Evaluating marginal likelihood is the most critical and computationally expensive task, when conducting Bayesian model averaging to quantify parametric and model uncertainties. The evaluation is commonly done by using Laplace approximations to evaluate semianalytical expressions of the marginal likelihood or by using Monte Carlo (MC) methods to evaluate arithmetic or harmonic mean of a joint likelihood function. This study introduces a new MC method, i.e., thermodynamic integration, which has not been attempted in environmental modeling. Instead of using samples only from prior parameter space (as in arithmetic mean evaluation) or posterior parameter space (as in harmonic mean evaluation), the thermodynamic integration method uses samples generated gradually from the prior to posterior parameter space. This is done through a path sampling that conducts Markov chain Monte Carlo simulation with different power coefficient values applied to the joint likelihood function. The thermodynamic integration method is evaluated using three analytical functions by comparing the method with two variants of the Laplace approximation method and three MC methods, including the nested sampling method that is recently introduced into environmental modeling. The thermodynamic integration method outperforms the other methods in terms of their accuracy, convergence, and consistency. The thermodynamic integration method is also applied to a synthetic case of groundwater modeling with four alternative models. The application shows that model probabilities obtained using the thermodynamic integration method improves predictive performance of Bayesian model averaging. The thermodynamic integration method is mathematically rigorous, and its MC implementation is computationally general for a wide range of environmental problems.

Soil microbial respiration pulses in response to episodic rainfall pulses (the “Birch effect”) are poorly understood. We developed and assessed five evolving microbial enzyme models against field measurements from a semiarid savannah characterized by pulsed precipitation to understand the mechanisms to generate the Birch pulses. The five models evolve from an existing four‐carbon (C) pool model to models with additional C pools and explicit representations of soil moisture controls on C degradation and microbial uptake rates. Assessing the models using techniques of model selection and model averaging suggests that models with additional C pools for accumulation of degraded C in the dry zone of the soil pore space result in a higher probability of reproducing the observed Birch pulses. Degraded C accumulated in dry soil pores during dry periods becomes immediately accessible to microbes in response to rainstorms, providing a major mechanism to generate respiration pulses. Explicitly representing the transition of degraded C and enzymes between dry and wet soil pores in response to soil moisture changes and soil moisture controls on C degradation and microbial uptake rates improve the models' efficiency and robustness in simulating the Birch effect. Assuming that enzymes in the dry soil pores facilitate degradation of complex C during dry periods (though at a lower rate) results in a greater accumulation of degraded C and thus further improves the models' performance. However, the actual mechanism inducing the greater accumulation of labile C needs further experimental studies.

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[1] Analysts are often faced with competing propositions for each uncertain model component. How can we judge that we select a correct proposition(s) for an uncertain model component out of numerous possible propositions? We introduce the hierarchical Bayesian model averaging (HBMA) method as a multimodel framework for uncertainty analysis. The HBMA allows for segregating, prioritizing, and evaluating different sources of uncertainty and their corresponding competing propositions through a hierarchy of BMA models that forms a BMA tree. We apply the HBMA to conduct uncertainty analysis on the reconstructed hydrostratigraphic architectures of the Baton Rouge aquifer-fault system, Louisiana. Due to uncertainty in model data, structure, and parameters, multiple possible hydrostratigraphic models are produced and calibrated as base models. The study considers four sources of uncertainty. With respect to data uncertainty, the study considers two calibration data sets. With respect to model structure, the study considers three different variogram models, two geological stationarity assumptions and two fault conceptualizations. The base models are produced following a combinatorial design to allow for uncertainty segregation. Thus, these four uncertain model components with their corresponding competing model propositions result in 24 base models. The results show that the systematic dissection of the uncertain model components along with their corresponding competing propositions allows for detecting the robust model propositions and the major sources of uncertainty.

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