Theory

In the frequency domain, reverse time migration of a shot gather ${d}(\textbf{x}_g\vert\textbf{x}_s)$ can be expressed as

$${m}_{mig}(\textbf{x}\vert\textbf{x}_s)=\sum_{\omega}\sum_{g} \ome... ...bf{x}_s){G}^* (\textbf{x}\vert\textbf{x}_g) {d}(\textbf{x}_g\vert\textbf{x}_s),$$ (42)

where ${m}_{mig}(\textbf{x}\vert\textbf{x}_s)$ is the migration image of the shot at $\textbf{x}_s$ , ${W}(\omega)$ is the source spectrum, $\textbf{x}_g$ indicates the receiver location, ${G}(\textbf{x}\vert\textbf{x}_s)$ is the Green's function from a source at $\textbf{x}_s$ to $\textbf{x}$ ; This Green's function is computed by a finite-difference solution to the wave equation. The $*$ indicates complex conjugate. For simplicity, the angular frequency $\omega$ is silent in the Green's function ${G}$ and data function ${d}$ .

For the velocity model in Figure (a), referred to as the L model, the recorded data contain prism waves. The yellow arrows in Figure (a) indicate the ray path for a prism wave excited at $(x,z)=(4.5,0)~km$ and recorded at $(x,z)=(2.5,0)~km$ , and Figure (b) depicts the wavepath (Luo and Schuster, 1991) of the prism wave generated by a source with a 20-Hz Ricker wavelet. The recorded trace is plotted in Figure (c) with a red window outlining the reflection from the horizontal reflector and the prism wave. For simplicity, I mute the direct wave and diffractions from the trace to keep only the part in the red window

$$d(\textbf{x}_g\vert\textbf{x}_s) = {d}_1(\textbf{x}_g\vert\textbf{x}_s) + {d}_2(\textbf{x}_g\vert\textbf{x}_s),$$ (43)

where $d_1(\textbf{x}_g\vert\textbf{x}_s)$ and ${d}_2(\textbf{x}_g\vert\textbf{x}_s)$ denote the first-order scattering reflection wave and the doubly scattered prism wave, respectively. When the horizontal reflector is extracted from the migration images and embedded in the migration velocity model (Figure (a)), conventional RTM can correctly migrate the prism waves to image the vertical reflector (Jones et al., 2007). In this case, the Green's function calculated with the migration velocity in Figure (a) contains two arrivals: a direct wave arrival and a reflection from the horizontal reflector as shown in Figure . Therefore, the Green's functions in equation can be decomposed into two parts:

$${G}(\textbf{x}\vert\textbf{x}_s)={G}_{o}(\textbf{x}\vert\textbf{x}_s) + {G}_{1}(\textbf{x}\vert\textbf{x}_s),$$ (44)

and

$${G}(\textbf{x}\vert\textbf{x}_g)={G}_{o}(\textbf{x}\vert\textbf{x}_g) + {G}_{1}(\textbf{x}\vert\textbf{x}_g),$$ (45)

where ${G}_o$ and ${G}_1$ denote the direct and the reflected waves, respectively. Note that in this case ${G}_o$ is a downgoing wave and ${G}_1$ is an upgoing wave.

Figure 4.2: (a) A two-layer velocity model. The star and triangle indicate the source and receiver locations. The yellow arrow is the ray path for the direct wave and the red arrows show the ray path for the reflected wave. (b) The trace recorded at the triangle. It is simulated with a 20-Hz Ricker wavelet.

When the data in the red window of Figure (c) are migrated with the velocity model in Figure (a), the migration image is shown in Figure (b), and is mathematically described by

$${m}_{mig}(\textbf{x}\vert\textbf{x}_s)$$

$$= \sum_{\omega}\omega^2 {W}^* (\omega) [{G}_o^* (\textbf{x}\vert\te... ...)][{d}_1 (\textbf{x}_g\vert\textbf{x}_s)+{d}_2 (\textbf{x}_g\vert\textbf{x}_s)]$$

$$= \overbrace{\sum_{\omega}\omega^2 {W}^* (\omega) {G}_o^* (\textbf{... ...rt\textbf{x}_g) {d}_1 (\textbf{x}_g\vert\textbf{x}_s)}^{First~Ellipse\sim O(r)}$$ (46)

$$+ \overbrace{\sum_{\omega}\omega^2 {W}^* (\omega){G}_o^* (\textbf{x... ...textbf{x}_g) {d}_2 (\textbf{x}_g\vert\textbf{x}_s)}^{Second~Ellipse\sim O(r^2)}$$ (47)

$$+ \overbrace{\sum_{\omega}\omega^2 {W}^* (\omega){G}_1^* (\textbf{x... ...extbf{x}_g) {d}_1 (\textbf{x}_g\vert\textbf{x}_s)}^{Left~Rabbit~Ear\sim O(r^2)}$$ (48)

$$+ \overbrace{\sum_{\omega}\omega^2 {W}^* (\omega){G}_o^* (\textbf{x... ...xtbf{x}_g) {d}_1 (\textbf{x}_g\vert\textbf{x}_s)}^{Right~Rabbit~Ear\sim O(r^2)}$$ (49)

$$+ \overbrace{\sum_{\omega}\omega^2 {W}^* (\omega){G}_1^* (\textbf{x... ..._g) {d}_2 (\textbf{x}_g\vert\textbf{x}_s)}^{First~Prism~Wave~Kernel\sim O(r^3)}$$ (50)

$$+ \overbrace{\sum_{\omega}\omega^2 {W}^* (\omega){G}_o^* (\textbf{x... ...g) {d}_2 (\textbf{x}_g\vert\textbf{x}_s)}^{Second~Prism~Wave~Kernel\sim O(r^3)}$$ (51)

$$+ other~terms.$$ (52)

Note that the summation over the receiver $g$ is omitted because there is only one trace in this example. With the assumption that the reflection coefficient is the angle-independent value $r$ , the amplitude of the direct wave Green's function $G_o$ is on the order of $O(1)$ and the amplitude of the reflection wave $G_1$ is on the order of $O(r)$ . Similarly, $d_1$ is with strength of $O(r)$ . The prism wave $d_2$ is a doubly scattered wave and its amplitude is on the order $O(r^2)$ . As an example, the first prism wave term in equation has $O(r^3)$ because it is a product of the $d_2$ term with amplitude $O(r^2)$ and the migration kernel $G_1\times G_o$ with strength $O(r)$ . With these assumptions, the amplitude of each term in the above equation can be expressed in terms of $r$ as shown in the labels.

Figure: (a) The homogeneous velocity ($2~km/s$ ) with a horizontal reflector embedded ($2.5~km/s$ ); (b) the migration image of the data within the red window in Figure 4.1(a) with the velocity model in panel (a).

Figure (b) shows two ellipses. The first one corresponds to the migration kernel in equation with the strongest amplitude $O(r)$ . When the prism wave is migrated as a primary wave (the term in equation ), it shows up as the second ellipse in Figure (b) with an amplitude $O(r^2)$ . This ellipse is an artifact. The migration kernels in equations and correspond to these two ``rabbit ears'' with the strength $O(r^2)$ . Equations and contain the migration kernels for the prism waves corresponding to these near-vertical curves in Figure (b) and their amplitudes are on the order of $O(r^3)$ , which are much weaker than other kernels, so in the migration image, the vertical reflector is of weaker amplitude compared to the horizontal ones.