Differential Evolution Adaptive Metropolis Algorithm Research Articles

Probabilistic Predictions of Ecologically Relevant Hydrologic Indices Using a Hydrological Model

AbstractThis study evaluated the effects of prior knowledge obtained from multi‐objective optimization on the uncertainty quantification of simulated ecologically relevant hydrologic indices (ERHIs) using a hydrological model. Two experiments were formulated considering contrasting conditions in prior knowledge when implementing Bayesian parameter estimation. In the first experiment, we reproduced a common modeling practice in the absence of previous information where uniform priors are considered when calibrating hydrological and error model parameters. In the second experiment, near‐optimal Pareto solutions from multi‐objective calibration were used to build a multivariate prior distribution for estimating hydrological and error model parameters using Bayesian calibration. In both experiments, a likelihood function considering heteroscedasticity and autocorrelation effects was employed. The Unified Non‐dominated Sorting Algorithm III was used for multi‐objective calibration, whereas the multiple‐try differential evolution adaptive metropolis algorithm was used for Markov Chain Monte Carlo sampling. The experiments were tested using a hydrologic model. We selected a subset of ERHIs comprised of 32 central tendency Indicators of Hydrologic Alteration and seven additional indices describing basic streamflow statistical properties. In general, using near‐optimal Pareto parameter distributions as prior knowledge in Bayesian calibration reduced both the bias and variability ranges in ERHIs prediction. Furthermore, there was no significant loss in the reliability of streamflow predictions (from 2% to 6%, p = 0.07) when targeting the selected ERHIs, while improving precision (from 66% to 58%, p < 0.05) and reducing the bias (from 6% to 4%, p < 0.05). Moreover, parametric uncertainty substantially shrank by 72% when linking multi‐objective calibration and Bayesian parameter estimation.

Parameter identification and uncertainty quantification of a non‐linear pump‐turbine governing system based on the differential evolution adaptive Metropolis algorithm

Pumped storage units (PSUs) are now widely used for energy storage. However, the uncertainty of the identification results of the pump-turbine governing system (PTGS) caused by the random observation noises and the lack of prior knowledge remains an unaddressed issue. In recent years, the differential evolution adaptive Metropolis algorithm (DREAM) based on the Bayesian theory has been extensively used for parameter estimation and uncertainty analysis, but its application to the uncertainty analysis of PTGS has been rare. This study systematically evaluates the applicability of DREAM in the parameter identification and uncertainty quantification of PTGS. A real PSU in China has been employed as a case study for numerical experiments. Four groups of control experiments with different proportions of observation noises and different prior search spaces have been constructed in this study. It can be concluded from this study that: (a) accurate point identification results and effective uncertainty quantification can be obtained simultaneously using DREAM, (b) a lower proportion of observation noises can enhance the efficiency and effectiveness of the DREAM algorithm when applying to PTGS simulation. The DREAM method can narrow the prior search space of PTGS parameters effectively and thus help the hydropower engineers to make decisions.

Stochastic inversion method for dynamic constitutive model of rock materials based on improved DREAM

The difference in dynamic mechanical properties of different rock samples leads to a random process of deformation and stress. Uncertainties exist in the parameters of the rock material dynamic constitutive model. Unlike traditional inversion analysis methods, this paper treats model parameters as random variables. Bayesian theory and Differential Evolution Adaptive Metropolis algorithm (DREAM) are used to accurately quantify the uncertainty of the dynamic constitutive model parameters. The error function of the DREAM algorithm is divided into prediction error and model error to mitigate the effect of parameter compensation and improve the prediction ability. Peak stress error term is added to the DREAM prediction error to increase the peak stress fitting accuracy. Experiments of shock compression under freezing and high temperature cooling damage are presented to illustrate the proposed method. Different optimization algorithms such as Genetic Algorithm and Ant Colony Optimization are used to compare the accuracy of inversion parameters. The results show that the obtained peak stress fitting accuracy of the dynamic constitutive model is significantly improved. The 95% confidence interval considering model errors almost completely covers the experimental observations, indicating that the parameter probability distribution interval can accurately cover the true value while reflecting the model uncertainty. Using inversion parameters to predict other working conditions, 95% confidence interval considering model errors has high coverage. Thus, stochastic inversion method can accurately fit and predict the deformation and failure law of rock under impact load.

Improving Simulation Efficiency of MCMC for Inverse Modeling of Hydrologic Systems With a Kalman‐Inspired Proposal Distribution

AbstractBayesian analysis is widely used in science and engineering for real‐time forecasting, decision making, and to help unravel the processes that explain the observed data. These data are some deterministic and/or stochastic transformations of the underlying parameters. A key task is then to summarize the posterior distribution of these parameters. When models become too difficult to analyze analytically, Monte Carlo methods can be used to approximate the target distribution. Of these, Markov chain Monte Carlo (MCMC) methods are particularly powerful. Such methods generate a random walk through the parameter space and, under strict conditions of reversibility and ergodicity, will successively visit solutions with frequency proportional to the underlying target density. This requires a proposal distribution that generates candidate solutions starting from an arbitrary initial state. The speed of the sampled chains converging to the target distribution deteriorates rapidly, however, with increasing parameter dimensionality. In this paper, we introduce a new proposal distribution that enhances significantly the efficiency of MCMC simulation for highly parameterized models. This proposal distribution exploits the cross covariance of model parameters, measurements, and model outputs and generates candidate states much alike the analysis step in the Kalman filter. We embed the Kalman‐inspired proposal distribution in the DiffeRential Evolution Adaptive Metropolis algorithm during burn‐in and present several numerical experiments with complex, high‐dimensional, or multimodal target distributions. Results demonstrate that this new proposal distribution can greatly improve simulation efficiency of MCMC. Specifically, we observe a speedup on the order of 10–30 times for groundwater models with more than 100 parameters.

Remotely sensed ET for streamflow modelling in catchments with contrasting flow characteristics: an attempt to improve efficiency

An efficient calibration with remotely sensed (RS) data is important for accurate predictions at ungauged catchments. This study investigates the advantages of streamflow-sensitive regionalization on calibration with RS evapotranspiration (ET). Regionalization experiments are performed at 28 catchments in Australia. The catchments are classified into three groups based on annual rainfall and runoff coefficients. Streamflow, RS ET, and a multi-objective RS ET-streamflow calibration are performed using the DiffeRential Evolution Adaptive Metropolis algorithm in each catchment. Simplified Australian Water Resource Assessment-Landscape model is calibrated for a selection of five parameters. Posterior probability distributions of parameters from three calibrations performed at donor catchments in each group are inspected to find the parameter for regionalization in the individual group. In group 1 of wetter catchments, regionalization of parameter FsoilEmax (soil evaporation scaling factor) helps to simplify the calibration without any deterioration in ET, soil moisture (SM) and streamflow predictions. Regionalization of parameter Beta (coefficient describing rate of hydraulic conductivity increase with water content) in group 2 assists to improve the streamflow predictions with no decrement in ET and SM predictions. However, regionalization is not able to provide satisfactory results in group 3. Group 3 includes low-yielding catchments, with average annual rainfall below 1000 mm/year and runoff coefficient less than 0.1, where traditional streamflow calibration also fails to produce accurate results. This study concludes that streamflow-sensitive regionalization is effective for improving the efficacy of RS ET calibration in wetter catchments.

Water level observations from unmanned aerial vehicles for improving estimates of surface water–groundwater interaction

AbstractIntegrated hydrological models are usually calibrated against observations of river discharge and piezometric head in groundwater aquifers. Calibration of such models against spatially distributed observations of river water level can potentially improve their reliability and predictive skill. However, traditional river gauging stations are normally spaced too far apart to capture spatial patterns in the water surface, whereas spaceborne observations have limited spatial and temporal resolution. Unmanned aerial vehicles can retrieve river water level measurements, providing (a) high spatial resolution; (b) spatially continuous profiles along or across the water body, and (c) flexible timing of sampling. A semisynthetic study was conducted to analyse the value of the new unmanned aerial vehicle‐borne datatype for improving hydrological models, in particular estimates of groundwater–surface water (GW–SW) interaction. Mølleåen River (Denmark) and its catchment were simulated using an integrated hydrological model (MIKE 11–MIKE SHE). Calibration against distributed surface water levels using the Differential Evolution Adaptive Metropolis algorithm demonstrated a significant improvement in estimating spatial patterns and time series of GW–SW interaction. After water level calibration, the sharpness of the estimates of GW–SW time series improves by ~50% and root mean square error decreases by ~75% compared with those of a model calibrated against discharge only.

An Improved Bayesian Structural Identification Using the First Two Derivatives of Log-Likelihood Measure

The posterior density of structural parameters conditioned by the measurement is obtained by a differential evolution adaptive Metropolis algorithm (DREAM). The surface of the formal log-likelihood measure is studied considering the uncertainty of measurement error to illustrate the problem of equifinality. To overcome the problem of equifinality, the first two derivatives of the log-likelihood measure are proposed to formulate a new informal likelihood measure for the sake of improving the accuracy of the estimator. Moreover, the proposed measure also reduces the standard deviation (uncertain range) of the posterior samples. The benefit of the proposed approach is demonstrated by simulations on identifying the structural parameters with limit output data and noise polluted measurements.

Assessment of the different sources of uncertainty in a SWAT model of the River Senne (Belgium)

Although rainfall input uncertainties are widely identified as being a key factor in hydrological models, the rainfall uncertainty is typically not included in the parameter identification and model output uncertainty analysis of complex distributed models such as SWAT and in maritime climate zones. This paper presents a methodology to assess the uncertainty of semi-distributed hydrological models by including, in addition to a list of model parameters, additional unknown factors in the calibration algorithm to account for the rainfall uncertainty (using multiplication factors for each separately identified rainfall event) and for the heteroscedastic nature of the errors of the stream flow. We used the Differential Evolution Adaptive Metropolis algorithm (DREAM(zs)) to infer the parameter posterior distributions and the output uncertainties of a SWAT model of the River Senne (Belgium). Explicitly considering heteroscedasticity and rainfall uncertainty leads to more realistic parameter values, better representation of water balance components and prediction uncertainty intervals.

Bayesian Inference for Source Reconstruction: A Real-World Application.

This paper applies a Bayesian probabilistic inferential methodology for the reconstruction of the location and emission rate from an actual contaminant source (emission from the Chalk River Laboratories medical isotope production facility) using a small number of activity concentration measurements of a noble gas (Xenon-133) obtained from three stations that form part of the International Monitoring System radionuclide network. The sampling of the resulting posterior distribution of the source parameters is undertaken using a very efficient Markov chain Monte Carlo technique that utilizes a multiple-try differential evolution adaptive Metropolis algorithm with an archive of past states. It is shown that the principal difficulty in the reconstruction lay in the correct specification of the model errors (both scale and structure) for use in the Bayesian inferential methodology. In this context, two different measurement models for incorporation of the model error of the predicted concentrations are considered. The performance of both of these measurement models with respect to their accuracy and precision in the recovery of the source parameters is compared and contrasted.

Approximate Bayesian Computation using Markov Chain Monte Carlo simulation: DREAM(ABC)

AbstractThe quest for a more powerful method for model evaluation has inspired Vrugt and Sadegh (2013) to introduce “likelihood‐free” inference as vehicle for diagnostic model evaluation. This class of methods is also referred to as Approximate Bayesian Computation (ABC) and relaxes the need for a residual‐based likelihood function in favor of one or multiple different summary statistics that exhibit superior diagnostic power. Here we propose several methodological improvements over commonly used ABC sampling methods to permit inference of complex system models. Our methodology entitled DREAM(ABC)uses the DiffeRential Evolution Adaptive Metropolis algorithm as its main building block and takes advantage of a continuous fitness function to efficiently explore the behavioral model space. Three case studies demonstrate that DREAM(ABC)is at least an order of magnitude more efficient than commonly used ABC sampling methods for more complex models. DREAM(ABC)is also more amenable to distributed, multi‐processor, implementation, a prerequisite to diagnostic inference of CPU‐intensive system models.

Distributed Soil Moisture from Crosshole Ground‐Penetrating Radar Travel Times using Stochastic Inversion

Geophysical methods offer several key advantages over conventional subsurface measurement approaches, yet their use for hydrologic interpretation is often problematic. We developed the theory and concepts of a novel Bayesian approach for high‐resolution soil moisture estimation using travel‐time observations from crosshole ground‐penetrating radar experiments. The recently developed Multi‐Try Differential Evolution Adaptive Metropolis algorithm with sampling from past states, MT‐DREAM(ZS), was used to infer, as closely and consistently as possible, the posterior distribution of spatially distributed vadose zone soil moisture and porosity under saturated conditions. Two differing and opposing model parameterization schemes were considered, one involving a classical uniform grid discretization and the other based on a discrete cosine transformation (DCT). We illustrated our approach using two different case studies involving geophysical data from a synthetic water tracer infiltration study and a real‐world field study under saturated conditions. Our results demonstrate that the DCT parameterization yields superior Markov chain Monte Carlo convergence rates along with the most accurate estimates of distributed soil moisture for a large range of spatial resolutions. In addition, DCT is admirably suited to investigate and quantify the effects of model truncation errors on the MT‐DREAM(ZS) inversion results. For the field example, lateral anisotropy needed to be enforced to derive reliable soil moisture estimates. Our results also demonstrate that the posterior soil moisture uncertainty derived with the proposed Bayesian procedure is significantly larger than its counterpart estimated from classical smoothness‐constrained deterministic inversions.

Integrated analysis of waveguide dispersed GPR pulses using deterministic and Bayesian inversion methods

ABSTRACTGround‐penetrating radar (GPR) data affected by waveguide dispersion are not straightforward to analyse. Therefore, waveguide dispersed common midpoint measurements are typically interpreted using so‐called dispersion curves, which describe the phase velocity as a function of frequency. These dispersion curves are typically evaluated with deterministic optimization algorithms that derive the dielectric properties of the subsurface as well as the location and depth of the respective layers. However, these methods do not provide estimates of the uncertainty of the inferred subsurface properties. Here, we applied a formal Bayesian inversion methodology using the recently developed DiffeRential Evolution Adaptive Metropolis algorithm. This Markov Chain Monte Carlo simulation method rapidly estimates the (non‐linear) parameter uncertainty and helps to treat the measurement error explicitly. We found that the frequency range used in the inversion has an important influence on the posterior parameter estimates, essentially because parameter sensitivity varies with measurement frequency. Moreover, we established that the measurement error associated with the dispersion curve is frequency dependent and that the estimated model parameters become severely biased if this frequency‐dependent nature of the measurement error is not properly accounted for. We estimated these frequency‐dependent measurement errors together with the model parameters using the algorithm. The posterior distribution of the model parameters derived in this way compared well with inversion results for a reduced frequency bandwidth which is an alternative, yet subjective method to reduce the bias introduced by this frequency‐dependent measurement error. Altogether, our inversion procedure provides an integrated and objective methodology for the analysis of dispersive GPR data and appropriately treats the measurement error and parameter uncertainty.

How efficient are one-dimensional models to reproduce the hydrodynamic behavior of structured soils subjected to multi-step outflow experiments?

Summary In this paper we investigate numerically the three following questions: (1) Do MSO data show the impact of soil structure on water flow? (2) If so, to what extent are the common one-dimensional hydraulic models able to apprehend any preferential flow features? and (3) What are the predictive capabilities of these models parameterized with MSO data? A Bayesian framework was used to infer the hydraulic models under virtual MSO conditions, for soil samples with different levels of heterogeneity. This, by coupling the HYDRUS-1D model with the DiffeRential Evolution Adaptive Metropolis algorithm (DREAM). Regarding questions 1 and 2, our findings indicate that (i) large outflow observed during the first steps of MSO may express the behavior of a real macropore, or of structural heterogeneity inside the soil core; (ii) this behavior cannot be characterized with the MV ( Mualem, 1976 , van Genuchten, 1980 ) and DR ( Durner, 1994 ) models whereas mobile–immobile ( Philip, 1968 , van Genuchten and Wierenga, 1976 ) and dual-permeability ( Gerke and van Genuchten, 1993a , Gerke and van Genuchten, 1993b ) preferential flow models can provide excellent fits depending on the soil architecture type; (iii) in the presence of macropores, the DUAL model performed excellently despite frequent convergence problems of the HYDRUS-1D code. Furthermore, neglecting the first MSO steps can result in a perfect match of the soil matrix behavior by the MV model. Regarding question 3, a virtual infiltration front experiment reveals that predictive capabilities of the MIM model parameterized with MSO are not satisfactorily. This indicates that the MIM model underlying concept induce excellent MSO fits for wrong reasons. Similar findings hold for the DUAL model and soil architectures other than macroporous. For a macroporous soil, i.e., the conceptual structure for which the DUAL model was designed, the latter model parameterized with MSO data can provide consistent results under infiltration conditions. This, however, should be verified with real soils. Lastly, neglecting the first MSO steps to calibrate the MV model may induce, in the presence of macropores, significant errors when predicting the matrix behavior under infiltration. This because of the macropore-matrix water transfer.