In this chapter, it is assumed that there are two components present in the

recorded signal - the desired component x and one coherent interference u. In existing SSP procedures, stacking is performed to take advantage of sig- nal redundancy and allow SNR enhancement of a desired signal component.

In reflection seismology, an image of the subsurface region is built up by enhancing and interpreting reflected components at increasing depths. As outlined in section 1.3, the redundancy necessary for stacking is due to low lateral variation rate for earth parameters combined with the relatively high spatial sampling rate of the sensors.

The generation of seismology data from survey work is considered to be deterministic. If the same excitation wavelet is applied in consecutive ex- periments, the same return signals containing componentsx, uare produced. This must be the case for redundancy to be possibly useful across separate data gathers. While the received signals have undergone a frequency selec- tive attenuation which is dependent on travel time (i.e. distance), the spatial oversampling amongst sufficiently few sensor groups combined with large sig- nal travel distances allows one to argue that the effects of propagation are the same for adjacent traces. If this did not occur, relatively unsophisticated methods such as stacking could not work. Although the waveform of the ex- citation signal is unknown, various methods are currently in use to estimate it. Statistical signal models have been proposed as a means of encapsulating uncertainty in algorithms to estimate time delay (see for example [28, 29]) and wavelet shape ([9] gives an overview of some approaches). In general, waveform shape and time delay estimation is a non-trivial task and there is much current research activity.

modelled by x and u are assumed to have been previously estimated. As the spectral content or time domain waveform may not be precisely known

for x, u, this work makes no assumptions other than that x, u possess a finite bandwidth and can therefore be sampled and accurately represented in discrete time18. Within the limits of the Nyquist rate, the signals are

considered to be broadband. Arguments in support of these assumptions include the fact that current seismic imaging techniques already depend on accurately determining the arrival times of reflections in order to locate the position of layer boundaries. Wavelet processing of original traces facilitates concentration of the wavelet energy into a smaller temporal region thereby allowing accurate ‘time picking’ and magnitude measurement at the ‘picked’ time because of the very high SNR present. Some cross checking of this information is also possible as any seismic imaging must be consistent with other geophysical information available.

The success of wavelet processing techniques also illustrates the validity of these assumptions. Recall from Chapter 1 that NMO correction can involve cross correlation analysis to determine relative arrival time delays between traces collected from adjacent sensors. The cross correlation of trace data that has undergone wavelet processing of individual traces (or spiking) is an example of Generalised Cross Correlation where the wavelet processing may be regarded as a frequency weighting operation of each channel. As

18_{Recall from Chapter 1 that wavelets possess finite time duration / finite energy}

mentioned earlier, the filtering of each wavelet can assist visual measurement of trace parameters or enhance the effectiveness of automatic methods.

Existing stacking can also be used to enhance the coherent interference providing yet another method of refining magnitude and arrival time infor- mation. In a similar way, the array processing techniques to be presented here are also able to ‘self boot’, i.e. approximate signal parameter estimates allow array filtering to enhance the interference u and obtain data for the design of a final array filter to null u.

In summary, the signal environment for the signal processing presented in this chapter will include a desired signal x and a coherent interference19

u, both possessing a known magnitude and arrival time. Signals x, u are baseband and broadband, i.e. all energy exists within a fixed bandwidth (usually in the 5Hz to 100Hz range), and energy is spread across all frequen- cies within the band. When u results from multipath propagation effects such as reflection, it may share the same waveform as x.

For an array sensor n= 1. . . N, the received signal will be written

yn(t) =anx(t−ξn) +bnu(t−ρn) (2.1) where{an, bn}will be referred to as signal magnitudes20 and{ξn, ρn}will be referred to as signal arrival times. This possible lack of spectral difference

19_{In Chapter 3, this noise model is extended to multiple coherent interferences and}

random noise.

20_{Strictly, these quantities are gains. Although no reference magnitude is known for}_{x}

oru, this work uses the termmagnitudein order to remain consistent with other work in this field.

between the interference and the desired signal, in combination with a desire to assume as little as possible about the actual waveforms present, suggests that directional source diversity be utilised. As an aid in developing Optimal Array Processing, narrowband beamforming is now introduced.