is one that applies only with sinusoids.Actually, destructive interference arises anytime a polarity-reversed waveform bounces back and crosses itself.Figure~\ref{fig:super} shows two waves of opposite polarity crossing each other.\sideplot{super}{width=3in,height=3in}{  Superposition of plane waves of opposite polarity.}Observe that one seismogram has a zero-valued signal,while its neighbors have anomalously higher amplitudesand higher energies than are found far away from the interference.The situation shown in Figure \ref{fig:super} does not occur easily in nature.Reflection naturally comes to mind,but usually the reflected wavecrosses the incident wave at a later time and then they don't extinguish.Approximate extinguishing occurs rather easily when waves are quasi-monochromatic.We will soon see, however,that methodologies for finding scales all begin with deconvolutionand that eliminates the monochromatic waves.\subsection{Computing the proper scale factor for a seismogram}With data like Figure~\ref{fig:super},rescaling traces to have equal energy would obviously be wrong.The question is, ``How can we determine the proper scale factor?''As we have seen, a superposition of N plane waves exactlysatisfies an N-th order (in $x$) difference equation.Given a 2-D wave field,we can find its PEFby minimizing output power.Then we ask the question,could rescaling the traces give a lower output power?To answer this, we set up an optimization goal:Given the leveler (be it a cubic PEF or two planar ones),find the best trace scales.(After solving this,we could return to re-estimate the leveler,and iterate.)To solve for the scales,we need a subroutine that scales tracesand the only tricky part is that the adjoint shouldbring us back to the space of scale factors.This is done by \texttt{scaletrace}\opdex{scaletrace}{trace scaling}{40}{45}{user/gee}Notice that to estimate scales,the adjoint forms an inner product of the raw dataon the previously scaled data.Let the operator implemented by \texttt{scaletrace}be denoted by $\bold D$,which is mnemonic for ``data'' and for ``diagonal matrix,''and let the vector of scale factors be denoted by $\bold s$ andthe leveler by $\bold A$.Now we consider the fitting goal $\bold 0\approx \bold A \bold D \bold s$.The trouble with this fitting goal is that the solutionis obviously $\bold s = \bold 0$.To avoid the trivial solution $\bold s = \bold 0$,we can choose from a variety of supplemental fitting goals.One possibility is that for the $i$-th scale factorwe could add the fitting goal $s_i\approx 1$.Another possibility, perhaps better if some of the signalshave the opposite of the correct polarity,is that the sum of the scales should be approximately unity.I regret that time has not yet allowed meto identify some interesting examples and work them through.\section{LOCAL MONOPLANE ANNIHILATOR}\sx{local monoplane annihilator}\sx{monoplane annihilator}\sx{annihilator, local monoplane}%       An interpreter looking at a migrated section containing%       two dips in the same place knows that something is wrong.%       To minimize the presence of multiple dipping events in the same place,%       we can design a rejection filter%       to remove the best fitting local plane.%       This is called a LOMOPLAN (LOcal MOno PLane ANnihilator).%       The output of a LOMOPLAN contains only other dips%       so minimizing that output should enable us%       to improve estimation of model parameters and missing data.%       Although the LOMOPLAN concept applies to models, not data,%       experience shows that processing field data with a LOMOPLAN%       quickly identifies data quality goals.\bx{LOMOPLAN} (LOcal MOno PLane ANnihilator)is a data-adaptive filter that extinguishes a local monoplane,but cannot extinguish a superposition of several planes.We presume an ideal sedimentary modelas made of (possibly curved) parallel layers.Because of the superposition principle,data can be a superposition of several plane waves,but the ideal model should consist locally of only a single plane.Thus, LOMOPLAN extinguishes an ideal model, but not typical data.I conceived of LOMOPLAN as the ``ultimate'' optimization criterionfor inversion problems in reflection seismology\shortcite{Claerbout.sep.73.409}but it has not yet demonstrated that it can attain that lofty goal.Instead,however, working in two dimensions,it is useful in data interpretationand in data quality inspection.\parThe main way we estimate parameters in reflection seismologyis that we maximize the coherence of theoretically redundant measurements.Thus, to estimate velocity and statics shifts,we maximize something like the power in the stacked data.Here I propose another optimization criterionfor estimating model parameters and missing data.An interpreter looking at a migrated section containingtwo dips in the same placesuspects wave superposition more likely than bedding texture superposition.To minimize the presence of multiple dipping events in the same place,we should use the mono plane annihilator (\bx{MOPLAN}) filteras the weighting operator for any fitting goal.Because the filter is intended for use on images or migrated data,not on data directly,I call it a {\it plane} annihilator, not a {\it planewave} annihilator.(A time-migration or merely a stack, however, might qualify as an image.)We should avoid using the word ``wavefront''because waves readily satisfy the superposition principle,whereas images do not,and it is this aspect of images that I advocate and formulateas ``prior information.''\parAn example of a MOPLAN in two dimensions,$(\partial_x + p_x \partial_\tau)$,is explored in Chapter 4 of \bx{PVI}\cite{Claerbout.blackwell.92},where the main goal is to estimate the$(\tau ,x)$-variation of $p_x$.Another family of MOPLANs arise from multidimensionalprediction-error filteringdescribed earlier in this book andin PVI, Chapter 8.\parHere I hypothesize that a MOPLAN may be a valuable weighting functionfor many estimation problems in seismology.Perhaps we can estimate statics, interval velocity, and missing dataif we use the principle of minimizing the power outof a LOcal MOno PLane ANnihilator (LOMOPLAN) on a migrated section.Thus, those embarrassing semicircles that we have seen for yearson our migrated sections may hold one of the keysfor unlocking the secrets of statics and lateral velocity variation.I do not claim that this concept is as powerful as our traditional methods.I merely claim that we have not yet exploited this concept in a systematic wayand that it might prove useful where traditional methods break.\par\noindent\boxit{For an image model of nonoverlapping curved planes,a suitable choice of weighting function for fitting problemsis the local filter that destroys the best fitting local plane.}\subsection{Mono-plane deconvolution}The coefficients of a 2-D monoplane annihilator filterare defined to be the same as those of a 2-D PEFof spatial order unity; in other words,those defined by either (\ref{eqn:spatialpred}) or (\ref{eqn:twoDwhite}).The filter can be lengthened in time but not in space.The choice of exactly two columns is a choiceto have an analytic form that can exactly destroya single plane, but cannot destroy two.Applied to two signals that are statistically independent,the filter (\ref{eqn:twoDwhite})reduces to the well-known prediction-error filterin the left column and zeros in the right column.If the filter coefficients were extendedin both directions on $t$ and to the right on $x$,the two-dimensional spectrum of the input would be flattened.\subsection{Monoplanes in local windows}\inputdir{sep73}The earth dip changes rapidly with location.In a small region there is a local dip and dip bandwidththat determines the best LOMOPLAN (LOcal MOPLAN).To see how to cope with the edge effects of filteringin a small region,and to see how to patch together these small regions,recall subroutine \texttt{patchn()} \vpageref{/prog:patch} and the weighting subroutines that work with it.\parFigure~\ref{fig:sigmoid} shows a synthetic modelthat illustrates local variation in bedding.Notice dipping bedding, curved bedding, unconformity between them,and a fault in the curved bedding.Also, notice that the image has its amplitude taperedto zero on the left and right sides.After local monoplane annihilation (LOMOPLAN),the continuous bedding is essentially gone.The fault and unconformity remain.\plot{sigmoid}{width=6.00in,height=3.5in}{  Left is a synthetic reflectivity model.  Right is the result of local monoplane annihilation.}\par\noindent\boxit{The local spatial prediction-error filters contain the essenceof a factored form of the inverse spectrum of the model.}%\par%The local spatial prediction-error filters contain the essence%of a factored form of the inverse spectral matrix of the model.%A spectral matrix has two aspects:%a spectral aspect in the prediction-error filter%and an amplitude aspect as in gain leveling.%Thus a possible additional step is gaining by the inverse%square root of the local power.%Thus the final step in local whitening is to divide the filters%by the amplitude of the filter output.%This is like applying automatic \bx{gain} adjustment%(\bx{AGC}) to the residuals%in Figure~\FIG{sigmoid0}.%Doing this gives Figure~\FIG{sigmoid1}.%\activeplot{sigmoid1}{width=6.00in,height=3.5in}{}{%       Left is a synthetic reflectivity model.%       Right is the result of local monoplane annihilation%       with inverse amplitudes (AGC).%       }%%Notice the model weakens in amplitude along the sides.%(The designer of this model (me) evidently planned%to use the model for diffraction and migration.)%These damping amplitudes are strong on the LOMOPLAN%because a damping plane wave has angular bandwidth%greater than zero.%Two ways to go beyond the monoplane model are (1) to bend the plane,%or (2) to allow amplitude variation along the plane.%A one-point filter will%predict perfectly an exponentially growing or decaying%amplitude along a plane wave.%But I prefer a bad prediction of the damping plane%as an indication that the planar model is breaking down.%(Here I took a hint from one-dimensional filter analysis.%Recall Burg's old ``forward and backward'' residual trick%(\bx{FGDP}, p.~134).%By forcing a 1-D two-term filter to predict both backward and forward,%\bx{Burg} avoids perfect predictability on growing exponentials.)%To achieve imperfect prediction of exponentially growing planes,%I require one filter%to extinguish two copies of the data,%the original copy and a second copy with both $\tau $ and $x$ reversed.%Note that dips on the reversed copy are the same as on the original%but amplitude growth on one is decay on the other.%To give these ideas concrete form,%I prepared subroutine \LPROG{burgf2} which applies convolution%with \GPROG{icaf2} to both forward and reversed data.%\progdex{burgf2}{filter forward and back}%Monoplane annilhilation can be done with subroutine \GPROG{pef2}.%Actually, the figures shown here were done%with a version of {\tt pef2()} that invokes the bidirectional%filtering in \GPROG{burgf2}.%I hypothesize that this invocation of symmetry is not worth%the extra clutter and I plan to repeat the calculations%with {\tt pef2()} to check.%%\progdex{moplan}{monoplane annihilate}%Additionally,%including the forward and backward residuals%produces a ``backward'' residual%that I could not find any reason to display.%%Also, the display does not explicitly show the patch size.%%My default values were 20 points on time and 6 channels.%%The number of patches was designed for a 50\% overlap.%%The time axes are sampled at 4 ms.%%I also incorporated a gain leveling ({\tt agc}) option%%though in principle,%%gain leveling is a separate process%%whose natural patch size%%might differ from that of the MOPLANs.%%%%\progdex{lomoplan}{local monoplane filter}%%Again, \GPROG{lomoplan} seems a substitute for \GPROG{lopef}%%so I may as well abandon it.%Because the plane waves are local,the illustrations were made with module \texttt{lopef} \vpageref{/prog:lopef}.\subsection{Crossing dips}\plot{conflict}{width=6.00in,height=2.25in}{  Conflicting dips before and after  application of a local monoplane annihilator.%  Press button for movie.%  The movie sequence is:%  1: data,%  2: data after LOMOPLAN,%  3: like previous but windows not overlapping,%  4: predicted data}Figure~\ref{fig:conflict} deserves careful study.The input frame is dipping events withamplitudes slowly changing as they cross the frame.The dip of the events is not commensurate with the mesh,so we use linear interpolationthat accounts for the irregularity along an event.