estimation of aquifer parameters under transient and steady state conditions 2 uniqueness stability and solution algorithms paper 2 of this three part series starts with a discussion of the question under what conditions is the aquifer inverse problem well posed after defining the terms uniqueness identifiability and stability theoretical considerations and synthetic examples are used to demonstrate that ill posedness can be mitigated by including prior information about the parameters in the estimation criterion to be minimized at the same time the inclusion of such information is shown to be insufficient to guarantee uniqueness and stability in all cases several test problems in the recent literature which have resulted in pessimistic conclusions about the solvability of the aquifer inverse problem are shown to be ill posed a question is thus raised about the validity of these conclusions in the general case various conjugate gradient algorithms coupled with the adjoint state finite element method for computing the gradient of the estimation criterion and with newton s method for determining the optimum step size downgradient are compared a marked improvement in the rate at which these algorithms converge is shown to be achieved by switching from one method to another when the former slows down or fails to converge

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

on model selection criteria in multimodel analysis 1 hydrologic systems are open and complex rendering them prone to multiple conceptualizations and mathematical descriptions there has been a growing tendency to postulate several alternative hydrologic models for a site and use model selection criteria to 1 rank these models 2 eliminate some of them and or 3 weigh and average predictions and statistics generated by multiple models this has led to some debate among hydrogeologists about the merits and demerits of common model selection also known as model discrimination or information criteria such as aic aicc bic and kic and some lack of clarity about the proper interpretation and mathematical representation of each criterion we examine the model selection literature to find that 1 all published rigorous derivations of aic and aicc require that the true model having generated the observational data be in the set of candidate models 2 though bic and kic were originally derived by assuming that such a model is in the set bic has been rederived by cavanaugh and neath 1999 without the need for such an assumption and 3 kic reduces to bic as the number of observations becomes large relative to the number of adjustable model parameters implying that it likewise does not require the existence of a true model in the set of alternatives we explain why kic is the only criterion accounting validly for the likelihood of prior parameter estimates elucidate the unique role that the fisher information matrix plays in kic and demonstrate through an example that it imbues kic with desirable model selection properties not shared by aic aicc or bic our example appears to provide the first comprehensive test of how aic aicc bic and kic weigh and rank alternative models in light of the models predictive performance under cross validation with real hydrologic data

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

estimation of spatial covariance structures by adjoint state maximum likelihood cross validation 1 theory this series of three papers describes a cross validation method to estimate the spatial covariance structure of intrinsic or nonintrinsic random functions from point or spatially averaged data that may be corrupted by noise any number of relevant parameters including nugget effect can be estimated the theory described in this paper is based on a maximum likelihood approach which treats the cross validation errors as gaussian various a posteriori statistical tests are used to verify this hypothesis and to show that in many cases correlation between these errors is weak the log likelihood criterion is optimized through a combination of conjugate gradient algorithms an adjoint state theory is used to efficiently calculate the gradient of the estimation criterion optimize the step size downgradient and compute a lower bound for the covariance matrix of the estimation errors issues related to the identifiability stability and uniqueness of the resulting adjoint state maximum likelihood cross validation asmlcv method are discussed this paper also describes the manner in which asmlcv allows one to use model structure identification criteria to select the best covariance model among a given set of alternatives practical aspects of asmlcv and its application to synthetic data are presented in paper 2 samper and neuman this issue a applications to real hydrogeological data transmissivities and groundwater levels have been presented elsewhere while hydrochemical and isotopic data are analyzed by asmlcv in paper 3 samper and neuman this issue b

estimation of root water uptake parameters by inverse modeling with soil water content data 1 in this paper we have tested the feasibility of the inverse modeling approach to derive root water uptake parameters rwup from soil water content data using numerical experiments for three differently textured soils and for an optimal drying period the rwup of interest are the rooting depth and the bottom root length density in a first step a thorough sensitivity analysis was performed this showed that soil water content dynamics is relatively insensitive to rwup and that the sensitivity depends on the texture of the considered soil for medium fine textured soil the sensitivity is particularly low due to relatively high unsaturated hydraulic conductivity values these ones allow a compensating effect to occur i e vertical unsaturated water fluxes overshadowing in some way the root water uptake in a second step we analyzed the well posedness of the solution stability and nonuniqueness when only rwup are optimized for this case the inverse problem is clearly ill posed except for the estimation of the rooting depth parameter for coarse and the very fine textured soils in a third step we addressed the case where rwup are estimated simultaneously with additional parameters of the system i e with soil hydraulic parameters for this case our study showed that the inverse problem is well posed for the coarse and very fine textured soils allowing for the estimation of both rwup of interest provided that a powerful global optimization algorithm is used on the contrary the estimation of rwup is unfeasible for medium fine textured soil due to the compensating effect of the vertical unsaturated water flows in conclusion we can state that the inverse modeling approach can be applied to derive rwup for some soils coarse and very fine textured and that the feasibility is strongly improved if the rwup are simultaneously optimized with additional parameters nevertheless more detailed research is needed to apply the inverse modeling approach to real cases for which additional issues are likely to be encountered such as soil heterogeneity and root dynamics

comparisons for measure valued processes with interactions this paper considers some measure valued processes xt dvtxt 0 t based on an underlying critical branching particle structure with random branching rates in the case of constant branching these processes are dawson watanabe processes sufficient conditions on functionals of the process are given that imply that the expectations e xt are comparable to the constant branching case applications to hitting estimates and regularity of solutions are discussed the result is established via the martingale optimality principle of stochastic control theory key steps which are of independent interest are the proof of a version of ito s lemma for xt suitable for a large class of functions of measures theorem 3 and the proof of various smoothing properties of the dawson watanabe transition semigroup section 3

necessary and sufficient conditions for balancing unstable systems necessary and sufficient conditions are given for the existence of balancing transformations for minimal state space realizations a b c where a may be unstable these conditions are expressed in terms of the real diagonalizability of the product of the reachability gramian and the observability gramian for symmetric realizations these conditions can be reformulated in terms of the real diagonalizability of the cross gramian and we show that minimal symmetric systems can be internally balanced if the associated hankel matrix is positive semidefinite examples are given of minimal systems including symmetric systems which cannot be balanced

uniqueness and observability of conceptual rainfall runoff model parameters the percolation process examined many researchers have expressed concerns regarding the uniqueness of parameter estimates for conceptual rainfall runoff r r models obtained through calibration recent studies sorooshian et al this issue sorooshian and gupta this issue have revealed that even though stochastic parameter estimation techniques can help the problems are not all due to inefficiencies in the calibration techniques used but are caused by the manner in which the model is structurally formulated thus even when calibrated under ideal conditions simulation studies it is often impossible to obtain unique estimates for the parameters it is possible to resolve this problem at least in part by appropriate reparameterizations of the pertinent model equations in this paper the percolation equation of the soil moisture accounting model of the national weather service river forecast system sma nwsrfs will be discussed it is shown that a logical reparameterization of this equation can result in conditions that improve the chances of obtaining unique parameter estimates it is believed that these results have implications for other conceptual r r models in which similar approaches are used in the representation of the percolation infiltration process

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

a multisignal automatic calibration methodology for hydrochemical models a case study of the birkenes model most hydrochemical models employ immeasurable or artificial parameters that need to be estimated from calibration data a methodology is presented here for the automatic calibration of artificial parameters which improves parameter identifiability through increasing the information available for determining the parameter values instead of calibrating the model using only one signal e g the hydrograph multiple signals e g chemical signals as well as the hydrograph are considered simultaneously either a simple least squares or a weighted least squares objective function may be used the methodology is applied to the hydrologic module of the birkenes hydrochemical model using artificial data from a wide range of starting points a gradient search optimization technique is able to consistently locate the correct parameter values uniformly better when using two signals the hydrograph and a conservative tracer than when only one signal the hydrograph is used

improving the shuffled complex evolution scheme for optimization of complex nonlinear hydrological systems application to the calibration of the sacramento soil moisture accounting model 1 an innovative algorithm shuffled complexes with principal components analysis sp uci is developed to overcome a critical deficiency of the shuffled complex evolution scheme population degeneration population degeneration means that during the evolutionary search process the population of search particles may degenerate into a subspace of the full parameter space thereby missing the capacity of fully exploring the parameter space being confined in a subspace may even lead the particle population to converge to nonstationary points which is a fatal malfunction to overcome this problem sp uci employs the principal components analysis to detect the occurrence of population degeneration and remedy the adverse effects the ensemble of calibrations of the sacramento soil moisture accounting model with the sp uci method over the leaf river basin mississippi retrieves the optimal parameter values with the lowest recorded root mean squared error of simulated daily runoff against the observation moreover the result also provides consistent narrow ranges model parameter distribution which results in a better understanding of the model s behavior given the watershed s hydrologic features

adaptive calibration of a conceptual model for flash flood forecasting the adaptive use of a conceptual model for real time flow forecasting is investigated maximum likelihood and ordinary least squares estimation criteria are considered and the performance of maximum likelihood techniques for autocorrelated amle and heteroscedastic hmle errors is analyzed jointly with that provided by the commonly used ordinary least squares estimation olse technique streamflow forecasts are compared for three rivers in central italy obtained by amle hmle and olse adaptive calibration of a simple conceptual model describing the rainfall runoff transformation by accounting for hortonian infiltration and linear basin response to rainfall excess although model residuals display both autocorrelation and heteroscedasticity olse is found to provide a rather satisfactory performance because the olse technique also requires less computational effort compared to that for amle and hmle one could consider olse as a suitable option for real time model operation

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

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

water quality modeling a review of the analysis of uncertainty this paper reviews the role of uncertainty in the identification of mathematical models of water quality and in the application of these models to problems of prediction more specifically four problem areas are examined in detail uncertainty about model structure uncertainty in the estimated model parameter values the propagation of prediction errors and the design of experiments in order to reduce the critical uncertainties associated with a model the review is rather lengthy and it has therefore been prepared in effect as two papers there is a shorter largely nontechnical version which gives a quick impression of the current and future issues in the analysis of uncertainty in water quality modeling enclosed by this shorter discussion is the main body of the review dealing in turn with 1 identifiability and experimental design 2 the generation of preliminary model hypotheses under conditions of sparse grossly uncertain field data 3 the selection and evaluation of model structure 4 parameter estimation model calibration 5 checks and balances on the identified model i e model verification and model discrimination and 6 prediction error propagation much time is spent in discussing the algorithms of system identification in particular the methods of recursive estimation and in relating these algorithms and the subject of identification to the problems of prediction uncertainty and first order error analysis there are two obvious omissions from the review it is not concerned primarily with either the development and solution of stochastic differential equations or the issue of decision making under uncertainty although clearly some reference must be made to these topics in brief the review concludes not surprisingly that much work has been done on the analysis of uncertainty in the development of mathematical models of water quality and much remains to be done a lack of model identifiability has been an outstanding difficulty in the interpretation and explanation of past observed system behavior and there is ample evidence to show that the larger more comprehensive models are easily capable of generating highly uncertain predictions of future behavior for the future of the subject it is speculated that there is the possibility of progress in the development of novel algorithms for model structure identification a need for new questions to be posed in the problem of prediction and a distinct challenge to the conventional views of this review in the new forms of knowledge representation and manipulation now emerging from the field of artificial intelligence

a nonparametric wet dry spell model for resampling daily precipitation a nonparametric wet dry spell model is developed for resampling daily precipitation at a site the model considers alternating sequences of wet and dry days in a given season of the year all marginal joint and conditional probability densities of interest e g dry spell length wet spell length precipitation amount and wet spell length given prior to dry spell length are estimated nonparametrically using at site data and kernel probability density estimators procedures for the disaggregation of wet spell precipitation into daily precipitation and for the generation of synthetic sequences are proffered an application of the model for generating synthetic precipitation traces at a site in utah is presented

comparison of interacting diffusions and an application to their ergodic theory a general comparison argument for expectations of certain multitime functionals of infinite systems of linearly interacting diffusions differing in the diffusion coefficient is derived as an application we prove clustering occurs in the case when the symmetrized interaction kernel is recurrent and the components take values in an interval bounded on one side the technique also gives an alternative proof of clustering in the case of compact intervals

super brownian motion path properties and hitting probabilities sample path properties of super brownian motion including a one sided modulus of continuity and exact hausdorff measure function of the range and closed support are obtained analytic estimates for the probability of hitting balls lead to upper bounds on the hausdorff measure of the set of k multiple points and a sufficient condition for a set to be polar

comparison of newton type and direct search algorithms for calibration of conceptual rainfall runoff models an examination of the calibration aspect of conceptual rainfall runoff models was undertaken using the sacramento soil moisture accounting model and a study comparing the performance of a newton type optimization algorithm with that of a direct search algorithm results indicate that the direct search algorithm is the more robust of the two because the newton type algorithm is more susceptible to poor conditioning of the response surface graphical studies of the response surface of the model s parameter space confirmed the presence of discontinuities and a rough textured surface particularly in the derivatives

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

quantile hydrologic model selection and model structure deficiency assessment 1 theory a theory for quantile based hydrologic model selection and model structure deficiency assessment is presented the paper demonstrates that the degree to which a model selection problem is constrained by the model structure measured by the lagrange multipliers of the constraints quantifies structural deficiency this leads to a formal definition of model structure deficiency or rigidity model structure deficiency introduces a bias in the prediction of an observed quantile which is often not equal across quantiles structure deficiency is therefore diagnosed when any two quantile predictions for a given model structure cross since unequal bias across quantiles result in quantile predictions crossing the analysis further suggests that the optimal value of quantile specific loss functions order different model structures by its structure deficiencies over a range of quantiles in addition to such novelties quantile hydrologic model selection is a frequentist approach that seeks to complement existing bayesian approaches to hydrological model uncertainty

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

parameter space methods in joint parameter estimation for groundwater flow models the joint estimation of model parameters using hydraulic head and tracer concentration data is examined using methods based on parameter space analysis response surfaces and confidence regions can indicate how multiple data sets interact to reduce parameter uncertainty each data set produces a unique response surface and confidence region if these surfaces are oriented differently for the two data sets parameter uncertainties are significantly reduced when the second data set is included in the inversion if these surfaces are similar for the two data sets the second data set will not reduce the parameter uncertainty significantly the axes of the linearized confidence ellipsoid are analyzed to determine the difference in orientation of the confidence regions for the two data sets the use of confidence regions can be extended to predict the value of a second data set in reducing the uncertainty of parameter estimates before the data are collected parameter space approaches are also introduced for selecting the relative weights for the individual data sets in joint parameter estimation the conventional method based on the analysis of data residuals is compared to three other methods with the weights selected to maximize parameter stability to minimize the volume of the confidence region or to minimize the longest axis of the confidence region each criterion can lead to substantially different weights applied to each of the data sets the application of these methods is demonstrated using hydraulic head measurements and 14c concentrations to calibrate a model of groundwater flow in the san juan basin new mexico

on hydrological model complexity its geometrical interpretations and prediction uncertainty knowledge of hydrological model complexity can aid selection of an optimal prediction model out of a set of available models optimal model selection is formalized as selection of the least complex model out of a subset of models that have lower empirical risk this may be considered equivalent to minimizing an upper bound on prediction error defined here as the mathematical expectation of empirical risk in this paper we derive an upper bound that is free from assumptions on data and underlying process distribution as well as on independence of model predictions over time we demonstrate that hydrological model complexity as defined in the presented theoretical framework plays an important role in determining the upper bound the model complexity also acts as a stabilizer to a hydrological model selection problem if it is deemed ill posed we provide an algorithm for computing complexity of any arbitrary hydrological model we also demonstrate that hydrological model complexity has a geometric interpretation as the size of model output space the presented theory is applied to quantify complexities of two hydrological model structures sac sma and sixpar it detects that sac sma is indeed more complex than sixpar we also develop an algorithm to estimate the upper bound on prediction error which is applied on five different rainfall runoff model structures that vary in complexity we show that a model selection problem is stabilized by regularizing it with model complexity complexity regularized model selection yields models that are robust in predicting future but yet unseen data

bayesian inversion of mualem van genuchten parameters in a multilayer soil profile a data driven assumption free likelihood function this paper introduces a hierarchical simulation and modeling framework that allows for inference and validation of the likelihood function in bayesian inversion of vadose zone hydraulic properties the likelihood function or its analogs objective functions and likelihood measures are commonly assumed to be multivariate gaussian in form however this assumption is not possible to verify without a hierarchical simulation and modeling framework in this paper we present the necessary statistical mechanisms for utilizing the hierarchical framework we apply the hierarchical framework to the inversion of the vadose zone hydraulic properties within a multilayer soil profile conditioned on moisture content observations collected in the uppermost four layers the key result of our work is that the goodness of fit validated likelihood function form provides empirical justification for the assumption of multivariate gaussian likelihood functions in past and future inversions at similar sites as an alternative the likelihood function need not be assumed to follow a parametric statistical distribution and can be computed directly using nonparametric methods the nonparametric methods are considerably more computationally demanding and to demonstrate this approach we present a smaller dimension synthetic case study of evaporation from a soil column the main drawback of our work is the increased computational expense of the inversion

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

improved parameter inference in catchment models 2 combining different kinds of hydrologic data and testing their compatibility often some of the parameters of catchment models fitted to runoff data are poorly determined thereby making the task of developing useful regionalization relationships more difficult the bayesian methodology developed in part 1 is extended to utilize several kinds of hydrologic data in parameter inference the goal being to improve the precision of poorly determined parameters the concept of compatibility is developed using statistical hypothesis tests different kinds of data are said to be compatible if differences between their fitted parameters are not statistically significant the pooling of incompatible data may undermine the model s ability to predict runoff and also induce bias in the parameters a hierarchy of three levels of information is introduced to enable systematic checking for compatibility finally a case study is presented using data on runoff soil moisture and interception it is shown that substantial reductions in parameter uncertainty can be realized also the importance of compatibility testing is demonstrated

stochastic identification of transmissivity and effective recharge in steady groundwater flow 2 case study the study is a continuation and extension of a previous work dagan 1985a whose aim was to identify the values of the log transmissivity y for steady flow the common basic assumptions are that y is a normal and stationary random space function the aquifer is unbounded and a first order approximation of the flow equation is adopted the expected value of the water head h as well as the y unconditional autocovariance are supposed to have analytical expressions which depend on a parameters vector the proposed solution of the inverse problem consists of identifying with the aid of the model and of the measurements of y and h and subsequently computing the statistical moments of y conditioned on the same data the additional features of the present study are 1 incorporation of a constant but random effective recharge and its identification and 2 accounting for the fact that estimation is associated with some uncertainty whereas before was assumed to be identified with certainty analytical expressions are derived for the y and h covariances for an exponential autocovariance of y paper 2 rubin and dagan this issue of the study illustrates the applications of the method to a real life case

stochastic simulation of nonstationary oscillation hydroclimatic processes using empirical mode decomposition 1 nonstationary oscillation nso processes are observed in a number of hydroclimatic data series stochastic simulation models are useful to study the impacts of the climatic variations induced by nso processes into hydroclimatic regimes reproducing nso processes in a stochastic time series model is however a difficult task because of the complexity of the nonstationary behaviors in the current study a novel stochastic simulation technique that reproduces the nso processes embedded in hydroclimatic data series is presented the proposed model reproduces nso processes by utilizing empirical mode decomposition emd and nonparametric simulation techniques i e k nearest neighbor resampling and block bootstrapping the model was first tested with synthetic data sets from trigonometric functions and the rossler system the north atlantic oscillation nao index was then examined as a real case study this nao index was then employed as an exogenous variable for the stochastic simulation of streamflows at the romaine river in the province of quebec canada the results of the application to the synthetic data sets and the real world case studies indicate that the proposed model preserves well the nso processes along with the key statistical characteristics of the observations it was concluded that the proposed model possesses a reasonable simulation capacity and a high potential as a stochastic model especially for hydroclimatic data sets that embed nso processes

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

a continuous time inverse operator for groundwater and contaminant transport modeling model identifiability general identifiability conditions are derived for the model coefficients of the spatially discretized continuous time diffusion equation these conditions in turn determine some sufficient and necessary conditions for the existence and uniqueness of the continuous time inverse operator described in a previous article the identifiability conditions are in terms of the initial conditions and the flux vector containing boundary condition information and the eigenspace of the spatially discretized system the derivation of these conditions gives some insight into the location of the solution of the inverse problem in terms of the direct operator under investigation in the space of the laplace transform and limited insight into the stability of the solution near that location

estimation of mountain front recharge to regional aquifers 2 a maximum likelihood approach incorporating prior information in this two part series a stochastic estimation procedure applicable to the analytic streamflow model derived in the companion is introduced the parameter estimation problem is posed in the framework of maximum likelihood theory where prior information about the model parameters and a suitable weighting scheme for the error terms in the estimation criterion are included various optimization methods are combined for parameter estimation the issues of model and parameter identifiability uniqueness and stability are addressed and strategies to mitigate identifiability problems in our modeling are discussed the seasonal streamflow model is applied to a mountainous watershed in southern arizona and maximum likelihood estimates of mountain front recharge and other model and statistical parameters are obtained the analysis of estimation errors is performed in both the eigenspace and the original space of the parameters

automatic calibration of conceptual rainfall runoff models optimization algorithms catchment conditions and model structure from 32 crr catchment cases combinations from four conceptual rainfall runoff models crr and eight catchments calibrated with either two or three optimization methods 1 the shuffle complex evolution method sce ua 2 the multiple start simplex msx and 3 the local simplex it seems that all three methods produced parameter sets of comparable local optimum quality even with comparable performance among the models some parameter values derived by the three optimization methods for the same crr catchment cases are surprisingly different from each other in addition parameter sets of sce ua or msx which often produce marginally better results than the local simplex at the calibration stage could end up with worse results at the validation stage apparently given the inherent limitations of calibration data model inadequacies and identifiability problems it is impossible to achieve global convergence in the parameter search however other than those for dry catchments such as ihimbu or bird creek the parameter sets obtained are generally feasible both sce ua and the local simplex are viable optimization tools while msx is inefficient computationally sce ua can complete the parameter search in one run while the local simplex often requires multirun operations to get good results

accurate approximations for posterior moments and marginal densities abstract this article describes approximations to the posterior means and variances of positive functions of a real or vector valued parameter and to the marginal posterior densities of arbitrary i e not necessarily positive parameters these approximations can also be used to compute approximate predictive densities to apply the proposed method one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima nevertheless the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third order expansions of the likelihood and requiring the evaluation of third derivatives the approximate marginal posterior densities behave very much like saddle point approximations for sampling distributions the principal regularity condition required is that the likelihood times prior be unimodal

quantile hydrologic model selection and model structure deficiency assessment 2 applications quantile hydrologic model selection and structure deficiency assessment is applied in three case studies the performance of quantile model selection problem is rigorously evaluated using a model structure on the french broad river basin data set the case study shows that quantile model selection encompasses model selection strategies based on summary statistics and that it is equivalent to maximum likelihood estimation under certain likelihood functions it also shows that quantile model predictions are fairly robust the second case study is of a parsimonious hydrological model for dry land areas in western india the case study shows that an intuitive improvement in the model structure leads to reductions in asymmetric loss function values for all considered quantiles the asymmetric loss function is a quantile specific metric that is minimized to obtain a quantile specific prediction model the case study provides evidence that a quantile wise reduction in the asymmetric loss function is a robust indicator of model structure improvement finally a case study of modeling daily streamflow for the guadalupe river basin is presented a model structure that is least deficient for the study area is identified from nine different model structures based on quantile structural deficiency assessment the nine model structures differ in interception routing overland flow and base flow conceptualizations the three case studies suggest that quantile model selection and deficiency assessment provides a robust mechanism to compare deficiencies of different model structures and helps to identify better model structures in addition to its novelty quantile hydrologic model selection is a frequentist approach that seeks to complement existing bayesian approaches to hydrological model uncertainty

parameterizations for natural exponential families with quadratic variance functions abstract parameterizations for natural exponential families nef s with quadratic variance functions qvf s are compared according to the nearness to normality of the likelihood and posterior distribution nonnormality of the likelihood posterior is measured using two criteria the first is the magnitude of a standardized third derivative of the log likelihood logposterior density the second is a comparison of the probability of particular tail regions under the normalized likelihood posterior distribution and under the corresponding normal approximation a relationship is given that links these two criteria sample sizes are recommended for adequate normality in the likelihood for various parameterizations of the nef qvf models and these results are extended to bayesian models with a conjugate prior

a new set of invariants for linear systems application to reduced order compensator design a new set of invariants for linear systems weighting the contribution of each state component to the inherent closed loop lqg behavior of the system is presented together with applications to model order reduction and reduced order compensator design

the analysis of structural identifiability theory and application to conceptual rainfall runoff models the identification of a conceptual hydrologic model involves two stages selection of a suitable model structure and parameter estimation these two stages cannot be treated as distinct and it is important that the limitations of available parameter estimation methodologies be given serious consideration during the procedure of model structure selection this paper focuses on the concepts and tools necessary for the evaluation of whether the structure of a hydrologic model is such that its parameters can be successfully identified a definition of structural identifiability is advanced that is based purely on model properties and is independent of the stochastic nature of the output observation errors the approach is used to develop a measure of local structural identifiability which is useful in the analysis of parameter sensitivity a measure called the sensitivity ratio is introduced which is suitable for studying parameter interactions in the multidimensional parameter space also the use of allowable reparameterizations to improve identifiability is discussed the utility of the concepts discussed in this paper are demonstrated by a simulation example using a conceptual rainfall runoff model

rates of convergence for empirical processes of stationary mixing sequences classical empirical process theory for vapnik cervonenkis classes deals mainly with sequences of independent variables this paper extends the theory to stationary sequences of dependent variables it establishes rates of convergence for mixing and mixing empirical processes indexed by classes of functions the method of proof depends on a coupling of the dependent sequence with sequences of independent blocks to which the classical theory can be applied a uniform o ns 1 s rate of convergence over v c classes is established for sequences whose mixing coefficients decay slightly faster than o n s

on the solution of a nonlinear matrix equation for mimo symmetric realizations abstract a result concerning a particular solution of a nonlinear solution matrix equation reported in a previous paper 2 and interesting for studying siso systems is extended to mimo symmetric realizations for such a class of systems other important siso results can be generalized