%\def\SEPCLASSLIB{../../../../sepclasslib}\def\CAKEDIR{.}\title{Dip and offset together}\author{Jon Claerbout}\maketitle\label{paper:dpmv}\def\vhalf{v_{\rm half}}\par\footnote{%	Jim Black prepared many figures%	and the geometrical derivations at the end of the chapter.	Matt Schwab prepared a draft of the Gardner DMO derivation.	Shuki Ronen gave me the ``law of cosines'' proof.	}Whendip and offset are combined,some serious complications arise.For many years it was common industry practiceto ignore these complications and to handle dip and offset separately.  Thus offset was handled by velocity analysis, normal moveout and stack (chapter~\ref{vela/paper:vela}).And dip was handled by zero-offset migration after stack (chapters~\ref{krch/paper:krch} and \ref{dwnc/paper:dwnc}).This practice is a good approximation only whenthe dips on the section are small.We need to handle large offset angles at the sametime we handle large dip anglesat the same time we are estimating rock velocity.It is confusing!Here we see the important stepsof bootstrapping yourself towards both the velocity and the image.%In fact, using this {\it separable} style%of seismic processing generally leads to seismic sections on which%the steeply-dipping reflectors are artificially attenuated relative %to the horizontal reflectors.%The easiest way to understand how things fail %is to consider the case of a planar dipping reflector at finite offset.\section{PRESTACK MIGRATION}\inputdir{XFig}%One solution to the problems pointed out in the previous section%is to forget about NMO, stack, and zero-offset migration%and replace them by prestack migration.Prestack migration creates an image of the earth's reflectivitydirectly from prestack data.It is an alternative tothe ``\bx{exploding reflector}'' conceptthat proved so useful in zero-offset migration.In \bx{prestack migration},\sx{migration!prestack}we consider both downgoing and upcoming waves.\parA good starting point for discussing prestack migration is a reflecting point within the earth.A wave incident on the point from any directionreflects waves in all directions.This geometry is particularly important becauseany model is a superposition of such point scatterers.The point-scatterer geometry for a pointlocated at $(x,z)$ is shown in Figure~\ref{fig:pgeometry}. %\sideplot{pgeometry}{width=3.5in}{  Geometry of a point scatterer.}The equationfor travel time  $t$  is the sum of the two travel paths is\begin{equation}t\,v\  \eq \  \sqrt { z^2\ +\ {( s \ -\  x ) }^2}  \ +\  \sqrt { z^2 \ +\ {( g \ -\  x )}^2} \label{eqn:dsrsg}\end{equation}We could model field datawith equation~(\ref{eqn:dsrsg}) by copying reflections from any pointin $(x,z)$-space into $(s,g,t)$-space.The adjoint program would form an image stacked overall offsets.This process would be called prestack migration.The problem here is that the real problemis estimating velocity.In this chapter we will see that it is not satisfactoryto use a horizontal layer approximation to estimate velocity,and then use equation~(\ref{eqn:dsrsg}) to do migration.Migration becomes sensitive to velocity when wide angles are involved.Errors in the velocity would spoil whatever benefit couldaccrue from prestack (instead of poststack) migration.\subsection{Cheops' pyramid}\sx{Cheops' pyramid}\inputdir{Math}\parBecause of the importance of the point-scatterer model,we will go to considerable lengths to visualize the functional dependenceamong $t$, $z$, $x$, $s$, and $g$ in equation (\ref{eqn:dsrsg}).This picture is more difficult---by one dimension---than isthe conic section of the exploding-reflector geometry.\parTo begin with,suppose that the first square root in (\ref{eqn:dsrsg}) is constantbecause everything in it is held constant.This leaves the familiar hyperbola in  $(g,t)$-space,except that a constant has been added to the time.Suppose instead that the other square root is constant.This likewise leaves a hyperbola in  $(s,t)$-space.In  $(s,g)$-space, travel time is a function of  $s$  plus a function of  $g$.I think of this as one coat hanger, which is parallel to the $s$-axis, being hung from another coat hanger,which is parallel to the $g$-axis.\parA view of the traveltime pyramid on the  $(s,g)$-planeor the  $(y,h)$-plane is shown in Figure~\ref{fig:cheop}a.\plot{cheop}{height=3.4in}{	Left is a picture of the traveltime pyramid	of equation (\protect(\ref{eqn:dsrsg}))	for fixed  $x$  and  $z$.	The darkened lines are constant-offset sections.	Right is a cross section through the pyramid	for large  $t$  (or small  $z$).  (Ottolini)	}%\newslideNotice that a cut through the pyramid atlarge  $t$  is a square, the corners of which have been smoothed.At very large  $t$,a constant value of  $t$  is the square contoured in  $(s,g)$-space,as in Figure~\ref{fig:cheop}b.Algebraically, the squareness becomes evident for a point reflectornear the surface, say,  $z  \to  0$.Then (\ref{eqn:dsrsg}) becomes\begin{equation}v\,t\  \eq \ | s \ -\  x |\ \ +\ \ | g \ -\  x |\label{eqn:2.4}\end{equation}The center of the square is located at  $(s,g) = (x,x)$.Taking travel time  $t$  to increase downwardfrom the horizontal plane of  $(s,g)$-space,the square contour is like a horizontal slice through the Egyptian pyramidof Cheops.To walk around the pyramid at a constant altitude is to walk around a square.Alternately,the altitude change of a traverse over$g$  (or $s$) at constant$s$  (or $g$) is simply a constant plus an absolute-value function.\parMore interesting and less obvious are the curveson common-midpoint gathers and constant-offset sections.Recall the definition that the midpoint between the shot and geophone is  $y$.Also recall that  $h$  is half the horizontal offsetfrom the shot to the geophone.\begin{eqnarray}y\ \ \ \ &=&\ \ \ \ {g \ +\  s  \over 2 }\label{eqn:2.5a}\\h\ \ \ \ &=&\ \ \ \ {g \ -\  s  \over 2 }\label{eqn:2.5b}\end{eqnarray}A traverseof $y$ at constant $h$ is shown in Figure~\ref{fig:cheop}.At the highest elevation on the traverse,you are walking along a flat horizontalstep like the flat-topped hyperboloids of Figure~\ref{fig:twopoint}.Some erosion to smooth the top and edges of the pyramidgives a model for nonzero reflector depth.\parFor rays that are near the vertical,the traveltime curves are far from the hyperbola asymptotes.Then the square roots in (\ref{eqn:dsrsg}) may be expanded in Taylor series,giving a parabola of revolution.This describes the eroded peak of the pyramid.\subsection{Prestack migration ellipse}\inputdir{dmovplot}Denoting the horizontal coordinate $x$ of the scattering point by $y_0$Equation (\ref{eqn:dsrsg}) in $(y,h)$-space is\begin{equation}t\,v\  \eq \  \sqrt { z^2\ +\ {( y \ -\  y_0  \  - \  h) }^2}       \ +\  \sqrt { z^2\ +\ {( y \ -\  y_0   \ + \  h) }^2} \label{eqn:dsryh}\end{equation}A basic insight into equation (\ref{eqn:dsrsg}) is to noticethat at constant-offset  $h$  and constant travel time  $t$the locus of possible reflectors isan ellipse in the $(y ,z)$-plane centered at $y_0$.The reason it is an \bx{ellipse}follows from the geometric definition of an ellipse.To draw an ellipse,place a nail or tack into $s$ on Figure~\ref{fig:pgeometry}and another into $g$.Connect the tacks by a stringthat is exactly long enough to go through  $(y_0 ,z)$.An ellipse going through  $(y_0 ,z)$  may be constructedby sliding a pencil along the string,keeping the string tight.The string keeps the total distance  $tv$  constant as is shown inFigure~\ref{fig:ellipse1}\sideplot{ellipse1}{width=3.5in}{	Prestack migration ellipse, the locus of all scatterers with	constant traveltime for	source S and receiver G.	}%\newslide\parReplacing depth $z$ in equation~(\ref{eqn:dsryh})by the vertical traveltime depth$\tau = 2z/v=z/\vhalf$ we get\begin{equation}t  \eq   {1 \over 2}\       \left(	\sqrt { \tau^2\ +\ [( y-y_0)-h]^2 / \vhalf^2 } 	\ +\  	\sqrt { \tau^2\ +\ [( y-y_0)+h]^2 / \vhalf^2 } 	\ 	\right)\label{eqn:dsryhtau}\end{equation}\subsection{Constant offset migration}\inputdir{matt}\sx{constant-offset migration}\sx{migration!constant-offset}Considering $h$ in equation~(\ref{eqn:dsryhtau})to be a constant,enables us to write a subroutine for migrating constant-offset sections.%Subroutine \texttt{flathyp()} \vpageref{/prog:flathyp} is easily prepared%from subroutine \texttt{kirchfast()} \vpageref{/prog:kirchfast} by%replacing its hyperbola equation with equation~(\ref{eqn:dsryhtau}).%\progdex{flathyp}{const offset migration}%The amplitude in subroutine {\tt flathyp()}%should be improved when we have time to do so.Forward and backward responses to impulses%of subroutine {\tt flathyp()}are found in Figures~\ref{fig:Cos1} and \ref{fig:Cos0}.\sideplot{Cos1}{width=3.in,height=3.in}{	Migrating impulses on a constant-offset section.%	with subroutine {\tt flathyp()}.	Notice that shallow impulses	(shallow compared to $h$)	appear ellipsoidal	while deep ones appear circular.	}%\newslide\sideplot{Cos0}{width=3.in,height=3.in}{	Forward modeling	from an earth impulse. % with subroutine {\tt flathyp()}.	}%\newslide\parIt is not easy to show that equation (\ref{eqn:dsryh}) can becast in the standard mathematical form of an ellipse, namely, a stretched circle.But the result is a simple one,and an important one for later analysis.Feel free to skip forward over the following verificationof this ancient wisdom.To help reduce algebraic verbosity,define a new  $y$  equal to the old one shifted by  $y_0 $.Also make the definitions\begin{eqnarray}t\,v \ \ \ \ &=&\ \ \ \ 2\ A\  			\label{eqn:mymajor}\\\alpha\ \ \ \ &=&\ \ \ \ z^2 \ \ +\ \  (y\ +\ h)^2   \nonumber\\\beta\ \ \ \ &=&\ \ \ \ z^2 \ \ +\ \  (y\ -\ h)^2   \nonumber\\\alpha\ \ -\ \ \beta\ \ \ \ &=&\ \ \ \ 4\ y\ h  \nonumber\end{eqnarray}With these definitions, (\ref{eqn:dsryh}) becomes\begin{displaymath}2\ A\  \eq \  \sqrt \alpha \ \ +\ \   \sqrt \beta  \end{displaymath}Square to get a new equation with only one square root.\begin{displaymath}4\ A^2 \ \ -\ \ (\alpha\ +\ \beta) \  \eq \ 2\  \sqrt{ \alpha \beta }\end{displaymath}Square again to eliminate the square root.\begin{eqnarray*}16\ A^4 \ \ -\ \ 8\ A^2 \, (\alpha\ +\ \beta) \ \ +\ \ (\alpha\ +\ \beta)^2 \ \ \ \ &=&\ \ \ \ 4\ \alpha\ \beta\\16\ A^4 \ \ -\ \ 8\ A^2 \, (\alpha\ +\ \beta) \ \ +\ \ (\alpha\ -\ \beta)^2 \ \ \ \ &=&\ \ \ \ 0\end{eqnarray*}Introduce definitions of  $\alpha$  and  $\beta$.\begin{displaymath}16\ A^4 \ \ -\ \  8\ A^2 \ [\,2\,z^2 \ +\ 2\,y^2 \ +\  2\,h^2 ] \ \ +\ \ 16\ y^2 \, h^2 \  \eq \ 0 \end{displaymath}Bring  $y$  and  $z$  to the right.\begin{eqnarray}A^4 \ \ -\ \  A^2 \, h^2 \ \ \ \ &=&\ \ \ \ 		\nonumberA^2 \, ( z^2 \ +\  y^2 ) \ \ -\ \  y^2 \, h^2\\A^2 \, ( A^2 \ -\  h^2 ) \ \ \ \ &=&\ \ \ \ A^2 \,  z^2 \ +\   ( A^2 \ -\   h^2 ) \, y^2		\nonumber\\A^2 \ \ \ \ &=&\ \ \ \  {z^2  \over 1 \ -\  {h^2  \over A^2}}\ \ +\ \ y^2							\label{eqn:torick}\end{eqnarray}Finally, recalling all earlier definitions and replacing $y$ by $y-y_0$, weobtain the canonical form of an ellipse with semi-major axis $A$ and semi-minor axis $B$:\begin{equation}{(y\ -\ y_0)^2 \over A^2} \ +\ {z^2 \over B^2} \eq 1 \ \ \ ,\label{eqn:canellipse}\end{equation}where\begin{eqnarray} A &\eq& {v\ t \over 2}    \\ B &\eq& \sqrt{A^2\ -\ h^2}\end{eqnarray}\parFixing $t$, equation (\ref{eqn:canellipse}) is the equation for a circle witha stretched $z$-axis.The above algebra confirms that the``string and tack'' definition of an \bx{ellipse}matches the ``stretched circle'' definition.An \bx{ellipse} in earth model space correspondsto an impulse on a constant-offset section.\section{INTRODUCTION TO DIP}\inputdir{XFig}We can consider a data space to be a superposition of pointsand then analyze the point response,or we can consider data spaceor model space to be a superposition of planesand then do an analysis of planes.Analysis of points is often easier than planes,but planes, particularly local planes,are more like observational data and earth models.\parThe simplest environment for reflection datais a single horizontal reflection interface,which is shown in Figure~\ref{fig:simple}.\plot{simple}{height=2.5in}{	Simplest earth model. 	}%\newslideAs expected, the zero-offset section mimics the earth model.The common-midpoint gather is a hyperbolawhose asymptotes are straight lineswith slopes of the inverse of the velocity  $v_1$.The most basic data processing is called{\em \bx{common-depth-point stack}}or \bx{CDP stack}.In it, all the traces on the common-midpoint (CMP) gatherare time shifted into alignmentand then added together.The result mimics a zero-offset trace.