Wave field extrapolation into subsurface

In prestack imaging, the first step is the extrapolation of the source and received wave fields into subsurface. The source and received wave fields are extrapolated recursively for each depth level as follows:  

$${\bf s}(z_n) = W(z_n,z_m) {\bf s}(z_m)$$ (46)

and  

$${\bf g}(z_n) = W^\ast(z_n,z_m) {\bf g}(z_m),$$ (47)

where $\ast$ denotes the adjoint and implies that the received wave field is extrapolated backward in time. By substituting equations () and () into the forward model (equation ()), we have  

$${\bf g}(z_n) = W^\ast(z_n,z_0) W(z_0,z_n) R(z_n) W(z_n,z_0) {\bf s}(z_0).$$ (48)

If we assume that the extrapolation operator W is a unitary operator, equation () becomes  

$${\bf g}(z_n) = R(z_n) {\bf s}(z_n).$$ (49)

Then using the above relation, we perform the imaging to retrieve reflectivity matrix R(z_n) from ${\bf g}(z_n)$ and ${\bf s}(z_n)$for each depth level.

Therefore, one of the most important properties of the extrapolation operator is the unitary that makes equation () valid. Even though many algorithms based on the wave equation are accepted as unitary operators, each algorithm has a different property in terms of closeness to unitary. Among the many extrapolation algorithms which allows lateral velocity changes like finite-difference Claerbout (1985), PSPI Gazdag and Sguazzero (1984) and split-step Fourier Stoffa and Fokkema (1990). I chose the split-step Fourier method Stoffa and Fokkema (1990) for this thesis because of its pseudounitary property (Appendix A).

To show the closeness to unitary of the split-step Fourier extrapolation, I tested a plane wave extrapolation followed by its adjoint operator in the Marmousi velocity model. The input is a plane wave which is horizontally aligned spikes at time equals zero (Figure (a)). Figure (b) shows synthesis operator obtained by extrapolating a plane wave from depth level 800 m to surface. Figure (c) shows source wave field obtained by applying its adjoint operator to the synthesis operator which is extrapolating the synthesis operator at surface to depth level 800 m. Figure  is amplitude of the source wave field (Figure (c)) along the plane wave.

 

unitary
Figure 3 Unitary property of the Split-step Fourier method. Top : Input plane wave ( ${\bf s}(z_n)$. Middle : Synthesis operator ${\bf s}(z_0) = W^\ast(z_0,z_n) {\bf s}(z_n)$. Bottom : source wave field at the datum ${\bf \tilde s}(z_n) = W(z_n,z_0) W^\ast(z_0,z_n) {\bf s }(z_n)$.

 

unitary-amp
Figure 4 Amplitude of source wave field at datum level.