To illustrate the slowness error introduced by the frequency averaging of the cross-power spectral density, we first consider synthetically generated data and then examine real (observed) data. We generate seismic waves to simulate multiple seismic background sources impinging on the array from different directions. The seismic wavefield is simulated with
y(~r,t) =
N
Â
i=1cos
(wi(~si·~ri t) +fi) (2.17)
forNsources, random phase shiftfi and amplitude of unity. We add white noise to each seismic
trace for a more realistic simulation. For a realistic appraisal of the IAS method, we choose our array station configurations to match the real inter station spacings of WRA (Warramunga Array, Australia) and ASAR (Alice Springs Array, Australia) as the aperture and station configuration for these two arrays are contrasting cases. WRA has widely spaced stations along two distinct arms whereas ASAR has closer spaced stations in an irregular arrangement (Figure S2.1). We further apply IAS Capon to real ambient seismic data from these arrays to illustrate the resolution capabilities.
For the synthetic test case, 25 randomly generated sources in a frequency range of 0.4 1.1 Hz and a velocity distribution between 2.5 5.5 km/s are constructed. The discretised time step
is set to 0.025 seconds and the trace length is 144000 npts in the case of WRA spacing and 0.05 seconds and 72000 npts for the ASAR spacings which amounts to a 1 hour recording with the true station parameters. For the analysis we choose a frequency range of 0.575±0.115 Hz, which yields reasonable resolution capabilities for the WRA configuration taking into account the array response function and the frequency content of the microseisms background. Out of 25 sources, 13 are located in the frequency range of interest (see Table S2.1 for list of parameters in the supplementary material). We perform approximate fk and Capon analysis by dividing the 1 hour long seismic signal into 23 non-overlapping time segments (temporal sub-windows) in order to obtain a reasonably accurate representation of the frequency spectrum. The cross-power spectral density is calculated as an average of the 23 temporal sub-windows to increase stability and the projection frequency is set to the middle of the frequency range of interest, namelyfp=0.575 Hz.
To reduce frequency leakage and side lobes, all sub-windows are tapered with the Hann window function. The results for the slowness spectra obtained by the approximate method are displayed, for the fk and Capon methods, together with the true source locations, in Fig. 2.1(a,c).
We repeat the fk and Capon analysis (eq. 2.11) with the same parameters using the IAS approach which sums over all possible frequencies in the computed frequency range and in the case of Capon we also implement diagonal loading with a constant factor ofasynth=104a(eq. 2.14) to
stabilise the results. The minimum frequency step size is the inverse of the sub-window time length and amounts toDf˜=6.4·10 3Hz. This results in a slowness spectrum that is averaged over 37
discrete frequencies (0.46 0.69 Hz). The calculated spectra for the IAS methods are displayed in Fig. 2.1(b,d). For the approximate methods we find that the Capon algorithm shows an increased resolution compared to the fk method for all sources as found in previous studies, but owing to the slowness error introduced by the frequency averaging of the cross-power spectral density, the slowness recovered is not the true slowness of the sources. The IAS methods preserves the true slowness values and reduces spectral leakage owing to the implementation of diagonal loading.
As a further illustration, we show synthetic analysis for an array which matches the response of ASAR and obtain the results in shown Fig. 2.1(e,f). For the approximate Capon method we do not find isolated peaks but a smeared spectrum, which would make analysis of real data difficult. The smearing is a direct result of the frequency smoothing, as multiple sources enter the projection procedure on a single frequency. This is related to the smaller aperture of ASAR and is discussed in a later section. The ASAR array response function in the chosen spectrum prevents an accu- rate estimation of sources with the approximate method. The IAS method is able to resolve the spectrum well, because sources with different frequencies do not interfere during the projection procedure. The numerical results for backazimuth, slowness and relative power for these calcula- tions, and the difference between inferred and true values for backazimuth and velocity for each maxima are given in the supplementary material. In all four synthetic slowness spectra Fig. 2.1(c- f), an additional source aroundbaz=320 degrees is estimated, which was not generated in the frequency range of interest. This is due to frequency leakage and will be discussed in Section 5 in more detail.
We now apply the approximate and IAS methods to real data for the case of WRA (Fig. 2.2) and ASAR (Fig. 2.3) showing the results for the Capon method only. We average over 37 discrete frequency bins for all real data WRA calculations and use a diagonal loading parameter
b WRA
e ASAR f ASAR
c WRA
a WRA
d WRA
Figure 2.1: a)b) Synthetic data fk analysis with WRA inter array spacings for the approximate and IAS algorithm: c)d) Synthetic data Capon analysis for WRA and (e,f) for ASAR inter array spacings. The red circles denote the true position of the synthetic sources. In both cases the IAS method accurately corrects the slowness vector to the true values. We use the same colour
WRA 2010-183-4 d v=3.33km/s v=4.16km/s WRA 2010-183-4 c v=3.33km/s v=4.16km/s 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 2.2:a)b) Real (observed) data comparison between the approximate Capon analysis and the IAS Capon analysis with diagonal loading for WRA. The bottom right label in each subfigure indicates the date and time of the one hour long data sample (year - Julian day - start hour , UTC); c)d) Comparison between methods for a different day and time. We observe an increase in the
resolution of seismic wave arrivals which we associate withRgandLgwaves.
ofareal =103a. For real data, this value is an order of magnitude smaller than is required for
the synthetic data owing to the greater signal content in the frequency range of interest. We display spectra from two different days to illustrate the improvement in performance using the IAS method, Fig. 2.2(a-d). In both cases we refine the observed peak showing multiple wave arrivals. The loaded IAS method finds main arrivals around v=3.33 km/s and v=4.16 km/s which can be associated withRgandLgphases [Koper et al. (2010)]. Further we observe a second
body wave arrival which was not detected by the approximate approach, Fig. 2.2(b). For the second WRA example we observe another peak split into Rg andLg phases, Fig. 2.2(d). We
compute the seismic data with the same parameters for ASAR and obtain Fig. 2.3(a,b). The resolution of the approximate and loaded IAS estimates is similar in this case, except that the slowness vector associated with the Rg wave is correctly recovered by the loaded IAS method
at 3.41 km/s (the approximate method gives 3.76 km/s). ASAR’s resolution capabilities in the analysed frequency range are limited by the array response function due to the small inter array spacings, therefore we evaluate the data at a frequency range suitable for the ASAR configuration, Fig. 2.3(c,d). We select 34 temporal sub-windows which amounts to a window length of 105.9 seconds and average over 49 discrete frequency bins in order to extract more information from the data. The approximate method detects a body wave arrival withv=12.33 km/s and a backazimuth
baz=340.56 degrees. The loaded IAS Capon method estimates the velocityv=10.9 km/s with a backazimuth ofbaz=348.69 degrees. It further detectsRgarrivals at a velocityv=3.03 km/s
d ASAR 2010-183-4 v=3.03km/s b ASAR 2010-183-4 v=3.33km/s v=4.16km/s c ASAR 2010-183-4 v=3.03km/s a ASAR 2010-183-4 v=3.33km/s v=4.16km/s
Figure 2.3: a)b) Real (observed) data comparison between methods for ASAR using the same parameters as in Fig. 2.2. c)d) Estimation for a different frequency range that is more suitable
for the detection capabilities of ASAR.
In improving the understanding of ambient seismic noise (microseisms) it is desirable to determine a slowness spectrum that provides information on simultaneous sources from multiple directions. This is usually not required for earthquake or nuclear event analysis. We therefore com- pare the IAS Capon estimation without temporal averaging, hence a rank-1 cross-power spectral density (Kværna and Doornbos, 1986), and a temporal averaged full-rank matrix, as we propose in this implementation, Fig. 2.4(a,b). We suggest that a well structured full rank matrix reveals a more detailed slowness spectrum, and therefore highlights the main arrivals and reveals further ambient seismic phases previously undetected. Temporal averaging is therefore a useful improve- ment, used in the IAS implementation, with particular benefit in the analysis of seismic noise.