Bayesian analysis of individual and systematic multiplicative errors for estimating ages with stratigraphic constraints in optically stimulated luminescence dating

Many dating techniques include significant error terms which are not independent between samples to date. This is typically the case in Optically Stimulated Luminescence (OSL) dating where the conversion from characteristic equivalent doses to the corresponding ages using the annual dosimetry data includes error terms that are common to all produced datings. Dealing with these errors is essential to estimate ages from a set of datings whose chronological ordering is known. In this work, we propose and we study a Bayesian model to address this problem. For this purpose, we first consider a multivariate model with multiplicative Gaussian errors in a Bayesian framework. This model relates a set of characteristic equivalent doses to the corresponding ages while taking into account for the systematic and non-systematic errors associated to the dosimetry. It thus offers the opportunity to deal properly with stratigraphic constraints within OSL datings, but also with other datings possessing errors which are independent from systematic errors of OSL (e.g. radiocarbon). Then, we use this model to extend an existing Bayesian model for the assessment of characteristic equivalent doses from Single Aliquot and Regenerative (SAR) dose measurements. The overall Bayesian model leads to the joint estimation of all the variables (which include all the dose–response functions and characteristic equivalent doses) of a sequence of, possibly heterogeneous, datings. We also consider a more generic solution consisting in using directly the age model from a set of characteristic equivalent dose estimates and their associated standard errors. We finally give an example of application on a set of five OSL datings with stratigraphic constraints and observe a good adequacy between the two approaches.     

A Bayesian hierarchical modeling approach for analyzing observational data from marine ecological studies

We introduce the Bayesian hierarchical modeling approach for analyzing observational data from marine ecological studies using a data set intended for inference on the effects of bottom-water hypoxia on macrobenthic communities in the northern Gulf of Mexico off the coast of Louisiana, USA. We illustrate (1) the process of developing a model, (2) the use of the hierarchical model results for statistical inference through innovative graphical presentation, and (3) a comparison to the conventional linear modeling approach (ANOVA). Our results indicate that the Bayesian hierarchical approach is better able to detect a “treatment” effect than classical ANOVA while avoiding several arbitrary assumptions necessary for linear models, and is also more easily interpreted when presented graphically. These results suggest that the hierarchical modeling approach is a better alternative than conventional linear models and should be considered for the analysis of observational field data from marine systems.     

A Bayesian hierarchical age model for optical dating of single grains of quartz

In optical dating, the last time that a sample of sediment was exposed to sunlight is determined by dividing its equivalent dose (D_e) by the dose rate. For single-grain dating, the sample D_e is based on the statistical analysis of the distribution of D_e values estimated for individual grains, whereas the dose rate is usually determined from measurements of the environmental radioactivity of the bulk sample, together with allowances for radiation sources internal to the grains and cosmic rays. Conventionally, the D_e and dose rate are measured and analysed separately to produce an estimate of the depositional age of a sample, but this approach may result in loss of information because distributions of single-grain D_e values are influenced by several factors. Existing statistical models do not incorporate all the key information contributing to age estimation, such as the pattern and scale of dispersion of single-grain D_e values and dose rates, the associated measurement uncertainties, the effect of natural variability among grains, and the outlier probabilities of D_e and dose rate estimates. Here we propose an empirical Bayesian hierarchical age model (BHAM) for optical dating of quartz samples that incorporates the above information to estimate their depositional ages. The BHAM is based on the implementation of standardised growth curve and L_nT_n methods to integrate information on the full distribution of single-grain D_e values, sources of measurement uncertainty, beta-dose heterogeneity (observed or modelled), and detection of outliers. We present the results of validation tests using data sets of optically stimulated luminescence measurements and dose rates obtained for quartz samples dated previously from Denisova Cave (Russia) and for simulated samples. We conclude that the BHAM represents a robust and flexible approach to dealing with data for single grains of quartz within a Bayesian hierarchical framework and is suitable for application to sediments deposited in a variety of depositional settings.     

Dimensionality reduction for efficient Bayesian estimation of groundwater flow in strongly heterogeneous aquifers

We focus on the Bayesian estimation of strongly heterogeneous transmissivity fields conditional on data sampled at a set of locations in an aquifer. Log-transmissivity, Y, is modeled as a stochastic Gaussian process, parameterized through a truncated Karhunen–Loève (KL) expansion. We consider Y fields characterized by a short correlation scale as compared to the size of the observed domain. These systems are associated with a KL decomposition which still requires a high number of parameters, thus hampering the efficiency of the Bayesian estimation of the underlying stochastic field. The distinctive aim of this work is to present an efficient approach for the stochastic inverse modeling of fully saturated groundwater flow in these types of strongly heterogeneous domains. The methodology is grounded on the construction of an optimal sparse KL decomposition which is achieved by retaining only a limited set of modes in the expansion. Mode selection is driven by model selection criteria and is conditional on available data of hydraulic heads and (optionally) Y. Bayesian inversion of the optimal sparse KLE is then inferred using Markov Chain Monte Carlo (MCMC) samplers. As a test bed, we illustrate our approach by way of a suite of computational examples where noisy head and Y values are sampled from a given randomly generated system. Our findings suggest that the proposed methodology yields a globally satisfactory inversion of the stochastic head and Y fields. Comparison of reference values against the corresponding MCMC predictive distributions suggests that observed values are well reproduced in a probabilistic sense. In a few cases, reference values at some unsampled locations (typically far from measurements) are not captured by the posterior probability distributions. In these cases, the quality of the estimation could be improved, e.g., by increasing the number of measurements and/or the threshold for the selection of KL modes.     

Modelling dose rate to single grains of quartz in well-sorted sand samples: The dispersion arising from the presence of potassium feldspars and implications for single grain OSL dating

Single grain OSL has become a widely used approach in Quaternary geochronology. However, the origins of D_e distributions and the sources of variation in individual dose estimates are still poorly understood. The amount of scatter in these distributions on top of the known uncertainties in measurement and analysis is defined by overdispersion and this quantity is generally used for weighting individual D_e values to calculate a central equivalent dose. In this study, we address the nature and amount of different sources of dispersion in quartz single grain D_e estimates, by (i) using appropriate statistical tools to characterize D_e populations and (ii) modelling, with a specifically designed Geant4 code, dose rate distributions arising from the presence of potassium feldspar grains in well-sorted sands. The model uses Monte Carlo simulations of beta emissions and interactions in a random close packing of quartz and feldspar spheres representing a sand sample. Based on the simulation results, we explain the discrepancy between intrinsic and natural overdispersion values in a well-bleached sample, thus validating the model. The three parameters having the most influence on dispersion in dose rate distributions, and modelled in this study, appear to be grain size, potassium content and total dose rate.Finally an analysis of measurement uncertainties and other sources of variations in equivalent dose estimates lead us to conclude that all age models (both logged and unlogged) which include an overdispersion value to weight individual D_e values rely mainly on unknown parameters; this ignorance may lead to an inadvertent bias in D_e estimates. Assuming counting statistics make a small contribution to dispersion (as is often the case), we suggest that in some cases it is most appropriate to use unweighted averages of equivalent doses when dividing by commonly measured average dose rates.     

Application of the SPARROW model in watersheds with limited information: a Bayesian assessment of the model uncertainty and the value of additional monitoring

How can spatially explicit nonlinear regression modelling be used for obtaining nonpoint source loading estimates in watersheds with limited information? What is the value of additional monitoring and where should future data‐collection efforts focus on? In this study, we address two frequently asked questions in watershed modelling by implementing Bayesian inference techniques to parameterize SPAtially Referenced Regressions On Watershed attributes (SPARROW), a model that empirically estimates the relation between in‐stream measurements of nutrient fluxes and the sources/sinks of nutrients within the watershed. Our case study is the Hamilton Harbour watershed, a mixed agricultural and urban residential area located at the western end of Lake Ontario, Canada. The proposed Bayesian approach explicitly accounts for the uncertainty associated with the existing knowledge from the system and the different types of spatial correlation typically underlying the parameter estimation of watershed models. Informative prior parameter distributions were formulated to overcome the problem of inadequate data quantity and quality, whereas the potential bias introduced from the pertinent assumptions is subsequently examined by quantifying the relative change of the posterior parameter patterns. Our modelling exercise offers the first estimates of export coefficients and delivery rates from the different subcatchments and thus generates testable hypotheses regarding the nutrient export ‘hot spots’ in the studied watershed. Despite substantial uncertainties characterizing our calibration dataset, ranging from 17% to nearly 400%, we arrived at an uncertainty level for the whole‐basin nutrient export estimates of only 36%. Finally, we conduct modelling experiments that evaluate the potential improvement of the model parameter estimates and the decrease of the predictive uncertainty if the uncertainty associated with the current nutrient loading estimates is reduced. Copyright © 2012 John Wiley & Sons, Ltd.     

Estimation and prediction of the spatial occurrence of fish species using Bayesian latent Gaussian models

A methodological approach for modelling the occurrence patterns of species for the purpose of fisheries management is proposed here. The presence/absence of the species is modelled with a hierarchical Bayesian spatial model using the geographical and environmental characteristics of each fishing location. Maps of predicted probabilities of presence are generated using Bayesian kriging. Bayesian inference on the parameters and prediction of presence/absence in new locations (Bayesian kriging) are made by considering the model as a latent Gaussian model, which allows the use of the integrated nested Laplace approximation ( INLA ) software (which has been seen to be quite a bit faster than the well-known MCMC methods). In particular, the spatial effect has been implemented with the stochastic partial differential equation (SPDE) approach. The methodology is evaluated on Mediterranean horse mackerel (Trachurus mediterraneus) in the Western Mediterranean. The analysis shows that environmental and geographical factors can play an important role in directing local distribution and variability in the occurrence of species. Although this approach is used to recognize the habitat of mackerel, it could also be for other different species and life stages in order to improve knowledge of fish populations and communities.     

Joint AVO inversion in the time and frequency domain with Bayesian interference

Amplitude variations with offset or incident angle (AVO/AVA) inversion are typically combined with statistical methods, such as Bayesian inference or deterministic inversion. We propose a joint elastic inversion method in the time and frequency domain based on Bayesian inversion theory to improve the resolution of the estimated P- and S-wave velocities and density. We initially construct the objective function using Bayesian inference by combining seismic data in the time and frequency domain. We use Cauchy and Gaussian probability distribution density functions to obtain the prior information for the model parameters and the likelihood function, respectively. We estimate the elastic parameters by solving the initial objective function with added model constraints to improve the inversion robustness. The results of the synthetic data suggest that the frequency spectra of the estimated parameters are wider than those obtained with conventional elastic inversion in the time domain. In addition, the proposed inversion approach offers stronger antinoising compared to the inversion approach in the frequency domain. Furthermore, results from synthetic examples with added Gaussian noise demonstrate the robustness of the proposed approach. From the real data, we infer that more model parameter details can be reproduced with the proposed joint elastic inversion.     

Bayesian autoregressive spatiotemporal model of PM<Subscript>10</Subscript> concentrations across Peninsular Malaysia

Rapid industrialization and haze episodes in Malaysia ensure pollution remains a public health challenge. Atmospheric pollutants such as PM₁₀ are typically variable in space and time. The increased vigilance of policy makers in monitoring pollutant levels has led to vast amounts of spatiotemporal data available for modelling and inference. The aim of this study is to model and predict the spatiotemporal daily PM₁₀ levels across Peninsular Malaysia. A hierarchical autoregressive spatiotemporal model is applied to daily PM₁₀ concentration levels from thirty-four monitoring stations in Peninsular Malaysia during January to December 2011. The model set in a three stage Bayesian hierarchical structure comprises data, process and parameter levels. The posterior estimates suggest moderate spatial correlation with effective range 157 km and a short term persistence of PM₁₀ in atmosphere with temporal correlation parameter 0.78. Spatial predictions and temporal forecasts of the PM₁₀ concentrations follow from the posterior and predictive distributions of the model parameters. Spatial predictions at the hold-out sites and one-step ahead PM₁₀ forecasts are obtained. The predictions and forecasts are validated by computing the RMSE, MAE, R² and MASE. For the spatial predictions and temporal forecasting, our results indicate a reasonable RMSE of 10.71 and 7.56, respectively for the spatiotemporal model compared to RMSE of 15.18 and 12.96, respectively from a simple linear regression model. Furthermore, the coverage probability of the 95% forecast intervals is 92.4% implying reasonable forecast results. We also present prediction maps of the one-step ahead forecasts for selected day at fine spatial scale.     

Sea-surge and wind speed extremes: optimal estimation strategies for planners and engineers

Accurate and precise estimation of return levels is often a key goal of any extreme value analysis. For example, in the UK the British Standards Institution (BSI) incorporate estimates of ‘once-in-50-year wind gust speeds’—or 50-year return levels—into their design codes for new structures; similarly, the Dutch Delta Commission use estimates of the 10,000-year return level for sea-surge to aid the construction of flood defence systems. In this paper, we briefly highlight the shortcomings of standard methods for estimating return levels, including the commonly-adopted block maxima and peaks over thresholds approach, before presenting an estimation framework which we show can substantially increase the precision of return level estimates. Our work allows explicit quantification of seasonal effects, as well as exploiting recent developments in the estimation of the extremal index for handling extremal clustering. From frequentist ideas, we turn to the Bayesian paradigm as a natural approach for building complex hierarchical or spatial models for extremes. Through simulations we show that the return level posterior mean does not have an exceedance probability in line with the intended encounter risk; we also argue that the Bayesian posterior predictive value gives the most satisfactory representation of a return level for use in practice, accounting for uncertainty in parameter estimation and future observations. Thus, where feasible, we propose a Bayesian estimation strategy for optimal return level inference.     

Exposure estimation using repeated blood concentration measurements

Physiologically based toxicokinetic (PBTK) modeling has been well established to study the distributions of chemicals in target tissues. In addition, the hierarchical Bayesian statistical approach using Markov Chain Monte Carlo (MCMC) simulations has been applied successfully for parameter estimation. The aim was to estimate the constant inhalation exposure concentration (assumed) using a PBTK model based on repeated measurements in venous blood, so that exposures could be estimated. By treating the constant exterior exposure as an unknown parameter of a four-compartment PBTK model, we applied MCMC simulations to estimate the exposure based on a hierarchical Bayesian approach. The dataset on 16 volunteers exposed to 100 ppm (≅0.538 mg/L) trichloroethylene vapors for 4 h was reanalyzed as an illustration. Cases of time-dependent exposures with a constant mean were also studied via 100 simulated datasets. The posterior geometric mean of 0.571, with narrow 95% posterior confidence interval (CI) (0.506, 0.645), estimated the true trichloroethylene inhalation concentration (0.538 mg/L) with very high precision. Also, the proposed method estimated the overall constant mean of the simulated time-dependent exposure scenarios well with slightly wider 95% CIs. The proposed method justifies the accuracy of exposure estimation from biomonitoring data using PBTK model and MCMC simulations from a real dataset and simulation studies numerically, which provides a starting point for future applications in occupational exposure assessment.     

Spatial reconstruction of rainfall fields from rain gauge and radar data

Rainfall is a phenomenon difficult to model and predict, for the strong spatial and temporal heterogeneity and the presence of many zero values. We deal with hourly rainfall data provided by rain gauges, sparsely distributed on the ground, and radar data available on a fine grid of pixels. Radar data overcome the problem of sparseness of the rain gauge network, but are not reliable for the assessment of rain amounts. In this work we investigate how to calibrate radar measurements via rain gauge data and make spatial predictions for hourly rainfall, by means of Monte Carlo Markov Chain algorithms in a Bayesian hierarchical framework. We use zero-inflated distributions for taking zero-measurements into account. Several models are compared both in terms of data fitting and predictive performances on a set of validation sites. Finally, rainfall fields are reconstructed and standard error estimates at each prediction site are shown via easy-to-read spatial maps.     

Stochastic modelling of multi-grain equivalent dose (De) distributions: Implications for OSL dating of sediment mixtures

A number of recent optically stimulated luminescence (OSL) studies have cited post-depositional mixing as a dominant source of equivalent dose (D_e) scatter across a range of sedimentary environments, including those previously considered ‘best suited’ for OSL dating. The potentially insidious nature of sediment mixing means that this problem may often only be identifiable by careful statistical analysis of D_e data sets. This study aims to address some of the important issues associated with the characterisation and statistical treatment of mixed D_e distributions at the multi-grain scale of analysis, using simulated D_e data sets produced with a simple stochastic model. Using this Monte Carlo approach we were able to generate theoretical distributions of single-grain D_e values, which were then randomly mixed together to simulate multi-grain aliquot D_e distributions containing a known number of mixing components and known corresponding burial doses. A range of sensitivity tests were undertaken using sediment mixtures with different aged dose components, different numbers of mixing components, and different types of dose component distributions (fully bleached, heterogeneously bleached and significantly overdispersed D_e distributions). The results of our modelling simulations reveal the inherent problems encountered when dating mixed sedimentary samples with multi-grain D_e estimation techniques. ‘Phantom’ dose components (i.e. discrete dose populations that do not correspond to the original single-grain mixing components) are an inevitable consequence of the ‘averaging’ effects of multi-grain D_e analysis, and prevent the correct number of mixing components being identified with the finite mixture model (FMM) for all of the multi-grain mixtures tested. Our findings caution against use of the FMM for multi-grain aliquot D_e data sets, even when the aliquots consist of only a few grains.     

OSL dating of tsunami deposits from Phra Thong Island,Thailand

To evaluate the potential of optical dating (OSL) in establishing a proper tsunami chronology for Phra Thong Island (SW Thailand), the method was applied to a suite of tsunamigenic and littoral sandy deposits, for which independent age control was available. Small aliquots of coarse grained quartz were used for measurements, and processed statistically by means of appropriate age models. Based on the equivalent dose distributions, the well bleached littoral deposits were analysed with the central age model (CAM); the tsunamigenic samples revealed poor bleaching, thus, the minimum age model (MAM) was applied. The cross-check with independent age data showed good agreement between luminescence ages and the existing radiocarbon chronology for the littoral deposits. The poorly bleached deposits of the 2004 Indian Ocean tsunami revealed residuals of less than 40 years, which are insignificant for older deposits and demonstrate the general suitability of the dating technique for tsunamites on Phra Thong. Afterwards, the approach was extended to tsunamigenic and littoral sediments of unknown age. Since those revealed properties similar to their reference deposits, the procedures of statistical D_e determination were adopted. The resulting ages were in agreement with the stratigraphical position and (largely) with the wider chronological context.     

Point process-based modeling of multiple debris flow landslides using INLA: an application to the 2009 Messina disaster

We develop a stochastic modeling approach based on spatial point processes of log-Gaussian Cox type for a collection of around 5000 landslide events provoked by a precipitation trigger in Sicily, Italy. Through the embedding into a hierarchical Bayesian estimation framework, we can use the integrated nested Laplace approximation methodology to make inference and obtain the posterior estimates of spatially distributed covariate and random effects. Several mapping units are useful to partition a given study area in landslide prediction studies. These units hierarchically subdivide the geographic space from the highest grid-based resolution to the stronger morphodynamic-oriented slope units. Here we integrate both mapping units into a single hierarchical model, by treating the landslide triggering locations as a random point pattern. This approach diverges fundamentally from the unanimously used presence–absence structure for areal units since we focus on modeling the expected landslide count jointly within the two mapping units. Predicting this landslide intensity provides more detailed and complete information as compared to the classically used susceptibility mapping approach based on relative probabilities. To illustrate the model’s versatility, we compute absolute probability maps of landslide occurrences and check their predictive power over space. While the landslide community typically produces spatial predictive models for landslides only in the sense that covariates are spatially distributed, no actual spatial dependence has been explicitly integrated so far. Our novel approach features a spatial latent effect defined at the slope unit level, allowing us to assess the spatial influence that remains unexplained by the covariates in the model. For rainfall-induced landslides in regions where the raingauge network is not sufficient to capture the spatial distribution of the triggering precipitation event, this latent effect provides valuable imaging support on the unobserved rainfall pattern.     

Comparison of sparse‐grid geometric and random sampling methods in nonlinear inverse solution uncertainty estimation

A new uncertainty estimation method, which we recently introduced in the literature, allows for the comprehensive search of model posterior space while maintaining a high degree of computational efficiency. The method starts with an optimal solution to an inverse problem, performs a parameter reduction step and then searches the resulting feasible model space using prior parameter bounds and sparse‐grid polynomial interpolation methods. After misfit rejection, the resulting model ensemble represents the equivalent model space and can be used to estimate inverse solution uncertainty. While parameter reduction introduces a posterior bias, it also allows for scaling this method to higher dimensional problems. The use of Smolyak sparse‐grid interpolation also dramatically increases sampling efficiency for large stochastic dimensions. Unlike Bayesian inference, which treats the posterior sampling problem as a random process, this geometric sampling method exploits the structure and smoothness in posterior distributions by solving a polynomial interpolation problem and then resampling from the resulting interpolant. The two questions we address in this paper are 1) whether our results are generally compatible with established Bayesian inference methods and 2) how does our method compare in terms of posterior sampling efficiency. We accomplish this by comparing our method for two electromagnetic problems from the literature with two commonly used Bayesian sampling schemes: Gibbs’ and Metropolis‐Hastings. While both the sparse‐grid and Bayesian samplers produce compatible results, in both examples, the sparse‐grid approach has a much higher sampling efficiency, requiring an order of magnitude fewer samples, suggesting that sparse‐grid methods can significantly improve the tractability of inference solutions for problems in high dimensions or with more costly forward physics.     

Significance of AIC differences for precipitation intensity distributions

Chain dependent models for daily precipitation typically model the occurrence process as a Markov chain and the precipitation intensity process using one of several probability distributions. It has been argued that the mixed exponential distribution is a superior model for the rainfall intensity process, since the value of its information criterion (Akaike information criterion or Bayesian information criterion) when fit to precipitation data is usually less than the more commonly used gamma distribution. The differences between the criterion values of the best and lesser models are generally small relative to the magnitude of the criterion value, which raises the question of whether these differences are statistically significant. Using a likelihood ratio statistic and nesting the gamma and mixed exponential distributions in a parent distribution, we show indirectly that generally the superiority of the mixed exponential distribution over the gamma distribution for modeling precipitation intensity is statistically significant. Comparisons are also made with a common-a gamma model, which are less informative.     

Optical dating of sediments in Wadi Sabra (SW Jordan)

At Wadi Sabra (SW Jordan) human occupation dates back to the Palaeolithic and Epipalaeolithic. Although there is stratigraphic correlation based on archaeological finds of Ahmarian origin, numerical age estimates are lacking. We applied single-aliquot optical dating of coarse grained quartz of wadi deposits and investigated the luminescence properties in detail to achieve more accurate age information about the time of human occupation. Weak luminescence signals and scattered dose distributions characterise the multi-grain aliquots. The residual doses of the investigated modern wadi sediment are between 0 and 7 Gy. Moreover, comparison of equivalent dose (De) values of 1 mm and 8 mm aliquots shows higher equivalent doses for the large aliquots. Both experiments indicate that the luminescence signal is partially bleached prior to deposition. The dose distributions of all samples are broadly scattered and have overdispersion values between 25 and 43%, some samples are significantly skewed. The shape of the dose distributions points to other sources of scatter, in addition to partial bleaching. Comparison of 1 mm multi-grain and single-grain data demonstrates that the luminescence signal of one multi-grain aliquot most likely is from a single grain. For this reason, variation in the number of photon counts due to the weak luminescence intensity and variations in beta microdosimetry have a bigger impact on the spread of dose distributions. However, we cannot quantify the particular impact of partial bleaching, weak luminescence intensity and beta microdosimetry. To account for the spread of the dose distribution, we use the central age model to calculate equivalent doses. Age calculations yield results in the range of 30–48 ka.     

Soil moisture estimation using tomographic ground penetrating radar in a MCMC–Bayesian framework

In this study, we focus on a hydrogeological inverse problem specifically targeting monitoring soil moisture variations using tomographic ground penetrating radar (GPR) travel time data. Technical challenges exist in the inversion of GPR tomographic data for handling non-uniqueness, nonlinearity and high-dimensionality of unknowns. We have developed a new method for estimating soil moisture fields from crosshole GPR data. It uses a pilot-point method to provide a low-dimensional representation of the relative dielectric permittivity field of the soil, which is the primary object of inference: the field can be converted to soil moisture using a petrophysical model. We integrate a multi-chain Markov chain Monte Carlo (MCMC)–Bayesian inversion framework with the pilot point concept, a curved-ray GPR travel time model, and a sequential Gaussian simulation algorithm, for estimating the dielectric permittivity at pilot point locations distributed within the tomogram, as well as the corresponding geostatistical parameters (i.e., spatial correlation range). We infer the dielectric permittivity as a probability density function, thus capturing the uncertainty in the inference. The multi-chain MCMC enables addressing high-dimensional inverse problems as required in the inversion setup. The method is scalable in terms of number of chains and processors, and is useful for computationally demanding Bayesian model calibration in scientific and engineering problems. The proposed inversion approach can successfully approximate the posterior density distributions of the pilot points, and capture the true values. The computational efficiency, accuracy, and convergence behaviors of the inversion approach were also systematically evaluated, by comparing the inversion results obtained with different levels of noises in the observations, increased observational data, as well as increased number of pilot points.     

A Bayesian hierarchical approach for spatial analysis of climate model bias in multi-model ensembles

Coupled atmosphere–ocean general circulation models are key tools to investigate climate dynamics and the climatic response to external forcings, to predict climate evolution and to generate future climate projections. Current general circulation models are, however, undisputedly affected by substantial systematic errors in their outputs compared to observations. The assessment of these so-called biases, both individually and collectively, is crucial for the models’ evaluation prior to their predictive use. We present a Bayesian hierarchical model for a unified assessment of spatially referenced climate model biases in a multi-model framework. A key feature of our approach is that the model quantifies an overall common bias that is obtained by synthesizing bias across the different climate models in the ensemble, further determining the contribution of each model to the overall bias. Moreover, we determine model-specific individual bias components by characterizing them as non-stationary spatial fields. The approach is illustrated based on the case of near-surface air temperature bias in the tropical Atlantic and bordering regions from a multi-model ensemble of historical simulations from the fifth phase of the Coupled Model Intercomparison Project. The results demonstrate the improved quantification of the bias and interpretative advantages allowed by the posterior distributions derived from the proposed Bayesian hierarchical framework, whose generality favors its broader application within climate model assessment.