Subspace modelling for structured noise suppression

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Abstract

The problem of structured noise suppression is addressed by (i) modelling the subspaces hosting the components of the signal conveying the information and (ii) applying a nonlinear non-extensive technique for effecting the right separation. Although the approach is applicable to all situations satisfying the hypothesis of the proposed framework, this work is motivated by a particular scenario, namely, the cancellation of low frequency noise in broadband seismic signals.

Introduction

The problem of structured noise suppression concerns the elimination of signal components produced by phenomena interfering with the observations of interest. This problem can be addressed by linear techniques provided that the subspaces hosting the signal components are complementary and well separated [1], [2], [3], [4]. More precisely, if a signal represented by the ket |f is produced as the superposition of two components |f1S1 and |f2S2, provided that S1S2={0}, the components of the superposition |f=|f1+|f2 can be separated by an oblique projection. Even when this condition is theoretically fulfilled, if the subspaces S1 and S2 are not well separated, the concomitant linear problem for extracting one of the signal components may be ill posed, which causes the failure to correctly split the signal by a linear operation. Hence, nonlinear techniques for determining a subspace VS1, such that |f1V, and the projection onto V along S2 is well posed, have been considered [3], [4], [5]. In those publications the theoretically complementary subspaces S1 and S2 are assumed to be known. Nevertheless, the condition S1S2={0} is strong and the possibility of meeting it depends on the ability to generate the right model for the subspaces. Unfortunately, the modelling of the complementary subspaces by pure physical considerations is not always possible and one needs to relay on more general mathematical modelling.

Although the technique for subspace modelling we introduce here is applicable to different situations, the work is motivated by a particular problem relevant to the processing of seismic signals. In the nearshore these signals may be affected by a population of low frequency waves called infragravity waves[6]. This type of noise may be also unavoidable in bottom broadband seismic observations [7]. The interested reader is referred to Ref. [8] for explanations on how infragravity waves are generated. We restrict our consideration to the problem of reducing that type of structured low frequency noise from broadband seismic signals.

Our purpose is twofold. We aim at (i) mathematically modelling the subspaces to represent the signal components (ii) providing a sparse enough representation of the signals so as to make sure that the correct splitting can be realized.

Under the hypothesis that one of the signal components lies in the subspace of low frequency signals, we determine the subspace of the other component in an adaptive manner. We assume that such a component belongs to an unknown spline space and determine the knots characterising the space by taking into account the curvature points of the signal in hand. In that sense, the space is ‘adapted’ to the particular signal being analysed. In line with [4] we tackle the problem of finding the representation of this component through the minimisation of the q-norm-like quantity, which is closely related to the non-extensive entropy introduced as ingredient of a thermodynamic framework in the seminal paper by Tsallis [9] and ever since broadly applied in physics [9], [10], [11], [12], [13], [14], [15], [16] and other disciplines [17].

The paper is organised as follows: In Section 2 we address the problem of subspaces modelling and discuss the nonlinear non-extensive technique yielding the right signal splitting. A numerical simulation concerning the filtering of low frequency noise from a seismic signal is presented in Section 3. The conclusions are drawn in Section 4.

Section snippets

Adaptive subspace modelling for structured noise filtering

As already mentioned, we are concerned with the problem of separating from a signal those components which are not relevant to the phenomenon of interest. For simplicity we consider that a signal |f is the superposition of only two components and the goal is to find a suitable model for the subspaces hosting such component. Since our work is motivated by the specific problem of filtering low frequency noise from a seismic signal, we further assume that the subspace representing that type of

Application to filtering of structured low frequency noise from a seismic signal

We apply here the proposed approach for filtering low frequency noise from a seismic signal. As already mentioned, a common interference with broadband seismic signals is produced by long waves, generated by known or unknown sources, called infragravity waves [6]. This interference is referred to as low frequency noise as it falls in a frequency range of up to 0.05 Hz. Thus, the model for the subspace of that type of structured noise, on a signal given by L=403 samples, is W=span{eı2πn(i1)L,i=

Conclusions

The problem of structured noise suppression has been considered by modelling the subspaces of the signal components and applying a nonlinear non-extensive technique for separating them. The work was motivated by the problem of filtering infragravity waves from broadband seismic signals. For this, the noise subspace was modelled using low frequency Fourier functions and that of the other component by a dedicated spline space (adapted to the signal in hand). A simulation involving a piece of

Acknowledgements

Support from the Engineering and Physical Sciences Research Council (EPSRC), UK, grant EP/D06263/1, is acknowledged. This work was initiated during Z. Xu’s stay at the Mathematics Department of Aston University, UK.

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