Principles

The horizontal velocity is calculated for various frequency bands. The raw signals are first divided in short time windows the length of which may depend upon the considered frequency band. The optimum window length is discussed in sections 6.1.2 and 6.1.3 from synthetic signal analysis. Eventually, a pre-processing method may be used to reject certain parts of the measured signals (transient or saturated signals, Bard (1998)). A Fourier transform is calculated for the signal of each sensor after a proper cutting of the current time window (a 10% cosine taper is applied). The frequency-wavenumber transformation itself is calculated in the frequency domain on the cut signals.

Frequency-wavenumber (f-k, Lacoss et al. (1969); Kvaerna and Ringdahl (1986)) analysis assumes horizontal plane waves to travel across the array of sensors laid out at the surface. Considering a wave with frequency $f$ , a direction of propagation and a velocity (or equivalently $k_x$ and $k_y$ , wavenumbers along X and Y horizontal axis, respectively) the relative arrival times are calculated at all sensor locations and the phases are shifted according to the time delays. The array output is calculated by the summation of shifted signals in the frequency domain. If the waves effectively travel with the given direction and velocity, all contributions will stack constructively, resulting in a high array output. The array output divided by the spectral power is called the semblance (Lacoss et al. (1969); Asten and Henstridge (1984)). The location of the maximum of semblance in the plane ($k_x$ , $k_y$ ) provides an estimate of the velocity and of the azimuth of the travelling waves across the array.

The velocity corresponding to the maximum of semblance is searched between limits which depend upon each particular software implementation. This part is detailed on page . For each time window, a velocity value is calculated, and an histogram is generally constructed for each frequency band. Examples of such results are found in chapter 6.

In the case of waves travelling simultaneously in various directions (usual situation for ambient vibrations), the assumption of uncorrelated signals may not be satisfied, leading to incorrect velocity estimates (Goldstein and Archuleta (1987)). With a limited number of sensors, stacking during a long enough period of time (a few tens of minutes) is then necessary to obtain correct velocity values. This issue will also be detailed in chapter 6.