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Next: Synthetic Data Example Up: Shan: Migration of multiples Previous: Introduction

Theory

Traditional source-receiver migration assumes that the source is an impulse function and the migration is based on survey sinking. Source-receiver migration downward continues the CMP gather $P(x,h,z=0,\omega)$ into the subsurface by the Double Square Root equation
\begin{displaymath}
\frac{\partial}{\partial z}P=\left(\frac{i\omega}{v_s}
\sqrt...
 ...c{v_r^2}{\omega^2}\frac{\partial^2 }{\partial x_r^2}}
\right)P,\end{displaymath} (1)
where x is midpoint, h is half-offset, xs is the source point, xr is the receiver point, and velocity vs=v(xs,z) and vr=v(xr,z). It images by extracting the wavefield at zero subsurface offset $P(x,h=0,z,\omega)$ and adding along all frequencies. In generalized source-receiver migration, instead of using the CMP gather of the recorded data directly, the cross-correlation between the source wavefield $D(x_D,z=0,\omega,s)$ and the receiver wavefield $U(x_U,z=0,\omega,s)$ at the surface is extrapolated into the media Shan and Zhang (2003). Namely, the wavefield to be downward continued by the Double Square Root equation is  
 \begin{displaymath}
P(x,h,z=0,\omega)=\sum_s U(x_U,z=0,\omega,s)\bar{D}(x_D,z=0,\omega,s)\end{displaymath} (2)
where x=(xU+xD)/2, h=(xU-xD)/2 and s means an areal shot. When the source is an impulse function, the cross-correlation between source and receiver wavefields is exactly the CMP gather of the recorded data and the generalized source-receiver migration algorithm is exactly the same as the traditional source-receiver migration.

Source-receiver migration of multiple reflections is a special case of generalized source-receiver migration, in which the source wavefield is the primary reflection and the receiver wavefield is the corresponding multiple reflection. The wavefield to be downward continued is the cross-correlation between primaries and the corresponding multiples. Since it behaves very similarly to a primary, I call it pseudo-primary data. There are two steps for source-receiver migration of multiples. First, pseudo-primary data are calculated by cross-correlating the primary reflections with the corresponding multiple reflections at the surface. Second, a traditional source-receiver migration is run on the pseudo-primary data. Figure [*] illustrates the principle of source-receiver migration of multiples. The phase of the trace at (x,h) in the pseudo-primary data is exactly the same as a trace at (x,h) from the CMP gather of primary, if we would have put a source at xD and a receiver at xU.

 
mm
mm
Figure 1
Left: Two traces in originally recorded data. The trace at xD records the primary reflection traveltime t1 of $x_S\rightarrow R_1 \rightarrow x_D$. The trace at xU records the multiple reflection traveltime t1+t2 of $x_S \rightarrow R_1 \rightarrow X_D \rightarrow R_2 \rightarrow x_U$.Right: The trace of pseudo-primary data related to trace xD and xU. The trace at (x,h) is the cross-correlation between trace xD and trace xU, where x, h are mid-point and half offset of xD and xU, respectively.
view

 
post
post
Figure 2
Left: One trace in original data. The trace at x has two impulses. The first one records the traveltime of the primary reflection t1: $x_S\rightarrow R_1\rightarrow x$ and the second one records the traveltime of the multiple reflection t2: $x_S\rightarrow R_1 \rightarrow x\rightarrow R_2 \rightarrow x$.Right: Zero-offset dataset of pseudo-primary data. The trace in the pseudo-primary data is the cross-correlation of the trace in the left figure with itself. The traveltime of the impulse in the trace is double the traveltime between x and R2.


view

Zero-offset data is very important in amplitude work, but it is never recorded in a real survey. The zero-offset dataset can be obtained from the pseudo-primary data very easily. In equation (2), letting xU=xD=x, we can get the zero-offset surface dataset  
 \begin{displaymath}
P(x,h=0,z=0,\omega)=\sum_s U(x,z=0,\omega)\bar{D}(x,z=0,\omega).\end{displaymath} (3)
Figure [*] illustrates how to generate a zero-offset surface dataset from pseudo-primary data. The phase information of the zero-offset dataset of the pseudo-primary data is exactly the same as the zero-offset would be if we would have put a source and receiver at x.


next up previous print clean
Next: Synthetic Data Example Up: Shan: Migration of multiples Previous: Introduction
Stanford Exploration Project
7/8/2003