Reflection imaging without low frequencies
Frank Natterer
Department of Mathematics and Computer Science
University of Münster
Münster, Germany
Abstract
This paper is concerned with wave equation reflection imaging for
source pulses that do not have low frequencies. It is well known that
the lack of these low frequencies causes serious difficulties in the image
reconstruction. We show that under favorable circumstances good
images can be obtained nevertheless by a data completion procedure.
1
Introduction
One of the biggest difficulties in seismic reflection imaging is the lack
of low frequencies in the source pulses. As is well known (see e. g. [1],
[4]) and well understood in a mathematical context (see [11], [5]) this
makes it impossible to quantitatively recover the the seismic velocity
distribution. Recently much effort has been put into using sources
which have a relevant amount of low frequencies, i. e. frequencies
below 5 Hz [3]. Corresponding problems occur in medical ultrasound
if it is based on reflection measurements only.
In the present note we study a purely mathematical approach
which does not require extra measurements at low frequencies. We
rather use a data completion procedure to synthesize the missing low
frequency data. Our method is some sort of analytic continuation in
the spirit of the work of Landau, Slepian and Pollak [10]. It is clear
that one can’t expect miracles from such an approach due to the inherent instabilty of analytic continuation. However, we shall demonstrate
by numerical estimates of the eigenvalues of the extension operator
and by numerical simulation that quantitatively correct images can
1
be reconstructed with acceptable stability, provided the dip angle of
the velocity distribution is not too large. Another restriction of our
approach is that it uses the Born approximation.
The outline of the paper is as follows. In the next section we solve
the inverse scattering problem for the wave equation with sources and
receivers on a horizontal plane in the Born approximation. It is shown
that the Fourier transform of the velocity distribution is determined
by these reflection data outside a torus of radius k with a circle of
radius k around the origin in the horizontal plane. This fact was
recognized long ago by Wu and Toksöz [11] and Mora [5]. An elegant
way to see this is the plane wave decomposition of the free space
Green’s function, as was done in [2], [7]. In section 3 we describe
the data completion process by a second kind integral equation. The
spectrum of the integral operator reveals the same dichotomy as the
operators in the Landau-Slepian-Pollak theory [10]. By computing
the largest eigenvalue numerically we find the spatial and frequency
ranges for which the second kind integral equation can be solved with
acceptable stability. In section 4 we present numerical examples which
simulate simple seismic models.
2 The inverse scattering problem and
Fourier analysis
We study the following inverse problem: Consider the initial value
problem of the wave equation
∂2u
(x, t) = c(x)2 (∆u(x, t) + δ(x − s)q(t)) for t > 0, x ∈ Rn ,
∂t2
u = 0 for t < 0.
(1)
(2)
Here, c = c(x) is the sought-for speed of sound in the medium which
is situated in the half space xn > 0, n = 2, 3. We assume that with a
known constant background speed c0 , c2 = c20 /(1 + f ) with a certain
function f . s = (s0 , 0)> , r = (r0 , 0)> denote sources and receivers,
resp., sitting on xn = 0, and q is the source pulse which vanishes for
t < 0. The inverse problems consists in finding c, i. e. f , from the
measurements of u(x, t) for all s, r on xn = 0, t > 0; see Fig. 1.
In the following we make frequent use of the Fourier transform in
2
Figure 1: Sources and receivers are sitting on the plane/line xn =0.
Rn , for which we use the notation
fˆ(ξ) = (2π)−n/2
Z
e−ix·ξ f (x)dx
(3)
Rn
A 1D Fourier transform with respect to t takes (1,2) into
∆u(x, ω) +
ω2
u(x, ω) = δ(x − s)q̂(ω)
c(x)2
(4)
which has to be complemented by the outgoing Sommerfeld radiation condition. For each source position s0 , each receiver position r0
and each frequency ω we put gω (r0 , s0 ) = u(r0 , 0, ω). This is our data
function. It has been shown in [7] that, after linearisation the inverse
problem is solved in the Born approximation by
fˆ(ρ0 + σ 0 , a(ρ0 ) + a(σ 0 )) = −
c20
ĝω (ρ0 , σ 0 )
(2π)n/2 γn2 ω 2 a(ρ0 )a(σ 0 )q̂(ω)
(5)
where a(ξ 0 ) = k 2 − |ξ 0 |2 , k = ω/c0 , ĝ is the 2(n − 1) D Fourier
transform of the data function with respect to r0 and s0 , and γn = 1/4π
for n = 2 and 1/8π 2 for n = 3. Assuming that q̂(ω) is significantly
different from 0 for ωmin ≤ ω ≤ ωmax we find that the nD Fourier
transform fˆ of f is determined by the data for ξ inside the circle of
radius 2kmax and outside the domain W ; Fig. 2. In the present context
kmax is much bigger than kmin and can be considered as infinitely
large.
p
3
Figure 2: The Fourier transform fˆ is determined by the data in the blue
domain. For n = 2 this domain is the circle of radius 2kmax , except for the
two circles of radius kmin around (±kmin , 0), which together make up the
domain W . For n = 3 one has to rotate this picture around the ξn axis.
4
3
Data completion
Let f be a function in Rn whose Fourier transform is known outside
a domain W (e. g. the W depicted in Fig. 2). Then, by taking the
inverse Fourier transform, we obtain
f (x) = (2π)
−n/2
Z
ix·ξ
e
fˆ(ξ)dξ + (2π)−n/2
Z
eix·ξ fˆ(ξ)dξ.
Rn \W
W
The second term, which we denote by fW (x), is known. In the first
term we express fˆ(ξ) by the Fourier integral, obtaining
f (x) = (2π)−n
Z
Z
Rn
ei(x−y)·ξ f (y)dξdy + fW (x).
W
Putting
KW (x) = (2π)−n
Z
eix·ξ dξ
(6)
W
this can be written as the second kind integral equation
f = KW f + fW , (KW f )(x) =
Z
KW (x − y)f (y)dy.
(7)
We consider this integral equation only in a domain B that contains
the support of f . It is this integral equation on which our completion
process is based. In principle it can be solved by the obvious iteration
f ← KW f + fW .
(8)
A preliminary study has already been presented in [8]. Integral equations of this type occur in the Landau-Slepian-Pollak theory [10]. In
the simplest case n is 1 and W the interval [−k, k]. Then KW (x) =
(k/π)sinc(kx) and the integral equation is considered in the interval
B = [−r, r]. For this case the integral operator has been carefully studied in [10]. Its eigenfunctions turn out to be the prolate spheroidal
wave functions, and the eigenvalues are all in (0, 1), with about 2kr/π
of them being virtually 1, the other ones virtually 0. From this it
is clear that the solution of integral equation such as (7), although
unique, is very unstable.
However, due to the special geometry of the set W in Fig. 2, and
under restrictions on the lateral variation of the true solution f to be
explained below, we obtain a much more favorable result. First we
5
Figure 3: The half-moon shaped areas in the Fourier domain in which fˆ has
to be reconstructed for α = 45◦ . The Figure is for n = 2. For n = 3 the
figure has to be rotated around the ξ3 axis.
compute the kernel function KW (6) for the set W from Fig. 2. From
the formula
Z
Jn/2 (k|x|)
eix·ξ dξ = (2π)n/2 k n
(k|x|)n/2
|ξ|<k
(see e. g. formulas VII.1.2 and VII.1.3 in [6]) we conclude that for
n=2
J1 (kmin |x|)
2
KW (x) = 2(2π)−1 kmin
cos (kmin x0 )
.
kmin |x|
The restriction we impose on the lateral variation of f is as follows.
In the jargon of seismic imaging we assume that the dip angle of the
seismic profile which is represented by f is not too big. Let Dα , 0 ≤
α ≤ π/2 be the projection that annihilates the Fourier transform in
the cone |ξ 0 | ≥ tan(α)|ξn |. We say that f has a dip angle smaller than
α if Dα f is negligible. Then only the Fourier components of f in the
half-moon shaped areas depicted in Fig. 3 have to be computed. The
relevant operator for solving (7) is now Dα KW Dα . The eigenvalues of
this operator behave similarly to those of the 1D case, but the largest
eigenvalue is significantly smaller than 1. We computed the largest
eigenvalue by the power method, restricting f to a rectangle with
6
Table 1: Largest eigenvalues of Dα KW Dα for depth 2.
kmin
π
3π
5π
α = 15◦
0.198
0.274
0.341
α = 30◦
0.324
0.609
0.864
α = 45◦
0.396
0.856
0.977
depth 2 and width 8. Some typical values for the largest eigenvalue
are collected in Table 1. This table shows what we can reasonably
expect as convergence and stability properties of the iteration (8).
In the application to seismic imaging it means the following. Assume that we have a background speed of 2 km/sec down to a depth
of 2 km, and that the lowest frequency in the source pulse is 5 Hz.
Then ωmin = 2π × 5 Hz=10π/sec, kmin = ω/c0 = 5π/km and the
largest wavelength 2π/kmin = 400 m. Thus the depth is five times
the largest wavelength. From the last row of the table we see that for
a dip angle restricted to 30◦ we have a maximal eigenvalue of 0.864,
indicating reasonable convergence and stability properties of the data
completion process.
We do not have a theoretical explanation for the behavior of the
eigenvalues. For the time being the success of the method depends
entirely on numerical evidence.
4
Numerical experiments
We did some numerical experiments with the 2D seismic structure in
Fig. 1. We first computed fW to show what one can obtain without
data completion and corrupted it with 10% multiplicative Gaussian
noise. With the corrupted fW we did 15 steps of the iteration (8) using
0 as initial approximation, and applied Dα to the result. After 15 steps
the iteration became stationary. Cross sections along the vertical axis
in Fig. 1 of the original structure f , the corrupted structure fW
as obtained without the completion process, and the final result are
displayed in Fig. 4. It is obvious that the improvement achieved by
the completion procedure is substantial. The experiments were done
with the values of kmin = 5π/km , α = 30◦ and a depth of 2 km,
as mentioned above. For the fast Fourier transform we used the the
7
Figure 4: Vertical cross sections through center of original 2D structure from
Fig. 1 (green), without completion process (blue), and final result after 15
iterations (red)
8
Numerical Recipes [9].
5
Conclusions
The analysis and the numerical results of the previous sections are
valid only in the Born approximation. They show that, under this
restriction, stable reconstruction is possible under much weaker conditions than predicted by plain Fourier analysis. The restriction on
the dip angle is natural, since it is well known that in the Born approximation laterally invariant media can be reconstructed from spectrally
incomplete data, as can be seen from Fig. 2. It is not clear yet whether
or not, and in case yes how, this can be extended to the fully nonlinear
problem. Obvious procedures, such as projecting onto the the space
of functions with small dip angle, don’t seem to work. Also other
practical aspects, such as apertures and discrete sampling, may create
additional difficulties and have to be discussed in the future.
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