Addressing a variety of questions within Earth science disciplines entails the inference of the spatiotemporal distribution of parameters of interest based on observations of related quantities. Such estimation problems often represent inverse problems that are formulated as linear optimization problems. Computational limitations arise when the number of observations and/or the size of the discretized state space becomes large, especially if the inverse problem is formulated in a probabilistic framework and therefore aims to assess the uncertainty associated with the estimates. This work proposes two approaches to lower the computational costs and memory requirements for large linear space-time inverse problems, taking the Bayesian approach for estimating carbon dioxide (CO2) emissions and uptake (a.k.a. fluxes) as a prototypical example. The first algorithm can be used to efficiently multiply two matrices, as long as one can be expressed as a Kronecker product of two smaller matrices, a condition that is typical when multiplying a sensitivity matrix by a covariance matrix in the solution of inverse problems. The second algorithm can be used to compute a posteriori uncertainties directly at aggregated spatiotemporal scales, which are the scales of most interest in many inverse problems. Both algorithms have significantly lower memory requirements and computational complexity relative to direct computation of the same quantities (O(n(2.5)) vs. O(n(3))). For an examined benchmark problem, the two algorithms yielded massive savings in floating point operations relative to direct computation of the same quantities. Sample computer codes are provided for assessing the computational and memory efficiency of the proposed algorithms for matrices of different dimensions.

Confrontation of climate models with observationally-based reference datasets is widespread and integral to model development. These comparisons yield skill metrics quantifying the mismatch between simulated and reference values and also involve analyst choices, or meta-parameters, in structuring the analysis. Here, we systematically vary five such meta-parameters (reference dataset, spatial resolution, regridding approach, land mask, and time period) in evaluating evapotranspiration (ET) from eight CMIP5 models in a factorial design that yields 68 700 intercomparisons. The results show that while model-data comparisons can provide some feedback on overall model performance, model ranks are ambiguous and inferred model skill and rank are highly sensitive to the choice of meta-parameters for all models. This suggests that model skill and rank are best represented probabilistically rather than as scalar values. For this case study, the choice of reference dataset is found to have a dominant influence on inferred model skill, even larger than the choice of model itself. This is primarily due to large differences between reference datasets, indicating that further work in developing a community-accepted standard ET reference dataset is crucial in order to decrease ambiguity in model skill.

Data assimilation (DA) approaches, including variational and the ensemble Kalman filter methods, provide a computationally efficient framework for solving the CO2 source-sink estimation problem. Unlike DA applications for weather prediction and constituent assimilation, however, the advantages and disadvantages of DA approaches for CO2 flux estimation have not been extensively explored. In this study, we compare and assess estimates from two advanced DA approaches (an ensemble square root filter and a variational technique) using a batch inverse modeling setup as a benchmark, within the context of a simple one-dimensional advection-diffusion prototypical inverse problem that has been designed to capture the nuances of a real CO2 flux estimation problem. Experiments are designed to identify the impact of the observational density, heterogeneity, and uncertainty, as well as operational constraints (i.e., ensemble size, number of descent iterations) on the DA estimates relative to the estimates from a batch inverse modeling scheme. No dynamical model is explicitly specified for the DA approaches to keep the problem setup analogous to a typical real CO2 flux estimation problem. Results demonstrate that the performance of the DA approaches depends on a complex interplay between the measurement network and the operational constraints. Overall, the variational approach (contingent on the availability of an adjoint transport model) more reliably captures the large-scale source-sink patterns. Conversely, the ensemble square root filter provides more realistic uncertainty estimates. Selection of one approach over the other must therefore be guided by the carbon science questions being asked and the operational constraints under which the approaches are being applied.

Terrestrial biosphere models (TBMs) have become an integral tool for extrapolating local observations and understanding of land-atmosphere carbon exchange to larger regions. The North American Carbon Program (NACP) Multi-scale synthesis and Terrestrial Model Intercomparison Project (MsTMIP) is a formal model intercomparison and evaluation effort focused on improving the diagnosis and attribution of carbon exchange at regional and global scales. MsTMIP builds upon current and past synthesis activities, and has a unique framework designed to isolate, interpret, and inform understanding of how model structural differences impact estimates of carbon uptake and release. Here we provide an overview of the MsTMIP effort and describe how the MsTMIP experimental design enables the assessment and quantification of TBM structural uncertainty. Model structure refers to the types of processes considered (e.g., nutrient cycling, disturbance, lateral transport of carbon), and how these processes are represented (e.g., photosynthetic formulation, temperature sensitivity, respiration) in the models. By prescribing a common experimental protocol with standard spin-up procedures and driver data sets, we isolate any biases and variability in TBM estimates of regional and global carbon budgets resulting from differences in the models themselves (i.e., model structure) and model-specific parameter values. An initial intercomparison of model structural differences is represented using hierarchical cluster diagrams (a.k.a. dendrograms), which highlight similarities and differences in how models account for carbon cycle, vegetation, energy, and nitrogen cycle dynamics. We show that, despite the standardized protocol used to derive initial conditions, models show a high degree of variation for GPP, total living biomass, and total soil carbon, underscoring the influence of differences in model structure and parameterization on model estimates.

Many inverse problems in the atmospheric sciences involve parameters with known physical constraints. Examples include nonnegativity (e. g., emissions of some urban air pollutants) or upward limits implied by reaction or solubility constants. However, probabilistic inverse modeling approaches based on Gaussian assumptions cannot incorporate such bounds and thus often produce unrealistic results. The atmospheric literature lacks consensus on the best means to overcome this problem, and existing atmospheric studies rely on a limited number of the possible methods with little examination of the relative merits of each.

This study quantitatively estimates the spatial distribution of anthropogenic methane sources in the United States by combining comprehensive atmospheric methane observations, extensive spatial datasets, and a high-resolution atmospheric transport model. Results show that current inventories from the US Environmental Protection Agency (EPA) and the Emissions Database for Global Atmospheric Research underestimate methane emissions nationally by a factor of similar to 1.5 and similar to 1.7, respectively. Our study indicates that emissions due to ruminants and manure are up to twice the magnitude of existing inventories. In addition, the discrepancy in methane source estimates is particularly pronounced in the south-central United States, where we find total emissions are similar to 2.7 times greater than in most inventories and account for 24 +/- 3% of national emissions. The spatial patterns of our emission fluxes and observed methane-propane correlations indicate that fossil fuel extraction and refining are major contributors (45 +/- 13%) in the south-central United States. This result suggests that regional methane emissions due to fossil fuel extraction and processing could be 4.9 +/- 2.6 times larger than in EDGAR, the most comprehensive global methane inventory. These results cast doubt on the US EPA's recent decision to downscale its estimate of national natural gas emissions by 25-30%. Overall, we conclude that methane emissions associated with both the animal husbandry and fossil fuel industries have larger greenhouse gas impacts than indicated by existing inventories.

Robust estimates of hypoxic extent (both area and volume) are important for assessing the impacts of low dissolved oxygen on aquatic ecosystems at large spatial scales. Such estimates are also important for calibrating models linking hypoxia to causal factors, such as nutrient loading and stratification, and for informing management decisions. In this study, we develop a rigorous geostatistical modeling framework to estimate the hypoxic extent in the northern Gulf of Mexico from data collected during midsummer, quasi-synoptic monitoring cruises (1985-2011). Instead of a traditional interpolation-based approach, we use a simulation-based approach that yields more robust extent estimates and quantified uncertainty. The modeling framework also makes use of covariate information (i.e., trend variables such as depth and spatial position), to reduce estimation uncertainty. Furthermore, adjustments are made to account for observational bias resulting from the use of different sampling instruments in different years. Our results suggest an increasing trend in hypoxic layer thickness (p = 0.05) from 1985 to 2011, but less than significant increases in volume (p = 0.12) and area (p = 0.42). The uncertainties in the extent estimates vary with sampling network coverage and instrument type, and generally decrease over the study period.

Wetlands comprise the single largest global source of atmospheric methane, but current flux estimates disagree in both magnitude and distribution at the continental scale. This study uses atmospheric methane observations over North America from 2007 to 2008 and a geostatistical inverse model to improve understanding of Canadian methane fluxes and associated biogeochemical models. The results bridge an existing gap between traditional top-down, inversion studies, which typically emphasize total emission budgets, and biogeochemical models, which usually emphasize environmental processes. The conclusions of this study are threefold. First, the most complete process-based methane models do not always describe available atmospheric methane observations better than simple models. In this study, a relatively simple model of wetland distribution, soil moisture, and soil temperature outperformed more complex model formulations. Second, we find that wetland methane fluxes have a broader spatial distribution across western Canada and into the northern U.S. than represented in existing flux models. Finally, we calculate total methane budgets for Canada and for the Hudson Bay Lowlands, a large wetland region (50-60 degrees N, 75-96 degrees W). Over these lowlands, we find total methane fluxes of 1.80.24 Tg C yr(-1), a number in the midrange of previous estimates. Our total Canadian methane budget of 16.01.2 Tg C yr(-1) is larger than existing inventories, primarily due to high anthropogenic emissions in Alberta. However, methane observations are sparse in western Canada, and additional measurements over Alberta will constrain anthropogenic sources in that province with greater confidence.