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THREE-DIMENSIONAL INVERSION OF MAGNETOTELLURIC DATA FROM

THE COSO GEOTHERMAL FIELD, BASED ON A FINITE DIFFERENCE,

GAUSS-NEWTON METHOD PARALLELIZED ON A

MULTICORE WORKSTATION

by

Virginie Maris

A dissertation submitted to the faculty of

The University of Utah

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in

Geophysics

Department of Geology and Geophysics

The University of Utah

May 2011


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Copyright Virginie Maris 2011All Rights Reserved


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3

T h e U n i v e r s i t y o f U t a h G r a d u a t e S c h o o l

STATEMENT OF DISSERTATION APPROVAL

The dissertation of Virginie Maris

has been approved by the following supervisory committee members:

Michael S. Zhdanov , Chair 11/09/2010Date Approved

John M. Bartley , Member 11/11/2010Date Approved

Susan L. Halgedahl , Member 11/17/2010Date Approved

George R. Jiracek , Member 11/17/2010Date Approved

Philip E. Wannamaker , Member 11/9/2010Date Approved

and by D. Kip Solomon , Chair of

the Department of Geology and Geophysics

and by Charles A. Wight, Dean of The Graduate School.


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ABSTRACT

An existing 3-D magnetotelluric (MT) inversion program written for a single

processor personal computer (PC) has been modified and parallelized using OpenMP, in

order to run the program efficiently on a multicore workstation. The program uses the

Gauss-Newton inversion algorithm based on a staggered-grid finite-difference forward

problem, requiring explicit calculation of the Frchet derivatives. The most time-

consuming tasks are calculating the derivatives and determining the model parameters at

each iteration. Forward modeling and derivative calculations are parallelized by assigning

the calculations for each frequency to separate threads, which execute concurrently.

Model parameters are obtained by factoring the Hessian using the LDLT method,

implemented using a block-cyclic algorithm and compact storage.

MT data from 102 tensor stations over the East Flank of the Coso Geothermal

Field, California are inverted. Less than three days are required to invert the dataset for ~

55,000 inversion parameters on a 2.66 GHz 8-CPU PC with 16 GB of RAM. Inversion

results, recovered from a halfspace rather than initial 2-D inversions, qualitatively

resemble models from massively parallel 3-D inversion by other researchers and overall,

exhibit an improved fit. A steeply west-dipping conductor under the western East Flank is

tentatively correlated with a zone of high-temperature ionic fluids based on known well

production and lost circulation intervals. Beneath the Main Field, vertical and north-

trending shallow conductors are correlated with geothermal producing intervals as well.


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CONTENTS

ABSTRACT ....................................................................................................................... iii

ACKNOWLEDGMENTS ................................................................................................. vi

CHAPTERS

1 INTRODUCTION ........................................................................................................... 1

2 MT PRINCIPLES ............................................................................................................ 5

2.1 Introduction ......................................................................................................... 52.2 Maxwell's equations for MT theory .................................................................... 62.3 1-D model ........................................................................................................... 82.4 2-D model ......................................................................................................... 102.5 3-D model ......................................................................................................... 12

3 STATIC SHIFT ............................................................................................................. 15

3.1 Theory ............................................................................................................... 163.2 Correction techniques ....................................................................................... 203.3 Distribution of shifts ......................................................................................... 223.4 Impedance ratio frequency-dependence ........................................................... 233.5 Shift frequency-dependence.............................................................................. 253.6 Summary ........................................................................................................... 30

4 FORWARD MODELING .......................................................................................... 32

4.1 Finite difference solution .................................................................................. 334.2 Biconjugate gradient method ............................................................................ 354.3 Divergence correction ....................................................................................... 35

5 INVERSION SCHEME .............................................................................................. 37

5.1 Theory ............................................................................................................... 385.2 Frchet derivatives ............................................................................................ 435.3 Static shift ......................................................................................................... 455.4 Modifications from original .............................................................................. 485.5 Synthetic test ..................................................................................................... 49


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v

6 PARALLELIZATION OF THE INVERSION CODE .............................................. 58

6.1 Introduction ....................................................................................................... 586.2 Frchet derivatives ............................................................................................ 606.3 Parameter update ............................................................................................... 636.4 Summary ........................................................................................................... 67

7 INVERSION OF MT DATA FROM THE COSO GEOTHERMAL FIELD ............ 68

7.1 Conceptual model for high-temperature geothermal systems .......................... 717.2 Coso Geothermal Field ..................................................................................... 73

7.2.1 Geologic setting ........................................................................................ 747.2.2 CGF description ........................................................................................ 76

7.3 Inversion ........................................................................................................... 787.3.1 Data ........................................................................................................... 787.3.2 Inversion set #1: 100 Hz - 0.63 Hz .......................................................... 837.3.3 Inversion set #2: 100 Hz - 1.6Hz ............................................................. 92

7.3.4 Discussion ............................................................................................... 1037.4 Conclusions ..................................................................................................... 117

8 SUMMARY .............................................................................................................. 118

REFERENCES ............................................................................................................ 120

v

115


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ACKNOWLEDGMENTS

I would like to thank Dr. P. E. Wannamaker for support and guidance of this

work, and Dr. Y. Sasaki for providing the original program on which it is based.

I am honored and deeply grateful to have been the recipient of the Society of

Exploration Geophysicists Stan and Shirley Ward, and the Charlie and Jean Smith,

graduate student scholarships.

Dr. J. Moore of the Energy and Geoscience Institute kindly spent much time

providing insight into the Coso geothermal system.

I am grateful to Dr. G. Newman and Dr. E. Gasperikova for providing Coso well

production and injection intervals, and to Dr. B. R. Julian for earthquake epicenter data.

I would like to thank Dr. M. S. Zhdanov, chair of my supervisory committee, and

committee members Dr. J. M. Bartley, Dr. S. L. Halgedahl, Dr. G. R. Jiracek, and Dr. P.

E. Wannamaker for review of this work.

Work on this inversion method was supported by U.S. Dept. of Energy contract

DE-PS36-04GO94001 to the University of Utah, Energy and Geoscience Institute (P.

Wannamaker, P. I.).

MT Data collection at the Coso Geothermal Field was supported under U.S. Dept.

of Energy contract DE-PS07-00ID13913 and Dept. of the Navy contract N68936-03-P-

0303 to the Energy and Geoscience Institute.


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CHAPTER 1

1INTRODUCTION

As three-dimensional (3-D) magnetotelluric (MT) surveys investigating the

subsurface electrical structure are becoming more common, access to fast, accurate 3-D

MT inversion programs has become more important (e.g. Uchida and Sasaki, 2006). The

serial Fortran77 program developed by Sasaki (2001, 2004) for Gauss-Newton inversion

of MT data based on the finite-difference staggered grid method for forward modeling

has been restructured and parallelized under Linux using OpenMP 2.0 to allow it to run

efficiently on a multicore workstation. The modified program is used to invert MT data

from 102 soundings over the East Flank of the Coso Geothermal Field (CGF), imaging

conductive zones tentatively correlated to producing reservoirs.

MT is a geophysical technique whereby naturally occurring electromagnetic (EM)

waves are used as source fields for imaging Earths electrical resistivity structure at

depths ranging from tens of meters to hundreds of kilometers (Vozoff, 1991; Simpson

and Bahr, 2005). It is used in resource exploration and in earthquake and volcano studies.

Interpretation of MT data can be further complicated by static shifts. These shifts are the

frequency-independent, site- and source- specific responses of small-scale, near-surface

inhomogeneities, which can mask the response of deeper structures of interest. Chapters 2

and 3 provide a brief review of MT theory and static shifts.


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2

Estimating the 3-D subsurface electrical conductivity structure from observed MT

data is done by nonlinear regularized parametric inversion. Several 3-D MT inversion

computer codes have been developed (Zhdanov and Golubev, 2003; Sasaki, 2004;

Mackie and Watts, 2004; Siripunvaraporn et al., 2004, 2005; Wan et al., 2006; Green et

al., 2008; Zhdanov and Gribenko, 2008). Adifficult and computationally intensive task,

3-D MT inversion remains an active research area (Avdeev, 2005). One approach has

been to develop distributed computing clusters (e.g. Newman and Alumbaugh, 2000;

Hargrove et al., 2001; Wan et al., 2006; Green et al., 2008; Zhdanov and Gribenko,

2008), although these can require a substantial investment and facility footprint. Anattractive alternative is to exploit multicore designs; this presents the prospect of parallel

computing within an affordable, single-box, format.

Sasaki (2001, 2004) developed a Gauss-Newton MT inversion algorithm for serial

PCs using the staggered-grid finite-difference forward modeling method to solve

Maxwell's equations. More information on forward modeling is presented in Chapter 4.

Estimating the subsurface resistivity structure requires factoring the approximate

regularized Hessian, formed from a matrix of Frchet derivatives. The inversion

algorithm of Sasaki (2004) is formulated to solve the inverse problem simultaneously for

static shift and 3-D subsurface conductivity distribution parameters, and is discussed in

Chapter 5. Modifications made to the program other than parallelization are also

described in this Chapter.

The program has been modified to allow it to run efficiently on a multicore PC.

Parallelization is accomplished using OpenMP, an easy to use application program

interface developed for shared-memory platforms such as multicore PCs. The serial


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Fortran77 program has been restructured and parallelized under Linux using OpenMP 2.0

directives embedded in the source code and tested on an Intel Xeon 5355, 2.66 GHz, 8-

core PC, with 16 GiB of RAM. Two key areas of the program that were parallelized are

the frequency loop containing the forward modeling and Frchet derivative calculations,

and factoring the approximate regularized Hessian, both time-consuming tasks.

Parallelization is described in Chapter 6.

The MT method has been successfully used to image subsurface electrical

resistivity in complex geothermal systems, detecting electrical resistivity variations

related to fluid flow, including due to high fluid concentrations in fractures, and toconductive alteration minerals. The modified inversion program has been applied to

inverting MT tensor data from 102 sites collected at the East Flank of the Coso

Geothermal Field, located in southeast California (Figure 1.1). The Coso Geothermal area

is a high-temperature power-producing field in southeastern California (Monastero et al.,

2005). Previously published interpretations of the Coso data have included 3-D inversion

using a massively parallel computer, of a finely discretized model seeded with a starting

model incorporating 2-D inversion results (Newman et al., 2005a, 2005b; Newman et al.,

2008). An important structure appearing in these interpretations is a high-angle conductor

most prominent in the southwest East Flank sector correlated with fluid-filled fractures.

As is shown here, similar results can be obtained from inversion on a workstation,

starting from a halfspace. Additionally, a vertical conductor and a north-trending shallow

conductor are imaged beneath the Main Field, correlated with producing well intervals.

Further information can be found in Chapter 7.


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4

Chapters 2 and 3 provide a brief review of MT theory and static shifts. More

information on forward modeling is presented in Chapter 4. The inversion algorithm

along with modifications other than parallelization are discussed in Chapter 5. Chapter 6

covers the details of parallelization. Inversion of MT data from the Coso Geothermal

Field and its geothermal implications are discussed in Chapter 7. A brief summary is

provided in Chapter 8.

Figure 1.1 Map of California showing the approximate location of the Coso GeothermalField. Modified from USGS Physiographic provinces map of California,http://education.usgs.gov/california/maps/provinces_B&W1.htm, accessed June 2010.


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CHAPTER 2

2MT PRINCIPLES

2.1 IntroductionThe MT method was launched by Tikhonov (1950) and Cagniard (1953). It

consists of measuring the naturally occurring, time-varying, orthogonal electric (E) and

magnetic fields (H) penetrating the earth, at a stationary point on the earth's surface. The

fields are measured at frequencies typically in the range of 0.001 Hz to 1000 Hz (Vozoff,

1991). For frequencies below 1 Hz, the fields are due to current systems in the earth's

magnetosphere; worldwide thunderstorm activity generates frequencies higher than 1 Hz

(Vozoff, 1991). At the earth's surface, these fields are considered to be downward-

propagating plane waves (Madden and Nelson, 1964). The amplitude and phase scaling

relationships between the fields depend on the subsurface conductivity structure and field

frequency, and are independent of the source strength. These relationships are expressed

through the use of the MT impedance (Z), a rank 2 tensor determined from:

2.1

The units of E are V.m-1 and of H are A.m-1. The apparent resistivity and phase are

=

y

x

y

x

H

H

E

EZ


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6

defined from the components of the surface MT impedance measurements according to:

0

2

Z

a=

=

)Re()Im(tan 1

ZZ

where is the angular frequency (rad.s-1) and 0 is the magnetic permeability of free

space, 4x 10-7H.m-1.

The method has been thoroughly reviewed in numerous publications including,

but not limited to, Vozoff (1991), Jiracek et al. (1995), Madden and Mackie (1989), and

more recently, Simpson and Bahr (2005) and Zhdanov (2009). The basis of the method,

Maxwell's equations, can be found in electromagnetics textbooks such as, but not limited

to, Stratton (1941) and Harrington (1961). Section 2.2 is intended to provide a brief

review of Maxwell's equations formulated for the MT method. The concepts of MTimpedance, apparent resistivity and phase are developed for 1-D homogeneous and

horizontally layered models in section 2.3, for 2-D models in section 2.4, and for 3-D

models in section 2.5.

2.2 Maxwell's equations for MT theoryMT theory is based on Maxwell's equations and the assumptions of a quasi-static,

monochromatic field in a source-free region. The governing equations, formulated in the

frequency domain for an tie + time-dependence, where is the angular frequency

(rad.s-1), 1=i , and t is the time (s), are:

2.2

2.3


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EH =

HE i=

where Eis the electric field intensity (V.m-1) and His the magnetic field intensity (A.m-

1). The electric conductivity, (S.m-1) and the magnetic permeability, (H.m-1) are

material properties. At MT frequencies, contributions from dielectric displacement are

ignored. It is commonly assumed in MT theory that , and , are frequency-independent;

furthermore, that can be adequately represented as constant and equal to the free-space

value of = 0= 1.2566x10-6

H.m

-1

(Keller, 1991). Expressing the fields as:

HE =1

EH =

i1

equations 2.4 and 2.5 can be transformed into

( ) EE i=

HH

i= 1( ) 0= E

2.4

2.5

2.6

2.7

2.8

2.9

2.10

2.11.0= H


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8

In the presence of a sharp conductivity boundary, the components ofHandE tangential

to the boundary must be continuous across it, while E normal is discontinuous,

preserving continuity of current jwhere Ej = (e.g. Ward and Hohmann, 1988).

In the following sections, an e(+it) time dependence is assumed together with a

right-hand cartesian coordinate system with z positive downward.

2.3 1-D modelIn a homogeneous medium, is constant; 0= E . Equations 2.8 and 2.9 reduce

to the Helmholtz equations

( )( ) 0

02

2

==

H

Eii .

For a plane wave vertically incident on a homogeneous or horizontally layered halfspace,

field vectors always lie in horizontal planes, and are instantaneously equal over entire

plane, such that derivatives with respect to x and y directions are zero (e.g. Zhdanov,

2002). The Helmholtz equations reduce to:

0

0

2

2

2

22

2

=

+

=

+

H

E

kz

kz

2.12

2.13


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where the wavenumber ik = , )0,,( yxEE=E , and )0,,( yx HH=H . These are

homogeneous second-order ordinary differential equations with the general solution, for

equation 2.13 (Vozoff, 1991),

ikzikzee EEE + +=

where z is the distance from the surface, and -Eand +Eare the upgoing and downgoing

field amplitudes. In a uniform half-space, there is only a downgoing field,

ikze+= EE

with corresponding magnetic fields from equation 2.7,

kzy

yx eEiz

Ei

H +== 1

and

kz

xx

y eEi

z

E

iH +=

=

1

.

Off-diagonal elements of the impedance are equal, and can be expressed as:

iZZ yxxy ==

2.14

2.15

2.16

2.17

2.18


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independent of the source strength and coordinate definition. On-diagonal elements are

zero. The surface impedance is intrinsic and the apparent resistivity equals the half-space

resistivity. For an e(+it)time dependence, the phase will be in the first quadrant, at 45 for

the XY mode, and in the third quadrant, at 135 for the YX mode. The YX mode phase

can be represented in the first quadrant by the addition of 180 . The depth of exploration

is taken to be skin depth (), given by

( ) 12 = .

For a conductivity structure consisting of horizontal layers, the fields inside each

layer are governed by the Helmholtz equations, but must satisfy the boundary condition

thatEandHfields tangential to the interface between layers must be continuous across.

Fields at the surface can be determined by assuming a starting value for the amplitude of

+E in the basal half-space and, working upwards, calculating the fields recursively at

overlying interfaces (e.g. Ward and Hohmann, 1988). The surface impedance is not

intrinsic; the apparent resistivity will be a weighted average of the sampled conductivity

structure and the phase will digress from 45 for the XY mode, and 135 for the YX

mode.

2.4 2-D modelIn 2-D models, conductivity varies with depth and horizontally along one axis

only. Structures strike perpendicular to this vertical plane and are assumed to extend to

infinity; vertical planes perpendicular to the strike direction are identical. Consider, for

2.19


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11

example, a structure with conductivity variations only in the y-z plane. For measurement

axes coincident with the structural axes, field derivatives with respect to x, the strike

direction, will be zero and scalar Maxwell's equations yield

.1111

112

2

2

2

=

=

=

=

z

H

zy

H

yiz

E

y

E

iH

z

E

y

E

iz

H

y

HE

xxyz

x

xxyz

x

The MT response separates naturally into two modes which propagate independently:

transverse electric (TE), where ( )0,0,xTE

E=E , is parallel to strike, and ( )zyTE HH ,,0=H ;

and transverse magnetic (TM) where ( )zyTM EE,,0=E , and ( )0,0,xTM H=H , is parallel to

strike. (e.g. Vozoff, 1991; deLugao and Wannamaker, 1996; Zhdanov, 2002). At lateral

conductivity variations,ETEis tangential to and continuous across the interface,

TETEi EE =2

whileETMis normal to the interface, and discontinuous from charges accumulated at the

boundary maintaining current continuity (e.g. Wannamaker et al., 1984; Jiracek, 1990).

Hence, 0 TME and the galvanic term, ( )TME is required when formulating

equation 2.8 for the TM mode,

( ) TMTMTM i EEE =2 .

2.20

2.21

2.22


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Note thatHTM, parallel to strike, remains continuous and can be determined from

TMTMi HH

= 1 ,

from which auxiliary ETM components can be calculated (deLugao and Wannamaker,

1996). The TE mode is considered more prone to distortion from finite strike, 3-D effects

than the TM mode because boundary charge effects are absent from TE physics

(Wannamaker, 1999).

For a 2-D structure striking in the x-direction, the impedance consists of off-

diagonal elements only,

=

==

0

0

x

y

yx

y

xxy

H

EZ

H

EZ

Z .

The YX component of the impedance will correspond to the transverse magnetic (TM)

mode and the XY to the transverse electric (TE) mode in this coordinate convention. A

rotation matrix can be applied to data collected at an arbitrary angle, to align it with the

strike direction (e.g. Vozoff, 1972; Simpson and Bahr, 2005).

2.5 3-D model3-D conductivity structures can be represented as consisting of a domain of

anomalous conductivities, superimposed on 1-D, homogeneous or horizontally layered,

2.23

2.24


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background. The total fields (Et,Ht) can be represented as the sum of background fields

(Ep,Hp) generated by the idealized horizontally-layered background, and scattered fields

(Es,Hs), caused by the anomalous domain (Es,Hs):

spT

spT

HHH

EEE

+=

+=

Expressing equation 2.8 in terms of Ep and Es, and making use of equation 2.12,

describing the behavior of Ep in a 1-D model, the relationship between the background

and scattered fields can be expressed as:

ppss ii EEE )( =+

where p is the conductivity of the horizontally layered earth at the depth at which the

fields are evaluated. Analytic solutions have been published for specific geometries, but

for an arbitrary conductivity distributions, there are no analytical solutions, and scattered

fields must be numerically approximated.

To calculate the impedance in computer simulations, fields are measured for at

least two source directions, s1, and s2, preferably orthogonal to one another, where

HH

EHZ

ij

ij=

and

2.25

2.26

2.27


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14

.2121

2212

1221

1221

2112

s

x

s

y

s

y

s

x

s

x

s

y

s

x

s

yyy

s

y

s

x

s

y

s

xxx

s

y

s

y

s

y

s

yyx

s

x

s

x

s

x

s

xxy

HHHHHHHEHEEH

HEHEEH

HEHEEH

HEHEEH

==

=

=

=

Estimating the impedance from measured EM field time series invokes a statistical

generalization of the preceding equation where the source field polarizations are

presumed to vary over the recording interval (e.g. Gamble et al., 1979). In a regionally 3-

D environment, the on-diagonal elements of the impedance are non-zero no matter what

coordinate orientation is chosen except in special cases of symmetry.

2.28


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CHAPTER 3

3STATIC SHIFT

A static shift is, by definition, the frequency-independent distortion of an apparent

resistivity sounding (see review in Wannamaker, 1999). It is caused by a near-surfaceinhomogeneity of dimension significantly smaller than the shortest EM field wavelength

sampled in the sounding. The inhomogeneity is assumed to have a frequency-independent

galvanic-only response; the frequency-dependent inductive component of the response is

considered negligible (e.g. Groom and Bailey, 1989). In the case of 1-D and 2-D TE

mode regional structures, meeting this condition is sufficient in representing, and

removing, the data distortion using frequency-independent parameters. In the case of 3-D

regional structures, the data distortion may be frequency dependent, despite the near-

surface inhomogeneity generating only a galvanic response.

A theoretical discussion of shifts is provided in section 3.1, followed by a brief

review of different correction techniques in section 3.2. Shift distribution is discussed in

section 3.3. Shifts are simulated for a hypothetical near-surface inhomogeneity contained

within a regional 3-D model representing an idealized geothermal system and examined

in sections 3.4 and 3.5, followed by a discussion in section 3.6.


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3.1 TheoryConsider a regional subsurface conductivity structure, which may be 3-D, with

corresponding regional electric (ER) and magnetic (HR) fields related by the impedance

(ZR), where the superscript R denotes that these quantities are defined for the regional

subsurface conductivity structure. A "distorting" small-scale, near-surface inhomogeneity

is introduced in the regional structure. The measured electric field (Ed) can be expressed

as the sum of the regional field and of a scattered field, introduced by the presence of the

inhomogeneity:

.SRd

EEE +=

Following Wannamaker et al. (1984), assuming that the inhomogeneity is small enough

that the regional electric field is constant over the inhomogeneitys lateral extent, the

scattered field corresponding to the inhomogeneity can be written in terms of the incident

regional field as:

[ ] RRd

PEEE +=

assuming that the fields are linearly related. No assumptions are made regarding the

character of the entries in P at this time, allowing it to remain complex. Assuming that the

total magnetic field is unaffected by the inhomogeneity and remains essentially HR, the

"distorted" impedance,Zd, relating the fields in the presence of the inhomogeneity, can be

expressed in terms of the regional fields as:

3.1

3.2


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17

[ ] RdRR

P HZEE =+

where the entries of P depend on the inhomogeneity and are complex. Agarwal and

Weaver (2000) demonstrate the validity of this assumption for the frequency range

commonly used in MT soundings. The distorted and regional impedances are related by:

+

+=

+=

R

yy

R

yx

R

xy

R

xx

Rd

ZZ

ZZ

dIc

baI

P ZZ 1

where the entries of [ ]P , inhomogeneity parameters a, b, c and d, may be complex and

frequency-dependent. Individual modes can be expressed as:

.RyyRyy

R

xyd

yy

R

yxR

yx

R

xxd

yx

R

xyR

xy

R

yyd

xy

R

xxRxx

R

yxd

xx

ZdIZ

ZcZ

ZdIZ

ZcZ

ZZ

ZbaIZ

ZZ

Z

baIZ

++=

++=

++=

++=

If the inhomogeneity is of dimension and conductivity such that it is smaller than the skin

depth at the highest frequency of interest, then it can be assumed that the frequency-

dependent part of the scattered field can be neglected, leaving only the galvanic

3.3

3.4

3.5


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18

component. The inhomogeneity parameters are then real only and frequency-

independent.

The XY mode apparent resistivity and phase in the presence of the inhomogeneity

can be expressed as:

R

xyaR

xy

R

yy

R

xyR

xy

R

yy

d

xya

Z

Zba

ZZ

Zba

,

20

2

2

,

1

1

++=

++

=

( )( )

R

xy

R

xy

R

xyd

xyZ

Z

=

=

tanRe

Imtan

with similar expressions obtained for the remaining modes. The phase is unaffected by

the inclusion of the inhomogeneity in the regional structure. The apparent resistivity is

scaled by a multiplicative factor termed shift. Note than when displayed on a log-scale,

the shift appears additive.

For a regionally 1-D (horizontally planar) or 2-D structure with measurement axes

rotated parallel to strike, the impedances simplify to:

3.6

3.7


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19

( )

( )R

xy

d

yy

R

yx

d

yx

R

xy

d

xy

R

yx

d

xx

R

yy

R

xx

cZZ

ZdIZ

ZaIZ

bZZ

ZZ

=+=

+=

=

== 0

with Ryx

R

xy ZZ = for the 1-D case. For these situations, the apparent resistivity shift will be

frequency-independent provided the inhomogeneity parameters are frequency-

independent.

As developed above for the 3-D case, the apparent resistivity shift may vary with

frequency because of the inclusion of complex, frequency-dependence of the regional

impedance ratios. Depending on the location of the receiver with respect to regional

structure, the ratio of on- to off-diagonal impedances may be frequency-dependent; thus,

even if the inhomogeneity generates only galvanic fields, the frequency-dependence may

be re-introduced into the static shift.

Pellerin and Hohmann (1990) and Spitzer (2001) demonstrate that the magnitude

and direction of the shift depends on the location of the electric field measurement

electrodes with respect to the location of the inhomogeneity. In practice, the electric field

in a particular direction is obtained by measuring the voltage between a pair of electrodes

aligned in such a direction, and dividing by the distance between them. Accurate

modeling of shifts in the vicinity of the inhomogeneity requires including the location of

the electric field measurement electrodes with respect to the location of the

inhomogeneity. In most cases, the electric field measurements are simulated as entirely

3.8


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20

within or entirely outside the inhomogeneity. No consideration is given to the position of

the electrodes.

3.2 Correction techniquesNumerous techniques have been developed to address quantifying and possibly

removing static shift from MT data prior to inversion and interpretation. Identifying the

presence of and quantifying how regionally 1-D and 2-D data are affected by 2-D and 3-

D effects in small-scale overlying structure lead to the developing of distortion analysis

and phase tensor techniques (Bahr, 1988; Groom and Bailey, 1989; Caldwell et al.,2004). Wannamaker (1999) groups these techniques into invariants and averages, spatial

averaging, and impedance tensor decomposition. Reviews of these techniques can also be

found in Jiracek (1990). The majority of these techniques are primarily applicable to 1-D

and 2-D interpretations. Ogawa (2002) identifies three categories of techniques most

applicable to dealing with static shifts in 3-D interpretations: (1) spatial filtering, as in

electromagnetic array profiling (EMAP); (2) the use of independent information that is

free from galvanic distortion, such as obtained by time-domain (TEM) central loop

soundings; and (3) solving for static shifts as parameters in inversion.

The EMAP method, developed by Torres-Verdin and Bostick (1992a, 1992b),

consists in measuring electric fields using a continuous profile of electric dipoles placed

end-to-end. orresponding magnetic fields and, optionally, orthogonal electric field

measurements, are collected at discrete intervals. The static shift is removed from the

data by spatially low-pass filtering the electric field. It exploits the fact that the secondary

electric field due to the near-surface heterogeneity integrated along the profile path is


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21

zero mean. This type of survey requires extensive field operations and continuous

sampling of the electric field (difficult) (Sternberg et al., 1988). While similar to the

traditional MT method, Pellerin and Hohmann (1990) consider the differences in field

operation and data processing required for EMAP significant enough for it to be

considered a distinct method from traditional MT.

Of the techniques most applicable for correcting MT data in the context of

regional 3-D interpretations, TEM measurements are the most commonly used. TEM

central loop soundings are relatively free from galvanic distortion and can be used at the

MT sounding location to estimate an average 1-D shallow structure (Pellerin andHohmann, 1990). Sternberg et al. (1988) demonstrated using TEM central loop soundings

to shift MT soundings, by matching the curves in the frequency range of overlap, and by

joint inversion. Pellerin and Hohmann (1990) transform TEM sounding data into 1-D MT

sounding by calculating the MT response at high frequencies for the 1-D shallow

subsurface structure obtained from inversion of the TEM sounding, and shift the

observed MT response using the computed MT response. Meju (1996) advocates joint

inversion of TEM and MT phase data, similar to Sternberg et al. (1988). Alternative

methods for computing the MT response used for curve-shifting are to use the shallow

subsurface structure obtained from DC resistivity soundings (Spitzer, 2001), or to use a-

priori geologic information (e.g. Jones, 1988).

An alternative to correcting the data for the presence of static shift is to consider

static shifts as parameters to be solved for in the inversion for subsurface structure.

deGroot-Hedlin (1991) introduced the concept of including static shifts as parameters to

be solved for in 2-D inversion, along with subsurface conductivity structure. To stabilize


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22

the inversion, a zero-sum constraint was applied to the static shift. deGroot-Hedlin (1995)

modified the zero-sum stabilizer such that sum of the shifts can be set to any user-

specified constant, allowing for the possibility of bias. Ogawa and Uchida (1996) solve

for the static shifts and 2-D subsurface structure, imposing a Gaussian-distribution

constraint on the static shift. Sasaki (2004) applies the concept of solving for static shift

and subsurface conductivity distribution to 3-D, allowing the user the choice of whether

to use a zero-sum or gaussian-distribution, zero-mean stabilizer.

3.3

Distribution of shiftsNo theoretical basis has been published as to whether the distribution of static

shifts observed for a collection of MT soundings should be Gaussian-distributed, have a

mean of zero or should sum to zero. A zero-sum constraint assumes that static shifts are

zero-mean random perturbations affecting each MT site; as the number of MT sites

increases, the sum of the static shifts should approach zero (deGroot-Hedlin, 1991). This

is argued to be the case for the electric field along a profile (EMAP analysis). The

assumption is considered to be generally reasonable (deGroot-Hedlin, 1995), but assumes

the shifts are random and that there are a large number of sites (Ogawa and Uchida,

1996). It does not take into account the possibility of bias inherent in MT site selection

and does not imply any particular distribution on the static shifts. Ogawa and Uchida

(1996) interpret histograms of static shifts estimated from collocated TEM and MT

soundings, published by Sternberg et al. (1988), to support the assumption of Gaussian

distribution. Sasaki and Meju (2006) analyzed collocated MT and TEM soundings from

various surveys and found "good support" for Gaussian distributions, skewed depending


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23

on geological environment. Deeply weathered regions and sedimentary terrains with

conductive overburdens were found to have net negative shifts and regions with recent

volcanics to have net positive shifts.

Sasaki and Meju (2006) investigate the artifacts created by static shifts in MT data

on 3-D inversion results. Without including static shifts as inversion parameters, they

observe that artifacts generated by 3-D inversion are concentrated in the near-surface

structures, and conclude that large-scale conductivity models recovered by 3-D inversion

of MT data are more satisfactory than can be recovered by 1-D and 2-D inversions.

Including static shifts as parameters in the inversion creates fewer artifacts and allows fora model with lower misfit and smoother convergence. Sasaki and Meju (2006) found that

the Gaussian and the zero-sum stabilizers worked equally well in simulating the synthetic

data and their shifts, which were approximately Gaussian and zero-mean, hence zero-

sum.

3.4 Impedance ratio frequency-dependenceTo better understand the conditions under which a near-surface inhomogeneity in

a regionally 3-D structure generates a frequency-dependent shift, the response from a 3-D

geothermal system is simulated, using the idealized model developed by Pellerin et al.

(1996). The model consists of a 25 m reservoir overlain by a thin, 5 m clay cap, in a

200 m half-space host (Figure 3.1). The clay cap is 6 km by 4.8 km laterally, 375 m

thick, with its top at 300 m depth. The reservoir is 2.7 km by 2.7 km laterally, and 5.4 km

thick, immediately beneath the clay cap. Synthetic data were determined for this


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24

Figure 3.1 Idealized geoelectrical conductivity structure of a high-temperature

geothermal system (Pellerin et al., 1996), with receiver locations used in forward

modeling. A small inhomogeneity is introduced for modeling shifts.

4Plan View

Receiver locations I: : : : ::: : : u r f a inhomogenei ty

2

Reservoir250hm.mlay ap50hm.m)

Easting km)

o

6

Depth viewCCIft l liiiJlEllliiiiiji;ci Nea rsu rfa ce inhomogene ity

Reservoir250hm .mj

~ ~Easting km)


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25

model using the forward modeling option in the modified program version of Sasaki

(2004). The finite difference mesh consists of 77 (x) x 73(y) x 40(z) nodes, with a

minimum node spacing of 150 m horizontally and 35 m vertically. Because the regional

structure is symmetric, receivers are distributed throughout the NE quarter of the domain

only, and are concentrated near the horizontal boundaries of the clay cap and reservoir.

The response is estimated for 255 receivers, at 19 frequencies equally spaced

logarithmically from 1000 Hz to 0.03 Hz. While 1000 Hz is higher than what is typically

used in MT surveys for geothermal targets, this will allow us to see the transition from

including some inductive component to purely low-frequency galvanic-response of whena near-surface inhomogeneity is included in the regional model.

The regional impedance ratios are constructed and evaluated. Ratios of on- to off-

diagonal impedances were calculated using the synthetic data for the regional model

corresponding to an idealized geothermal system. These ratios are used when determining

the XY and YX mode apparent resistivity data. The frequency-dependence inherent in the

regional impedance ratios at each receiver is measured as the maximum difference across

all frequencies (Figure 3.2). The most frequency-dependent impedance ratios are

observed near the edge of the clay cap parallel to the direction of the measured magnetic

field along the eastern edge for the XY mode and along the northern edge for the YX

mode. The absolute value of the maximum impedance ratio is approximately 1.5.

3.5 Shift frequency-dependenceA near-surface, small inhomogeneity is placed near the northeast corner of the

clay cap, where the regional impedance ratios are highly frequency-dependent, and the


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26

Figure 3.2 Plan view of the maximum, across all frequencies modeled, of the impedanceratios used in calculating XY and YX impedances for the geothermal model of Pellerinet al. (1996). Boundaries of the clay cap and reservoir are shown.


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27

ensuing distorted response calculated. Hypothetical shifts are simulated by including a

small, near-surface inhomogeneity in the regional geothermal system model. The

inhomogeneity consists of a 35 m thick and 150 m x 150 m wide body of contrasting

resistivity located near the boundary of the clay cap (Figure 3.1). Synthetic data were

calculated for the inhomogeneity resistivity contrast of 10 with respect to the surrounding

half-space (20 m, 2000 m). The distorted soundings are shown in Figure 3.3.

Including the inhomogeneity in the regional model causes distortion of both the phase

and apparent resistivity soundings. Apparent resistivity soundings are distorted such that

the sounding for the conductive inhomogeneity is shifted downwards with respect to theundistorted sounding that for the resistive inhomogeneity is shifted upwards.

At frequencies where the inductive component of the small inhomogeneity

response is negligible, distorted soundings can be approximated numerically using the

regional structure and some estimate of the shift or inhomogeneity parameters. Shifts are

estimated by calculating the arithmetic average of the ratio of the modeled distorted and

regional apparent resistivity data at the three longest periods in the frequency range of

interest, where both the shift and the impedance ratio are approximately constant.

Assumptions inherent in this approximation are that the frequency-dependence of the

impedance ratios can be neglected, and that the inhomogeneity parameters are constant at

the frequencies of interest. The distorted sounding can be corrected to approximate the

regional sounding by dividing the apparent resistivity by the shift for each mode (Figure

3.4). The difference between distorted and regional soundings can be defined as

negligible when the distorted data fall within the envelope formed by the regional data

error. Arbitrarily, a data weight error of 0.01 log10 units is chosen for the apparent


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Figure 3.3 Distorted apparent resistivity soundings obtained by introducing a shallowinhomogeneity into the geothermal model, alternately set to be more resistive than thesurrounding near-surface layer or more conductive.


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29

Figure 3.4 Regional sounding and distorted sounding after being corrected using a

constant shift, and their normalized difference in apparent resistivity, for the receiverdirectly above the inhomogeneity introduced in the geothermal model. Normalized phase

differences are included to evaluate where the shifts are truly static.

Sounding distorted by Sounding distorted by_ 3 resistive inhomogeneity 10 J conductive inhomogeneityE E

02 102 0a:10

10 10 10Frequency Hz)

a:

10


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30

resistivity and 0.66 for the phase, corresponding to good quality MT data. The distorted

phase data are within the data error for frequencies lower than 20 Hz for the XY mode

data and lower than 100 Hz for the YX data (Figure 3.4). Both soundings exhibit phase

distortions at higher frequencies, suggesting that the inhomogeneity has a nonnegligible

inductive response and should not, at these frequencies, be considered to produce a static

shift. The corrected data most differ from the regional data at high frequencies, where

the inhomogeneity has a nonnegligible inductive component also seen in the phase. The

difference between the regional and the corrected apparent resistivity data is generally

less than the data error for frequencies where the inhomogeneity generates a static shift,despite the frequency-dependent regional impedance ratios observed for this sounding.

This suggests that the inhomogeneity parameters which scale the impedance ratios are

very small, rendering the frequency-dependent contribution of the regional impedance

ratio trivial for this test example.

3.6 SummaryNumerical modeling suggests that including a small, near-surface inhomogeneity

near the 3-D geothermal model developed by Pellerin et al. (1996) distorts neighboring

apparent resistivity soundings which would be measured in its absence. The distortion

can be simulated using inhomogeneity parameters and undistorted impedance ratios.

Provided the frequency of investigation is low enough that induction in the

inhomogeneity is negligible, phase measurements are unaffected and inhomogeneity

parameters are frequency-independent. Frequency dependence of the apparent resistivity

distortion due to varying impedances is observable, but not significant in this example.


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31

Soundings corrected using approximate static shifts are within the assigned data errors.

Two approximations are considered, both based on constant shifts. In the first estimate,

the impedance ratio is assumed frequency-independent and the distorted data are

corrected by applying the mean of the difference between the regional and distorted data.

In the second estimate, corrections are calculated using the on-diagonal inhomogeneity

parameters a and d, neglecting parameters b and c and the impedance ratios altogether.

Both estimates are similar to the modeled distorted data, particularly at the longer

periods. This reinforces the validity of the assumption of static shifts implicit when

solving for the shifts along with the 3-D conductivity structure. There may be othermodels that show greater frequency dependence of distortion, but these are difficult to

predict a priori.


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CHAPTER 4

4FORWARD MODELING

To calculate the MT apparent resistivity and phase at the earth's surface requires

solving Maxwell's equations (ME) for a particular subsurface conductivity structure.

Analytical solutions of ME for arbitrary 3-D conductivity structures do not exist;

numerical approximation is necessary, either in differential or integral form. The most

widely used solution methods are based on finite differences (e.g. Madden and Mackie,

1989; Smith, 1996a; Newman and Alumbaugh, 1995, 2002; Siripunvaraporn et al., 2004,

2005, among others), and on integral equations (e.g. Hohmann, 1975; Weidelt, 1975;

Wannamaker et al., 1984; Zhdanov, 2009, among others). The FD method is attractive

because of the apparent simplicity of its implementation compared to that of the IE

approach and because it generates sparse regular system matrices for solution. The IE

approach is attractive because it requires discretizing only the volume containing the

arbitrary 3-D conductivity structure; the FD method requires discretizing a much larger

volume including the background and extending into the air. A more thorough

description of the methods and their relative attributes can be found in, e.g., Zhdanov

(2009).

Sasaki's program is based on using the finite difference method to solve the

differential form of Maxwell's equations, using staggered grids. The forward problem and


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33

its solution using the finite-difference method are described in section 4.1. The resulting

system of equations is solved using a preconditioned biconjugate gradient method,

described in section 4.2, using the divergence-correction technique of Smith (1996b),

described in section 4.3. Sasaki (2001) verified the accuracy of the forward solution by

simluating airborne EM data and comparing them to results obtained using the finite-

difference solution of Newman and Alumbaugh (1995).

4.1 Finite difference solutionConductivity structures can be represented as consisting of a domain of

anomalous conductivities, superimposed on a horizontally-layered background. Total

fields can be determined by summing background fields, (Ep,Hp) analytically calculated,

and scattered fields (Es,Hs), caused by the anomalous domain (Es,Hs), which must be

numerically approximated. From equation 2.26, the scattered fields in ME in 3-D

domains can be written in terms of the background field:

) pps ik EE )(2 =+

To solve for the scattered fields, the conductivity structure is discretized into rectangular

cells; conductivity is assumed to be homogeneous inside each cell. Derivatives can then

be approximated as finite differences. The scattered electric fields are solved for, and

magnetic fields are calculated from the solution using ME.

The finite-difference scheme is implemented using staggered grids, pioneered by

Yee (1996) for use in electromagnetic modeling. Electric and magnetic field components

4.1


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34

are estimated at different locations forming grids staggered with respect to one another.

First-order central differences are used to represent differentiation operations. The appeal

of this method resides in discretized curl, divergence and gradient operators which hold

true (Smith, 1996a). For a given cell of homogeneous conductivity, electric fields are

measured at the centers of the cell edges; Exare measured at the centers of the cell edges

parallel to the x-direction, Eyat the centers of edges parallel to the y direction and Ezat

the centers of edges parallel to the z direction. Magnetic fields are measured at the centers

of the cell faces, in a similar manner. Note that it is also possible to sample electric fields

at the cell-face centers and the magnetic fields on the cell-edge centers (e.g. Smith,1996a). Sasaki (2004) tested both configurations and found no significant difference in

forward modeling responses between the two. Where adjacent cells are of different

conductivity, a weighted average conductivity is used. Weights are proportional to the

cross-sectional area of each cell that is normal to the sampled component (Sasaki, 2001).

The resulting system of equations expressed in matrix notation is:

sK =sf

where K is the system matrix and s is the source vector. The diagonal entries of Kare

complex; off-diagonal entries are real. The system matrix is made symmetric by scaling

each row and the source vector entry by the product of the grid spacings spanning thesampling position (Smith, 1996a). The source vector depends on the source polarization

and the boundary conditions (Sasaki, 2004). The condition applied is that the scattered

electric field is negligible at the edges of the finite-difference mesh, such that the total

4.2


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35

electric field, tangential to the edge of the mesh, is continuous across it. Implicit in this

condition is that the edge-most finite difference cells define the horizontally-layered

background structure, and that lateral conductivity variations must be sufficiently distant

from the boundary. Hence, the finite-difference method requires a large mesh that

extends far past the volume of interest (Zhdanov et al., 1997). Madden and Mackie

(1989) discuss the design of finite-difference meshes for the MT forward problem. In

order to satisfy zero boundary conditions, the distance to the boundaries of the finite-

difference domain must be greater than the largest skin depth considered.

4.2 Biconjugate gradient methodThe scattered field solution is obtained by solving the system of equations using

the biconjugate gradient (BiCG) method, preconditioned with incomplete Cholesky

decomposition (Mackie et al., 1994; Smith, 1996b). The system matrix is symmetric, but

because of the complex entries on the diagonal, it is not Hermitian and the CG method

cannot be directly applied. The BiCG method is a generalization of the CG method, with

the conjugate transpose replaced by a simple transpose (Smith, 1996b). Note that the

BiCG method is different from the CG method in that the residual is not monotonically

decreasing. To account for this, the scattered field solution is updated only when the

residual decreases.

4.3 Divergence correctionSolution for the scattered fields is improved by including Smith's (1996b)

correction procedure that enforces divergence-free conditions on current density.


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36

Convergence of the scattered field solution tends to degrade at low frequencies. As the

frequency falls, both the spatial derivatives in the system matrix and the frequency-

dependent source vector become small. The solution becomes computationally inexact,

introducing current sinks or sources throughout the finite-difference mesh. Smith (1996b)

developed a procedure which essentially cancels these artifacts. The procedure is applied

to the updated scattered field solution, within the outermost loop of the biconjugate

gradient scheme, prior to iterative refinement. The correction vector (Ec) is obtained

from:

( ) ( )sc EE =

The correction vector and the source vector are complex; the divergence is a linear

operator. Real and imaginary corrections are computed separately using the CG method,

with incomplete Cholesky decomposition preconditioning. The corrected scattered field

can then be calculated as the sum of the correction and the scattered field solution.

4.3


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CHAPTER 5

5INVERSION SCHEME

The subsurface conductivity structure sampled in MT surveying can be

represented approximately as a set of model block parameters with heterogeneously

distributed conductivities. These conductivities can be deduced by inversion of MT

measurements. The inversion process for MT measurements is an iterative process

consisting of calculating the predicted data for some model, determining the misfit

between the predicted and observed data, modifying the model to reduce the misfit, and

repeating these steps. The process is terminated when some minimum error or maximum

number of iterations is reached. The starting model can be estimated from a priori

structural information or data averages.

There have been several computer codes generated for 3-D MT inversion (e.g.

Sasaki, 2001; Zhdanov and Golubev, 2003; Mackie and Watts, 2004; Siripunvaraporn et

al., 2004, 2005). For example, Newman et al. (2005a) introduced a finite-difference-

based 3-D massively parallel inversion algorithm for MT data, which, however, requires

large computing resources and long run times. In recent years, CEMI has developed

several algorithms and computer codes for MT inversion based on fast forward-modeling

approximations and on rigorous integral equation (IE) solvers (Golubev et al., 1999;

Golubev and Zhdanov, 2000; Zhdanov, 2002; Wan, et al., 2006; Green et al., 2008;


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38

Zhdanov and Gribenko, 2008). Note that interpretation of MT data can be further

complicated by static shifts. The inversion algorithm of Sasaki (2004) is formulated to

solve the inverse problem simultaneously for static shift and 3-D subsurface conductivity

distribution parameters. The program runs on a personal computer, and can handle

moderate-sized data sets. The inversion is done using the Gauss-Newton method, which

requires calculating Frchet derivatives at each iteration.

Section 5.1 introduces the theory behind the MT inversion scheme implemented

in the program. Sensitivity calculations are described in section 5.2. The inclusion of

static shifts in the inversion scheme is described in section 5.3. In section 5.4, theprogram is demonstrated by inverting synthetic data for a model similar to that of the

idealized geothermal system presented by Pellerin et al. (1996).

5.1 TheoryThe inverse MT problem is that of approximating the subsurface conductivity

structure corresponding to observed MT measurements. Solving this problem consists of

finding some conductivity structure which minimizes the weighted residual,r

( )obspred

ddWr =

which is the difference between the observed data dobs, and thepredicted data dpre,

normalized by the reciprocal of the data standard deviation, stored in diagonal matrix Wd.

The data consist of the apparent resistivity, in log10-space, and the impedance phase. The

predicted data can be determined according to

5.1


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39

)A(md =pre

whereAis a nonlinear forward operator, based on Maxwells equations, andmis a vector

of subsurface conductivity model parameters. For the Sasaki-based inversion program

here, the predicted data are calculated using the finite-difference method outlined in

Chapter 4.

Minimizing the misfit is equivalent to minimizing the size of the residual

measured using the L2 norm, known as the misfit functional , where

min)(2

2== rm .

The normalized RMS misfit (nRMS) between predicted and observed data is calculated

using

( )N

nRMS m=

where N is the number of data. As the predicted data approach the observed data to

within the data standard deviation, the misfit should approach 1.

Minimizing the misfit functional is an ill-posed problem, rendered conditionally

stable by including stabilizing functionals. Popular stabilizing functionals are described

5.2

5.3

5.4


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40

in Tarantola (1987, 2005) and Zhdanov (2009). The parametric functional used in the

program includes two stabilizing functionals,

( ) ( )

( )

( ) ( ) ( )

min

2

21

2

21

2

212

2

212

2

21

=

+++

+

=

c

apriorizC

c

aprioriyC

c

apriorixC

m

apriorim

obspred

zyxmmCWmmCWmmCW

mmW

ddWm

Terms and are regularization parameters, which control the trade-off in fitting theresidual and the stabilizers. The first stabilizer, scaled by , is of the form m , and acts

to restrict the magnitude of the conductivity parameters in the new model to adhere to

those of the a priori model. This stabilizer may function as a Marquardt factor by using

the model from the previous iteration as a priori. The second stabilizer, scaled by , is of

the form ( )m where banded matrices Cx, Cyand Czperform first-order differencing

between adjacent parameters. This stabilizer acts on the conductivity gradient, or spatial

smoothness, to match that of the a priori model. Matriceszyx CCC

WWW ,, and mW aremodel weighting matrices. These stabilizers provide a maximum-smoothness type of

model.

Sasakis program uses the iterative Gauss-Newton inversion scheme.

Minimization of the residual can be expressed in the context of minimization of the

parametric functional, accomplished by calculating the first variation of the parametric

functional with respect to the change in model parameters m and setting it to zero

(Zhdanov, 2002). This yields the following equation:

5.5


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41

( )

( )( )( )

( )( )( )( )( )( )prioriazC

prioriayC

prioriaxC

apriorim

obs

T

m

z

y

x

A

+

++

+

+=

=

mmCWm

mmCWm

mmCWm

mmWm

dmmFm

mm

12

12

12

12

0

1

,2

,2

,2

,2

,2

0

0

where m0 is the previous model, and m1 the updated model. The forward operator is

linearized by introducing the Frchet derivative matrix F, a Jacobian:

)()()(000

mFmmm ++ mAA

where

( )00

0m

mF Am = .

This leads to the system of equations

[ ] bAmJJ TT =1

where

+

=

=

aprm

aprzC

apryC

aprxC

obspred

m

zC

yC

xC

d

z

y

x

z

y

x

mW

mCW

mCWmCW

ddmFW

b

W

CW

CWCW

FW

J

mm

0,000

; .

5.6

5.7

5.8

5.9

5.10


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42

As the approximate regularized Hessian, matrix JTJ, is square and symmetric, the system

of equations is solved using the LDLTmethod. The updated model can then be used at the

next iteration as the previous model. Implicit in linearizing the forward operator, the

change in model parameters is limited so the Frchet derivative matrix must be re-

calculated for each iteration. Inversion schemes other than Gauss-Newton such as

steepest descent and conjugate gradients have also been used successfully.

At each iteration, the regularization parameters control how closely the updated

model adheres to the class of models imposed by the stabilizers. Various techniques for

selecting regularization parameters have been developed (e.g. Zhdanov, 2002). Ratherthan using a full regularization approach in this implementation, regularization

parameters can be selected at each iteration by a line search to determine the parameter

which results in the lowest misfit (Sasaki, 2001; Sasaki and Meju, 2006). The nRMS

misfits of three models, each obtained for different regularization parameters, are

calculated. Starting from a model typically distant from the true, coupled with the

requirement for few iterations driving the choice of regularization parameters, the misfit

functional may not decrease monotonically with decreasing regularization parameter as

illustrated in Zhdanov (2002). The parameter estimated to provide the lowest nRMS

misfit is selected to be used for that iteration of the inversion. As this is not the rigorous

approach to regularization as described in Zhdanov (2009), there is no guarantee that the

parametric functional, normalized misfit and regularization parameter will decrease

monotonically (deGroot-Hedlin and Constable, 1990).

The non-zero diagonal entries of the model weighting matrix applied to the

magnitude stabilizer are calculated according to


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( ) 41FFdiag Tm=W

as recommended in Zhdanov (2002), with a floor of 0.05. The weighting matrices

zyx CCC WWW ,, applied to the gradient stabilizer are based on averaging the weights applied

to the magnitude stabilizer for inversion cells beneath the receiver.

5.2 Frchet derivativesThe Frchet matrix consists of the derivatives of the apparent resistivity and phase

with respect to the model parameters. These derivatives can be expressed in terms of the

derivative of the impedance:

( ) ( )( )( )

pre

ii

b

i

i

ii

b

pre

i

b

ic

i

b

pre

ia

Z

dm

dZZZZ

dm

d

dm

dZZ

dm

d

2

,

cosRe

ReReImIm

Re2

=

=

The derivative of the impedance can in turn be determined from the derivative of the

fields.

HH

dHHZdEH

dm

dZ ijij

b

ij =

where

5.11

5.12

5.13


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44

21212121

2121

s

x

s

y

s

x

s

y

s

y

s

x

s

y

s

x

s

x

s

y

s

y

s

x

dHHHdHdHHHdHdHH

HHHHHH

+=

=

and

22221212

12122121

12122121

21211212

s

x

s

y

s

x

s

y

s

x

s

y

s

x

s

yyy

s

y

s

x

s

y

s

x

s

y

s

x

s

y

s

xxx

s

y

s

y

s

y

s

y

s

y

s

y

s

y

s

yyx

s

x

s

x

s

x

s

x

s

x

s

x

s

x

s

xxy

dHEHdEdHEHdEdEH

dHEHdEdHEHdEdEH

dHEHdEdHEHdEdEH

dHEHdEdHEHdEdEH

+=

+=

+=

+=

Field derivatives can be obtained using the adjoint equation approach, formally derived

by, e.g., McGillivray and Oldenburg (1990). The method makes use of the reciprocity

condition of EM data (Parasnis, 1988), and a similar approach is described by deLugao

and Wannamaker (1996). In summary, derivatives at a receiver can be obtained by

determining the electric fields within the entire domain, due to fictitious horizontal

electric and magnetic dipole sources placed at the receiver. Calculating the full derivative

matrix requires finding the predicted data for four dipole sources, for each receiver, at

each frequency. The derivatives with respect to a change in the conductivity of a model

parameter are obtained by multiplying together the dipole field and the MT field, and

integrating over the parameter. To determine the electric fields in the subsurface due to

horizontal electric and magnetic dipoles at the surface, background and scattered fields

are solved for separately. The background field is calculated for a homogeneous

halfspace, of conductivity equal to that immediately beneath the dipole source, in order to

avoid singularities due to the source. Equations for dipoles in halfspaces and their

derivations can be found in, e.g., Spies and Frischknecht (1991). The scattered field can

5.14

5.15


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then be solved for using the same system matrix as used for the MT forward problem,

and following the same procedure. Once the scattered field solution is obtained, the total

fields are determined by summing scattered and background fields.

Calculating rigorous Frchet derivatives can be the most time-consuming portion

of the inversion process. Once rigorous Frchet derivatives have been calculated for the

first several iterations, they can be approximated at future iterations using Broyden's

method (Broyden, 2000). Developed in 1965, this method of approximation is used to

correct the Jacobian when the Jacobian is not expected to change significantly from one

iteration to the next. The approximation is given by

( ) ( )( ) ( )mm

mmFrrFF+= T

T

iiiii 111

where

ibi

ii

m

mmm

10

1

log==

To limit run-time, the user of the algorithm as modified herein can specify optionally

after a minimum of two model parameter updates that later iterations are calculated with

approximate Jacobians determined using Broyden's method.

5.3 Static shiftAssuming that near-surface, small-scale inhomogeneities generate frequency-

independent, real-only shifts, predicted data can be expressed in log space as;

5.16

5.17


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Gsm

Gsdd

+=

+=

)A(

R

pre

d

pre

where dpre

d is a vector of predicted data, composed of the logarithms of apparent

resistivities and phases; )A(md =Rpre are the logarithms of the predicted apparent

resistivities and phases, due to the regional structure; s is a vector of static shift

parameters; and Gis a book-keeping matrix of ones and zeros, that relates corresponding

predicted apparent resistivity values and static shifts (Sasaki, 2004). For each entry in dpre

which is an apparent resistivity, the corresponding row in G will contain a 1, with

remaining entries 0. The 1 will be in the column corresponding to the entry in swhich

contains the shift applied to that apparent resistivity and to all other apparent resistivity

entries in dpre, which are for the same receiver and mode, at different frequencies. For

entries in dprewhich are phases, the corresponding rows in Gcontain only zeros.

To solve the inverse problem for both subsurface conductivity model parameters

m and static shift s, and to constrain the solution, the following objective function is

defined:

( )

( ) ( ) ( )

( ) 2222212

2

21

2

21

2

212

2

2

smmW

mmCWmmCWmmCW

ddW

++

+++

=

aprm

prioriazCprioriayCprioriaxC

obs

d

d

zyx

pre

.

The fourth and final stabilizer is applied to the static shifts, enforcing a Gaussian

distribution with zero mean:

5.18

5.19


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( )2222 = ss .

Note that how strictly the stabilizer is enforced is determined by the regularization

parameter applied to the constraints. Increasing drives both the size of the static

shifts and their mean to zero as well as binding them more tightly to a Gaussian

distribution.

This objective function, minimized using a direct Gauss-Newton scheme, yields

the following system of equations (Sasaki, 2004):

[ ] [ ]bJs

mJJ

TT =

1

1

where

( )

+

=

=

0

;

0

0

0

0

01,111

aprm

aprzC

apryC

aprxC

obsipreid

m

zC

yC

xC

dd

z

y

x

i

z

y

x

i

mW

mCW

mCW

mCW

ddmFW

b

I

W

CW

CW

CW

GWFW

J

mm

The system of equations is solved using the LDLTmethod, as when static shifts are not

included.

5.20

5.21

5.22


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5.4 Modifications from originalModifications were made to the original program, other than restructuring and

parallelization. In the original program by Sasaki, the model parameter update is obtained

by solving a system of equations of the form

[ ] [ ]bs

mJ =

1

1

using the Gram-Schmidt method, rather than the system described in equation 5.21,

solved using the LDLTmethod. One advantage of working with the system in equation.

5.23 is that the matrix-matrix multiplication to calculate the approximate Hessian is not

required. A disadvantage is that the number of rows in Jis proportional to the number of

data as well as to the number of model parameters, while the Hessian is proportional to

the number of model parameters. The number of rows in both are proportional to the

number of parameters. Working with the LDLTin equation 5.21 allowed straightforward

application of parallelization concepts for the parameter step on a multicore workstation.

The parametric functional and stabilizers used in the modified program have been

altered from the original, which was of the form:

( ) ( )( ) ( )

2

21

2

211

2

2

apriorimaprioriobspre mmWmmmCddW +++= .

The smoothness stabilizer initially used was based on minimizing the rate of change

(Laplacian) between adjacent model parameters. This has been replaced by minimizing

5.23

5.24


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the change in adjacent model parameters with respect to the change in adjacent

parameters of the apriori model (equation. 5.5) (cf. Rodi and Mackie, 2001). The a priori

model could be set to either remain fixed using the starting model or to be updated during

the inversion. Smoothness and adherence stabilizers are decoupled in the modified

version and may refer to either fixed or updated a priori model.

The model weighting matrices used by the original program rely on a complicated

function based on model parameter sizes and depths. In the modified program, model

weighting matrices are based on the diagonal of the approximate Hessian (equation.

5.11), as recommended in Zhdanov (2002).A data weighting matrix based on data errors read from the input file was

introduced into the program. Data weights can then reflect the confidence interval of the

data. The user can exclude specific data points by flagging the data weight using a large

error. As well as carrying out joint XY and YX mode inversions, the program has been

modified to invert a single principal mode at a time, or any combination of apparent

resistivity and/or phase data.

Various other minor modifications have been made throughout the program, in an

effort to customize it as needs arose. For example, it has been modified to read in a 3-D

conductivity structure and carry out forward modeling only, a useful feature when

preparing synthetic models for testing.

5.5 Synthetic testTo demonstrate the inversion, a synthetic model based on the conceptual

geothermal system of Pellerin et al. (1996) was inverted. The model consists of a 5 m


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clay cap and 25 m reservoir, embedded in a 200 m halfspace (Figure 5.1). The

dimensions of the clay cap are 2.6 x 2.2 x 0.4 km, with the top at 300 m. The reservoir is

immediately beneath, with dimensions of 1.4 x 1.0 km x 3.3 km. XY and YX mode data

were predicted using the forward modeling option of the program, for 30 stations at 14

frequencies equally spaced logarithmically from 100 to 0.25 Hz. Errors with a Gaussian

distribution and standard deviation of 0.02 log10 (m) and 1.32 were applied to the

data.

Figure 5.1 Model used to generate synthetic data. MT stations are shown as dots in theplan view.


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The synthetic data were calculated using a finite difference mesh with 63 x 55 x

43 nodes. A minimum horizontal spacing of 100 m was used in a 4.2 km by 3.4 km

region centered on and encompassing the domain of anomalous conductivity, the spacing

doubling as distance away from this region increases (e.g. Madden and Mackie, 1989).

The edge-most nodes are 38.7 km from the central region. Vertically, node spacing

increases from 40 m immediately beneath the surface to 460 m at 6 km depth, thereafter

doubling until a maximum depth of 34.5 km is reached. Node spacing above the air-earth

interface increases from 40 m to 20.4 km. The mesh size was selected to be large enough

to satisfy the boundary conditions of the lowest frequency data while retainingsufficiently small node spacing in the near-surface zone of anomalous conductivity for

predicting higher-frequency data. For an average resistivity of 100 m, the depth of

penetration is approximately 500 m at 100 Hz and 10 km at 0.25 Hz.

The synthetic data were inverted using a similar finite difference mesh with 63 x

55 x 43 nodes, and an inversion domain containing 29 x 25 x 26 model parameters.

Model parameters have minimum dimensions of 200 m horizontally and 100 m

vertically. The smoothness stabilizer was fixed at 2 for the first model update, and

allowed to vary at later iterations. The adherence stabilizer was fixed at 1 for all

iterations. Rigorous Frchet derivatives were calculated for the starting model and

subsequent two iterations with Broyden approximations thereafter. The starting model

was a halfspace of 105 m, determined by the program based on the input apparent

resistivity data. An nRMS misfit of 1.03 was achieved after 6 model updates (Figure 5.2).

Misfit was observed to increase at the subsequent iteration and the inversion terminated.


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Predicted and observed soundings from the MT station in the southwest (#1) and

from near the center (#15) of the survey are shown in Figure 5.3. The recovered

conductivity structure agrees relatively well with the true model (Figure 5.4). The clay

cap appears to be the most prominent feature and the conductivity structure

corresponding to the reservoir is more diffuse than the true model, similar to observations

made by Pellerin et al (1996).

Static shifts with a Gaussian distribution and standard deviation of 0.12 log10(m)

were applied to the synthetic data, along with errors. These data were inverted using a

similar finite difference mesh and inversion domain as previously used. The

regularization parameter controlling the smoothness stabilizer was fixed to a value of 2

for the first model update, and allowed to vary at later iterations; that for the adherence

stabilizer was fixed to 1; and that applied to the static shift stabilizer was fixed to 0.7.

These values were determined by trial and error. Rigorous Frchet derivatives were

calculated for the starting model and subsequent two, with Broyden approximations

Figure 5.2 Convergence observed for inversion of the synthetic data.


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Figure 5.3 Observed and predicted soundings from MT stations 1 and 15.

Sounding for Central MT station (15)103- - --------------------

102 10 10Frequency (Hz)

~ 8

Observed data, XY ModeObserved data, YX ModePredicted response, XY ModePredicted response, YX Mode

n{l6 r =.-=,

40't 20

1 2 10 10uFrequency (Hz)

Sounding for SouthWest MT station (1)

10 10uFrequency (Hz)

Observed data, XY ModeObserved data, YX ModePredicted response, XY ModePredicted response, Y Mode

~ 8Cl6E ~ r ~ t 1140 I , 0_'t 20

1 2 10 1 0Frequency (Hz)


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thereafter. The starting model was a halfspace of 107 m, determined by the program

based on the input apparent resistivity data. The target nRMS misfit of 1.0 was achieved

after 7 model parameter updates (Figure 5.5). The increase in misfit for the 4th model

parameter update is attributed to the use of a nonrigorous regularization inversion

approach. Predicted and observed soundings from the MT station in the southwest (#1)

and from near the center (#15) of the survey are shown in Figure 5.6. The recovered

conductivity structure agrees relatively well with the true model (Figure 5.7) and the

predicted model from inversion of the synthetic data with errors only (Figur

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