Tau-p processing allows for the analysis of array data in intercept-slowness space. The transformation into the Tau-p domain allows one to resolve multiple arrivals and removes focussing and defocussing effects, yielding signals with plane wave amplitudes and phases (Phinney et al., 1981). In Tau-p and co-p domains the frequency content and the resolution in slowness of the signals crossing the array can be determined. Also, time, frequency and slowness filters/windows can be applied to the data. This method is employed in exploration geophysics (e.g. Turner, 1988) to eliminate unwanted signals, e.g. sideswipe (Rayleigh waves), but can be used on array data as a complete filtering technique which also gives information on the slowness vector of the signals.
This process is quite involved as it requires data manipulation in many different domains. A detailed outline on how the data is processed is described below and is summarised by the algorithm displayed in fig.2.9.
Working in and out of the Tau-p domain and x-t space the radon transform needs to be used. For continuous data in the time domain, the radon transform pair is defined as follows (Deans, 1983):
oo
- radon transform (slant stack) (2.11)
-oo oo
w here y+(p.t-px) = g fH (y ) oo '•y(p,T-px)j T t-t J (2-13)
In equations 2.11, 2.12 & 2.13 the transform ed w avefield y is a function o f the ray param eter p (horizontal slowness) and the intercept time t. The operator y+ which consists of a time derivative o f the Hilbert transform is commonly referred to as the convolutional filter in the radon transform. F or discrete data, integration is replaced by summation over the aperture o f the seismic array, the time window and the slowness range used in the forward transform. / Accurate \ inverse radon \tra n sfo rm S Filters / Windows Filters / Windows Time derivative Hilbert transform Spatial deconvolution Radon back projector R* x - co space Seismic section x - 1 space p - co space Tau - p space Radon forward projector
23 Initially, equations 2.11 & 2.12 are Fourier transformed with respect to time to give :
$(p,co) = |u(x,co)e1(°Pxdx (2.14)
u(x,cl>) = J^(p,co)e"1C0Pxdp (2.15) For discretised data in the frequency domain, the (forward) radon transform can be expressed as a matrix product (Greenhalgh et al., 1990):
#(co) = R(co)ü(cd) (2.16)
where the phase shift matrix R is
Rjk = eicopjxk (2.17)
j = l ,2 , Np (Np - number of p values) k = 1, 2, Ntr (Ntr - number of traces)
Having transformed the data into p-co space, which in turn can be expressed in the Tau-p domain by a fast fourier transform operation on $(p,co), time, frequency and slowness filters/windows can be applied to the data to eliminate unwanted signals.
The inverse transform, rearranging eqtn. 2.16 is of the form :
fl(CD) = R*(co)£(co) (2.18)
where R*, the Radon back projection operator, is simply the conjugate transpose of matrix R. Inverse fourier transformation of vector ü yields the inverse slant stack. Artifacts and edge effects lead to poor fidelity in the inverse transform, which results in the inaccuracy of the time derivative Hilbert transform approach, indicated by equation 2.13. A more accurate method is to deconvolve the Radon transform wavefield, whereby the best fit of the continuous radon transform to a discrete radon transform is calculated. This method deconvolves the spatial discretisation enforced by discontinuous and limited aperture sampling of the wavefield. From eqtn.2.18 we g e t:
fi = [R*R]-' R*£ (2.19) = ] Mn (2.20) where IN p Hit = y ei(0PKxk-*j) J 1=1 (2.21) This is now in the form of a standard deconvolution equation where the inverse filter H_1 is equivalent to the autocorrelation matrix (Greenhalgh et al., 1990). To stabilise the procedure
white noise is added. The deconvolution process is applied in k-co space, and a double Fourier transform with respect to k and co gets us back to x-t space (see algorithm in fig.2.9).
The range of p values that can be employed in the forward radon transform is restricted by the temporal and spatial sampling of the wavefield. To avoid aliasing, the maximum slowness must satisfy the following criterion (Turner, 1988)
< 1
Pmax - 2Axf„Lmax (2.22)
where Ax is the station spacing and fmax is the maximum frequency used in the analysis. Also for unambiguous reconstruction in the inverse approach, the maximum slowness increment allowed is (Turner, 1988):
Ap max “ ?xnf1 (2.23)
z,AR1max
where xR is the range of x (i.e. the aperture of the seismic array).
The Tau-p section for the New Ireland event for frequencies below 2Hz is shown in fig.2.10a. The plot shows that the resolution in p is low (as the amplitude of the traces for a given arrival are very similar), this is a limitation of the process, and is primarily due to the limited dimensions of the array aperture. The traces have the same form as in the beamforming techniques, but in this case they are simpler and smoother. By reversing the process we find that the recorded wavefield (fig.2.1) has been simplified to the extent where only the coherent phases remain (see fig.2.1 Ob).
An example of how the filters/windows work is shown by the case where all parts of the Tau-p domain are muted except for the region outlined by the box in fig.2.10a. After applying the reverse process we find that all sections of the wavefield outside this time window are non-existent, however at the edges the wavefield is not resolved properly (see fig.2.10c). These errors occur because the resolution in time is poor. Filtering in co-p space allows for the elimination of unwanted signals, e.g. Rayleigh waves, which have a different frequency to those being analysed.
Since I have used a plane wave development of the Tau-p process, slight errors will also occur and these will appear as secondary effects. A full cylindrical wave approach described by Harding (1985) will give proper amplitude and phase information.
S
lo
w
n
es
s
(s
/k
m
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0 .0 7 0.10 0.13 55 0 5 TIME (SECONDS) I i i i i i 11 i I I t I » I I I I I i i i I I i i I I I I i I I i I I I t t i I * I I t * * * I I * * ---wV Vm N - ■ 1 ... --- — —--- --- - . . - f r W V v ^ v --- --- $v/VVvV\/~'-*«~---— --- -vAtVvW v- 1 1 — ---— ---k/VVAtatv--- --- I t t I I I I I I I ) I I I I I t I I ) I ' I I I I t 1 1 1 I I I ! I I 1 ' ! 1 ! I T I i 1 ' t I [ - 23 40 4 3 SÖ 33 0 3 TO 13 20 TIME (SECONDS)'Figure 2.10 a) Tau-p plot for the New Ireland event. The box indicates the section which is not muted out when a time window is applied in this domain to give the results in fig.2.10c; b) the output after the initial data (fig.2.1) has been through the forward and reverse process; c) the final output for the muted section.
In exploration geophysics there are many more traces and the resolution in time is much better, so this method is more appropriate to this type of work. This method is the major process in controlled direction reception filters developed by Greenhalgh et al. (1990) for the separation of P and S phases.
The Tau-p technique is virtually equivalent to linear beamforming, but has filtering and windowing capabilities.
2.1.5 Discussion
None of the processes described above is able to cope with interfering wavelets of similar character, as the composite wavelet changes form across the array thus invalidating the simple plane wave model (Kennen, 1987). However most of the arrivals analysed in travel-time and velocity model studies are coherent plane wavefronts, that reflect on processes occurring well away from the array, either in the upper mantle or at the source. Neither the f-k or Tau-p method has been employed for the analysis of earthquakes in this project, as both techniques have limitations on how well they can analyse the seismic wavefield, and have proved less flexible than the linear and non-linear stack methods.