The main challenge of Markov chain Monte Carlo sampling is to define a proposal distribution that simultaneously is a good approximation of the posterior probability while being inexpensive to manipulate. We present a gradient‐based Markov chain Monte Carlo inversion of pre‐stack seismic data in which the posterior sampling is accelerated by defining a proposal that is a local, Gaussian approximation of the posterior model, while a non‐parametric prior is assumed for the distribution of the elastic properties. The proposal is constructed from the local Hessian and gradient information of the log posterior, whereas the non‐linear, exact Zoeppritz equations constitute the forward modelling engine for the inversion procedure. Hessian and gradient information is made computationally tractable by a reduction of data and model spaces through a discrete cosine transform reparameterization. This reparameterization acts as a regularization operator in the model space, while also preserving the spatial and temporal continuity of the elastic properties in the sampled models. We test the implemented algorithm on synthetic pre‐stack inversions under different signal‐to‐noise ratios in the observed data. We also compare the results provided by the presented method when a computationally expensive (but accurate) finite‐difference scheme is used for the Jacobian computation, with those obtained when the Jacobian is derived from the linearization of the exact Zoeppritz equations. The outcomes of the proposed approach are also compared against those yielded by a gradient‐free Monte Carlo sampling and by a deterministic least‐squares inversion. Our tests demonstrate that the gradient‐based sampling reaches accurate uncertainty estimations with a much lower computational effort than the gradient‐free approach.

中文翻译：

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基于梯度的马尔可夫链蒙特卡洛算法，用于弹性叠前反演，数据和模型空间减少

马尔可夫链蒙特卡洛采样的主要挑战是定义一个提案分布，该提案分布同时要很好地近似后验概率，同时又要便宜得多。我们提出了基于梯度的马尔可夫链蒙特卡罗反演的叠前地震数据，其中通过定义一个后验模型的局部，高斯近似的提议来加速后验采样，而对于后验模型则假定为非参数先验。弹性性能的分布。该建议由对数后验的局部Hessian和梯度信息构造而成，而非线性的精确Zoeppritz方程则构成了反演过程的正向建模引擎。通过离散余弦变换重新参数化减少数据和模型空间，可以使Hessian和梯度信息在计算上易于处理。此重新参数化充当模型空间中的正则化运算符，同时还保留了采样模型中弹性属性的空间和时间连续性。我们在观测到的数据中，在不同信噪比下，对合成叠前反演的已实现算法进行了测试。当雅可比计算使用计算上昂贵（但准确的）有限差分方案时，我们还将本方法提供的结果与精确Zoeppritz方程的线性化推导出雅可比方法时获得的结果进行比较。还将所提方法的结果与无梯度蒙特卡洛采样和确定性最小二乘反演得出的结果进行比较。我们的测试表明，与无梯度方法相比，基于梯度的采样能够以低得多的计算量实现准确的不确定性估计。