ABSTRACT Mei Han (Master of Science in Petroleum Engineering) Application of EM algorithms for facies classiﬁcation and measurement error estimation Directed by Dr Gaoming Li and Second Member 103 pp., Chapter 5: Conclusions (507 words) Facies identiﬁcation is important to reservoir characterization. Traditionally, facies classiﬁcation is done by inspection and judgment of available seismic, core or log data. The process requires experience and it is time-consuming. This thesis discusses the application of the EM algorithms to automatically identify geological facies from seismic data. An extension on the application of the EM algorithms is to measurement error estimation is also discusses. When seismic data are used for history-matching, measurement error covariance is used as the weight to the data. As this information is usually unknown, it is estimated by ﬁrst dividing the seismic data into relatively smooth regions and then a smoothing algorithm is applied to each region to separate the signal and measurement error. In this way, it avoids smoothing across some natural boundaries (facies boundaries, or ﬂuid content boundaries such WOC or GOC), which show high contrast in seismic data and hence reduce the possibility of overestimating measurement error. The traditional EM algorithm has long been used for data clustering, in which the data assumed to be samples from a Gaussian mixture model. The EM algorithm is designed to iteratively estimate the unknown the Gaussian mixture model parameters (mean, variance and proportionality of each Gaussian model) for given data by i
maximizing a log-likelihood function. In each iteration of the EM algorithm, there are two steps: Expectation-step and Maximization-step. In the E-step, the membership matrix, in which the entries represent the probability that each datum samples from each Gaussian model (group), is estimated. The M-step then calculates the Gaussian mixture model parameters by maximizing the log-likelihood function using the estimated the membership matrix from the E-step. After convergence, the data are divided into groups according the ﬁnal membership matrix. However, the traditional EM algorithm clusters data without considering spatial relationship. When it is applied to seismic data, it may divide data into groups that are spatially discontinuous. This thesis improves the spatial continuity using two spatially constrained EM algorithms: Spatial EM and Neighborhood EM (NEM). Both algorithms enhance the spatial relationship among data in a group by modifying the membership matrix using spatial information. In general, the number of groups in not know a priori. Unlike the traditional EM algorithm, these two EM algorithms consider the uncertainty on the initial number of groups. They start with a large number groups and then gradually delete groups during the iteration and ﬁnally obtain approximately the right number of groups. First the Em algorithms are applied to a synthetic 1-D case, in which there are 500 log-derived porosity and permeability pairs along a well. There are ﬁve layers out of three rock types (facies). Then the algorithms are applied some 2-D synthetic seismic examples for facies classiﬁcation. The seismic data include acoustic impedance and Poisson’s ratio data. Finally they are applied a synthetic seismic example and a ﬁeld case for measurement error estimation. In all the examples, three EM algorithms: traditional EM, Spatial EM and NEM are compared. The sensitivity to the initial number of groups and the parameters used in the algorithms are tested.