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.     

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.     

Simulating solute transport in a structured field soil: uncertainty in parameter identification and predictions

Dual-permeability models have been developed to account for the significant effects of macropore flow on contaminant transport, but their use is hampered by difficulties in estimating the additional parameters required. Therefore, our objective was to evaluate data requirements for parameter identification for predictive modeling with the dual-permeability model MACRO. Two different approaches were compared: sequential uncertainty fitting (SUFI) and generalized likelihood uncertainty estimation (GLUE). We investigated six parameters controlling macropore flow and pesticide sorption and degradation, applying MACRO to a comprehensive field data set of bromide andbentazone [3-isopropyl-1H-2,1,3-benzothiadiazin-4(3H)-one-2,2dioxide] transport in a structured soil. The GLUE analyses of parameter conditioning for different combinations of observations showed that both resident and flux concentrations were needed to obtain highly conditioned and unbiased parameters and that observations of tracer transport generally improved the conditioning of macropore flow parameters. The GLUE "behavioral" parameter sets covered wider parameter ranges than the SUFI posterior uncertainty domains. Nevertheless, estimation uncertainty ranges defined by the 5th and 95th percentiles were similar and many simulations randomly sampled from the SUFI posterior uncertainty domains had negative model efficiencies (minimum of -3.2). This is because parameter correlations are neglected in SUFI and the posterior uncertainty domains were not always determined correctly. For the same reasons, uncertainty ranges for predictions of bentazone losses through drainflow for good agricultural practice in southern Sweden were 27% larger for SUFI compared with GLUE. Although SUFI proved to be an efficient parameter estimation tool, GLUE seems better suited as a method of uncertainty estimation for predictions.     

STATISTICAL IDENTIFICATION OF STORAGE MODELS WITH APPLICATION TO STOCHASTIC HYDROLOGY1

ABSTRACT: This paper describes a method for the statistical identification of storage models for daily riverflow time series, together with numerical results. The first step in the identification process is to obtain a discrete time version of a storage model using a local linearization approach. It is shown that the discrete time version thus obtained may be utilized in the identification of the original storage model. A statistical method for the identification of daily rainfall time series models used in simulation is also presented. This identification procedure employs the maximum likelihood method for point process data analysis and is illustrated by means of numerical examples.     

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.     

PERIODIC AUTOREGRESSIVE-MOVING AVERAGE (PARMA) MODELING WITH APPLICATIONS TO WATER RESOURCES1

Results involving correlation properties and parameter estimation for autoregressive-moving average models with periodic parameters are presented. A multivariate representation of the PARMA model is used to derive parameter space restrictions and difference equations for the periodic autocorrelations. Close approximation to the likelihood function for Gaussian PARMA processes results in efficient maximum-likelihood estimation procedures. Terms in the Fourier expansion of the parameters are sequentially included, and a selection criterion is given for determining the optimal number of harmonics to be included. Application of the techniques is demonstrated through analysis of a monthly streamflow time series.     

ESTIMATING PARAMETERS OF EV1 DISTRIBUTION FOR FLOOD FREQUENCY ANALYSIS1

ABSTRACT: The parameters of the extreme value type 1 distribution were estimated for 55 annual flood data sets by seven methods. These are the methods of (1) moments, (2) probability weighted moments, (3) mixed moments, (4) maximum likelihood estimation, (5) incomplete means, (6) principle of maximum entropy, and (7) least squares. The method of maximum likelihood estimation was found to be the best and the method of incomplete means the worst. The differences between the methods of principle of maximum entropy, probability weighted moments, moments, and least squares were only minor. The difference between these methods and the method of maximum likelihood was not pronounced.     

Research on an Online Identification Algorithm for a Thevenin Battery Model by an Experimental Approach

To improve the estimation accuracy of battery’s inner state for battery management system, an online parameters identification algorithm for Thevenin battery model is researched. The Thevenin model and parameters identification algorithm based on recursive least square adaptive filter algorithm was built with the Simulink/xPC Target. The results of hardware-in-loop experiment, which uses Federal Urban Driving Schedule test to verify the parameters identification approach, show the proposed approach can accurately identify the model parameters within 1% maximum terminal voltage estimation error, and the State of Charge error which calculated by the open circuit voltage estimates can be efficiently reduced to 4%.     

THE EFFECT OF DATA INDEPENDENCE IN MODEL CALIBRATION AND MODEL TESTING1

ABSTRACT: With the increased use of models in hydrologic design, there is an immediate need for a comprehensive comparison of hydrologic models, especially those intended for use at ungaged locations (i.e., where measured data are either not available or inadequate for model calibration). But some past comparisons of hydrologic models have used the same data base for both calibration and testing of the different models or implied that the results of model calibration are indicative of the accuracy at ungaged locations. This practice was examined using both the regression equation approach to peak discharge estimation and a unit hydrograph model that was intended for use in urban areas. The results suggested that the lack of data independence in the calibration and testing of regression equations may lead to both biased results and misleading statements about prediction accuracy. Additionally, although split-sample testing is recognized as desirable, the split-samples should be selected using a systematic-random sampling scheme, rather than random sampling, because random sampling with small samples may lead to a testing sample that is not representative of the population. A systematic-random sampling technique should lead to more valid conclusions about model reliability. For models like a unit hydrograph model, which are more complex and for which calibration is a more involved process, data independence is not as critical because the data fitting error variation is not as dominant as the error variation due to the calibration process and the inability of the model structure to conform with data variability.     

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.     

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.     

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。     

REGIONALIZATION OF LOW-FLOW FREQUENCY ESTIMATES: AN ALABAMA CASE STUI)Y1

ABSTRACT: Low-flow estimates, as determined by probabilistic modeling of observed data sequences, are commonly used to describe certain streamflow characteristics. Unfortunately, however, reliable low-flow estimates can be difficult to come by, particularly for gaging sites with short record lengths. The shortness of records leads to uncertainties not only in the selection of a distribution for modeling purposes but also in the estimates of the parameters of a chosen model. In flood frequency analysis, the common approach to mitigation of some of these problems is through the regionalization of frequency behavior. The same general approach is applied here to the case of low-flow estimation, with the general intent of not only improving low-flow estimates but also illustrating the gains that might be attained in so doing. Data used for this study is that which has been systematically observed at 128 streamflow gaging sites across the State of Alabama. Our conclusions are that the log Pearson Type 3 distribution is a suitable candidate for modeling of Alabama low-flows, and that the shape parameter of that distribution can be estimated on a regional basis. Low-flow estimates based on the regional estimator are compared with estimates based on the use of only at-site estimation techniques.     

KALMAN FILTER ESTIMATION AND PREDICTION OF DAILY STREAM FLOWS: II. APPLICATION TO THE POTOMAC RIVER1

Results are reported from an application of the state space formulation and the Kalman filter to real-time forecasting of daily river flows. It is shown that the application of filtering techniques improves the overall forecasting performance of the model. As is true for most hydrologic systems, the model is not completely known. Therefore, the procedures pertaining to on-line parameter and noise statistics estimation, as presented in the first paper, are implemented. The example in this paper shows that these techniques also perform satisfactorily when applied to a real-world situation.     

A NEW LOOK AT FLOOD RISK DETERMINATION1

ABSTRACT: The probability distributions of annual peak flows used in flood risk analysis quantify the risk that a design flood will be exceeded. But the parameters of these distributions are themselves to a degree uncertain and this uncertainty increases the risk that the flood protection provided will in fact prove to be inadequate. The increase in flood risk due to parameter uncertainty is small when a fairly long record of data is available and the annual flood peaks are serially independent, which is the standard assumption in flood frequency analysis. But standard tests for serial independence are insensitive to the type of grouping of high and low values in a time series, which is measured by the Hurst coefficient. This grouping increases the parameter uncertainty considerably. A study of 49 annual peak flow series for Canadian rivers shows that many have a high Hurst coefficient. The corresponding increase in flood risk due to parameter uncertainty is shown to be substantial even for rivers with a long record, and therefore should not be neglected. The paper presents a method of rationally combining parameter uncertainty due to serial correlation, and the stochastic variability of peak flows in a single risk assessment. In addition, a relatively simple time series model that is capable of reproducing the observed serial correlation of flood peaks is presented.     

A SURROGATE-PARAMETER APPROACH TO MODELING GROUNDWATER BASINS1

ABSTRACT: A surrrogate-prarmeter approach to modeling groundwater basins is presented, which has the following advantages over current simulation-type methods: (i) conducivness to modeling nonhomogeneous and nonisotropic basins; (ii) there is no need to guess boundary conditions if accurate information is not available; (iii) the model is amenable to systematic calibration or identification through the use of optimization techniques; and (iv) compatibility with systematic algorithms for analyzing a wide range of management strategies. Since the parameter identification problem is large-scale and nonconvex, it is decomposed through application of generalized duality theory, into several subproblems of smaller size which are solved independently a number of times in order to achieve an overall solution. Results are presented for a hypothetical system of four interacting wells.     

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.     

Approximating SWAT Model Using Artificial Neural Network and Support Vector Machine1

Abstract: With the popularity of complex, physically based hydrologic models, the time consumed for running these models is increasing substantially. Using surrogate models to approximate the computationally intensive models is a promising method to save huge amounts of time for parameter estimation. In this study, two learning machines [Artificial Neural Network (ANN) and support vector machine (SVM)] were evaluated and compared for approximating the Soil and Water Assessment Tool (SWAT) model. These two learning machines were tested in two watersheds (Little River Experimental Watershed in Georgia and Mahatango Creek Experimental Watershed in Pennsylvania). The results show that SVM in general exhibited better generalization ability than ANN. In order to effectively and efficiently apply SVM to approximate SWAT, the effect of cross‐validation schemes, parameter dimensions, and training sample sizes on the performance of SVM was evaluated and discussed. It is suggested that 3‐fold cross‐validation is adequate for training the SVM model, and reducing the parameter dimension through determining the parameter values from field data and the sensitivity analysis is an effective means of improving the performance of SVM. As far as the training sample size, it is difficult to determine the appropriate number of samples for training SVM based on the test results obtained in this study. Simple examples were used to illustrate the potential applicability of combining the SVM model with uncertainty analysis algorithm to save efforts for parameter uncertainty of SWAT. In the future, evaluating the applicability of SVM for approximating SWAT in other watersheds and combining SVM with different parameter uncertainty analysis algorithms and evolutionary optimization algorithms deserve further research.     

基于贝叶斯网络分类的土壤盐渍化遥感监测

贝叶斯网络是一种将贝叶斯概率方法和有向无环图的网络拓扑结构有机结合的概率模型.采用贝叶斯网络分类对具有典型干旱特征的库车县土壤盐渍化情况进行监测,首先应用条件独立性测试原理建立贝叶斯网络结构,把研究区遥感数据进行离散化,然后应用贝叶斯定理作为分类原则,将每个像元分为像元最大概率的类别.研究结果表明该方法来分类6种地类的整体分类精度达到96%,并为该区盐渍地面积,空间分布等特征监测提供较好的依据.