Difference between revisions of "Array signal processing"
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* Van Trees (2002) <ref name="Van Trees (2002)">H.L. Van Trees, Optimum Array Processing, Part IV of 'Detection, Estimation, and Modulation Theory, John Wiley | * Van Trees (2002) <ref name="Van Trees (2002)">H.L. Van Trees, Optimum Array Processing, Part IV of 'Detection, Estimation, and Modulation Theory, John Wiley | ||
and Sons, Inc., New York, 2002.</ref> | and Sons, Inc., New York, 2002.</ref> | ||
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| + | '''[[Warangps|warangps]]''' is a graphical interface for simple visualization | ||
| + | of the array response function for a specific array geometry. | ||
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| + | The array reponse function is computed on a regular cartesian grid | ||
| + | in the wavenumber domain <math>\vec{k}</math> by evaluating: | ||
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| + | <math>AR(\vec{k}-\vec{k}_0) = \| \frac{1}{N} \sum_{i=1}^{N} \exp(-j\vec{r}_i(\vec{k}-\vec{k}_0)) \|</math> | ||
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| + | Here, <math>\vec{r}_i</math> is the position vector of sensor i in | ||
| + | a cartesian coordinate system, <math>j</math> the imaginary unit and | ||
| + | <math>\vec{k}_0</math> the ''true'' wavenumber vector of a single | ||
| + | plane wave. | ||
| + | |||
| + | As most seismological array applications are restricted to the | ||
| + | deployment of sensors on the earth's surface, the above equation | ||
| + | is evaluated in the horizontal plane, i.e. <math>\vec{r}_i = (r_{ix}, r_{iy})</math> | ||
| + | and <math>\vec{k} = (k_x, k_y)</math> (horizontal wavenumber vector). | ||
| + | |||
| + | The evaluation of the array reponse function is simple, | ||
| + | still the limits of the wavenumber range as well as the desired | ||
| + | resolution for the grid computation <math>(\delta k_x, \delta k_y)</math> | ||
| + | needs to be specified. The resolution needs to be sufficient high in | ||
| + | order to not miss small details of the peaked surface and the | ||
| + | size of the grid should allow to determine the position of | ||
| + | grating lobes or strong secondary side lobes in the 2D surface. | ||
| + | |||
| + | '''[[Gpfksimulator|gpfksimulator]]''' is another tool for ... | ||
== Frequency wavenumber power spectrum == | == Frequency wavenumber power spectrum == | ||
Revision as of 09:22, 10 March 2010
Contents
Basic principle of array processing - delay and sum / shift and sum
Array response function
some books and papers for reference
- Johnson and Dudgeon (1993) [1]
- Mykkelveit et al. (1983) [2]
- Rost and Thomas (2002) [3]
- Schweitzer et al. (2002) [4]
- Van Trees (2002) [5]
warangps is a graphical interface for simple visualization of the array response function for a specific array geometry.
The array reponse function is computed on a regular cartesian grid in the wavenumber domain by evaluating:
Here, is the position vector of sensor i in a cartesian coordinate system, the imaginary unit and the true wavenumber vector of a single plane wave.
As most seismological array applications are restricted to the deployment of sensors on the earth's surface, the above equation is evaluated in the horizontal plane, i.e. and (horizontal wavenumber vector).
The evaluation of the array reponse function is simple, still the limits of the wavenumber range as well as the desired resolution for the grid computation needs to be specified. The resolution needs to be sufficient high in order to not miss small details of the peaked surface and the size of the grid should allow to determine the position of grating lobes or strong secondary side lobes in the 2D surface.
gpfksimulator is another tool for ...
Frequency wavenumber power spectrum
Modified Spatial Autocorrelation method (MSPAC)
The Modified Spatial Autocorrelation (MSPAC) was introduced by Bettig et al. (2001) [6] after pioneer paper of Aki (1957)[7]. This method allows computing spatial autocorrelation coefficients for any arbitary array configurations.
BLA BLA BLA
The co-array is defined as the set of all possible combinations of two array sensors (Haubrich, 1968 [8]): the array can thus be divided into several semicircular sub-arrays (hereafter called rings) as described in Bettig et al. (2001) [6]. 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.
References
- ↑ D.H.Johnson & D.E.Dudgeon, Array Signal Processing - concepts and techniques, Prentice Hall Signal Processing Series, Alan v. Oppenheim, Series Editor, 1993.
- ↑ Mykkeltveit, S., Astebol, K., Doornbos, D.J., and Husebye, E.S. (1983). Seismic Array configuration Optimization, BSSA, 73(1), pp. 173-186.
- ↑ Rost, S. and Ch. Thomas, Array Seismology, Reviews of Geophysics, 2002.
- ↑ Schweitzer, J., Fyen, J., Mykkeltveit, S. and Kvaerna, T. (2002). Chapter 9: Seismic Arrays, In: Bormann, P. (Ed.), IASPEI New Manual of Seismological Observatory Practice, GeoForschungsZentrum Potsdam, Vol. 1, 51pp.
- ↑ H.L. Van Trees, Optimum Array Processing, Part IV of 'Detection, Estimation, and Modulation Theory, John Wiley and Sons, Inc., New York, 2002.
- ↑ 6.0 6.1 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.
