Tu P4 15 Enhancement of Microseismic Events Using ... - Earthdoc

Tu P4 15 Enhancement of Microseismic Events Using Tensor Decomposition and Time-frequency Representation N. Iqbal* (King Fahd University of Petroleum & Minerals), E. Liu (Georgia Institute of Technology), J.H. McClellan (Georgia Institute of Technology), A. Al-Shuhail (King Fahd University of Petroleum & Minerals), S.I. Kaka (King Fahd University of Petroleu

Summary Analysis of passive microseismic data is usually a challenging task due to low signal-to-noise ratio environment. This study introduces an approach for enhancing the microseismic events using tensor decomposition and timefrequency representation. The proposed method shows promising results when applied on microseismic data set.

79th EAGE Conference & Exhibition 2017 Paris, France, 12-15 June 2017

Introduction Microseismic events induced during hydraulic fracturing are characterized by small magnitudes. Furthermore, microseismic data is noisy, especially if sensors are located at the surface due to overwhelming surface waves. These noisy events may result in inaccurate detection and estimation. While this field is still evolving, better methods continue to emerge to enhance the signal-to-noise ratio (SNR). Rank reduction is a popular approach for denoising (Iqbal et al., 2016). The basic idea behind this approach is that properly sampled seismic data, in the absence of noise, is low rank. Additive noise increases the rank of the seismic data matrix. Hence, denoising can easily be implemented using rank reduction. Recently, rank reduction techniques are applied on tensors to solve the multi-dimensional (3D or higher) data completion problem (Kreimer and Sacchi, 2011). Tensors are used to represent high dimensional data and to extract useful information from high dimensions rather than from the 2D matrix (De Lathauwer et al., 2000; Kolda and Bader, 2009; Bergqvist and Larsson, 2010). Tensors are extensively used for data completion of higher dimension (3D, 4D and 5D) active seismic data (see (Ely et al., 2013) and references therein). In this paper, we propose a method for enhancing microseismic events using tensor decomposition. We consider microseismic data as a tensor of order 3. Microseismic data is collected by an array of geophones and interpreted as 2D data (time and trace number), unlike active seismic data. Microseismic data has one trace per receiver (2D), which motivates us to transform it to 3D by moving into the timefrequency (T-F) domain with the hope that the desired denoising results can be achieved by using SVD on a tensor of order 3. Here, we will use the short time Fourier transform (STFT) (Benesty et al., 2008; Boashash, 2003; Dutoit and Marques, 2009) to transform a trace to the T-F domain, since it is easy to compute and invertible. The three dimensions of this tensor represent frequency, temporal, and spatial information. The reason that we expect good performance with this approach is that the desired signal will be recovered from a 3D tensor (more information) instead of a 2D matrix (less information). Furthermore, SVDs are applied on various components of a tensor which gives better denoising result, this is in contrast to the method of applying SVD on the entire 2D microseismic data set. In this study, we use an SVD-like tensor decomposition, tSVD (Kilmer and Martin, 2011) for denoising, since it retains the orientation information and it is useful for time-series applications.

Theory and Method The STFT is computed by applying fast Fourier transform (FFT) on a subset of data points N. This subset of data is selected using a window w = [w0 , w1 , ∙ ∙ ∙ , wl−1 ]. First , the FFT is computed for data points of length l, then the window is moved h data points and again the FFT is calculated. Thus, l − h is the window overlap. This procedure is repeated until the window covers the last l data points. The time series is recovered from the STFT by applying an inverse FFT (IFFT) at each time instant and then the overlap-add method is used to get the final time-domain data. After taking the STFT of each trace in the data set, we express the data set as a 3rd -order tensor, i.e., X ∈ CI1 ×I2 ×I3 , where I1 , I2 , and I3 denote the total number of time samples, frequency samples, and traces, respectively. The tensor data structure is shown schematically in Figure 1, which depicts a pictorial view of the 3rd -order tensor representing the T-F representation of three traces. The system model in terms of tensor terminology is represented as follows: Y = X +N

(1)

where X corresponds to a low-rank signal component and N to the noise component of tensor Y . Each frontal slice of Y is the T-F representation of the time-series trace on a certain sensor. A conventional way for denoising (via low-rank approximation) is to solve the following optimization problem: min kX k∗

subject to kY − X k2F ≤ δ

(2)

where k ∙ k∗ and k ∙ kF represent the nuclear norm and Frobenius norm, respectively, of a tensor. For a matrix k ∙ k∗ is the sum of its singular values, whereas, the nuclear norm of a tensor is sum of the nuclear 79th EAGE Conference & Exhibition 2017 Paris, France, 12-15 June 2017

Figure 1 An example of a 3rd -order tensor, representing time-frequency representation of three traces norms of the 2nd -order tensors (i.e., a matrices) on which the SVD is applied. Candès and Recht (2009) demonstrated that low-rank matrices are perfectly recovered by solving the nuclear norm minimization problem. Moreover, Cai et al. (2010) showed that the low-rank approximation of a matrix using nuclear norm minimization together with data fidelity with respect to the Frobenius norm can easily be solved using soft-thresholding on singular values of the matrix.

Denoising using tSVD tSVD is based on two operations: t-product and t-transpose (Kilmer and Martin, 2011). t-product is a multiplication of two tensors, and t-transpose is a transposition operation for a tensor. These are extensions of matrix multiplication and transposition. In the denoising operation of the microseimic event using tSVD, first the 3rd -order tensor Yˆ is obtained by applying FFT on the third dimension of Y which can be written in a MATLAB-like syntax as Yˆ = fft(Y , [ ], 3).

(3)

Next, the SVD of each frontal slice of Yˆ is computed ˆ ::i = U ˆ ::i Sˆ ::i V ˆH Y ::i

for i = 1, . . . , I3

(4)

ˆ ::i is the ith frontal slice of Yˆ . The nuclear norm of a tensor A is given as where Y M

kA k∗ = ∑ kA::i k∗ ,

(5)

i=1

rank(A)

where the nuclear norm of a matrix A is defined as kAk∗ = ∑i=1 σi , and rank(A) is the number of nonzero singular values, σi . Hence, minimizing (2) is equivalent to applying soft thresholding (Bachmayr and Schneider, 2016) on the singular values (i.e., shrinking the singular values) as si = (σi − τ )+ ,

(6)

where si is the soft thresholding operator applied to the ith singular value σi and (∙)+ = max{0, (∙)} removes the negative part of (∙) where σi ≤ τ . After applying soft thresholding on Sˆ ::i , we get ˆH ˆ 0::i = U ˆ ::i Sˆ 0::i V X ::i .

(7)

Along the 3rd dimension, the component vectors are converted back to the time-domain via the IFFT ˆ 0 , [ ], 3) U = ifft(Uˆ , [ ], 3), V = ifft(Vˆ , [ ], 3), S 0 = ifft(S

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and finally, X 0 is obtained using t-product (∗) and t-transpose (Kilmer and Martin, 2011) as X 0 = U ∗ S 0 ∗ V H.


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Figure 2 (a) Noiseless microseismic traces, (b) noisy traces with SNR =−10 dB, (c) T-F representation of trace 22, (d) denoised traces using tSVD.

Results The waveform used in this study is obtained from the the Northern California Earthquake Data Center (NCEDC). The sampling time is 4 msec. The waveform is repeated 40 times with a shift that simulates multiple traces recorded at distances far from the source. Noise is added to this data set in order to make SNR=−8.14 dB. This makes it difficult to detect the microseismic event. The noiseless and noisy data sets are shown in Figure 2a and 2b, respectively. To detect this event, trace is transformed to the time-frequency domain and the event can be clearly seen in the low frequency range of Figure 2c. After the detection stage, tSVD is applied for denoising and the near-to-exact matching of denoised traces and noiseless traces can be seen in Figure 2d. One of the important steps in denoising is to shrink the singular values using soft thresholding, which requires a parameter τ . To define the threshold τ , singular values obtained for each frontal slice are averaged and then the rate of change of the Averaged Singular Values (ASV) is taken into account. The threshold is defined based on the fact that the rate of change of the singular values belonging to the noise sub-space is low. The ASV and its rate of change are shown in Figure 3a and 3b, respectively. From Figure 3b, it can be observed that the rate of change is very small above the value of −0.1. Therefore, the threshold lies between the 5th and 6th singular value, which from Figure 3a suggests a threshold value τ of 95 for the data shown in Figure 2.

Conclusions In this study, an enhancement method for microseismic events using tensor decomposition is presented. By considering the T-F representation of the traces as a third-order tensor, the singular value decomposition is applied on the tensor. The denoised traces are obtained by shrinking the singular values. Test on the field data set validate the claim made in the study.

Acknowledgments We appreciate the support of this work by the Center for Energy and Geo Processing (CeGP) at by King Fahd University of Petroleum and Minerals (KFUPM) and Georgia Tech under project number GTEC1311. 79th EAGE Conference & Exhibition 2017 Paris, France, 12-15 June 2017

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Figure 3 (a) Singular values averaged over all frontal slices (ASV), (b) rate of change of averaged singular values (derivative of ASV).

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