Principles

The spatial auto-correlation function between two sensors is defined by (Aki (1957))

$$\phi(\xi)=\frac{1}{T} \int_0^T v_0(t) v_{\xi}(t) dt$$ (1.10)

where $v_0$ and $v_{\xi}$ are the signals recorded during $T$ seconds at two stations separated by a distance $\xi$ . If the signals are filtered with a narrow frequency band around $\omega_0$ , the auto-correlation ratios defined by

$$\rho(\xi,\omega_0)=\frac{\phi(\xi,\omega_0)}{\phi(0,\omega_0)}$$ (1.11)

are calculated for all pairs of receivers. For a given inter-distance $\xi$ , Aki (1957) demonstrated that the azimuthal average of $\rho(\xi,\omega_0)$ has the shape of Bessel functions (same as equation (3.49)).

$$\overline{\rho(\xi,\omega_0)}=J_0\left(\frac{\omega_0 \xi}{c(\omega_0)}\right)$$ (1.12)

where $J_0$ is the Bessel function of the first order and $c(\omega_0)$ is the dispersion curve. Equation (1.11) is computed in the time domain on filtered signals (a taper in the frequency domain is used to ensure a zero phase filter). Another expression is also available in the frequency domain which avoids one computation of the Fourier transform, but its results are not as precise as equation (1.11) (Metaxian (1994)).

Like for the f-k method, the raw signals are cut in smaller time windows (section 1.1.1) on which the auto-correlation ratios are computed. Consequently, for each frequency band, for each range of inter-distance, and for each individual time windows, an azimuthally averaged auto-correlation ratio is calculated. The results are generally presented under the form of auto-correlation curves with error bars plotted against frequency or inter-distance (e.g. figure 6.14).