Complex lithology prediction using mean impact value, particle swarm optimization, and probabilistic neural network techniques

Abstract

Lithology prediction is a fundamental problem because the outcome of lithology prediction is the critical underlying data for some basic geological work, e.g., establishing stratigraphic framework or analyzing distribution of sedimentary facies. As the geological formation generally consists of many different lithologies, the lithology prediction is always viewed as a tough work by geologists. Probabilistic neural network (PNN) shows high efficiency when solving pattern recognition problem since learning data do not need to do any pre-training of learning data and calculation results are universally reliable, and then, this model could be considered as an effective solution. However, there are two factors that seriously limit the PNN’s performance: One is existence of the interference variables of learning samples, and the other is selection of the window length of probability density distribution. In view of adverse impact of those two factors, two techniques, mean impact value (MIV) and particle swarm optimization (PSO), are introduced to improve the PNN’s calculation capability. Thus, a new prediction method referred as MIV–PSO–PNN is proposed in this paper. The proposed method is validated by three well-designed experiments, and the corresponding experiment data are recorded by two cored wells of the LULA oilfield. For the three experiments, prediction accuracies of the results provided by the proposed method are 81.67%, 73.34% and 88.34%, respectively, all of which are higher than those provided by other comparative approaches including backpropagation (BP), PNN, and MIV-PNN. The experiment results strongly demonstrate that the proposed method is capable to predict complex lithology.

Appendices

Appendix 1

$$ \begin{aligned} \overbrace {{{\mathbf{A}}_{mn} = \left[ {\begin{array}{*{20}c} {a_{11} } & {a_{12} } & , & \cdots & , & {a_{1n} } \\ {a_{21} } & {a_{22} } & , & \cdots & , & {a_{2n} } \\ , & , & , & {} & {} & , \\ \vdots & \vdots & {} & \ddots & {} & \vdots \\ , & , & {} & {} & , & , \\ {a_{m1} } & {a_{m2} } & , & \cdots & , & {a_{mn} } \\ \end{array} } \right]\;and\;{\mathbf{B}}_{fn} = \left\lfloor {\begin{array}{*{20}c} {b_{11} } & {b_{12} } & , & \cdots & , & {b_{1n} } \\ {b_{21} } & {b_{22} } & , & \cdots & , & {b_{2n} } \\ , & , & , & {} & {} & , \\ \vdots & \vdots & {} & \ddots & {} & \vdots \\ , & , & {} & {} & , & , \\ {b_{f1} } & {b_{f2} } & , & \cdots & , & {b_{fn} } \\ \end{array} } \right\rfloor }}^{{{\text{Input}}\;{\text{layer}}}}\mathop{\longrightarrow}\limits{\substack{ {\text{Euclidean}}\;{\text{distance}} \\ {\text{calculation}} } } \hfill \\ \overbrace {{{ = }\left| {\begin{array}{*{20}c} {d_{11} } & {d_{12} } & , & \cdots & , & {d_{1m} } \\ {d_{21} } & {d_{22} } & , & \cdots & , & {d_{2m} } \\ , & , & , & {} & {} & , \\ \vdots & \vdots & {} & \ddots & {} & \vdots \\ , & , & {} & {} & , & , \\ {d_{f1} } & {d_{f2} } & , & \cdots & , & {d_{fm} } \\ \end{array} } \right| = {\mathbf{D}}_{fm} \mathop{\longrightarrow}\limits {\substack{ {\text{Probability}}\;{\text{density}} \\ {\text{calculation}} } }\left[ {\begin{array}{*{20}c} {p_{11} } & {p_{12} } & , & \cdots & , & {p_{1m} } \\ {p_{21} } & {p_{22} } & , & \cdots & , & {p_{2m} } \\ , & , & , & {} & {} & , \\ \vdots & \vdots & {} & \ddots & {} & \vdots \\ , & , & {} & {} & , & , \\ {p_{f1} } & {p_{f2} } & , & \cdots & , & {p_{fm} } \\ \end{array} } \right] = {\mathbf{P}}_{fm} }}^{{{\text{Pattern}}\;{\text{layer}}}}\mathop{\longrightarrow}\limits ^{\text{sum}} \hfill \\ \overbrace {{\left[ {\begin{array}{*{20}c} {\sum\limits_{k = 1}^{{v_{1} }} {p_{1k} } } & {\sum\limits_{{k = v_{1} + 1}}^{{v_{2} }} {p_{1k} } } & , & \cdots & , & {\sum\limits_{{k = v_{z - 1} + 1}}^{{v_{z} }} {p_{1k} } } \\ {\sum\limits_{k = 1}^{{v_{1} }} {p_{2k} } } & {\sum\limits_{{k = v_{1} + 1}}^{{v_{2} }} {p_{2k} } } & , & \cdots & , & {\sum\limits_{{k = v_{z - 1} + 1}}^{{v_{z} }} {p_{2k} } } \\ , & , & , & {} & {} & , \\ \vdots & \vdots & {} & \ddots & {} & \vdots \\ , & , & {} & {} & , & , \\ {\sum\limits_{k = 1}^{{v_{1} }} {p_{fk} } } & {\sum\limits_{{k = v_{1} + 1}}^{{v_{2} }} {p_{fk} } } & , & \cdots & , & {\sum\limits_{{k = v_{z - 1} + 1}}^{{v_{z} }} {p_{fk} } } \\ \end{array} } \right]{ = }\left[ {\begin{array}{*{20}c} {s_{11} } & {s_{12} } & , & \cdots & , & {s_{1z} } \\ {s_{21} } & {s_{22} } & , & \cdots & , & {s_{2z} } \\ , & , & , & {} & {} & , \\ \vdots & \vdots & {} & \ddots & {} & \vdots \\ , & , & {} & {} & , & , \\ {s_{f1} } & {s_{f2} } & , & \cdots & , & {s_{fz} } \\ \end{array} } \right] = {\mathbf{S}}_{fz} }}^{{{\text{sum}}\;{\text{layer}}}}\mathop{\longrightarrow}\limits ^{\text{compete}} \hfill \\ \overbrace {{ = \frac{{s_{ij} }}{{\sum\limits_{k = 1}^{z} {s_{ik} } }}(i = 1,2, \ldots ,f) = \left[ {\begin{array}{*{20}c} {c_{11} } & {c_{12} } & , & \cdots & , & {c_{1z} } \\ {c_{21} } & {c_{22} } & , & \cdots & , & {c_{2z} } \\ , & , & , & {} & {} & , \\ \vdots & \vdots & {} & \ddots & {} & \vdots \\ , & , & {} & {} & , & , \\ {c_{f1} } & {c_{f2} } & , & \cdots & , & {c_{fz} } \\ \end{array} } \right] = {\mathbf{C}}_{fz} \mathop{\longrightarrow}\limits ^ {{{{ \hbox{max} }\;{\text{in}}\;{\text{row}}}}}{\mathbf{O}}_{f} }}^{{{\text{Output}}\;{\text{layer}}}} \hfill \\ \end{aligned} $$

“Euclidean distance” is determined by the power exponent component of Eq. (2).“max in row” means extracting the maximum value of each row.

Appendix 2

$$ \begin{aligned} & {\mathbf{A}}_{mn} \times \left[ {\begin{array}{*{20}c} {\beta_{1}^{1} } & 0 & , & \cdots & , & 0 \\ 0 & 1 & , & \cdots & , & 0 \\ , & , & , & {} & {} & , \\ \vdots & \vdots & {} & \ddots & {} & \vdots \\ , & , & {} & {} & , & , \\ 0 & 0 & , & \cdots & , & 1 \\ \end{array} } \right]_{n \times n} = \left[ {\begin{array}{*{20}c} {a'_{11} } & {a_{12} } & , & \cdots & , & {a_{1n} } \\ {a'_{21} } & {a_{22} } & , & \cdots & , & {a_{2n} } \\ , & , & , & {} & {} & , \\ \vdots & \vdots & {} & \ddots & {} & \vdots \\ , & , & {} & {} & , & , \\ {a'_{m1} } & {a_{m2} } & , & \cdots & , & {a_{mn} } \\ \end{array} } \right] = {\mathbf{A}}'_{mn} \\ & {\mathbf{A}}_{mn} \times \left[ {\begin{array}{*{20}c} {\beta_{1}^{2} } & 0 & , & \cdots & , & 0 \\ 0 & 1 & , & \cdots & , & 0 \\ , & , & , & {} & {} & , \\ \vdots & \vdots & {} & \ddots & {} & \vdots \\ , & , & {} & {} & , & , \\ 0 & 0 & , & \cdots & , & 1 \\ \end{array} } \right]_{n \times n} = \left[ {\begin{array}{*{20}c} {a''_{11} } & {a_{12} } & , & \cdots & , & {a_{1n} } \\ {a''_{21} } & {a_{22} } & , & \cdots & , & {a_{2n} } \\ , & , & , & {} & {} & , \\ \vdots & \vdots & {} & \ddots & {} & \vdots \\ , & , & {} & {} & , & , \\ {a''_{m1} } & {a_{m2} } & , & \cdots & , & {a_{mn} } \\ \end{array} } \right] = {\mathbf{A}}''_{mn} \\ \end{aligned} $$

Appendix 3

$$ \left| {\begin{array}{*{20}c} {d_{11} } & {d_{12} } & , & \cdots & , & {d_{1m} } \\ {d_{21} } & {d_{22} } & , & \cdots & , & {d_{2m} } \\ , & , & , & {} & {} & , \\ \vdots & \vdots & {} & \ddots & {} & \vdots \\ , & , & {} & {} & , & , \\ {d_{f1} } & {d_{f2} } & , & \cdots & , & {d_{fm} } \\ \end{array} } \right|\mathop{\longrightarrow}\limits ^{{{\text{substituting}}\;{\varvec{\upsigma}}_{i} }}\left[ {\begin{array}{*{20}c} {p_{{_{11} }}^{i} } & {p_{{_{12} }}^{i} } & , & \cdots & , & {p_{{_{1m} }}^{i} } \\ {p_{{_{21} }}^{i} } & {p_{{_{22} }}^{i} } & , & \cdots & , & {p_{{_{2m} }}^{i} } \\ , & , & , & {} & {} & , \\ \vdots & \vdots & {} & \ddots & {} & \vdots \\ , & , & {} & {} & , & , \\ {p_{{_{f1} }}^{i} } & {p_{{_{f2} }}^{i} } & , & \cdots & , & {p_{{_{fm} }}^{i} } \\ \end{array} } \right] = {\mathbf{P}}_{fm}^{i} \mathop{\longrightarrow}\limits ^{{{\text{sum}}\;{\text{and}}\;{\text{compete}}}}{\mathbf{O}}_{if} $$