An Abstract Formulation of Differential Semblance

The general definition of differential semblance presented here owes much to ideas introduced by Hua Song and Mark Gockenbach in their theses Gockenbach et al. (1995); Song (1994).

The (reference or background) velocity v includes the slowly varying components of velocity and perhaps other fields. The reflectivity r is a field (or vector of fields) encompassing the rapidly varying components of the model. Linearized scattering treats r as a perturbation of v. Thus in the forward modeling operator F[v]r the dependence on r is linear, whereas the dependence on v is (quite!) nonlinear.

Minimal data sets are those on which the kinematic relations in the data are bijective. Minimal data sets include common shot and common offset gathers, and - for layered models - single traces. For a few models, such as constant density acoustics, the forward modeling operator F[v] is invertible (modulo smoothing operators) on minimal data sets. This paper deals only with constant density acoustics.

Denote by G[v] an approximate inverse operator for F[v] on each (minimal) data bin (common source, common offset, single trace,...). Thus G[v] applied to the data produces a prestack reflectivity volume. Similarly, understand by F[v] the application of forward modeling independently for each reflectivity bin.

Each binning scheme also implies a notion of neighboring bins: that is, neighboring source positions, offsets,... Denote by W an operator approximating the derivative or gradient in the bin parameter(s). Generally the definition of F[v] necessarily incorporates a cutoff or mute, as does that of G[v]. Differentiation in the bin direction across this mute produces edge artifacts. To control these, introduce an additional mute $\phi$ slightly more severe than the mutes built into F and G. Since the edge effects are localized, application of this secondary mute $\phi$ eliminates them.

Differentiation enhances high frequency content. To keep the spectrum of the differential semblance output comparable to that of the data, employ a smoothing operator H. An appropriate choice is the inverse square root of the Helmholtz operator $(I - \nabla^2)^{-\frac{1}{2}}$in all of the variables on which the data depends, i.e. both within-bin and cross-bin variables.

With these notations, define differential semblance J₀[v] by:

$$J_0[v]=\frac{1}{2}\left\Vert H \phi F[v] W G[v] S\right\Vert^2$$

Here S denotes the data, and the vertical double bars denote the L² norm or root mean square, i.e. summation of the square of the quantity inside over all variables, followed by square root.