% copyright (c) 1997 Jon Claerbout\title{Plane waves in three dimensions}\author{Jon Claerbout}\label{paper:lmn}\maketitle\inputdir{XFig}In this chapter we seek a deeper understandingof \bx{plane wave}s in {\it three} dimensions,where the examples and theory typically refer to functionsof time $t$ and two space coordinates $(x,y)$,or to 3-D migration images where the $t$coordinate is depth or traveltime depth.As in Chapter \ref{pch/paper:pch},we need to decompose data volumes into subcubes,shown in Figure \ref{fig:rayab3D}.\sideplot{rayab3D}{width=3.00in}{  Left is space of inputs and outputs.  Right is their separation during analysis.}\parIn this chapter we will see thatthe wave model implies the 3-D whitener is not a cube filterbut two planar filters.The wave model allows us to determinethe scale factor of a signal,even where signals fluctuate in strength because of interference.Finally, we examine the local-monoplane conceptthat uses the superposition principleto distinguish a sedimentary model cube from a data \bx{cube}.%\begin{notforlecture}\section{THE LEVELER:  A VOLUME OR TWO PLANES?}\sx{leveler}\parIn {\it two} dimensions,levelers were taken to be PEFs,small rectangular planes of numbers in which the time axisincluded enough points to include reasonable stepouts were includedand the space axis contained one level plus another space level,for each plane-wave slope supposed to be present.\parWe saw that a whitening filter in {\it three} dimensionsis a small volume with shape defined by subroutine \texttt{createhelix()}.%\vpageref{/prog:createhelix_mod}.  % unresolved referenceIt might seem natural that the number of points on the $x$-and $y$-axes be related to the number of plane waves present.Instead, I assert that if the volume contains plane waves,we don't want a {\it volume} filter to whiten it;we can use a {\it pair} of {\it planar} filters to do soand the order of those filters is the number of planesthought to be simultaneously present.I have no firm mathematical proofs,but I offer you some interesting discussions, examples,and computer tools for you to experiment with.It seems that some applications call for the volume filterwhile others call for the two planes.Because two planes of numbers generally containmany fewer adjustable values than a volume,statistical-estimation reasons also favor the planes.\par\noindent\boxit{        What is the lowest-order filter that,        when applied to a volume,        will destroy        one and only one slope of plane wave?        }\parFirst we seek the answer to the question,``What is the lowest order filter that will destroyone and only one plane?''To begin with, we consider that plane to be horizontalso the volume of numbers is$f(t,x,y) = b(t)$where $b(t)$ is an arbitrary function of time.One filter that has zero-valued output (destroys the plane)is $\partial_x \equiv \partial / \partial x$.Another is the operator$\partial_y \equiv \partial / \partial y$.Still another is the \bx{Laplacian operator} which is$\partial_{xx}+\partial_{yy}\equiv\partial^2/\partial x^2+\partial^2/\partial y^2$.\parThe problem with $\partial / \partial x$ is thatalthough it destroys the required plane,it also destroys$f(t,x,y) = a(t,y)$ where $a(t,y)$ is an {\it arbitrary} function of $(t,y)$such as a cylinder with axis parallel to the $x$-axis.The operator $\partial / \partial y$ has the same problembut with the axes rotated.The Laplacian operator not only destroys our desired plane,but it also destroys the well known nonplanar function$e^{ax}\cos(ay)$,which is just one example of the many other interesting shapesthat constitute solutions to Laplace's equation.\parI remind you of a basic fact:When we set up the fitting goal $\bold 0\approx \bold A \bold f$,the \bx{quadratic form} minimized is $\bold f'\bold A' \bold A \bold f$,which by differentiation with respect to $\bold f'$gives us (in a constraint-free region) $\bold A'\bold A\bold f = \bold 0$.So, minimizing the volume integral (actually the sum)of the squares of the components of the gradientimplies that Laplace's equation is satisfied.\parIn any volume,the lowest-order filterthat will destroy level planes and no other wave slopeis a filter that has one input and {\it two outputs}.That filter is the gradient,$(\partial / \partial x, \partial / \partial y)$.Both outputs vanish if and only ifthe plane has the proper horizontal orientation.Other objects and functions are not extinguished(except for the non-wave-like function $f(t,x,y)= {\rm const}$).It is annoying that we must deal with {\it two} outputsand that will be the topic of further discussion.\parA wavefield of tilted parallel planes is$f(t, x,y)= g(\tau -p_x x -p_y y)$, where $g()$ isan arbitrary one-dimensional function.The operator that destroys these tilted planesis the two-component operator$(\partial_x + p_x \partial_t,\   \partial_y + p_y \partial_t)$.\par\noindent\boxit{\noindentThe operator that destroys a family of dipping planes$$f(t, x,y) \eq g(\tau -p_x x -p_y y)$$is$$\left[\begin{array}{c}        {\partial \over \partial x} \ +\  p_x \,{\partial \over \partial t} \\        \ \\        {\partial \over \partial y} \ +\  p_y \,{\partial \over \partial t} \end{array}\right]$$}\subsection{PEFs overcome spatial aliasing of difference operators}The problem I found withfinite-difference representations of differential operatorsis that they are susceptible to \bx{spatial aliasing}.Even before they encounter spatial aliasing,they are susceptible to accuracy problems knownin finite-difference wave propagation as ``frequency dispersion.''The aliasing problem can be avoided by the use of spatial prediction operatorssuch as\begin{equation}   \begin{array}{cc}      \cdot &a     \\      \cdot &b     \\      1     &c     \\      \cdot &d     \\      \cdot &e     \end{array}\label{eqn:spatialpred}\end{equation}where the vertical axis is time;the horizontal axis is space; andthe ``$\cdot$''s are zeros.Another possibility is the 2-D whitening filter\begin{equation}   \begin{array}{cc}      f     &a     \\      g     &b     \\      1     &c     \\      \cdot &d     \\      \cdot &e      \end{array}\label{eqn:twoDwhite}\end{equation}Imagine all the coefficients vanished but $d=-1$ and the given 1.Such filters would annihilate the appropriately sloping plane wave.Slopes that are not exact integers are also approximately extinguishable,because the adjustable filter coefficients can interpolate in time.Filters like (\ref{eqn:twoDwhite})do the operation $\partial_x + p_x \partial_t$,which is a component of the gradient in the plane of the wavefront,{\it and}they include a temporal deconvolution aspectand a spatial coherency aspect.My experience shows that the operators (\ref{eqn:spatialpred}) and(\ref{eqn:twoDwhite})behave significantly differently in practice,and I am not prepared to fully explain the difference,but it seems to be similar to the gapping of one-dimensional filters.\parYou might find it alarmingthat your teacher is not fully preparedto explain the difference between a volume and two planes,but please remember that we are talking aboutthe factorization of the volumetric spectrum.Spectral matrices are well known to have structure,but books on theory typically handle them as simply $\lambda \bold I$.Anyway, wherever you see an $\bold A$ in a three-dimensional context,you may wonder whether it should be interpreted as a cubic filterthat takes one volume to another,or as two planar filtersthat take one volume to two volumesas shown in Figure~\ref{fig:rayab3Doper}.\sideplot{rayab3Doper}{width=1.50in}{  An inline 2-D PEF and a crossline 2-D PEF  both applied throughout the volume.  To find each filter,  minimize each output power independently.}\subsection{My view of the world}\parI start from the idea that the four-dimensional world $(t,x,y,z)$is filled with expanding spherical waves and with quasispherical wavesthat result from reflection from quasiplanar objectsand refraction through quasihomogeneous materials.We rarely, if ever seein an observational data cube,an entire expanding spherical wave,but we normally have a two- or three-dimensional slicewith some wavefront curvature.We analyze data subcubes that I call \bx{bricks}.In any brick we see only local patches of apparent plane waves.I call them \bx{platelets}.From the microview of this brick,the platelets come from the ``great random-point-generator in the sky,''which then somehow convolves the random pointswith a platelike impulse response.If we could deconvolve these platelets back to their random source points,there would be nothing left inside the brickbecause the energy would have gone outside.We would have destroyed the energy inside the brick.If the platelets were coin shaped,then the gradient magnitude would convert each coin to its circular rim.The plate sizes and shapes are all differentand they damp with distance from their centers, as do Gaussian beams.If we observed {\it rays} instead of wavefront plateletsthen we might think of the world as being filled with noodles,and then.\ .\ .\ .\parHow is it possible that in a small brick we can do something realisticabout deconvolving a spheroidal impulse responsethat is much bigger than the brick?The same way as in one dimension,wherein a small time interval we can estimate the correct deconvolution filterof a long resonant signal.A three-point filter destroys a sinusoid.\parThe inverse filter to the expanding spherical wave might be a huge cube.Good approximations to this inverse at the brick levelmight be two small planes.Their time extent would be chosen to encompass the slowest waves,and their spatial extent could be two or three points,representingthe idea that normally we can listen to only one person at a time,occasionally we can listen to two,and we can never listen to three people talking at the same time.\section{WAVE INTERFERENCE AND TRACE SCALING}\sx{interference}\sx{scaling a trace}Although neighboring seismometers tend to show equal powers,the energy on one seismometer can differ greatlyfrom that of a neighbor for both theoretical reasonsand practical ones.Should a trace ever be rescaledto give it the same energy as its neighbors?Here we review the strong theoretical arguments against rescaling.In practice,however, especially on land where coupling is irregular,scaling seems a necessity.The question is,what can go wrong if we scale traces to have equal energy,and more basically,where the proper \bx{scale factor} cannot be recorded,what should we do to get the best scale factor?A related question is how to make good measurements of amplitude versus offset.To understand these issues we reviewthe fundamentals of wave interference.\inputdir{scale}\parTheoretically, a scale-factor problem arisesbecause {\it locally,} wavefields, not energies, add.Nodes on standing waves are familiar from theory,but they could give you the wrong idea that the concept of node