3-D inversion of induced polarization data3-D Inversion of IP Data

TLDR: In this paper, an algorithm for inverting induced polarization (IP) data acquired in a 3D environment is presented based upon the linearized equation for the IP response, and the inverse problem is solved by minimizing an objective function of the chargeability model subject to data and bound constraints.

Abstract: We present an algorithm for inverting induced polarization (IP) data acquired in a 3-D environment. The algorithm is based upon the linearized equation for the IP response, and the inverse problem is solved by minimizing an objective function of the chargeability model subject to data and bound constraints. The minimization is carried out using an interior-point method in which the bounds are incorporated by using a logarithmic barrier and the solution of the linear equations is accelerated using wavelet transforms. Inversion of IP data requires knowledge of the background conductivity. We study the effect of different approximations to the background conductivity by comparing IP inversions performed using different conductivity models, including a uniform half-space and conductivities recovered from one-pass 3-D inversions, composite 2-D inversions, limited AIM updates, and full 3-D nonlinear inversions of the dc resistivity data. We demonstrate that, when the background conductivity is simple, reasonable IP results are obtainable without using the best conductivity estimate derived from full 3-D inversion of the dc resistivity data. As a final area of investigation, we study the joint use of surface and borehole data to improve the resolution of the recovered chargeability models. We demonstrate that the joint inversion of surface and crosshole data produces chargeability models superior to those obtained from inversions of individual data sets.

Figures

FIG. 1. A synthetic 2-D conductivity model is shown in (a). Surface electrodes are spaced at 10 m over the interval x = (-100, 100) m. Current is input at each electrode site in turn, and potentials are observed at the remaining 20 sites. The pole-pole potentials <P<T are converted to apparent conductivity ITa = I1(211'r<P<T)' where r denotes the distance between current and potential electrodes and are plotted in (b). The apparent conductivity value is plotted midway between the current and potential electrodes at a (pseudo) depth of z = 0.86r. The grey scales indicate conductivity in mS/m. The synthetic chargeability model is given in (c) and the apparent chargeability, plotted in the pseudosection format is shown in (d). The grey scale indicates chargeability in percent.

FIG. 4. Inversion results using Method I. The true and apparent chargeability models are shown in (a) and (b). An inversion that uses sensitivities obtained from the inverted conductivity in Figure 3(c) produces the chargeability model in (c). Discrepancies between (a) and (c) that are related to aspects of the subspace algorithm are inferred from (d). That panel shows the results of inverting synthetic data obtained by multiplying the sensitivity matrix J by the true chargeability model in (a) and inverting the resultant data. The chargeability models in (e) and (f) are the inversion results by respectively using the sensitivity matrix developed from the true conductivity model and the sensitivity matrix from a half-space.

FIG. II. The dipole-dipole apparent chargeability phase pseudosection ~s gi,:en in (a). Va~ues are in f!lrad. Three inversions carried out at successively smaller misfit levels are grven 10 (b)-(d). Chi-squared misfits, assuming an initi~ standard deviation of each datum of 0.13 mrad, are, respectively, 1500, 1400,and 1330for the three inversions.

FIG. 5. Inversion results from Method I using m = In TJ as the variable in the objective function . The recovered logarithmic chargeability is shown in (a) and it has been replotted in linear form in (b). For comparison, the inversion carried out using m = TJ and setting U s = 0 is shown in (c). Thus (a) and (c) are a direct comparison of the results of using the same objective function form but applying it to m = In 1] and to m = TJ, respectively.

FIG. 10. A dipole-dipole apparent conductivity pseudosection is shown in (a). The inverted conductivity is presented in (b). Gray scales indicate the conductivity in mS/m.

FIG. 6. Inversion results from Method II: The recovered conductivity structures obtained by inverting <!>IT and <!>Tj are shown in (a) and (b), respectively. The difference is shown in (c), and the recovered chargeability is shown in (d). The gray scales indicate conductivity in mSlm and chargeability in percent.

Full text

GEOPHYSICS, VOL. 59,
NO.9
(SEPTEMBER
1994);
P. 1327-1341, II FIGS.
Inversion
of
induced
polarization
data
Douglas W. Oldenburg* and Yaoguo Li*
ABSTRACT
We develop three methods to invert induced polar-
ization (IP) data. The foundation for our algorithms is
an assumption that the ultimate effect of chargeability
is to alter the effective conductivity when current is
applied. This assumption, which was first put forth by
Siegel and has been routinely adopted in the literature,
permits the IP responses to be numerically modeled by
carrying out two forward modelings using a DC resis-
tivity algorithm. The intimate connection between DC
and IP data means that inversion of IP data is a
two-step process. First, the DC potentials are inverted
to recover a background conductivity. The distribu-
tion of chargeability can then be found by using any
one of the three following techniques:
(I) linearizing
the IP data equation and solving a linear inverse
problem, (2) manipulating the conductivities obtained
after performing two DC resistivity inversions, and (3)
INTRODUCTION
Induced Polarization (IP) data are routinely collected in
mineral exploration surveys and they are also finding their
niche in environmental surveys (Barker,
1990).
Excellent
reviews on the IP method and case histories can be found in
Sumner (1976), Bertin and Loeb
(1976),
Fink et al.
(1990),
and Ward
(1990).
The difficulty with interpreting IP data is
the lack of flexible, efficient, and robust inversion algo-
rithms. This is reflected in the remark in Hohmann
(1990)
regarding the availability of forward algorithms and the lack
of general inversion techniques for the IP method. Conse-
quently, much of today's interpretation is still carried out by
working with pseudosections. Yet only in very simplistic
circumstances will the images on the pseudosections emu-
late geologic structure, and consequently, inferences made
about the substructure directly from the data are often
solving'a nonlinear inverse problem. Our procedure for
performing the inversion is to divide the earth into
rectangularprisms and to assume that the conductivity
(J"
and chargeability
TJ
are constant in each cell. To emulate
complicatedearth structure we allowmany cells, usually
far morethan there are data. The inverse problem, which
has many solutions, is then solved as a problem in
optimization theory. A model objective function is de-
signed,and a "model" (either the distribution of
(J" or
TJ)
is soughtthat minimizesthe objectivefunction subject to
adequately fitting the data. Generalized subspace meth-
odologiesare used to solve both inverse problems, and
positivity constraints are included. The IP inversion
procedures we design are generic and can be applied to
I-D, 2-D,or 3-Dearth modelsand withany configuration
of current and potential electrodes. We illustrate our
methods by inverting synthetic DC/IP data taken over a
2-D earth structure and by inverting dipole-dipole data
taken in Quebec.
incorrect. As an illustration, we present a fairly simple
example of DC/IP data taken over a 2-D earth structure. The
true conductivity and chargeability models and the pseu-
dosection plots obtained by carrying out a pole-pole DC/IP
survey are shown in Figure 1. Except for the region near the
surface, there is little compelling evidence in the IP pseu-
dosection to indicate a chargeable body at depth.
Methods for inverting IP data do exist but literature on this
subject is sparse. In defining IP data, most authors adopt a
presentation given in Siegel
(1959)
that the ultimate effect of
a chargeable body is to alter its effective conductivity. As
such, the IP and DC resistivity problems are intimately
linked, and the inversion of IP data is a two-step process. In
the first stage, the DC potentials are inverted to recover the
background conductivity
(J" b : The second step accepts (J" b as
the true conductivity of the medium and attempts to find a
chargeability that satisfies the data. This is usually done by
Presented at the 63rd Annual International Meeting, Society of Exploration Geophysicists. Manuscript received by the Editor AprilS, 1993;
revised manuscript received March 1, 1994.
*Geophysical Inversion Facility, Department of Geophysics and Astronomy, The University of British Columbia, Vancouver, BC V6T 124,
Canada.
© 1994Society of Exploration Geophysicists. All rights reserved.
1327
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1328
Oldenburg
and Li
linearizing the equations about ITb to produce a system of
equations that can be solved for the chargeability distribu-
tion.
Early algorithms for inverting DC and IP data generally
parameterized the earth model into a relatively smallnumber
of blocks and kept the same parameterization for inverting
the DC and IP data (Pelton et al.,
1978;
Sasaki,
1982;
Rijo,
1984).
Overdetermined systems of equations were solved
and algorithm convergence was judged on the basis of data
misfit alone. There are practical difficulties that arise with
this approach. The electrical conductivity structure of the
earth is complicated and rarely does a representation by a
few blocks adequately represent the true distribution of this
physical property. Also, anomalous regions of high or low
conductivity do not necessarily correspond to regions of
highchargeability. If only a few blocks are used, keeping the
same parameterization may preclude the possibility of find-
inga meaningfulsolution. In addition, the restriction of using
only a few model cells does not allow insight about the
nonuniqueness that is inherent in the inverse problem. The
difficulties with respect to parameterization can be overcome
by discretizing the earth into a large number of cells. The
inverse problem is then solved as an optimization problem
where an objective function of the model is minimized
subject to adequately fittingthe data. An archetypal example
applied to a geophysical inverse problem was presented by
Constable et al.
(1987).
LaBrecque
(1991)
has applied this
methodology to carry out a 2-D inversion of IP data in a
cross-borehole tomography experiment, and Beard and
Hohmann
(1992)
have provided an approximate inversion of
IP data that is valid when resistivity contrasts are small.
In this paper, we use Siegel's
(1959)
formulation and
develop three methods by which to invert IP data. In the first
method, we assume that the chargeability is small and
linearize the equations. The technique is similar to that
presented in LaBrecque
(1991),
but we use a more general
model objective function, we incorporate a subspace meth-
odology to bypass the large computations normally required
to invert the full matrix system, and we work with charge-
abilities directly in the inverse problem rather than with their
logarithms. The second method makes use of formal map-
pings connecting DC/IP voltages with a conductive and
chargeable earth. Two DC resistivity inversions of different
data are carried out, and the chargeability is obtained by
manipulatingthe recovered conductivities. The third method
makes no assumptions about the size of
TJ
and solves a
nonlinear inverse problem to recover the chargeabilities.
The resultant algorithm is essentially the same as that used
to invert the DC resistivity data. All of our techniques are
applicable to any dimension of earth structure and to any
configurationof electrodes. Our paper begins by definingthe
forward mappingfor the IP data. Next, we introduce the 2-D
synthetic example and invert pole-pole DC potentials to
recover the background conductivity. This is presented in
some detail because the subspace method used to invert the
DC data also plays the dominant role in inverting IP data.
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FIG.
1. A synthetic 2-D conductivity model is shown in (a). Surface electrodes are spaced at 10m over the interval x =
(-100,
100) m. Current is input at each electrode site in turn, and potentials are observed at the remaining 20 sites. The pole-pole
potentials
<P<T
are converted to apparent conductivity ITa =
I1(211'r<P<T)'
where r denotes the distance between current and
potential electrodes and are plotted in (b). The apparent conductivity value is plotted midway between the current and potential
electrodes at a (pseudo) depth of
z =
0.86r.
The grey scales indicate conductivity in mS/m. The synthetic chargeability model
is given in (c) and the apparent chargeability, plotted in the pseudosection format is shown in (d). The grey scale indicates
chargeability in percent.
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Inversion
of
Induced
Polarization
Data
1329
Three methods for inverting IP data are explained and
illustrated with the synthetic example. We address the issues
of positivity, of how well the background conductivity needs
to be known, and practical considerations pertaining to the
data to be inverted. We also examine the merits of using
TJ
or
In
TJ
as the variable in the inversion and show how minimiz-
ing different objective functions can be useful in hypothesis
testing and in exploring nonuniqueness. A field data set of
dipole-dipole data is inverted, and the paper concludes with
a discussion about the relative merits of the inversion
algorithms.
and the boundary conditions are
a<l>cr/an
= 0 at the earth's
surface and
<l>cr
~
0 as Ir - rsI
~
00,
where rs denotes the
location of the current electrode.
If the ground is chargeable, then the potential
<1>'1]'
re-
corded after the constant current is applied, will differ from
<l>cr'
According to Siegel's formulation, the effect of the
chargeability of the ground is modeled by using the DC
resistivity forward mapping
?:F
de but with the conductivity
replaced by
0"
=
0"(1
-
TJ).
Thus
(3)
FORWARD MODEL
or
or
The IP datum, which we refer to as apparent chargeability, is
defined by
(6)
(5)
Equation (6) shows that the apparent chargeability can be
computed by carrying out two DC resistivity forward mod-
elings with conductivities
0"
and
0"(1
-
TJ).
Equation (6) defines the forward mapping for the IP data.
The data can be inverted by: (1) linearizing this equation and
solving a linear inverse problem, (2)introducing a formal DC
inverse operator
?:F;;'}
and obtaining
TJ
by manipulating the
conductivity models obtained after applying
?:F
icl to
<1>'1]
and
to
<l>cr'
or (3) solving a nonlinear inverse problem that
involves linearization but iterates until data predicted from
equation (6) are in agreement with the observations. Irre-
spective of the method, the inversion of the IP data requires
that the potentials
<l>cr
be inverted to recover the background
conductivity. We address this issue first.
Complete understanding of the microscopic phenomenon
that result in the macroscopic IP response has not been
achieved. Here we adopt a macroscopic representation of
the physical property governing the IP response that was put
forth in Siegel (1959).Basically, he introduces a macroscopic
physical parameter called chargeability to represent all of the
microscopic phenomena. As such, our earth model is de-
scribed by the two quantities: conductivity
O"(x, y, z) and
chargeability
TJ(x, Y, z). Both are positive, but while
conductivity varies over many orders of magnitude, charge-
ability is confined to the region [0,
1). We note that Siegel's
model refers only to the volumetrically distributed polariza-
tion and does not apply to highly conductive targets in which
surface polarization dominates the IP effect. Fortunately,
this is not an important limitation for the majority of practi-
cal situations.
A typical IP experiment involves inputting a current
I to
the ground and measuring the potential away from the
source. In a time-domain system, the current has a duty
cycle that alternates the direction of the current and has
off-times between the current pulses at whichthe IP voltages
are measured. A typical time-domain signature is shown in
Figure 2. In that figure,
<l>cr
is the potential that is measured
in the absence of chargeability effects. This is the "instan-
taneous" potential measured when the current is turned on.
In mathematical terms
(1)
INVERSION OF DC POTENTIALS
FIG.
2. Definition of the three potentials associated with the
IP survey.
where the forward mapping operator
?:F
de is defined by the
equations
Equation (1) is a nonlinear relationship between the ob-
served potentials
<l>cr
and the conductivity
0".
The goal of the
inverse problem is to find the function
0"
which gave rise to
those observations. There have been many attacks on this
problem.
For
the example problem in this paper where the
structure is presumed to be 2-D, we use the subspace
inversion method given in Oldenburg et al. (1993). This
methodology will be used for inverting both DC data and IP
data, and we therefore outline essential details of the ap-
proach.
Let the data be denoted generically by the symbol
d and
the model by
m. To carry out forward modeling to generate
theoretical responses, and also to attack the inverse prob-
lem, we divide our model domain into
M rectangular cells
and assume that the conductivity is constant within each
cell. Our inverse problem is solved by finding the vector
m
= {m I , m
2,
•••
, m M} which adequately reproduces the
observations
do = (dOl' d
o
2,""
dON)'
(2)
Time
T
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(7)
(9)
Oldenburg
and Li
The solution of equation (11) requires that yT
Oy
~
Wm +
I1J
T.vV
be inverted and the numerical efficiency of the
inversion is therefore realized since this is a
q x q matrix. At
each iteration in the inversion, we desire a model perturba-
tion that minimizes
Iji
m and alters
Iji
d so that it achieves a
specific target value
Iji~.
To prevent the buildup of unnec-
essary roughness, the target misfit begins at an initial value
(usually a fraction of the misfit generated by the starting
model) and decreases with successive iterations towards a
final value selected by the interpreter. Convergence is
reached when the data misfit reaches this final target and no
further reduction in the model norm is obtained with succes-
sive iterations.
In the IP survey carried out to produce Figure 1, surface
electrodes are located every
10m
in the interval x =
(-
100,
100 m). Each of the
21
electrode positions can be activated
as a current site and when it is, electric potentials are
recorded at the remaining electrodes. The observed data set
consists of 420 potential values, each of which has been
contaminated by Gaussian noise having a standard deviation
equal to
5 percent of the true potential. The data are
generated using a finite-difference code (McGillivray,
1992),
and the mesh, used both for forward modeling and for the
inversion, consists of
1296
elements.
Because of the nonuniqueness of the inverse solution,
the character of the final model is heavily influenced by the
model objective function. Our choice for
Iji
m is guided by the
fact that we often wish to find a model that has minimum
structure in the vertical and horizontal directions and at the
same time is close to a base model
mo. This model, because
it is "simple" in some respect, may well be representative of
the major earth structure; however, other earth models
might be closer to reality. Also, even if a geologically
reasonable model has been found, it is insightful to generate
different models that fit the data. This can provide under-
standing about whether features observed in the constructed
model are required by the observations or if they are merely
the result of minimizing a particular model objective func-
tion. An objective function that has the flexibility to accom-
plish these goals is
q
(lm
=
L:
aivi
==
ya.
i=
I
1330
The inverse problem is posed as a standard optimization:
minimize
!jJm(m,
mo) =
IIWm(m
- mo)!!2
subject to
ljid(d, do) = IIWd(d - d
o)ll2
= Iji'd.
In equation (7), mo is a base model and Wm is a general
weighting matrix that is designed so that a model with
specific characteristics is produced. The minimization of the
model objective function
!jJ
m yields a model that is close to
mo with the metric defined by Wmand so the characteristics
of the recovered model are directly controlled by these two
quantities. The choice of
Iji
mis crucial to the solution of the
inverse problem, but we defer the details until later. W
d is a
datum weighting matrix. We shall assume that the noise
contaminating the
jth
observation is an uncorrelated
Gaussian random variable having zero mean and standard
deviation
Ej'
As such, an appropriate form for the N x N
matrix is
w,
= diag I
l/s
, ,
"',
liEN}' Wirh this choice.w,
is the random variable distributed as chi-squared with N
degrees of freedom. Its expected value is approximately
equal to
N and accordingly,
Iji~,
the target misfit for the
inversion, should be about this value. The appropriate
objective function to be minimized is
Iji(m)
= ljim(m, mo) + 11(ljid(d, do)
-lji'd)'
(8)
where
11
is a Lagrange multiplier.
The inverse problem is nonlinear and is generally attacked
by linearizing equation
(8) about the current model m(n),
differentiating with respect to parameters mj and solving the
resultant
M x M system of equations for a perturbation
om.
This can be computationally intensive when M becomes
large and hence the use of the subspace methods. In the
subspace method, the
"model"
perturbation is restricted to
be a linear combination of search vectors
{Vi}
i = I, q. Thus
The linearized objective function, obtained by substituting
min) +
ya
into equation (8), is
(13)
(12)
+ JJ
{axwxC(m
a~
mo)f
+
azwz(a(m
~
m
o
»)
2}
dx dz:
In equation (12), the functions
w"
w
x
'
W
z
are specified by
the user, and the constants
a"
ax,
a
z
control the impor-
tance of closeness of the constructed model to the base
model
mo and control the roughness of the model in the
horizontal and vertical directions. The discrete form of
equation (12) is
Ijim=(m-mo)T{a
s
W;Ws
+a
x
W;Wx
+ a
z
W;Wz}(m-mo)
11{ljid
+
'Y~Ya
+
~
aTyTW~Wdya
-lji'd}'
(10)
where
'Y
m = Vm
Iji
m and
'Y
d = V
m!jJ
d are gradient vectors,
and
Vmis the operator
(alam),
...
, alamM)T. In equation
(10),
ljim
is understood to be
!jJm(m(n)
, mO)'!jJd is
!jJd(d(n)
,
do), and the sensitivity matrix J has elements Jij =
ad,.(m(n»/amj' Differentiating equation (10) with respect to
a and setting the resultant equations equal to zero yields
The solution of this system requires that a line search be
carried out to find the value of the Lagrange multiplier
11
so
that a specific target value
Iji~
is achieved. This involves an
initial guess for
11,
solving equation (11) by SVD for the
vector
a,
computing the perturbation
om,
carrying out a
forward modeling to evaluate the true responses and misfit,
and adjusting
11.
The estimation of an acceptable value of
11
typically requires three or four forward modelings.
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Inversion of Induced Polarization Data
1331
12.0
10
.0
17••
6
.32
14 .6
4.'"
20 .8
5.75
e
.n
(17)
(16)
(14)
(15)
us .
100 . -
.e.
.e.
.e.
-20
. 20 .
X (m)
-2e.
21
.
X (m)
-ze. 2e.
X (m)
-b0
.
-
M"
M
TJa;
L JijTJj, i = 1,
...
,
N,
j = I
1.70
2.14
4
.10
1
.37
g
.lg
15 .3 23 .8 37 .0 57 .4
(J'j
a<j>
a In
<j>
TJa
= - L - -
TJj
= - L
--
TJj'
j
<j>
aaj j a In
aj
o.
20 .
]
'0
.
N
.0
.
eo.
llUL
-1''''.
o.
20.
]
N
'0
.
0
'tl
.0
.
"
~
~
0..
eo.
,
,,,,
.
·10
0.
o.
20.
]
.e.
N
. 0 .
eo.
Substituting into equation (6) yields
Thus the ith datum is
M
a<j>
<j>T]
=
<!>(a
- TJa) =
<j>(a)
- L - TJjaj +
H.O.T.
j = I aaj
This can be written approximately as
We shall develop three procedures for inverting IP data.
Method I assumes that the chargeability
TJ
-e:
1.0.
Equation (6) is linearized about the conductivity
a b recov-
ered from the inversion of the DC potentials, and a linear
inverse problem is solved to recover
TJ.
In Method II we
rearrange the mapping defined by equation (6) and compute
TJ
by manipulating the conductivities obtained by performing
two DC resistivity inversions. In Method III we present a
general approach that does not require that chargeability be
small and solves the IP problem as a nonlinear inverse
problem.
INVERSION OF IP DATA
and this defines the matrix Wm in equation (7).
For
the
inversion of the synthetic data, we have set
w
s'
W
x'
W z
equal to unity, and
Sv
in equation (13)
Ws
is a diagonal
matrix with elements
AxAz, where Axis the length of the
cell and
Az is its thickness;
Wx
has elements ±YAz/ox,
where ox is the distance between the centers of horizontally
adjacent cells; and
Wz has elements ± YAx/oz, where
OZ
is
the distance between the centers of vertically adjacent cells.
In defining the objective function, we can choose
m to be a
or In a. Because earth conductivities typically vary
over
many orders of magnitude, we choose m = In a. This choice
also ensures that the recovered conductivity will remain
positive. In addition, we specify
as
= .0002, ax = 1.0,
a
z
= 1.0 and the reference model mo corresponding to a
background conductivity of ao
= 5.0 mS/m.
Our
search vectors for the subspace inversion are ob-
tained by partitioning
IjJ
d into
data
sets associated with
individual current electrodes. Steepest descent vectors as-
sociated with these 21
data
objective functions are combined
with the steepest descent vector for the model gradient and
a constant vector to form a basis for the subspace. The
inversion begins with a halfspace of conductivity 5.0 mS/m.
At every iteration we ask for a 50 percent decrease in the
misfit objective function until a final misfit
IjJd
= N is
achieved, where
N is the number of data. A line search using
forward modeling ensures that this is achieved, or in cases
where it is not achievable, the line search is used to find that
value of
f.lwhich provides the greatest decrease in the misfit.
Once the target misfit has been obtained, the line search
ensures
that
the misfit remains at the target value, and hence
subsequent iterations alter only
IjJ
m'
The desired misfit
IjJd
= 420 is achieved by iteration 13, but a few more
iterations are carried out until no further decrease in the
model objective function is obtained. The model obtained at
iteration 20 is shown in Figure 3c.
It
compares favorably
with the true model in Figure 3a. The surface variation is
well defined and so is the conductive anomaly in the center
of the figure. There is no manifestation of the resistive ledge
at the bottom left of the picture, but this might have been
expected since its depth is 67 m and the electrodes span the
region
(-100,
100) m.
Method I: Linearization of the data equations
2
.56
3.4 5 4 .6 5 6 . 25 8 .41 1
1.3
15 .2 20. 5 27 .8
Let
the earth model be partitioned into M cells and let
TJi
and a i denote the chargeability and electrical conductivity of
the ith cell. Linearizing the potential
<j>T]
about the conduc-
tivity model
a yields
FIG.
3. The true and apparent conductivities are plotted
respectively in (a) and (b). Inversion
of
the DC potentials
yields the recovered model in (c). The grey scales indicate
conductivity in mS/m.
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