paper.tex

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\lefthead{Fomel}
\righthead{Velocity continuation}
\footer{SEP--92}

\title{Velocity continuation and the anatomy of \\
residual prestack time migration}

\email{[e-mail]}
\author{Sergey Fomel}

\maketitle

\begin{abstract} 
  Velocity continuation is an imaginary continuous process of
  seismic image transformation in the post-migration domain. It generalizes
  the concepts of residual and cascaded migrations. Understanding the laws
  of velocity continuation is crucially important for a successful application
  of time migration velocity analysis.  These laws predict the changes
  in the geometry and intensity of reflection events on migrated images with
  the change of the migration velocity.  In this paper, I derive kinematic
  and dynamic laws for the case of prestack residual migration from simple
  geometric principles. The main theoretical result is a decomposition
  of prestack velocity continuation into three different components
  corresponding to residual normal moveout, residual dip moveout, and residual
  zero-offset migration. I analyze the contribution and properties of each of
  the three components separately. This theory forms the basis for
  constructing efficient finite-difference and spectral algorithms for
  time migration velocity analysis.
\end{abstract}

\section{Introduction}
%%%%%%%%%%%%%%%%%%

\begin{comment}
Migration velocity analysis is a routine part of prestack time
migration applications. It serves both as a tool for velocity
estimation \cite[]{FBR08-06-02240234} and as a tool for optimal stacking
of migrated seismic sections and modeling zero-offset data for depth
migration \cite[]{GEO62-02-05680576}. In the most common form, migration
velocity analysis amounts to residual moveout correction on CRP
(common reflection point) gathers. However, in the case of dipping
reflectors, this correction does not provide optimal focusing of
reflection energy, since it does not account for lateral movement of
reflectors caused by the change in migration velocity. In other words,
different points on a stacking hyperbola in a CRP gather can
correspond to different reflection points at the actual reflector. The
situation is similar to that of the conventional NMO velocity
analysis, where the reflection point dispersal problem is usually
overcome with the help of DMO \cite[]{FBR04-07-00070024,dmo}. An
analogous correction is required for optimal focusing in the
post-migration domain. In this paper, I propose and test velocity
continuation as a method of migration velocity analysis. The method
enhances the conventional residual moveout correction by taking into
account lateral movements of migrated reflection events.
\end{comment}

The conventional approach to seismic migration theory
\cite[]{Claerbout.blackwell.85,Berkhout.mig.14A.1985} employs the
downward continuation concept. According to this concept, migration
extrapolates upgoing reflected waves, recorded on the surface, to the
place of their reflection to form an image of subsurface
structures.
%In order to understand the concept of velocity continuation, we need
%to look at the fundamentals of seismic time migration. 
Post-stack time migration possesses peculiar properties, which can
lead to a different viewpoint on migration.  One of the most
interesting properties is an ability to decompose the time migration
procedure into a cascade of two or more migrations with smaller
migration velocities. This remarkable property is described by
\cite{GEO50-01-01100126} as {\em residual migration}.
\cite{GEO52-05-06180643} generalized the method of residual
migration to one of {\em cascaded migration.} Cascading
finite-difference migrations overcomes the dip limitations of
conventional finite-difference algorithms \cite[]{GEO52-05-06180643};
cascading Stolt-type {\em f-k} migrations expands their range of
validity to the case of a vertically varying velocity
\cite[]{GEO53-07-08810893}. Further theoretical generalization sets
the number of migrations in a cascade to infinity, making each step in
the velocity space infinitesimally small. This leads to a partial
differential equation in the time-midpoint-velocity space, discovered
by \cite{Claerbout.sep.48.79}. Claerbout's equation describes the
process of {\em velocity continuation,} which fills the velocity space
in the same manner as a set of constant-velocity migrations. Slicing
in the migration velocity space can serve as a method of velocity
analysis for migration with nonconstant velocity
\cite[]{shurtleff,SEG-1984-S1.8,Fowler.sepphd.58, GEO57-01-00510059}.

\new{The concept of velocity continuation was introduced in the earlier
publications} \cite[]{me,SEG-1997-1762}.  \cite{hubral} and
\cite{GEO62-02-05890597} \new{
use the term \emph{image waves} to describe a
similar idea.} \cite{adler} \new{generalizes it to the case of variable
background velocities under the name \emph{Kirchhoff image propagation}.  The
importance of this concept lies in its ability to predict changes in the
geometry and intensity of reflection events on seismic images with the change
of migration velocity. While conventional approaches to migration velocity
analysis methods take into account only vertical movement of reflectors}
\cite[]{FBR08-06-02240234,GEO60-01-01420153}, 
\new{velocity continuation attempts to describe both
vertical and lateral movements, thus providing for optimal focusing in
velocity analysis applications} \cite[]{SEG-2001-11071110,second}. 

In this paper, I describe the velocity continuation theory for the case of
prestack time migration, connecting it with the theory of prestack residual
migration \cite[]{Al-Yahya.sep.50.219,Etgen.sepphd.68,GEO61-02-06050607}. By
exploiting the mathematical theory of characteristics, a simplified kinematic
derivation of the velocity continuation equation leads to a differential
equation with correct dynamic properties. 
%In the post-stack case, the
%solution of the boundary-value problem, associated with this equation,
%coincides precisely with the operators of Kirchoff migration, traditionally
%derived from a completely different prospective.  
In practice, one can
accomplish dynamic velocity continuation by integral, finite-difference, or
spectral methods.
\begin{comment}
For practical
applications, I chose the Fourier spectral method. The method has its
limitations \cite[]{Fomel.sep.97.sergey2}, but looks optimal in terms of
the accuracy versus efficiency trade-off.
 
Applying velocity continuation to migration velocity analysis involves
the following steps: 
\begin{enumerate}
\item prestack common-offset (and common-azimuth) migration - to
  generate the initial data for continuation,
\item velocity continuation with stacking across different offsets -
  to transform the offset data dimension into the velocity dimension,
\item picking the optimal velocity and slicing through the migrated
  data volume - to generate an optimally focused image.
\end{enumerate}
The final part of this paper includes a demonstration of all three
steps on a simple two-dimensional dataset.
\end{comment}
The accompanying paper \cite[]{second} introduces one of the possible
numerical implementations and demonstrates its application on a \new{field}
data example.

\new{
The paper is organized into two main sections. First, I derive the kinematics
of velocity continuation from the first geometric principles. I identify three
distinctive terms, corresponding to zero-offset residual migration, residual
normal moveout, and residual dip moveout. Each term is analyzed separately to
derive an analytical prediction for the changes in the geometry of 
traveltime curves (reflection events on migrated images) with the change of
migration velocity. Second, the dynamic behavior of seismic images is
described with the help of partial differential equations and their
solutions. Reconstruction of the dynamical counterparts for kinematic
equations is not unique. However, I show that, with an appropriate selection of
additional terms, the image waves corresponding to the velocity continuation
process have the correct dynamic behavior. In particular, a special 
boundary value problem with the zero-offset velocity continuation equation
produces the solution identical to the conventional Kirchoff time migration.}

\begin{comment}
It is important to note that although the velocity continuation result
could be achieved in principle by using prestack residual migration in
Kirchhoff \cite[]{Etgen.sepphd.68} or Stolt \cite[]{GEO61-02-06050607}
formulation, the first is evidently inferior in efficiency, and the
second is not convenient for velocity analysis across different
offsets, because it mixes them in the Fourier domain.
\end{comment}

\section{KINEMATICS OF VELOCITY CONTINUATION}

From the kinematic point of view, it is convenient to describe the
reflector as a locally smooth surface $z = z(x)$, where $z$ is the
depth, and $x$ is the point on the surface ($x$ is a two-dimensional
vector in the 3-D problem). The image of the reflector obtained after
a common-offset prestack migration with a half-offset $h$ and a
constant velocity $v$ is the surface $z = z(x;h,v)$. Appendix A
provides the derivations of the partial differential equation
describing the image surface in the depth-midpoint-offset-velocity
space. The purpose of this section is to discuss the laws of kinematic
transformations implied by the velocity continuation equation. Later
in this paper, I obtain dynamic analogs of the kinematic
relationships in order to describe the continuation of migrated
sections in the velocity space.

The kinematic equation for prestack velocity continuation, derived in
Appendix A, takes the following form:
\begin{equation}
{{\partial \tau} \over {\partial v}} = 
v\,\tau\,\left({{\partial \tau} \over {\partial x}}\right)^2 +
{{h^2} \over {v^3\,\tau}}\,- 
\frac{h^2 v}{\tau}\,
\left({{\partial \tau} \over {\partial x}}\right)^2\,
\left({{\partial \tau} \over {\partial h}}\right)^2\;.
\label{eq:eikonal} 
\end{equation}
Here $\tau$ denotes the one-way vertical traveltime $\left(\tau = {z
\over v}\right)$. The right-hand side of equation (\ref{eq:eikonal})
consists of three distinctive terms. Each has its own geophysical meaning. The first term is the only one remaining
when the half-offset $h$ equals zero. This term corresponds to the procedure of
{\em zero-offset residual migration}.  Setting the traveltime dip to
zero eliminates the first and third terms, leaving the second,
dip-independent one. One can associate the second term with the process of
{\em residual normal moveout}. The third term is both dip- and offset-
dependent. The process that it describes is {\em residual dip
moveout}. It is convenient to analyze each of the three processes
separately, evaluating their contributions to the cumulative process
of prestack velocity continuation.

\subsection{Kinematics of Zero-Offset Velocity Continuation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The kinematic equation for zero-offset velocity continuation is
\begin{equation}
{{\partial \tau} \over {\partial v}} = 
v\,\tau\,\left({{\partial \tau} \over {\partial x}}\right)^2\;.
\label{eq:POMeikonal} 
\end{equation}
The typical boundary-value problem associated with it is to find the
traveltime surface $\tau_2(x_2)$ for a constant velocity $v_2$, given the
traveltime surface $\tau_1(x_1)$ at some other velocity $v_1$. Both surfaces
correspond to the reflector images obtained by time migrations with the
specified velocities.  When the migration velocity approaches zero, post-stack
time migration approaches the identity operator. Therefore, the case of $v_1 =
0$ corresponds kinematically to the zero-offset (post-stack) migration, and
the case of $v_2 = 0$ corresponds to the zero-offset modeling (demigration).
\new{The variable $x$ in equation~(\ref{eq:POMeikonal}) describes both the surface
midpoint coordinate and the subsurface image coordinate. One of them is
continuously transformed into the other in the velocity continuation process.}

The appropriate mathematical method of solving the kinematic
problem posed above is the method of characteristics \cite[]{kurant2}. The
characteristics of equation (\ref{eq:POMeikonal}) are the trajectories
followed by individual points of the reflector image in the velocity
continuation process. These trajectories are called {\em velocity rays} 
\cite[]{me,them,adler}. Velocity rays are defined by the system of ordinary
differential equations derived from (\ref{eq:POMeikonal}) according to the
\new{Hamilton-Jacobi theory}:
\begin{eqnarray}
{{{dx} \over {dv}} = - 2\,v\,\tau\,\tau_x} & , &
{{{d\tau} \over {dv}} = - \tau_v}\;,
\label{eq:velray1} \\
{{{d\tau_x} \over {dv}} = v\,\tau_x^3} & , &
{{{d\tau_v} \over {dv}} = \left(\tau + v\,\tau_v\right)\,\tau_x^2}\;,
\label{eq:velray2} 
\end{eqnarray}
\new{where $\tau_x$ and $\tau_v$ are the phase-space parameters}.
An additional constraint for $\tau_x$ and $\tau_v$
follows from equation (\ref{eq:POMeikonal}), rewritten in the form
\begin{equation}
\tau_v = v\,\tau\,\tau_x^2\;. 
\label{eq:equiveikonal} 
\end{equation}
%One can easily solve the system of equations (\ref{eq:velray1}) and
%(\ref{eq:velray2}) by the classic mathematical methods for ordinary
%differential equations. 
The general solution of the system of
equations~(\ref{eq:velray1}-\ref{eq:velray2}) takes the
parametric form
\begin{eqnarray}
x(v) & = & A - C v^2\;,\quad
\tau^2(v) = B - C^2\,v^2\;,
\label{eq:velrayg1} \\ 
\tau_x(v) & = & {C \over {\tau(v)}}\;,\quad
\tau_v(v) = {{C^2\,v} \over {\tau(v)}}\;,
\label{eq:velrayg2} 
\end{eqnarray}
where $A$, $B$, and $C$ are constant along each individual velocity
ray. These three constants are determined from the boundary conditions
as
\begin{equation}
A = x_1 + v_1^2\,\tau_1\,{{\partial \tau_1} \over {\partial x_1}} = x_0\;,
\label{eq:a} 
\end{equation}
\begin{equation}
B = \tau_1^2\,\left(1 + v_1^2\,
\left({{\partial \tau_1} \over {\partial x_1}}\right)^2\right) = \tau_0^2\;,
\label{eq:b} 
\end{equation}
\begin{equation}
C = \tau_1\,{{\partial \tau_1} \over {\partial x_1}} = 
\tau_0\,{{\partial \tau_0} \over {\partial x_0}}\;,
\label{eq:c} 
\end{equation}
where $\tau_0$ and $x_0$ correspond to the zero velocity (unmigrated
section), while $\tau_1$ and $x_1$ correspond to the velocity $v_1$.
% Equations (\ref{eq:a}), (\ref{eq:b}), and (\ref{eq:c}) have a clear
%geometric meaning illustrated in Figure \ref{fig:vlczor}. 
The
simple relationship between the midpoint derivative of the vertical
traveltime and the local dip angle $\alpha$ (appendix A),
\begin{equation}
{{\partial \tau} \over {\partial x}} = 
{{\tan{\alpha}} \over v}\;,
\label{eq:dtaudx} 
\end{equation}
shows that equations (\ref{eq:a}) and (\ref{eq:b}) are precisely equivalent
to the evident geometric relationships (Figure~\ref{fig:vlczor})
\begin{equation}
x_1 + v_1\,\tau_1\,\tan{\alpha} = x_0\;,
\;{\tau_1 \over {\cos{\alpha}}} = \tau_0\;.
\label{eq:evident}
\end{equation}
Equation (\ref{eq:c}) states that the points on a velocity ray correspond
to a single reflection point, constrained by the values of $\tau_1$,
$v_1$, and $\alpha$.  As follows from equations (\ref{eq:velrayg1}), the
projection of a velocity ray to the time-midpoint plane has the
parabolic shape $x(\tau) = A + (\tau^2 - B) / C$, which has been
noticed by \cite{GEO46-05-07170733}. On the depth-midpoint plane, the
velocity rays have the circular shape $z^2(x) = (A - x)\,B / C - (A -
x)^2$, described by \cite{them} as ``Thales circles.''

\inputdir{XFig}

\sideplot{vlczor}{width=0.9\textwidth}{Zero-offset reflection in a
  constant velocity medium (a scheme).}

For an example of kinematic continuation by velocity rays, let us
consider the case of a point diffractor. If the diffractor location in
the subsurface is the point ${x_d,z_d}$, then the reflection traveltime at
zero offset is defined from Pythagoras's theorem as the hyperbolic
curve
\begin{equation}
\tau_0(x_0) = {{\sqrt{z_d^2 + (x_0 - x_d)^2}} \over v_d}\;,
\label{eq:dift0}
\end{equation}
where $v_d$ is half of the actual velocity. Applying equations
(\ref{eq:velrayg1}) produces the following mathematical expressions
for the velocity rays:
\begin{eqnarray}
x(v) & = & x_d\,{v^2 \over v_d^2} + 
x_0\,\left(1 -  {v^2 \over v_d^2}\right)\;,
\label{eq:difrayg1} \\ 
\tau^2(v) & = & \tau_d^2 + {{(x_0 - x_d)^2} \over v_d^2}\,
\left(1 -  {v^2 \over v_d^2}\right)\;,
\label{eq:difrayg2} 
\end{eqnarray}
where $\tau_d = {z_d \over v_d}$.
Eliminating $x_0$ from the system of equations (\ref{eq:difrayg1}) and
(\ref{eq:difrayg2}) leads to the expression for the velocity continuation
``wavefront'': 
\begin{equation}
\tau(x)=\sqrt{\tau_d^2 + {{(x - x_d)^2} \over {v_d^2 - v^2}}}\;.
\label{eq:diffront}
\end{equation}
For the case of a point diffractor, the wavefront corresponds precisely
to the summation path of the residual migration operator
\cite[]{GEO50-01-01100126}. It has a hyperbolic shape when $v_d > v$
(undermigration) and an elliptic shape when $v_d < v$
(overmigration). The wavefront collapses to a point when the velocity
$v$ approaches the actual effective velocity $v_d$. At zero
velocity, $v=0$, the wavefront takes the familiar form of the post-stack migration
hyperbolic summation path. The form of the velocity rays and wavefronts
is illustrated in the left plot of Figure \ref{fig:vlcvrs}.

\inputdir{Math}