Seismic trace interpolation with approximate message passing Navid Ghadermarzy and Felix Herrmann and Özgür Yılmaz, University of British Columbia

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1 Seismic trace interpolation with approximate message passing Navid Ghadermarzy and Felix Herrmann and Özgür Yılmaz, University of British Colmbia SUMMARY Approximate message passing (AMP) is a comptationally effective algorithm for recovering high dimensional signals from a few compressed measrements. In this paper we se AMP to solve the seismic trace interpolation problem. We also show that we can exploit the fast AMP algorithm to improve the recovery reslts of seismic trace interpolation in crvelet domain, both in terms of convergence speed and recovery performance by sing AMP in Forier domain as a preprocessor for the ` recovery in Crvelet domain. INTRODUCTION Reglar sampling along the time axis is common approach for seismic imaging. However, seismic data are often spatially ndersampled de to economical reasons as well as grond srface limitations. Too large spatial sampling interval may reslt in spatial aliasing. Operational conditions might also reslt is noisy traces or even gaps in coverage and irreglar sampling which often leads to image artifacts. Seismic trace interpolation aims to interpolate missing traces in an otherwise reglarly sampled seismic data to a complete reglar grid. Compressed sensing (CS) is a data acqisition techniqe to efficiently acqiring and recovering high-dimensional signals from seemingly incomplete linear measrements, provided that the signal is sparse or compressible in some transform domain (Donoho (006); Candès et al. (006)). Several works have shown that the strctre of seismic wavefronts can be captred by a few significant transform coefficients (e.g., crvelet or Forier coefficients) and the problem of seismic trace interpolation can be cast as a sparse recovery problem once the data is transformed into a sitable domain, e.g., discrete Forier transform (DFT) (Sacchi and Ulrych (996); Li and Sacchi (004)), Radon transform (Thorson and Claerbot (985); Hampson et al. (986); Herrmann et al. (000)), or crvelet transform (Hennenfent and Herrmann (005, 006, 008); Naghizadeh et al. (00)). Hence, the interpolation problem becomes that of finding transform domain coefficients with the smallest ` norm that satisfy the sbsampled measrements. In this paper we first consider sing crvelet transform as the sparsifying transform (see, e.g., Hennenfent and Herrmann (006)). Crvelet transform is a redndant transform which expands the size of the model and has been shown to be very effective in captring the strctre of the data with very few significant coefficients. Then, we recover the reslting compressible signal by solving a convex minimization problem. Next we consider sing Forier transform as the transform operator. The reslting transform is not as effective as crvelet transform (in terms of sparsity of the reslting transform). However, the transformed data can still be approximated by a solving a sparse recovery problem which is mch faster than the crvelet case. This is de to two reasons. First one is the availability of very fast DFT transforms and second is the applicability of approximate message passing (AMP) for this problem. AMP (Donoho et al. (009)) is a fast (needs few matrix-vector mltiplications) iterative ` recovery algorithm which can be sed to recover sparse Forier coefficients (The crrent variants of AMP can not recover crvelet coefficients). Also notice that AMP can be sed to speed p interpolation techniqes that work on small high dimensional windows. In this paper we se AMP with Forier transform coefficients to generate relatively good approximations of the seismic data which is sed to enhance the speed and recovery performance of seismic trace interpolation in the crvelet domain. First we formlate the problem of seismic trace interpolation as a sparse recovery problem. Next we explain the approach we take to solve this problem both from the crvelet transform and Forier transform coefficients and then we present the -stage algorithm which combines these two approaches. PROBLEM FORMULATION Consider a seismic line with N s sorces located on earth srface which send sond waves into the earth and N r receivers record the reflection in N t time samples. Rearranging the seismic line, we have a signal f R N, where N = N sn rn t. Assme x = Sf where x is the sparse representation of f in some transform domain. We want to recover a very high dimensional seismic data volme x by interpolating between a smaller nmber of measrements b = RMf, where RM is a sampling operator composed of the prodct of a restriction matrix R, which specifies the sample locations that have data and a measrement basis matrix M. To overcome the problem of high dimensionality, recovery is performed by first partitioning the seismic data volmes into freqency slices and then solving a seqence of individal sbproblems (Mansor et al. (03)). For each partition j, let b (j) be the sbsampled measrements of data f (j) in partition j and the corresponding measrement matrix is given by A (j) = R (j) S H (Notice that the sperscript H denoted the Hermitian transpose and for simplicity, we have as-

2 smed the measrement matrix M to be the identity). Then for each partition we need to estimate the soltion of the following minimization problem independently: ex (j) := arg min kk 0 s.t. A (j) = b (j) ; () and then f (j) is approximated by S H ex (j). Recovery in crvelet domain with convex optimization In this case S C C P N is the crvelet transform which is sed as the sparsifying transform S. For each partition we se the -D crvelet transform where N P 8 which is highly redndant and allows for sparse representations of the data in the transformed domain. Notice that in this case A (j) = R (j) SC H. The minimization problem () is a combinatorial problem and therefore we approximate the soltion by the following convex relaxation: ex (j) ` := arg min kk s.t. A (j) = b (j) ; () and then f (j) is approximated by SC H ex (j). Here kk ` = NP j i j and () can be efficiently solved by algorithms i= like SPGL (Van Den Berg and Friedlander (007)). Mansor et al. (03) noted that althogh partitioning the data sing and -D crvelets redces the size of the problem, it doesn t tilize the continity across partitions. Therefore they sggest sing spport set of each partition as a spport estimate for the next one. For partition j > assme T j := spp(s C SC H ex jk) which is the locations of the largest k entries of S C SC H ex in magnitde and define the weight vector w R N sch that wj = for j = T j and wj =! < for j T j, where k is chosen to be the nmber of largest transform-domain coefficients that contribte 95% of the signal energy. Mansor et al. (0) prove that solving the following weighted ` minimization reslts in better recovery compared to () if the spport estimate T j is at least 50% accrate: ex (j) := argmin kk ;wj s.t. A (j) = b (j) ; (3) w` where kk ;wj = NP wij i j is the weighted ` norm of. i= Recovery in Forier domain with AMP In this case we se DFT matrix S F C N N as the sparsifying operator. Hence, A (j) = R (j) S F is the sbsampled -D DFT matrix. To solve () the AMP algorithm (Donoho et al. (009)) starts from an initial x 0 amp and an initial threshold ^ 0 = and iteratively goes by x t+ amp = (x t amp + A (j)h z t ; t ); z t = y A (j) x t amp + z t h 0 (x t amp + A (j)h z t ; t )i; (4) and ex (j) tf inal amp = xamp where t final is the nmber of times we iterate the algorithm. Here SF H xt amp is the crrent estimate of f (j), z t is the crrent residal, h i is the arithmetic mean, and is the soft thresholding fnction [(a;s)]j = sign(a j )(a j s) +. This fnction acts on the coefficients of the signal a and zeroes ot any vale which is less than the scalar s in magnitde. Similar to the previos case we can tilize the continity of seismic data across partitions to estimate the spport set of each partition. As before T j := spp(s F SF H x jk) and the weight vector w is defined sch that wj = for j = T j and wj =! < for j T j. Ghadermarzy and Yilmaz (03) incorporate this prior spport information into the AMP algorithm by the following iterations: x t+ wamp = (x t wamp + A (j)h z t ; t w j ); z t = y A (j) x t wamp + z t h 0 (x t wamp + A (j)h z t ; t w j )i: (5) The only difference between AMP and weighted AMP (WAMP) is in the vale of the thresholds which is set to be a smaller vale for the coefficients in the spport estimate which makes them more likely to remain nonzero. In Figres.a-f we compare the performance of AMP and WAMP (in Forier domain) with weighted ` (in crvelet domain) on recovering a seismic line from the Glf of Sez with 50% randomly sbsampled receivers. The seismic line at fll resoltion has N s = 78 sorces, N r = 78 receivers, and Nt = time samples acqired with a sampling interval of 4 ms. Conseqently, the seismic line contains samples collected in a s temporal window with a maximm freqency of 5 Hz. To access the freqency slices we take the -D DFT of the data along time axis and interpolate the missing receivers by solving the seismic trace interpolation with weighted `, AMP and weighted AMP algorithms explained above (Reslts of ` minimization is omitted as the weighted ` minimization is always giving better reslts (Mansor et al. (03)). Notice that reconstrction reslts of weighted AMP is better than those of AMP. -STAGE ALGORITHM As mentioned before AMP algorithm is a significantly fast algorithm and Figre shows that althogh the reslts of weighted ` is better than those of weighted AMP, the reslts of weighted AMP is still reasonably good (considering that weighted AMP is abot 40 times faster than the weighted ` algorithm). Weighted ` minimization in the crvelet domain gives mch better reslts which is de to the better choice of sparsifying transform domain. On the other hand AMP is a very fast algorithm that approximates the tre data reasonably good. Notice that both these algorithms are solving the same problem in different transform domains.

3 Weighted L error image in SR 0. Weighted L minimization in SR (a) Recovery by weighted ` in crvelet domain AMP error image in SR (c) Recovery by AMP in Forier domain (d) AMP error Weighted AMP error image in SR Weighted AMP in SR (b) Weighted ` error AMP in SR (e) Recovery by weighted AMP in Forier domain (f) WAMP error WAMP+W L minimization in SR WAMP+W L error image in SR (g) -stage reconstrction (h) -stage error Figre : Comparison of the different recovery methods. Figres show recovered shotgather nmber 84 from sbsampled seismic line from the Glf of Sez. Figres on the left show the reconstrctions and figres on the right show the recovery errors for weighted ` (a, b), AMP (c, d), WAMP (e,f), and the -stage algorithm (g, h). f is approximated by: Therefore, we can combine these algorithms to se both the fast convergence of AMP algorithm and sperior reslts of weighted `. Notice that at each partition j, e e H weighted ` : f SC xw` weighted AM P 3 : f S H F xwamp (6)

4 Hence the otcome of the weighted ` algorithm can be approximated by the final otcome of the weighted AMP algorithm (which is in Forier domain) throgh: S H C ex (j) SF H w` ex (j) wamp: (7) Hence, we propose a -stage algorithm, that for each partition j first we apply the weighted AMP algorithm (5) in the Forier domain to get the estimate SC H ex (j) w`. Next we se the largest coefficients of S C SF H ex (j) wamp as a spport estimate for ex (j) and solve a weighted ` w` algorithm in the crvelet domain. Figre shows an example of the accracy achieved to predict the spport of the crvelet coefficients in a seismic line when we jst se the adjacent freqency slices (the weighted ` case) and when we combine the information of the previos freqency slice with the approximation gained by WAMP. Here the spport is chosen to be the set of largest transform-domain coefficients that contribte 95% of the signal energy. The spport estimate gained by the -stage algorithm is always better than the estimate gained by the previos slice. We also feed the approximation S C SF H ex (j) wamp into the weighted ` algorithm as a good initial point for the SPGL algorithm. As the crvelet transform is a redndant transform, we can not mathematically jstify why this choice of initial point improves the convergence rate. However, empirical reslts show significant improvement when we feed the reslt of WAMP into the ` algorithm. Figres.gh show the recovery reslts by the -stage algorithm. In Table we have compared the nmber of matrix-vector mltiplications and average time it takes for the above algorithms to recover different freqency slices. The - stage algorithm presented ses fewer crvelet transforms compared to the weighted ` which makes it faster. Figre 3 shows the SNRs achieved by `, weighted `, AMP, Weighted AMP, and the -stage algorithms for different shot gathers. The reslts of the -satge algorithm is always better than the weighted `. recovery method Comparison of different methods # of DFT # of crvelet transforms transforms average rntime weighted ` s AMP s WAMP s -stage WAMP+w` s Table : Comparison of approximate average comptational complexity for different recovery methods in Figre averaged over freqency slices. Fraction of accrate detected spport 0. -stage Weighted l Freqency (Hz) Figre : Example of the accracy achieved in estimating the spport of a freqency slice by sing the -stage algorithm and the weighted ` algorithm. SNR Shot gather nmber W L on freq. in SR L on freq. in SR AMP on freq. in SR Figre 3: Comparison of the SNRs achieved by different methods explained above in recovering shot gathers from the Sez seismic line in the sorce-receiver domain. CONCLUSIONS In this abstract we showed how we can se the approximate message passing algorithm for the problem of seismic trace interpolation. De to fast natre of AMP algorithm and the DFT transform we can get a good approximation of the seismic shot gathers mch faster than doing the interpolation in the crvelet domain. Hence, AMP can be sed as a fast preprocessor for interpolation in crvelet domain. This reslt shows that doing this not only improves the recovery reslts significantly and bt also the -stage algorithms need less SPGL iterations (with crvelets) to get to a good approximation once the fast WAMP algorithms is done in the Forier domain. Weighted AMP on freq. in SR Two stage WAMP+Wspgl 4

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