MULTIVARIATE EXTREME VALUE DISTRIBUTION WITH MIXED GUMBEL MARGINALS1

ABSTRACT: Bivariate and trivariate distributions have been derived from the logistic model for the multivariate extreme value distribution. Marginals in the models are extreme value type I distributions for two-component mixture variables (mixed Gumbel distribution). This paper is a continuation of the previous works on multivariate distribution in hydrology. Interest is focused on the analysis of floods which are generated by different types of storms. The construction of their corresponding probability distributions and density functions are described. In order to obtain the parameters of such a bivariate or trivariate distribution, a generalized maximum likelihood estimation procedure is proposed to allow for the cases of samples with different lengths of record. A region in Northern Mexico with 42 gauging stations, grouped into two homogeneous regions, has been selected to apply the models. Results produced by the multivariate distributions have been compared with those obtained by the Normal, log-Normal-2, log-Normal-3, Gamma-2, Gamma-3, log-Pearson-3, Gumbel, TCEV and General Extreme Value distributions. Goodness of fit is measured by the criterion of standard error of fit. Results suggest that the proposed models are a suitable option to be considered when performing flood frequency analysis.     

Regional Scale Stressor‐Response Models in Aquatic Ecosystems1

Abstract: A systematic method for identification and estimation of regional scale stressor‐response models in aquatic ecosystems will be useful in monitoring and assessment of aquatic resources, determination of regional nutrient criteria and for increased understanding of the differences between regions. The model response variable is chlorophyll a, a measure of algal density, while the stressors include nutrient concentrations from the USEPA Nutrient Criteria Database (NCD) for lakes/ponds and reservoirs of the continental United States. The NCD has observations for both stressors and biological responses determined using methods that are not consistently available at the continental scale. To link multiple environmental stressors to biological responses and quantify uncertainty in model predictions, we take a multilevel modeling approach to the estimation of a linear model for prediction of log Chlorophyll a using predictors log TP and log TN. The multilevel modeling approach allows us to adjust the impact of covariates at all levels (observation, higher level groups) for the simultaneous operation of contextual and individual variability in the outcome. Here, we wish to allow separate regression coefficients for inference regarding similarities and differences between each of 14 ecoregions, and between the two water‐body types, lakes/ponds and reservoirs. We are also interested in the nuisance effects of the categorical variables indicating the type of nitrogen measurements (three levels) and the type of chlorophyll a measurements (four levels) used. Model‐based determination of nutrient criteria points to an apparent incompatibility of criteria developed for nutrient stressors and eutrophication responses using current Environmental Protection Agency’s guidance.     

Nonstationarity: Flood Magnification and Recurrence Reduction Factors in the United States1

Vogel, Richard M., Chad Yaindl, and Meghan Walter, 2011. Nonstationarity: Flood Magnification and Recurrence Reduction Factors in the United States. Journal of the American Water Resources Association (JAWRA) 47(3):464‐474. DOI: 10.1111/j.1752‐1688.2011.00541.x Abstract: It may no longer be reasonable to model streamflow as a stationary process, yet nearly all existing water resource planning methods assume that historical streamflows will remain unchanged in the future. In the few instances when trends in extreme events have been considered, most recent work has focused on the influence of climate change, alone. This study takes a different approach by exploring trends in floods in watersheds which are subject to a very broad range of anthropogenic influences, not limited to climate change. A simple statistical model is developed which can both mimic observed flood trends as well as the frequency of floods in a nonstationary world. This model is used to explore a range of flood planning issues in a nonstationary world. A decadal flood magnification factor is defined as the ratio of the T‐year flood in a decade to the T‐year flood today. Using historical flood data across the United States we obtain flood magnification factors in excess of 2‐5 for many regions of the United States, particularly those regions with higher population densities. Similarly, we compute recurrence reduction factors which indicate that what is now considered the 100‐year flood, may become much more common in many watersheds. Nonstationarity in floods can result from a variety of anthropogenic processes including changes in land use, climate, and water use, with likely interactions among those processes making it very difficult to attribute trends to a particular cause.     

Approaches to Evaluate Water Quality Model Parameter Uncertainty for Adaptive TMDL Implementation1

Abstract: The National Research Council recommended Adaptive Total Maximum Daily Load implementation with the recognition that the predictive uncertainty of water quality models can be high. Quantifying predictive uncertainty provides important information for model selection and decision‐making. We review five methods that have been used with water quality models to evaluate model parameter and predictive uncertainty. These methods (1) Regionalized Sensitivity Analysis, (2) Generalized Likelihood Uncertainty Estimation, (3) Bayesian Monte Carlo, (4) Importance Sampling, and (5) Markov Chain Monte Carlo (MCMC) are based on similar concepts; their development over time was facilitated by the increasing availability of fast, cheap computers. Using a Streeter‐Phelps model as an example we show that, applied consistently, these methods give compatible results. Thus, all of these methods can, in principle, provide useful sets of parameter values that can be used to evaluate model predictive uncertainty, though, in practice, some are quickly limited by the “curse of dimensionality” or may have difficulty evaluating irregularly shaped parameter spaces. Adaptive implementation invites model updating, as new data become available reflecting water‐body responses to pollutant load reductions, and a Bayesian approach using MCMC is particularly handy for that task.     

Magnitude,Frequency, and Duration Relations for Suspended Sediment in Stable (“Reference”) Southeastern Streams1

Abstract: Sediment is listed as one of the leading causes of water‐quality impairments in surface waters of the United States (U.S.). A water body becomes listed by a State, Territory or Tribe if its designated use is not being attained (i.e., impaired). In many cases, the prescribed designated use is aquatic health or habitat, indicating that total maximum daily loads (TMDL) targets for sediment should be functionally related to this use. TMDL targets for sediment transport have been developed for many ecoregions over the past several years using suspended‐sediment yield as a metric. Target values were based on data from “reference” streams or reaches, defined as those exhibiting geomorphic characteristics of equilibrium. This approach has proved useful to some states developing TMDLs for suspended sediment, although one cannot conclude that if a stream exceeds the target range, the aquatic ecosystem will be adversely impacted. To address this problem, historical flow‐transport and sediment‐transport data from hundreds of sites in the Southeastern U.S. were re‐examined to develop parameters (metrics) such as frequency and duration of sediment concentrations. Sites determined as geomorphically stable from field evaluations and from analysis of gauging‐station records were sorted by ecoregion. Mean‐daily flow data obtained from the U.S. Geological Survey were applied to sediment‐transport rating relations to determine suspended‐sediment load for each day of record. The frequency and duration that a given concentration was equaled or exceeded were then calculated to produce a frequency distribution for each site. “Reference” distributions were created using the stable sites in each ecoregion by averaging all of the distributions at specified exceedance intervals. As with the “reference” suspended‐sediment yields, there is a broad range of frequency and duration distributions that reflects the hydrologic and sediment‐transport regimes of the ecoregions. Ecoregions such as the Mississippi Valley Loess Plains (#74) maintain high suspended sediment concentrations for extended periods, whereas coastal plain ecoregions (#63 and 75) show much lower concentrations.     

Modes of Variability of Annual and Seasonal Rainfall in Mexico

The aim of this study is to identify temporal and spatial variability patterns of annual and seasonal rainfall in Mexico. A set of 769 weather stations located in Mexico was examined. The country was divided into 12 homogeneous rainfall regions via principal component analysis. A Pettitt test was conducted to perform a homogeneity analysis for detecting abrupt changes in mean rainfall levels, and a Mann‐Kendall test was conducted to examine the presence of monotonically increasing/decreasing patterns in the data. In total, 14.4% of the annual series was deemed nonstationary. Fourteen percent of the samples were nonstationary in the winter and summer, and 9% were nonstationary in the spring and autumn. According to the results, the nonstationarity of some seasonal rainfall series may be associated with the presence of atmospheric phenomena (e.g., El Niño/Southern Oscillation, Pacific Decadal Oscillation, Atlantic Multidecadal Oscillation, and North Atlantic Oscillation). A rainfall frequency analysis was performed for the nonstationary annual series, and significant differences in the return levels can be expected for the scenarios analyzed. The identification of areas that are more susceptible to changes in rainfall levels will improve water resource management plans in the country, and it is expected that these plans will take into account nonstationary theory.     

Load Estimation Method Using Distributions with Covariates: A Comparison with Commonly Used Estimation Methods

Load estimates obtained using an approach based on statistical distributions with parameters expressed as a function of covariates (e.g., streamflow) (distribution with covariates hereafter called DC method) were compared to four load estimation methods: (1) flow‐weighted mean concentration; (2) integral regression; (3) segmented regression (the last two with Ferguson's correction factor); and (4) hydrograph separation methods. A total of 25 datasets (from 19 stations) of daily concentrations of total dissolved solids, nutrients, or suspended particulate matter were used. The selected stations represented a wide range of hydrological conditions. Annual flux errors were determined by randomly generating 50 monthly sample series from daily series. Annual and interannual biases and dispersions were evaluated and compared. The impact of sampling frequency was investigated through the generation of bimonthly and weekly surveys. Interannual uncertainty analysis showed that the performance of the DC method was comparable with those of the other methods, except for stations showing high hydrological variability. In this case, the DC method performed better, with annual biases lower than those characterizing the other methods. Results show that the DC method generated the smallest pollutant load errors when considering a monthly sampling frequency for rivers showing high variability in hydrological conditions and contaminant concentrations.     

SWAT‐LUT: A Desktop Graphical User Interface for Updating Land Use in SWAT

Long‐term simulations of agricultural watersheds have often been done assuming constant land use over time, but this is not a realistic assumption for many agricultural regions. This paper presents the soil and water assessment tool (SWAT)‐Landuse Update Tool (LUT), a standalone, user‐friendly desktop‐based tool for updating land use in the SWAT model that allows users to process multi‐year land use data. SWAT‐LUT is compatible with several SWAT model interfaces, provides users with several options to easily prepare and incorporate land use changes (LUCs) over a simulation period, and allows users to incorporate past or emerging land use categories. Incorporation of LUCs is expected to provide realistic model parameterization and scenario simulations. SWAT‐LUT is a public domain interface written in Python programming language. Two applications at the Fort Cobb Reservoir Experimental Watershed located in Oklahoma and pertinent results are provided to demonstrate its use. Incorporating LUCs related to implementation of recommended conservation practices over the years reduced discharge, evapotranspiration, sediment, total nitrogen, and total phosphorus loads by 59%, 9%, 68%, 53%, and 88%, respectively. The user’s manual is included in this article as Supporting Information. The SWAT‐LUT executable file and an example SWAT project with three land use rasters and the user’s manual are available at the United States Department of Agriculture‐Agricultural Research Service Grazinglands Research Laboratory website under Software. Editor’s note : This paper is part of the featured series on Optimizing Ogallala Aquifer Water Use to Sustain Food Systems. See the February 2019 issue for the introduction and background to the series.     

GEOSTATISTICAL ESTIMATION OF HORIZONTAL HYDRAULIC CONDUCTIVITY FOR THE KIRKWOOD‐COHANSEY AQUIFER1

ABSTRACT: The Kirkwood‐Cohansey aquifer has been identified as a critical source for meeting existing and expected water supply needs for southern New Jersey. Several contaminated sites exist in the region; their impact on the aquifer has to be evaluated using ground water flow and transport models. Ground water modeling depends on availability of measured hydrogeologic data (e.g., hydraulic conductivity, for parameterization of the modeling runs). However, field measurements of such critical data have inadequate spatial density, and their locations are often clustered. The goal of this study was to research, compile, and geocode existing data, then use geostatistics and advanced mapping methods to develop a map of horizontal hydraulic conductivity for the Kirkwood‐Cohansey aquifer. Spatial interpolation of horizontal hydraulic conductivity measurements was performed using the Bayesian Maximum Entropy (BME) Method implemented in the BMELib code library. This involved the integration of actual measurements with soft information on likely ranges of hydraulic conductivity at a given location to obtain estimate maps. The estimation error variance maps provide an insight into the uncertainty associated with the estimates, and indicate areas where more information on hydraulic conductivity is required.     

A SCALE‐INVARIANT GAUSS‐MARKOV MODEL FOR DESIGN STORM HYETOGRAPHS1

Abstract. Hyetographs are essential to many hydrological designs. Many studies have shown that hyetographs are specific to storm types and durations. Recent work presented evidence that dimensionless hyetographs are scale invariant. We show that the simple scaling property of rainfall guarantees that the normalized rainfall rates of different storm durations are identically distributed and propose a nonstationary Gauss‐Markov model based on the annual maximum events that arise from the dominant storm type. We derive the unique estimators for the parameters of the Gauss‐Markov model under two constraints that: (a) the typical peak rainfall rate is preserved, and (b) the most likely hyetograph is obtained. One attractive feature of this model is that it allows translating hyetographs between storms of different durations. Two examples illustrate our model.     

AN ILLUSTRATION OF MODEL STRUCTURE IDENTIFICATION1

ABSTRACT: The purpose of this article is to discuss the importance of uncertainty analysis in water quality modeling, with an emphasis on the identification of the correct model specification. A wetland phosphorus retention model is used as an example to illustrate the procedure of using a filtering technique for model structure identification. Model structure identification is typically done through model parameter estimation. However, due to many sources of error in both model parameterization and observed variables and data, error-in-variable is often a problem. Therefore, it is not appropriate to use the least squares method for parameter estimation. Two alternative methods for parameter estimation are presented. The first method is the maximum likelihood estimator, which assumes independence of the observed response variable values. In anticipating the possible violation of the independence assumption, a second method, which coupled a maximum likelihood estimator and Kalman filter model, was presented. Furthermore, a Monte Carlo simulation algorithm is presented as a preliminary method for judging whether the model structure is appropriate or not.     

A HYBRID OPTIMIZATION APPROACH TO THE ESTIMATION OF DISTRIBUTED PARAMETERS IN TWO-DIMENSIONAL CONFINED AQUIFERS1

ABSTRACT: In using non-linear optimization techniques for estimation of parameters in a distributed ground water model, the initial values of the parameters and prior information about them play important roles. In this paper, the genetic algorithm (GA) is combined with the truncated-Newton search technique to estimate groundwater parameters for a confined steady-state ground water model. Use of prior information about the parameters is shown to be important in estimating correct or near-correct values of parameters on a regional scale. The amount of prior information needed for an accurate solution is estimated by evaluation of the sensitivity of the performance function to the parameters. For the example presented here, it is experimentally demonstrated that only one piece of prior information of the least sensitive parameter is sufficient to arrive at the global or near-global optimum solution. For hydraulic head data with measurement errors, the error in the estimation of parameters increases as the standard deviation of the errors increases. Results from our experiments show that, in general, the accuracy of the estimated parameters depends on the level of noise in the hydraulic head data and the initial values used in the truncated-Newton search technique.     

The Regionalization of National‐Scale SPARROW Models for Stream Nutrients1

Schwarz, Gregory E., Richard B. Alexander, Richard A. Smith, and Stephen D. Preston, 2011. The Regionalization of National‐Scale SPARROW Models for Stream Nutrients. Journal of the American Water Resources Association (JAWRA) 47(5):1151‐1172. DOI: 10.1111/j.1752‐1688.2011.00581.x Abstract: This analysis modifies the parsimonious specification of recently published total nitrogen (TN) and total phosphorus (TP) national‐scale SPAtially Referenced Regressions On Watershed attributes models to allow each model coefficient to vary geographically among three major river basins of the conterminous United States. Regionalization of the national models reduces the standard errors in the prediction of TN and TP loads, expressed as a percentage of the predicted load, by about 6 and 7%. We develop and apply a method for combining national‐scale and regional‐scale information to estimate a hybrid model that imposes cross‐region constraints that limit regional variation in model coefficients, effectively reducing the number of free model parameters as compared to a collection of independent regional models. The hybrid TN and TP regional models have improved model fit relative to the respective national models, reducing the standard error in the prediction of loads, expressed as a percentage of load, by about 5 and 4%. Only 19% of the TN hybrid model coefficients and just 2% of the TP hybrid model coefficients show evidence of substantial regional specificity (more than ±100% deviation from the national model estimate). The hybrid models have much greater precision in the estimated coefficients than do the unconstrained regional models, demonstrating the efficacy of pooling information across regions to improve regional models.     

Improving Predictive Models of In‐Stream Phosphorus Concentration Based on Nationally‐Available Spatial Data Coverages

Spatial data are playing an increasingly important role in watershed science and management. Large investments have been made by government agencies to provide nationally‐available spatial databases; however, their relevance and suitability for local watershed applications is largely unscrutinized. We investigated how goodness of fit and predictive accuracy of total phosphorus (TP) concentration models developed from nationally‐available spatial data could be improved by including local watershed‐specific data in the East Fork of the Little Miami River, Ohio, a 1,290 km² watershed. We also determined whether a spatial stream network (SSN) modeling approach improved on multiple linear regression (nonspatial) models. Goodness of fit and predictive accuracy were highest for the SSN model that included local covariates, and lowest for the nonspatial model developed from national data. Septic systems and point source TP loads were significant covariates in the local models. These local data not only improved the models but enabled a more explicit interpretation of the processes affecting TP concentrations than more generic national covariates. The results suggest SSN modeling greatly improves prediction and should be applied when using national covariates. Including local covariates further increases the accuracy of TP predictions throughout the studied watershed; such variables should be included in future national databases, particularly the locations of septic systems.     

A Multilevel Model of the Impact of Farm‐Level Best Management Practices on Phosphorus Runoff1

Abstract: Multilevel or hierarchical models have been applied for a number of years in the social sciences but only relatively recently in the environmental sciences. These models can be developed in either a frequentist or Bayesian context and have similarities to other methods such as empirical Bayes analysis and random coefficients regression. In essence, multilevel models take advantage of the hierarchical structure that exists in many multivariate datasets; for example, water quality measurements may be taken from individual lakes, lakes are located in various climatic zones, lakes may be natural or man‐made, and so on. The groups, or levels, may effectively yield different responses or behaviors (e.g., nutrient load response in lakes) that often make retaining group membership more effective when developing a predictive model than when working with either all of the data together or working separately with the individuals. Here, we develop a multilevel model of the impact of farm level best management practices (BMPs) on phosphorus runoff. The result of this research is a model with parameters which vary with key practice categories and thus may be used to evaluate the effectiveness of these practices on phosphorus runoff. For example, it was found that the effect of fertilizer application rate on farm‐scale phosphorus loss is a function of the application method, the hydrologic soil group, and the land use (crop type). Further, results indicate that the most effective method for controlling fertilizer loss is through soil injection. In summary, the resultant multilevel model can be used to estimate phosphorus loss from farms and hence serve as a useful tool for BMP selection.     

Watershed Regressions for Pesticides (warp) Models for Predicting Atrazine Concentrations in Corn Belt Streams1

Stone, Wesley W. and Robert J. Gilliom, 2012. Watershed Regressions for Pesticides (WARP) Models for Predicting Atrazine Concentrations in Corn Belt Streams. Journal of the American Water Resources Association (JAWRA) 48(5): 970‐986. DOI: 10.1111/j.1752‐1688.2012.00661.x Abstract: Watershed Regressions for Pesticides (WARP) models, previously developed for atrazine at the national scale, are improved for application to the United States (U.S.) Corn Belt region by developing region‐specific models that include watershed characteristics that are influential in predicting atrazine concentration statistics within the Corn Belt. WARP models for the Corn Belt (WARP‐CB) were developed for annual maximum moving‐average (14‐, 21‐, 30‐, 60‐, and 90‐day durations) and annual 95th‐percentile atrazine concentrations in streams of the Corn Belt region. The WARP‐CB models accounted for 53 to 62% of the variability in the various concentration statistics among the model‐development sites. Model predictions were within a factor of 5 of the observed concentration statistic for over 90% of the model‐development sites. The WARP‐CB residuals and uncertainty are lower than those of the National WARP model for the same sites. Although atrazine‐use intensity is the most important explanatory variable in the National WARP models, it is not a significant variable in the WARP‐CB models. The WARP‐CB models provide improved predictions for Corn Belt streams draining watersheds with atrazine‐use intensities of 17 kg/km² of watershed area or greater.     

Estimating Lag to Peak between Rainfall and Peak Streamflow with a Mixed‐Effects Model

We test the use of a mixed‐effects model for estimating lag to peak for small basins in Maine (drainage areas from 0.8 to 78 km²). Lag to peak is defined as the time between the center of volume of the excess rainfall during a storm event and the resulting peak streamflow. A mixed‐effects model allows for multiple observations at sites without violating model assumptions inherent in traditional ordinary least squares models, which assume each observation is independent. The mixed model includes basin drainage area and maximum 15‐min rainfall depth for individual storms as explanatory features. Based on a remove‐one‐site cross‐validation analysis, the prediction errors of this model ranged from ?42% to +73%. The mixed model substantially outperformed three published models for lag to peak and one published model for centroid lag for estimating lag to peak for small basins in Maine. Lag to peak estimates are a key input to rainfall–runoff models used to design hydraulic infrastructure. The improved accuracy and consistency with model assumptions indicates that mixed models may provide increased data utilization that could enhance models and estimates of lag to peak in other regions.     

Using Instream Water Temperature Forecasts for Fisheries Management: An Application in the Pacific Northwest1

Huang, Biao, Christian Langpap, and Richard M. Adams, 2011. Using Instream Water Temperature Forecasts for Fisheries Management: An Application in the Pacific Northwest. Journal of the American Water Resources Association (JAWRA) 47(4):861‐876. DOI: 10.1111/j.1752‐1688.2011.00562.x Abstract: Water temperature is an important factor affecting aquatic life within the stream environment. Cold water species, such as salmonids, are particularly susceptible to elevated water temperatures. This paper examines the potential usefulness of short‐term (7 to 10 days) water temperature forecasts for salmonid management. Forecasts may be valuable if they allow the water resource manager to make better water allocation decisions. This study considers two applications: water releases from Lewiston Dam for management of adult Chinook salmon (Oncorhynchus tshawytscha) in the Klamath River and leasing water from agriculture for management of steelhead trout (Oncorhynchus mykiss) in the John Day River. We incorporate biophysical models and water temperature distribution data into a Bayesian framework to simulate changes in fish populations and the corresponding opportunity cost of water under different levels of temperature forecast reliability. Simulation results indicate that use of the forecasts results in increased fish production and that marginal costs decline as forecast reliability increases, suggesting that provision and use of such stream temperature forecasts would have potential value to society.