - 1 8 0 ) ) TBEN

i r (R E Y (J ) .C T .O ) TBEN R E Y (J)— REY(J)

E N D irIF ( A IE Y (J ) . L T .O ) TBEN

A IE Y IJ )— AZEY(J)CNDtr

E L S E ir < (P H E Y (J1 .C E .1 S 0 .AND. P B E Y (J ) .L T .2 7 0 ) .OR ./ (P B E Y (J) .G E .-1 8 0 .AND. P B E Y ( J ) .L T . - 9 0 ) ) TBEN

i r (X E Y (J).G T .O ) TBEN R E Y < J)« -R E Y (J)

ENDZFi r (A IE Y (J ) .G T .O ) TBEN

AX EY( J ) •**AZEY ( J )ENDir

ELSEXF | ( PREY(J ) . L T . 0 .AND. P B E Y (J ) .G E .-9 0 ) .OR./ (P H E Y fJ) .G E .2 7 0 .AND. P B E Y {J). L T .3 6 0 ) ) TBEN

I F (X E Y (J ) . LT.O ) TBEN R C Y (3 )» R C Y (J)

ENDIFI F (A Z E Y (J).C T .O ) THEN

ENDIF


9 /2 5 / 9 1| gbnosplit TIME 8:17:0 S pm

LINE * TEXT

8X -X <4)-X <3>

8 3 — 1 -< X (3 )» X (4 ) )

8 " { X ( 1 ) * R E B X (J ) ) ~ ( X ( 3 ) * A X M H X (J ) ) D - < X ( 1 ) » R S B Y ( J ) M X ( 3 ) * A X M B Y ( J ) ) e - < X < 2 ) * R E B X ( J ) ) + ( X ( l > « A I M B X ( 3 ) ) r» < X m « R E g y (J ) ) + < X ( l)» M M B Y tJ i)

rX** ( 8 1 * B ) * ( 8 3 * 0 ) ra-(8i*c)*(22*F)F3a (8 3 * B )+ (8 4 * D )

A L P 8 A (1 « 1 )» (Z X * R E 8 X (J))+ (Z 3 * R E 8 ¥ (J)) ALPBA(1« 2 )» (-2 1 * M M B X ( J ) )**( Z3*AXMHY(<7) ) ALPBA(1 « 3 ) “ (B * (~ 1 ) )+ (0 * (® X < 4 ) ) )

A X fB A (3 ,l )— AXJ>BA(1»3)ALPBA(2«3 )"A 1P S A (1 ,1 ) MPHM2.3)-(e*l-X))+trM- <4)))A L P B A (3 ,1 )» (8 3 * R E B X (J))+ < 8 4 * R E 8 Y (J))A L P B A (3 ,2 )-(-8 3 * A IM B X (J))-(8 4 * A X K H Y (J))M P B A (3 ,3 )» (B « < -X (4 ) ) )+ D

ALP8A<4 , X) — ALP8A(3 , 3 ) A L F 8 A (4 ,2 )> A L P B A (3 ,1 ) A L P B A < 4 ,3 )« < E * (-X (4 ) ) )+ r A L P B A (4 ,4 )« (E * (-X (3 )1 1 + F

BETA< X) — ( n - ( REX ( J ) ) ) 8ETA<3 ) — ( P 2 - <AXEX(J ) ) ) B E T A (3)— (T 3 - ( R E Y ( J ) ) ) B E T A ( 4 ) « - ( r 4 - ( » I E Y ( J ) ) l

T o a v o i d p r o b le m s o £ u n i t s 6TY72 I s m u l t i p l i e d b y T 0 tr« 0 .O X , w h ic h v i l l m a te E 8 R f » l \ o £ t h e E l e c t r i c f i e l d , r e g a r d l e s s o f t h o u n i t s i n y h l e h l s . g l v B . _ H T e I s ETYP3 ( p g _ E _ t y p i c s l _ a q u s r o ) . ________ . . . .

E T r P 2 « ( (A B S (E X (J ) ) )» « 3 ) - f ( (A B S (E Y < J ) )* ;2 H ____


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T b la p r o g r a a c o a p u t s s t b a t w i s t ( t ) , s p l i t ( a ) , p h a s e ( p h i ) , a n d a a g n l t u d a o f t b a 10 la p a d a o o a t Cor a s a t oC CM C la ld d a t a . T b a d a o o a - p o a i t l o o u s a d b a r e o a a Da r e p r e s e n t e d b y t b a a a t r l x a q u a t l o o t

e w b lc b I s e q l v a l e n t t o p s r t o r a t b a f o l l o w i n g s t a p s i oc 1 ) C lo d T E T (b ) t b a t r o t a t e s b p o l a r i s a t i o n a l l i p s a t o c o o r d , a x i sc 2 ) " TCT<e) “ • a • " * * "c 3 ) e o a p u ta TWISTi T E T (b ) - T £ T ( e ) - P I /3e 4 ) e o a p u ta PHASE w . r . t . b i P H ( z ) - P R ( e ) - P 8 ( b )o 5 ) u a a e - t h t o s o l v e ( o r t l a n d SPLIT < s)

PARAMETER (N r - 2 0 )PARAMETER ( P I - J .1415*2454>PARAMETER ( PERMFREESPACE"4. 0 » P I* 0 .0 0 0 0 0 0 1 )REAL ANUME, ANUMM, DENOE, 0EN 08, UMAJ, HMtN, EMAJ, EMIN INTEGER N R T .M /J.EDIMENSION RERX(NP), AIMBX(NF), REHY(NF), AZM BY(NF),REX(NF),

/A Z E X (N F),R E Y (N F),A IE Y (N F),PH E X (N F),PH E Y (N F),/T E T E (N F ), TETU(NF) , LM IN(NF)«IMAJ (NT) ,S P L X T (N F )«/E L L IP H (N F ) <ELLXPE(NF) ,TN1SY(NF) «ANGFREQ(NF) ,E 1 (N F ) ,E 2 ( N F ) , /B l ( N F ) ,H 2{N F),PH X E X (N F)»/PH IB Y (N F )»R E E X P R l(N F ), REEYPRI(HP), A IE X PR I(N T )«/A IEY PR I(N T ),R E O X PR Z (N F),R E B Y PR I(N F), A IH X PR I(N F), A IH Y PR I(N F), /P E X (N F ),P H Y (N F ), APPMAJ(NF),APPNIN(NF)«PHAROT(NF)<PBAOR(NF)

COMMON /T E S T 2 / FREQ(NF)«NRFCOMMON /T E S T 4 / IIX(NF) ,B Y (N F )«E X (N F )»C Y (N F ),P B l(N F ),P B 2(N F ) ,

/P U 3 (N F ) ,P B 4 (N F ) , PBS(NF) o

10 W RITE(S• * ) *Bow a a n y s t a t i o n s i n t b i s p r o f i l e ? } 'REAO(3«*)M

e0 0 100 K -l.M

oo E n t a r t h a o r . o ( ( r a q u a n c i a s a t s o u n d in gQ * « « • « « •

W R IT E (6 ,• j ' I n p u t t b a o r . oC ( r a q u a n c i a s a t s t a t l o n i ' , K REA D (S,* )NRF

C M I H t M t t M d M . M K . M M M M I t M i t t M M I t t l l l M M U M t M H M M ' o R aad i n t b a d a t a

CALL READ

OPEN(UNIT-23,STATUS*'UNKNOWN', F I L E - 'S O l t r a r s . d a t ' , /ACCESS*'APPENO*)

OPEN( U N IT -34 .ST A T U S-• UNKNONN•/A C C ESS*'A PPEN D ')

OPEN( U N IT -33 .ST A T U S-' UNKNOWN •/A C C E S S -'A P P E N D ')

OPCN<UNIT-23,STATUS-• UNKNOWN*, F IL E - ' r o t C o n . d a t ' , /A C C E S S -'A P P E N D ')

F X L E - 'a n g l e s . d a t ' ,

F l L E - 'o r l c o o p . d a t ' ,

o C o a p u ta t h a LO i a p a d a n c a a o d d i s t o r t i o n p a r a a a t a r a C o r a a e b f r e q u e n c y

W RITE(3S, • ) * DISTORTION PARAMETERS/ STATION N O .: ' ,KWRITE ( 2 5 , • ) 'F R E Q (J) APPM AJ(J) APPM IN(J) PB A SE (J) T N IS T (J)

/ S P L IT ( J ) E L L IP H (J) E L L IP E (J ) 'WRZTE(24,M 'ROTATION ANGLES/ STATION N O .i ' .X WRITE( 2 4 , * ) 'TETAE TETAB PB10RIWRZTE(23, • ) 'ORIGINAL COMPONENTS/ STATION N O .t ' .X WRITE( 2 3 , • ) ' REX SEX RET SET REX

/IB T *W R IT B (22,* ) 'ROTATED COMPONENTS/ STATION N O .t ' .K W R IT E (32 .« ) 'R E X IEX RET 1ST RBX

/ I BY'

PBZROT'

XBX 1

ZBX

DO 6 * 0 J -L .N R F

WRITE ( « . * ) 'WRITE ( « , • ) 'PROCESSING FREQUENCYt' , PREQ(J)

c T b a p b a s a oC a s w a s c o n s i d e r e d t c b e z e r o , t h u s

PHUx— p h i PBHy - P H 3 PHEX-PH3-PH3 PB E y-P B 2-P B l

R E H X (J )-S Q R T (( (H X (J ))* *2 ) /( ( (TAND(PHl( J ) ) ) * « 2 ) * l ) ) AIM HX(J) — REBX(J) *TAND(PBL(J) )REIIY(J)-SQRT( ( (t(Y(<J)) • • ! ) / ( ( (TAN D (PB l(J) ) | • • 2 ) + l ) )AIMHY(J ) — REHY( J ) «TAND(PB3( J ))R B X (J)-S Q R T((E X (J) * * 2 ) / ( ( (T A N D (P B 3(J)-P B 3(J)) ) * * 2 ) + l ) ) A IE X (J )-(T A N D (P H 5 (J )-P B 3 (J )) )*REX(J)R E Y (J )-S Q R T ((E Y (J )* *2 ) /( ( (T A N D (P B 2 (J )-P B 1 (J ) ) ) * "2 )+ l))

^ i m J l ; l T ANDi P H 2 t J i - P B l ( j m ?REY(J l ________________________

IF ( ( P R l ( J ) .G E .O .AND. P H 1 (J ) .L T .9 0 ) .OR./ ( P B l ( J ) . G E . - J 6 0 .AND. P B l ( J ) . L T . - 2 7 0 ) ) TBEN

I f {R C U X (J).L T .O ) TBEN R ERX(J) — REHX(J)

E N D IFt r (A IM B X (J).G T .O ) THEN

AIMHX(J) — AIMHX(J)E N O IF

ELSEZF ( ( P B L ( J ) ,C E .90 .AND. P B l ( J ) .L T .1 8 0 ) .OR./ ( P H i ( J ) .G E .-2 7 0 . A N D .P H l(J ) . L T .- 1 8 0 ) ) TBEN

IT (R E n X (S ).G T .O ) TBEN REBX(J) — REHX(J)

ENDITIF (A IM H X (J).G T .O ) TBEN

AZM8XU) — A IK B X (J)E N O tr

E L S E ir ( ( P R l ( J ) .G E . lS O .AND. P H 1 (J ) . L T .2 7 0 ) .OR./ (P R 1 ( J ) .G E .*180 .AND. P B l ( J ) . L T .* » 0 ) ) TBEN

i r (R E H X (J).G T .O ) TBEN RERX(3 ) -* R E llX (J )______ _______


:U n

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tNotr: r (A IM B X (J).C T .O ) THEN

M M flX (J )— M M HX(J)ENOtr

E LS E ir ( (P tU ( J ) •LT.O .ANO. P B 1 (J ).G E .-9 0 ) .OR./ (P H 1 (J ) .O C.270 .ANO. P H 1(J) . L T .3 6 0 )) THEN

t r (R E ItX (J ). L T .0) TBEN R E B X (J)--R E »X (J)

ENOtrt r (A IM HX(J) .G T.O ) TBEN

AtMBX ( J ) a~MMBX (J )ENOtr.-S IS IS __________________ ___________

I P ( < P 8 3 ( J ) . C E . O .AND. P B J ( J ) . L T . 9 0 ) .O R ./ ( P B 3 ( J ) . G E . - 3 6 0 .A N O . P 8 3 ( J ) . I T . - 2 7 0 ) ) TBEN

t r ( R E B Y ( J ) .L T .O ) TBEN R E H Y (J )* - R E H Y (J )

E N O trt r ( A Z X B Y ( J ) .L T .O ) TBEN

A Z K B Y (J) — A IK M Y (J)EN D ir

E L S E i r < < P H 3 ( J ) .G E .9 0 .A N O . P B 3 ( J ) . L T .1 8 0 ) .OR./ ( P H 3 ( J ) . G E . - 2 7 0 , A N D . P B 3 ( J ) . I T . - 1 8 0 ) ) TBEN

t r ( R E U Y ( J ) .G T .O ) TBEN R E B Y < J )" -R E H Y (J )

E N O trt r ( A I M B Y ( J ) .L T .O ) THEN

A IM H Y (J)— A IM H Y (J)ENOtr

ELSEir ( ( P B 3 < J ) . G E . U 0 ,AND. PH3<J).LT.270) ,08./ ( P H 3 ( J ) .G E . - I B O .A N O . P B 3 ( J ) . L T . - 9 0 ) ) TBEN

t r ( R E H Y ( J ) .G T .O ) TBEN R E H Y (J )— R E H Y <J)

E N O irt r ( A I M B Y ( J ) .G T .O ) TBEN

AZMHY( J ) — A IH B Y ( J )E N O tr

E L S E i r ( ( P B S ( J ) . L T . O .A N O . P B 3 ( J ) . C E . - 9 0 ) .O R ./ ( P B 3 ( J ) . G E . 2 7 0 .A N D . P H 3 ( J ) . L T . 3 6 0 ) ) TBEN

t r ( R E H Y ( J ) .L T .O ) TBEN R E H Y (J )« - R E H Y (J )

E N O IFt r ( A Z M B Y (J ) .G T .O ) TBEN

A1KBY ( 3 >«-AXKHY { 3 >ENOtr

E N O tr

P S E X (J )> P 8 5 < J ) -P B 3 (J )P H E Y tJ ) " P B 3 (J ) -P H l( J )

ir ( (P B E X (J ) .G E .O .AND. P B E X (J ) .L T .9 0 ) .OR ./ (P H E X (J) .C E .-3 6 Q .AND. P H E X (J ) .L T .- 3 7 0 ) ) THDI

i r (R E X (J ) .L T .O ) TBEN REX<J) — R EX (J)

ENDZFI F (A IE X (J ) .L T .O ) THEN

AXEX{J)— M E X tJ )ENOtr

ELSEIF ( (F H E X < J).G E .9 0 .AND. P B E X (J ) .L T .1 8 0 ) .OR./ (P B E X (J ) .G E .-2 7 0 .A N D .P B E X (J ) .L T .-1 B 0 )) TBEN

t r (R E X (J ) .G T .O ) TBEN R E X (J)* -R E X (J)

E N O trir (A X E X (J).L T .O ) TBEN A Z E X (J)» -A IE X (J)

ENDITE L S E ir ( (F H E X (J ) .G E .1 8 0 .AND. P B E X < J) .L T .2 7 0 ) .OR .

/ (P B E X (J ) .G E .-1 8 Q .ANO. P B E X ( J ) .L T .- 9 0 ) ) THENI F (R E X (J ) .G T .O ) TBEN

R E X (J)— R EX (J)ENOtri r (A I E X ( J ) .G T .0 ) TBEN

A 1 E X (J)— A IE X (J)ENOZF

E L S E ir ( (P R E X (J) .L T .O .AND. P B E X (J ) .G E .-9 0 ) .OR./ (P B E X (J ) .G E .2 7 0 .AND. P B E X (J ) .L T .3 6 0 ) ) TBEN

I F (R E X (J ) .L T .O ) TBEN REX<J)— REX<J)

ENDIFt r (A IE X (J ) .G T .O ) TBEN

A ZEX (J) *-A IE X ( J )ENOtr

ENDIT

tr ( (P B E Y (J).G E .O .AND. P 8E Y < J).L T .9 0 ) .OR./ (P B E Y (J). G E .-360 .AND. P B E Y (J ) .L T .-2 7 0 )) THEN

i r (R E Y (J ).L T .O ) TBEN R EY(J)— REY(J)ENOtr

t r (A IE Y (J ) .L T .O ) TBEN A IE ¥ (J )— AZEY(J)

ENOtrE L S E ir ( IP B E Y tJ ) .G E .9 0 .AND. P B E Y tJ ).L T .1 6 0 ) .OR.

/ (P B C Y (J).G E .-270 . AND.PBEY(J). L T .-1 9 0 ) ) TBENi r (R E Y (J).G T .O ) TBEN

R EY(J)— REY(J)ENOtrZ r (A Z E Y (J ).L T .O ) TBEN

AZEY(J) — A tE Y (J)E N D ir

ELSEIF ((P B E Y (J ).G E .1 8 0 .AND. P H EY(J). L T .270 ) .OR./ . P B E Y (J).G E .-180 .AND. P B E Y (J ) .L T .-9 0 )) TBEN

IP (R C Y (J).C T .O ) TBEN R E Y (J)*-R E Y (J)

ENOtrt r (A Z E Y (J).G T .O ) TBEN

AXEY( J ) a -AIEY(<1)ENOIF

ELSEZF {(P B E Y (J ).L T .O .AND. P 8 E Y (J ).G E .-9 0 ) .OR./ (P R E Y (J).G E .270 .ANO. P B E Y (J ).L T .3 6 0 )) THQt

i r (R E Y (J ).L T .O ) THEN REY(J) — REY<J»ENOtr

IF (A tE Y (J ).G T .O ) THEN _______ A tE Y (J ) — A i r i ( J ) ______________________________________


U N IX™ , v'it"text File: Listing

FILE BATt 9/25/91gbnoshear TIM8 8:16:40 pm

92

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P C X (J)a ATAM2D(MEX(J) .R C X (J ) )PHY ( J ) * XTANaD ( M MKY t J ) « XEVY (J ) )aP N AO* (J | • P B X ( J ) - P IfY ( J )

AMUMEa 3* ( (R E X (J) »REY (J) )•*( AXEX(J) ’ A I E Y tJ ) ) ) .......................DENOBa (< R tY < J) » • • » ! ♦ ( CAXCY(J ) ) • * * M (***<■*>) <M EX (<J)) •

OENOBa ( {REMY(J) ) , # 3)*< (AIM BY{J) ) J ) “ ( <RCBX<<J J ) * * 3 ) a

T E T C t-l)« 0 .5 * <ATAN3Q< ANUME.OCNOEl)I 5 I MI J i " 0 '(A T A H lO tA WUMEtPEWQHj}nr__________________________

T g < n a tg cw «d c o « p o a « n ta

MEEXPRI(J J*EEX(<Jl *COS(TETE(J J J* R E Y (J1 •S X N (T E T E (Jl) AXEXPRX(J)a A IE X <J)»C O S<TE T E (.I)}-A IE Y <J)«SIN tT E T E <.l)» REEYPRX(J)a REX<J) •SXN(TETE(«J) )*REY( J ) •6 0 3 (T E T E (J J ) AXEYPRI(J J a A X E X (J}•SX N (T E T E (J)J*A X E Y (J)»C O S<T E T E (J)) REBXPRX(J)a R E R X (J)*C O S (T E T fl(J))-R E H Y (J)*S X N (T E T 8< J)) AXBXPRX(J)-AIM HX(J) •C O S (T E T B < J))-A X M B ¥(J)•S IN (T E T B (J)) REH Y PM t(J)a REIIX( J ) * S X N (T 2 T 9 (J))4 R E B Y (J)« C 0 S (T E T B (J)) AXHYPRI (J ) a AXMBX(«X) *SIN (T C T B (J) )+AIKHY( J ) * C 0 3 (T E T 8 (J ) )

(N fy _ p b < 8 « .4 B g 4 fT r ib a f o n M d a a g l o s

PBXih>a P B X tf ,_ ) -P B X th , n

PBIBX(<7)a ATAN3D( AXEXPRX( J)» R E E X P R X (J))PBX BY (J)a ATAN2D(AXHYPRX(J)«REBYPRX(J))

P8A R 0T (JJ-P B IE X (JJ-P B X H Y (J)

IT | (M M K fft J ) .O T .JO i .AMD. tPHAROTtO) . 1 8 .5 7 0 ) )TBEH PBAROT(J)-PBAROT(J > -180

ELSEXP ( (PUAROT(<X). L T .-9 0 ) .AND. (PHAROT(J) .C E .“ 370)JTHEN PBAROT(3)a PHABOT(J)+l®0

E L S E ir ( (P R A R O T (J).S T .3 7 0 ) .AN O .<PB A R O T(J).LE .X tO ))TH EN PB A H O T(J)-PU A R O T(J)-360

E L S E ir t tP ttW tO T < J).X ,T .-3 7 0 ).A N D .tP R A ItO T t3 ).G E .-J€ 0 )} T H E N PB A RO T(J)*PBA X O T(J)+J*0CNDtr

S p l i t ,

E l { J ) a <REEX PR X (J)**3)+(A X B X PR X (J)**3) B 3 ( J ) a <REEYPRI( J J • « 3 )+ (A X E Y P R I(J) * * 3 J H l ( J J a (R E 8 X P R X (J |‘ ‘ Xl+tAXHXPRX(J I **31 B a ( J ) a (X C H Y 7 R I(J )* * 3 )+ (A IB Y P R l(J )* * 2 )

X r ( B 1 ( J ) .O T .E 3 ( J J JTBEH E H A J -E l(J )EK IN -C 3<J)

ELSEEHAJa E 3 ( J )EMINa E l ( J )ENDir

x r (B 1 < J ) .g t . b 3 ( J ) ) tbenBMAJa R l ( J )H M IN -B 3(J)

ELSERMAJa B 3 (J )BMINa H l ( J )

ENDXF

S P L X T (J) a ( EMAJ«HMAJ+EMXN•BNIN) / _/tEM AJ;BNAJrENXN?HHIN).

c p ^ u t« ^ H K A j / R N !N .ap d _ p tA J/E N IN . t o haT p « p l d o a o t t a p • l l l p t l c i t y

C L L IP B < J)a BMXN/HMAJi EX.LI P E ( J i « EMXN/£MAJ _____ ____ . . . __________________

c A p p l l t o d o o f t p l p p p d a n c p

tM A J \ J ) a \ EMAJ/ EMAJ)*/ ( ( 1 + ( S P L X T ( J ) ) * * 3 J / ( l + S P L X T ( J ) J )

Z N IN (J )a (EMXN/RMIN)«■ <ttUtSPLXTt3V ?3>/t\*SPLITt3tANCrRCQt J ) a 3 .0 * P I TR E Q (J)APPNAJt3>alLMAJ<3)**3)/tANCmCQt )*PCEMmCSPACE) APPNXN(J)a (lM X N (J)**3}/(A N CrREQ (J)'PERM FREESPA CE) T N IS T (J ) a 9 0 * ((A B S (T E T E (J ))-(A B S (T E T fl< J )) ) ) )

W r i t* s o l u t i o n * t o t r s r x . d a t

WRITE ( 2 5 . • 1FREQ(J1»APPMM<J) »APPKXN< JJ »PEAROT(J)»/TWXST( J J , SPLXTf J J ,ELLXPB( J | ,ELLXPE(J)

WRITE {3 4 . • ) TETE (J ) , TETB ( J ) , PHAOR ( J ) r PBAJtOT( J )H R IT e (3 J ,* )R E X (J) ,A tE X (J t.R E Y (J} .A X C ¥ (3 U R E E X (J l,A X K E X (JK R E 8 Y < S > ,

/A IM B Y (J)WRTTE{3 3 . * )REEXPRI(J) . AXEXPRI(J), REEYPRX(J).AIEYPRI (J),REHXPRX ( J ) #

/AXBXPRX(J)« REHYPRl( J )« AXBYPRI ( J )

100100 CONTINUECONTINUE

CLOSE(3S) CLOSE(3 4 | CLOSE(33) C LO SE(33) END


9 /2 5 /9 1 93™E 8:16;40 pm

3 1

r»i364

13651366 .367 _368

369 -370 1371 _372 “ 373 1374 “ 375 1378 -377

378379360361 382'J»J

.38 4 1385 J88"

387 386 .389 ‘ 90

91392393394395396

3 9 7 ;396399400.401

3 0 2 -403404

3 o s406

3 0 7308 309-4.1ft.

pIU:

B ;i|lBIS:I^2o; 32'1 3 2 2 j 3233 2S3 2 0 .3 2 7328.311:3 3 2 :

4 3 34 3 4

_ 435 ' ^ 3 6337

3 3 8 439 440

441342' 443“ V 4 4345

446447'3 * *‘ 449' 450*

4SI

SUBROUTINE REM)

S u b r o u t i n e t o r o o d l o t h o C lo ld d a t a C or a e a a x t s t a t i o n a t t h o d lC C o r o a t M a n u r e d C z o q u o a o le a ( u a u a l l y 14 )

N R r* n u e b e r o f C ro q u o D o to a F R E Q « a e tu a l C ro q u e n o le a ( t o 8 a )C U R R « ap p llo d o u r r o a t t o t h o g r o u n d ( A s p ) . U aod t o a o r a a l l a o t h o

a m p l i t u d e o f t h o E l a o d 8 1 t l o l d a (P h o o a ix d a t a U a l r e a d y a o n i a l i a o d )

8 1 s o a q a l t u d o oC t h o 1 t h e o a p o a e a t o f t h o a a g n e t l c C lo ld E l* o a q a l t u d o oC t h o 1 th o o a p o a e o t oC th o e l o c t r l o C lo ld

SXQB1* a t a a d a r d d e v i a t i o n oC t h o a a g a e t l o C lo ld * 1* e o a p o n o a t SXGEl* • " ■ e l e o t r l o " * *

P H I- p h a a o Ha-HX ( l a d ey * w « a)M l*P H I-PH 4-P 8 J -PB «-

Ey-B x a a -a y a x -a y ex - ay E y -E x

(f • )(P 8 2 l a a y a o t e a ) (P 8 1 1 a a y a o to a )

SPH n- o o r r o a p o a d l a q a t a r d a r d d e v i a t i o n oC t h o p h a a o a a q l o a

PARAMETER (NP-15.NF-20)CHARACTER*20 FXLENAMKEAL rRCQ(NF) .CURR(NF) ,BX (N F) ,SX G 8X (N r) «HY(NF) .SICBY(NT) ,B 2 (N F ) ,

+*jlG B£(N F) iE X (N F) »S2GEX(NF) »EY(NF) 'SX Q EY (N F)«PH l(N F) »3PH 1(N F) » + P H 2 (N F )» S P B 2 (N F ), P H 3 (N F ), S P B 3 (N T )» P B 4 (N F )|3 P H 4 (N F )« * P H 3 (N F )» S P H S (N F ).P 8 6 (N F ),8 P B ft(N F )

COMMON /T E S T 2 / FREQ(NF)iNRFCOMMON /T E S T 4 / B X (N F ),H T (N F )* E X (N F )< E T (N F ),P 8 1 (N F ),P B 2 (N F ),

+ P B 3 < N r) ,P B 4 (N F ),P H S < h T ),P a « (N F l W RZTE(C.«) ‘ ENTER NAME OF T8E OATA FX LEt*READ(3< 5 ) FILENAM FORMAT(A20)N-ITRMLM(FI LENAM)0PEN(UNXT-7» FXLE-FXLENAM (liN), STATUS-'O LD1 )

l o o p t o r o a d d a t a C or NRF C ro q u e a e ie a

DO 10 J -L ,N R FW R IT E (I ,« ) JREA D (7»*) F R E Q (J)*C U R R (J)»H X (J)«S IG H X (J) « H Y (J)> S IC B Y (J)«

♦ H E ( J ) ,S I C B 2 < J ) , E X ( J ) ,S I C E X ( J ) , C T ( J ) .S I C r Y ( J ) , P H l ( J ) ,S P H l ( J ) , + P 8 2 ( J ) ,S P B 2 ( J ) ,P B 1 < J ) .S P H 3 < . ) ) ,P 8 4 ( J ) ,5 P H 4 ( J ) » P B 5 ( J ) , + S P 8 9 ( J ) ,P B 6 < J ) ,S P H < (J )

w r l t o d a t a o n t h o a e z e e a

NRXTC(t,«) mQtJ),CCRRtJ),EX(J),aiCBX(J),HT(3)#SXCHT<J), +BZ<J),SXCHZ(J),EX(J)i3XGEX<J),ET(J),SXGEY(J),pai(J),SPHl<J)i +PB2(J),5PB2(J),PH3<J),SPB3<J),PB4(J),SPB4(J),PB9(J), +SPHS(J).PB6(J),SPH6(J)CONTINUEC tO SE<7)ENDrUNCTZON XTRMtM (STRING)

T h l a C u n o t lo a r o t u r a a t h o l o a g t h oC a o b a r a e t o r a t r l o g w i t h a l l t r a l l l a q b l a a h a r e a o v e d . I t r o a d a th o c h a r a c t e r a t z l a g b a c tw a r d a u n t i l a 00 0 - b l a a X c h a r a c t e r l a o a c o u a to r o d .

C8ECK THE LENGTH OF THE CHARACTER STRING

L - LEN(STRING)

ZF (L .L E .O ) TBEN XTRMLM-0 RETURN

END ZF

I- 1XF (STRZN G (ZiZ) .N E . ‘ * ) GO TO 20

I • I - 1 GO TO 10

CONTINUE ZF (Z .G T .O ) TBEN

ZTRMLM • Z ELSE

ZTRMLM ■ 0 END z r PCTURN


A p p e n d ix B

Rotation Angles of Polarization Ellipses


95

Let h be the horizontal magnetic field. In general, A is a. complex vector which can be expressed in terms of its x and y components as:

hx = AXr "f“hy = hyr 4“ ffiyi, (0*1)

where the subcripts r and i stand for real and imaginary parts respectively. In terms of the phase and amplitude these equations are

hx = he** hy = hei6>,

with h — \Jhx2 + hy2, <j>x = and <py = tan-1^ .

In order to transform hx and hy to a new coordinate system such as they are 90° out of phase and such that the plane of the resulting polarization ellipse lies in the new system, we apply the rotation operator R -1 to A, i.e.

or

h'x = hxcos0h — hysinOk= hXTcos0h + ihxicosdh — kyrsinOh — ihyisindh = hxrCosOh — hyTsin0h + i(hxicos0h — hyisinOh), (0.4)

and similarly

hy = hxrsin8h + hyrcosdk + i(hxisindk — hyiCosQh)• (0.5)

(0.2)


96

Because h'x and h'y are related through the relation

h'y = iahx, (0.6)

with a a scalar, then

hxrsin0k + hyrCosOh + i(hxisin0h — hyicos0h) = ict(hXTcos0h — hyTsin0h + i(halpka(ihxrcos6h

—hyrSinOh — (hxicos0h — hyisinOh)), (0.7)

therefore we have that

hxr sin9h + hyr cos&h — —a(hxicos0h — hyisindh) hxisinBh + hyiCosOh = a{hxrcos9h — hyTsin0h). (0.8)

Dividing the first of these two equations by the second we get

hXTsin0h + hyrCosBh _ —hx{Cos9h — hyisin9h . .hxisinBk + hyiCosBh hxrcos9h — hyrsin9h ’

which rearranging terms gives the expression

a _ -1 2(A*rAyr + hxihyi) /n in\(V ’ + 0 - 0 ' (0'10)

This is the equation for the angle needed to rotate h to the new reference system. In a similar way, it can be found that the required angle to perform a transformation of the polarization ellipse defined by the electrical horizontal field e is given by

n________- l 2(erreyr + e x , -e y i ) /n n \( v ! + V ! ( °' n )


References

97

Bahr, I<., 19S8, In terpre tation of the M agnetote llu ric Im pedance Tensor: Regional Induction and Local Telluric D istortion , J. Geophys., v.62, p.119-127

Banos, A., 1966, D ipole Radiation in the presence of a conducting Half-space, Pergamon Press, Inc.

Bartel, L.C., and Jacobson, R.D., 1987, Results of a C S A M T survey at the Puhim au Therm al area, K ilauea Volcano, Hawaii, Geophysics, 52, 655-677.

Berdichevsky, M.N., and Dmitriev, V .I., 1976a, Basic principles of in terpretation of M agnetotelluric curves, in Geoelectric and Geothermal studies, A. Adam, Ed., Akademini Kiado, 165-221

Cagniard, L., 1953, Basic Theory of the M agnetote llu ric M ethod of Geophysical Prospecting, Geophysics, 18, p.60{5-635

Conte, S.D., and deBoor C., 1980, E lem entary N um erica l Analysis, McGraw Hill Co.

Edwards, R.N., 1980, A grounded vertical long w ire source for Plane W ave M agnetotelluric analog modeling, Geophysics, 45, 1523-1529.

Eggers, D.E., 1982, An Eigenstate form ulation of the M agnetote lluric Im pedance Tensor, Geophysics, 47, 1.204-1214.

Frishknecht, F.C., 19S7, Electrom agnetic physical scale modeling, in Nabighian, M.N., Ed., E.M. Methods in Applied Geophvsics, v .l, 365-441.


9S

Gromm, R.W., 19SS, The effects of inhomogeneities on Magne- totellurics, Ph.D. Thesis, Univ. of Toronto.

Groom, R.W . and Bailey, R.C., 19S9, Decomposition of Magnetote lluric Im p e - dance Tensors in the Presence of Local Three- Dim ensional Galvanic D istortion , J. Geophys, Res., 94B, p.1913- 1925

Goldstein, M .A ., 1971, M agnetote lluric Experim ents employing an artific ia l D ipole Source, Ph.D. thesis, University of Toronto.

Goldstein, M .A. and Strangway, D.W ., 1975, Audio-frequency M ag- netotellurics w ith a grounded E lectrica l D ipole Source, Geophysics, 40, 669-6S3.

Kaufman, A.A. and Keller, G.V., 19S3, Frequency and Transient Soundings, Elsevier Science Publ. Co., Inc.

LaTorraca, G.A., Madden, T.R. and Korringa, J., 19S6, A n A nalysis of the M agnetotelluric Im pedance for three-dim ensional conductivity structures, Geophysics, 51, 1S19-1S29.

Parasnis, D.S., 1988, Resiprocity theorem s in Geoelectric and Geoelectrom agnetic W ork, Geoexploration, 25, 177-198.

Phoenix Geophysics Ltd., 1989, R ep o rt on the Tensor Controlled Source A udio -M agnetotelluric test survey in M idw est area Saskatchewan, Canada, Part 1.

Press, W .H., Flannery, B.P., Teukolsky, S.A. and Vetterling, W .T., 198S, N um erica l Recipies, Cambridge Univ. Press.

Price, A .T.. 1962, The Theory of M agneto -Te llu ric methods when the source field is considered, J. Geophys. Res., 67, 1907-1918.

Sandberg, S.K. and Hohmann, G.W., 1982, Controlled-source A u- diomagnetotellurics in geotherm al exploration, Geophysics, 47, 100-116.


99

Sims, W.E., Bostick, F.X. and Smith, H.W., 1971, The estim ation of M agnetotelluric Im pedance Tensor elements from measured data, Geophysics, 36, 938-942.

Schmucker, U., 1970, Anomalies o f Geomagnetic variation in the South-W estern Uninated States, Bull. Scripps Init. Ocean., r.’niv. of Calif., San Diego, 13.

Swift, C.M., Jr., 1967, A M agnetote llu ric investigation o f an electrical conductivity anom aly in the Southwestern U n ited States, Ph.D. thesis, Mass. Inst, of Technol., Cambridge.

Wait, J.R., 1953c, Propagation of radio waves over stratified ground, Geophysics, 18, 416-422.

Wait, J.R., 1954, On the relation between Telluric currents and the E a rth ’s M agnetic F ield , Geophysics, 19, 281-2S9.

Wait, J.R., 1962, Theory of M agneto -Tellu ric Fields, J. Res. Nat.B. Stan. Rad. Pro., 66D, 509-541.

Wait, J.R., 1982, Geo-electromagnetism, Academic Press, Inc.

Wannamaker, P.E., Hohmann, G.W. and Ward, S.H., 1984b, M agnetotelluric responses of three-dim ensional bodies in layered earth, Geophysics, 49, 1517-1533.

Ward, S.H. and Hohmann, G.W., 1987, Electrom agnetic theory for geophysical applications, in Nabighian, M .N ., Ed., E.M. Methods in Applied Geophysics, V .l, 131-31TT

Yamashita, M ., Phoenix Geophysics Ltd., personal com munication

Yfo, E. and Paulson, P.V., 19S7, T h e Cannonical Decom position r 1 it relationship to other forms of M agnetotelluric Im pedance ^ “nsor analysis, J. Geophys., 61, 173-189.

/ongt. \<.L. and Hughes L.J., 19S8, Controlled Source Audio-frequency 'agnetotelhirics, in Nabighian, M .N ., Ed., E.M. Methods in Applied

v; ■ .physics, V.2.


collinear o-'*bo

35 IS broad side

55

Figure 1.- Far field CSAMT measurement zonesgenerated by a single dipolar source (after Zonge and Hughes, 1988).


TransverseLines

RxConductive

ZoneLake

BipolarTransmitter

Tx

2 km

/Figure 2.- Plan view of the survey lines and

transmitter-receiver configuration used (modified from Phoenix Geophysics, Ltd., 1989).


4175 4225 4275 4325 4375 4425 4475 4525

200m WirtSPtyS,

Friable zone 1 ' i.'.'.'

Ore zone

Basement

Fault

approx. scale

Figure 3.- Geological cross section of the surveyed test area (modified from Phoenix Geophysics, Ltd., 1989).


Conductive Zone

a)>VzIUDO'utOSu .

b)

>-uzui=>ouia

2048 -

128 -

4S2S 4475 4425 4375 4325 4275 4225 4175

STATION (m l

Conductive Zone

4096 -

2048 -

1024 -

512 -

256

128

64

32

16

84 -

2 -4525 4475 4425 4375 4325 4275 4225 4175

P hase App Res

0.B22a .7S4 Q . 6 9 5

0 . 6 6 0

0 . 6 3 2

0 .612 0 . 5 9 5

0 . 5 7 3

0 . 5 5 3

0 . 5 3 9

0 . 5 2 2

0 . 5 0 3 0 . 4 3 4

0 . 4 6 50 . 4 4 7

0 . 4 2 5 0 . 4 0 4 0 . 3 3 2 0 . 361 0 . 3 4 70 . 5 3 9

0 . 3 2 8

0 . 3 1 7

0 . 3 0 4

0 . 2 9 2

0 . 2 7 8

0 . 2 6 5

0 . 2 5 1

0 . 2 3 6

0 . 2 2 4

0.211 0 - 1 9 7

0 . 1 8 3

0 . 1 S3

0 . 1 5 4

0 . 1 4 0

0 . 121 0 . 0 9 3

0 . 0 5 6

0 . 9 7 5 0

0 . 9 5 0 0

0 . 3 3 1 7

0 . 3 1 6 1

0 . 2 9 6 0

0 . 2 7 9 3 0 . 2 5 4 0

0 . 2 4 6 3

0 . 2 3 1 0

0 . 2 1 4 2

0 . 1 9 7 6

0 . 1 8 1 2 0 . 1 5 4 9

0 . 1 4 3 1

0 . 1 3 1 7

0 . 1 1 6 1

0 . 1 0 1 2 0 . 0 8 7 3

0 . 0 7 5 8 0 . 0 6 4 3

0 . 0 5 4 1

0 . 0 4 5 3 0 . 0 3 7 4

0 . 0 3 0 5

0 . 0 2 4 9

0 . 0 2 1 7

0 . 0 1 9 7

0 . 0 1 8 1

0 . 0 1 6 3

0 . 0 1 4 4

0 . 0 1 2 8

0 . 0 1 1 10 . 0 0 9 2

0 . 0 0 7 2

0 . 0 0 5 2

0 . 0 0 3 20 . 0 0 1 7

0 . 0 0 0 9

0 . 0 0 0 5

STATION (n i)

F ig u r e 4 . - 2 -D n o r m a l iz e d a p p a r e n t r e s i s t i v i t y w i t h o u t d i s t o r t i o n ( a ) , and n o r m a l iz e d p h a s e (b ) .


Conductive Zone

<ZZWXc

a)

4096 H

2048 -

1024 -

128 -

LfMX.!S>

b )

4525 4475 4425 4375 4325 4275 4225 4175

S T A T IO N (m )

Conductive Zone

>•uzUIDOId02u.

4096 H

2048 H

1024

512 -j

256 -

128 -

64 -

32 -

16 -

8 - 4 -

4525 4475 4425 4375 4325 4275 4225 41

P hase App Res

j . O V J . d ’ 50■J . s * '. ' o . - s ; ? .

J . 9 - 7 j . •;

l i . O l f . 0 . M O -3

. 5 • * 0 $ ■j j

0 - ft . I ’ S ' u

r . ft ft 7

y . ft d o 1 . : : f . 6

c . o ’ : J . . ' i l l ft0 . 8 * 4 0 . 1 3 6

c . a s s 0 . 1 7 0 0o . { w ? u . I b b t .

i'l . t . 1 31

0 .1 3 0 3

f t . 5 3 1 i • o . j : 4 C i

o . h : 5!

I

0 . : J f ; 4

o . e i v. 0 . 0 0 • ■’C: . SOQ *1. 0 0 0 :0 . - 3 ? _ : 1. I ' t i : !' j . '<‘ 8 6 _ j . (1 s 6

:j . 7 7 0 _ 0 . 0 4 9 ?

0 . 7 5 : 0 . 0 - 1 1 4D . V J 4 O' 0 . 0 3 4 4

D . 1 6 n . o o f i i

0 . 6 ^ 6 0 . 0 : 3 10 . 6 3 0 n . o i a o

U . & b i Q . G l f i f .0 . 6 - 1 5 J . U 1 4 3

0 . 6 3D 0 . U 1 3 4

0 . 6 1 b 0 . 0 1 : 1

0 - 6 3 2 0 . 0 1 0 7

o . s r. i : 0 . 0 0 9 4■r . 5 7 7 0 . 0 0 5 0J . S o 3 0 . 0 0 6 5

0 . 6 - 1 3 0 .Cl Oc.O

0 . 5 J 3 0 . 0 0 3 5n . 4 3 7 O . O O : :

J . 4 S 9 0 . 0 0 1 1r ; . 4 17 0 . o n o t j■ J . : 7 7 D JJO ’j i

S T A T IO N ( m l

F ig u r e 5 . - 2 -D n o r m a l iz e d a p p a r e n t r e s i s t i v i t y w i t h o u t d i s t o r t i o n , r o t a t e d 90 d e g re e s w i t h r e s p e c t t o t h e p r i n c i p a l a x e s( a ) , an d n o r m a l iz e d p h a s e ( b ) . A p h a s e s h i f t o f 180 d e c re e s i s o b s e rv e d .


Conductive Zone

P hase

4625 4475 4425 4375 4325 4275 4225 4175

S TA TIO N (m )

0 . 1 4 9 ?rJ * 1 3 2 ri 3 . 1 1 5 !

3 . 1 0 0 5

o. ossa0 .075.' 3 . 0 S 6 8

O .D b U t 0 * 0 5 5 5

0 . 0 5 1 0

0*04? s* 0 . 0 4 1 ?

3 . 0 0 c :

0 . 0 3 0 6 3 .0 : 1 9

3 . 0 1 2 9

0 . 0 1 4 3 3 . 0 1 0 5

3 . 0 0 ? :

0 . 0 0 4 5

0 . 0 0 3 9

Conductive Zone

2 'i i r4525 4475 4425 4375 4325 4275 4225 4175

STATION (m )

F ig u r e 6 . - Z e r o - s p l i t t i n g s t r i k e in d e p e n d e n t n o r m a l iz e d a p p a re n t r e s i s t i v i t y ( a ) , an d p h a s e (b ) c o r r e s p o n d in g t o t h e s m a l le r m a g n itu d e o f t h e im p e d a n c e t e n s o r . P eak v a lu e s u s e d f o r n o r m a l i z a t io n w e re 1 6 ,5 9 6 Ohm-m an d 8 9 .1 d p .o re e s r e s o e c t i v e l v .


Conduct lv* 7.tin«

I'hass

STATION (ml

Conduct I v* Zun*

4525 4475 4428 4378 4328 4278 4228 4178

STATION (ml

F ig u r e 7 . - Z e r o - s p l i t t i n g s t r i k e in d ep en d en t n o rm a li* ed ap p aren t r e s i s t i v i t y ( a ) , and p h a se c o r r e sp o n d in g t o th e la r g e r m agn itu d e o f t h e im pedance t e n s o r .T h e peak v a lu e s used in t h e n o r m a l is a t io n s w ere 4 9 6 ,9 0 9 Ohm-m and 6 9 .6 d e a r ee n ra n n e n t.lv e l v .


C onductive Zone

a )

b)

>•Oz,UiDOU)06u.

4525 4475 4425 4375 4325 4275 4225 4175

STATION <m)

Conductive Zone

u2 .UiL3Ouios

2048

P hase A pp Res

0 . 7 7 6 0 . 3 9 1 2

0 . 7 6 0 0 . 2 5 0 9

0 . 7 3 7 0 . 1 3 2 3

0 . 7 J 2 0 . 1 4 4 2

0 . 6 8 8 0 . 1 1 6 0

0 . 6 6 7 0 . 0 9 9 90 . 6 4 5 0 . 0 8 4 4

0 . 6 2 6 0 . 0 7 0 7

0 . 6 0 6 0 . 0 6 0 2

0 . 6 8 1 0 . 0 5 3 0

0 . 5 5 3 0 . 0 4 8 3

0 . 5 2 1 0 . 0 4 0 6

0 . 4 6 9 0 . 0 3 4 6

0 . 4 5 6 | 0 . 0 2 9 20 . 4 2 4 0 . 0 2 4 4

0 . 3 7 9n

0 . 0 2 0 7

0 . 3 2 2 E j 0 . 0 1 7 4

0 . 2 7 4r H

0 . 0 1 4 5

0 . 2 3 9 L J 0 . 0 1 2 0

0 . 1 8 6r H 0 . 0 1 0 0

0 . 1 2 1 ED 0 . 0 0 3 2

0 . 0 4 1 i Sy 0 . 0 0 6 6

• 0 . 0 3 2 s 0 . 0 0 5 5

- 0 . 1 0 0 0 . 0 0 4 3

• 0 . 1 6 9 0 . 0 0 3 3

- 0 . 2 3 5 0 . 0 0 2 5- 0 . 2 9 4 0 . 0 0 1 9- 0 . 3 4 7 0 . 0 0 1 5

- 0 . 4 0 2 0 . 0 0 1 0

- 0 - 4 5 8 0 . 0 0 0 9- 0 . 5 0 0 0 . 0 0 0 8

- 0 . 5 4 8 0 . 0 0 0 7

- 0 . 5 9 8 0 . 0 0 0 6

- 0 . 6 4 9 0 . 0 0 0 5

- 0 . 7 0 2 0 . 0 0 0 4

- 0 . 7 6 1 0 . 0 0 0 3

- 0 . 8 2 6 0 . 0 0 0 2

- 0 . 8 8 9 0 . 0 0 0 1

- 0 . 9 4 6 0 . 0 0 0 0

4525 4475 4425 4375 4325 4275 4225 4175

S TA T IO N lm>

F ig u r e 8 . - Z e r o - s h e a r n o r m a l iz e d a p p a r e n t r e s i s t i v i t y (a ) and p h a s e( b ) . P eak v a lu e s u s e d t o n o r m a l iz e th e s e p r o f i l e s w e re m n # 5 ^ n h m —m an d 9 1 .7 d e a re e s r e s D e c t i v e l v .


Conductive Zone

<Z5&Z

a )

b)

4525 4475 4425 4375 4325 4275 4225 4175

S TA T IO N (m l

Conductive Zone

u z Iu D O' u cc bu

4096

S h e a r T w is t

o.:oi- 0 . 0 2 9

- 0 . 1 6 )

• 0 . 2 3 5

- 0 . 3 U

•o.ss:- o . 4 4 6 - 0 . 4 8 3

- 0 . 5 0 9

- 0 . 5 3 3

- 0 . 5 5 5

- 0 . 5 8 0

-0.6U: -0.G21- 0 . 6 3 4

- 0 . 6 4 4

- 0 - 5 5 4

- 0 . 6 6 6 - 0 . 6 7 0

- 0 . 5 6 8

- 0 . 6 9 8

- 0 . 7 0 8

- 0 . 7 I 9

- 0 . 7 2 !

- 0 . 7 3 f

- 0 . 7 4 0

- 0 . 7 5 5

- 0 . 7 8 5

- 0 . 7 7 :

- U . 7 7 8

- 0 . 7 8 3

- 0 . 7 8 7

- 0 . 7 9 0

- 0 . 7 9 2 - 0 . 7 9 4

- 0 . 7 9 C

• 0 . 7 9 0

- 9 . 3 0 1

- u . b u V

0 . 8 0 5

n.eoiU . B O O0. 940 . 7 8 60 . 7 7 ?

0 . 7 6 2

Q . 7 5 1

0 . 7 3 9

0 . 7 3 0

0 . 7 2 2

a.7i? 0.707 0 . 6 9 9

0 . 6 0 0

0.68:0 . 6 7 2

0 . 6 6 1

0 . 6 4 9

0 . 6 3 7

0 . 6 2 9

0 . 6 1 9

0 . 6 0 4

0 . 5 5 9

0 . 5 5 3

0 . 5 2 0

0 . 4 8 4

0 . 4 3 1

0 . 3 6 1

U . 2 6 0

0 . 1 5 0

0 - 0 5 4

- 0 . 0 7 C J

- 0 . 1 9 8

- 0 . 3 2 7

- 0 . 4 6 0

- 0 . 6 0 0

- 0 . 7 2 1

-w» . 0*JU

4525 4475 4425 4375 4325 4275 4225 4175

S T A T IO N (m l

F ig u r e 9 . - N o r m a l iz e d T w is t (a ) an d S h e a r (b ) f o r t h e " s m a l le r "s o l u t i o n o f t h e Z e r o - s p l i t t i n g 1 -D d i s t o r t i o n a p p ro a c h .P p p V v a l l l P R W P T P r P R n P r H v p l V U I A a n r t 1 1 9 A A a n r o a a


C onductive Zone

STATION (m l

Conductive Zone

4525 4475 4425 4375 4325 4275 4225 4175

S T A T IO N (m l

F ig u r e 1 0 . - N o rm a liz e d T w is t (a ) a n d S h e a r (b ) f o r t h e " b ig g e r " s o l u t i o n o f t h e Z e r o - s p l i t t i n g 1 -D d i s t o r t i o n a p p ro a c h . P eak v a lu e s u s e d w e re 1 0 1 .3 an d 9 9 .0 d e g re e s r o e s n p r t * i v p I v .


FREQ

UEN

CY

(Hz)

FR

EQU

ENC

Y I H

z)C onductive Zone

4525 4475 4425 4375 4325 4275 4225 4175

S TA T IO N (m )

Conductive Zone

4096

S p l i t t i n g T w is t

o.reoo.:u: o . t s s 0 . H 3

I ) . 1 6 4

0 . 1 S 6

0 . I S 2

0 . 1 4 6

D.UO 0 . 1 3 5

0 . 1 3 0

o.i :6u . l 2 'i

. 1 2 0

. 1 16

.1 l b

. 1 1 4

. 112

.lit 0 . 1 0 9

0.10B0 . 1 0 7

0.10$ 0 . 1 0 4

0 ♦ 10 4

0 . 1 0 b 0.102 0.101 o.ioo0 . 0 0 9

0 . 0 9 9

0 . 0 9 8

0 . 0 0 8

0 . 0 9 5

0 . 0 9 7 0 . 0 9 7

0.03$ 0 . 0 92

0 . 7 7 60 . 7 4 9

0 . 7 3 4

0.700 0 . 7 0 90 . 7 0 -

0 . 6 9 3 0 . 6 0 4

0 . 6 7 3

0 . 6 6 3

0 . 6 9 3

0 . 6 4 3

0 . 6 3 3

0 . 6 2 2

0 . 0 0 9 0 . 6 0 8

0 . 5 3 6 0 . 5 7 2

0 . 5 5 9

0 . 5 4 4

0.628 0 . 8 1 2

0 . 4 9 6

0 . 4 8 1

U . 4 6 5

0 . 4 4 9

0 . 4 3 5

0 . 4 2 5 0 . 4 1 4

0 . 4 0 3 0 . 3 9 2

0 . 3 8 2 0 . 3 7 2

0 . 3 6 7

0 . 3 5 1

0 . 3 4 0

0 . 3 2 8

0 . 3 1 7

0.-JU6

4525 4475 4425 4375 4325 4275 4225 4175

S TA T IO N (n il

F ig u r e 1 1 . - N o rm a liz e d T w is t (a ) a n d S p l i t t i n g (b ) o b ta in e d u s in g t h e Z e r o - s h e a r 1 -D d i s t o r t i o n a p p ro a c h . T he p e a k v a lu e s u s e d i n t h e n o r m a l i z a t io n w e re 9 4 .6 an d 9 1 .7 d e g re e s r e s o e c t i v e l v .


C onductive Zone

a)

>■UzUIDOui06u.

b)

4096

4525 4475 4425 4375 4325 4275 4225 4175

S T A T IO N (m )

Conductive Zone

uzUJooUI06U.

2046

(b) (a)5 “ . 0 1 0

n . : 5 0

3 1 . 3 0 0

2 5 . 3 1 0

21 .-190 1 8 . 5 2 0

1 5 . 2 3 0

1 4 . 2 9 0

1 3 . 2 2 0

12.020 1 0 . 7 1 0

9 ..170 8 . 3 5 0

7 . 3 3 0

5 . 4 6 0

3 . 5 4 2

4 . 6 5 0

4 . 1 5 0

3.470 3 . 1 1 0

2 .640 2 . 3 6 0

2 . 2 8 0

2 . 0 2 0 1 . 7 5 0

1 . 5 3 0

1 . 3 0 0

1 . 0 9 0

0 . 8 3 0

0 . 6 7 0

4 7 0

3001700 700500 5 0

0 3 0

ro 010

0 .4

0 . 1

0 . 0

Q . 0 2

I 2 2 . 5 0 0

1 5 . 5 4 0

1 3 . 0 3 0

1 1 . 2 1 0

I 9 . 7 7 0

3 . 5 9 0

7 . 5 9 0

I S . 7 3 0

5 . 9 3 0

1 4 . 9 5 0

4 . 0 5 0

i 3 . 2 5 0

| 2 . 5 7 0

2 . 3 4 0

2 . 1 I D

1 . 6 4 0

1 . 5 9 0

1 . 3 5 0

1 .iso 0 . 9 7 0

o.eoo0 . 6 5 0

0 . 5 3 0

0 . 4 2 0

0 . 3 1 0

0 . 2 1 0 I 0 . 1 4 0

Q . U 8 0

0 . 0 5 D

| 0 . 0 3 0

I 0 . 0 3 0

0 .0 2 0 I 0 . 0 2 0 J 0 . 0 1 0

I 0.010 0 .00 0 0 . 0 0 0

) 0.000 I 0.000

4525 4475 4425 4375 4325 4275 4225 4175

S TA T IO N (m l

F ig u r e 1 2 . - F i t t i n g e r r o r s g iv e n i n p e r c e n ta g e , f o r t h e " b ig g e r " ( a ) a n d " s m a l l e r " (b ) s o l u t io n s o f t h e Z e r o - s p l i t t i n g d e c o m p o s i t io n . L a r g e r e r r o r s a r e o b s e rv e d f o r t h e " s m a l l e r " s o l u t i o n .


Norm

alize

d Tw

ist

Norm

alize

d Tw

ist

GROOM-BAILEY/Twist (Line 8)

0.6

0.2

- 0.2

-0.4- 0.6

- 0.8

1CP102

Frequency (Hertz)

10l

(a)

TRARZ/Twist (Line 8)1 ■ 1 ■i i i i i i i i i 1 i t nun 1i—i 11 m iii- i—r rmn

0.9 - -

0.8•

••

-

0.7 ' + •

+ +-

0.6X

. o • .

0 0 * + •

0.5 - ° X 0 + •

•• • -

0.4 . x + 0 * +* + * v° * .

X i xo 0 +

0.3 -0

S 0 -

0.2 - -

0.1 - -............. ....

10° 10* 102 103 10

Frequency (Hertz)

(C)

GROOM-BAILEY/Shear (Line 8)

0.8

0.6

0.4WC3OJ

JZ 0.2co•o

cdsu - 0.2

-0.4- 0.6

- 0.8

102

Frequency (Hertz)

10>

(b)

TRARZ/Split (Line 8)

0.90.8

0.7txCO■o

0.6

0.5cd

0.40.30.2

0.1

102 103

Frequency (Hertz)

10*

F ig u r e 1 3 . - F r e q u e n c y d e p e n d e n c e o f d i s t o r t i o n p a r a m e te r s T w i s t (a ) a n d S h e a r ( b ) f o r t h e Z e r o - s p l i t t i n g 1 -D d i s t o r t i o n a p p r o a c h , a n d T w i s t ( c ) a n d S p l i t t i n g ( d ) f o r t h e Z e r o - s h e a r s o u r c e p o l a r i z a t i o n a p p r o a c h . ( * ) r e p r e s e n t s s t a t i o n 4 5 2 5 , ( o ) s t a t i o n 4 4 2 5 , ( x ) s t a t i o n 4 2 7 5 , a n d


a surface distortion decomposition for vector csamt … ja 1991.pdf · single source csamt data,...

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