The synthetic data results displayed in Fig. 2.1 and in supplementary material (TablesS2.2 and
S2.3) suggest that the IAS implementation of the fk and Capon methods will be of considerable utility. In the following section we first outline the limitations of the IAS method under various parameters, then re-evaluate where the approximate approach is favourable and discuss the results obtained by the IAS method for synthetic and real data.
For the case of synthetic data, we find an increase in the background power level for IAS fk and IAS Capon compared to the approximate approaches. This can be explained by the fact that the approximate methods project on a single frequency and therefore a single array response function. For the IAS methods, we average over multiple array response functions with different frequencies
WRA 2010-107-3 a v=3.33km/s v=4.16km/s WRA 2010-107-3 b v=3.33km/s v=4.16km/s
Figure 2.4: a) Real (observed) data slowness spectrum without temporal averaging (rank-1 cross-power spectral density) compared to a temporal averaged slowness spectrum b) (full-rank cross-power spectral density). The full rank approach increases the resolution of the IAS method and estimates wave arrivals in more detail. All IAS Capon spectra shown throughout this contri-
bution (except Fig. 4a) are estimated using a full-rank matrix.
which leads, in the case of synthetic data, to a higher power spectrum background level. We find that the length of the temporal sub-windows and diagonal loading parameter in the case of IAS Capon strongly influence the accuracy of the method. For smaller temporal sub-windows the Fourier transform is less accurate and we observe frequency leakage (Fig. 2.5a, 138 temporal sub-windows), which can be seen as multiple sources with a constant backazimuth. Sources from neighbouring frequencies ’leak’ into the cross-power spectral density and are estimated with an incorrect slowness vector according to eq. (2.10), which is equivalent to frequency smoothing and results in radial smearing (Woods and Lintz, 1973). It should be noted that frequency leakage is present in the approximate methods as well, but owing to the averaging over frequencies and projection onto a single frequency, the effect is suppressed. It is unclear which frequencies in the averaged frequency range contribute most to the resulting approximate power spectrum as it has been shown that strongest signals do not necessarily dominate (Gibbons et al., 2010). For this reason setting the projection frequency to the peak of the frequency range of interest might not yield the correct slowness for the main microseism sources. On the other hand, selecting short temporal sub-windows will ensure a robust backazimuth estimation of the sources as phase information are divided into small portions (amount of temporal sub-windows) and added to the cross-power spectral density. This averaging procedure of the cross-power spectral density matrix is desirable as the robustness of the Capon algorithm increases with more phase information stored in the matrix. An increase in temporal sub-window length is therefore desirable to minimise the effect of frequency leakage but will result in a less averaged cross-power spectral density and therefore less stable estimates (Fig. 2.5b, 23 temporal sub-windows). The figure displays an estimate that is corrupted by the array response pattern. The estimate is corrupted because we sum over frequency bins with low information content which falsifies the resulting averaged slowness spectrum (Wang and Kaveh, 1985). Fortunately, the microseismic background is continuous and does not display any spectral gaps, which minimises the risk of noise dominated spectra falsifying the final result.
a WRA b WRA
Figure 2.5:a) An example of synthetic data analysis using the IAS method with an insufficiently accurate Fourier spectrum and no diagonal loading correction. We observe frequency leakage which can be seen as source locations on constant backazimuth paths. b) Spectrum generated by frequency bins with a sufficiently accurate Fourier spectrum. In this synthetic case some bins are devoid of information which results in a noise dominated estimate. The noise influence can
be reduced by diagonal loading as shown in Fig 2.1(d).
and noise contributing to the final averaged spectrum.
In our analysis we found that diagonal loading will decrease spectral leakage and reduce the error introduced by frequency bins with low SNR as can be seen in Fig. 2.5(b) where IAS Capon is displayed without diagonal loading and Fig. 2.1(d) with diagonal loading where all sources are identified with great accuracy and the noise influence is suppressed. The diagonal loading constant has to be chosen by the user in this simple model although methods that operate in the spatial domain have been developed to automate this calculation (Du et al., 2010). We have found that the influence of the DL parameter is dependent on the number of temporal sub-windows and its application is therefore important for lower frequencies as their accurate spectral representation demands longer temporal sub-windows.
The Capon method demands the existence of the inverse cross-power spectral density matrix. This criterion would not be fulfilled in the IAS Capon approach if the number of windows is less than the number of sensors as the matrix is not full rank. Diagonal loading adds a constant parameter to all diagonal elements (this can be seen as adding spatially white noise to the data prior processing) and therefore ensures the existence of the inverse even in the case of less windows than sensors. We select 22 temporal sub-windows for WRA which has 23 stations and find a strongly biased slowness spectrum (Fig. 2.6a-d, for different DL parameters). For a diagonal loading parameter of 101a, Fig. 2.6(a), we observe a highly localised spectrum. The arrival
maxima of these highly localised spectra match well the backazimuth and slowness vector of the robust estimates. This behaviour is also observed for a smaller number of temporal sub windows, but slowness estimates coincide more with the robust estimates in the near full rank matrix case. Averaging over fewer temporal sub-windows than array sensors can therefore be used to better estimate the location of body waves although artefact arrivals may also be produced. Increasing the DL parameter will reduce the bias and we obtain a good agreement with previous calculations (Fig. 2.2b) although multiple peaks are no longer observed. We observe a performance decrease of the IAS method if the window length is further increased. This can be explained as less phase
d WRA 2010-107-3 a WRA 2010-107-3 b WRA 2010-107-3 c WRA 2010-107-3
Figure 2.6: a) IAS spectra, where the number of temporal sub-windows is lower by 1 than the number of array sensors, for the case of WRA. With this setting the estimated spectrum becomes highly dependent on the diagonal loading parameter and displays highly resolved arrivals. For
(b-d) the parameter is increased to induce stability into the estimations.
information being stored in the cross-power spectral density (lower rank). IAS therefore shows best performance when the number of temporal windows is equal or greater then the number of sensors.
The approximate approach remains favourable in situations with low SNR as frequency smoothing increases the robustness of the Capon method. Signals with low signal to noise ra- tio can still be used to calculate slowness and backazimuth, but error introduction according to eq. (2.10) should be considered. Computationally, the approximate approach outperforms IAS as the cross-power spectral density is projected only on a single frequency, while for IAS this step is performed for each frequency bin.
As an additional advantage of using IAS, we find that averaging over narrowband calcula- tions minimises the smearing effect which is connected to the array response function (Fig. 2.1e). This results in a better resolved, more accurate estimation of the arrival maxima. In studies that were previously limited by the array response function owing to smearing effects between closely spaced sources, the frequency bounds may be extended. We have further attempted to implement leakage free frequency estimations with multi taper coherency measurements between the inter array seismic traces (Thomson, 1982; Park et al., 1987; Vernon et al., 1991; Prieto et al., 2007). These narrowband measurements may be combined in future research to yield an alternate, accu- rate direction of arrival estimation in a broader frequency range.
In the case of real ambient noise data we observe better resolved slownesses to the equivalent of eitherRg orLg wavespeeds, which gives us confidence that the IAS method is working as in-
a velocity equivalent toRgwaves. We have produced an open source code that implements this
technique (see acknowledgments).