Modeling Hydrologic Change: Statistical Methods - Chapter 12

12 Sensitivity Analysis 12.1 INTRODUCTION Modeling is central to many hydrologic decisions, both in analysis and in design. Decisions in hydrologic engineering often require the development of a new model or the use of an existing model. When developing a new model, the response of the model should closely mimic the response of the real-world system being modeled. Before using an existing model, the modeler should verify that the model is appro-priate for solving the type of problem being addressed. When a modeler uses an existing model developed by someone else, the modeler should be sure that he or she completely understands the way that the model responds to the input. In all of these cases, the modeler should understand the model being used. Sensitivity analysis is a modeling tool that is essential to the proper use of a model because it enables the model user to understand the importance of variables and the effects of errors in inputs on computed outputs. Questions such as the following can be answered by performing a sensitivity analysis: · Which are the most important variables of a model? · What is the effect of error in inputs on the output predicted with the model? · What physical hydrologic variables are most likely related to fitted model coefficients? · In calibrating a model, are all of the model coefficients equally important to the accuracy of the model? Each of these questions suggests a relationship between two or more variables or the effect of one factor on another. In modeling terms, they are concerned with sensitivity, whether it is the sensitivity to errors, the sensitivity of goodness-of-fit criteria, or the sensitivity to watershed characteristics. Questions such as those presented above can be answered by applying an important modeling tool, specifi-cally sensitivity analysis. Of special interest in hydrologic analysis is the detection of effects of land use change. Watershed change can cause abrupt or gradual change in measured flood data or in characteristics of the data, such as the 100-year peak discharge. Just as the amount of watershed change is important, the spatial pattern of land use also can significantly influence measured hydrologic data. Specific questions that may be of interest include: © 2003 by CRC Press LLC · How sensitive are annual maximum discharges to gradual or abrupt land use change? · How sensitive are the moments of an annual flood series to historic information? · How sensitive is the 100-year peak discharge to the existence of an outlier in the sample data? · How sensitive is the 100-year peak discharge to changes in channel roughness? Modeling studies are often used to answer such questions. The clarity of the answers to such questions depends, in part, on the way that the results are presented. Better decisions will be made when the results are organized and systematically presented. An understanding of the foundations of sensitivity analysis should ensure that the results are presented in ways that will ensure that the modeling efforts lead to the best possible decisions. 12.2 MATHEMATICAL FOUNDATIONS OF SENSITIVITY ANALYSIS 12.2.1 DEFINITION Sensitivity is the rate of change in one factor with respect to change in another. Although such a definition is vague in terms of the factors involved, nevertheless it implies a quotient of two differentials. Stressing the nebulosity of the definition is important because, in practice, the sensitivity of model parameters is rarely recog-nized as a special case of the concept of sensitivity. The failure to recognize the generality of sensitivity has been partially responsible for the limited use of sensi-tivity as a tool for the design and analysis of hydrologic models. 12.2.2 THE SENSITIVITY EQUATION The general definition of sensitivity can be expressed in mathematical form using a Taylor series expansion of the explicit function: O = f(F, F ,K, F ) (12.1) where O is often a model output or the output of one component of a model and the Fi are factors that influence O. The change in factor O resulting from change in a factor Fi is given by the Taylor series: f(F + F, F j¹i ) = O0 + ¶O0 i 2 F + 2! ¶F 0 F2 +L (12.2) in which O0 is the value of O at some specified level of each Fi. If the nonlinear terms of Equation 12.2 are small in comparison with the linear terms, then © 2003 by CRC Press LLC Equation 12.2 reduces to f(F + F, F j¹i ) = O0 + ¶O0 F (12.3) i Thus, the incremental change in O is O0 = f(F + F, F j¹1) − O0 =  ¶O0  F (12.4) i Since it is based only on the linear terms of the Taylor series expansion, Equation 12.4 is referred to herein as the linearized sensitivity equation. It measures the change in factor O that results from change in factor Fi. The linearized sensitivity equation can be extended to the case where more than one parameter is changed simulta-neously. The general definition of sensitivity is derived from Equations 12.1 and 12.4: S = ¶O0 = f(F + i F, F j¹i ) − f(F, F2,K, Fn ) (12.5) i 12.2.3 COMPUTATIONAL METHODS The general definition of sensitivity, which is expressed in mathematical from by Equation 12.5, suggests two methods of computation. The left-hand side of Equation 12.5 suggests that the sensitivity of O to changes in factor Fi can be estimated by differentiating the explicit relationship of Equation 12.1 with respect to factor Fi: ¶O0 ¶F (12.6) Analytical differentiation is not used extensively for evaluating the sensitivity of hydrologic models because the complexity of most hydrologic models precludes analytical differentiation. The method of factor perturbation, which is the computational method suggested by the right side of Equation 12.5, is the more commonly used method in hydrologic analysis. The right-hand side of Equation 12.5 indicates that the sensitivity of O to change in Fi can be derived by incrementing Fi by an amount Fi and computing the resulting change in the solution O. The sensitivity is the ratio of the two changes and can be expressed in finite difference form as follows: S = O f(F + F, Fj j¹i)− f(F, F ,K, F ) F i (12.7) © 2003 by CRC Press LLC 12.2.4 PARAMETRIC AND COMPONENT SENSITIVITY A simplified system or a component of a more complex system is described by three functions: the input function, the output function, and the system response, or transfer, function. The transfer function is the component(s) of the system that transforms the input function into the output function. In a simple form, it could be a probability distribution function that depends on one or more parameters. As another example, the transfer function could be an empirical time-of-concentration formula that depends on inputs for the length, slope, and roughness. In a more complex form, the transfer function could consist of all components of a continuous simulation model. Sensitivity analyses of models can be used to measure the effect of parametric variations on the output. Such analyses focus on the output and response functions. Using the form of Equation 12.5 parametric sensitivity can be mathematically expressed as Spi = ¶O = f(P + i P; P j¹i )− f(P, P ,K, P ) P (12.8) where O represents the output function and Pi is the parameter of the system response function under consideration. Unfortunately, the general concept of sensitivity has been overshadowed by parametric sensitivity. As hydrologic models have become more complex, the der-ivation of parametric sensitivity estimates has become increasingly more difficult and often impossible to compute. However, by considering the input and output functions, the general definition of sensitivity (Equation 12.5) can be used to define another form of sensitivity. Specifically, component sensitivity measures the effect of variation in the input function I on the output function: Sc = ¶O = O I (12.9) Combining component and parameter sensitivity functions makes it feasible to estimate the sensitivity of parameters of complex hydrologic models. For example, in the simplified two-component model of Figure 12.1, the sensitivity of Y to vari-ation in P1 and the sensitivity of Z to variation in P2 are readily computed using sensitivity as defined by Equation 12.6: S = ¶P and S2 = ¶P (12.10) Component 1 X h1(P ) Y Component 2 h2 (P ) Z FIGURE 12.1 Two-component model. © 2003 by CRC Press LLC However, the sensitivity of the output from component 2 to change in the parameter of component 1 cannot always be estimated directly from the differential ¶Z/¶P1. In such case, the component sensitivity function of component 2 can be used with the parametric sensitivity function S to estimate the sensitivity of Z to change in P . Specifically, the sensitivity of ¶Z/¶P equals the product of the component sensitivity function ¶Z/¶Y and the parametric sensitivity function ¶Y/¶P1: ¶Z ¶Z ¶Y ¶P ¶Y ¶P (12.11) Whereas the differentials ¶Z/¶Y and ¶Y/¶P are often easily derived, an explicit sensitivity function ¶Z/¶P1 can be computed only for very simple models. When a solution cannot be obtained analytically, the numerical method of Equation 12.7 must be used. Example 12.1 Consider the simple two-component model of Figure 12.1 as a hydrologic model with X being the rainfall input, h1(P1) being a unit hydrograph representation of the watershed processes, Y being the flow from the watershed into the channel system, h2(P2) being the channel system, and Z being the discharge at the watershed outlet. The unit hydrograph component is represented by Zoch’s (1934) simple, linear storage (or single linear reservoir) unit hydrograph: Y = Ke−KXt (12.12) where Xt and Yt are the input (rainfall) and output (surface runoff) at time t, and K is the unit hydrograph parameter. The transfer of Yt into Zt, the discharge at the watershed outlet, uses the Convex routing (U.S. Soil Conservation Service, 1974) method: Zt = wYt + (1 − w)Yt−1 (12.13) in which w is the parameter of the component and corresponds to the routing coefficient of the Convex routing procedure. In performing a complete sensitivity analysis of this model, two component sensitivities (¶Z /¶Y and ¶Y /¶X), three parametric sensitivities (¶Z /¶K, ¶Z /¶w, and ¶Yt/¶K), and one model sensitivity (¶Zt/¶Xt) would be of interest. For this simple model, they can be computed either analytically or numerically. Analytically, the two component sensitivities are ¶Y = −K2e−KXt t ¶Zt ¶Y (12.14) (12.15) © 2003 by CRC Press LLC ... - tailieumienphi.vn