\title{Field recording geometry}\author{Jon Claerbout} \label{paper:fld}\maketitle\pagenumbering{arabic}  % hopefully, this causes page 1 to be on chapter 1 page.%{\em \today .  This chapter is owned by JFC.}\parThe basic equipment for reflection seismic prospectingis a source for impulsive sound waves,a geophone (something like a microphone),and a multichannel waveform display system.A survey line is defined along the earth's surface.It could be the path for a ship,in which case the receiver is called a hydrophone.About every 25 meters the source is activated, andthe echoes are recorded nearby.The sound source and receiver have almost no directional tuning capabilitybecause the frequencies that penetrate the earthhave wavelengths longer than the ship.Consequently, echoes can arrive from several directions at the same time.It is the joint task of geophysicists and geologiststo interpret the results.Geophysicists assume the quantitative, physical, and statistical tasks.Their main goals, and the goal to which thisbook is mainly directed, is to make good pictures of the earth's interior from the echoes.\section{RECORDING GEOMETRY}\parAlong the horizontal $x$-axis we define two points,$s$, where the source (or shot or sender) is located,and  $g$, where the geophone (or hydrophone or microphone) is located.Then, define the \bx{midpoint}  $y$  between theshot and geophone, anddefine  $h$  to be half the horizontal \bx{offset}between the shot and geophone:%.ne 1.2i\begin{eqnarray}y\ \ &=&\ \ {g \ +\  s  \over 2 }\label{eqn:0.1a}\\h\ \ &=&\ \ {g \ -\  s  \over 2 }\label{eqn:0.1b}\end{eqnarray}The reason for using{\em half}the offset in the equationsis to simplify and symmetrize many later equations.Offset is definedwith  $g - s$  rather than with  $s - g$  so thatpositive offset means wavesmoving in the positive  $x$ direction.In the marine case, this means the ship is presumed to sailnegatively along the  $x$-axis.In reality the ship may go either way,and shot points may either increase or decrease as the survey proceeds.In some situations you can clarify matters by setting thefield observer's shot-point numbers to negative values.\parData is defined experimentally in the space of  $(s,\,g)$.Equations~(\ref{eqn:0.1a}) and (\ref{eqn:0.1b})represent a change of coordinates to the space of $(y,\,h)$.Midpoint-offset coordinates are especially usefulfor interpretation and data processing.Since the data is also a function of the travel time  $t$,the full dataset lies in a volume.Because it is so difficult to make a satisfactory display of such a volume,what is customarily done is to display slices.The names of slices vary slightly from one company to the next.The following names seem to be well known and clearly understood:\begin{center}\begin{tabular}{lp{2.75in}}$(y,\ h = 0,\ t)$       & zero-offset section \\$(y,\ h = h_{\rm min\,,}\  t)$ & near-trace section \\$(y,\ h = \hbox{const} ,\  t)$ &        constant-offset section \\$(y,\ h = h_{\rm max\,,}\  t)$ & far-trace section \\$(y = \hbox{const} ,\  h,\  t)$ & common-midpoint gather \\$(s = \hbox{const} ,\  g,\  t)$ & field profile (or common-shot gather) \\$(s,\ g = \hbox{const} ,\  t)$ & common-geophone gather \\$(s,\ g,\ t = \hbox{const} )$ & time slice \\$(h,\ y,\ t = \hbox{const} )$ & time slice \\%\end{tabbing}\end{tabular}\end{center}\sx{time slice}\sx{zero-offset section}\sx{near-trace section}\sx{section!near-trace}\sx{section!zero-offset}\par\inputdir{XFig}A diagram of slice names is in Figure~\ref{fig:sg}.Figure~\ref{fig:cube} shows three slices from the data volume.The first mode of display is ``engineering drawing mode.''The second mode of display is on the faces of a cube.But notice that although the data is displayed on the surfaceof a cube, the slices themselves are taken from the interior of the cube.The intersections of slices across one another are shown by dark lines.\plot{sg}{width=6.0in}{        Top shows field recording of marine seismograms from        a shot at location  $s$  to geophones at locations labeled  $g$.        There is a horizontal reflecting layer to aid interpretation.        The lower diagram is called a \bx{stacking diagram}.        (It is        {\em not}        a perspective drawing).        Each dot in this plane depicts a possible seismogram.        Think of time running out from the plane.        The center geophone above (circled)        records the seismogram (circled dot) that        may be found in various geophysical displays.        Lines in this $(s,g)$-plane are planes in the $(t,s,g)$-volume.	Planes of various orientations have the names given in the text.        }%\activeplot{rick}{height=4.5in}{}{%       Slices from a cube of data from the Grand Banks.%       Left is ``engineering drawing'' mode.%       At the right slices from within the cube are shown as faces on the cube.%       (Data from Amoco.%       Display via Rick Ottolini's movie program).%       }\inputdir{cube}\plot{cube}{width=6.00in,height=8.0in}{        Slices from within a cube of data.        Top: Slices displayed as a mechanical drawing.        Bottom: Same slices shown on perspective of cube faces.%       (Push button to interact with Steve Cole's Cubeplot program.)        }\parA common-depth-point (CDP) gather is definedby the industry and by common usageto be the same thing as a common-midpoint (CMP) gather.But in this booka distinction will be made.A \bx{CDP gather} is a \bx{CMP gather} with its timeaxis stretched according to some velocity model, say,$$(y=\hbox{const},\ h,\ \sqrt{t^2 - 4h^2 /v^2 }) \quad \quad\hbox{common-depth-point gather}$$This offset-dependent stretching makes the time axis of the gather become more like a{\em depth}axis, thus providing the{\em D} in CDP.The stretching is called {\em \bx{normal moveout} correction}(NMO).Notice that as the velocity goes to infinity, the amount of stretchinggoes to zero.\parThere are basically two ways to get two-dimensional informationfrom three-dimensional information.The most obvious is to cut out the slices defined above.A second possibility is to remove a dimension by summing over it.In practice, the offset axis is the best candidate for summation.Each CDP gather is summed over offset.The resulting sum is a single trace.Such a trace can be constructed at each midpoint.The collection of such traces, a function of midpoint and time,is called a CDP stack.Roughly speaking, a CDP stack is like a zero-offset section,but it has a less noisy appearance.\parThe construction of a CDP stack requires that a numerical choicebe made for the moveout-correction velocity.This choice is called the {\em stacking velocity.}The stacking velocity may be simply someone's guess of the earth's velocity.Or the guess may be improved by stacking with some trial velocitiesto see which gives the strongest and least noisy CDP stack.\parFigures~\ref{fig:yc02} and~\ref{fig:yc20} showtypical marine and land \bx{profile}s(common-shot gathers).\inputdir{yc}\sideplot{yc02}{height=4.0in,width=3.5in}{        A seismic land profile.        There is a gap where there are no receivers near the shot.        You can see events of three different velocities.%        Press button to change plot parameters.        (Western Geophysical).        }\sideplot{yc20}{height=4.0in,width=3.5in}{        A marine profile off the Aleutian Islands.%        Press button to change plot parameters.        (Western Geophysical).        }The land data has geophones on both sides of the source.The arrangement shown is called an{\em uneven \bx{split spread}.}The energy source was a vibrator.The marine data happens to nicely illustratetwo or three head waves.\todo{define head waves?}The marine energy source was an air gun.These field profiles were each recorded with about 120 geophones.\subsection{Fast ship versus slow ship}\inputdir{sg}For marine seismic data,the spacing between shots $\Delta s$ is a function of the speedof the ship and the time interval between shots.Naturally we like $\Delta s$ small(which means more shots)but that means either the boat slows down,