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This study aimed to construct a kernel Fisher discriminant analysis (KFDA) method from well logs for lithology identification purposes. KFDA, via the use of a kernel trick, greatly improves the multiclassification accuracy compared with Fisher discriminant analysis (FDA). The optimal kernel Fisher projection of KFDA can be expressed as a generalized characteristic equation. However, it is difficult to solve the characteristic equation; therefore, a regularized method is used for it. In the absence of a method to determine the value of the regularized parameter, it is often determined based on expert human experience or is specified by tests. In this paper, it is proposed to use an improved KFDA (IKFDA) to obtain the optimal regularized parameter by means of a numerical method. The approach exploits the optimal regularized parameter selection ability of KFDA to obtain improved classification results. The method is simple and not computationally complex. The IKFDA was applied to the

China’s tight clastic rock reservoir is considerably wide, containing sediments that were deposited during the Carboniferous, Permian, Triassic, and Jurassic periods. The reservoirs in Western Sichuan and Erdos are more representative. The West Sichuan depression is located in the Western Sichuan Basin, which belongs to the western depression belt that is located in Yangtze Platform or Longmenshan Fault Zone. A tight gas reservoir was discovered in the Xujiahe and Shaximiao formations that occur in this area. Tight clastic reservoirs are noted for their low porosity, because of their dense, multilayered stacking and strong heterogeneous characteristics that are caused by their complexity and particularity. These characteristics complicate the identification of the lithology, which would enable the prediction of the properties of a reservoir. Previous research [

The KFDA has its roots in Fisher discriminant analysis (FDA) and is the nonlinear scheme for two-class and multiclass problems [

The principle that underlies KFDA is that input data are mapped into a high-dimensional feature space by using a nonlinear function, after which FDA is used for recognition or classification in feature space. KFDA requires factorization of the Gram matrix into the kernel within-class scatter matrix

This paper proposes a new approach for the selection of the regularized parameter

Let

KFDA significantly improves the classification ability for the nonlinear separable sample of FDA via the use of a kernel trick. To adapt to nonlinear cases,

Lithology identification by KFDA can be attributed to the optimization of kernel Fisher criterion function as follows:

According to the theory of reproducing a kernel [

If

The solution of practical problems requires the use of

The

Scatter gram plots of

The following experiments involve a comparative study of the different selection schemes of

The experiment was processed within a MATLAB 7.0 environment running on a PC powered by a Pentium 3.3 GHz CPU. The experimental results are shown in Figures

Value of the determinant versus the value of regularized

Classification results with different value of regularized

This is further illustrated in Figure

The AC region is located in the central segment of the West Sichuan depression and was formed during Upper Triassic and Jurassic periods. The Triassic-Jurassic stratum is the part of the thick entity of the Western Sichuan foreland basin, with a total thickness of 6500 m. The AC region is located in a large uplift belt of the West Sichuan depression, which shows NEE trend [

The Xujiahe formation consists of alternating layers of sandstone, siltstone, mudstone, shale, and coal series, in which the rock types are more complex. According to logging data obtained from wells and the extraction of the physical parameters, the rock in the Xujiahe formation can be divided into mudstone, siltstone, and sandstone based on the physical intersection diagram and histogram analysis of rocks.

Logging parameters provide a comprehensive reflection of lithology, and their sensitivity is different for lithology identification. The sensitivity of these parameters for lithology identification was studied using the method of correlation analysis and finally determined acoustic (AC), natural gamma ray (GR), density (DEN), and compensated neutron logging (CNL) as the characteristic variables. Training and testing sets in the standard layer, each of which contained 50 samples, were obtained. The cross-sectional plot is displayed in Figure

Cross-sectional plots of the logging parameters: green O (sandstone); red (siltstone); blue (mudstone).

In the following experiments, FDA and KFDA are compared on logging attribute data sets. The FDA method is used to extract the optimal and suboptimal discriminant vector of the training sets, and then the testing sets of sample are projected onto vectors (Figure

FDA identification result.

Experiments were conducted on logging attribute sets using IKFDA. The kernel function employed in IKFDA is the Gaussian kernel function, and a numerical method was used to obtain the optimal value of parameter

IKFDA identification result.

The optimal kernel Fisher projection of KFDA can be expressed as a generalized characteristic equation by using a regularized method. For multiclass problems, the value of the regularized parameter is a key factor in the application effect, which is largely influenced by human experience. This paper proposes a novel method to optimize the regularized parameter using a numerical method. Thus, by selecting an optimal value for the regularized parameter, it becomes possible to solve the generalized characteristic equation, thereby eliminating the human factor. The effectiveness of the IKFDA was demonstrated by applying it to the nonlinearly separable

In this paper, the selection of the regularized parameter depended on the numerical method. We analyzed the applied validity of the improved Kernel Fisher discriminant analysis without performing deep theoretical analysis. This needs to be addressed by conducting further research in the future.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are most grateful for the valuable advice on the revision of the paper from the reviewers. The work was supported by the Foundation of Geomathematics Key Laboratory of Sichuan Province.