Mutes

The linearized model accurately predicts only precritical primary reflections. For layered media, precritical reflections have downgoing incident rays. Along downgoing rays, time is an increasing function of depth. It follows that if t₀ is to be a depth variable, then T must be an increasing function of t₀. This is generally true only in a subset of the t,x plane, i.e. only part of this plane contains data accurately modeled by linearized acoustics. Therefore the rest of the data must be muted out.

Define the stretch factor

$$s(t,x)= \frac{\partial T_0}{\partial t}(t,x)=\left(\frac{\partial T}{\partial t_0} (T_0(t,x),x)\right)^{-1}$$

Then the condition that T(t₀,x) be monotone increasing as a function of t₀ is equivalent to demanding that for large enough t

$$0<s(t,x)<C_{\rm stretch}$$

where $C_{\rm stretch}$ is a user-specified parameter larger than one. Define $T_{\rm mute}(x)$ (the mute boundary) to be the infimum of all t for which the above inequality is satisfied on the interval $(t,T_{\rm max})$. Then the support of the mute function $\phi$ should be contained in the set $\{(t,x): t \ge T_{\rm mute}(x)\}$.

Define a corresponding t₀,x domain mute by $\phi_0(t_0,x) =\phi(T(t_0,x),x)$.