Principles

If the ambient wavefield is recorded with $n$ sensors located at $\overrightarrow{r_i}$ , let $X(\overrightarrow{r_i},\omega)$ being the spectra calculated for station $i$

$$X(\overrightarrow{r_i},\omega)=\sum_{m=1}^q S_m(\omega) e^{j(\overrightarrow{k_m}\cdot \overrightarrow{r_i})}+\eta (\overrightarrow{r_i},\omega)$$ (1.4)

where $\omega=2\pi f$ is the angular frequency, $S_m(\omega)$ is the complex spectrum and $\overrightarrow{k_m}$ is the wavenumber vector of the plane wave triggered by source $m$ , and $\eta$ is the uncorrelated part of the signal ("the noise of the ambient vibrations"). The array output is

$$R(\overrightarrow{k},\omega)=\sum_{i=1}^n W_i(\omega) X(\overrightarrow{r_i},\omega)e^{-j \overrightarrow{k}\cdot \overrightarrow{r_i}}$$ (1.5)

where $W_i(\omega)$ are arbitrary weighting functions. The f-k method presented in section 1.1.1 uses constant weighting functions equal to 1. In this case, equations 1.3 and 1.5 are equivalent.

Estimates of the wave velocity at frequency $\omega$ ( $\overrightarrow{k}(\omega)$ ) are hence obtained by maximizing the complex modulus of $R(\overrightarrow{k},\omega)$ in the wavenumber plane. At the maximum, $\overrightarrow{k}$ equals to $\overrightarrow{k_m}$ , the wavenumber of the dominant plane wave. Using matrix notations,

$$R=AWX$$ (1.6)

where,

$$A = \left[ e^{-j\overrightarrow{k}\cdot \overrightarrow{r_1}}, \ldots, e^{-j\overrightarrow{k}\cdot \overrightarrow{r_n}} \right]$$

$$W = \left[ \begin{array}{ccccc} W_1(\omega) & 0 & \ldots & 0 \\ 0 & & & \ldots \\ \ldots & & & 0 \\ 0 & \ldots & 0 & W_n(\omega) \end{array} \right]$$ (1.7)

$$X = \left[ X_1(\omega), \ldots, X_n(\omega) \right]$$

The frequency-wavenumber cross-spectrum is hence

$$P=AW C W^H A^H$$ (1.8)

where $C=E[XX^H]$ is the cross spectral matrix evaluated using frequency or spatial smoothing, and $^H$ denotes hermitian conjugate operator.

Capon (1969) introduced particular weighting functions optimized by minimizing the signal power of $WCW^H$ for all wavenumbers differing from the considered $\overrightarrow{k}$ , which leads to

$$W=\frac{C^{-1} A}{A^HC^{-1}A}$$ (1.9)

Theoretically, this high-resolution method allows higher resolution. This assertion is checked for a simulated and a real case in chapter 6.