We have introduced a Bayesian neural network in quantitative log prediction studies with the goal of improving the petrophysical characterization and quantifying the uncertainty of model predictions. Neural network (NN) methods are gaining popularity in the petrophysics and geophysics communities; however, uncertainty quantification in model predictions is often neglected in the available literature, where the prediction is frequently performed in a deterministic setting. Determination of the uncertainty of the petrophysical model requires the estimation of the posterior distribution of the neural parameters that is generally mathematically intractable; for this reason, we adopt a variational approach to approximate the posterior model of the Bayesian network. To represent the uncertainty, we randomly draw samples from the posterior distribution of neural parameters to predict the model variables given the input data, leading to a learned-ensemble predictor. The proposed approach combines the ability of the NN of extracting hidden relations within the data set that physical relations cannot describe and the probabilistic framework for uncertainty quantification. We apply the proposed method to a well-log data set from the Volve Field, offshore Norway, to predict well logs in intervals where the data are incomplete or missing due to operational issues in the drilling procedure. The proposed approach is validated in intervals where the true data are available but not included in the training process. In the proposed application, the correlation coefficient between predictions and true data is greater than 0.9. In terms of accuracy, the results are comparable to those obtained using a traditional NN approach; however, the proposed method also provides a quantification of the uncertainty in the results, which offers additional information on the confidence in the predictions.

中文翻译：

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贝叶斯神经网络中的变分推理用于测井预测

我们已经在定量测井预测研究中引入了贝叶斯神经网络，目的是改善岩石物理特征并量化模型预测的不确定性。神经网络（NN）方法在石油物理学和地球物理学界越来越流行。但是，模型预测中的不确定性量化通常在现有文献中被忽略，在这种文献中，预测经常是在确定性的环境中进行的。确定岩石物理模型的不确定性需要估算神经参数的后验分布，这在数学上通常是难于处理的。因此，我们采用变分方法来近似贝叶斯网络的后验模型。为了表示不确定性，我们从神经参数的后验分布中随机抽取样本，以预测在给定输入数据的情况下的模型变量，从而得出学习型整体预测器。所提出的方法结合了NN提取物理关系无法描述的数据集中的隐藏关系的能力以及不确定性量化的概率框架。我们将提出的方法应用于挪威海上Volve油田的测井数据集，以预测由于钻井过程中的操作问题而导致数据不完整或缺失的时间段内的测井记录。在提供真实数据但不将其包含在训练过程中的时间间隔中，对提出的方法进行验证。在提出的应用中，预测与真实数据之间的相关系数大于0.9。在准确性方面，结果与使用传统NN方法获得的结果相当；但是，提出的方法还提供了结果不确定性的量化，从而提供了有关预测置信度的其他信息。