Spectral Analysis of Surface Waves SASW (SASW.m, saswv.m)

Prior to the multi-channel recording of surface waves, the SASW method was developed to employ just two geophones and spectrum analyzer instrumentation. The method has also be used with geophone signals processed with the Fast Fourier Transform (FFT) (21). A typical survey involves taking a source effort on both sides of the geophones (only left side shown in Figure 49). Sounding is performed by expanding the geophone and source separation, X, often changing the source so that low frequencies are radiated at larger spacings, X. Geophones are placed symmetrically about the center line for all expansions. In this example, we take the first two traces of our synthetic seismogram shown in Figure 45. Shown in Figure 49 are blow-ups of the first two traces. Octave program, SASW.m, is used to compute the phase velocity dispersion.

Figure 49: SASW recording places two geophones about a center line. The FFT is used to perform a cross correlation between the two signals in the frequency domain. The phase velocity dispersion curve is computed from the phase of the cross correlation and knowledge of the geophone spacing. Unwrapping of phase is required to compute dispersion beyond the spatial Nyquist frequency.

Program SASW.m requires that function files segyinfo.m and bsegin.m be located in the same directory as the data file and SASW.m. Upon execution, the program asks the user to select a *.seg file and two signals from the *.seg data file. The FFT is then computed for both signals, $S_j(f)$ and $S_k(f)$. The cross-spectrum is computed in the frequency domain by multiplying one spectrum with the complex conjugate of the other spectrum:

$$G_{jk}(f)=S_j(f)\cdot \tilde{S}_k(f)~~~~.$$ (40)

The cross spectrum, $G_{jk}(f)$ is complex. In polar coordinates, the phase is given by $\phi(f)$. A time delay is computed at each frequency by

$$t=T \cdot \phi(f) / 2 \pi~~~~,$$ (41)

where $T$ is the period ($T=1/f$) for frequency, f. The phase velocity at each frequency, f, is computed by

$$V=\dfrac{X}{t}~~~~,$$ (42)

where $X$ is the spacing between the two geophones. The coherence, $C$, is computed from the ratio of the square of the cross-spectrum amplitude squared to the product of the auto-spectra of each signal,

$$C=\dfrac{\lvert G_{jk}(f) \rvert ^2}{G_{jj}(f)\cdot G_{kk}(f)} ~~~~.$$ (43)

The coherence is computed using the pwelch function in Octave. This function divides a signal into a number of overlapping intervals and averages the resulting spectrum (ie. the modified peridogram method).

Some cautions are warranted in using the SASW method. First, it is assumed that only the fundamental mode is in the analysis window. Higher modes will have an adverse affect on the computations. This can be avoided by using sources which radiate low frequencies at large geophone separations. Using a broad-band source at large geophone separations increases the risk that higher modes will enter the analysis window. The dispersion curve computed by SASW.m can then be inverted using the Octave program invR1.m, as was illustrated in the example above. Thus, SASW.m is an alternative to bvax for computing a phase velocity dispersion curve.

Program saswv.m is for non-impulses sources. A sample data set is included in /usr/local/octave/site-m/dx32f.txt (T-Rex Shaker data shared in an ASCE Geophysical Engineering Committee project). See file Dataformat.pdf in the same directory that documents this included sample.

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