CE 504 Computational Hydrology (Spring 2004) Introduction Fritz R. Fiedler

Transcription

1 CE 504 Computational Hydrology (Spring 2004) Introduction Fritz R. Fiedler 1. Computational Methods a) Overview b) Steps in developing c) Steps in applying 2) Computational Hydrology a) Hydrologic cycle b) Application to components and integrated modeling c) Uncertainty, scale, and variability 3) Course Objectives 4) Course Structure Computational Methods Computational methods are computer-based implementations of methods to solve mathematical equations that describe physical processes. Computers are routinely used to solve almost all problems of practical significance, using tools ranging from generic spreadsheet programs to highly complex and specialized codes. While there is no reason to re-create generic tools, there is often a need to develop and apply specialized codes (simple or complex). There are many solution methods that can be implemented on a computer, including analytical, numerical, and empirical (observational). Analytical methods are exact or approximate solutions, in the form of an equation (or set of equations). These methods include laplace transforms used to solve ODEs, separation of variables for PDEs, and variational principles. Numerical methods are approximate solutions in the form of a set of discrete values. Finite differences/elements and cellular automata are numerical methods. Empirical methods involve model inference from observations. Examples include least squares function fitting (regression), transforms (such as wavelets), and probability density estimation. Implementation involves instructing the computer to execute the calculations that comprise the solution method. For any given method, there are numerous ways computers can do the work for you. For example, to solve a hydrologic routing problem (say, using a fourth-order Runge-Kutta method to integrate the governing ODE), the generic tools Excel and MathCad are useful. If this calculation needed to be repeated many times, it may be more efficient to write your own computer program or better yet, find a specialized code that is already written and tested. Each type of problem and relevant solutions method(s) poses unique challenges, and the development and application of specific computational methods varies accordingly. Often, the approach is somewhat dictated by the class of problem and

2 relevant methods. There are some subtle but important differences between the general approach to developing a computational method (which may involve writing a computer program) and applying a computational method (usually using a program someone else has written). Steps in Computational Method Development include: Understand the important aspects of the physical system to be modeled. Identify the appropriate equations. Select a solution method. Write a program to solve the equations with the selected method. Verify that the equations are solved correctly. Validate that the equations/method adequately represent the process. Calibrate the model if necessary. Use the model to simulate the physical system. Critically evaluate results. Steps in Computational Method Application include: Understand the important aspects of the physical system to be modeled. Identify the appropriate equations. Select software that solves these equations (rely on previous verification and validation). Calibrate the model if necessary. Use the model to simulate the physical system. Critically evaluate results. Computational Hydrology Broadly speaking, computational hydrology encompasses all computational methods applied to hydrologic systems (or parts of these systems), both natural and manmade, and the interaction of hydrologic systems with the earth and its inhabitants. We will not cover every aspect of computational hydrology. The hydrologic cycle is a convenient way to depict the global hydrologic system and the role of the hydrologic processes that govern the behavior of hydrologic systems; note that humans interact with and modify the natural cycle in many ways.

3 The movement of water throughout the system is governed by fundamental physical laws (e.g., conservation of mass, momentum, and energy; the ideal gas law). These laws are represented by equations, which may vary in complexity based on assumptions about the system. For example, streamflow could be simulated using the three-dimensional Navier-Stokes equations, which describe fluid flow at a fundamental level. However, this degree of complexity is seldom warranted in hydrology, and the equations are simplified by reducing the dimension of the problem. For example, in deriving the depth-averaged Navier-Stokes equations, it is assumed that vertical acceleration is negligible (among other assumptions), resulting in the St. Venant equations. Or, simplification can be achieved by assuming certain components in the equations are negligible. The St. Venant equations can be simplified by assuming that the friction slope equals the bed slope (local acceleration, convective acceleration, and unbalanced pressure negligible) resulting in the kinematic wave approximation. To simulate hydrologic systems, there must be a way to couple the equations that describe the individual processes, or to model the integrated effects of the processes that comprise the system. In one way to model a watershed, precipitation is used as an input, and various equations (such as those that describe infiltration and evapotranspiration) are used to describe the transformation of precipitation into streamflow; once in the stream, the movement of water to the basin outlet is simulated by, for example, the St. Venant equations. Clearly there must be some assumption about how the runoff enters the stream, i.e., how the streams and hillslopes are coupled. Alternatively, precipitation can be transformed directly into streamflow at the outlet, lumping the infiltration, evapotranspiration, and streamflow processes into a single function. This is part of the difference between distributed and lumped hydrologic models. Distributed models also

4 typically simulate, to some degree, the effects of spatially variable physical characteristics for a given process. For example, soil type variability causes infiltration and runoff generation to vary across the landscape, thus runoff generation from various soil units and the interaction between units can be modeled. However, we are never able to completely describe or simulate the variability, so even distributed hydrologic models are lumped at some spatial scale. It is essential to have a basic appreciation for and understanding of the effects of temporal and spatial variability in hydrology when developing or applying computational methods. It is not possible to perfectly describe or simulate variability, which is one reason why there is always some degree of uncertainty in model results. Other sources of uncertainty include imperfect data, discrepancies between how the component or system actually behaves and how we simulate it (model structure error), and the propagation of input and data errors through the model. The temporal and spatial scale at which hydrologic processes are simulated interacts with uncertainty. These are active areas of hydrologic research. Course Objectives This course aims to provide students with: A general knowledge of the types of models available for simulating hydrologic processes at various scales; An introduction to some of the mathematical and numerical methods used in hydrologic modeling, concentrating on mechanistic methods; Knowledge of how model parameters are determined, including issues of scale and spatial variability; An understanding of how to calibrate hydrologic models and evaluate the quality of calibration; and Experience using select widely available models. Course Structure The objectives will be achieved through a combination of lectures, activities, homework assignments, self-study, a modeling project, and a final exam. There is no text for this class, so a wide range of resources will be used. While I will provide you with much information, I will also expect you to be resourceful; being able to find information about a particular subject on your own is a valuable skill (and I will provide assistance with this when asked). As with any course, you should expect to spend at least 2-3 hours working outside of class for every hour of class. Exam (~20%) There will be one comprehensive final exam, likely in take-home format. Assignments (~40%) You will be given assignments throughout the semester to enforce and expand upon what we discuss in class. These will involve reading and summarizing articles, solving hydrologic problems using the methods presented in class and generic software

5 tools (e.g., Excel, Mathcad, etc.), and writing small programs using a computer language of your choosing. The programming requirements will not be large, and I expect that student abilities will vary widely, but I will expect everyone to improve upon their current ability. Project (~40%) For the semester long project, you are to develop or apply a hydrologic model to system of your choosing. If you choose to develop your own, the model may be quite simple. If you use an existing model, what and how you model will be more sophisticated. In either case, I expect you to address the steps in model development and application. For example, if you develop a code to model infiltration with the Green- Ampt equation, I expect you to compare your model results to other calculations to show that the model solves the equation correctly. The point is not simply to gain specific experience, but also to learn the general approach to modeling anything.