Characterizing Uncertainty in Daily Streamflow Estimates at Ungauged Locations for the Massachusetts Sustainable Yield Estimator

Hydrologic characterization at ungauged locations is one of the quintessential challenges of hydrology. Beyond simulation of historical streamflows, it is similarly important to characterize the level of uncertainty in hydrologic estimates. In tandem with updates to Massachusetts Sustainable Yield Estimator, this work explores the application of global uncertainty estimates to daily streamflow simulations. Expanding on a method developed for deterministic modeling, this approach produces confidence intervals on daily streamflow developed through nonlinear spatial interpolation of daily streamflow using flow duration curves; the 95% confidence is examined. Archived cross‐validations of daily streamflows from 66 watersheds in and around Massachusetts are used to evaluate an approach to uncertainty characterization. Neighboring sites are treated as ungauged, producing relative errors that can be resampled and applied to target sites. The method, with some modification, is found to provide appropriately narrow confidence intervals that contain 95% of the observed streamflows in cross‐validation. Further characterizing uncertainty, multiday means of daily streamflow are evaluated. Working through cross‐validation in Massachusetts, two‐ to three‐month averages of daily streamflow show the best performance. These two approaches to uncertainty characterization inform how streamflow simulation produced for prediction in ungauged basins can be used for water resources management.     

Inexact rough-interval two-stage stochastic programming for conjunctive water allocation problems

An inexact rough-interval two-stage stochastic programming (IRTSP) method is developed for conjunctive water allocation problems. Rough intervals (RIs), as a particular case of rough sets, are introduced into the modeling framework to tackle dual-layer information provided by decision makers. Through embeding upper and lower approximation intervals, rough intervals are capable of reflecting complex parameters with the most reliable and possible variation ranges being identified. An interactive solution method is also derived. A conjunctive water-allocation system is then structured for characterizing the proposed model. Solutions indicate a detailed optimal allocation scheme with a rough-interval form; a total of [[1048.83, 2078.29]:[1482.26, 2020.60]] would be obtained under the pre-regulated inputs. Comparisons of the proposed model to a conventional and an interval two-stage stochastic programming model are also conducted. The results indicate that the optimal objective function values of TSP and ITSP always fall into the range of , while they are sometimes out of the range of ; the optimal solutions of decision variables also present this feature. This implies the reliability of IRTSP in handling conjunctive water allocation problems.     

CONFIDENCE INTERVALS FOR FLOOD EVENTS UNDER A PEARSON 3 OR LOG PEARSON 3 DISTRIBUTION1

ABSTRACT: The Pearson type 3 (P3) and log Pearson type 3 (LP3) distributions are very frequently used in flood frequency analysis. Existing methods for constructing confidence intervals for quantiles (X_p) of these two distributions are very crude. Most of these methods are based on the idea of adjusting confidence intervals for quantiles Y_p of the normal distribution to obtain approximate confidence inervals for quantiles X_p of the P3/LP3 distribution. Since there is no theoretical reason why this “base” distribution, Y, should be taken to be normal, we search in the present study for the best possible base distribution for producing confidence intervals for P3/LP3 quantiles. We consider a group of base distributions such as the normal, log normal, Weibull, Gumbel, and exponential. We first assume that the skew coefficient, γ of X, to be known, and develop a method for adjusting confidence intervals for Y_p to produce approximate confidence intervals for X_p. We then compare this method (Method A) with another method (Method B) introduced by Stedinger. Simulation shows that the performance of each of these two methods depends on the base distribution Y that is being used, but as a whole, the normal distribution appears to be the best-fit distribution for producing confidence intervals for P3/LP3 quantiles when γ is assumed to be known. We then extend our method (Method A) to the more important case of unknown coefficient of skewness. It is shown that by taking Y to be Weibull, fairly accurate confidence intervals for P3/LP3 quantiles can be obtained for quite a wide range of sample sizes and coefficients of skewness commonly found in hydrology. The case of the P3 distribution with negative skewness needs further research.     

Two-stage planning for sustainable water-quality management under uncertainty

In water-quality management problems, uncertainties may exist in a number of impact factors and pollution-related processes (e.g., the volume and strength of industrial wastewater and their variations can be presented as random events through identifying a statistical distribution for each source); moreover, nonlinear relationships may exist among many system components (e.g., cost parameters may be functions of wastewater-discharge levels). In this study, an inexact two-stage stochastic quadratic programming (ITQP) method is developed for water-quality management under uncertainty. It is a hybrid of inexact quadratic programming (IQP) and two-stage stochastic programming (TSP) methods. The developed ITQP can handle not only uncertainties expressed as probability distributions and interval values but also nonlinearities in the objective function. It can be used for analyzing various scenarios that are associated with different levels of economic penalties or opportunity losses caused by improper policies. The ITQP is applied to a case of water-quality management to deal with uncertainties presented in terms of probabilities and intervals and to reflect dynamic interactions between pollutant loading and water quality. Interactive and derivative algorithms are employed for solving the ITQP model. The solutions are presented as combinations of deterministic, interval and distributional information, and can thus facilitate communications for different forms of uncertainties. They are helpful for managers in not only making decisions regarding wastewater discharge but also gaining insight into the tradeoff between the system benefit and the environmental requirement.     

Value of information for climate observing systems

The Interagency Working Group Memo on the social cost of carbon is used to compute the value of information (VOI) of climate observing systems. A generic decision context is posited in which society switches from a business as usual (BAU) emissions path to a reduced emissions path upon achieving sufficient confidence that a trigger variable exceeds a stipulated critical value. Using assessments of natural variability and uncertainty of measuring instruments, it is possible to compute the time at which the required confidence would be reached under the current and under a new observing system, if indeed the critical value is reached. Economic damages (worldwide) from carbon emissions are computed with an integrated assessment model. The more accurate observing system acquires the required confidence earlier and switches sooner to the reduced emissions path, thereby avoiding more damages which would otherwise be incurred by BAU emissions. The difference in expected net present value of averted damages under the two observing systems is the VOI of the new observing system relative to the existing system. As illustration, the VOI for the proposed space-borne CLARREO system relative to current space-borne systems is computed. Depending on details of the decision context, the VOI ranges from 2 to 30 trillion US dollars.     

TWO PRACTICAL APPROACHES FOR DETERMINING THE CONFIDENCE INTERVAL OF RAINFALL DEPTH PERCENTILES1

ABSTRACT: A simple simulation type approach and a statistical method are proposed for determining the confidence interval of the T‐year frequency rainfall percentiles (or precipitation extremes) for generalized extreme value (GEV) distributions. The former method is based on the Monte Carlo testing procedure. To generate realizations, the covariance structure of the three parameters of GEV is investigated using an observed information matrix of the likelihood function. For distributions with realistic parameters, the correlation between the location and the scale parameters is practically constant when the shape parameter varies around values close to its optimal value. The latter method is based on likelihood ratio statistics. In the case where the joint confidence surface for shape parameters and estimates is plotted with lines of best estimates, the region where the estimated best percentile value can be chosen as a possible estimate is part of the joint confidence surface. The projection of this bounded region on axis of percentile is defined as the effective confidence interval in this research. The use of this effective interval as the confidence interval of the percentile of T‐year frequency rainfall is particularly recommended because it is stable for T and it reflects variations in all three parameters of GEV appropriately.     

REGIONAL FREQUENCY ANALYSIS OF HYDROLOGIC MULTIYEAR DROUGHTS1

This paper is concerned with regional frequency analysis of hydrologic multiyear droughts. A drought event is defined by three parameters: severity, duration, and magnitude. A method is proposed here to standardize drought severities with a duration adjustment to enable comparison among drought events. For purposes of a regional study, the index drought method is selected and applied to standardized droughts to give a regional frequency curve. However, the recurrence intervals of the drought events obtained from index drought method are limited to the historic period of record. Therefore, by taking advantage of random variations of droughts in both time and space, a multivariate simulation model is used to estimate exceedence probabilities associated with regional drought maxima. This method, named the regional extreme drought method, is capable of generating a series of drought events which, although they have not occurred historically, are more severe than historic events. By combining the results of the index drought method and regional extreme drought analysis, a regional drought probability graph is constructed which ranges from severe droughts to more frequent droughts. This procedure is applied to the mean annual flow records of streams located in the San Joaquin Valley of California, and drought-severity-frequency plots are prepared for 1-year, 2-year, and 3-year durations.     

Regression modeling of streamflow, baseflow, and runoff using geographic information systems

Regression models for predicting total streamflow (TSF), baseflow (TBF), and storm runoff (TRO) are needed for water resource planning and management. This study used 54 streams with >20 years of streamflow gaging station records during the period October 1971 to September 2001 in Pennsylvania and partitioned TSF into TBF and TRO. TBF was considered a surrogate of groundwater recharge for basins. Regression models for predicting basin-wide TSF, TBF, and TRO were developed under three scenarios that varied in regression variables used for model development. Regression variables representing basin geomorphological, geological, soil, and climatic characteristics were estimated using geographic information systems. All regression models for TSF, TBF, and TRO had R(2) values >0.94 and reasonable prediction errors. The two best TSF models developed under scenarios 1 and 2 had similar absolute prediction errors. The same was true for the two best TBF models. Therefore, any one of the two best TSF and TBF models could be used for respective flow prediction depending on variable availability. The TRO model developed under scenario 1 had smaller absolute prediction errors than that developed under scenario 2. Simplified Area-alone models developed under scenario 3 might be used when variables for using best models are not available, but had lower R(2) values and higher or more variable prediction errors than the best models.     

ASSESSMENT OF SAMPLING ERROR ASSOCIATED WITH SOIL MOISTURE ESTIMATION DESIGNS1

A spectral formalism was developed and applied to quantify the sampling errors due to spatial and/or temporal gaps in soil moisture measurements. A design filter was developed to compute the sampling errors for discrete measurements in space and time. This filter has as its advantage a general form applicable to various types of sampling design. The lack of temporal measurements of the two‐dimensional soil moisture field made it difficult to compute the spectra directly from observed records. Therefore, the wave number frequency spectra of soil moisture data derived from stochastic models of rainfall and soil moisture were used. Parameters for both models were estimated using data from the Southern Great Plains Hydrology Experiment (SGP97) and the Oklahoma Mesonet. The estimated sampling error of the spatial average soil moisture measurement by airborne L‐band microwave remote sensing during the SGP97 hydrology experiment is estimated to be 2.4 percent. Under the same climate conditions and soil properties as the SGP97 experiment, equally spaced ground probe networks at intervals of 25 and 50 km are expected to have about 16 percent and 27 percent sampling error, respectively. Satellite designs with temporal gaps of two and three days are expected to have about 6 percent and 9 percent sampling errors, respectively.     

THE USE OF A SIMPLE MODEL AND UNCERTAINTY ANALYSIS IN LAKE MANAGEMENT1

ABSTRACT: A simple, black-box lake model was developed for phosphorus, using nonlinear regression analysis on a data base of north temperate lakes. The uncertainty associated with the model was then combined with the parameter uncertainty and the independent variable uncertainty to provide an estimate of the confidence limits associated with a predicted value. The prediction uncertainty is often neglected, yet it is an important measure of the usefulness of a model. Prediction uncertainty reflects the modeler's confidence in the model, and it should be used by a decision maker as a weight indicating the value of the model prediction. A procedure is outlined that combined lake modeling and uncertainty analysis for use in lake quality assessment and lake management. An example is provided illustrating the use of this procedure in nutrient budget sampling design, data analysis, and the evaluation of lake management strategies for a 208 program in New Hampshire.     

A Sensitivity and Error Analysis Framework for Lake Eutrophication Modeling1

ABSTRACT: A framework for sensitivity and error analysis in mathematical modeling is described and demonstrated. The Lake Eutrophication Analysis Procedure (LEAP) consists of a series of linked models which predict lake water quality conditions as a function of watershed land use, hydrolgic variables, and morphometric variables. Specification of input variables as distributions (means and standard errors) and use of first-order error analysis techniques permits estimation of output variable means, standard errors, and confidence ranges. Predicted distributions compare favorably with those estimated using Monte-Carlo simulation. The framework is demonstrated by applying it to data from Lake Morey, Vermont. While possible biases exist in the models calibrated for this application, prediction variances, attributed chiefly to model error, are comparable to the observed year-to-year variance in water quality, as measured by spring phosphorus concentration, hypolimnetic oxygen depletion rate, summer chlorophyll-a, and summer transparency in this lake. Use of the framework provides insight into important controlling factors and relationships and identifies the major sources of uncertainty in a given model application.     

USING A CHOICE MODELLING APPROACH FOR CUSTOMER SERVICE STANDARDS IN URBAN WATER1

ABSTRACT: A series of reforms in the water industry in Australia has created a demand from the industry and regulators for objective methodologies to evaluate incremental changes in the customer service standards. In this paper, the use of choice modeling for estimating implicit prices associated with urban water supply attributes is explored. Results from multinomial logit (MNL) and random parameters logit (RPL) models show that increases in annual water bills and the frequency of future interruptions were the most important attributes. Implicit price confidence intervals based on the best models suggest that people are willing to pay positive amounts to achieve a water supply that is less frequently interrupted. The provision of alternative water supplies during an interruption and notification of the interruption were found to be unimportant to respondents. Choice modeling proved to be a useful technique and provided the industry and regulators with additional information for standard setting.     

Incorporating Uncertainty Into the Ranking of SPARROW Model Nutrient Yields From Mississippi/Atchafalaya River Basin Watersheds1

Abstract: Excessive loads of nutrients transported by tributary rivers have been linked to hypoxia in the Gulf of Mexico. Management efforts to reduce the hypoxic zone in the Gulf of Mexico and improve the water quality of rivers and streams could benefit from targeting nutrient reductions toward watersheds with the highest nutrient yields delivered to sensitive downstream waters. One challenge is that most conventional watershed modeling approaches (e.g., mechanistic models) used in these management decisions do not consider uncertainties in the predictions of nutrient yields and their downstream delivery. The increasing use of parameter estimation procedures to statistically estimate model coefficients, however, allows uncertainties in these predictions to be reliably estimated. Here, we use a robust bootstrapping procedure applied to the results of a previous application of the hybrid statistical/mechanistic watershed model SPARROW (Spatially Referenced Regression On Watershed attributes) to develop a statistically reliable method for identifying “high priority” areas for management, based on a probabilistic ranking of delivered nutrient yields from watersheds throughout a basin. The method is designed to be used by managers to prioritize watersheds where additional stream monitoring and evaluations of nutrient‐reduction strategies could be undertaken. Our ranking procedure incorporates information on the confidence intervals of model predictions and the corresponding watershed rankings of the delivered nutrient yields. From this quantified uncertainty, we estimate the probability that individual watersheds are among a collection of watersheds that have the highest delivered nutrient yields. We illustrate the application of the procedure to 818 eight‐digit Hydrologic Unit Code watersheds in the Mississippi/Atchafalaya River basin by identifying 150 watersheds having the highest delivered nutrient yields to the Gulf of Mexico. Highest delivered yields were from watersheds in the Central Mississippi, Ohio, and Lower Mississippi River basins. With 90% confidence, only a few watersheds can be reliably placed into the highest 150 category; however, many more watersheds can be removed from consideration as not belonging to the highest 150 category. Results from this ranking procedure provide robust information on watershed nutrient yields that can benefit management efforts to reduce nutrient loadings to downstream coastal waters, such as the Gulf of Mexico, or to local receiving streams and reservoirs.     

Nonparametric Monte Carlo Simulation for Flood Frequency Curve Derivation: An Application to a Korean Watershed1

Abstract: A mix of causative mechanisms may be responsible for flood at a site. Floods may be caused because of extreme rainfall or rain on other rainfall events. The statistical attributes of these events differ according to the watershed characteristics and the causes. Traditional methods of flood frequency analysis are only adequate for specific situations. Also, to address the uncertainty of flood frequency estimates for hydraulic structures, a series of probabilistic analyses of rainfall‐runoff and flow routing models, and their associated inputs, are used. This is a complex problem in that the probability distributions of multiple independent and derived random variables need to be estimated to evaluate the probability of floods. Therefore, the objectives of this study were to develop a flood frequency curve derivation method driven by multiple random variables and to develop a tool that can consider the uncertainties of design floods. This study focuses on developing a flood frequency curve based on nonparametric statistical methods for the estimation of probabilities of rare floods that are more appropriate in Korea. To derive the frequency curve, rainfall generation using the nonparametric kernel density estimation approach is proposed. Many flood events are simulated by nonparametric Monte Carlo simulations coupled with the center Latin hypercube sampling method to estimate the associated uncertainty. This study applies the methods described to a Korean watershed. The results provide higher physical appropriateness and reasonable estimates of design flood.     

Determination of Frequency for Remeasuring Ground and Vegetation Cover Factor Needed for Soil Erosion Modeling

Determining a remeasurement frequency of variables over time is required in monitoring environmental systems. This article demonstrates methods based on regression modeling and spatio-temporal variability to determine the time interval to remeasure the ground and vegetation cover factor on permanent plots for monitoring a soil erosion system. The spatio-temporal variability methods include use of historical data to predict semivariograms, modeling average temporal variability, and temporal interpolation by two-step kriging. The results show that for the cover factor, the relative errors of the prediction increase with an increased length of time interval between remeasurements when using the regression and semivariogram models. Given precision or accuracy requirements, appropriate time intervals can be determined. However, the remeasurement frequency also varies depending on the prediction interval time. As an alternative method, the range parameter of a semivariogram model can be used to quantify average temporal variability that approximates the maximum time interval between remeasurements. This method is simpler than regression and semivariogram modeling, but it requires a long-term dataset based on permanent plots. In addition, the temporal interpolation by two-step kriging is also used to determine the time interval. This method is applicable when remeasurements in time are not sufficient. If spatial and temporal remeasurements are sufficient, it can be expanded and applied to design spatial and temporal sampling simultaneously.     

An Interval-Parameter Waste-Load-Allocation Model for River Water Quality Management Under Uncertainty

A simulation-based interval quadratic waste load allocation (IQWLA) model was developed for supporting river water quality management. A multi-segment simulation model was developed to generate water-quality transformation matrices and vectors under steady-state river flow conditions. The established matrices and vectors were then used to establish the water-quality constraints that were included in a water quality management model. Uncertainties associated with water quality parameters, cost functions, and environmental guidelines were described as intervals. The cost functions of wastewater treatment units were expressed in quadratic forms. A water-quality planning problem in the Changsha section of Xiangjiang River in China was used as a study case to demonstrate applicability of the proposed method. The study results demonstrated that IQWLA model could effectively communicate the interval-format uncertainties into optimization process, and generate inexact solutions that contain a spectrum of potential wastewater treatment options. Decision alternatives can be generated by adjusting different combinations of the decision variables within their solution intervals. The results are valuable for supporting local decision makers in generating cost-effective water quality management strategies.     

Assessing Uncertainty in Estimates of Nitrogen Loading to Estuaries for Research, Planning, and Risk Assessment

Parametric (propagation for normal error estimates) and nonparametric methods (bootstrap and enumeration of combinations) to assess the uncertainty in calculated rates of nitrogen loading were compared, based on the propagation of uncertainty observed in the variables used in the calculation. In addition, since such calculations are often based on literature surveys rather than random replicate measurements for the site in question, error propagation was also compared using the uncertainty of the sampled population (e.g., standard deviation) as well as the uncertainty of the mean (e.g., standard error of the mean). Calculations for the predicted nitrogen loading to a shallow estuary (Waquoit Bay, MA) were used as an example. The previously estimated mean loading from the watershed (5,400 ha) to Waquoit Bay (600 ha) was 23,000 kg N yr⁻¹. The mode of a nonparametric estimate of the probability distribution differed dramatically, equaling only 70% of this mean. Repeated observations were available for only 8 of the 16 variables used in our calculation. We estimated uncertainty in model predictions by treating these as sample replicates. Parametric and nonparametric estimates of the standard error of the mean loading rate were 12–14%. However, since the available data include site-to-site variability, as is often the case, standard error may be an inappropriate measure of confidence. The standard deviations were around 38% of the loading rate. Further, 95% confidence intervals differed between the nonparametric and parametric methods, with those of the nonparametric method arranged asymmetrically around the predicted loading rate. The disparity in magnitude and symmetry of calculated confidence limits argue for careful consideration of the nature of the uncertainty of variables used in chained calculations. This analysis also suggests that a nonparametric method of calculating loading rates using most frequently observed values for variables used in loading calculations may be more appropriate than using mean values. These findings reinforce the importance of including assessment of uncertainty when evaluating nutrient loading rates in research and planning. Risk assessment, which may need to consider relative probability of extreme events in worst-case scenarios, will be in serious error using normal estimates, or even the nonparametric bootstrap. A method such as our enumeration of combinations produces a more reliable distribution of risk.     

SOIL MOISTURE ACCOUNTING COMPONENT OF THE USDAHL-74 MODEL OF WATERSHED HYDROLOGY1

ABSTRACT: Soil moisture in two layers of a soil near Chickasha, Oklahoma, was simulated, using USDAHL-74 Model of Watershed Hydrology. Weekly values computed for both layers compared well with those observed during the 15-month period. Certain key parameters required adjustments in the model which illustrate the need for accurate input information. The experiment demonstrates that the model, which has previously given good results in continuous streamflow prediction on watersheds up to 100 square miles, can also compute soil moisture continuously at a site. This capability suggests other model uses, for example, in monitoring the disposition of applied chemicals.     

SHORT-TERM FORECASTING OF SNOWMELT RUNOFF USING ARMAX MODELS1

ABSTRACT: Time series models of the ARMAX class were investigated for use in forecasting daily riverflow resulting from combined snowmelt/rainfall. The Snowmelt Runoff Model (Martinec-Rango Model) is shown to have a form similar to the ARMAX model. The advantage of the ARMAX approach is that analytical model identification and parameter estimation techniques are available. In addition, previous forecast errors can be included to improve forecasts and confidence limits can be estimated for the forecasts. Diagnostic checks are available to determine if the model is performing properly. Finally, Kalman filtering can be used to allow the model parameters to vary continuously to reflect changing basin runoff conditions. The above advantages result in improved flow forecasts with fewer model parameters.     

A STRIP MODEL APPROACH TO PARAMETERIZE A COUPLED GREEN‐AMPT KINEMATIC WAVE MODEL1

ABSTRACT: Infiltration processes at the plot scale are often described and modeled using a single effective hydraulic conductivity (Kg) value. This can lead to errors in runoff and erosion prediction. An integrated field measurement and modeling study was conducted to evaluate: (1) the relationship among rainfall intensity, spatially variable soil and vegetation characteristics, and infiltration processes; and (2) how this relationship could be modeled using Green and Ampt and a spatially distributed hydrologic model. Experiments were conducted using a newly developed variable intensity rainfall simulator on 2 m by 6 m plots in a rangeland watershed in southeastern Arizona. Rainfall application rates varied between 50 and 200 mm/hr. Results of the rainfall simulator experiments showed that the observed hydrologic response changed with changes in rainfall intensity and that the response varied with antecedent moisture condition. A distributed process based hydrologic simulation model was used to model the plots at different levels of hydrologic complexity. The measurement and simulation model results show that the rainfall runoff relationship cannot be accurately described or modeled using a single Kg value at the plot scale. Multi‐plane model configurations with infiltration parameters based on soil and plot characteristics resulted in a significant improvement over single‐plane configurations.