Conceptual Model Uncertainty Research Articles

Non-isothermal consolidation: A systematic evaluation of two implementations based on multiphase and Richards equations

In this paper, the governing equations of non-isothermal two-phase flow in unsaturated, deformable porous media are presented based on different representations of the gaseous phase along with their implementations in the open-source FEM code OpenGeoSys 6. As one implementation utilises the Richards equation to represent the unsaturated system whereas the second implementation relies on a two-component two-phase flow formulation, the impact of a free gaseous phase on model predictions is illustrated by a series of tests of increasing complexity, motivated by similar investigations in Task C of the DECOVALEX 2023 project. We demonstrate that the governing equations of both implementations collapse and become identical in fully saturated regions and that the numerical implementations remain stable in the special cases. Furthermore, the implicit assumptions made in the Richards equation are explicitly applied to the physically more complex and comprehensive TH2M model. Thus, the impact of a constant residual gas pressure on the model predictions is investigated. Finally, the examples chosen for this work feature thermal consolidation and pore fluid pressurisation effects in the Opalinus Clay due to heat release from a nuclear waste canister as well as dry-out with subsequent resaturation of a bentonite buffer around the heat source. All test cases are evaluated on the level of the governing equations of each model as well as on the level of results obtained by each model for a discussion of conceptual model uncertainty.

Impacts of Uncontrolled Operator Splitting Methods on Parameter Identification, Prediction Uncertainty, and Subsurface Flux Representation in Conceptual Hydrological Models

AbstractThe proper numerical representation of physical processes in mechanistic hydrological models is essential to produce robust predictions. A common problem with numerical schemes in hydrological models is that multiple concurrent fluxes are calculated sequentially. Although the importance of errors introduced by inappropriate numerical schemes is well recognized in the literature, many hydrological models calculate concurrent fluxes sequentially. Here, two versions of the HYPE model are used to investigate the limitations of sequential calculations. A fourth order Gear‐Nordsieck solution of the continuous state‐space formulation of HYPE (I‐HYPE) is developed to provide a robust solution, and a fixed‐step implicit Euler scheme (IE‐HYPE) is implemented to provide a computationally efficient and robust approximation of the I‐HYPE simulations. In contrast to I‐HYPE, results show that the original HYPE and the sequential calculation implemented in the continuous state‐space formulation of HYPE (SQ‐HYPE) typically simulate no interflow when soil moisture levels exceed the field capacity. The discrepancy between SQ‐HYPE and I‐HYPE grows with the size of the computation time step, and this implies a compromised representation of flow paths by sequential schemes. IE‐HYPE provides responses comparable with I‐HYPE for both daily and hourly time steps. IE‐HYPE and SQ‐HYPE are compared in terms of their groundwater representation, parameter identifiability, and predictive skills for two catchments. The sequential models have larger groundwater contributions to flow than IE‐HYPE because the splitting errors in SQ‐HYPE limit the interflow flux. IE‐HYPE estimates of the groundwater flux are more consistent with literature values of groundwater contributions to flow for the basins studied.

Application of stochastic time dependent parameters to improve the characterization of uncertainty in conceptual hydrological models

• We test stochastic processes to improve modeling of uncertainty in a real case study. • Bayesian inference and cross-validation are used to calibrate and assess performances. • Results highlight clear benefits for appropriate modeling choices. • We make recommendations to uncover/avoid issues due to overparameterization. • We test a novel parallel framework that simplifies access to HPC resources. The traditional description of a hydrological system with a deterministic, conceptual model and a lumped output error term does not explicitly consider the main mechanisms of uncertainty generation due to approximate process representation, unobserved variability in processes and influence factors, and input uncertainty. In this study, we test the description of such intrinsic uncertainty in conceptual models by making process rates stochastic through stochastic, time-dependent rate parameters. We analyze the advantages and challenges of this approach by using Bayesian inference to jointly estimate model parameters, parameters of the stochastic processes, and the time series of the stochastic parameters. Numerically, we use a particle filter to infer the stochastic time series and to approximate the marginal likelihood for Markov chain sampling of the constant parameters. Compared to the lumped error formulation, we achieve a more realistic description of uncertainty, as we obtain larger errors in intrinsic states, larger uncertainty during prediction than calibration periods (a feature missing for simple lumped error models), and autocorrelated model outputs. However, the additional degrees of freedom introduced with stochastic parameters can lead to an unintentional compensation for model deficits or input errors. This problem is symptomatic of model structure inadequacy or poor selection of the parameters that are made stochastic, and can be diagnosed through cross-validation and careful posterior analysis. The proposed approach is computationally more demanding and its implementation more challenging than the traditional description of hydrological systems using a deterministic model with a lumped error term. However, its advantages both in providing a more realistic representation of uncertainties and in diagnosing model deficiencies suggest its adoption and further development in future studies.

An Approach to Include Hydrogeologic Barriers With Unknown Geometric Properties in Groundwater Model Inversions

AbstractHydrogeological barriers can significantly impact groundwater model predictions. They are, however, often excluded from, or misrepresented in, groundwater models if their presence is unknown or their properties are poorly constrained. Here we show that sharp barriers can be included in groundwater model inversion, even where their presence is uncertain. We describe an approach utilizing "phantom structures"—randomly located, linear groups of model cells assigned a unique hydraulic conductivity value—to improve identifiability of barriers. Algorithmic parameter estimation using model‐independent parameter estimation code (PEST) is implemented to determine model structures that best match the hydraulic head and groundwater age observation data from a hypothetical aquifer. Our results show that for a series of case studies, this method was successful in inferring the appropriate location and properties of hydrogeological barriers, when that barrier was not aligned with the dominant flow direction. We compare these results to model inversion using traditional pilot points. The phantom structures approach shows promise in identifying hydrogeological structures and in reproducing groundwater flow across a model domain. Our results demonstrate that the geometric properties of geological structures can remain flexible in a model inversion. This is a step toward reducing conceptual model uncertainty where the presence and properties of hydrogeologic barriers are undefined.

Episodic extreme rainfall events drive groundwater recharge in arid zone environments of central Australia

Understanding the factors controlling groundwater recharge in arid vadose zones is critical for sustainable groundwater management in the face of climate variability and change. Here we estimated groundwater recharge in data sparse regions under arid climate by implementing a multi-model approach using HYDRUS-1D for a century-long time series of meteorological data and site-specific regolith hydraulic properties of a bare soil and a Mulga savanna-type soil in central Australia. Grain‐size analysis provided a contiguous 12-m deep profile of hydraulic properties that were generated by means of pedotransfer functions. To account for conceptual model uncertainty in hydraulic properties, eleven pedotransfer functions were applied. Climate data from three stations accounted for spatial heterogeneity in local climate of the Ti Tree Basin study area. Simulated water fluxes in the vadose zone indicated that only rainfall events of more than 150–200 mm resulted in groundwater recharge. Recharge was linked to extreme rainfall associated with monsoonal cyclones. Based on the 130-year climate record, average recharge for the savanna-type vegetation ranged from 4.3 to 7.4 mm/a across the three climate stations, with an overall mean of 4.6 mm/a. The bare soil had an overall mean recharge of 29.5 mm/a, ranging from 23.5 to 35.8 mm/a. Results from this study provide a better understanding of the highly episodic recharge in arid environments and are critical input to sustainably manage groundwater resources.

Quantification of Conceptual Model Uncertainty in the Modeling of Wet Deposited Atmospheric Pollutants.

Conceptual model uncertainty and parameter uncertainty are dominant contributors to the total uncertainty of a radioecological model output. In the present study the focus is on conceptual model uncertainty, which is often not acknowledged. Conceptual model uncertainty is assessed by subtracting from the total uncertainty of the model output the propagated parameter uncertainty, obtained by means of Bayesian inference analysis. The conceptual model uncertainty is quantified for two process-based models, which describe the interception of wet deposited pollutants under equilibrium and kinetic conditions, respectively. The natural variability due the chemical valence of the elements considered is accounted for in both models. Quantitative evidence has been obtained that the conceptual model uncertainty can contribute to the total uncertainty budget of the models for interception of wet deposited pollutants at least as much as, if not more than, parameter uncertainty.

Application of Indicator Kriging to hydraulic head data to test alternative conceptual models for spring source aquifers

Hydraulic head distributions can inform spring source-water characterisation by determining whether aquifers meet the thresholds required to sustain spring flow. Assessing hydraulic head data can be challenging in areas where data are sparsely distributed and subject to variable measurement uncertainty. Geostatistical methods can be used to estimate hydraulic head values at unmeasured locations with quantitative uncertainty estimates. While these methods have been applied extensively for hydraulic head estimation in management contexts, no studies have used these approaches for spring source-water conceptual model testing. In this study, an investigation into the hydraulic head distribution was conducted through the application of Ordinary Indicator Co-Kriging. Interpolated hydraulic head estimates were used to quantitatively assess the plausibility of source-aquifer conceptual models for the Doongmabulla Springs Complex (DSC), Queensland, Australia. The results offer insights into the likelihood of alternative source aquifers having adequate head to support flow to eight springs within the DSC. Probabilities of adequate head to support the springs ranged from 0.03 to 0.12 for the Permian Formations, compared to <0.01 to 0.49 for the Triassic Formations. Analyses indicated that the Triassic Formations are more likely to have adequate hydraulic head to support spring flow. However, significant uncertainty exists in the conceptual model assessment due to hydraulic head measurement scarcity, particularly in the vicinity of the springs. These findings have important implications given that the Permian Formations will be dewatered by the operation of the nearby Carmichael coal mine. The techniques employed here can inform conceptual model uncertainties arising from the interpretation of sparsely distributed hydraulic head datasets, a major benefit over traditional interpolation methods.

基于概念模型不确定性框架的滨海岩溶含水层咸淡水界面数值模拟

基于概念模型不确定性框架的滨海岩溶含水层咸淡水界面数值模拟

Advances in ecological risk assessment of soil in contaminated sites

With the rapid urbanization and industrial structure adjustment in China, many contaminated sites have been left for remediation. It is essential to develop and implement ecological risk assessment (ERA) before remediating contaminated sites at a large scale as well as sequential management. In this review, we discussed the key problems in ecological risk assessment of soils in contaminated sites focusing on scientific principles, frameworks, techniques, and approaches, including 1) the site-specific framework, 2) uncertainty of conceptual model, 3) toxic mechanisms of combined contamination in soil, 4) screening of assessment endpoints, and 5) development of assessing approaches and frameworks. Then, two perspectives were addressed: the toxicological mechanism of soil combined pollution including bioavailability of contaminants in soil and their joint effect is the scientific problem in ecological risk assessment of soil in contaminated site; and weight of evidence approach based on USEPA four-step approach and EU Tier approach is applicable for ecological risk assessment in field conditions. Future studies should focus on: 1) the coordination of ecological risk assessment (ERA) framework and risk management framework, 2) conceptual mo-del, 3) process-based reactive transport models for exposure evaluation, 4) ecotoxicological mechanism of combined contamination in site soil, and 5) high ecological level endpoints. The aim of this review was to provide theoretical base and framework for the establishment of local guideline of ecological risk assessment in China.

Evaluating two multi-model simulation–optimization approaches for managing groundwater contaminant plumes

Addressing conceptual model uncertainty using multi-model approaches has become an important topic in groundwater hydrology. Model uncertainty is of practical importance when designing a pump-and-treat system for groundwater remediation, as inappropriate consideration of model uncertainty may ultimately lead to ineffective system designs. This study introduces two multi-model approaches for addressing model uncertainty at a site contaminated by chlorinated hydrocarbons and BTEX compounds. We use a multi-objective simulation–optimization method to design a multiple-well pump-and-treat system to contain contaminant plumes within the site perimeter, while minimizing the pumping rate. The design parameters include the coordinates of the pumping well locations and pumping rates. We consider four groundwater models to address uncertainty in conceptualizing site geology, boundary condition, and recharge. Using two multi-model approaches, we evaluate different design concepts. The first approach is a model selection approach that identifies the critical models, which particularly govern the design process, within the multi-model ensemble. The second approach is a model aggregation approach that identifies system designs that meet the remediation objectives for all models of the ensemble. Results show that the model aggregation approach is a more conservative option, which results in reliable designs at the expense of higher pumping rates. This study evaluates these two approaches within the context of multi-model simulation–optimization, providing insights on multi-model simulation–optimization in groundwater management.

Natural springs protection and probabilistic risk assessment under uncertain conditions

The study introduces a comprehensive framework for natural springs' protection and probabilistic risk assessment in the presence of uncertainty associated with the characterization of the groundwater system. The methodology is applied to a regional-scale hydrogeological setting, located in Northern Italy and characterized by the presence of high-quality natural springs forming a unique system whose preservation is of critical importance for the region. Diverse risk pathways are presented to constitute a fault tree model enabling identification of all basic events, each associated with uncertainty and contributing to an undesired system failure. The latter is quantified in terms of hydraulic head falling below a given threshold value for at least one amongst all active springs. The workflow explicitly includes the impact of model parameter uncertainty on the evaluation of the overall probability of system failure due to alternative groundwater extraction strategies. To cope with conceptual model uncertainty, two models based on different reconstructions of the aquifer geological structure are considered. In each conceptual model, hydraulic conductivities related to the geomaterials composing the aquifer are affected by uncertainty. It is found that (a) the type of conceptual model employed to characterize the aquifer structure strongly affects the probability of system failure and (b) uncertainties associated with the ensuing conductivity fields, even as constrained through model calibration, lead to different impacts on the variability of hydraulic head levels at the springs depending on the conceptual model adopted. The results of the study demonstrate that the proposed approach enables one to (i) quantify the risk associated with springs depletion due to alternative strategies of aquifer exploitation; (ii) quantify the way diverse sources of uncertainty (i.e., model and parameter uncertainty) affect the probability of system failure; (iii) determine the optimal exploitation strategy ensuring system functioning; and (iv) identify the most vulnerable springs, where depletion first occurs.

Precipitation phase uncertainty in cold region conceptual models resulting from meteorological forcing time-step intervals

Abstract Precipitation phase determination is a known source of uncertainty in surface-based hydrological, ecological, safety, and climate models. This is primarily due to the surface precipitation phase being a result of cloud and atmospheric properties not measured at surface meteorological or hydrological stations. Adding to the uncertainty, many conceptual hydrological models use a 24-h average air temperature to determine the precipitation phase. However, meteorological changes to atmospheric properties that control the precipitation phase often substantially change at sub-daily timescales. Model uncertainty (precipitation phase error) using air temperature (AT), dew-point temperature (DP), and wet-bulb temperature (WB) thresholds were compared using averaged and time of observation readings at 1-, 3-, 6-, 12-, and 24-h periods. Precipitation phase uncertainty grew 35–65% from the use of 1–24 h data. Within a sub-dataset of observations occurring between AT −6 and 6 °C representing 57% of annual precipitation, misclassified precipitation was 7.9% 1 h and 11.8% 24 h. Of note, there was also little difference between 1 and 3 h uncertainty, typical time steps for surface meteorological observations.

Reducing uncertainty in conceptual prior models of complex geologic systems via integration of flow response data

A major source of uncertainty in subsurface flow modeling is related to the conceptual model of geologic continuity that is adopted by geologists and hydrogeologists. Traditional model calibration methods usually rely on a given model of geologic continuity that is used to constrain the spatial distribution of aquifer properties. Flow response data, however, may also be used as additional source of information to constrain and revise conceptual geologic models or parameters that control them. The problem is more complicated when models that exhibit complex non-Gaussian spatial connectivity patterns are considered. We formulate the problem of flow data integration to select conceptual geologic scenarios with complex lithofacies as a regularized least-squares problem. The solution approach combines learning with model selection regularization techniques, where the former imposes geological feasibility while the latter eliminates scenarios that do not contribute to reproducing the flow response data. Geologic feasibility constraint to preserve complex connectivity patterns is enforced through supervised machine learning while geologic scenario selection is implemented via a mixed l1/l2-norm regularization term. The least-squares minimization is performed using an alternating directions algorithm that involves a sequence of model selection steps followed by geologic feasibility mapping, the latter implemented by combining k nearest neighbor (k-NN) algorithm with a pattern-learning method. Several examples with complex fluvial channel models are used to illustrate the performance of this approach.

Reduction of conceptual model uncertainty using ground-penetrating radar profiles: Field-demonstration for a braided-river aquifer

Hydrogeological flow and transport strongly depend on the connectivity of subsurface properties. Uncertainty concerning the underlying geological setting, due to a lack of field data and prior knowledge, calls for an evaluation of alternative geological conceptual models. To reduce the computational costs associated with inversions (parameter estimation for a given conceptual model), it is beneficial to rank and discard unlikely conceptual models prior to inversion. Here, we demonstrate an approach based on a quantitative comparison of ground-penetrating radar (GPR) sections obtained from field data with corresponding simulation results arising from various geological scenarios. The comparison is based on three global distance measures related to wavelet decomposition, multiple-point histograms, and connectivity that capture geometrical characteristics of geophysical reflection images. Using field data from the Tagliamento braided river system, Italy, we demonstrate that seven out of nine considered geological scenarios can be discarded as they produce GPR sections that are incompatible with those observed in the field. The retained scenarios reproduce important features such as cross-stratified deposits and irregular property interfaces. The most convenient distance measure of those considered is the one based on wavelet-decomposition. Direct analysis of the distances is the most intuitive and fastest way to compare scenarios.

Assessing titanium dioxide nanoparticles transport models by Bayesian uncertainty analysis

With the rapid growth of nanotechnology industry, nanomaterials as an emerging pollutant are gradually released into subsurface environments and become great concerns. Simulating the transport of nanomaterials in groundwater is an important approach to investigate and predict the impact of nanomaterials on subsurface environments. Currently, a number of transport models are used to simulate this process, and the outputs of these models could be inconsistent with each other due to conceptual model uncertainty. However, the performances of different models on simulating nanoparticles transport in groundwater are rarely assessed in Bayesian framework in previous researches, and these will be the primary objective of this study. A porous media column experiment is conducted to observe the transport of Titanium Dioxide Nanoparticles (nano-TiO2). Ten typical transport models which consider different chemical reaction processes are used to simulate the transport of nano-TiO2, and the observed nano-TiO2 breakthrough curves data are used to calibrate these models. For each transport model, the parameter uncertainty is evaluated using Markov Chain Monte Carlo, and the DREAM(ZS) algorithm is used to sample parameter probability space. Moreover, the Bayesian model averaging (BMA) method is used to incorporate the conceptual model uncertainty arising from different chemical reaction based transport models. The results indicate that both two-sites and nonequilibrium sorption models can well reproduce the retention of nano-TiO2 transport in porous media. The linear equilibrium sorption isotherm, first-order degradation, and mobile-immobile models fail to describe the nano-TiO2 retention and transport. The BMA method could instead provide more reliable estimations of the predictive uncertainty compared to that using a single model.

A stable computation of log‐derivatives from noisy drawdown data

AbstractPumping tests interpretation is an art that involves dealing with noise coming from multiple sources and conceptual model uncertainty. Interpretation is greatly helped by diagnostic plots, which include drawdown data and their derivative with respect to log‐time, called log‐derivative. Log‐derivatives are especially useful to complement geological understanding in helping to identify the underlying model of fluid flow because they are sensitive to subtle variations in the response to pumping of aquifers and oil reservoirs. The main problem with their use lies in the calculation of the log‐derivatives themselves, which may display fluctuations when data are noisy. To overcome this difficulty, we propose a variational regularization approach based on the minimization of a functional consisting of two terms: one ensuring that the computed log‐derivatives honor measurements and one that penalizes fluctuations. The minimization leads to a diffusion‐like differential equation in the log‐derivatives, and boundary conditions that are appropriate for well hydraulics (i.e., radial flow, wellbore storage, fractal behavior, etc.). We have solved this equation by finite differences. We tested the methodology on two synthetic examples showing that a robust solution is obtained. We also report the resulting log‐derivative for a real case.

Assessing the relative importance of parameter and forcing uncertainty and their interactions in conceptual hydrological model simulations

Predictions of river flow dynamics provide vital information for many aspects of water management including water resource planning, climate adaptation, and flood and drought assessments. Many of the subjective choices that modellers make including model and criteria selection can have a significant impact on the magnitude and distribution of the output uncertainty. Hydrological modellers are tasked with understanding and minimising the uncertainty surrounding streamflow predictions before communicating the overall uncertainty to decision makers. Parameter uncertainty in conceptual rainfall-runoff models has been widely investigated, and model structural uncertainty and forcing data have been receiving increasing attention. This study aimed to assess uncertainties in streamflow predictions due to forcing data and the identification of behavioural parameter sets in 31 Irish catchments. By combining stochastic rainfall ensembles and multiple parameter sets for three conceptual rainfall-runoff models, an analysis of variance model was used to decompose the total uncertainty in streamflow simulations into contributions from (i) forcing data, (ii) identification of model parameters and (iii) interactions between the two. The analysis illustrates that, for our subjective choices, hydrological model selection had a greater contribution to overall uncertainty, while performance criteria selection influenced the relative intra-annual uncertainties in streamflow predictions. Uncertainties in streamflow predictions due to the method of determining parameters were relatively lower for wetter catchments, and more evenly distributed throughout the year when the Nash-Sutcliffe Efficiency of logarithmic values of flow (lnNSE) was the evaluation criterion.

Solute-Transport Predictive Uncertainty in Alternative Water Supply, Storage, and Treatment Systems.

Alternative water supply, storage, and treatment (AWSST) systems, which utilize aquifers to supply, store, and naturally treat water, are increasingly being implemented globally to address water scarcity and safety. The failure of some AWSST systems to meet water quality expectations was caused by conceptual model error, in which local hydrogeological conditions were less favorable than recognized or considered during project feasibility assessments, economic analyses, and design. More successful implementation of AWSST projects requires that conceptual model error be explicitly and rigorously addressed. Recommended approaches to addressing conceptual model uncertainty include more detailed aquifer characterization, recognition of a wide suite of possible alternative conceptual models, and then screening the models as to whether or not they are plausible and relevant in terms of materially impacting predictive results. Subjective professional judgement remains the basis for assigning probabilities to relevant conceptual models (contingencies).

A Bayesian belief network approach for assessing uncertainty in conceptual site models at contaminated sites

A key component in risk assessment of contaminated sites is in the formulation of a conceptual site model (CSM). A CSM is a simplified representation of reality and forms the basis for the mathematical modeling of contaminant fate and transport at the site. The CSM should therefore identify the most important site-specific features and processes that may affect the contaminant transport behavior at the site. However, the development of a CSM will always be associated with uncertainties due to limited data and lack of understanding of the site conditions. CSM uncertainty is often found to be a major source of model error and it should therefore be accounted for when evaluating uncertainties in risk assessments. We present a Bayesian belief network (BBN) approach for constructing CSMs and assessing their uncertainty at contaminated sites. BBNs are graphical probabilistic models that are effective for integrating quantitative and qualitative information, and thus can strengthen decisions when empirical data are lacking. The proposed BBN approach facilitates a systematic construction of multiple CSMs, and then determines the belief in each CSM using a variety of data types and/or expert opinion at different knowledge levels. The developed BBNs combine data from desktop studies and initial site investigations with expert opinion to assess which of the CSMs are more likely to reflect the actual site conditions. The method is demonstrated on a Danish field site, contaminated with chlorinated ethenes. Four different CSMs are developed by combining two contaminant source zone interpretations (presence or absence of a separate phase contamination) and two geological interpretations (fractured or unfractured clay till). The beliefs in each of the CSMs are assessed sequentially based on data from three investigation stages (a screening investigation, a more detailed investigation, and an expert consultation) to demonstrate that the belief can be updated as more information becomes available.

Optimal observation network design for conceptual model discrimination and uncertainty reduction

AbstractThis study expands the Box‐Hill discrimination function to design an optimal observation network to discriminate conceptual models and, in turn, identify a most favored model. The Box‐Hill discrimination function measures the expected decrease in Shannon entropy (for model identification) before and after the optimal design for one additional observation. This study modifies the discrimination function to account for multiple future observations that are assumed spatiotemporally independent and Gaussian‐distributed. Bayesian model averaging (BMA) is used to incorporate existing observation data and quantify future observation uncertainty arising from conceptual and parametric uncertainties in the discrimination function. In addition, the BMA method is adopted to predict future observation data in a statistical sense. The design goal is to find optimal locations and least data via maximizing the Box‐Hill discrimination function value subject to a posterior model probability threshold. The optimal observation network design is illustrated using a groundwater study in Baton Rouge, Louisiana, to collect additional groundwater heads from USGS wells. The sources of uncertainty creating multiple groundwater models are geological architecture, boundary condition, and fault permeability architecture. Impacts of considering homoscedastic and heteroscedastic future observation data and the sources of uncertainties on potential observation areas are analyzed. Results show that heteroscedasticity should be considered in the design procedure to account for various sources of future observation uncertainty. After the optimal design is obtained and the corresponding data are collected for model updating, total variances of head predictions can be significantly reduced by identifying a model with a superior posterior model probability.