Stationarity is undead: Uncertainty dominates the distribution of extremes

The increasing effort to develop and apply nonstationary models in hydrologic frequency analyses under changing environmental conditions can be frustrated when the additional uncertainty related to the model complexity is accounted for along with the sampling uncertainty. In order to show the practical implications and possible problems of using nonstationary models and provide critical guidelines, in this study we review the main tools developed in this field (such as nonstationary distribution functions, return periods, and risk of failure) highlighting advantages and disadvantages. The discussion is supported by three case studies that revise three illustrative examples reported in the scientific and technical literature referring to the Little Sugar Creek (at Charlotte, North Carolina), Red River of the North (North Dakota/Minnesota), and the Assunpink Creek (at Trenton, New Jersey). The uncertainty of the results is assessed by complementing point estimates with confidence intervals (CIs) and emphasizing critical aspects such as the subjectivity affecting the choice of the models’ structure. Our results show that (1) nonstationary frequency analyses should not only be based on at-site time series but require additional information and detailed exploratory data analyses (EDA); (2) as nonstationary models imply that the time-varying model structure holds true for the entire future design life period, an appropriate modeling strategy requires that EDA identifies a well-defined deterministic mechanism leading the examined process; (3) when the model structure cannot be inferred in a deductive manner and nonstationary models are fitted by inductive inference, model structure introduces an additional source of uncertainty so that the resulting nonstationary models can provide no practical enhancement of the credibility and accuracy of the predicted extreme quantiles, whereas possible model misspecification can easily lead to physically inconsistent results; (4) when the model structure is uncertain, stationary models and a suitable assessment of the uncertainty accounting for possible temporal persistence should be retained as more theoretically coherent and reliable options for practical applications in real-world design and management problems; (5) a clear understanding of the actual probabilistic meaning of stationary and nonstationary return periods and risk of failure is required for a correct risk assessment and communication.     

Modeling groundwater velocity uncertainty in nonstationary composite porous media

Despite the intensive research over the past decades in the field of stochastic subsurface hydrology, our ability to analyze and model heterogeneous groundwater systems remains limited. Most existing theories are either too restrictive to handle practical complexity or too expensive to be applied to realistic problem sizes. In this paper we present approximate, closed-form equations that allow modeling 2D nonstationary flows in statistically inhomogeneous aquifers, including composite aquifers containing multiple zones characterized by different statistical models. The composite representation has the effect of decreasing the variance of deviations from the mean, relaxing the limitation of the small-perturbation assumption. The simple formulas are illustrated with a number of examples and compared with a corresponding first-order nonstationary numerical analysis and Monte Carlo simulation. The results show that, despite the gross simplifications, the closed-form equations are robust and able to capture complex variance dynamics, reproducing surprisingly well the first-order numerical solutions and the Monte Carlo simulation even in highly nonstationary, variable situations.     

Conceptual fuzzy neural network model for water quality simulation

Artificial neural networks (ANNs) have been applied successfully in various fields. However, ANN models depend on large sets of historical data, and are of limited use when only vague and uncertain information is available, which leads to difficulties in defining the model architecture and a low reliability of results. A conceptual fuzzy neural network (CFNN) is proposed and applied in a water quality model to simulate the Barra Bonita reservoir system, located in the southeast region of Brazil. The CFNN model consists of a rationally‐defined architecture based on accumulated expert knowledge about variables and processes included in the model. A genetic algorithm is used as the training method for finding the parameters of fuzzy inference and the connection weights. The proposed model may handle the uncertainties related to the system itself, model parameterization, complexity of concepts involved and scarcity and inaccuracy of data. The CFNN showed greater robustness and reliability when dealing with systems for which data are considered to be vague, uncertain or incomplete. The CFNN model structure is easier to understand and to define than other ANN‐based models. Moreover, it can help to understand the basic behaviour of the system as a whole, being a successful example of cooperation between human and machine. Copyright © 2006 John Wiley & Sons, Ltd.     

Entropy-based inference of simple physical models for regional flood analysis

 Regional flood analysis is formulated as a physical-modelling problem consisting in the inference of meaningful physical models for a set of observable uncertain quantities representing floods, given the observed data separately associated with them. It is argued that physical modelling suitable for representing causality relationships should involve the use of models comprising functional dependences of the observable uncertain quantities with regard to other quantities which are unobservable. The regional physical-modelling problem becomes the selection, from any proposed space of candidate models, of a probability distribution for the unobservable uncertain quantities together with a functional-dependence model connecting the observable to the unobservable uncertain quantities. Due to the need to coherently represent observational data and to express precisely the available evidences, the physical modelling problem is formalized in a plausible logic language, within the logical probability framework. A logical inference procedure called the relative entropy method with fractile constraints (REF) is formulated within this framework and extended to solve the regional physical-modelling problem. Contrary to the current statistical methods, it allows the selection and validation of inferred models and can be applied whatever it is the number of observational data. The complete solution to the problem using the relative entropy procedure is presented. This method is applied to the regional modelling of annual maximum floods of a set of separate rivers in the Iberian Peninsula. For this application the space of candidate models includes several types of two-parameter probability distributions for the unobservable uncertain quantities and the class of linear homogeneous functional-dependence models connecting the observable to the unobservable quantities.     

Testing linear models of sea-floor topography

Stochastic models can generate profiles that resemble topography by taking uncorrelated, zero-average noise as input, introducing some correlation in the time series of noise, and integrating the resulting correlated noise. The output profile will depict a nonstationary, randomly rough surface. Two models have been chosen for comparison: a fractal model, in which the noise is correlated even at large distances, and an autoregressive model of order 1, in which the correlation of the noise decays rapidly. Both models have as an end-member a random walk, which is the integration of uncorrelated noise. The models have been fitted to profiles of submarine topography, and the sample autocorrelation, power spectrum and variogram have been compared to the theoretical predictions. The results suggest that a linear system approach is a viable method to model and classify sea-floor topography. The comparison does not show substantial disagreement of the data with either the autoregressive or the fractal model, although a fractal model seems to give a better fit. However, the amplitudes predicted by a nonstationary fractal model for long wavelengths (of the order of 1000 km) are unreasonably large. When viewed through a large window, ocean floor topography is likely to have an expected value determined by isostasy, and to be stationary. Nonstationary models are best applied to wavelengths of the order of 100 km or less.     

A nonstationary runoff frequency analysis for future climate change and its uncertainties

With the intensification of climate change, its impact on runoff variations cannot be ignored. The main purpose of this study is to analyse the nonstationarity of runoff frequency adjusted for future climate change in the Luanhe River basin, China, and quantify the different sources of uncertainties in nonstationary runoff frequency analysis. The advantage of our method is the combination of generalized additive models in location, scale, and shape (GAMLSS) and downscaling models. The nonstationary GAMLSS models were established for the nonstationary frequency analysis of runoff (1961–2010) by using the observed precipitation as a covariate, which is closely related to runoff and contributes significantly to its nonstationarity. To consider the nonstationary effects of future climate change on future runoff variations, the downscaled precipitation series in the future (2011–2080) from the general circulation models (GCMs) were substituted into the selected nonstationary model to calculate the statistical parameters and runoff frequency in the future. A variance decomposition method was applied to quantify the impacts of different sources of uncertainty on the nonstationary runoff frequency analysis. The results show that the impacts of uncertainty in the GCMs, scenarios, and statistical parameters of the GAMLSS model increase with increasing runoff magnitude. In addition, GCMs and GAMLSS model parameters have the main impacts on runoff uncertainty, accounting for 14% and 83% of the total uncertainty sources, respectively. Conversely, the interactions and scenarios make limited contributions, accounting for 2% and 1%, respectively. Further analysis shows that the sources of uncertainty in the statistical parameters of the nonstationary model mainly result from the fluctuations in the precipitation sequence. This result indicates the necessity of considering the precipitation sequence as a covariate for runoff frequency analysis in the future.     

Parameter identification in particle models

For the simulation of the transport of dissolved matter particle models can be used. In this paper a technique is developed for the identification of uncertain parameters in these models. This model calibration is formulated as an optimization problem and is solved with a gradient based algorithm. Here adjoint particle tracks are used for the calculation of the gradient of the cost function. The performance of the calibration method is illustrated by simulations and an application to a river Rhine water quality calamity in November 1986.     

Parameter identification in particle models

For the simulation of the transport of dissolved matter particle models can be used. In this paper a technique is developed for the identification of uncertain parameters in these models. This model calibration is formulated as an optimization problem and is solved with a gradient based algorithm. Here adjoint particle tracks are used for the calculation of the gradient of the cost function. The performance of the calibration method is illustrated by simulations and an application to a river Rhine water quality calamity in November 1986.     

Analysis and integrated modelling of groundwater infiltration to sewer networks

Infiltration of groundwater to sewer systems is a problem for the capacity of the system as well as for treatment processes at waste water treatment plants. This paper quantifies the infiltration of groundwater to a sewer system in Frederikshavn Municipality, Denmark, by measurements of sewer flow and novel model set‐up, which simulates the interaction between groundwater and sewer flow. The study area has a separate waste water sewer system, but the discharged volumes from the system are approximately twice the volumes from a tight system without infiltration. The model set‐up makes use of two commercial models: mike she for simulation of groundwater transport and mike urban (mouse ) [DHI, Hørsholm, Denmark] for simulation of sewer flow. By simulating the groundwater level and calibrating infiltration coefficients against sewer flow measurements, it has been possible to estimate the average infiltration to the sewer system with satisfying results. The infiltration processes are indeed complicated and to a large degree heterogeneous throughout the sewer system. The paper shows contribution from both saturated and unsaturated groundwater zones, which makes the modelling process complex. Copyright © 2016 John Wiley & Sons, Ltd.     

Time series analysis of water quality parameters at Stillaguamish River using order series method

This study applied the time series analysis approach to model and predict univariate dissolved oxygen and temperature time series for four water quality assessment stations at Stillaguamish River located in the state of Washington. The order series method was applied to fulfill the normality assumption for modeling the univariate time series. Then, the AR(I)MA models were applied to study the stationary and nonstationary time series, the Auto-Regressive Fractionally Integrated Moving Average model was applied to study the time series with long memory. The results showed there existed three different structures for the univariate water quality time series at Stillaguamish River watershed. The identified time series model for each univariate water quality time series was found to be capable of predicting future values with reasonable accuracy. Overall, the time series modeling approach may be an efficient tool in assessment of the water quality in the river system.     

Advance first order second moment (AFOSM) method for single reservoir operation reliability analysis: a case study

Reservoir system reliability is the ability of reservoir to perform its required functions under stated conditions for a specified period of time. In classical method of reservoir system reliability analysis, the operation policy is used in a simple simulation model, considering the historical/synthetic inflow series and a number of physical bounds on a reservoir system. This type of reliability analysis assumes a reservoir system as fully failed or functioning, called binary state assumption. A number of researchers from various research backgrounds have shown that the binary state assumption in the traditional reliability theory is not extensively acceptable. Our approach to tackle the present problem space is to implement the algorithm of advance first order second moment (AFOSM) method. In this new method, the inflow and reservoir storage are considered as uncertain variables. The mean, variance and covariance of uncertain variables are determined using moment values of reservoir state variables. For this purpose, a stochastic optimization model developed based on the constraint state formulation is applied. The proposed model of reliability analysis is used to a real case study in Iran. As a result, monthly probabilities of water allocation were computed from AFOSM method, and the outputs were compared with those from Monte Carlo method. The comparison shows that the outputs from AFOSM method are similar to those from the Monte Carlo method. In term of practical use of this study, the proposed method is appropriate to determine the monthly probability of failure in water allocation without the aid of simulation.     

Identification of Glacial Isostatic Adjustment in Eastern Canada Using S Transform Filtering of GPS Observations

Over the years, a number of different models and techniques have been proposed to both quantify and explain the glacial isostatic adjustment (GIA) process. There are serious challenges, however, to obtaining accurate results from measurements, due to noise in the data and the long periods of time necessary to identify the relatively small-magnitude signal in certain regions. The primary difficulty, in general, is that most of the geophysical signals that occur in addition to GIA are nonstationary in nature. These signals are also corrupted by random as well as correlated noise added during data acquisition. The nonstationary characteristic of the data makes it difficult for traditional frequency-domain denoising approaches to be effective. Time–frequency filters present a more robust and reliable alternative to deal with this problem. This paper proposes an extended S transform filtering approach to separate the various signals and isolate that associated with GIA. Continuous global positioning system (GPS) data from eastern Canada for the period from June 2001 to June 2006 are analyzed here, and the vertical velocities computed after filtering are consistent with the GIA models put forward by other researchers.     

A hybrid artificial neural network-numerical model for ground water problems

Numerical models constitute the most advanced physical-based methods for modeling complex ground water systems. Spatial and/or temporal variability of aquifer parameters, boundary conditions, and initial conditions (for transient simulations) can be assigned across the numerical model domain. While this constitutes a powerful modeling advantage, it also presents the formidable challenge of overcoming parameter uncertainty, which, to date, has not been satisfactorily resolved, inevitably producing model prediction errors. In previous research, artificial neural networks (ANNs), developed with more accessible field data, have achieved excellent predictive accuracy over discrete stress periods at site-specific field locations in complex ground water systems. In an effort to combine the relative advantages of numerical models and ANNs, a new modeling paradigm is presented. The ANN models generate accurate predictions for a limited number of field locations. Appending them to a numerical model produces an overdetermined system of equations, which can be solved using a variety of mathematical techniques, potentially yielding more accurate numerical predictions. Mathematical theory and a simple two-dimensional example are presented to overview relevant mathematical and modeling issues. Two of the three methods for solving the overdetermined system achieved an overall improvement in numerical model accuracy for various levels of synthetic ANN errors using relatively few constrained head values (i.e., cells), which, while demonstrating promise, requires further research. This hybrid approach is not limited to ANN technology; it can be used with other approaches for improving numerical model predictions, such as regression or support vector machines (SVMs).     

Error control of iterative linear solvers for integrated groundwater models

An open problem that arises when using modern iterative linear solvers, such as the preconditioned conjugate gradient method or Generalized Minimum RESidual (GMRES) method, is how to choose the residual tolerance in the linear solver to be consistent with the tolerance on the solution error. This problem is especially acute for integrated groundwater models, which are implicitly coupled to another model, such as surface water models, and resolve both multiple scales of flow and temporal interaction terms, giving rise to linear systems with variable scaling. This article uses the theory of "forward error bound estimation" to explain the correspondence between the residual error in the preconditioned linear system and the solution error. Using examples of linear systems from models developed by the US Geological Survey and the California State Department of Water Resources, we observe that this error bound guides the choice of a practical measure for controlling the error in linear systems. We implemented a preconditioned GMRES algorithm and benchmarked it against the Successive Over-Relaxation (SOR) method, the most widely known iterative solver for nonsymmetric coefficient matrices. With forward error control, GMRES can easily replace the SOR method in legacy groundwater modeling packages, resulting in the overall simulation speedups as large as 7.74×. This research is expected to broadly impact groundwater modelers through the demonstration of a practical and general approach for setting the residual tolerance in line with the solution error tolerance and presentation of GMRES performance benchmarking results.     

一种通过地形指数提取流域自由水蓄水容量空间分布的方法

对于分布式水文模型而言,如何获得参数的空间分布是模型应用的重点和难点问题.本文将分水源参数中的敏感参数——自由水蓄水容量为研究对象.建立地形指数与自由水蓄水容量的函数关系,以此提取流域内的自由水蓄水容量空间分布.最后利用本方法提取了陕西省陈河流域的自由水蓄水容量空间分布,并将之作为栅格型新安江模型的参数进行洪水模拟演算.应用结果表明本文提出的方法得到了理想的模拟结果.该方法以物理规律为基础能较为准确地计算出流域内自由水蓄水容量的空间分布,为分布式模型的发展奠定了坚实的基础.     

Parameter identification in tidal models with uncertain boundary conditions

In this paper a parameter estimation algorithm is developed to estimate uncertain parameters in two dimensional shallow water flow models. Since in practice the open boundary conditions of these models are usually not known accurately, the uncertainty of these boundary conditions has to be taken into account to prevent that boundary errors are interpreted by the estimation procedure as parameter fluctuations. Therefore the open boundary conditions are embedded into a stochastic environment and a constant gain extended Kalman filter is employed to identify the state of the system. Defining a error functional that measures the differences between the filtered state of the system and the measurements, a quasi Newton method is employed to determine the minimum of this functional. To reduce the computational burden, the gradient of the criterium that is required using the quasi Newton method is determined by solving the adjoint system.     

Uncertainty assessment for watershed water quality modeling: A Probabilistic Collocation Method based approach

Watershed water quality models are increasingly used in management. However, simulations by such complex models often involve significant uncertainty, especially those for non-conventional pollutants which are often poorly monitored. This study first proposed an integrated framework for watershed water quality modeling. Within this framework, Probabilistic Collocation Method (PCM) was then applied to a WARMF model of diazinon pollution to assess the modeling uncertainty. Based on PCM, a global sensitivity analysis method named PCM-VD (VD stands for variance decomposition) was also developed, which quantifies variance contribution of all uncertain parameters. The study results validated the applicability of PCM and PCM-VD to the WARMF model. The PCM-based approach is much more efficient, regarding computational time, than conventional Monte Carlo methods. It has also been demonstrated that analysis using the PCM-based approach could provide insights into data collection, model structure improvement and management practices. It was concluded that the PCM-based approach could play an important role in watershed water quality modeling, as an alternative to conventional Monte Carlo methods to account for parametric uncertainty and uncertainty propagation.     

Parameter identification in tidal models with uncertain boundary conditions

In this paper a parameter estimation algorithm is developed to estimate uncertain parameters in two dimensional shallow water flow models. Since in practice the open boundary conditions of these models are usually not known accurately, the uncertainty of these boundary conditions has to be taken into account to prevent that boundary errors are interpreted by the estimation procedure as parameter fluctuations. Therefore the open boundary conditions are embedded into a stochastic environment and a constant gain extended Kalman filter is employed to identify the state of the system. Defining a error functional that measures the differences between the filtered state of the system and the measurements, a quasi Newton method is employed to determine the minimum of this functional. To reduce the computational burden, the gradient of the criterium that is required using the quasi Newton method is determined by solving the adjoint system.     

Uncertainty assessment of a model for biological nitrogen and phosphorus removal: Application to a large wastewater treatment plant

In the last few years, the use of mathematical models in WasteWater Treatment Plant (WWTP) processes has become a common way to predict WWTP behaviour. However, mathematical models generally demand advanced input for their implementation that must be evaluated by an extensive data-gathering campaign, which cannot always be carried out. This fact, together with the intrinsic complexity of the model structure, leads to model results that may be very uncertain. Quantification of the uncertainty is imperative. However, despite the importance of uncertainty quantification, only few studies have been carried out in the wastewater treatment field, and those studies only included a few of the sources of model uncertainty. Seeking the development of the area, the paper presents the uncertainty assessment of a mathematical model simulating biological nitrogen and phosphorus removal. The uncertainty assessment was conducted according to the Generalised Likelihood Uncertainty Estimation (GLUE) methodology that has been scarcely applied in wastewater field. The model was based on activated-sludge models 1 (ASM) and 2 (ASM2). Different approaches can be used for uncertainty analysis. The GLUE methodology requires a large number of Monte Carlo simulations in which a random sampling of individual parameters drawn from probability distributions is used to determine a set of parameter values. Using this approach, model reliability was evaluated based on its capacity to globally limit the uncertainty. The method was applied to a large full-scale WWTP for which quantity and quality data was gathered. The analysis enabled to gain useful insights for WWTP modelling identifying the crucial aspects where higher uncertainty rely and where therefore, more efforts should be provided in terms of both data gathering and modelling practises.     

The Baltic Sea Experiment BALTEX: A brief overview and some selected results of the authors

The mechanisms responsible for the transfer of energy and water within the climate system are under worldwide investigation within the framework of the Global Energy and Water Cycle Experiment (GEWEX) to improve the predictability of natural and man-made climate changes at short and long ranges and their impact on water resources. Five continental-scale experiments have been established within GEWEX to enable a more complete coupling between atmospheric and hydrological models. One of them is the Baltic Sea Experiment (BALTEX).In this paper, the goals and structure of BALTEX are outlined. A short overview of measuring and modelling strategies is given. Atmospheric and hydrological model results of the authors only are presented. These include also the validation of precipitation using station measurements as well as validation of modelled cloud cover with cloud estimates from satellite data. Furthermore, results of a large-scale grid based hydrological model to be coupled to atmospheric models are presented.This research has never been possible without the contribution of research groups and operational institutions from all 10 member countries. We concentrate here on results obtained at the GKSS research center.