Reply on RC1

With the rise of affordable, autonomous sensors and IoT (Internet-of-Things) technology, it is possible to monitor soil moisture in a field online and in real time. This offers opportunities for real-time model calibration for irrigation scheduling. A framework is presented where realtime sensor data are coupled with a soil water balance model to predict soil moisture content and irrigation requirements at field scale. SWIM², Sensor Wielded Inverse Modelling of a Soil Water Irrigation Model, is a framework based on the DREAM inverse modelling approach to estimate 12 model parameters (soil and crop growth parameters) and their uncertainty distribution. These parameter distributions result in soil moisture predictions with a prediction uncertainty estimate, which enables a farmer to anticipate droughts and estimate irrigation requirements. The SWIM² framework was validated based on three growing seasons (2021-2023) in about 30 fields of vegetable growers in Flanders. Kullback–Leibler divergence (KLD) was used as a metric to quantify information gain of the model parameters starting from non-informative priors. Performance was validated in two steps, i.e. the calibration period and prediction period, which is in correspondence with the real-world implementation of the framework. The RMSE, correlation (R, NSE) and Kling-Gupta efficiency (KGE) of soil moisture were analyzed in function of time, i.e. the amount of available sensor data for calibration. Soil moisture can be predicted accurately after 10 to 20 days of sensor data is available for calibration. The RMSE during the calibration period is generally around 0.02 m³/m³, while the RMSE during the prediction period decreases from 0.04 to 0.02 m³/m³ when more calibration data is available. Information gain (KLD) of some parameters (e.g. field capacity and curve number) largely depends on the presence of dynamic events (e.g. precipitation events) during the calibration period. After 40 days of sensor data, the KGE and Pearson correlation of the calibration period become stable with median values of 0.8 and 0.9, respectively. For the validation period, the KGE and Pearson correlation are increasing in time, with median values from 0.3 to 0.7 (KGE) and from 0.7 to 0.95 (R). These good results show that, with this framework, we can simulate and predict soil moisture accurately. These predictions can in turn be used to estimate irrigation requirements. Precipitation radar data was primarily considered as an input without uncertainty. As an extension, precipitation forcing error can be treated in DREAM by applying rainfall multipliers as additional parameters that are estimated in the inverse modelling framework. The multiplicative error of the radar data was quantified by comparison of radar data to rain gauge measurements. The prior uncertainty of the logarithmic multipliers was described by a Laplace distribution and was implemented in DREAM. The extended framework with rainfall multipliers shows better convergence and acceptance rate compared to the main framework. The calibration period shows better performance with higher correlations and lower RMSE values, but a decrease in performance was found for the validation period. These results suggest that the implementation of rainfall multipliers leads to overfitting, resulting in lower predictive power.

An autoregressive moving average (ARMA) is a time series model that is applied in everyday life for pattern recognition and forecasting. The ARMA model contains a noise which is assumed to have a specific distribution. The noise is often considered to have a Gaussian distribution. However in applications, the noise is sometimes found that does not have a Gaussian distribution. The first objective is to develop the ARMA model in which noise has a Laplacian distribution. The second objective is to estimate the parameters of the ARMA model. The ARMA model parameters include ARMA model orders, ARMA model coefficients, and noise variance. The parameter estimation of the ARMA model is carried out in the Bayesian framework. In the Bayesian framework, the ARMA model parameters are treated as a variable that has a prior distribution. The prior distribution for the ARMA model parameters is combined with the likelihood function for the data to get the posterior distribution for the parameter. The posterior distribution for parameters has a complex form so that the Bayes estimator cannot be determined analytically. The reversible jump Markov chain Monte Carlo (MCMC) algorithm was adopted to determine the Bayes estimator. The first result, the ARMA model can be developed by assuming Laplacian distribution noise. The second result, the performance of the algorithm was tested using simulation studies. The simulation shows that the reversible jump MCMC algorithm can estimate the parameters of the ARMA model correctly.

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.

<p>Climate change will have a major impact on Ireland’s water resources. It will pose significant risks to water management and exacerbate existing pressures in terms of water supply, quality, flooding and drought. Early detection of these pressures in hydrological regimes is key to informing adaptation strategies and minimizing adverse environmental and societal impacts. In response to this risk, a framework was developed to enable short-term forecasting of groundwater levels, and to quantify and analyse the impact that climate change will have on Irish groundwater systems.</p><p>A key element of the framework is the use of robust hydrological models to simulate groundwater levels. The framework was developed in Python, with a particular focus on groundwater flooding, and incorporates two modelling approaches: 1) mathematical transfer functions (combination of exponential decay, gamma distribution, and linear decay functions), and 2) a physically based lumped model (reservoir model). For both modelling approaches, precipitation data is converted to effective rainfall based on soil moisture deficit and evapotranspiration data, and the model parameters are calibrated using a Bayesian Markov Chain Monte Carlo algorithm.</p><p>The framework was tested and implemented with synthetically generated groundwater level time series, and with a selection of 12 groundwater dependent wetlands covering a wide range of hydrological behaviours. The tested approaches have proof successful to: 1) produce viable numerical models for those systems, with Nash-Sutcliffe Efficiency (NSE) and Kling-Gupta Efficiency (KGE) values between 0.85 and 0.98 for most of the sites calibration and validation datasets, and 2) inform on how the groundwater systems operate (i.e. multiple outflows, changes in catchment area). The models are now used for forecasting groundwater levels and assessing the potential impact of climate change in ecologically important wetlands.</p><p>The outputs of this project will improve the national ability to understand how groundwater resources respond to climatic stresses and improve the reliability of adaptation planning and predictions in the groundwater sector.</p>

Towards robust quantification and reduction of uncertainty in hydrologic predictions: Integration of particle Markov chain Monte Carlo and factorial polynomial chaos expansion

A joined multi-metric calibration of river discharge and nitrate loads with different performance measures

The Stochastic Volatility Jump Diffusion (SVJD) model is a common tool used for option pricing. Existing literature has employed many techniques in order to estimate the parameters of this model, including both Bayesian and frequentist approaches. In 2010, Duan and Yeh [Jump and volatility risk premiums implied by VIX. J Econ Dyn Control. 2010;34(11):2232–2244] used a frequentist approach with maximum likelihood estimation that included the VIX, the Chicago Board Options Exchange (CBOE) Volatility Index, which carries real time S&P 500 option price information in addition to the S&P 500 data. In this paper, we combine the previous Bayesian approach in Cape et al. [Estimating Heston's and Bates' models parameters using Markov Chain Monte Carlo simulation. J Stat Comput Simul. 2015;85(11):2295–2314] with the inclusion of VIX by Duan and Yeh. We provide the derivations of the conditional posterior distributions of the SVJD model's parameters and a guide to a Markov Chain Monte Carlo (MCMC) algorithm used to estimate the parameters. We apply the algorithm to the historical S&P 500 and VIX data and provide the subsequent parameter estimates. We offer these tools for others performing similar research with our R code available upon request.

Hydrologic modeling in Nigeria is plagued by non-existent or paucity of hydro-metrological/morphological records, which has detrimental impacts on sustainable water resource management and agricultural production. Nowadays, freely accessible remotely sensed products are used as inputs in hydrologic modeling, especially in regions with deficient observed records. Therefore, it is appropriate to utilize the fine-resolution spatial coverage offered by these products in a parameter regionalization method that supports sub-grid variability. This study assessed the transferability of optimized model parameters from a gauged to an ungauged basin using the mesoscale Hydrologic Model (mHM)—Multiscale Parameter Regionalization (MPR) technique. The ability of the fifth generation European Centre for Medium-Range Weather Forecasts Reanalysis product (ERA5), Climate Hazards Group InfraRed Precipitation with Station data (CHIRPS), Global Precipitation Climatology Centre (GPCC), and Multi-Source Weighted-Ensemble Precipitation (MSWEP) gridded rainfall products to simulate observed discharge in three basins was first assessed. Thereafter, the CHIRPS rainfall product was used in three multi-basin mHM setups. Optimized model parameters were then transferred to independent basins, and the reproduction of observed discharges was assessed. Kling–Gupta Efficiency (KGE) scores showed improvements when mHM runs were performed using optimized parameters in comparison to using default parameters for discharge simulations. Optimized mHM runs performed reasonably (KGE > 0.4) for all basins and rainfall products. However, only one basin showed a satisfactory KGE value (KGE = 0.54) when optimized parameters were transferred to an ungauged basin. This study underscores the utility of the mHM-MPR tool for parameter transferability during discharge simulation in data-scarce regions.

Calibration of rainfall-runoff models is made complicated by uncertainties in data, and by the arbitrary emphasis placed on various magnitudes of the model residuals by most traditional measures of fit. Current research highlights the importance of driving model identification by assimilating information from the data. In this paper, we evaluate the potential use of an entropybased measure as an objective function or as a model diagnostic in hydrological modelling, with particular interest in providing an appropriate quantitative measure of fit to the flow duration curve (FDC). The proposed Conditioned Entropy Difference (CED) metric is capable of characterising the information in the flow frequency distribution and thereby constrain the model calibration to respect this distributional information. Four years of hourly data from the 46.6 km2 Mahurangi catchment, NZ, are used to calibrate the 6-parameter Probability Distributed Moisture model. Results are analysed using three measures: the proposed entropy-based measure, the Nash-Sutcliffe (NSE), and the recently proposed Kling-Gupta efficiency (KGE). We also examine a conditioned entropy metric that trades-off and reweights different segments of the FDC to drive model calibration in a way that is based on modelling objectives. Overall, the entropy-based measure results in good performance in terms of NSE but poor performance in terms of KGE. This entropy measure is strongly sensitive to the shape of the flow distribution and is, from some viewpoints, the single best descriptor of the FDC. By conditioning entropy to respect multiple segments of the FDC, we can reweight entropy to respect those parts of the flow distribution of most interest to the modelling application. This approach constrains the behavioural parameter space so as to better identify parameters that represent both the “fast” and “slow” runoff processes. Use of this importance-weighted, conditioned entropy metric can constrain high flow predictions equally well as the NSE and KGE, while simultaneously providing wellconstrained low flow predictions that the NSE or KGE are unable to achieve.

Due to the increasing injection of intermittent power sources (solar+wind) into a common grid, dispatchable sources such as hydro power should be able to help reduce the variability in load and the variability in generation caused by the intermittent sources. A hydro generator should be able to operate short-term beyond its thermal capability limit. This requires the monitoring of internal temperatures in the hydro generator. In this paper, a thermal model of an air-cooled synchronous generator is presented, emphasizing the various aspects of parameter estimation and identifiability using Bayesian inference. Inferences are drawn from the posterior distributions of the parameters and initial conditions, dispersion (spreading) of particles and sampling efficiency, practical parameter identifiability, and model mismatch with experiments. Results show extremely narrow parameter distributions. It is early to generalize about the posterior distribution of air-related and metal-related parameters of the air-cooled synchronous generator based on the single experimental data presented here.

Distributed Markov Chain Monte Carlo for Bayesian Hierarchical Models

Bayesian model updating has received considerable attention and has been extensively used in structural damage detection. It provides a rigorous statistical framework for realizing structural system identification and characterizing uncertainties associated with modeling and measurements. The Markov Chain Monte Carlo (MCMC) is a promising tool for inferring the posterior distribution of model parameters to avoid the intractable evaluation of multi-dimensional integration. However, the efficacy of most MCMC techniques suffers from the curse of parameter dimension, which restricts the application of Bayesian model updating to the damage detection of large-scale systems. In addition, there are several MCMC techniques that require users to properly choose application-specific models, based on the understanding of algorithm mechanisms and limitations. As seen in the literature, there is a lack of comprehensive work that investigates the performances of various MCMC algorithms in their application of structural damage detection. In this study, the Differential Evolutionary Adaptive Metropolis (DREAM), a multi-chain MCMC, is explored and adapted to Bayesian model updating. This paper illustrates how DREAM is used for model updating with many uncertainty parameters (i.e., 40 parameters). Furthermore, the study provides a tutorial to users who may be less experienced with Bayesian model updating and MCMC. Two advanced single-chain MCMC algorithms, namely, the Delayed Rejection Adaptive Metropolis (DRAM) and Transitional Markov Chain Monte Carlo (TMCMC), and DREAM are elaborately introduced to allow practitioners to understand better the concepts and practical implementations. Their performances in model updating and damage detection are compared through three different engineering applications with increased complexity, e.g., a forty-story shear building, a two-span continuous steel beam, and a large-scale steel pedestrian bridge.

Abstract. A better understanding of the reasons why hydrological model performance is unsatisfying represents a crucial part of meaningful model evaluation. However, current evaluation efforts are mostly based on aggregated efficiency measures such as Kling–Gupta efficiency (KGE) or Nash–Sutcliffe efficiency (NSE). These aggregated measures provide a relative gradation of model performance. Especially in the case of a weak model performance it is important to identify the different errors which may have caused such unsatisfactory predictions. These errors may originate from the model parameters, the model structure, and/or the input data. In order to provide more insight, we define three types of errors which may be related to their source: constant error (e.g. caused by consistent input data error such as precipitation), dynamic error (e.g. structural model errors such as a deficient storage routine) and timing error (e.g. caused by input data errors or deficient model routines/parameters). Based on these types of errors, we propose the novel diagnostic efficiency (DE) measure, which accounts for these three error types. The disaggregation of DE into its three metric terms can be visualized in a plain radial space using diagnostic polar plots. A major advantage of this visualization technique is that error contributions can be clearly differentiated. In order to provide a proof of concept, we first generated time series artificially with the three different error types (i.e. simulations are surrogated by manipulating observations). By computing DE and the related diagnostic polar plots for the reproduced errors, we could then supply evidence for the concept. Finally, we tested the applicability of our approach for a modelling example. For a particular catchment, we compared streamflow simulations realized with different parameter sets to the observed streamflow. For this modelling example, the diagnostic polar plot suggests that dynamic errors explain the overall error to a large extent. The proposed evaluation approach provides a diagnostic tool for model developers and model users and the diagnostic polar plot facilitates interpretation of the proposed performance measure as well as a relative gradation of model performance similar to the well-established efficiency measures in hydrology.

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> We develop a hierarchical Bayesian approach for estimating defect signals from noisy measurements and apply it to nondestructive evaluation (NDE) of materials. We propose a parametric model for the shape of the defect region and assume that the defect signals within this region are random with unknown mean and variance. Markov chain Monte Carlo (MCMC) algorithms are derived for simulating from the posterior distributions of the model parameters and defect signals. These algorithms are then utilized to identify potential defect regions and estimate their size and reflectivity parameters. Our approach provides Bayesian confidence regions (credible sets) for the estimated parameters, which are important in NDE applications. We specialize the proposed framework to elliptical defect shape and Gaussian signal and noise models and apply it to experimental ultrasonic <formula formulatype="inline"><tex>$C$</tex> </formula>-scan data from an inspection of a cylindrical titanium billet. We also outline a simple classification scheme for separating defects from nondefects using estimated mean signals and areas of the potential defects. </para>

The X-ray Earth occultation sounding (XEOS) is an emerging method for measuring the neutral density in the lower thermosphere. In this paper, the X-ray Earth occultation (XEO) of the Crab Nebula is investigated by using the Insight-HXMT. The pointing observation data on the 30th September, 2018 recorded by the Low Energy X-ray telescope (LE) of Insight-HXMT are selected and analyzed. The extinction lightcurves and spectra during the X-ray Earth occultation process are extracted. A forward model for the XEO lightcurve is established and the theoretical observational signal for lightcurve is predicted. A Bayesian data analysis method is developed for the XEO lightcurve modeling and the atmospheric density retrieval. The posterior probability distribution of the model parameters is derived through the Markov Chain Monte Carlo (MCMC) algorithm with the NRLMSISE-00 model and the NRLMSIS 2.0 model as basis functions and the best-fit density profiles are retrieved respectively. It is found that in the altitude range of 105–200 km, the retrieved density profile is 88.8 % of the density of NRLMSISE-00 and 109.7 % of the density of NRLMSIS 2.0 by fitting the lightcurve in the energy range of 1.0–2.5 keV based on XEOS method. In the altitude range of 95–125 km, the retrieved density profile is 81.0 % of the density of NRLMSISE-00 and 92.3 % of the density of NRLMSIS 2.0 by fitting the lightcurve in the energy range of 2.5–6.0 keV based on XEOS method. In the altitude range of 85–110 km, the retrieved density profile is 87.7 % of the density of NRLMSISE-00 and 101.4 % of the density of NRLMSIS 2.0 by fitting the lightcurve in the energy range of 6.0–10.0 keV based on XEOS method. The measurements of density profiles are compared with the NRLMSISE-00/NRLMSIS 2.0 model simulations and the previous retrieval results with RXTE satellite. Finally, we find that the retrieved density profile from Insight-HXMT based on the NRLMSISE-00/NRLMSIS 2.0 models is qualitatively consistent with the previous retrieved results from RXTE. This study demonstrate that the XEOS from the X-ray astronomical satellite Insight-HXMT can provide an approach for the study of the upper atmosphere. The Insight-HXMT satellite can join the family of the XEOS. The Insight-HXMT satellite with other X-ray astronomical satellites in orbit can form a space observation network for XEOS in the future.