%\def\SEPCLASSLIB{../../../../sepclasslib}\def\CAKEDIR{.}\title{Zero-offset migration}\author{Jon Claerbout}\maketitle%\chapter{Kirchhoff migration}\label{paper:krch}\sx{Kirchhoff migration}\sx{migration!Kirchhoff}\parIn chapter~\ref{vela/paper:vela} we discussedmethods of imaging horizontal reflectorsand of estimating velocity $v(z)$from the offset dependence of seismic recordings.In this chapter,we turn our attention to imaging methods for {\em dipping} reflectors.These imaging methodsare usually referred to as ``migration'' techniques.   \parOffset is a geometrical nuisance when reflectors have dip.For this reason,we develop migration methods here and in the next chapterfor forming images from hypothetical {\em zero-offset} seismic experiments.  Although there is usually ample data recorded near zero-offset,we never record purely zero-offset seismic data.However, when we consider offset and dip together in chapter~\ref{dpmv/paper:dpmv} we will encountera widely-used technique (dip-moveout) that often converts finite-offset data into a useful estimate of the equivalent zero-offset data.For this reason,\bx{zero-offset migration}methods are widely used today in industrial practice.Furthermore the concepts of zero-offset migrationare the simplest starting point for approachingthe complications of finite-offset migration.\section{MIGRATION DEFINED}\parThe term ``migration'' probably got its namefrom some association with movement.A casual inspection of migrated and unmigrated sectionsshows that migration causes many reflection eventsto shift their positions.These shifts are necessary because the {\em apparent} positionsof reflection events on unmigrated sections are generally not the {\em true}positions of the reflectors in the earth.It is not difficult to visualize why such ``acoustic illusions'' occur.An analysis of a zero-offset sectionshot above a dipping reflector illustrates most of the key concepts.\subsection{A dipping reflector}\inputdir{vplot}\parConsider the zero-offset seismic survey shown in Figure~\ref{fig:reflexpt}.This survey  uses one source-receiver pair,and the receiver is always at the same location as the source.At each position, denoted by $S_1, S_2, \mbox{and} S_3$ in the figure,the source emits waves and the receiver records the echoesas a single seismic trace.After each trace is recorded,the source-receiver pair is moved a small distanceand the experiment is repeated.%\sideplot{reflexpt}{width=3.5in}{  Raypaths and wavefronts for a zero-offset seismic line shot  above a dipping reflector.  The earth's propagation velocity is constant.}%\parAs shown in the figure,the source at $S_2$ emits a spherically-spreading wavethat bounces off the reflectorand then returns to the receiver at $S_2$.The raypaths drawn between $S_i$ and $R_i$ are orthogonalto the reflector and hence are called {\em normal rays}.\sx{normal rays}\sx{rays, normal}These rays reveal how the zero-offset section misrepresents the truth. For example,the trace recorded at $S_2$ is dominated by the reflectivity near reflection point $R_2$,where the normal ray from $S_2$ hits the reflector.   If the zero-offset section corresponding to Figure~\ref{fig:reflexpt} is displayed, the reflectivity at $R_2$ will be falsely displayedas though it were directly beneath $S_2$,which it certainly is not.This lateral mispositioning is the first part of the illusion.The second part is vertical:if converted to depth,the zero-offset section will show $R_2$ to be deeper than it really is.The reason is that the slant path of the normal rayis longer than a vertical shaft drilled from the surface down to $R_2$.\subsection{Dipping-reflector shifts}\parA little geometry gives simple expressionsfor the horizontal and vertical position errors on the zero-offset section,which are to be corrected by migration.Figure~\ref{fig:reflkine} defines therequired quantities for a reflection event recorded at $S$ correspondingto the reflectivity at $R$.  %\sideplot{reflkine}{width=3.5in}{  Geometry of the normal ray of length $d$ and the vertical ``shaft''   of length $z$ for  a zero-offset experiment above a dipping reflector.}%The two-way travel time for the event isrelated to the length $d$ of the normal ray by\begin{equation}t \eq {2\,d \over v}\label{eqn:dipt0} \ \ ,\end{equation}where $v$ is the constant propagation velocity.Geometry of the triangle $CRS$ showsthat the true depth of the reflector at $R$ is given by\begin{equation}z \eq d\ \cos\theta \ \ ,\label{eqn:dipz}\end{equation}and the lateral shift between true position $C$ and false position $S$is given by\begin{equation}\Delta x \eq d\ \sin\theta  \eq {v\,t \over 2}\ \sin\theta \ \ .\label{eqn:dipdx}\end{equation}It is conventional to rewrite equation~(\ref{eqn:dipz}) in terms of two-way{\em vertical} traveltime $\tau$:\begin{equation}\tau \eq {2\,z \over v} \eq t\, \cos\theta  \ \ .\label{eqn:diptau}\end{equation}Thus both the vertical shift $t - \tau$ and the horizontal shift $\Delta x$are seen to vanish when the dip angle $\theta$ is zero.  \subsection{Hand migration}\sx{hand migration}\sx{migration!hand}\parGeophysicists recognized the need to correct these positioning errorson zero-offset sections long before it was practical to use computersto make the corrections.Thus a number of hand-migration techniques arose.It is instructive to see how one such scheme works.  Equations~(\ref{eqn:dipdx}) and (\ref{eqn:diptau}) require knowledge of three quantities:$t$, $v$, and $\theta$.Of these, the event time $t$ is readily measured on the zero-offset section.The velocity $v$ is usually {\em not} measurable on the zero offset sectionand must be estimated from finite-offset data,as was shown in chapter~\ref{vela/paper:vela}.That leaves the dip angle $\theta$.This can be related to the reflection slope $p$ of the observed event,which is measurable on the zero-offset section: \begin{equation}p_0 \eq {\partial t \over \partial y}  \ \ ,\label{eqn:pdefn}\end{equation}where $y$ (the midpoint coordinate) is the location of the source-receiver pair.The slope $p_0$ is sometimes called the {\em ``time-dip of the event''} or \sx{time dip}more loosely as the {\em ``dip of the event''}.It is obviously closely related to Snell's parameter,which we discussed in chapter~\ref{wvs/paper:wvs}.   The relationship between the measurable time-dip $p_0$ and the dip angle $\theta$ is called``\bx{Tuchel's law}'':\begin{equation}\sin\theta \eq {v\,p_0 \over 2}   \ \ .\label{eqn:tuchel}\end{equation}This equation is clearly just another versionof equation~(\ref{wvs/eqn:5iei4a}),in which a factor of $2$ has been inserted to account for the two-way traveltimeof the zero-offset section.%\parRewriting the migration shift equations in terms of the measurablequantities $t$ and $p$ yields usable ``hand-migration'' formulas:\begin{eqnarray}\Delta x &\eq& {v^2\ p\ t \over 4}\label{eqn:laydxp}\\\tau &\eq& t\ \sqrt{1\ -\ {v^2 p^2 \over 4} }  \ \  .\label{eqn:laytaup}\end{eqnarray}Hand migration divides each observed reflection eventinto a set of small segments for which $p$ has been measured.This is necessary because $p$ is generally not constant along real seismic events.But we can consider more generalevents to be the union of a large number of very small dipping reflectors.Each such segment is then mapped from its unmigrated $(y,t)$ locationto its migrated $(y,\tau)$ location based on the equations above.  Such a procedure is sometimes also known as ``map migration.''  \parEquations~(\ref{eqn:laydxp}) and (\ref{eqn:laytaup}) are useful for giving an idea of what goes on in zero-offset migration.But using these equations directly for practical seismic migration can be tedious and error-prone because of the need to provide the time dip $p$ as a separateset of input data values as a function of $y$ and $t$.  One nasty complication is that it is quite commonto see {\em crossing events} on zero-offset sections.This happens whenever reflection energy coming from two different reflectorsarrives at a receiver at the same time.  When this happens the time dip $p$becomes a {\em multi-valued} function of the $(y,t)$ coordinates.Furthermore, the recorded wavefield is now the sum of two different events.It is then difficult to figure out which part of summed amplitudeto move in one direction and which part to move in the other direction.\parFor the above reasons,the seismic industry has generally turned away from hand-migration techniquesin favor of more automatic methods.These methods require as inputs nothing more than \begin{itemize}\item The zero-offset section \item The velocity $v$  \ \ .\end{itemize}There is no need to separately estimate a $p(y,t)$ field.The automatic migration program somehow ``figures out''which way to move the events,even if they cross one another.Such automatic methods are generally referred to as{\em ``wave-equation migration''}  techniques,and are the subject of the remainder of this chapter.But before we introduce the automatic migration methods,we need to introduce one additional conceptthat greatly simplifies the migration of zero-offset sections.\subsection{A powerful analogy}\inputdir{XFig}\parFigure~\ref{fig:expref} shows two wave-propagation situations. %\plot{expref}{width=5.0in}{  Echoes collected with a source-receiver pair  moved to all points on the earth's surface (left)  and the ``exploding-reflectors'' conceptual model (right).}%The first is realistic field sounding.The second is a thought experiment in whichthe reflectors in the earth suddenly explode.Waves from the hypothetical explosion propagateup to the earth's surface where they areobserved by a hypothetical string of geophones.\parNotice in the figure that the ray paths in thefield-recording case seem to be thesame as those in the \bxbx{exploding-reflector}{exploding reflector} case.It is a great conceptual advantage to imagine that the two wavefields,the observed and the hypothetical, are indeed the same.If they are the same,the many thousands of experimentsthat have really been done can be ignored,and attention can be focused on the one hypothetical experiment.One obvious difference between the two cases is that in thefield geometry waves must first go downand then return upward along the same path,whereas in the hypothetical experiment they just go up.Travel time in field experiments could be divided by two.In practice, the data of the field experiments (two-way time)is analyzed assuming the sound velocity to be half its true value.\subsection{Limitations of the exploding-reflector concept}\parThe exploding-reflector concept is a powerful and fortunate analogy.It enables us to think of the data of many experimentsas though it were a single experiment.Unfortunately,the exploding-reflector concept has a serious shortcoming.No one has yet figured out how to extend the conceptto apply to data recorded at nonzero offset.Furthermore, most data is recorded at rather large offsets.In a modern marine prospecting survey,there is not one hydrophone,but hundreds, which are strung out in a cable towed behind the ship.The recording cableis typically 2-3 kilometers long.Drilling may be about 3 kilometers deep.So in practice the angles are big.Therein lie both new problems and new opportunities,none of which will be considered until chapter~\ref{dpmv/paper:dpmv}.\parFurthermore, even at zero offset,the exploding-reflector concept is not quantitatively correct.For the moment, note three obvious failings:First, Figure~\ref{fig:fail} shows rays that are not predictedby the exploding-reflector model.\plot{fail}{}{  Two rays, not predicted by the exploding-reflector model,  that would nevertheless be found on a zero-offset section.}%was \activeplot{fail}{height=1.6in}{NR}{}These rays will be present in a zero-offset section.Lateral velocity variation is required for thissituation to exist.  \parSecond, the exploding-reflector concept fails with \bx{multiple reflection}s.For a flat sea floor with a two-way travel time  $t_1$, multiple reflectionsare predicted at times  $2t_1$,  3$t_1$,  4$t_1$, etc.In the exploding-reflector geometry the first multiplegoes from reflector to surface,then from surface to reflector,then from reflector to surface, for a total time  3$t_1$.Subsequent multiples occur at times  5$t_1$,  7$t_1$, etc.Clearly the multiple reflections generated on the zero-offset sectiondiffer from those of the exploding-reflector model.\parThe third failing of the exploding-reflector modelis where we are able to see wavesbounced from both sides of an interface.The exploding-reflector model predicts