Reduction In Parameter Uncertainty Research Articles

Identifying More Realistic Model Structures by Electrical Conductivity Observations of the Karst Spring

AbstractElectrical conductivity (EC) of karst spring discharge has always been a fundamental variable to characterize karst systems. However, to incorporate EC into the lumped hydrologic modeling is challenging but has a huge potential since EC observations are widely collected. In this study, we present a new framework to integrate EC into lumped karst hydrological models for model structure identification and parameter uncertainty reduction. Our framework is tested in a small, well‐instrumented karst catchment near Guilin city (China) where EC dynamics are mostly controlled by the dissolution of carbonate rock and dilution by event water. Four karst models with different structures were equipped to consider the linear growth of EC with the carbonate rock dissolution and its mixing within the karst system. Applying a parameter estimation framework that accounts for uncertainty in discharge and EC simulations, we find that all hydrologic models obtain similar performances concerning spring discharge. However, their performances in simulating spring EC strongly differ permitting us to identify the most realistic model out of the four model structures. Our analysis further shows that including EC into parameter estimation reduces parameter uncertainty concerning concentrated flow in the conduits and helps to estimate immobile water storage in the system. Consequently, our new framework to include EC in karst models provides new directions for the more realistic simulation that can easily be transferred to other karst systems where EC dynamics are controlled by rock dissolution and dilution by event water.

Parameter Optimization for Uncertainty Reduction and Simulation Improvement of Hydrological Modeling

Hydrological modeling has experienced rapid development and played a significant role in water resource management in recent decades. However, modeling uncertainties, which are propagated throughout model runs, may affect the credibility of simulation results and mislead management decisions. Therefore, analyzing and reducing uncertainty is of significant importance in providing greater confidence in hydrological simulations. To reduce and quantify parameter uncertainty, in this study, we attempted to introduce additional remotely sensed data (such as evapotranspiration (ET)) into a common parameter estimation procedure that uses observed streamflow only. We undertook a case study of an application of the Soil Water Assessment Tool in the Guijiang River Basin (GRB) in China. We also compared the effects of different combinations of parameter estimation algorithms (e.g., Sequential Uncertainty Fitting version 2, particle swarm optimization) on reduction in parameter uncertainty and improvement in modeling precision improvement. The results indicated that combining Sequential Uncertainty Fitting version 2 (SUFI-2) and particle swarm optimization (PSO) can substantially reduce the modeling uncertainty (reduction in the R-factor from 0.9 to 0.1) in terms of the convergence of parameter ranges and the aggregation of parameters, in addition to iterative optimization. Furthermore, the combined approaches ensured the rationality of the parameters’ physical meanings and reduced the complexity of the model calibration procedure. We also found the simulation accuracy of ET improved substantially after adding remotely sensed ET data. The parameter ranges and optimal parameter sets obtained by multi-objective calibration (using streamflow plus ET) were more reasonable and the Nash–Sutcliffe coefficient (NSE) improved more rapidly using multiple objectives, indicating a more efficient parameter optimization procedure. Overall, the selected combined approach with multiple objectives can help reduce modeling uncertainty and attain a reliable hydrological simulation. The presented procedure can be applied to any hydrological model.

Bayesian Information-Theoretic Calibration of Radiotherapy Sensitivity Parameters for Informing Effective Scanning Protocols in Cancer

With new advancements in technology, it is now possible to collect data for a variety of different metrics describing tumor growth, including tumor volume, composition, and vascularity, among others. For any proposed model of tumor growth and treatment, we observe large variability among individual patients’ parameter values, particularly those relating to treatment response; thus, exploiting the use of these various metrics for model calibration can be helpful to infer such patient-specific parameters both accurately and early, so that treatment protocols can be adjusted mid-course for maximum efficacy. However, taking measurements can be costly and invasive, limiting clinicians to a sparse collection schedule. As such, the determination of optimal times and metrics for which to collect data in order to best inform proper treatment protocols could be of great assistance to clinicians. In this investigation, we employ a Bayesian information-theoretic calibration protocol for experimental design in order to identify the optimal times at which to collect data for informing treatment parameters. Within this procedure, data collection times are chosen sequentially to maximize the reduction in parameter uncertainty with each added measurement, ensuring that a budget of n high-fidelity experimental measurements results in maximum information gain about the low-fidelity model parameter values. In addition to investigating the optimal temporal pattern for data collection, we also develop a framework for deciding which metrics should be utilized at each data collection point. We illustrate this framework with a variety of toy examples, each utilizing a radiotherapy treatment regimen. For each scenario, we analyze the dependence of the predictive power of the low-fidelity model upon the measurement budget.

Robust Inverse Modeling of Growing Season Net Ecosystem Exchange in a Mountainous Peatland: Influence of Distributional Assumptions on Estimated Parameters and Total Carbon Fluxes

AbstractWhile boreal lowland bogs have been extensively studied using the eddy‐covariance (EC) technique, less knowledge exists on mountainous peatlands. Hence, half‐hourly CO2 fluxes of an ombrotrophic peat bog in the Harz Mountains, Germany, were measured with the EC technique during a growing season with exceptionally dry weather spells. A common biophysical process model for net ecosystem exchange was used to describe measured CO2 fluxes and to fill data gaps. Model parameters and uncertainties were estimated by robust inverse modelling in a Bayesian framework using a population‐based Markov Chain Monte Carlo sampler. The focus of this study was on the correct statistical description of error, i.e. the differences between the measured and simulated carbon fluxes, and the influence of distributional assumptions on parameter estimates, cumulative carbon fluxes, and uncertainties. We tested the Gaussian, Laplace, and Student's t distribution as error models. The t‐distribution was identified as best error model by the deviance information criterion. Its use led to markedly different parameter estimates, a reduction of parameter uncertainty by about 40%, and, most importantly, to a 5% higher estimated cumulative CO2 uptake as compared to the commonly assumed Gaussian error distribution. As open‐path measurement systems have larger measurement error at high humidity, the standard deviation of the error was modeled as a function of measured vapor pressure deficit. Overall, this paper demonstrates the importance of critically assessing the influence of distributional assumptions on estimated model parameters and cumulative carbon fluxes between the land surface and the atmosphere.

Improving energy model calibration of multi-unit residential buildings through component air infiltration testing

Building infiltration rates are one of the most uncertain parameters among multi-unit residential building (MURB) energy model inputs and have the potential to greatly impact building energy consumption. Infiltration rates exhibit high spatial and temporal variability and are highly building-specific making them difficult to estimate from published data. Reduction of parameter uncertainty using on-site measurements has traditionally been prohibitive, both from a cost and a logistical standpoint.Window component infiltration rate testing was conducted at two MURBs to develop component-weighted infiltration rates, which were input into whole-building energy models and compared with models that used a single building-level infiltration rate. The component-weighted infiltration rate allowed for the incorporation of building-specific measurements, which helped to reduce parameter uncertainty during model calibration. The component-weighted approach also helped to account for the variation in zone infiltration rates based on type and quantity of leakage paths. The discretization of the infiltration rate by building component can help during the modeling of energy retrofits that specifically target building infiltration, by setting bounds on the maximum possible improvement. Implementation of a component-weighted approach will help to improve model calibration and energy retrofit analysis in whole-building energy models, allowing owners and operators to make better-informed decisions regarding energy retrofits.

Estimating Soil Hydraulic Parameters by Inverse Modeling with PEST

Core Ideas Coupling SWAP with PEST, soil moisture and ET are included in inverse modeling of soil water flow. Multi‐objective optimization could improve model predictions and reduce parameter uncertainty. Observation errors and frequency and spatial arrangement impact predictions and uncertainty. The accurate estimation of soil hydraulic properties is an important part of the application of hydrological models for quantifying water transport in the vadose zone. Various inverse models have been developed to solve hydraulic properties optimization problems, and the accuracy of the results obtained by different algorithms is still debated because of the inherent ill‐posedness of such problems. In this study, we coupled an agrohydrological Soil–Water–Atmosphere–Plant (SWAP) model with an independent parameter estimation program (PEST) to calibrate the soil hydraulic parameters and investigate the uncertainty in the parameter estimations. The objectives of this study were to assess to what extent the SWAP model can be calibrated from a single observation (only soil moisture, θ) and further to investigate whether involving additional evapotranspiration fluxes including actual evapotranspiration (ETa), actual evaporation (Ea), or actual transpiration (Ta) can lead to an improvement in model predictions and a reduction in parameter uncertainty. Extensive synthetic experiments were conducted to achieve the objectives. Both double (winter wheat [Triticum aestivum L.]–summer maize [Zea mays L.]) and single (only winter wheat or only summer maize) cropping systems were considered in the multi‐objective optimizations. The results indicate that although the addition of evapotranspiration fluxes does not necessarily improve the accuracy of the soil moisture prediction, it can reduce the parameter uncertainty for both single‐ and double‐cropping systems. Moreover, the parameter estimation of the topsoil layer could greatly benefit from the addition of Ea, whereas the addition of Ta could help reduce the parameter uncertainty of the lower layers.

An information theoretic approach to use high-fidelity codes to calibrate low-fidelity codes

For many simulation models, it can be prohibitively expensive or physically infeasible to obtain a complete set of experimental data to calibrate model parameters. In such cases, one can alternatively employ validated higher-fidelity codes to generate simulated data, which can be used to calibrate the lower-fidelity code. In this paper, we employ an information-theoretic framework to determine the reduction in parameter uncertainty that is obtained by evaluating the high-fidelity code at a specific set of design conditions. These conditions are chosen sequentially, based on the amount of information that they contribute to the low-fidelity model parameters. The goal is to employ Bayesian experimental design techniques to minimize the number of high-fidelity code evaluations required to accurately calibrate the low-fidelity model. We illustrate the performance of this framework using heat and diffusion examples, a 1-D kinetic neutron diffusion equation, and a particle transport model, and include initial results from the integration of the high-fidelity thermal-hydraulics code Hydra-TH with a low-fidelity exponential model for the friction correlation factor.

Optimum Experimental Design for Patient Specific Mathematical Leukopenia Models

Abstract: Mathematical models are essential for simulation-driven decision support for clinical doctors. For an estimation of parameters for patient specific models, values such as the number of certain blood cells need to be measured. In this paper we focus on leukopenia, a clinically important side effect arising from the treatment of leukemia with chemotherapy. A mathematical leukopenia model is presented describing the dynamics of leukocytes and we show that the standard deviations of the parameter estimates depend strongly on the timing of the measurements. We discuss the issue of measurement time points for two patients being in the consolidation phase of acute myeloid leukemia and provide optimal solutions. Optimized measurement time points and the thus enabled accurate simulations have a large impact: drug treatments can be adapted individually and patients may safely leave the hospital for longer and more convenient time intervals. The dynamics of leukocytes are modeled by a system of ordinary differential equations and the chemotherapy with cytarabine is described by a pharmacokinetics/pharmacodynamics model consisting of two compartments and a log-linear function representing the drug effect. The measurement time points are optimized by optimal experimental design. With optimal experimental design an average parameter uncertainty reduction of 57% ( Patient 1 ) and 80% ( Patient 2 ) can be achieved compared to the clinical experimental designs, with the same total number of measurements. These encouraging results motivate further research and an extension of the data basis to more patients.

Streamflow rating uncertainty: Characterisation and impacts on model calibration and performance

Common streamflow gauging procedures require assumptions about the stage-discharge relationship (the ‘rating curve’) that can introduce considerable uncertainties in streamflow records. These rating uncertainties are not usually considered fully in hydrological model calibration and evaluation yet can have potentially important impacts. We analysed streamflow gauge data and conducted two modelling experiments to assess rating uncertainty in operational rating curves, its impacts on modelling and possible ways to reduce those impacts. We found clear evidence of variance heterogeneity (heteroscedasticity) in streamflow estimates, with higher residual values at higher stage values. In addition, we confirmed the occurrence of streamflow extrapolation beyond the highest or lowest stage measurement in many operational rating curves, even when these were previously flagged as not extrapolated. The first experiment investigated the impact on regional calibration/evaluation of: (i) using two streamflow data transformations (logarithmic and square-root), compared to using non-transformed streamflow data, in an attempt to reduce heteroscedasticity and; (ii) censoring the extrapolated flows, compared to no censoring. Results of calibration/evaluation showed that using a square-root transformed streamflow (thus, compromising weight on high and low streamflow) performed better than using non-transformed and log-transformed streamflow. Also, surprisingly, censoring extrapolated streamflow reduced rather than improved model performance. The second experiment investigated the impact of rating curve uncertainty on catchment calibration/evaluation and parameter estimation. A Monte-Carlo approach and the nonparametric Weighted Nadaraya-Watson (WNW) estimator were used to derive streamflow uncertainty bounds. These were later used in calibration/evaluation using a standard Nash-Sutcliffe Efficiency (NSE) objective function (OBJ) and a modified NSE OBJ that penalised uncertain flows. Using square-root transformed flows and the modified NSE OBJ considerably improved calibration and predictions, particularly for mid and low flows, and there was an overall reduction in parameter uncertainty.

Evaluation of mixed forest evapotranspiration and soil moisture using measured and swat simulated results in a hillslope watershed

This study evaluated SWAT model when measured Evapotranspiration (ET) and Soil Moisture (SM) in addition to streamflow (Q) data were used to calibrate the simulated values in a 8.54 km2 hillslope watershed of South Korea. By using 7 years (2003–2009) daily Q at the watershed outlet, 3 years (2007–2009) daily ET and SM data measured in the mixed forest, the SWAT model was calibrated and validated with the average Nash-Sutcliffe model efficiency (E) for Q of 0.76, and the coefficient of determination (R2) for SM and ET of 0.55 and 0.59, respectively. The use of measured hydrological state variables such as ET or SM in SWAT modeling provided more reliable interpretation of model parameters by inducing the reduction of parameter uncertainty and understanding the behavior of the watershed water balance. When the watershed has only Q data for model calibration, it is encouraged to consider soil (ESCO) and plant related parameters (EPCO and CANMX) in addition to surface runoff (CN2, and Surlag) and groundwater related parameters (GW_ DELAY and GW_ REVAP).

A Tree-Ring-Based Reconstruction of Delaware River Basin Streamflow Using Hierarchical Bayesian Regression

AbstractA hierarchical Bayesian regression model is presented for reconstructing the average summer streamflow at five gauges in the Delaware River basin using eight regional tree-ring chronologies. The model provides estimates of the posterior probability distribution of each reconstructed streamflow series considering parameter uncertainty. The vectors of regression coefficients are modeled as draws from a common multivariate normal distribution with unknown parameters estimated as part of the analysis. This leads to a multilevel structure. The covariance structure of the streamflow residuals across sites is explicitly modeled. The resulting partial pooling of information across multiple stations leads to a reduction in parameter uncertainty. The effect of no pooling and full pooling of station information, as end points of the method, is explored. The no-pooling model considers independent estimation of the regression coefficients for each streamflow gauge with respect to each tree-ring chronology. The full-pooling model considers that the same regression coefficients apply across all streamflow sites for a particular tree-ring chronology. The cross-site correlation of residuals is modeled in all cases. Performance on metrics typically used by tree-ring reconstruction experts, such as reduction of error, coefficient of efficiency, and coverage rates under credible intervals is comparable to, or better, for the partial-pooling model relative to the no-pooling model, and streamflow estimation uncertainty is reduced. Long record simulations from reconstructions are used to develop estimates of the probability of duration and severity of droughts in the region. Analysis of monotonic trends in the reconstructed drought events do not reject the null hypothesis of no trend at the 90% significance over 1754–2000.

Parameter Uncertainty Reduction for SWAT Using Grace, Streamflow, and Groundwater Table Data for Lower Missouri River Basin1

Abstract: This study incorporates the newly available Gravity Recovery and Climate Experiment (GRACE) water storage data and water table data from well logs to reduce parameter uncertainty in Soil and Water Assessment Tool (SWAT) calibration using a SUFI2 (sequential uncertainty fitting) framework for the Lower Missouri River Basin. Model evaluations are performed in multiple stages using a multiobjective function consisting of multisite streamflow and GRACE water storage data as well as a groundwater component. Results show that (1) a model calibrated with both streamflow and GRACE data simultaneously can maintain the water balance for the whole basin, but may improperly partition surface flow and base flow. Additional inclusion of the groundwater constraint can significantly improve the model performance in groundwater hydrological processes. In our case, the estimation of specific yield of shallow aquifers has been increased to 10−2 from previous much underestimated level (<10−3). (2) The daily streamflow data are needed to confine the parameters related to water flow in channels such as the Manning’s coefficient, which are less sensitive to the monthly simulations. (3) Parameters are nonuniformly sensitive for different goal variables, and thus, proper specification of a prior distribution of parameters may be the key factor for global optimization algorithms to obtain stable and realistic model performance.

Network inference via adaptive optimal design

BackgroundCurrent research in network reverse engineering for genetic or metabolic networks very often does not include a proper experimental and/or input design. In this paper we address this issue in more detail and suggest a method that includes an iterative design of experiments based, on the most recent data that become available. The presented approach allows a reliable reconstruction of the network and addresses an important issue, i.e., the analysis and the propagation of uncertainties as they exist in both the data and in our own knowledge. These two types of uncertainties have their immediate ramifications for the uncertainties in the parameter estimates and, hence, are taken into account from the very beginning of our experimental design.FindingsThe method is demonstrated for two small networks that include a genetic network for mRNA synthesis and degradation and an oscillatory network describing a molecular network underlying adenosine 3’-5’ cyclic monophosphate (cAMP) as observed in populations of Dyctyostelium cells. In both cases a substantial reduction in parameter uncertainty was observed. Extension to larger scale networks is possible but needs a more rigorous parameter estimation algorithm that includes sparsity as a constraint in the optimization procedure.ConclusionWe conclude that a careful experiment design very often (but not always) pays off in terms of reliability in the inferred network topology. For large scale networks a better parameter estimation algorithm is required that includes sparsity as an additional constraint. These algorithms are available in the literature and can also be used in an adaptive optimal design setting as demonstrated in this paper.

Calibrating Vadose Zone Models with Time‐Lapse Gravity Data

The vadose zone plays an important role in the hydrologic cycle. Various geophysical methods can determine soil water content variations in time and space in volumes ranging from a few cubic centimeters to several cubic meters. In contrast to the established methods, time‐lapse gravity measurements of changes in soil water content do not rely on a petrophysical relationship between the measured quantity and the water content but give a direct measure of the mass change in the soil. Only recently has the vadose zone been systematically incorporated when ground‐based gravity data are used to infer hydrologic information. In this study, changes in the soil water content gave rise to a measurable signal in a forced infiltration experiment on a 107‐m2 grassland area. Time‐lapse gravity data were able to constrain the van Genuchten soil hydraulic parameters in both a synthetic example and a field experiment with forced infiltration. The most significant reduction in parameter uncertainty was achieved for the saturated water content, while gravity data had some ability to constrain the saturated hydraulic conductivity and the van Genuchten n Cross‐borehole ground penetrating radar data were used to support the interpretation and control of the gravity data.

Interval Uncertainty Reduction and Single-Disciplinary Sensitivity Analysis With Multi-Objective Optimization

Sensitivity analysis has received significant attention in engineering design. While sensitivity analysis methods can be global, taking into account all variations, or local, taking into account small variations, they generally identify which uncertain parameters are most important and to what extent their effect might be on design performance. The extant methods do not, in general, tackle the question of which ranges of parameter uncertainty are most important or how to best allocate Investments to partial uncertainty reduction in parameters under a limited budget. More specifically, no previous approach has been reported that can handle single-disciplinary multi-output global sensitivity analysis for both a single design and multiple designs under interval uncertainty. Two new global uncertainty metrics, i.e., radius of output sensitivity region and multi-output entropy performance, are presented. With these metrics, a multi-objective optimization model is developed and solved to obtain fractional levels of parameter uncertainty reduction that provide the greatest payoff in system performance for the least amount of “Investment.” Two case studies of varying difficulty are presented to demonstrate the applicability of the proposed approach.

Data-worth analysis for multiobjective optimal design of pump-and-treat remediation systems

The design and the management of pump-and-treat (PAT) remediation systems for contaminated aquifers under uncertain hydrogeological settings and parameters often involve decisions that trade off cost optimality against reliability. Both design objectives can be improved by planning site characterization programs that reduce subsurface parameter uncertainty. However, the cost for subsurface investigation often weighs heavily upon the budget of the remedial action and must thus be taken into account in the trade-off analysis. In this paper, we develop a stochastic data-worth framework with the purpose of estimating the economic opportunity of subsurface investigation programs. Since the spatial distribution of hydraulic conductivity is most often the major source of uncertainty, we focus on the direct sampling of hydraulic conductivity at prescribed locations of the aquifer. The data worth of hydraulic conductivity measurements is estimated from the reduction of the overall management cost ensuing from the reduction in parameter uncertainty obtained from sampling. The overall cost is estimated as the expected value of the cost of installing and operating the PAT system plus penalties incurred due to violations of cleanup goals and constraints. The crucial point of the data-worth framework is represented by the so-called pre-posterior analysis. Here, the tradeoff between decreasing overall costs and increasing site-investigation budgets is assessed to determine a management strategy proposed on the basis of the information available at the start of remediation. The goal of the pre-posterior analysis is to indicate whether the proposed management strategy should be implemented as is, or re-designed on the basis of additional data collected with a particular site-investigation program. The study indicates that the value of information is ultimately related to the estimates of cleanup target violations and decision makers’ degree of risk-aversion.

Utility of different data types for calibrating flood inundation models within a GLUE framework

Abstract. To translate a point hydrograph forecast into products for use by environmental agencies and civil protection authorities, a hydraulic model is necessary. Typical one- and two-dimensional hydraulic models are able to predict dynamically varying inundation extent, water depth and velocity for river and floodplain reaches up to 100 km in length. However, because of uncertainties over appropriate surface friction parameters, calibration of hydraulic models against observed data is a necessity. The value of different types of data is explored in constraining the predictions of a simple two-dimensional hydraulic model, LISFLOOD-FP. For the January 1995 flooding on the River Meuse, The Netherlands, a flow observation data set has been assembled for the 35-km reach between Borgharen and Maaseik, consisting of Synthetic Aperture Radar and air photo images of inundation extent, downstream stage and discharge hydrographs, two stage hydrographs internal to the model domain and 84 point observations of maximum free surface elevation. The data set thus contains examples of all the types of data that potentially can be used to calibrate flood inundation models. 500 realisations of the model have been conducted with different friction parameterisations and the performance of each realisation has been evaluated against each observed data set. Implementation of the Generalised Likelihood Uncertainty Estimation (GLUE) methodology is then used to determine the value of each data set in constraining the model predictions as well as the reduction in parameter uncertainty resulting from the updating of generalised likelihoods based on multiple data sources.

Optimal experimental design for parameter estimation in column outflow experiments

Inverse modeling is increasingly used for water flow and reactive solute transport in the unsaturated zone to determine unknown parameters, e.g., soil hydraulic parameters, degradation rates or dispersion coefficients. However, experimental data are frequently not suitable for parameter estimation leading instead to practically singular estimation problems. Optimal experimental design aims at identifying experimental conditions and sampling schemes that deliver measurement data which are most sensitive to unknown parameters. A new numerical approach to optimize experimental designs for parameter estimation in coupled partial differential equations (PDEs) and ordinary differential equation (ODEs) arising from transport and degradation processes is presented. The optimal experimental design problem is embedded in the framework of optimal control theory. This results in an intricate, nonlinear, constrained optimization problem that is solved by a direct approach based on a structured SQP method. These methods are used to investigate several scenarios for a column outflow experiment in a simulation study. Simultaneously, experimental conditions, such as irrigation schemes, substance concentrations, and sampling schemes are optimized. Result is an enormous reduction of parameter uncertainty. Additionally, it is demonstrated that it is possible to estimate both soil hydraulic and solute transport parameters together in one experiment on the basis of leachate data only. The usefulness of the numerical approach for practical applications is discussed.

Automatic calibration of conceptual rainfall-runoff models: sensitivity to calibration data

The identification of hydrologic models requires that appropriate data be selected for model calibration. In the research presented here, the shuffled complex evolution (SCE-UA) global optimization method was used to calibrate the NWSRFS-SMA conceptual rainfall-runoff flood forecasting model of the US National Weather Service, using a 40-year record of historical data. Based on 344 calibration runs using different lengths of data from different sections of the historical record, we conclude that approximately 8 years of data are required to obtain calibrations that are relatively insensitive to the period selected. Further, the reduction in parameter uncertainty is maximal when the wettest data periods on record are used. A residual analysis is used to compare the performance of the daily root mean square (DRMS) and heteroscedastic maximum likelihood error (HMLE) objective functions. The results suggest that the factor currently limiting model performance is the unavailability of strategies that explicitly account for model error during calibration.