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\markright{Common-focus-point gathers --- W.S. Harlan} 
\title{Comments on Common-Focus-Point Gathers}

\author{William S. Harlan}

\date{Feb 1999}

\maketitle
\section{Introduction}
At the 1996 SEG Annual Meeting in Denver,
three papers first introduced the concept of velocity-independent
imaging with common-focus-point (CFP) gathers.
I came to some rather negative conclusions about the method
at the time and haven't seen any reason to change this opinion
since.  Here are my remarks after first seeing the papers.
\section{Common-focus-point gathers}
Three papers discussed common-focus-point (CFP) gathers:
``Seismic processing between two focusing
steps,'' (MIG 1.1) by A.J. Berkhout, ``Migration velocity
analysis using the common focus point technology,'' (MIG 1.2)
by M.M. Nurul Kabir and D.J. Verschuur, and ``Automating
prestack migration analysis using common focal point gathers,''
(MIG 1.3) by Scott A.\ Morton and Jan Thorbecke.
These gathers are equivalent
to those used by conventional depth-focusing analysis
\cite{GEO57-12-16081622,SEG-1986-S7.6},
but with a slightly different use.  

Scott A.\ Morton of Cray Research defined a CFP gather 
simply with a Kirchhoff implementation. (A.J.\ Berkhout
used his operator notation, with less explicit arguments.)
\def \xv {{\svector x}} 
\begin{equation}
	{\rm CFP} (\xv_s , \xv_m , \tau) = \int \int 
	{\rm data}[ \xv_s , \xv_r, 
		t = \tau + T(\xv_m , \xv_s) + T(\xv_m , \xv_r) ] 
	d^2 \xv_r ,
\end{equation}
where $\xv_s , \xv_r , \xv_m $ are the spatial coordinates of
the source, receiver, and focus point, $t$ is the recorded time,
and $\tau$ is a downward-continued time.  The functions $T$ 
give the one-way traveltime between two points for a particular
velocity model.   Each output CFP extrapolates
receivers down to the depth of the focus (focal) point and subtracts
the time to the source.  (The source shift was in Scott Morton's slide
presentation, but not abstract.)
Ideally, a good velocity model should produce a flat consistent phase
at zero time for different sources.  A conventional Kirchhoff
depth migration would produce an amplitude at the CFP location
$\xv_m$ by summing over all source positions ($\xv_s$) at zero time.  

Most CFP gathers are not perfectly flat at zero time
because of suboptimum velocities.
Conventional depth-focusing analysis locates the flattest 
reflections at earlier or later times and then displays this
error as an equivalent depth or average velocity correction.  
The correct depth of a flat event at non-zero time $\tau$ is expected 
to fall halfway between the CFP depth $\xv_m$ and the 
depth at which the event would migrate to zero time without flattening.
These depth or velocity errors give only an average correction
to the velocity model from the surface down to the reflector
depth.  Some sort of tomographic back-projection is
necessary to distribute these velocity errors correctly in the overlying
model and to reconcile with the errors for other reflections.

The new CFP papers assume that velocity models will be layered and
that layer boundaries will produce reflections that
can be identified in unstacked CFP gathers.  Velocity models
are optimized by layer-stripping---one layer velocity and boundary 
at a time.   

At this point CFP analysis begins to depart from depth-focusing
analysis.
A user identifies the next significant reflection,
chooses an initial velocity for the overlying layer, and proposes
corresponding depths for the reflector (perhaps from the
depth image for the previous iteration).  The user examines
CFP gathers at the proposed depths and then looks for 
the unflattened reflection that was expected to image at this
depth---or for any other reflection that might now appear 
easier to pick.
Because the mislocated reflection is not flat, the coherence
cannot be identified as automatically as for depth-focusing analysis. 
The reflection may also lie several cycles away from the CFP zero time,
so snapping would appear impossible.  

Instead of attempting to use this imaging error to update velocities,
these authors update the traveltimes for the Kirchhoff operator 
by adding half the picked time errors to the traveltimes
used previously for this CFP position. They 
produce a new CFP gather without a more expensive remodeling
of traveltimes.  Again, the procedure has converged when the CFP's
are flat at zero time. 
Although the abstracts do not say, I expect
the CFP depth positions are also revised by half the difference 
with the image depth of the intended reflection.
(Otherwise the final CFP's will not track the reflection.)

This splitting of time errors would appear to assume that 
velocity errors are well behaved in the lateral direction
from near to far offset.
Conventional focusing analysis makes the same assumption to split depth errors.

Scott Morton states that the final unimplemented
step of the algorithm is to revise velocities by a tomographic
inversion of the updated Kirchhoff operators.  There is
no guarantee that revised traveltimes can be fit by
a single velocity model.  

Hans Tieman of GDC pointed out an interesting degenerate case to me.
For a single layer beginning at the surface, one could imagine 
that the data had been migrated
with a zero velocity at zero depth.  The CFP gathers then become
identical to the original shot profiles.  Picking residual
moveout amounts to picking the raw prestack moveouts.
The data could be stacked and imaged perfectly in the next
iteration. The final nontrivial step is to convert all
these picked traveltimes into a velocity model (tomography).
CFP's would remain at zero depth
until we revised our reference velocity model.

A constant-offset implementation
would better avoid artifacts from the limited range of
offsets present in common-source profiles, but might violate 
some of the (unstated) assumptions in these three papers.

All in all, I find it difficult to extract a practical algorithm
from these details, assuming that we desire to arrive at a meaningful
depth section.  The authors do not say
how to revise CFP depths for the intended reflection.  
Without revision, why should a reflection be forced to produce
a flat CFP at an arbitrarily chosen depth?  
Nevertheless, some features are 
interesting, and many listeners were enchanted by the idea that
the imaging operator could be revised directly without
a physically consistent revision of the velocities.  

Two admirers of the CFP approach, who read my description above, 
believe the method is not intended to estimate meaningful depths
directly.  They stress that the method uses downward continuation to 
simplify the coherence and improve the signal-to-noise ratio
of reflections before picking.  The revised traveltime operators are the
final objective: these picks provide a robust estimate of reflection
moveouts for input to tomography.  

I have already used several forms of prestack moveout picking as
input to reflection tomography: moveouts after constant
offset depth or time migration, after DMO only, from combinations
of prestack moveouts and poststack picks, and other
gathers which appear conveniently during processing.
The moveouts of all such picks are modeled to invert geometrically 
the effects of the imaging and produce equivalent
tables of unmigrated traveltimes.  After
conversion, the same reflection tomography program 
inverts them all.  It would not be difficult to add
CFP picks to this list and use them as a new alternative.
Nevertheless, I find few advantages.
Shot profile migration produces too many artifacts,
compared to constant-offset migration.  Picking residual
moveouts is easy unless we are expected to track
specific reflections before and after imaging.  I would
prefer to pick the moveouts of the flattest reflections in 
a CFP gather, as preferred by conventional depth focusing
analysis.  Unstacked prestack depth migration with a reference
model enhances the signal-to-noise ratio.

Imaging algorithms cannot leave velocity
estimation as an exercise for the reader.  A solid
tomography algorithm probably takes an order of magnitude
more computer code than an imaging algorithm.  Velocities
are the hard part.
It would be convenient if we could produce depth images
without velocities, but we would be obliged to accept
an arbitrarily scaled depth axis.

\section{Later remarks}
In 1998, this method continues to be discussed, although I have
yet to see anyone estimate a velocity model from recorded
data.  The fatal flaw remains the same.

One must choose a CFP gather for a particular image depth, 
then identify, at a non-zero image time, the reflection that 
one expected to see at zero time.  This seems fundamentally
impractical.  The mislocated reflection will not be flat or
have any other distinctive coherence.  Instead,
one must recognize a reflection
that one has seen before imaging.  Not surprisingly, the
only examples I have seen use synthetic data with a few isolated
strong reflections.  On Gulf Coast data, with many weak reflections,
such picking would be impossible.

Conventional depth-focusing analysis 
uses similar image gathers, but allows one to pick 
the flattest reflection at a non-zero imaging time.  
Recognizing flatness is easy with 
numerical tools like semblance.  It is not necessary to know
where this reflection came from before imaging.

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