DERIVATION OF THE GAMMA DISTRIBUTION BY USING THE PRINCIPLE OF MAXIMUM ENTROPY (POME)1

The principle of maximum entropy (POME) was used to derive the two-parameter gamma distribution used frequently in synthesis of instantaneous or finite-period unit hydrographs. The POME yielded the minimally prejudiced gamma distribution by maximizing the entropy subject to two appropriate constraints which were the mean of real values and the mean of the logarithms of real values of the variable. It provided a unique method for parameter estimation. Experimental data were used to compare this method with the methods of moments, cumulants, maximum likelihood estimation, and least squares.     

PARAMETRIC ESTIMATION OF COVARIANCES OF REGIONALIZED VARIABLES1

ABSTRACT: The usefulness of stochastic models in describing the spatial variability of hydrogeologic quantities, such as permeability, storativity, piezometric head, seepage velocity, and solute concentrations is now widely recognized. In practice, these quantities are represented as the sum of a well-structured component, or drift, and a more erratic fluctuation component which is described statistically through its covariance function. This paper reviews some of the most recent and most promising methods for the estimation of parameters of these covariances from existing data. They are maximum likelihood, restricted maximum likelihood, minimum-variance unbiased quadratic estimation, and minimum-norm (weighted least squares) estimation. The applicability of such methods to conditional and unconditional probability problems is discussed.     

PARAMETER ESTIMATION FOR TPLN DISTRIBUTION FOR FLOOD FREQUENCY ANALYSIS1

ABSTRACT: The principle of maximum entropy (POME) was used to derive an alternative method for parameter estimation for the three parameter lognormal (TPLN) distribution. Six sets of annual peak discharge data were used to evaluate this method and compare it with the methods of moments and maximum likelihood estimation.     

SIMULTANEOUS EQUATION SYSTEMS: A CONSISTENT ESTIMATOR FOR UNKNOWN PARAMETERS IN CONFINED AQUIFERS1

ABSTRACT: This paper presents criteria for establishing the identification status of the inverse problem for confined aquifer flow. Three linear estimation methods (ordinary least squares, two-stage least squares, and three-stage least squares) and one nonlinear method (maximum likelihood) are used to estimate the matrices of parameters embedded in the partial differential equation characterizing confined flow. Computational experience indicates several advantages of maximum likelihood over the linear methods.     

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.     

COMPARISON OF PARAMETRIC TAIL ESTIMATORS FOR LOW‐FLOW FREQUENCY ANALYSIS1

ABSTRACT: In recent years, several approaches to hydrologic frequency analysis have been proposed that enable one to direct attention to that portion of an overall probability distribution that is of greatest interest. The majority of the studies have focused on the upper tail of a distribution for flood analyses, though the same ideas can be applied to low flows. This paper presents an evaluation of the performances of five different estimation methods that place an emphasis on fitting the lower tail of the lognormal distribution for estimation of the ten‐year low‐flow quantile. The methods compared include distributional truncation, MLE treatment of censored data, partial probability weighted moments, LL‐moments, and expected moments. It is concluded that while there are some differences among the alternative methods in terms of their biases and root mean square errors, no one method consistently performs better than the others, particularly with recognition that the underlying population distribution is unknown. Therefore, it seems perfectly legitimate to make a selection of a method on the basis other criteria, such as ease of use. It is also shown in this paper that the five alternative methods can perform about as well as, if not better than, an estimation strategy involving fitting the complete lognormal distribution using L‐moments.     

熵法估参在环境监测数据频率分析中的应用

以最大信息熵原理为理论基础的熵法估参方法，是一种具有严格物理和数学意义的新型参数估计方法，本文针对珠江广州河段主要污染物含量长年监测数据，对比熵法与传统方法矩法对四参数Г分布的估参结果，并以频率绝对离盖和最小为准则进行判定，结果表明，熵法估参结果与矩法总体上相当接近，且大部分样本的熵法估计参数优于矩法，在环境监测数据频率分析中具有实用性和推广价值。     

UNIT HYDROGRAPHS – A COMPARATIVE STUDY1

ABSTRACT. Unit hydrographs derived by using two methods, linear programming and least squares, are compared. Test data comprise rainfall and runoff information from four storms over the North Branch Potomac River near Cumberland, Maryland. The mathematical bases of these methods for unit-hydrograph derivation are explained. The linear programming method minimizes the sum of absolute deviations, and the least squares method minimizes the sum of the squares of deviations. Computer subroutines are readily available for application of these methods. The unit hydrographs derived with the two methods are practically the same for storms 2 and 3, but differ somewhat for storms 1 and 4. However, the reconstituted direct surface runoff hydrographs are similar to those observed with the exception of the hydrograph for storm 4 which had a relatively more non-uniform rainfall excess of a considerably larger duration.     

DYNAMIC MODELING OF STREAM QUALITY BY INVARIANT IMBEDDING1

ABSTRACT. The estimator equations obtained using invariant imbedding is used to estimate the parameters in river or stream pollution. By using these equations, the parameters can be estimated directly from differential equations representing the pollution model and from measured noisy data such as BOD and DO. Another advantage of this approach is that a sequential estimation scheme is obtained. By using this sequential scheme, only current data are needed to estimate current or future values of the unknown parameters. Consequently, a large amount of computer time and computer memory can be saved. Furthermore, not only the parameters but also the concentrations of pollutants can be estimated. Thus, it also forms an effective forecasting technique. The classical least squares criterion is used in the estimation. Several examples are solved to illustrate the technique. (KEY WORDS: dynamic modeling; water pollution; invariant imbedding; forecasting; least squares criterion; estimation)     

LINEAR PROGRAMS FOR NONLINEAR HYDROLOGIC ESTIMATION1

ABSTRACT: The minimization of the sum of absolute deviations and the minimization of the absolute maximum deviation (mini-max) were transformed into equivalent linear programs for the estimation of parameters in a transient and linear hydrologic system. It is demonstrated that these two methods yield viable parameter estimates that are globally optimal and reproduce properly the timing and magnitude of hydrologic events and associated variables such as total runoff. The two linear estimation methods compared favorably with the popular least-squares nonlinear estimation method. The generality of the theoretical developments shows that linear program equivalents are adequate competitors of nonlinear methods of hydrologic estimation and parameter calibration.     

NONLINEAR MODELING AND PREDICTION OF A RIVER FLOW SYSTEM1

ABSTRACT: Model estimation and prediction of a river flow system are investigated using nonlinear system identification techniques. We demonstrate how the dynamics of the system, rainfall, and river flow can be modeled using NARMAX (Nonlinear Autoregressive Moving Average with eXogenuous input) models. The parameters of the model are estimated using an orthogonal least squares algorithm with intelligent structure detection. The identification of the nonlinear model is described to represent the relationship between local rainfall and river flow at Enoree station (inputs) and river flow at Whitmire (output) for a river flow system in South Carolina.     

OPTIMAL IDENTIFICATION OF MUSKINGUM ROUTING COEFFICIENTS1

Traditionally, identification of the Muskingum routing coefficients has been based on observations of the linearity of a loop formed by graphically plotting a forward and a reverse path. This graphical procedure is time-consuming and may not minimize the error of estimation. A procedure was developed to improve the drawbacks of the graphical method. This procedure calls for (a) the use of least square regression on the forward and reverse paths to determine their respective slopes, and (b) the use of statistical t-test to evaluate the hypothesis that these two slopes are equal. The computational procedure is repeated, using incremental values of the flow weighting coefficient, x. A graph of the computed t-value versus x can be constructed. The optimal value of x, as read from the graph, occurs at the minimum computed t-value. The procedure has been demonstrated superior to the graphical method for three illustrative examples, resulting in a reduction of the error squares by factors ranging from 5 to 6.     

CATCHMENT RESPONSE TIME AS A PREDICTOR OF FLOOD QUANTILES1

ABSTRACT: A regression analysis using a generalized least squares approach on flow data from the driftless area of Wisconsin indicates that the ratio of drainage area to time-to-peak is a good predictor of flood quantiles. The estimation of time-to-peak (or some other measure of basin response time) requires direct measurement of river stage and possibly rainfall at the site of which the quantiles are to be estimated. The cost-effectiveness of such an approach must yet be determined.     

Effects of Spatial Distribution of Prairie Vegetation in an Agricultural Landscape on Curve Number Values

The curve number (CN) method is used to calculate runoff in many hydrologic models, including the Soil and Water Assessment Tool (SWAT). The CN method does not account for the spatial distribution of land cover types, an important factor controlling runoff patterns. The objective of this study was to empirically derive CN values that reflect the strategic placement of native prairie vegetation (NPV) within row crop agricultural landscapes. CNs were derived using precipitation and runoff data from a seven‐year period for 14 small watersheds in Iowa. The watersheds were planted with varying amounts of NPV located in different watershed positions. The least squares and asymptotic least squares methods (LSM) were used to derive CNs using an initial abstraction coefficient (λ) of 0.2 and 0.05. The CNs were verified using leave‐one‐out cross‐validation and adjustment for antecedent moisture conditions (AMC) was tested. The asymptotic method produced CN values for watersheds with NPV treatment that were 8.9 and 14.7% lower than watersheds with 100% row crop at λ = 0.2 and λ = 0.05, respectively. The derived CNs produced Nash‐Sutcliffe efficiency values ranging from 0.4 to 0.7 during validation. Our analyses show the CNs verified best for the asymptotic LSM, when using λ of 0.05 and adjusting for AMC. Further, comparison of derived CNs against an area weighted CN indicated that the placement of vegetation does impact the CN value. Editor's note : This paper is part of the featured series on SWAT Applications for Emerging Hydrologic and Water Quality Challenges. See the February 2017 issue for the introduction and background to the series.     

LAKE QUALITY DISCRIMINANT ANALYSIS1

ABSTRACT: Despite the fact that lake phosphorus loading criteria have proven to be valuable tools in lake management, they are generally subjective in nature or incomplete in form. In order to address these shortcomings, the oxic-anoxic transition point was selected as an objective quality criterion and discriminant analysis was used to construct a lake classification function. This function is dependent upon lake phosphorus loading, mean depth, and overflow rate. The value of the function may be expressed as a probability of classification (as either oxic or anoxic). When used in prediction, inclusion of the input error permits the estimation of the change in classification probability as input uncertainty is reduced. Further, the form of the discriminant function suggests that the annual volumetric loading is a more informative term for the expression of phosphorus loading than is the annual areal loading.     

ESTIMATING THE MEAN AND VARIANCE OF CENSORED PHOSPHORUS CONCENTRATIONS IN FLORIDA RAINFALL1

ABSTRACT: The total phosphorous (TP) concentrations in the South Florida rainfall have been recorded in weekly intervals with a detection limit (D_L) of 3.5 μg/L. As a large amount of the data is reported as below the D_L, appropriate statistical methods are needed for data analysis. Thus, an attempt was made to identify an appropriate method to estimate the mean and variance of the data. In particular, a method to separate the statistics for the below D_L portion from the estimated population statistics is proposed. The estimated statistics of the censored data are compared with the statistics of the uncensored data available from the recent years’ laboratory records. It was found that the one-step restricted maximum likelihood method is the most accurate for the wet TP data, and that the proposed method to combine the estimated statistics for TP < D_L portion and the sample statistics for TP ≥ D_L portion improves estimates compared to the conventional maximum likelihood estimates.