Interference analysis

If we have two waves with different apparent velocities that interfere with each other, then it is reasonable to apply wave-field continuation into the homogeneous model because these waves can be separated as a result of propagation in a continued field. The consideration is very simple. If both waves are observed in the interval $-a\leq x\leq a$ and the k-th one has travel time $\tau_k(x) = t_0 + x/v_K$ ($k = 1,2, \ v_2<v_1,$ see Figure a), then as the result of forward continuation into the medium with velocity, v_c<min{v₁,v₂,} we obtain waves with eiconals

$$\tau_K^{(+)} (x, z) = t_0 + {z\over v_c} \sqrt{1 - \eta_k^2} +{x-d_k\over v_k},\quad \mid x-d_k \mid \leq a$$

where $\eta_k = v_c/ v_k,\ d_u = z\eta_k / \sqrt{1 -\eta_k^2}$,see Figure b. It is easy to check that $\tau_k^{(+)}$ satisfies to the eiconal equation and to the boundary condition $\tau_k^{(+)}(x, 0) =\tau_k(x)$.

The choice of velocity v_c does not depend on the velocity in the true model of the medium, but only on the condition of wave resolution. We have time resolution if $\tau_2^{(+)} (-a+d_2, z) \gt\tau_1^{(+)}(-a_2+d_2, z) + 2T_0$(where T₀ is the duration of the pulse) and spatial resolution if $-a+d_2 \geq a+d_1$. It is very easy to derive proper conditions in terms of z, a,v₁, v₂ and V_c. For example, we have spatial resolution if $z/a \gt 2\sqrt{1 - \eta^2}/ \eta$ where $\eta = V_0/V_n,\ V_0 = V_c^{-1} - V_1^{-1},\ V_n=V_2^{-1} - V_1^{-1}$. But we shall not develop these considerations since there are more powerful solution. Let us return to the condition V_c <min{V₁, V₂}. Does it have a sense to refuse from the condition?

Let one of the waves be nondesirable (for example, one which has less value of apparent velocity: v₂ < v₁ ). Then the best choice of the velocity for continuation is v=v_c:v₂ <v_c<v₁. It is well known that apparent velocity is always more than the velocity in the medium. When v=v_c, then the wave with velocity v₂<v_c will not be continued into a medium at all! To be exact, it will propagate in the model as an evanescent wave (along the surface $\Sigma$)whose amplitude very quickly (exponentially) diminishes with increasing |z|.

In fact, the operators ${\bf P}^{\pm} (v_{c})$ act as fan filters for the fan of velocities

|v|>v_c.

In order to improve the quality of resolution, we can introduce preliminary time shifts $\triangle t=-x/v_1$.Afterward waves have new apparent velocities $v'_1 =\infty$and v'₂ =v_n. Now we may apply the operator $P^\pm(v_c)$at v_c<v_n. The big advantage of this approach is that it is not necessary that apparent velocities v₁ and v₂ (or v'₁ and v'₂) not be constant along a profile. The only essential condition is v₁(x)<v_c<v₂(x).

The Figure illustrates interference analysis for the situation when V₁ changes in the interval (5 km/s, 6.3 km/s) and V₂ changes in the interval (3 km/s, 4.1 km/s). Figure a shows the original record, Figure b shows the result of standard fan filtration in the velocity fan (4.6 km/s, 7.1 km/s), and Figure c shows the result of the forward continuation with V_c = 4.9 km/s.