Admissible Models

In this section I introduce admissible sets $\cal{A}$ of models, on which the convolutional model as defined above is reasonably well behaved. Note that the constraints imposed on the models by membership in the admissible sets are very natural from the physical or geological point of view.

First of all, the velocity must be smooth, as noted above in the section on errors. The restriction of v to a bounded subset of $C^{\infty}$ implies bounds (maximum absolute value, mean square,...) on any derivative of v.

Second, impose smooth upper and lower ``envelope'' velocities as hard constraints: $v_{\rm min}(t_0)\le v(t_0) \le v_{\rm max}(t_0)$.It is natural to assume that the velocity is known at the surface, so assume that $v_{\rm min}(0)= v_{\rm max}(0)$. These bounds derive from geophysical measurements and general knowledge about rock physics, so should be regarded as distinct from the bounds implied by the first condition (membership in a bounded set in $C^{\infty}$).

The set of velocities satisfying the constraints just outlined form the admissible set $\cal{A}$.

An important consequence is that the mute $\phi \in C^{\infty}_0({\bf R}^2)$ may be chosen uniform over $\cal{A}$, as uniform bounds then exist for every value of the stretch factor s(t,x). These bounds follow from the equations of geometric optics. However they are even more simply derived for the hyperbolic moveout approximation to traveltime, which I will eventually adopt, so I do not give a derivation here.