Collaborative Research: Nonlinear Dynamics of Streamflow: Classification, Predictability and Forecasting

[9508085] Lall了解不同时间和空间尺度下水流的动态（或时间演化）、可预测性、因果性和变异性是水文学研究的核心问题。拟议的研究旨在利用美国地质调查局（USGS）长期记录（60至114年）的每日流量，开发美国各地点流量的可预测性和动态相似性的定量概念，这些记录被认为几乎没有人为影响。直接从时间序列预测水流的方法（基于非线性动力学）将是这项研究的副产品。在气候变化研究的背景下，未来许多年可能需要使用外源模式气候数据预估河流流量。考虑到不确定的输入，重要的是评估这些预测在何种情况下以多快的速度退化为随机轨迹，而不考虑模型的形式主义。至少在定性上，了解水流对不同致病因素的敏感性也可能是令人感兴趣的。在不同的空间和时间尺度上对水流进行建模意味着什么？什么时候追求确定性、分布式或集总模型是有用的，什么时候必须诉诸纯粹的统计方法？笼统地说，径流是大尺度大气环流与缓慢变化或固定的地表条件相互作用的结果。随着感兴趣的空间尺度（如流域面积）的增加，后者可能会使水流有组织，从而限制了具有主要影响的“动态”因素的有效数量。我们将研究的一个想法是，一个更大的流域在空间上平均了气候强迫中的许多动态过程——特定地点的降雨、流域的地理特征、蒸发和土壤性质，这些在任何流域都肯定是异质的。这种平均可以减少响应的维数，我们通过溪流取样。同样，具有长时间尺度“结构”的气候波动（如厄尔尼诺-南方涛动）。可能增加蒸汽流量的可预测性。非线性动力学的最新进展，特别是非线性过程的时间序列分析，为深入了解这些过程提供了一种有趣的方法。其思想是，动力系统的单个状态变量（例如，流）的时间序列可以用于几何上重建包含潜在系统的可预测性和复杂性的基本信息的“状态空间”。可预测性是通过李雅普诺夫指数在信息理论中测量的，该指数测量状态空间中附近轨迹的散度率。通过检查状态空间的填充密度以及这种密度在状态空间上的变化，通过广义维度来测量复杂性。设计了在重构状态空间中预测状态变量的策略。我们使用这些方法对大盐湖体积的分析非常富有成果，并使我们坚信，对河流过程的本质的重要见解和不同尺度的河流理论模型的发展将是可能的。本文提出的方法是系统地分析各种美国长流量数据集，以(1)查看是否可能从时间序列中重建底层动态，(2)估计李雅普诺夫指数作为可预测性的度量，(3)估计广义维度以描述底层动态的复杂性，(4)制定有效的日流量预测策略。(5)识别可预测性、复杂性和预测能力对气候属性和流域属性（如流域面积）的影响。特别令人感兴趣的是物理阈值，在该阈值处，系统的响应经历了对不同性质动力学的变化。在使用非参数回归方法从时间序列中恢复系统不变量后，对这些阈值的存在性进行了探讨。

9508085 Lall An understanding of the dynamics (or time evolution) of streamflow, its predictability, causality and variability at different time and space scales is a central issue in hydrologic research. The proposed research seeks to develop quantitative notions of predictability and dynamical similarity of streamflow at U.S. sites, using long USGS records (60 to 114 years) of daily streamflow that are presumed to have little human impact. Methods (based on nonlinear dynamics) for forecasting streamflow directly from the time series will be a byproduct of the research. In the context of climate change research, projections of streamflow using exogenous model climate data may be needed for many years in the future. Given uncertain inputs, it is important to assess how quickly, and under which conditions such predictions deteriorate into random traces, irrespective of model formalism. It may also be of interest to know, at least qualitatively, the sensitivity of streamflow to different causative factors. What are the implications for modeling streamflow at different space and time scales? When is it useful to pursue deterministic, distributed or lumped models, and when must one resort to a purely statistical approach? Loosely speaking, streamflow results from the interaction of large scale atmospheric circulation with slowly varying or fixed surface conditions. The latter may lend organization to streamflow as the spatial scale of interest (e.g., drainage area) increases, thus limiting the effective number of "dynamical" factors that have major influence. An idea which we will investigate is that a larger basin spatially averages over the many dynamical processes in the climatic forcing--rainfall at specific locations, geographical features in the basin, evaporation and soil properties which are certainly heterogeneous across any watershed. This averaging may reduce the dimension of the response which we sample by streamflow. Similarly, climatic fluctuations that have "structure " at long time scales, (e.g., El Nino Southern Oscillation). may increase stearmflow predictability. An interesting methodology for insights into such processes is provided by recent advances in nonlinear dynamics, and in particular for time series analysis from nonlinear processes. The idea is that time series for a single state variable (e.g., streamflow) for the dynamical system can be used to geometrically reconstruct a "state space" that contains the essential information on predictability and complexity of the underlying system. Predictability is measured in information theoretic terms through Lyapunov exponents that measure the rate of divergence of nearby trajectories in state space. Complexity is measured through generalized dimensions by an examination of how densely the state space is filled and variations in such a density over the state space. Strategies for forecasting the state variable in the reconstructed state space are devised. Our analyses of the Great Salt Lake volume using these methods have been very fruitful, and have given us the firm conviction that significant insights into the nature of the streamflow process and development of theoretical models for streamflow at different scales will be possible. The approach proposed here is to systematically analyze variety of long US streamflow data sets, to (1) see if a reconstruction of the underlying dynamics is possible from the time series, (2) estimate Lyapunov exponents as a measure of predictability, (3) estimate generalized dimensions to describe the complexity of the underlying dynamics, (4) to develop effective forecasting strategies for daily streamflow, and (5) identify how predictability and complexity and forecasting ability vary the climatic attributes and basin attributes such as drainage area. Of particular interest are physical thresholds at which the response of the system undergoes a change to qualitatively different dynamics. The existence of such thresholds is probed after the recovery of system invariants from the time series using nonparametric regression methods with respect to the parameters of interest.