In this work, an empirical Bayes method was applied to estimate highly parameterized transmissivity fields in 2D aquifers under conditions of steady flow. The Bayesian inverse procedure was coupled with the Akaike’s Bayesian information criterion to identify both the main transmissivity field and the hyperparameters of the prior distribution. The forward problem, solved with a version of MODFLOW, consists in computing hydraulic heads at monitoring points considering fully known boundary conditions, and the transmissivity field. As for the required observations for the inverse problem, the monitored hydraulic head data were used. Due to the nonlinear relationship between the observed data (hydraulic heads) and the unknowns (log transmissivity values in each finite difference cell), the inverse approach is based on a successive linearization method coupled with an adjoint state methodology. At the end, the posterior distribution of the unknowns allows quantifying their uncertainty. The methodology was tested on a well-known literature case study consisting of a confined aquifer, with both Dirichelet- and Neumann-type boundary conditions and considering different degrees of heterogeneities. The inverse approach showed robust, efficient results fully consistent with other methods available in the literature. The methodology was implemented in a free and user-friendly code named ebaPEST.

中文翻译：

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结合经验贝叶斯和Akaike的贝叶斯信息准则估计含水层的透射率场

在这项工作中，采用了经验贝叶斯方法来估算稳态流动条件下二维含水层中高度参数化的透射率场。贝叶斯逆过程与Akaike的贝叶斯信息准则相结合，可以识别主要透射率场和先验分布的超参数。用MODFLOW版本解决的前向问题在于，在考虑已知边界条件和透射率场的情况下，在监控点计算液压头。至于反问题所需的观测，则使用了监控的液压头数据。由于观测数据（液压头）和未知数（每个有限差分像元中的对数透射率值）之间存在非线性关系，逆方法基于连续线性化方法和伴随状态方法。最后，未知数的后验分布可以量化其不确定性。该方法论在一个著名的文献案例研究中进行了测试，该案例包括密闭含水层，同时具有Dirichelet型和Neumann型边界条件，并考虑了不同程度的异质性。逆方法显示出鲁棒，有效的结果，与文献中提供的其他方法完全一致。该方法是通过名为ebaPEST的免费且用户友好的代码实现的。同时具有Dirichelet型和Neumann型边界条件，并考虑了不同程度的异质性。逆方法显示出鲁棒，有效的结果，与文献中提供的其他方法完全一致。该方法是通过名为ebaPEST的免费且用户友好的代码实现的。同时具有Dirichelet型和Neumann型边界条件，并考虑了不同程度的异质性。逆方法显示出鲁棒，有效的结果，与文献中提供的其他方法完全一致。该方法是通过名为ebaPEST的免费且用户友好的代码实现的。