Array signal processing

Array response function

some books and papers for reference

• Johnson and Dudgeon (1993)
• Mykkelveit et al. (1983)
• Rost and Thomas (2002)
• Schweitzer et al. (2002)
• Van Trees (2002)

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 ${\vec {k}}$ by evaluating:

$AR({\vec {k}}-{\vec {k}}_{0})=\|{\frac {1}{N}}\sum _{i=1}^{N}\exp(-j{\vec {r}}_{i}({\vec {k}}-{\vec {k}}_{0}))\|$

Here, ${\vec {r}}_{i}$ is the position vector of sensor i in a cartesian coordinate system, $j$ the imaginary unit and ${\vec {k}}_{0}$ 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. ${\vec {r}}_{i}=(r_{ix},r_{iy})$ and ${\vec {k}}=(k_{x},k_{y})$ (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 $(\delta k_{x},\delta k_{y})$ 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 ...

Modified Spatial Autocorrelation method (MSPAC)

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.

BLA BLA BLA

The co-array is defined as the set of all possible combinations of two array sensors (Haubrich, 1968 ): the array can thus be divided into several semicircular sub-arrays (hereafter called rings) as described in Bettig et al. (2001) . 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.