Ambient vibration array

Common pre-requisites for array processing - getting ready

For any array processing of ambient vibration data there are some common steps / pre-requisites regarding your signal recordings (selection of simultaneous recordings, coordinate insertion, signal grouping). Learn about all these options/preparatory steps in the following sub-chapters (linking to other parts of this wiki)

Loading and viewing signals (time series)

links to loading and viewing signals

Group signals

links to groups

Insert / edit station coordinates

links to set receivers

F-K Toolbox (conventional f-k)

The conventional frequency wavenumber technique as implemented in Geopsy is based on the simple idea of delay and sum (or shift and sum). This technique may be effectuated equivalently in time domain or frequency domain. In Geopsy we follow the frequency domain approach, as it is the most convenient and effective way to use this approach for determination of frequency dependent apparent velocity estimation (i.e dispersion curve estimation under the assumption of the wave field being composed of surface waves only).

The simultaneous waveform recordings of a group of spatially distributed stations are analyzed in many narrow (mostly overlapping) frequency bands for individual analysis windows cut from the overall recordings. For each analysis window and frequency band, a grid search is performed in the wavenumber domain to effectively find the propagation properties of the most coherent and/or powerful plane wave arrival in the analysis window. Given the assumption of surface waves dominating the wave field, the apparent velocity equals the phase velocity of the surface wave at this particular frequency.

Details of the signal processing can be found in the array signal processing page of this wiki or following one of the links below in section see also

For learning about the detailed use of the f-k toolbox (geopsy plugin) in a tutorial like fashion follow this link to FK.

High resolution frequency wavenumber Toolbox (Capon's method)

links to HRFK

Modified Spatial Autocorrelation (MSPAC) Toolbox

The Modified Spatial Autocorrelation (MSPAC) was introduced by Bettig et al. (2001) after pioneer paper of Aki (1957). This method allows computing spatial autocorrelation coefficients for any arbitary array configurations. Indeed, MSPAC was developed in order to circumvent - especially in urban areas - non perfect shaped arrays as required when one wants to apply SPAC (Aki (1957)) or ESAC (XXX) techniques.

As for the original SPAC technique, MSPAC relies on a stochastic ambient noise wavefield stationary in both time and space. Aki (1957) showed, that, given this assumption, the existing relation between the spectrum densities in space and time can be used to derive the following expression for a plane wave narrow-band filtered around $\omega _{0}$:

${\overline {\rho (\omega _{0},r)}}={\frac {1}{\pi }}\int _{\pi }^{0}\rho (\omega _{0},r,\varphi )d(\varphi )=J_{0}({\frac {\omega r}{c(\omega _{0})}})$

${\overline {\rho (\omega _{0},r)}}$ represents the averaging over azimuth of spatial autocorrelations ${\rho (\omega _{0},r,\varphi )}=cos({\frac {\omega _{0}r}{c(\omega _{0})}}cos(\theta -\varphi ))$

where ${\theta }$ is the wave azimuth and ${\varphi }$ the direction azimuth between stations pairs.

Application of the SPAC technique (as well as further derived techniques like ESAC) requires perfect shaped arrays (circular, semi-circular, nested triangles) which may be difficult to achieve in urban environment. To overcome these difficulties, Bettig et al. (2001) suggested to

divide the array into several semicircular sub-arrays (hereafter called rings). The co-array is defined as the set of all possible combinations of two array sensors (Haubrich, 1968 ). Since sensors paris of each sub-arrays not being a perfect semicircle Since, for each ring, an azimuthal and radial integration is performed when computing spatial autocorrelation values, the design of rings results from a compromise between a number of sensors pair per ring as large as possible and a ring thickness as small as possible.

For learning the usage of the MSPAC Toolbox in detail in a tutorial like fashion, please follow this link to MSPAC.

See also

• Wikipedia page on array signal processing
• geopsy.org
• etc.