Commit 0469cf1b authored by thomas.forbriger's avatar thomas.forbriger
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[TASK][DOC] (libstfinv): revise description of Fourier domain least squares

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Procedure: Least squares in the frequency domain
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This procedure calculates a least squares fit in the Fourier domain. A
waterlevel as a fraction of the signal energy of the input synthetics is
applied. If per receiver scaling is selected, the receivers will be weighted
in the deconvolution.
This procedure calculates a least squares fit in the Fourier domain, which
equals a waterlevel-deconvolution. A waterlevel as a fraction of the signal
energy of the input synthetics is applied. If selected, offset dependent
weights will be applied to the signals from different receivers.
Options and parameters:
waterlevel=l waterlevel to be applied for regularization.
The procedure is implemented in the Fourier domain. Fourier coefficients for
the correction filter are constructed such that the residual between Fourier
coefficients of recorded signals and Fourier coefficients of synthetic signals
after application of the correction filter are minimized in a least-squares
sense. The least-squares solution is subject to a damping constraint. If the
weighted power spectral density of the synthetics at a given frequency drops
below the l-th fraction (waterlevel parameter) of the average (over all
frequencies) power spectral density, the filter coefficients are damped
(artificially made smaller) by the damping constraint (i.e. waterlevel as seen
from the perspective of deconvolution).
Start with l=0.01 as a reasonable initial value for the waterlevel. The best
fitting waterlevel must be searched by trial and error and depends of signal
noise and on how well the synthetics describe the actually observed wave
propagtion.
The theory behind the Fourier domain least squares procedure is outlined by
Lisa Groos (2013, Appendix F, page 146). She also describes a way to find an
approrpiate water level by application of the L-curve criterion (Groos, 2013,
approrpiate waterlevel by application of the L-curve criterion (Groos, 2013,
Appendix G, page 148).
Lisa Groos. 2013. 2D full waveform inversion of shallow seismic Rayleigh waves.
Dissertation, Karlsruher Institut für Technologie.
Dissertation, Karlsruhe Institute of Technology.
http://nbn-resolving.org/urn:nbn:de:swb:90-373206
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