新编《参数估计与逆问题》教材：理论与应用深度解析

inverse

problems

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To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal business use only by the author(s), editor(s),

reviewer(s), Elsevier and typesetter diacriTech. It is not allowed to publish this proof online or in print. This proof copy is the copyright property of the publisher

and is conﬁdential until formal publication.

“Aster Ch01-9780123850485” — 2011/11/8 — page 11 — #11

1.3. Examples of Inverse Problems 11

either require speciﬁc strategies or, more commonly, can by solved by iterative methods

that may rely on local linearization.

•

Example 1.6

A classic pedagogical inverse problem is an experiment in which an angular distribution

of illumination passes through a thin slit and produces a diffraction pattern, for which

the intensity is observed (Figure 1.6; [141]).

The data, d (s), are measurements of diffracted light intensity as a function of the

outgoing angle −π/2 ≤ s ≤ π/2. Our goal is to ﬁnd the intensity of the incident light

on the slit, m(θ), as a function of the incoming angle −π/2 ≤ θ ≤ π/2.

The forward problem relating d and m can be expressed as the linear mathematical

model,

d(s) =

π/2

−π/2

(cos(s) +cos(θ ))



sin(π(sin(s) +sin(θ)))

π(sin(s) +sin(θ))



m(θ)dθ . (1.26)

•

Example 1.7

Consider the problem of recovering the history of groundwater pollution at a source

site from later measurements of the contamination at downstream wells to which the

contaminant plume has been transported by advection and diffusion (Figure 1.7). This

Slit

d(s)

m(θ)

Figure 1.6 The Shaw diffraction intensity problem (1.26).

To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal business use only by the author(s), editor(s),

reviewer(s), Elsevier and typesetter diacriTech. It is not allowed to publish this proof online or in print. This proof copy is the copyright property of the publisher

and is conﬁdential until formal publication.

“Aster Ch01-9780123850485” — 2011/11/8 — page 13 — #13

1.3. Examples of Inverse Problems 13

acoustic), or source intensity (e.g., positron emission). Here, we will use examples from

seismic tomography. Although tomography problems originally involved determining

models that were two-dimensional slices of three-dimensional objects, the term is now

commonly used in situations where the model is two- or three-dimensional. Tomog-

raphy has many applications in medicine, engineering, acoustics, and Earth science.

One important geophysical example is cross-well tomography, where seismic sources

are installed in a borehole, and the signals are received by sensors in another borehole.

Another example is joint earthquake location/velocity structure inversion carried out

on scales ranging from a fraction of a cubic kilometer to global [158, 159, 160].

The physical model for tomography in its most basic form (Figure 1.8) assumes that

geometric ray theory (essentially the high-frequency limiting case of the wave equation)

is valid, so that wave energy traveling between a source and receiver can be considered to

be propagating along inﬁnitesimally narrow ray paths. The density of ray path coverage

in a tomographic problem may vary signiﬁcantly throughout a section or volume, and

provide much better constraints on physical properties in densely sampled regions than

in sparsely sampled ones.

In seismic tomography, if the slowness at a point x is s(x), and the ray path ` is

known, then the travel time for seismic energy transiting along that ray path is given by

the line integral along `

t =

s(x(l))dl. (1.30)

Receivers

Sources

Unknown

internal

structure

Figure 1.8 Conceptual depiction of ray path tomography. Sources and receivers may, in general, be

either at the edges or within the volume, and ray paths may be either straight, as depicted, or curved.

To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal business use only by the author(s), editor(s),

reviewer(s), Elsevier and typesetter diacriTech. It is not allowed to publish this proof online or in print. This proof copy is the copyright property of the publisher

and is conﬁdential until formal publication.

“Aster Ch01-9780123850485” — 2011/11/8 — page 14 — #14

14 Chapter 1 Introduction

Note that (1.30) is just a higher-dimensional generalization of (1.21), the forward prob-

lem for the vertical seismic proﬁling example. In general, seismic ray paths will be

bent due to refraction and/or reﬂection. In cases where such effects are negligible, ray

paths can be usefully approximated as straight lines (e.g., as depicted in Figure 1.8),

and the forward and inverse problems can be cast in a linear form. However, if the ray

paths are signiﬁcantly bent by slowness variations, the resulting inverse problem will be

nonlinear.

1.4. DISCRETIZING INTEGRAL EQUATIONS

Consider problems of the form

d(x) =

g(x, ξ)m(ξ)dξ . (1.31)

Here d(x) is a known function, typically representing observed data. The kernel g(x, ξ )

is considered to be given, and encodes the physics relating an unknown model m(x)

to observed data d(s). The interval [a, b] may be ﬁnite or inﬁnite. The function d(x)

might in theory be known over an entire interval, but in practice we will only have

measurements of d(x) at a ﬁnite set of points.

We wish to solve for m(x). This type of linear equation is called a Fredholm inte-

gral equation of the ﬁrst kind, or IFK. A surprisingly large number of inverse prob-

lems, including all of the examples from the previous section, can be written as IFKs.

Unfortunately, IFKs have properties that can make them very challenging to solve.

To obtain useful numerical solutions to IFKs, we will frequently discretize them into

forms that are tractably solvable using the methods of linear algebra. We ﬁrst assume

that d(x) is known at a ﬁnite number of points x

, x

, . . . , x

. We can then write the

forward problem as

= d(x

) =

g(x

, ξ )m(ξ )dξ i = 1, 2, . . . , m (1.32)

or as

(x)m(x)dx i = 1, 2, . . . , m, (1.33)

where g

(x) = g(x

, ξ ). The functions g

(t) are referred to as representers or data

kernels.

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