Soil Moisture, Subsurface Storage and Runoff as a Low Dimensional Dynamical System

该研究将验证降雨径流过程可以被表示为一个低维动力系统的假设，该系统受地形、土壤、地质和气候变化的影响。所谓“低维”是指将过程近似为非线性常微分方程系统所需的最小状态变量数。我们研究了降雨径流由多孔土壤和浅层地下水循环控制的情况，以及集水区储存水和产生径流的能力取决于系统的储存-通量关系的性质。蒸散发作为土壤水分储存的参数函数的作用也将被研究。“低维”模型将作为代表局部过程的非线性偏微分方程（理查德方程）的物理替代品。在构建流域动力学模型时，关键问题是分离或区分流域动力学的空间和时间组成部分，以便阐明所涉及的过程的重要机制，而不丢失关键的非线性结构。本研究包括以下内容：(1)基于饱和-非饱和流动偏微分方程的有限元解进行综合数值试验，建立地形积分本构关系（如蓄能-流量关系）。拟议的研究将建立在一系列稳态数值实验（Lee, 1993; Duffy, 1994）上，对具有均匀土壤特性的二维山坡几何形状进行研究。先前的研究发现，地下水位的补给和地下水流是至少两个状态变量的非线性函数：综合土壤湿度和综合饱和蓄水量。目前的目标是将这些实验扩展到完全三维和时变流量的情况下，并测试土壤分层和变异性对非线性储存-通量关系和径流响应的作用。(2)开发和测试PA中部页岩山流域（8公顷，森林，集水区）状态变量和通量的尺度和空间整合程序（J. Lynch等，1976）。20世纪70年代，页岩山进行了一项独特的实验，以评估先前的土壤湿度对风暴流量和时间的影响。人工降雨8次，初始湿度从干燥到非常潮湿。对整个流域多个深度的土壤水分和饱和储量进行了综合计算。虽然这似乎是一个直截了当的问题，但分散的现场观测的空间积分需要一个适当的加权函数。Duffy（1994）提出了从次对称分布中导出的加权函数，以及对每个山坡或零阶盆地的山坡轨迹进行局部重新标度。这种缩放和平均方法将用于Shale Hills数据库。现场估计的存储-通量关系将与(1)中的数值实验进行比较。(3)一种称为适当正交分解（POD）的独立方法允许直接从随机场数据或控制的pde重建动力学的基本空间结构。该方法被广泛应用于探测流体动力学湍流（Lumley, 1967）、气候场演化（North等，1982）和非线性振动（Cusumano, and Bai, 1993， Cusumano等，1993，Lin和Cusumano， 1993， Cusumano等，1994）中的相干结构。POD方法将应用于Shale Hills实验的现场数据和Richard’s方程，以提供对系统建模所需的维数或状态变量数量的测量。

9418674 Duffy This research will test the hypothesis that the rainfall-runoff process can be represented as a low-dimensional dynamical system, forced by topographic, soil, geologic and climatic variability. By "low-dimensional " we mean the minimum number of state variables required to approximate the processes as a system of nonlinear ordinary differential equations. We examine the case where rainfall-runoff is controlled by porous soils and shallow groundwater circulation, and where the ability of the catchment to store water and yield runoff depends on the nature of the storage-flux relationships of the system. The role of evapotranspiration as a parametic function of soil-moisture storage will also be examined. The "low-dimensional" model will serve as a physically-based alternative to the nonlinear partial differential equations (Richard's equation) representing the local processes. In constructing the dynamical model, the essential problem is to separate or distinguish among spatial and temporal components of watershed dynamics, such that the important mechanics of the processes involved are elucidated, without loss of critical nonlinear structure. The research has the following elements: (1) Comprehensive numerical experiments based on finite element solutions to the partial differential equations for saturated-unsaturated flow will be performed To establish terrain-integrated constitutive relations (e.g. storage-flux relations). the proposed research will build on a series of steady-state numerical experiments (Lee, 1993; Duffy, 1994) for two dimensional hillslope geometry with uniform soil properties. This previous work found that recharge to the water table and subsurface flow to the stream were nonlinear functions of at least two state variables: the integrated soil moisture and integrated saturated storage. The present objective is to extend these experiments to the case of fully three dimensional and time varying flow, and test the role of soil stratification and variability on nonlinear storage-flux relations and runoff response. (2) Develop and test procedures for scaling and spatial integration of the state variables and fluxes for the Shale Hills watershed, 8 hectare, forested, catchment in central PA (J. Lynch et al, 1976). Shale Hills was the site of a unique experiment in the 1970's to evaluate the effects of antecedent soil moisture on stormflow volume and timing. Rainfall was artificially applied for 8 events with initial moisture ranging from dry to very wet. a comprehensive accounting of soil moisture and saturated storage at multiple depths was made over the entire watershed. Although if may seem to be a straight forward problem, spatial integration of scattered field observations requires an appropriate weighting function. Duffy (1994) has proposed weighting function derived from the hypsometric distribution, and a local rescaling of hillslope trajectories for each hillslope or zero-order basin. This scaling and averaging method will be carried out for the Shale Hills data base. Field-estimated storage-flux relations will be compared with the numerical experiments in (1). (3) An independent method known as proper orthogonal decomposition (POD), allows the essential spatial structure of the dynamics to be reconstructed directly from random field data or from the governing pde's. The method is widely applied to detecting coherent structures in hydrodynamics turbulence (Lumley, 1967), the evolution of climatic fields (North et al, 1982), and nonlinear vibration (Cusumano, and Bai, 1993, Cusumano et al, 1993, Lin and Cusumano, 1993, Cusumano et al, 1994). The POD method will be applied to the field data of the Shale Hills experiment and the Richard's equation to provide a measure of the dimensionality, or number of state variables required to model the system.