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\markright{Automatic moveout picking--- W.S. Harlan}
\title{Constrained automatic moveout picking from semblances}

\author{William S. Harlan}

\date{September 2001}

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\section{Introduction}
I first implemented this method of automatic picking in 
1987-88 while working at the Osservatorio Geofisico
Sperimentale in Trieste Italy and at Stanford University
as a visiting scholar.  The implementation was
an obvious application and simplification of the work
of John Toldi \cite{toldi85} \cite{Toldi89} which I
had closely followed for several years.  That implementation
was in a computer language I now try to forget.

Over the years, I applied that algorithm to many varieties of
data, including noisy data with weak reflections, with large
lateral velocity anomalies, and with residual moveouts
after prestack time or depth imaging.  I've rarely resorted
to hand-picking tools except as a Q.C. tool for this algorithm.

Recently I reimplemented this method to take advantage 
of a newer C++ Gauss-Newton optimization algorithm, 
with fewer restrictions on the number of spatial dimensions.
But the numerical properties are exactly the same.

\section{The objective function}
First one must perform a stacking velocity analysis to construct 
a hypercube of stacking semblances $S( m, \xv )$
as a function of a moveout parameter $m$ and
dimensions $\xv$ including zero-offset time and a spatial X and Y.

The moveout parameter should be chosen so that resolution
is approximately constant for large or small moveouts.
I prefer to use the squared reciprocal of a conventional stacking
velocity.  That moveout parameter also conveniently includes
flat or negative moveouts.

The goal of this algorithm is to find a smooth surface $m(\xv)$
that gives a single-valued moveout as a function of all
coordinates.  The surface has a limited number of degrees
of freedom to make the surface stiff.  The best surface
maximizes the integral of the semblance over all coordinates.
\begin{equation}
\max_{m({\bf x} )} \int S [ m(\xv ), \xv ] d \xv 
\end{equation}

This is essentially the same objective function as used
by John Toldi, but without his tomographic constraints
on moveout.  I assume that stiffness and hard bounds
alone will be sufficient to keep the moveouts reasonable.
Zhang Lin \cite{lin:pick} similarly dropped Toldi's physical
constraints, but made more extensive changes to the objective
function and optimization.  I make no significant changes
to Toldi's objective function.  I use a very conventional
Gauss-Newton optimization, with Toldi's iterative 
smoothing and relaxation to ensure convergence.

\section{Motivation}
Stiffness prevents the moveout surface from jumping easily
from primary reflections to a multiple and back again.
Hard bounds may be necessary in areas where multiples
and other coherent noise are more common than primaries.
A hard constraint is implemented by simply muting that
part of the semblance that falls outside an allowed range.
Ranges of moveouts can be specified relative to reference
moveouts, or independently.

The moveout surface is stiff over all spatial dimensions,
including time.  This allows more redundancy over more dimensions
than available to a human interpreter.  A surface integral
includes the contribution of many weak reflections rather than
just the few strong reflections visible to the eye on a 
contour plot.

Weak lateral velocity anomalies can cause stacking ``velocities''
to swing wildly by 50\% or more as inner and outer offsets
are delayed relative to each other.  
A human interpreter
will reject such swings as unphysical, although the 
characteristic signature of such anomalies 
can be seen clearly when neighboring ensembles are included.
An automatic picker sees more data and finds the swings
are essential for consistency.

Some automatic pickers attempt first to identify peaks and fit
them second.  Unfortunately all such peaks influence the
final result, even when they are inconsistent with neighbors.
An integral, on the other hand, does not increase the penalty
of moving farther from a multiple once the peak has been
found inconsistent.

\section{Optimization}

A Gauss-Newton algorithm finds model parameters that
best fit some data after a non-linear transform.
This particular objective function does not look
like an inversion at first glance: we have an objective
function and a model, but no data.  Let our data
be a single value that represents the maximum normalized
integral of semblances.

A Gauss Newton algorithm requires three transforms.

First we need a full non-linear calculation of the data from the
model.  Forward integrate semblance $S( m, \xv ) $ over moveouts $m$ 
and positions $\xv$:
\begin{equation}
d = \int S [ m(\xv ), \xv ] d \xv .
\end{equation}

Next, we need a linearized perturbation of the data $\Delta d$
for a perturbation of the model $\Delta m$ -- i.e. a gradient:
\begin{equation}
\Delta d = 
\int {\textstyle \frac{\partial}{\partial m}} S [ m(\xv ), \xv ] 
\, \Delta m(\xv ) d \xv  .
\end{equation}

Finally, we need the adjoint of the linearized forward transform:
\begin{equation}
\Delta m (\xv ) = {\textstyle \frac{\partial}{\partial m} }
	S [ m(\xv ), \xv ] \, \Delta d .
\end{equation}
Set data $d$ to the maximum possible normalized sum
and plug into your favorite Gauss-Newton solver.

Actually, the conventional solver requires one modification.
In early iterations, the moveout surface is far from 
the optimum peaks.  The
surface needs to be close enough to a peak for the gradient
to push the solution in the correct direction.  Fortunately,
John Toldi had a clever solution.
In early iterations the surface should be extremely stiff
with perhaps only a few basis functions over the range
of each dimension.  The semblance cube should be heavily
smoothed in the direction of the moveout parameter ---
up to half the allowed range of values in the first iteration.
The locations of the peaks are less accurate but the surface
always finds itself on a broadened flank with the gradient 
pointing in the correct direction.
After this small number of degrees of freedom has been
allowed to converge, then later iterations can reduce the
smoothing and increase the number of basis functions.
In the final iteration, the semblance volume is not smoothed
at all, and the surface is as flexible as the user allows.

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