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

合作研究:CMG:多尺度随机场和污染迁移

基本信息

项目摘要

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.
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. 只有一小部分含水层可以被取样和测试。 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,以预测流量和输运。 这些复制品必须包含可能存在于所有感兴趣的尺度上的结构。 副本应荣誉真实的数据测量(包括可能的非高斯分布),并应代表相关性,可能会被限制到非常狭窄的方向窗口,例如,离散裂缝集方向。 研究人员正在研究一个模型,该模型可以从许多经过充分研究的含水层(颗粒和/或裂缝)中获得数据。该模型使用运营商分数噪声和运动,因为缩放,或分形,含水层的属性随方向而变化。 算子分数运动是基于研究者最近定义的3-D分数积分微分。 算符分数运动在地球科学中被称为“广义标度不变性”,其中标度指数取决于方向。 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。 他们也在研究通过运营商-分数含水层副本的运输与分数阶运输方程和广义连续时间随机游走中体现的运输的简单解析描述之间的对应关系。污染在含水层中的扩散受沉积物结构的控制,这些沉积物结构存在于很大的尺度范围内。 微小的淤泥和粘土可能会保留污染物多年,而埋藏(残留)的砾石河道可能会在短时间内将一些污染物移动数英里。 为了对饮用水的脆弱性或最终净化进行现实的预测,重要的是构建具有这种多尺度结构的含水层模型。 在另一个更像核废料储存的环境中,同时存在的小裂缝和大裂缝将解释放射性从未来的储存库中扩散的缓慢和快速途径。这些含水层的代表性模型必须是“可建造的”和可测试的,因此研究人员首先浇筑混凝土地基。 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学,流体力学,电气工程和金融的应用研究。 当观察股票价格的图表时,金融应用就显现出来了:每分钟的波动与每年的收益和损失相似。有些股票是紧密相连的,有些则不是。没有哪个市场的反应速度是一样的,变化的幅度也不容易描述。 最后,一些研究生和使徒博士研究员也得到了资助。 他们每个人都在接受物理科学和数学方面的广泛训练。 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。

项目成果

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Mark Meerschaert其他文献

Mark Meerschaert的其他文献

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{{ truncateString('Mark Meerschaert', 18)}}的其他基金

Travel Support for 7th International Conference on Levy Processes
第七届征税程序国际会议的差旅支持
  • 批准号:
    1310224
  • 财政年份:
    2013
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
CMG Collaborative Research: Tempered stable models for preasymptotic pollutant transport in natural media
CMG 合作研究:自然介质中渐近污染物传输的稳定模型
  • 批准号:
    1025486
  • 财政年份:
    2010
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Collaborative Research: Geomorphic transport laws, landscape evolution, and fractional calculus
合作研究:地貌传输定律、景观演化和分数阶微积分
  • 批准号:
    0823965
  • 财政年份:
    2008
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Stochastic Models for Anomalous Diffusion
反常扩散的随机模型
  • 批准号:
    0803360
  • 财政年份:
    2008
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Collaborative Research: CMG: Multi-scaling Random Fields and Pollution Migration
合作研究:CMG:多尺度随机场和污染迁移
  • 批准号:
    0706440
  • 财政年份:
    2006
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Collaborative Research: Stochastic Methods for Fractional Partial Differential Equations
合作研究:分数阶偏微分方程的随机方法
  • 批准号:
    0139927
  • 财政年份:
    2002
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Mathematical Sciences: Norming Operators for Generalized Domains of Attraction
数学科学:广义吸引域的规范算子
  • 批准号:
    9103131
  • 财政年份:
    1991
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Mathematical Sciences: Exponents and Symmetries of Operator-Stable Laws
数学科学:算子稳定定律的指数和对称性
  • 批准号:
    8923068
  • 财政年份:
    1990
  • 资助金额:
    --
  • 项目类别:
    Standard Grant

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Cell Research
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Research on the Rapid Growth Mechanism of KDP Crystal
  • 批准号:
    10774081
  • 批准年份:
    2007
  • 资助金额:
    45.0 万元
  • 项目类别:
    面上项目

相似海外基金

CMG Collaborative Research: Tempered Stable Models for Preasymptotic Pollutant Transport in Natural Media
CMG 合作研究:自然介质中渐进前污染物传输的稳定模型
  • 批准号:
    1460319
  • 财政年份:
    2014
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    --
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    Standard Grant
Collaborative Research: CMG--Analysis and Modeling of Rotating Stratified Flows
合作研究:CMG--旋转层流分析与建模
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    1025166
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    2010
  • 资助金额:
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CMG Collaborative Research: Tempered Stable Models for Preasymptotic Pollutant Transport in Natural Media
CMG 合作研究:自然介质中渐进前污染物传输的稳定模型
  • 批准号:
    1025417
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    2010
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CMG COLLABORATIVE RESEARCH: Quantum Monte Carlo Calculations of Deep Earth Materials
CMG 合作研究:地球深部材料的量子蒙特卡罗计算
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    1024936
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CMG Collaborative Research: Non-assimilation Fusion of Data and Models
CMG协同研究:数据与模型的非同化融合
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CMG COLLABORATIVE RESEARCH: Nonlinear elastic-wave inverse scattering and tomography- from cracks to mantle convection
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