Chapter 7 Integral Geometry and Tomography 192
1.4 Tomography and the inverse seismic problem
The task ofintegral geometry, or (in applications)tomography, is to find a function f given the integrals
γ f dγ (1.4.1)
over a family of manifolds{γ}. The case when theγare straight lines inR2is quite important because it models X-rays. Then the integrals (1.4.1) are available from medical measurements. Uniqueness of recovery of f and an explicit reconstruction formula were due to Radon in 1917, so often this problem is called after him.
1.4. Tomography and the inverse seismic problem. 11 But the applied importance of this problem has been made clear by Cormack and Hounsfield, who developed in the 1960s an effective numerical and medical technique for exploring the interior of the human body for diagnostic purposes. In 1979 they received the Nobel Prize for this work.
If a seismic (elastic) wave propagates in the earth, it travels along geodesics γ(x,y) of the Riemannian metrica2(x)|d x|2. In simplest case,ais the density of the earth. The travel time fromxtoyis then the integral
τ(x,y)=
γ(x,y)
dγ, (1.4.2)
which is available from geophysical measurements.
Theinverse seismic problemis to findagivenτ(x,y) forx,y∈that is a part of∂.
Seismic waves can be artificially incited by some perturbations (microexplo- sions) on part of the surface of the Earth, and seismic measurements can be implemented with high precision. The spherically symmetric model of earth (is a ball andadepends only on the distance to its center) was considered by Herglotz in the 1910s, who developed one of the first mathematical models in geophysical prospecting.
We will consider the even simpler (but still interesting) case when is the half-space{x3<0}inR3anda=a(x3). Sinceadoes not depend onx2, the curve γ(x,y) will be contained in the plane{x2=0}, provided that both xand yare in this plane. Later on we will drop the variablex2. It is known (and not hard to show) that the functionτ satisfies the following eikonal equation:
a2(x3)((∂1τ)2+(∂3τ)2)−1=0, or
∂3τ +
a−2−(∂1τ)2=0,
where∂j is partial differentiation with respect toxj, and√can be with+or− depending on the part ofγ. It is clear thatτ(x,y)=τ(y,x), so later on we will fixy=(0,0) and consider travel time only as a function ofx, which we will treat as an arrival point. The known theory of nonlinear partial differential equations of first order [CouH, p. 106] is based on the following system of ordinary differential equations for characteristics:
d x1
d x3
=p1(a−2−p12)−1/2,d p1
d x3
=0,
where p1 =∂1τ. Whena is known, a solutionτ to the eikonal equation inis uniquely determined by the initial data on the line{x3=0}, and according to the known theory of differential equations of first order it is formed from characteristics that are original geodesics. Whena is an increasing function ofx3so that it goes to zero whenx3 goes−∞, these characteristics consist of two symmetric parts, wherex1is monotone with respect tox3, which have a common point (x1m,x3m) with the minimum ofx3over the geodesics achieved atx3m. Integrating the first
of our differential equations for characteristics over the interval (0,x3m), we will obtain a half–travel time along the geodesics
τ((2x1m,0),y)=2 x3m
0
p1(a−2(s)−p21)−1/2ds=2 p
α t1/2(p−t)−1/2g(t)dt when we use the substitution of the inverse functiont =a2(s) so thats=g(t) and let p= p1−2, α=a2(0). The upper limit is pbecause at the point (x1m,x3m) the geodesic is parallel to thex3-axis, and therefore the denominator in both integrals is zero. Now,αcan be considered as a known function as well aspas a function of x1m because these quantities are measured at {x3=0}. So we arrive at the following integral equation:
p
α (p−t)λ−1f(t)dt=F(p), α < p< β, (1.4.3)
λ=1/2, with respect to f(t)(=t1/2g(t)), which is the well-known Abel integral equation. It arises also in other inverse problems (tomography (see Section 7.1) and determining the shape of a hill from travel times of a heavy ball up and down (see paper of J. Keller [Ke])). Equation (1.4.3) is one of the earliest inverse problems.
It was formulated and solved by Abel around 1820.
Exercise 1.4.1. Show that the Abel equation (1.4.3) has the unique solution f(t)= sinπλ
π d dt
t
α(t−p)−λF(p)d p, α <t < β, (1.4.4)
provided that 0< λ <1 and f ∈C[α, β] exists.
{Hint: Multiply both sides of (1.4.3) by (s−p)−λ, integrate over the interval (α,s), change the order of integration in the double integral on the left side, and make use of the known identity
1
0
θ−λ(1−θ)λ−1dθ= π sinπλ to calculate the interior integral with respect to p.}
More general equations of Abel type as well as their theory and applications can be found in the book of Gorenflo and Vessella [GorV].
It is interesting and important to consider the more general problem of finding f andafrom the integrals
γ(x,y)
ρ(, γ)f dγ, (1.4.5)
whereρis a partially unknown (weight) function that reflects diffusion (attenua- tion) in applied problems. Not much is known about this general problem. We will describe some results about this problem in Chapter 7.
1.4. Tomography and the inverse seismic problem. 13 The problems of integral geometry are closely related to inverse problems for the hyperbolic equation
a0∂t2u+b0∂tu−div(a∇u)+cu = f in×(0,T) (1.4.6)
with zero initial data
u=∂tu =0 on× {0} (1.4.7)
and the lateral Neumann boundary data
a∂νu=g1on∂×(0,T).
(1.4.8)
The initial boundary value problem (1.4.6)–(1.4.8) has a unique solutionu for any (regular) boundary data, provided thata,b0,c, f are given and sufficiently smooth.
The inverse problem is to finda,b0,c, f (or some of them) from the additional boundary data
u=g0onγ×(0,T), (1.4.9)
whereγis a part of∂. We will discuss this problem in Chapters 7 and 8. It is far from a complete solution in the case of one boundary measurement. But once the lateral Neumann-to-Dirichlet mapl :g1→g0is given, the problem was recently solved in several important cases. Under reasonable assumptions one can guarantee uniqueness and stability of recovery ofb0,cwhenT andare sufficiently large andg0is given for all smoothg1. The situation withais more complicated: it can be uniquely determined only up to a conformal transformation of a corresponding Riemannian manifold, and the stability of a known hypothetical reconstruction is quite weak. In any case, ifa=1,b0=c=0 there is a uniqueness theorem due to Belishev that is valid for anyγand guarantees uniqueness of recovery ofa0in the domain that can be reached by waves initiated and observed onγ. If timeT is large enough, this domain is the whole of.
In the isotropic case, behavior of elastic materials and elastic waves is governed by the elasticity system for the displacement vectoru=(u1,u2,u3),
ρ∂t2u−div(A((u))=fin×(0,T), (1.4.10)
where(u) is the stress tensor with the components 12(∂lum+∂mul) andAis the elastic tensor with the componentsaj klm(x). In the general case these compo- nents satisfy the symmetry conditionsaj klm =alm j k=ak jlm, and in the important simplest case of classical elasticity,
aj klm=λδj kδlm+à(δjlδj k+δj mδkl).
The system (1.4.10) is considered together with the initial conditions that pre- scribe initial displacements and velocities and a lateral boundary condition, e.g., prescribing normal components of the stress tensor A(u) on ∂×(0,T). As in electromagnetic scattering one can consider time-periodic elastic vibrations and elastic scattering problems. Only recently has there been some progress in understanding inverse problems in elasticity, and we report on certain results in
Sections 5.8 and 8.2. The contemporary state of the inverse seismic problem based on general linear system of (anisotropic) elasticity is described by de Hoop [I2]
and de Hoop and Stolk [DS].
The inverse problems for hyperbolic equations and problems of integral geom- etry are closely related. One can show that the data of the inverse problem for the hyperbolic equation determine the data for tomographic and seismic problems.
To do so one can use special high-frequency (beam) solutions or propagation of singularities of nonsmooth (in particular) fundamental solutions.
Sometimes tomographic approximation is not satisfactory for applications (in particular, it does not properly describe diffusion), while multidimensional inverse problems for hyperbolic equations are hard to solve. As a good compromise one can consider inverse problems for the transport equation.
∂tu+vã ∇u+b0u =
W
K(,v,w)u(,w)dw+ f (1.4.11)
in a bounded convex domain⊂Rn, whereu(x,t,v) is the density of particles andK(x,v,w) is the so-called collision kernel. Let∂vbe the “illuminated” part {x∈∂:ν(x)ãv<0}. One can show that the initial boundary value problem (with data on∂v) for the nonstationary transport equation (1.4.11) has a stable unique solution (in appropriate natural functional spaces) under some reasonable assumptions. The inverse problem is to find the diffusion coefficientb0, the col- lision kernel K, and the source term f fromu given on ∂for some or for all possible boundary data and zero initial conditions. Not much is known about the general problem, though there are some partial results. Quite important is a station- ary problem when one dropst-dependence and the initial conditions. The inverse problem is more difficult, and even the simplest questions have no answers yet.
We discuss these problems in Section 7.4. Observe that ifb0=0,K =0, and f is unknown, we arrive at tomography over straight lines, which is satisfactorily un- derstood. But whenb0≥0 is not zero there are many challenging open questions including the fundamental one about the uniqueness of f andb0.