DIFFERENTIAL TELLURICS
with applications to mineral exploration
and crustal resistivity monitoring
by
Gerald Alan LaTorraca
S.B. Northeastern University (1965)
S.M. Massachusetts Institute of Technology
(1972)
SUBMITTED TO THE DEPARTMENT OF
EARTH AND PLANETARY SCIENCES
IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 1981
1
Massachusetts Institute of Technology 1981
Signature of Author
Department of Earth and Planetary Sciences
September 25, 1981
S//
Certified by
C
.
by
/
, .
Theodore R. Madden
Thesis Supervisor
Accepted by
Theodore R. Madden
MA Chairman, Deartmental Committee on Graduate Students
JA
.'
". .
h wnes
DIFFERENTIAL TELLURICS
with applications to mineral exploration
and crustal resistivity monitoring
by
Gerald Alan LaTorraca
Submitted to the Department of Earth and
Planetary Sciences on September 25, 1981
in partial fulfillment of the requirements
for the degree of Doctor of Philosophy
in Geophysics
ABSTRACT
The intent of this thesis is to infer the fine
structure of the telluric field from differential telluric
measurements. From the frequency dependence of small scale
differential measurements, we can infer the presence of
of
induced polarization targets. From the stability
differential measurements, we can infer variations in the
state of stress and strain in the crust and from the
frequency dependence of large scale differential telluric
measurements, we can infer the spatial variation of the
thickness and apparent conductivity of the upper crust.
Tellurics are the electric fields induced in the
earth by the large scale motions of charged particles
outside the earth's atmosphere. The fine structure of the
telluric field is contained in the tensor relationships
vector electric field measurements. To gain
between
tensor
these
of
properties
the
into
insights
relationships, we apply the shifted eigenvalue analysis of
Lanczos(1961) to not only the telluric tensor but also to
the associated impedance tensor relating magnetotelluric
fields.
For the telluric tensor, the products of the
eigenvalues and one set of electric field eigenvectors
represent the maximum and minimum electric fields that can
be produced by unit electric fields aligned with the
second set of electric eigenvectors. Similarly, for the
impedance tensor, the products of the eigenvalues and
electric field eigenvectors represent the maximum and
minimum electric fields that can be produced by unit
magnetic fields. The magnetic eigenvectors represent the
magnetic field geometry that yields the electric field
extrema. The conventional skew is shown to be the tangent
of the angular deviation of the electric and magnetic
eigenvectors from perpendicular.
telluric
differential
of
sensitivity
The
in
crustal
variations
induced
stress
measurements to
stability of the telluric
conductivity is studied. The
are found,
parameters, eigenvalues and skew,
tensor
respectively, to be sensitive measures of the stability of
anisotropy of
the crustal conductivity and the effective
the degree of
conductivity. Additionally,
crustal
the
crust is
upper
the
telluric current saturation within
found to be the most important factor in determining the
sensitivities of differential telluric measurements.
induced polarization
an
The conductivity of
target (ore body) varies with frequency relative to its
periods)
frequency (<10 second
Low
surroundings.
telluric measurements are used to infer the
differential
field
telluric
the
frequency dependence of
relative
and within .the effective boundaries of the ore
outside
in
expressed
body. This relative frequency dependence is
terms of telluric tensor eigenstates.
Thesis Supervisor: Prof. Theodore R. Madden
Title: Professor of Geophysics
iii
_
ACKNOWLEDGEMENTS
been
I have
years,
past eleven
the
During
fortunate to make many friends in the Dept. of Earth and
to
Planetary Sciences and I shall miss them all. I came
EPS to work for Gene Simmons. Gene has given me a great
deal of trust and responsibility and has encouraged me to
get my Phd. I thank him for his friendship, assistance and
advice.
I worked for many years with Frank Miller. Frank
has been a close and loyal friend as well as a continuing
for all the
I thank him
technical advice.
source of
kindness and supporst he has shown me.
Bill Fairing,
I wish to'thank Larry Bannister,
for Space
Center
the
of
Baker
and Dick
Calileo,
Dan
Bob
advice.
technical
and
friendship
Research for their
Hirst,
Jock
Riach,
Dave
Keough,
George
Walsh,
Stevens, Joe
and Jimmy Byrne have all been helpful and friendly to me
and I thank them.
help
her
I wish to thank Debby Gillett for all
upcoming
her
in
well
her
wish
and
years
the past
over
marriage to Steve Roecker.
I wish to thank the following professors in the
department who have contributed greatly to my education in
Brace, C. Burchfiel, J.
W.
K. Aki,
the geosciences:
and
Dickey, H. Fairbairn, S. Hart, W. Pinson, S. Solomon,
N. Toksoz.
My officemates and their spouses have taught me
a great deal and I shall miss the give and take of ideas
contributed so
(plus the occasional argument) which have
much to my education. My officemates in the order in which
they left are: Rambabu Ranganayaki, Adolfo Figueroa-Vinas,
Frances Bagenal, and Earle
Olu Agunloye, Dale Morgan,
Williams. Steve Park and John Williams are still here. To
A special thank you
all, my most heartfelt thanks.
you
Park and
Steve
Dale Morgan,
goes to Earle Williams,
for their assistance in my field studies
Frances Bagenal
in California and Massachusetts.
Many other graduate students have contributed to
I wish
my education and the enjoyment of my stay at MIT.
Shamita
Evans,
Brian
,
Fehler
Mike
Roecker,
to thank Steve
Karl
Suarez,
Gerardo
Shakal,
Tony
Zandt,
George
Das,
and
Coyner, John Bartley, Alan Zindler, Hubert Staudigel,
Julie Morris.
Rosenstein,
I wish to thank Judy Stein, Shelley
help and
their
all
for
Roos
Judy
and
Brydges
Sara
years.
the
through
friendship
Pam and Stan Hart have been very kind to my wife
Ana and me. Thanks to you both. I have spent many a Friday
evening with Lynn and John Dickey at their home in Beacon
I wish to thank them for their hospitality and
and
Hill
friendship.
Ted Madden has been my advisor and friend since
I returned to graduate school. Ted, his wife Halima and
their children have extended their warm hospitality to me
Ted's influence
through the years and I shall miss them.
used the
and I have
is considerable
thesis
on this
thesis to reflect this
the
throughout
"we"
editorial
than I thought
further
influence. Ted has extended me
I am very grateful for the education he has
possible aud
given me.
Beatrice and John
I wish to thank my parents,
their loyal support and encouragement. I
LaTorraca, for
they have
all
for
wish to dedicate this thesis to them
done for me.
Ana
I wish to thank my dearest friend and wife,
during the
Silfer, for her $support and encouragement
are
writing of this thesis. Her patience and selflessness
deeply appreciated.
the
the USGS and
aid from NASA
Financial
Planetary Sciences has been
and
of Earth
department
and I gratefully
accorded to me as a graduate student
acknowledge this assistance.
TABLE OF CONTENTS
Title
Abstract
Acknowledgements
Contents
List of Figures
List of Tables
Glossary of terms
ii
iv
vi
ix
x
xi
1
Chapter 1: Introduction
1.0 History and development of magnetotellurics
pertinent to this study
4
1.1 Induced Polarization (IP) techniques
5
1.2 Thesis content by chapter
6
Chapter 2: Eigenstate Analysis
2.0 Introduction
2.1 The Eigenstates of the impedance and telluric
7
tensors
16
2.2 Telluric tensors
Chapter 3: The Sensitivity of Telluric Field Measurements
21
to Stress
3.0 Introduction
3.0.0 History and overview of
resistivity monitoring with tellurics
3.0.1 Chapter Content
26
3.1 Electrical Properties of Rocks under Stress
3.1.0 Content
3.1.1 Electrical Conductivity Mechanisms
3.1.2 Conductivity: Stress-Strain
27
relationships
31
3.2 Telluric Cancellations
3.2.0 Nature of the low frequency
telluric field
3.2.1 Electronic determination of
telluric tensor relationships
3.2.2 Hollister and Palmdale array
32
measurements
3.2.3 Telluric cancellations and
34
magnetotelluric eigenstates
36
variations
Interpreting
3.2.4
39
Analysis
Sheet
Thin
Generalized
3.3
3.3.0 Introduction
3.3.1 Theoretical Basis
41
3.3.2 The Numerical Grid
43
3.3.3 Block Conductance versus stress
3.3.4 Telluric Current Saturation
44
conditions
11.-~1111~.-I.II_~LYI I
~~_l__~
~__~_
X1~
^ _l___l_~_lmlll^~_il^--
3.4 The Palmdale Thin Sheet Model
3.4.0 Content
3.4.1 Data Constraints
3.4.2 Conductance Assignments and the"
Base Crustal Model
3.4.3 Base Model Eigenstates
3.5 Crustal Model Stress Sensitivity
3.5.0 Introduction
3.5.1 Structural Control of the
Telluric Currents
3.5.2 Eigenstate Sensitivity Measures
3.6 Sensitivity Analysis: Results
3.6.0 Introduction
3.6.1 General Results
3.6.2 Isotropic versus Anisotropic
Conductivity variations
3.6.3 Conclusions
Induced Polarization with Telluric Fields
4.0 Introduction
4.1 Telluric Measurement Geometries for
discerning IP targets
4.2 Telluric Field Measurements near an IP tar get
4.2.1 Previous geoelectric measureme nts
near Harvard, Mass.
4.2.2 Data Acquisition
4.2.3 Data Analysis
4.3 Telluric Field Measurements near Salinas, CA
4.4 Summary and Conclusions
4.4.1 Summary
4.4.2 Conclusions
Chapter 5: Thesis Summary and Extensions
5.0 Summary
5.1 Lateral Variations in Crustal Conductivity
5.2 Suggestions for Future Study
Chapter 4:
REFERENCES
Appendix A: Equipment and Field Procedures
A.1 Field Equipment
A.2 Telluric Field Measurements
A.3 Magnetic Field Measurements
Appendix B: Impedance Eigenstate formalisms with a
numerical example
B.0 Introduction
B.1 Orthogonality Conditions
B.2 Algebraic form of the Impedance Eigenstates
B.3 A numerical example of the
eigenstate formulation
Calculations; An Approximate Form
Impedance
C:
Appendix
behaviour
phase
In
C.0
C.1 Impedance calculations
Appendix D: Poloidal mode response to embedded ellipsoids
D.0 Introduction
D.1 Telluric Tensors near a conducting spheroid
D.2 Tensor eigenstates
vii
46
49
54
60
62
66
68
73
78
84
88
90
98
98
103
103
104
108
110
112
112
113
114
120
120
120
122
124
135
135
137
154
154
156
162
D.3 Prolate and oblate spheroid calculations
D.4 The current saturation condition
Appendix E: Crustal Conductance Calculations
E.O Introduction
E.1 Block Conductance Calculations
E.2 Block Conductance versus Stress
Biography
Viii
165
168
173
173
173
176
182
I--
List of Figures
Page
22
23
Palmdale Array
33
Palmdale Array Signals
Approximate Eigenvector Directions
37
42
3.5 Thin SheetNumerical Grid
47
3.6 Magnetotelluric data (from Reddy et al.)
51
3.7 Conductivity Model for Palmdale
52
3.8 Perturbed Region
55
3.9 Thin Sheet Model Eigenvector Directions
56
3.10 Magnetic Eigenvectors
58
3.11 Telluric Eigenstates
69
3.12 Isotropic Eigenvalue Sensitivities
70
3.13 Anisotropic Eigenvalue Sensitivities
71
3.14 Anisotropic Skew Sensitivities
74
4.1 Equivalent Circuit for mineralized rock
80
4.2 Four dipole telluric tensor geometry
80
4.3 Three dipole measurement geometry
83
4.4 Measurement Sensitivities
85
4.5 Location Map for Harvard, MA
4.6 Magnetotelluric Survey Map (from Davis,1979) 87
89
4.7 Dipole location at Harvard, MA
91
4.8 High Coherency Recording
91
4.9 Low Coherency Recording
4.10 Eigenvalue frequency dependence for
95
Harvard, MA telluric data
96
4.11 Eigenvector directions versus frequency
99
4.12 Location Map for Salinas, CA
99
4.13 Salinas Telluric Data
4.14 Eigenvalue frequency dependence for
100
Salinas, CA data
100
4.15 Eigenvector directions for Salinas site
107
signals
Cancellation
Frequency
High
5.1
116
A. 1 Telluric Cancellation System
117
A.2 Electric Field Preamplifier
118
A.3 Bessel Filter
119
A.4 Magnetic Field Preamplifier
134
B. 1 Magnetotelluric Eigenstates
145
C.1 Inpedance Estimates for LH2
147
C.2 MT Eigenstates for LH2
148
C.3 Raw Signals at LH2 site
150
C.4 Impedance Estimates for PH1
151
C.5 MT Eigenstates for PH1
152
C.6 Raw Signals at PH1 site
173
E.1 Block Geometries
177
E.2 Stressed Block
177
E.3 Crack Response to Stress
179
E.4 Sub Block Stress
Figure
3.1
3.2
3.3
3.4
#
Hollister
II1__________
_i~
rjiC nl.LXTI.~YPLIF
IIII~-~
List of Tables
3.1 Estimated Resistivity: Stress-Strain
relationships
D.1 Prolate Spheroid
D.2 Oblate Spheroid
29
170
170
_Y-^---L--I
LI~-_i__
-~~i~^_t~il
-1I--~-~1------~I I-~Y_~X~L~-I-~--~
GLOSSARY OF TERMS AND ABBREVIATIONS
1D, 2D, 3D
the number of model dimensions along which the
conductivity can vary
p
a
resistivity
6
can be either the skin depth or a variational
depending upon context
eigenvalue or, in the ellipsoid analysis, a dummy
variable
conductivity
X
(ohm-meters)
(ohm-meters)-1
A
eigenvalue matrix
e.
electric field eigenvectors
h.
magnetic field eigenvectors
1eigenvector
Ui
matrix
eigenvector matrix
(~)
complex conjugate transpose
(*)
( )T
conjugate
transpose
J
current density .(amps/m 2 )
E
electric (telluric) field
H
magnetic field (amps/meter)
a..
-1
elements of conductivity tensor (ohm-meters)
Z
impedance
T
telluric tensor (mv/mv)
MT
magnetotellurics
H
time derivative of the magnetic field (amps/meter/sec)
Z
modified impedance (mv/km/y/sec)
IP
induced polarization
DC
Direct Current
(ohms) or
(volts/m)
(mv/km/y)
(electrostatic)
CHAPTER 1
Introduction
pertinent
1.0 History and development of magnetotellurics
to this study
to the western world by showing that for
magnetotellurics
conductivity
fields could be used to infer the electrical
beneath the measurement system. Improvements in
structure
the
the development of computational
with
along
interaction
magnetotellurics (MT)
schemes necessary for the making
Cantwell (1960),
the
modelled
as
elements
between
the
is
Additionally,
tensor
the
minimizes
the
diagonal
of
terms
principal
of
the
the
axis
rotation
coordinate
terms
of
field
magnetic
and
electric
calculation
in
defined
directions are defined in terms of a
which
be
can
two dimensional, the impedance is recognized
impedance
measurements.
by
and
Sims
structure
conductivity
earth's
as requiring a tensor description and the
coherencies
a
and Swift (1967) to name a few. With these
Bostick(1962),
studies,
Nelson (1964),
and
Madden
published
been
have
tool
geophysical
practical
source-earth
the
of
nature
the
of
understanding
the
magnetic
to
electric (telluric)
the
of
ratios
electromagnetic
earth's
the fluctuating component of the
field,
of
study
the
initiated
Cagniard (1953)
impedance
tensor.
introduced
Berdichevskii (1960)
geophysical
literature
the
idea
to
the
of using low frequency
differential telluric measurements to
variation
of
the
conductivity
the
infer
thickness
as
product
the
conductance of the upper crust. He modelled
field
spatial
or
telluric
the response of the earth to a constant current
source implicitly assuming that no resistive
coupling
of
current exists between the upper crust and mantle.
Two
more
Madden (1979)
and
extended
our
dimensional
recent
papers
Eggers(1981)
understanding
structure
on
of
telluric
and
Ranganayaki
have,
the
the
established the need for an
impedance
by
respectively,
effects
of
three
magnetotelluric field and
eigenstate
analysis
tensors.
Ranganayaki
Madden (1979) have introduced
and
a
generalized
of
thin
the
and
sheet
approach to model the earth's crust. In their studies they
point
low
out the effects of regional structure on the local,
frequency
telluric
magnetotelluric
leaking
currents
field.
to
and
They
from
find
in
the
telluric
field.
and
is
equal
to
conductance of
the
upper
crust
thickness
distortions
The distance required for these
distortions to diminish by 1/e is
distance
mantle at
the
lateral changes in crustal conductivity cause
that
called
the
the
square
times
the
adjustment
root
of
resistivity
product of the lower crust. Two consequences of
this result are that the telluric current system is not
constant
the
current
Berdichevskii (1960)
frequency MT data,
source
on
a
assumed
and
that
large
to
a
scale
as
analyze
low
the crustal model must be of dimensions
larger than the adjustment distance. As part of this
much
to
measurements
telluric
crustal
induced
stress
need
conductivity variations. Accordingly, we
to
use
a
model of the magnetotelluric response which the
realistic
sheet
analysis
pertinent
to
thin
generalized
recent
differential
of
sensitivity
thesis we seek to infer the
paper
approach
computational
which
involves
studies
our
ability
our
improves
second
The
provides.
a
to
discern fine structure in the telluric field.
Eggers (1981),
has
Geophysics,
pointed
rotational
approach
directions
of
to
in
a
out
the
submitted
paper
determine
incompleteness of the
the impedance tensor and has suggested the
the
of
impedance
tensor.
conventional eigenvalue approach is the
the
real tensors. Because the impedance and
fortuitously
are
only
the
completely
determine
general
in
the
Implicit
requirement
telluric
approach
of
that
development
computational
telluric
use.
tensors
Lanczos (1961)
to
of both tensors. We feel that
in
the
magnetotelluric
and
our eigenstate analysis is the next logical
widespread
the
Hermitian, we have elected to use
eigenstates
the
differential
analyze
be analyzed be Hermitian or symmetric for
to
tensor
axis
principal
the
use of a conventional eigenvalue approach to
properties
to
of
the
techniques
and
step
should
find
The eigenstate scheme is useful also for
the inference of buried induced polarization (IP)
from the fine structure of the telluric field.
targets
1.1
Induced Polarization (IP) techniques
of
resistivity
as
rocks
the
of
indicator
an
bearing minerals.
ore
of
presence
electrical
an
technique which uses the frequency dependence
prospecting
of the
is
Polarization
Induced
The IP technique was
first used extensively by the geophysical group of Newmont
Exploration, Ltd. (Cantwell and Madden,1967) in the
important
an
ore
of
in
the
In
the
the IP technique, active sources are used
of
to measure the frequency dependence
presence
tool
mineralization.
sulfide
prospecting for copper
application
is
IP
Presently,
1950s.
early
associated
with
The active source technique is
bodies.
limited by inductive coupling at frequencies greater
10
the
than
Hz and by telluric noise for frequencies less than 0.1
Hertz. In order to find ore bodies
meters,
attempts
have
deeper
than
tens
of
made to reduce the telluric
been
noise (Halverson,1981) in the active measurements. Instead
Madden (1979)
of removing the telluric fields,
suggested
using the telluric field directly to infer the presence of
buried
IP
targets.
In
this
thesis,
we
consider
feasibility of Madden's hypothesis and develop
to
implement
techniques
IP prospecting with tellurics. We feel that
the inference of IP targets
shows
the
considerable
promise
with
differential
tellurics
for detecting deeper targets
than can be inferred with active measurements and may also
lead to the discrimination between minerals because of the
extension of the frequency bandwidth to much lower periods
_~__j___l____ll^__slL~_il~C--iYi--I~B1~ -i~111-TI*~--ll~-4LI.
(Morgan ,1981).
than can be used with active measurements
1.2 Thesis content by chapter
of
analysis
sensitivity
present
of
we
4,
our
measurements
the
of
studies
the
differential telluric measurements to infer
of
ore
of
conductivity.
crustal
in
present
analyses
our
telluric
differential
stress induced variations
Chapter
to the impedance and telluric
Lanczos (1961)
tensors. In Chapter 3, we
eigenvalue
shifted
the
In Chapter 2, we apply
use
the
to
In
of
presence
bodies and in Chapter 5, we describe our progress
in determining large
telluric
scale
measurements,
summarize
differential
from
structure
results
our
and make
suggestions for further study.
and
we
procedures
present
procedures
used in our field studies. In Appendix B,
the
eigenstate
relationship
between the
a
numerical
example
and
establish
the
of
conventional skew and the angle between the
magnetic
equipment
field
In Appendix A, we describe the
In
eigenvectors.
Appendix
approximate technique for determining
C,
the
electric
and
present an
we
low
frequency
impedance tensor and describe noise suppression techniques
used
in
the analysis of bandlimited MT data. In Appendix
D, we present our three dimensional model of
IP
target
an
embedded
and in Appendix E, we describe procedures used
to model the effective conductivity and stress sensitivity
of crustal blocks used
Chapter 3.
in
the
thin
sheet
analysis
of
Chapter 2
2.0 Introduction
Throughout
the
of
properties
thesis
this
earth's
field.
earth properties are the
and telluric tensors which relate respectively.
the electric
E
to
separated
the
magnetic
vector
E
H
vector
fields.
The
properties of these tensors can be expressed in
tensor
electrical
magnetotelluric
The numerical expressions of the
spatially
infer
crust based on models of the
interaction of the crust with the
impedance
we
fields
and
fundamental
terms
of
eigenstates: eigenvalues and eigenvectors. In this
chapter, we formulate the eigenstates of the impedance and
telluric and tensors using the shifted eigenvalue analysis
of Lanczos (1961).
The concept of tensor eigenstates is the
thread
through
each
common
part of the thesis. The eigenstates
yield insights into the physical
meaning
of
the
tensor
elements and allow the study of variations in the telluric
tensor otherwise hidden.
.
-_-L-^e-I111
1L
XIYLI~-l III-IIPX~~-lli~i--LI--
Illi~l~.~-~-*----
2.1 The Eigenstates of the Impedance and Telluric Tensors
is
The magnetotelluric surface impedance, Z,
a
which relates the horizontal magnetic and electric
tensor
separations from the order
measurements
to
tens
measurements
such
as
near
arrays
of
of
one
kilometer
for
kilometers
Madden's
Hollister
require electrode
measurements
field
electric
However,
resistivity
Ca.
and Palmdale,
point.
a
at
defined
are
tensors
components. Normally
is
field
electric
horizontal
the
relate
to
formulated
T
tensor
fields at the earth's surface. The telluric
for
local
large
scale
monitoring
Kasameyer(1974)
and Swift(1967) have analyzed the difficulties in applying
tensor analysis to impedances
using
long
not spanning both sides of a two dimensional contact,
impedance
could be treated as a tensor.
analyzed separately the rows of the
telluric
each
with
line
other.
techniques
associated
line when each line covered diferent
associated
authors
used
two
dimensional
modelling
to analyze their data but pointed out that the
approximation of considering long line telluric fields
point
measures
axes.
as
can lead to full tensors in 2D structures
even when the impedance tensor has
principal
with
constrain the estimates associated with the
to
Both
the
Kasameyer (1974)
impedance
structures. He then used the Z estimates
one
line
Swift(1967) showed that if the lines were
data.
telluric
obtained
In
this
been
rotated
to
its
chapter we shall consider the
eigenstates of the impedance and telluric tensors with the
of
extension
its
use.
normal
this
consider the effects of
the
of
use
our
that
realization
In
tensor
3 we shall
Chapter
our
on
assumption
and
direct relation between the magnetotelluric
first
consider
us
let
an
of long line telluric data. Because of the
interpretation
tensors,
is
tensor
term
telluric
the eigenstates of the
eigenstate
these
impedance tensor. Later we shall extend
concepts to the telluric tensor.
For
model,
in
earth
the
the
which
as a function of depth, the 1D
only
varies
conductivity
of
model
a
the impedance degenerates to a simple
For
scalar.
the model of a 2D earth for which conductivity varies with
depth
one lateral direction. the impedance tensor is
and
parallel
impedance
to
perpendicular
and
conductivity
direction along which the
is
strike
the
constant.
are
often
to infer geologic structure
sufficient
lower
from estimates of the impedance tensor. However, at
and
frequencies
in
geologically
three dimensional modelling
of
heterogeneous regions,
magnetotelluric
data
is
Additionally, Eggers(1981) has pointed out the
need for a more general approach to the
impedance
the
of
than about 1 Hertz, one or two dimensional models
necessary.
In
homogeneous areas and at frequencies greater
geologically
earth
the
representing
elements
diagonal
off
reduced to two
tensor
Z
by
showing
that
approach produces ambiguous principal
resistivities
because
much
of
the
analysis
of
the
the 2D rotational
axes
and
information
apparent
in the
__/__1__ICL1~
Specifically,
ignored.
impedance tensor is
Eggers(1981)
notes that the rotationally defined apparent resistivities
are
to
insensitive
the
along
diagonal
set
parameter
the addition of an abitrary constant
of
that
and
Z
conventional
the
incomplete. A more general analysis of
is
the impedance tensor can be accomplished by application of
the "shifted eigenvalue"
by Swiftt1967).
suggested
apply
actually
to
first
of
analysis
Lanczos(1961),
as
Eggers(1981). however, was the
eigenstate
analysis
to
the
impedance tensor. Eggers' paper has not yet been published
and
may be changed. Presently, he is using a conventional
eigenstate
analysis
approach
valid
a
is
which
for
Hermitian matrices but can lead to defective matrices when
applied to non Hermitian matrices. The impedance tensor is
rarely
Hermitian.
Accordingly,
eigenstate
idea of
using
shifted
eigenvalue
we
analysis
approach
of
shall follow Eggers'
but
shall
Lanczos(1961)
use
the
which is
completely general and can be applied to all matrices.
In the frequency domain, the impedance tensor
Z
is a complex, non Hermitian matrix relating the horizontal
electric
and
magnetic fields on the earth's surface such
that:
(2-1)
E = Z H
The tensor itself is found from
statistical
averages
of
fields measured in a specific coordinate system e.g. X and
Y and can be written as:
ZZxx
Zxy
Zxy
Zyx
Zyy
(2-2)
self
not
Because Z is non Hermitian. it is also
adjoint
i.e.
Z
4 Z
(2-3)
where the tilde represents complex conjugate transpose. To
find the eigenstates of a non Hermitian matrix such as
Z,
Lanczos suggests the use of the augmented matrix form:
0
(2-4)
S =
where
S
is
a
can
transformation
matrix
Hermitian
be
found
principal
whose
the
through
axis
eigenvalue
equation:
1 w
Sw=
where (A)
is
corresponding
a
real
(2-5)
eigenvalue
eigenvector.
S
of
each
eigenvalue.
We
designate
two
the
eigenvector as (u) and the magnetic field
(v).
Then.
(w) can be written as:
w
is
the
with equation 4,
Consistent
the augmented eigenvector w consists of
for
and
eigenvectors
electric field
eigenvector
as
U
w =
(2-6)
V
with (u) an eigenvector in the column space of Z
and
(v)
an eigenvector in the row space of Z. From 4 and 5 we note
that:
Z v
Xu
(2-7)
Z u
v
=
Multiplying both sides of 7 respectively by
Z
and
Z
we
find
ZZ v :
v
(2-8)
2
ZZ u=
Thus,
the
(u)
independently.
and
u
(v)
Arranging,
eigenvectors
the
columns in the matrices U and V
can
be
found
normalized (u) and (v) as
and
the
eigenvalues
elements of the diagonal matrix (--) we can expand 7 to:
Z V = uA=j
(2-9)
Z U = V.
as
as
orthogonal
and
normalized
two
the
B,
As shown in Appendix
are
eigenvectors
(u)
are the (v) eigenvectors.
Accordingly:
These eigenvectors form complete sets.
(2-10)
V V = I
Combining 9 and 10, the formal eigenstructure of Z is:
v
u, u2
V =
Z = U
(2-11)
One problem not addressed explicitly by Lanczos is how the
formulation. This phase problem arises because
eigenstate
in
the u and v eigenvectors
to
results of equation 8 in equation 7
between
the
find
phases
u and v eigenvectors as suggested implicitly
the
by Aki and Richards (1980).
a
require
would
still
these
phases
suggest
in
complete this analysis, we could use the
To
8.
equation
eigenvectors
the
decouple
we
when
lost
truly
not
are
6
equation
constraints that exist between u
phase
The
independent.
and v are
the
in
assigned
phases of complex tensors such as Z are
to
assigning
to
eigenvectors
However,
set of conventions for assigning
eigenvectors.
the
the
the
approach
an
such
phases
Alternatively,
between
eigenvalues.
the
The
u
and
we
v
resultant
eigenstructure has the natural separation of the magnitude
and
phase
of
the
impedance from the principal axis and
polarization ellipticity information in the eigenvectors.
requires
Allowing the eigenvalues to be complex
a
simple modification of the Lanczos analysis. Equation 5
is modified to:
S w =
w
(2-12)
where (*) denotes complex conjugate. As before the u and v
must
eigenvectors
in
described
'obey
the
B.
Appendix
condition
orthogonality
complex
With
eigenvalues,
equation 7 becomes:
Z v =
u
(2-13)
Z u = )v
and, consequently. equation 8 is changed to:
Z
v = vIV
(2-14)
Z Z u =
I, u
we
With equations 12 and 13,
eigenvalues
retaining
ellipticity
functions
the
information
in
the
invariant to coordinate transformation.
phase
shift
To
end
this
at
ellipse.
The
four
the
of
its
then
are
peak
eigenvectors
defined in terms of four points in space and one point
time.
relative
we
the eigenvectors calculated from equation 14
so that at t=O each eigenvector is
polarization
these
of
phases
the earth properties and
of
only
to
the
eigenvectors. We wish also to make
eigenvalues
phase
assign
can
Coordinate
positions
transformations
of
these
points
change
nor
neither
the
in
the
phase
_~I_^_L___ _1*_ _1~__ _(~I_~
_
differences
shifted
phase
these
With
them.
between
eigenvectors, the phases of the eigenvalues are calculated
the
reflect
difference
phase
(u) and
the
between
eigenvectors at their respective peak magnitudes
.therefore,
well.
invariant
formulation
we
are
as
three dimensional measures of
infer
can
(v)
eigenstate
modified
this
with
Additionally,
and
transformation
coordinate
to
phases
eigenvalue
The
13.
or
11
with either equation
structure from the eigenvectors.
expect
In general, we can
eigenvectors
structures and the
to be controlled by local
magnetic field eigenvectors controlled both by
regional
structures.
Chapter 3, we find that
field
electric
the
local
and
our theoretical 3D modelling in
In
to the
current funnelling parallel
eigenvectors
coastline causes the near coast magnetic
to
be aligned perpendicular and parallel to the coastline but
further
control and
electric
the magnetic eigenvectors return to local
inland
field
near
perpendicular
relationships
the
eigenvectors.
For a 2D earth model, the electric and
eigenvectors
with
are
eigenvector
(u,)
counterpart
(vi).
linearly
is
perpendicular
Accordingly,
individual eigenvectors and the
perpendicularity
each electric
and
polarized
to
magnetic
its
magnetic
the ellipticities of the
skew
or
deviation
from
of the electric and magnetic eigenvector
directions are 3D measures of structure.
This eigenstate
formalism is
14
consistent
with the
notion that resistivity structures can cause the deviation
known extremum properties of these eigenvectors
the
From
field.
magnetic
the
to
of current away from the normal
(Lanczos, 1961),
we
eigenvectors
as
the
possible for
a
unit
eigenvectors
as
the
electric
the
interpret
can
maximum and minimum electric fields
and
field
magnetic
the
magnetic
magnetic field geometry that yields
the electric field 6xtrema.
The calculation of
conventional
the
with
asignments
the
of
its
relationship
included in Appendix B.
is
skew
conventions
the
Additionally. ellipticity and
for
and
skew
necessary
and ellipticity phases to
signs
individual eigenvectors are included in Appendix
B
along
with a numerical example of the eigenstate procedures. The
formalism
algebraic
of
eigenstates in terms of the
the
elements of Z are included in Appendix B as
applications can be found in Chapter
thin
sheet
analysis
California as well
telluric
the
shifted
is
eigenvalue
tensors.
15
for
Hollister
as in Appendix C
MT field data from Palmdale
extend
near
3
well. Further
the
and
crustal
Palmdale,
where the analysis of
described.
Now
let
us
analysis to the study of
2.2 Telluric Tensors
The telluric
tensor
T
is
formulated
in
the
frequency domain to relate two electric fields either from
single region or from separate regions. T is a function
a
of
the
geologic
measurements.
beneath
structure
The
of
form
T
both
field
induced from the
be
can
E
For a vector set of E,H measurements:
impedance tensor.
E, =
Z, HI
(2-15)
and for a second vector set:
Ez =
ZZ HZ
which
current
the
reflects
horizontal magnetic
spatial
field
due
and
finite
channelling
can
and (H2)
The magnetic fields (H,)
tensor
(2-16)
be
variation
structurally
to
by
related
of
a
the
induced
source wavelengths. The
form of this relationship is:
H, =
TH H
(2-17)
z
Babour et al.(1976), and Swift(1967) have
low
frequencies
and
middle
shown
latitudes,
varying
magnetic field tends to be a slowly
the
that
at
regional
function
of
position. Thus, for moderate measurement site separations,
(TH)
is
essentially
diagonal
and
nearly
equal to the
identity matrix. The telluric tensor equation relating
and (E
) ,
then can be written in
the form:
(E,)
-1
-I
E, = Z, TH Z- E 2 = T E 2
Z,
and
Zz
properties
are
of
the
the nature of
eigenstate
functions
the
form
of
earth.
the
the
regional
and
local
To gain further insights into
telluric
of
(2-18)
tensor
T
let
(Z,
impedances
)
us
and
use
the
(Za) in
equation 15 with:
z
: U M XMV "
(2-19)
Z z = UM-
where (U
m
)
\ VM
is the E field eigenvector matrix,
eigenvalue matrix and (V m ) is
matrix.
Here
the
H
field
(P4)
is
the
eigenvector
subscript M designates magnetotelluric
the
eigenstates. With equation 19, we can expand
equation
15
to the form:
I
U-M:
d
"J
I
".J
Vm TM V,(Aa)U
EE
(2-20)
When the magnetic field and magnetic eigenvectors
do
not
vary with position, the eigenstates of the telluric tensor
become:
z,
t , MA--(Am
l- m
(A
mMM
(2-2t)
with:
-T
where I is the identity
maximum
the
and
T
-
matrix.
(z-22)
Additionally,
when
the
minimum resistivity directions correspond to
directions
of
maximum
and
minimum
contrast
in
resistivity:
I
(2-23)
I
and
Z
C2 -24)
= C(M7
and the telluric eigenvalues are related to the
ratio
of
their magnetotelluric counterparts such that:
M
7
(iL~~)
The telluric tensor eigenstates
subscript
eigenstates.
T
is
2-
'C
are expressed as:
L
-f~Ci
where the
(
used
( 2 -
to
designate
C)
telluric
This eigenvector correspondence is true for
two dimensional structures
within
which
the
horizontal
magnetic field does not vary.
eigenvalues
(-A
of
T)
are
For
such
ratios
structures,
of impedances and the
eigenvectors at one position are orthogonal to each
and
parallel
to
the
other
their counterparts at another position.
Additionally,
the
2D
perpendicular
and
parallel to the strike direction, i.e.
along the directions
contrast.
eigenvectors
of
maximum
Correspondence
can
will
and
also
be
minimum
occur
aligned
impedance
when
the
direction of the maximum change in resistivity is
aligned
with
the
maximum
locally two
However,
resistivity
dimensional
for
more
but
direction. Such cases are
can
complicated
exhibit
small
skews.
structures, the magnetic
eigenvectors can
vary
eigenvectors
the magnetotelluric and telluric tensors
of
with
position
and
the
electric
need no longer be in one to one correspondence.
The spatial
eigenvectors
can
variation
tensors
for points
within
centered
near
the
magnetotelluric
be seen in our theoretical modelling of
the eigenstates of the
Impedance
of
impedance
and
an
tensor
in
Chapter
their eigenstates were calculated
area
Palmdale, CA.
of
270
kilometers
squared
For the tensor calculations
we used the generalized thin sheet approach,
a
quasi
analysis, devised by Ranganayaki and Madden (1979).
this
geologically
eigenvectors
3.
complicated
tend to vary with
region,
position
the
along
3D
Within
magnetic
with
the
electric eigenvectors but when current funnelling occurred
the
magnetic
eigenvectors did not vary spatially and the
telluric
eigenvectors
corresponded
locally
to
their
apply
this
magnetotelluric counterparts.
In
eigenstate
the
following
analysis
chapters,
we
to infer the sensitivity of telluric
tensors to variations in crustal conductivity and to infer
the presence of induced polarization targets based on
the
frequency dependence of the telluric tensor eigenstates.
CHAPTER 3
The Sensitivity of Telluric Field Measurements to Stress
3.0 Introduction
with
3.0.0 History and overview of resistivity monitoring
tellurics
Since
19.72,
investigating
Prof.
means
of
Madden
of
predicting
MIT
has
been
earthquakes
using
changes in the electrical properties as
precursors.
these
a
studies,
Madden
has
devised
From
technique
for
monitoring resistivity in the crust based on the stability
of
the
tensor
measurements.
relationships
Currently,
between
he
has
telluric
field
two arrays of telluric
measurement dipoles operating on a continuous
basis.
array
of
is
centered
in
California where the
the
San
Hollister
Andreas
merge as shown Figure 3.1.
area
and
One
central
Calaveras
Faults
The second array is centered on
the San Andreas Fault near Palmdale in southern California
as depicted in Figure 3.2.
In
this
chapter,
our
primary
goal
is
to
investigate the sensitivity of telluric array measurements
to
changes
Specifically,
dimensional
in
crustal
we
shall
model
of
conductivity
infer
the
an
due
to
approximate
conductivity
and
relate
the
three
structure
Palmdale, calculate the magnetotelluric response
model
stress.
to
near
this
telluric field tensors calculated
SALINAS
/
Figure 3. 1 HOLLISTER ARRAY
St6
iS
RO6
3
nLONIT
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rvlci
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-
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sei:.1Tr.
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reRosamond
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avLAX
AcS
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and Jakes &Ilot which are cenerally dry.
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ISnIr.
o
MAP orF wESTERN M\0.J.AVE DSE.R" RL(;ON. C.'Li )lORNIA
SHOWING M.\1.JOR IIIYSIOG(;I.\I'I I IC .\AND GEOGR.\PillC F.V1L'R S
10
FIGURE 3.2
Palmdale Array.
0
10
20 MiLES
-
from this response to the tensor relationships between the
dipole signals of Madden's array. Perturbations in crustal'
seen
as
briefly
the
be
can
then,
stress,
to
due
conductivity
variations in the telluric tensor relationships.
3.0.1 Chapter content
In sections 3.1 and 3.2, we review
properties
electrical
of rocks and Madden's (1978) model
of the stress and strain sensitivities of fault zones. The
electrical
in
variations
are
precise
very
temporal
can be expected as
properties
precursors to earthquakes and
small
that
are
implications of Madden's model
measurements
needed to monitor changes in crustal conductivity. In
Section 3.2, we describe how
measures
of
telluric
the
of
coherency
stability
the
between
relationships
Madden
has
field
to
the
of
high
the
used
produce
precise
telluric
tensor
dipole measurements. Additionally,
we shall relate the nature of the telluric field
response
near Palmdale to the eigenstate analysis of Chapter 2.
To infer
Madden's
array
a
realistic
sensitivity
stress
near Palmdale, we must use a conductivity
major
model of the crust which incorporates the
of
the
response
regional
of
dimensional
control
of
and
simpler
earth
the
for
features
local geology. The magnetotelluric
geological
model)
telluric
(e.g.
models
a
two
would not reflect properly the
current
system
features (Ranganayaki and Madden,1979).
thin sheet approach of Ranganayaki and
The
by
regional
generalized
Madden (1979)
not
only reflects the regional control of the telluric current
system
but
also
yields
an
approximate
measure of the
magnetotelluric response to a three dimensional
the
earth.
Consequently,
in
Sections
3.3
describe and apply the thin sheet analysis
magnetotelluric
model
of
and 3.4, we
to
infer
the
response of a large area (270km by 270km)
of southern California centered near Palmdale.
In order to determine
Palmdale
array
to
stress
the
related
sensitivity
of
changes
crustal
conductivi ty, in Section 3.5 we perturb the
of
in
the
conductivity
crustal blocks within our numerical grid and infer the
sensitivities of the telluric response in terms of
eigenstates.
Finally,
in
Section
3.6,
we
tensor
relate the
eigenstate sensitivities for our crustal model to Madden's
Palmdale array measurements and
the
efficacy
of
using
reach
telluric
conclusions
measurements
about
to infer
variations in the state of stress in
the
earth's
crust.
Sections
descriptions
of the
3.1-3.5
contain
detailed
techniques and constraints placed on our model. I
passing
suggest
over these sections in the initial reading of the
chapter and using 3.1-3.5 as a reference for Section 3.6.
3.1 Electrical Properties of rocks under stress
3.1.0 Content
In
section
upper crustal rocks. This
providing
of
purposes
chapter and to show why
background
is
for
changes
small
properties of
rest of the
the
are
the
for
included
expected
earthquakes.
between
conductivity
crustal
electrical
the
control
which
mechanisms
the
briefly,
describe
we
section,
this
in
Most of the
information in this section can be found in greater detail
in the paper of Madden(1978).
3.1.1 Electrical Conductivity Mechanisms
The
controlled
largely
by
upper
of
conductivity
the
pore
rocks
crustal
is
fluid salinity and the
volume and geometry of interconnected pores and cracks. In
terms
of
crustal
their
electrical
can
rocks
be
and
mechanical
classified
properties,
three
in
groups (Madden,1978): igneous and metamorphic, sedimentary
and fault zone rocks. Because little is
known
about
the
properties of fault rocks, we shall assume that fault rock
properties can be inferred from studies on sedimentary and
igneous and metamorphic rocks.
The bulk conductivity of sedimentary
rocks
can
be described by Archie's law:
The conductivity
controlled
by
of
igneous
and
metamorphic
rocks
is
crack sizes and geometry. For small cracks
-illi~liil-PI-.
LI_~IIIPI
L___I^_~_LIILILYII_1__~_9-~.-
on
and pores less than .01 micron across, conduction
crack
the
becomes
surfaces
controlling
of ions attracted electrostatically to the
a
factor for thg
This surface conduction is due to an excess
conductivity.
where
the
net
exists on the mineral surfaces. The
charge
potential
caused
potential
which
by
is
this
typically -50
zeta
the
called
is
charge
to -70 my for silicate
12.52
temperature (MIT,
minerals at room
surface
crack
course
notes,
and
cracks
1974).
The effect of the geometry of pores
on the conductivity is not well established. However, from
studies using embedded networks to model cracks
numerical
and pores, Madden (1974) has found that narrow cracks have
an influence on the rock conductivity
contribution
their
to
the
far
pore
total
in
excess
of
volume. Because
expect
narrow cracks are the most easily deformed, we can
rocks to be sensitive to strain.
3.1.2 Conductivity: Stress-Strain Relationships
the
With
largely
by
and
Brace
of
use
laboratory
data,
collected
coworkers, Madden(1978) has shown
that the sensitivity of crustal rocks to stress and strain
depends
on
extent
the
conductivity.
High
that
cracks
metamorphic
the
porosity sedimentary rocks have been
shown to exhibit little stress sensitivity
and
control
rocks
while
igneous
have been shown to exhibit higher
sensitivities.
Additionally,
Madden
stress-strain
sensitivities
tend
has
inferred
that
to
decrease
with
increasing confining stress but exhibit a reversal of this
trend at the onset of dilatancy. Based on
and
the
assumption
properties
crustal
of
that
fault
sensitivity
cracks
rocks,
models
these
findings
control the electrical
Madden(1978)
presented
amplification factors are the changes in
proposed
the
as Table 3.1.
The
resistivity
per
microstrain and the stress changeSare in percentage change
of resistivity per bar of deviatoric stress.
Earthquakes along the San Andreas system tend to
occur at depths of 3 to 12 kilometers. Applying his
to
the
San
Andreas,
conductivity per bar
factors
of
Madden
.03
to
predicted
.1%
and
model
changes
amplification
of 80 to 150 for effective porosities of 3%.
these sensitivity
estimates
bar/year
change,
stress
conductivity changes
of
and
t he
assumption
Madden (1978)
.03
to
.1%
From
of
concluded
per
in
year
1
that
can
be
expected in active fault zones. However, he argued further
that
crustal
heterogeneities
could cause unequal stress
distributions resulting in accelerated local variations of
up to 1% prior to earthquakes.
Perhaps the most
important
conclusions
to
be
drawn from Madden's studies are that only small changes in
crustal
conductivity
impending earthquakes
monitoring
system
stable measurements.
can
be
expected
and that any
must
be
as precursors of
electrical
capable
properties
of very precise and
In the next section, we describe how
Madden uses telluric cancellations
to
achieve
the
high
~II
Table 3.1
Stress-Strain Relationships
Estimated Resistivity:
Amplification Factor (Ap/p)/Au
Porosity in
Depth, km
1
3
30
500
100
7
400
100
7
30.0
80
7
200
60
750
200
10
500
200
8
400
150
7
300
100
non-dilatant strain region
dilatant strain region
__
Stress Sensitivity %Ap/bar
Porosity in %
1
3
10
30
0
.4
.3
.04
.02
1
.2
.15
.04
.02
3
.07
.10
.03
.02
10
.03
.03
.02
.5
.4
.1
.03
.3
.2
.1
.03
.1
.10
.05
.03
.05
.03
Depth, km
.05
non-dilatant strain region
dilatant strain region
(from Madden. 1978)
29
sensitivities necessary to monitor the state of stress and
strain in the crust.
3.2 Telluric Cancellations
3.2.0 Nature of the low frequency telluric field
of
The elements
shift exists between
phase
negligible
and
Madden,1979)
skin
telluric fluctuations for which the
tend
wavelengths
negligible
phase
to
also
very
be
much
Source
producing
large
telluric
between
shifts
is
depth
thickness.
crustal
upper
the
than
and
(Ranganayaki
crust
lower
resistive
the
by
the conductive upper
in
trapped
are
currents
telluric
the
frequencies,
low
at
However,
frequency dependent.
larger
relating
field
fields at the surface of the earth are generally
electric
crust
telluric
the
measurements
separated by as much as 400 kilometers as indicated by the
high coherency (>.999) between
from
the
simultaneous
measurements
Palmdale and Hollister arrays (Madden, Personal
Communication,1980).
3.2.1
of
determination
Electronic
telluric
tensor
relationships
Consistent
with
observations,
these
three
telluric field measurements are related accurately by real
constants such that:
(3-2)
A 5 bB + cC
These components can be
wide
bandwidth
combined
electronically
over
a
of low frequencies to produce a near null
or residual signal R such that:
R = A - bB - cC
(3-3)
producing
This process of
is
measurements
telluric
from
residuals
called
more
or
two
telluric cancellations"
and
(Madden,1976). The residual signal R can be amplified
its
content studied with greater sensitivity than similar
individual
studies of the
signals
a
for
given
The constants b and c are measures
digitization accuracy.
of
dipole
the integrated crustal conductivity under the telluric
measurements. The stability of these constants then can be
used as a measure of the temporal variation of the crustal
signals
conductivity (Madden,1978). Recordings
of
these
included
as
Figure
SB1
and
over
period
day
5
a
represents
a
single
relative to H is
dipole
cancellation
scalar
represent
are
signal.
residuals
shown on the graph.
3.3. H
SB2
and their gain
The remaining signals
represent tensor cancellations and their gains relative to
H are listed as well.
Dipoles A and F have pre-gains of 11
and 6 respectively.
3.2.2 Hollister and Palmdale array measurements
To implement the telluric
uses telephone lines to connect distant electrodes
Madden
to central stations in Hollister and Palmdale,
(Figures
3.1,3.2).
These
residuals
residuals reflect the incompleteness of the
cancellations.
stability
California
Combining three or four dipole signals
at a time, he has produced and recorded sets of
R..
1
scheme,
cancellation
To
relate
the
residuals
Ri
to
the
of the tensor elements, two independent signals
SA and S B with which any of the dipole signals can
be
_.1
11
l_ ~_y__L______L__^C^_L1
1_1 Iill~e_^~l
5 DAY PERIOD
PB7180
P07280
I
P07580
P07380
P07480
-------------------
-------------------.
. _ - --
-
Sc-H
gain = 1
gain = 1
gain = 2
DxCySb 1
gain = 7
gain = 7
gain = 2
F-x H-ySb6
gain = 2
Sb2*xSb1
S 62 xS 6l
gain = 12
FIGURE 3.3
Palmdale Array Signals
*U1-~L-.~LI.._ i_-.
*i~111-141~11~11~111.1
recorded as well. Temporal variations in
are
represented
the amounts of S A and S B
used
as
measures
By
elements.
the changes
in the
telluric tensor
of
combinations
using
are
then
residuals
the
cancellation
the relative variation of an individual dipole
residuals,
set
or
signal
of
in
be
can
signals
determined
to tensor
the sensitivities
scheme,
this
With
uniquely.
dipole
of
element variations are presently better than
over
.1%
a
period of a year.
Thus, Madden has developed a technique
in
to variations
of high
sensitivity
Stable
circuitry
and
calibration
precise
of
tensor
telluric
this
information
models,
Palmdale array to constrain our crustal
the
to
establish
relationships
between
the
MT
and the telluric cancellation scheme.
eigenstates
3.2.3
are
appears to be the dependence on the integrity of
telephone lines. To use the
need
schemes
weakness
cancellation scheme but the only major
we
conductivity.
crustal
in the practical implementation of the telluric
necessary
from
is
and simple to implement yet achieves the goal
inexpensive
approach
that
Telluric cancellations
and
magnetotelluric
tensor
eigenstates
Before establishing the tensor cancellations
Palmdale,
Madden
dipole signals,
tendency
is
had
noted
a
consistent
that
dominant
all
of
common
in
the low frequency
component.
This
with a nearly linearly polarized
regional telluric field. This near linear polarization
is
~ I-iil~-_-_X~III
_~-_1I.-1-
a
consequence of the regional crustal structure which, on
terms
the
crustal conductivity produces a large spread in
effective
a
the impedance eigenvalue magnitudes and causes
described
Section
in
of
modelling
sheet
thin
our
to strike. From
California,
3.3,
the
in
spread
eigenvalues
particularly
found
we
Additionally,
San Gabriel
the
for
southern
we found a wide
spread in the eigenvalues for the area and a
Mountains.
general
the eigenvectors perpendicular and parallel
of
alignment
large
in
anisotropy
large
this
eigenstates,
of
and
highly anisotropic effective conductivity. In
a
produces
coastline
the
to
parallel
strikes
average,
the
that
the
telluric
eigenvectors for the impedance tensors in the San Gabriels
to
be
aligned
strike of the
the
spanning
closely perpendicular and parallel to the
mountains.
San
Thus,
Gabriel
we
Mountains
infer
will
that
be
dipoles
strongly
linearly polarized reflecting the large anisotropy in
the
eigenstates.
In Madden's Palmdale array, two dipoles span the
San Gabriel
Mountains (D and H)
indistinguishable
virtually
as
and
their
predicted
sheet analysis. Madden chose the larger of
as
his
independent
signal
SA
of
are
from the thin
these
dipoles
which we infer to be a
related closely to the major eigenstate of
tensor.
signals
the
impedance
When Madden used electrcnic differencing on pairs
dipole
signals from the Palmdale array, he found that
the residual signals all looked alike but were independent
---X~CCI^YL
I1111I1.- -IIU-II^--C
X
of
these
interpreted
He
.
polarizations of the telluric
as the
signals
independent
SA
signal
primary
the
and
field and chose two difference signals to be his SB1
Each
SB2.
signal
in
a
linear
combination of S A
as
interpreted
SB1
or SB2.
was
then,
array,
his
dipole
and either
The two difference signals used were:
= H + x1 C
SB1
(3-4)
= B -
SB2
Both of these signals
addition of
weighted vector
the
by
set
for the areas spanned by dipoles B,C,
procedure
the
and
directions
SB1
of
directions
satisfactory result for SB2 as depicted in
The model principal
and H, we
eigenstate
minor
model
the
find near alignment between
this
Following
dipoles.
corresponding
the
dipoles
can be considered as pseudo
directions
along
oriented
x2 C
a
less
Figure
3.4.
but
axes are the solid lines in each block
and the dashed lines represent the principal axes inferred
from array measurements.
a
In the block containing dipole H,
dipole signal will be dominated by the major eigenstate
for
except
a narrow
dipole
However,
sufficiently
Madden
'
directions
the
correspondence
to make
s independent signals
between
us feel
are
to the major
that
good
from
direction
inaccurate
an
is
signal
close
SB 2
the
estimating
eigenstate. Thus,
perpendicular
band nearly
process.
directions
the directions
approximations
is
of
to
the electric field eigenvector directions of the impedance
APPROXIMATE EIGENVECTOR DIRECTIONS
FIGURE 3.4
^ i__LLI~_____PI_~II)li_-~ia^lt
In sections 3.3 and 3.4,
tensors for the Palmdale region.
we
use
relationships
eigenstate
approximate
these
*II~-C
constrain our crustal conductivity model of
to'
Palmdale
the
region.
3.2.4 Interpreting variations
answer
two
elements
and
conductivity
expect
we
should
changes
for
these
should
how
interpreted? Madden has
variations
What
questigns..
hypothesized
that
seek
we
For the remainder of this chapter,
the tensor
in
crustal
in
variations
telluric
the
change
spanning a finite region, produces measures of the
effective anisotropy of the crustal conductivity under
the array. In the last section of this chapter,
reconsider
our
be
approach, which involves an array of dipoles
cancellation
in
to
we
shall
Madden's hypothesis in terms of the results of
numerical
models
of
the
sensitivity
measurements to changes in stress.
of
telluric
-~IIIIXXlr--~X-
_-..i (I-.--~.
__
1_IX_
il_LT-f~X
3.3 Generalized Thin Sheet Analysis
3.3.0 Introduction
treatment
The development and extensions of the
of
a conducting sheet from Price's
as
crust
upper
the
and
Ranganayaki
Madden(1980).
paper
the
in
described
original analysis(1949) are
of
In this section, we shall
outline qualitatively the basis for their
and
model
its
applicability to telluric resistivity monitoring along the
San
Andreas
fault
this
modelling technique in in
near Palmdale. This region is a
geology
crustal
is
technique
the
account
to
necessary
good
example
of
varied
three dimensional MT modelling
a
where
predict
to
chapter
in Southern California centered
field
telluric
regional
We use, the thin sheet
in California.
for
scale
large
constraints on the local field.
3.3.1 Theoretical Basis
In the generalized thin sheet model,
crust
upper
the
is considered to be a thin conductive layer and the
and
lower crust
resistive
layer
part
approximation
by
underlain
thin
generalized
the
of
that
sheet
most
earth
of
upper
mantle
a
thicker
a conductive mantle. This
model
the
is
based
on
the
change in the magnetic
field occurs across the conductive sheet while most of the
change in the elctric field occurs
sheet.
ratio
across
the
resistive
For low frequency waves penetrating the earth, the
of
skin depth to upper crustal thickness is large.
39
the
Ranganayaki
generated.
term
distance"
"adjustment
in
crustal
used to describe the spatial
away
from
a
lateral
conductivity.
Near
such
a
lateral
E
contrast.
the
from
far
distance.
its
from
The distance required to
diminish this distortion by a factor of 1/e is called
adjustment
an
introduced
Madden
and
conductivity contrast, the E field is distorted
value
lateral
field
variation of the surface
change
of
regions
in
contrasts where vertical E (electric) fields
conductivity
are
mantle
the
leakageto and from
of
is confined to the upper crust with'
current
induced
most
resistive,
Because the lower crust is much more
the
The form of the adjustment distance
is:
are
and (/)
where ()
,
-
adjustment
)
1
of
functions
(3-5)
position
on
the
surface (X,Y)
("A3)
is
the conductivity
thickness product
of the upper crust
(ca-z) is the resistivity thickness product
of the lower crust
From an ocean continent boundary, the adjustment
may
be
as
much
km over the ocean and 100s of
1000
as
kilomters over the continent.
the
adjustment
distance
However,
concept
for
the importance
us
is
frequency E fields in areas exhibiting variety in
geology,
structure.
are rarely
"in
distance
that
of
low
crustal
adjustment" with the conductivity
)I
~__~I~_____
I
Y___~_I/~_ULI_
3.3.2 The Numerical Grid
In
I
thesis,
this
use
(1979)
Ranganayaki's
numerical formulation of the magnetotelluric (MT) response
for
thin sheet earth model. A copy of her
generalized
a
T.R.
Prof.
by
Program
Research
to
Report
Technical
Semi-Annual
Park,
Steve
by
revised
program,
is
the
in
included
USGS
the
Geothermal
In
Madden(1980).
the
present thin sheet formulation, the conductivity thickness
product is allowed to vary as a function of position (X,Y)
but the resistive sheet is modelled as a homogeneous layer
overlying
a
layered model of the mantle. Figure
3.5 is a
Each
graphical representation of the numerical grid used.
grid
block can span electrically varied crustal sections.
block
each
The average conductivity thickness product of
is represented as a conductance tensor. Homogeneous blocks
are
To
conductances.
isotropic
assigned
assume
conductances to the heterogeneous blocks, we
each
can
block
represented
be
equal
homogeneous sub blocks of
conductances
are
treated
as
assign
as geometric mix of two
The
volume.
circuit
sub
block.
choice
The
initially
on
the
the
of
anisotropy
of
block
geology
conductance
resistivity
data.
the
for each full
conductivities
is
based
within each block as inferred
from regional geology maps, known depths to
available
block
like elements and
combined as series and parallel averages to determine
effective
that
basement
and
Our procedure for assigning
anisotropy to blocks is illustrated by example in Appendix
IL~U---llt4_I1CSC 1~_
CROSS SErTION
U
(Es, Hs)
L
V
(Es, H=.)
L
L
(Es,
conductive thin sheet
,
O(xYv)
res-.ttive thin sheet
p(x,Y) aJ.
Hs)
-----3
-- --------
Space//////i////i//////i//i//i
////////////////HaIf
Es and Hs are the horizontal electric and magnetic fields at
the top (U) and the bottom (L) of the thin sheet sandwich
Es = 0 across the conductive thin sheet
Hs = 0 across the resistive thin sheet
Generalized thin sheet geometry
MAP VIEI
Y
I
I
I
I
I
I
1
I
I
I
I
I
I
I
I
II
I
_
I
I
I
1
I
I
I
I
I
I
I
I
I
I
I
I
I
___
I
I
_
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
If
CROSS
SECTION
Z
THIN SHFET NUMERICAL GRIP
42
FIGURE 3.5
i~.l .~Pi--*
_ .....
*YII~I*-^--IPCI-I~CI
3.3.3 Block Conductance versus Stress
sensitivity
Ultimately, we wish to know the
array measurements to changes in stress and
Palmdale
the
of
Consequently,
strain (Madden, 1978).
consider
shall
we
changes in the crustal conductance consistent with changes
in stress. Given our base model, we perturb the stress and
infer
conductance tensor. The perturbed
the
in
changes
conductivity model is used with the thin sheet program
a
determine
set of impedance tensors. The impedance
new
which
tensors are then used to calculate telluric tensors
relate
block
the
to
electric fields for each block to a reference
spanning
middle
the
The
array.
Madden's
of
sensitivity of the array is then inferred from the changes
in
tensor eigenstates for stress related changes in block
conductances.
of
The calculation
block
conductances
to
stress
assumption that the electrical
near
Palmdale
are
crack
the
sensitivities
properties
the
based on the
is
changes
of
of
the
crust
controlled (Madden,1978).
Accordingly, changes in block conductance
are
associated
with the opening and closing of old cracks and the opening
of
new cracks. Cracks are opened along the applied stress
and
direction (increasing conductivity)
perpendicular
to
applied
the
stress (decreasing conductivity). Under
closed
uniaxial
stress,
the conductance of a homogeneous block becomes anisotropic
..~-- i_-nlXI--LII*CIIYW~
II_-_-_1II~
I^L~LII~L---i Il~i~-U-
the
and
anisotropic conductance of a heterogeneous block
is modified. Examples of
model
changes
tht
in
blocks due to changes in stress
crustal
of
conductances
we
how
are included in Appendix E.
3.3.4 Telluric Current Saturation Conditions
How the
block
telluric
current
responds
system
conductance , variations will be an important factor
in the use of telluric measurements to monitor
of
to
stress
the
state
in the crust. The electric field response to a
block conductance change depends on the current saturation
condition for that block. Perturbations in current density
J can be expressed as:
- 0-
+
-:::: <j-
E(3-6)
For complete current saturation (Appendix D),
no change in
J occurs and:
cr
(3-7)
E
For a totally unsaturated current condition, no change
in
E occurs and:
----
(3-8)
In general, we can expect the current saturation condition
for a crustal block to be between these two
extremes
and
~-LL..~..^-~IXIII~~--~_IP--
_-~^~~l--^-; r__-- ;i rarxrrrr-~*Y
...__..
the
electric field to reflect some fraction of the change
in block conductance
Thus,
the
sensitivity
the
of
in
change
a
by
produced
electric
field to block
conductance changes depends on the role of that
establishing
regional
the
thin
to
sheet
geophysical
block
distribution.
current
section 3.5, we compute the sensitivity
fields
stress;
of
the
in
In
electric
changes in block conductance using generalized
models,
and
-but
first
geological
we
shall
describe
the
constraints on our model and
present our base crustal thin sheet model for the Palmdale
area.
LPmi;n~
I~L^l-r~
~______~l~lj/XI_______~I
3.4 The Palmdale Thin Sheet Model
3.4.0 Content
on
constraints
the
consider
we
In this section,
geophysical
our model of the crustal conductance near
along
constraints
Palmdale. We use these
with
sediment
thicknesses determined from gravity and geological mapping
to
model MT
for
which
eigenstates
telluric
and
references
model. We present as well the base
initial
an
infer
are
used
as
our stress sensitivity analysis presented
in section 3.5.
3.4.1 Data Constraints
telluric
models
our
on
Constraints
measurements
magnetotelluric
and
are
provided
as
well
by
as
The MT data of Reddy
gravity and geological measurements.
et. al.(1979) are useful not only as constraints but serve
a
also to illustrate the need for
regional
approach
to
interpret local measurements. They obtained MT data in the
San Gabriel mountains over the wide frequency band of .001
to
10 Hertz. From their data, presented as Figure
infer that the low frequency current in the
to
perpendicular
essentially
basin contact
with
the
E
the
field
3.6, we
mountains
mountain-Los
perpendicular
is
Angeles
to
the
contact much larger than the parallel E field. Analysis of
the
small
E
parallel
field, based on a two dimensional
model of the earth, has lead Reddy
interpretation
that
the
San
et
to
the
mountains
are
al.(1979)
Gabriel
'0000
--
r
'I
T TTT
1-LT- TT
I 1
T
T
1000
TT
-- 4--'- r
IT
-
T
--
iIIIi
II
i00
f
40,
rr~
4
If~~zr
rff Is-I-I
~R"-
Ct
10 -
0.1
1
10
100 '
PERIOD, sec
Thc magnetotelluric tensor apparent resistivity curves in the
principal directions with their 95% confidence limits for the San Ghbriel
Mountains, from Reddy et al. (1977)
FIGURE 3.6
1000
~L-.----~-~l---rri--i~--~~-
the7
that
assumption
the
is
schemes
dimensional
two
such
in
Implicit
underlain by a conductive lower crust.
with
electric fields parallel to strike are in adjustment
the
s~Y*
--XII~~ (..-~IP~I~F-~~I.I.Y-L-I--I-I-*~L~.~__LII
geology i.e. the parallel field is set only by
local
Ranganayaki
geology.
local
the local magnetic field and
and Madden(1980) using the generalized thin sheet approach
have
the low frequency telluric field is set
that
shown
Reddy
al.
et
to be consistent with the San Gabriels as a
crust
lower
window to the
with
Consistent
is
field
regional geology.
resistivity
model
a
are
on
constraints
Further
telluric measurements used
to
regional
our
array
Madden's
by
provided
resistivity
monitor
this
the complicated
of
manifestation
a
frequency
interpretation, the large anisotropy in the low
electric
for
requirement
no
with
basement.
conductive
shallow
of
data
MT
the
interpret
their thin sheet approach we
Using
geology.
not only by the local but also the regional
of
near
Palmdale, California and local bandlimited MT measurements
also obtained near Palmdale.
As described
directions
of
the
in
previously
independent
3.2,
Section
the
S A andS B signals are
oriented respectively roughly parallel to
the
major
and
minor electric field principal axes of the magnetotelluric
tensor. We compare the directions of these signals in each
blocx
spanned
by
dipoles
in
the
array with the model
principal axis directions and adjust the model to
achieve
correspondence. An example of this correspondence has been
48
II--~LY;-~ ~-~-~-~-PC~ -PIPI~.
____LIIYII_____III__
shown in Figure 3.4.
As a consequence of
cancellation'
telluric
the
tensor scheme described earlier, we can relate each dipole
in
two other dipoles i.e. one dipole
to
array
Madden's
signal is a linear combination of two others. Thus, if the
the
impedance tensor relating any two of our dipoles with
field
magnetic
known, we can determine the impedance
is
tensors for the remaining
simultaneously telluric signals from the
set
that
From
measurements
of
impedance eigenstates for
of
pairs
array.
Palmdale
have
we
recorded
we
Palmdale-,
near
measurements
field
MT
conducting
While
dipoles.
dipoles
inferred
the
within
the
array. Descriptions of our field procedures and details on
the
analytical approaches used can be found in Appendices
A and C.
3.4.2 Conductance Assignments and the Base Crustal Model
The MT and telluric data from the Palmdale array
model
plus the data of Reddy et al. are used to constrain
conductances
the
of
near
blocks
Mojave
conductances
inferred
the
of
from
measurements
a
San Gabriel Mountain blocks and the
Palmdale
data (Hanna et al.,1974) and
western
Mojave
of
our
bandlimited
geological
(Dibblee, 1967).
mapping
Gravity
Joaquim
Valley
and
49
Los
Angeles
MT
of
the
and geological
for
the
Basin.
The
mapping data are used also to infer conductances
San
The
gravity
Mojave,
the
in
Lancaster.
Mojave desert blocks are
remaining
combination
and
Nevada, San Gabriel and San Bernardino
Sierra
Tehachapi,
mountains were all
the
of
rocks
Mesozoic
and
Precambrian
metamorphosed
assigned the same conductance.
Following the procedures outlined in Appendix E,
were
conductances
block
Ocean
regions.
heterogeneous
spanning
conductances
anisotropic
assigned
we
assumption
determined from the depth of the ocean and the
of 3.3 mhos/meter for sea water.
for
Our base model
the
3.7.
Palmdale region is shown as Figure
are square, 30 km on a side,
We
the conductivity
blocks
The
and members of a 9 by 9 grid.
matrix notation to designate each block such that
use
effective
The
grid.
9
by
9
within
the block
and column of
I,J corresponds to the row
the
structure of
of
thickness
the
conductive thin sheet is 3 km. Beneath the conductive thin
sheet,
the earth is modelled as layered with the resistive
upper crust as the top layer. Conductivities
listed
conductivity
Palmdale
array
Figure 3.8.
are
This
Gabriel
the
delineated
by
region
spans
blocks.
translation
The
are
Blocks
50
of the
lines
heavy
structure. Blocks 5,4-5,6
and
most
blocks
sensitivity
to
in
the array and includes a
Precambrian and Mesozoic
Mountains
homogeneous
map.
of
quite varied geological
metamorphic
their
with
the
below
analysis
our
to
important
form
map
in
arranged
indices
block conductivity
and thicknesses in meters. The
are
are in mhos/m
terrain
modelled
4,3-4,6
span
of
as
part
span the
the
San
resistive
of
the
I ~
i*~~i
^ry__~~;~.r rr----l^rr~-r~------ i3-l'l-~-*Elarr-ri .-..
Y=9
X=9
NUMBER OF BLOCKS:
0.3000E+05X0.3000E+05 METERS
INDIVIDUAL BLOCK DIMENSION:
3 KM
THIN SHEET THICKNESS
CONDUCTIVITIES AND THICKNESSES FOR CAGNIARD LAYERED
MODEL BELOW THIN SHEET
0.3000E+05
0.1000E-03
0.7000E+05
0.1000E-01
0.0
0.1000E+00
PERIOD=1000 SECONDS
CONDUCTIVITY MAP
3 3 3' 3. 3 3
8
8 8
3
3
3 3
3 3
3
8 8
3
3 3 3
5 3
2
8 4
8 8
3 3
3 3
2
5 2
5
5 5
8
8
5 8
4 4
3
3 4
3
3 3 4
3
3
4
3 8
1
5 1 1
1 3
2
1 1
0
0 0 1
0 0
1
1
0 0
0 0 1
0 0
INDEX
0
1
2
3
4
5
6
7
8
9
CDXX
0.3000E+01
0.4000E+00
0. 1000E+00
0.5000E-01
0.1750E-01
0.5500E-02
0.3300E-01
0.3300E-01
0.1000E-02
0.1000E+00
CDXY
0.0
0.0
0.0
0.0
0.0
0.0
0. 3000E-01
CDYY
CDYX
0.3000E+01
0.0
0.4000E+00
0.0
0.1000E+00
0.0
0.5000E-01
0.0
0.2000E-02
0.0
0.2000E-02
0.0
0.3000E-010.3300E-01
-.
-. 3000E-010.3300E-01
3000E-01
0.0
0.0
0.0
0.0
0.1000E-02
0.1000E-02
Conductivities in nmho/m; thicknesses in meters
Figure 3.7
Conductivity Model for Palmdale Array
~.~
liPrP~"slL
rulll*r~_-*-rr,,uPUrr~-----YI
--rii~..uur-,,~--r~
!ffYc'
i
~-~u~
'St---'
3-'_
FIGURE 3.8
PERTURBED REGION
i____^_yX__I1____I__I____1_
~^I I_~
~1_~
__i____X
sediments of the Western Mojave Desert and are modelled as
(5,3)
and
(5,2)
(4,2),
and homogeneous. Blocks
conductive
represent the heterogeneous transitional zone between
San
and
Tehachapi
than
blocks
represent part of the Los
Angeles
upthrusted
marine
conductive
modelled as anisotropic and more
Blocks 6,2-6,6
5,4-5,6.
Fernando
San
Basin,
blocks are
These
Mountains.
Gabriel
the
and
Valley,
sediments and are modelled as conductive and homogeneous.
adjustments to our base model were made by
Final
as
used
the
with
consistent
The
we
difficulty
only
data
(MT)
impedance
bandlimited
constraints.
impedances.
and data
for our base model are largely
eigenstates
impedance
The
for our model
the eigenstates
between
correspondence
achieve
to
conductances
block
changing
had
we
in
adjusting the model block conductances to fit the model to
data
band
narrow
the
associated
with
mostly
Pearblossom
was
a
factor
from
C
dipole
predicted with a reasonable set of
Palmdale
to
larger than could be
two
of
the
eigenstate
the
where
(4,5)
block
Lancaster-Palmdale
in
occurred
eigenstates
for
conductances
the
block based on local MT measurements and gravity profiles.
We
attribute this
anomalously large eigenstate
resistive zone along the usually
Fault
in
system
possibility is
sufficiently
the
conductive
to a local
San
Andreas
which current may be funnelled. Another
that the 24 km long dipoles B and C spanned
different structures that the description
impedance
as
a
tensor
was
of
unwarranted
1I 10L~I
i--.--Ixn---rr--- ----r-i--i~u i LI--711~**rll-*--i-
(Kasameyer, 1974).
3.4.3 Base Model Eigenstates
With our base model, we
for
eigenstates
tensor
relative
eigenstates
resultant
impedance
the
each grid block and the telluric
The
5,4 (row,column).
block
to
all essentially linearly
are
eigenvectors
MT
compute
polarized (small ellipticities) with appreciable skew only
in the ocean blocks where
current
occurs.
funneling
We
graphically, the MT eigenstates for each block
represent,
Figure
in two parts.
3.9
is
a
the
of
representation
electric field eigenvectors multiplied by their respective
amplitudes and Figure 3.10 is a representation
eigenvalue
representations
of the magnetic field eigenvectors. These
are drawn on a geological map of the region which includes
the
Madden's telluric array. The major and
of
locations
minor axes of the ellipses in
amplitudes
field
electric
Figure
produced
3.9
by
the
represent
the
magnetic
eigenvectors of Figure 3.10. As a point of distinction the
We
ellipses of Figure 3.9 are not polarization ellipses.
have
left
out
the
ellipticity
and
phase
eigenvalue
information, because the ellipticities are small and
little
with
vary
conductance perturbations and the eigenvalue
phases depend largely on the mantle conductivity which for
our model does not vary with position.
The ellipses point
toward the directions of maximum electric fields
block
and
reflect
the
effects
of
regional
geology on the telluric current system. Our major
54
in
each
and local
use
of
11~11~1^1
-1~LBI*-ll~)
ELECTRIC EIGENVECTOR PRINCIPAL.AXIS ELLIPSES FOR THE IMPEDANCE TENSOR
FIGURE 3.9 THIN SHEET MODEL EIGENVECTOR DIRECTIONS
,. ...~
y--xl~~y~
-~yY~sr~C-~r~---rsic--.r~---I-i-- ..L*~.~n~~n~-
MAGNETIC EIGENVECTOR PRINCIPAL AXIS DIRECTIONS FOR THE IMPEDANCE-TENSOR
t-i
FIGUR -!3.1 M-N I-C--~ ~~~~
)
,L
-B
. .
FIGUR
... ....
leEIENVETOR
.g MGNETC
Lo-,,A 3.10
"~i
.KvoFIGUE
3.1yANEIIENETR
__
our'
of
constraints
the
within
model
crustal
reasonable
a
infer
to
was
eigenstates
impedance
the
magnetotelluric measurements. However, the telluric tensor
can be used directly to infer the sensitivity
eigenstates
of Madden's arrays. Consequently, for
chapter,
our
concentrate
shall
we
this
of
rest
the
on
efforts
the
characteristics of the telluric eigenstates.
The telluric eigenstates
are
maximum and minimum electric fields
the
represent
and
D
dipoles
where
5,4
H
more
than 4 degrees
out
left
Gabriel
and
phase
Again,
Mountains.
ellipticity
differences between the
we
information. The
largely
eigenvalues represent
phases of the telluric
phase
rarely
are
5,4
from being parallel and perpendicular
to the strike of the San
have
The
located.
are
directions of the eigenvectors for block
reference
in
fields
for that block due to unit electric
block
each
for
The telluric ellipses
shown as Figure 3.11.
block
for the Palmdale region
the
impedance eigenvalue phases
and are consequently nearly zero. The ellipticities of the
eigenvectors
are
also
small
conductance perturbations.
block
point
toward
the
and
little
vary
with
The telluric ellipses for each
maximum contrast in resistivity
between that block and the reference block, whereas the MT
ellipses point toward the maximum
The correspondence
resistivity
of these directions, then,
direction.
is a measure
of the alignment of the maximum resistivity direction
with
the
maximum change in resistivity.
57
In the
(MT)
Palmdale
~ .~ ._..rxi.n.i~.r-..r~-r--X~---r---LL-_-^~-
R4
D
jr
19~
InLs
-N
le%
L
n?
U
~PL
JI"-LstnF~L'
~
L~
Al
~NtLh~;~-~m
7~3~~J,---I()
34
K~Jv
-1~
-A
I
FIGURE 3.11
TELLURIC EIGENSTATES
gl_~l
region, the geological structure is aligned
the
coastline
with
the
change in resistivity and maximum
tend
and
to
be
telluric
directions.
that
result
largely
with
both the maximum'
resistivity
directions
perpendicular to the coast and the impedance
eigenstates
exhibit
similar
principal
_1___Y5~_~__rYll__k
3.5 Crustal Model Stress Sensitivity
3.5.0 Introduction
In this section, we discuss how
the
geological
control of the telluric current system is reflected in the
MT
response
of the Palmdale region. We consider then the
implications of the current controls on the sensitivity of
the telluric eigenspates to changes in block conductances.
parameters,
Next, we present the sensitivity
and
which we use in our studies of the sensitivity
skew,
of the telluric eigenstates to stress induced
block
changes
in
conductance. As noted previously, earthquakes occur
we
are
In
crust.
this
considering only the upper portion of the
procedures
of
because
earthquake depth range,
modelling
the
in
at depths of 3 to 12 kilometers
study,
eigenvalues
and
limitations
in
our
expected insensitivity of
the
telluric measurements to variations in the resistivity
of
the lower crust.
3.5.1 Structural Control of the Telluric Currents
From Figure 3.9, we notice that the San
blocks,
though
anisotropic
because
modelled
This
responses.
isotropic,
large
strongly
have
anisotropy
occurs
the current flowing between the Mojave desert and
the Los Angeles Basin
causing
as
Gabriel
large
must
electric
pass
fields
through
across
the
the
mountains
mountains.
Conversely, most of the current parallel to the strike
of
the San Gabriels can bypass the mountains and be channeled
-~u--r~.x~~--~-8.---.-i-m^
in
Mojave
the
The"
telluric
the
of
control
the
then, is a consequence of
a
for the mountain blocks,
eigenstates
the
in
anisotropy
mountains.
the
in
resultant small parallel E field
with
basin
Angeles
Los
and
desert
~-*
,x*-~
currents by regional structure.
where
The western Mojave is another region
The
structure.
wedge
conductive
a
is
Mojave
western
regional
by
controlled
are
levels
current
telluric
the
between the relatively resistive San Gabriel and Tehachapi
bisector need pass only through the resistive tip
wedge
of
the
San Joaquim Valley and consequently will
the
into
wedge
the
to
parallel
currents
Telluric
mountains.
tend to be larger than currents perpendicular to the wedge
Tehachapi
through
pass
bisector which must
Conductive
mountains.
control the current levels in the
blocks
the exceptions
within
a
of
resistive
the
tend
block
blocks,
resistive
isolated
to
be
currents
unsaturated and
block
conductance
result that the electric field within the block
will tend to
conductance
resistive
while
crust
and
then, tend to
blocks,
decrease in response to an increase in
with
Gabriel
the distribution of currents. With
influence
can
San
the
remain
change
constant.
in
of
one
blocks need not be expected
to
Thus,
a
stress
induced
the San Gabriel Mountain
cause
a
change
in
the
electric fields of that block but can be expected to cause
changes in the electric fields of the adjacent Los Angeles
Basin
and
Mojave
Desert blocks because of the change in
61
ill~iP~Y
C~li~---*~YP
--l-r~uLlr~*-ur~irLL;----~-sr--l--.a----,---- I-~-II
current. Additionally, conductive blocks are more
telluric
the
setting
blocks
will
to
eigenstates
the
Thus,
tend
resistive
the
blocks
current levels, we can
telluric
Madden's
in
H
and
D
dipoles
expect
crustal
of
control
conductive
of resistive blocks.
With the assumption that
little
of
sensitive to conductance
more
be
changes than the eigenstates
have
in-
telluric current levels than do the
average
resistive blocks.
role
greater
apt to be saturated because they play a
array (Figure 3.2) to be largely insensitive to variations
Conversely, we can expect
in mountain block conductances.
dipoles A,B,C, and F, which are located in more conductive
areas,
to be
sensitive not only to conductance variations
in their own blocks
but
also,
to
a
lesser
degree,
to
variations in adjacent unsaturated blocks.
3.5.2 Eigenstate Sensitivity Measures
The telluric
tensor eigenstates are derived from
the elements of the telluric tensor. Thus,
the sensitivity
of the telluric eigenstates to conductivity variations
a
direct
meassure
of
the
sensitivity
elements. Another measure of the telluric
skew
i.e.
the
formulation,
the
tensor
tensor
is
the
difference between the directions of the
Similar to the magnetotelluric
corresponding eigenstates.
skew
of
is
the
telluric
represented as:
62
skew
angle
can
be
I--L'."P-L--^I~---1YIII^~IYY~^I--L~ ~-~XII4~--.--.
ID
-IUi-IT~-LIL.
T
T(3-9)
where the telluric tensor is represented as:
-yrT
"
(3-10)
In terms of the tensor eigenstates, the angle TS is
angle
between
the
U (1) and UT (2)
When the directions of UT (1) is
direction
of
uT1(2)
TS
eigenvectors.
clockwise
is
the
from
positive.
the
For
counterclockwise rotation, TS is negative. A measure
of
skew can be determined even when only three dipole signals
are used. The tensor relating three dipoles that have been
rotated
to
a Cartesian coordinate system with one dipole
aligned with a coordinate axis can be expressed as:
Ij-
j
n
-- -
C
The skew angle for this tensor when all the
(3-11)
elements
are
real is:
-1
(3-12)
is
it
because
We introduce the telluric skew,
sensitive to anisotropic changes in conductivity which can
to deviatoric stress. A simple example of this
due
occur
behavior can be inferred from our analysis in Chapter 4 of
the tensor relating electrical fields inside and outside a
The
earth.
conducting ellipsoid embedded in a conducting
tensor relating these fields is symmetric with a resultant
changes in conductivity do not
Isotropic
zero.
of
skew
change the symmetry of the tensor. Thus, no change in skew
occurs. However, anisotropic changes in conductivity cause
the tensor to become asymmetric
deduce
we
Thus,
the
effective conductivity of
change
in
result
will
crust
the
in
change
anisotropic
an
that
skew.
finite
a
causing
a
the telluric skew while isotropic conductivity
in
changes will cause no change in
of
modelling
the
of
effects
In
skew.
numerical
our
perturbations
in crustal
conductivity, we shall determine the sensitivities of
the
of
the
skew
and
eigenstates
telluric
as
measures
measurements
sensitivity of Madden's telluric array
near
Palmdale.
We
conductivity
present
perturbations.
two
of
results
the
first
The
hydrostatic
anisctropic
changes
of
the
in
stress
and
second
variations
conductivity
deviatoric stress.
telluric
the
eigenvalues
64
of
involves
set
isotropic conductivity changes associated with
in
sets
variations
involves
set
associated
with
We define the sensitivity
to
changes
in
block
conductance as a relative percentage change such that:
(3-13)
T
change in conductivity such that:
angle for a percentage
:
-
conductance
eigenvalue
100% means
of
skew
and
that the
in
skew.
in
Changes
skew
the
in
one means that a one
percent change in the conductivity produces a
change
change in
completely
reflected
is
variation
(3-14)
0
An eigenvalue sensitivity of
block
skew
in
and we define the skew sensitivity as the change
degree
one
can be thought of as
percentage changes relative to one radian such that a
degree change in skew is equivalent to a 1% change.
final
section
results of our
of
this
chapter
sensitivity
(3.6),
studies
using
we
.57
In the
present the
the
telluric
tensor skew and eigenvalues as sensitivity parameters.
~^~LI
--L~I
1_~II___IIYYI_____~I~II
OI-~ I -...^IP--^^IIII-Y----I-
3.6 Sensitivity Analysis: Results
3.6.0 Introduction
First
This section is divided into three parts.
we
make general observations applicable to both isotropic
Then,
and anisotropic variations in crustal conductivity.
we
sensitivities
the
discuss
peculiar to isotropic and
results
anisotropic variations and finally we apply these
to the Palmdale array measurements.
perturb
We
blocks
the
conductivity
crustal
the
of
delineated in heavy ink in Figure 3.8. This region
includes the Palmdale array and spans a varied geology
Section
in
described
skew
and
Eigenvalue
3.4.
as
sensitivities are calculated for the blocks which span the
Palmdale array within the perturbed region. Variations
the
are
eigenvalues
The
from the model electric
determined
fields for a spatially invariant
magnetic
of the change in apparent resistivity of that
dipoles
D
reference
our
to
and
sensitivities
source.
field
sensitivities for each block are measures
eigenvalue
respect
in
H
are
in
the
block (5,4)
Palmdale
block
which
array.
with
includes
The
skew
measures of anisotropic changes in the
resistivity contrast between the measurement and reference
block:s.
3.6.1 General Results
The general pattern of eigenvalue sensitivity is
a manifestation of the role of the
66
individual
blocks
in
11~~
i 1---~Pi--lp---rr~-~i--1..UII~ -~--PIC
Or~^~
Tlt~PIll~
-LI~
determining
the
levels
and distribution of the telluric
current system. The eigenvalue sensitivities of
conductive
much
blocks
higher
mountain
small
more
(4,3)-(4,6) and (6,2)-(6,6) tend to be
than
the
sensitivities
blocks (5,4)-(5,6).
changes
the
in
current
In
of
the
occur
the
resistive
conductive blocks,
for
a
change
in
conductivity with the result that most of the conductivity
change
is
reflected
in
the
electric
fields.
For the
mountain blocks, changes in conductivity cause the current
levels to change with the result that no change occurs
the
in
electric fields for those blocks. However, the actual
current change is so small
current
relative
to
the
unperturbed
levels that little effect is seen in the electric
fields of surrounding current saturated blocks.
Conductivity variations in
blocks
which
cause
the current levels to change in the reference block result
in
a
diffused
eigenvalue
sensitivity. All of the block
eigenvalues change proportionally to
the
in
change
the
apparent conductivity of the reference block.
Blocks in the
Basin
Mojave
Desert
and
Los
Angeles
tend to be partially saturated with the result that
in
variations in one block affect the current levels
nearby
blocks
and
implication of this
inference
of
a
all the block eigenvalues change. The
sensitivity
diffusion
is
that
completely
the
variation in many of the Desert or Basin
blocks from telluric measurements is not unique. In
only
the
fact,
current saturated blocks will produce an
-YL4L_
eigenvalue variation that can be determined uniquely
from
telluric measurements.
3.6.2 Isotropic versus Anisotropic Conductivity Variations
Isotropic conductivity variations are consistent
with variations in hydrostatic
consistent
are
variations
anisotropic
while
with deviatoric variations in
both
variations,
isotropic
For
E).
stress (Appendix
stress
eigenvalues of the telluric tensor tend to change together
while
for anisotropic changes the eigenvalues can vary in
skew occurs but for anisotropic changes the
in
variation
negligeable
changes,
For isotropic
opposite directions.
block
skew varies as much as .4 degrees/percent change in
isotropic
sensitivities are less than their
because
the eigenvalue
changes,
anisotropic
For
conductivity.
counterparts
part of the anisotropic variation is reflected in
the change in skew. Thus, changes in skew and the signs of
the eigenvalue
between
hydrostatic
Figures
3.12-3.14
relative
can
variations
sensitivities
to
distinguish
stress
variations.
used
deviatoric
and
are,
be
respectively,
matrices
of
the
the isotropic eigenvalues and
of
anisotropic eigenvalues and skews. Each row represents the
sensitivity
of
one
block
to
variations
in
the
corresponding blocks in the columns of the matrix.
3.6.3 Conclusions
From our sensitivity analysis
region,
of
the
Palmdale
we have found that the telluric tensor parameters
68
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_
determine
eigenvalue and skew are important to
the
conductivity
of
levels
_~ L___(____Li_~~~ll
(IIX~Y~LI
not
only
variations but also the the'
type of variation. We find that measurements in the Mojave
Desert
(dipoles
Additionally,
C)
should
not
only
dipoles
A
be
and
F
sensitive
to
under the dipoles but
areas
surrounding
the
in
and
variations
conductivity
also
B
are
of
the
located
current saturated regions and should be sensitive
Mojave.
in nearly
largely
to local variations although dipole F may also be affected
by changes in adjoining regions.
Dipoles
unsaturated
San
D
and
H
span
the
largely
current
Gabriel Mountains and are insensitive to
variations in conductivity. One of these dipoles is useful
as a reference but the other appears redundant.
ili~iV~-ll--
-~
__
Chapter 4
INDUCED POLARIZATION WITH TELLURIC FIELDS
4.0 Introduction
The term induced polarization (IP) is applied to
an electrical prospecting technique designed to detect the
presence of metallic minerals
polarization
by
effects induced on the metals by currents in
the
the ground. For an in depth review of
read
paper
the
only
present
surface
the
measuring
by
those
Cantwell
aspects
IP
technique,
Here we
and Madden (1967).
of
our
for
necessary
IP
studies.
Electrical
in
conduction
due
is
rocks
predominantly to the flow of ions through pore spaces. The
presence
adds an
of
metallic minerals in and near the rock pores
electrochemical
barrier
to
electrical
current
because of a change in the conduction mechanism from ionic
to electronic. This added electrochemical impedance is due
to the depletion or excess of current carrying ions at the
metal solution interface. The ion imbalance is compensated
by diffusion to or from the solution. Because diffusion is
time
dependent
(or
equivalently
frequency
dependent)
current through a metal solution interface will require
finite
time
to
equilibrate
after
an electric field is
applied and will decay accordingly when the applied
is removed.
a
field
rL-rirrr~i-L-----L~-IYurri-r-^l~---~-c--~ II *~~
The impedance of
rock
a
bearing
mineral
metallic
be represented by purely resistive paths due to'
can
mineralized
partially
A
dependent.
Figure
in
time/frequency
are
which
circuit
simplified
depicted
is
impedance
paths
the
of
analog
4.1
with
parallel
in
conduction through barren pores acting
IP
from Cantwell and
Madden (1967).
Zm M
m
f-1/4
Figure 4.1 Equivalent Circuit for mineralized rock
where Ro represents the barren pore paths
the
represents
metal
mineralized
in the band
.01
shifts typically less than
(Rrn
+
ZIM)
pore paths. The impedance of
bearing rocks varies from 1 to 100%
frequency
and
to 100 Hz with
1 degree per
per
decade
of
associated phase
decade
(Cantwell
and M adden,1967).
In practice,
applied
to
a time varying
source
current
is
the ground and the potential is measured as a
I~-^(l-'iYr
function of position. For time domain methods, the
current
is
switched
on
and
off and the rise and decay
techniques
Both
noise
by
limited
are
of
presence
times of the potential used to infer the
targets.
source
IP
from
electromagnetic coupling at high frequencies (>100 Hz) and
tellurics at low frequencies (<1
frequency (Cantwell.1960). The presence
then,
field,
decreasing
with
rapidly
increases
field
telluric
the
where
Hz)
telluric
the
of
limits the depth to which IP targets can be
discerned using active measurements. To alleviate this low
problem,
noise
frequency
used
has
(1981)
Halverson
telluric cancellations to remove tellurics from his active
Also,
measurements.
IP
SEG
of
infer the presence
frequency
an
IP
dependence
target
of
by
an
meeting
oral
in
feasibility
of
applying
analyzing
the
telluric measurements
within and near the target. In this chapter,
the
in
proposed using the telluric fields directly to
Australia,
relative
biennial
first
the
to
presentation
Madden(1979),
we
consider
Madden's premise to actual
field data.
We model the IP target as an embedded
of
revolution
whose
conductivity varies with frequency.
Then, using an approximate extension of the
field
electrostatic
solutions for an embedded ellipsoid, we predict the
frequency dependence of the telluric
Next,
ellipsoid
we
calculate
the
frequency
tensor
eigenstates.
dependence
telluric tensor eigenstates for field data
of
obtained
the
near
an IP target,
a pyritic
and data analyses and set bounds on the
our model
compare
schist near Harvard, Mass. We then
applicability of our models and our ability to discern the
IP frequency effects in telluric
noise.
data corrupted by
As a control study, we analyze telluric data from Salinas,
Ca. where no IP effect is expected.
as
The results of our studies can be summarized
follows:
(1)
target.
the
of
dependence
unambiguously
discern
To
field associated with an IP
telluric
the coherency in predicting one field component in
terms of two others must be high (the order of
(2) For low noise measurements,
analysis
the
frequency
the
allows
us to
beneath
earth
the
the
.995).
eigenstate
study the electrical properties of
measurements
source
from
free
effects.
(3)
presence
of
Frequency
an
effects
consistent
the
with
IP target were inferred from the Harvard
data
data and no such frequency effects were found in the
from Salinas.
(4)
heterogeneity
The
ellipsoidal
provides
useful
approximation
3D
a
to
but
insights
qualitative
care must be used in its quantitative application.
The rest of this chapter is
sections.
In
section
4.1.
we
divided
consider
into
four
the effects
measurement geometry on our ability to discern IP
of
effects
_~_~_ __~_II _r~Ym
__
the
in
field
telluric
and
develop
a
strategy
conducting field measurements. In section 4.2. we
analysis
our
section
4.3,
in
Harvard,
Mass.
we present the analysis of similar small
scale telluric measurements
where
present
of telluric data from the vicinity of an IP
target, a pyritic schist near the town of
In
for
obtained
near
Salinas.
Ca.
no IP related frequency dependence was expected and
section
4.4.
we
summarize
recommendations for further study.
our
results
and
make
11_1_
for
geometries
measurement
4.1 Telluric
discerning
IP
targets
4.1.1 The ellipsoidal model (Appendix D)
In Appendix D we present the algebraic formalism
for the electrostatic response of a
of
revolution
(spheroid)
ellipsoid
conducting
embedded in a conducting earth
and the approximate extension of this analysis to the
field
telluric
frequency
response to a buried spheroid.
With this approximate analysis, we establish
the
telluric
low
model
a
of
relating the telluric fields across
tensor
the boundary of the embedded spheroid. As a consequence of
field,
our
sets
symmetric and the directions of the two parallel
tensor
not
eigenvectors
conductivity
contrast
surroundings
but
the
DC
the telluric tensor is shown to be
of
model
of
nature
spheroidal symmetry and the curl free
of
also
only
the
to
insensitive
with
ellipsoid
of
the
its
always normal and parallel to the
surface of the ellipsoid. Additionally, we find
that
for
thin spheroids exhibiting moderate conductivity contrasts,
measurement
positions
exist
for which one of the tensor
eigenstates can be insensitive to the frequency dependence
of the spheroidal IP target. Thus. the directions
tensor
eigenvectors
of
the
are set only by geometry whereas the
eigenvalues and their frequency
dependences
conductivity contrast and target shape.
are
set
by
4.1.2 Array locations and configurations
our model of the
In the preceding discussion of
calculated
was
each side of an
the
situation
electric field measurements at
two
from
of the
ore body is unknown and a
strategy for prospecting with telluric
we
Accordingly.
established.
be
question, what measurement
practical
a
In
body.
ore
ellipsoidal
location
tensor
the
telluric tensor, we have assumed tacitly that
need
measurements
seek now to answer the
configurations
locations
and
are necessary to infer the frequency dependent eigenstates
telluric tensor when the IP target has the shape of
of the
a
or
prolate
spheroid? Additionally. we wish to
oblate
coverage
streamline our measurement system to allow rapid
regions
of
ore
an
where
body is expected to exist.
"normal"
of
method
inferring
telluric tensor using
the
three dipole
a
under which
to infer
this
system and investigate
simpler
logistically
geometry
the
four dipole measurements. Then. we simplify
to
the
first
present
accordance with these goals, we shall
In
the conditions
system
can
tensor
at
be
used
the presence of an IP target.
To calculate the
telluric
field
a
site,
an array of four dipoles with a common center can be
used
as
depicted
in Figure 4.2.
In general,
between dipoles can be somewhat arbitrary and
usually
to
are
chosen
conform with the terrain, the road system. or
suspected target boundaries.
C
the angles
In Figure 4.2,
are chosen roughly parallel
as are
dipoles A
dipoles B and D.
and
The
B
C
set 1
e
t2
ZD
Figure
4.2
Four dipole telluric
X
tensor geometry
7,C
Y
Figure 4.3
Three dipole measurement geometry
2 (C and D) are calculated and rotated to the
set
dipole
eigenstates
Cartesian coordinate axes X and Y. The tensor
are
in
calculated
then
dipole
A
use
a
to
is
approach
system.
a common center as depicted in
with
array
coordinate
this
logistically simpler measurement
three
and
(A and B)
1
tensor relationships between dipole set
calculate
can
Figure 4.3. With only three dipoles, we
a
tensor relationship of the form:
j=
]
[T
=T
(4- 1)
and after rotation to X and Y coordinates, the eigenstates
of this three dipole tensor
three
four
and
dipole
can
tensors
be
within
are
the
can be analyzed for the
presence of an IP target when the centers
arrays
Both
inferred.
dipole
the
of
or near the boundaries of the target.
The most important aspect of the array location is that at
least one dipole must exhibit sensitivity to the frequency
of the IP
dependence
little
exhibit
no
a
sensitivity
second
to
dipole
center
can
must
frequency
this
Now let us consider how the position
dependence.
array
or
target while
of
the
affect the frequency dependence of the
tensor eigenstates.
Using three dipoles as shown in Figure
least
one
4.3.
at
of the tensor eigenstates will be sensitive to
the presence of an IP target when one of
the
dipoles
is
within
frequency dependence of the dipole
all
three
dipoles
are
the relative
of
because
boundaries
target
the
However.
signals.
if
within the IP target boundaries,
usually there will be no indication of the presence of the
IP target because all three dipoles will have the same
or
close to the same sensitivity to conductivity changes with
The one exception to this rule occurs when the
frequency.
current saturation condition, described in Appendix D,
anisotropic
IP target. For example, when the
the
within
to
saturated
is
target
current within the
consequently,
and,
anisotropic
field
telluric
the frequency dependence of the
at
dipoles
more
the
span
tensor
will
eigenstates
dependences of the IP
Figure
4.4,
targets
of
with
a
telluric
mix
and
its
surroundings.
of
As
detection
of
fields. Additionally. we feel
using
a
four
dipole
array
three dipole array is the reduction of the
possibility that none
confines
of frequency
the
present a summary of the important three
we
that the major advantage of
instead
If
the
because
reflect
target
be
boundaries, our
target
dipole array geometries associated with the
IP
in
the
of
frequency.
diminished
ability to discern the target is
will
one
least
telluric eigenstates will be a function of
or
changes
in one direction and unsaturated in another,
conductivity
one
is
the
of
the
dipoles
lies
within
the
IP target, but that this advantage does
not outweigh the logistical simplicity of the three dipole
system.
Three dipoles outside target
No measureable IP effect
One dipole inside,
two outside
At least one eigenstate will exhibit
port or all of the frequency dependence
of the IP target
Two dipoles
inside, one outside
Both eigenstates will exhibit the full
effects, unless an anisotropic current
saturation condition exists
One dipole partially inside, two outside
One eicenstate will exhibit partial
frequency dependence of the IP target
Three dipoles inside
No observable frequency dependence
unless on anisotropic current
saturation condition exists
FIGURE 4.4
MEASUREMENT
SENSITIVITIES
ll~_
I-LPIIL-~YII~- .---III~I~WL~---~ ~7iX
with
Our strategy for IP prospecting
then.
is,
tellurics
to use three dipole arrays over a region where
the presence of an IP target is suspected. In particularly
promising areas, we suggest orienting two colinear dipoles
normal to
maximize
the
the
frequency
of
boundary
suspected
IP
the
to
zone
dependence of the related tensor
element. The number of measurements required to
delineate
the IP zone will depend on complexity of its boundary, the
of
size
the
zone
and the degree of a priore geological
control. Let us now apply these concepts to the
inference
of the boundary of a pyritic schist near Harvard, Mass.
4.2 Telluric field measurements near an IP site
4.2.1
Previous
geoelectric
measurements
near
Harvard,
Mass.
Harvard, MA is a town located
miles northwest of Boston (Figure 4.5).
approximately
40
The basement rock
in the Harvard area is largely Ayer granite with dispersed
metamorphosed
remnants
One of these remnants is
Worcester formation.
of
of
a
older sedimentary structures.
mica
schist
facies
of
the
This schist contains varying amounts
pyrite and graphite and exhibits an IP response.
East
of the center of Harvard, the schist is a topographic high
upon which the Harvard Observatory is located.
past
Over
the
two decades, geophysical surveys have been conducted
within and near the observatory's
pioneering
magnetotelluric
grounds
studies
of
including
the
Cantwell (1960).
J_
100
200
kilometers
Figure 4.5 Location Map for Harvard, MA
Recently a study of the lateral
heterogeneity of the crust
in
and
1979
IP
and
resistivity
source
active
1980
and
(1979),
in the Harvard area was conducted by Bob Davis
measurements were conducted on a smaller scale by students
in
MIT Geophysics Field Course under the tutelage of
the
Park.
Prof. Madden and Steve
source
active
the
From
formation
conductive
(AI10-100SI-rm)
to
10%/decade
of frequency.
(Ayer
approximately
of
effect
IP
an
exhibit
of pyrite.
more
because of the presence
and,
granite. F=1000-10,000fL-rm)
be
to
surroundings
its
than
the
inferred
been
has
Worcester
measurements,
From geologic maps of the area,
the surficial expression of the Worcester
is
formation
a
narrow (1-1.5 km) schist extending tens of kilometers in a
(1979).
second
from Davis
4.6
Figure
in
depicted
as
direction
NE-SW
Using magnetotelluric measurements in the
band,
schist was not
Davis inferred that the
Bob
only a shallow feature (~1.5 km) relative
50-150
to
its
length
but was pinched out by Ayer granite near Whitney Road
Figure
can
be
4.6).
Thus.
conducting
body
the Worcester formation near Harvard
an
as
modelled
(see
which
elongated
exhibits
an
three
IP
dimensional
response and is
embedded in a more resistive granite host.
To test Madden's hypothesis
fields
inside
and
outside
an
IP
that
the
target would exhibit
relative frequency dependence. we have conducted
fielc
measurements
in and near
telluric
telluric
the schist. Additionally.
00
97
49
App (A-nM)
o0
Z kt
MAGNETOTELLURIC SURVEY MAP (From Oavia. 1979)
FIGURE 4.6
____1_)_____1____1________
____~ ~CI__I~*I^XPI__RII_
with the realization that IP targets are
dimensional
generally
three
bodies, we developed the approximate embedded
ellipsoid analysis outlined previously in this chapter and
discussed in detail
in Appendix D.
Now let us consider the
acquisition and analysis of telluric field data
vicinity
from
the
of the Harvard Observatory.
4.2.2 Data Acquisition
Near Harvard, MA we deployed an array
dipoles
as depicted in
of
Figure 4.7. Dipole A is inside the
schist approximately perpendicular to the strike
direction
of
schist
and
or
long
schist. Dipole B is outside the schist
the
also roughly perpendicular to strike. Dipole C
the
three
oriented
roughly
is
parallel to
within
strike. A
detailed description of our field equipment and procedures
can be found in Appendix A.
The telluric signals were amplified and filtered
second band and
in the 10-120
period.
sampling
road
power lines,
potentials
at
the
much higher degree
surburban
our
digitized
data contained noise
traffic,
and
a
at
1
second
due to nearby
particularly
high
self
electrode sites. We were plagued by a
of
cultural
at
noise
Harvard than at measurement
our
site
in
sites in desert and
farming regions in California. The local
power ground
for
a high voltage line was located near our center electrode.
Rectification
caused
the
of
high
current
self
through
potential
this
noise
ground may have
we
measured.
,.oC
4
/,-C .++
Ayer Granite
Fgr
>7 /
a
@
0;-
0
4
0
4P
"Q0
Whitney Road
Harvard D
Fiur
47Diol
-\
O
Harvard, MA
kilometers
Ictin
Hrar,4I
much as 5 volts AC was measured between
as
Additionally,
electrodes separated by 500 meters. which
suppression
of
the
required
strong
AC in our input circuitry to prevent
saturation of our amplifiers.
The noise
in the records was limited
to
a
few
points or was impulsive decaying at the rate of the filter
time
An
constant.
coherency between signals is
a
of
example
with high
recording
given as Figure
4.8
and
unacceptably noisy recording is shown as Figure 4.9.
of
consist
recordings
three
electronically filtered and amplified
of
combinations
in
4.8 is .992
Figure
signals in Figure 4.9
band
represent
or
signals
slight at a
coherency
between
noise) while for the
(12%
the coherency is
.975 (23%
noise).
Data Analysis
the telluric data was
Our aim in the analysis of
telluric
to infer a relative frequency dependence between
fields
inside
decade
of
and
outside
of
When
frequency.
expected
differentiate
dependence.
estimates
between
noise
the
noise
and
of the eigenstate
90
frequency
we
not
could
relative
we were able
Consequently,
or
levels
approached or exceeded
dependences,
frequency
per
the schist of 10-20%
incoherencies between these signals
their
Both
The difference between
signals.
However the broad
first glance.
4.2.3
dipole
levels in the two recordings appears
the noise
signals
dipole
which
signals
an
to infer
dependences
frequency
reliable
for
only
__~
A
A-xC
Figure 4.8 High Coherency Recording
Figure 4
L
Coherency Recording
Figure 4.9 Low Coherency Recording
_j_ L~I__1
p__
~~_L_~~_I__CI_
__I
_III___III___~- ~-IC~
LLi
-~IIL
~_;___I^__~CL~ ~_^
__~nWIIIU~LI
one set of recordings.
Two numerical approaches can be
the
used
to
infer
frequency dependence of the telluric tensor. The data
can be transformed to the frequency domain
tensor
elements
and
the
real
calculated as a function of frequency in
the least squares sense.
Alternatively.
constant Q digital
filters can be applied to the recorded time series and the
tensor elements estimated in the least squares
each
band.
For
both
domains
tensor prediction of one
signal
sense
for
the coherency between the
dipole
signal
and
the
actual
was used as a measure of goodness of fit. We tried
both approaches but
series
approach
suppression simply
ultimately
we
because
by
the
used
the
could
removal
filtered
accomplish
of
time
noise
incoherent
data
points. Noise suppression in the frequency domain was more
Least
complicated.
domain
time
squares
described in detail in Appendix C.
analysis
Let us present
now
is
a
example of how we estimate and reduce the noise in
simple
the elements of our telluric tensors.
We
seek
a
least
squares
solution
to
the
equation:
Bpr
(t)
= a A(t ) + c C(t
which relates the dipole signals
oritnted
as
in Figure 4.7.
A
(4-2)
i=1 to N
i)
and
C
to
dipole
B
The three dipole time series
can ce thought of as column vectors of length N.
In terms
of the dot products between these column vectors,
a
least
squares solution for the tensor elements a and c is of the
92
_.lli _LI _I~_~_____LIII__I____LIC~U*~.
form:
A*A AC
a
A*B
(4 -
C*B
C*A C*C
c
-
-
I
L-
3)
signal
prediction of
The coherency between the
and the actual signal B is determined with the
(Bpred)
B
equation:
B*B
B
B
Bpred Bpred
ch(B"BPred) =
For
the
Harvard,
elemen ts
and
MA
coherency
data,
1/2ed
(4
the
calculated
we
4-1)
(equation
for
noisy
spline fit.
filter
ed
the
With
data
data
this
in
noise
initial
four
we
for
which
the
error
exceed ed a predetermined value.
set,
we
recomputed the
in
suppression,
discarded
then
the
With
we
least
this
data
squares fit
reduced
data
tensor values and coherencies and
iterated on this procedure reducing the
squares
isolated
and recalculated the
bands
tensor elements and cohe rencies. We
points
tensor
we replaced with a
which
points
4)
the complete
freque ncy band. With the se tensor estimates,
spurio usly
-
acceptable
least
error with each step. The iterations were stopped
93
I1__I__~
_IrXICIO____Ci___*_~-1L~ _
__1_____11
in
reduction
percentage
the
exceeded
half
incoherency. Typically. more than
of
number
the
in
when the percentage reduction
level
noise
data
the
points
or
points
improved
were used in the last iteration and the coherency
from an initial value of approximately .992 to better than
.998
for the final
The tensor values from the
iteration.
coordinate
Cartesian
a
to
final iteration were rotated
system (X.Y) whose axes aligned perpendicular and parallel
eigenstates were then calculated for each
to dipole C. The
versus
(HV296)
is
as Figure 4.10 and the eigenvector orientations
presented
are shown as Figure 4.11.
From Figures 4.10 and
4.11.
we
eigenvalue is consistent with a
larger
the
that
notice
eigenvalues
recording
coherent
most
our
frequency for
the
of
plot
A
band.
frequency
more than 50 to 1 conductivity contrast between the schist
and the surrounding granite and the smaller eigenvalue
along the strike of the
consistent with some heterogeneity
eigenvalues
Both
schist.
is
exhibit
frequency dependence
consistent with the presence of an IP target. In
fact, the
telluric tensor frequency dependence is greater
than
10%/decade
source measurements
smaller
IP
effect
inferred
dependence
frequency
10
at
and
1
second
from
over
the
telluric field
the
active
active
periods.
schist
that
has
little
surface
effect on the
but can dilute the frequency dependence
source
The
for active source measurements may be
caused by the presence of a pyrite free weathered
layer
the
measurements.
Alternatively.
of
the
~___~
60
-
1 55
1
-- 7
12
*
11
x xX2
1
50
I.003
.003
.01
Frequency
Figure 4.10
.03
(Hertz)
Eigenvalue frequency dependence
for Harvard, Mass telluric data.
95
OUTSIDE 41 5
20
o
10
dipole C
dipole
C
INSIDE
8.60
8.6
20 -
40
41. 40
\t
dipole C
dipole C
5.7"
5.70
40 - 80
41
8
5.548
dipole C
dipole C
5.5*
I0
- 160
41
5.8
dipole C
1
dipole C
5.8*
Figure 4.11
Eigenvector directions versus frequency
II
__I_~1_I I__CIII~~WII1__YI~
__~~_(~~_C_
frequencies
increased IP effect at the lower telluric
is
pyrite
by
We notice also that the frequency dependence
of
laboratory
with
consistent
on
measurements
Morgan (1981).
the
eigenvalues is consistent with the ellipsoidal models
of a prolate IP target but unlike the ellipsoid model. the
not
with frequency but also are not parallel.
vary
only
schist
the
corresponding eigenvectors inside and outside
the
in
This variation
with
directions
axis
principal
frequency could be due either to noise in the data causing
the
in
uncertainty
eigenvector
heterogeneity within the schist.
to local
or
estimates
possibility
Another
is
dipoles B and C may have spanned the boundary of the
that
of
the
The most significant points of this example
are
their
schist and
signals
mixture
a
exhibited
internal and external frequency dependences.
consistent
that
a
able to infer
with Madden's hypothesis (1979). we were
frequency
dependence
for fields in and near an IP target. Additionally,
tensor
consistent with Morgan's observations, the
end
telluric
the
in
of
the
telluric
spectrum
may
frequency
low
be
discriminating between the presence of pyrite
helpful
and
in
copper
sulfide deposits. To add credence to our IP field studies,
we
applied
a
similar
analysis to telluric measurements
near Salinas. CA where no IP effects were either
or found.
97
expected
4.3 Telluric Field Measurements near Salinas, CA
an
We present in this section,
telluric
a
of
analysis
tensor relating fields measured near the site in
Salinas. CA depicted in Figure 4.12. The small scale array
and
ocean
is within 10 miles of the Pacific
3
involves
dipoles less than one kilometer in length. Near coastlines
the
telluric
tends to
be
is dictated largely by the ocean and
field
be
can
polarization
linear
toward
This- tendency
polarized.
linearly
in a 1024 second long
seen
recording of the dipole signals presented as Figure 4.13.
described
Applying the same procedures
previous
we
section,
computed
the
tensor
eigenstates plotted in Figures 4.14 and 4.15.
is a plot of the tensor eigenvalues versus
4.15
Figure
is
a
of
plot
eigenvector directions.
the
tensor
and
Figure 4.14
frequency
frequency
and
independent
IP
target
and
the
dipole lengths used and, in fact, should be typical
short
of small scale telluric tensor analyses.
in
this
The
data
greater
sets
example were extremely coherent. Typically
the coherency between the predicted and actual fields
was
than .9995 and unlike the IP example little noise
suppression was required.
4.4 Summary and Conclusions
4.4.1
the
This lack of frequency dependence
is consistent with the absence of an
used
in
Summary
Il ^~_
_____1__1~_1 I__~
San Francisco
Salinas
Los Angeles
Figure 4.12 Location Map for Salinas, CA
H31062
Ch1
Ch2
Ch3
Figure 4.13 Salinas Telluric Data
I___eYll__~lf___LIIII
_~I~~~_ -.ilf_~l__l-.~--L-lI
1. 1-
-.7
1 1.0-
-7
2
2
10.9-
.6
.003
i
1
.01
.03
Frequency (Hertz)
Figure 4.14
Eigenvalue frequency dependence
for Salinas, California data.
26diole
dipole 2
dipole 2
Figure 4.15 Eigenvector directions for Salinas site
100
- -r..~L-i~L
-illll~ II~~.-4UILLIII~
IX-^~
-I-~I*-~~Y~-3~
III --I1P
_)Ij_*^~I_~____l_
model
approximate
telluric field
frequency
low
the
of
an
presented
have
we
chapter,
this
In
body)
(ore
response to an embedded ellipsoidal IP target
and have described an experimental procedure for measuring
the
telluric
dependent
of
the
an
IP
target.
such
eigenstate analysis of Chapter 2 to
telluric data from two
from
far
sites; one near an IP target and the other
known
the
applied the
have
We
in
tensor
telluric
eigenstates
of
vicinity
fields necessary to determine the frequency
any
IP targets. We have inferred a frequency dependence
for the tensor eigenstates consistent with the
an IP target at the first
presence of
site and found no such frequency
dependence at the second site.
4.4.2 Conclusions
We conclude from our studies that the use of the
telluric field and the telluric tensor eigenstate analysis
for
as a prospecting tool
shows
considerable
promise.
of
inference
the
We feel
IP
targets
that more extensive
this
studies are necessary to determine the generality of
and
approach
explore
the possibility of differentiating
between pyrite and copper sulfide targets.
We find that the
target
is
eigenstate
ellipsoidal
model
a qualitatively useful tool for
frequency
applicability
appears
dependences
limited
formations.
101
but
to
of
an
IP
predicting the
its
simple
quantitative
geological
_~ I__-rri----ili.l--QL.-1---11-m ~-r*-~----Y---p.-- iB-L~~ I-~YYII ILl
Finally. we feel that the ease
can
be
which
noise
removed using the time domain analysis along with
the requirement
filtered
at
time
of
highly
series
coherent
approach
domain analysis.
102
signals
preferable
makes
the
to frequency
CHAPTER 5
Thesis Summary and Extensions
5.0 Summary
tellurics
differential
infer
to
geophysical information
have
We
field.
from the fine structure of the telluric
found the small
of
concept
the
apply
to
sought
have
We
frequency dependence in the fine structure
of local telluric field measurements to be consistent with
of
stability
the
crust
determined
a
of
conductance
of the stability of the
have
and
is
measurements
telluric
differential
sensitive measure
that the
IP target. We have ascertained
of an
the presence
that the sensitivity
is
dependent on the degree of current saturation.
field fine
telluric
structure with the shifted eigenvalue
analysis of Lanczos(1961).
relationships
tensor
We can
telluric
the
separate
two
into
representing the structurally
electric
the
into
We have obtained considerable insight
modes
modes
natural
imposed maximum and
minimum
directions and use the skew of the tensor
field
as a measure of the spatial variation of these directions.
Thus, with the eigenstate analysis, we can
structure
of
the
parameters
which
properties
of
the
telluric
are
field
reflections
of
terms
in
the
of
fine
study the
a
few
physical
earth. The application of the Lanczos
analysis to the impedance tensor also has yielded insights
into
the
electromagnetic
response
103
of
the
earth.
~---r~Y*c-~l'~~llsrr--~--~^XL~
--- -__~~1
Consistent with the heterogeneous nature of the earth, the
and magnetic eigenstates need not be orthogonal.
electric
The
skew
a
as
usually
considered
dimensional
"three
parameter", can be interpreted as the structurally imposed
of
deviation
from
fields
magnetic
and
electric
the
perpendicularity.
measurements,
telluric
differential
sensitivity
of
have considered
telluric
tensor
are independent of frequency even for large
measurement separations
Additionally,
we
the
that
source frequencies low enough
relationships
stress
the
of
In our studies
our
in
varied
over
structure.
have
considered
we
studies
IP
crustal
that
measurement separations small enough
any
frequency
dependence
in
the telluric field fine structure could be
attributed
to
a
frequency
dependent.
while
is
sensitivity
study
telluric
tensor
for
limit
represents the low frequency
relationships
the
Thus,
conductivity
whose
feature
local
the IP study represents the limiting
case of small separations
between
measurements.
In
the
next section, we describe our progress in the study of the
structure of the telluric field for frequencies high
fine
enough
enough and separations large
dependence
change
in
of
the
upper
telluric
crustal
tensor
atructure
that
the
frequency
is a measure of the
between
telluric
measurements.
5.1
Lateral Variations in Crustal Conductivity
For frequencies greater than about
104
.1
Hz,
the
IIUPIU__^l_~___ls~_
i_~I~l__l__~L__ t
crustal
upper
of
ratio
small and the frequency dependence
longer be considered
the telluric field becomes a
start to "skin out"
position.
between
difference between the
the
in
described
As
frequency responses.
D,
Appendix
telluric current system can be separated into poloidal
and
surface field
At
dependent.
frequencies,
low
to
contributor
dominant
the poloidal mode is the
surface
the
the
and
field
telluric
dependence of large scale differential
frequency
measurements can be attributed to the skinning out of
thin sheet approximation begins to break down, the
separated
relating
eigenstates
field
associated
phase shift. Minimal
phase
tensor
measurements will
frequency
exhibit amplitude variations with
the
high enough that the
frequencies
At
currents.
poloidal
and
frequency
generally
are
modes
both
the
systems contribute to the
current
Both
modes.
toroidal
For
currents will
measurements
telluric
exhibit
will
regions
of
at lower frequencies than in resistive
differential
regions and
these
of
function
in conductive regions the telluric
example,
can no
to skin depth
thickness
with
little
shifts are expected
initially with the poloidal mode because the low frequency
the
phase at each telluric site is set largely by
conductivity
is
which
position on the earth's
be
as
much
as 45
telluric
field
known
to
vary
rapidly with
surface. The eigenvalue
phase
can
degrees for the limiting case when the
telluric field at one
the
not
mantle
site is
at
completely skinned out while
another
105
measurement
site
is
l-zi~lClil-r~L
--~~-~---~r.iafl--r~--
;I-~-~~--jl
conductance.
is
frequency
At higher frequencies, the
telluric
tensor
to
be real.
At even
and
complex
is
tends
and
dependent
tensor
telluric
the
frequencies,
higher
measure of the crustal
a
and
frequency
of
independent
regimes,
then,
and
thickness
relative
the
of
diagnostic
be
be
to
tends
by the apparent conductivity of the
set
upper crust. The transition between these
should
real,
is
tensor
telluric
the
then,
frequencies,
low
At
conductance.
crustal
the
to
only
sensitive
conductivity of the upper crust.
telluric fields for frequencies up to
we have measured the
.3
the
of
Notice
5.1.
between
the tensor relationships
determine
example
telluric cancellation scheme to
the
applied
and
Hz
crustal
in
change
relative
consistent
conductivity
measurement dipoles. We suggest that with
should
crustal
able
be
to get both estimates of
conductance
conductivity
under
the
and
dipole
the
the
suggest
section,
conductivity
further
study
and
on
upper
this
such
with
the
under
we
data,
the contrast in
apparent
measurements.
From these
spatial
crustal
subject.
variations
and
thickness
In
the
next
we make specific suggestions for future study
106
a
in
contrast
estimates, we should be able to infer
in
contains
signal
residual
increasing levels of high frequencies
is
cancellation
of
degree
resultant
the
increased,
the
as
An
dipoles.
signals is given as Figure
cancellation
that
dipoles,
and Palmdale array
Using the Hollister
in
Pd offico 092779
4 diek 7
trk#
Pd offico 0927 79
trk#
3 doek
7
gain=l
a in = 1
59-x3C=Sbl
qoin=-5. 87AV
x=. 774
gain=-1. 1
H+x3C
gain=-l. 50
gain=- 1.50
x=. 874
x-. 874
H-xO-ySb2
gain=-20.86
x=.
-"
902
gain=-20. 6
x=. 902
y=. 0591
y=. 0591
Figure 5. 1
High Frequency Cancellotions
1024 second sweeps
*C----~~LI(-~--~PIR
~__I___Lr
_~~X1
Il
resistivity
to
the application of differential tellurics
and IP prospecting as well as consider further
monitoring
the application of eigenstate analysis
impedance
the
to
tensor.
5.2 Future Studies
of
the
succinctly
the
use
In Chapter 2, we have proposed the
and
properties of the telluric
How
to
tensors.
magnetotelluric
ellipticity
the
interpret
represent
to
formulation
eigenstate
of
impedance
the
eigenvectors is still problematic. One possibility is that
the
represents
ellipticity
vicinity
of
induction
effects
magnetic
fields.
vertical
large
this
to
given
be
consideration should
in
the
Further
interpretation,
however.
In Chapter 3, we have presented our study of the
sensitivity of differential telluric measurements
Palmdale
southern
of
region
California.
studies of Madden's Hollister array but the
yet
finished.
We
in
the
We have begun
work
is
not
have suggested that the inference from
where
telluric measurements as to
a
stress
change
has
occurred is not unique but further consideration should be
given to this problem.
Studies
differential
of
the
stability
of
telluric measurements have been initiated by
Prof. Madden. Such small scale measurements
and
can
scale
small
are
portable
be deployed rapidly. From preliminary studies of
the sensitivities of small
scale
108
measurements,
we
feel
that
the
of the telluric tensor should be the most
skew
because
of
heterogeneities
to
results
of
sensitive tensor parameter to stress changes,
the
local
of
response
anisotropic
deviatoric stress.
In Chapter 4, we have presented the
our
of an IP target from the fine structure in
inference
the telluric field. We have presented only one result from
an IP zone and feel that
field
more
should
studies
We have noted that the eigenv-ectors associated
conducted.
frequency
with our IP target, a pyritic schist, vary with
while
be
eigenvectors
the
associated
with the surrounding
granites are invariant with frequency (Figure
eigenvectors as a manifestation of an
conductivity
of
dependence
frequency
the
interpret
4.12).
the
anisotropy
We
schist
in
the
of the schist but feel that further study is
warranted.
Another approach to using differential tellurics
to infer the presence of IP targets is to establish a base
telluric
station
(Berdichevskii,1960)
and
a
roving
telluric
station
and through a telemetry link monitor
tellurics simultaneously from both stations. With such
approach,
we
should be able to detect the presence of an
IP target even when
effective
an
the
boundaries
of
roving
station
is
within
the
the IP target. Further study is
warranted here as well.
109
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34,
p313-320.
Swift,C.M.,(1967), A Magnetotelluric Investigation of an Electrical
Conductivity Anomaly in the Southwestern United States.
Phd thesis, MIT,pp 211.
111
rr --- ^aru~ 'i-Y~-n'x~-
APPENDIX A
Field Equipment and Procedures
A.1
Field Equipment
by
obtained
are
signals
Small scale telluric
measuring the voltage difference between two electrodes in
contact
separated by distances the
and
earth
the
with
are
measured,
signals
approximately
dipoles
with
usually
dipole
three
order of a kilometer. As a minimum
orthogonal as depicted in Figure A-i.
with
silver
chloride immersed in a saturated solution of
potassium chloride and enclosed
electrode
pair
or
dipole
is
a low noise
by
buffered
Each
pot.
porous
a
in
instrumentation amplifier with a variable gain of
201
coated
mesh
silver
The electrodes are made of
to
1.2
followed by a 500 sec high pass filter with a gain of
dipoles
signals
are
combining a fraction
then
of
one
with DC amplifiers
cancelled
another
with
signal
consequent
The
A-2.
10 as shown schematically in Figure
then
combining these scalar signals in a similar manner to form
a tensor cancellation as depicted in Figure A-1.
The output or residual of these cancellations is
then filtered and amplified before
digital
acquisition
system
Tibor Lukac and packaged for
being
designed
field
use
recorded
on
a
and constructed by
by
Steve
Park.
Figure A-3 is the schematic of the two pole Bessel filters
used to amplify the telluric signals in the band 120 to 10
112
---L------.yln
.---^i-ra.- - -x~-rrr--
direct data transfer to
system
computer
our
a
Hewletf
computer with which data analysis
desktop
9825T
Packard
system was designed for
acquisition
data
Our
seconds.
~-~~-~YII~-~
ll)-* 1X
-,_l,_l-i-i.r--.llllhep~ )-
was performed.
To measure magnetic fields we use
built
coils
and
Cantwell(1960)
by
described
and
for
magnetic
the
The
packaged in PVC pipe by Bob Davis(1979).
are
coils
five and six feet long with 90,000 turns of magnet wire on
a cylindrical core of high magnetic permeability. The coil
constants are, respectively, for the long and short coils:
3.51 gammas/mv/Hz (long coil)
5.37 gammas/mv/Hz (short coil)
The coil outputs tend to be a fraction of a
the
10
100
to
of
use
the
accomplished
by
stabilized
operational
in
and preamplification is
periods
second
microvolt
the
low
amplifier
schematically as Figure A-4. All of the
chopper
noise
drawn
circuit
electronics
were
battery operated.
A.2 Telluric Field Measurements
To obtain
the
three
or
four
dipole
signals
necessary to perform telluric cancellations long wires are
laid
out
to
connect distant electrodes to the centrally
located circuitry. We have used number 22 magnet
wire
as
well as PVC coated number 28 stranded copper wire for this
purpose.
To
minimize
electric
field
noise
induced by
motion of the wire in the earth's magnetic field, care
113
is
~i
taken
ground. The need for
the
on
wire
the
keep
the
a
with
seen
immobilizing the dipole wire can be
.----_I- -.^IL~VI.^^-IUltUI
-YIY131
L-l--ll---i-.-
simple
oscillation
of 5 centimeters
over a 5 meter length of wire in a 40,000
gamma field will
100
A
calculation.
second
signal on the wire. Since the
produce a 10 microvolt noise
100
typical electric fields are approximately 1 mv/km for
noise.
1% amplitude
represents
microvolts
10
periods,
second
the
Other major sources of telluric noise are from
or
themselves
electrodes
measuring
from
self
local
potentials affecting the electrodes. To minimize electrode
All electrodes
noise we use the following procedures.
be
inspected for proper
are
measurements
the
in
used
the
plating (indicated usually by a grey-black coating on
silver
field.
At
each
installed
being
before
potential and impedance
for consistency in
pit
a
in
tested
and
mesh)
to
the
in
site a pit approximately 30 cm
electrode
deep is dug and saturated with water . A local (Imeter) SP
survey
installation
minimize drying of
temperature
electrode
the
of
the
soil
,
the
Additionally
variations.
pit is covered to
the
around
After
pit.
electrode
the
around
made
is
,
electrode
and
when possible,
each electrode is left in its hole overnight to allow
for
chemical equilibration with its surroundings.
A.3 Magnetic Field Measurements
Potentials can be
magnetic
coils
by
induced at the output
relative
114
motion
of
of their ends in
the
the
earth's magnetic field (a v X B term).
importantly,
within a hundred meters of our coils car
vehicles
moving
More
cause unacceptably high magnetic variations in our
period
second
stable positions as far
as
traffic
the coils must be placed in
Thus,
range.
and
automobile
from
pedestrian
possible. To this end, we buried the coils at
distances of 50-100 meters from
our
recording
equipment
far from existing roads as possible. The need for
as
and
10-120
large separations between the coils
and-
traffic
can
be
seen also by a simple example.
Assume a vehicle to have a magnetic dipole field
typical
of 10,000 gammas at 1 meter. The
field
measured
100 seconds is less than 1 gamma. The
at
inverse
field of a dipole decreases as the
separation,
so
that
when
the
the
coils
cube
gammas will be
signal
to
recording equipment.
noise
in
the
sensed
approximately 1% noise amplitude.
or
magnetic field preamp was placed near the coils (10
maximize
of
vehicle moves within 100
meters of the coils, a field of .01
by
magnetotelluric
the
long
cables
m)
The
to
to the
Buffer Amplifiers
Differenoing
Amplifiers
T=9 3(S2+zS1)
S2= 9 1 (A+xB)
Sl= 9 2 (C+yD)
ASSUMED:
IB1 > IAI
IDI > ICI
IS1 I > IS21
DIPOLE GEOMETRY
TELLURIC CANCELLATION SYSTEM
FIGURE
A-1
out
hi
I0
ivi
i
to
30K
out
1th
CVt1
.0
IA-instrumentation amplifier Analog AD522A
CA-oporational amplifier Preoision Monolithios OP-15
resistora-1% (Cermet)
capacitors <Imf 10% ceramic
1jm 1% (Component
Research)
GAIN=10 to 1000
ELECTRIC FIELD PREAMPLIFIER
FIGURE
A-2
1.58M
5K
OUT
1.58M
3. 83M
1% (Component Researoh)
oapacitore
resistore 1% cermet
operational
amplifiere Preoision Monolithios OP-15 or equivalent
Bandpass 120-10 seoonde, gains .2 to 200
BESSEL FILTER
FIGURE
A-3
1M
1K
F-"
GAIN=1001
270K
.27
resistors
capacito r8
T-
1% cermet
1% Component Research
Chopper stabilized operational amplifier Analog (AD261K)
MAGNETIC FIELD PREAMPLIFIER
FIGURE
A-4
Appendix B
Impedance
Eigenstate formalisms with a numerical example
B.0 Introduction
In this appendix we
relationships
for
the
u
present
and
v
establish an algebraic formalism for
the
impedance
the
orthogonality
space
eigenvectors,
the
eigenstates
of
tensor and give a numerical example of the
eigenstate-calculations described in Chapter 2.
Lanczos'
book "Linear Differential
out of print and difficult to
Richards (1980)
have
locate.
included
the
Operators"
However,
Lanczos
Aki
is
and
analysis in
their book along with a numerical example which emphasizes
the utility of his analysis
matrices.
and
do
discussed
appendix.
They
not
applied
to
defective
use real matrix elements in their example
address
in
when
Chapter
Let
us
the
2
phase
convention
problem
we
and describe by example in this
consider
now
the
eigenvector
orthogonality conditions.
B.1 Orthogonality Conditions
From Chapter 2. the eigenvalue
equation
the form:
S w =
w
(B-1)
120
is
of
where S is the augmented Hermitian matrix, (A) is
eigenvalue
and
w
is
the
augmented
a
real
eigenvector
with
components u and v such that:
S
Additionally,
w =
;
1
we
(B-2)
modified
equation
1
to
allow
the
eigenvalues to be complex such that:
=
S w'
X* w'
(B-3)
For both equations 1 and
3.
any
two
distinct
eigenvectors of S are orthogonal. Therefore:
u .u!
+
: 0
v2 V
For a non zero eigenvalue ( hk
j
(B-4)
) not only does the solution
) exist but the solution
to (4) (u,v,
as well.
i #
(u,-v,- A )
As a consequence of the second solution:
uu
u!-
v
vi
=
0
i
# j
(B-5)
which together with (5) requires that:
u,
u
=
0
i
V
* VI
0
121
#
j
(B-6)
exists
B.2 Algebraic form
of the Impedance Eigenstates
As indicated in Chapter 2, the impedance
in
a
given
coordinate
written as:
Z
system
Zxx
Zxy
Zyx
Zyy
X,Y
Z down can be
(2-2)
To establish eigenstate magnitudes
apply
with
tensor
equation (2-14),
Z Z v =
and relative phases, we
rewritten below, to equation (2-2).
1i
v
(2-14)
u
i
Z Z
The matrix product ZZ takes the form:
SZ2
!Zxxl + IZxyl
ZxxZyx + ZxyZyy
+ ZxZy
2
Z
iZyxi + jZyyI
ZxxZyx + ZxyZyy
ZZ
Labelling the determinant of ZZ as
det(ZZ)
(B-7)
and
defining
the variable B as:
B = {
the
2
a
2
+
Zxx
IZxyj
+
eigenvalue equation
S= B +
{B
-
ZyxI
2
+
jZyy1 }/2,
(B-8)
can be expressed as:
det(ZZ)
(B-9)
}V
For the degenerate model of a 1D earth:
Zxx = Zyy = 0
(B-10)
; Zxy = Zyx
122
and the eigenvalues become:
(B-11)
A11
Izyxl
For a 2D earth:
Zxx = Zyy = 0
(B-12)
and the eigenvalues become:
S
Zxy
Z
ZLYx
(B-13)
2yx1
perpendicular
and
where Zxy
and Zyx are
parallel
to the strike direction. In general, however, no
the
impedances
simplifications occur and the eigenvalues
are
calculated
with equation 9.
With
equations
7
and
9,
the
unnormalized
eigenvectors for the electric field take the form:
ZxxZyx + ZxyZyy
(B-14a)
Zyx +iZyyl-
Zxxi-lZxy }/2
Zxy +I Zxx -I Zyx I-
Zyy I1/2
E =I
{B -
-
det(ZZ)}
{B - det(ZZ)l
.
(B-14b)
ZxxZyx + ZxyZyy
Similarly,
the unnormalized eigenvectors for
the
magnetic
field have the form:
ZxxZxy + ZyxZyy
{IZxy +Zyy
Sdet((B-15a)
l-|Zxx I-Zyx 1/2 + {B-det(ZZ)I
123
r
H :
H
{B-det(ZZ))
-Zxxl+IZyxl-!Zxy'l-iZyyi}/2
(B-15b)
ZxxZxy+ZyxZyy
where the eigenvalues are calculated with equation 9.
The
eigenvectors paired with each eigenvalue are:
H,
E
with
H , E
with
(B-16)
,
and consistent with equation 6:
H-H
+ E*E
= 0
(B-17)
= 0
but the E and H eigenvectors need
each
other.
In
the next section,
illustrating
example
not
be
to
we present a numerical
procedures
eigenstate
our
orthogonal
and
conventions.
B.3 A numerical example of the eigenstate formulation
Let
us
consider
the
eigenstates
for
the
following impedance tensor.
Z=
.4314exp(-i68.29)
S7.
5.481exp(-i57.34)
8 9 6 exp(i13.46)
(B-18)
.9775exp(-i58.57)
Forming the matrix products
necessary
to
determine
eigenstates we find:
62.53
ZZ =
, -5.359exp(+i6.65)'
I
-5.359exp(-i6.65)
31.00
124
(B-19)
the
1.964exp(-i6.40)
From
j
1.964exp(+i6.40)
30.23
ZZ
application
of
63.30
equation
(2-14),
(B-20)
we
find
the
magnitudes of the eigenvalues to be:
1
S,= 7.964
(B-21)
= 5.488
Again using equation (2-14). we find
the
ratios
of
the
eigenvector components to be:
With equation 22, the eigenvector component magnitudes are.
found. To assign the signs and phases to
components,
we
eigenvector.
determined
must
Then,
choose
the
the
eigenvector
arbitrarily the form of one
remaining
eigenvectors
can
be
from the orthogonality conditions and Poynting
vector requirements.
We consider first the v, eigenvector and
to
associate
the
phase
difference and negative sign of
equation 22 with the
Y
component.
assignments,
bo th
e
X
negative
satisfy
to
choose
With
these
initial
and Y components of u, must be
the
Poynting
vector
requirements:
Re{u x v' } > 0
for Z positive down.
(B-23)
Allowing for the phase shift required
to put v, at the peak of its polarization ellipse at
we find v,
to be of the form:
125
t=0.
e-A
+,q8G3e
vI-
0VY
O(B-2L4)
where
the
To find go
= w to.
l.
v,
polarization
the time domain form
and the sense of
rotation
of
ellipse. we need to determine first
of
the
complex
v
vector
Then,
.
following a procedure outlined by Eggers(1981) we find the
time
at which the magnitude v,(t) is a maximum and noting
the eigenvector position at
and
,
,o
+
*
we
find
the
sense of rotation of v,.
The frequency domain representation of v, can be
written as:
v,
a + i b
=
(B-25)
where a and b are vectors in the
X.Y
coordinate
system.
The time domain representation of v, is:
v,(t) = Re{exp(-iwt)[a +ib]
(B-26)
= a cos(wt)
+ b sin(wt)
Taking the derivative of the squared
with
respect
for which v, (t)
to
is
magnitude
of
v, (t)
(wt) we find that the phase shift (wto)
at
the
maximum
of
its
polarization
ellipse can be calculated with the equation:
tan(2wt) = 2 a-b/(
lal -
Ib2 )
Applying equations 25-27 to equation 24.
(B-27)
we find the phase
shift Po to be:
po =Wto = -. 18 degrees
(B-28)
126
and the final form for v,
is:
Vix = +.9863 exp(+i.18)
= -. 1648 exp(-i6.47) 0
VI
(B-29)
Following similar procedures for finding the phase
at
v,
its
of
v,
polarization ellipse maximum and requiring v, and
to be Hermitianally orthogonal, we find v. to be:
-Vx=
.1648
exp(+i6.47)
Consistent
with
components
and
equation
assign
in
the
polarization
assignment,
the
same
and
we
calculate
ellipticity
u,
and
v,
direction
about
As
result
ellipses.
uz
23,
the
component which causes
rotate
(B-30)
exp(-i.18)
Vz = .9863
v.
a
also
have
the
phase
of
to
respective
this
same
phase
sense
rotation. The phase at the polarization maximum
to be essentially zero and u,
to the X
eigenvectors
their
the
u,
is
of
found
can be written as:
o
ix, = -. 05907
exp(+i6.40)
(B-31)
-. 9983
Similarly,
u z is found to be:
u,
= +.9983
Q7-s
= -. 05907 exp(-i6.40)
o
To calculate the
eigenvalue
(B-32)
phases,
we
apply
equation
(2-13) such that:
Z v
where o(
is
=
J exp(im) u i
i = 1,2
the phase of the eigenvalue.
33 to equations 29-32,
we find the
be:
127
(B-33)
Applying equation
eigenvalue
phases
to
01
= -66.5
0z = -57.4
(B-34)
o
and the impedance can be written as:
0
-.
oso e
q
0-
3
oq3\
j
-,qOS90o q9383
L
.s3 e
148 e
.48e
(B-3s)
The
eigenvalue
minimum
magnitudes
electric
fields
represent
that
the
maximum
and
can be produced with unit
magnetic fields. The eigenvalue phases represent the phase
shift between the E and H eigenvector fields caused by the
presence of the conducting earth. In this example,
and
v
eigenvectors
are
nearly
the
u
linearly polarized with
principal directions within 10 degrees
of
the
X
and
Y
axes.
The angle
makes
with
equation
respect
each
to
of
the
polarization
ellipses
the X axis can be found with the
(Stratton,1941) for the electric eigenvectors:
anx 2Vj
co
128
and for the magnetic eigenvectors:
2-V",IV I
t=
where (Kl)
.. Co(
(
are the phase differences between the X
the phase differences between the X and
magnetic
and (#~)
field
Y
Y
are
components
of
eigenvectors. The quadrants for (9.)
are determined from considerations of
of the eigenvector components.
the
signs
For this example:
90 € °=
4=94 -
As
and
and ( 1)
of the electric eigenvectors (u)
components
the
3c)
mentioned
in
Chapter
2,
the
three
dimensional
parameter. skew, is the deviation from perpendicularity of
the
u
and
v
eigenvectors.
We
define
our
coordinate
invariant skew as:
S = 90-1
where (0)
directions
is either of the u
and
-
I
(B-39)
eigenvector
axis
(0) is the corresponding v principal axis
direction. Negative skew corresponds to an
than
principal
angle
greater
90 degrees and positive skew for angles less than 90
degrees. For our example:
S = +6 degrees
With
this
sign
convention,
129
the
(B-4I0)
tangent
of
our
skew
angle (S)
equal to the conventional skew (Sc).
is
We can
derive this relationship by noting first that:
=
'v
l
Re
(e -41)
Cos
and:
B- 42)
(Ux v(I
Re
where (
) is the angle between the unit eigenvectors (u;)
and (v). From equations (2-2) and (2-7). we find that:
-43)
V'C
-FtA
, ,- V
x
and:
ZX9j
-+~n
:z~x
Combining (43) and (44),
represented in terms
,~
2 ~jQ 2 uv2 )45 (L -41)\
the conventional skew (Sc) can be
of
the
eigenvector
products
such
that:
-
L xx
+
z
CIr
;i
From (141)
and (42),
(u xt4V2)
L41 V I
C , Z()X izx
I"I
the real part of the conventional skew
is related to the angle between
Re {Sc} =
Because the angle
shift
between (u)
(S-4-5-
v).'
(S )
and
(u
cot(
represents
(v)
structures minus the skew (S).
) and (v,)
(B-46)
)
the
associated
90
degree
usually
we can write
130
such that:
(C
)
phase
with 2D
in
the
form:
= 90 - S
(B-47)
and:
RejX
Thus,
our
- -2
-
skew
and
2
the
conventional
skew
the
have
relationship:
S = arc tan[Re{Sc}]
(B-49)
Sc = .105exp(il.2 )
(B-50)
For our example:
and:
tan[Re{Sc}]
= 6 degrees = S
Thus, the conventional skew is
skew
definition
normal
of
the
and
is
electric
related
(B-51)
directly
to
our
a measure of the deviation from
and
magnetic
principal
axis
directions.
The
other
magnetotelluric
eigenvector
orthogonality
3D
studies
parameter
is
polarization
the
aspect
ellipses.
interest
ratio
Because
of
of
in
the
the
condition, the aspect ratio is the same for
both the electric polarization
magnetic
of
eigenvector
ellipses.
ellipses
Similarly.
the
are described by a single
aspect ratio. The aspect ratio for an ellipse is the ratio
of its
suggests
minor
axis
to
its
major
axis.
Eggers (1981)
assigning a sign to the aspect ratio to indicate
131
the rotation direction of the eigenvector or handedness of
the polarization state.
Using
Eggers (1981)
ellipticity
formulation, the signed aspect ratio is:
(+r
where:
t
-
-
for the u eigenvectors and:
lvixl 4 V441 - 2T, CvA,
for the v eigenvectors.
of
the
ellipticity
negative sign
This formulation
representation
positive sign corresponds
the
L()
is an
of Jackson (1975).
corresponds
to
clockwise
For our example, the aspect ratios are:
.o349
A
to counterclockwise rotation and
viewed against the direction of propagation
Cv=
outgrowth
+
evz
132
o00"T9
rotation
as
(Eggers,1981).
the
To reiterate
ZZ
form
first
we
procedure.
ratios.
component
eigenvector
the
normalize
our
ZZ with which we find the eigenvalue
and
magnitudes and
of
steps
and
magnitudes
component
one
arbitrarily, the phase and sense of
(v)
then
We
assign,
eigenvector
which the second (v) eigenvector is calculated to be
with
Hermitianally
orthogonal.
(u)
corresponding
vector
signs
of
(u)
the
are then determined such that the
components
eigenvector
The
and
eigenvectors
(v)
obey
Poynting
The phases are assigned so that the
requirements.
corresponding (u) and (v) eigenvectors rotate in the 'same
around
direction
their
polarization
The
ellipses.
individual polarization ellipses are phase shifted so that
each eigenvector is at its maximum at (t = 0).
After these
assignments, the phases of the eigenvalues are calculated.
The skew is then calculated from the spatial directions of
the
(u)
corresponding
and
(v)
The
eigenstates.
eigenstates
are
calculated from the eigenvector magnitudes at (wt=O)
and
ellipticities
(Wt
of
(u)
the
and
: /Z ).
As
individual
Figure
B.1.
we
present
magnetic
fields
graphically
polarization
eigenvector
represent the electric fields (,u,
unit
(v)
(v,)
ellipses
) and
and (v,).
(C,u
z )
the
which
for
the
The arrows point to
the positions on the polarization ellipses of each of
the
eigenstates at (t = 0).
To summarize this appendix,
133
we
have
presented
the
eigenvector
form
of the impedance eigenstates,
of our procedures
orthogonality
and
conditions, the algebraic
and a numerical example
conventions
for
calculating
the
eigenstates of the impedance tensor.
lu
1-1
t=O
t=o
22
2 U2
V1 at
V
-2
FIGURE B. 1
at
t=O
MAGNETOTELLURIC EIGENSTATES
134
t=O
Appendix C
Impedance Calculations: An Approximate Form
C.0 In phase behaviour
To
and
structure
regional
field at sites
measured the magnetotelluric
the San Andreas Fault
Wrightwood,
periods
system
The
data
50
150
seconds.
and
we
along and near
Frazier
between
California.
between
patterns.
density
current
geologic
in
variations
lateral
infer
Park
and
bandlimited
was
For
these
to
long
periods, the relationship between E and H is controlled by
the
mantle
conductivity
which increases with increasing
depth. Accordingly. the horizontal E
within
20-30
derivatives
phase
degrees
of
being
in
tend
to Ted Madden
(personal
an
be
This
in
best by an example
due
communication).
Consider a toroidal E
to
to
phase with the time
of their magnetic field counterparts.
behaviour can be illustrated
responding
fields
field
source
(Ez
=
0)
earth for which conductivity increases
with increasing depth. Then,
the
curl
of
E
in
the
X
direction is:
(C-1)
d -t
With
the
conductivity
increasing
with
diminish to zero at some depth z and :
135
depth
Ex
will
or
(c-3)
Conversely. if the conductivity decreases with
some
depth.
the magnetic field will diminish to zero and.
point
from the curl of H. the surface H will be proportional
and
in
at
the
with
phase
to
surface E. Straddling these two
the
extremes is the case of a homogeneous earth for which
phase between E and H is 45 degrees.
At low frequencies.
field
magnetotelluric
the
samples the upper mantle and the deviation from zero phase
between
E
H
and
is
due largely to the mantle's finite
increasing
conductivity. For this finite but
the
conductivity,
elements
the
of
depth
with
modified
impedance
relating E and H have a common phase. Ignoring this common
phase and using real analysis has
calculation
and
reduces
little
slightly
only
we
As
the
estimates
the
due
to
the
of
show later in this appendix, larger
biases of the calculation of the impedance
occur
on
of the principal axes of the impedance tensor
eigenvalues.
can
effect
tensor
itself
noise in the measurements and we feel
that the use of the approximate real analysis is justified
especially at the low frequencies of our measurements.
136
C.1
Impedance
calculations
As indicated in Appendix
we
A.
is proportional to H.
consisted of two
horizontal
H
magnetic
The output of these
coils to measure the magnetic fields.
coils
used
Thus. our field measurements
E
horizontal
measurements.
field
channels
and
two
In view of the approximately
in phase relationships between our E and H measurements we
chose to simplify our analysis to estimating the
modified
impedance tensor Z for which:
E = Z H
(C-4)
where 2 has the units mv/km//sec.
limited
frequency
independent
of
band
frequency
We note that
estimates
of
which
allows
our
for
Z will tend to beus
to
form
Let
data comparisons between our MT stations.
consistent
us consider now the techniques used to infer estimates
Two impedance estimation procedures can be
to
find
Z.
For
of noise.
are
i
channels
estimate Z.
have
noise.
we
Because both
E
use both approaches to
Then, based on the calculated coherencies and
spread of tensor element values between the
estimates,
assumed
With both procedures the impedance tensor
is determined in the least squares sense.
and
used
one procedure the E fields are assumed
free of noise and for the other the H fields
free
of
two
sets
of
we infer an intermediate impedance tensor as a
coherency weighted geometric mean of
the
two
tensors and estimate a signal to noise ratio.
137
calculated
Finally, we
two
inverse
The stochastic
inverse.
tensor
a
using
estimates
tensor
both
recompute
stochastic
provides damping
of
the
estimates toward the intermediate model.
The
level of damping is based on the
to
assumed signal
noise
ratio and model undertainty.
The major
the
that
requirement
not degrade
damping does
intermediate Z or signal
degradation
of the E,H
of
these
examples of impedance
to
coherency.
estimation
tensors
the
markedly
the
Poor choices of an
Let us
in
now
the
followed
by
consider
procedures
calculated
result
can
ratio
noise
is
scheme
this
between the E and H fields.
coherency
details
on
constraint
from
data
taken
near the San Andreas fault system in southern California.
After
content
data
is
usually
filtered
inspection
two
of
frequency
with a four
the
data
bands
pole Bessel
138
for
spectral
are chosen and the
digital
filter.
For these filtered records. we assume the
E
and
H
data
sets are related by real constants such that:
00
*
°
where
Ei
and
Hi
(C-5)
.
are
the
filtered
time
series.
Alternatively we can write 5 in the more compact form:
E
= H Z
(C-6)
For the assumption of no noise in the magnetic
fields
we
pre-multiply both sides of (6) by the filter time series H
and solve for Z such that:
H E
= H HZ
(C-7)
and:
Z
where A-B is the vector dot product.
Hi
or
H2
is
additive
--
E H [H HT] -,
in
Note that
I
(C-8)
noise
in
their auto products with the
result that the estimates of i are biased
139
downward.
For
the
assumption of no noise in
the E fields we premultiply
both sides of (6) by E and solve for Z such that:
E E
T
&r . -r
(C-9)
=E H Z
and:
Z = E E[H ET]
(C-I0)
E?-Et E?- E-
E
Here noise in E is additive in the auto products with
result
that
the
also that if
polarized,
the
estimates of Z are biased upward.
the E fields
the inverse
are
[HET]
close
in
to
(10)
being
Note
linearly
can become singular
which also tends to bias the estimate of Z upward.
Thus we see that in the presence
Z
actual
of
noise
the
tensor lies
somewhere between the estimates of
equations (8) and (10).
Because of the strong tendency of
low frequency E fields to be coherent
coherencies
to
upward
bias
model
for
Accordingly.
E
field
is
we
closer
to
the
than those of (10)
choose
an
intermediate
the stochastic inverse as a coherency weighted
geometric mean of the two
consider
Z
of equation (8) (no i noise)
(no E noise).
high
the impedance estimates. we
feel that in most cases the actual
estimates
and
now.
the
impedance
details
140
estimates.
associated
with
Let
us
applying
stochastic
constraints on
the
two
impedance
estimating
procedures.
In
Z
in
the application of the stochastic inverse
equation (8).
simultaneously
which is
biased by noise in
to minimize the following
to
H. we seek
quantities
with
respect to changes in Z.
where
and CP
uncertainties
weighted
in
are respectively noise in the
the
model
for Z,and Zo
data
and
is a coherency
average of estimates of Z from equations (8)
and
(10).
The resulting stochastic estimation of Z
is
of
the form:
S
and similarly for equation (10)
7
.
we seek to minimize:
D(1
141
(C-12)
which results in the stochastic inverse
;L
\-
-z
Tr
"T
I
where
,
and
(C-1I4)
i7- are respectively estimates of
the noise in the E field and the uncertainty in the
of
Y
model
Y0 is the inverse of Zo the intermediate model
and
estimate.
Usually a signal to noise
assumed
and
an
intermediate
Z.
ratio
the
(10).
least
squares
solutions
If a particular choise of Z
ratio
resulted
in
significant
equations
estimates
and
signal
change
was
in
filtering
the
the
tensor.
impedance
a
similar
tensor
the
H
Thus
by
of
Although
a
real
approach can be used for a
modifying
our
digital
of the data to include a quadrature output time
addition
of
quadrature time series is
in
E,
8-14 we have a procedure for estimating a
series as well as an in phase time
Then
Z
noise
chosen.
the above analysis is based on the assumption
complex
of
to
modified form of the impedance relating E and H.
impedance
was
of equations (8) and
coherencies, a new intermediate model
with
three
was determined using a
coherency weighted geometric mean of the
from
of
frequency
Swift (1967).
dot
series
(Swift. 1967).
products of the in phase and
equivalent to complex arithmetic
domain.
For
example.
following
let the subscript (i) represent in phase and
142
~X_
l______~XI___I____X__ __
represent
(q)
.tl- ~..l^i--L . ~-l~-Bi----LI
a~
---
Then the equivalence between
quadrature.
the
power is of
the frequency domain and time domain auto
form:
2
z
^
z,
(C-15)
and the cross powers have the equivalent forms:
Re
E
"j,
H-t
(
I
W
(
where
vector
) represents
dot
-
HI
E-H
(r.
complex conjugate
approach
requires more
Fourier
Transform
note
We
products.
(C-i6)
H
)
and
A-B
However.
approach.
does
we
domain approach because we have found the
discrimination
To
no relative
examples
conclude
of
the
use
task
fast
the
the time
of
noise
in the time domain. especially for
simpler
signals which exhibit
time series
this
that
time than
computer
represents
this
frequency
appendix let
us
dependence.
consider
two
of the impedance tensor from
inference
time series and the determination of the principal axes of
the
impedance
tensor
the
using
eigenstate
analysis
described in Appendix B.
Our data are bandlimited (50
measurements
from
the
region
143
of
to 150 seconds)
MT
the San Andreas fault
system near Palmdale. CA.
This data was taken
before
we
had developed a digital recording system requiring the
use
of
Analog
Rustrak
recorders.
The
ignominious
task of
digitization was accomplished with an HP plotter/digitizer
coupled
to
an
correspondence
HP9825
desk
computer,
between
hand
digitized
Although
data can produce
errors in the analysis. coherencies of up to
E
field
measurements
were
coherencies were .5 to .8.
error
is
that
found
Another
considerable
and
Fortunately,
however.
between
typically
E,H
source
of
exists in the 30-50
second band near the low pass poles of
filters.
.996
potential
energy
time
our analog
our
filters
circuit
were well
enough matched to allow us to use this data.
Our first
example
is
butte north of Lake Hughes
electric
fields
are
from
in
the
nearly
H).
larger
(10)
Using an intermediate model near
impedance
estimates without
coherences
as
shown
in
(no noise
Figure
the
elements
predict
to
fields.
The least
and
(10)
are included with the
on
equations
(12)
(8)
the
in
E)
geometric
a more consistent
C-I.
The
As
coherencies
mean
set of
of the
Figure C-I.
and
(14).
the E fields from
stochastic estimates
Zo
144
E,
we
impedance
the H
squares analyses based on equations
and
is
(no noise in
a major degradation
present the analysis for
calculated
Desert.
polarized and our
than that based on equation
between the two estimates we find
infested
Mojave
linearly
estimate of Z based on equation
much
a rattlesnake
(8)
based
is the intermediate model
LEAST SQUARES
STOCHASTIC INVERSE
MTstat
LH2001
band = 100 to 150 sec
E1/E2 coh'
-. 82 0.996
MTstat
LH2001
sig/noise = 3
band = 100 to 150
E1/E2
-. 82 0.996
El = aHl+bH2
a
b
0.55
0.37
coh=0.634
a
b
1.12
0.78
con=0.525
El = aHl+bH2
b
a
0.53
0.37
coh=0.634
a
b
1.25
0.97
coh=0.525
E2 = aH2+bHl
a
b
-0.48
-0.60
coh=0.631
a
b
-1.34
-1.23
coh=0.546
MTstat
LH2001
band = 60 to 100
El/E2 coh
-. 82 0.989
El = aHl+bH2
a
b
0.44
0.34
coh=0.713
a
b
1.66
0.43
coh=0.371
E2 = aH2+bHl
a
b
-0.50
-. 0.60
coh=0.631
a
b
-1.08
-1.18
coh=0.540
3ec
MTstat
LH2001
sig/noise = 3
band = 60 to 100
0.8
0.6
-0.8
-0.6
sec
El/E2
coh
-. 82 0.989
El = aHl+bH2
a
b
0.51
0.34
coh=0.712
a
b
0.88
0.57
coh=0.329
E2 = aH2+bHl
,a
b
-. 43
-0.52
coh=0.723
a
b
-. 77
-1.15
coh=0.623
FIGURE C-1
sec
E2 + aH2+bH1
a
I
-0.45
-0.55
coh=0.723
a
b
-0.79
-0.84
cch-0.600
IMPEDANCE ESTIMATES FOR LH2
145
for Z used in the stochastic estimates.
estimates
or (12)
of
of
and E2 are determined from equations
(8)
and are biased by noise in H. The bottom
estimates
or (14)
El
sets
the
E
field
with
the
and
components.
consistent sets of estimates of
of
stochastic
amplitude
differences
the
coherency
elements
inverse
exist
high
As noted before. more
of
Z
are
without appreciable
degradation of the E, H coherencies.
Even
between
though
large
the two stochastic
weighted estimates of Z. the directions of
axes
sets
of El and E2 are determined from equations (10)
and are biased by noise in E
between
found
The top
the
principal
of the impedance tensor are quite consistent between
estimates as shown in Figure C-2.
In
the
E
Figure C-2 we have plotted two
fields,
the
axes
principal
directions of maximum and minimum
corresponding
unit
Figure
The
C-2.
assumption
of
no
H
of
ellipses
which
resistivities
outer
ellipse
the
for
the
corresponds
noise in E (equation 14)
(equation
high
infer
that
the
correspondence between the principal axis
to
the
and the inner
no
We
are
field eigenvectors also plotted in
ellipse corresponds to the assumption of
12).
for
noise
in
degree
H
of
directions
and
the large variation in amplitudes are.for the same reason.
the
nearly
linear
(coherency .996).
constrained
singular.
an'd
polarization
of
The electric field
the
impedance
direction
estimates
The raw data associated with
146
the electric fields
this
are
is
well
nearly
example
is
MT STATION LH2
H FIELD
E FIELD
PRINCIPAL AXES
PRINCIPAL AXES
Ha
387, 17
LH201
653,51
E=.386. 58atE2-60
Eb= 16. 94atE2 3
Ha at H2 43
Hb at. H2-47
LH20~ 1
En=653. 20atE2-61
EL= 51. 5c.t. E2 29
Ho ,t H2 43
Hb ,at, H2-47
FIGURE C-2
C-2
MT EIGENSTATES FOR LH2
147
LH-2 SITE
LAKE HUGHES QUAD 9/26/78
160TH SOUTH OF FAIRMOtIT BUTTE
E n-a (00)
1 , 288 mv/km/cm
225 sec/inch
H o-w (090)
.000626 v/sec/cm
E
4-w (090)
I
144 mv/km/cm
FIGURE
FIGURE C-3
. 000409 <Y/sec/cm
presented as Figure C-3.
For our second example. we present data from
MT
site
near
Phelan, CA in the Mojave Desert.
impedance is nearly
eigenstate
isotropic.
As
predicted
an
Here the
from
the
analysis of Appendix B. the orientation of the
principal axes becomes indeterminate.
As we did
for
the
first example. we present the least squares and stochastic
analyses
of
Z as Figure 4.
the eigenstate of Z as Figure
C-5 and the raw data as Figure C-6.
We observe
that
the
spread- of values for Z is less pronounced than in example
one
but
the
principal
directions
of
the
impedance
eigenstates are poorly resolved.
The procedures outlined
used
to
this
is
The interpretation of
where El.
related
are
procedures
E2.
Chapter
3.
can be used to relate telluric fields
the horizontal electric
to
these
used to provide constraints on the thin sheet
modelling of southern California described in
Similar
appendix
analyze MT data from southern California between
Frazier Park and Wrightwood.
results
in
field
at
site
1.
is
the field at site 2 through the telluric
tensor T e.g.
E
= T Ez
For the assumption of no noise in E2.
149
(C-17)
T is of the form
~_---~--_
/__j~X
_~X1-_ll~-~~
~~I_
LEAST SQUARES ANALYSIS
STOCHASTIC INVERSE
MTstat
PHI001
band = 80 to 120 sec
E1/E 2
1.31
sig/noise = 3
coh
0.449'
E1 = a H + b H
2
a
b
0.05
0.57
coh = .414
a
-0.30
0.449
El = a H1 + b H
2
a
b
0.82
0.19
coh = .414
b
a
2.46
0.13
coh = .410
a
b
3.28
0.08
coh = .410
E2 = a H
2
coh
E 1/E 2
1.31
+ b H1
E 2 = a H 2 + b H1
a
b
-0.32
0.12
coh = .576
a
b
-0.61
0.74
coh = .625
b
0.10
coh = .576
a
b
-0.75
1.06
coh = .626
r1.2
.18
L-.46
.17j
0 =
sig/noise = 3
band = 30 to 50 sec
E /E 2
-1.03
coh
E 1 /E 2
coh
0.078
-1.01
0.113
El = a H I + b 82
E 1 = a H1 + b H
2
a
b
0.72
0.18
coh = .656
a
b
0.19
0.93
coh = .674
a
b
2.01
0.18
coh = .573
a
b
1.62
0.19
coh = .591
E2 = a H2 + b H1
E 2 = a H2
b
a
0.11
-0.40
coh = .832
a
b
-0.39
0.05
coh = .823
a
b
-0.5o4
con
+ b H1
S
b
5V
coh = .864
.856
FIGURE C-4 Impedance Estimates
150
for PHI
n1~-~
11--1~
~111~--1
MT STATION PH1
E FIELD
PRINCIPAL AXES
H FIELD
PRINCIPAL AXES
HA -58
74.56
(INNER)
113.84
PH1901
Ea=7380AT
E2 AT 90 WRT NORTH
H2
AT
0 WRT NORTH
2-14
vi=55.84AT E2 76
HA AT H2 0
HB AT H2-90
PH1001
EA=113.18 AT E2 39
EB= 83.61 AT E2-51
AT
!A
B AT LZ
FIGURE C - 5
MT EIGENSTATES FOR PHI
151
(OUTER)
.1~~----
Pi1-1 PlIELAN OFF SHFEP CREEK RD AND RAtCIMO
E no (000)
RD 10/7/76 N-80014 E-800M
I
225 SEC/INCH
H o-w (2090)
1
1.86x10 -2 MV/KM/CM
.1
5,.26x10 - 4 Y/SEC/CM
E o-w (090)
-NII,
I
'
3,70x1q -2 MV/KM/CM
H n-o (000)v"1
P
3.44x10^ 4
FIGURE C - 6
Y/SEC/CM
and for the assumption of no noise in El. T is of the form
-l
Similar
to
estimates.
estimates
the
noise
noise
in
downward
correlation
E, E,
C,
between
(C-19)
E2
while
will
noise
the
of
corruption
bias
in
impedance
the telluric tensor
El
and/or
high
E2 and El will bias the estimates of
(19) upward.
153
Appendix D
Poloidal mode response to embedded ellipsoids
D.0 Introduction
present the eigenstates of the electric field
We
revolution
of
ellipsoid
conducting
outside
of
a
embedded
in
a
and
inside
tensor relating static fields
conducting earth. We extend these static
to
polarization
induced
3D
embedded
an
field response to
target.
material for modelling the telluric
background
necessary
the
provide
The purpose of this appendix is to
solutions
field
low frequency telluric field response to a buried
the
Also.
ellipsoid.
the
on
saturation
we
the
discuss
sensitivity
of
current
of
effects
eigenstates
the
to
changes in ellipsoid conductivity with frequency.
presence
The telluric field is distorted locally
by
the
heterogeneities
in
the
three
of
dimensional
resistivity structure. The current systems associated with
these
distortions
toroidal.
currents
poloidal
The
classified
be
can
currents induced in the earth
particle
motions
currents
are
induction
outside
largely
vortices
or
and
large scale
large
scale
earth's atmosphere. These
horizontal.
current
poloidal
the
similarly
by
the
are
as
The
loops
scale
is
the
continents (Berdichevskii,1960). Because of the
of
the
order of
resistive
lower crust. even at low frequencies the poloidal currents
154
and outside as:
AS EO, x
Vz:= va+(-t
I
3
+
C4
-I
-
E4 Y
~- I
0(CB
(D-4)
where the conductivity contrast is represented as:
2
(D-5)
and the geometric coefficients
(A,
and By,)
are
determined
from:
r
tac
ooa
z +L)A
-
(D-6)
w i
r
rD
du
where 1 is the positive root of:
2.
x2
a2
A o and Bo
The
zt
bJz
(D-7)
-q
represent the geometric
electric
field
is
coefficients
for
derived from the gradient
potential and is represented as:
158
1=0.
of the
are
local
around
and
through
pass
currents
the
currents,
electrostatic
to
Similar
Madden,1979).
poloidal
and
mostly to the upper crust (Ranganayaki
confined
are
heterogeneities. The distortions of the surface field
thereby coupled resistively with buried heterogeneities.
distorts
loops
heterogeneities. The existence of current
local
conductive
to
largely
confined
loops
current
of
consists
The toroidal current system
the surface field inductively.
overburden
As long as the skin depth in the
for
enough
small
currents to penetrate to the
telluric
locally
heterogeneity, the surface field will be affected
through
poloidal
the
measurement
separations,
mode. At low frequencies and small
toroidal
current
is
system
with
associated
is
current
electrostatic
The frequency dependence
system Berdichevskii (1960).
the
an
to
system
current
the poloidal
approximately
equivalent
is
of
the
conductivity, dimensions, and shape of the conducting body
(Kaufmann,1978a).
In general both modes contribute to the
surface field. However, at low
frequencies
the
poloidal
mode is the dominant contributor.
Here we consider source frequencies
and
mode
telluric
can
poloidal
be
low
enough
line lengths short enough that the toroidal
ignored
response
to
and
we
analyze
heterogeneities
the
in
approximate
the
form
of
ellipsoids of revolution (circular crossection).
When the low
frequency
155
poloidal
mode
of
the
essentially equivalent to a DC field,
is
.field
telluric
by
approximated
the
of
solutions
the wave
Laplace's
of
solutions
electrostatic
be
can
equation
Helmholtz
equation in the boundary value study of buried or embedded
useful
yet
simple,
heterogeneities (Lee,1977). A mathematically
model of an embedded heterogeneity is an ellipsoid
be
of revolution (spheroid). The range of shapes that can
with ellipsoids of revolution varies from that
considered
for
Solutions
spheroid).
(prolate
cigar
long
a
conducting
medium
These
electric
into
the
in
mathematical
ellipsoid
telluric
near buried heterogeneities in the
and
and
fields
conductivity
first
us
Let
formalism
the local electrical
and
fields
telluric
structure.
resistivity
Lee,1977).
Jaeger,1952,and
solutions provide us with insights
field
expected
relationships
the
well known (Stratton,1941,
are
and
Sommerfeld,1952,Carslaw
a
embedded
electrostatic response of a conducting ellipsoid
in
to
sphere
a
of a thin disk (oblate spheroid) through
present
the
then consider the effects on
of
a
consistent
perturbation
with
the
in
the
frequency
dependent conductivity of an IP target.
D.1
Telluric tensors near a conducting spheroid
We formulate the electric fields in and near
an
analysis
of
ellipsoid
using
the
embedded
conducting
Lee(1977).
We place an ellipsoid of conductivity (0)
homogeneous medium of conductivity ().
156
The axes
of
in a
the
ellipse (a.b and c) are aligned respectively with the X-Y.
and
Z directions and the fields in and near the ellipsoid
are formulated in terms of a distant source field in the X
where
ellipsoid
prolate
are
cases
and Y directions. Two
a>b=c
the
then
and
the
first
considered,
oblate
ellipsoid where a<b=c.
General Solution
The equation at the point (X.Y.Z) on the surface
of an ellipsoid whose axes are aligned
with
Cartesian
a
coordinate system is:
a-_
-(D-1)
-
ellipsoid
Let the potential far from the
be
represented
as:
-
oX Y
x -
(D-2)
Then
the
potential
inside
the
represented as:
_E(D-3)
a
o
I, -
--
7
157
ellipsoid
for
z=0O
is
-V V.
(D-8)
Inside the ellipsoid. the electric field is:
x -
CI+
Eox
Bo
I
6-
(D-9)
while outside the ellipsoid, the electric field is:
K1
L
+
Lay
LC
(D-1 0)
where:
CO
E2x
~BP
~ 3
(c-i)
c~
(D-11)
3e
J
Y1t-L~Bj
L <Bo-Ba 0 (El
-±B !
-.
i
X+
I + Ao3
B(E
<
-l)
(e -,
cf-
159
h~
As a consequence of (9) the tensor
between
field
the
at
relationship
infinity and the field within the
ellipsoid is:
So-
EI
-----
(D-12)
Ea
and from (12) the tdnsor relationship between the field at
infinity and the field near the ellipsoid is:
I+
A- < - ,)
EA,
(D-13)
Otuo(&
fz
I
Combining (13)
+tB
and (14) we find
~l
the
tensor
relationship
between the field inside and out to be:
E z = TzT,
-I
160
E,= T E,
(D-14)
where:
/
(D-15)
-9,
XA
As we shall show in Section D.3:
(D-16)
-
with the result that T
characteristic
a
is
tensor.
symmetric
T. which we shall show in Section D.2.
of
is that the principal axis directions are the same
and
Another
inside
outside the ellipsoid and independent of conductivity
we
contrast. In fact,
fields
inside
to
find
relating
ellipsoid,
the
perpendicular
and
the
always
are
tensors
for
outside
those
principal axis directions
that
parallel to the surface of the ellipsoidal conductor.
Equations (1) through (16)
electrostatic responses of both
spheroids.
The
calculation
the
are applicable to the
prolate
and
oblate
of the elements of T for the
prolate and oblate spheroids is described in Section D.3.
161
D.2 Tensor eigenstates
In this section, we use the analysis of
2
to
Chapter
establish the eigenstates of the Hermitian tensor T
which relates the electric fields inside and outside of an
embedded
directions
ellipsoid.
of
the
We
show
that
the
principal
axis
eigenvectors are always perpendicular
and parallel to the ellipsoid boundaries.
Because T is Hermitian,
have
the
rows
and
columns
same eigenvectors. Thus, the eigenvector matrix
representing the
parallel
its
eigenvectors
inside
the
ellipsoid
is
to the external eigenvector matrix. The combined
eigenstates of T can be expressed as:
zCOJL
The U
(D-17)
eigenvector matrix contains the eigenvectors within
the ellipsoid
contains
and
the
parallel
matrix
eigenvector
U
the eigenvectors at some point P external to the
ellipsoid as depicted in Figure D.1.
The
matrix
product
U 1-_represents the electric fields at point P produced by
the
unit
eigenvector
fields
U .
Consistent
impedance formulation, the components of
yuz,
represent
respectively
the
UJ ,
maximum
electric fields that can be produced at point
electric fields within the ellipsoid.
162
and
P
with
the
tu
and
minimum
by
unit
The eigenvalues of T can be written as:
=- + (E-1)
(D-18)
and the ratios of the eigenvector components which
define
the principal axis directions can be expressed as:
A-I
-i=1.2
(D-19)
where:
X
~4,a,)(R~cR)
JA
a'z
aBP
~---,5
+~a~i-
57>
faR
lyZ
,-
2:IC
2I
j
X-
J
(D-20)
--
a
We notice from equations 19 and 20 that the principal axis
directions are. as stated -initially,
conductivity
contrast
(E)
and
163
are
independent
of
functions
only
the
of
position (X,Y) and ellipsoid geometry. In
use
parallel to the ellipsoid boundary, we shall make
of
insensitivity
this
are
axes
principal
point
external
maximum and
the
with
aligned
of
the
to
axes
principal
the
conductivity contrast and the fact at any
the
show
principal axis directions are perpendicular and
the
that
to
order
minimum electric field directions.
conductivity of the ellipsoid tend
If we let'the
we see
infinity.
toward
eigenvalues
will
from
to
boundary
the
to satisfy continuity of normal current
(,u for
field.
electric
maximum
minimum
and
the
the
ellipsoid
directions
electric field ,
boundary.
been
have
to
in
general
be
aligned
electric
field
be
the
independent
principal
of
axes
normal and parallel to the
ellipsoid boundary. The significance
that
axis
principal
the
conductivity contrast (equation 19),
must
boundary
uZ must be parallel to
Because
shown
a unit internal
ellipsoid
the
electric field must be normal to
Thus,
field.
density and continuity of parallel electric
the
parallel
electric
normal to the boundary and the minimum
must be
field
electric
external
ellipsoid
the
Near
unaffected.
maximum
the
the
that
toward infinity but the principal
tend
axis directions will be
boundary
18-20
equations
of
this
result
is
measurements near the boundaries of
ellipsoidal heterogeneities can be used to infer the shape
of the heterogeneity. Additionally. as we show in
Section
in
prolate
D.4.
the
effects
of
current
164
saturation
spheroids can be relegated to the eigenstates parallel
the
surface
ellipsoid
to
perpendicular to their minor'
and
axes.
D.R Prolate and oblate spheroid calculations
In
necessary
this
to
section.
calculate
embedded prolate and
we
present
the
equations
the telluric tensor elements for
oblate
spheroids.
For
a
prolate
spheroid with its long axis (a) parallel to the X axis:
a > b = c
and introducing the ellipticity terms (e)
and
(e')
such
that:
e = (a
-c
)/aZ
(D-21)
and
e
= (aL-c')/(az+1)
(D-22)
we find that the geometric coefficients can be written
terms of the ellipticities
such that:
t-eD
P-
14 e
ia,(t
165
in
and the spatial derivatives of the geometric
coefficients
become:
I-,3
13
ee
-
where:
D
=
21 +a +c -
(x
+y
+z
)
(D-25)
From inspection of (24). we see that:
(D-16)
and
T
= TzI
in
(15)
as indicated earlier.
(25)
by using equations (22) through
through
(15)
we
relationships between
conducting
prolate
the
have
the
E
ellipsoid
in
associated
imbedded
earth- We present now a similar analysis
166
equations
electrostatic
fields
Finally,
in
for
(12)
tensor
with
a
a conducting
the
oblate
ellipsoid of revolution.
For an
oblate
spheroid
with
its
short
axis'
parallel to the X direction:
a < b =c
(D-26)
Defining the ellipticity terms (e) and (e')
.
as:
(b 2-a)/b
(D-27)
e
=(bZ-a
)/(b +1)
we find that the geometric coefficients can be written
in
terms of the oblate spheroid ellipticities as:
- -e
(D-28)
e2e
-
3
and the spatial derivatives of the geometric
coefficients
become:
_A
_
X.
Ak
( (,
l)
/
&
-eL
-
-k
1
e
(D-29)
D 3
~3
"3
e
167
2
where:
DI =
7
21 +a-+b
-
(x
)
+y +z
(D-30)"
(14)
Equations (28) and (29) can be used directly in
(15)
and
to determine the tensor relating the electric fields
inside and outside of an embedded oblate ellipsoid. In the
limit of zero ellipticity
resp6nses.
electrostatic
symmetric. the u
eigenvectors
and
the
can
the
calculate
are
character
and
is
the
ellipsoid are parallel to their
anisotropic
solutions
field
we
embedded ellipsoids. In the next section.
anisotropic
same
the
prolate
spheroid
oblate
With these sets of
external counterparts.
we
spaces
v
inside
the
for
tensor
the
solutions,
same
the
the
with
Consistent
prolate
and
oblate
exhibit
and
spheres
become
spheroids
the
both
of
these
response
now
of
relate
the
to
the
solutions
sensitivity of the telluric tensor eigenstates to
changes
in the conductivity of the ellipsoid.
D.4 The current saturation condition
For a constant source field.
current
the
electrostatic
in and near an ellipsoid is determined by
system
its ellipticity and by the contrast in conductivity of the
ellipsoid with its surroundings.
the
long
direction
of
prolate
Current is funnelled
ellipsoids
in order to
satisfy the continuity of parallel electric field
perpendicular
to
the
long
direction
no
whereas
funneling
necessary to satisfy the continuity of the normal
168
in
is
current
density
J.
Similarly, current is funnelled perpendicular
to the minor axis of
satisfy
the
an
oblate
continuity
ellipsoid
in
order
of the. tangential'electric field
but no current enhancement is required for the
of
to
continuity
J parallel to the minor axis. Thus. the electric field
tensor TJ
ellipsoid,
relating fields
will
be
anisotropy dependent on
and
ellipsoid
inside
to
those
outside
generally
anisotropic
position,
conductivity
the
with
the
contrast
shape. Additionally, when the current in a
particular direction does not increase as the conductivity
of the ellipsoid is increased that direction is said to be
saturated (Ness.1959). Conversely, when the current
particular
direction
changes
with
conductivity, that direction is said
To
a
to
in
variation
be
a
in
unsaturated.
illustrate the saturation condition we have calculated
the electric fields over a wide range of ellipticities and
conductivity contrasts. As Tables D.1 and D.2. we
the
calculated
electric
fields
inside
oblate spheroids for a source field of
of
present
prolate and
magnitude
one
in
both the X and Y directions.
For the
prolate
spheroid
(Table
D.1)
the
X
direction is parallel to the major (long) axis and for the
oblate
spheroid
(Table D.2).
the X direction is parallel
to the minor axis. From Table D.1.
we notice that
ratio
is
of
major
to
minor
conductivity contrast is
field
Ex
into
axis
needed
saturation.
169
to
On
as
the
increased, a higher
drive
the
the
other
parallel
hand
the
Y
C
PROLATE SPHEROID
TABLE D.1
x
E
(a/c)
_10
I
,3
i
S100
Ex
--EY
,8214
,5287
10
--
I
Ex
E
30
I
Ex
100
(
EY
Ex
EY
,9610
.5051
.9932 ,5009
.9991
,5001
,5055 .1996 ,8456
,2408 ,0718 ,6296
.1848
,9699 ,1823
.9961
,1819
,0650
,9092
,9877
.0645
,3324
.0202
,7457
,9592
,0198
.0222
,0850
,0647
.0199
TABLE D,2 OBLATE SPHEROID
(a/b)
Ex
3
100
3
Ev
i n
,4404 ,7328 .3674
,1488 ,3787 .1143
,0515
,1591
,0385
,0156
,0525
.0116
innin
Ex
Ex
,8778
,3449
,9522
Ex
EY
,3368
,1014
,9847
.9348
,1047 ,8157
.3313 ,0350 .5786 ,0338 ,8164
,1267 .0105 .2869 ,0102 .5657
,6149
weak
of
function
from
Similarly
to
parallel
we
D.2.
Table
is
of
dependent
of
axes
condition
saturation
of the elements
in the ellipsoid conductivity.
ellipsoid has an important implication for
IP
with the poloidal mode
targets
of the
field.
telluric
of
IP
an
a function of frequency varying roughly 1-100%
is
to
per decade of frequency in the range of 0.01
and
(Cantwell
Madden.1967).
10
Hertz
of the approximate
Because
equivalence between the electrostatic
poloidal
and
mode
to an embedded heterogeneity. we can relate the
responses
frequency dependence of the telluric
the
for
the search
conductivity
the
4.
As described in Chapter
target
electrostatic
the
of
variation in the conductivity of an embedded
a
to
a
an ellipsoid will
within
contrast
The anisotropic response
field
major
Thus.
ellipsoid.
the
result in an anisotropy in the sensitivity
of T to changes
the
of
to either
on the conductivity contrast and ratio
minor
to
major
field
the
that
notice
saturation of the field parallel
axes
minor axes.-
to
major
very
axis is easily saturated whereas
minor
the
of
ratio
the
a
and is
perpendicular field Ey is easily saturated
tensor
elements
to
sensitivity of the electrostatic tensor to changes in
contrast
a
Additionally,
conductivity.
within
the
saturation
ellipsoid results
condition
in an anisotropic
frequency dependence of the telluric tensor eigenstates.
In Section D.2.
consist
of
a pair
we showed that
the
eigenstates
of colinear eigenvectors normal to the
171
eigenvalue
ellipsoid surface and related through a single
a
and
parallel
eigenvectors
of
pair
to the ellipsoid
boundary and related by a single eigenvalue.
are
eigenvectors
the
to conductivity contrasts.
insensitive
directions
axis
principal
corresponding
their
Because
are
insensitive to conductivity variations with frequency. The
eigenvalue for the colinear eigenstate is always sensitive
to
ellipsoid conductivity but the parallel
changes in the
conductivity
eigenstate can be insensitive to
when
eigenvectors
its
are
aligned
with an
variations
unsaturated
current direction.
The essential
as
summarized
follows.
of
shape
position
this
section
can
be
The relative frequency responses
near an ellipsoidal IP target depend on
of telluric fields
the
of
points
the
and
ellipsoid
the
orientation
of the telluric measurements. The eigenstates
and
of
the telluric tensor relating fields inside and outside the
ellipsoid are directed parallel
surface
of
the
and perpendicular
to
the
ellipsoid. The perpendicular eigenstates
are always sensitive to the frequency
dependence
of
the
ellipsoid whereas the frequency dependence of the parallel
eigenstates
is a function of the saturation condition.
Chapter 4. we apply these concepts to the field
pyrite bearing schist which has been shown
IP response.
172
to
In
study of a
exhibit
an
Appendix E
E.0 Introduction
In
heterogeneous
of
conductance
apparent
determining
the
blocks (E.1)
used in the thin sheet models of
Additionally,
E.2
in
describe
we
for
methods
present
we
appendix,
this
Chapter
procedures
our
relating changes in block conductances to uniaxial
3.
for
stress
perturbations.
E.1 Block Conductance Calculations
Consider the block geometries
drawn
as
Figure
E.1.
Figure E.1
b
a
Examples (a).
and
a
(b),
the
and
c
(c) are blocks spanning a
mountain
where the valley alluvium, sub block 2. is
valley
more conductive
general
Block Geometries
than
the
mountain,
sub
block
1.
In
relationship between the horizontal electric
field E and the current density J is:
Ix xCTXY
jYj
GYX
J
173
The conductance tensor is Hermitian for
used.
Using- the
can find
J is related
the
the
density
current
to the electric field E by a diagonal
For model (a)
to
model
block
the eigenstate analysis of Chapter 2,we
axes for which the
principal
each
sub block contact is
the
tensor.
parallel
X axis and the principal axes of the conductivity
tensor are aligned with our coordinate axes X
and
Y.
In
the Y direction. sub block conductances 1 and 2 combine in
series
such
) the average conductance in the Y
that (%
direction associated with a Y directed E field is:
2
l a(2
1
+
2
and in the X direction, the sub block conductances combine
in parallel such that:
a1+
XX
As
02
2
2
for the geometry of model
indicated previously,
(a)
no
at
45
coupling occurs between the X and Y currents and:
S=
xy
0
yx =0
the sub block contact
For model (b),
degrees
with
respect
(E-4)
is
to the X and Y axes. Consequently,
the principal axes of the
conductance
tensor
which
are
parallel and perpendicular to the contact are rotated from
174
the
X
contact as
did
we
X
component
Y
and
for
to the sub block
find
we
the
(a)
.Then
E
and
J fields in terms of
combine
we
Finally,
components.
expressions
the
model
parallel
and
perpendicular
their
and parallel
perpendicular
conductances
find
we
tensor,
conductance
full
resultant
the elements of the
determine
To
axes.
Y
and
the
J and E to determine the full
for
our X and Y coordinate system.
conductance tensor in
the
to
Perpendicular
sub
block
contact
the
conductance is:
2 01
rl
+
2
2
and parallel to the sub block contact:
-
a1 +
G2
(E-C
S =+ 2
2
The
perpendicular
and
parallel
E
and
J
fields
expressed in terms of their X and Y components as:
/2
E= /2
/2
(Ex + E )
E =-/2
(E - Ex
E
where:
J, =
a, E,
J11 = cr E
175
8)
are
-~I
-. ...II~^Y~--
tensor relating E and
and in the X and Y directions the
J
is:
al+
a
2
2
- 2
2,-o
2
tensor,
similar
diagonal
off
the
bit
(c),
model
Following the same procedure for
find
we
a
elements have a
change in sign.
o + 0
0-
2
2
c -
+2
2
2
a
1
Thus, with combinations of and variations
shown
in
Figure
model
within the smallest scale of our numerical
electrical
conductivity
blocks
can account for heterogeneity
we
E.1,
the
in
product
thickness
of
the
of the upper
crust.
E.2
Block Conductance versus Stress
We
conductivity
in
E.2.
the
manner.
in
stress
changes
to
Consider
indicated
in
the
in
the Y direction and
in the X direction. Along Y cracks will
to open and along
depicted
following
change
stress
in
The maximum stress is in
the minimum stress
tend
changes
compressive
horizontal
Figure
relate
X cracks
in Figure E.3.
will
tend
to
close
as
Accordingly. in regions where the
176
li(.llllll
_ _1____*I_
_F~~1__~~_~__U_
FIGURE E. 2
Stressed Block
COMPRESSIONAL STRESS ALONG Y
(:D
o-o
FIGURE E. 3
== =.
C
Crack Response to Stress
177
~_
i~~YL~II~L-~
--
i^_j~_j~~ l~l__~_ __1_______Y__
the
the Y direction will increase while
in
conductivity
the conductivity in the X
The
decrease.
will
direction
for an initially isotropic block then
tensor
conductance
Madden(1978),
controlled
electrical properties are crack
takes the form:
(-
" + a0
aI
a + 6a'
0.
cracks
Similarly, for a tensile change in stress along Y,
X will tend to open and cracks along Y will tend to
along
close.
Figure E.3
represents
of
case
simple
the
a
single homogeneous block subjected to a compressive change
in stress. In general, each block is heterogeneous and the
of
axis
maximum change in stress is not aligned with the
principal axis of the block conductivity tensor.
the
stress
conductance
perturbed
sub
block, first we find the
conductance
block
tensors
oriented along the maximum change in stress direction;
rotate
these
perpendicular
directions
to
tensors
find
heterogeneous
a
for
To
we
and
parallel to the sub block interface; combine the sub block
conductances with the
and,finally,
condition
current
of
we rotate the combined tensor to the X and Y
directions. With this procedure. then.
the
effects
continuity;
of
stress
we
can
establish
induced changes at the sub block
level on the conductance tensor of
178
the
full
block,
our
Ijq_
minimum
no
further
scale
the
accomodate
to
need
combine
but
blocks
individual
follow the same procedure for the
have
the regional level. we
At
scale.
computational
the block tensors to
thin
requirements
of
the
let
us
consider
sheet
program.
As
an
heterogeneous
illustration.
block
directed at an angle
subjected
of
to
degrees
45
now
a
a stress perturbation
to
the
sub
block
contact as shown in Figure E.4.
X
S - stress
direction
Y"
Figure E.'4
In both sub blocks we find
Sub Block Stress
the
perturbed
conductivities
perpendicular and parallel to the maximum stress direction
such that:
i = ai + 6i
(-12
179
)
Rotating the sub block conductivity tensors to
parallel
perpendicular
and
directions
the sub block contact wde
to
find:
=. 0
=a
Xa
yyi
Ixxi
1
(E- )
cai -
xyi.=
+
=
6
&i
2
Oi/i
2
Finally, we apply boundary conditions
the
on
sub blocks
with the result:
a
xxl
xx
S
Y
+a
xx2
CY
x
2
2a
yyl
yyl
xl
a
xy
yy2
+ Cr
a
+ a 2
yyl
yy
yy2
a
xy2 yyl
yx
+yy2
yy2
(E-14)
180
I_ I~~~_~ _I_~C(___
The new block conductance tensor
plus
a
The
perturbation.
equals
the
old
Ily~L-_-.~~
tensor
portion is of the
perturbed
form:
-xx
S
F-,r, -t- E
Q-x 4
'k
X 2-
6E -/ -
+c
a-
a-
C~g' 2 SO '_
y
Thus, we have
a
perturbation
of
cLI
r
c4irLt cJ2
simple
the
_x(l~~lb;y
procedure
for
conductance
subjected to a homogeneous deformation.
181
IY
of
calculating
crustal
the
blocks
- II~1LL--P
~- I~C-.
_ _~^--II
-~-I~X~-~
-tl.1~.~--~-I-~-
BIOGRAPHY
The author was born in Boston on July
attended Boston public schools graduating from Boston
and
Latin School in 1960. He attended Northeastern
in
1942
13,
Boston
graduating
in
student technician at the
MIT
University
In 1961 he started as a
1965.
Instrumentation
Laboratory
at
and remained as an engineer after graduation. Between
1969 and 1974, he worked for Prof. Simmons of the Dept. of
Earth and
Masters
Planetary
Sciences.
In
1972- he
obtained
degree in Aeronautics and Astronautics at MIT and
became interested in Geophysics enrolling in the Dept.
Earth
a
of
and Planetary Sciences in 1975. After defending his
thesis, he will
commence
work
with
Research Co. in La Habra, California.
182
Chevron
Oil
Field
L- ~-_C-.~L_