# 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)

Before processing your ambient vibration data, you need to load your signal files into geopsy and build a database. For checking the data quality of recordings and potential timing problems while recording you might want to use the graphic viewer and the chronogram viewer. It is often also a good idea to compute Spectra and or H/V spectral ratios (in case of 3C data) from your recordings to check the spectral energy content, problematic installations and 1-D-ness of your region.

### Group signals

An array is defined by grouping a set of stations with common time base and recording simultaneously the properties of the wavefield at different, but closely co-located positions in space. After having built a database with many recordings, you may group individual arrays from the full set of signals. Both single component and three component arrays can be bundled together causing the array processing toolboxes to work on single or three component data, respectively.

For details on how to group signals for array processing take a look in groups

### Insert / edit station coordinates

The final step in the preparation of your data set consists in attaching the stations coordinates to your recordings. Note that some waveform formats store the station coordinates and are eventually already included while importing the signal files. For most formats, however, this will be not the case. So, prepare you coordinate information from yor field book or your measurement devices (theodulite / (D)GPS device / etc.) and prepare some multi-column ASCII file which can be read into geopsy. Details how to set coordinates can be found in section set receivers

Since July 2018, high resolution and conventional FK toolboxes have been merged into a single FK toolbox. The text below should be reworked.

FK processing can be achieved inside Geopsy graphical interface or through a command line geopsy-fk.

## 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, please follow this link to FK.

## High resolution frequency wavenumber Toolbox (Capon's method)

The high resolution frequency wavenumber (hrfk) algorithm can be viewed as a generalized beamforming algorithm. It is implemented in Geopsy following the ideas of Capon (1969) [1], using an auto-adaptive (optimal) complex spatial weighting scheme for analysis of narrowband stationary signals. It is one of the most common and preferred frequency wavenumber techniques applied to ambient vibration analysis.

The Capon beamformer relies on a very simple formula:

${\displaystyle BP_{Capon}(\omega ,{\vec {k}})={\frac {1}{{\vec {e}}^{\dagger }(\omega ,{\vec {k}}){\underline {R}}^{-1}(\omega ){\vec {e}}(\omega ,{\vec {k}})}}}$

where ${\displaystyle {\underline {R}}^{-1}(\omega )}$ is simply the inverse of the cross spectral matrix estimate and ${\displaystyle {\vec {e}}}$ is the so-called steering vector summarizing the shift times for the harmonic plane wave with wavenumber vector ${\displaystyle {\vec {k}}}$ for each station within the array.The ${\displaystyle \dagger }$ symbol presents the conjugate transpose operation. In the same notation, the conventional f-k algorithm would be formulated as:

${\displaystyle BP_{conv.}(\omega ,{\vec {k}})={\vec {e}}^{\dagger }(\omega ,{\vec {k}}){\underline {R}}(\omega ){\vec {e}}(\omega ,{\vec {k}})}$

The required computation of the inverse of the cross-spectral matrix results in some potential instabilities for the HRFK estimates.

The corresponding toolbox can be opened using the icon . It is nearly equivalent to the conventional frequency wavenumber toolbox and the processing flow is exactly equivalent to the conventional technique.

For learning about the detailed use of the HRFK toolbox (geopsy plugin) in a tutorial like fashion, please follow this link to HRFK

## Modified Spatial Autocorrelation (MSPAC) Toolbox

The Modified Spatial Autocorrelation (MSPAC) was introduced by Bettig et al. (2001) [2] after pioneer paper of Aki (1957)[3] and allows to compute average spatial autocorrelation coefficients for any arbitary array configurations. As 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 ${\displaystyle \omega _{0}}$:

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

${\displaystyle {\overline {\rho (\omega _{0},r)}}}$ represents the averaging over azimuth of spatial autocorrelations ${\displaystyle {\rho (\omega _{0},r,\varphi )}=cos({\frac {\omega _{0}r}{c(\omega _{0})}}cos(\theta -\varphi ))}$ where ${\displaystyle {\theta }}$ is the wave azimuth and ${\displaystyle {\varphi }}$ 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) [2] suggested to use the co-array - which is defined as the set of all possible combinations of two array sensors (Haubrich, 1968 [4]) - to divide the array into several semicircular sub-arrays. Each sub-array (hereafter called ring) is thus composed of several sensors pairs. To account for ring thickness, an azimuthal and radial integration is then needed to compute averaged spatial autocorrelation values [2]:

${\displaystyle {\overline {\rho _{r_{1},r_{2}}(\omega )}}={\frac {2}{r_{2}^{2}-r_{1}^{2}}}\int _{r_{2}}^{r_{1}}r.J_{0}({\frac {\omega r}{c(\omega )}})dr={\frac {2}{r_{2}^{2}-r_{1}^{2}}}{\frac {c(\omega )}{\omega }}[r.J_{1}({\frac {\omega r}{c(\omega )}})]_{r_{2}}^{r_{1}}}$

where ${\displaystyle r_{1}}$ and ${\displaystyle r_{2}}$ are the inner and outer radius of the ring, respectively.

This method allows computing spatial autocorrelation coefficients for any arbitary array configurations. 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.

## References

1. Capon, J., High-Resolution Frequency-Wavenumber Spectrum Analysis, Proceedings of the IEEE, 57, No. 8, 1408-1419, 1969.
2. Bettig B., P.-Y. Bard, F. Scherbaum, J. Riepl, F. Cotton, C. Cornou, D. Hatzfeld, 2001. Analysis of dense array measurements using the modified spatial auto-correlation method (SPAC). Application to Grenoble area., Boletin de Geofisica Teoria e Applicata, 42, 3-4, 281-304.
3. Aki, K., 1957. Space and Time Spectra of Stationary Stochastic Waves, with Special Reference to Microtremors, Bull. Earthq. Res. Inst. Tokyo, 35, 415-457.
4. Haubrich, R.A., 1968. Array Design, Bull. seism. Soc. Am., 58(3), 977–991.