Kinematics in iterated correlations of a passive acoustic experiment

Example of Green's function retrieval by conventional interferometry

To aid interpretation of iterated interferometry in later sections, we study the kinematics of conventional interferometry for a medium containing a scatterer. The background velocity is $c_0=2000$ m/s. Stations $A$ and $B$ are positioned $200$ m distant from each other, and the scatterer is positioned $125$ m above and in between the stations. The stations and scatterer are surrounded by $512$ sources on a circle with a radius of $800$ m, centered between the two stations; see Figure 3.

We simulate the measurements at stations $A$ and $B$ using the single-scatterer Born approximation (see appendix equation A-11). We assume all sources emit a zero-phase Ricker wavelet, $s(t)$ (see appendix equation A-8). For each source location separately, the responses recorded at stations A and B are cross-correlated, and their contribution to the integral on the right-hand side of equation 2 is shown as a function of angle in the correlogram in Figure 4(a).

The correlogram contains three events labeled $1$, $2$ and $3$. These correspond to the first three terms respectively in the correlation product in equation 4. Term $1$ is associated with the direct event between stations $A$ and $B$. It has two stationary points at angles of $\phi=0$ and $\phi=\pi$ radians, where the stations and source are aligned on a line as $\mathbf{x}_s \rightarrow \mathbf{x}_B \rightarrow \mathbf{x}_A$ and $\mathbf{x}_s \rightarrow \mathbf{x}_A \rightarrow \mathbf{x}_B$, respectively. For all other angles, the correlation peak resides at a lag that is smaller than the actual travel time between the stations. The second and third terms correspond to correlations of recorded events that are either scattered at A and direct at B or vice versa. Both events have two stationary phases. Event $2$, for example, has a stationary phase for a source positioned close to $\phi=3/4\pi$ where $\mathbf{x}_s \rightarrow \mathbf{x}_c \rightarrow \mathbf{x}_B$, and at approximately $\phi=4/3\pi$ where $\mathbf{x}_s \rightarrow \mathbf{x}_B \rightarrow \mathbf{x}_c$. The total correlogram is summed over all angles and multiplied by a factor $-\frac{2i\omega}{c_0}$, according to equation 5, to match the asymmetrized true Green's function on the left-hand side of equation 2. The asymmetrized Green's function is multiplied with the auto-correlation of the Ricker wavelet to match the source function after correlations. Although the calculation matches before normalization, the Green's functions are normalized to have a peak value of 1. The comparison between the retrieved result (dashed green line) found by evaluating the right-hand side of equation 2 and the directly modeled result (solid blue curve) found by computing the left-hand side of equation 2 is shown in Figure 4(b), they match exactly.

Three contributions of stationary angles are isolated from all other source contributions and compared to the fully retrieved result. These stationary angles have events arriving with the correct travel time but incorrect phase. They also have events with incorrect travel times. However the contribution from a source positioned close to $\phi=4/3\pi$ radians seems to have an event with a travel time approximately corresponding to the acausal direct event. It is non-stationary and associated with the acausal scattered events as can be seen in Figure 4(a).

Kinematics in iterated correlations of a passive acoustic experiment