[TASK][DOC] (libstfinv): revise description of Fourier domain least squares

src/libs/libstfinv/usage/stfinvfdleastsquares_description_usage.txt

@@ -3,20 +3,36 @@
 #
 Procedure: Least squares in the frequency domain
 ------------------------------------------------
-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
 # ----- END OF stfinvfdleastsquares_description_usage.txt -----