Derivation of some frequency distributions using the principle of maximum entropy (POME)

The principle of maximum entropy (POME) was employed to develop a procedure for derivation of a number of frequency distributions used in hydrology. The procedure required specification of constraints and maximization of entropy, and is thus a solution of the classical optimization problem. The POME led to a unique technique for parameter estimation. For six selected river gaging stations, parameters of the gamma distribution, the log-Pearson type III distribution and extreme value type I distribution fitted to annual maximum discharges, were evaluated by this technique and compared with those obtained by using the methods of moments and maximum likelihood estimation. The concept of entropy, used as a measured of uncertainty associated with a specified distribution, facilitated this comparison.     

Practical Postcalibration Uncertainty Analysis: Yucca Mountain, Nevada

The values of parameters in a groundwater flow model govern the precision of predictions of future system behavior. Predictive precision, thus, typically depends on an ability to infer values of system properties from historical measurements through calibration. When such data are scarce, or when their information content with respect to parameters that are most relevant to predictions of interest is weak, predictive uncertainty may be high, even if the model is "calibrated." Recent advances help recognize this condition, quantitatively evaluate predictive uncertainty, and suggest a path toward improved predictive accuracy by identifying sources of predictive uncertainty and by determining what observations will most effectively reduce this uncertainty. We demonstrate linear and nonlinear predictive error/uncertainty analyses as applied to a groundwater flow model of Yucca Mountain, Nevada, the United States' proposed site for disposal of high-level radioactive waste. Linear and nonlinear uncertainty analyses are readily implemented as an adjunct to model calibration with medium to high parameterization density. Linear analysis yields contributions made by each parameter to a prediction's uncertainty and the worth of different observations, both existing and yet-to-be-gathered, toward reducing this uncertainty. Nonlinear analysis provides more accurate characterization of the uncertainty of model predictions while yielding their (approximate) probability distribution functions. This article applies the above methods to a prediction of specific discharge and confirms the uncertainty bounds on specific discharge supplied in the Yucca Mountain Project License Application.     

Parameter estimation for 2-parameter generalized pareto distribution by POME

The principle of maximum entropy (POME) was employed to derive a new method of parameter estimation for the 2-parameter generalized Pareto (GP2) distribution. Monte Carlo simulated data were used to evaluate this method and compare it with the methods of moments (MOM), probability weighted moments (PWM), and maximum likelihood estimation (MLE). The parameter estimates yielded by POME were comparable or better within certain ranges of sample size and coefficient of variation.     

The influence of parameter distribution uncertainty on hydrological and sediment modeling: a case study of SWAT model applied to the Daning watershed of the Three Gorges Reservoir Region, China

Parameter uncertainty involved in hydrological and sediment modeling often refers to the parameter dispersion and the sensitivity of the parameter. However, a limitation of the previous studies lies in that the assignment of range and specification of probability distribution for each parameter is usually difficult and subjective. Therefore, there is great uncertainty in the process of parameter calibration and model prediction. In this study, the impact of probability parameter distribution on hydrological and sediment modeling was evaluated using a semi-distributed model—the Soil and Water Assessment Tool (SWAT) and Monte Carlo method (MC)—in the Daning River watershed of the Three Gorges Reservoir Region (TGRA), China. The classic types of parameter distribution such as uniform, normal and logarithmic normal distribution were involved in this study. Based on results, parameter probability distribution showed a diverse degree of influence on the hydrological and sediment prediction, such as the sampling size, the width of 95% confidence interval (CI), the ranking of the parameter related to uncertainty, as well as the sensitivity of the parameter on model output. It can be further inferred that model parameters presented greater uncertainty in certain regions of the primitive parameter range and parameter samples densely obtained from these regions would lead to a wider 95 CI, resulting in a more doubtful prediction. This study suggested the value of the optimized value obtained by the parameter calibration process could may also be of vital importance in selecting the probability distribution function (PDF). Such cases, where parameter value corresponds to the watershed characteristic, can be used to provide a more credible distribution for both hydrological and sediment modeling.     

Regionalisation of the parameters of the log‐Pearson 3 distribution: a case study for New South Wales,Australia

The index flood method is widely used in regional flood frequency analysis (RFFA) but explicitly relies on the identification of ‘acceptable homogeneous regions’. This paper presents an alternative RFFA method, which is particularly useful when ‘acceptably homogeneous regions’ cannot be identified. The new RFFA method is based on the region of influence (ROI) approach where a ‘local region’ can be formed to estimate statistics at the site of interest. The new method is applied here to regionalize the parameters of the log‐Pearson 3 (LP3) flood probability model using Bayesian generalized least squares (GLS) regression. The ROI approach is used to reduce model error arising from the heterogeneity unaccounted for by the predictor variables in the traditional fixed‐region GLS analysis. A case study was undertaken for 55 catchments located in eastern New South Wales, Australia. The selection of predictor variables was guided by minimizing model error. Using an approach similar to stepwise regression, the best model for the LP3 mean was found to use catchment area and 50‐year, 12‐h rainfall intensity as explanatory variables, whereas the models for the LP3 standard deviation and skewness only had a constant term for the derived ROIs. Diagnostics based on leave‐one‐out cross validation show that the regression model assumptions were not inconsistent with the data and, importantly, no genuine outlier sites were identified. Significantly, the ROI GLS approach produced more accurate and consistent results than a fixed‐region GLS model, highlighting the superior ability of the ROI approach to deal with heterogeneity. This method is particularly applicable to regions that show a high degree of regional heterogeneity. Copyright © 2014 John Wiley & Sons, Ltd.     

Derivation of the Pearson type (PT) III distribution by using the principle of maximum entropy (POME)

The principle of maximum entropy (POME) was used to derive the Pearson type (PT) III distribution. The POME yielded the minimally prejudiced PT III distribution by maximizing the entropy subject to two appropriate constraints which were the mean and the mean of the logarithm of real values about a constant >0. This provided a unique method for parameter estimation. Historical flood data were used to evaluate this method and compare it with the methods of moments and maximum likelihood estimation.     

Input data resolution‐induced uncertainty in watershed modelling

The temporal‐spatial resolution of input data‐induced uncertainty in a watershed‐based water quality model, Hydrologic Simulation Program‐FORTRAN (HSPF), is investigated in this study. The temporal resolution‐induced uncertainty is described using the coefficient of variation (CV). The CV is found to decrease with decreasing temporal resolution and follow a log‐normal relation with time interval for temperature data while it exhibits a power‐law relation for rainfall data. The temporal‐scale uncertainties in the temperature and rainfall data follow a general extreme value distribution and a Weibull distribution, respectively. The Nash‐Sutcliffe coefficient (NSC) is employed to represent the spatial resolution induced uncertainty. The spatial resolution uncertainty in the dissolved oxygen and nitrate‐nitrogen concentrations simulated using HSPF is observed to follow a general extreme value distribution and a log‐normal distribution, respectively. The probability density functions (PDF) provide new insights into the effect of temporal‐scale and spatial resolution of input data on uncertainties involved in watershed modelling and total maximum daily load calculations. This study exhibits non‐symmetric distributions of uncertainty in water quality modelling, which simplify weather and water quality monitoring and reducing the cost involved in flow and water quality monitoring. Copyright © 2011 John Wiley & Sons, Ltd.     

A time dependent mixing model to close PDF equations for transport in heterogeneous aquifers

Probability density function (PDF) methods are a promising alternative to predicting the transport of solutes in groundwater under uncertainty. They make it possible to derive the evolution equations of the mean concentration and the concentration variance, used in moment methods. The mixing model, describing the transport of the PDF in concentration space, is essential for both methods. Finding a satisfactory mixing model is still an open question and due to the rather elaborate PDF methods, a difficult undertaking. Both the PDF equation and the concentration variance equation depend on the same mixing model. This connection is used to find and test an improved mixing model for the much easier to handle concentration variance. Subsequently, this mixing model is transferred to the PDF equation and tested. The newly proposed mixing model yields significantly improved results for both variance modelling and PDF modelling.     

地下水观测数据拟合与预测的支持向量机方法

影响地下水位变化因素有很多,在正常情况下,地下水位的变化实际上反应了气压、固体潮和降雨这些因素的变化,但是这些影响因子与地下水位之间有着较强的非线性关系。该文使用支持向量机方法建立起崇明中学观测站地下水位与气压、固体潮和降雨这些因素之间的非线性关系模型,并用于地下水观测数据拟合与预测,得到了较理想的结果,明显优于逐步回归方法。研究结果表明,支持向量机方法在地震前兆数据处理中有着广泛的应用前景。文中还对支持向量机方法在实际应用中的有关问题进行了讨论。     

Parameter estimation for 3-parameter log-logistic distribution (LLD3) by Pome

The principle of maximum entropy (POME) was employed to derive a new method of parameter estimation for the 3-parameter log-logistic distribution (LLD3). Monte Carlo simulated data were used to evaluate this method and compare it with the methods of moments (MOM), probability weighted moments (PWM), and maximum likelihood estimation (MLE). Simulation results showed that POME's performance was superior in predicting quantiles of large recurrence intervals when population skew was greater than or equal to 2.0. In all other cases, POME's performance was comparable to other methods.     

Emulation of simulated earthquake catalogues

In earthquake occurrence studies, the so-called q value can be considered both as one of the parameters describing the distribution of interevent times and as an index of non-extensivity. Using simulated datasets, we compare four kinds of estimators, based on principle of maximum entropy (POME), method of moments (MOM), maximum likelihood (MLE), and probability weighted moments (PWM) of the parameters (q and τ ₀) of the distribution of inter-events times, assumed to be a generalized Pareto distribution (GPD), as defined by Tsallis (1988) in the frame of non-extensive statistical physics. We then propose to use the unbiased version of PWM estimators to compute the q value for the distribution of inter-event times in a realistic earthquake catalogue simulated according to the epidemic type aftershock sequence (ETAS) model. Finally, we use these findings to build a statistical emulator of the q values of ETAS model. We employ treed Gaussian processes to obtain partitions of the parameter space so that the resulting model respects sharp changes in physical behaviour. The emulator is used to understand the joint effects of input parameters on the q value, exploring the relationship between ETAS model formulation and distribution of inter-event times.     

Parameter estimation for 3-parameter log-logistic distribution (LLD3) by Pome

The principle of maximum entropy (POME) was employed to derive a new method of parameter estimation for the 3-parameter log-logistic distribution (LLD3). Monte Carlo simulated data were used to evaluate this method and compare it with the methods of moments (MOM), probability weighted moments (PWM), and maximum likelihood estimation (MLE). Simulation results showed that POME's performance was superior in predicting quantiles of large recurrence intervals when population skew was greater than or equal to 2.0. In all other cases, POME's performance was comparable to other methods.     

Parameter Uncertainty Propagation in a Rainfall–Runoff Model; Case Study: Karoon-III River Basin

Conceptual hydrological models are popular tools for simulating land phase of hydrological cycle. Uncertainty arises from a variety of sources such as input error, calibration and parameters. Hydrologic modeling researches indicate that parametric uncertainty has been considered as one of the most important source. The objective of this study was to evaluate parameter uncertainty and its propagation in rainfall-runoff modeling. This study tried to model daily flows and calculate uncertainty bounds for Karoon-III basin, Southwest of Iran, using HEC-HMS (SMA). The parameters were represented by probability distribution functions (PDF), and the effect on simulated runoff was investigated using Latin Hypercube Sampling (LHS) on Monte Carlo (MC). Three chosen parameters, based on sensitivity analysis, were saturated-hydraulic-conductivity (Ks), Clark storage coefficient (R) and time of concentration (t _c ). Uncertainty associated with parameters were accounted for, by representing each with a probability distribution. Uncertainty bounds was calculated, using parameter sets captured from LHS on parameters PDF of sub-basins and propagating to the model. Results showed that maximum reliability (11%) resulted from Ks propagating. For three parameters, underestimation was more than overestimation. Maximum sharpness and standard deviation (STD) was resulted from propagating Ks. Cumulative Distribution Function (CDF) of flow and uncertainty bounds showed that as flow increased, the width of uncertainty bounds increased for all parameters.     

A univariate model for long-term streamflow forecasting

This paper, the first in a series of two, employs the principle of maximum entropy (POME) via maximum entropy spectral analysis (MESA) to develop a univariate model for long-term streamflow forecasting. Three cases of streamflow forecasting are investigated: forward forecasting, backward forecasting (or reconstruction) and intermittent forecasting (or filling in missing records). Application of the model is discussed in the second paper.     

Numerical Study on Behavior of Groundwater Flow in Single Rough Fractures Under Mechanical Effect

This paper numerically investigates the characteristics of groundwater flow in spatially correlated variable aperture fractures under the mechanical effect. Spatially correlated aperture distributions are generated using the geostatistical method (i.e., turning bands algorithm in this study). To represent a nonlinear relationship between the effective normal stress and the fracture aperture, a simple mechanical formula is combined with a local flow model. Numerical results indicate that the groundwater flow is significantly affected by the geometry of aperture distribution, varying with the applied effective normal stress as well as the spatial correlation length of aperture distribution. Moreover, using the flow results simulated in this study, two empirical formulae are proposed: (1) the first one (modified Louis formula) is to represent the relationship between the effective normal stress and the effective permeability of fracture and (2) the second one is to represent the relationship between relative roughness and effective permeability.     

Performance evaluation of soil moisture profile estimation through entropy-based and exponential filter models

ABSTRACT

In this study we analyzed two models commonly used in remote sensing-based root-zone soil moisture (SM) estimations: one utilizing the exponential decaying function and the other derived from the principle of maximum entropy (POME). We used both models to deduce root-zone (0–100 cm) SM conditions at 11 sites located in the southeastern USA for the period 2012–2017 and evaluated the strengths and weaknesses of each approach against ground observations. The results indicate that, temporally, at shallow depths (10 cm), both models performed similarly, with correlation coefficients (r) of 0.89 (POME) and 0.88 (exponential). However, with increasing depths, the models start to deviate: at 50 cm the POME resulted in r of 0.93 while the exponential filter (EF) model had r of 0.58. Similar trends were observed for unbiased root mean square error (ubRMSE) and bias. Vertical profile analysis suggests that, overall, the POME model had nearly 30% less ubRMSE compared to the EF model, indicating that the POME model was relatively better able to distribute the moisture content through the soil column.     

Latin hypercube approach to estimate uncertainty in ground water vulnerability

A methodology is proposed to quantify prediction uncertainty associated with ground water vulnerability models that were developed through an approach that coupled multivariate logistic regression with a geographic information system (GIS). This method uses Latin hypercube sampling (LHS) to illustrate the propagation of input error and estimate uncertainty associated with the logistic regression predictions of ground water vulnerability. Central to the proposed method is the assumption that prediction uncertainty in ground water vulnerability models is a function of input error propagation from uncertainty in the estimated logistic regression model coefficients (model error) and the values of explanatory variables represented in the GIS (data error). Input probability distributions that represent both model and data error sources of uncertainty were simultaneously sampled using a Latin hypercube approach with logistic regression calculations of probability of elevated nonpoint source contaminants in ground water. The resulting probability distribution represents the prediction intervals and associated uncertainty of the ground water vulnerability predictions. The method is illustrated through a ground water vulnerability assessment of the High Plains regional aquifer. Results of the LHS simulations reveal significant prediction uncertainties that vary spatially across the regional aquifer. Additionally, the proposed method enables a spatial deconstruction of the prediction uncertainty that can lead to improved prediction of ground water vulnerability.     

Estimation of atmospheric predictability for multivariable system using information theory in nonlinear error growth dynamics

To estimate atmospheric predictability for multivariable system, based on information theory in nonlinear error growth dynamics, a quantitative method is introduced in this paper using multivariable joint predictability limit (MJPL) and corresponding single variable predictability limit (SVPL). The predictability limit, obtained from the evolutions of nonlinear error entropy and climatological state entropy, is not only used to measure the predictability of dynamical system with the constant climatological state entropy, but also appropriate to the case of climatological state entropy changed with time. With the help of daily NCEP-NCAR reanalysis data, by using a method of local dynamical analog, the nonlinear error entropy, climatological state entropy, and predictability limit are obtained, and the SVPLs and MJPL of the winter 500-hPa temperature field, zonal wind field and meridional wind field are also investigated. The results show that atmospheric predictability is well associated with the analytical variable. For single variable predictability, there exists a big difference for the three variables, with the higher predictability found for the temperature field and zonal wind field and the lower predictability for the meridional wind field. As seen from their spatial distributions, the SVPLs of the three variables appear to have a property of zonal distribution, especially for the meridional wind field, which has three zonal belts with low predictability and four zonal belts with high predictability. For multivariable joint predictability, the MJPL of multivariable system with the three variables is not a simple mean or linear combination of its SVPLs. It presents an obvious regional difference characteristic. Different regions have different results. In some regions, the MJPL is among its SVPLs. However, in other regions, the MJPL is less than its all SVPLs.     

Sensitivity analysis of prior model probabilities and the value of prior knowledge in the assessment of conceptual model uncertainty in groundwater modelling

A key point in the application of multi‐model Bayesian averaging techniques to assess the predictive uncertainty in groundwater modelling applications is the definition of prior model probabilities, which reflect the prior perception about the plausibility of alternative models. In this work the influence of prior knowledge and prior model probabilities on posterior model probabilities, multi‐model predictions, and conceptual model uncertainty estimations is analysed. The sensitivity to prior model probabilities is assessed using an extensive numerical analysis in which the prior probability space of a set of plausible conceptualizations is discretized to obtain a large ensemble of possible combinations of prior model probabilities. Additionally, the value of prior knowledge about alternative models in reducing conceptual model uncertainty is assessed by considering three example knowledge states, expressed as quantitative relations among the alternative models. A constrained maximum entropy approach is used to find the set of prior model probabilities that correspond to the different prior knowledge states. For illustrative purposes, a three‐dimensional hypothetical setup approximated by seven alternative conceptual models is employed. Results show that posterior model probabilities, leading moments of the predictive distributions and estimations of conceptual model uncertainty are very sensitive to prior model probabilities, indicating the relevance of selecting proper prior probabilities. Additionally, including proper prior knowledge improves the predictive performance of the multi‐model approach, expressed by reductions of the multi‐model prediction variances by up to 60% compared with a non‐informative case. However, the ratio between‐model to total variance does not substantially decrease. This suggests that the contribution of conceptual model uncertainty to the total variance cannot be further reduced based only on prior knowledge about the plausibility of alternative models. These results advocate including proper prior knowledge about alternative conceptualizations in combination with extra conditioning data to further reduce conceptual model uncertainty in groundwater modelling predictions. Copyright © 2009 John Wiley & Sons, Ltd.