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.

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Abbreviations

\( p({\mathbf{y}}|{\mathbf{X}}_{i} ) \) :

The probability density value of \( {\mathbf{y}} \) for the cluster \( {\mathbf{X}}_{i} \)

\( n \) :

The number of variables, or the number of samples in Eq. (2), or the number of output units in Eq. (6)

\( {\mathbf{x}}_{ik} \) :

kth sample of \( {\mathbf{X}}_{i} \)

\( l \) :

The dimension of sample, or the number of hidden units in Eq. (6) and this number is an integer

\( \sigma \) :

The smooth factor, also named as “spread”

\( {\mathbf{B}}_{fn} \) :

The test data

\( m \) :

The number of learning samples, or the number of input units in Eq. (6)

\( f \) :

The number of test samples

\( {\mathbf{D}}_{fm} \) :

The matrix of the Euclidean distance

\( {\mathbf{P}}_{fm} \) :

The matrix of the probability density

\( {\mathbf{S}}_{fz} \) :

The matrix of the total probability density

\( {\mathbf{C}}_{fz} \) :

The ratio matrix containing the pattern recognition ratio of each test sample for each cluster

\( \beta_{1}^{1} \) :

The enlarged coefficient

\( \beta_{1}^{2} \) :

The reduced coefficient

\( {\mathbf{H}} \) :

The learning data without interference variables

\( {\mathbf{R}} \) :

The vector of MIV results

\( f_{\text{MIV}} ( \cdot ) \) :

The MIV results

\( {\varvec{\Gamma}} \) :

The spread population

\( {\varvec{\upsigma}}_{i} \) :

ith spread seed

\( {\mathbf{P}}_{fm}^{i} \) :

The matrix of the probability density scaled by ith spread seed

\( {\varvec{\Delta}}_{i}^{j} \) :

The iteration result of ith spread seed; its corresponding predicted result has the smallest error in the first j iteration times

\( {\varvec{\upsigma}}_{i}^{j} \) :

The iteration result of ith spread seed; its corresponding predicted result has the smallest error in jth iteration time

\( {\mathbf{O}}_{if}^{j} \) :

The predicted result corresponding to the ith spread seed in jth iteration time

\( {\mathbf{O}}_{{f\_{\text{std}}}} \) :

The standard result

\( t \) :

The iteration time

\( {\varvec{\Delta}}_{g}^{j} \) :

The iteration result of one spread seed; its corresponding predicted result has the smallest error in the first j iteration times

\( {\varvec{\upsigma}}_{\varGamma }^{j} \) :

The iteration result of one spread seed; its corresponding predicted result has the smallest error in jth iteration time

\( {\mathbf{O}}_{\varGamma f}^{j} \) :

The predicted result corresponding to one spread seed and has the smallest error in jth iteration time

\( {\mathbf{W}}_{i}^{j} \) :

The iteration step of ith spread seed in jth iteration time

\( \omega \) :

The inertia weight

\( c_{1} \) :

The acceleration coefficient

\( c_{2} \) :

The acceleration coefficient

\( r_{1} \) :

Random value limited in [0,1]

\( r_{2} \) :

Random value limited in [0,1]

\( \sigma_{\text{max} \_1} \) :

The left limit of spread seed

\( \sigma_{\text{max} \_2} \) :

The right limit of spread seed

\( W_{\text{max} } \) :

The limit of iteration step

\( \hbox{min} ( \cdot ) \) :

The function acquiring the minimum

\( \left\| \cdot \right\|_{2} \) :

The function calculating L2 norm

\( \omega_{\text{max} } \) :

The maximum inertia

\( \omega_{\text{min} } \) :

The minimum inertia

\( t_{\text{max} } \) :

The maximum iteration time

\( v_{\text{nor}} \) :

The normalized logging value

\( v \) :

The logging value

\( v_{\text{max} } \) :

The maximum value of one log

\( v_{\text{min} } \) :

The minimum value of one log

\( a \) :

An integer value and its taking range is [1,10]

\( {\varvec{\upsigma}}_{\# 1} \) :

The spread vector used for prediction of #1 well

\( {\varvec{\upsigma}}_{\# 2} \) :

The spread vector used for prediction of #2 well

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Acknowledgements

This work was supported by grants from the National Key Oil and Gas Project of China (No. 2016ZX05033-002-005) and the Sinopec Funded Project (No. P19020-2).

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Correspondence to Yufeng Gu.

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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

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

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Gu, Y., Zhang, Z., Zhang, D. et al. Complex lithology prediction using mean impact value, particle swarm optimization, and probabilistic neural network techniques. Acta Geophys. 68, 1727–1752 (2020). https://doi.org/10.1007/s11600-020-00504-2

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Keywords

  • Lacustrine carbonate formation
  • Complex lithology prediction
  • Backpropagation
  • Probabilistic neural network
  • Mean impact value
  • Particle swarm optimization