Inverse Problems 4 (1988) 625-642. Printed in the UK
Three-dimensional inverse scattering: layer-stripping
formulae and ill-posedness results
Margaret Cheneyf and Gerhard Kristensson
Division of Electromagnetic Theory, Royal Institute of Technology, 100 44 Stockholm,
Sweden
Received 7 August 1987, in final form 4 December 1987
Abstract. We consider the three-dimensional direct and inverse scattering problems for
the Schriidinger equation and for the reduced wave equation with variable velocity. The
scatterer is probed with either point sources or plane waves of fixed frequency. We ask the
question. 'How does the wave field change when the scatterer is truncated'!' Simple
formulae for the derivative of the wave field with respect to the truncation parameter are
obtained. Similar formulae are obtained for the scattering amplitudes. These formulae are
used to derive ill-posedness results for various inverse scattering problems. The illposedness results apply when data are collected over a range of frequencies.
1. Introduction
The inverse scattering problem is to obtain information about an inaccessible region
of space from scattering data. The problem is important in a variety of fields, such as
medical imaging, non-destructive testing and seismic prospecting.
In the one-dimensional case, the inverse scattering problem can be solved by a
number of algorithms. The most successful of these have been the so called layerstripping algorithms, which are based on the idea of reconstructing the scatterer layer
by layer. This can be done using either of two methods. The first is downward
continuation, in which the wave field is reconstructed at successively deeper layers
(Bruckstein et a1 1985, Symes 1986, Yagle and Levy 1984). The second is invariant
embedding, in which the scatterer is embedded in a family of scatterers (Corones et a1
1983, 1984, Corones and Krueger 1983). Both of these methods use the idea of
decomposing the wave into components travelling in different directions. Algorithms
based on these ideas tend to be fast and robust; similar algorithms are even useful in
computing solutions of the direct scattering problem.
Because the one-dimensional invariant embedding and downward continuation
algorithms have been so useful, a number of researchers have looked for multidimensional analogues. Yagle and Levy (1986) and Yagle (1986) have proposed inverse
scattering algorithms based on splitting the Schrodinger equation into upward- and
downward-going wave components. Weston (1987) and Yagle and Levy (1985) have
found similar wave decompositions in the case of a three-dimensional wave propagating in a medium that depends on only one dimension.
i- Pcrmanent address: Department of Mathematics. Duke University, Durham. NC 27706, USA.
0266-5611/88/030625
+ 18 $02.50 @ 1988 IOP Publishing Ltd
625
626
M Cheney and G Kristensson
In this paper we consider fully three-dimensional scattering problems for the
Schrodinger equation and the reduced wave equation with variable wave velocity. We
first derive a simple layer-stripping formula that tells how the wave field changes when
the scatterer is truncated. We then use this formula to show that various inverse
scattering problems are ill-posed.
This work differs from the previous multidimensional work in that we do not use
the idea of wave decomposition. Instead, we mathematically truncate the scatterer. In
addition, we require no assumptions about the symmetry of the scatterer or dependence on fewer dimensions. Our work has the added advantage of being able to treat
the Schrodinger equation and wave equation simultaneously. We carry out our
analysis for the three-dimensional case, but similar results should hold in other
dimensions.
This work also differs from previous work in that we use our layer-stripping
formula not to solve inverse problems but rather to show that some of them are illposed. Specifically, we show that scatterers differing only in a ball of size a give rise to
the same scattering field as a-0. For the Schrodinger equation case, we obtain a
result involving all frequencies; for the wave equation, we are restricted to finite
intervals of frequencies. These results go beyond the fixed-frequency ill-posedness
results that can be obtained from the theory of homogenisation (Sanchez-Palencia
1980).
This paper is organised as follows. In 8 2 we recall the necessary facts about
scattering with point sources. In $ 3 we derive the layer-stripping formula, which is
then used in $4to prove a number of ill-posedness results. The paper ends with two
appendices containing technical details and an appendix containing some suggestions
for future work.
2. Scattering with point sources
The scattering problems we consider are those in which waves are generated with
‘point sources’ and propagate according to the equation
[V’
+ k2- V ( k ,x ) ] G ( k ,X , x , ) = d(x -x,).
(2.1)
Here k is a real scalar corresponding to the frequency of the wave; x and x, are in R’
with x, denoting the position of the ‘point source’. We assume that V is real-valued,
bounded and has support in the ball B,?.
We denote by g the Green function
g ( k , 1x1) = - exp(ik/xl)/4nlx1.
(2.2)
It is a fundamental solution for O’+ k’:
(02
+ k’)g(k. 1x1) = d(x).
We use g as follows to obtain an integral equation for solutions of equation (2.1).
We write equation (2.1) as
(V’+ k’)G = d
+ VG.
(2.4)
Then G can be given as convolution with the right-hand side of equation (2.4):
G ( k ,x, x,) =g(k. lx-x,l)
+
g ( k , lx - y l ) V ( k , y ) C ( k ,y. x,) dy.
(2.5)
Layer-stripping formulae and ill-posed results
627
Equation (2.5), which is called a Lippmann-Schwinger equation, is an integral
equation for G . It incorporates equation (2.1) as well as the ‘boundary conditions’ at
infinity. The behaviour at infinity can be seen as follows.
Using equation (2.2) we expand equation (2.5) for large 1x1. If V has compact
support, we obtain
G ( k ,x , x,) = g ( k , Ix -xd) + T ( k ,.f, x , ) exp(iklxl)lk + o(ixl-’)
where i= x / ~ xand T is the ‘scattering amplitude’
(2.6)
(2.7)
Equation (2.6) shows that G is the sum of the incident field g and an outgoing
spherical wave.
Since equation (2.5) incorporates both equation (2.1) and ‘boundary conditions’,
we take equation (2.5) to define our solution G.
In the two cases V ( k , x ) = q ( x ) and V ( k , x ) = k ’ q ( x ) , existence and uniqueness
theorems for solutions of equation (2.5) can be stated simply. Proofs for the case of
incident plane waves have been given by Simon (197l), Agmon (1975), Newton
(1977) and Reed and Simon (1979). We sketch a proof for the point source case here
because the results are crucial in proving the layer-stripping formula (theorem 3.1).
Theorem 2.1. Suppose V ( k , x ) = q ( x ) , where q is a real-valued, bounded function
with compact support. Then equation (2.5) has a unique solution G in the pointwise
sense (for x f x , ) with /V11’2Gin L 2as a function of the variable x . Moreover, 1/1Vl”2G//2
is uniformly bounded for all x , and for all k S k , > O . If V supports no zero-energy
bound or half-bound states, then ///V)”2G1/2
is uniformly bounded for all k and x,.
Proof. Define
K ( k ,X ,
Z)
= /q(x)/’”g(k,Ix- zl)q(z)lq(z)I-1’2.
This definition allows equation (2.5) to be written as
G ( k ,X ,x,) = Iq(x)/”’g(k,
IX - ~ , l )
+
i
K ( k , X , z ) G ( k ,Z , x,) dz
(2.9
where G=lq/”’G. We show in appendix 1 that 1 - X is invertible on L’ for every
non-zero, real k . This implies that equation (2.9) has a unique solution 6. G, in turn,
gives rise to a solution G of equation (2.5) by
(2.10)
To prove uniqueness i n the pointwise sense, we suppose there are two solutions of
equation ( 2 . 5 ) , G I and Cl. Then the difference satisfies
(2.11)
However, 6,= Iql”’G, and G2=1q)’”G2both solve equation (2.9) and must therefore
be equal in the L’ sense. The Schwarz inequality applied to equation (2.11) then
shows that G, = G2pointwise.
628
M Cheney and C Kristensson
The uniform boundedness of ~ ~ ( ~ - . l L ) - follows
’~~
from the continuity in k of
( Z - X - ’ and the fact that II7LIl-+O as k-+
(Newton 1982).
QED
M , depends only on V and k , , .
We denote by Mi the uniform bound for I//VIi’2G//2.
Corollary 2.2. Suppose V ( k , s) = k’q(x), where q is a real-valued, bounded function
with compact support. Then equation (2.5) has a unique solution C in the pointwise
sense ( x f x , ) with IVI”% in L’ as a function of the variable x . Moreover, on any finite
interval J of k values, /I/Vl”’G/12
is uniformly bounded for all x, and for all k in J .
Proof. The proof of theorem 2.1 holds in this case as well. Furthermore, it can be
strengthened to hold for k = 0, because 3C = 0 there.
QED
We denote by M 2 the uniform bound for /1IV/’?G/I2.
M 2 depends only on V and J .
Finally, we need reciprocity.
Proposition2.3 (Simon 1971, Ikebe 1960). Suppose V satisfies the hypotheses of
theorem 2.1 or corollary 2.2. Then
(2.12)
C ( k ,x,x,) = C ( k ,x,, x).
3. Layer-stripping formulae
In this section we investigate the change in C when the medium V is changed. More
specifically, suppose {r,}
is a one-parameter family of C’ surfaces that foliates the ball
B, containing the support of V . We denote by the characteristic function that is one
on the side of r , that corresponds to smaller values of a and zero on the other side.
Then we truncate V by merely multiplying the potential by x(l.
xu
3.1. Layer stripping with point sources
We denote by C ( k , x, x,, U ) the solution of equation (2.5) in which V has been
replaced by xi,V. Explicitly, we have
To see how C changes with a , we differentiate equation (3.1) with respect to a.
Formally we get
a,G(k, x,x,,a ) =
L“
g ( k , 1~ - s / ) V ( k s)G(k,
,
S, x,, a ) ds
r
where s E r, and ds is a surface measure on r, defined by dy = du ds. We note that
equation (3.2) is the same as equation (3.1) except that the inhomogeneous terms are
different. However, the inhomogeneous term of (3.2) can be written as a linear
Layer-strippirzg formulae and ill-posed results
629
combination of copies of the inhomogeneous term of equation (3.1). By the superposition principle and by uniqueness of the solutions of equation (3. l ) , the solution
a,,G of equation (3.2) must be the same linear combination of copies of G. In other
words,
8,,G(k,x,x,,a ) =
Jrc,
G ( k ,x,s, n ) V ( k ,.s)C(k, x,,a ) ds.
$9
(3.3)
We summarise this argument in the following theorem.
Theorem 3. I . Suppose {r((}
is a one-parameter family of C’ surfaces that foliates the
ball B,+ Moreover, for fixed a we assume that the surfaces in a neighbourhood of r,
can be locally represented as r,,+(={s+tn(s,t ) : s ~ r , , where
},
n(s, t ) is normal to r,
and is continuously differentiable in t. ( n ( s , t ) does not necessarily have length one.)
Assume V is continuous, real-valued and supported in BR. Then, for k f 0, equation
(3.3) holds for all x,in the L:c,csense as a function of x. It holds pointwise for all xZx,.
Furthermore, it holds for x=x, whenever xer,,n supp V.
Proof. To make the foregoing argument rigorous, we proceed as follows. We define
to be the kernel of the operator
We then take the difference
X<,.
Then we can write equation (3.1) as
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M Cheney and G Kvistenssorz
We denote the first integral on the right-hand side of equation (3.6) by
H , ( k ,x , xs,a) and the left-hand side by F,(k, x, x,, a). Since I -X,is invertible (see
appendix 1) we have
F,= ( I - x o ) - ' H E .
(3.7)
In appendix 2 we show that HE+ H,, as e+O in the L' sense, where
&(k, x , xs, a) =
i
IV(k x)l'",
Ix -sl)V(k, s)G(k, 5 , x,, a ) ds.
(3.8)
r,,
Since ( ~ - - X ~ , )is- 'continuous, F,, satisfies
F,, = H,,+ X,F,,.
(3.9)
We now define a,,G by
(3.10)
in the LE,' sense for all x, and in the pointwise sense provided x and x, are not on
rtjn supp V and equal to each other. The function a,,G defined by equation (3.10)
satisfies equation (3.2) in both senses. The theorem results from the fact that the
solution of equation (3.1) is unique in both senses even for other inhomogeneous
terms.
QED
Remarks. (i) It is important to emphasise that it is V that is assumed to be continuous,
not x,V. (ii) The theorem holds whenever ( I - K) is invertible. Thus it holds for the
zero-k Schrodinger equation when there are no zero-energy bound states or halfbound states. (iii) The heuristic argument above theorem 3.1 resembles an argument
used by Kailath et a1 (1978). (iv) Equation (3.3) is an invariant embedding equation in
the sense that it gives information about how the scattering problem is embedded in a
family of similar problems.
By reciprocity. we can write equation (3.3) as
a,G(k,x , x,,a ) =
J
G ( k ,x,s, u ) V ( k ,s)G(k,x,, s ,
U) ds.
(3.11)
r,,
Equation (3.11) tells us about the change in G as more layers of V are stripped away.
This change can be computed by a nonlinear expression that involves putting point
sources on the truncated surface and measuring the resulting field at x and x,.
Equation (3.11) may be useful in developing efficient algorithms for solving the direct
and inverse scattering problems. See appendix 3 for further remarks in this direction.
Two important special cases occur when r,,is a family of concentric spheres or a
family of translated planes. In the first case equation (3.11) becomes
G ( k ,x , a6, a ) V ( k ,a6)G(k,x s ,a6, a ) d e
3,G(k,x, x,, a)= a 2
(3.12)
S2
where S' is the unit sphere and 8 is a point on S'. In the second case equation (3.11) is
631
Layer-stripping formulae and ill-posed results
where e is a unit vector normal to the family of planes parametrised by a .
We note here an integrated version of equation (3.3):
G(k,x,x,, a) = g ( k , (x - x 4 +
1I,,#
G ( k ,x,s, a ' ) V ( k ,s ) G ( k , s, x,, a ' ) ds da'.
(3.14)
We also obtain a layer-stripping equation for the scattering amplitude defined by
equation (2.7). We denote by T ( k , R,x,, a) the scattering amplitude corresponding to
the potential X ( ~ V .
Corollary 3.2. Suppose the hypotheses of theorem 3.1 hold. Then, for all i E S' and all
x,.we have
a,,T(k,R , x,,a )
exp( - i k i s)
+ T ( k ,i ,s, a)
1
V ( k ,s ) G ( k ,xs,s, a) ds.
(3.15)
3.2. Layer stripping for plane waves
Readers Rho are familiar with plane wave scattering might wonder whether a formula
similar to equation (3.3) holds in the plane wave case. Below we state such a formula.
We recall that the field q corresponding to an incident plane wave satisfies
[ V'
+ k' - V(k , X)J w = 0.
(3.16)
We define scattering solutions corresponding to an incident plane wave with direction
eES'by
~ ( kx,, e) = exp(ike * x ) +
i
G ( k ,x , y)V(k, y) exp(ike * y) dy.
(3.17)
As before, we denote by ~ ( kx , e, a) the quantity in equation (3.17) when V is
replaced by xtiV. The layer-stripping result we obtain is as below.
Corollary 3.3. Suppose the hypotheses of theorem 3.1 hold. Then, for all e E S' and all
x. we have
a,,q(k, x. e , a ) =
G ( k ,x,s, a ) V ( k ,s ) ~ ( ks,, e , a) ds.
(3.18)
Similarly, we get a layer-stripping result for the plane wave scattering amplitude,
which is defined by
(3.19)
where e' E S'. The scattering amplitude can be obtained from the large-1x1 behaviour of
?$ as follows:
y ( k , x. e) = exp(ike x) + A ( k , R,e)
exp(iklxl)
1x1
+
o(~x~-l)
(3.20)
632
M Cheney und G Kristerisson
where 2=xXi~xl.We write A ( k , e’, e , U ) for the scattering amplitude corresponding to
the potential xI,V.Then we have the following.
Corollary 3.4. Suppose the hypotheses of theorem 3.1 hold. Then, for all e , e’ E S’,
we have
&A(k,e’, e. a)
--
exp( - ike’ s)
+ T ( k , e‘, s, a)
V ( k ,s ) v ( k ,s, e, a) ds.
(3.21)
4. Ill-posedness of the inverse problem
In this section we use equation (3.11) to give rigorous proofs of ill-posedness results
for various inverse scattering problems.
For simplicity, let us consider the case in which sources are placed at positions x,
outside BR and measurements of G are also made at positions x outside B R . In this
geometry, we consider the case in which the surfaces r,,of 5 3 are concentric spheres of
radius a.Note that we do not assume V ( k , x ) to have any symmetry. We take xcfto be
the characteristic function of the exterior of the surface r((.Roughly speaking, we are
taking ‘bites’ out of the middle of V . Here xl, and r,,are not parametrised quite the
same way as in 0 3 , but the results of 93 still hold.
Theorem 4.1. Suppose V ( k , x ) = q ( x ) is a continuous real-valued function with
support in B R . Then for a< R , we have
for all x and x,outside the ball of the radius R . The constants M , and M depend on q
but not on a or k ( M , depends on ko in the sense of theorem 2.1).
Proof. By the fundamental theorem of calculus,
lG(k,x,xs, a ) - G ( k ,x,xs, 011 a max IarG(k, x,xs, Y)I.
rE[O a]
(4.1)
We use equation (3.12) to estimate the right-hand side of equation (4.1):
max la,G(k, x,xs,r)i d max ?
re[O.a]
I
€10. a ]
i2
IG(k,x,r e , r)q(re)G(k,r e , xI,r)i de.
(44
To estimate each G in equation (4.2) we use equation (3. l ) , where we take x,= r e
and a = r . We apply the Schwarz inequality to the integral in equation (3. l ) , writing
the integrand as the product g ( k , Ix -yl)xr(y)Iq(y)11’2 times q(y)xr(y)/q(Y)l-”*
G ( k ,y , xs,Y ) . The L2-norm of gxrlq11/2we bound by 11g1q/”2112=M . We estimate
x , ~ ~ ~ ’ by
/ ’ /Iql’”~G1,
G~
which by theorem 2.1 is bounded in L’ by M , .
QED
This theorem tells us that the data G ( k , x,x,,0) for the potential q look very much
like the data G ( k , x,x,, a ) for the different potential xI,qif a is small enough. In other
words, large changes in q inside r,,can lead to very small changes in the data.
633
L uy er -stripsing formu lue and ill-posed resu Its
In particular, theorem 4.1 shows that the following problem is ill-posed.
Problem A . G ( k , x,x,) is a solution to
[O’+ k’- q ( x ) ] C ( k ,X ,1,)=d(x-x,)
where q(x) is bounded, real-valued, and has support in BR. G ( k , x, x,) also satisfies a
radiation condition. i.e. satisfies
From the knowledge of { G ( k ,x, x,): all x and x, outside B,?, all k a k , , > 0 } we try to
determine q(x) inside BI+ If q has no zero-energy bound or half-bound states, then we
take k,,= 0.
Problem A is ill-posed in the following sense.
Corollury 4.2. Let q be a potential as above that is continuous and non-zero at the
origin, having corresponding data { G ( k ,x,x,)= G ( k ,x,x,,0) : all x and x,outside B R ,
all k s k , , > 0 } . Then there is a sequence of potentials {q,J}={x(Jq}
such that there is
some E > O independent of a for which l/qn- q/lx> F but the corresponding data
{ G , } = { G ( k ,x , x,,a ) } converge to G ( k , x,x,,0) in L“ as a-tO. Specifically, we have
the estimate
IlG(k, x,x,,0) - G ( k ,X ,x,,O)ll,
CO’
where the sup is taken over all k a k,,> 0 and x and x,outside BR and where C depends
only on k,,,R and q. If q supports no zero-energy bound or half-bound states, then the
sup can be taken over all k 2 O .
Yet the following theorem shows that a subset of the data for problem A uniquely
determines the potential.
Theorem 4.3. For any p>O, the data {G(k,x,x i ) = C ( k ,x,x,.0): all x and x, outside
B,,, all k S k , , > O } uniquely determine the potential q in problem A.
Proof. We expand G(k, x , x,) in equation (4.3) for large 1x1 and lx51
G ( k ,x,
-g(k,
/b-x,l>
where x = i l x ( and x , = i , l x , i , and where Ixlix,lr(k,x, x,)+O as 1xl+a
(Rose et a1 1985).
Moreover, we have r ( k , x,x,)-+O as k - t
This shows that the Fourier transform %q(z)is given by
%q(z)= r x<,k--ca
Iim 25’27t1’2[G(k,
x,x,) - g ( k , / x - x s ~ ) ] ~ xexp[
~ ~ x5 ~ik((x/+ lx,/)].
I= k(r
or l x , l - t m .
(4.5)
+ r,)
QED
634
M Cheney and G Kristensson
Equation (4.5) could be used to solve the inverse problem if G were known for all
k . However, measurements for all k can never be made in real experiments. Inversion
with equation (4.5) is therefore impractical.
We proved theorem 4.1 explicitly for the Schr6dinger equation; however, the
same result holds for the wave equation case V ( k ,x) = k'q(x) if one keeps in mind that
M z , which depends on k , must replace M , . We thus obtain an ill-posedness result for
the wave equation inverse problem in which data are only collected for a bounded
range of frequencies. Again, the ill-posedness is in the sup-norm.
Problem B. G ( k , x, x,) is a solution to
[V'+ k 2 - k ' q ( x ) ] G ( k , x , x , ) = ~ ( x - x , )
where q(x) is bounded, real-valued, and has support in BI(.G ( k ,x, x,) also satisfies a
radiation condition, i.e. satisfies
From the knowledge of { G ( k ,x,x,): all x and x,outside BR, all k in some finite interval
J } we try to determine q(x) inside BR.
Problem B is ill-posed in the following sense.
Corollary 4.4. Let q be a potential as above that is continuous and non-zero at the
origin, having corresponding data { G ( k ,x,x,)= G ( k , x,x,,0): all x and x, outside B,?,
all k in some finite interval J } . Then there is a sequence of potentials { q , }= {x/,q}such
that there is some E > 0 independent of a for which liq,,- 411- > E but the corresponding
data { G , } = { G ( kx,
, x,,a ) } converge to G ( k , x. x,,0) in L" as a-0. Specifically, we
have the estimate
IlG(k,X, xS,a ) - G ( k ,X, x,, 0)1l2 Ca3
where the sup is taken over all k E J and x,x,outside
J , R and q .
BR,
and where C depends only on
Again, a subset of these data uniquely determines the potential.
Theorem 4.5 (Ramm 1986). The knowledge of { G ( k ,x , x,):all x and x,outside BR, all
k in some interval around k = 0) uniquely determines q of problem B.
Sketch ofproof. We write G ( k ,x,x,)= g ( k , / x - x , l > + k'G"(k, x,x,).Then G" satisfies
V'G
= qg - k'(
1 - q)G".
At k = 0 equation (4.7) becomes
Q2GSc
= qg.
This linearises the inverse problem:
(4.7)
Layer-stripping formulae and ill-posed results
635
Ramm (1986) has shown that the transform q+ Gcof equation (4.9) can be inverted.
QED
Equation (4.9) could be used to solve the inverse problem if k - ? ( G - g ) were
known for k=O. However, such measurements can rarely be made in real experiments. Even if they could be made, inversion of equation (4.9) is more ill-posed than
problem B, as the following result shows.
Theorem 4.6. Let q be any integrable function with compact support and with corresponding data G"(0, x,x,). Then there is a sequence of potentials qa such that for any
P E (1, C O ] , we have ljqa-q/)p-+CO,but the corresponding data {Ga}={G"(O, x, x,,a ) }
converge to G"(O, x,xs)in L" as a-+O, where the convergence is uniform for all x and
x , outside BR.
Proof. For this proof we let xi,= 1 in the interior of B , and 0 on the closure of the
exterior.
u
O<a<
In the case p < C O , we let E = p - 1 and choose qu= q + a - 3 + a ~ where
3 d ( l + E ) . Note that q r , = q . From equation (4.9) we have the estimate
a'a-?+"
I C ( 0 ,x,x,,a ) - G"(0. x,x,)i
for x,x,$BR.
(4.10)
1 2 4 R -U)'
However,
as a-+O.
In the case p
= CC
, we can choose a = 2. Then from equation (4.10) we again have
I C ( 0 , x,x,, a ) - G"(0, x. x,)1+0
as a+O.
But
/ / q- qt,llr=a-'-+
CO
as a+O.
QED
The result shows that recovery of q from k - ? ( G - g ) at k=O is impractical.
5 . Conclusion
The main result of this paper is the invariant embedding equation (3.3). It is an
invariant embedding equation in the sense that it gives information about how the
scattered problem is embedded in a family of similar problems. Equation (3.3) is in
this respect similar to the one-dimensional formulae studied by Corones et a1 (1983,
1984) and Corones and Krueger (1983) in the time domain.
Equation (3.3) has led us to ill-posedness results for certain inverse problems.
These ill-posedness results show that scatterers differing only in a ball of size a give
636
M Cherzey and G Kvistenssori
rise to the same scattered field as u+O. This is the case even though the scattered field
determines the scatterer uniquely.
Let us consider these results in the light of the physical principle that one needs
large frequencies to resolve fine detail. This physical principle is exhibited in the
Schrodinger equation case by theorem 4.3, which shows that the large-k limit of the
data determines the potential. However. these large-k data are taken into account in
the ill-posedness result. In particular, corollary 4.2 shows that if potentials differ only
in a ball of size a , then as a-0, they give rise to the same scattered field, even at
arbitrarily large k . This tells us that, even if we have large-k data, we should not
expect to reconstruct the potential in the pointwise sense.
The situation is somewhat different in the wave equation case. Here, the validity
of the large-k-for-fine-detail principle is not so clear. Indeed, one expects that the
high-frequency signals alone would not suffice to determine the scatterer, because
they might not propagate into some regions. The only uniqueness result, theorem 4.5,
is in fact a low-frequency result. Here again, the ill-posedness result, corollary 4.4,
incorporates these low-frequency data. High-frequency data, however, are not
included in the ill-posedness result; whether they can be included is an open question.
Also open are the questions concerning conditions under which these inverse
problems can be well-posed. Are they well-posed in integral-type norms, or must
additional a priori information be included? Other open questions that need to be
investigated are the possibilities of using the embedding equation (3.3) to solve the
direct and inverse problem (see appendix 3 for comments about this). Work is in
progress to answer these questions.
Acknowledgments
The work of MC was supported by the Office of Naval Research, Young Investigator
Grant No N00014-85-K-0224. She would also like to thank the Division of
Electromagnetic Theory, Royal Institute of Technology for their hospitality during
the summers of 1986 and 1987. The work of GK was supported by the National
Swedish Board for Technical Development and their support is gratefully acknowledged. The authors are indebted to Robert Kohn for pointing out that ill-posedness
results can be obtained from homogenisation theory, to Adrian Nachman for suggesting the ideas in appendix 1 and to Rainer Kress for some helpful comments.
Appendix 1
In this appendix we show that equation (2.9) is uniquely solvable. This proof
combines elements of the proofs of Agmon (1975) and Newton (1977). We include it
here because it is particularly simple and seems to be new. Proofs requiring weaker
conditions on V can be found in Agmon (1975), Amrein er a1 (1977), Hormander
(1983) and Reed and Simon (1978).
Theorem A . l . Let V be a bounded, real-valued function of compact support. Then
the operator I - X is invertible for real k#O, where ‘3% is defined by the kernel
IX
K ( k , X,Z ) = JV(k,~ ) l ” ’ g ( k , - z / ) V ( k ~
, ) l V ( kz)I-”’.
,
(A.1)
Layer-stripping formulae and ill-posed results
637
Proof. The kernel K defines a Hilbert-Schmidt operator (Reed and Simon 1973) on
L'(R'). Therefore, if ( I - X ) is not invertible on L2 for some real k f O , then the
equation
9 = Xy?
('4.2)
must have a non-trivial solution p in L'. A solution 9 of equation (A.2) gives rise to a
solution q of
by means of the correspondence
Equation (A.3) implies
+
(02 k")W = vq.
We denote the right-hand side of equation (A.5) b y f . We then compute
where (., - ) denotes the L2 inner product and $ 2 denotes the imaginary part of the
complex number z .
The right-hand side of equation (A.6) is zero because V is real. To compute the
left-hand side, we use the Parseval theorem and equation (V.4) of Reed and Simon
(1973):
6
where = [/I61 and $f denotes the Fourier transform of f . This computation shows
that $ f ( E ) = 0 almost everywhere on 161 = k k . In other words,
i
V ( k ,x)ly(k,x) exp( - ike x) CFX = 0
(A.8 )
for almost all e E S'.
We combine equation (A.8) with Newton's result on the large-1x1 asymptotics of q
(Newton 1977)
where h ( k . EL' and e =xIIxI.
We have now shown that any non-trivial solution of equation (A.2) that is in L2
must give rise to a q also in L2.But Kato's theorem (Kato 1959) shows that equation
(A.5) has no non-trivial L2 solutions for k f O when V ( k ,x) = O(/x/-'-') at infinity.
Therefore, equation (A.2) has no non-trivial solutions.
QED
e)
638
M Cheney and G Kristensson
Appendix 2
In this appendix we show that
Recall that H , is defined by the first integral on the right-hand side of equation (3.6)
and H,, is defined in equation (3.8).
We have assumed that a point y on T,+,
can be written y = s t n ( s , t ) , where s E r,,
and n(s, t ) is normal to r, and points in the direction of increasing a. With this
notation, we can write H , as
+
x V ( k ,s + tn(s, t ) ) G ( k ,s
+ tn(s, t ) , x,,a ) ds(s, t ) dt
(A.lO)
where we have explicitly put in the dependence of ds on s and t . Similarly, we write
We recall that V is continuous and C = g + h , where h is continuous in the variable
x. Because of these facts and because ds depends continuously on t , the main
contribution to IIH, - H& comes from the term
1
jx - s - m(s, t)i jx,- s - tn(s, t)l
-
Jx- SI Ix,-SI
ds dt
1’
dx. (A.12)
In this main contribution (A.12), ds no longer depends on t.
We bound IV(k, x ) V ( k ,s)l by IlVlll, which we move outside the integrals of (A.12).
We then write out the square in expression (A.12) as a product of two integrals,
obtaining an integral over s, s f , t , t ’ , x of
Layer-stripping formulae and ill-posed results
639
where n’ =n(s’,t‘). We add and subtract terms so that we finally bound (A.12) by the
sum of the following four terms:
1
Ix,- s - tn/
Ix-s-tn/
- 2)
ds ds’ dt dt’ dx
Ix,- S I
(A.13)
L)
ds ds’ dt dt’ dx
/x-ssJ
(A.14)
-
)
1
)
ds ds’ dt dt’ dx
(A.15)
dsds‘dtdt‘du.
(A.16)
The integrals in (A.16) will be treated the same way as (A.14);below we will carry out
the computations for (A.14). Similarly, treatment of the integrals in (A.13) and
(A. 15) will be illustrated by carrying out the computations for (A. 13).
To estimate (A.l4), we interchange the order of integration so that we do the x
integral first. We move the factors involving xs outside the x integral, and then apply
the Schwarz inequality to the remaining two factors. The resulting factor
is a constant (independent of s’ and t’) and can be moved outside.
Next we show that the remaining factor, for small t , satisfies
(A.17)
640
M Cheney and G Kristensson
where C is independent of s and t . We introduce the letters r = Ix-sl and u =
n (x - s ) / ~ x - s ~ ~With
n ~ . this notation the integral in (A.17) is bounded by
-
2nI:RJ-l
1
2r'+ t'n'
-
+
2rtlnju - 2 r d r ' t'n'- 2rtlnIu
du dr.
r'+ t'n' - 2rt/n/u
(A. 18)
The expression (A. IS) can be evaluated analytically; it is equal to
2n
[ (
--
ti:,
2R---
'
In-
2R - tin/
2R+2tlnl .
2R+tt/nl
1
(A.19)
Expanding the logarithm of (A.19) gives the result in (A.17).
We have now estimated (A.14) by
(A.20)
which clearly goes to zero as &-+U.
We now turn our attention to (A.13). Again, we do the x integral first; it is
bounded by a constant (independent of s, s', t and t ' ) and can be pulled outside. Next
we do the s' and the t' integrals, they are both finite and can be pulled outside. We are
left with
(A.21)
If x,$T,, then the integrand in (A.21) is bounded by a constant times t . This shows
that in the case x,$T,, (A.21) goes to zero as E-0.
If x,E r,, we split the r, integral of (A.21) into two pieces, one over a small disc D6
of radius 6 about x, and the other over r,\Dd. In the latter, the integral can be treated
as above. The former we write in polar coordinates with r = /x,- in this notation the
integral becomes
SI;
(A.22)
where we have used the fact that
by
is locally planar. Equation (A.22) can be bounded
(A.23)
where In(s, t)l S C . The r integral in (A.23) we evaluate analytically; (A.23) is equal to
E
E
(6 -
m2
+ tc) dt.
,,
J II
This integral goes to zero as E-O.
(A.24)
64 1
Layer-stripping formulae and ill-posed results
We have now shown that (A.21), and therefore (A.13), go to zero as E-O. Thus
we now have shown that the integrals in (A.13)-(A.16) go to zero in the E-O limit.
Finally, we must consider the remainder terms, the most singular of which is
This expression can be estimated by multiplying out the squared term to obtain
s, s’, f , t‘ integrals. We do the x integral first. It is bounded by a constant, which can be
+
moved outside. What remains goes to zero because V ( k , s t n ) - V ( k ,s ) is bounded
by a constant that goes to zero as E+O.
The other remainder terms can be estimated by similar arguments.
QED
Appendix 3
This appendix contains speculations about possible uses of equation (3.3) in solving
the inverse problem. Equation (3.3) suggests the followng algorithm.
(1) Measure G ( k , Ra, RP, R ) for all a , /3 in S’.
(2) Determine V ( R a ) for all a in S’.
(3) Use a finite-difference approximation to equation (3.3) to obtain, for all a , P
in S’,
G ( k ,R u , RP, R - A )
= G ( k ,R a , RP, R )
- AR2
1
G ( k ,R a , RO, R ) V ( k ,RO)G(k,RO, RP, R ) de.
S2
(4) Use the free equation (i.e. equation (2.1) with V=O) to propagate the sources
and receivers inward. Thus obtain G ( k , ( R - A ) a , ( R - A)@, R - A) for all a , b, in S?.
( 5 ) Repeat, starting with step (2), using R - A instead of R .
However, the following difficulties must be overcome in order to make this
algorithm computationally feasible:
(i) G is singular when the source and receiver coincide;
(ii) step 4 is presumably unstable;
(iii) there must be a consistency condition in step 2.
The implementation of the algorithm or a modified version of it will undoubtedly
be tricky, because the problem is ill-posed. More work is needed to determine
whether some modification of this algorithm will be useful.
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