Collaborative Research: CMG: Multi-scaling Random Fields and Pollution Migration

地下水含水层的性质在几乎所有的物理位置都是不确定的。含水层中只有一小部分可以进行取样和测试。必须选择和实施一些统计模型来描述控制地下水和溶解污染物运动的参数（主要是水力传导率）的分布和相关结构。在统计模型的基础上，构造了含水层的复制品，对含水层的流量和输运进行了预测。这些复制品必须包含可能出现在所有感兴趣的尺度上的结构。副本应该尊重测量的真实数据（包括可能的非高斯分布），并且应该表示可能限制在非常狭窄的方向窗口的相关性；例如，离散裂缝定位。研究人员正在研究一种模型，该模型可以从许多研究得很好的含水层（颗粒层和/或裂缝层）中获取数据。该模型使用算子分数噪声和运动，因为含水层的尺度或分形性质随方向变化。算子-分数运动是基于研究者最近对三维分数积分-微分的定义。算符分数运动在地球科学中被称为“广义标度不变性”，其中标度指数依赖于方向。研究人员正在严格定义这些运动的数学性质及其数值实现。他们还研究了通过算子分数含水层副本的输运与分数阶输运方程和广义连续时间随机游走中体现的输运的简单解析描述之间的对应关系。污染在含水层中的扩散是由存在于大范围尺度上的沉积物结构控制的。细小的淤泥和粘土可以保留污染物数年，而掩埋的（残余的）砾石河道可能在短时间内将一些污染物移动数英里。建立具有这种多尺度结构的含水层模型，是对饮用水脆弱性进行预测的重要手段。在另一种与核废料储存更为密切的环境中，同时存在的大小裂缝将解释放射性物质从未来储存库中扩散的快慢途径。这些含水层的代表性模型必须是“可建造的”和可测试的，因此研究人员首先要建立具体的数学基础。虽然这项工作主要涉及数学和水文学，但其结果也有助于物理、流体动力学、电气工程和金融方面的应用研究。在看股票价格的图表时，金融应用就显露出来了：每分钟都有波动，类似于年复一年的收益和损失。一些股票是紧密耦合的；另一些则不然。不同市场的反应速度不同，变化的幅度也不容易确定。最后，几名研究生和博士后研究员也得到了资助。他们每个人都在物理科学和数学方面接受了广泛的训练。这种交叉训练使理论科学和应用科学之间的合作更加富有成效。

The properties of groundwater aquifers are uncertain at almost allphysical locations. Only a small fraction of an aquifer can be sampledand tested. Some statistical model must be chosen and implemented thatdescribes the distribution and correlation structure of the parameters(primarily the hydraulic conductivity) that control the motion ofgroundwater and dissolved pollutants. Based on the statistical model,replicas of the aquifers are constructed to make predictions of flow andtransport. These replicas must contain the structures that may bepresent across all scales of interest. The replicas should honor thereal data where measured (including a possibly non-Gaussiandistribution), and should represent correlation that might be restrictedto very narrow directional windows; for example, discrete fracture setorientations. The investigators are working on one model that can honorthe data from many well-studied aquifers (granular and/or fractured). The model uses operator-fractional noise and motions, since the scaling,or fractal, properties of aquifers vary with direction. The operatorfractional motions are based on the investigators' recent definitions of3-D fractional integro-differentiation. The operator-fractional motionshave what is known in the earth sciences as "generalized scaleinvariance" in which the index of scaling depends on direction. Theinvestigators are rigorously defining the mathematical properties ofthese motions and their numerical implementations. They are alsostudying the correspondence between transport through theoperator-fractional aquifer replicas and simple analytic descriptions oftransport embodied in fractional-order transport equations andgeneralized continuous time random walks. The spreading of pollution in an aquifer is controlled by sedimentstructures that are present across a huge range of scales. Tiny bits ofsilt and clay may retain pollutants for years, while buried (remnant)gravelly stream channels might move some pollutants for miles in a shorttime. To make realistic predictions of drinking water vulnerability oreventual cleanup, it is important to construct models of aquifers thathave this kind of multi-scale structure. In another setting moregermane to nuclear waste storage, the simultaneous presence of small andlarge fractures will account for both the slow and fast pathways for thespreading of radioactivity away from future repositories. Representative models of these aquifers must be "buildable" andtestable, so the investigators are first pouring a concrete mathematicalfooting. While the work deals primarily with mathematics and hydrology,the results also contribute to applied studies in physics, fluiddynamics, electrical engineering, and finance. The financialapplications are revealed when looking at a graph of a stock price:there are minute-to-minute fluctuations that are similar to year-to-yeargains and losses. Some stocks are tightly coupled; others are not. Notall markets respond at the same rate, nor are the magnitudes of thechanges easily characterized. Finally, several graduate students and apost-doctoral researcher are also supported by the grant. Each of themis receiving extensive training in both physical sciences andmathematics. This cross-training engenders more fruitful cooperationbetween the theoretical and applied sciences.