Theory of Generalized Diffraction-stack Migration

In this section, I will first derive the equation for generalized diffraction-stack migration (GDM) and then show how it can be interpreted as a dot-product of the migration kernel with the data.

For weak scattering, the scattered data $D({\bf {r}}\vert{\bf {s}})$ in the frequency domain can be migrated (McMechan, 1983; Stolt and Benson, 1986; Claerbout, 1992) using the Born formula

$$m_{mig}({\bf {x}}) = \int \int_{B} \omega^2 G^*({\bf {x}}\vert{\bf {s}})~[ G^*({\bf {r}}\vert {\bf {x}}) D({\bf {r}}\vert{\bf {s}}) ]~dx_r d \omega,$$

$$= \int \int_{B} {g({\bf {x}},t\vert{\bf {s}},0) ~[ g({\bf {r}},-t\vert {\bf {x}},0) \ast \ddot d({\bf {r}}, t\vert{\bf {s}},0)} ]~ dx_r dt,$$ (3)

where $\omega$ is the angular frequency, the frequency integration is over the bandwidth of the source, $G({\bf {x}}' \vert {\bf {x}})$ represents the space-frequency Green's function for the Helmholtz equation with a source at ${\bf {x}}$ and a receiver at ${\bf {x}}'$ , and $g({\bf {x}}' ,t\vert {\bf {x}},0)$ is the corresponding Green's function in the space-time domain. The term $d({\bf {r}}, t\vert{\bf {s}},0)$ represents the bandlimited data in the time domain for a source at ${\bf {s}}$ and a receiver at ${\bf {r}}$ . Here I have assumed a wideband source spectrum $W(\omega)=1$ in the Born modeling, the time integration is over the duration time of the trace, and the $dx_r$ integration is over the receiver coordinate ${\bf {x}}_r=(x_r,0)~\epsilon~B$ associated with receivers on a horizontal surface denoted by $B$ .

The double dot symbol represents the trace differentiated twice in time, and $g({\bf {r}},-t\vert {\bf {x}},0) \ast \ddot d({\bf {r}},t\vert{\bf {s}},0)$ represents convolution of the time reversed Green's function traces with the recorded trace. This operation backpropagates the trace energy at ${\bf {r}}$ to the subsurface at ${\bf {x}}$ . In contrast, the Green's function $g({\bf {x}},t\vert{\bf {s}},0)$ forward propagates the energy at the source point ${\bf {s}}$ to the subsurface point ${\bf {x}}$ , and the migration image at ${\bf {x}}$ is formed by taking the zero-lag temporal correlation of $g({\bf {x}},t\vert{\bf {s}},0)$ with the backpropagated trace at ${\bf {x}}$ . Traditional reverse-time migration simulates backpropagation by a finite-difference solution to the acoustic wave equation, where the point sources are at the traces located on the surface and the traces act as the time histories for backpropagating seismic wavefields at the receiver locations (McMechan, 1983).

A different implementation of reverse-time migration can be obtained by left shifting the square brackets in equation  to get

$$m_{mig}({\bf {x}}) = \int \int_{B} \omega^2~[G({\bf {x}}\vert{\bf {s}}) G({\bf {r}}\vert {\bf {x}}) ]^*~D({\bf {r}}\vert{\bf {s}})~dx_r d \omega,$$

$$= \int \int_{B} \overbrace{[g({\bf {x}},t\vert{\bf {s}},0) \ast g({... ... {\bf {x}},0)]}^{GDM~kernel}~ \ddot d({\bf {r}},t\vert{\bf {s}},0) ~dx_r dt.$$ (4)

Figure: Formation of the GDM kernel. a). Migration kernel ${\mathcal G}({\bf{r}},{\bf{s}},{\bf{x}},t)$ for a fixed image point at $\bf x$ decomposed into the two modeling kernels: b) one $g({{\bf{x}}},t \vert {\bf s},0)$ for a source at ${\bf {s}}$ and the c) other $g({{\bf{x}}},t \vert {\bf r},0)$ for a source at ${\bf {r}}$ . Convolving the trace in b) with the trace in c) gives the far-right trace shown in a). The other traces in a) represent the migration kernels at other receiver positions.

The bracketed term

$${\mathcal G} ({\bf {r}},{\bf {s}},{\bf {x}},t) = g({\bf {x}},t\vert{\bf {s}},0) \ast g({\bf {r}},t\vert {\bf {x}},0),$$ (5)

is the migration kernel that refocuses reflection energy recorded at ${\bf {r}}$ (for a source at ${\bf {s}}$ ) back to the scatterer at ${\bf {x}}$ . As illustrated in Figure , ${\mathcal G}({\bf{r}},{\bf{s}},{\bf{x}},t)$ in Figure a is obtained by computing the Green's function $g({\bf {x}}' ,t\vert {\bf {x}},0)$ for a source at ${\bf {x}}$ and receivers at ${\bf {x}}' \epsilon B$ , and convolving $g({\bf {x}},t\vert{\bf {s}},0)$ in Figure b with $g({\bf {x}},t\vert{\bf {r}},0)=g({\bf {r}},t\vert{\bf {x}},0)$ in Figure c.

Equation  says that the migration image at ${\bf {x}}$ is computed by taking the dot-product of the shot gather $\ddot d({\bf {r}},t\vert{\bf {s}},0)$ with the migration kernel ${\mathcal G}({\bf{r}},{\bf{s}},{\bf{x}},t)$ in Figure a. This is similar to the interpretation of Kirchhoff migration (KM), except only primary events are accounted for in standard KM, while GDM takes into account both primaries and multiples (Figure ).