Multiple-point simulations have been introduced over the past decade to overcome the limitations of second-order stochastic simulations in dealing with geologic complexity, curvilinear patterns, and non-Gaussianity. However, a limitation is that they sometimes fail to generate results that comply with the statistics of the available data while maintaining the consistency of high-order spatial statistics. As an alternative, high-order stochastic simulations based on spatial cumulants or spatial moments have been proposed; however, they are also computationally demanding, which limits their applicability. The present work derives a new computational model to numerically approximate the conditional probability density function (cpdf) as a multivariate Legendre polynomial series based on the concept of spatial Legendre moments. The advantage of this method is that no explicit computations of moments (or cumulants) are needed in the model. The approximation of the cpdf is simplified to the computation of a unified empirical function. Moreover, the new computational model computes the cpdfs within a local neighborhood without storing the high-order spatial statistics through a predefined template. With this computational model, the algorithm for the estimation of the cpdf is developed in such a way that the conditional cumulative distribution function (ccdf) can be computed conveniently through another recursive algorithm. In addition to the significant reduction of computational cost, the new algorithm maintains higher numerical precision compared to the original version of the high-order simulation. A new method is also proposed to deal with the replicates in the simulation algorithm, reducing the impacts of conflicting statistics between the sample data and the training image (TI). A brief description of implementation is provided and, for comparison and verification, a set of case studies is conducted and compared with the results of the well-established multi-point simulation algorithm, filtersim. This comparison demonstrates that the proposed high-order simulation algorithm can generate spatially complex geological patterns while also reproducing the high-order spatial statistics from the sample data.

中文翻译：

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基于空间勒让德矩的高阶随机模拟新计算模型。

过去十年中引入了多点模拟，以克服二阶随机模拟在处理地质复杂性、曲线模式和非高斯性方面的局限性。然而，其局限性在于有时无法生成符合可用数据统计的结果，同时保持高阶空间统计的一致性。作为替代方案，已经提出了基于空间累积量或空间矩的高阶随机模拟；然而，它们对计算的要求也很高，这限制了它们的适用性。目前的工作推导了一种新的计算模型，以基于空间勒让德矩的概念，将条件概率密度函数（cpdf）数值近似为多元勒让德多项式级数。此方法的优点是模型中不需要显式计算矩（或累积量）。cpdf 的近似被简化为统一经验函数的计算。此外，新的计算模型计算局部邻域内的 cpdf，而无需通过预定义模板存储高阶空间统计数据。利用该计算模型，开发了cpdf估计算法，使得可以通过另一种递归算法方便地计算条件累积分布函数（ccdf）。除了计算成本显着降低外，新算法与原始版本的高阶模拟相比还保持了更高的数值精度。还提出了一种新方法来处理模拟算法中的重复，减少样本数据和训练图像（TI）之间统计冲突的影响。提供了实现的简要描述，并且为了进行比较和验证，进行了一组案例研究，并将其与完善的多点仿真算法 Filtersim 的结果进行了比较。这种比较表明，所提出的高阶模拟算法可以生成空间复杂的地质模式，同时还可以从样本数据再现高阶空间统计数据。