# Array signal processing

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

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

Here, ${\displaystyle {\vec {r}}_{i}}$ is the position vector of sensor i in a cartesian coordinate system, ${\displaystyle j}$ the imaginary unit and ${\displaystyle {\vec {k}}_{0}}$ 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. ${\displaystyle {\vec {r}}_{i}=(r_{ix},r_{iy})}$ and ${\displaystyle {\vec {k}}=(k_{x},k_{y})}$ (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 ${\displaystyle (\delta k_{x},\delta k_{y})}$ needs to be specified. The resolution needs to be sufficiently high in order not to 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 ...

## References

1. D.H.Johnson & D.E.Dudgeon, Array Signal Processing - concepts and techniques, Prentice Hall Signal Processing Series, Alan v. Oppenheim, Series Editor, 1993.
2. Mykkeltveit, S., Astebol, K., Doornbos, D.J., and Husebye, E.S. (1983). Seismic Array configuration Optimization, BSSA, 73(1), pp. 173-186.
3. Rost, S. and Ch. Thomas, Array Seismology, Reviews of Geophysics, 2002.
4. 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.
5. H.L. Van Trees, Optimum Array Processing, Part IV of 'Detection, Estimation, and Modulation Theory, John Wiley and Sons, Inc., New York, 2002.