Proceedings of Meetings on Acoustics

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1 Proceedings of Meetings on Acoustics Volume 19, ICA 2013 Montreal Montreal, Canada 2-7 June 2013 Physical Acoustics Session 3aPA: Borehole Acoustics Logging for Hydrocarbon Reservoir Characterization I 3aPA3. Space-time methods for robust slowness estimation for monopole logging while drilling Shuchin Aeron*, Sandip Bose and Henri-Pierre Valero *Corresponding author's address: Electrical and Computer Engineering, Tufts University, 161 College Ave, Medford, MA 02155, shuchin@ece.tufts.edu In this paper we present methods for interference cancelation for robust slowness estimation from noisy Monopole Logging While Drilling (LWD) data. The main contributions are two fold. First, we show via tests on real data sets presence of systematic propagative interferences in Monopole LWD data, which is the primary reason for loss of compression and shear semblance in the Slowness Time Coherence (STC) processing of the LWD data. This interference in turn is mostly dominated by Stoneley type propagative component, which, unlike the main Stoneley mode, is time persistent over the entire acquisition interval. In addition, we also show that in fast formations the shear wave can significantly interfere with the compressional wave making the compressional slowness estimates quite bad. Second, based on these observations we propose a Successive Interference Cancellation (SIC) algorithm to estimate and cancel these interferences leading to STC enhancement and improved slowness estimation of the head waves. The algorithm exploits a novel representation of borehole acoustic signals using a dictionary of space-time propagators and is more robust compared to traditional slowness filtering methods, especially for the low aperture borehole acoustic array. We show the superior performance of the proposed algorithm on synthetic and real data sets. Published by the Acoustical Society of America through the American Institute of Physics 2013 Acoustical Society of America [DOI: / ] Received 19 Jan 2013; published 2 Jun 2013 Proceedings of Meetings on Acoustics, Vol. 19, (2013) Page 1

2 INTRODUCTION In this paper we present computationally efficient and robust algorithms for semblance or slowness time coherence (STC) processing [1] of monopole LWD data [2]. We first carried out a case study of the LWD noise in the monopole channel. We compared the STC plots of the noise acquisitions with the STC plots for the monopole firing at the same depths. In addition to the presence of heavy additive noise, we observed that LWD interference in the monopole channel has a modal content that is similar to that of a typical monopole firing, i.e. consists of propagative processes with dispersion similar to the Stoneley and the head waves. However, unlike the time-compact waves generated by the transmitter firing, this LWD interference is persistent over the window of acquisition. In addition, there are reflections that interfere with the forward propagating head waves. Moreover, in fast formations, there could be significant inter-modal (shear on compressional) interference in STC processing especially after filtering. We addressed the problem of combating additive noise and reflections in a previous paper [3] where we exploited the transform domain coefficient sparsity (compactness) of monopole modes in the discrete Radon transform (DRT) domain to propose the use of shrinkage (see [4]) to suppress additive noise and reflections. The novelty of the work in [3] lies in the use of semblance for deriving near optimal space-time shrinkage factors for the DRT coefficients for de-noising. Although effective in dealing with additive noise and reflections, the shrinkage-based approach fails to adequately recover head wave semblance especially in fast formations. This is due to heavy forward propagative and inter-modal interferences. One possible approach to combat these propagative interferences is to employ the method proposed in [5], which consists of applying a slowness-time mask to the forward DRT before taking the inverse. Although effective in dealing with interferences that are well separated in slowness and time, this strategy is not effective when they are close in slowness and time. Further, this approach is not suitable for low-aperture arrays as used in LWD tools. In this paper, we propose a novel method which consists of robustly estimating the interference(s) using time-frequency compact space-time propagators for signal representation and subsequently subtracting it from the data. This approach is motivated by our recent work [6] in which we used space-time methods for dispersion extraction from LWD dipole data. Finally we combine the shrinkage in the DRT domain ([3]) with the interference cancellation methods to jointly suppress all major LWD interferences and noise. MATHEMATICAL PRELIMINARIES AND PROBLEM SET-UP We begin with the following simplified signal model for monopole compressional and shear (P & S) acquisition in a fluid filled borehole, appropriate for non-dispersive propagation across the sonic array borehole tool, as used in [1]: y l (t) = s c (t p c (z l z 1 ) τ c ) + s sh (t p sh (z l z 1 ) τ sh ) + s St (t p St (z l z 1 ) τ St ) + w l (t) (1) where z l denotes the location of the l th receiver, s c (t) denotes the temporal waveform at the first receiver corresponding to the compressional head wave with arrival time, τ c, and slowness (or moveout), p c.. Similarly s sh (t) and s ST (t) denote the shear and Stoneley arrivals respectively. Finally, w l (t) models the additive receiver noise and interference process at receiver l 1. It is useful to keep in mind the following equivalent (frequency-wavenumber) f k domain signal model, (also known as the sum of exponentials model of [7], with wavenumber defined as k = pf): Y l (f ) = S c (f )e i2πp c f (z l z 1 ) + S sh (f )e i2πp sh f (z l z 1 ) + S St e i2πp Stf (z l z 1 ) +W l (f ) (2) 1 We ignore the modal dispersion and attenuation, regarding them as being negligible in the processing band of interest Proceedings of Meetings on Acoustics, Vol. 19, (2013) Page 2

3 where S c (f ), S sh (f )ands St (f ) denote the Fourier transform of the respective time domain signals and W l (f ) denotes the Fourier transform of the noise. LWD noise - The LWD noise W l (f ) can be studied by analyzing data from noise only acquisitions. Figure 1 shows slowness semblance logs from STC processing as well as from a slowness frequency analysis of noise only waveforms encompassing both positive and negative slownesses. It is seen that the LWD noise includes, in addition to an additive background noise process, time-persistent forward- and backward-propagating components with modal content and slownesses similar to the data generated by the monopole excitation at the same depths. Similar to the monopole excitation, this LWD interference is dominated by a Stoneley-type propagative process. Apart from the Stoneleytype interference, another major interfering component on the weaker compressional signal in fast formations is the shear-type interference. Both these types of inteferences can be seen in STC logs of the noise waveforms as shown by the plots in Figure 1. Overall signal model - We state the overall model in the frequency wavenumber domain, but the reader should bear in mind the space-time properties of the modes and the LWD noise as stated above, namely that the modes are time compact while the interferences and noises are not: Y 1 (f ) W add 1 (f ) Y 2 (f ) S c (f ) W add 2 (f ). = [v c (f ),v sh (f ),v st (f )] S sh (f ) + S. st (f ) }{{} Y L (f ) S(f ) }{{} Y(f ).. W add (f ) L }{{} W add (f ) + v D+ (f )S D+ (f ) + v n D (f )S D (f ) }{{} W sys (f ) where v m (f ) = [e i2πf p m(z l z 1 ),...,e i2πf p m(z L z 1 ) ] T is the exponential vector for mode m with slowness p m. Here D + and D denote, respectively, the propagation vectors for the LWD forward- and backward-propagating interference components. Note that according to the observation about the systematic LWD noise v D+ (f ) = [v St (f ) v sh (f )] and similarly for v D+ (f ). As pointed out above, the additive noise and reflections can be well handled by shrinkage in the DRT domain as shown in [3]. In this paper, we focus on dealing with mitigation of forwardpropagating and inter-modal interferences for robust slowness estimates. In particular, we present interference cancellation strategies for STC enhancement and then combine them with the de-noising algorithm of [3] for significant STC enhancement of head waves. COMBATING STONELEY AND SHEAR LWD INTERFERENCES We begin by first outlining a framework for acoustic data representation using space-time propagators. Representation of modes using space-time propagators - To construct a space time propagator, we begin with a waveform, say ψ z 0 τ (t), at a reference receiver, z 0, with time concentration (or support), T, around a central time location, τ, and with effective frequency concentration (or support) in the band F. There can be several choices of this waveform, ψ, depending on the scenario at hand, which we will outline later. Then, given a dispersion curve specifying the wavenumber as a function of frequency, k(f ), and a fixed τ, we propagate ψ z 0 τ (t) with dispersion k(f ) as follows: φ zl (t) = ψ z 0 τ (f )e j2πk(f )(z l z 0 ) e j2πft df (4) (3) where φ zl (t) represents the propagated waveform at receiver location z l. complex in general for attenuative modes. Note that k(f ) can be Proceedings of Meetings on Acoustics, Vol. 19, (2013) Page 3

4 (a) (b) (c) (d) FIGURE 1: Figure showing the characteristics of LWD monopole noise. (a) Shows the STC log of a monopole firing showing the monopole content at each depth, compressional 70 80μs/ft, shear μs/ft and Stoneley 220μs/ft. (b) Shows the STC log of pure noise acquisitions at the same depths. Note the dominating Stoneley and shear type interferences. (c) Shows the STC log concentrating only on the back propagating waves. Note that the back propagating waves also have a content similar to a monopole firing. (d) Shows the slowness frequency projection (SFA) vs depth as described in [12]. The projection clearly indicates that significant energy is propagative at slownesses around the shear and Stoneley. Proceedings of Meetings on Acoustics, Vol. 19, (2013) Page 4

5 Definition 3.1 A space-time propagator with signature waveform ψ, and central time location τ at the reference receiver z 0 propagating with dispersion k(f ) is given by the collection of propagated waveforms as: π z0 (τ, k(f )) = [ φ z0 (τ, k(f ));φ z1 (τ, k(f ));... ;φ zl (τ, k(f )) ] (5) Since the waveform ψ has an effective frequency concentration F, the propagator in the above definition is to be interpreted as approximating the dispersion k(f ) in band F. This interpretation is not critical for non-dispersive (and non-attenuating) space-time propagators at a given slowness p (say) that can be constructed using the above construction by enforcing k(f ) = pf, f. Ψ τ Dispersion k(f), f F Ψ φ φ zm (τ,k( f )) φ z (τ,k( f )) φ z0 (τ,k( f )) FIGURE 2: A space-time propagator π z0 (τ, k(f )) using the Morlet wavelet as the time-frequency compact waveform at reference receiver 1. An example of a waveform ψ(t) and the corresponding propagator is shown in Figure 2. There are several choices of the time-frequency compact waveform ψ depending on the application and the available information about the spectral content of the data. For a time-sampled system we enlist several obvious choices here: (a) Morlet wavelets, [8] (b) prolate spheroidal wave functions (PSWF), [9] and (c) waveforms with coefficients equal to the finite-impulse response (FIR) filter coefficients, where the FIR filter is designed for the pass band F. If the data is pre-filtered using an FIR filter prior to processing, then it makes sense to use the corresponding coefficients to construct the space-time propagators. Indeed the choice of ψ may be optimized given the spectral and time constraints on the system and many algorithms exist in the literature to achieve such an optimization. Given an appropriate choice of ψ, let us denote the collection of π z0 (τ, k(f )) of equation (5) over a given time support T as follows: Π z0 (T, k(f )) = {π z0 (τ, k(f ))} τ T, f F (6) Mode representation - We say that a mode S with a given dispersion k(f ) in a given band F has a time-compact representation over T using space-time propagators in Π z0 (T, k(f )) if, S = τ π z0 (τ, k(f ))x τ,z0, f F,τ T (7) where x τ,z0 can be identified with the "mode coefficients" in the given representation at the reference receiver. Clearly for a finite time-sampled system, as assumed in this work, x τ,z0,τ T {1,2,..., N} can be obtained by solving the system of equations Ψx τ,z0 = s z0 (τ) where Ψ = [ψ z0 (1),ψ z0 (2),...,ψ z0 (N)] is the matrix formed of shifted versions of ψ(t) at the reference receiver. If s z0 (t) is strictly time compact and ψ(t) is also strictly time compact then the coefficients x τ,z0 will also be strictly time compact. Also one can verify that if s z0 (τ) is approximately time compact in the sense that the signal envelope decays rapidly to zero around a peak value, then the coefficients x τ,z0 will also have the same property. It is important to bear in mind that the time support T of the signature waveform ψ is different from the time support T for the modal representation and may be chosen almost independently. In [10] such an approach was used for dispersion extraction, i.e. for estimation of Proceedings of Meetings on Acoustics, Vol. 19, (2013) Page 5

6 k(f ), in disjoint frequency bands where in each band, the data were modeled as superposition of such broadband space-time propagators parameterized by k(f ) k(f 0 ) + k (f )(f f 0 ), f, f 0 F. In this work, we use this construction for interference cancellation when an estimate of k(f ) for the interfering mode is already provided. Interference cancellation using space-time propagators - Interference cancellation using space-time propagators consists of robustly estimating the interfering mode and then subtracting it from the data. To this end, assume that we are given an estimate k(f ) of the dispersion of the interfering mode in the processing band F. Note that for the monopole firing with non-dispersive modes k( f ) = ˆpf where ˆp is the estimated slowness for the stronger interfering mode. Using this estimate we construct a dictionary of space-time propagators Π z0 (T, p) for an estimated time support T and for a set of moveouts p P = { ˆp JΔp,.., ˆp,..., ˆp + JΔp} around the estimated moveout ˆp and for an appropriately chosen value of Δp. Denote this dictionary by D, i.e., D = [ Π z0 (T, p 1 ),Π z0 (T, p 2 ),...,Π z0 (T, p 2J+1 ) ], p j P (8) Then the estimate ŜI of the interfering mode can be obtained by finding a regularized solution to the system of equations, Y = D X T,P + W (9) where the coefficients X T,P can be thought of as representing a local DRT around the slownesses and time locations in P and T, respectively, and W represents additive noise and interference. Therefore the idea is to reliably estimate these local DRT coefficients in the presence of noise and then subsequently use these estimates to estimate and subtract out the interference. For a robust estimation of the local DRT coefficients, we employ two commonly used regularization methods: Tikhonov regularization and truncated pseudo inverse solution. See [11] for details of these methods. The Tikhonov regularization solution consists of the following steps: ˆX T,P = argmin X T,P Y D X T,P 2 + λ X T,P 2 (10) Ŝ I = D ˆX T,P (11) for an appropriate choice of the regularization parameter λ. For implementation purposes, it is useful to note the following closed-form solution to the regularization problem. Let [UΣV ] = SVD(D) denote the singular value decomposition (SVD) of D, then ˆX T,P = UΣ(Σ Σ + λi) 1 Σ U Y (12) It is clear that the computation of the right singular vectors V is not required, thereby saving computations. On the other hand, the truncated pseudo inverse solution consists of Ŝ I = U1 ΣT 0U Y (13) where Σ T = T(Σ,Γ), where T(,Γ) is a truncation operation that truncates the elements of the matrix (in the argument) to zero if the absolute value falls below Γ and 1 denotes the indicator function. Choice of regularization parameter - Of course the above regularization techniques require the choice of the regularization parameters. There are many methods by which one can choose these parameters, e.g., see [11]. The relation between the two regularization methods can be understood in terms of the following mappings. For Tikhonov regularization the overall operation amounts to Proceedings of Meetings on Acoustics, Vol. 19, (2013) Page 6

7 1 dampening the projection on the vector U i by a factor of, whereas the truncated pseudo inverse 1+ λ σ 2 i solution truncates the corresponding projection for σ i < Γ and does not dampen otherwise. Joint versus successive interference cancellation (SIC) - Following the development described above, to cancel the Stoneley and shear interferences, we construct two dictionaries D St and D sh around the estimated Stoneley and shear move-outs and the time locations at which to cancel them (so as to improve the compressional processing). Then we form a system of equations: ] XTst,P Y = [D st D sh ][ st + W (14) X Tsh,P sh We want to solve the following: ˆX T,P = argmin X T,P Y [D st D sh ] [ XTst,P st X Tsh,P sh ] 2 + λ 1 X Tst,P st 2 + λ 2 X Tsh,P sh 2 (15) Unfortunately for the joint estimation there remains a computational bottleneck. After some basic algebra (which we omit here due to lack of space), we see that there is a need to recompute ( ) 1 ( 1, Σ st U st P U D sh,λ 2 st Σ st + λ 1 I and Σ sh U sh P U D st,λ 1 sh Σ sh + λ 2 I) where [Ust Σ st V st ] = SVD(D st ) and [U sh Σ sh V sh ] = SVD(D sh ) and P = U D st,λ 1 st (I Σ st (Σ st Σ st+λ 1 I) 1 Σ st )U st and P = U D sh,λ 2 sh (I Σ sh (Σ sh Σ sh+ λ 2 I) 1 Σ sh )U for each depth, which is computationally expensive. To remove this computational bottleneck, we propose to use successive interference cancelation (SIC) wherein we exploit the slowness sh separation between shear and Stoneley in fast formations to successively estimate and remove the Stoneley interference followed by the shear. We first estimate the Stoneley interference using, D st ˆX Tst,P st = U st Σ st ( Σ st Σ st + λ 1 I ) 1 Σ st U st Y (16) followed by estimation of the shear interference on the residual as, ( D sh ˆX Tsh,P sh = U sh Σ sh Σ sh Σ sh + λ 2 I ) 1 ( ) Σ sh U sh Y Dst ˆX Tst,P st (17) ρ FIGURE 3: Algorithm for processing the Monopole LWD waveforms. which is then subtracted from the data for overall LWD interference suppression. Note that here we can pre-compute ( Σ st Σ st + λ 1 I ) 1 ( and Σ sh Σ sh + λ 2 I ) 1 while the left singular vectors are obtained using a fast time shift operation. It turns out that using SIC instead of a joint estimation and cancellation yields acceptable solutions with reasonable computational costs. It is also possible to iterate this, i.e. solve for the Stoneley interference again after subtracting out the shear interference analogously to Equation 17 and repeat to refine the estimates of the interferences. In practice, due to the slowness separation between shear and Stoneley, this successive approach can approximate the performance of the joint estimation and cancellation even after one or Proceedings of Meetings on Acoustics, Vol. 19, (2013) Page 7

8 (a) (b) (c) (d) FIGURE 4: Figure showing the performance of the proposed method(s) on a real data set from a fast formation (a) Shows the STC log of the waveforms with the standard processing. Note the poor detection (coverage) of the compressional mode. (b) Shows the STC log after Stoneley interference removal followed by applying the shrinkage based de-nosing. Note the increase in semblance of both shear and compressional. (c) Shows the STC log after Stoneley + shear interference cancelation. Note the improvement in coverage for the compressional log. (d) Shows the STC log after Stoneley + shear LWD interference cancelation and de-noising. Note further improvement in the compressional semblance as well as coverage compared to (b) & (c). Proceedings of Meetings on Acoustics, Vol. 19, (2013) Page 8

9 two iterations, with a considerable saving of cost over that of the joint estimation. Figure 3 presents a high level view of the algorithm to process a LWD monopole frame where we have combined the shrinkage-based de-noising algorithm of [3] with the successive interference cancellation (SIC) approach presented in this paper for robust detection of head waves. Performance on real data - Figure 4 shows the performance on a real LWD data set from a deviated well in a fast formation. The monopole data is slightly more complicated than usual as the tool is passing through a boundary with two different rock matrices with different compressional and shear slownesses. Note the clear performance improvements obtained by applying the proposed algorithms. Not only is the compressional as well as shear log coverage dramatically improved compared to standard processing results, but in fact the complexity of this section of the dataset covering a region close to a bed boundary is highlighted with the enhancement of all the arrivals, which is invaluable for proper interpretation. REFERENCES [1] C. V. Kimball and T. L. Marzetta, Semblance processing of borehole acoustic array data, Geophysics 49, (1984). [2] T. Kinoshita, T. Endo, H. Nakajima, H.Yamamoto, A. Dumont, and A. Hawthorne, Next generation lwd sonic tool, in The 14th Formation Evaluation Symposium of Japan (2008). [3] S. Aeron, S. Bose, and H.-P. Valero, Robust detection of waek acoustic signals in noise using near optimal shrinkage in the radon domains, in IEEE Statistical Signal Processing workshop (2012). [4] E. J. Candés, Modern statistical estimation via oracle inequalities, Acta Numerica 1 69 (2006). [5] G. Beylkin, Discrete radon transform, IEEE Transactions on Acoustics, Speech and Signal Processing ASSP-35, (1987). [6] S. B. Bose, S. Aeron, and H.-P. Valero, Joint multi-mode dispersion extraction in fourier and space time domains, Submitted to Geophysics Journal International (GJI), under review [7] S. Lang, A. Kurkjian, J. McClellan, C. Morris, and T. Parks, Estimating slowness dispersion from arrays of sonic logging waveforms, Geophysics 52, (1987). [8] I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics) (1992), isbn [9] C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, CA) (1957). [10] S. Aeron, S. Bose, H.-P. Valero, and V. Saligrama, Automatic dispersion extraction of multiple time overlapped acoustic signals, (2010), URL Patent number: WO/2010/ [11] P. C. Hansen, Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion (Society for Industrial and Applied Mathematics, Philadelphia, PA, USA) (1998). [12] T. Plona and S.-K. Chang, Slowness-frequency projection display and animation, (Published 2010), patent Number - US B2. Proceedings of Meetings on Acoustics, Vol. 19, (2013) Page 9