%\def\SEPCLASSLIB{../../../../sepclasslib}\def\CAKEDIR{.}\title{Waves in strata}\author{Jon Claerbout}\label{paper:wvs}\maketitle\def\RMS{{{\sc rms}}}   % Global RMS.  Always type this one, even in equations.                        % Then we can change it to suit final editor.                        % This definition dies in equations.\def\RMS{{\rm RMS}}     % Global RMS.  Always type this one, even in equations.The waves of practical interest in reflection seismology are usually complicatedbecause the propagation velocities are generally complex.In this book, we have chosen to build up the complexity of the waves weconsider, chapter by chapter.  The simplest waves to understand are simpleplane waves and spherical waves propagating through a constant-velocity medium.In seismology however, the earth's velocity is almost never well approximatedby a constant.A good first approximation is to assume that the earth's velocity increases with depth.In this situation, the simple planar and circular wavefronts are modified by the effects of $v(z)$.In this chapter we study the basic equations describing plane-like and spherical-like waves propagating in media where the velocity $v(z)$ is a function only of depth.This is a reasonable starting point,even though it neglects the even more complicated distortions that occurwhen there are lateral velocity variations.We will also examine data that shows plane-like wavesand spherical-like waves resulting when wavesfrom a point source bounce back from a planar reflector.\section{TRAVEL-TIME DEPTH}\parEcho soundings give us a picture of the earth.A zero-offest section, for example,is a planar display of traces where the horizontal axis runs along the earth's surfaceand the vertical axis, running down, seems to measure depth,but actually measures the two-way echo delay time.Thus, in practice the vertical axis is almost never depth $z$;it is the{\em vertical travel time}$ \tau $.In a constant-velocity earththe time and the depth are relatedby a simple scale factor, the speed of sound.  This is analogous to theway that astronomers measure distances in light-years, always referencingthe speed of light.The meaning of the scale factor in seismic imaging is that the  $(x, \tau$)-planehas a vertical exaggeration compared to the  $(x,z)$-plane.In reconnaissance work,the vertical is often exaggerated by about a factor of five.By the time prospects have been sufficiently narrowed for a drillsite to be selected,the vertical exaggeration factor in use is likely to be about unity(no exaggeration).\todo{[JON, THIS IS PROBABLY TRUE BUT IT IS NOT CONVINCINGUNLESS WE SPECIFY THEDISPLAY SCALES THAT ARE TYPICALLY USED: KM/INCH, SEC/IN.  SEE MY RELATED COMMENTS BELOW THE NEXT PARAGRAPH]  [IF WE WISH TORETAIN THIS ``EXAGGERATION'' MATERIAL, WHICH I AM INCLINED TO DO, THENWE REALLY NEED A FIGURE THAT GRAPHICALLY ILLUSTRATES THE PHENOMENON WITHACTUAL DISPLAY SCALES AND AN ACTUAL FUNCTION v(z)].  THE READER SHOULDBE ABLE TO ``PLAY WITH'' THE DISPLAY PARAMETERS AND WITH v(z) AND SEEHOW THE EXAGGERATION CHANGES.  WE COULD GENERATE A SET OF SYNTHETIC HORIZONTALREFLECTORS AT DEPTHS OF 1.0, 2.0, 3.0 KM, FOR EXAMPLE}\parIn seismic reflection imaging, the waves go down and then up,so the \bx{traveltime depth} $\tau$is defined as two-way vertical travel time.\begin{equation}\label{eqn:twowaytau}                \tau \eq  {2\,z  \over  v }  \ \ .\end{equation}This is the convention that I have chosen to use throughout this book.\subsection{Vertical exaggeration}\parThe first task in interpretation of seismic datais to figure out the approximate numerical valueof the \bx{vertical exaggeration}.The vertical exaggeration is $2/v$ because it is the ratio ofthe apparent slope$\Delta \tau / \Delta x$to the actual slope$\Delta z/ \Delta x$where $\Delta \tau = 2\ \Delta z / v$.Sincethe velocity generally{\em  increases} with depth,the \bx{vertical exaggeration} generally{\em  decreases} with depth.\parFor velocity-stratified media,the time-to-depth conversion formula is\begin{equation}\tau (z) \ \ =\ \  \int_0^z \  {2\ dz \over v (z) }\ \ \ \ \ \ \ \ \  {\rm or} \ \ \ \ \ \ \ \ { d \tau   \over dz }\ \ =\ \  {2 \over v }\label{eqn:3.2}\end{equation}\todo { [JON, IF WE WANT TORETAIN THIS ``EXAGGERATION'' MATERIAL, WHICH I REALLY THINK WE SHOULD, THENWE MUST HAVE A FIGURE THAT GRAPHICALLY ILLUSTRATES THE PHENOMENON WITHACTUAL DISPLAY SCALES AND AN ACTUAL FUNCTION v(z)].  THE READER SHOULDBE ABLE TO ``PLAY WITH'' THE DISPLAY PARAMETERS AND WITH v(z) AND SEEHOW THE EXAGGERATION CHANGES.  WE COULD GENERATE A SET OF SYNTHETIC HORIZONTALREFLECTORS AT DEPTHS OF 1.0, 2.0, 3.0 KM, FOR EXAMPLE}\section{HORIZONTALLY MOVING WAVES}\inputdir{headray}In practice, horizontally going waves are easy to recognizebecause their travel time is a linear functionof the offset distance between shot and receiver.There are two kinds of horizontally going waves,one where the traveltime line goes through the origin,and the other where it does not.When the line goes through the origin,it means the ray path is always near the earth's surfacewhere the sound source and the receivers are located.(Such waves are called ``\bx{ground roll}'' on landor ``\bx{guided wave}s'' at sea;sometimes they are just called ``\bx{direct arrivals}''.)\parWhen the traveltime line does not pass through the originit means parts of the ray path plunge into the earth.This is usually explained bythe unlikely looking rays shown in Figure~\ref{fig:headray}which frequently occur in practice.%\sideplot{headray}{width=3in,height=.8in}{  Rays associated with \bx{head wave}s.} %Later in this chapter we will see that Snell's lawpredicts these rays in a model of the earth with two layers,where the deeper layer is faster and the ray bottomis along the interface between the slow medium and the fast medium.Meanwhile, however, notice that these ray pathsimply data with a linear travel time versus distancecorresponding to increasing ray length along the ray bottom.Where the ray is horizontal in the lower medium,its wavefronts are vertical.These waves are called ``\bx{head wave}s,''perhaps because they are typically fastand arrive {\em  ahead} of other waves.\subsection{Amplitudes}\parThe nearly vertically-propagating waves (reflections)spread out essentially in three dimensions,whereas the nearly horizontally-going wavesnever get deep into the earth because,as we will see,they are deflected back upward by the velocity gradient.Thus horizontal waves spread out in essentially two dimensions, so that energy conservation suggeststhat their amplitudes should dominate the amplitudes of reflectionson raw data.This is often true for \bx{ground roll}.Head waves, on the other hand,are often much weaker, often being visible only becausethey often arrive before more energetic waves.The weakness of \bx{head wave}sis explained bythe small percentage of solid angle occupied by the waves leaving a sourcethat eventually happen to match up with layer boundaries and propagate ashead waves.I %\footnote{  %      In this chapter, and perhaps throughout the book,  %      the word ``I'' refers to the two authors working  %      in consultation, and the word ``we'' refers  %      to readers and authors.  %      }selected the examples below because of the strong headwaves.They are nearly as strong as the guided waves.To compensate for diminishing energy with distance,I scaled data displays by multiplying by the offset distancebetween the shot and the receiver.\parIn data display, the slowness (slope of the time-distance curve)is often called the \bx{stepout}  $p$.  Other commonly-used names forthis slope are \bx{time dip} and \bx{reflection slope}.The best way to view waves with \bx{linear moveout}is after time shifting to remove a standard linear moveoutsuch as that of water.An equation for the shifted time is\begin{equation}\tau \eq t - p x\label{eqn:lmo}\end{equation}where $p$ is often chosen to be the inverse of the velocity of water,namely, about 1.5 km/s, or $p=.66 {\rm s/km}$and $x=2h$ is the horizontal separation betweenthe sound source and receiver, usually referred to as the \bx{offset}.\inputdir{head}\par\bxbx{Ground roll}{ground roll}and \bx{guided wave}s are typically slowbecause materials near the earth's surface typically are slow.Slow waves are steeply sloped on a time-versus-offset display.It is not surprising that marine guided wavestypically have speeds comparable to water waves(near 1.47 km/s approximately 1.5 km/s).It is perhaps surprising that \bx{ground roll}also often has the speed of sound in water.Indeed, the depth to underground water is often determinedby seismology before drilling for water.Ground roll also often has a speedcomparable to the speed of sound in air,0.3 km/sec, though, much to my annoyance I could not finda good example of it today.%(Record wz.25 seems to have both water speed and air speed head waves,%but the given parameters imply two speeds exactly twice as fast%as those two speeds and I am suspicious.)Figure~\ref{fig:wzl-34} is an example of energetic \bx{ground roll} (land)that happens to have a speed close to that of water.\plot{wzl-34}{width=6.00in,height=3.0in}{  Land shot profile (Yilmaz and Cumro) \#39 from the Middle East  before (left) and after (right)  linear moveout at water velocity.}\parThe speed of a ray traveling along a layer interfaceis the rock speed in the faster layer (nearly always the lower layer).It is not an average of the layer above and the layer below.% The discussion below is based on a JLB fantasy figure.%  I doubt this discussion belongs here.  Meanwhile, I commented it out. -Jon%Rays are not directly observable; they are theoretical.%To better understand what is happening, think of %the wavefronts for the \bx{head wave} in Figure~\FIG{horzrays}.%A ray along an interface is perpendicular%to a vertical wavefront (horizontal ray) in the lower medium%and a wavefront dipping at angle $\theta_c$ in the upper medium.%So the horizontal slowness in the upper medium,%$\sin\theta_c /v_1$, matches the horizontal slowness $1/v_2$%of the front in the lower medium.  This fact, known as \bx{Snell's law}%will be proven in Section~{dipping} of this chapter.\parFigures~\ref{fig:wzl-20} and~\ref{fig:wzl-32}are examples of energetic marine guided waves.In Figure~\ref{fig:wzl-20}at $\tau=0$ (designated {\tt t-t\_water}) at small offsetis the wave that travels directly from the shot to the receivers.This wave dies out rapidly with offset(because it interferes with a wave of opposite polarityreflected from the water surface).At near offset slightly later than $\tau=0$ is the water bottom reflection.At wide offset, the water bottom reflection is quickly followed by multiple reflections from the bottom.Critical angle reflection is defined as where the \bx{head wave}comes tangent to the reflected wave.Before (above) $\tau=0$ are the \bx{head wave}s.There are two obvious slopes,hence two obvious layer interfaces.Figure \ref{fig:wzl-32} is much like Figure \ref{fig:wzl-20}but the water bottom is shallower.\plot{wzl-20}{width=6.00in,height=3.0in}{  Marine shot profile (Yilmaz and Cumro) \#20 from the Aleutian Islands.}\plot{wzl-32}{width=6.00in,height=3.0in}{  Marine shot profile (Yilmaz and Cumro) \#32 from the North Sea.}\parFigure~\ref{fig:wglmo} shows data where the first arriving energyis not along a few straight line segments,but is along a curve.This means the velocity increases smoothly with depthas soft sediments compress.\plot{wglmo}{width=6.00in,height=2.7in}{  A common midpoint gather from the Gulf of Mexico  before (left) and after (right) linear moveout  at water velocity.  Later I hope to estimate  velocity with depth  in shallow strata.  Press button for \bx{movie} over midpoint.}\subsection{LMO by nearest-neighbor interpolation}To do \bx{linear moveout} (\bx{LMO}) correction, we need to time-shift data.Shifting data requires us to interpolate it.The easiest interpolation method is the nearest-neighbor method.We begin with a signal given at times {\tt t = t0+dt*(it-1)}where {\tt it} is an integer.Then we can use equation~(\ref{eqn:lmo}),namely $\tau=t-px$.Given the location {\tt tau} of the desired valuewe backsolve for an integer, say {\tt itau}.In Fortran, conversion of a real value to an integer is done bytruncating the fractional part of the real value.To get rounding up as well as down,we add{\tt 0.5}before conversion to an integer,namely {\tt itau=int(1.5+(tau-tau0)/dt)}.This gives the nearest neighbor.Theway the program works is to identify two points,one in $(t,x)$-space and one in $(\tau,x)$-space.Thenthe data value at one point in one space is carried to the other.The adjoint operation copies $\tau$ space back to $t$ space.%The subroutine used in the illustrations above is%\texttt{lmo()} \vpageref{/prog:lmo}%with {\tt adj=1}.%\progdex{lmo}{linear moveout}\parNearest neighbor rounding is crude but ordinarily very reliable.I discovered a very rare numerical roundoff problempeculiar to signal time-shifting, a problemwhich arises in the linear moveout applicationwhen the water velocity, about 1.48km/sec is approximated by 1.5=3/2.The problem arises only where the amount of the time shiftis a numerical value (like 12.5000001 or 12.499999)and the fractional part should be exactly 1/2 butnumerical rounding pushes it randomly in either direction.We would not care if an entire signal was shiftedby either 12 units or by 13 units.What is troublesome, however, is if some random portionof the signal shifts 12 units while the rest of it shifts 13 units.Then the output signal has places which are empty whileadjacent places contain the sum of two values.Linear moveout is the only applicationwhere I have ever encountered this difficulty.A simple fix here was to modify the\texttt{lmo()} \vpageref{lst:lmo}subroutine changingthe ``1.5'' to ``1.5001''.The problem disappears if we use a more accurate sound velocityor if we switch from nearest-neighbor interpolationto linear interpolation.\subsection{Muting}\inputdir{vscan}Surface waves are a mathematician's delightbecause they exhibit many complex phenomena.Since these waves are often extremely strong,and since the information they contain about the earthrefers only to the shallowest layers,typically,considerable effort is applied to array design in field recordingto suppress these waves.Nevertheless, in many areas of the earth,these pesky waves may totally dominate the data.\parA simple method to suppress \bx{ground roll} in data processingis to multiply a strip of data by a near-zero weight (the mute).To reduce truncation artifacts,the mute should taper smoothly to zero (or some small value).Because of the extreme variability from place to place