fast decay of solutions for linear wave equations with dissipation localized near infinity in an exterior domain abstract uniform energy and l 2 decay of solutions for linear wave equations with localized dissipation will be given in order to derive the l 2 decay property of the solution a useful device whose idea comes from ikehata matsuyama sci math japon 55 2002 33 is used in fact we shall show that the l 2 norm and the total energy of solutions respectively decay like o 1 t and o 1 t 2 as t for a kind of the weighted initial data

appraisal of the generalized likelihood uncertainty estimation glue method 1 recent research documents that the widely accepted generalized likelihood uncertainty estimation glue method for describing forecasting precision and the impact of parameter uncertainty in rainfall runoff watershed models fails to achieve the intended purpose when used with an informal likelihood measure in particular glue generally fails to produce intervals that capture the precision of estimated parameters and the difference between predictions and future observations this paper illustrates these problems with glue using a simple linear rainfall runoff model so that model calibration is a linear regression problem for which exact expressions for prediction precision and parameter uncertainty are well known and understood the simple regression example enables us to clearly and simply illustrate glue deficiencies beven and others have suggested that the choice of the likelihood measure used in a glue computation is subjective and may be selected to reflect the goals of the modeler if an arbitrary likelihood is adopted that does not reasonably reflect the sampling distribution of the model errors then glue generates arbitrary results without statistical validity that should not be used in scientific work the traditional subjective likelihood measures that have been used with glue also fail to reflect the nonnormality heteroscedasticity and serial correlation among the residual errors generally found in real problems and hence are poor metrics for even simple sensitivity analyses and model calibration most previous applications of glue only produce uncertainty intervals for the average model prediction which by construction should not be expected to include future observations with the prescribed probability we show how the glue methodology when properly implemented with a statistically valid likelihood function can provide prediction intervals for future observations which will agree with widely accepted and statistically valid analyses

bayesian estimation of uncertainty in runoff prediction and the value of data an application of the glue approach this paper addresses the problem of evaluating the predictive uncertainty of topmodel using the bayesian generalised likelihood uncertainty estimation glue methodology in an application to the small ringelbach research catchment in the vosges france the wide range of parameter sets giving acceptable simulations is demonstrated and uncertainty bands are presented based on different likelihood measures it is shown how the distributions of predicted discharges are non gaussian and vary in shape through time and with discharge updating of the likelihood weights using bayes equation is demonstrated after each year of record and it is shown how the additional data can be evaluated in terms of the way they constrain the uncertainty bands

on constraining the predictions of a distributed model the incorporation of fuzzy estimates of saturated areas into the calibration process distributed hydrological models are generally overparameterized resulting in the possibility of multiple parameterizations from many areas of the parameter space providing acceptable fits to observed data in this study topmodel parameterizations are conditioned on discharges and then further conditioned on estimates of saturated areas derived from ers i synthetic aperture radar sar images combined with the in tan topographic index and compared to ground truth saturation measurements made in one small subcatchment the uncertainty associated with the catchment wide predictions of saturated area is explicitly incorporated into the conditioning through the weighting of estimates within a fuzzy set framework the predictive uncertainty associated with the parameterizations is then assessed using the generalized likelihood uncertainty estimation glue methodology it is shown that despite the uncertainty in the predictions of saturated area the methodology can reject many previously acceptable parameterizations with the consequence of a marked reduction in the acceptable range of a catchment average transmissivity parameter and of improved predictions of some discharge events

development of a formal likelihood function for improved bayesian inference of ephemeral catchments 1 the application of formal bayesian inferential approaches in hydrologic modeling is often criticized for requiring explicit assumptions to be made about the distribution of the errors via the likelihood function these assumptions can be adequate in some situations but often little attention is paid to the selection of an appropriate likelihood function this paper investigates the application of bayesian methods in modeling ephemeral catchments we consider two modeling case studies including a synthetically generated data set and a real data set of an arid australian catchment the case studies investigate some typical forms of the likelihood function which have been applied widely by previous researchers as well as introducing new implementations aimed at better addressing likelihood function assumptions in dry catchments the results of the case studies indicate the importance of explicitly accounting for model residuals that are highly positively skewed due to the presence of many zeros zero inflation arising from the dry spells experienced by the catchment specifically the form of the likelihood function was found to significantly impact the calibrated parameter posterior distributions which in turn have the potential to greatly affect the uncertainty estimates in each application the likelihood function that explicitly accounted for the nonconstant variance of the errors and the zero inflation of the errors resulted in 1 equivalent or better fits to the observed discharge in both timing and volume 2 superior estimation of the uncertainty as measured by the reliability and sharpness metrics and 3 more linear quantile quantile plots indicating the errors were more closely matched to the assumptions of this form of the likelihood function

probabilistic optimization for conceptual rainfall runoff models a comparison of the shuffled complex evolution and simulated annealing algorithms automatic optimization algorithms are used routinely to calibrate conceptual rainfall runoff crr models the goal of calibration is to estimate a feasible and unique global set of parameter estimates that best fit the observed runoff data most if not all optimization algorithms have difficulty in locating the global optimum because of response surfaces that contain multiple local optima with regions of attraction of differing size discontinuities and long ridges and valleys extensive research has been undertaken to develop efficient and robust global optimization algorithms over the last 10 years this study compares the performance of two probabilistic global optimization methods the shuffled complex evolution algorithm sce ua and the three phase simulated annealing algorithm sa sx both algorithms are used to calibrate two parameter sets of a modified version of boughton s 1984 sfb model using data from two australian catchments that have low and high runoff yields for the reduced well identified parameter set the algorithms have a similar efficiency for the low yielding catchment but sce ua is almost twice as robust although the robustness of the algorithms is similar for the high yielding catchment sce ua is six times more efficient than sa sx when fitting the full parameter set the performance of sa sx deteriorated markedly for both catchments these results indicated that sce ua s use of multiple complexes and shuffling provided a more effective search of the parameter space than sa sx s single simplex with stochastic step acceptance criterion especially when the level of parameterization is increased examination of the response surface for the low yielding catchment revealed some reasons why sce ua outperformed sa sx and why probabilistic optimization algorithms can experience difficulty in locating the global optimum

improved parameter inference in catchment models 1 evaluating parameter uncertainty a bayesian methodology is developed to evaluate parameter uncertainty in catchment models fitted to a hydrologic response such as runoff the goal being to improve the chance of successful regionalization the catchment model is posed as a nonlinear regression model with stochastic errors possibly being both autocorrelated and heteroscedastic the end result of this methodology which may use box cox power transformations and arma error models is the posterior distribution which summarizes what is known about the catchment model parameters this can be simplified to a multivariate normal provided a linearization in parameter space is acceptable means of checking and improving this assumption are discussed the posterior standard deviations give a direct measure of parameter uncertainty and study of the posterior correlation matrix can indicate what kinds of data are required to improve the precision of poorly determined parameters finally a case study involving a nine parameter catchment model fitted to monthly runoff and soil moisture data is presented it is shown that use of ordinary least squares when its underlying error assumptions are violated gives an erroneous description of parameter uncertainty

catchments as simple dynamical systems a case study on methods and data requirements for parameter identification in many rainfall runoff models at least some calibration of model parameters has to take place especially for ungauged or poorly gauged basins this can be problematic because there is little or no data available for calibration a possible solution to overcome the problems caused by data scarcity is to set up a measurement campaign for a limited time period in this study we determine the minimum amount of data required to determine robust parameter values for a simple model with two parameters the model is constructed such that the parameters can be determined not only with automatic calibration but also by recession analysis and a priori from boussinesq theory the model has been applied to a research catchment in switzerland for automatic calibration and recession analysis one season 5 months is found to be sufficient to give robust parameters for simulation of high flows over the full observation period for automatic calibration this should be the season with the highest precipitation for recession analysis the season with least evapotranspiration the boussinesq equation is able to give good parameter estimates for modeling high flows but detailed in situ knowledge of the catchment is required automatic calibration outperforms recession analysis and boussinesq theory by far when it comes to parameter estimation with a focus on prediction of low flows it was shown that a single set of parameters cannot simultaneously describe high and low flows with a reasonable accuracy suggesting that more than two parameters are needed to characterize subsurface properties

on maximum likelihood estimators for the multisite lag one streamflow model complete and incomplete data cases two practical problems may arise when using the traditional estimators of the parameters in the multisite lag one streamflow model 1 the estimated covariance matrix may not be positive definite thereby preventing its decomposition a necessary step for synthetic streamflow generation and 2 if there are missing observations streamflow data must be truncated to the shortest record resulting in a loss of useful information this note draws attention to the existence of maximum likelihood estimators which overcome both problems in the complete data case explicit maximum likelihood estimators exist however in the incomplete data case iterative maximum likelihood procedures must be used in particular the expectation maximization em algorithm is considered in a state space framework compatible with the multisite streamflow model this algorithm has robust convergence properties is simple to implement and produces smoothed estimates of the missing data an example is presented illustrating problems with decomposition of the traditional covariance matrix estimate and also illustrating application of the em algorithm

bayesian spectral likelihood for hydrological parameter inference this paper proposes a spectral domain likelihood function for the bayesian estimation of hydrological model parameters from a time series of model residuals the spectral domain error model is based on the power density spectrum pds of the stochastic process assumed to describe residual errors the bayesian spectral domain likelihood bsl is mathematically equivalent to the corresponding bayesian time domain likelihood btl and yields the same inference when all residual error assumptions are satisfied and all residual error parameters are inferred however the bsl likelihood function does not depend on the residual error distribution in the original time domain which offers a theoretical advantage in terms of robustness for hydrological parameter inference the theoretical properties of bsl are demonstrated and compared to btl and a previously proposed spectral likelihood by montanari and toth 2007 using a set of synthetic case studies and a real case study based on the leaf river catchment in the u s the empirical analyses confirm the theoretical properties of bsl when applied to heteroscedastic and autocorrelated error models where heteroscedasticity is represented using the log transformation and autocorrelation is represented using an ar 1 process unlike mtl the use of bsl did not introduce additional parametric uncertainty compared to btl future work will explore the application of bsl to challenging modeling scenarios in arid catchments and indirect calibration with nonconcomitant input output time series

critical exponents for semilinear dissipative wave equations in rn abstract we shall present new critical exponents 1 2m n with m 1 2 to the cauchy problem utt u ut u p 1u with the initial data u 0 u 1 h 1 r n l m r n l 2 r n l m r n that is the small data global existence property can be derived to the cauchy problem above with power 1 2m n

inversion of the borden tracer experiment data investigation of stochastic moment models inversion of dagan s two and three dimensional stochastic models using freyberg s 1986 rajaram and gelhar s 1988 1991 and barry and sposito s 1990 moment data from the borden experiment is carried out to examine 1 the validity of the two dimensional dagan 1982 and three dimensional dagan 1988 models and 2 the reduction in uncertainty of the spatial moments over nonconditioned estimates a direct application of bayesian statistical inference in conjunction with monte carlo integration is used to produce posterior probability density functions for the parameters the parameter ranges from all methods show horizontal integral scales between 1 85 and 4 04 m vertical integral scales z between 0 144 and 0 459 m and log hydraulic conductivities in k between 0 120 and 0 197 these results compare well to the earlier estimates of woodbury and sudicky 1991 and robin et al 1991 we show that moment prediction uncertainty is substantially reduced when both tracer moment data and prior estimates of the in k geostatistical parameters are incorporated into theoretical formulae based on stochastic dispersion theory

a multiple criteria decision support system for testing integrated environmental models abstract spatial models of ecological and hydrological processes are widely used tools for studying natural systems over large areas however these models lack specific mechanisms for reporting output uncertainty contributed by model structure and so testing their suitability for studying a large range of problems is difficult this paper describes a method of evaluating the uncertainty contributed by underlying assumptions used in constructing integrated environmental models from two or more sub models that were developed for different purposes integrated environmental models are typically constructed from many individual process based models conflicting assumptions between these sub models e g spatial scale differences are easily overlooked during model development and application this semantic error cannot be predicted prior to simulation as it may only emerge through the interaction of sub models applied to a particular set of data used to drive a simulation model agreement is proposed and demonstrated as a way to detect problems of model integration at the state variable level within an integrated ecosystem model this model agreement is then propagated to model response variables using multiple criteria to examine their sensitivity predictability and synchronicity to the measured uncertainty in state variables these three properties are combined under fuzzy logic in order to provide decision support on where for a given time during simulation the sub models agree on a particular response variable this paper describes the details of the approach and its application using an existing integrated environmental model the results show that for a given set of model inputs and application integrated environmental models may have spatially variable levels of agreement at the sub model level the results using rhessysd a spatially integrated ecosystem hydrology model indicate that semantic error in estimates of plant available soil moisture are consistent with observations of the need for resetting events such as flooding to initialize the model to a point where further simulation results can be trusted these results suggest that a dynamic selection of sub models may be warranted given a reasonable method of determining sub model disagreement during simulation fuzzy set theory may be a useful tool in arriving at such a model selection process as it allows for a relatively straightforward synthesis of numerous model evaluation criteria with a large quantity of output from the model

assessing the impacts of parameter uncertainty for computationally expensive groundwater models 1 computational requirements often limit the assessment of uncertainty in complex environmental models monte carlo based techniques such as generalized likelihood uncertainty estimation glue and bayesian uncertainty assessment are limited in applicability because of their heavy computational demands this paper presents a computationally efficient method that combines automatic calibration using function approximation based global optimization with importance weighting for assessing parametric uncertainty in model outputs for computationally expensive models we call this method automatic calibration and uncertainty assessment using response surfaces acuars the bias introduced during optimization because of oversampling of high goodness of fit regions of the parameter space is corrected using a multidimensional cell declustering technique the proposed method is applied to two hypothetical examples in enhanced bioremediation of chlorinated ethenes in groundwater that differ in computational requirements and the number of parameters that were calibrated the results from the simpler example in which uncertainty of three model parameters in model outputs was considered showed that for a given acceptability threshold and a given number of permissible model simulations acuars finds many more representative parameter sets than glue glue produced only half the number of acceptable parameter sets for uncertainty assessment even after 10 times as many simulations as acuars was performed for glue further the median output of acuars predicted the true value of the outputs more closely than glue for a more complicated numerical example where noisy data were used for calibration of seven model parameters glue required 2400 simulations to find 31 acceptable parameter sets while acuars found 46 acceptable parameter sets in only 300 simulations for a moderately stringent threshold this enabled a more accurate estimate of uncertainty by providing closer estimates of the true outputs

critical exponent for semilinear damped wave equations in the n dimensional half space abstract we generalize a previous result of ikehata math methods appl sci in press which studies the critical exponent problem of a semilinear damped wave equation in the one dimensional half space to the general n dimensional half space case that is to say one can show the small data global existence of solutions of a mixed problem for the equation utt u ut u p with the power p satisfying p n 1 2 n 1 if we deal with the problem in the n dimensional half space

multivariate calibration of a water and energy balance model in the spectral domain 1 the objective of this paper is to explore the possibility of using multiple variables in the calibration of hydrologic models in the spectral domain a simple water and energy balance model was used combined with observations of the energy balance and the soil moisture profile the correlation functions of the model outputs and the observations for the different variables have been calculated after the removal of the diurnal cycle of the energy balance variables these were transformed to the frequency domain to obtain spectral density functions which were combined in the calibration algorithm it has been found that it is best to use the square root of the spectral densities in the parameter estimation under these conditions spectral calibration performs almost equally as well as time domain calibration using least squares differences between observed and simulated time series incorporation of the spectral coefficients of the cross correlation functions did not improve the results of the calibration calibration on the correlation functions in the time domain led to worse model performance when the meteorological forcing and model calibration data are not overlapping in time spectral calibration has been shown to lead to an acceptable model performance overall the results in this paper suggest that in case of data scarcity multivariate spectral calibration can be an attractive tool to estimate model parameters

a new bayesian recursive technique for parameter estimation 1 the performance of any model depends on how well its associated parameters are estimated in the current application a localized bayesian recursive estimation lobare approach is devised for parameter estimation the lobare methodology is an extension of the bayesian recursive estimation bare method it is applied in this paper on two different types of models an artificial intelligence ai model in the form of a support vector machine svm application for forecasting soil moisture and a conceptual rainfall runoff crr model represented by the sacramento soil moisture accounting sac sma model support vector machines based on statistical learning theory slt represent the modeling task as a quadratic optimization problem and have already been used in various applications in hydrology they require estimation of three parameters sac sma is a very well known model that estimates runoff it has a 13 dimensional parameter space in the lobare approach presented here bayesian inference is used in an iterative fashion to estimate the parameter space that will most likely enclose a best parameter set this is done by narrowing the sampling space through updating the parent bounds based on their fitness these bounds are actually the parameter sets that were selected by bare runs on subspaces of the initial parameter space the new approach results in faster convergence toward the optimal parameter set using minimum training calibration data and fewer sets of parameter values the efficacy of the localized methodology is also compared with the previously used bare algorithm

efficient prediction uncertainty approximation in the calibration of environmental simulation models 1 this paper is aimed at improving the efficiency of model uncertainty analyses that are conditioned on measured calibration data specifically the focus is on developing an alternative methodology to the generalized likelihood uncertainty estimation glue technique when pseudolikelihood functions are utilized instead of a traditional statistical likelihood function we demonstrate for multiple calibration case studies that the most common sampling approach utilized in glue applications uniform random sampling is much too inefficient and can generate misleading estimates of prediction uncertainty we present how the new dynamically dimensioned search dds optimization algorithm can be used to independently identify multiple acceptable or behavioral model parameter sets in two ways dds could replace random sampling in typical applications of glue more importantly we present a new practical and efficient uncertainty analysis methodology called dds approximation of uncertainty dds au that quantifies prediction uncertainty using prediction bounds rather than prediction limits results for 13 14 26 and 30 parameter calibration problems show that dds au can be hundreds or thousands of times more efficient at finding behavioral parameter sets than glue with random sampling results for one example show that for the same limited computational effort dds au prediction bounds can simultaneously be smaller and contain more of the measured data in comparison to glue prediction bounds we also argue and then demonstrate that within the glue framework when behavioral parameter sets are not sampled frequently enough latin hypercube sampling does not offer any improvements over simple random sampling

effective and efficient global optimization for conceptual rainfall runoff models the successful application of a conceptual rainfall runoff crr model depends on how well it is calibrated despite the popularity of crr models reports in the literature indicate that it is typically difficult if not impossible to obtain unique optimal values for their parameters using automatic calibration methods unless the best set of parameters associated with a given calibration data set can be found it is difficult to determine how sensitive the parameter estimates and hence the model forecasts are to factors such as input and output data error model error quantity and quality of data objective function used and so on results are presented that establish clearly the nature of the multiple optima problem for the research crr model sixpar these results suggest that the crr model optimization problem is more difficult than had been previously thought and that currently used local search procedures have a very low probability of successfully finding the optimal parameter sets next the performance of three existing global search procedures are evaluated on the model sixpar finally a powerful new global optimization procedure is presented entitled the shuffled complex evolution sce ua method which was able to consistently locate the global optimum of the sixpar model and appears to be capable of efficiently and effectively solving the crr model optimization problem

corruption of accuracy and efficiency of markov chain monte carlo simulation by inaccurate numerical implementation of conceptual hydrologic models conceptual rainfall runoff models have traditionally been applied without paying much attention to numerical errors induced by temporal integration of water balance dynamics reliance on first order explicit fixed step integration methods leads to computationally cheap simulation models that are easy to implement computational speed is especially desirable for estimating parameter and predictive uncertainty using markov chain monte carlo mcmc methods confirming earlier work of kavetski et al 2003 we show here that the computational speed of first order explicit fixed step integration methods comes at a cost for a case study with a spatially lumped conceptual rainfall runoff model it introduces artificial bimodality in the marginal posterior parameter distributions which is not present in numerically accurate implementations of the same model the resulting effects on mcmc simulation include 1 inconsistent estimates of posterior parameter and predictive distributions 2 poor performance and slow convergence of the mcmc algorithm and 3 unreliable convergence diagnosis using the gelman rubin statistic we studied several alternative numerical implementations to remedy these problems including various adaptive step finite difference schemes and an operator splitting method our results show that adaptive step second order methods based on either explicit finite differencing or operator splitting with analytical integration provide the best alternative for accurate and efficient mcmc simulation fixed step or adaptive step implicit methods may also be used for increased accuracy but they cannot match the efficiency of adaptive step explicit finite differencing or operator splitting of the latter two explicit finite differencing is more generally applicable and is preferred if the individual hydrologic flux laws cannot be integrated analytically as the splitting method then loses its advantage

corrigendum to conical plurisubharmonic measure and new cross theorems j math anal appl 365 2010 429 434

generalized likelihood uncertainty estimation glue and approximate bayesian computation what s the connection 1 there has been a recent debate in the hydrological community about the relative merits of the informal generalized likelihood uncertainty estimation glue approach to uncertainty assessment in hydrological modeling versus formal probabilistic approaches some recent literature has suggested that the methods can give similar results in practice when properly applied in this note we show that the connection between formal bayes and glue is not merely operational but goes deeper with glue corresponding to a certain approximate bayesian procedure even when the generalized likelihood is not a true likelihood the connection we describe relates to recent approximate bayesian computation abc methods originating in genetics abc algorithms involve the use of a kernel function and the generalized likelihood in glue can be thought of as relating to this kernel function rather than to the model likelihood two interpretations of glue emerge one as a computational approximation to a bayes procedure for a certain error free model and the second as an exact bayes procedure for a perturbation of that model in which the truncation of the generalized likelihood in glue plays a role the intent of this study is to encourage cross fertilization of ideas regarding glue and abc in hydrologic applications the connection we outline suggests the possibility of combining a formal likelihood with a kernel based on a generalized likelihood within the abc framework and also allows advanced abc computational methods to be used in glue applications the model based interpretation of glue may also be helpful in partially illuminating the implicit assumptions in different choices of generalized likelihood

stochastic capture zone delineation within the generalized likelihood uncertainty estimation methodology conditioning on head observations a stochastic methodology to evaluate the predictive uncertainty in well capture zones in heterogeneous aquifers with uncertain parameters is presented the approach is based on the generalized likelihood uncertainty estimation methodology the hydraulic conductivity is modeled as a random space function allowing for the uncertainty that stems from the imperfect knowledge of the parameters of the correlation structure parameters are sampled from prior distributions and are used for the generation of a large number of hydraulic conductivity fields which are subsequently used to solve the groundwater flow equation a likelihood is calculated for every simulation based on some goodness of fit measure between simulated heads and available observations using inverse particle tracking a capture zone is determined which is assigned the likelihood calculated for that particular simulation statistical analysis of the ensemble of all simulations enables the predictive uncertainty of the well capture zones to be defined results are presented for a hypothetical test case and different likelihood definitions used in the conditioning process the results show that the delineated capture zones are most sensitive to the mean hydraulic conductivity and the variance whereas the integral scale of the variogram is the parameter with the smallest influence for all likelihood measures the prior uncertainty is reduced considerably by introducing the observation heads but the reduction is most effective for the very selective likelihood definition the method presented can be used in real applications to quantify the uncertainty in the location and extent of well capture zones when little or no information is available about the hydraulic properties through the conditioning on head observations

evaluation of maximum likelihood parameter estimation techniques for conceptual rainfall runoff models influence of calibration data variability and length on model credibility the success of an automatic calibration procedure is highly dependent on the choice of the objective function and the nature quantity and quality of the data used the objective function should be selected on the basis of the stochastic properties of the errors present in the data and in the model also the data should be chosen so as to contain as much valuable information about the process as possible in this paper we compare the performance of two maximum likelihood estimators the amle which assumes the presence of first lag autocorrelated homogeneous variance errors and the hmle which assumes the presence of uncorrelated inhomogeneous variance errors to the commonly used simple least squares criterion sls the model calibrated was the soil moisture accounting model of the u s national weather service s river forecast system sma nwsrfs the results indicate that a properly chosen objective function can enhance the possibility of obtaining unique and conceptually realistic parameter estimates furthermore the sensitivity of the estimation results to various characteristics of the calibration data such as hydrologic variability and length are substantially reduced

global existence of solutions for semilinear damped wave equation in 2 d exterior domain abstract we consider a mixed problem of a damped wave equation utt u ut u p in the two dimensional exterior domain case small global in time solutions can be constructed in the case when the power p on the nonlinear term u p satisfies p 2 for this purpose we shall deal with a radially symmetric solution in the exterior domain a new device developed in ikehata matsuyama sci math japon 55 2002 33 plays an effective role

shcherbina s theorem for finely holomorphic functions we prove an analogue of sadullaev s theorem concerning the size of the set where a maximal totally real manifold m can meet a pluripolar set m has to be of class c1 only this readily leads to a version of shcherbina s theorem for c1 functions f that are defined in a neighborhood of certain compact sets k subset mathbb c if the graph f k is pluripolar then frac partial f partial bar z 0 in the closure of the fine interior of k

practical identifiability analysis of large environmental simulation models large environmental simulation models are usually overparameterized with respect to given sets of observations this results in poorly identifiable or nonidentifiable model parameters for small models plots of sensitivity functions have proven to be useful for the analysis of parameter identifiability for models with many parameters however near linear dependence of sensitivity functions can no longer be assessed graphically in this paper a systematic approach for tackling the parameter identifiability problem of large models based on local sensitivity analysis is presented the calculation of two identifiability measures that are easy to handle and interpret is suggested the first accounts for the sensitivity of model results to single parameters and the second accounts for the degree of near linear dependence of sensitivity functions of parameter subsets it is shown how these measures provide identifiability diagnosis for parameter subsets how they are able to guide the selection of identifiable parameter subsets for parameter estimation and how they facilitate the interpretation of the correlation matrix of the parameter estimate with respect to parameter identifiability in addition we show how potential bias of the parameter estimates due to a priori fixing of some of the parameters can be analyzed finally two case studies are presented in order to illustrate the suggested approach

comparison of joint versus postprocessor approaches for hydrological uncertainty estimation accounting for error autocorrelation and heteroscedasticity the paper appraises two approaches for the treatment of heteroscedasticity and autocorrelation in residual errors of hydrological models both approaches use weighted least squares wls with heteroscedasticity modeled as a linear function of predicted flows and autocorrelation represented using an ar 1 process in the first approach heteroscedasticity and autocorrelation parameters are inferred jointly with hydrological model parameters the second approach is a two stage postprocessor scheme where stage 1 infers the hydrological parameters ignoring autocorrelation and stage 2 conditionally infers the heteroscedasticity and autocorrelation parameters these approaches are compared to a wls scheme that ignores autocorrelation empirical analysis is carried out using daily data from 12 us catchments from the mopex set using two conceptual rainfall runoff models gr4j and hbv under synthetic conditions the postprocessor and joint approaches provide similar predictive performance though the postprocessor approach tends to underestimate parameter uncertainty however the mopex results indicate that the joint approach can be nonrobust in particular when applied to gr4j it often produces poor predictions due to strong multiway interactions between a hydrological water balance parameter and the error model parameters the postprocessor approach is more robust precisely because it ignores these interactions practical benefits of accounting for error autocorrelation are demonstrated by analyzing streamflow predictions aggregated to a monthly scale where ignoring daily scale error autocorrelation leads to significantly underestimated predictive uncertainty and by analyzing one day ahead predictions where accounting for the error autocorrelation produces clearly higher precision and better tracking of observed data including autocorrelation into the residual error model also significantly affects calibrated parameter values and uncertainty estimates the paper concludes with a summary of outstanding challenges in residual error modeling particularly in ephemeral catchments

maximum likelihood bayesian averaging of spatial variability models in unsaturated fractured tuff 1 hydrologic analyses typically rely on a single conceptual mathematical model yet hydrologic environments are open and complex rendering them prone to multiple interpretations and mathematical descriptions adopting only one of these may lead to statistical bias and underestimation of uncertainty bayesian model averaging bma hoeting et al 1999 provides an optimal way to combine the predictions of several competing models and to assess their joint predictive uncertainty however it tends to be computationally demanding and relies heavily on prior information about model parameters neuman 2002 2003 proposed a maximum likelihood version mlbma of bma to render it computationally feasible and to allow dealing with cases where reliable prior information is lacking we apply mlbma to seven alternative variogram models of log air permeability data from single hole pneumatic injection tests in six boreholes at the apache leap research site alrs in central arizona unbiased ml estimates of variogram and drift parameters are obtained using adjoint state maximum likelihood cross validation samper and neuman 1989a in conjunction with universal kriging and generalized least squares standard information criteria provide an ambiguous ranking of the models which does not justify selecting one of them and discarding all others as is commonly done in practice instead we eliminate some of the models based on their negligibly small posterior probabilities and use the rest to project the measured log permeabilities by kriging onto a rock volume containing the six boreholes we then average these four projections and associated kriging variances using the posterior probability of each model as weight finally we cross validate the results by eliminating from consideration all data from one borehole at a time repeating the above process and comparing the predictive capability of mlbma with that of each individual model we find that mlbma is superior to any individual geostatistical model of log permeability among those we consider at the alrs

ensemble evaluation of hydrological model hypotheses it is demonstrated for the first time how model parameter structural and data uncertainties can be accounted for explicitly and simultaneously within the generalized likelihood uncertainty estimation glue methodology as an example application 72 variants of a single soil moisture accounting store are tested as simplified hypotheses of runoff generation at six experimental grassland field scale lysimeters through model rejection and a novel diagnostic scheme the fields designed as replicates exhibit different hydrological behaviors which yield different model performances for fields with low initial discharge levels at the beginning of events the conceptual stores considered reach their limit of applicability conversely one of the fields yielding more discharge than the others but having larger data gaps allows for greater flexibility in the choice of model structures as a model learning exercise the study points to a leaking of the fields not evident from previous field experiments it is discussed how understanding observational uncertainties and incorporating these into model diagnostics can help appreciate the scale of model structural error

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

a shuffled complex evolution metropolis algorithm for optimization and uncertainty assessment of hydrologic model parameters author s vrugt ja gupta hv bouten w sorooshian s abstract markov chain monte carlo mcmc methods have become increasingly popular for estimating the posterior probability distribution of parameters in hydrologic models however mcmc methods require the a priori definition of a proposal or sampling distribution which determines the explorative capabilities and efficiency of the sampler and therefore the statistical properties of the markov chain and its rate of convergence in this paper we present an mcmc sampler entitled the shuffled complex evolution metropolis algorithm scem ua which is well suited to infer the posterior distribution of hydrologic model parameters the scem ua algorithm is a modified version of the original sce ua global optimization algorithm developed by duan et al 1992 the scem ua algorithm operates by merging the strengths of the metropolis algorithm controlled random search competitive evolution and complex shuffling in order to continuously update the proposal distribution and evolve the sampler to the posterior target distribution three case studies demonstrate that the adaptive capability of the scem ua algorithm significantly reduces the number of model simulations needed to infer the posterior distribution of the parameters when compared with the traditional metropolis hastings samplers

calibration of rainfall runoff models application of global optimization to the sacramento soil moisture accounting model conceptual rainfall runoff models are difficult to calibrate by means of automatic methods one major reason for this is the inability of conventional procedures to locate the globally optimal set of parameters this paper investigates the consistency with which two global optimization methods the shuffled complex evolution sce ua method developed by the authors and the multistart simplex msx method are able to find the optimal parameter set during calibration of the sacramento soil moisture accounting model sac sma of the national weather service river forecast system nwsrfs in the first phase of this study error free synthetic data are used to conduct a comparative evaluation of the algorithms under ideal conditions in 10 independent trials of each algorithm in which 13 parameters of the sac sma model were optimized simultaneously the sce ua method achieved a 100 success rate in locating the precise global optimum i e the true parameter values while the msx method failed in all trials even with more than twice the number of function evaluations in the second phase historical data from the leaf river watershed are used to conduct a comparative evaluation of the algorithms under real conditions using two different estimation criteria drms and hmle the sce ua algorithm obtained consistently lower function values and more closely grouped parameter estimates while using one third fewer function evaluations than the msx algorithm

conical plurisubharmonic measure and new cross theorems we study the boundary behavior of the conical plurisubharmonic measure as an application we establish some new extension theorems for separately holomorphic mappings defined on cross sets

a markov chain monte carlo scheme for parameter estimation and inference in conceptual rainfall runoff modeling a fully bayesian approach to parameter estimation and inference in conceptual rainfall runoff models crrms is presented computations are performed using a markov chain monte carlo mcmc method based on the metropolis hastings algorithm single site and block updating schemes are used for model parameters subject to nonnegativity restrictions as well as interval equality and order constraints diagnostics for the convergence of the markov chain and crrm assessment are also considered the mcmc approach produces samples from the joint posterior distribution of the model parameters this provides more information than single point estimates and avoids the need to use a normal approximation to the posterior as the basis for inference the methodology is demonstrated using an eight parameter conceptual rainfall runoff model and two case studies from southeastern australia the first case study considers a watershed with high runoff yield over a 12 year period the second case study considers a watershed with low yield over a 17 year period the results indicate that 1 bayesian methods provide an objective framework for model criticism and choice 2 the proposed strategies for handling constraints on model parameters are effective 3 the model parameters are sensitive to likelihood function selection 4 the conventional approach of using a power transformation and an autoregressive process to stabilize error variance and model dependence in the residuals may have limited success and 5 some care is required in the implementation of the mcmc approach and reliable results will be difficult to obtain when crrm complexity exceeds the limitations of the rainfall runoff data at hand a key finding is that the mcmc scheme presented herein provides a powerful means of identifying specific inadequacies in the structure of crrms

energy decay for the linear and semilinear wave equations in exterior domains with some localized dissipations we derive the total energy decay e t leq i 0 1 t 1 and l 2 boundedness u t 2 leq ci 0 for the solutions to the initial boundary value problem for the wave equation in an exterior domain omega u tt delta u a x u t 0 mbox in omega times 0 infty with u x 0 u 0 x u t x 0 u 1 x mbox and u partial omega 0 where i 0 u 0 h 1 u 1 2 and a x is a nonnegative function which is positive near some part of the boundary partial omega and near infinity we apply these estimates to prove the global existence of decaying solutions for semilinear wave equations with nonlinearity f u like u alpha u alpha 0 we note that no geometrical condition is imposed on the boundary partial omega

model selection on solid ground rigorous comparison of nine ways to evaluate bayesian model evidence bayesian model selection or averaging objectively ranks a number of plausible competing conceptual models based on bayes theorem it implicitly performs an optimal trade off between performance in fitting available data and minimum model complexity the procedure requires determining bayesian model evidence bme which is the likelihood of the observed data integrated over each model s parameter space the computation of this integral is highly challenging because it is as high dimensional as the number of model parameters three classes of techniques to compute bme are available each with its own challenges and limitations 1 exact and fast analytical solutions are limited by strong assumptions 2 numerical evaluation quickly becomes unfeasible for expensive models 3 approximations known as information criteria ics such as the aic bic or kic akaike bayesian or kashyap information criterion respectively yield contradicting results with regard to model ranking our study features a theory based intercomparison of these techniques we further assess their accuracy in a simplistic synthetic example where for some scenarios an exact analytical solution exists in more challenging scenarios we use a brute force monte carlo integration method as reference we continue this analysis with a real world application of hydrological model selection this is a first time benchmarking of the various methods for bme evaluation against true solutions results show that bme values from ics are often heavily biased and that the choice of approximation method substantially influences the accuracy of model ranking for reliable model selection bias free numerical methods should be preferred over ics whenever computationally feasible

toward a generalization of the topmodel concepts topographic indices of hydrological similarity preliminary studies of the application of topmodel to the 36 ha ringelbach catchment suggested that the original form of exponential transmissivity function leading to the in a tan topographic index and first order hyperbolic base flow recession curve is not appropriate to this catchment two alternative forms of topographic index and soil topographic index are developed based on parabolic and linear transmissivity functions leading to the more frequently observed second order hyperbolic and exponential recession curves respectively it is shown how these can be used in the same way as the original to relate catchment average water table depths to local water table depths so that patterns of saturation can be evaluated two companion ambroise et al this issue freer et al this issue papers show how the new parabolic index is used in the prediction of ringelbach discharges and how the limitations of the model are reflected in the estimated predictive uncertainties using the generalised likelihood uncertainty estimation glue approach

a bayesian approach for inverse modeling data assimilation and conditional simulation of spatial random fields 1 this paper addresses the inverse problem in spatially variable fields such as hydraulic conductivity in groundwater aquifers or rainfall intensity in hydrology common to all these problems is the existence of a complex pattern of spatial variability of the target variables and observations the multiple sources of data available for characterizing the fields the complex relations between the observed and target variables and the multiple scales and frequencies of the observations the method of anchored distributions mad that we propose here is a general bayesian method of inverse modeling of spatial random fields that addresses this complexity the central elements of mad are a modular classification of all relevant data and a new concept called anchors data types are classified by the way they relate to the target variable as either local or nonlocal and as either direct or indirect anchors are devices for localization of data they are used to convert nonlocal indirect data into local distributions of the target variables the target of the inversion is the derivation of the joint distribution of the anchors and structural parameters conditional to all measurements regardless of scale or frequency of measurement the structural parameters describe large scale trends of the target variable fields whereas the anchors capture local inhomogeneities following inversion the joint distribution of anchors and structural parameters is used for generating random fields of the target variable s that are conditioned on the nonlocal indirect data through their anchor representation we demonstrate mad through a detailed case study that assimilates point measurements of the conductivity with head measurements from natural gradient flow the resulting statistical distributions of the parameters are non gaussian similarly the moments of the estimates of the hydraulic head are non gaussian we provide an extended discussion of mad vis a vis other inversion methods including maximum likelihood and maximum a posteriori with an emphasis on the differences between mad and the pilot points method

improving probabilistic prediction of daily streamflow by identifying pareto optimal approaches for modeling heteroscedastic residual errors reliable and precise probabilistic prediction of daily catchment scale streamflow requires statistical characterization of residual errors of hydrological models this study focuses on approaches for representing error heteroscedasticity with respect to simulated streamflow i e the pattern of larger errors in higher streamflow predictions we evaluate 8 common residual error schemes including standard and weighted least squares the box cox transformation with fixed and calibrated power parameter and the log sinh transformation case studies include 17 perennial and 6 ephemeral catchments in australia and usa and two lumped hydrological models performance is quantified using predictive reliability precision and volumetric bias metrics we find the choice of heteroscedastic error modelling approach significantly impacts on predictive performance though no single scheme simultaneously optimizes all performance metrics the set of pareto optimal schemes reflecting performance trade offs comprises box cox schemes with of 0 2 and 0 5 and the log scheme 0 perennial catchments only these schemes significantly outperform even the average performing remaining schemes e g across ephemeral catchments median precision tightens from 105 to 40 of observed streamflow and median biases decrease from 25 to 4 theoretical interpretations of empirical results highlight the importance of capturing the skew kurtosis of raw residuals and reproducing zero flows paradoxically calibration of is often counterproductive in perennial catchments it tends to overfit low flows at the expense of abysmal precision in high flows the log sinh transformation is dominated by the simpler pareto optimal schemes listed above recommendations for researchers and practitioners seeking robust residual error schemes for practical work are provided this article is protected by copyright all rights reserved

ancient numerical daemons of conceptual hydrological modeling 2 impact of time stepping schemes on model analysis and prediction 1 despite the widespread use of conceptual hydrological models in environmental research and operations they remain frequently implemented using numerically unreliable methods this paper considers the impact of the time stepping scheme on model analysis sensitivity analysis parameter optimization and markov chain monte carlo based uncertainty estimation and prediction it builds on the companion paper clark and kavetski 2010 which focused on numerical accuracy fidelity and computational efficiency empirical and theoretical analysis of eight distinct time stepping schemes for six different hydrological models in 13 diverse basins demonstrates several critical conclusions 1 unreliable time stepping schemes in particular fixed step explicit methods suffer from troublesome numerical artifacts that severely deform the objective function of the model these deformations are not rare isolated instances but can arise in any model structure in any catchment and under common hydroclimatic conditions 2 sensitivity analysis can be severely contaminated by numerical errors often to the extent that it becomes dominated by the sensitivity of truncation errors rather than the model equations 3 robust time stepping schemes generally produce better behaved objective functions free of spurious local optima and with sufficient numerical continuity to permit parameter optimization using efficient quasi newton methods when implemented within a multistart framework modern newton type optimizers are robust even when started far from the optima and provide valuable diagnostic insights not directly available from evolutionary global optimizers 4 unreliable time stepping schemes lead to inconsistent and biased inferences of the model parameters and internal states 5 even when interactions between hydrological parameters and numerical errors provide the right result for the wrong reason and the calibrated model performance appears adequate unreliable time stepping schemes make the model unnecessarily fragile in predictive mode undermining validation assessments and operational use erroneous or misleading conclusions of model analysis and prediction arising from numerical artifacts in hydrological models are intolerable especially given that robust numerics are accepted as mainstream in other areas of science and engineering we hope that the vivid empirical findings will encourage the conceptual hydrological community to close its pandora s box of numerical problems paving the way for more meaningful model application and interpretation

bayesian analysis for uncertainty estimation of a canopy transpiration model 1 a bayesian approach was used to fit a conceptual transpiration model to half hourly transpiration rates for a sugar maple acer saccharum stand collected over a 5 month period and probabilistically estimate its parameter and prediction uncertainties the model used the penman monteith equation with the jarvis model for canopy conductance this deterministic model was extended by adding a normally distributed error term this extension enabled using markov chain monte carlo simulations to sample the posterior parameter distributions the residuals revealed approximate conformance to the assumption of normally distributed errors however minor systematic structures in the residuals at fine timescales suggested model changes that would potentially improve the modeling of transpiration results also indicated considerable uncertainties in the parameter and transpiration estimates this simple methodology of uncertainty analysis would facilitate the deductive step during the development cycle of deterministic conceptual models by accounting for these uncertainties while drawing inferences from data

extending the applicability of the generalized likelihood function for zero inflated data series proper uncertainty estimation for data series with a high proportion of zero and near zero observations has been a challenge in hydrologic studies this technical note proposes a modification to the generalized likelihood function that accounts for zero inflation of the error distribution zi gl we compare the performance of the proposed zi gl with the original generalized likelihood function using the entire data series gl and by simply suppressing zero observations gly 0 these approaches were applied to two interception modeling examples characterized by data series with a significant number of zeros the zi gl produced better uncertainty ranges than the gl as measured by the precision reliability and volumetric bias metrics the comparison between zi gl and gly 0 highlights the need for further improvement in the treatment of residuals from near zero simulations when a linear heteroscedastic error model is considered aside from the interception modeling examples illustrated herein the proposed zi gl may be useful for other hydrologic studies such as for the modeling of the runoff generation in hillslopes and ephemeral catchments

two dimensional exterior mixed problem for semilinear damped wave equations abstract we consider two dimensional exterior mixed problems for a semilinear damped wave equation with a power type nonlinearity u p for compactly supported initial data which have a small energy we shall derive global in time existence results in the case when the power of the nonlinearity satisfies 2 p this generalizes a previous result of j differential equations 200 2004 53 68 which dealt with a radially symmetric solution

time varying sensitivity analysis clarifies the effects of watershed model formulation on model behavior 1 lumped rainfall runoff models are widely used for flow prediction but a long recognized need exists for diagnostic tools to determine whether the process level behavior of a model aligns with the expectations inherent in its formulation to this end we develop a comprehensive exploration of dominant parameters in the hymod hbv and sacramento soil moisture accounting sac sma model structures model controls are isolated using time varying sobol sensitivity analysis for twelve mopex watersheds in the eastern united states over a 10 year period sensitivity indices are visualized along gradients of observed precipitation and streamflow to identify key behavioral differences between the three models and to connect these back to the models underlying assumptions results indicate that the models dominant parameters strongly depend on time varying hydroclimatic conditions parameters associated with surface processes such as evapotranspiration and runoff generally dominate under dry conditions when high evaporative fluxes are required for accurate simulation parameters associated with routing processes typically dominate under high flow conditions when performance depends on the timing of flow events the results highlight significant inter model differences in performance controls even in cases where the models share similar process formulations the dominant parameters identified can be counterintuitive even these simple models represent complex nonlinear systems and the links between formulation and behavior are difficult to discern a priori as complexity increases scrutinizing the links between model formulation and behavior becomes an important diagnostic approach particularly in applications such as predictions under change where dominant model controls will shift under hydrologic extremes

a generalized grid connectivity based parameterization for subsurface flow model calibration 1 we develop a novel method of parameterization for spatial hydraulic property characterization to mitigate the challenges associated with the nonlinear inverse problem of subsurface flow model calibration the parameterization is performed by the projection of the estimable hydraulic property field onto an orthonormal basis derived from the grid connectivity structure the basis functions represent the modal shapes or harmonics of the grid are defined by a modal frequency and converge to special cases of the discrete fourier series under certain grid geometries and boundary assumptions therefore hydraulic property updates are performed in the spectral domain and merge with fourier analysis in ideal cases dependence on the grid alone implies that the basis may characterize any grid geometry including corner point and unstructured is model independent and is constructed off line and only once prior to flow data assimilation we apply the parameterization in an adaptive multiscale model calibration workflow for three subsurface flow models several different grid geometries are considered in each case the prior hydraulic property model is updated using a parameterized multiplier field that is superimposed onto the grid and assigned an initial value of unity at each cell the special case corresponding to a constant multiplier is always applied through the constant basis function higher modes are adaptively employed during minimization of data misfit to resolve multiscale heterogeneity in the geomodel the parameterization demonstrates selective updating of heterogeneity at locations and spatial scales sensitive to the available data otherwise leaving the prior model unchanged as desired

a maximum likelihood criterion for use with data collected at unequal time intervals a new maximum likelihood nmle criterion suitable for model calibration using data which are recorded at unequal time intervals and contain autocorrelated errors is derived validation of the nmle criterion has been carried out both on a simple two parameter reservoir model using synthetically generated data and on a more complicated hillslope model using real data from the pukeiti catchment in new zealand comparison between the nmle criterion and the simple least squares criterion reveals the superiority of the former over the latter comparison between the nmle criterion and the maximum likelihood criterion for the autocorrelated case proposed by sorooshian 1978 has shown that both criteria would yield results with no practical difference if equal time interval data were used however the nmle criterion can work on variable time interval data which provide more information than equal time interval data and therefore produce better visual results in hydrologic simulations

review of parameter identification procedures in groundwater hydrology the inverse problem the purpose of this survey is to review parameter identification procedures in groundwater hydrology and to examine computational techniques which have been developed to solve the inverse problem parameter identification methods are classified under the error criterion used in the formulation of the inverse problem the problem of ill posedness in connection with the inverse problem is addressed typical inverse solution techniques are highlighted the review also includes the evaluation of methods used for computing the sensitivity matrix statistics which can be used to estimate the parameter uncertainty are outlined attempts have been made to compare and contrast representative inverse procedures and direction for future research is suggested

subsets of full measure in a generic submanifold in mathbb c n are non plurithin in this paper we prove that if i subset m is a subset of measure 0 in a c 2 smooth generic submanifold m subset mathbb c n then m setminus i is non plurithin at each point of m in mathbb c n this result improves a previous result of a edigarian and j wiegerinck who considered the case where i is pluripolar set contained in a c 1 smooth generic submanifold m subset mathbb c n edigarian and wiegernick in math z 266 2 393 398 2010 the proof of our result is essentially different

estimating the uncertain mathematical structure of a water balance model via bayesian data assimilation 1 when constructing a hydrological model at the macroscale e g watershed scale the structure of this model will inherently be uncertain because of many factors including the lack of a robust hydrological theory at that scale in this work we assume that a suitable conceptual model structure for the hydrologic system has already been determined that is the system boundaries have been specified the important state variables and input and output fluxes to be included have been selected the major hydrological processes and geometries of their interconnections have been identified and the continuity equation mass balance has been assumed to hold the remaining structural identification problem that remains then is to select the mathematical form of the dependence of the output on the inputs and state variables so that a computational model can be constructed for making simulations and or predictions of the system input state output behavior the conventional approach to this problem is to preassume some fixed and possibly erroneous mathematical forms for the model output equations we show instead how bayesian data assimilation can be used to directly estimate construct the form of these mathematical relationships such that they are statistically consistent with macroscale measurements of the system inputs outputs and if available state variables the resulting model has a stochastic rather than deterministic form and thereby properly represents both what we know our certainty and what we do not know our uncertainty about the underlying structure and behavior of the system further the bayesian approach enables us to merge prior beliefs in the form of preassumed model equations with information derived from the data to construct a posterior model as a consequence in regions of the model space for which observational data are available the errors in preassumed mathematical form of the model can be corrected improving model performance for regions where no such data are available the prior theoretical assumptions about the model structure and behavior will dominate the approach entitled bayesian estimation of structure is used to estimate water balance models for the leaf river basin mississippi at annual monthly and weekly time scales conditioned on the assumption of a simple single state variable conceptual model structure inputs to the system are uncertain observed precipitation and potential evapotranspiration and outputs are estimated probability distributions of actual evapotranspiration and streamflow discharge results show that the models estimated for the annual and monthly time scales perform quite well however model performance deteriorates for the weekly time scale suggesting limitations in the assumed form of the conceptual model

bayesian inference of uncertainties in precipitation streamflow modeling in a snow affected catchment 1 bayesian inference is used to study the effect of precipitation and model structural uncertainty on estimates of model parameters and confidence limits of predictive variables in a conceptual rainfall runoff model in the snow fed rudback catchment 142 ha in southern finland the ihacres model is coupled with a simple degree day model to account for snow accumulation and melt the posterior probability distribution of the model parameters is sampled by using the differential evolution adaptive metropolis dream zs algorithm and the generalized likelihood function precipitation uncertainty is taken into account by introducing additional latent variables that were used as multipliers for individual storm events results suggest that occasional snow water equivalent swe observations together with daily streamflow observations do not contain enough information to simultaneously identify model parameters precipitation uncertainty and model structural uncertainty in the rudback catchment the addition of an autoregressive component to account for model structure error and latent variables having uniform priors to account for input uncertainty lead to dubious posterior distributions of model parameters thus our hypothesis that informative priors for latent variables could be replaced by additional swe data could not be confirmed the model was found to work adequately in 1 day ahead simulation mode but the results were poor in the simulation batch mode this was caused by the interaction of parameters that were used to describe different sources of uncertainty the findings may have lessons for other cases where parameterizations are similarly high in relation to available prior information

consistent detection of a monotonic trend superposed on a stationary time series summary consider a time series made up of a signal and a stationary autocorrelated error series a statistic is proposed for examining the hypothesis that the signal term is constant versus the hypothesis that it is monotonic in time the statistic is the ratio of a linear combination of the time series values with coefficients introduced by abelson tukey 1963 to an estimate of the standard error of the linear combination the statistic has asymptotic power 1 for a broad class of monotonic alternatives the procedure is illustrated for the series of river heights at a location on the rio negro in brazil where there is concern that the height is rising due to deforestation of the amazon basin the significance level obtained is 0025

quantitative comparison of canopy conductance models using a bayesian approach 1 a quantitative model comparison methodology based on deviance information criterion a bayesian measure of the trade off between model complexity and goodness of fit is developed and demonstrated by comparing semiempirical transpiration models this methodology accounts for parameter and prediction uncertainties associated with such models and facilitates objective selection of the simplest model out of available alternatives which does not significantly compromise the ability to accurately model observations we use this methodology to compare various jarvis canopy conductance model configurations embedded within a larger transpiration model against canopy transpiration measured by sap flux the results indicate that descriptions of the dependence of stomatal conductance on vapor pressure deficit photosynthetic radiation and temperature as well as the gradual variation in canopy conductance through the season are essential in the transpiration model use of soil moisture was moderately significant but only when used with a hyperbolic vapor pressure deficit relationship subtle differences in model quality could be clearly associated with small structural changes through the use of this methodology the results also indicate that increments in model complexity are not always accompanied by improvements in model quality and that such improvements are conditional on model structure possible application of this methodology to compare complex semiempirical models of natural systems in general is also discussed

lp lq estimates of solutions to the damped wave equation in 3 dimensional space and their application it has been asserted that the damped wave equation has the diffusive structure as t in this paper we consider the cauchy problem in 3 dimensional space for the linear damped wave equation and the corresponding parabolic equation and obtain the lp lq estimates of the difference of each solution which represent the assertion precisely explicit formulas of the solutions are analyzed for the proof the second aim is to apply the lp lq estimates to the semilinear damped wave equation with power nonlinearity if the power is larger than the fujita exponent then the time global existence of small weak solution is proved and its optimal decay order is obtained

estimating epistemic and aleatory uncertainties during hydrologic modeling an information theoretic approach 1 with growing interest in understanding the magnitudes and sources of uncertainty in hydrological modeling the difficult problem of characterizing model structure adequacy is now attracting considerable attention here we examine this problem via a model structureindependent approach based in information theory in particular we a discuss how to assess and compute the information content in multivariate hydrological data b present practical methods for quantifying the uncertainty and shared information in data while accounting for heteroscedasticity c show how these tools can be used to estimate the best achievable predictive performance of a model for a system given the available data and d show how model adequacy can be characterized in terms of the magnitude and nature of its aleatory uncertainty that cannot be diminished and is resolvable only up to specification of its density and its epistemic uncertainty that can in principle be suitably resolved by improving the model an illustrative modeling example is provided using catchment scale data from three river basins the leaf and chunky river basins in the united states and the chuzhou basin in china our analysis shows that the aleatory uncertainty associated with making catchment simulations using this data set is significant 50 further estimated epistemic uncertainties of the hymod sac sma and xinanjiang model hypotheses indicate that considerable room for model structural improvements remain

uncertainty assessment of hydrological models with fuzzy extension principle evaluation of a new arithmetic operator the most important drawback of standard fuzzy arithmetic is unrealistic accumulation of input uncertainties which results in divergence of fuzzy outputs some of the currently available methods of simulating fuzzy systems provide the results which tend to naive large values after several time steps of the system simulation in this paper a new fuzzy arithmetic operator based on fuzzy extension principle has been proposed for simulation and assessment of uncertainty in hydrological systems implementing the concept of fuzzy approximate reasoning in the proposed approach in this study represents acceptable behavior in uncertainty propagation from the parameters and structures of the models to the outputs to show the efficiency of the proposed fuzzy arithmetic operator in the context of hydrologic modeling two nonlinear monthly water balance models have been examined and their outputs have been compared with the results obtained by standard fuzzy arithmetic and the vertex method one small humid basin in france and a middle size basin in a semiarid region in iran have been the case studies of this research in this paper the lower and upper bounds and the most frequent values of the model parameters inferred from the sampling simulation procedure have been used to define triangular fuzzy membership functions three statistical indicators have been used to evaluate efficiency of the methods based on the bracket observations and coverage of the uncertainty bounds the estimated values of these indicators have shown that both vertex and the proposed methods outperform standard fuzzy arithmetic also the proposed method has provided better or roughly equal efficiencies compared with vertex method over both basins

a formal likelihood function for parameter and predictive inference of hydrologic models with correlated heteroscedastic and non gaussian errors estimation of parameter and predictive uncertainty of hydrologic models has traditionally relied on several simplifying assumptions residual errors are often assumed to be independent and to be adequately described by a gaussian probability distribution with a mean of zero and a constant variance here we investigate to what extent estimates of parameter and predictive uncertainty are affected when these assumptions are relaxed a formal generalized likelihood function is presented which extends the applicability of previously used likelihood functions to situations where residual errors are correlated heteroscedastic and non gaussian with varying degrees of kurtosis and skewness the approach focuses on a correct statistical description of the data and the total model residuals without separating out various error sources application to bayesian uncertainty analysis of a conceptual rainfall runoff model simultaneously identifies the hydrologic model parameters and the appropriate statistical distribution of the residual errors when applied to daily rainfall runoff data from a humid basin we find that 1 residual errors are much better described by a heteroscedastic first order auto correlated error model with a laplacian distribution function characterized by heavier tails than a gaussian distribution and 2 compared to a standard least squares approach proper representation of the statistical distribution of residual errors yields tighter predictive uncertainty bands and different parameter uncertainty estimates that are less sensitive to the particular time period used for inference application to daily rainfall runoff data from a semiarid basin with more significant residual errors and systematic underprediction of peak flows shows that 1 multiplicative bias factors can be used to compensate for some of the largest errors and 2 a skewed error distribution yields improved estimates of predictive uncertainty in this semiarid basin with near zero flows we conclude that the presented methodology provides improved estimates of parameter and total prediction uncertainty and should be useful for handling complex residual errors in other hydrologic regression models as well

effect of different uncertainty sources on the skill of 10 day ensemble low flow forecasts for two hydrological models the aim of this paper is to investigate the effect of uncertainty originating from model inputs parameters and initial conditions on 10 day ensemble low flow forecasts two hydrological models namely gr4j and hbv are applied to the moselle river and performance in the calibration validation and forecast periods and the effect of different uncertainty sources on the quality of low flow forecasts are compared the forecasts are generated by using ecmwf meteorological ensemble forecasts as input to the gr4j and hbv models the ensembles each consisting of 51 members provided the uncertainty range for the model inputs the generalised likelihood uncertainty estimation glue approach is used to estimate parameter uncertainty part of the glue behavioural parameter set containing parameters directly linked to storage estimates and the observed discharge on the forecast issue day are used to update the model storages and reflect the uncertainty from the initial conditions the quality of the probabilistic low flow forecasts has been assessed by the relative confidence interval reliability and hit false alarm rates the daily observed low flows are captured by the 90 confidence interval for both models most of the time however the gr4j model usually overestimates low flows whereas hbv is prone to underestimate low flows this is particularly the case if the parameter uncertainty is included in the forecasts the total uncertainty in the gr4j model outputs is higher than in the hbv model outputs the forecasts issued by the hbv model incorporating input uncertainty resulted in the most reliable forecast distribution the number of hits is about equal for both models if only the input uncertainty is considered the parameter uncertainty was the main reason reducing the number of hits the number of false alarms in gr4j model is twice the number of false alarms in hbv when considering all uncertainty sources the results of this study in general showed the parameter uncertainty has the largest effect whereas the input uncertainty had the smallest effect on the medium range low flow forecasts

stochastic subsurface hydrology from theory to applications research on stochastic analysis of subsurface flow has developed rapidly in the last decade but applications of this approach have been very limited the purpose of this paper is to illustrate how currently available techniques and results can be used to answer important questions about the large scale behavior of naturally heterogeneous aquifers perturbation based spectral theory which presumes local statistical homogeneity provides generic theoretical results for the head variance effective conductivity tensor and macrodispersivity tensor in a field these results emphasize the key role of the variance and spatial correlation scales of the log hydraulic conductivity field field information of variances and correlation scales of natural materials is summarized the validity of some of the generic stochastic results is evaluated through comparisons with monte carlo simulations and field observations a specific field application example is developed to illustrate how the stochastic results are used to estimate large scale parameters and determine the reliability of three dimensional numerical simulations using typical log conductivity covariance parameters the effective hydraulic conductivity tensor and the macrodispersivity tensor are estimated the calculated head variance based on the simulated mean hydraulic gradient is used as a measure of the adequacy of the calculation of the steady state flow model discussion emphasizes limitations and extensions of this approach and ongoing field evaluations of the results