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R184 (P, Web version) rules for responsible modelling |
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OPTIMAL COMPLEXITY, RELIABILITY, ERROR ANALYSIS, PARAMETER OPTIMIZATION, ACCURACY AND SENSITIVITY ANALYSIS FOR LARGE-SCALE, LONG-TERM, CONTINUOUS, DETERMINISTIC SURFACE WATER QUALITY MODELLING by William James
This web paper, abstracted from an obsolete version of my booklet of the same title, is an attempt to review sensitivity analysis, its purpose, methods, and its use in long-term deterministic surface water quality modelling. Optimal complexity, reliability, error and uncertainty analysis, parameter optimization, accuracy and sensitivity analysis appropriate for large-scale, ultra-long-term, continuous modelling are briefly reviewed. A heuristic methodology is briefly described. Towards the end, approaches using principles of fuzzy reasoning are developed in order to reduce, on a logical basis, the amount of computing required. Finally a framework and some simple rules for what is here termed reliable modelling are presented.
Unfortunately the word model is used as a verb, adjective and noun, and means anything from an idea (as in paradigm) to a tall, skinny person who walks peculiarly. To be more precise, in this paper when the term model is applied to a real area, it should be taken to mean the application of a deterministic surface water quality model (WQM). This is what we get when a generalized computer code or program has been attached to a specific, hydro-topographic input-data-file, calibrated and executed. Once the water quality code is tied directly to a tract of landscape, whether already existing, or a proposed development, it becomes a model of that tract. In this sense, HSPF is better labelled than is SWMM. SWMM does not arrive firmly attached to one and only one particular datafile. The point is significant because the data file determines which processes are to be deleted, or rendered relatively inactive. Only this locally-applied model, in this restricted sense of the word, can be analyzed for sensitivity, parameter optimization and error (engineers often refer to model sensitivity. Many program packages - take for instance the U.S.EPA Stormwater Management Model SWMM (Huber and Dickinson,1988), or the U.S.EPA Hydrological Simulation Program in Fortran (HSPF, Johansen et al., 19xx)- comprise hundreds of files, hundreds of routines, and tens of thousands of lines of source code. Thus SWMM, for example, is not a simple, single model, as its name implies, but actually many separate executable and FORTRAN source-code programs together with a vast array of data files. Once applied to, shall we say, Foxran Estates, the active code together with the datafile that activates some of the processes programmed, may be called a Foxran Estates Stormwater Management Model. Nevertheless both HSPF and SWMM are loosely referred to as WQMs in the literature, and thus also in this paper. Potentially, WQMs could become more-or-less permanent fixtures in the engineering office: Fantastic as it may seem to some readers, in fact precisely this scenario is being increasingly attempted, and so model reliability issues are becoming more important than ever. At the risk of being repetitive, it does not make sense to test a WQM per se for sensitivity, parameter optimization, or uncertainty, if the WQM is a large set of generalized programs, because individual applications are likely to be radically different, by virtue of the different processes and parameter values actively used. (Later on we argue that the output objective functions and thus the post-processor also forms an inherent part of a model.) Models are used to fill in missing data or information. The only purpose of using WQMs as a design tool, is to provide reliable information about the performance of alternate arrangements of, e.g. water management facilities. Proper analysis of the available information provides knowledge of the system. If the engineering community had innate knowledge of the long-term environmental performance of large, complex, urban drainage and pollutant removal systems, then modelling would of course not be necessary. But urban stormwater drainage systems are sometimes extensively large, incorporating more conveyances and devices than can be comprehended. They are often exceedingly complicated, starting with atmospheric pollutant and storm processes at the top and ending with effluents mixing in creeks and at lake outfalls at the bottom, perhaps hundreds of kilometres apart. Furthermore, there are long-term water quality and environmental impacts that are difficult to predict, such as flooding and water quality violations at various places in the drainage system, or the gradual displacement of cold-water aquatic ecosystems at various places. It is now generally agreed that large-scale deterministic surface water quality models can better compute the almost-infinite number of calculations needed to deal with such complexity, than manual methods can. But of course, computer models are very restricting - they can only compute three things: produce a run, perform a sensitivity analysis, and optimize a parameter; they cannot predict anything! (If you disagree, check any dictionary definition or usage of the word predict. I will thank you to use the more honest word compute.) For a schematic of these and other terms and ideas, click here. Nevertheless models are often extremely useful. Decision-makers expect to deal with increasingly complex problems of water quality management by means of WQMs. Some modellers, perhaps too many, are however beginning to believe that the output from their WQMs is more-or-less reliable. A concern that is perhaps not explicitly dealt with, is the reliability of the interpretation of the model output. Few models report the uncertainty of their computed response.
I remind you that all models are wrong! How then should a model be used? The relationship between design questions, model uncertainty and output interpretation is the kernel of this paper; the point is to develop a robust, computable method for reliable, long-term, deterministic, stormwater quality management modelling. The effort starts with a clear statement of the design problems to be addressed. The general design problem considered here may be formulated as: find the optimum cost-effective array of BMPs to solve a list of external design problems(James and Robinson, 1981 a, b). To come to terms with the task of relating the modelling to the design problems, the problems may at the outset be formulated as a list of questions - each with a question mark, e.g.:
Other questions are suggested in the Conceptual Workbook, pp 50-52 (James, 1993b). The precise questions should be formally established by mutual discussions between the client, engineering committee, and the modellers. When design questions are formulated in this simple form, it is much easier to relate the design problems to the required model output, to select the best objective functions and evaluation criteria, and to provide quantitative solutions or decision support. (When the design problems are not precise, fuzzy approaches may be better - examples of fuzzy design questions are given later in this paper.) It has become essential, for a professional, careful design, to integrate as many existing, related, process codes as possible or feasible into large executable codes, and to carefully apply these programs to the existing (as-is) and proposed (to-be) urban landscapes. In this way, the best-informed decisions can be made. Unfortunately, where no suitable code is available (e.g., for an innovative, proposed BMP), model users often "fudge" the input for a similar but existing process, to approximate the new processes, so far as they are known. This is where output interpretation becomes especially problematical, and where the model-critics may successfully impute unreliability into the whole modelling exercise. Further enhancements needed in WQMs. It should be clear from the foregoing discussion, that a comprehensive WQM requires proven code, not just for all the processes involved in an as-is urban drainage system, but also for a wide range of to-be BMPs. It is in the latter capability that existing WQMs have their greatest shortcomings: code is at present supplied for only a small number of common BMPs, such as detention storage. In particular, we need new codes for processes relevant to the long-term sustainability of cold water fisheries (or at least reduction of their unsustainability). These include:
With the passage of time, WQMs like SWMM should have had more and more code appended, dealing with BMPs designed to meet new and emerging stormwater management practices and policies. User-group meetings are a vital way of keeping informed about the additional code written by various users. But the rapid upgrading of public-domain code with proven, reliable, additional process code has not happened, and is becoming an urgent concern.
Introduction In the surface water quality modelling litany, models that disregard all dry-weather processes are referred to as event models, while models that include code for processes that are active during dry weather, such as pollutant build-up, evapo-transpiration, storage depletion, recovery of loss rates, and so on, are termed continuous models. Continuous models also usually include processes associated with winter seasons. The difficulty with event modelling is that every model run is governed by arbitrary assumptions of startup conditions, which are themselves not subject to careful modelling scrutiny, such as sensitivity analysis, calibration, and error analysis. Event models are obviously only run for short durations. To the extent that the effect of these initial conditions persists through the model run, the computed results may be unreliable. There is a great deal of hydrology literature asserting that these start-up effects are indeed important (James and Robinson, 1986a, provide a review). Event modelling evolved in bygone times before computing, and it is simply no longer appropriate to adopt such simplistic methodology (James and Shivalingaiah, 1986). In this present paper, modelling is taken to be continuous, except in so far as short runs for both dry and wet events, and events that are combinations thereof, are recommended for analysis of sensitivity, parameter estimates and error. (In these cases there is no start-up error, because the initial state is given by the observed record.) Southerland (1982) examined the types of analyses that should be performed with continuous models for non-point-source control strategies to meet downstream water quality goals. BMPs simulated included multipurpose detention basins and infiltration trenches for the time span 1980 to 2005 for a 479 sq km catchment in suburban Washington, DC, using a precursor of HSPF. Her work convincingly demonstrated that assessments of nonpoint source control strategies require the concentration-frequency information that only continuous models can provide, and, more ominously, that disparate source controls may produce more water quality violations than an uncontrolled catchment, due to synergistic concentration releases. Again, only continuous models can provide the necessary information. The major difficulty associated with continuous models, on the other hand, relates to the copious amounts of input, state, and output information that must be managed. Most continuous models provide code for this purpose, but inexperienced continuous model users can soon run out of computer storage capacity. Although storage capacity for the computed response functions is probably the most stringent constraint - requiring gigabytes for moderate design problems - the literature has focused rather on computer time. This is because there are limits on the duration of the simulation that inexpensive workstations can handle in the span of a routine day in the design office. In our research, we consider runs that exceed eight hours on an inexpensive desktop microcomputer (taken to be a workstation with ordinary software costing say $5K), not to be cost-effective, or not computable (Kuch and James, 1993; Kuch, 1994). Nevertheless, many continuous model studies are being reported, and increasingly so. Chaudhury (1992) built a continuous SWMM model of the City of Providence, Rhode Island, using the RAIN, RUNOFF and EXTRAN blocks to estimate the CSO loadings to the Providence River and Upper Narragansett Bay. At the other end of the scale, Wigmosta (1991) at the University of Washington developed a continuous hydrologic model for a 37-hectare forested catchment, and a 17-hectare urbanized catchment, obtaining excellent results for storm peaks using a 12 month record of 15-minute data. James (1993a), citing work by Kuch (1994), [it's not often one gets a chance to write a statement like that] argues that 75-year continuous modelling has now become feasible, indeed desirable, in order to address concerns of sustainability. In a landmark case, the Supreme Court of Canada upheld a decision in favour of downstream riparian owners suffering fluviological impacts resulting from the urbanization of a large city. The arguments that helped to convince the judges were based on continuous modelling, whereas the losing side based their analyses on event hydrology (James, in press). Event hydrology cannot be used to evaluate fluvial morphology downstream (James and Robinson, 1986). Short-term calibration dataset. Click here for a schematic on model parameter optimization. Contrary to what is evidently a widespread belief, it is absolutely unnecessary to have very long observed time series to develop a reliable, continuous model; there need only be sufficient, good, observed time series to cover the number of events needed for the parameter optimization. It turns out that the total field-data collection effort (for calibration) is not significantly different for event and continuous models. This is because the dry-weather inter-event records can be obtained at virtually no increase in cost. Sometimes, when good data is available only in nearby drainage systems, or elsewhere in the system, two models are developed, and the optimized parameters transposed to the model lacking data. In either case, the observed time series, which must be accurate, say for a duration of a year or two, must be searched for the requisite events for calibration. Long-term dataset for inference. Of course, a credible long input time series is required for the inferential runs, when all the alternative arrays of BMPs are to be compared. But this long input time series need only be transposed from a similar hydro-meteorologic region, or it may be generated synthetically from a reasonably long observed time series in the same hydrologic region. The important point here is that the continuous input time series driving the continuous model, will be used for comparing various scenarios. It is only necessary, then, for the long-term input time series to be entirely plausible; the test is: could this input time series have equally likely or reasonably occurred at this point? Or: will the decision-makers and the public accept it? The implicit design problem is to find the optimum cost-effective array of BMPs. The solution may be stated: if the 75-year rainfall time series that occurred at the International Airport, had in fact occurred at Foxran Estates, then plan 126 would have been the most cost-effective of the 329 plans examined. Assuming, of course, that they all existed over this 75-year period of time. Once the long term continuous rain record has been developed (i.e., processed, transposed, or generated) it is available for all studies in that hydrologic region, much like standard design storms. It is not often recognized that the basic information is exactly the same as that originally used to derive the design storms for that area, and that no additional data collection effort is required, unless the local design storms are in fact entirely arbitrary. Of course, the derivation procedure for design storms always results in a loss of information: the variability is reduced to simplistic representations. Despite almost 30 years' of arguments in favour of continuous modelling, the method has not yet become routine in design offices. Amazingly, most stormwater and flood design manuals that have been published over the past year or two still do not recommend continuous over event modelling. One assumes that this is due to design inertia, by which is probably meant the fear that designers have of provoking critical reactions (for upsetting the modus operandi). There need be no fear of inertia here. Adoption of a long-term time-series should not be the insurmountable challenge that it seems to have been. My advice is: instead of using the one-in-fifty-year-storm, use one fifty-year-storm instead, which at least ensures that the inherent variability is correctly modelled.
Introduction Continuous models are now reaching a level of complexity that requires that increasing attention be paid to code that manages the modelling (Orlob, 1992). Special user-friendly graphics-oriented codes are being integrated into pre- and post-processors for WQMs. Many of these codes are known as expert systems (see e.g. the PhD thesis by Taymour El Hossieny in my list of graduate students). Orlob calls this integrated code a decision support system (DSS), and I have used his term here. Orlob (1991) does not include control of model reliability in his definition of a DSS. But procedures to manage model error have also been aggregated under the term decision support systems, which is unfortunate in a sense, because error-management procedures relate to the control of model reliability at the model input phase, rather than output interpretation, as the phrase decision support may imply. Uber et al. (1992) strongly recommend that mathematical programming techniques be incorporated into DSSs for WQMs, especially those that have graphical user interfaces (GUIs) in a windows, interactive, menu-driven, pointing-device (WIMP) shell - such as XP-SWMM (Dickinson and Thompson, 1993) - in order to evaluate the larger number of alternatives that can now be generated. Model builders and model users should quantify and present the uncertainty of their model output, to allow end-users the opportunity to evaluate the results, and the confidence that should be placed in them. Numerous DSSs in the form of shells have been and are now being written for SWMM (Dickinson and Thompson,1993; James and James,1993; TenBroek, pers. comm.; Windows SWMM). In the case of SWMM and HSPF, most applications make arbitrary judgements about parameter selection and estimation, and use deficient data, and so the uncertainty and sensitivity to these assumptions are critical to end-users, but seldom reported. This is the principal argument for embedding the PC-TOOLBOX (Kuch, 1994) sensitivity ideas into PCSWMM4. PC-TOOLBOX is considered to be research code; PCSWMM4 is considered to be an instructional shell. Its genesis is described by James and James (1994). Both exceed Orlob's definition, but may be called DSSs. Ideally, decision support systems (DSSs) should provide: Such DSSs must become a normal part of the modelling procedures, so that uncertainty due to poorly defined processes and information are duly and responsibly revealed to the end-users. Kuch (1994) has developed the first tool of a series to form part of a DSS known as PC-TOOLBOX. There seems to be no doubt that our deterministic surface water quality models could benefit from more statistical tools. More statistical manipulations are necessary to build 75 years of data, and to present the results of 75 years of flow and many pollutants at many points. End-users of three-generation (3G) modelling will not be able to comprehend such results without a fair amount of statistical manipulation. The melding of statistics with deterministic models will have another advantage: removal of the artificial distinction between data gatherers on the one hand and data consumers (modellers) on the other (or between monitoring people and analytical laboratory people on the one hand, and the modellers on the other). Provided such models incorporate suitable sensitivity analysis, very-long-term, complex, deterministic models with comprehensive statistical tools are useful management tools for data collection programs. They are the best means for filling in missing data. Ranking the sensitive parameters helps rank priority for selecting chemicals, sampling frequencies and locations, and accuracies of determination. Widely distributed, integrated databases are about to become useful. Our surface water quality models should be tied in with suitable data and user networks, such as the Great Lakes Information Network (GLIN). We also need to encourage widespread use of our models, in education, decision making, engineering and research. We should write code that helps make our models available in different languages. Continuous modelling requires a sequence of two main sets of modelling activities: The calibration activities involve parameter estimation and optimization against short-term, accurate, observed input functions; the inference activities involve long-term, continuous, synthetic or transposed input functions, and error analysis. The figure below shows the relationship between some of these activities: short-term calibration input functions (IFs); model; response functions (RFs); objective functions (OFs); performance evaluation functions (EFs); sensitivity analysis; parameter optimization; long-term continuous input functions; error analysis; long-term, continuous "fuzzy" response functions; and output interpretation or inference. The figure 3.1 is meant to be schematic and conceptual, and only to show the average sequence of activities in the broadest of terms. It is clear that the modelling activities shown in the figure will benefit from a well-written DSS. More detailed discussion on each of these activities follows. For a schematic of the relationship between the various modelling activities proposed in this paper, click here [in this paper and in the schematic, many new terms describing modelling activities are introduced, and some are used slightly differently from normal use].
Introduction In general, in this paper, we try to use certain words in a technically accurate way. Thus the driving input hydro-meteorological time series for a surface WQM is termed the input function, or input variable. Typical examples are rainfall, evapo-transpiration, wind speed, wind direction, snowfall, radiation, humidity, temperature, and some pollutant generating mechanisms such as traffic. Other coefficients that are also input, but less likely to form very long times series at a fine time resolution, and generally control independent component processes, are termed parameters. It is sometimes not clear how to distinguish between input variables and environmental parameters. The computed hydrological and water quality time series, output by the WQM, is termed the response function. An objective function is a statistic or a representative number derived from the response function. Programs such as HSPF and SWMM may be fed a large number of very long input functions (IFs), and can then provide very many response functions (RFs), eg pollutographs (pollutant concentration), loadographs (pollutant flux), hydrographs, cost estimates, and capacities and geometries of storages and conveyances and at many points. Many other responses can be simply inferred from the model output. Post-processors such as the STATISTICS module in SWMM compute several objective functions (OFs), eg event peaks, event means, numbers and durations of exceedances and deficits, etc, for most of these input and response functions. There are of course also similar objective functions for the equivalent measured or observed time series. The OF should not be confused with the evaluation function (EF), used as the measure of agreement between the observed OFs and the computed OFs. An example of an EF is the sum of the squares of the deviations between the observed event peak flows and the computed event peak flows in a calibration plot. EFs are discussed in the next Chapter. Response Functions and Statistical Objective Functions in SWMM The response functions computed by SWMM include both complete and summaries of hydrographs and pollutographs, placed at the end of each output file. These summaries for selected nodes include certain TS statistics such as flow-weighted-average, standard deviation, maximum, minimum flow rates and total volume of runoff for the hydrographs and flow weighted average, standard deviation, maximum, minimum concentrations and total load for pollutants and constituents. Such statistical derivatives from the computed response functions are called objective functions herein, because they do not depend on the error when compared to an observed record. Table 4.1 is a sample output file summary.
Except for the flow-weighted standard deviation, the above output objective functions are commonly used, the four most common being the peak and total (flows and pollutants). Each of the eight objective function types (four each for hydrographs and pollutographs) should be user selectable in a general DSS. Figure 4.1 illustrates the short list of simple OFs given in Table 4.2, by depicting a typical cycle of the input and response function (IF and RF) time series. Although the literature stresses RFs, a similar analysis applies to all input hydrometeorological time series such as air temperature, wind speed, etc.
Cycle 1
RF(t), IF(t)
RFcrit, IFcrit
t1,1 t1,2 t1,3 t1,4 t2,1 t2,2
The functions may be observed, synthetic or computed. RFcrit and IFcrit are arbitrary.
. The four OFs in Table 4.2 that require integration are:
1
2
3
4 In Table 4.2 only fourteen simple OFs are shown, yet for (say) sixteen pollutants, the total number of possible response function OFs would be 238 per location (of which there may be several hundred). Warning: most studies use the term OFs for what we call in this paper EFs, and are more complicated than the OFs in Table 4.2, as described later. Careful selection of the best OF is necessary, a point which is perhaps seldom stressed enough. Users must them-selves thoughtfully choose both the RFs and the OFs that most closely relate to the questions to be answered. Also, the number of OFs should be minimized, if the amount of computed output time series is to be kept within manageable limits. Some guidance is provided below. In the literature, some researchers have reported that it is difficult to calibrate certain WQMs to more than one OF (Seo, 1991). It is important to realise that not all OFs are relevant to all parameters, or design questions. In fact the applicability is quite restricted. In matching them, Table 4.3 (refers to just the OFs listed in Table 4.2) may be used as a first guide:
Depending on the application, the choice of OF may predicate which of the model parameters are eventually optimized, and their absolute values. A further discussion of OFs is given by Rivera-Santos (1988) and Han (1981). In an event-oriented, water-only study to select the best OF using ILLUDAS, SWMM and MINNOUR, Han found that the sum of the squares of the deviations between the observed and the computed hydrograph ordinates gave the best overall performance. This measure is used for an EF in the method developed in this paper. Jewell et al. (1978) chose the standard error of estimate (SEE, listed as EF11 in Chapter 5) as a representative statistic to measure the accuracy of fit between observed and SWMM-computed data: [4.1]
In which: n = number of predicted and measured data points; COFi = predicted value of the ith data point; and OOFi = measured value of the ith data point. Warwick and Wilson (1990) used a total error statistic (EFt) similar to EF20 to quantify overall goodness of fit: [4.2]
where: EFt = total error statistic (m3/s); W = weighting factor; n = number of measured hourly flows; OOF = measured flow (m3/s); COF = computed flow (m3/s); OPF = measured peak flow (m3/s); and CPF = computed peak flow (m3/s).Assigning a value of 1.0 to W results in a complete focusing upon matching the peak flow. Reasonable balances between hydrograph shape and peak were obtained with W = 0.2. Han and Rao (1980) showed that, for SWMM model calibration purposes, the sum of the squared deviations between the observed and computed flows gave the best overall performance out of a total of seventeen objective functions reviewed: [4.3]
where: W = the sum of squared deviations. OOFi = the observed hydrograph ordinates COFi = the calculated hydrograph ordinatesThis is the same as EF1 in the next section.
Introduction In this section we list some common functions that have been used to evaluate the goodness of fit between a computed and observed series of RFs and/or OFs (Rivera-Santos, 1988). In the literature, the tests for goodness-of-fit are usually carried out between the computed and observed hydrograph, denoted a RF in this paper, and the functions listed below are often referred to as objective functions. Hence in the following list, reference is made to high or low flows. In the methodology developed here, the goodness-of-fit tests are carried out on simple statistics such as peak flow, derived from the response function, and these simple statistics are denoted objective functions. So in this paper the following functions are denoted performance evaluation functions (EF) in this paper. In the list below, the computed objective function is COF, the observed objective function OOF, and the difference between them OOFi - COFi is the error ei. The number of points compared is n.
Introduction Click here for a schematic on discretization, and model development. In this paper: Click here for a schematic on types of uncertainty. There are several sources of error associated with modelling, and all should be kept under control during design applications. In practical applications, the most common errors may be traced to wrong data in the input files, caused by blunders, data entry errors, and user/modeller misconceptions. But the most serious errors are probably those that are made well after the model runs are completed: poor interpretation of the results, their inherent error, and reliability, by model builders, users and decision-makers alike. This point perhaps needs to be kept in mind, because the errors most commonly dealt with in the literature exclude them, but cover others such as: The structural error due to disaggregation and poor component process models, is also called framework error. The propagated error is also known as parameter error, and is the focus of this paper and also of PCSWMM4. The framework error is difficult to estimate, and is therefore generally taken to be the difference between the total model error and the parameter error. There is some evidence that the effect of several of these sources of error can be mitigated by careful parameter estimation. Certainly the perhaps unintentional effect in much surface water quality modelling, is to correct for what are sometimes gross errors in all seven categories, by a substantial parameter optimization effort. To put it crudely: poor rain data is often corrected by a thoughtless quick fix, e.g. by decreasing parameters controlling soil surface infiltration capacities. This is an example of what we term in this paper unreliable modelling. Perhaps dishonest is a better word.
Introduction There are two types of user-controlled model complexity to be considered here: On the other hand, it is desirable to include as many relevant processes, at as fine a spatial resolution as possible, to improve the accuracy and reliability of the model. It is, however, by no means certain that model reliability continues to increase with model complexity, because of the difficulty of getting good parameter estimates, and their combined effect on the computed response (Seo,1991). The optimal order of model complexity is very much dependent on the evaluation function chosen, eg for cost-effectiveness. The function should exhibit a minimum when plotted against a number representing the complexity, e.g. a combination of input variables and parameters. Also, the function should penalise model inaccuracy for very simple, coarse and inaccurate models, and penalise cost for very complex models. It is possible to estimate the design-office costs as a function of collecting various types of field data, but it is not nearly so easy to place a value on model reliability. The evaluation function is here written: [cost + f(e)] where cost represents the design office costs, including computing, and f(e) is a model reliability function (of model error), being large for unacceptable error, small when the model achieves the requisite accuracy, and increases with large complexity, due to propagated error. Figure 7.1 depicts the relationship sought. evaluation function model error design costs term term optimum model complexity
Introduction. Click here for a schematic on model parameter optimization.
Interestingly, only three types of modelling experiments are possible (James, 1993b): Background review A rather quick review of the freely-available serial and academic literature revealed that a small number of papers have appeared on how to calibrate SWMM (Alvarado,1982; Baffaut and Delleur,1989 & 1990; El-Sharkawy and Kummler,1984; Huber,1993; Irvine et al.,1993 Jewell et al.,1978; Liong et al.,1990; Udhiri,1984; Zaghloul,1981; Kuch and James, 1993; James and Robinson,1985; Dunn and James,1985; Bonema and Sangal,1994,). In this search, none were found for HSPF. Several dissertations and other academic publications have also appeared that include extensive discussion of SWMM and similar model calibration (Jewell,1974; Dunn, 1986; Kuch,1994; Baffaut,1988; James,1993b). A modest number of helpful papers is now available on the subject of calibration of general hydrological models where the difficulties are somewhat similar to those of SWMM and HSPF (Ibrahim and Liong,1993; Walker,1982; Leavesley et al.,1983; Stephenson,1989; Thompson,1989; Mein and Brown,1978). Additionally, recent work on Monte Carlo and other methods is useful (Dilks,1987; Beck,1976 & 1983;Hwang,1985; Qaisi,1985; Rivera-Santos,1989; Scavia,1980; Seo,1991). State variables Many of these publications are examined in this paper. Further papers are listed in the bibliography at the end of this paper, for readers wishing to read further into the subject. One point needs to be clearly understood by users: it is the individual parameter values that are optimized, not the model per se. Each parameter is associated with a process, which is active only under certain circumstances, or states of the model. To estimate the best value of a parameter, the only states that need to be examined are the states or events when the related processes are active. These causative events need to be established, and selected from the short, good, continuous record, to be used for calibration; the specific, observed events are then used to calibrate the related, specific, active processes and parameters. If I may be permitted another circular argument here: if done correctly, the processes will then more accurately output the required response. Explain Table 8.1 here!
Introduction Sensitivity analysis consists of State variables From the standpoint of engineering hydrology, it is entirely reasonable to calibrate distinct processes against distinct events, since all WQMs were deliberately formulated such that their processes do not mutually interfere; the processes are not mutually destructive. The obvious example is snowmelt: tweaking (say) the sewer erosion process in the model, by perturbing one of those parameters, e.g. a particle fall velocity, clearly cannot interfere with the snowmelt process in the model. The rest of the component processes have been structured in the same independent way. Thus the scope for parameter interaction is apparently limited. The inherent structure of a WQM is very much dependent on the model builder's understanding of the precise meaning of the technical words and phrases used to describe the processes involved. The model builder uses rules that are not always obvious to the user from the documentation; indeed several expressions may appear to a reader to be ambiguously used in a single volume of documentation for a specific WQM. One of the inherent rules seems to be that model builders so define their processes that each process is active only in limited states of the model, and that the states can be predetermined and simply related to an input variable. Of course many processes may be simultaneously active, but the processes are all described by independent algorithms. If fuzzy logic is not being used, there is a clear and distinct boundary to the state-variable (SV) space in which each process is active. This limited SV space is called the SV sub-space herein (see Figure 9.1 and 10.2). This is important because model users must be able to: State variable sub-spaces To clarify the foregoing discussion, consider for example overland flow from impervious surfaces, which becomes active as soon as light rain exceeds the impervious area depression storage. It then dominates the runoff process until the rain exceeds infiltration capacities and runoff is contributed from adjacent pervious areas (if any - the relative sensitivity is clearly affected by the percent imperviousness). The state variables here are both rate-of-rain (must not exceed infiltration capacities of pervious areas) and duration of rain (total rain must exceed impervious area depression storage). Since two dimensions of rain are being used, we refer to state-variable space. Relevant SVs are rain, snow, evapo-transpiration, wind, temperature, radiation, etc. Figure 10.1 illustrates the point. This sub-spatial effect, shown in Figure 10.1, where certain processes dominate, is readily translated to a calibration plot, such as that shown in Figure 9.1. Figure 9.1 can be explained in simplest terms as follows: consider peak flows as the objective function, in which case the state-variable is rainfall (product of rate and duration). A then denotes light rains, when infiltration capacities are not exceeded, C denotes heavy rains when all surfaces contribute and infiltration capacities have reached their asymptotic lowest values, B denotes intermediate events, when infiltration capacities are of the same order of magnitude as rains, and D denotes the fuzzy overlapping zones. Similar illustrations can be provided for processes such as, e.g. pollutant washoff, erosion, or snowmelt. Aside: At this point the reader may well wonder whether the intellectual effort required to follow this methodology is worth the gains, if any. The answer is yes! - there is a substantial payoff in reducing the computational effort in sensitivity, calibration and error analysis. Also, it pays to be better informed, cynics notwithstanding. This is part of the case for an honest modelling ethic, developed at the end of this paper. Fortunately it is not necessary to precisely determine entire boundaries of SV sub-spaces manually, because the sensitivity analysis routines developed herein are useful for this purpose. Also, it seems likely that fuzzy logic could be helpful in dealing with the fringes of the applicable SV sub-spaces (fuzzy logic is also discussed further later). For the sensitivity analysis, artificial TS are used, with constant intensities. The intensities and durations are chosen so that they relate in a fuzzy way to the scale of the model problem. These artificial state variable spaces are like simple design storms or design droughts; they are named according to their fuzzy zones as listed in Table 9.1 below (as you have no doubt by now noticed, dialogue with gui-wimps is necessarily simple; they talk only in easy-to-remember acronyms - symbols in scientific journals are the antithesis).
(OFi)c A represents "small" events B represents "medium" events C represents "big" events D represents fuzzy overlaps (OFi)o c denotes computed; o denotes observed; OFi denotes the objective function chosen. Square sub-spaces have been chosen for mathematical simplicity. As a guide, to start the discussion, it may be helpful to relate dominant processes to SV sub-spaces as shown in Table 9.2.
Parameter uncertainty We can categorize parameters to be optimized under four groups:
The three values are shown schematically in Figure 9.2.
<--------------------parameter range--------------------------> <--sa range----> Emin -0.25 Emed +0.25 Emax computer run: 3 1 2 s = 0.25; the origin may be far to the left.
Because the three estimates are estimated by the user, based on the user's own logic, the method has been denoted heuristic in this paper. At this point it is worth recalling that the total number of runs required for n parameters is 2n + 1 so that, for (say) 49 of the parameters of the SWMM-RUNOFF module, less that 100 short-event runs are required. Monte Carlo analysis on the other hand, would require perhaps hundreds of thousands of runs of the full continuous dataset, clearly not computable, as we define the term here. Sensitivity gradients Sensitivity analysis reveals the inner structure and rules of the model, beyond those invented by the model builder, because it aggregates all processes that are active in various limited SV sub-spatial domains. Moreover, the sensitivity gradients can be used to estimate the propagated error, and to optimize the input parameters against observed OFs. As discussed later, there must be in each limited SV sub-space at least as many observed OFs as there are parameters to be optimized. Excess observations, if any, will improve the reliability of the parameter estimates, especially if they cover the full range of the sub-space, also discussed later. It is a relatively mechanical matter for the PCSWMM4 code to generate and integrate the required 99 input datafiles into one run. The shell accumulates a large sensitivity output file, by appending results from subsequent runs. The dimensionless sensitivity gradients (DSGs) are then presented in a screen by a family of plots, one plot for each parameter in an identifiable process or SV sub-space, one such family of plots for each process or sub-space per screen, as shown in Figure 9.3. The DSGs are also ranked. Note that this procedure correctly ranks the parameters that are both most sensitive and have the greatest uncertainty, and vice versa.
There is evidence that certain urban drainage processes are non-linear: doubling the rate of rain does not double the runoff at a point if it flows through a surcharged sewer system, especially one that incorporates active or remotely and arbitrarily controlled diversions. On the other hand, there have been six decades of largely successful numerical hydrologic modelling based on the assumption that the runoff and transport processes are almost linear. I would submit that this is sufficient prima facie evidence to justify examining linear decision support systems for analysis of sensitivity, complexity, reliability, parameter optimization and error. However, the widespread practice of simply assigning an arbitrary perturbation of each input parameter based on its expected value, e.g. 25% of its absolute value, is probably not appropriate, as it very likely would carry the sensitivity analysis into non-linear situations, which in turn will be difficult to manage. This is because the sensitivity of many parameters changes with large perturbations - they are curved as shown in Figure 9.3. The trick to the success of this methodology is the structure of the logic underlying the limitation of the analyses to just the minimum number of runs absolutely essential for the analysis. Insensitive processes and parameters are eliminated at the outset, and only the most sensitive parameters are subject to further parameter optimization, certainty and error analysis. The next section uses fuzzy logic to explain the approach.
Three methods are widely reported in the literature: Kalman filters, first-order sensitivity-error-analysis, and Monte Carlo. No use of Kalman and/or Monte Carlo methods on long-term, continuous applications of these surface WQMs has been reported, so far as this literature search has revealed. On the other hand, it seems that for most applications, first-order approximations are adequate, if not preferable, because: 1. the input variable error and model error estimates are themselves approximate, and 2. the first-order procedure readily ranks parameter sensitivity and uncertainty (Walker, 1982), which are useful for aiding in understanding the model performance. The output function OFi is a function of the n model parameters p1, p2, , , pn: OFi = fj(p1,p2, , ,pn) (13.1) where: fj denotes a function dependent on the state j of the model. In general, the absolute error in the computed objective function caused by parameter variation is: e(OFij) = DSG1.?p1 + DSG2.?p2 + ..+ DSGn.?pn (13.2) where: DSGi is the dimensionless sensitivity gradient for parameter pi, and ?pi is the half-range in the estimates, (Emax - Emed)i for positive error, and (Emed - Emin)i for negative error. As expressed here, the error arises by virtue of reasonable estimates by different users, and, if both the negative and positive errors are computed, will represent a range of about 4 standard deviations for each term (assuming the parameter estimates were normally distributed; this may also apply to other distributions). As a reminder, our basic approach here is to deal with a range of OFs likely to have been computed by a number of different users, rather than a very wide range of OFs computed using one user's estimates of the widest possible range for each of very many parameters (PC-TOOLBOX currently - 1993 - analyzes 48 parameters in the RUNOFF module alone). At this point a significant difficulty arises: how do we deal with the probability that a user may have over- or under- estimated every one of the parameters involved, or estimated all of them to compound the effect of the sign of the DSG - overestimating positive DSGs and underestimating negative DSGs? Or have estimated half of them up and half down, thus producing no difference ultimately between the error-computed OF and the OF computed using all the median expected parameter estimates? A simple Monte Carlo technique is suggested here: generate a random sequence of binary positive/negative decisions, apply them serially to all of the parameter half-ranges and then substitute them into the first-order error equation. This will be extraordinary fast, because the error equation is explicit and simple, so that a large number of runs will present no difficulty. This modified Monte-Carlo/first-order-error procedure, and the requisite routines, have not yet been developed in PCSWMM4.
By using the logical approach and software described in this paper, the number of runs for sensitivity and parameter optimization can be reduced to manageable, computable amounts, given modern inexpensive workstations (1994). The methodology has been found to be helpful, workable, and essential for honest modelling. It is summarised here in the form of a framework, useful for informed, purposeful surface water quality modelling. The framework is based on the reasonable premises that the WQM has been deliberately constructed such that component processes do not interfere with one another, and that the overall model behaves as if it were largely linear. Fuzzy logic is helpful in reducing the analysis.
Framework for continuous modelling:
Recommendations: The following rules, taken from a personal catechism for honest, very-long term, continuous surface water quality modelling, have been espoused in this paper. Try them!
Rule 1: Do not calibrate all parameters simultaneously against a long-term continuous observed record, notwithstanding any early advice to the contrary in the literature.
Rule 2: Transpose or synthesize a long-term, hydro-meteorologic input time-series from the same hydrologic region, and use this for inferring comparative performance of various arrays of BMPs. Many records of 50 years duration or longer are available.
Rule 3: Carefully choose the best objective functions that represent the design questions and the model variability. Get the advisory committee to justify the selections in writing.
Rule 4: In order to control the amount of computing, associate the input parameters with processes, and processes with causative events, and causative events with limited state-variable sub-spaces. For this activity, sensitivity analysis code in PCSWMM4 is helpful. Do not analyze parameters outside these spaces.
Rule 5: Use three estimates of the most likely parameter values. It is more meaningful to compare the computed response from several reasonable models, rather than responses computed using extreme values.
Rule 6. Assume that the WQM is approximately linear, for the purposes of optimizing parameters, and estimating the propagated error. Then analyze for sensitivity near the mean expected values of all input parameters.
Rule 7: Calibrate only sensitive parameters, and then only against relevant events for which you have good, short-term observed data. And that must include good rate-of-rain with adequate coverage and spatial resolution.
Rule 8: Use first-order linear error analysis, and report the estimated propagated error in your recommended design solution.
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GLOSSARY
best management practice (BMP) Structural devices that temporarily store or treat urban stormwater runoff to reduce flooding, remove pollutants, and provide other amenities. catchment That area determined by topographic features within which falling rain will contribute to runoff at a particular point under consideration. channel A natural stream that conveys water; a ditch or drain excavated for the flow of water. channel erosion The widening, deepening, and headward cutting of small channels and waterways, due to erosion caused by moderate to large floods. cold water fishery A fresh water, mixed fish population, including some salmonids. combined sewer A sewer intended to carry surface runoff, sewage and industrial wastes allowed by sewer by-laws. combined sewer overflow Flow from a combined sewer, in excess of the sewer capacity, that is discharged into a receiving water. computable A simulation that can be performed in the temporal space of a working day (eight hours) and the output of the simulation to require normal hard drive working space of a typical engineering office work station (in the order of 200 Mb). continuous modelling A simulation that models both the dry and wet processes of hydrology with a continuous record of atmospheric data. In contrast event modelling is a simulation of short defined storm events with subjective startup conditions. dissaggregation The degree to which the components of a physical system are modelled by increasing the number of defined processes. discretization The number of components selected to represent the physical system that has been dissaggregated into processes on those components and the degree to which the physical parameters are lumped as spatial and temporal averages. detention The slowing, dampening, or attenuating of flows either entering the ewer system or within the sewer system, by temporarily holding the water on a surface area, in a storage basin, or within the sewer itself. detention time The amount of time a parcel of water actually is present in a BMP. Theoretical detention time for a runoff event is the average time parcels of water reside in the basin over the period of release from the BMP. dominant processes When the hydrological processes coded in the program and coupled with the input file (forming a model of the physical system) are active and represent large percentages of contribution to the selected objective function. For example infiltration is a process coded in SWMM and during low rainfall intensities this process would be dominant in extracting water from the surface and reducing runoff volumes and peak flows. drainage 1. To provide channels, such as open ditches or dosed drains, so that excess water can be removed by surface flow or internal flow. 2. To lose water (from the soil) by percolation. dry weather flow Combination of domestic, industrial and commercial wastes found in sanitary sewers during dry weather not affected by recent or current rain. erodibility (of soil) The susceptibility of soil material to detachment and transportation by wind or water. erosion 1. The wearing away of the land surface by running water, wind, ice or other geological agents, including such processes as gravitational creep. 2. Detachment and movement of soil or rock fragments by water, wind, ice or gravity. error The difference between a computed and an observed value, event mean concentration (EMC) The average concentration of an urban pollutant measured during a storm runoff event. The EMC is calculated by flow-weighting each pollutant sample measured during a storm event. first flush The condition, often occurring in storm sewer discharges and combined sewer overflows, in which an unusually high pollution load is carried in the first portion of the discharge or overflow. flood frequency A measure of how often a flood of given magnitude should, on an average, be equalled or exceeded. fuzzy process A process that has different levels of dominance which is dependent on the state or input variable. For example the modelling of pollutant buildup and washoff is a fuzzy process when the rainfall intensity oscillates from zero to very low levels; is pollutant buildup or pollutant washoff occuring? hydrograph A graph showing variation in stage (depth) or discharge of a stream of water over a period of time. impervious area Impermeable surfaces, such as pavement or rooftops, which prevent the infiltration of water into the soil. infiltration The seepage in dry or wet weather or both of groundwater or vadose water into any sewer (storm, sanitary, combined). Generally, infiltration enters through cracked pipes, poor pipe joints or cracked or poorly jointed manholes; also the loss of surface runoff into pervious ground. infiltration (of soils) Movement of water from the ground surface into a soil. input variable space/state variable space The combinations of input time series that alter the dominance of and trigger the processes coded in the program. input function, or input variable The driving input hydro-meteorological time series for a surface WQM. Typical examples are rainfall, evapo-transpiration, wind speed, wind direction, snowfall, radiation, humidity, temperature, and some pollutant generating mechanisms such as traffic. non-point source An area from which pollutants are exported in a manner not compatible with practical means of pollutant removal (e.g. crop lands). model complexity A measure of the number of uncertain parameters in a model. model uncertainty The degree to which the output of a simulation represents the actual outcome of the physical system. The model uncertainty comprises the uncertainty of many sources including the parameter estimation uncertainty and the degree to which the code of the program models the physical system. objective function A statistic or a representative number derived from the response function. objective, water quality A designated concentration of a constituent, based on scientific judgements, that, when not exceeded will protect an organism, a community of organisms, or a prescribed water use with an adequate degree of safety. optimal complexity The level of discretization and dissagregation that yeilds the minimum modelling cost for a given level of model accuracy. parameters. Coefficients that are also input, but less likely to form very long times series at a fine time resolution, and generally control independent component processes parameter estimation/calibration A procedure to discover the global optimum of an array of modelling parameters that can only be discovered with estimation and are not directly measurable in the field. Calibration is completed when an objective function defined as a degree of fit between measured and computed output is minimized. parameter sensitivity The influence of a parameter's value on the model output. Parameters are very sensitive when small changes in the value of a parameter have a significant effect on an output objective function such as peak concentration or total constituent load. peak discharge (flow) The maximum instantaneous flow at a specific location resulting from a given storm condition. pollutant Dredged soil, solid waste, incinerator residue, sewage, garbage, sludge, chemical wastes, biological materials, radioactive materials, heat, wrecked or discarded equipment, rock, sand, dirt and industrial, municipal and agricultural waste discharged into water. recurrence interval (return period) The average interval of time within the magnitude of a particular event (e.g. storm or flood) which will be equalled or exceeded. e.g 1 in 5 year frequency of 1:5 AEP. resolution 1. The scale of spatial and temporal disscretization. 2. The size of the time step in a continuous simulation. response function The computed hydrological and water quality time series, output by the WQM, return period See recurrence interval. riparian A relatively narrow strip of land that borders a stream or river, often coincides with the maximum water surface elevation of the 100 year storm. runoff That portion of the precipitation on a drainage area that is discharged from the area into stream channels. sanitary sewer A sewer that carries liquid and water-borne wastes from residences, commercial buildings, industrial plants, and institutions, together with relatively low quantities of ground, storm, and surface waters that are not admitted intentionally. sediment Solid material, both mineral and organic, that is in suspension, is being transported, or has been moved from its site of origin by air, water, gravity, or ice, and has come to rest on the earth's surface either above or below sea level. sedimentation The process of subsidence and deposition of suspended matter carried by water, sewage, or other liquids, by gravity. sewershed The area of a municipality served by a given sewer network. For example, the area tributary to a given combined sewer overflow or a given WPCP would be termed the sewershed tributory to the overflow or WPCP. simulation Representation of physical systems and phenomena by mathematical models. stormflow The portion of flow which reaches the stream shortly after a storm event. storm sewer A sewer that carries storm water and surface water, street wash and other wash waters or drainage, but excludes sewage and industrial wastes. stormwater Water resulting from precipitation which either percolates into the oil, runs off freely from the surface, or is captured by storm sewer, combined sewer, and to a limited degree, sanitary sewer facilities. streamflow Water flowing in a natural channel, above ground. surcharge The flow condition occurring in closed conduits when the sewer is pressurized or the hydraulic grade line is above the crown of the sewer. time of concentration (hydraulics) The shortest time necessary for all points n a catchment area to contribute simultaneously to flow past a specified point. uncertainty A possible value an error may have; urban runoff Surface runoff from an urban drainage area that reaches a stream or other body of water or a sewer. urbanized area Central city, or cities, and surrounding closely settled territory. Central city (cities) have populations of 50,000 or more. Peripheral areas with a population density of one person per acre or more are included (United States city definition). variability The different values that a parameter may have; variance The square of the standard deviation (a measure of uncertainty); and watercourse A natural or constructed channel for the flow of water. watershed The region drained by or contributing water to a stream, lake, or other body or water. waterway A natural or man-made drainage way. Commonly used to refer to a channel which has been shaped to a parabolic or trapezoidal cross-section and stabilized with grasses (and sometimes legumes), and which is designed to carry flows at a velocity that will not induce scouring. wet weather flow A combination of dry weather flows, infiltration and inflow which occurs as a result of rain and storms.
LIST OF ABBREVIATIONS AND ACRONYMS
3G three generation(s) (approx. 75 years) 3GM three generation modelling AES Atmospheric Environment Services ANSI American National Standards Institute AOC area of concern ASCE American Society of Civil Engineers BBS bulletin board system BMP best management practice BOD biological oxygen demand CAD computer aided design/drafting CAE computer aided engineering CBSQMP combinations of better stormwater quality management proposals CDM Camp Dresser McKee (a consulting engineering company) CF continuous flow CFS cubic feet per second CPU central processing unit CSCE Canadian Society of Civil Engineers CSO combined sewer overflow CSWQMM continuous storm water quality management modelling DEIS draft environmental impact statement dpi dots per inch DSS decision support system DWF dry weather flow EPA Environmental Protection Agency GRU grouped response units GUI graphical user interface HRU homogeneous response units HSI habitat suitability indices IAHR International Association for Hydraulic Ressearch IAWPRC International Association for Water Pollution Research and Control IBM International Business Machines IDF Intensity-Duration-Frequency curves IET inter-event time Mb Mega bytes (millions of bytes) MDP master drainage plan MIPS million instruction sets per second MOEE (Ontario) Ministry of Environment and Energy NPDES National Pollution Discharge Elimination System NPS nonpoint source NURP Nationwide Urban Runoff Program OMNR Ontario Ministry of Natural Resources PC personal computer PCP pollution control plan pdf's probability density functions PF plug flow pH negative log of hydrogen ion concentration PWQO Provincial Water Quality Objectives RTC real time control SCS Soil Conservation Service SS suspended solids TBRG tipping bucket rain gages TS time series TSM time series management TSS total suspended solids USEPA United States Environmental Protection Agency USGS United States Geological Survey UWRRC Urban Water Resources Research Council UZS upper zone storage VGA video graphics array WIMP windows-icons-menus-pointing devices ZUM zones of uniform meteorology
LIST OF MODELS AND PROGRAMS
ARCINFO a GIS program AutoCAD(r) an automated computer aided drafting package BASIC a programming language BMP-Planner an OOP for hydrology and planning C, C++ a programming language DBMS database management system EXTRAN Extended transport program FORTRAN a programming language GIS Geographic Information System HEC Hydrologic Engineering Center (US Army Corps of Engineers) HSPF Hydrologic Simulation Program-Fortran HYMO a hydrologic model LEAP a program to compute eutrophication effects in small lakes PCSWMM Personal Computer version of SWMM PRMS Precipitation-Runoff Modelling System QuattroPRO a spreadsheet program QuickBASIC a programming language RUNOFF hydrology module in SWMM STATISTICS a post-processor in SWMM STORM Storage Treatment Overflow Runoff Model SWMM Storm Water Management Model TRANSPORT a drainage system of pipes, conduits and structures module in SWMM TurboVision an application framework for Borland's C++ compiler WATFLOOD a hydrologic program XP-SWMM SWMM Graphic based platform
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