Bayesian flood frequency analysis in the light of model and parameter uncertainties

The specific objective of the paper is to propose a new flood frequency analysis method considering uncertainty of both probability distribution selection (model uncertainty) and uncertainty of parameter estimation (parameter uncertainty). Based on Bayesian theory sampling distribution of quantiles or design floods coupling these two kinds of uncertainties is derived, not only point estimator but also confidence interval of the quantiles can be provided. Markov Chain Monte Carlo is adopted in order to overcome difficulties to compute the integrals in estimating the sampling distribution. As an example, the proposed method is applied for flood frequency analysis at a gauge in Huai River, China. It has been shown that the approach considering only model uncertainty or parameter uncertainty could not fully account for uncertainties in quantile estimations, instead, method coupling these two uncertainties should be employed. Furthermore, the proposed Bayesian-based method provides not only various quantile estimators, but also quantitative assessment on uncertainties of flood frequency analysis.     

György Kovacs 1925–1988

Abstract

There has been a trend in recent years towards the development and popularity of physically-based deterministic models. However, the application of such models is not without difficulties. This paper investigates the usefulness of a conceptual single-event model for simulating floods from catchments covering a wide variety of climatic and physiographic areas. The model has been calibrated on a group of catchments and the calibrated parameter values related to physical catchment indices. The resulting quantitative relationships are assessed with respect to their value for estimating the parameter values of the model when calibration is not possible. The results indicate that the technique is likely to provide flood estimations for medium sized catchments (5–150 km²) that are more reliable than several flood estimation methods currently in use in South Africa.     

Uncertainty issues of a conceptual water balance model for a semi‐arid watershed in north‐west of China

Hydrological models are useful tools for better understanding the hydrological processes and performing the hydrological prediction. However, the reliability of the prediction depends largely on its uncertainty range. This study mainly focuses on estimating model parameter uncertainty and quantifying the simulation uncertainties caused by sole model parameters and the co‐effects of model parameters and model structure in a lumped conceptual water balance model called WASMOD (Water And Snow balance MODeling system). The validity of statistical hypotheses on residuals made in the model formation is tested as well, as it is the base of parameter estimation and simulation uncertainty evaluation. The bootstrap method is employed to examine the parameter uncertainty in the selected model. The Yingluoxia watershed at the upper reaches of the Heihe River basin in north‐west of China is selected as the study area. Results show that all parameters in the model can be regarded as normally distributed based on their marginal distributions and the Kolmogorov–Smirnov test, although they appear slightly skewed for two parameters. Their uncertainty ranges are different from each other. The model residuals are tested to be independent, homoscedastic and normally distributed. Based on such valid hypotheses of model residuals, simulation uncertainties caused by co‐effects of model parameters and model structure can be evaluated effectively. It is found that the 95% and 99% confidence intervals (CIs) of simulated discharge cover 42.7% and 52.4% of the observations when only parameter uncertainty is considered, indicating that parameter uncertainty has a great effect on simulation uncertainty but still cannot be used to explain all the simulation uncertainty in this study. The 95% and 99% CIs become wider, and the percentages of observations falling inside such CIs become larger when co‐effects of parameters and model structure are considered, indicating that simultaneous consideration of both parameters and model structure uncertainties accounts sufficient contribution for model simulation uncertainty. Copyright © 2012 John Wiley & Sons, Ltd.     

Development of probability distributions for urban hydrologic model parameters and a Monte Carlo analysis of model sensitivity

This paper proposes an approach to estimating the uncertainty related to EPA Storm Water Management Model model parameters, percentage routed (P_R) and saturated hydraulic conductivity (K_sat), which are used to calculate stormwater runoff volumes. The methodology proposed in this paper addresses uncertainty through the development of probability distributions for urban hydrologic parameters through extensive calibration to observed flow data in the Philadelphia collection system. The established probability distributions are then applied to the Philadelphia Southeast district model through a Monte Carlo approach to estimate the uncertainty in prediction of combined sewer overflow volumes as related to hydrologic model parameter estimation. Understanding urban hydrology is critical to defining urban water resource problems. A variety of land use types within Philadelphia coupled with a history of cut and fill have resulted in a patchwork of urban fill and native soils. The complexity of urban hydrology can make model parameter estimation and defining model uncertainty a difficult task. The development of probability distributions for hydrologic parameters applied through Monte Carlo simulations provided a significant improvement in estimating model uncertainty over traditional model sensitivity analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

Reducing Uncertainty in Calibrating Aquifer Flow Model with Multiple Scales of Heterogeneity

Modeling and calibration of natural aquifers with multiple scales of heterogeneity is a challenging task due to limited subsurface access. While computer modeling plays an essential role in aquifer studies, large uncertainty exists in developing a conceptual model of an aquifer and in calibrating the model for decision making. Due to uncertainties such as a lack of understanding of subsurface processes and a lack of techniques to parameterize the subsurface environment (including hydraulic conductivity, source/sink rate, and aquifer boundary conditions), existing aquifer models often suffer nonuniqueness in calibration, leading to poor predictive capability. A robust calibration methodology is needed that can address the simultaneous estimations of aquifer parameters, source/sink, and boundary conditions. In this paper, we propose a multistage and multiscale approach that addresses subsurface heterogeneity at multiple scales, while reducing uncertainty in estimating the model parameters and model boundary conditions. The key to this approach lies in the appropriate development, verification, and synthesis of existing and new techniques of static and dynamic data integration. In particular, based on a given set of observation data, new inversion techniques can be first used to estimate aquifer large‐scale effective parameters and smoothed boundary conditions, based on which parameter and boundary condition estimation can be refined at increasing detail using standard or highly parameterized estimation techniques.     

A critical examination of seismic response uncertainty analysis in earthquake engineering

The last decade of performance‐based earthquake engineering (PBEE) research has seen a rapidly increasing emphasis placed on the explicit quantification of uncertainties. This paper examines uncertainty consideration in input ground‐motion and numerical seismic response analyses as part of PBEE, with particular attention given to the physical consistency and completeness of uncertainty consideration. It is argued that the use of the commonly adopted incremental dynamic analysis leads to a biased representation of the seismic intensity and that when considering the number of ground motions to be used in seismic response analyses, attention should be given to both reducing parameter estimation uncertainty and also limiting ground‐motion selection bias. Research into uncertainties in system‐specific numerical seismic response analysis models to date has been largely restricted to the consideration of ‘low‐level’ constitutive model parameter uncertainties. However, ‘high‐level’ constitutive model and model methodology uncertainties are likely significant and therefore represent a key research area in the coming years. It is also argued that the common omission of high‐level seismic response analysis modelling uncertainties leads to a fallacy that ground‐motion uncertainty is more significant than numerical modelling uncertainty. The author's opinion of the role of uncertainty analysis in PBEE is also presented. Copyright © 2013 John Wiley & Sons, Ltd.     

Characterizing parameter sensitivity and uncertainty for a snow model across hydroclimatic regimes

The National Weather Service (NWS) uses the SNOW17 model to forecast snow accumulation and ablation processes in snow-dominated watersheds nationwide. Successful application of the SNOW17 relies heavily on site-specific estimation of model parameters. The current study undertakes a comprehensive sensitivity and uncertainty analysis of SNOW17 model parameters using forcing and snow water equivalent (SWE) data from 12 sites with differing meteorological and geographic characteristics. The Generalized Sensitivity Analysis and the recently developed Differential Evolution Adaptive Metropolis (DREAM) algorithm are utilized to explore the parameter space and assess model parametric and predictive uncertainty. Results indicate that SNOW17 parameter sensitivity and uncertainty generally varies between sites. Of the six hydroclimatic characteristics studied, only air temperature shows strong correlation with the sensitivity and uncertainty ranges of two parameters, while precipitation is highly correlated with the uncertainty of one parameter. Posterior marginal distributions of two parameters are also shown to be site-dependent in terms of distribution type. The SNOW17 prediction ensembles generated by the DREAM-derived posterior parameter sets contain most of the observed SWE. The proposed uncertainty analysis provides posterior parameter information on parameter uncertainty and distribution types that can serve as a foundation for a data assimilation framework for hydrologic models.     

基于背向散射脉冲传播时间统计特性的随机界面参数反演研究

研究了利用背向散射脉冲传播时间统计特性反演随机界面参数的问题.首先扩展了Fuks & Godin关于随机界面背向散射脉冲传播时间统计特性理论,给出了当随机界面高度-斜率统计相关情形下,最早返回的两个脉冲传播时间及其时延的概率密度函数（PDF）表达式.进而提出了一种基于背向散射脉冲传播时间统计特性的随机界面参数反演算法：采用遗传算法匹配前两个背向散射脉冲传播时间及其时延的PDF实现随机界面高度—斜率相关系数的绝对值|ρ|和无量纲参数T的反演.最后利用参数的后验概率分析反演结果的不确定性.数值模拟结果表明：该算法可有效反演随机界面的参数,参数|ρ|相对于T反演精度较高;|ρ|和T之间存在较强的参数耦合.与前人认为可忽略高度—斜率相关性（|ρ|）的观点不同：此类反演问题中应同时考虑|ρ|和T.     

Comparison of ensemble-based state and parameter estimation methods for soil moisture data assimilation

Model parameters are a source of uncertainty that can easily cause systematic deviation and significantly affect the accuracy of soil moisture generation in assimilation systems. This study addresses the issue of retrieving model parameters related to soil moisture via the simultaneous estimation of states and parameters based on the Common Land Model (CoLM). The state-parameter estimation algorithms AEnKF (Augmented Ensemble Kalman Filter), DEnKF (Dual Ensemble Kalman Filter) and SODA (Simultaneous optimization and data assimilation) are entirely implemented within an EnKF framework to investigate how the three algorithms can correct model parameters and improve the accuracy of soil moisture estimation. The analysis is illustrated by assimilating the surface soil moisture levels from varying observation intervals using data from Mongolian plateau sites. Furthermore, a radiation transfer model is introduced as an observation operator to analyze the influence of brightness temperature assimilation on states and parameters that are estimated at different microwave signal frequencies. Three cases were analyzed for both soil moisture and brightness temperature assimilation, focusing on the progressive incorporation of parameter uncertainty, forcing data uncertainty and model uncertainty. It has been demonstrated that EnKF is outperformed by all other methods, as it consistently maintains a bias. State-parameter estimation algorithms can provide a more accurate estimation of soil moisture than EnKF. AEnKF is the most robust method, with the lowest RMSE values for retrieving states and parameters dealing only with parameter uncertainty, but it possesses disadvantages related to increasing sources of uncertainty and decreasing numbers of observations. SODA performs well under the complex situations in which DEnKF shows slight disadvantages in terms of statistical indicators; however, the former consumes far more memory and time than the latter.     

Effects of uncertainty in rock-physics models on reservoir parameter estimation using seismic amplitude variation with angle and controlled-source electromagnetics data

This paper investigates the effects of uncertainty in rock-physics models on reservoir parameter estimation using seismic amplitude variation with angle and controlled-source electromagnetics data. The reservoir parameters are related to electrical resistivity by the Poupon model and to elastic moduli and density by the Xu-White model. To handle uncertainty in the rock-physics models, we consider their outputs to be random functions with modes or means given by the predictions of those rock-physics models and we consider the parameters of the rock-physics models to be random variables defined by specified probability distributions. Using a Bayesian framework and Markov Chain Monte Carlo sampling methods, we are able to obtain estimates of reservoir parameters and information on the uncertainty in the estimation. The developed method is applied to a synthetic case study based on a layered reservoir model and the results show that uncertainty in both rock-physics models and in their parameters may have significant effects on reservoir parameter estimation. When the biases in rock-physics models and in their associated parameters are unknown, conventional joint inversion approaches, which consider rock-physics models as deterministic functions and the model parameters as fixed values, may produce misleading results. The developed stochastic method in this study provides an integrated approach for quantifying how uncertainty and biases in rock-physics models and in their associated parameters affect the estimates of reservoir parameters and therefore is a more robust method for reservoir parameter estimation.     

Ground water model calibration using pilot points and regularization

Use of nonlinear parameter estimation techniques is now commonplace in ground water model calibration. However, there is still ample room for further development of these techniques in order to enable them to extract more information from calibration datasets, to more thoroughly explore the uncertainty associated with model predictions, and to make them easier to implement in various modeling contexts. This paper describes the use of "pilot points" as a methodology for spatial hydraulic property characterization. When used in conjunction with nonlinear parameter estimation software that incorporates advanced regularization functionality (such as PEST), use of pilot points can add a great deal of flexibility to the calibration process at the same time as it makes this process easier to implement. Pilot points can be used either as a substitute for zones of piecewise parameter uniformity, or in conjunction with such zones. In either case, they allow the disposition of areas of high and low hydraulic property value to be inferred through the calibration process, without the need for the modeler to guess the geometry of such areas prior to estimating the parameters that pertain to them. Pilot points and regularization can also be used as an adjunct to geostatistically based stochastic parameterization methods. Using the techniques described herein, a series of hydraulic property fields can be generated, all of which recognize the stochastic characterization of an area at the same time that they satisfy the constraints imposed on hydraulic property values by the need to ensure that model outputs match field measurements. Model predictions can then be made using all of these fields as a mechanism for exploring predictive uncertainty.     

Comparison of regional index flood estimation procedures based on the extreme value type I distribution

 A comparison of different methods for estimating T-year events is presented, all based on the Extreme Value Type I distribution. Series of annual maximum flood from ten gauging stations at the New Zealand South Island have been used. Different methods of predicting the 100-year event and the connected uncertainty have been applied: At-site estimation and regional index-flood estimation with and without accounting for intersite correlation using either the method of moments or the method of probability weighted moments for parameter estimation. Furthermore, estimation at ungauged sites were considered applying either a log-linear relationship between at-site mean annual flood and catchment characteristics or a direct log-linear relationship between 100-year events and catchment characteristics. Comparison of the results shows that the existence of at-site measurements significantly diminishes the prediction uncertainty and that the presence of intersite correlation tends to increase the uncertainty. A simulation study revealed that in regional index-flood estimation the method of probability weighted moments is preferable to method of moment estimation with regard to bias and RMSE.     

An Estimation Method of Pore Structure and Mineral Moduli Based on Kuster-Toksöz (KT) Model and Biot’s Coefficient

Pore structure and mineral matrix elastic moduli are indispensable in rock physics models. We propose an estimation method of pore structure and mineral moduli based on Kuster-Toksöz model and Biot’s coefficient. In this technique, pore aspect ratios of five different scales from 100 to 10^?4 are considered, Biot’s coefficient is used to determine bounds of mineral moduli, and an estimation procedure combined with simulated annealing (SA) algorithm to handle real logs or laboratory measurements is developed. The proposed method is applied to parameter estimations on 28 sandstone samples, the properties of which have been measured in lab. The water saturated data are used for estimating pore structure and mineral moduli, and the oil saturated data are used for testing these estimated parameters through fluid substitution in Kuster-Toksöz model. We then compare fluid substitution results with lab measurements and find that relative errors of P-wave and S-wave velocities are all less than 5%, which indicates that the estimation results are accurate.     

Analyzing hydrologic uncertainty and its impact upon decision making in water resources

Bayesian Inference and Decision Theory tools are applied to the problem of synthetic hydrology when model and parameter uncertainty exist. Issues such as optimal parameter estimation, use of synthetic generation in design problems, and the effects of parameter uncertainty on statistical estimation are discussed and applied to the problem of reservoir slorage-yield analysis.     

Uncertainty and multiple objective calibration in regional water balance modelling: case study in 320 Austrian catchments

We examine the value of additional information in multiple objective calibration in terms of model performance and parameter uncertainty. We calibrate and validate a semi‐distributed conceptual catchment model for two 11‐year periods in 320 Austrian catchments and test three approaches of parameter calibration: (a) traditional single objective calibration (SINGLE) on daily runoff; (b) multiple objective calibration (MULTI) using daily runoff and snow cover data; (c) multiple objective calibration (APRIORI) that incorporates an a priori expert guess about the parameter distribution as additional information to runoff and snow cover data. Results indicate that the MULTI approach performs slightly poorer than the SINGLE approach in terms of runoff simulations, but significantly better in terms of snow cover simulations. The APRIORI approach is essentially as good as the SINGLE approach in terms of runoff simulations but is slightly poorer than the MULTI approach in terms of snow cover simulations. An analysis of the parameter uncertainty indicates that the MULTI approach significantly decreases the uncertainty of the model parameters related to snow processes but does not decrease the uncertainty of other model parameters as compared to the SINGLE case. The APRIORI approach tends to decrease the uncertainty of all model parameters as compared to the SINGLE case. Copyright © 2006 John Wiley & Sons, Ltd.     

Parametric uncertainty assessment of hydrological models: coupling UNEEC-P and a fuzzy general regression neural network

Due to the complicated nature of environmental processes, consideration of uncertainty is an important part of environmental modelling. In this paper, a new variant of the machine learning-based method for residual estimation and parametric model uncertainty is presented. This method is based on the UNEEC-P (UNcertainty Estimation based on local Errors and Clustering – Parameter) method, but instead of multilayer perceptron uses a “fuzzified” version of the general regression neural network (GRNN). Two hydrological models are chosen and the proposed method is used to evaluate their parametric uncertainty. The approach can be classified as a hybrid uncertainty estimation method, and is compared to the group method of data handling (GMDH) and ordinary kriging with linear external drift (OKLED) methods. It is shown that, in terms of inherent complexity, measured by Akaike information criterion (AIC), the proposed fuzzy GRNN method has advantages over other techniques, while its accuracy is comparable. Statistical metrics on verification datasets demonstrate the capability and appropriate efficiency of the proposed method to estimate the uncertainty of environmental models.     

Ground motion spatial correlation fitting methods and estimation uncertainty

Ground shaking intensity varies spatially in earthquakes, and many studies have estimated correlations of intensity from past earthquake data. This paper presents a framework for quantifying uncertainty in the estimation of correlations and true variability in correlations from earthquake to earthquake. A procedure for evaluating estimation uncertainty is proposed and used to evaluate several methods that have been used in past studies to estimate correlations. The results indicate that a weighted least squares algorithm is most effective in estimating spatial correlation models and that earthquakes with at least 100 recordings are needed to produce informative earthquake-specific estimates of spatial correlations. The proposed procedure is also used to distinguish between estimation uncertainty and the true variability in model parameters that exist in a given data set. The estimation uncertainty is seen to vary between well-recorded and poorly recorded earthquakes, whereas the true variability is more stable.     

Coupling basin- and site-scale inverse models of the Española aquifer

Large-scale models are frequently used to estimate fluxes to small-scale models. The uncertainty associated with these flux estimates, however, is rarely addressed. We present a case study from the Espa?ola Basin, northern New Mexico, where we use a basin-scale model coupled with a high-resolution, nested site-scale model. Both models are three-dimensional and are analyzed by codes FEHM and PEST. Using constrained nonlinear optimization, we examine the effect of parameter uncertainty in the basin-scale model on the nonlinear confidence limits of predicted fluxes to the site-scale model. We find that some of the fluxes are very well constrained, while for others there is fairly large uncertainty. Site-scale transport simulation results, however, are relatively insensitive to the estimated uncertainty in the fluxes. We also compare parameter estimates obtained by the basin- and site-scale inverse models. Differences in the model grid resolution (scale of parameter estimation) result in differing delineation of hydrostratigraphic units, so the two models produce different estimates for some units. The effect is similar to the observed scale effect in medium properties owing to differences in tested volume. More important, estimation uncertainty of model parameters is quite different at the two scales. Overall, the basin inverse model resulted in significantly lower estimates of uncertainty, because of the larger calibration dataset available. This suggests that the basin-scale model contributes not only important boundary condition information but also improved parameter identification for some units. Our results demonstrate that caution is warranted when applying parameter estimates inferred from a large-scale model to small-scale simulations, and vice versa.     

Evaluation of analytical methodologies used to derive vulnerability functions

The recognition of fragility and vulnerability functions as a fundamental tool in seismic risk assessment has led to the development of more and more complex and elaborate procedures for their computation. Although these functions have been traditionally produced using observed damage and loss data, more recent studies propose the employment of analytical methodologies as a way to overcome the frequent lack of post‐earthquake data. The variation of the structural modelling approach on the estimation of building capacity has been the target of many studies in the past; however, its influence on the resulting vulnerability model for classes of buildings, the impact in loss estimations or propagation of the uncertainty to the seismic risk calculations has so far been the object of limited scrutiny. In this paper, an extensive study of static and dynamic procedures for estimating the nonlinear response of buildings has been carried out to evaluate the impact of the chosen methodology on the resulting capacity, fragility, vulnerability and risk outputs. Moreover, the computational effort and numerical stability provided by each approach have been evaluated and conclusions drawn regarding the optimal balance between accuracy and complexity. Copyright © 2013 John Wiley & Sons, Ltd.