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Regional extreme value models for estimation of extreme rainfall intensities are widely applied, but their underlying assumption of stationarity is challenged. Many recent studies show that the rainfall extremes worldwide exhibit a non-stationary behavior. This paper presents a spatio-temporal model of extreme rainfall. The framework is built on a Partial Duration Series approach with a non-stationary, regional threshold value. The model is based on Generalized Linear Regression solved by Generalized Estimation Equations. It allows a spatial correlation between the stations in the network and accounts furthermore for variable observation periods at each station and in each year. Marginal regional and temporal regression models solved by Generalized Least Squares are used to validate and discuss the results of the full spatio-temporal model. The model is applied on data from a large Danish rain gauge network for four durations ranging from 10 minutes to 24 hours. The observation period differs between stations, and the number of stations with more than 10 years of observations has increased over the years. A spatio-temporal model for the threshold is suggested, applying the Mean Annual Precipitation and time as the explanatory variables in the regional and temporal domain, respectively. Further analysis of Partial Duration Series with non-stationary and regional thresholds shows that the mean exceedances also exhibit a significant variation in space and time for some rainfall durations, while the shape parameter is found to be constant. This article is protected by copyright. All rights reserved.

More intense and frequent rainfalls have increased the number of urban flooding events in recent years, prompting adaptation efforts. Economic optimization is considered an efficient tool to decide on the design level for adaptation. The costs associated with a flooding to the T-year level and the annual capital and operational costs of adapting to this level are described with log-linear relations. The total flooding costs are developed as the expected annual damage of flooding above the T-year level plus the annual capital and operational costs for ensuring no flooding below the T-year level. The value of the return period T that corresponds to the minimum of the sum of these costs will then be the optimal adaptation level. The change in climate, however, is expected to continue in the next century, which calls for expansion of the above model. The change can be expressed in terms of a climate factor (the ratio between the future and the current design level) which is assumed to increase in time. This implies increasing costs of flooding in the future for many places in the world. The optimal adaptation level is found for immediate as well as for delayed adaptation. In these cases, the optimum is determined by considering the net present value of the incurred costs during a sufficiently long time-span. Immediate as well as delayed adaptation is considered.

[1] Changes in the properties of extreme rainfall events have been observed worldwide. In relation to the discussion of ongoing climatic changes, it is of high importance to attribute these changes to known sources of climate variability. Focusing on spatial and temporal changes in the frequency of extreme rainfall events, a statistical model is tested for this purpose. The model is built on the theory of generalized linear models and uses Poisson regression solved by generalized estimation equations. Spatial and temporal explanatory variables can be included simultaneously, and their relative importance can be assessed. Additionally, the model allows for a spatial correlation between the measurements. Data from a Danish rain gauge network are used as a case study for model evaluation. Focusing on 10 min and 24 h rainfall extremes, it was found that regional variation in the mean annual precipitation could explain a significant part of the spatial variability. Still, this variable was found to be of minor influence in comparison to explanatory variables in the temporal domain. The identified significant temporal variables comprise the East Atlantic pattern, the average summer precipitation, and the average summer temperature. The two latter showed a high relative importance. The established link will be beneficial when predicting future occurrences of precipitation extremes.

A regional estimation procedure that combines the index-flood concept with an empirical Bayes method for inferring regional information is introduced. The model is based on the partial duration series approach with generalized Pareto (GP) distributed exceedances. The prior information of the model parameters is inferred from regional data using generalized least squares (GLS) regression. Two different Bayesian T-year event estimators are introduced: a linear estimator that requires only some moments of the prior distributions to be specified and a parametric estimator that is based on specified families of prior distributions. The regional method is applied to flood records from 48 New Zealand catchments. In the case of a strongly heterogeneous intersite correlation structure, the GLS procedure provides a more efficient estimate of the regional GP shape parameter as compared to the usually applied weighted regional average. If intersite dependence is ignored, the uncertainty of the regional estimator may be seriously underestimated and erroneous conclusions with respect to regional homogeneity may be drawn. The GLS procedure is shown to provide a general framework for a reliable evaluation of parameter uncertainty as well as for an objective appraisal of regional homogeneity. A comparison of the two different Bayesian T-year event estimators reveals that generally the simple linear estimator is adequate.

Two different models for analyzing extreme hydrologic events, based on, respectively, partial duration series (PDS) and annual maximum series (AMS), are compared. The PDS model assumes a generalized Pareto distribution for modeling threshold exceedances corresponding to a generalized extreme value distribution for annual maxima. The performance of the two models in terms of the uncertainty of the T-year event estimator is evaluated in the cases of estimation with, respectively, the maximum likelihood (ML) method, the method of moments (MOM), and the method of probability weighted moments (PWM). In the case of ML estimation, the PDS model provides the most efficient T-year event estimator. In the cases of MOM and PWM estimation, the PDS model is generally preferable for negative shape parameters, whereas the AMS model yields the most efficient estimator for positive shape parameters. A comparison of the considered methods reveals that in general, one should use the PDS model with MOM estimation for negative shape parameters, the PDS model with exponentially distributed exceedances if the shape parameter is close to zero, the AMS model with MOM estimation for moderately positive shape parameters, and the PDS model with ML estimation for large positive shape parameters. Since heavy-tailed distributions, corresponding to negative shape parameters, are far the most common in hydrology, the PDS model generally is to be preferred for at-site quantile estimation.

A regional index-flood method based on the partial duration series model is introduced. The model comprises the assumptions of a Poisson-distributed number of threshold exceedances and generalized Pareto (GP) distributed peak magnitudes. The regional T-year event estimator is based on a regional estimate of the GP shape parameter. Analytical approximations of bias and variance of the regional T-year event estimator are deduced. The performance of the regional method is evaluated in the case of regional heterogeneity and intersite dependence. For small to moderate sample sizes, the regional estimator is superior to the at-site estimator even in extremely heterogenous regions, the performance of the regional estimator being relatively better in regions with a negative shape parameter. When the record length increases, the relative performance of the regional estimator decreases, but it is still preferable to at-site estimation in moderately heterogeneous and homogeneous regions for large sample sizes. Modest intersite dependence has only a small effect on the performance of the regional index-flood estimator.

An application of an inverse flow and transport model to a contaminated aquifer is presented. The objective of the study is to identify physical and nonreactive flow and transport parameters through an optimization approach. The approach can be classified as a statistical procedure, where a flow and transport simulation model is combined with nonlinear least squares multiple regression. The U.S. Geological Survey method of characteristics model is used to simulate flow and transport, and the optimization part is solved using a Levenberg-Marquardt algorithm. The sensitivity of the optimization approach to steady state versus transient flow conditions and to the amount of hydraulic and solute data used is investigated. The flow parameters, transmissivity and leakage factor, are estimated simultaneously with the transport parameters: source strength, porosity, and longitudinal dispersivity. This paper is the first in a two-paper series describing contaminant transport at a waste residue site. In the second paper, reactive transport at the site is investigated.

Received 1 June 2011; revised 1 May 2012; accepted 9 May 2012; published 20 June 2012. [1] This paper presents a groundwater management model, considering the interaction between a confined aquifer and an unlooped Water Distribution Network (WDN), conveying the groundwater into the Water Works distribution mains. The pumps are controlled by regulating the characteristic curves. The objective of the management is to minimize the total cost of pump operations over a multistep time horizon, while fulfilling a set of time-varying management constraints. Optimization in groundwater management and pressurized WDNs have been widely investigated in the literature. Problem formulations are often convex, hence global optimality can be attained by a wealth of algorithms. Among these, the Interior Point methods are extensively employed for practical applications, as they are capable of efficiently solving large-scale problems. Despite this, management models explicitly embedding both systems without simplifications are rare, and they usually involve heuristic techniques. The main limitation with heuristics is that neither optimality nor suboptimality bounds can be guarantee. This paper extends the proof of convexity to mixed management models, enabling the use of Interior Point techniques to compute globally optimal management solutions. If convexity is not achieved, it is shown how suboptimal solutions can be computed, and how to bind their deviation from the optimality. Experimental results obtained by testing the methodology in a well field located nearby Copenhagen (DK), show that management solutions can consistently perform within the 99.9% of the true optimum. Furthermore it is shown how not considering the Water Distribution Network in optimization is likely to result in unfeasible management solutions.

[1] The estimation of hydrological model parameters by calibration to field data is a critical step in the modeling process. However, calibration often fails because of parameter correlation. Here it is shown that time-lapse gravity data can be combined with hydraulic head data in a coupled hydrogeophysical inversion to decrease parameter correlation in groundwater models. This is demonstrated for a model of riverbank infiltration where combined inversion successfully constrains hydraulic conductivity and specific yield in both an analytical and a numerical groundwater model. A sensitivity study shows that time-lapse gravity data are especially useful to constrain specific yield. Furthermore, we demonstrate that evapotranspiration, and riverbed conductance are better constrained by coupled inversion to gravity and head data than to head data alone. When estimating the four parameters simultaneously, the six correlation coefficients were reduced from unity when only head data were employed to significantly lower values when gravity and head data were combined. Our analysis reveals that the estimated parameter values are not very sensitive to the choice of weighting between head and gravity data over a large interval of relative weights.

[1] We define a Markov-modulated autoregressive model with exogenous input (MARX) to generate runoff scenarios using climatic information. Runoff parameterization is assumed to be conditioned on a hidden climate state following a Markov chain, where state transition probabilities are functions of the climatic input. MARX allows stochastic modeling of nonstationary runoff, as runoff anomalies are described by a mixture of autoregressive models with exogenous input, each one corresponding to a climate state. We apply MARX to inflow time series of the Daule Peripa reservoir (Ecuador). El Nino-Southern Oscillation (ENSO) information is used to condition runoff parameterization. Among the investigated ENSO indexes, the NINO 1+2 sea surface temperature anomalies and the trans-Nino index perform best as predictors. In the perspective of reservoir optimization at various time scales, MARX produces realistic long-term scenarios and short-term forecasts, especially when intense El Nino events occur. Low predictive ability is found for negative runoff anomalies, as no climatic index correlating properly with negative inflow anomalies has yet been identified.

As a generalization of the common assumption of exponential distribution of the exceedances in partial duration series the generalized Pareto distribution has been adopted. Estimators for the parameters are presented using estimation by both method of moments and probability-weighted moments. The corresponding estimators for the T-year event are given and approximate expressions for bias and variance of the estimators are derived in both cases. Using the mean square error of the T-year event estimator as a performance index it is shown that the method of moments is preferable to the probability-weighted moments. Maintaining the generalized Pareto distribution as the parent exceedance distribution the T-year event is estimated assuming the exceedances to be exponentially distributed. For moderately long-tailed exceedance distributions and small to moderate sample sizes it is found, by comparing mean square errors of the T-year event estimators, that the exponential distribution is preferable to the correct generalized Pareto distribution despite the introduced model error and despite a possible rejection of the exponential hypothesis by a test of significance. For moderately short-tailed exceedance distributions (with physically justified upper limit) the correct exceedance distribution should be applied despite a possible acceptance of the exponential assumption by a test of significance.

In order to obtain a good description of the exceedances in a partial duration series it is often necessary to divide the year into a number (2–4) of seasons. Hereby a stationary exceedance distribution can be maintained within each season. This type of seasonal models may, however, not be suitable for prediction purposes due to the large number of parameters required. In the particular case with exponentially distributed exceedances and Poissonian occurrence times the precision of the T year event estimator has been thoroughly examined considering both seasonal and nonseasonal models. The two-seasonal probability density function of the T year event estimator has been deduced and used in the assessment of the precision of approximate moments. The nonseasonal approach covered both a total omission of seasonality by pooling data from different flood seasons and a discarding of nonsignificant season(s) before the analysis of extremes. Mean square error approximations (bias second order, variance first and second order) were employed as measures for prediction uncertainty. It was found that optimal estimates can usually be obtained with a nonseasonal approach.

Four different methods of parameterizing spatially varying log transmissivities in an inverse approach are compared with respect to prediction accuracy of simulated flow and transport. Transport parameter estimation is included by two-stage feedback optimization. In stage one the log transmissivities are estimated by fitting both head and concentration data, given initial values of the source concentration and the dispersivities. In stage two, the source concentration and the dispersivities are estimated by fitting the concentration data. With the updated transport parameters, final estimates of the log transmissivities are obtained by repeating the optimization of stage one. The formulated objective functions are minimized using Levenberg-Marquardt's algorithm. The models are applied to synthetic two-dimensional transport problems in steady state flow regimes. The “true” log transmissivity fields are generated by the turning bands method, thereby incorporating spatial variability. The test cases differ in the input variances of the generated fields and with respect to the amount and accuracy of “observed” transmissivity data.

[1] Flowmeter-measured hydraulic conductivities from the heterogeneous MADE site have been used predictively in advection-dispersion models. Resulting simulated concentrations failed to reproduce even major plume characteristics and some have concluded that other mechanisms, such as dual porosity, are important. Here an alternative possibility is investigated: that the small-scale flowmeter measurements are too noisy and possibly too biased to use so directly in site-scale models and that the hydraulic head and transport data are more suitable for site-scale characterization. Using a calibrated finite element model of the site and a new framework to evaluate random and systematic model and measurement errors, the following conclusions are derived. (1) If variations in subsurface fluid velocities like those simulated in this work (0.1 and 2.0 m per day along parallel and reasonably close flow paths) exist, it is likely that classical advection-dispersion processes can explain the measured plume characteristics. (2) The flowmeter measurements are possibly systematically lower than site-scale values when the measurements are considered individually and using common averaging methods and display variability that obscures abrupt changes in hydraulic conductivities that are well supported by changes in hydraulic gradients and are important to the simulation of transport.

[1] A general framework for regional analysis and modeling of extreme rainfall characteristics is presented. The model is based on the partial duration series (PDS) method that includes in the analysis all events above a threshold level. In the PDS model the average annual number of exceedances, the mean value of the exceedance magnitudes, and the coefficient of L variation (LCV) are considered as regional variables. A generalized least squares (GLS) regression model that explicitly accounts for intersite correlation and sampling uncertainties is applied for evaluating the regional heterogenity of the PDS parameters. For the parameters that show a significant regional variability the GLS model is subsequently adopted for describing the variability from physiographic and climatic characteristics. For determination of a proper regional parent distribution L moment analysis is applied for discriminating between the exponential distribution and various two-parameter distributions in the PDS model. The resulting model can be used for estimation of rainfall intensity-duration-frequency curves at an arbitrary location in a region. For illustration, the regional model is applied to rainfall data from a rain gauge network in Denmark.

The estimation of design floods is, in practice, often based on small samples of data, which may cause a severe uncertainty. For a particular version of the partial duration series (exponentially distributed exceedances and Poissonian occurrence times) the distribution of the T-year design estimate xˆT is derived along with the distribution of RT; defined as the true risk of exceeding xˆT within a given disposal period. For a fixed flood level the distributions of the return period estimator Tˆ and the estimator of the risk in lifetime Rˆ are also presented. Analytical closed-form expressions for mean value and standard deviation are derived for these variables, except for Tˆ, which does not possess moments. The concept of “expected risk” is introduced, and an analytical expression describing this property is derived. A risk-based design technique, which is essentially different from the traditional procedure, is presented, and its applicability is verified using Monte Carlo simulation.