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

on dirac s conjecture for systems having only first class constraints dirac s conjecture is proved for a general class of systems having only first class constraints in other words it is shown for that kind of problem that all the primary or secondary first class constraints generate equivalence transformations between physical states

recursive bayesian estimation of conceptual rainfall runoff model errors in real time prediction of streamflow conceptual rainfall runoff models account for the spatial dynamics of hydrological processes in a basin using simple spatially lumped storage flow relations such rough approximations introduce model errors that are often difficult to characterize here we develop and apply a methodology that recursively estimates and accounts for model errors in real time streamflow prediction settings by adding time dependent random noise to the internal states storages of the hydrological model magnitude of the added noise depends on a precision inverse variance parameter that is estimated from rainfall runoff data a recursive bayesian technique is used for estimation posteriors of hydrological parameters and states are updated through time with an ensemble kalman filter whereas the posterior of the precision parameter is updated recursively using a novel gamma density approximation technique applying this algorithm to different model error scenarios allows identification of the main source of model errors the methodology is applied to short term streamflow prediction with the hymod rainfall runoff model in a semi cold semi humid basin in iran results show that i streamflow prediction in this snow dominated basin is more affected by model errors in the slow flow than the quick flow component of the model ii accounting for model errors in the slow flow component improves both low and high flow predictions and iii predictive performance further improves by accounting for hymod parameter uncertainty in addition to model errors overall accounting for model errors increased nash sutcliffe efficiency by 1 5 reduced mean absolute error by 2 43 and improved probabilistic predictive performance by 50 80

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

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

generic error model for calibration and uncertainty estimation of hydrological models 1 because of the necessary simplification of the complex natural processes and the limited availability of observations model simulations are always uncertain and this uncertainty should be quantified in this contribution the model error is quantified using a combined procedure for the uncertainty of discharge due to meteorological input a stochastic simulation method is used to quantify the effect of process representation and parameterization a sensitivity analysis is carried out it is assumed that the model error due to process uncertainty is proportional to the sensitivity the final model error variance can thus be calculated from the stochastic errors and the process sensitivities the coefficients used for the quantification are estimated simultaneously with the model parameters the methodology presented produces error series that are normally distributed and that represent the varying importance of different processes in time this uncertainty time series can be used as a weighting factor to normalize the model residuals during calibration so that the assumptions of least squares optimization are fulfilled calibration and uncertainty estimation are demonstrated with an example application of a distributed hydrological bureau waterbalance hbv model of three watersheds in the neckar basin in southwest germany the model residual distributions are presented and compared to a standard calibration method further it is shown that the new methodology leads to more realistic confidence intervals for model simulations although applied to the hbv model as an example the method is general and can be applied to any model and also in conjunction with other uncertainty estimation techniques

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

effects of error covariance structure on estimation of model averaging weights and predictive performance 1 when conducting model averaging for assessing groundwater conceptual model uncertainty the averaging weights are often evaluated using model selection criteria such as aic aicc bic and kic akaike information criterion corrected akaike information criterion bayesian information criterion and kashyap information criterion respectively however this method often leads to an unrealistic situation in which the best model receives overwhelmingly large averaging weight close to 100 which cannot be justified by available data and knowledge it was found in this study that this problem was caused by using the covariance matrix c of measurement errors for estimating the negative log likelihood function common to all the model selection criteria this problem can be resolved by using the covariance matrix cek of total errors including model errors and measurement errors to account for the correlation between the total errors an iterative two stage method was developed in the context of maximum likelihood inverse modeling to iteratively infer the unknown cek from the residuals during model calibration the inferred cek was then used in the evaluation of model selection criteria and model averaging weights while this method was limited to serial data using time series techniques in this study it can be extended to spatial data using geostatistical techniques the method was first evaluated in a synthetic study and then applied to an experimental study in which alternative surface complexation models were developed to simulate column experiments of uranium reactive transport it was found that the total errors of the alternative models were temporally correlated due to the model errors the iterative two stage method using cek resolved the problem that the best model receives 100 model averaging weight and the resulting model averaging weights were supported by the calibration results and physical understanding of the alternative models using cek obtained from the iterative two stage method also improved predictive performance of the individual models and model averaging in both synthetic and experimental studies

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

an adaptive sparse grid high order stochastic collocation method for bayesian inference in groundwater reactive transport modeling 1 bayesian analysis has become vital to uncertainty quantification in groundwater modeling but its application has been hindered by the computational cost associated with numerous model executions required by exploring the posterior probability density function ppdf of model parameters this is particularly the case when the ppdf is estimated using markov chain monte carlo mcmc sampling in this study a new approach is developed to improve the computational efficiency of bayesian inference by constructing a surrogate of the ppdf using an adaptive sparse grid high order stochastic collocation asg hsc method unlike previous works using first order hierarchical basis this paper utilizes a compactly supported higher order hierarchical basis to construct the surrogate system resulting in a significant reduction in the number of required model executions in addition using the hierarchical surplus as an error indicator allows locally adaptive refinement of sparse grids in the parameter space which further improves computational efficiency to efficiently build the surrogate system for the ppdf with multiple significant modes optimization techniques are used to identify the modes for which high probability regions are defined and components of the asg hsc approximation are constructed after the surrogate is determined the ppdf can be evaluated by sampling the surrogate system directly without model execution resulting in improved efficiency of the surrogate based mcmc compared with conventional mcmc the developed method is evaluated using two synthetic groundwater reactive transport models the first example involves coupled linear reactions and demonstrates the accuracy of our high order hierarchical basis approach in approximating high dimensional posteriori distribution the second example is highly nonlinear because of the reactions of uranium surface complexation and demonstrates how the iterative asg hsc method is able to capture multimodal and non gaussian features of ppdf caused by model nonlinearity both experiments show that asg hsc is an effective and efficient tool for bayesian inference

bayesian calibration of a large scale geothermal reservoir model by a new adaptive delayed acceptance metropolis hastings algorithm 1 the aim of this research is to estimate the parameters of a large scale numerical model of a geothermal reservoir using markov chain monte carlo mcmc sampling within the framework of bayesian inference all feasible parameters that are consistent with the measured data are summarized by the posterior distribution and hence parameter estimation and uncertainty quantification are both given by calculating expected values of statistics of interest over the posterior distribution it appears to be computationally infeasible to use the standard metropolis hastings algorithm mh to sample the high dimensional computationally expensive posterior distribution to improve the sampling efficiency a new adaptive delayed acceptance mh algorithm adamh is implemented to adaptively build a stochastic model of the error introduced by the use of a reduced order model this use of adaptivity differs from existing adaptive mcmc algorithms that tune proposal distributions of the metropolis hastings algorithm mh though adamh also implements that technique for the 3 d geothermal reservoir model we present here adamh shows a great improvement in the computational efficiency of the mcmc sampling and promising results for parameter estimation and uncertainty quantification are obtained this algorithm could offer significant improvement in computational efficiency when implementing sample based inference in other large scale inverse problems

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

inference of reactive transport model parameters using a bayesian multivariate approach parameter estimation of subsurface transport models from multispecies data requires the definition of an objective function that includes different types of measurements common approaches are weighted least squares wls where weights are specified a priori for each measurement and weighted least squares with weight estimation wls we where weights are estimated from the data together with the parameters in this study we formulate the parameter estimation task as a multivariate bayesian inference problem the wls and wls we methods are special cases in this framework corresponding to specific prior assumptions about the residual covariance matrix the bayesian perspective allows for generalizations to cases where residual correlation is important and for efficient inference by analytically integrating out the variances weights and selected covariances from the joint posterior specifically the wls and wls we methods are compared to a multivariate mv approach that accounts for specific residual correlations without the need for explicit estimation of the error parameters when applied to inference of reactive transport model parameters from column scale data on dissolved species concentrations the following results were obtained 1 accounting for residual correlation between species provides more accurate parameter estimation for high residual correlation levels whereas its influence for predictive uncertainty is negligible 2 integrating out the co variances leads to an efficient estimation of the full joint posterior with a reduced computational effort compared to the wls we method and 3 in the presence of model structural errors none of the methods is able to identify the correct parameter values

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

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

toward diagnostic model calibration and evaluation approximate bayesian computation the ever increasing pace of computational power along with continued advances in measurement technologies and improvements in process understanding has stimulated the development of increasingly complex hydrologic models that simulate soil moisture flow groundwater recharge surface runoff root water uptake and river discharge at different spatial and temporal scales reconciling these high order system models with perpetually larger volumes of field data is becoming more and more difficult particularly because classical likelihood based fitting methods lack the power to detect and pinpoint deficiencies in the model structure gupta et al 2008 has recently proposed steps amongst others toward the development of a more robust and powerful method of model evaluation their diagnostic approach uses signature behaviors and patterns observed in the input output data to illuminate to what degree a representation of the real world has been adequately achieved and how the model should be improved for the purpose of learning and scientific discovery in this paper we introduce approximate bayesian computation abc as a vehicle for diagnostic model evaluation this statistical methodology relaxes the need for an explicit likelihood function in favor of one or multiple different summary statistics rooted in hydrologic theory that together have a clearer and more compelling diagnostic power than some average measure of the size of the error residuals two illustrative case studies are used to demonstrate that abc is relatively easy to implement and readily employs signature based indices to analyze and pinpoint which part of the model is malfunctioning and in need of further improvement 2013 american geophysical union all rights reserved

on bayesian modeling of fat tails and skewness abstract we consider a bayesian analysis of linear regression models that can account for skewed error distributions with fat tails the latter two features are often observed characteristics of empirical datasets and we formally incorporate them in the inferential process a general procedure for introducing skewness into symmetric distributions is first proposed even though this allows for a great deal of flexibility in distributional shape tail behavior is not affected applying this skewness procedure to a student t distribution we generate a skewed student distribution which displays both flexible tails and possible skewness each entirely controlled by a separate scalar parameter the linear regression model with a skewed student error term is the main focus of the article we first characterize existence of the posterior distribution and its moments using standard improper priors and allowing for inference on skewness and tail parameters for posterior inference with this model we suggest

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

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

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

efficient posterior exploration of a high dimensional groundwater model from two stage markov chain monte carlo simulation and polynomial chaos expansion 1 this study reports on two strategies for accelerating posterior inference of a highly parameterized and cpu demanding groundwater flow model our method builds on previous stochastic collocation approaches e g marzouk and xiu 2009 and marzouk and najm 2009 and uses generalized polynomial chaos gpc theory and dimensionality reduction to emulate the output of a large scale groundwater flow model the resulting surrogate model is cpu efficient and serves to explore the posterior distribution at a much lower computational cost using two stage mcmc simulation the case study reported in this paper demonstrates a two to five times speed up in sampling efficiency

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

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

asymptotic theory for clustered samples abstract we provide a complete asymptotic distribution theory for clustered data with a large number of independent groups generalizing the classic laws of large numbers uniform laws central limit theory and clustered covariance matrix estimation our theory allows for clustered observations with heterogeneous and unbounded cluster sizes our conditions cleanly nest the classical results for i n i d observations in the sense that our conditions specialize to the classical conditions under independent sampling we use this theory to develop a full asymptotic distribution theory for estimation based on linear least squares 2sls nonlinear mle and nonlinear gmm

estimating information entropy for hydrological data one dimensional case there has been a recent resurgence of interest in the application of information theory to problems of system identification in the earth and environmental sciences while the concept of entropy has found increased application little attention has yet been given to the practical problems of estimating entropy when dealing with the unique characteristics of two commonly used kinds of hydrologic data rainfall and runoff in this paper we discuss four important issues of practical relevance that can bias the computation of entropy if not properly handled the first zero effect arises when precipitation and ephemeral streamflow data must be viewed as arising from a discrete continuous hybrid distribution due to the occurrence of many zero values e g days with no rain no runoff second in the widely used bin counting method for estimation of pdf s significant error can be introduced if the bin width is not carefully selected the third measurement effect arises due to the fact that continuously varying hydrologic variables can typically only be observed discretely to some degree of precision the fourth skewness effect arises when the distribution of a variable is significantly skewed here we present an approach that can deal with all four of these issues and test them with artificially generated and real hydrological data the results indicate that the method is accurate and robust

multiresponse multilayer vadose zone model calibration using markov chain monte carlo simulation and field water retention data 1 in the past two decades significant progress has been made toward the application of inverse modeling to estimate the water retention and hydraulic conductivity functions of the vadose zone at different spatial scales many of these contributions have focused on estimating only a few soil hydraulic parameters without recourse to appropriately capturing and addressing spatial variability the assumption of a homogeneous medium significantly simplifies the complexity of the resulting inverse problem allowing the use of classical parameter estimation algorithms here we present an inverse modeling study with a high degree of vertical complexity that involves calibration of a 25 parameter richards based hydrus 1d model using in situ measurements of volumetric water content and pressure head from multiple depths in a heterogeneous vadose zone in new zealand we first determine the trade off in the fitting of both data types using the amalgam multiple objective evolutionary search algorithm then we adopt a bayesian framework and derive posterior probability density functions of parameter and model predictive uncertainty using the recently developed differential evolution adaptive metropolis dreamzs adaptive markov chain monte carlo scheme we use four different formulations of the likelihood function each differing in their underlying assumption about the statistical properties of the error residual and data used for calibration we show that amalgam and dreamzs can solve for the 25 hydraulic parameters describing the water retention and hydraulic conductivity functions of the multilayer heterogeneous vadose zone our study clearly highlights that multiple data types are simultaneously required in the likelihood function to result in an accurate soil hydraulic characterization of the vadose zone of interest remaining error residuals are most likely caused by model deficiencies that are not encapsulated by the multilayer model and can not be accessed by the statistics and likelihood function used the utilization of an explicit autoregressive error model of the remaining error residuals does not work well for the water content data with hydrus 1d prediction uncertainty bounds that become unrealistically large