Model output uncertainty of a coupled pathogen indicator–hydrologic catchment model due to input data uncertainty

A conceptual model of Escherichia coli behaviour in catchments, the EG model, coupled with a standard hydrological model has been previously developed and tested. This paper presents an analysis of the uncertainty of the modelled pathogen concentrations and loads due to uncertainties in the models data inputs. The data collected at three different large Australian catchments were used. Firstly, uncertainties in the models input data, i.e. hourly rainfall, monthly potential evapotranspiration, catchment size and daily surface pathogen deposition rates, were assessed. Random and systematic sources of errors were taken into account. It was found that systematic errors in rainfall and random errors in pathogen deposition rates have the biggest impact on uncertainty in the models output.     

Seasonal hydrological and nutrient loading forecasts for watersheds over the Southeastern United States

We show useful seasonal deterministic and probabilistic prediction skill of streamflow and nutrient loading over watersheds in the Southeastern United States (SEUS) for the winter and spring seasons. The study accounts for forecast uncertainties stemming from the meteorological forcing and hydrological model uncertainty. Multi-model estimation from three hydrological models, each forced with an ensemble of forcing derived by matching observed analogues of forecasted quartile rainfall anomalies from a seasonal climate forecast is used. The attained useful hydrological prediction skill is despite the climate model overestimating rainfall by over 23% over these SEUS watersheds in December–May period. The prediction skill in the month of April and May is deteriorated as compared to the period from December–March (zero lead forecast). A nutrient streamflow rating curve is developed using a log linear tool for this purpose. The skill in the prediction of seasonal nutrient loading is identical to the skill of seasonal streamflow forecast.     

Numerical and visual evaluation of hydrological and environmental models using the Monte Carlo analysis toolbox

The detailed evaluation of mathematical models and the consideration of uncertainty in the modeling of hydrological and environmental systems are of increasing importance, and are sometimes even demanded by decision makers. At the same time, the growing complexity of models to represent real-world systems makes it more and more difficult to understand model behavior, sensitivities and uncertainties. The Monte Carlo Analysis Toolbox (MCAT) is a Matlab library of visual and numerical analysis tools for the evaluation of hydrological and environmental models. Input to the MCAT is the result of a Monte Carlo or population evolution based sampling of the parameter space of the model structure under investigation. The MCAT can be used off-line, i.e. it does not have to be connected to the evaluated model, and can thus be used for any model for which an appropriate sampling can be performed. The MCAT contains tools for the evaluation of performance, identifiability, sensitivity, predictive uncertainty and also allows for the testing of hypotheses with respect to the model structure used. In addition to research applications, the MCAT can be used as a teaching tool in courses that include the use of mathematical models.     

Variance-based sensitivity analysis of the probability of hydrologically induced slope instability

Analysis of the sensitivity of predictions of slope instability to input data and model uncertainties provides a rationale for targeted site investigation and iterative refinement of geotechnical models. However, sensitivity methods based on local derivatives do not reflect model behaviour over the whole range of input variables, whereas methods based on standardised regression or correlation coefficients cannot detect non-linear and non-monotonic relationships between model input and output. Variance-based sensitivity analysis (VBSA) provides a global, model-independent sensitivity measure. The approach is demonstrated using the Combined Hydrology and Stability Model (CHASM) and is applicable to a wide variety of computer models. The method of Sobol’, assuming independence between input variables, was used to identify interactions between model input variables, whilst replicated Latin Hypercube Sampling (LHS) is used to investigate the effects of statistical dependence between the input variables. The SIMLAB software was used, both to generate the input sample and to calculate the sensitivity indices. The analysis provided quantified evidence of well-known sensitivities as well demonstrating how uncertainty in slope failure during rainfall is, for the examples tested here, more attributable to uncertainty in the soil strength than to uncertainty in the rainfall.     

Moving SWAT model calibration and uncertainty analysis to an enterprise Hadoop-based cloud

With enhanced availability of high spatial resolution data, hydrologic models such as the Soil and Water Assessment Tool (SWAT) are increasingly used to investigate effects of management activities and climate change on water availability and quality. The advantages come at a price of greater computational demand and run time. This becomes challenging to model calibration and uncertainty analysis as these routines involve a large number of model runs. For efficient modelling, a cloud-based Calibration and Uncertainty analysis Tool for SWAT (CUT-SWAT) was implemented using Hadoop, an open source cloud platform, and the Generalized Likelihood Uncertainty Estimation method. Test results on an enterprise cloud showed that CUT-SWAT can significantly speedup the calibration and uncertainty analysis processes with a speedup of 21.7–26.6 depending on model complexity and provides a flexible and fault-tolerant model execution environment (it can gracefully and automatically handle partial failure), thus would be an ideal method to solve computational demand problems in hydrological modelling.     

Comparison of variance-based and moment-independent global sensitivity analysis approaches by application to the SWAT model

Global Sensitivity Analysis (GSA) is an essential technique to support the calibration of environmental models by identifying the influential parameters (screening) and ranking them.In this paper, the widely-used variance-based method (Sobol') and the recently proposed moment-independent PAWN method for GSA are applied to the Soil and Water Assessment Tool (SWAT), and compared in terms of ranking and screening results of 26 SWAT parameters. In order to set a threshold for parameter screening, we propose the use of a “dummy parameter”, which has no influence on the model output. The sensitivity index of the dummy parameter is calculated from sampled data, without changing the model equations. We find that Sobol' and PAWN identify the same 12 influential parameters but rank them differently, and discuss how this result may be related to the limitations of the Sobol' method when the output distribution is asymmetric.     

SWAT-DayCent coupler: An integration tool for simultaneous hydro-biogeochemical modeling using SWAT and DayCent

Process-based numerical models in environmental science can help understand and quantify terrestrial material cycles in nature. However, the existing models usually focus on the cycles of one or more elements (e.g., water, carbon, or nitrogen). For example, hydrological models such as Soil and Water Assessment Tool (SWAT) focus on the water cycle and nutrient loadings at watershed scale, whereas biogeochemical models such as DayCent (i.e., daily CENTURY) emphasize carbon/nitrogen storage and fluxes of ecosystems at landscape scale. Therefore, using either one of the two categories of models is not enough for understanding/solving the current complex environmental issues that involve multiple aspects. Although use of both models (SWAT and DayCent) could be an expedient way toward treating the problem, creating separate model projects for a single area could be challenging and time consuming, and integration/analyses of model results have some limitations due to the non-uniformity of input spatial data between models. To overcome this issue, we developed an integrated model implementation coupler that aims to drive SWAT and DayCent—the two representative models in hydrology and biogeochemistry, respectively—just using a user's SWAT project without the need of any extra efforts such as developing a framework or preparing input data for DayCent modeling. This software is easy to use and would be promising for conducting comprehensive environmental impact assessment involving hydrological and biogeochemical cycles at watershed scale.     

A parameter estimation and identifiability analysis methodology applied to a street canyon air pollution model

Mathematical models are increasingly used in environmental science thus increasing the importance of uncertainty and sensitivity analyses. In the present study, an iterative parameter estimation and identifiability analysis methodology is applied to an atmospheric model – the Operational Street Pollution Model (OSPM^®). To assess the predictive validity of the model, the data is split into an estimation and a prediction data set using two data splitting approaches and data preparation techniques (clustering and outlier detection) are analysed. The sensitivity analysis, being part of the identifiability analysis, showed that some model parameters were significantly more sensitive than others. The application of the determined optimal parameter values was shown to successfully equilibrate the model biases among the individual streets and species. It was as well shown that the frequentist approach applied for the uncertainty calculations underestimated the parameter uncertainties. The model parameter uncertainty was qualitatively assessed to be significant, and reduction strategies were identified.     

A bootstrap approach to assess parameter uncertainty in simple catchment models

Catchment models simulate water and solute dynamics at catchment scales and are invaluable tools for natural resource management. Parameters for catchment models can provide useful information about the importance of the hydrological processes involved. We propose and demonstrate a bootstrap approach to assess parameter uncertainty in dynamic catchment models. This approach, which is non-Bayesian and essentially non-parametric, requires no distributional assumptions about parameters and only weak assumptions about the distributional form of the model residuals. It is able to handle autocorrelated model errors which are very common in the application of dynamic hydrological models at catchment scales. The ability of our bootstrap approach to assess parameter uncertainty is demonstrated using numerical experiments with the abc hydrological model and an application of a conceptual model of salt load from an irrigated catchment in southeastern Australia.     

Using a parallelized MCMC algorithm in R to identify appropriate likelihood functions for SWAT

Markov Chain Monte Carlo (MCMC) algorithms allow the analysis of parameter uncertainty. This analysis can inform the choice of appropriate likelihood functions, thereby advancing hydrologic modeling with improved parameter and quantity estimates and more reliable assessment of uncertainty. For long-running models, the Differential Evolution Adaptive Metropolis (DREAM) algorithm offers spectacular reductions in time required for MCMC analysis. This is partly due to multiple parameter sets being evaluated simultaneously. The ability to use this feature is hindered in models that have a large number of input files, such as SWAT. A conceptually simple, robust method for applying DREAM to SWAT in R is provided. The general approach is transferrable to any executable that reads input files. We provide this approach to reduce barriers to the use of MCMC algorithms and to promote the development of appropriate likelihood functions.     

Parameter uncertainty and temporal dynamics of sensitivity for hydrologic models: A hybrid sequential data assimilation and probabilistic collocation method

In this study, a hybrid sequential data assimilation and probabilistic collocation (HSDAPC) approach is proposed for analyzing uncertainty propagation and parameter sensitivity of hydrologic models. In HSDAPC, the posterior probability distributions of model parameters are first estimated through a particle filter method based on streamflow discharge data. A probabilistic collocation method (PCM) is further employed to show uncertainty propagation from model parameters to model outputs. The temporal dynamics of parameter sensitivities are then generated based on the polynomial chaos expansion (PCE) generated by PCM, which can reveal the dominant model components for different catchment conditions. The maximal information coefficient (MIC) is finally employed to characterize the correlation/association between model parameter sensitivity and catchment precipitation, potential evapotranspiration and observed discharge. The proposed method is applied to the Xiangxi River located in the Three Gorges Reservoir area. The results show that: (i) the proposed HSDAPC approach can generate effective 2nd and 3rd PCE models which provide accuracy predictions; (ii) 2^nd-order PCE, which can run nearly ten time faster than the hydrologic model, can capably represent the original hydrological model to show the uncertainty propagation in a hydrologic simulation; (iii) the slow (R_s) and quick flows (R_q) in Hymod show significant sensitivities during the simulation periods but the distribution factor (α) shows a least sensitivity to model performance; (iv) the model parameter sensitivities show significant correlation with the catchment hydro-meteorological conditions, especially during the rainy period with MIC values larger than 0.5. Overall, the results in this paper indicate that uncertainty propagation and temporal sensitivities of parameters can be effectively characterized through the proposed HSDAPC approach.     

Design of robust reliable H∞ output feedback control for a class of uncertain linear systems with sensor failure

The problem of robust reliable H_∞ output feedback controller design is investigated for uncertain linear systems with sensor failures within a prespecified subset of sensors. The uncertainty considered here is time-varying norm-bounded parameter uncertainty in the state matrix. The output of a faulty sensor is assumed to be any arbitrary energy-bounded signal. An observer-based output feedback control design is presented which stabilizes the plant and guarantees an H_∞ norm bound on attenuation of augmented disturbances, for all admissible uncertainties as well as sensor failures. The construction of the observer-based output feedback control law requires the positive-definite solutions of two algebraic Riccati equations. The result can be regarded as an extension of existing results on robust H_∞ control and reliable H_∞ control of uncertain linear systems.     

A General Probabilistic Framework for uncertainty and global sensitivity analysis of deterministic models: A hydrological case study

The present study proposes a General Probabilistic Framework (GPF) for uncertainty and global sensitivity analysis of deterministic models in which, in addition to scalar inputs, non-scalar and correlated inputs can be considered as well. The analysis is conducted with the variance-based approach of Sobol/Saltelli where first and total sensitivity indices are estimated. The results of the framework can be used in a loop for model improvement, parameter estimation or model simplification. The framework is applied to SWAP, a 1D hydrological model for the transport of water, solutes and heat in unsaturated and saturated soils. The sources of uncertainty are grouped in five main classes: model structure (soil discretization), input (weather data), time-varying (crop) parameters, scalar parameters (soil properties) and observations (measured soil moisture). For each source of uncertainty, different realizations are created based on direct monitoring activities. Uncertainty of evapotranspiration, soil moisture in the root zone and bottom fluxes below the root zone are considered in the analysis. The results show that the sources of uncertainty are different for each output considered and it is necessary to consider multiple output variables for a proper assessment of the model. Improvements on the performance of the model can be achieved reducing the uncertainty in the observations, in the soil parameters and in the weather data. Overall, the study shows the capability of the GPF to quantify the relative contribution of the different sources of uncertainty and to identify the priorities required to improve the performance of the model. The proposed framework can be extended to a wide variety of modelling applications, also when direct measurements of model output are not available.     

Data-based mechanistic modelling of stochastic rainfall-flow processes by state dependent parameter estimation

Due to the inherent nonlinearity in the process of transformation of rainfall into river flow, a simple direct input-output transfer function (TF) model may not sufficiently capture the catchment's hydrological dynamics. This paper presents an application of state dependent parameter (SDP) models for nonlinear, stochastic dynamic system to identify the location and form of the nonlinearity in the rainfall-effective rainfall dynamics. The objective was to develop an effective rainfall input time series that was then used to improve the performance of an originally developed direct input-output TF model of daily rainfall-flow relationship. The CAPTAIN Toolbox in the MATLAB^® environment was used in the model identification in which the recursive filtering and smoothing procedures formulated within a stochastic state space setting were applied to the time series data in order to identify the location and form of nonlinearities within a generic TF model. The nonparametric estimation as well as the parametric optimisation of the resulting nonlinear models was done using the Curve Fitting Toolbox in MATLAB^®. The results showed an improved and more parsimonious TF model. The model improved from explaining only 13% of the data to 56% presenting an improvement of 43% in the model fit. The study demonstrates that simple stochastic but robust tools can be successfully applied to develop and improve applicable hydrological models.     

An enhanced SWAT wetland module to quantify hydraulic interactions between riparian depressional wetlands,rivers and aquifers

This study develops a modified version of the Soil and Water Assessment Tool (SWAT) designed to better represent riparian depressional wetlands (SWATrw). It replaces existing unidirectional hydrological interactions between a wetland and a river/aquifer with a more robust bidirectional approach based on hydraulic principles. SWATrw incorporates a more flexible wetland morphometric formula and a connecting channel concept to model wetland-river interactions. SWAT and SWATrw were tested for the Barak-Kushiyara River Basin (Bangladesh and India). Although the two models showed small differences in simulated stream flow, SWATrw outperformed SWAT in reproducing river stages and the pre-monsoon river-spills into riparian wetlands. SWATrw showed that the observed presence of dry season water in the wetland was due to reduced seepage to the local groundwater table whilst continuous seepage simulated by SWAT resulted in the wetland drying out completely. The new model therefore more closely simulates the hydrological interactions between wetlands, rivers and groundwater.     

Setting up a hydrological model of Alberta: Data discrimination analyses prior to calibration

Failure to setup a large-scale hydrological model correctly may not allow proper calibration and uncertainty analyses, leading to inaccurate model prediction. To build a model with accurate accounting of hydrological processes, a data discrimination procedure was applied in this study. The framework uses a hydrological model of Alberta built with the Soil and Water Assessment Tool (SWAT) program. The model was used to quantify the causes and extents of biases in predictions due to different types of input data. Data types represented different sources of errors, including input data (e.g., climate), conceptual model (e.g., potholes, glaciers), and control structure (e.g., reservoirs, dams). The results showed that accounting for these measures leads to a better physical accounting of hydrological processes, significantly improving the overall model performance. The procedure used in this study helps to avoid unnecessary and arbitrary adjustment of parameters to compensate for the errors in the model structure.     

Calibrating hourly rainfall-runoff models with daily forcings for streamflow forecasting applications in meso-scale catchments

The absence of long sub-daily rainfall records can hamper development of continuous streamflow forecasting systems run at sub-daily time steps. We test the hypothesis that simple disaggregation of daily rainfall data to hourly data, combined with hourly streamflow data, can be used to establish efficient hourly rainfall-runoff models. The approach is tested on four rainfall-runoff models and a range of meso-scale catchments (150–3500 km²). We also compare our disaggregation approach to a method of parameter scaling that attains an hourly parameter-set from daily data.Simple disaggregation of daily rainfall produces hourly streamflow models that perform almost as well as those developed from hourly rainfall data. Rainfall disaggregation performs at least as well as parameter scaling, and often better. For the catchments and models we test, simple disaggregation is a very straightforward and effective way to establish hydrological models for continuous sub-daily streamflow forecasting systems when sub-daily rainfall data are unavailable.     

Simultaneous fault estimation and fault-tolerant tracking control for uncertain nonlinear discrete-time systems

This paper deals with simultaneous fault estimation and control for a class of nonlinear systems with parameter uncertainty, which is described by Takagi–Sugeno (T–S) fuzzy model with parameter uncertainties and unknown disturbance. In this paper, a fuzzy reference model is used to generate error dynamic for tracking control. By considering actuator fault as an auxiliary state vector, we construct an augmented error system and propose a fault estimator/controller to achieve simultaneous fault estimation and fault-tolerant tracking control. H_∞ approach is used in the design of estimator/controller to attenuate the effect of the unknown disturbance and parameter uncertainties. The design conditions are formulated into a set of linear matrix inequalities (LMIs), which can be efficiently solved. Finally, a pitch-axis nonlinear missile model is used to illustrate the effectiveness of the proposed method.     

基于DRFNN的共振频率自适应反推控制

针对共振破碎机频率控制系统的不确定性问题,提出基于动态递归模糊神经网络的自适应反推控制策略。建立了破碎机频率控制系统的数学模型,在忽略不确定性项的前提下,设计了基于自适应Back-stepping方法控制律。其次将电液系统中影响频率控制性能的不确定性因素定义为待估计项,采用动态递归模糊神经网络对其进行实时估计,给出了基于动态递归模糊神经网络的参数自适应律,并通过了Lyapunov的稳定性分析。仿真实验和车载测试结果表明,对于系统参数的不确定性,该方法具有较好地频率控制性能。