Asymptotic Approximation of Differential Semblance

The convolutional offset trace model is one of those for which the forward modeling operator on a minimal gather, ie. a single trace, is invertible. The inverse operator is

$$G[v]S(t_0,x)=\frac{S(T(t_0,x),x)}{a(T(t_0,x),x)}$$

The operator measuring semblance differentially is

$$W= \frac{\partial}{\partial x}$$

Then

$$F[v]WG[v]S(t,x)=a(t,x)\left[\frac{\partial}{\partial x} \frac{S(T(t_0,x),x)}{a(T(t_0,x),x)}\right]_{t_0=T_0(t,x)}$$

$$= \left(\frac{\partial T}{\partial x}(T_0(t,x),x) \frac{\partial}{\partial t} +\frac{\partial}{\partial x}\right)S(t,x) + ...$$

$$=\left(p(t,x)\frac{\partial}{\partial t} +\frac{\partial}{\partial x}\right)S(t,x) + ...$$

where

$$p(t,x) = \frac{\partial T}{\partial x}(T_0(t,x),x)$$

is the arrival (horizontal) slowness of the ray passing offset x at time t, and the elided terms involve the amplitude a, but do not involve derivatives of the data S. Thus these terms are of lower frequency content than the leading term (explicitly displayed), and are of the same relative order in frequency as terms neglected in the derivation of the convolutional model from the acoustic wave equation. Therefore they can be dropped: this leads to the remarkable conclusion that the differential semblance objective is independent of the amplitude at least to leading order in frequency.

This observation is due to Hua Song. As a result, within accuracy limitations already built into the asymptotic linearized model, a might as well be replaced by 1!. That is, to leading order in frequency, differential semblance is insensitive to wave dynamics (amplitude), and responds only to kinematic model changes, i.e. changes in traveltime. Thus minimization of differential semblance will amount to a sort of traveltime tomography.

Fons ten Kroode (personal communication) has pointed out that replacement of G[v] by an asymptotically unitary operator with the same kinematics also yields an asymptotocally identical objective without leading order amplitude dependence, and without application of the forward modeling operator, thus at lower computational cost.

The computations above are correct when the map $(t_0,x) \mapsto (t,x)$ is smooth and invertible. This is so inside the mute zone defined above, uniformly for $v \in \cal{A}$. Therefore application of the inverse square root Helmholtz operator following will bring the spectral content back into alignment with that of the data, uniformly over $v \in \cal{A}$. Thus

$$H \phi F[v]WG[v]S = H \phi \left(\frac{\partial S}{\partial x}+p \frac{\partial S}{\partial t}\right) + O(\lambda)$$

The ray slowness p is locally a smooth function of the velocity v in any fixed open subset of the mute zone, hence J₀ (which is the mean square of the above expression) is a smooth function of $v \in \cal{A}$as well.