Markov chain Monte Carlo algorithms are commonly employed for accurate uncertainty appraisals in non‐linear inverse problems. The downside of these algorithms is the considerable number of samples needed to achieve reliable posterior estimations, especially in high‐dimensional model spaces. To overcome this issue, the Hamiltonian Monte Carlo algorithm has recently been introduced to solve geophysical inversions. Different from classical Markov chain Monte Carlo algorithms, this approach exploits the derivative information of the target posterior probability density to guide the sampling of the model space. However, its main downside is the computational cost for the derivative computation (i.e. the computation of the Jacobian matrix around each sampled model). Possible strategies to mitigate this issue are the reduction of the dimensionality of the model space and/or the use of efficient methods to compute the gradient of the target density. Here we focus the attention to the estimation of elastic properties (P‐, S‐wave velocities and density) from pre‐stack data through a non‐linear amplitude versus angle inversion in which the Hamiltonian Monte Carlo algorithm is used to sample the posterior probability. To decrease the computational cost of the inversion procedure, we employ the discrete cosine transform to reparametrize the model space, and we train a convolutional neural network to predict the Jacobian matrix around each sampled model. The training data set for the network is also parametrized in the discrete cosine transform space, thus allowing for a reduction of the number of parameters to be optimized during the learning phase. Once trained the network can be used to compute the Jacobian matrix associated with each sampled model in real time. The outcomes of the proposed approach are compared and validated with the predictions of Hamiltonian Monte Carlo inversions in which a quite computationally expensive, but accurate finite‐difference scheme is used to compute the Jacobian matrix and with those obtained by replacing the Jacobian with a matrix operator derived from a linear approximation of the Zoeppritz equations. Synthetic and field inversion experiments demonstrate that the proposed approach dramatically reduces the cost of the Hamiltonian Monte Carlo inversion while preserving an accurate and efficient sampling of the posterior probability.

中文翻译：

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结合离散余弦变换和卷积神经网络以加快叠前地震数据的哈密顿蒙特卡洛反演

马尔可夫链蒙特卡罗算法通常用于非线性逆问题的准确不确定度评估。这些算法的缺点是，需要大量样本才能实现可靠的后验估计，尤其是在高维模型空间中。为了克服这个问题，最近引入了哈密顿蒙特卡罗算法来解决地球物理反演问题。与经典的马尔可夫链蒙特卡罗算法不同，该方法利用目标后验概率密度的导数信息来指导模型空间的采样。但是，它的主要缺点是微分计算（即围绕每个采样模型的雅可比矩阵的计算）的计算成本。减轻此问题的可能策略是减少模型空间的维数和/或使用有效的方法来计算目标密度的梯度。在这里，我们将注意力集中在通过非线性振幅与角度反演从叠前数据估计弹性特性（P，S波速度和密度）中，其中使用哈密顿量的蒙特卡洛算法对后验概率进行采样。为了减少反演过程的计算成本，我们采用离散余弦变换对模型空间进行重新参数化，并训练卷积神经网络来预测每个采样模型周围的雅可比矩阵。在离散余弦变换空间中还对网络的训练数据集进行了参数化，因此减少了学习阶段要优化的参数数量。训练后，网络可用于实时计算与每个采样模型相关的雅可比矩阵。将该方法的结果与汉密尔顿蒙特卡罗反演的预测进行了比较和验证，其中使用了计算量大但精确的有限差分方案来计算雅可比矩阵，以及通过用矩阵算子替换雅可比矩阵获得的结果从Zoeppritz方程的线性逼近导出。合成和现场反演实验表明，该方法大大降低了汉密尔顿蒙特卡罗反演的成本，同时保留了准确而有效的后验概率采样。