Phase Angles: Multi-station spectral analysis of surface wave (MSASW)

Similar to the SASW method, the frequency-offset data can provide the dispersion

relation from the difference of the phase angle. Due to the multi-channel records, the phase velocity can be directly estimated in φ-x domain without making use of the cross-spectral density (CSD) of signals. The phase velocity in (2-55) can be rewritten in form of (4-7):

x f f f

v

i i i

Δ

= Δ

) ( ) 2

( φπ

(4-7)

For any specific frequency fi, the phase velocity depends on the increasing rate of phase angle with offset (i.e. the slope (Δφ/Δx)) in φ-x domain by the definition of (4-7). As shown in Fig. 4-1, for a synthetic single frequency wave with f=10 Hz and v=200 m/sec, the phase angle in φ-x domain oscillates only between π to –π. After unwrapping in φ-x domain (not the phase angle of CSD in φ-f domain in SASW), the slope can be estimated and the phase velocity of the specific frequency is obtained.

Estimations of the slopes (Δφ/Δx) may also be affected by some inherent natural effects like near and far field effects, higher mode participation and ambient noises. The numerous samplings in space domain of the multi-channel record here provide some advantage. The slope (Δφ/Δx) can be determined by the linear regression of the unwrapped Δφ; as shown in Fig. 4-1. The data quality at each sampling in space domain can be evaluated by the R-square statistic (R²) of the regression analysis. Using the image of real part and the energy spectrum of f-x data, a technique, named “optimum offset range selection” (see detailed in Sec. 4.1.4), can be applied to screen out poor data before the regression of further dispersion analysis. The phase–offset regression is a multi-station extension of the SASW method, and is referred to as the multi-station spectral analysis of surface wave (MSASW) hereafter. The MSASW can provide a better credibility due to its numerous samplings in the space domain rather than two samplings of SASW.

It has been shown that errors may arise in experimental dispersion curves when usual SASW test and data analysis procedures are followed, in particular the phase unwrapping

procedure. Sources that contain significant energy in very low frequencies and receivers with very low natural frequency are necessary to avoid erroneous un-wrapping of phase angles at low frequencies. Hence, the data acquisition system of a SASW test is typically different from that of a refraction survey although they share many things in common. Unwrapping errors may occur for sites where, across the frequency range used, there is a shift from one dominant surface wave propagation mode to another, a phenomenon termed ‘mode jumping’.

Furthermore, the use of only a pair of receivers leads to the necessity of performing the test using several testing configuration and the so-called common receiver midpoint geometry. For each receiver spacing, multiple measurements are necessary for evaluating the data coherence.

This results in a quite time-consuming procedure on site for the collection of all the necessary data and on data reduction for combining the dispersion data points from records obtained at all spacings. Since many non-trivial choices need to be made based on the data quality and testing configuration, the test requires the expertise of an operator and automation of the data reduction is difficult.

Compared to dispersion analysis based on wavefield transformations, MSASW and SASW methods do not allow identification and separation of multiple modes. When tests are performed on complex strata with higher mode wave propagation, the effects of higher modes may result in non-linear φ(x) relation. The dispersion relation obtained from φ-x regression is considered as an “effective” dispersion curve which means a combined result of dispersion curves of several different modes. Only in the case of applying the sufficiently long geophone spreading L, the phase velocity of the dominant mode can be found by the φ-x relation. Thus, the MSASW is not to be used to replace 2-D wavefield transformation. It can be seen as a by-product and supplement of the unified dispersion analysis. The R² of MSASW can be useful information for data quality and identification of existence of multiple modes.

A field case was demonstrated at a test site located at the court yard of Min Ann temple

in Yuan Lin Township in middle Taiwan. Two spreads of 24 geophones were placed roughly perpendicular to each other, one array 23 m long (Δx = 1 m) and the other 11.5 m long (Δx = 0.5 m). A sledgehammer impacting on a steel plate was used as the seismic source, with a near offset 15 m.

Fig. 4-2 shows the results of the dispersion analysis for the short array. It is insightful to examine the surface wave data in various domains. In the time–space domain, the raw data of the shot gathers shows rich ground roll energy without much contamination of body wave or ambient noise (Fig. 4-2a). In the f–x domain, the amplitude spectrum does not show much variation with offset because of the short array used (Fig. 4-2b). The linearity of the phase spectrum with respect to the offset is presented as the R² of φ-x regression analysis as shown in Fig. 4-2c. Low R² values at low frequencies indicate the near field effect while low values at high frequencies reveal far field effect or mode jumping. In this case, the spectrum amplitude of high frequencies does not significantly decrease with increasing offsets. Hence, the low R² values at high frequencies are signs of mode jumping or multiple dominant modes rather than far field effect. Fig. 4-2d shows the f-v spectrum and the associated (maximum) peaks at each frequency. The results of the MSASW analysis are also shown in Fig. 4-2d. For short geophone arrays, separate peaks associated with adjacent modes may smear or even disappear because the spectrum main lobes associated with each mode interfere with each other due to leakage in the space domain. The frequencies at which the results of MSASW and f-v spectrum are significantly different coincide with those frequencies with low R² values.

The differences are due to mode jumping (i.e. the phase–offset relation becomes nonlinear) and possibly further due to unwrapping errors resulting from noise or mode jumping. The experimental dispersion curve should approach the dominant mode for long geophone arrays.

For short geophone array, as is in this case, the experimental dispersion curve may not represent the ‘true’ answer for any modes at frequencies where multiple modes dominate. In

this case, the inversion interpretation must be conducted considering the apparent phase velocity that is associated with mode superposition and the method of analysis.

The four-plot figure as shown in Fig. 4-2 can be obtained on site in a fraction of seconds automatically, making it a very powerful tool for data quality control in the field. Necessary adjustments to the testing program may be made immediately after the initial test. The results of the analysis of the adjusted geophone arrays (i.e. 23-m long array) are shown in Fig. 4-3.

The experimental (apparent) dispersion curve becomes more representative of the dominant mode as the offset range increases. Furthermore, the f-v spectrum clearly shows separate modes in this case.

Fig. 4-2 Results of the dispersion analysis of the short array (11.5 m) at the verification test site (a) raw data in the time–space domain, (b) amplitude spectrum in the frequency–space domain, (c) R² statistics

of the linear regression in the phase–space domain, and (d) amplitude spectrum in the f–v domain.

Fig. 4-3 Results of the dispersion analysis of the short array (23 m) at the verification test site

(a) raw data in the time–space domain, (b) amplitude spectrum in the frequency–space domain, (c) R² statistics of the linear regression in the phase–space domain, and (d) amplitude spectrum in the f–v domain.