Globally optimized Fourier finite-difference migration method
Lian-Jie Huang
and Michael C. Fehler, Los Alamos Seismic Research Center, Los Alamos National Laboratory
Summary
To accurately image complex structures with strong lateral ve-
locity variations and steep dips, we develop a globally opti-
mized Fourier finite-difference method that uses a rational ap-
proximation of the square-root operator in the one-way wave
equation. The method uses a split-step Fourier operator cou-
pled with a one-term optimized finite-difference operator. The
two coefficients in the rational approximation are obtained by
an optimization scheme that maximizes the maximum dip an-
gle of the method for a given model. Our opt imized method
uses the same coefficients throughout a model in contrast
to Ristow-R¨uhl’s locally optimized Fourier finite-difference
scheme which uses an optimized coefficient that varies with the
lateral velocity contrast and hence is cumbersome during im-
plementation. For small lateral velocity contrasts, our method
is accurate for dip angles of up to
90
Æ
like other Fourier finite-
difference methods. For large lateral velocity contrasts, max-
imum dip angles for our method are approximately
65
Æ
–
67
Æ
,
which are approximately
16
Æ
–
20
Æ
larger than those for Ristow-
R¨uhl’s unoptimized Fourier finite-difference method, while
Ristow-R¨uhl’s locally optimized scheme can handle approxi-
mately
16
Æ
larger dip angles than their unoptimized scheme.
The computational cost of our method is the same as the other
Fourier finite-difference methods.
Introduction
Migration methods that use fi nite-difference of the one-way
wave equation can handle arbitrarily large velocity contrasts but
are only accurate up to a fixed dip angle even in a homogeneous
region (Claerbout, 1985). The phase-shift migration (Gazdag,
1978) is accurate for dip angles of up to
90
Æ
but it requires
a laterally homogeneous velocity model. A hybrid approach,
termed the Fourier finite-difference (FFD) method proposed
by Ristow and R¨uhl (1994), has the advantage of both finite-
difference and phase-shift methods. The FFD method uses a
Taylor expansion of the square-root operator in the one-way
wave equation and the expansion is recombined into a rational
approximation. Its implementation uses the split-step Fourier
(SSF) propagator (Stoffa et al., 1990; Huang and Fehler, 1998)
followed by a finite-difference scheme to increase the accu-
racy for imaging structures with large lateral velocity contrasts.
Some other Fourier transform based methods, such as the ex-
tended local Born Fourier (Huang et al., 1999b), the extended
local Rytov Fourier (Huang et al., 1999a), and the quasi-Born
Fourier (Huang and Fehler, 2000) methods, are more accurate
than the SSF method, but they are less accurate than the FFD
method for large lateral velocity contrasts. Another version of
FFD (Xie and Wu, 1998) is based on the approximation of the
square-root operator using the first order Pad ´e approximation
which is the same as Claerbout ’s
45
Æ
or Muir’s
R
2
approxima-
tions (Claerbout, 1985). It is less accurate than Ristow-R¨uhl’s
FFD method. We refer to the Pad´e-based method as the PFFD
method hereafter.
To increase the accuracy of the FFD methods, one can in princi-
pal add additional terms to the finite-difference operator. How-
ever, the computational cost of the finite-difference operator
increases proportionally to the number of terms added. To
increase the accuracy of the second-order FFD method while
using only one term in the finite-difference operator, Ristow
and R¨uhl (1994) proposed a locally optimized Four ier finite-
difference (LOFFD) scheme in which they use an optimized
expansion coefficient for each lateral velocity contrast. For a
heterogeneous model, the LOFF D method requires a large ta-
ble of the optimized coefficient. L ooking for a value of the
optimized coefficient for each grid point from a large table of
the coefficient makes the LOFFD method cumbersome during
implementation, especially in 3D cases.
We propose an FFD method based on a rational approxi mation
of the square-root operator. The form of our FFD scheme is
the same as that of the PFFD method. However, the two coef-
ficients in the rational approximation used in our FFD method
are determined using an optimization scheme that maximizes
the maximum dip angle for a given model, rather than us-
ing those of the one-term Pad ´e approximation. We give the
optimization algorithm, perform the error analysis of our op-
timized scheme, and compare impulse responses of different
methods. Our optimized FFD method does not require a table
of optimized coefficients because these coefficients are fixed
for a given model. Therefore, we call it the globally opti-
mized Fourier finite-difference (GOFFD) method. The GOFFD
method is accurate for dips with angles of approximately
16
Æ
–
20
Æ
larger than the FFD method, while the LOFFD scheme can
handle approximately
16
Æ
larger dip angles than their unopti-
mized scheme. The computational cost of the GOFFD method
is the same as the other FFD methods.
Expansion of Square-Root Operator
The one-way wave equation in the frequency-space domain is
@P
(
x; z
;
!
)
@z
=
iQ
(
x; z
;
!
)
P
(
x; z
;
!
)
;
(1)
where
P
is the pressure and the operator
Q
is defined by
Q
s
!
2
v
2
(
x; z
)
+
@
2
@x
2
=
!
v
(
x; z
)
R;
(2)
where
!
is the circular frequency,
v
is the velocity, and
R
is the
square-root operator given by
R
p
1
X
2
;
(3)
with
X
2
=
v
2
!
2
@
2
@x
2
:
(4)
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