Untying loops

If, at direct propagating of a wave in a medium with velocity $v=v({\bf r})$, one observes appearance of loops, then at reverse continuation of the wave into the same medium these loops will be untied. But usually the velocity function $v({\bf r})$ is unknown. Moreover, we have to untie loops just in order to obtain good conditions for velocity estimation. So the question arises: how many continuation velocity v_c can be different from real velocity without the loss of the untying effect?

I've obtained the following conditions:

1. The observed field $u_{0}({\bf r}_0,t) \ (r \in \Sigma )$ is not complicated by loops if

$$K\gt{\cos \alpha \over 2} \left[ {\cos (\alpha - \phi) \over h}+ {\cos (\alpha + \phi) \over h }\right]$$

holds for any ray, where K,h and $\phi$ are a curvature, a depth and a slope of the interface at the reflection point accordingly; $\alpha$ is an angle of incident.

2. The condition for the absence of loops in the field $u^{(-)}({\bf r};t)$ continued (with respect to receivers) onto the horizontal plane z=h_c with continued velocity $v_{c}\neq v$ becomes weaker:  

$$K\gt{\cos \alpha \over 2} \left[ {\cos (\alpha - \phi) \over h}+ {\cos (\alpha + \phi) \over h - \kappa h_{c}}\right]$$ (72)

where

$$\kappa = {v_{c} \over v} \cos ^{3} (\alpha + \phi) {\left[ 1... ...\over v^{2}} \sin ^{2} (\alpha + \phi)\right] }^{-{3 \over 2}}.$$

3. In the case of double (with respect to receivers and sources) continuation we have the weakest condition:

$$K\gt-{\cos \alpha \over 2}\left[{\cos (\alpha - \phi ) + \cos (\alpha + \phi ) \over h - \kappa h_{c}}\right].$$

4. For small reflection angles the condition (72) may be simplified:

$$0 \leq {3 \over 2} {h_{c} v_{c} \over hv} \left( 1-{{v_{c}}^... ...}v_{c} \over hv}) \\ 1- {h_{c}v_{c} \over hv}\end{array}\right.$$

where

$$\epsilon = {1 \over 2 \mu -1}, {\:}\mu = -\min {Kh \over \cos \phi} \gt.$$

This condition is shown in Figure . An example of untying loops is shown in Figure where data of physical modeling are represented.

We can interpret inverse wave-field continuation into the model with velocity v=v_c as a continuation in a fictitious layer above the surface z=0 (the layer with known velocity).

The fact that at z>0 we have direct propagation and at z<0 reverse continuation causes the modification of Snell's law at the interface z=0. This modification is shown in Figure .

$$\lambda_q=(-1)^{r_q}v^{-1}_q \hbox{grad} \tau(N_q), \ \ \ q=1,2, \ \ \ N_q\in D_q$$

$$r_q=\left\{ \begin{array} {ll} 0 & \hbox{at forward propaga... ...1 & \hbox{at reverse propagation at} \ D_q \end{array} \right.$$