Forward modeling operator

In forward modeling, the wavefield recorded at each geophone along an irregular surface is the wavefield propagated up to the depth level where the geophones are located. It is necessary to stop the wavefield propagation after recording because the reflection coefficient at surface is almost -1 Ji and Claerbout (1992). To do so, I formulate the forward modeling operator by propagating the wavefield upward with a filter between extrapolation steps to stop the wavefield propagation where it is recorded. Then all wavefields from all depth levels where the geophones are located are summed together to produce the wavefield along the irregular surface. In order to explain the algorithm clearly and schematically, I use a simple topography model that has only eight geophone groups on an irregular surface, as illustrated in Figure 1.

 

syngeometry Figure 1 Synthetic surface recording geometry. solid squares represent geophone location on an undulating surface.

Figure 2 schematically describes the forward modeling algorithm for the simple model. W_i represents upward extrapolation at the i-th depth level and F₁, F₂, and F₃ are spatial filters for grabbing the wavefield where the geophones are located at the corresponding depth levels. The operators I-F₃ and I-F₂-F₃ in Figure 2 stop the wavefield at the locations where it is recorded below or at the corresponding depth level and pass the wavefield at the locations where it is not yet recorded. Each small rectangle in Figure 2 represents an abstract vector that contains wavefields at the corresponding space location. The wavefield along the irregular surface is obtained by summing the wavefields that are grabbed at the various depth levels.

 

tpfrdschm
Figure 2 Forward modeling scheme: the schematic diagram for forward depth extrapolation when the surface is not flat. W_i represents the upward extrapolation operator at the i-th depth level. F₁, F₂, and F₃ are spatial filters shown in the text, and I is the identity matrix.

The forward modeling scheme shown in Figure 2 can be algebraically generalized, if we divide the topography into z levels, as follows:

$$\left[ \begin{array} {c} d_0 \end{array} \right] = \left[ \b... ...array} \right] \left[ \begin{array} {c} d_z \end{array} \right]$$ (1)

$$E_i = \left[ \begin{array} {cc} I&0 \\ 0&W_i\\ \end{array} \right]$$ (2)

$$G_i = \left[ \begin{array} {cc} I&F_i\\ 0&K_i\\ \end{array} \right]$$ (3)

where

In equation (1), d₀ and d_z are wavefields on the irregular surface and the datum level, respectively. The extrapolation operator E is followed by the spatial filter G at every depth level. We can see that the upward extrapolation operator W_i is applied to the wavefield that does not arrived at the surface because the operator K_i-1 remove the wavefield if it has arrived at any previous depth level. All wavefields that arrive at the surface are saved by the operator F_i-1 for the final output. For the simple geometry shown in Figure 1, F₁, F₂, and F₃ are just diagonal matrices whose elements are 1 where the geophones are located and elsewhere. Thus their diagonal elements are as follows: The operator W_i in equation (2) can be any extrapolation scheme including the Kirchhoff, phase-shift, split-step, or finite-difference method. If we use the phase-shift extrapolation algorithm for W_i, we need an additional inverse Fourier transform in every extrapolation step because the operator G is in the space domain. However, all other algorithms, such as the Kirchhoff, split-step, and finite-difference methods, don't need any additional computation except the operation by G, which is the multiplication of the extrapolated wavefield by the zero/one filter.