\documentclass[12pt]{article}
\usepackage{wsh}

%\documentstyle[seg]{revtex} 
%\documentstyle[seg,abstract]{revtex} 
%\documentstyle[seg,manuscript]{revtex} 
%\documentstyle{conocoplain} 
%\documentstyle{abstractfix} % This line modifies style for expanded abstracts.

\begin{document}
\bibliographystyle{plain}

\markright{Tomographic seismic amplitude correction --- W.S. Harlan}

\title{Tomographic correction of transmission 
distortions in reflected seismic amplitudes\\
{\normalsize (1994 SEG meeting, Los Angeles, SI2.2, Expanded Abstracts)}
}    

%\\
%\mbox { } \\
%{\sl Running head:} \\
%Tomographic seismic amplitude correction
%}

\author{William S. Harlan}

% \address{Conoco Inc.; P.O. Box 1267; Ponca City, OK 74602--1267}

\date{December 6, 1994}
\maketitle

\section{Summary}

A tomographic inversion adjusts seismic reflection amplitudes
to remove distortions caused by spatial variations in the
transmission properties of the overlying earth.
These variations create distinctive patterns on displays of 
reflection amplitude versus source-receiver offsets and midpoints.
These patterns are inverted for amplitude corrections that
remove the transmission distortions.
The methodology is demonstrated on a strong 
``bright spot'' reflection under a large filled channel
in the Gulf of Mexico.

Transmission anomalies are defined at each point in a depth model 
as a fractional increase or decrease in wave amplitude.
These changes in amplitude scale multiplicatively along raypaths.
An inversion of transmission anomalies
(i) minimizes errors between modeled and picked
amplitudes, (ii) uses a quadratic objective function for easy
optimization, and (iii) distinguishes reflectivity 
changes from transmission anomalies.
Reflection raypaths are estimated by reflection tomography
for interval velocities.

The effects of channel irregularities greatly obscured
the observed amplitude versus offset in the
Gulf of Mexico dataset.
The transmission anomaly model reconstructed 
recorded amplitudes accurately and removed the corresponding
interference patterns.  Most transmission anomalies were imaged near the top 
of the channel.  Local focusing and defocusing of waves 
by velocity variations can explain these perturbations of amplitudes.
An anomaly does have different effects on reflections at different depths.

\section{Introduction}
\label{sec:intro}

Einar Kjartansson \cite{kj:the} demonstrated that
surface reflection seismic data contain useful information
on transmission amplitudes between the surface
and reflector.  Assuming linear raypaths, he produced
images of transmission losses as a function of vertical traveltime,
with an inverse much like a slant-stack transform.
Since then, the analysis of reflectivity with reflection
angle has received wide application \cite{ca:avo}, but generally
without considering distortions of amplitudes during transmission.

I invert transmission amplitude anomalies more flexibly as a function
of depth and correct for these effects before interpreting changes
with angle. I do not assume any particular mechanism for the perturbation
of transmitted wave amplitudes, but I do expect strong contributions
from local focusing and defocusing of wavefronts by velocity anomalies.
The model assigns a single fractional increase
or decrease in amplitudes to each point in depth.   

\section{Observed transmission anomalies}

Figure 1 shows a migrated seismic image of reflectivities from the
Mississippi Canyon Area of the Gulf of Mexico, spanning 12km
and reaching an imaged depth of 2.5km.
A large producing gas sand creates a ``bright spot''
with anomalously large reflectivity at 2.2km depth
(approximately 2.3s).  
A reflection from an interface with
positive reflectivity should appear as a black filled peak.
From the water bottom reflection at 0.5km depth to approximately 1.2km depth is
a large filled channel with a very irregular
erosional discontinuity.
The interior contains poorly imaged
scatterers and few coherent reflectors.  
This depth section was
derived from tomographically estimated interval velocities, as
described in Harlan et al \cite{ha:91a,ha:91b}.

Unmigrated, unstacked seismic data were examined on a workstation 
as a three-dimensional volume.  Minimal preprocessing included
deconvolution, hyperbolic moveout corrections to flatten reflections
over offset (the distance between source and receiver),
and a gentle time-varying  gain to balance the amplitudes
of weak background reflections over time. Trace balancing
removed irregularities in source strengths and hydrophone receptivities.
Because of uncorrected lateral velocity anomalies, the strong gas reflection
showed non-hyperbolic residual moveouts on the order of a wavelength.

The strongest negative amplitude peak was tracked and picked
consistently over offset and midpoint (the center of sources and receivers).
Figure 2a plots these picked amplitudes over
midpoint (labeled Distance) and offset.  
(For comparison, the median absolute value of all preprocessed 
amplitudes is near 10.)  The ``bright spot'' appears
as the darker region over midpoints 4200m to 8000m.  Changes
depending only on midpoint are clearly due to changes in reflector
strength.  Most striking are diagonal streaks that change
with both offset and midpoint.  Most prestack analyses of amplitude 
would interpret each midpoint independently,
emphasize changes with offset or angle, and 
produce misleading results.

The lower part of figure 3 (based on Claerbout \cite{cl:iei}) 
represents possible anomalous streaks in figure 2a. 
The upper part of figure 3 shows interpreted
raypaths passing through isolated transmission anomalies.
(We assume straight lines and flat reflectors in this cartoon.)
The geometry on the left assumes an anomaly at
the surface, so surface-consistent streaks should appear over
offset and midpoint at a 60 degree angle,
along constant source and receiver coordinates.
Surface-consistent trace balancing should remove such anomalies.
An anomaly just above a reflector (indistinguishable from
an irregularity in the reflector) perturbs amplitudes
along a constant midpoint.  When an anomaly appears
between the surface and reflector, as on the right of figure
3, then the pair of streaks will make an angle less than 60 degrees, 
as in the picks of Figure 2a.
The width of a streak increases with the depth of the anomaly
because the Fresnel zone increases.  An anomaly smaller than a Fresnel
zone may not be detectable.  

\section{Inversion/correction of anomalies}
\def\xv{{\svector x}}

To invert the picked amplitudes in figure 2a, we choose a model that 
(i) minimizes errors between modeled and picked
amplitudes, (ii) uses a quadratic objective function for easy
optimization, and (iii) distinguishes reflectivity 
changes from transmission anomalies.
The methods of Harlan et al \cite{ha:91b} identify
common reflection points in depth and reconstruct their raypaths.
For each common reflection point, indexed $i$, let $a_{i,j}$ be the
picked prestack amplitude for a collection of source-receiver
pairs, indexed by $j$.  
For each of these picks, we construct
raypaths whose Cartesian coordinates
$\xv_{i,j} (s)$ are a function of distance
$s$ along the ray from source to reflector to receiver.
Let $r_i$ be chosen as a median reference amplitude for each reflection 
point $i$ so that the values $a_{i,j}/r_i$ are as close to 1 as possible
for all $j$.
Let $t(\xv)$ describe the transmission perturbations of amplitudes throughout
the region covered by raypaths.  Perturbations are assumed
multiplicative and linearized by logarithms.
A model of amplitudes $\hat a_{i,j}$ is defined by an integral
of perturbations along the raypath:
\input equationfix.tex
\begin{equation}
\hat a_{i,j} = r_i \exp \left \{ \int t[\xv_{i,j}(s)] ds \right \} .
\label{eq:mod} 
\end{equation}
The anomalies $t(\xv)$ are parameterized as a spatially continuous 
sum of smooth overlapping basis functions.  

A direct minimization of errors between the picked and modeled amplitudes
would introduce unnecessary non-linearity in the optimization.
The following damped, weighted least-squares objective function
is completely quadratic in $t(\xv)$ and allows fast, stable optimization
by the conjugate gradient algorithm.  (The gradient is linear.)
%\input equationfix.tex
\begin{equation}
\min_{\mbox{t($\xv$)}}
\sum_{i,j} \left [ r_i \log (a_{i,j}/r_i) 
- r_i \log ({\hat a}_{i,j}/r_i) \right ] ^2
+ \epsilon \int t( \xv )^2 d\xv .
\label{eq:obj} 
\end{equation}
The small damping factor $\epsilon$ is the ratio of the assumed variance
of noise (additive to $a_{i,j}$) divided by
the variance of $t(\xv)$. 
This optimization equivalently
minimizes errors between $a_{i,j}$ and ${\hat a}_{i,j}$
because $\log(x) \approx x -1$ when $x \approx 1$.

Figure 4 shows such a reconstructed image of transmission anomalies
$t(\xv)$ in gray values
as a function of midpoint/distance and depth. 
(Equation (\ref{eq:mod}) integrates these magnitudes in meters.)
Solid lines show the water bottom and picked reflector.
Strong anomalies appear in the upper portion of the channel fill.
Negative values, displayed as light grays, indicate a weakening
of amplitudes passing through these points.
Most weakening anomalies are also flanked by a pair of 
amplifying anomalies, in dark grays.
The combination of weakening and amplification argue for
velocity irregularities that defocus energy.

The corrected amplitudes $a'_{i,j}$ in figure 2c 
remove transmission effects, where
\input equationfix.tex
\begin{equation}
a'_{i,j} = a_{i,j} \exp \left \{ -  \int t[\xv_{i,j}(s)] ds \right \}.
\label{eq:cor}
\end{equation}
The corrected reflectivity of the bright spot becomes somewhat
less negative with offset (a strong class III reflection).  
Interference from weak multiple reflections
appear as horizontal stripes at constant offsets.

Although a reference value $r_i$ was assumed for the reflectivity,
the optimization can compensate for a poor choice with
a transmission anomaly just over the reflector.  
To avoid biasing the corrected
amplitude picks with the assumed value, 
the integral in the correction (\ref{eq:cor}) should not include 
the region just above the reflector.

Although not immediately obvious, a reflectivity that changes
with offset and angle, but not with midpoint, cannot be reproduced by 
the model (\ref{eq:mod}).
A false transmission anomaly that attempted to reproduce one increasing
amplitude with offset would cause erroneously
decreasing amplitudes at other reflection points.
Thus, amplitude changes that depend only on reflectivity
will be preserved.

A simultaneous inversion of many reflectors did not
improve the image of spatial variations in transmission properties.
Rather, the effect of a transmission
anomaly on different reflectors appears inconsistent.
The effects of local focusing and defocusing may change
with distance from the anomaly.  

\section{Conclusions}

Amplitude tomography can be a simple extension to existing
methods of reflection tomography for velocity. 
A quadratic objective function allows stable inversion of
multiplicative perturbations in transmitted amplitudes along
raypaths. These transmission anomalies should be routinely
examined and corrected before interpretation of reflectivity
versus angle.  
Images of anomalies potentially could delineate inhomogeneities associated
with gas, overpressure, or stratigraphy.

More rigorous modeling of transmission wave phenomena 
could incorporate methods of diffraction
tomography \cite{de:dif,pr:dif,wo:the}, but generalized for
an extra order of scattering, with Born and Rytov approximations  
for the reflection and transmission effects.
Moreover, the wave focusing by transmission anomalies
is frequency-dependent \cite{bi:eik}.

\section{Acknowledgments}
These data were provided by TGS-CALIBRE Geophysical and GECO-PRAKLA,
thanks to John A. Adamick and Kim Abdallah.  Software by Paul 
Hauge, now of Dresser Atlas, provided reliable picks of amplitudes.
Thanks also to Thom Cavanaugh, Dale Miller, and Bruce McClellan of 
Conoco Inc.\ for their interest and assistance.

\bibliography{wsh}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\section{Figure Captions}
{
%\textwidth 6.5in
%\setlength{\textwidth}{6.5in}
%\hoffset=-0.5in
%\hsize=6.5in
\parindent=0pt

FIG.\ 1. A migrated seismic image of reflectivities from 
Mississippi Canyon in the Gulf of Mexico, spanning 12km and 
and reaching an imaged depth of 2.5km.

\bigskip

FIG.\ 2. (a) Picked amplitudes as a function of midpoint and offset, 
along a strong ``bright spot'' reflection at 2.2km depth.
(b) A best-fitting reconstruction of
figure 2a, using equation~(\ref{eq:mod}).
(c) Picked amplitudes with transmission effects removed 
by equation~(\ref{eq:cor}).

\bigskip

FIG.\ 3. A geometric explanation of anomalous streaks in figures 2a and 2b.
Above, circles outline transmission anomalies that affect 
certain reflection raypaths.
Below are corresponding offsets and midpoints that would be affected
by these anomalies. (After Claerbout \cite{cl:iei}).

\bigskip


FIG.\ 4. Estimated transmission anomalies in depth
that reconstruct the modeled data in figure 2b.  
Strong anomalies appear near the top of the filled channel.
Solid lines show the water bottom and picked reflector.
Dimensions are identical to those of figure 1.  


}

\end{document}
%   %W%     %G%