Modeling Hydrologic Change: Statistical Methods - Chapter 10

10 Modeling Change 10.1 INTRODUCTION The analysis of hydrologic data is assumed to be capable of detecting the effects of watershed change. Graphical methods and statistical tests can be used to support hypotheses developed on the basis of independent information. However, detection is rarely the ultimate goal. Instead, the purpose of detection is generally to model the effect of the watershed change so that the effect can be extrapolated to some future state. For example, if the study design requires the specification of ultimate watershed development, knowing the hydrologic effect of partial development in a watershed may enable more accurate forecasts of the effects that ultimate development will have. Detecting hydrologic effects of watershed change is the first step. Modeling the effect is necessary in order to forecast the effects of future change. Therefore, one graph or one statistical test will probably be inadequate to detect hydrologic change. Several statistical tests may be required in order to understand the true nature of the effect. Is the change in the distribution of the hydrologic variable due to a change in the function itself or to a change in one or more of the moments? Does a statistically significant serial correlation reflect a lack of independence or is it due to an unidentified secular trend in the data? Questions such as these emphasize the need for a thorough effort in detecting the true effect. An inadequate effort may ultimately lead to an incorrect assessment of the hydrologic effect and, therefore, an incorrect forecast. Assuming that the correct hydrologic effect has been detected, the next step is to properly model the effect. Given the substantial amount of natural variation in hydrologic data, it is important to use the best process and tools to model the hydrologic effect of change. Using a simple model is best only when it is accurate. With the availability of computer-aided modeling tools, the ability to model non-stationary hydrologic processes has greatly improved. This chapter provides a discus-sion of the fundamentals of a few basic modeling tools. A complete introduction to all modeling tools would require several volumes. 10.2 CONCEPTUALIZATION The first phase of modeling, conceptualization, is often lumped with the second phase, formulation. While the two phases have some similarities and commonalities, they also differ in important ways. The conceptualization phase is more general and involves assembling resources, including underlying theory and available data mea-surements. It involves important decisions about the purpose of the model, approx-imate complexity of the model, physical processes to be modeled, and criteria that will be used to judge model accuracy. © 2003 by CRC Press LLC The first step is to decide on the purpose of the model, that is, the task for which the model is being designed. For example, a model may be designed to adjust measured annual maximum discharges in a nonstationary series for the effects of urbanization. It may be intended to provide a prediction of the effect but not the magnitude of the annual maximum discharge. By limiting the purpose of a model, the required complexity can usually be reduced, but the flexibility of the model is also limited. When identifying the purpose of a model, it is important to try to identify the range of potential model users to ensure that it will be adequate for their purposes. This may prevent its misuse. As part of the use of the model, the type of model must be identified. Models can be empirical, theoretical, or some combination of the two. Models can be real-time forecast models or design storm models. Continuous, multi-event, and single-event models are possible. Models can be deterministic or stochastic. A model may provide a single point estimate or the entire distribution of the output variable. Decisions about the type of model will influence its flexibility and quite likely its accuracy. The next aspect of model conceptualization is to identify the variables that will be involved, including both the criterion or output variables and the predictor or independent variables. The fewer variables involved, the less data required, both for calibration and for use. However, the fewer variables involved, the less flexible the model and the fewer situations where the model will be applicable. It is generally unwise to require inputs that are not readily available. For example, the lack of spatial land-use data from past decades makes it very difficult to accurately model the effects of land use on the annual flood series for a locale. Model variables largely reflect the physical processes to be modeled. Decisions about the hydrologic, meteorologic, and geomorphic processes to be represented in the model affect the required variables. For some very simple empirical models, decisions about the processes to be modeled are unnecessary. The available data are largely known and the primary decision concerns the functional form to be used. The expected availability of data is another consideration in model conceptual-ization. It makes little sense to propose a model for which data for the necessary inputs are rarely available. The quality, or accuracy, and quantity of data generally available are both factors to be considered. For each input variable, the range of measured data commonly available must be compared to the range over which the model is expected to function. For example, if data from watersheds with a range of impervious area from 0% to 20% are available, a model developed from such data may not be accurate for predictions on watersheds at 50% to 75% impervious-ness. This would require extrapolation well beyond the bounds of the calibrated data. In the conceptualization stage, it is also important to decide the level of accuracy required and the criteria that will be used to judge the adequacy of the model. Accuracy criteria such as correlation coefficients, model biases, and standard errors of estimate are typically used in empirical modeling, but factors such as the failure of correlation coefficients for small databases must be addressed. The selection of the level of significance to be used in hypothesis testing should be made at this stage before data are assembled and used to calibrate the model. Some thought must be given to the potential problem that the model will not meet the accuracy criteria © 2003 by CRC Press LLC established for its use. The feedback loop to revise the model or expend the resources necessary to collect more data should be considered. It may seem that this somewhat nonquantitative phase of modeling is not impor-tant, while in reality it is well known that an improperly conceived model may lead to poor decisions. Example 10.1 Consider the problem of revising the TR-55 (Soil Conservation Service, 1986) peak-discharge, model-adjustment factor for ponded storage. Currently, the adjustment uses only a multiplication factor Fp, where the value of Fp depends only on the percentage of the watershed area in ponds or swampy lands. Thus, the peak discharge is the product of four factors: Fp; the drainage area, A; the depth of runoff, Q; and the unit peak discharge, qu: qp = Fpqu AQ (10.1) This adjustment factor does not account for the depth of the pond dp or the surface area of the watershed that drains to pond Ac, both of which are important. In this case, the type of model is set by the type of model to which the new model will be attached. Since the discharge model of Equation 10.1 is a deterministic, single-event, design-storm model, the new method of adjusting for ponds will be classified as the same type of model. The adjustment should also be relatively simple, such as the multiplication factor Fp of Equation 10.1. The statement of the problem indicates that the model would be conceptually more realistic if both dp and Ac were used to determine the volume of required storage. If the objective is to provide a storage volume either at one location within the watershed or distributed throughout the watershed that can contain all of the runoff from the contributed area Ac for design depth Q, then the volume that needs to be stored is AcQ. If the pond or swamp lands have a total area of Ap, then the depth of storage would be: dp = (Ac/Ap)Q If all runoff during the design storm is contained within the pond, then the area that drains to the watershed outlet is A − Ac. Thus, the adjustment factor Fp of Equation 10.1 is not needed, and the peak discharge can be computed by qp = qu(A − Ac)Q (10.2) The conceptual development of this model (Equation 10.2) assumes that the surface area that drains to storage does not produce any runoff. The conceptual development of Equation 10.1 does not require that assumption. Instead, Equation 10.1 assumes that the entire watershed contributes flow to the watershed outlet, that the storage areas are effective in storing water, and that the depth of storage is not relevant. However, the depth of storage is relevant. Equation 10.1 assumes that the same value © 2003 by CRC Press LLC of Fp is used regardless of the available storage, which is conceptually faulty. The volume of storage available and the location of the storage are conceptually realistic, and the model should be formulated to incorporate such concepts. 10.3 MODEL FORMULATION In the conceptualization phase of modeling, specific forms of model components are not selected. The physical processes involved and the related variables are identified. In the formulation stage, the algorithm to be calibrated is formalized. Specific functional forms are selected, often after considering graphical summaries of the available data. Interactions between model components must be specified as well as ways that data will be summarized in order to compute the accuracy and rationality of the calibrated model considered. The main objective in this phase of modeling is the assembling of the model that will be calibrated, or fitted, to the measured data and then assessed. 10.3.1 TYPES OF PARAMETERS Functional forms used in models can vary in position, spread, and form. Specifically, the characteristics of a function depend on parameters of the function. Three general types of parameters are available: location, scale, and shape. Location parameters can be used to position a function on the ordinate, the abscissa, or both. Scale parameters control the spread or variation of the function. Shape parameters control the form of a function. Not all models include all types of parameters. Some models include just one parameter, while other models require one or more of each type of parameter to represent data. In the bivariate linear model ˆ = a + bX, the intercept a acts as a location parameter and the slope coefficient b acts as a scale parameter. The intercept positions the line along the y-axis, and it can thus be classed as a y-axis location parameter. The slope coefficient scales the relationship between Y and X. A more general model is Y = C + C3(X − C2 )C4 (10.3) in which C1 acts as a y-axis location parameter, C2 acts as an x-axis location parameter, C3 acts as a scale parameter, and C4 acts as a shape parameter. The two location parameters enable the function to shift location along the axes. The shape parameter C4 enables the function to take on a linear form (C4 = 1), a zero-sloped line (C4 = 0), an increasing function with a decreasing slope (0 < C4 < 1), or an increasing function with an increasing slope (C4 > 1). C4 could also be negative, which yields corresponding shapes below the axis created with C4 greater than 0. The point of Equation 10.3 is that greater flexibility in form can be achieved through more complex functions. The price that must be paid is the potential decrease in the degrees of freedom due to the greater number of coefficients that must be fit and the requirement of a fitting method that can be used to fit the more complex model structures. Increasing the flexibility of a model by adding additional parameters may create problems of irrationality. A more flexible structure may be problematic © 2003 by CRC Press LLC if nonsystematic variation in the data causes the function to produce irrational effects. Polynomial functions are one example. Flexibility is both an advantage and disad-vantage of complex functions. Polynomials can fit many data sets and produce good correlation coefficients, but they often suffer from polynomial swing. Example 10.2 Equation 10.3 includes two location parameters, a scale parameter, and a shape param-eter. The effect of variation in each of these is shown in Figure 10.1. In Figure 10.1(a), the y-axis location parameter C1 is varied, with the line moving vertically upward as C1 is increased. In Figure 10.1(b), the x-axis location parameter C2 is varied, with the line moving horizontally to the right as C2 increases. In Figure 10.1(c), the value of the scale parameter C3 is varied, with the line increasing in slope as C3 increases; note that the intercept remains fixed as C1 and C2 are constant. In Figure 10.1(d), the shape parameter C4 is varied. For C4 equal to 1, the function is linear. For C4 greater than 1, the function appears with an increasing slope (concave upward). For C4 less than 1, the function has a decreasing slope (concave downward). In Figure 10.1(e), both C3 and C4 are varied, with the shape and scale changing. 24 20 20 20 16 16 16 12 12 12 8 8 8 4 4 4 0 0 0 0 1 2 3 0 1 2 3 0 1 2 3 (a) C2 = –1; C3 = 2.5; C4 = 1.5 (b) C1 = 0.5; C = 2.5; C4 = 1.5 (c) C 1 = 0.5; C2 = −1; C4 = 1.5 20 20 16 16 12 C = 1 12 8 8 4 C = 0.5 4 0 0 0 1 2 3 0 1 2 3 (d) C1 = 0.5; C2 = −1; C3 = 2.5 (e) C1 = 0.5; C2 = −1 FIGURE 10.1 Effect of parameters of Equation 10.3. © 2003 by CRC Press LLC ... - tailieumienphi.vn