Texts such as [30, 31] give a current treatment of the use of filtering sections in place of complex multipliers when pursuing the attenuation of a broadband interference. The classical papers [35, 36, 37] introduce the signal processing structure used in this and other current work applied to the attenuation of
coherent interference encountered in seismic analysis. The array processing, which now includes linear FIR filters, can be viewed as a generalisation of the complex weighting of (2.9). As the seismic problem uses recorded data,
the need for algorithms which can adapt, perhaps even in real-time, does not arise. Further, the data may be reprocessed until accurate signal parameter estimates are obtained. This latter freedom has led to the use of a signal model (described earlier) in formulation and development of ‘Optimum Array Filters’ (OAF) and ‘Absolutely Optimum Array Filters’ (AOAF) presented respectively in [40, 41]. The more recent work on ‘Absolutely Optimum Array Filters’  is briefly described.
For each of N sensors, the output signal is modelled by
yn(t) =anx(t−ξn) +bnu(t−ρn) +wn(t) (2.11) where x(t) is the desired signal with magnitude an and arrival time ξn, u(t) is the coherent interference with magnitudebnand arrival timeρn, andwn(t) is random sensor noise received at the nth sensor. The signals are then time shifted and amplitude normalised so as to align x(t) giving
sn(t) = x(t) +αnu(t−τn) +vn(t) (2.12) where αn = bn/an, τ = ρn − ξn, and vn(t) = wn(t +ξn)/an. The array processor is now steered towards x(t). The noise component vn(t) is not included in any further analysis although in , a simulation is shown in which some random noise is included.
Figure 2.2: Array processing structure for absolutely optimal array filters.
The signalsyn(t) are passed through filtersFn(ω) and summed (see Figure 2.2) to give S(ω) = N X n=1 Fn(ω)X(ω) + N X n=1 αne−jωτnFn(ω)U(ω) . (2.13) In this description, the response of the array to the desired signal is denoted
D(ω) i.e. D(ω) = N X n=1 Fn(ω) (2.14)
while the response of the array to the coherent interference is given as
R(ω) = N
Rn(ω) =αne−jωτnFn(ω) (2.16) In , a problem formulation based on minimisingkR(ω)k2gave an underde-
termined system of equations. This problem was then avoided by minimising
i=1kR(ω)−Ri(ω)k2 instead. Note however that there is no guarantee that minimising the difference between each individual filter response and the overall filtering response will necessarily attenuate the interference response. However, examples were presented in  demonstrating such effect. The dif- ficulties of  in minimisingkR(ω)k2 were solved in  by generalising the
cost function using the parameter γ in (2.17). The filter transfer functions22
are adjusted to minimise
J(γ, ω) = 1 2 N X n=1 |γR(ω)−(1−γ)Rn(ω)|2 (2.17) subject to the desired signal all-pass constraint
D(ω) = 1 (2.18)
The filter coefficients for the AOAF are solved in (, equation (26) when
γ = 1) as Fn(ω) = 1/|an|2 −δ(1/a∗n) PN m=1(1/am) PN m=1(1/|am|2)−δ|PNm=1(1/am)|2 (2.19) where δ =N−1 ; an=an(ω) = anejωτn (2.20)
22Setting γ = 1/2 corresponds to what was initially called ‘Optimum Array Filters’,
and settingγ = 1 corresponds to minimisation ofkR(ω)k2 giving what was then termed
and a∗n is the complex conjugate of an.
The response R(ω) of the array to the coherent interference is shown in (, equations (29a,29b)) to be zero except for a case of esoteric signal conditions. As pointed out in , this rejection response, which can be plotted with a main lobe and zero sidelobe level, is not the same as the
spatial response of a beamformer. For the AOAF, the rejection operation is dependent on signal ratios αn and relative arrival times τn while for a beamformer, the rejection response is usually calculated for a plane wave signal with equal amplitude at each sensor.
A useful insight into the operation of the AOAF is given in  by con- sidering the case in which the interference and desired signal are both plane wave signals arriving at the sensors with equal amplitudes. Assuming equal
an, (2.19) may be written Fn(ω) = (1−N−1B(ω, θI)ejωτn) N−N−1|B(ω, θ I)|2 (2.21) where B(ω, θI) is the response of the array steered to a plane wave signal at angle θI. The operation of (2.21) may be viewed in terms of 2 beamformer sections: the first term in the numerator (unity) passes the desired signal; and the second term with its progressive phase shift ωτn subtracts off the response of the array to the interference. The denominator term scales the total difference ensuring that (2.18) is satisfied given the interaction of the 2 beamformer structures.
how many sensor signals are required to null completely an interference23.
Since the signal model assumes coherency between sensor signals, it is im- portant to be able to specify a minimum seismic signal count for successful filtering. In this way, the possibility of failure of the coherency assumption (i.e. that adjacent seismic signals have sampled similar subsurface regions and hence do contain the same x(t) andu(t)) may be minimised.
The filters Fn(ω) are not causal so block data processing is required. For recorded seismic signals, this is not a problem although implementation via Fast Fourier Transforms (FFT) does impose possible discrete data sample count constraints. Although digitally recorded field data usually has a sample count directly suitable for a radix-2 FFT, trace alignment of the x signals will modify the data samples available for processing. This means that either zero padding is required for implementation via a radix-2 FFT algorithm, or a specific sample count FFT algorithm be used.
Because the filter design is formulated and solved in the continuous do- main, implementation in the discrete domain requires some approximation (or sampling) of Fn(ω) to be formed of the actual filter functions. It is not clear how accurately the filter responsesFn(ω) must be discretised to ensure proper operation.
For AOAF, the discrete time implementation difficulties combined with
23In , it is shown that operation of an OAF requires at least 3 sensors (see equation
(28)) but an OAF is unable to null the interference completely. Neither does this require- ment possess an interpretation in any engineering sense related to signal properties.
the absence of a mechanism to limit failure of the coherency signal model assumption has led to the optimal array filtering now presented.