Frequency-wavenumber spectral estimation is a powerful technique for signal detection and waveform analysis, as it provides information on the amount of power distributed among different wave velocities and directions of approach. Providing the signal is stationary in time and space the wavefield u(x,t) can be represented in k-co space by the following relationship (Abrahamson and Bolt, 1987) :
oo oo
u(k,co) = ——-
J
ju(x,t)exp{i(k-x - cot)}dkdt (2.8) (2jt)3 -oo -ooThe power spectrum used for slowness and azimuth determination is simply lu(k,co)l2. The power spectrum consists of a main lobe with a 1/2-width roughly equal to the inverse of the array aperture, with sidelobes
1/V
m in amplitude of the main lobe, and both depend on the array geometry (Green et al., 1966). In estimating the power spectrum (see Aki & Richards, 1980; Abrahamson and Bolt, 1987) delays are inserted such that the array response can be steered in different directions (i.e. the estimation is calculated in much the same way as in the beamforming techniques, but in this case it is performed over a limited wavenumber interval). The f-k method similarly gives maximum SNR if all the sensors have identical signal waveforms and when the noise is independent. However, because interfering noise at a particular frequency does not arrive with uniform power over all k, but from particular regions of the k plane (i.e. noise between sensors are correlated so that the delay and sum isno longer optimum), amplitude weighting coefficients are added to the time delays in each seismometer's output to improve the estimate (Green et al, 1966).
A maximum-likelihood technique developed by Capon (1969) claims to have a higher resolution than the above weighting method. This process involves the addition of weights to every frequency over the band of interest (i.e. each frequency-wavenumber pair has different weighting coefficients), and although the method imposes no frequency distortion on the signal, it manages to steer the nulls and sidelobes, frequency by frequency, to maximise output power over the frequency band.
The f-k method is essentially similar to the appropriate beamforming techniques but it operates in a different domain, and is more restrictive as longer time intervals are required to obtain stable solutions. Consequently it is mainly used in the study of long-period waves recorded at large aperture arrays (Capon et al.,1969; Capon, 1969). However it is now a major tool at the small aperture Noress array (see NORSAR Semiannual Technical Reports) for the study of regional P and S phases, and has an advantage of being feasible in real time.
I have used a program written by T. Kvsema at Norsar to obtain the f-k solution for the group of arrivals between 43 and 47s on fig.2.1. The f-k solution is presented in terms of a contour plot and the position of the maxima is roughly equivalent to a slowness of 0.09s/km at an azimuth of 230°.
2.1.3 Adaptive processing
This process determines the arrival times of individual wavefronts at each sensor by cross- correlating the observed wavelet of interest with the corresponding wave on the beam trace. The new arrival times are then used to create a new and improved beam, and the whole operation is repeated a number of times in order for convergence to take place. This adaptive technique is a modification of the method introduced by King et al. in 1973 and includes the following steps :
(1) The array is steered relative to a reference point via delays obtained from a given slowness and azimuth. Since the solutions are dependent on the initial slowness and azimuth, two reference slownesses 0.005s/km either side of the expected slowness (e.g.
Ux(s/km)
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
[______________ I______________ |__ ____________ |______________ I______________ |______________ |
Figure 2.7 F-k solution for the group of arrivals between 43 & 47s on fig.2.1.
252.0 242.0 . 232.0 . 222.0 . 212.0 . 0.13 0.12 . 0.11 . 0.10 . 0.09 . 0.08 . 0.07 . 0 .0 6_________________________________________________________________________________________________ 35 40 45 50 55 0 5 10 15 20 TIME (SECO N D S)
Figure 2.8 Adaptive processing solutions for the New Ireland earthquake.
I I I i I I I I 1 I I I I I I I I I I I I I I I I 1 I 1 I I I I I I i I t I i 1 I I i 1 1 1 Azimuth ++ w * ' ++- ++~+ +*+‘ Slowness
I I I I I I I I I I I ! t f I I I I I I I I I I 1 I I I I I I I I 1 I I f 1 I__ 1 I__ I J __ I___I___1___1___L
252.0 . 242.0 . 232.0 . 222.0 . 212.0 . 0.13 . 0.12 . 0.11 . 0.10 . 0.09 . 0.08 . 0.07 . 0.06
obtained from the scanning technique) are employed along with the source-receiver azimuth. If one allows the initial azimuth to vary by up to 5° and the slowness by 0.015s/km accurate solutions still result.
(2) The array beam is formed (sum of all channels once phased together):
where ZR and ZB are beam sums of different sets of stations of an array. For WRA (see section 2.3.1) the two sums refer to beam sums over the red and blue arms of the array respectively. TAP measures the strength of the signal and hence determines which part of the beam to analyse.
(4) For each channel: the appropriately aligned channel is subtracted from the beam and is cross-correlated with the depleted beam. Once the maximum is located the channel is added back into the beam realigned.
(5) Step 4 is repeated eight times such that the difference between computed delays can reach some desired accuracy. This therefore allows for maximum correlation between the individual channels.
(6) The final time delays are put into Kelly's (1964) normal equations (Appendix 1) to obtain slowness and azimuth solutions. If the solutions for the two reference slownesses are within 0.002s/km and 1.5° of each other, the solutions are said to converge, and the average solution is the best solution possible that can be determined by this method. (7) The error of the final solution is related to the amount by which the time delays are refined through the iteration process (Appendix 1), combined with the difference between the 2 solutions. The time delay error is small, and so if the difference is too large the solutions are not plotted, as the solutions do not converge.
Figure 2.8 shows the adaptive processing solutions for the New Ireland event, and the solutions presented are roughly equivalent to those of the scanning technique. Strong coherent signals have small errors, and since the errors are closely dependent on correlation properties, many of the partially coherent signals detected by the scanning technique do not
BEAM = Sui(t + Tj) (3) The time average product (TAP) is calculated :
(2.9)
21
register here. By using the convergence of two reference slownesses, compared to King et al's employment of one, reduces the high scatter of the solutions and minimises the errors normally seen when inadequate initial parameters are used. Also steering delays do not need to be that accurately known to obtain resolution of the order of 0.002s/km and 1.5°. Simpson (1973) and King (1974) have used King et al.'s (1973) adaptive processing technique to resolve the times and slownesses of multiple arrivals in the distance range 15-30° from WRA.