Difference between revisions of "Ambient vibration array"
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== Modified Spatial Autocorrelation (MSPAC) Toolbox == | == Modified Spatial Autocorrelation (MSPAC) Toolbox == | ||
| − | The Modified Spatial Autocorrelation (MSPAC) was introduced by Bettig et al. (2001) <ref name="Bettig et al. (2001)"> 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.</ref> after pioneer paper of Aki (1957)<ref name="Aki (1957)."> Aki, K., 1957. Space and Time Spectra of Stationary Stochastic Waves, with Special Reference to Microtremors, Bull. Earthq. Res. Inst. Tokyo, 35, 415-457. </ref>. 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 <math>\omega_0</math>: | + | The Modified Spatial Autocorrelation (MSPAC) was introduced by Bettig et al. (2001) <ref name="Bettig et al. (2001)"> 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.</ref> after pioneer paper of Aki (1957)<ref name="Aki (1957)."> Aki, K., 1957. Space and Time Spectra of Stationary Stochastic Waves, with Special Reference to Microtremors, Bull. Earthq. Res. Inst. Tokyo, 35, 415-457. </ref> 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 <math>\omega_0</math>: |
<math> | <math> | ||
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<math>\overline{\rho(\omega_0 , r)}</math> represents the averaging over azimuth of spatial autocorrelations <math> {\rho(\omega_0 , r, \varphi)} = cos(\frac{\omega_0 r}{c(\omega_0)}cos(\theta-\varphi)) </math> where <math>{\theta}</math> is the wave azimuth and <math>{\varphi}</math> the direction azimuth between stations pairs. | <math>\overline{\rho(\omega_0 , r)}</math> represents the averaging over azimuth of spatial autocorrelations <math> {\rho(\omega_0 , r, \varphi)} = cos(\frac{\omega_0 r}{c(\omega_0)}cos(\theta-\varphi)) </math> where <math>{\theta}</math> is the wave azimuth and <math>{\varphi}</math> the direction azimuth between stations pairs. | ||
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| − | 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) <ref name="Bettig et al. (2001)"> 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.</ref> suggested to use the co-array - which is defined as the set of all possible combinations of two array sensors (Haubrich, 1968 <ref name="Haubrich (1968)"> Haubrich, R.A., 1968. Array Design, Bull. seism. Soc. Am., 58(3), 977–991. </ref>) - to divide the array into several semicircular sub-arrays. | + | 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) <ref name="Bettig et al. (2001)"> 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.</ref> suggested to use the co-array - which is defined as the set of all possible combinations of two array sensors (Haubrich, 1968 <ref name="Haubrich (1968)"> Haubrich, R.A., 1968. Array Design, Bull. seism. Soc. Am., 58(3), 977–991. </ref>) - 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 <ref name="Bettig et al. (2001)"> 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.</ref>: |
| − | + | <math> | |
| + | \overline{\rho_{r_1,r_2}(\omega)} = \frac{2}{r_2^2-r_1^2} \int^{r_1}_{r_2}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_1}_{r_2} | ||
| + | </math> | ||
| − | + | where <math> r_1 </math> and <math> r_2 </math> are the inner and outer radius of the ring, respectively. | |
| − | This method allows computing spatial autocorrelation coefficients for any arbitary array configurations. | + | 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. |
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For learning the usage of the MSPAC Toolbox in detail in a tutorial like fashion, please [[MSPAC| follow this link to MSPAC]]. | For learning the usage of the MSPAC Toolbox in detail in a tutorial like fashion, please [[MSPAC| follow this link to MSPAC]]. | ||
Revision as of 08:58, 11 March 2010
Contents
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) [1] after pioneer paper of Aki (1957)[2] 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 :
represents the averaging over azimuth of spatial autocorrelations where is the wave azimuth and 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) [1] suggested to use the co-array - which is defined as the set of all possible combinations of two array sensors (Haubrich, 1968 [3]) - 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 [1]:
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af9a2b6c819b058ec778207e51d41a812dfd33ff)
where and 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.
See also
References
- ↑ 1.0 1.1 1.2 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.
- ↑ Aki, K., 1957. Space and Time Spectra of Stationary Stochastic Waves, with Special Reference to Microtremors, Bull. Earthq. Res. Inst. Tokyo, 35, 415-457.
- ↑ Haubrich, R.A., 1968. Array Design, Bull. seism. Soc. Am., 58(3), 977–991.
