Journal of Applied Geophysics 120 (2015) 60–68

⁎

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Journal of Applied Geophysics

j ourna l homepage: www.e lsev ie r .com/ locate / j appgeo

Support Vector Machine (SVM) based prestack AVO inversion andits applications

Guangcai Li, Jiachun You, Xuewei Liu ⁎School of Geophysics and Information Technology, China University of Geosciences (Beijing), Beijing 100083, China

Corresponding author.E-mail addresses: lgcgood111@163.com (G. Li), liuxw@cu

http://dx.doi.org/10.1016/j.jappgeo.2015.06.0090926-9851/© 2015 Elsevier B.V. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 25 December 2014Received in revised form 24 June 2015Accepted 25 June 2015Available online 2 July 2015

Keywords:AVO inversionSupport Vector Machine (SVM)Bayesian method

AVO inversion can be used to estimate P-wave velocity, S-wave velocity, and density perturbations from reflec-tion seismic data. The inversion of the density term, however, due to its little sensitivity to amplitudes and thepaucity of large angle incident information, is usually difficult and unstable. The conventional method of linear-ized approximation is usually not accurate enough and tends to be affected by the background information, whilethe accurate method is more likely to be trapped in a local minimum and more computationally intensive. Thispaper delineates a novel method of AVO inversion based on Support Vector Machine (SVM). First, we describethe basic principle of SVM, and thenwe investigate an SVMprocedure for the three-termAVO inversion problem.To demonstrate its performance, we compare it with the conventional Bayesian method. From the inversion re-sults of both the synthetic and real data, we conclude that the algorithm of SVM leads to high-resolution P-wavevelocity, S-wave velocity, and density perturbation, moreover, the resolution of the density term has largeimprovement, compared to the Bayesian method. They all demonstrate the feasibility and application of SVMon both synthetic and real data.

© 2015 Elsevier B.V. All rights reserved.

1. Introduction

The objective of seismic AVO (Amplitude variation with offset) in-version is to achieve the physical and elastic properties of subsurfacelayers using the prestack gathers. Since the occurrence of AVO analysisproposed by Ostrander (1984), it has received considerable attentionin the field of seismic exploration (Aki and Richards, 1980; Smith andGidlow, 1987; Swan, 1993; Simmons and Backus, 1996; Buland andOmre, 2003; Riedel et al., 2003; Downton, 2005; Rabben et al., 2008;Theune et al., 2010; Alemie and Sacchi, 2011; Zhang et al., 2013; Zonget al., 2013).

Due to the band-limited nature and the noise levels of the seismicdata, it is known that the AVO inversion problem is ill-posed and unsta-ble for most common acquisition geometries. To constrain and stabilizethe AVO inversion problem, however, some scholars have endeavoredto adopt different prior probabilistic constraints following a Bayesianframework. Based on the classical work on geophysical probabilisticinverse theory (e.g., Tarantola, 1987; Menke, 1989), they employ theBayesian statistical method to formulate the AVO inversion problemsuccessfully (Gouveia, 1996; Sen and Stoffa, 1996; Gouveia and Scales,1998; Buland and Omre, 2003; Riedel et al., 2003; Downton, 2005;Rabben et al., 2008; Karimi et al., 2010; Theune et al., 2010; Alemieand Sacchi, 2011; Zong et al., 2013). In the literature listed above,some are using the linearized approximations (see Aki and Richards,

gb.edu.cn (X. Liu).

1980; Shuey, 1985; Smith and Gidlow, 1987; Gray et al., 1999; Ursinand Dahl, 1992; Wang, 1999; Ursenbach, 2002) and some areemploying the accurate Zeoppritz equations (Simmons and Backus,1996). Through various parameterizations and approximations of theaccurate Zeoppritz equations, they are popular in the AVO inversionproblems. However, both kinds of methods are unavoidably to facetheir own shortcomings. Themethod of using linearized approximationis usually not accurate enough and easy to be affected by the back-ground information, while the accurate method is more likely to betrapped in a local minimum and more computationally expensive. Inaddition, due to the paucity of large angle incident information andthe different sensitivity for the reflection coefficients to the bulk densityterm, usually the estimation of density is difficult (Tarantola, 1986;Beydoun and Mendes, 1989; Crase et al., 1990; Nicolao et al., 1993;Swan, 1993; Chen et al., 2001; Cambois, 2001; Lebrun et al., 2001;Djikpesse et al., 2006; Choi et al., 2008; Virieux and Operto, 2009;Zong et al., 2013).

Alternatively, we present a novel algorithm called Support VectorMachine (SVM) to address the AVO inversion problem. SVMwas intro-duced by Vapnik (1995) in the middle of 1990s. Derived from the ideaof statistical theory, SVM is especially effective to tackle nonlinearproblems with a small number of samples. In addition, known as anattractive tool for the classification and regression problems, SVMtechniques possess strong generalization ability. Since then, SVM hasdeveloped quickly and become popular in variable fields including geo-physics. Li and Castagna (2004) successfully adopt the SVM to the clas-sification of AVO attributes including intercept and gradient. However,

61G. Li et al. / Journal of Applied Geophysics 120 (2015) 60–68

SVMs can also be used for regression problems. Kuzma (2003) employsthe SVM to the application of AVO interpretation and performs a nonlin-ear AVO inversion (Kuzma and Rector, 2004). Also, we quantitativelycompare the inversion results and analyze the errors of the SVM algo-rithm with the classical Bayesian method utilized by Downton (2005)and give our viewpoints of the reasons behind them. It demonstratesthe feasibility and application of SVM, especially for the estimation ofdensity term,which has a great improvement in the accuracy. Addition-ally the SVM used as a nonlinear estimator is more robust than a least-squares estimator because it is insensitive to small changes.

The rest of this paper is organized as follows. Section 2 introducesthe theory of SVM; Section 3 gives a simulation of the AVO forwardmodeling, delineates the AVO inversion formulation based on SVMand compares the inversion results; Section 4 applies the SVM to thereal marine seismic data and then in Section 5 the discussion andconclusions are drawn.

2. Theory of Support Vector Machine (SVM)

The SVM theory has been described in a considerable literature(Vapnik, 1995; Burges, 1998). Here we just make a brief summary ofthe basic idea of SVM.

In distinction to the classical neural networks, the formulation of thelearning problem of SVM leads to the quadratic programming withlinear constraints. Based on the structured risk minimization (SRM)principle, SVMs seek to minimize an upper bound of the generalizationerror instead of the empirical error, thus the novel prediction modeleffectively avoids the over-fitting problem. In addition, the SVMmodelswork in the high dimensional feature space formed by the nonlinearmapping of the N-dimensional input vector x into a K-dimensionalfeature space (K N N) through the use of the nonlinear function φ(x).The SVM regression function is formulated as follows:

y xð Þ ¼XKj¼1

wjφ j xð Þ þ b ¼ wTφ xð Þ þ b ð1Þ

1400 1600 18000

100

200

300

400

500

600

700

800

Tim

e(m

s)

2000

100

200

300

400

500

600

700

800

(a) (

Fig. 1. (a) P-wave velocity, (b) S-wave velocity, and (c) de

where w = [w1, w2,…wK]T is the weight vector, b the bias and φ(x) =[φ1(x), φ2(x),…,φK(x)]T the bias function vector. The coefficients w andb are estimated by minimizing

R Cð Þ ¼ C1l

Xli¼1

Lε di; yið Þ þ 12

wk k2 ð2Þ

where

Lε d; yð Þ ¼ 0; d− y xð Þj j b εd−yj j−ε; d−y xð Þj j≥ε

�ð3Þ

where l is the number of samples, both C and ε are prescribed parame-ters, and C is called the penalty function,which determines the trade-offbetween the empirical risk and the smoothness of the model, while ε isthe error function which determines themargin within which the erroris neglected. The term Lε(d,y) is called the ε-intensive loss function. di isthe actual value. This function indicates that errors below ε are not

penalized. The term C 1l ∑

l

i¼1Lεðdi; yiÞ is the empirical error. The second

term, 12 kwk2 measures the smoothness of the function.The solution of the so defined optimization problem is solved by the

introduction of the Lagrangian function and the Lagrangemultipliers αi,αi′ (i = 1, 2,…, l) responsible for the functional constraints defined byEq. (2). The minimization of the Lagrangian function has been trans-formed to the so-called dual problem:

maxXli¼1

di αi−α 0i

� �−εXli¼1

αi−α 0i

� �−

12

Xli¼1

Xlj¼1

αi−α 0i

� �α j−α 0

j

� �K xi; xj� �8<

:9=;

ð4Þ

400 600 1.6 1.8 20

100

200

300

400

500

600

700

800

b) (c)

nsity. These data are used to test the novel algorithm.

0 10 20 30 40

100

200

300

400

500

600

700

angle(degree)

time(

ms)

0 10 20 30 40

100

200

300

400

500

600

700

angle(degree)

time(

ms)

(a) (b)

Fig. 2. Synthetic data (a) without noise and (b) with S/N = 5.

62 G. Li et al. / Journal of Applied Geophysics 120 (2015) 60–68

at the constraints

0 ≤ αi ≤ C;0 ≤ α0i ≤ CXp

i¼1

αi−α0i

� � ¼ 0

8><>: ð5Þ

where K(xi,xj)=φT(xi)φ(xj) is an inner-product kernel defined in accor-dance with the Mercer's theorem (Vapnik, 1998) for the learning dataset x.

-0.05 0 0.05

100

200

300

400

500

600

700

800

Tim

e(m

s)

-0.02 0 0.02 -0.5 0 0.5 -

(a) (b) (c)

Fig. 3. Inversion using SVM without noise. (a) The line in black denotes the inverted P-wave rinverted S-wave reflectivity, and the line in red indicates the true value; (e) The line in blackThe lines in cyan in (b), (d), and (f) are corresponding to the errors of P-wave reflectivity, S-wavmeanings in Figs. 4–6.(For interpretation of the references to color in this figure legend, the re

After solving the dual problemall weights are expressed through theNsv nonzero Lagrange multipliers ai, αi′ and the same number of learn-ing vectors xi associated with them. The network output signal y(x)can be then expressed in the form:

y xð Þ ¼XNsv

i¼1

αi−α0i

� �K x; xið Þ þ b ð6Þ

where b is a bias term.

0.2 0 0.2 -0.1 0 0.1 -0.02 0 0.02

(d) (e) (f)

eflectivity, and the line in blue indicates the true value; (c) The line in black denotes thedenotes the inverted density reflectivity, and the line in green indicates the true value.e reflectivity, and density reflectivity, respectively. They represent the same correspondingader is referred to the web version of this article.)

(a) (b) (c) (d) (e) (f)

-0.05 0 0.05

100

200

300

400

500

600

700

800

Tim

e(m

s)

-0.02 0 0.02 -0.5 0 0.5 -0.2 0 0.2 -0.1 0 0.1 -0.02 0 0.02

Fig. 4. Inversion using SVM with S/N = 5. The meanings of the lines are the same as in Fig. 3.

63G. Li et al. / Journal of Applied Geophysics 120 (2015) 60–68

Schölkopf et al. (1997) concluded that Radial Basis Function (RBF)kernel performs better when comparedwith other kernels such as line-ar, polynomial, sigmoid or spline, thus the RBF kernel function is select-ed in this paper. Mathematically, RBF Kernel function is expressed byEq. (7)

K x; xið Þ ¼ exp −xi−xk kγ2

2 !

ð7Þ

where γ is the kernel width.

(a) (b) (c)

-0.05 0 0.05

0

100

200

300

400

500

600

700

800

Tim

e(m

s)

-0.02 0 0.02 -0.5 0 0.5 -

Fig. 5. Inversion using the Bayesian method without nois

3. Simulation

3.1. AVO forward modeling

AVO forward modeling is first to calculate the exact P–P reflectioncoefficients using the Zeoppritz equations as

sin a cos β − sin a0

cos β0

cos a − sin β cos a0

sin β0

sin 2avp1vs1

cos 2βvp1vp2

v2s2v2s1

ρ2

ρ1sin 2a

0−

ρ2

ρ1

vp1vs2v2s1

cos 2β0

cos 2β −vs1vp1

sin 2β −ρ2

ρ1

vp2vp1

cos 2β0

−ρ2

ρ1

vs2vp1

sin 2β0

266664

377775

Rpp

Rps

Tpp

Tps

2664

3775 ¼

− sin acos asin 2a

− cos 2β

2664

3775

ð8Þ

(d) (e) (f)

0.2 0 0.2 -0.1 0 0.1 -0.02 0 0.02

e. The meanings of the lines are the same as in Fig. 3.

-0.05 0 0.05

0

100

200

300

400

500

600

700

800

Tim

e(m

s)

-0.02 0 0.02 -0.5 0 0.5 -0.2 0 0.2 -0.1 0 0.1 -0.02 0 0.02

(a) (b) (c) (d) (e) (f)

Fig. 6. Inversion using the Bayesian method with S/N = 5. The meanings of the lines are the same as in Fig. 3.

64 G. Li et al. / Journal of Applied Geophysics 120 (2015) 60–68

where a, β, a' and β' are defined in terms of the ray trajectories, reflec-tion and transmission angles of P-wave and S-wave, respectively. vp1,vs1, ρ1, vp2, vs2and ρ2 are the P-wave velocity, S-wave velocity and den-sity for the upper and lower media of the interface. Rpp and Rps are thereflection coefficients of P-wave and S-wave, and Tpp and Tpsdenotethe transmission coefficients of P-wave and S-wave.

Next we construct the time series reflectivity, i.e., the P-wave, S-wave, and density reflectivity derived from the true P-wave velocity,S-wave velocity, and density shown in Fig. 1. Note that here the modelparameters of interest are the P-wave, S-wave, and density reflectivity(readers are referred to Downton, 2005; Li, 2006; Rabben et al., 2008;Alemie and Sacchi, 2011; Zhang et al., 2013) given as:

Δvpvp

¼ vp2−vp112

vp2 þ vp1� � ;Δvsvs

¼ vs2−vs112

vs2 þ vs1ð Þ;Δρρ

¼ ρ2−ρ112

ρ2 þ ρ1ð Þ: ð9Þ

Also note that these true model parameters can be partiallyemployed for training as the output arguments.

To simulate the real case, then the synthetic AVO data are generatedby convolving the series of the reflection coefficients with a 40-HzRicker wavelet, and here the series of the reflection coefficients arecalculated using the accurate Zeoppritz equations displayed in Eq. (8).The generated synthetic data in the entire time window are shown inFig. 2a (free of noise) and Fig. 2b (with signal-to-noise-ratio [S/N] = 5).The sampling interval is 2 ms and the total number of samples is 376.The data consist of 40 traces with angles ranging from 1 to 40°. Strongevents caused by large contrasts in elastic parameters can be observedaround 410 ms.

Table 1The correlation coefficients between the inverted value and the true model with the SVMand Bayesian method for the data with and without noise, respectively.

Parameters SVM Bayesian

Free of noise S/N = 5 Free of noise S/N = 5

P-wave reflectivity 99.9% 93.2% 90.0% 72.0%S-wave reflectivity 95.2% 82.9% 82.1% 48.8%Density reflectivity 99.6% 91.6% 90.1% 78.9%

3.2. AVO inversion formulation based on SVM

According to the basic theory of SVM depicted in Section 2, (xi,di)(i = 1,2,…,l) denote the input/output training data pairs, wherexi ∈ ℜN, as the input arguments, they represent the known reflectedamplitudes, viz., the modeling seismic AVO data, and di ∈ ℜ, while asthe output arguments, they denote the three parameters of interest,i.e., the aforementioned P-wave, S-wave, and density reflectivity.Basically, via these training data pairs, our aim is to build the mappingrelationship between di and xi:

f : ℜN→ℜdi ¼ f xið Þ i ¼ 1;2;…; lð Þ: ð10Þ

Here 250 sample pairs are selected as the training data, as the angleranging from 1 to 40° in themodeling, so here the input arguments xi ofthe samples are assumed to have 40 features or elements. To acquirethe suitable hyperparameters, we set ε ∈ [0.001,1], C ∈ [0,1000], γ ∈(0,100) by experience, and these parameters are optimized with 5-fold cross validation (Picard and Cook, 1984). That means the data setis randomly divided into 5 disjoint subsets of equal size, and each subsetis used once as a validation set, while the other 4 subsets are put togeth-er to form a training set. In the simplest case, the average accuracy of the5 validation sets is used as an estimator for the accuracy of the method.Themethods often used to determine suitable hyperparameters includecross validation, random search, grid search, heuristic search, genetic al-gorithms, particle swarm optimization, pattern search, etc. The opti-mum values of ε, C, and γ are found to be 0.01002, 88.2094 and6.4039, respectively.

Table 2The mean absolute error (MAE) between the inverted value and the true model with theSVM and Bayesian method for the data with and without noise, respectively.

Parameters SVM Bayesian

Free of noise S/N = 5 Free of noise S/N = 5

P-wave reflectivity (×103) 0.22 1.54 1.76 3.01S-wave reflectivity (×103) 7.22 14.38 14.62 21.46Density reflectivity (×103) 0.40 2.21 2.38 3.60

Fig. 7.Migrated seismic profile across Site SH2.

65G. Li et al. / Journal of Applied Geophysics 120 (2015) 60–68

Subsequently the SVM model is trained using the training data togenerate the AVO inversion formulation as follows.

y xð Þ ¼XNsv

i¼1

αi−α0i

� �K x; xið Þ þ b

This is the regression express of Eq. (6). It is available while gettingthe nonzero Lagrange multipliers αi, αi′ and the corresponding supportvectors including the bias b. Finally, as the objective output arguments,the three parameters of interest expressed in Eq. (9)will be acquired viathe above formulation.

Fig. 8. The partial stacked

3.3. Results for the synthetic data

Fig. 3 shows the inversion results and the corresponding errors usingSVM for the data without noise. In Fig. 3a, c and e, the lines in blackdenote the inverted P-wave reflectivity, S-wave reflectivity, anddensity reflectivity, respectively, while the lines in color indicate thecorresponding true values; the lines in cyan in Fig. 3b, d, and f are corre-sponding to the errors of P-wave reflectivity, S-wave reflectivity, anddensity reflectivity, respectively.

Fig. 4 shows the inversion results and the corresponding errors usingSVM for the data with S/N = 5.

To further assess the performance of the SVMalgorithm, the Bayesianmethod is assigned and implemented for comparison. The forward

gather at CDP1095.

66 G. Li et al. / Journal of Applied Geophysics 120 (2015) 60–68

model used in the Bayesian inversion is identical to SVM to ensure thatwe adopt the same “observational data” to implement the inversionthus for the convenience of comparison.

Figs. 5, and 6 give the inversion results and the corresponding errorsusing the Bayesianmethod (Downton, 2005) for the data without noiseand with S/N = 5, respectively. Here the a priori distribution for themodel parameters is assumed to yield univariate Cauchy distributionand the random noise is Gaussian distributed.

Table 1 lists the correlation coefficients of the results between theinverted value and the true model with the SVM and Bayesian methodfor the data with and without noise, respectively. To further quantify

(a)

(c)

1400 1600 1800 2000 2200 2400

1750

1800

1850

1900

1950

2000

ms

0 50 100 150 200

1750

1800

1850

1900

1950

2000

ms

Fig. 9.The inverted results of P-wave velocity (a) anddensity (b) and their corresponding absolu(b) denote the well log data. In all panels, the red lines denote the results of SVMmethod, whiinterpretation of the references to color in this figure legend, the reader is referred to the web

the effect of the inversion methods, a mean absolute error (MAE) isalso calculated. Table 2 gives the MAE between the inverted value andthe truemodel with the above twomethods for the data with andwith-out noise, respectively.

The mean absolute error (MAE) is defined as follows:

MAE ¼ ∑n

i¼1

mtrue−minvertedj jn

: ð11Þ

Comparing the correlation coefficients and theMAE listed in Table 1and Table 2, it is not difficult to find that, the resultant estimation of the

(b)

(d)

1.6 1.8 2 2.2 2.4

1750

1800

1850

1900

1950

2000

ms

0 0.1 0.2 0.3 0.4

1750

1800

1850

1900

1950

2000

ms

te inversion errors of P-wave velocity (c) anddensity (d). The neutral black lines in (a) andle the blue ones indicate the results adopting the Bayesian method (Downton, 2005). (Forversion of this article.)

67G. Li et al. / Journal of Applied Geophysics 120 (2015) 60–68

model parameters is influenced by the random noise for both methods.While either for the clean or the noise contaminated data, the SVMmethod shows a great improvement in the accuracy of the three param-eters, especially for the S-wave reflectivity and the density terms.

4. Real data example

The field data set is from Guangzhou Marine Geological Survey 01(GMGS-01) in Shenhu area, the Pearl River Mouth (PRM) Basin, north-ern South China Sea. Fig. 7 shows a migrated seismic profile across SiteSH2 and a clear BSR (Bottom Simulating Reflector) can be easily identi-fied. Above the BSR, usually a typical gas hydrate-bearing formationaccompanies with high P-wave velocity. The black line denotes theposition of Site SH2.

After the regular AVO processing, including a band-pass filter, aspherical divergence correction, directivity correction and NMO correc-tion, we select the CDP gather at 1095 near SH2 to perform the inver-sion. The depth of water is about 1230 m and the base of the gashydrate stability zone 229 mbsf. The maximum incident angle is about42°. Fig. 8 shows the partial stacked seismic data. For the convenienceof comparison with the well data, the inversion is runwith a short win-dowof data starting at 1761ms and ending at 1989ms. Thenwe chooseto use a trace integratingmethod (for example, see Zhang et al., 2013) tocalculate the absolute valueswith the low frequency trend derived fromin situ petrophysical measurements. Unfortunately, due to the lack of S-wave velocity well log data, it is only available for the comparison of theP-wave velocity and density. Fig. 9a and b shows the inverted result ofP-wave velocity and density, respectively. Fig. 9c and d displays the cor-responding absolute inversion errors of P-wave velocity and density. Ascan be seen, the results of the twomethods are similar in awhole, whilein some areas depicted in the two green ellipses in Fig. 9a and b, the P-wave velocity and density acquired based on the SVMaremore accurateand stable than the ones using the Bayesian method. This also can beeasily observed and compared from the two error plots. To give a furtherquantitative analysis, the mean squared error (MSE), MAE and correla-tion coefficients (CC) of the SVM and Bayesian methods are listed inTable 3.

5. Discussion and conclusions

It is appropriate at this point to compare the approach describedabove to other recent work on AVO inversion approaches. Comparedwith the Bayesian method, it illuminates that the algorithm of SVMoutperforms the conventional Bayesian method, especially for theestimation of the density term, which has a great improvement in theaccuracy. We guess it is mainly derived from the following two aspects.On the one hand, SVM is based on the structured risk minimization,which acts like a regularization to release the serious nonuniquenessof the AVO inversion problem and stabilize it effectively, On the otherhand, for the Bayesianmethod, some local linearization unavoidably de-viates from the true geological and rock physical environment which isdifficult to know a priori, and this is likely to influence the resolution ofthe result. The above two examples demonstrate the feasibility andapplication of SVM.

For simplicity, however, this paper does not consider the effect ofNMO stretch for the synthetic case. We have not compared SVM withthe Bayesianmethod using the full Zoeppritz equations in the inversion

Table 3The mean squared error (MSE), MAE and correlation coefficients (CC) of the SVM andBayesian methods.

SVM Bayesian

MSE MAE CC MSE MAE CC

Vp 40.32 33.57 95.31% 48.65 35.37 92.58%Density 0.041 0.032 84.32% 0.057 0.039 67.72%

process yet, as considering the convenience of computation for theparameterization.

Additionally, one of the open and tough questions that remains ishow to set the reasonable and accurate hyperparameters of an SVMalgorithm, such as the penalty factor C and any parameters specifyingthe kernel function (for example the width of an RBF kernel), which isusually considered as the key point of a successful SVM application;however, this is beyond the scope of this paper and it can be thoughtas the model selection problem for SVMs. We will investigate thistopic in future work.

Acknowledgments

This work was financially supported by the International Science &Technology Cooperation Program of China (2010DFA21630) andthe National Basic Research Program of China (973 Program,2009CB219505). We would like to thank the science team of the gashydrate program expedition Guangzhou Marine Geological Survey-1(GMGS-1).

We also thank the reviewers and the editor for their helpful com-ments that largely improved the quality of this manuscript.

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