An Extension of Linear Inverse Scattering Methods for Absorptive Media to the Case of an Absorptive Reference

1. 70th
EAGE Conference & Exhibition — Rome, Italy, 9 - 12 June 2008
P175
An Extension of Linear Inverse Scattering
Methods for Absorptive Media to the Case of an
Absorptive Reference
K.A. Innanen* (University of Houston), J.E. Lira (University of Houston) & A.
B. Weglein (University of Houston)
SUMMARY
We cast and present inverse scattering quantities appropriate for the description of a two-parameter (P-
wave velocity and Q) absorptive medium given an absorptive reference, and present a tentative procedure
for carrying them out on measured seismic primary data. We note particularly that (1) this procedure
involves a Q compensation component, and therefore must be expected to require regularization in the
presence of noise, and (2) the formalism does not tend to our earlier non-absorptive reference procedure as
reference Q goes to infinity; the absorptive, or the non-absorptive reference case must be chosen at the
outset. These linear inverse results form part of a developing framework for direct non-linear Q
compensation, or data-driven enhancement of resolution lost due to absorptive processes.

2. Introduction
There exists a wide range of techniques for determining, and compensating for, Q from and
within reflection and transmission seismic data (Tonn, 1991). Most estimation techniques obtain
“Q information” from seismic data sets by observing the evolution of the spectra of echoes (or
direct waves) over an interval in time or space, whether directly (e.g., Rickett, 2007), or within
regularized inversion settings (Zhang and Ulrych, 2002).
The inverse scattering series (Weglein et al. , 2003), which admits a broad class of wave
models, including those associated with absorptive media, is being investigated as a means
to derive direct non-linear Q estimation and compensation algorithms (Innanen and Weglein,
2005). As a product of this investigation, a linear inverse scattering procedure for determining
(to first order) arbitrary multidimensional variations in P-wave velocity and Q from reflected
primary waves has recently been presented (Innanen and Weglein, 2007). In particular we noted
the distinct way in which these equations of inverse scattering demand that “Q information” be
detected in the data — through the variability of the reflection coefficient with frequency and/or
plane wave incidence angle. We also pointed out that in a recent description and parametrization
of absorptive-dispersive reflections of essentially the kind we must use, de Hoop et al. (2005)
have specifically advocated using these types of variations to drive inverse procedures.
The output of the above linear procedures may be used in either of two ways. First, if the
perturbations are small, and we are identifying a single interface below a well-characterized
overburden, it may be used as a means of direct Q estimation, i.e., absorptive medium identifi-
cation. Second, if the perturbations are large and sustained, the linear inverse output becomes
the input to higher order, non-linear algorithms, in which the data is used to directly construct
operators for Q-compensation. The latter can be accomplished in the form of full Q compensa-
tion, or a correction of dispersion only, which removes much of the sensitivity of the processing
to noise. Therefore it is correct to think of these procedures as first stages in a framework for
non-linear, direct recovery of the resolution lost through processes of absorption.
To date, these inverse scattering methods have involved reference media that are non ab-
sorptive, thereafter perturbing them such that the actual medium is properly absorptive. Since
the reference medium is assumed to be in agreement with the actual medium at and above
the source and measurement surfaces, this choice disallows at the outset any environment in
which the sources and receivers are embedded in an absorptive material. To complement, then,
the existing procedures, appropriate when the actual medium near the sources/receivers is non-
absorptive, we present a linear inverse procedure using an absorptive reference, appropriate
when the actual medium near the sources/receivers is absorptive.
Scattering quantities
The linear data equations will require forms for the absorptive reference Green’s functions and
an appropriate scattering potential. We use
G0(xg, zg, x , z , ω) =
1
2π
dkgeikg(xg−x ) eiqg|zg−z |
i2qg
,
G0(x , z , ks, zs, ω) = eiksxs
eiqs|z−zs|
i2qs
,
(1)
where q2
g = K2 − k2
g, etc., and K = ω
c0
1 + i
2Q0
− 1
πQ0
log ω
ωr
as per Aki and Richards
(2002). The scattering potential V is defined as the difference between reference and actual
absorptive differential operators. Defining F(ω) = i/2 − 1/π log ω
ωr
, we have
V =
ω2
c2
0
1 +
F(ω)
Q0
2
−
ω2
c2(x)
1 +
F(ω)
Q(x)
2
. (2)

3. We next require a suitable way of expressing the two medium variables, c and Q, in a perturba-
tional form. Defining
α(z) = 1 −
c2
0
c2(x)
β(z) = 1 −
Q0
Q(x)
,
(3)
and, noting (1) that even if the reference medium is highly attenuative, e.g., Q0 = 10, the terms
in 1/Q2
0 will be an order of magnitude smaller than those in 1/Q0, and (2) that terms in the
product αβ are generally small also, neglecting smaller terms, we have, upon substitution,
V ≈
ω2
c2
0
1 + 2
F(ω)
Q0
α(x) + 2
ω2
c2
0
F(ω)
Q0
β(x). (4)
In this form the component of V that is linear in the data, V1, is straightforwardly expressed in
terms of the components of α and β that are themselves also linear in the data, α1 and β1, as
V1 =
ω2
c2
0
1 + 2
F(ω)
Q0
α1(x) + 2
ω2
c2
0
F(ω)
Q0
β1(x). (5)
The quantities in equations (1) and (5) are next used to construct the linear data equations.
A procedure for linear inversion over a depth-varying perturbation
We proceed similarly to Clayton and Stolt (1981). We assume for present convenience (1) that
the linear component of the scattering potential is a function of depth z only, and (2) we have line
sources occupying the entire plane zs, and a single line receiver at (xg, zg). Upon substitution
of equations (1) and (5) into the first equation of the inverse scattering series, viz.
D (xg, zg, ks, zs, ω) = S(ω) dx dz G0(xg, zg, x , z , ω)V1(z )G0(x , z , ks, zs, ω), (6)
where S is the (known) source wavelet, we have
D(ks, ω) = α1(−2qs) + W(ω)β1(−2qs), (7)
where W(ω) = 2F(ω)
Q0
1 + 2F(ω)
Q0
−1
, and D is related to D by
D(ks, ω) = −4S−1
(ω) 1 +
2F(ω)
Q0
−1
q2
s c2
0
ω2
e−iksxg
eiqs(zg+zs)
D (xg, zg, ks, zs, ω). (8)
D should be thought of as the measured data, pre-processed as above to produce D. Equations
(7) are the heart of the inversion, and, c.f. Innanen and Weglein (2007), the variability of W
with temporal frequency for any given spectral component of the model parameters α1 and β1
determines the conditioning of the problem. Defining the depth wavenumber over which our
perturbations are to be solved to be kz = −2qs, the equations become
D(ks, ω) = α1(kz) + W(ω)β1(kz). (9)
At this stage we have several options. Ideally, we would subdivide the data into components
D(kz, θ) and solve the linear problem with sets of angles. However, the (kz, θ) parametrization
turns out to be inconvenient here, as there is no straightforward way of solving for ω(kz, θ). A
more convenient choice, since the data equations are independent directly in terms of ω already,
is to change variables from D(ks, ω) to D(kz, ω), and solve at each kz using a set of N >

4. 2 frequencies. To proceed in this way, we need to know what ks value is associated with a
particular pair kz, ω. From the plane wave geometry we have
k2
s + q2
s =
ω2
c2
0
1 +
F(ω)
Q0
2
, (10)
hence
ks(kz, ω) =
ω2
c2
0
1 +
F(ω)
Q0
2
−
k2
z
4
. (11)
We then have the following prescription for performing the linear inversion:
1. From experimental values and from its definition, determine a suitable (complex) wavenum-
ber vector kz.
2. Find in the data D (kz, ω) = dtdxse−iωte
−i
r
ω2
c2
0
h
1+
F (ω)
Q0
i2
−
k2
z
4
xs
D (xs, t).
3. Process from D → D using reference medium quantities.
4. Now D(kz, ω) = α1(kz) + W(ω)β1(kz) holds; solve for α1 and β1 for each kz using
pairs (or larger sets) of frequencies ω1 and ω2.
5. Invert for α1(z|ω1, ω2) = 1
2π dkzeikzzα1(kz|ω1, ω2) and β1(z|ω1, ω2)
= 1
2π dkzeikzzβ1(kz|ω1, ω2). This is expected to be an unstable process, and the re-
quirement of some dampening of large kz values should be anticipated, especially in the
presence of noise.
Conclusions
We present an extension of some recent linear inverse scattering methods for absorptive media;
here the reference medium too is considered absorptive. This procedure complements the earlier
linear inverse procedure for non-absorptive reference media. We see, importantly, that one or
other of these must be chosen at the outset; the current method does not tend to the previous
method as Q0 → ∞. In fact, if the actual Q values remain finite, the current theory does
not respond at all well in this limit, so, should a non-absorptive reference medium be deemed
necessary, the (entirely different) definition of the Q perturbation of Innanen and Weglein (2007)
must be invoked. The choice of one or the other reference medium will be determined by
the known nature of the material in which the sources and receivers are embedded; this is an
important choice, since we typically assume the reference medium and the actual medium to be
in agreement at the source and receiver depths. We further note that this current form of linear
inversion involves an amount of Q compensation, as evidenced in the inverse transformation
from the kz domain to the z domain. This sets it apart from its non-absorptive counterpart
method. However, in many ways the two remain of a kind. Both interrogate the data via the
frequency or angle dependence of the reflection strengths. And both represent frameworks,
and first steps, from within which to develop non-linear inverse algorithms with the capacity to
enhance resolution through direct, data driven operations.
Acknowledgments
We wish to thank the sponsors and personnel of M-OSRP. J. Lira was supported by Petrobras; K.
Innanen and A. Weglein were supported by U.S. D.O.E. Grant No. DOE-De-FG02-05ER15697;
A. Weglein was supported by NSF-CMG award DMS-0327778.

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