Menke W. Geophysical Data Analysis: Discrete Inverse Theory
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8
I
Describing
Inverse
Problems
If
for instance, one were attempting to determine the density of an
object by measuring its mass and volume, there would be two data-
mass and volume (say,
d,
and
d2,
respectively)-and one unknown
model parameter, density (say,
m,).
The model would be the statement
that density times volume equals mass, which can be written com-
pactly by the vector equation
d2mI
=
d,
.
In more realistic situations the data and model parameters are
related in more complicated ways. Most generally, the data and model
parameters might be related by one or more implicit equations such as
L(d,
m>
=
0
ad,
m)
=
0
(
1.2)
f,(d,
m>
=
0
where
L
is the number of equations. In the above examples concerning
the measuring of density,
L
=
1
and
d,m,
=
d,
would constitute the
one equation of the formfi(d,
m)
=
0.
These implicit equations, which
can be compactly written
as
the vector equation
f(d,
m)
=
0,
summa-
rize what is known about how the measured data and the unknown
model parameters are related. The purpose of inverse theory is to
solve, or “invert,” these equations for the model parameters, or what-
ever kinds of answers might be possible or desirable in any given
situation.
No
claims are made either that the equations
f(d,
m)
=
0
contain
enough information to specify the model parameters uniquely or that
they are even consistent. One of the purposes
of
inverse theory is to
answer these kinds of questions and to provide means of dealing with
the problems that they imply. In general,
f(d,
m)
=
0
can consist
of
arbitrarily complicated (nonlinear) functions of the data and model
parameters. In many problems, however, the equation takes on one
of
several simple forms. It is convenient to gyve names to some of these
special cases, since they commonly arise in practical problems; and we
shall give them special consideration in later chapters.
1.1.1
IMPLICIT LINEAR FORM
The function
f
is linear in both data and model parameters and can
therefore be written as the matrix equation
1.2
The Linear Inverse
Problem
LI
f(d,
m)
=
0
=
F
9
(1.3)
where
F
is an
L
X
(M
+
N)
matrix.
1.1.2
EXPLICIT FORM
In many instances it is possible to separate the data from the model
parameters and thus to form
L
=
N
equations that are linear in the
data (but still nonlinear in the model parameters through a vector
function
g).
f(d,
rn)
=
0
=
d
-
g(m)
1.1.3
EXPLICIT
LINEAR FORM
In the explicit linear form the function
g
is also linear, leading to the
(1.5)
N
X
A4
matrix equation (where
L
=
N)
f(d,
m)
=
0
=
d
-
Gm
Using this form is equivalent to saying that the matrix
F
in Section
1.1.1
is:
F
=
[-I
G]
1.2
The
Linear
Inverse
Problem
The simplest and best-understood inverse problems are those that
can be represented with the explicit linear equation
Gm
=
d.
This
equation, therefore, forms the foundation of the study of discrete
inverse theory.
As
will be shown below, many important inverse
problems that arise in the physical sciences involve precisely this
equation. Others, while involving more complicated equations, can
often be solved through linear approximations.
The matrix
G
is called the data kernel, in analogy to the theory of
integral equations, in which the analogs of the data and model parame-
ters are two continuous functions
d(x)
and
m(x),
where
x
is some
independent variable. Continuous inverse theory lies between these.
two extremes, with discrete
data
but
a
continuous model function:
10
1
Describing Inverse Problems
Discrete inverse theory:
M
di
=
2
Gumj
j-
1
Continuous inverse theory:
di
=
G,(x)rn(x)
dx
I
Integral equation theory:
(1.7a)
(I
.7b)
(1.7c)
The main difference between discrete inverse theory, continuous
inverse theory, and integral equation theory is whether the model
rn
and data dare treated as continuous functions or discrete parameters.
The data
di
in inverse theory are necessarily discrete, since inverse
theory is concerned with deducing information from observational
data, which always has a discrete nature. Both continuous inverse
problems and integral equations can be converted to discrete inverse
problems by approximating the integral as a summation using the
trapezoidal rule or some other quadrature formula.
1.3
Examples
of
Formulating
1.3.1
Suppose that Ntemperature measurements
Ti
are made at depths
zi
in
the earth. The data are then a vector
d
of
N
measurements
of
temperature, where
d
=
[TI,
T2,
T3,
. .
.
,
TJT.
The depths
zi
are
not, strictly speaking, data. Instead, they provide some auxiliary infor-
mation that describes the geometry of the experiment. This distinction
will be further clarified below.
Suppose that we assume a model in which temperature
is
a linear
function of depth:
T
=
a
+
bz.
The intercept
a
and slope
b
then form
Inverse Problems
EXAMPLE
1:
FITTING
A
STRAIGHT LINE
1.3
Examples
of
Formulating Inverse Problems
11
the two model parameters
of
the problem,
m
=
[a,b]*. According to
the model, each temperature observation must satisfy
T
=
a
+
bz:
T,
=
u
+
bzl
T2
=
a
+
bz2
T,=
a
4-
bz,
These equations can be arranged as the matrix equation
Gm
=
d:
(1.9)
1.3.2
EXAMPLE
2:
FITTING
A PARABOLA
If
the model in Example
1
is
changed
to
assume a quadratic varia-
tion
of
temperature with depth of the form T
=
a
+
bz
+
czz,
then a
new model parameter is added to the problem,
m
=
[a,b,cIT. The
number
of
model parameters
is
now M
=
3.
The data are supposed to
satisfy
T,
=
a
+
bzl
+
CZ:
T2
=
a
+
bz2
+
CZ;
(1.10)
12
1
Describing Inverse Problems
These equations can be arranged into the matrix equation
1
21
z:
1
z2
2:
.. .
.. .
..
.
ZN
d‘,
“I
(1.11)
This matrix equation has the explicit linear form
Gm
=
d.
Note
that, although the equation is linear in the data and model parameters,
it is not linear in the auxiliary variable
z.
The equation has a very similar form to the equation of the previous
example, which brings out one of the underlying reasons for employ-
ing matrix notation: it can often emphasize similarities between su-
perficially different problems.
1.3.3
EXAMPLE 3: ACOUSTIC TOMOGRAPHY
Suppose that a wall is assembled from a rectangular array of bricks
(Fig.
1.1)
and that each brick is composed of a different type of clay. If
the acoustic velocities of the different clays differ, one might attempt to
Fig.
1
.I.
The travel time
of
acoustic waves (dashed lines) through the rows and columns
of
a square array
of
bricks is measured with the acoustic source
S
and receiver
R
placed
on the edges
of
the square. The inverse problem is to infer the acoustic properties
of
the
bricks (which are assumed to be homogeneous).
1.3
Examples
of
Formulating Inverse Problems
13
distinguish the different kinds
of
bricks by measuring the travel time
of
sound across the various rows and columns
of
bricks in the wall. The
data in this problem are
N
=
8
measurements
of
travel times,
d
=
[TI,
TI,
T3,
. .
.
,
T,IT.
The model assumes that each brick
is
composed
of
a
uniform material and that the travel time
of
sound across each brick
is proportional to the width and height
of
the brick. The proportion.
ality factor is the brick’s
slowness
si,
thus giving
hf=
16
model
parameters
rn
=
[sI,
s,,
s3,
. . .
,
s,6IT,
where the ordering
is
according to the numbering scheme
of
the
figure
as
(1.12)
and the matrix equation is
‘
=h
1
!
0001000100010001
1111000000000000
0000111100000000
................
................
................
SI
s2
(1.13)
Here the bricks
are
assumed to be
of
width
and
height
h.
1.3.4
EXAMPLE
4:
X-RAY
IMAGING
Tomography is the process of forming images of the interior of an
object from measurements made along rays passed through that object
(“tomo” comes from the Greek word for “slice”). The Computerized
Axial Tomography (CAT) scanner is an x-ray imaging device that has
revolutionized the diagnosis of brain tumors and certain other medical
conditions. The scanner solves an inverse problem for the x-ray opacity
14
1
Describing Inverse Problems
of body tissues using measurements of the amount of radiation ab-
sorbed from many criss-crossing beams of
x
rays (Fig. 1.2).
The basic physical model underlying this device is the idea that the
intensity
of
x
rays diminishes with the distance traveled, at a rate
proportional to the intensity
of
the beam, and an absorption coeffi-
cient that depends on the type of tissue:
dZ/ds
=
-
c(x,~)Z
(1.14)
Here
Z
is the intensity
of
the beam,
s
the distance along the beam, and
c(x,y)
the absorption coefficient, which varies with position. If the
x-ray source has intensity
I,,
then the intensity at the ith detector is
(
1.15a)
(
1.1
5b)
Note that
Eq.
(1.15a) is a nonlinear function of the unknown absorp-
tion coefficient
c(x,y)
and that the absorption coefficient varies contin-
uously along the beam. This problem is a nonlinear problem in contin-
uous inverse theory. However, it can be linearized by taking the
logarithm
of
both sides or, for small net absorption, by approximating
the exponential with the first two terms in its Taylor series expansion.
We
convert this problem to a discrete inverse problem of the
form
Zi
=
I,,
exp
(-
/
c(x,y)
ds)
buuni
In
I,,
-
In
Zi
=
/-ic(x,Y)
ds
Fig.
1.2.
An idealized x-ray tomography device measures x-ray absorption along lines
(dashed) passing through body. After a set
of
measurements is made, the source
S
and
receivers
R,,
R,,
R,,
.
. .
are rotated and the measurements are repeated,
so
data
along many crisscrossing lines are collected. The inverse problem is to determine the
x-ray opacity as a function
of
position in the body.
1.3
Examples
of
Formulating Inverse Problems
15
Gm
=
d.
The first step is to assume that the net absorption of
x
rays is
small; this allows the exponential to be approximated by the first two
terms in its Taylor series, exp(-x)
=
1
-
x.
The second step is to
assume that the continuously varying absorption coefficient can be
adequately represented by a grid of many small square boxes, each of
which has a constant absorption coefficient. With these boxes num-
bered
1
-
M,
the model parameters are then the vector
m
=
[cI
,
c,,
c,,
.
. .
,
cMIT.
The integral can then be written as the sum
(1.16)
Here the data
AZi
represent the differences between the x-ray intensi-
ties at the source and at the detector, and
Asij
is the distance the ith
beam travels in the jth box.
The inverse problem can then be summarized by the matrix equa-
tion
(1.17)
Since each beam passes through only a few of the many boxes, many
Asi;s
are zero, and the matrix is very sparse.
1.3.5
EXAMPLE
5:
GAUSSIAN
CURVE
FITTING
Not every inverse problem can be adequately represented by the
discrete linear equation
Gm
=
d.
Consider, for example, an experi-
ment in which the intensity of the
x
rays scattered from a crystal lattice
is measured as a function of Brag angle. Near one of the diffraction
peaks the x-ray intensity
I(@
varies as
(1.18)
Here
8
is the Bragg angle,
8
the Bragg angle of the peak‘s maximum
(Fig. 1.3),
A
the amplitude of the peak, and
o
a measure of its width.
16
I
Describing Inverse Problems
Bragg
angle
8
Fig.
1.3.
A
hypothetical x-ray diffraction experiment. Measurements
of
x-ray intensity
I
against Bragg angle
0
form
a peak that is assumed to have a Gaussian shape. The
inverse problem is to determine the peak area, width, and position.
In a typical experiment
I(@
is measured at many different
8’s,
and
the model parameters
A,
8,
and
0
are to be determined. The data and
model are therefore related by the nonlinear explicit equation
d
=
g(m).
Furthermore, since the argument
of
the exponential is large
in the vicinity of the peak, the problem cannot be linearized.
1.3.6
EXAMPLE
6:
FACTOR ANALYSIS
Another example
of
a nonlinear inverse problem is that
of
deter-
mining the composition of chemical end members on the basis
of
the
chemistry of a suite of mixtures of the end members. Consider a
simplified “ocean” (Fig.
1.4)
in which sediments are composed of
mixtures
of
several chemically distinct rocks eroded from the conti-
nents. One expects the fraction of chemical
j
in the ith sediment
sample
S,
to
be related to the amount
of
end-member rock in sediment
ocean
sediment
Fig.
1.4.
Sediment on the floor
of
this idealized ocean is a simple mixture
of
rocks
eroded
from
several sources
S,.
The inverse problem is to determine the number and
composition
of
the sources.
1.4
Solutions
to
Inverse
Problems
17
sample
z
(c,k)
and to the amount of the Jth chemical in the end-
member rock
(
Fk,)
as
1
sample
]
=
c.
[
amount of
]
[end member
composition
end
end member composition
members
P
s,=
2
elk&,
Or
s=c
F
(1.19)
In a typical experiment, the number
of
end members
p,
the end-
member composition
F,
and the amount of end members in the
samples
C
are all unknown model parameters. Since the data
S
are on
one side of the equations, this problem is also of the explicit nonlinear
type. Note that basically the problem is to factor a matrix
S
into two
other matrices
C
and
F.
This factoring problem is a well-studied part of
the theory of matrices, and methods are available to solve it.
As
will be
discussed in Chapter
10,
this problem (which is often called
factor
analysis)
is very closely related to the algebraic eigenvalue problem of
linear algebra.
k=
I
1.4
Solutions to Inverse Problems
We shall use the terms “solution” and “answer” to indicate broadly
whatever information we are able to determine about the problem
under consideration.
As
we shall see, there are many different points of
view regarding what constitutes a solution to an inverse problem. Of
course, one generally wants to know the numerical values of the model
parameters (we call this kind of answer an estimate of the model
parameters). Unfortunately, only very infrequently can an inverse
problem be solved in such a way as to yield this lund of exact
information. More typically, the practitioner of inverse theory is
forced to make various compromises between the kind of information
he or she actually wants and the lund of information that can in fact be
obtained from any given dataset. These compromises lead to other
kinds of “answers” that are more abstract than simple estimates of the
model parameters. Part of the practice of inverse theory is the identify-
ing of what features of a solution are most valuable and malung the
compromises that emphasize these features. Some of the possible
forms an “answer” to an inverse problem might take are described
below.