Discerning connectivity features and scaling behaviour of spatial random fields through the Method of Anchored Distributions (MAD).

由于具有高度的空间变异性以及信息的稀缺性，诸如电导率或透过率之类的水文地质量通常很难以精确的方式表示。因此，这些量通常被建模为随机场，在高斯过程的假设下，由它们的期望值和它们的协方差函数（方差函数）分别定义。该变差函数通常通过将模型函数拟合到由电导率测量得出的实验变差函数来参数化。典型的变异函数模型包括指数型、高斯型和球面型。尽管存在差异，但这些不同的函数通常可以以相似的精度拟合到实验变异图上。上述模型函数的选择通常对随后进行的流动和输运模拟几乎没有影响，这一事实证实了这一观察结果。然而，在本研究中，我们将研究两种情况，其中这种变异函数模型可能被误认为结构上不同的变异函数模型，从而导致非常不同的结果。第一个场景是高斯随机场与高斯变差函数与非高斯随机场的比较，该随机场的极值具有高度连通性。已经证明，这些场可以具有非常相似的变差函数，使得它们仅根据实验变差函数几乎无法区分。然而，由于不同的连通性特征，所产生的流动和传输行为将有很大的不同。第二种情况是高斯随机场的比较，它们要么具有指数函数，要么具有所谓的截断幂律变差函数。与上述情况类似，这两个场具有相似的变差函数，但在缩放行为方面存在很大差异。如果错误地忽略这一事实，就会在同化不同尺度的数据或将这样发现的结果转移到其他空间尺度时导致错误。这两种情况都有一个共同的事实，即，仅基于均值和方差图的特征，它们很难区分，但如果用于进一步分析它们所代表的属性，可能会导致非常不同的结果。因此，有必要使用附加数据来辨别随机场的原始结构。在本研究中，我们将使用锚定分布方法，这是一种用于空间随机场逆表征的新工具。该方法在使用数据方面非常通用，具有模块化结构，并且不假设目标变量（测井水力导率）与用于反演过程的数据之间存在任何形式关系。

Hydrogeolocial quantities like conductivity or transmissivity are usually hard to represent in a precise manner due to having both a high degree of spatial variability as well as exhibiting a scarcity of information. As a result these quantities are commonly modeled as random fields, which, under the assumption of a Gaussian process, are defined by their expectation value and their covariance function respectively the variogram. This variogram is usually parametrized by fitting a model functions to an experimental variogram derived from measurements of the conductivity. Typical variogram model include the exponential, Gaussian and the spherical variogram. Despite their differences these different functions can often be fitted to the experimental variogram with similar accuracy. This observation is substantiated by the fact that the choice of the aforementioned model functions often has little impact on subsequently performed flow and transport simulations. In this study we will however, investigate the two scenarios, where such variogram models can be mistaken for structurally different variogram models therefore leading to very different results.The first scenario is a comparison of a Gaussian random field with a Gaussian variogram function vs. a non-Gaussian random field with a high degree of connectivity of the extreme values of the field. It has been shown that such fields can have very similar variogram functions making them nearly indistinguishable based on the experimental variogram alone. Due to the different connectivity features the resulting flow and transport behavior will however, differ strongly.The second scenario is a comparison of Gaussian random fields having either an exponential or a so called truncated power-law variogram function. In analogy to above scenario both these fields have a similar variogram function but strongly differ with respect to the scaling behavior. Falsely ignoring this fact will leads to errors if data from different scales are assimilated or the so found results are transferred to other spatial scales.These two presented scenarios both share the fact, that, based on characterization of mean and variogram alone, they are hard to discriminate yet can lead to very different results if used for further analyses of the properties they represent. As a result it is necessary to use additional data in order to discern the original structure of the random field.In this study we will use the Method of Anchored Distributions, which is a novel tool for the inverse characterization of spatial random fields. The method is very versatile with respect to the used data, has a modular structure and does not assume any formal relationship between the target variable (log hydraulic conductivity) and the data used for the inversion process.