Depth extrapolation and the v(x) challenge

Can the constant-velocity result help us achieve the challenging goal of a stable implicit depth extrapolation through media with lateral velocity variations?

The first idea that comes to mind is to replace the space-invariant helix filters with a precomputed set of spatially varying filters, which reflect local changes in the velocity fields. This approach would merely reproduce the conventional practice of explicit depth extrapolators, popularized by Holberg (1988) and Hale (1991b). However, it hides the danger of losing the property of unconditional stability, which is obviously the major asset of implicit extrapolators.

Another route, partially explored by Nichols (1991), is to implement the matrix inversion in the three-dimensional implicit scheme by an iterative method. In this case, the helix inversion may serve as a powerful preconditioner, providing an immediate answer in constant velocity layers and speeding up the convergence in the case of velocity variations. To see why this might be true, one can write the variable-coefficient matrix $\tilde{\bold{A}}$ in the form  

$$\tilde{\bold{A}} = \bold{B} + \bold{D}\;,$$ (19)

where matrix $\bold{B}$ corresponds to some constant average velocity, and $\bold{D}$ is the matrix of velocity perturbations. The system of linear equations that we need to solve is then  

$$\left(\bold{B} + \bold{D}\right) \bold{m} = \bold{d}\;,$$ (20)

where $\bold{m}$ is the vector of extrapolated wavefield, and $\bold{d}$ is an appropriate righthand side. The helix transform provides us with the operator $\bold{B}^{-1}$, which we can use to precondition system (20). Introducing the change of variables  

$$\bold{m} = \bold{B}^{-1} \bold{x}\;,$$ (21)

we can transform the original system (20) to the form  

$$\bold{d} = \left(\bold{B} + \bold{D}\right) \bold{B}^{-1} \bold{x} = \bold{x} + \bold{D}\,\bold{B}^{-1} \bold{x}\;.$$ (22)

When the velocity perturbation is small, even the simple iteration

$$\begin{eqnarray} \bold{x}_0 & = & \bold{d}\;; \ \bold{x}_k & = & \bold{d} - \bold{D}\,\bold{B}^{-1} \bold{x}_{k+1}\;\end{eqnarray}$$ (23) (24)

will converge rapidly to the desired solution. This interesting possibility needs thorough testing.

The third untested possibility (Papanicolaou, personal communication) is to implement a clever patching in the velocity domain, applying a constant-velocity filter locally inside each patch. Recently developed fast wavelet transform techniques Vetterli and Kovacevic (1995), in particular the local cosine transform , provide a formal framework for that approach.