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IEEE GEOSCIENCE AND REMOTE SENSING LETTERS 1

Optimal Seismic Reflectivity Inversion: Data-Driven�p-Loss-�q-Regularization Sparse RegressionFangyu Li , Rui Xie, Student Member, IEEE, Wen-Zhan Song, Senior Member, IEEE, Hui Chen

Abstract— Seismic reflectivity inversion is widely applied toimprove the seismic resolution to obtain detailed undergroundunderstandings. Based on the convolution model, seismic inver-sion removes the wavelet effect by solving an optimizationproblem. Taking advantage of the sparsity property, the �1-normis commonly adopted in the regularization terms to overcomethe noise/interference vulnerability observed in the � p-lossesminimization. However, no one has provided a deterministicconclusion that �1-norm regularization is the best choice forseismic reflectivity inversion. Instead of using an unproved fixedregularization norm, we propose an optimal seismic reflectivityinversion approach. Our method adaptively adopts an � p-loss-�q -regularization (i.e., � p,q -regularization) for p = 2, 0 < q < 1to estimate a more accurate and detailed reflectivity profile.In addition, we employ a K -fold cross-validation (CV)-basedapproach to obtain the optimal damping factor λ to furtherimprove the seismic inversion results. The letter starts with theintroduction of nonconvex constraint for seismic inversion and thenecessity of the �q-norm regularization. Then, the majorization-minimization and CV algorithms are briefly described. Theperformance of the proposed seismic inversion approach isevaluated through synthetic examples and a field example fromthe Bohai Bay Basin, China.

Index Terms— cross-validation (CV), damping factor,� p,q -regularization, optimal regularization, seismic reflectivityinversion.

I. INTRODUCTION

RESOLUTION is an unavoidable topic in seismic inter-pretation, which is the ultimate goal for all seismic

processing steps. High seismic resolution helps to characterizethe correlation between the geological structures and geophys-ical images, which is important for the reservoir depositionanalysis, especially thin-bed reservoirs [1].

Based on the seismic convolution model, Chen andWang [2] improved the seismic resolution via a waveletscaling method. Likewise, Zhou et al. [3] extended the seismicspectrum using a nonstationary wavelet estimation. A seismic

Manuscript received August 6, 2018; revised October 28, 2018; acceptedNovember 8, 2018. This work was supported in part by NSF under Grant NSF-CNS-1066391, Grant NSF-CNS-0914371, Grant NSF-CPS-1135814, GrantNSF-CDI-1125165, and Grant NSF-DMS-1222718, in part by NIH underGrant NIH-R01GM122080 and Grant NIH-R01GM113242, and in part bySouthern Company. (Corresponding author: Fangyu Li.)

F. Li, R. Xie, and W.-Z. Song are with the Center for Cyber-Physical Systems, University of Georgia, Athens, GA 30602 USA (e-mail:(fangyu.li@uga.edu; ruixie@uga.edu; wsong@uga.edu).

H. Chen is with the Geomathematics Key Laboratory of SichuanProvince, Chengdu University of Technology, Chengdu 610059, China(e-mail: huichencdut@cdut.edu.cn).

Color versions of one or more of the figures in this letter are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/LGRS.2018.2881102

reflectivity obtained from seismic amplitude inversion has beenused in the applications with little/weak well controls [4].For regularized inversion, besides the �2-norm constraint,�1-norm regularization is more popular in sparse spike inver-sion [5], [6]. Theoretically, �0-norm regularization should alsobe a good sparsity measurement, but the actual seismic signalsdo not always show 0 values, so finding exact p nonzeroelements in signal, i.e., �x�0 = p is not a computationallyfeasible and proper constraint in the real world [7]. Signalsparsity properties show different highlights [8], so a data-driven sparsity measurement is more suitable for the seismicreflectivity inversion problem [9], [10]. The convergence of�q (0 < q < 1)-norm regularization has been proven by [11]and its advantages were also demonstrated. There are somewell-established algorithms to solve the nonconvex or con-cave optimization problem related to �q -regularized inver-sion, for examples, cyclic descent algorithm [12], reweighted�1 minimization [13], iteratively reweighted algorithms [14],and so on.

However, to our best knowledge, the sparsity regulariza-tion selection for seismic reflectivity inversion has not beengenerally discussed. Although there are discussions with thedamping factor λ selection, such as [10], they are only relatedto either �1- or �2-norm. In the previous work [15], a �q -norm(0 < q < 1) regularized optimization has been applied to theseismic reflectivity inversion. Compared to [15], this letterproposes a general data-driven seismic reflectivity inversionapproach. We extend �q -norm (0 < q < 1) to a �p-loss-�q -regularization ( �p,q regularization) framework, whichallows us to consider a wide range of loss functions. Thepossible choices of loss functions include the �p-losses andprediction losses [11]. In this letter, we study the �2-loss asan illustration. Here, we also explicitly define the way to obtainthe optimal q , where the majorize-minimization (MM) algo-rithm is described. In addition, K -fold cross-validation (CV)is adopted for generality. We establish an adaptive seismicinversion approach including the optimal regularization termselection as well as the adaptive damping factor determination.We introduce the theories and algorithms first. Then, the effec-tiveness of the proposed method is proven through syntheticexamples and a field application from Bohai Bay Basin,China.

II. THEORY

A. Seismic Convolution Model

After removing the undesirable interferences, seismic datacan be modeled as the convolution between the source wavelet

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2 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS

w(t) and reflectivity series r(t) in the form as [10], [15], [16]

s(t) =∫ ∞−∞

w(τ)r(t − τ )dτ + �(t) (1)

where �(t) is an additive random noise. Using a convolutionsymbol, we can rewrite it as

s(t) = w(t) ∗ r(t)+ �(t). (2)

Then, a discrete matrix format realized at samplest = 1, 2, . . . , M can also be formulated [17]

s =Wr + � (3)

where s = [s1, . . . , sM ]T is the observation, W is the waveletkernel matrix which can be estimated from the seismic powerspectrum [16], and r = [r1, . . . , rM ]T is the reflectivity series.

B. Seismic Inversion

The seismic convolution model forms a linear relationshipbetween the recorded seismic data and reflectivity series.The solution of (3) is nonunique, which necessitates certainconstraint to achieve stable. In general, the objective functioncan be formulated as

φ(q, λ) = ||s−Wr||pp + λ�r�q (4)

where ||s −Wr||pp = ∑Mt=1 |s(t) − w(t) ∗ r(t)|pp and �r�q =

(∑M

j=1 |r j |q)1/q (q indicates the constraint condition), and λis the damping factor [12]. The common choice for the lossfunction is the �2-loss, i.e., p = 2.

C. �q-Norm Regularization

Seismic inversion usually adopts either �1- or �2-normregularization. This nonconvex constraint is based on thecommon assumption that the reflectivity series r(t) is sparse.However, accurately identifying the sparsity of the signal isnot easy.

The �0-norm, which measures the number of nonzero of r:�r�0 = {r j �= 0, j = 1, . . . , N}, is not put into consideration.It asks to find the “best subset” among all possible sparsitycandidates that make it impractical for seismic reflectivity dueto intractable computational cost and difficulty in choosingnumber of nonzero elements. To loosen the constraint, the�1-norm regularization, i.e., Lasso regression [18], is easilycalculated with sparsity constraint. Frank and Friedman [19]built the connection between �0- and �2-regularizations.Because the convex �q -regularization with q > 1 does not holdthe sparsity property, which is necessary for seismic inversion,we need to pick an optimal q from (0, 1). As shown in Fig. 1,different regularization norms produce various constraint onsparsity.

In order to obtain the optimal choice of q , for a given λ,we can formulate the objective function as

q̂ = arg minq∈Q

φ(q, λ)⇒ q̂ = arg minq∈Q

φ(q) (5)

where Q is a predefined nonempty set containing possible qvalues. This problem can be solved through a coordinate-wiseoptimization method.

Fig. 1. Geometric interpretation of different regularizations. (Left)�1-regularization (blue) and �2-regularization (red). (Right) �q -regularizationwith q = 0.5 (green). The ellipses are contours of the �p -loss errors of thetoy signal r = [r1, r2]T .

Algorithm 1 Majorization-Minimization Algorithm1: Initialize q0 ∈ Q and set k = 02: repeat3: Majorization step:4: Construct g(·|qk) satisfying the upperbound

property: g(q|qk) ≥ φ(q)+ ck , ∀q ∈ Q5: Minimization step:6: Update x as qk+1 = arg min

q∈Qg(q|qk)

7: k ← k + 18: until certain convergence criterion is met

D. Majorize-Minimization Algorithm

We propose to use the MM algorithm [20] to solvethe nonconvex optimization problem in equation (4) with0 < q < 1. As an iterative optimization method, the MMalgorithm consists of two steps in one iteration. In the firstmajorization step, a surrogate function g(·|qk) is employed tolocally approximate the objective function with their differenceminimized at the current point. In other words, the surrogateupperbounds the objective function up to a constant ck . Then,in the minimization step, the surrogate function is mini-mized. The sequence φ(qk) is nonincreasing since φ(qk+1) ≤g(qk+1|qk) − ck ≤ g(qk|qk) − ck = φ(qk). The procedure isshown in Algorithm 1. The computational complexity of theMM algorithm is O(M3).

E. Damping Factor Selection

The damping factor λ in (4) controls the sparsity of therecovered signal. We propose to use the CV approach [21] tochoose the optimal value of the damping factor, which worksfor parameter selection according to its statistical performance.CV provides the optimal bias-variance tradeoff under thesetting of (4). The K -fold CV is adopted roughly as follows.First, randomly split the data into K equal clusters. Second,the kth cluster (k = 1, 2, . . . , K ) is used for testing theoptimization model fit using the data from other cluster. Third,the CV prediction error for each λ is calculated. Then, repeatthe random selection. In the end, combine all the K prediction

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LI et al.: OPTIMAL SEISMIC REFLECTIVITY INVERSION 3

TABLE I

SIMILARITY BETWEEN THE GROUND TRUTH AND INVERTEDREFLECTIVITY PROFILE FROM 100 TIMES MONTE CARLO

EXPERIMENTS OF EVERY SITUATION. (“STD” STANDS

FOR STANDARD DEVIATION)

errors together to obtain the CV estimate

CV (λ) = 1

K

K∑k=1

(sk −Wk r̂(−k)(λ))2 (6)

leading to the chosen optimal λ̂ = arg minλ∈[a,b]CV (λ),where [a, b], a, b ∈ R+ is the given range of λ.

III. EXPERIMENTS

A. Synthetic Example

We first test the proposed method in numerical experi-ments. Compared with synthetic experiments in [10] and [15],we designed the sparse reflectivity series randomly insteadof using fixed positions, which makes more sense for thenonstationary seismic applications.

Table I shows the comparisons of the similarity measure-ment between ground truth and inverted reflectivity withdifferent SNRs. �2-, �1-, and �p,q -regularizations are applied toinvert the random reflectivity profile, respectively. SNR variesfrom 20 to −10 dB, which covers possible SNR situations offield applications, and Monte Carlo experiments are conducted100 times for every different situation. Note that the pro-posed �p,q method achieves the highest inversion accuraciesat different noise levels with relatively stable performances.Specifically, when the data have a low SNR, the proposedregularization constraint produces the most robust results.

To visualize the results, Fig. 2(a) shows one of syn-thetic Monte Carlo reflectivity models consisting of randomlylocated, sparse reflectivity series. Based on the convolutionmodel, we obtain the seismic data by convolving the reflectiv-ity profile with a 30-Hz Ricker wavelet. The synthetic seismictrace is shown in Fig. 2(b). Fig. 2(c) displays a noisy seismictrace with 10-dB random noise. Fig. 2(d)–(f) demonstratesthe inverted results from the noisy seismic data based onthe regularizations of �2-, �1-, and proposed �q -norms. Ourproposed �q method and �1-norm successfully recovered thesparsity of the reflectivity, where �2-norm failed to do so.Compared to �1-norm, the proposed �q method captured moredetailed information, such as small-amplitude picks, succes-sive picks, than �1-norm method. Due to the high nonconvexityof the �q -regularization, our proposed method distinguishes the

Fig. 2. Synthetic example. (a) Random reflectivity. (b) Synthetic seismictrace. (c) Noisy seismic trace with 10-dB noise. (d)–(f) Inverted reflectivityfrom �2-norm, �1-norm, and the proposed �q method, respectively.

Fig. 3. Field seismic data from Bohai Bay Basin, China.

true nonzero reflectivity components from noises by projectedobserved data onto a �q -ball. By searching the �q -ball usingthe MM algorithm, the true reflectivity can be efficientlyrecovered from the noisy data. The results demonstrated that,under the noise contamination, the �q method outperforms thepopular seismic inversion techniques such as �1- or �2-normregularized inversion.

B. Field Application

We apply the proposed method to seismic field data from theBohai Bay Basin, China. The target reservoir in this area is thesiliciclastic reservoir, where the thin-bed structure develops,so the high-resolution interpretation, especially the reflectivityinversion, is of great importance.

We display the original seismic amplitude in Fig. 3.Fig. 4 shows the inverse result of �2-norm regularizationwith fivefold CV. We see that it fails to remove the noise

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4 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS

Fig. 4. Seismic inversion using �2-norm.

Fig. 5. Seismic inversion using �1-norm.

Fig. 6. Seismic inversion using �q -norm.

from observations or recover the sparse reflectivity series.Compared to �2 method, �1 and �q methods shown in Figs. 5–7provide more sparse recovered reflectivity series. The optimal�q inverse result with q = 0.7 is shown in Fig. 6. We see thatour proposed �q method recovered more detailed informationwhen comparing to the l1 method, and more thin beds canbe characterized now. Note that the damping factors for bothmethods are chosen using fivefold CV.

To demonstrate the effectiveness of choice of dampingfactors, we compare the optimal �q inversion (Fig. 6) withan over regularized �q inversion (Fig. 7). We see that a proper

Fig. 7. Seismic inversion using �q -norm without optimal regularizationparameter selection.

Fig. 8. Comparison among the reflectivities calculated from the sonic logs,the inverted reflectivity based on �1- and �q -norms. (The trace number is 209.)

regularization, i.e., finding the optimal damping factor usingCV, will help keeping the continuity of the inversion resultwhile maintaining the sparsity.

To further evaluate the inverted results, we compare the�1- and �q -norms inversion reflectivity profiles with the reflec-tivity calculated from the sonic logs, shown in Fig. 8. Theresults from the proposed �q -norm inversion show more detailsand have better correspondence with the reflectivity from welllogs. In addition, we also calculate the relative similarity coef-ficient, ρ(·, Refllog)/ρ(Seismic, Refllog), between the invertedreflectivity and the sonic reflectivity with ρ(·) as the crosscorrelation function, which are 0.5843 with the �q -norm and0.4347 with the �1-norm. Therefore, from both qualitative andquantitative comparisons, the proposed method is better thanthe existing methods.

IV. CONCLUSION

We propose a novel sparse reflectivity inversion methodwith �p,q-norm regularization and optimal damping factorselection. The nonconvex constraints have drawn a lot ofattention because it satisfies the mathematical assumptions ofseismic reflectivity series. Illustrating the case of p = 2,we adaptively adopt the �q -norm (0 < q < 1) constraint.In addition, because the estimates of q and λ value are fullydata driven, we believe the inverted results are relativelyoptimal. The synthetic examples verify that the proposedmethod can eliminate the effects of wavelet effectively and

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LI et al.: OPTIMAL SEISMIC REFLECTIVITY INVERSION 5

can obtain the stable reflectivity series, even with moderaterandom noises. Furthermore, the derived reflectivity in thefield example shows high correspondence with the reflectivitycalculated from well logs.

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