Shot-profile migration

Shot-profile migration reconstructs the subsurface reflectivity profile by approximately reconstructing the physics of wave-propagation and scattering that generated individual shot records. Central to this formulation is the notion of two independent wavefields: a source wavefield, S, that interacts with discontinuous structure to generate a scattered receiver wavefield, R. The shot-profile migration algorithm consists of two processing steps. The first step is the independent propagation of the S and R wavefields. The second step combines wavefields S and R in a physical imaging condition to generate a map of subsurface reflectivity.

The first shot-profile migration step is an independent extrapolation of S and R, which requires the recursive solution of Claerbout (1985),

$$\begin{eqnarray} S(x_s,z_s+\Delta z_s;\omega) =& S(x_s,z_s;\omega)\; \rm{e}^{ ... ... R(x_r,z_r;\omega)\; \rm{e}^{-i k_{z_r} \Delta z_r}, \nonumber \end{eqnarray}$$ (1)

where k_zs and k_zr are the source and receiver vertical wavenumbers from the wave-equation dispersion relationship, $\Delta z_s$ and $\Delta z_r$ are depth step intervals, and $\omega$ is angular frequency. Successive applications of the complex exponential operators in Equation (1) generate the full S and R wavefield volumes, $S(\mathbf{x_s};\omega)$ and $R(\mathbf{x_r};\omega)$, at all source, $\mathbf{x_s}$, and receiver, $\mathbf{x_r}$, points in model space.

The source wavefield extrapolation operator in Equation (1) includes symbol $\pm$ to distinguish between forward- and backscattering migration scenarios. This parameter explains the causality arguments illustrated in Figure . The four panels represent the forward (i.e., modeling) and adjoint (i.e., migration) propagation of wavefields for both the forward- and backscattered scenarios. Causal propagation is indicated with a forward time-arrow and a positive sign in the extrapolation operator.

 

FSBS
Figure 1 Sketch representing causality arguments for forward- and backscattering scenarios. Time-arrows and positive extrapolation operators indicate causal propagation. Upper left: Backscattered modeling; lower left: backscattered migration; upper right: forward-scattered modeling; and lower right: forward-scattered migration. Note the differing extrapolation operators required for migration that arise from causality arguments.

In backscattered modeling (upper left), a surface-excited source wavefield propagates to a point scatterer and then diffracts as a scattered wavefield, R, upward to the surface. Migrating backscattered wavefields (lower left) propagates R backward in time into the subsurface, which requires reversing the direction of the receiver time arrow and the sign of the receiver extrapolation operator. In forward-scattered modeling (upper right), an upgoing source wavefield impinging from below interacts with the point scatterer, again generating an upgoing scattered wavefield, R. Migrating forward-scattered waves (lower right) requires propagating both S and R backward in time into the subsurface, which reverses the direction of the two time arrows and the signs of both extrapolation operators.

The second shot-profile migration step generates an image, I, of subsurface reflectivity through an evaluation of a physical imaging condition Claerbout (1971). The most basic imaging condition extracts the zero-lag coefficient of the correlation of wavefields S and R. An important extension includes an additional image-space dimension, subsurface half-offset $\mathbf{h}$, generated by shifting S and R in opposing directions by distance $\mathbf{h}$ prior to correlation Rickett and Sava (2002). We emphasize that $\mathbf{h}$ is not equivalent to the surface offset parameters often encountered in shot-geophone or Kirchhoff migration approaches. We write this step with the following equation,  

$$I(\mathbf{x},\mathbf{h}) = \sum_{\omega} \left[\delta(\mat... ...ega)} \ast \delta (\mathbf{x_r}-\mathbf{x}+\mathbf{h})\right],$$ (2)

where $\mathbf{x}$ is the spatial variable of image-space, $\overline{R}$ is the complex conjugate of R, and $\ast$ indicates convolution. The resulting image volume is termed an offset-domain common-image gather (ODCIG). In general, the shifting operation can be oriented in any direction; however, generating a complete 2-D ODCIG volume requires shifts in only two orthogonal directions. For computational simplicity, this is usually done along the horizontal (HODCIGs) and vertical (VODCIGs) axes Biondi and Symes (2004).