Frequency-Time Analysis (FTAN)

The third step in the processing algorithm focuses on measuring the phase and group ve- locities as a function of period by applying the FTAN method to the estimates of Green’s functions found in the second step. A calculation of surface wave dispersion allows for a description of surface waves in terms of phase angle, amplitude, and group arrival times (Feng and Teng, 1983). FTAN was originally used to develop dispersion curves for group velocities. Phase velocity dispersion curves are obtained simultaneously, though, and may also be analyzed. Dispersion curves bring us closer to the final goal of most surface wave analysis studies: the construction of a 2- or 3-dimensional velocity model of the subsurface. The evolution of FTAN as a distinct method came from two different representations of

the same idea: first, the moving window analysis of Landismanet al.(1969), and second,

the multiple filter technique of Dziewonskiet al.(1969). Both methods were developed as

a solution to the problem of determining dispersion data for frequencies with low SNR as prior methods had been limited to only the frequencies with higher SNR (Feng and Teng, 1983). The moving window analysis is a time-domain convolution of the signal with a filter of the form

g_ω(t) =h_ω(t)exp(iωt), (3.3)

whereω is frequency, hω is a symmetric window function, and the convolution produces

a complex function, C(ω,t). The multiple filter technique instead calculates C(ω,t) by

applying a symmetric bandpass filter to the spectrum and then taking its inverse Fourier transform, using the center frequency of the filter bandwidth. It is easy to show that the two methods are equivalent (Feng and Teng, 1983), and this proves that the filter application in the frequency domain or the time domain produces the same result. Since both methods are interchangable, they are collectively referred to as FTAN.

FTAN is applied to signals to first calculate the group arrival time as a function of fre-

quency by finding the point where the absolute value of the signal,C(ω,t), as determined

by either filtering in the time or frequency domain, reaches its maximum value (Nyman and Landisman, 1977). Then, the value of the phase angle and amplitude for the particular

frequency of interest are obtained from the phase and amplitude ofC(ω,t). The group ve-

locity dispersion is measured from the amplitude, whereas the phase velocity dispersion is measured from the progression of the phase of the signal as it travels, which is comprised of a propagation term, a phase ambiguity term, and an initial source phase. The phase am- biguity term may be resolved through the use of a global 3-D model, although the accuracy of phase velocity dispersion measurements is affected by geographical distance between

the stations and frequency (Bensenet al., 2007).

The FTAN operation smooths the dispersion data and thus enhances the resolution of the dispersion, even for frequencies where SNR is low. A disadvantage of FTAN is that it is poorly suited to areas where the amplitude or phase spectrum behavior changes rapidly over a narrow bandwidth and, in this region, systematic errors are introduced (Feng and Teng, 1983). One solution to the problem is to apply the residual dispersion method (RDM), where a theoretical signal designed to have a dispersion curve that approximates the curve for the actual data is generated. The original signal is cross correlated with the synthetic signal to produce what is called the “residual” signal, upon which FTAN is applied. The method of measuring the residual signal is called match-filtering. Since match-filtering eliminates the sidelobes in the cross correlation, narrowband signals cannot be resolved by this method. The match-filtering option is well-suited only to broadband measurements

(Bensen et al., 2007). The outcome of the RDM is a reduction of errors, as the residual

The method of selecting filter bandwidth has varied throughout the literature to suit the purpose of investigation and preference of the researchers. The most common type of

filter is the constant relative bandwidth filter (CRBF) of Dziewonski et al.(1969), where

bandwidth of the filter is the same at all frequencies. A second choice is the display- equalized filter (DEF) of Nyman and Landisman (1977), where the signal is averaged over an elliptical region of group velocity and period where the area of the ellipse is allowed to vary in size to prevent the disparity between the region of high time resolution and low frequency resolution at short periods and low velocities and low time resolution but high frequency resolution at long periods and high velocities in dispersion plots. This method defines a bandwidth that is dependent on both period and group velocity.

Both the CRBF and the DEF require no prior knowledge of the behavior of the disper- sion. The third choice for the definition of the bandwidth parameter of the filter is known

as the optimum bandwidth filter (OBF) of Instonet al.(1971) and Cara (1973). The benefit

of this method is the optimization of the temporal resolution of the dispersion curve across the entire frequency band, but the implementation is based upon an understanding of the approximate dispersion behavior of the signal to be analyzed, which must first be obtained by a simpler method such as CRBF (Feng and Teng, 1983).

Part of the appeal of FTAN is the ease with which the calculations may be automated.

By using the formulation of FTAN by Levshinet al.(1972), it is simple to create a series

of narrowband filters that can be applied (with bandwidth determined according to the method that suits the user’s preference) one at a time to the signal. Then, the generation of dispersion curves is almost instantaneous as the maximum of the signal at each frequency is plotted on a frequency- or period-traveltime graph. Based on the distance between array elements, which is always known (either from GPS positioning of the sensors in permanent

IRIS arrays or by the investigator’s own choice survey geometry), the traveltime axis can then be converted to velocity. The characterization of these plots as a matrix of elements was particularly eloquently expressed by Nyman and Landisman (1977): “Consecutive rows [of the array] correspond to equal increments in...velocity, while consecutive columns correspond to equal increments in log period.” Examples of applying FTAN with more

specific steps and plots from earthquake seismology studies are included in Bensenet al.

(2007).

Given that FTAN is a method of interpreting the Green’s function obtained in the sec- ond processing step for determining phase and group velocity, the accuracy of the FTAN results relies on the accuracy of the extracted Green’s function. When the Green’s function is obtained for station pairs under the assumption that the source of the noise recorded at each is omnidirectional for times when this assumption is invalid, the resultant phase and group velocity measurements will be inaccurate. To improve the reliability of subsurface velocity profiles for cases where there exists persistent directional noise, one must incor- porate the azimuth of the noise source when calculating the travel times between stations. As is demonstrated in Chapters 4 and 5, strongly directional noise will produce an offset in the peak of the cross correlation at positive or negative lag, depending on whether the first term in the cross correlation is closer to or farther away from the source of the incoming di- rectional energy. It is easy to see why a directional source will contaminate the calculation of velocity profiles if the azimuthal approach is not incorporated into the calculations.