1.1444077

GEOPHYSICS, VOL. 61, NO. 6 (NOVEMBER-DECEMBER 1996); P. 1575-1588,8 FIGS.
Waveform-based AVO inversion and
AVO prediction-error
James L. Simmons, Jr.* and Milo M. Backus$
ABSTRACT
A practical approach to linear prestack seismic inver-
sion in the context of a locally 1-D earth is employed to
use amplitude variation with offset (AVO) information
for the direct detection in hydrocarbons.
The inversion is based on the three-term linearized
approximation to the Zoeppritz equations. The normal-
incidence compressional-wave reflection coefficient Ro
models the background reflectivity in the absence of
hydrocarbons and incorporates the mudrock curve
and Gardner's equation. Prediction-error parameters,
ARsh and ORp , represent perturbations in the normal-
incidence shear-wave reflection coefficient and the
density contribution to the normal incidence reflectivity,
respectively, from that predicted by the mudrock curve
and Gardner's equation. This prediction-error approach
can detect hydrocarbons in the absence of an overall in-
crease in AVO, and in the absence of bright spots, as
expected in theory.
Linear inversion is applied to a portion of a young,
Tertiary, shallow-marine data set that contains known
hydrocarbon accumulations. Prestack data are in the
INTRODUCTION
Considerable research has gone into attempting to extract
earth parameters (or relative changes in parameters) from
seismic data. One-step linearized inversion of prestack seismic
reflection data is possible, given a good background velocity
estimate, a good wavelet estimate, and processed data domi-
nated by primary reflections. Linearized inversion attempts to
estimate parameter changes relative to a smooth background
that is known and held fixed (Clayton and Stolt, 1981; Wiggins
et al., 1984; Stolt and Weglein, 1985; Smith and Gidlow, 1987).
A fundamental weakness of these simple weighted-stack ap-
proaches is that many effects which are present in real data
form of angle stack, or constant offset-to-depth ratio,
gathers. Prestack synthetic seismograms are obtained by
primaries-only ray tracing using the linearized approxi-
mation to the Zoeppritz equations to model the reflec-
tion amplitudes. Where the a priori assumptions hold, the
data are reproduced with a single parameter R0 . Hydro-
carbons are detected as low impedance relative to the
surrounding shales and the downdip brine-filled reser-
voir on R0 , also as positive perturbations (opposite po-
larity relative to R0 ) on ARsh and ARp . The maximum
perturbation in ARsh from the normal-incidence shear-
wave reflection coefficient predicted by the a priori as-
sumptions is 0.08. Hydrocarbon detection is achieved,
although the overall seismic response of a gas-filled
thin layer shows a decrease in amplitude with offset
(angle).
The angle-stack data (70 prestack ensembles, 0.504-
1.936 s time range) are reproduced with a data residual
that is 7 dB down. Reflectivity-based prestack seismo-
grams properly model a gas/water contact as a strong
increase in AVO and a gas-filled thin layer as a decrease
in AVO.
cannot be comprehended by the primitive forward modeling
implicit to these methods. Extreme sensitivity to the time align-
ment of the primary reflections and failure to comprehend ef-
fects such as the seismic wavelet, source and receiver array
response, and thin layering have hampered attempts at abso-
lute rock property recovery. The more successful applications
of AVO have resulted when used in a relative sense to detect
anomalies (Smith and Gidlow, 1987; Chiburis, 1987).
Background velocity refinement is a potential advantage of
iterative inversion approaches that use more exact forward
modeling. These iterative approaches typically use reflectivity
or finite-difference methods for prestack modeling. The veloc-
ities can adjust to better account for the reflection traveltimes.
Manuscript received by the Editor March 6, 1995; revised manuscript received December 5, 1995.
*Formerly Department of Geological Sciences, The University of Texas at Austin; presently Bureau of Economic Geology, University Station,
Box X, Austin, TX 78713-7908.
$Department of Geological Sciences, The University of Texas at Austin, P.O. Box 7909, Austin, TX 78713-7909.
© 1996 Society of Exploration Geophysicists. All rights reserved.
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1576

Simmons and Backus
McAulay (1985) used reflectivity modeling and was able to re-
cover the background velocities albeit for acoustic synthetic
data in which the wavelet was known. Attempts at recover-
ing the background velocity from real data inversions have not
been particularly successful (Mora, 1987; Assous et al., 1989;
Pica et al., 1990; Wood, 1993).
Another potential advantage of inversion methods that use
nonlinear elastic forward modeling is the exploitation of long-
period converted shear waves and surface multiple reflections.
However, the low-frequency shear-wave velocity is generally
unknown, making it most unlikely that the long-period shear
waves can be modeled accurately. In our opinion, it is doubt-
ful that surface multiple reflections can be modeled adequately
without comprehending the 3-D nature (bedding plane dip and
curvature) of the subsurface (Tsai, 1981; Backus and Simmons,
1984). Huston and Backus (1989) show real data examples
where primary reflections tie at line intersections but multiple
reflections do not. However, to our knowledge, perstack non-
linear inversion of 3-D data sets has not yet been attempted.
McAulay (1985) and Wood (1993) have shown that the high-
frequency changes in rock properties are recovered primarily
after a single iterate, assuming a reasonably logical background
starting model. The Frechet derivatives are predominantly con-
trolled by the high-frequency perturbations in the rock prop-
erties. Wood (1993) applied nonlinear inversion in the r—p do-
main to several real seismic data sets. A main conclusion of his
work is that the same information could have been obtained
using a single-iterate, strictly linear inversion.
We present a practical approach to maximum-likelihood, lin-
ear, prestack seismic inversion and apply the technique to real
seismic data in the context of a locally 1-D earth. Develop-
ment of the algorithm is identical to waveform-based inversion
methods that directly compute the Frechet derivatives.
The three-term linearized approximation to the Zoeppritz
equations describes the reflection amplitudes as a function of
angle. This expression is reformulated to consider a term that
models the background reflectivity based on the a priori re-
lationships between the compressional and shear velocity, and
between the compressional velocity and density. One term is
sufficient to model the reflection amplitudes where the a priori
assumptions hold. Two terms describe the perturbations rela-
tive to the a priori assumptions and detect the presence of hy-
drocarbons as anomalies relative to the background reflectivity.
A simple, computationally efficient, forward-modeling
method produces the Frechet derivatives. Prestack modeling
uses primaries—only ray tracing and a linearized approxima-
tion to the Zoeppritz equations to model the response of a
stack of thin layers. The effects of thin layering, the seismic
wavelet, normal-moveout stretch, and the receiver array re-
sponse are incorporated into the full-waveform modeling and
the calculation of the Frechet derivatives.
A diagonal prior model covariance matrix is more defensi-
ble with the prediction-error form of model parameterization.
A single-iterate maximum-likelihood solution incorporates the
expected changes in the model parameters as well as specifica-
tion of the noise in the data.
FIG. 1. Constant angle-stack, or constant offset-to-depth ratio (ODR), sections. The trace spacing is 50 m. (a) 18° section, b) 38°
section. A growth fault produces the anticlinal feature that serves as the structural trap for hydrocarbons within the 1.45-2.0 s time
window over CDP's 30-70.

AVO Inversion

1577
Application of the algorithm to real seismic data shows that
hydrocarbons are detected directly in the model parameters
as deviations from the a priori assumptions caused by their
anomalous amplitude variation with offset (AVO) behavior.
We also present several approaches for evaluating the inversion
results by comparing the predicted (modeled) data with the
observed data.
PARTIAL-RANGE STACK DATA
We analyze our prestack data in the form of angle-stack,
or offset-to-depth ratio (ODR) gathers. Elements of the par-
tial offset-range stacking procedure are discussed in Todd and
Backus (1985) and Huston and Backus (1989). Partial offset-
range stacking produces a suite of six traces for each common
depth-point (CDP) gather. Each trace approximates a con-
stant local-incidence angle and a constant ODR. The signal-to-
noise ratio is increased by the partial stacking, and the offset-
dependent information is retained out to a local incidence angle
of 46°. The partial-range stack design provides a roughly uni-
form multiple reduction for all traces. Long period multiples
and converted waves are attenuated by the partial stacking,
thus, producing a prestack data set that better conforms to our
linearized forward modeling assumptions.
Figure 1 shows a portion of a dip line in shallow marine, a
Tertiary clastic basin that contains several hydrocarbon accu-
mulations. Growth faulting produces a rollover anticline that is
the trapping mechanism for hydrocarbons within the 1.45-2.0 s
time range. The geologic section is composed of sands and
shales. The data are two ODR sections obtained by applying
time- and offset-variant mutes to the normal moveout-
(NMO-)corrected CDP gathers. Gain proportional to t 2 is ap-
plied to the CDP gathers prior to NMO correction and partial
offset-range stacking. The t2 gain roughly corrects for geomet-
ric divergence and anelastic attenuation (Claerbout, 1985).
Figure 1a is the 18° section, and the 38° section is shown in
Figure lb. Gapped deconvolution is applied after the partial
offset-range stacking to attenuate the water-layer reverbera-
tions (Simmons, 1994). The data are unmigrated. A gas-water
contact is apparent as a flat spot at 1.46 s on CDP's 45-70.
A thin gas-filled layer produces a bright spot near 1.72 s on
CDP's 30-70. Backus and Chen (1975) analyzed the producing
hydrocarbons using stacked seismic data and well logs.
Given the gain treatment applied to the data, most of the
reflections show a decrease in amplitude versus angle (AVA).
Notice that the flat spot shows a strong increase in AVA, while
the bright spot shows a slight decrease in AVA. In general,
the fluid anomaly signal produced by the change in pore fluids
from brine to hydrocarbons is expected to increase in AVA and
must be separated from the background reflectivity for AVA
analysis to be most successful at detecting hydrocarbons.
THE INVERSE PROBLEM
In general, for prestack seismic reflection data, nonlinear
forward modeling relates the model parameters, m, to the data,
d, as
d=g(m)+n, (1)
where g is the nonlinear forward modeling operator, and n
is noise. In the following development, we assume that we are
dealing with vectors and matrices as a result of having discretely
sampled data and a finite-dimensional model space.
We follow the approach of Tarantola (1987). by assuming
that the probability density functions describing the a priori
data errors, model parameter uncertainties, and errors in the
forward modeling are Gaussian. We seek a model m, such that
we minimize the objective function, S(m),
S(m) = 2 [(g(m) — d)T Cd t (S(m) — d) + (m — mprior)
T
X Cyn l (m — mprior)]• (2)
The a priori values of the model parameters are contained in
mprior . The data covariance matrix Ca describes the noise in
the data. The model covariance matrix ,1, describes the un-
certainties in the a priori model.
We assume that the forward modeling can be linearized
around an initial model mprior as
g(m) — g(mprior) + G(m — mprior), (3)
where G represents the Frechet derivatives evaluated at the
initial model.
Our model parameter estimate is obtained as
m = mprior + [G T Cd 1 G + Cnil ]-1
x G T Cd t (dohs — g(mprior)) , (4)
where d„h, now denotes the observed data.
Equation (4) is the most general form for the least-squares
solution to a linearized problem. We provide the specific solu-
tion to our problem after discussing the model parameteriza-
tion and the covariance matrices.
Model parameterization
Relationships between the model parameters have impor-
tant implications for the inverse problem. When the model
parameters are related, off-diagonal terms are implied in the
model covariance matrix. Specified relationships between pa-
rameters can be incorporated into the model parameterization
resulting in new model parameters that are more nearly in-
dependent. The model covariance matrix of the new model
parameters is more truly diagonal.
The three-term linearized approximation to the Zoeppritz
equations (Aki and Richards, 1980) is most commonly written
in terms of the fractional changes in the elastic properties. The
linearized approximation expresses the reflection coefficient as
a function of the incident angle R(0) as
1 Oa ^p^ 1 ^Da ^p
1
x sine ^9 + as) sing tan2 , (5)
where 4 = 4(8/a)2, 0 is the incident angle, and a, $, and p rep-
resent the local average background compressional velocity,
shear velocity, and density, respectively. The fractional changes
in compressional velocity, shear velocity, and density are de-
noted as Aa/a, 0^/,B, and Lip/p, respectively. We assume an
isotropic locally 1-D earth in which a, ,B, and p vary with ver-
tical traveltime (depth).

1578

Simmons and Backus
We define the normal-incidence compressional-wave and
shear-wave reflection coefficients, Rp and Rh, as
Rp = Ra + Rp , and Rsh = Rp + Rp , (6)
where Ra = 1 (Aa/a), R# = 1 (D$/,B), and Rp = 2 (Op/p).
Equation (5) can be written in terms of Rp, Rsh, and Rp as
R(t) = [1 + sin2 09 + sin2 t tan2 1]Rp — 2T sin2 ^9Rsh
+ [W — 1 — tan2 ] sin2 ^9Rp . (7)
We note that there are no prior assumptions regarding relation-
ships between any of the parameters involved in equation (7).
We now assume that Gardner's equation (Gardner et al.,
1974) relates density to the compressional-wave velocity. In
terms of the fractional changes in a and p, Gardner's equation
is (Fatti et al., 1994)
^p = 0.25 ^a , (8)
p a
and in terms of the reflection coefficients
Rp = 0.25Ra . (9)
Following equation (6), we define a normal-incidence compres-
sional-wave reflection coefficient Ro as
The offset-dependent reflection coefficient produced by
these a priori assumptions is denoted as RCG (>g). The mudrock
curve and Gardner's equation are incorporated into the coef-
ficient that weights R0 , and ideally are valid in areas where
hydrocarbons are not present. These assumptions model the
background reflectivity and predict RPA and R$ hA (and RaA)
from R0 .
The presence of hydrocarbons produces deviations from the
background reflectivity. Lithological variations may also pro-
duce an anomalous AVA response relative to the background.
To account for these effects, we express Rh and R P in terms
of RPA , RShA, and the prediction-error components ARsh, and
ARP , as
Rsh = RShA + OR,sh, and Rp = RPA + ORp , (19)
where
RShA = R,iA +RPA , and ARsh = AR,^+ARp, (20)
with
ARS = Rp — RPA , or ORf = R,g — 0.8kR0 , (21)
and
ARP = Rp — RPA , or ARP = Rp — 0.2Ro . (22)
where
and consequently
Ro = RaA + RPA , (10)
RPA = 0.2R0 , (11)
RaA = 0.8R0 . (12)
The extent to which ARsh 0 and/or ARP zA 0 suggests that the
prior assumptions are violated. As a result, the AVA response
of the observed data deviates from the AVA response of the
background that is predicted by the smooth compressional-
wave velocity, the mudrock curve, and Gardner's equation.
The angle-dependent reflection coefficient R() can now be
expressed in terms of RcG(t ), AR.rh, and ARP as
The additional subscript A indicates that these parameters are
estimated from Ro using a priori assumptions. RPA is simply
a scaled version of RaA . Nonzero values of Ro are distributed
among RaA and RPA according to equations (11) and (12).
We also assume that the mudrock curve (Castagna et al.,
1985) relates the shear-wave velocity to the compressional-
wave velocity as
_ a-1360
1.16
(13)
where the velocities are expressed in m/s.
The mudrock relation produces a relationship between Rp
and Ra as
R, = kRa , (14)
where k = 0.86(a/fi). Making use of equation (12)
R A = 0.8kR0 , (15)
and following equation (10) we obtain

RShA = RPA + RPA, (16)
RShA = (0.8k + 0.2)R0 . (17)
Invoking Gardner's equation and the mudrock relation,
R(i) can be expressed solely in terms of Ro as
RCG(1) _ [1 + (0.8 — 0.2W — 1.6kW) sine ^9
+ 0.8 sin2 tan2 9]Ro . (18)
R(#) = RCG(#) — 2W sin2 Rsh
+ [W — 1 — tan2 0] sin i90Rp . (23)
R(?9) is modeled as a linear combination of RCG() which
obeys the a priori assumptions, and deviations from our expec-
tations of the shear-wave reflectivity and density component
of the reflectivity, 1Rsh and ORp , respectively. Note that in
the case of ARP = 0, OR,sh = ARC which is the fluid factor of
Smith and Gidlow (1987).
Assume that the mudrock curve and Gardner's equation
hold exactly. At shallow depths where the compressional ve-
locity is near water velocity (WY ~ 0, since ,B ti 0), Rcc(#)
increases strongly with angle (roughly as 1 + 0.8 tan 2 t). With
increasing depth, the T sine 0 term increases in relative magni-
tude, and RCG (0) decreases with angle. At producing depths in
Tertiary clastic basins where these assumptions hold, we expect
the background reflectivity to decrease in amplitude with off-
set (angle). When the background reflectivity decreases with
offset, hydrocarbons should become more readily apparent as
deviations from the background.
Prior model covariance: C.
The choice of model parameters affects the Frechet
derivatives and the model covariance matrix. Differences
in the model covariance matrices occur because different
model parameterizations imply different correlations between
parameters. Certain parameterizations require off-diagonal

AVO Inversion

1579
terms while others result in a diagonal model covariance ma-
trix.
Consider a three-parameter inverse problem such as that im-
plied by equation (5). The most general form of the model co-
variance matrix for a single layer appropriate for equation (5) is
Ua ffap bap
Cm = ffga ff CTP
(24)
apa aPj ffP
where o- . is the variance in parameter m 1 relative to the a priori
model, and ami „2 . is the covariance between parameters m, and
m1. Off-diagonal terms are usually neglected in C,^, which er-
roneously implies that the model parameters are independent.
The exact relationship between the model parameters, and
thus the covariance, is not known exactly. We can express the
degree of implied correlation and determine the covariance
through a correlation coefficient as
ffm.m. ffmimJSmi m^ _ = (25)
62-r ffm.ffmJ
where E„^,.,,,^ is the correlation coefficient bounded as —1 <
Em m1 < 1. The covariances are determined from the a pri-
ori expected standard deviations of the parameters 5mi and
0m1 , and the expected correlation between parameters Em . m1 .
If E,,,imJ = 1, 0, or —1, the parameters are perfectly corre-
lated, not correlated, or are perfectly anticorrelated, respec-
tively. Wang (1990) also discusses this point.
While the specification of the off-diagonal terms is relatively
straightforward, an assumption as to the expected degree of
correlation (validity of the a priori assumptions) is required.
An alternate approach is to use the model parameterization
of equation (23). Perturbations from the mudrock curve and
Gardner's equation, ARsh and AR., are more nearly indepen-
dent from each other and from changes in the compressional
reflectivity Ro . This leads to a diagonal model covariance ma-
trix as
ffRo 0 0
Cm = 0 ffoRsh 0 (26)
0 0 6AR
P
A diagonal model covariance matrix is much simpler to deal
with. If the a priori assumptions implicit in R o do not hold
exactly, deviations from these assumptions appear in the ARSh
and ARP terms.
Prior data covariance: ed
For noise in the data that is uniform and uncorrelated, the
data covariance is simply
Cd = ffd I (27)
where I is the identity matrix. The noise variance ad must be
specified in units that are meaningful with respect to the data
being inverted. Here, ad is defined as an apparent equivalent
reflectivity for the background noise. For example, if we assume
that a known reflection event corresponds to a particular re-
flection coefficient, the magnitude of the noise can be specified
relative to this reference reflector.
In this case, the reference reflection is the bright spot
discussed in Backus and Chen (1975). This event is a low-
impedance thin layer that has reflection coefficients of —0.1
and 0.1 at the top and base, respectively, and has a normal-
incidence two-way traveltime thickness of 11 ms. The apparent
layer thickness is modified for an 18° angle of incidence. This
layer response convolved with our seismic wavelet estimate
serves as the reference event. A single scalar is applied to the
real data so that the amplitude of the bright-spot reflection at
a reference location agrees with that of the reference event.
We note that this amplitude treatment is somewhat crude, is
model based, and does not comprehend the angle dependence
of the seismic wavelet.
Forward modeling
Primaries-only ray tracing using equation (23) to describe
the variation in reflection coefficient as a function of offset (in-
cidence angle) provides the basis for prestack modeling of the
ODR gathers. The earth model is parameterized as a stack of
thin layers. Traveltimes are obtained by ray tracing through the
smooth background compressional velocity. We have a reflec-
tion from the top and the base of each thin layer, and the layer
time thickness decreases with increasing offset. The top and
base reflection amplitude are +R(), controlled by the three
model parameters for that layer [equation (23)]. R () relates to
the contrast between the thin layer and the background, and is
equal in magnitude and opposite in sign for the top and base of
the layer. This approximately accounts for the locally converted
shear waves within the layer (Simmons and Backus, 1994).
For a stack of thin layers, each individual layer is modeled
independently relative to the smooth background, and the re-
sults are superposed to produce the complete seismogram. For
example, two adjacent layers with the same properties will thus
interfere to produce the seismic response of a single layer hav-
ing twice the thickness.
Convolution of the spike seismograms with an estimate of
the source wavelet produces the reflection seismogram as a
function of offset and traveltime. Receiver arrays are simu-
lated by calculating the seismograms at a fine spatial sampling
and then stacking over the receiver array length. ODR gathers
are then obtained by partial stacking of the normal-moveout
corrected seismograms. The effects of thin layering, the seis-
mic wavelet, the receiver array, NMO stretch, and the partial-
stacking response of each ODR are now incorporated into the
forward modeling.
The gain treatment applied to the real data consists of a
t2 gain applied prior to NMO correlation followed by the
model-based calibration to bring the real data amplitudes into
agreement with the modeled data amplitudes. Consequently,
the effect of geometric divergence is not included in the for-
ward modeling and the t 2 gain correction is not applied to the
modeled data.
Inversion and the G matrix
We seek to find adjustments relative to a smooth initial model
that reproduce the observed data. Since the initial earth model
is smooth, the model parameters Ro , ARsh, and ARP for each
layer are zero, thus,
mpnor = 0, and g(mprior) = 0,

1580

Simmons and Backus
because no reflections are produced. It is important to note
that the initial model implicitly contains the smooth back-
ground compressional velocity used for ray tracing and NMO,
the smooth background shear velocity, and the smooth back-
ground density. We solve for the adjustments relative to the
smooth initial model as
Am = [GT G + 0,2
C-1] GT dobs• (28)
The G matrix contains the Frechet derivatives that indicate
how the data are expected to change because of a perturbation
in a particular model parameter. A most attractive quality of
the linearized approximation is that it is linear with respect
to the unknown model parameters. For a smooth background,
the changes in R() for perturbations in each of the model
parameters R0 , ORsh, and ARp in equation (23) are simply
the coefficients that weight those terms. Seismograms for each
of these perturbations are obtained by ray tracing, convolution
with the source wavelet, stacking to simulate the receiver array,
NMO correction, and partial offset-range stacking.
Figure 2 shows the Frechet derivatives [determined using
equation (23)] associated with model parameters located at
three two-way vertical traveltimes. The seismograms are dis-
played in the form of ODR gathers. The earth model consists
of a stack of 180 layers spanning the two-way vertical travel-
time interval of 0.504-1.936 s. Each layer has a two-way vertical
traveltime thickness of 8 ms.
The seismic wavelet is approximately symmetric, therefore,
the reflection response from a thin layer produces an asym-
metric wavelet. A perturbation in Ro for a specified layer
(denoted by the superscript) generally produces a reflection
13 18 46 18 46
ODR 1 6 1 6
that decreases in magnitude with angle. The magnitude of
the overall decrease in AVA increases with increasing trav-
eltime (depth). Meanwhile, perturbations in ARsh and ARp
produce reflection events that increase in magnitude with an-
gle. For ORch, the reflection grows stronger with increasing
traveltime since increases (the ratio of shear-wave velocity
to compressional-wave velocity increases). The magnitude of
the reflection response of A R p does not increase continuously
with increasing vertical traveltime because of the tang term.
Perturbations in ARsh and AR, show a polarity opposite to
that of R0 .
Note the similarity of ARsh and ARp . This leads to poor
resolution between these two parameters, which we will note
in the real data example.
Figure 2 also diagramatically depicts the inverse problem.
We attempt to reproduce the observed data (Figure 2d) as a
weighted linear combination of the Frechet derivatives. The
model parameters R0 , ORS y,, and AR,o are the unknown weights
that are estimated by equation (28). In Figure 2, only three sets
of Frechet derivatives are shown. The G matrix for the real
data inversion contains 180 sets of Frechet derivatives.
These Frechet derivatives are comparable to those that
would be obtained using a reflectivity method for the forward
modeling, although obtained more efficiently. Differences in-
clude the omission of transmission losses, some converted shear
waves, and some multiple reflections.
Determining Cd
Ideally, Qawould represent an equivalent reflectivity for the
background noise, and in essence, define the signal-to-noise
18 46 18 46
1 6 1 6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
•1wifi+ =
ttttR ItttttARt ItttAR =
ftR38 38}38
itFIG. 2. Depiction of the inverse problem. (a), (b), and (c) are the Frechet derivatives for layers 13, 63, and 138, respectively. The
observed data (Figure 2d) are reproduced as a weighted linear combination of the Frechet derivatives and the unknown model
parameters R0 , ARsh, and ARP . ODR traces 1 and 6 correspond to local incident angles i9 =18, and = 46 0.

AVO Inversion

1581
ratio. The band-limited nature of the seismic wavelet causes
the true signal-to-noise ratio to vary as a function of frequency.
Relating a to the actual signal-to-noise ratio is beyond the
scope of this work. Wang (1990) explored this issue to some
extent. We proceed clearly recognizing that in a practical sense
v simply acts as a damping parameter.
An appropriate value for ad is determined empirically, and
artfully, based on the results of the linearized inversion applied
to a portion of the real data. The data are the ODR gather from
CDP 61 in Figure 1. ODR's 1-6 have approximate local inci-
dence angles of 18°, 26°, 32°, 38°, 42°, and 46°, respectively.
The smooth background compressional velocity used to gen-
erate the ODR gathers is adjusted independently at each CDP
location using the approach of Simmons (1994). Background
shear velocities and densities are derived from the compres-
sional velocity using the mudrock curve and Gardner's equa-
tion, respectively. A time- and angle-invariant minimum-phase
wavelet is estimated statistically from the 18° data and modified
based on a blue reflectivity assumption (Simmons and Backus,
1996). The amplitude of the seismic wavelet is calibrated using
the bright-spot reflection as a reference.
The earth model is parameterized with 180 layers spanning
the two-way traveltime interval from 0.504 to 1.936 s. Each
layer has a normal-incidence, two-way traveltime thickness of
8 ms. The smooth background parameters are held fixed in
the inversion. The unknown model parameters for each layer
are Ro , ARSh, and ARP . Consequently, there are 540 model
parameters to be estimated for the ODR gather.
Specification of the model covariance matrix is based on
a priori knowledge. The diagonal values represent the vari-
ance of the expected perturbations from the smooth starting
model. The model covariance matrix is diagonal [equation (26)]
and has values describing the expected standard deviations as
cR0 = 0.05, aoR,rh
= 0.05, and ooRP = 0.0125.
Since the starting model is reflection free, these values rep-
resent the expected range of reflection coefficients. The largest
reflection coefficients in the section are expected to be on the
order of JR,J = 0.1 (Backus and Chen, 1975). The standard
deviation in Ro is set to one half of this value, QRQ = 0.05. Ex-
pected perturbations in ARsh are less well known, thus we set
ooRsh = aRo . Density is the most poorly resolved parameter
(Drufuca and Mazzotti, 1995). The expected perturbation from
Gardner's equation is set to be small at cLRp = 0.0125.
The specification of the model covariance matrix assumes
that we have compressional impedance variations of +10%
rms about the background, that the shear impedance variations
are predictable from the compressional impedance variations
with an rms error of +10%, and that the density variations are
predictable from compressional impedance variations with an
rms error of +2.5%. Ideally, both the mean and the variance
for the model parameters should be estimated from local well
control, if available.
Figure 3 shows the resulting model parameter estimates for
each trial value of a. Equation (28) produces estimates of R0 ,
AR,,h, and ARP at every other time sample (8 ms sampling for
4 ms data) from 0.504-1.936 s. The chosen values for are
indicated.
Large values of a effectively damp the model parameter
perturbations. For values of od > 0.01, perturbations in Ro are
larger than perturbations in ARA and ARP . The second pa-
rameter, ARrh, begins to contribute noticeably for va = 0.01.
As yr decreases, the magnitude of the model parameter per-
turbations increases, particularly ARsh. An increase in tem-
poral bandwidth appears as we reduce the damping, with the
2
6d 0.1 0.05 0.01 0.0075 0.005 0.0025 0.001 0.0005 0.0001
7 1 1 7 q , 7 'i 1 7 7 , 7 'a 1 7 'i 1 1) 'a 1 7 7 , 1 9
0.50
0.60
0.70
0.80
0.90
T too
do
E tm
t3o
t40
s50
1.60
1.70
1.80
1.90
FIG. 3. Model parameter estimates for variable od. Traces 1-3 are R0 , ARsh, and ARP , respectively.

1582

Simmons and Backus
expansion in ARA trailing that of Ro by about 6 dB in the Qa
damping progression.
The models in Figure 3 are remarkably independent from ad
except for the gain, and at very small damping, an overzealous
amplification of the high frequencies.
Figure 4 shows the observed data, predicted data, and the
data misfit as a function of a. At a value of od = 0.01, the
data are reproduced reasonably well. The data are reproduced
to about the same degree for 0.001 < od < 0.01, with a slight
decrease in the data misfit for od < 0.001
We show inversion results using a value of ad = 0.0075 for
the prestack inversion. Here, od crudely allows for uncertain-
ties in the seismic wavelet, inappropriate parameterization of
the earth model, etc. We elaborate on our reasons for using a
relatively large ad in the discussion.
RESULTS
Linearized AVO inversion is applied to the ODR gathers
associated with the data seen in Figure 1. Inversion results
are shown for r = 0.0075. All other inversion parameters
are unchanged. The model parameter estimates are shown in
Figure 5. Figures 5a and 5b show the Hilbert transform of the
Ro and ARsh estimates, respectively. Impedance is the time
integral of reflectivity. A time integration would overempha-
size the low-frequency components of the data. The Hilbert
transform provides the 90° phase shift without affecting the
frequency content of the data. For simplicity, the model param-
eter estimates will be referred to as Ro and ORS h, recognizing
that these data have been Hilbert transformed.
These sections are calibrated at the same gain. A one-trace
deflection indicates a reflection coefficient of 0.06. Figure 5b
shows the AVA behavior that deviates from the background re-
flectivity predicted by the a priori assumptions. Hydrocarbons
appear as low impedance layers on Ro (Figure 5a) and as pos-
itive perturbations (peaks) from the mudrock curve on ORsh
(Figure 5b). The strongest perturbations from the mudrock
relation on ORS h occur in the hydrocarbon-bearing regions.
The flat-spot reflection (FS), which shows an overall increase
in AVA, is emphasized on the ARsh section. The bright-spot
reflection (BS), although decreasing in overall AVA response,
is also detected nicely as a perturbation relative to the back-
ground reflectivity.
Parameter ARP is not shown because it is essentially a scaled
version of AR,.h. The magnitude of AR. is smaller than ARsh
because of the specification of C,,. For this example, AR,h and
AR,o are not resolved. Though ARsh and AR, are uncorrelated
in our prior model for Cm , they are strongly correlated in the
solution. This implies no real correlation, but rather our inabil-
ity to resolve two model parameters that have a very similar
effect on the observations.
Figure 6 shows the data (observed, predicted, and misfit)
for CDP's 11, 21, 31, 41, 51, and 61. The data are reproduced
nicely. The flat-spot reflection increases in amplitude with in-
creasing angle and is apparent near 1.46 s on CDP's 51 and 61.
A slight data misfit is apparent at CDP 61 immediately preced-
ing and following the contact reflection. The bright spot occurs
on CDP's 41, 51, and 61 at approximately 1.72 s. The bright
spot is detected and modeled reasonably well. However, the
angle dependence of the flat spot and bright spot is slightly
underestimated.
The data misfit includes events with large residual moveout.
For example, note the events near 0.73 s and 0.82 s on CDP
31, 1.25 s on CDP 51, and 1.82 s on CDP 21. These events
may be locally converted shear waves, and/or multiples that are
not attenuated completely by the partial stacking and gapped
deconvolution. Noise present in the data includes both residual
additive noise and convolutional noise as discussed in Huston
and Backus (1989).
Use of equation (23) for modeling the reflection ampli-
tudes makes available several techniques that can be used to
analyze the inversion results. The model estimates obtained
in the linear inversion can be used to emphasize anomalous
AVA behavior in the data prediction-error. Figure 7a shows
the data prediction-error for the inversion with ad = 0.0075.
These data are the difference between the observed data
and the data predicted using only the Ro term. The pre-
dicted data obey the mudrock curve and Gardner's relation
exactly. Displayed in Figure 7a are those components modeled
by ARsh and ORp and energy not predictable by the linear
inversion.
The background reflectivity modeled by Ro decreases with
increasing angle. The fluid-anomaly signal produced by a
change in pore fluids from brine to hydrocarbons is expected to
show an increase with angle. Hydrocarbons now appear in the
data prediction error as anomalous relative to the background
reflectivity. The bright spot and flat-spot reflections indicate
increasing amplitude with angle relative to the background
response. This display, and/or a stack of this display, can be
suitable for direct detection purposes.
There is also a substantial amount of energy having large
residual moveout that is not modeled in the inversion. Ef-
fects of shallow gas are apparent near CDP's 45-51 from
0.5-0.6 s.
Figure 7b shows the AVA signal obtained by modeling the
data with only ORsh and ARP . These are the components that
deviate from the a priori assumptions that are predicted in the
inversion. The flat spot at 1.45 s is now most obvious. The bright
spot also shows an increase in AVA.
A portion of the final data misfit resulting from the three-
term modeling (R0 , ORsh, and ORp ) is shown in Figure 6.
These data are the components that are not modeled. The data
misfit for the entire 70 CDP ensemble over the 0.504-1.936 s
time range is 7 dB down from the observed data.
Figures 6 and 7 are quite informative for assessing the inver-
sion results. Events that show residual moveout may produce
some of the apparent AVA response not associated with hy-
drocarbons. Comparison of these data for the entire data set
suggests that this effect is probably of second-order importance
in this data set. Regions that show confusing moveout relation-
ships in Figures 6 and 7a do not necessarily show large AVA
responses in Figure 7b.
MODEL VALIDATION: COMPARISON WITH
REFLECTIVITY MODELING
Our processing emphasizes the detection of anomalous AVA
responses relative to the background reflectivity. The problem
of detection is much easier than that of recovering absolute
rock properties. A comparison of the observed data with the
data predicted by the linear inversion, and the associated data
misfit, helps to validate the credibility of the model parameter

AVO Inversion

1583
FIG. 4. Seismograms for variable a. The observed data are from CDP 61. The observed data are traces 1-6, the predicted (modeled)
data are traces 7-12, and the data misfit are traces 13-18.

1584

Simmons and Backus
FIG. 5. Model parameter estimates for od = 0.0075. The model estimates are Hilbert transformed such that they can be interpreted
in terms of relative impedances. (a) R0 estimates. A low-impedance layer should appear as a trough flanked by two equal amplitude
peaks. (b) AR,h estimates. An anomalous gas-filled layer should appear as a peak flanked by two equal amplitude troughs.
Ft. 6. Observed data (traces 1 6), predicted data (traces 7 12), and data misfit (traces 13-18) from the inversion with Qa= 0.0075.

AVO Inversion

1585
estimates. Our inversion makes use of approximate forward
modeling. A further step in model and procedure validation
is to convert the model parameter estimates from reflection
coefficients to perturbations in compressional velocity, shear
velocity, and density, and then use a reflectivity method to gen-
erate a fully-elastic prestack synthetic seismogram.
The smooth background parameters and the model param-
eter estimates account for all of the terms in equation (23).
Simmons (1994) provides the expressions for converting the
model parameter estimates to perturbations in compressional
velocity, shear velocity, and density, which are then added to the
smooth background parameters. The result is an earth model
comprised of a stack of isotropic layers suitable for input to a
reflectivity method.
We use the reflectivity method developed in Sherwood et al.
(1983). We assume a point source and that the source wavelet
is an impulse in pressure. The compressional and shear-wave
attenuation factors are set to provide high but finite Q. The
seismogram is recorded without the effects of a free surface.
Layer properties are smoothly varying to a normal incidence
two-way traveltime of 0.504 s. Seismograms are generated at
a fine spatial sampling in offset-traveltime, convolved with the
seismic wavelet estimate, and then the effect of the receiver
array is simulated by substacking. Gain proportional to t2 is
applied prior to NMO correction. Partial stacking of the NMO-
corrected data produces the synthetic ODR gather.
Figure 8 compares the ODR gather produced by the re-
flectivity modeling with the results of the linear inversion.
Figure 8a shows the observed data for CDP 61. Figures 8b
and 8c show the data predicted by the linear and reflectivity
modeling, respectively. The misfit between the observed and
predicted data are shown in Figures 8d and 8e.
In general, the agreement in waveform character and AVA
behavior between Figures 8b and 8c is quite striking. It appears
that the use of the linear modeling (Figure 8b) is defensible for
these data. Reflectivity modeling does predict an AVA increase
for the flat spot, and a slight AVA decrease for the bright spot.
There are some differences between the modeling ap-
proaches. The event at 1.6 s shows a very different AVA
behavior between Figures 8b and 8c. Both modeling meth-
ods underestimate the AVA behavior of the flat-spot reflection
(1.45 s) relative to Figure 8a.
DISCUSSION
One-step linearized inversion holds the background model
fixed. Iterative linearized inversion allows the background to
adjust and typically uses more elaborate nonlinear forward
modeling such as reflectivity or finite-difference methods. In
either case, it is reasonable to estimate the background com-
pressional velocity prior to the full waveform inversion.
Our approach to prestack full-waveform modeling of the
ODR data is less than theoretically exact. The partial stacking
inherent to the ODR process attenuates modes of propagation
that have large residual moveout relative to the primary reflec-
tions. These modes are typically longer-period converted shear
waves and surface multiples that cannot be modeled accurately
FIG. 7. (a) Data misfit between the observed data and the data predicted using only the R0 term in the forward modeling. (b) Predicted
data obtained by using only the ARI, and ARF, terms in the forward modeling. Every third CDP within the range indicated is shown.
The tic marks locate ODR 6 for the CDP locations shown.

1586

Simmons and Backus
with our uncertainty in the background shear-wave velocity
and in the context of a locally 1-D earth. These modes are not
included in our prestack modeling.
Thin-layer modeling using the linearized approximation to
Zoeppritz roughly accounts for the locally converted shear
wave. Incorporation of the seismic wavelet, thin layering, NMO
stretch, the receiver array, and the partial-range stacking re-
sponse into the Frechet derivatives differentiates this method
from a conventional weighted stack. Our approach is likely less
sensitive to NMO stretch and residual normal moveout than
conventional methods, although further examination is neces-
sary to quantify this claim.
More importantly, the effect of geometric divergence was
crudely treated, and Q-type attenuation effects were neglected
in the forward modeling. In the general spirit of the approach,
geometric divergence and a reasonable Q could be applied in
the forward modeling. A t2 gain could then be applied to the
modeled data to mimic the real data processing.
The selected model parameterization results in a prior
model covariance matrix that is diagonal. This makes the
problem more readily interpretable. Reflections that fit the
prior assumptions can be described with a single parameter,
R0 . The other parameters recognize perturbations relative to
the prior assumptions. Hydrocarbon zones are recognized as
low impedance on R0 . Perturbations of opposite polarity in
the shear reflectivity, and perturbations from Gardner's re-
lation are indicated where hydrocarbons occur. The fluid-
anomaly signals show the expected increases in amplitude with
angle.
In our inversion approach, Cd provides some cover for in-
adequacies in the model parameterization, inadequacies in
estimating the background compressional-wave velocity, eight
millisecond layering, assumption of isotropy, angle and time
invariance of the seismic wavelet, neglect of attenuation, and
the rough treatment of geometric divergence. Until all of these
factors are more properly comprehended, accounted for, and
modeled, Cd is set at the largest value that provides ade-
quate detection capability and results in a reasonable data fit
(ad — 0.0075).
The linear inversion is decidedly nonunique. An infinite
number of combinations of Cd and Cm produce a synthetic re-
sponse using the simplified modeling that fits the data equally
well. The model estimates, however, are different. The specifi-
cation of Cd and CM is somewhat qualitative based on a priori
expectations as to the accuracy of the starting model and the
expected parameter perturbations.
Figure 3 illustrates the tradeoff between model parameter
resolution and model parameter variance. For large values of
ad, the variance in the model parameters is small, but the pa-
rameters are poorly resolved. As ad decreases, the parameters
become better resolved (the magnitude and frequency content
increases) with an associated increase in the model parameter
variance. The interpreter must determine the optimum accept-
able tradeoff between resolution and variance.
Signal is distributed among the model parameters Ro , ARsh,
and ARP according to the specification of Cd and C7,. For a
given C,,, the magnitude of Ca influences the data fit and the
model parameter perturbations from the a priori values. For
large Cd,Ro at the center of the seismic pass band is initially
recovered. As Cd decreases, Ro within the entire pass band
is resolved. The recovery of ARsh follows a similar pattern,
however, it is lower frequency than Ro . If voRsh is increased,
FIG. 8. Seismogram results for CDP 61. (a) Observed data. (b) Data predicted by the linear inversion. (c) Data computed by a
reflectivity method using the R0 , ORsh, and ARP estimates to derive the compressional velocities, shear velocities, and densities.
(d) Data misfit between the observed data and the data predicted by the linear inversion. (e) Data misfit between the observed
data and the data predicted by the reflectivity method.

AVO Inversion

1587
one can cause ARsh to be recovered at larger values of Cd prior
to the recovery of R0 .
A single parameter, R0 , accounts for most of the data fairly
well. The prediction error anomalies are most pronounced
where there are hydrocarbons and are similar to the fluid factor
of Smith and Gidlow (1987). As might be expected, we find a
strong posterior covariance between the two perturbation pa-
rameters ARsh and AR,,. Cm controls the relative importance
of the model parameters. With the relatively large Cd used in
the inversion and a maximum incident angle of 46°, these two
parameters are not resolved, so they simply share in a single
parameter with the relative weights determined by the prior
variance specification.
Smith and Gidlow (1987) show impressive results using an
empirical relationship between the background compressional
and shear velocities. Initial attempts at using empirical rela-
tionships between the Ro and ARsh sections in Figure 5 did not
improve the hydrocarbon detection threshold over that seen
in Figure 5b. Further work addressing this issue is warranted.
An attractive feature of performing the inversion in the
time domain is that the prior assumptions relating the back-
ground velocities and density can be time and spatially vari-
ant as a function of depositional system, lithology, and depth.
Equations (8) and (13) can be temporally and spatially depen-
dent. A more accurate treatment would allow the method to
be a better predictor in regions where hydrocarbons are not
present. Hydrocarbons would then have an even better chance
of appearing as anomalies in AR,.h relative to R0 . A time-,
and/or angle-dependent seismic wavelet can be readily incor-
porated into the inversion. This should result in an improved
treatment of attenuation and divergence effects.
Our choice of model parameterizaton and forward mode-
ling makes a quantitative assessment of the inversion re-
sults possible by examining the predicted data and data misfit
(Figure 6), the data prediction-error relative to the background
(Figure 7a), and the predicted AVA signal (Figure 7b).
We can compare the use of AVO inversion to a comparison of
different angle-range stacks. Compare Figure 1 with Figure 5.
In this data set, either display reveals the gas sands and an
unambiguous winner cannot be declared.
Reflectivity methods are the most theoretically exact form of
modeling for 1-D isotropic earth models. Reflectivity methods
account for all modes of propagation, including all multiples
and converted waves. Figure 8 suggests that little is lost by our
simplified forward modeling when the data are compared in
ODR form.
Our modeling underestimates the AVA behavior of the flat-
spot reflection. This is likely due mainly to the conservative
approach to damping which attenuates the anomalies. How-
ever, more research is required to explain the underestimation
of the AVA behavior of the flat spot.
First-order sources of error in our examples are in the layer
sampling and the seismic wavelet estimate. Both of these fac-
tors likely contribute to the underestimation of the hydrocar-
bons. The 8-ms layer sampling is rather coarse and is by no
means inherent to the algorithm. A wavelet extracted from
the data and modified for a blue reflectivity spectrum is rea-
sonable, and defensible, at our current level of understanding.
The wavelet does include the partial-stack response of the 18°
data from which it is estimated but is otherwise angle- and time-
independent. The amplitude scale factor is calibrated based on
assumed reflection coefficients for the bright-spot reflection.
This calibration is inexact and ultimately affects the magni-
tudes of the elastic property perturbations.
The final data misfit in Figure 6 is the information that would
modify these initial models in a second iteration of a nonlin-
ear inversion. Given the specified seismic wavelet, layer pa-
rameterization, and our other approximations, the additional
information that could be reliably recovered using more exact
modeling, and iterative inversion, is in question.
Our estimate of rock properties is actually an estimate of
the pseudo properties. The earth is undeniably anisotropic..
Transverse isotropy has a first-order effect on the sin e 09 terms
in the linearized approximation (Blangy, 1994). Knowledge
regarding lateral velocity variations (Simmons and Backus,
1992; Huston and Backus, 1989), attenuation, the time-, space-,
and angle-dependent seismic wavelet, and the low-frequency
shear velocity and density is poorly known. An approach
where detecting lateral changes (anomalies) is emphasized is
more reasonable, given our current level of understanding,
than the attempt to recover absolute rock properties.
CONCLUSIONS
We extract information from amplitude variation versus off-
set using a practical approach to maximum-likelihood prestack
inversion. A prediction-error model parameterization is de-
signed to reproduce the background reflectivity, in the ab-
sence of hydrocarbons, with a single parameter. Reflection
events with an anomalous amplitude variation-versus-offset
dependence relative to the background are then detected as
anomalies.
Our approach is designed to incorporate the characteristics
of reflectivity-based inversion methods, yet be more computa-
tionally practical. Major features of the approach include the
incorporation of NMO stretch, signal changes produced by the
noise-reduction methods used, and user control over the com-
promise between resolution and variance.
This approach to prestack seismic inversion is believed to be
competitive with many of the approaches currently in use. The
simplified forward modeling results in a full-waveform inver-
sion that is computationally practical in a workstation environ-
ment.
Initial attempts at using absolute rock properties obtained
from the linear inversion model estimates for fully-elastic
prestack reflectivity modeling are encouraging, at least qualita-
tively, and will be pursued in a future paper. However, we feel
that given the current state of the art, one should emphasize
the detection of anomalies, rather than the recovery of absolute
rock properties.
ACKNOWLEDGMENTS
Critical reviews by Nick Bernitsas, Chris Finn, and Bill
Harlan increased the signal-to-noise ratio of the original
manuscript.
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