Application Of Maximum Likelihood Method To Reservoir Predition With Seismic Data

Currently,most works on reservoir prediction with themethod of pattern recognition in seismic prospecting are donewith the use of multi-component linear regression method orthe method of artificial neural networks. To explore moreefficient method, the authors of this article apply themaximum likelihood method discriminating function toreservoir prediction in seismic prospecting. Since there is alittle work using the maximum likelihood methoddiscriminating function, the study is still lack of experience.Since one can make better classification and may extendprediction basing multi-well with the use of such method, andhence make the reservoir prediction more accurate. Based onthe above reasons, the authors of this article started thisstudy.According to the requirement for the study, this articleperforms the following study:⒈Summarizing and discussing the maximum likelihoodmethod discriminating function and the forms ofdiscriminating functions of various normal distributionparameters which include linear and nonlinear discriminatingfunctions;also discussing the estimates to some relatedparameters.Discriminant functions of parameters of differentnormal distributions are given as follows:a. If probability density function of pattern recognition ismultivariate normal density functionp XXMiTiXMiimi( |ω i )= ( 2π)n/1 2Σ1/2exp[?12(?)Σ?1 (?)],=1,2,...,,(1)where M i, Σ i are expected vectors and covariace matrix ofclass ω i,Σ i is determinant of Σ i,Σ ?1 is inverse matrix of Σ ,then discriminant function is()()]2exp[1(2)( )(/ 2)1/2iTi1iiniDi X= π Pω Σ?X?MΣ?X?M, (2)or ()()2ln12ln21D i ( X)= lnP(ω i)?n2π?Σi?X?MiTΣi?1 X?Mi, (3)ignoring unrelated constants of class, we still write it asDi (X)()()2ln12Di ( X)= lnP(ω i)?1Σi?X?MiTΣi?1 X?Mi, (4)which is a quadratic function of X,so it is a nonlineardiscriminant function。BecauseiiTiiiTiTiiTiiiTiTiiTiiTiXXXMMMXMXMXXMXXMMM111111112()()????????=Σ?Σ+Σ? Σ?=Σ?Σ?Σ+Σ, (5)So (4) can be written asiiTiiiTiTD i ( X)= lnP(ω i)?12lnΣi?12XΣ?1 X+XΣ?1M?12MΣ?1M, (6)When covariace matrixes of all classes are equal,Σ 1 = Σ2==Σm=Σ,the second term and the third term in (6) is unrelated to classand ignored,we still denote by Di (X),theniT1iiT1D i (X)= lnp(ωi)+XΣ? M?21MΣ?M, (7)which is a linear discriminant function。When prior probabilities of all classes are equal,P(ω1 )= P(ω2)==P(ωm)=P(ω ),discriminant function can be written asiTiiTDi ( X)= XΣ?1 M?12MΣ?1M, (8)i.e.if X T Σ ?1 Mi?12 MiTΣ?1Mi≥XTΣ?1Mj?12MjTΣ?1Mj,j=1,2,,m, (9)then X ∈ ωib. Parameter estimationLet n dimensional space,m pattern class,Nk(k=1,2,…,m)is sample number of each class pattern, trained datas aredescribed in the following table。sample of the kth class patternParameter estimation of mean vector is∑=∧=NKlklkk XM N1 1 (), k=1,2,…,m,let component forms of M∧ k, X l(k) isM? k = (M?1(k)M?n(k))T,covariace matrix of class isvariablesample1 2 … … nNk21()()()()2()22()21()1()12()1112kNkNkNknkkknkkkkknxxxxxxxxx∑Σ== ??=×NklnnkijTkkklklk? k N1 1 [(X( )M?)(X()M?)](σ 2()),whereXMXMijnNkjkljNlkiklikkijk1 ()(()()),,1,2,,12 ()= ∑()?()?==σ .Estimation of covariace matrix isNm[(XM)(XM)]Σmk1Nl1Tk(k)kl(k)lk???= ∑∑= =??? ,or,1i,jnNm(xm)(xm)σmk1Nl1(k)j(k)lj(k)i(k)li2ijk?≤≤??= ∑∑= =??? ,Where j, k are numbers of column and row of eachcomponent of Σ? , i.e. σ? ik。⒉ Stating the current situation of applying seismicattributes to perform reservoir prediction and the fundamentalprinciple for performing reservoir prediction with the use ofmaximum likelihood method discriminating classifications. Onthe other hand, the article discusses the abstract of seismicattributes and efficiency analyses, which are also importantcontents in seismic attribute techniques.⒊Compiling the program to perform reservoir predictionwith the use of maximum likelihood method and uploading theprogram to the Landmark seismic data interpretation system.At the same time, the authors perform classifying analysis to3 given groups of experimenting data for verifying theaccuracy of the program;⒋Performing study on practical experimental data bychoosing a seismic area at the north of Songliao Basin.According to the data from 3 prospecting wells in this area,the authors apply the program to perform classifyingprediction analysis to deep layer gas and obtain the results ofclassifying prediction with the use of nonlinear and linearmaximum likelihood discriminating functions respectively.By above study, the main results obtained are thefollowing:⒈In predicting reservoir classification with seismic data,the authors introduce successfully nonlinear maximumlikelihood method discriminating function which has beenproved to be applicable in practically and theoretically.⒉Abstracting 3 classes with a total of 23 seismicattributes parameters from seismic wave energy, frequencyand fractal geometry information etc. laid solid foundation forapplying reservoir prediction with maximum likelihoodmethod.⒊Compiling computer program with C language forreservoir prediction with maximum likelihood method. Makingcomputer program module in Landmark seismic datainterpretation system for predicting reservoir with the methodwhich are convenient for the use by individuals and raised theworking efficiency and the scope of application of the datafrom seismic and log data.⒋Known data indicate that the classification accuracy ofnonlinear maximum likelihood method discriminating functionis better than that of linear maximum likelihood methoddiscriminating function.⒌The practical data show that it is feasible to predictreservoir with the use of maximum likelihood method and theaccuracy is better, but there is no obvious difference for theresults of classifying predictions with the two discriminatingfunctions.Practical results as follows:D ash1D ash2D ash 3Dash1Dash3Dash2Figure 5-1 Predicted results using nonlinear maximumLikelihood method discriminating functionFigure 5-2 Predicted results using linear maximumLikelihood method discriminating functionBye the study, one may see that, the reservoir predictiontechnique with maximum likelihood method is good forextending classifying prediction basing multi-well in seismicareas, which is one of its advantage, but such technique issemi-quantity prediction technique, which is one of itsdisadvantage. In practical use, it may have better effect if onecombines such technique with other reservoir predictiontechniques.