The sample data set and other scripts can be obtained by downloading Dist.tgz and then unpacking with the command
gunzip -c Dist.tgz | tar xf - cd EMPIRICAL_GREEN/DIST
In that folder you will find the following files:
EMPIRICAL_GREEN/DIST/
|--/RANDOM4/
| |--DOIT
| |--MFTDOOVERLAY
| |--PHVDOOVERLAY
| |--DOCLEAN
| |--CUS.mod
|
|--/EXAMPLE1/
| |--DOSRF
| |--MFTDOOVERLAY
| |--PHVDOOVERLAY
| |--DOCLEAN
| |--SIUCBHZBLOBHZ.WSTK
| |--CUS.mod
|
|--/EXAMPLE.GRN/
|--CUS.mod
|--DOIT
|--DOCLEAN
This discussion uses the files in the sub-directory
EMPIRICAL_GREEN/DIST/EXAMPLE1. The tutorial on the use
of do_mft is here.
The critical relationship for the determination of phase velocities is the phase correction applied in Equation (6) of the document MFT.pdf.
Consider the subdirectory EMPIRICAL_GREEN/DIST/RANDOM4.
The DOIT script creates a data set consisting of MAXSRC time
segments, each of which has 20 source events distributed randomly
in a source region 1600 km x 1600 km. Two stations are at
coordinates (-200,0) and (200, 0 ) on this grid. The random
source coordinates and station coordinates in km are converted to
latitude and longitude in order to create a map. For the
demonstration script, the values of these parameters are
MAXSRC=100 and SUBSOURCE takes the vales 00, 01, ..., 19. A larger
value of MAXSRC will yield better results, but the computations
will take longer.
The idea is to create MAXSRC time series, representing the
different noise segments used with real data. Each of this time
series consists of the waveforms or 20 randomly distributed
sources. This approach is use to emulate processing of real noise
data.
The current DOIT script does the following:
do_mft -G -IG -T *.symleads to the following comparison plots:
Running do_mft with the same arguments, leads to the following images:#mt to ZNE AZ ${AZ2} BAZ ${BAZ2} FN $FN FE $FE FD $FD FILE $B
mt to ZNE AZ ${AZ2} BAZ ${BAZ2} STK 0 DIP 80 RAKE 10 MW 2 FILE $B
In running the script for MAXSRC=2500 and 20 subsources
consisting of randomly directed forces, 9 inter-component
cross-correlations can be computed. The previous discussion
focused on the E1E2, N1N2 and Z1Z2. The next figure displays all
nine.
These are displayed with the gsac command ylim all
so that the same amplitude scale is used for each. As
expected for a uniform medium, the Z1N2, N1Z2, N1E2 and E1N2
cross-correlations are very small. From the theoretical
development of Wapenaar (2004), we would expect the Z1N2, for
example, to be related to the wavefield at the second sensor in
the +N direction (negative transverse) due to a point upward
vertical force at the first. Theoretically zero transverse
motion is generated by this symmetric vertical source. The
N1E2, the radial motion generated by a transverse source should
also be zero because of the radiation pattern when the observation
point is in a direction perpendicular ot the direction of the
horizontal force.
To test an inverted velocity model, synthetics can be generated
for comparison to the empirical Greens functions. The waveforms
will not be similar because of a different frequency content. Both
can be whitened for comparison and then similarly bandpass
filtered.
Consider the DOIT script in DIST/EXAMPLE.GRN. Green's
functions are computed for the fundamental mode surface wave for a
source depth of 0.01 km and a receiver at the surface. The gsac
command mt is used to make three component synthetics for
a vertical force, Z1Z2 Z1N2 Z1E2, a radially directed horizontal
force, E1Z2 E1N2 E1E2, and a transversely directed force, N1Z2
N1N2 N1E2.
Next given the Z1Z2, etc files in EXAMPLE.GRN and the symmetric
empirical Greens functions in RANDOM4, run the following
commands:
for i in Z1Z2 N1N2 E1E2
do
gsac << EOF
r EXAMPLE.GRN/$i RANDOM4/$i.sym
whiten freqlimits 0.01 0.02 0.25 0.5
w $i.G $i.E
q
EOF
done
#####
# make a plot
#####
gsac << EOF
r E* N* Z*
xlim o 80 o 180
ylim all
fileid name
bg plt
p
q
EOF
plotnps -BGFILL -F7 -W10 -EPS -K < P001.PLT > t.eps
convert -trim t.eps EMPGRN.png
The whiten command with the zero-phase bandpass filter gives the
same amplitude spectrum to both. The resultant plot is
The .G are from the synthetics and the >E are from the
cross-correlation. The waveform comparison is very good. The
difference in absolute amplitude is no problem.
The demonstrates that synthetics can be generated to mimic the
noise cross-correlation results. This may be valuable to
understand the limitations of the group and phase velocity
analysis, expecially at very short distance.
Bensen, G. D., M. H. Ritwoller, M. P. Barmin, A. L. Levshin, F. Lin, M. P. Moschetti, N. M. Shapiro and Y. Yang (2007). Processing seismic ambient noise data to obtain reliable broad-band surface wave dispersion measureements, Geophys. J. Int. 169, 1239-1260. doi: 10.1111/j.1365-246X.2007.03374.x
Herrmann, R. B. (1973). Some aspects of band-pass filtering of
surface waves, Bull. Seism. Soc. Am. 63, 663-671.
Lin, Fan-Chi and Moschetti, Morgan P. and Ritzwoller, Michael H.
(2008). Surface wave tomography of the western United States from
ambient seismic noise: {Rayleigh} and Love wave phase velocity
maps, Geophys. J. Int. 173, 2810298, doi
10.1111/j.1365-246X.2008.03720.x
Snieder, R. (2004). Extracting the Green's function from the
correlation of coda waves: A derivation based on stationary phase,
Physical. Rev E 69, 046610-1,10
Wapenaar, K. (2004). Retrieving the elastodynamic Greens
function of an arbitrary inhomogeneous medium by cross
correlation, Phys. Rev. Lettrs. 93, 254301-1 - 254301-4.