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Transcript of 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
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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
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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
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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.
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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
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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.
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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
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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.
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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
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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.
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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
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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
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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
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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,
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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)
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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-
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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
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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
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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
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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 \( f ) = - iwDi(r) + <rEi(r) + j i ( r ) (2.6)
B1{ r ) = p H i {r) (2.7)
and
D 1(r) = e £ ( f ) (2.8)
Similarly, for some other distribution 2 of the transmitter currents at
the same point r
V x Ez{f) = iuBz{r) (2.9)
V x Hzir) = —iu D iir ) + aE i{r) + .7 2 (f) (2.10)
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20
&(#D = t*H * (?) (2.11)
and
A ( r ) = eE2(f). (2.12)
Combining these two sets of equations by taking the scalar products of
equation (2.5) and (2.6) by H 2 and E 2 respectively and equations (2.9)
and (2.10) by H \ and E\ then combining them and rearranging we get
( tf2V x i a - i i V x H 2) + ( i 2V x H i - t f xV x E 2) +
— H 2E \) -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)
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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
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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
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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
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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
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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
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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
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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)
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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 = °
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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)
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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
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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
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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 \
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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)
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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 =
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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).
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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
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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
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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
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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
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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)
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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,
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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.
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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),
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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.
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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)
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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)
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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
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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
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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.
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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
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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
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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
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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
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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' ^
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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)
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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
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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
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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
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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.
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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
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61
for 1-D earth. A test of these methods using experimental field data is
discussed in the following chapter.
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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.
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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
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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
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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
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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
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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
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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
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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.
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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
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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.
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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
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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.
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A p p e n d ix A
Programs for 1-D Galvanic Distortion
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■vU IN I AITextF ile : Listing:, jte-fl gbn°spl»t 8:17:05 pw
10
------ 5 6
to;“TO-
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
U N J,xText* File. Listing
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. i»1 154 155 ISO 157 ISO 159 .00 lot .02 J03 .04. i«s 00 07 : 08 .. 09 70i n;
" 73 74III 77 70: 79 00
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 )
90 ' 91 '92 1939495 _ 9ft197
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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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)_________________________ ____________
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
U N I X ™Text .File Listing
FILE date 9 /2 5 /9 1 7Sgbnospiit ■nwe 8:17:05 pm
LINE $
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36) * 364
36S >"366 . .367
166 _ 389 _ 370
371 372>.373 _ J74
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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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 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
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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)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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—.033 034zoos— 030— 037_03B
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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_____________
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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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 )
3*1.84284]8446458461847846849850851852853
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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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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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
gbnosplitU N t *
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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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 )_________________________ _
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
> 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) ____________________
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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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
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. 6 2 4 . 625.
. 6 2 6 1 6 2 7 .6 2 8 .6 2 9
630 1 6 3 1 .. 632 .6 3 3
634 . 635
636 1637
638 . 639
640 1641 1842
643 - 644 ^ 8 4 5
1646 “ 1647 e
1646 e “ 1649 ' “ 1650
(651 “ 165 2 3 6 5 3
654 “ 1655 " (6 5 6
1657 “ 1656 ' 659
660 661 662 66 J
“ (6 6 4* I 66S ~ 666
667 1 6 6 8 1669
“ 1670 ’ 671
672673674
’ 67516761677 678
1 6 7 9I8 6 0
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 ____
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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FILE DATt 9/25/91g b n o s h e a r TtMt 8:16:40 pm 90
text
a i
PROGRAM TRAMS
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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 )______ _______
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
:U n
HR l a s t i n g gbnoshearDATt 9 /2 5 /91time 8:16:40 pm 91
22 23.2425 202728 29.3031
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T86 87 88'
. 89 90
1.91 . 9 2. . 93 194 1.95"196197198
9911200" “ 2 o r —202 1_203
204 Z20S
206' _207
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T 2 i r jH.2I2hIT2I3"" 2 1 4 "21 S' “ 216 “ 1217 "2 1 8 "2 1 9 22011221".222
_223"2 2 4"22S"2 2 6
2271226""2 2 9"2 3 0"231"2 3 2
233"2 3 4"2 3 511236"2 3 7
238239 24Q
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 ) ______________________________________
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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)) •
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(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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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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 )
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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 «-
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(f • )(P 8 2 l a a y a o t e a ) (P 8 1 1 a a y a o to a )
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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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A p p e n d ix B
Rotation Angles of Polarization Ellipses
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
95
Let h be the horizontal magnetic field. In general, A is a. com- plex 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)
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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 )
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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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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.
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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).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Conductive Zone
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b)
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0 . 2 3 1 0
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0 . 1 4 3 1
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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
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0 . 0 1 8 1
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0 . 0 1 4 4
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STATION (n i)
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.