interpolation (Spitz, 1991). Prediction filters are often estimated from small spatio-temporal win- dows where waveforms can be approximated by events with ...
Adaptive linear prediction filtering for random noise attenuation Mauricio D . Sacchi* and Mostafa Naghizadeh, University of Alberta SUMMARY xk = a1 xk−1 + a2 xk−2 + a3 xk−3 + εk . We propose an algorithm to compute time and space variant prediction filters for signal-to-noise ratio enhancement. Prediction filtering for seismic signal enhancement is, in general, implemented via filters that are estimated from the inversion of a system of equations in the t − x or f − x domain. In addition, prediction error filters are applied in small windows where the data can be modeled via a finite number of plane waves. Our algorithm, on the other hand, does not require the inversion of matrices. Furthermore, it does not require spatio-temporal windowing; the algorithm is implemented via a recursive scheme where the filter is continuously adapted to predict the signal. We postulate the prediction problem as a local smoothing problem and use a quadratic constraint to avoid solutions that model the noise. The algorithm uses a t − x recursive implementation where the prediction filter for a given observation point is estimated via a simple rule. It turns out that the proposed algorithm is equivalent to the LMS (Least Mean Squares) filter often used for adaptive filtering. It is important to mention, however, that our derivation follows the framework that it is often used to solve underdetermined linear inverse problems. The latter involves the minimization of a cost function that includes a quadratic constraint to guarantee a stable solution.
(1)
It is important to point out that one could have also proposed an analysis in terms of prediction error filters xk − a1 xk−1 − a2 xk−2 − a3 xk−3 = εk ,
(2)
where by calling f0 = 1, and fk = −ak , k = 1, 2, 3, equation (1) becomes xk f0 + f1 xk−1 + f2 xk−2 + f3 xk−3 = εk .
(3)
We now assume that the coefficients of the autoregressive model are varying in time or space. In other words, each output sample k is predicted via coefficients that depend on k. The latter can be mathematically expressed as follows xk = ak1 xk−1 + ak2 xk−2 + ak3 xk−3 + εk .
(4)
or, in matrix form
Synthetic and real data examples are used to test the algorithm. In particular, a field data test shows that adaptive t −x filtering could offer an efficient and versatile alternative to classical f − x deconvolution filtering.
ak1 k xk−3 ) a2 ak3
xk
=
(xk−1
=
Mk ak .
xk−2
(5)
INTRODUCTION Prediction filters play an important role in seismic data processing with applications ranging from seismic deconvolution, signal-to-noise-ratio enhancement (Canales, 1984; Gulunay, 1986; Abma, 1995) and trace interpolation (Spitz, 1991). Prediction filters are often estimated from small spatio-temporal windows where waveforms can be approximated by events with constant dip. The latter is required for the optimal performance of the prediction filter. We propose to avoid windowing via a recursive algorithm where one prediction filter is estimated for each data sample. We show that the aforementioned problem is underdetermined and, as it is wellknown, admits an infinite number of solutions. A unique and stable solution if found by formulating the problem in terms of a regularization constraint. The filter required to smooth a given data point is constrained to be similar to the filter used to smooth an adjacent data point. Our formulation follows the classical approach used to solve an underdetermined problem. The final algorithm is equivalent to the LMS (Least Mean Squares) algorithm described in Widrow and Stearns (1985) and Hornbostel (1989).
It is clear that for each sample k we have p = 3 unknown coefficients. This is an underdetermined problem that admits an infinite number of solutions. Uniqueness of the solution is guaranteed by introducing a quadratic regularization term. The local prediction filter for sample k is found by minimizing the following cost function Jk = ||xk − Mk ak ||22 + µ 2 ||ak − a0 ||22 .
(6)
The first term in the right hand side of equation (6), the misfit function, measures how well the filter predicts the signal. The second term, the regularization term, measures the closeness of the local prediction filter to an a priori filter a0 . The regularization parameter µ determines the importance of the regularization term relative to the data misfit. Taking the derivatives with respect to the unknown filter coefficients and, after setting them to zero, leads to the following system of equations ak
=
2 −1 MH M ) a (I − (MH K 0 k Mk + µ I) k
(7) +(MH M + µ 2 I)−1 MH k xk .
ADAPTIVE PREDICTION FILTERS 1-D case We start to formulate our problem by introducing an auto-regressive (AR) or linear prediction operator modeling operator of order p. To avoid notational clutter we first consider a 1D problem with p = 3
Where (.)H is used to indicate the conjugate transpose. Notice that our analysis considers the general case where the signal xk is complex. We now adopt the following identity to simplify equation (7) 2 −1 H H H 2 −1 (MH k Mk + µ I) Mk = Mk (MM + µ I)
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(8)
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Adaptive filtering where ηk =
and after considering that δk2 = Mk MH k =
Mk = (xn−1 xn−2 xn−3 ),
2D (t − x) algorithm
3
∑ |xk−n |2 .
(9)
n=1
After a few algebraic manipulations we arrive to the following expression
aˆ k
=
MH
MH M
k (I − δ 2k+µ 2k )a0 + δ 2 +µ 2 xk . k
The 1-D algorithm can be easily extended to process 2-D and 3-D data. For that purpose, we first replace the 1-D inner product MH k ak = ∑ j xk− j akj by its multidimensional counterpart which, for a 2D prediction filter, is given by
∑ ∑ xk−i,l− j ak,l i, j .
(10)
i
k
It is clear that the locally smoothed signal (equation (5)) is now given by xˆk
1 . δk2 +µ 2
=
Mk aˆ k
=
k k (1 − δ 2 +µ 2 )Mk a0 + δ 2 +µ 2 xk
δ2
δ2
k
(11)
j
With this rule in mind, it is easy to generalize (15) to 2-D and 3-D algorithms. For instance, the 2-D t − x adaptive algorithm can be implemented via the following expressions
p
q
2 δk,l
=
∑n=1 ∑m=1 |xk−n,l−m |2
ηk,l
=
1 2 +µ 2 δk,l
ak,l i, j
=
p k,l ak−1,l−1 − ηk,l xk−i,l− j (∑i=1 ∑ j=1 xk−i,l− j ai, j − xk,l ) i, j
xˆk,l
=
∑i=1 ∑ j=1 xk−i,l− j ai, j
k
The trade-off parameter µ controls the degree of adaptability to changes in the signal. In particular, we examine the influence of µ in the following two situations:
q
p
q
k,l
1 Minimum norm solution with perfect data fitting µ → 0,
xˆk = xk .
(12)
where we have considered prediction filters of size p × q. Tests
The filter is predicting the original data and therefore, signal and noise are simultaneously modeled. This is often referred in inverse problems as over-fitting. 2 No adaptability
µ → ∞,
xˆk = Mk a0 = xk−1 a1,0 + xk−2 a2,0 + xk−3 a3,0 . (13) The prediction filter cannot adapt to changes in xk and, therefore, smoothing becomes independent of k. Last equation is equivalent to conventional non-adaptive f − x or t − x prediction filtering (Gulunay, 1986; Abma, 1995) where the filter a0 is estimated from the data via the least-squares method.
An optimal value of µ needs to be selected in order to achieve noise suppression and adaptability to signal changes (e.g. changes in dip in the t − x domain). We will return to this point in the section devoted to tests.
Synthetic example Figure 1 portrays a synthetic data composed of 5 events with parabolic moveout. The data were contaminated with Gaussian band-limited noise with signal-to-noise ratio SNR = 1. Figure 1a portrays the noisy data. Figure 1b portrays the filtered data obtained via the algorithm described in this paper. We have adopted a parameter µ = 20 and 2D (t − x) adaptive prediction filters of size 20 × 20. Figure 1c displays the noise section (data minus filtered data). The tradeoff parameter is obtained by inspecting the noise section. The optimal µ yields a noise section with little amount of coherent energy. In this example the adaptive filter was run in space-time starting from the first observation (t = 0, x = 0) and moving toward the last observation (t = tmax , x = xmax ) with initial prediction filter a0,0 k,l = 0. The algorithm is rerun starting from the last observation and moving the recursion toward the first observation. The initial filter in this case is also given max ,xmax by atk,l = 0. By this procedure, two smoothed signals were estimated. The average signal was adopted as the final estimate of the filtered data.
Recursive algorithm
We have also tried to initialize the algorithm with a least-squares t − x filter computed in a small window (Abma and Claerbout, 1995). The aforementioned initialization did not provide important improvements.
We will assume that the signal varies slowly in time or space and, consequently, we propose
Field data example
a0 = ak−1 .
(14)
The latter leads to the following algorithm Initialize a0 and for k = 1, 2, 3 . . . ak
=
ak−1 − ηk MH k (Mk ak−1 − xk )
xˆk
=
Mk ak
(15)
We continue with our tests by filtering a real data set from the Western Canadian Basin. The data consist of a seismic section with an appreciable deterioration in SNR for the deep part of the image. Figure 2 shows the mean normalized spectra of the data after and before filtering. Figure 3a shows the section prior to filtering. Figure 3b provides the filtered section via f − x deconvolution. The f − x filter was computed using the least-squares solution to the forward/backward prediction filtering problem. The method avoids finite aperture artifacts by considering non-Toeplitz covariance matrices ( Marple, 1987). The length of the f − x prediction filter is L = 4, the
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Adaptive filtering (a)
0
(b)
(c)
0.1
time (s)
0.2 0.3 0.4 0.5 0.6
0
50
100
150
Figure 1: a) Synthetic seismic gather contaminated with Gaussian band-limited noise (SNR = 1). b) Output of the t − x adaptive prediction filtering algorithm. c) Difference between the data and the filtered data.
covariance matrix was inverted with a tradeoff parameter that represents 0.1% of its trace. Figure 3c present the filtered data after applying the t − x adaptive prediction filter with an operator of size 20 × 10 (time × space). The tradeoff parameter for this example is set to µ = 15. The tradeoff parameter was found by examining the residual section. In other words, the algorithm is run with different tradeoff parameters; the optimal tradeoff is the one that leads to a residual section with a minimum amount of coherent energy. 1 Input f−x decon t−x adaptive
0.9 0.8 0.7
Power
0.6 0.5
strategies to cope with spatial changes in dip. As a final remark, it is interesting to point out that the adaptive method can be easily used to design 2D algorithms that operate in t − x or in f − x − y for f slices. It can also be extended to the 3D case to cope with noise attenuation in t − x − y. In summary, adaptive filtering offers a versatile design that can be easily accommodated to noise attenuation in different domains.
CONCLUSIONS We have presented an inversion-based approach to the design of adaptive filters. Our approach involves considering adaptive filtering as an undetermined linear inverse problem where a set of filter coefficients are used to predict one observation point at the time. The problem, clearly, involves estimating more than one parameter from a single observation. A quadratic regularization constraint was used to retrieve a filter that can locally predict the signal and, at the same time, adapt to temporal and spatial changes in the data. When the a priori filter (initial filter) is the filter compute from a nearby observation point, the method leads to a recursive algorithm that is equivalent to the LMS filter. Synthetic and real data tests were used to validate the adaptive algorithm as a tool for noise suppression. In particular, our field data tests show that adaptive filtering is a viable alternative to classical f − x filtering for noise suppression.
0.4 0.3 0.2
ACKNOWLEDGMENTS
0.1
This research has been supported by the sponsors of the Signal Analysis and Imaging Group, at the University of Alberta and the National Sciences and Engineering Research Council of Canada via a Discovery Grant to MDS. Discussions with Sam Kaplan and Tad Ulrych were instrumental in the preparation of this paper.
0
20
40
60 80 Frequency (Hz)
100
120
Figure 2: Power spectra of the data and filtered data in Figure 3. The yellow boxes in Figure3a,b, and c were redisplayed in Figures 4 a, b and c, respectively. In addition, Figures 4d and e show the estimators of the noise. The field data example clearly shows that both method have lead to similar results. The main advantage of the adaptive procedure roots in its simplicity. In addition, one does not have to worry about inverting large matrices (Abma and Clearbout, 1995) or to specify windowing
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Adaptive filtering
Figure 3: a) Seismic section from a survey in the Western Canadian Basin. b) Seismic section after f − x deconvolution. c) Seismic section after noise attenuation via adaptive t − x prediction filtering. The boxes are redisplayed in Figure 4.
(a)
(b)
(c)
(d)
(e)
Figure 4: a) Original data. b) Data after signal-to-noise ratio enhancement via f − x deconvolution. c) Data after signal-to-noise ratio enhancement via adaptive t − x prediction filtering. d) Noise removed from the data via f − x deconvolution. e) Noise removed from the data via adaptive t − x prediction filtering. The data correspond to every second trace in the boxes in Figure 3.
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EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2009 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES Abma, R., and J. Claerbout, 1995, Lateral prediction for noise attenuation by t-x and F-X techniques: Geophysics, 60, no. 6, 1887–1896. Canales, L. L., 1984, Random noise reduction: 54th Annual International Meeting, SEG, Expanded Abstracts, 525–527. Gulunay, N., 1986, Fx decon and complex Wiener prediction filter: 56th Annual International Meeting, SEG, Expanded Abstracts, 279–281. Hornbostel S.,1989, Spatial prediction filtering in the t-x and f-x domains: Geophysics, 63, 1618–2026. Marple, Lawrence S., 1997, Digital Spectral Analysis: With Applications, Prentice Hall. Spitz, S., 1991, Seismic trace interpolation in the F-X domain: Geophysics, 56, no. 6, 785–794. Widrow, B., and S. D. Stearns, 1985 Adaptive Signal Processing, Prentice Hall.
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