%\def\SEPCLASSLIB{../../../../sepclasslib}\def\CAKEDIR{.}\title{Downward continuation}\author{Jon Claerbout}\maketitle\label{paper:dwnc}%{\em \today . This chapter is owned by JFC.}\section{MIGRATION BY DOWNWARD CONTINUATION}Given waves observed along the earth's surface,some well-known mathematical techniquesthat are introduced hereenable us to extrapolate(\bx{downward continue})these waves down into the earth.Migration is a simple consequence of this extrapolation.\subsection{Huygens secondary point source}\inputdir{XFig}\parWaves on the ocean have wavelengths comparableto those of waves in seismic prospecting (15-500 meters),but ocean waves move slowly enough to be seen.Imagine a long harbor barrier parallel to the beachwith a small entrance in the barrier for the passage of ships.This is shown in Figure~\ref{fig:storm}. %%\activeplot{storm}{width=5.9in}{NR}{\plot{storm}{width=5.0in}{        Waves going through a gap in a barrier        have semicircular wavefronts        (if the wavelengt h is long compared to the gap size).} %%\newslideA plane wave incident on the barrier from the open oceanwill send a wave through the gap in the barrier.It is an observed fact that the wavefront in the harborbecomes a circle with the gap as its center.The difference between this beam of water wavesand a light beam through a windowis in the ratio of wavelength to hole size.\inputdir{phasemod}\parLinearity is a property of all low-amplitude waves(not those foamy, breaking waves near the shore).This means that two gaps in the harbor barriermake two semicircular wavefronts.Where the circles cross, the wave heights combine by simple linear addition.It is interesting to think of a barrier with many holes.In the limiting case of very many holes, the barrier disappears,being nothing but one gap alongside another.Semicircular wavefronts combine to make only the incident plane wave.Hyperbolas do the same.Figure~\ref{fig:stormhole} shows hyperbolasincreasing in density from left to right. %\plot{stormhole}{width=6.0in,height=2.5in}{        A barrier with many holes (top).        Waves, $(x , t)$-space, seen beyond the barrier (bottom).        } %%\newslideAll those waves at nonvertical angles must somehow combinewith one another to extinguish all evidence of anything but the plane wave.\par\inputdir{vphyp}A Cartesian coordinate system has been superimposedon the ocean surface with  $x$  going along the beachand  $z$  measuring the distance from shore.For the analogy with reflection seismology,people are confined to thebeach (the earth's surface) where they makemeasurements of wave height as a function of  $x$  and  $t$.From this data they can make inferences aboutthe existence of gaps in the barrier out in the $(x , z)$-plane.  The first frame ofFigure \ref{fig:dc} shows the arrival time at the beachof a wave from the ocean through a gap.\plot{dc}{width=6.0in,height=2in}{        The left frame shows the hyperbolic wave arrival time        seen at the beach.         Frames to the right show arrivals        at increasing distances out in the water.        The $x$-axis is compressed        from Figure~\protect\ref{fig:storm}.        }%\newslideThe earliest arrival occurs nearest the gap.What mathematical expression determinesthe shape of the arrival curve seen in the $(x , t)$-plane?\parThe waves are expanding circles.An equation for a circle expanding with velocity  $v$  abouta point  $( x_3 , z_3 )$  is\begin{equation}{ ( x - x_3 ) }^2\ \ +\ \ {( z - z_3 )}^2\ \ =\ \ v^2 \, t^2\label{eqn:1.1}\end{equation}Considering  $t$  to be a constant,i.e.~taking a snapshot, equation~(\ref{eqn:1.1}) is that of a circle.Considering  $z$  to be a constant,it is an equation in the $(x , t)$-plane for a hyperbola.  Considered in the $(t , x , z)$-volume,equation~(\ref{eqn:1.1}) is that of a cone.Slices at various values of  $t$  show circles of various sizes.Slices of various values of  $z$  show various hyperbolas.Figure~\ref{fig:dc} shows four hyperbolas.The first is the observation made at the beach  $z_0 = 0$.  The second is a hypothetical set of observations atsome distance  $z_1$  out in the water.The third set of observations is at  $z_2$,an even greater distance from the beach.The fourth set of observations is at  $z_3$,nearly all the way out to the barrier,where the hyperbola has degenerated to a point.All these hyperbolas are from a family of hyperbolas,each with the same asymptote.The asymptote refers to a wave that turns nearly 90$^\circ$ at the gap andis found moving nearly parallel to the shore at thespeed  $dx/dt$  of a water wave.(For this water wave analogy it is presumed---incorrectly---thatthe speed of water waves is a constant independent of water depth).\parIf the original incident wave was a positive pulse,the Huygens secondary source must consist of both positiveand negative polarities to enable the destructiveinterference of all but the plane wave.So the Huygens waveform has a phase shift.In the next section,mathematical expressions will be found for the Huygens secondary source.Another phenomenon, well known to boaters,is that the largest amplitude of the Huygens semicircleis in the direction pointing straight toward shore.The amplitude drops to zero for waves moving parallel to the shore.In optics this amplitude drop-off with angle is called the{\em obliquity factor.}\subsection{Migration derived from downward continuation}\parA dictionary gives many definitions for the word{\em  run.}They are related, but they are distinct.Similarly,the word{\em  migration}in geophysical prospecting has aboutfour related but distinct meanings.The simplest is like the meaning of the word{\em  move.}When an object at some location in the $(x , z)$-planeis found at a different location at a later time  $t$,then we say it {\em  moves.}Analogously, when a wave arrival (often called an %{\em  event%} )at some location in the $(x , t)$-space of geophysical observationsis found at a different position for a different survey lineat a greater depth  $z$, then we say it {\em  migrates.}\parTo see this more clearly,imagine the four frames of Figure~\ref{fig:dc}being taken from a movie.During the movie, the depth  $z$  changesbeginning at the beach (the earth's surface)and going out to the storm barrier.The frames are superimposed in Figure \ref{fig:dcretard}(left).\sideplot{dcretard}{width=3in}{        Left shows a superposition of the hyperbolas        of Figure~\protect\ref{fig:dc}.        At the right the superposition incorporates a shift,        called retardation  $t' \,=\, t + z / v $,        to keep the hyperbola tops together.        }%\newslideMainly what happens in the movie is thatthe event migrates upward toward  $t=0$.To remove this dominating effect of vertical translationwe make another superposition,keeping the hyperbola tops all in the same place.Mathematically, the time $t$ axis is replaced by a so-called{\em  retarded}time axis  $t  '   = t + z/v$, shown in Figure \ref{fig:dcretard}(right).The second, more precise definition of{\em  migration}is the motion of an event in $ ( x , t  '   )$-space as  $z$  changes.After removing the vertical shift,the residual motion is mainly a shape change.By this definition, hyperbola tops, or horizontal layers, do not migrate.\parThe hyperbolas in Figure~\ref{fig:dcretard} really extend to infinity,but the drawing cuts each one off at a time equal  $\sqrt{2}$  timesits earliest arrival.Thus the hyperbolas shown depict only raysmoving within 45$^\circ$ of the vertical.It is good to remember this,that the ratio of first arrival time on a hyperbolato any other arrival timegives the cosine of the angle of propagation.The cutoff on each hyperbola is a ray at 45$^\circ$.Notice that the end points of the hyperbolas on the drawingcan be connected by a straight line.Also, the slope at the end of each hyperbola is the same.In physical space, the angle of any ray is$\tan \, \theta \,=\, dx /dz$.For any plane wave(or seismic event that is near a plane wave),the slope  $ v\,dt / dx$  is  $\sin \, \theta$,as you can see by considering a wavefront intercepting the earth's surfaceat angle  $\theta$.So, energy moving on a straight line in physical $(x , z)$-spacemigrates along a straight line in data $(x , t)$-space.As  $z$  increases, the energy of all angles comes together to a focus.The focus is the exploding reflector.It is the gap in the barrier.This third definition of migration is that it is theprocess that somehow pushes observational data---waveheight as a function of  $x$  and  $t$ ---from the beach to the barrier.The third definition stresses not so much the motion itself,but the transformation from the beginning point to the ending point.\parTo go further, a more general example is neededthan the storm barrier example.The barrier example is confined to making Huygens sourcesonly at some particular  $z$.Sources are needed at other depths as well.Then, given a wave-extrapolation processto move data to increasing  $z$  values,exploding-reflector images are constructed with\begin{equation}\hbox{Image}\ ( x , z )\ \eq\ \ \hbox{Wave}\ ( t=0 , x , z )\label{eqn:1.2}\end{equation}The fourth definition of migration also incorporates the definition of{\em  diffraction}as the opposite of migration.\begin{center}\begin{tabular}{ccc}  {\rm observations} & & {\rm model} \\  \begin{tabular}{|c|} \hline    \\    \ $z = 0$ \ \\    \\  all$\ t$  \\   \\    \hline  \end{tabular}  &\begin{tabular}{c}    {\rm migration} \\ $\longrightarrow$ \\ $\longleftarrow$       \\ \rm{diffraction}  \\   \end{tabular}  &\begin{tabular}{|c|} \hline   \\    \ $t = 0$ \  \\    \\  all$\ z$  \\   \\   \hline   \end{tabular}\end{tabular}\end{center}\par\vspace{1.0\baselineskip}Diffraction\sx{diffraction}is sometimes regarded as the natural process that createsand enlarges hyperboloids.{\em  Migration}\sx{migration, defined}is the computer process that does the reverse.\parAnother aspect of the use of the word{\em  migration}ariseswhere the horizontal coordinate can be either shot-to-geophone midpoint  $y$,or offset  $h$.Hyperboloids can be downward continuedin both the  $(y , t)$-  and the $(h , t)$-plane.In the $(y , t)$-plane this is called{\em  migration}or {\em  imaging,}and in the  $(h , t)$-plane it is called{\em  \bx{focus}ing}or{\em velocity analysis.}\section{DOWNWARD CONTINUATION}Given a vertically upcoming plane waveat the earth's surface,say $u(t,x,z=0)=u(t) {\rm const}(x)$,and an assumption that the earth's velocity isvertically stratified, i.e.~$v=v(z)$,we can presume that the upcoming wave down in the earthis simply time-shifted from what we see on the surface.(This assumes no multiple reflections.)Time shifting can be represented as a linear operator in the time domainby representing it as convolution with an impulse function.In the frequency domain, time shifting is simply multiplyingby a complex exponential.This is expressed as\begin{eqnarray}u( t    ,z) &=& u(     t,z=0) \ast \delta( t+z/v) \\U(\omega,z) &=& U(\omega,z=0) \ e^{-i\omega z/v}\end{eqnarray}Sign conventions must be attended to,and that is explained more fully in chapter~\ref{ft1/paper:ft1}.\subsection{Continuation of a dipping plane wave.}Next consider a plane wave \bx{dip}ping at some angle $\theta$.It is natural to imagine continuing such a wave back along a ray.Instead, we will continue the wave straight down.This requires the assumption that the plane wave is a perfect one,namely that the same waveform is observed at all $x$.Imagine two sensors in a vertical well bore.They should record the same signal except fora time shift that depends on the angle of the wave.Notice that the arrival time difference between sensorsat two different depths is greatest for vertically propagating waves,and the time difference drops to zero for horizontally propagating waves.So the time shift $\Delta t$ is $v^{-1} \cos\theta\,\Delta z$where $\theta$ is the angle between the wavefront and the earth's surface(or the angle between the well bore and the ray).Thus an equation to downward continue the wave is\begin{eqnarray}U( \omega , \theta ,z+\Delta z) &=&U( \omega , \theta ,z) \  \exp ( \,  -i \omega \, \Delta t)                                                        \\U( \omega , \theta ,z+\Delta z) &=&U( \omega , \theta ,z) \  \exp \left(         \,  -i \omega \,         {\Delta z \cos\theta \over v}\  \right)\label{eqn:dctheta}\end{eqnarray}Equation~(\ref{eqn:dctheta})is a downward continuation formula for any angle $\theta$.Following methods of chapter~\ref{wvs/paper:wvs} we can generalizethe method to media where the velocity is a function of depth.Evidently we can apply equation~(\ref{eqn:dctheta})for each layer of thickness $\Delta z$,and allow the velocity vary with $z$.This is a well known approximation that handles the timing correctlybut keeps the amplitude constant (since $|e^{i\phi}|=1$)when in real life,the amplitude should varybecause of reflection and transmission coefficients.Suffice it to say that in practical earth imaging,this approximation is almost universally satisfactory.\parIn a stratified earth,it is customary to eliminate the angle $\theta$ which is depth variable,and change it to the Snell's parameter $p$ which is constant for all depths.Thus the downward continuation equation for any Snell's parameter is\begin{equation}U( \omega , p,z+\Delta z) \eqU( \omega , p,z) \  \exp \left( \,  -\  {i \omega \Delta z \over v(z) } \\sqrt{1-p^2v(z)^2}  \right)