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Accurate accounting of emissions and removals of CO2 is critical for the planning and verification of emission reduction targets in support of the Paris Agreement. Here, we present a pilot dataset of country-specific net carbon exchange (NCE; fossil plus terrestrial ecosystem fluxes) and terrestrial carbon stock changes aimed at informing countries' carbon budgets. These estimates are based on “top-down” NCE outputs from the v10 Orbiting Carbon Observatory (OCO-2) modeling intercomparison project (MIP), wherein an ensemble of inverse modeling groups conducted standardized experiments assimilating OCO-2 column-averaged dry-air mole fraction (XCO2) retrievals (ACOS v10), in situ CO2 measurements or combinations of these data. The v10 OCO-2 MIP NCE estimates are combined with “bottom-up” estimates of fossil fuel emissions and lateral carbon fluxes to estimate changes in terrestrial carbon stocks, which are impacted by anthropogenic and natural drivers. These flux and stock change estimates are reported annually (2015–2020) as both a global 1∘ × 1∘ gridded dataset and a country-level dataset and are available for download from the Committee on Earth Observation Satellites' (CEOS) website: https://doi.org/10.48588/npf6-sw92 (Byrne et al., 2022). Across the v10 OCO-2 MIP experiments, we obtain increases in the ensemble median terrestrial carbon stocks of 3.29–4.58 Pg CO2 yr−1 (0.90–1.25 Pg C yr−1). This is a result of broad increases in terrestrial carbon stocks across the northern extratropics, while the tropics generally have stock losses but with considerable regional variability and differences between v10 OCO-2 MIP experiments. We discuss the state of the science for tracking emissions and removals using top-down methods, including current limitations and future developments towards top-down monitoring and verification systems.

WOMBAT (the WOllongong Methodology for Bayesian Assimilation of Trace-gases) is a fully Bayesian hierarchical statistical framework for flux inversion of trace gases from flask, in situ, and remotely sensed data. WOMBAT extends the conventional Bayesian synthesis framework through the consideration of a correlated error term, the capacity for online bias correction, and the provision of uncertainty quantification on all unknowns that appear in the Bayesian statistical model. We show, in an observing system simulation experiment (OSSE), that these extensions are crucial when the data are indeed biased and have errors that are spatio-temporally correlated. Using the GEOS-Chem atmospheric transport model, we show that WOMBAT is able to obtain posterior means and variances on non-fossil-fuel CO2 fluxes from Orbiting Carbon Observatory-2 (OCO-2) data that are comparable to those from the Model Intercomparison Project (MIP) reported in Crowell et al. (2019). We also find that WOMBAT's predictions of out-of-sample retrievals obtained from the Total Column Carbon Observing Network (TCCON) are, for the most part, more accurate than those made by the MIP participants.

Climate models produce output over decades or longer at high spatial and temporal resolution. Starting values, boundary conditions, greenhouse gas emissions, and so forth make the climate model an uncertain representation of the climate system. A standard paradigm for assessing the quality of climate model simulations is to compare what these models produce for past and present time periods, to observations of the past and present. Many of these comparisons are based on simple summary statistics called metrics. In this article, we propose an alternative: evaluation of competing climate models through probabilities derived from tests of the hypothesis that climate-model-simulated and observed time sequences share common climate-scale signals. The probabilities are based on the behavior of summary statistics of climate model output and observational data over ensembles of pseudo-realizations. These are obtained by partitioning the original time sequences into signal and noise components, and using a parametric bootstrap to create pseudo-realizations of the noise sequences. The statistics we choose come from working in the space of decorrelated and dimension-reduced wavelet coefficients. Here, we compare monthly sequences of CMIP5 model output of average global near-surface temperature anomalies to similar sequences obtained from the well-known HadCRUT4 data set as an illustration.

Sea surface temperature (SST) in the Pacific Ocean is a key component of many global climate models and the El Niño–Southern Oscillation (ENSO) phenomenon. We shall analyse SST for the period November 1981–December 2014. To study the temporal variability of the ENSO phenomenon, we have selected a subregion of the tropical Pacific Ocean, namely the Niño 3.4 region, as it is thought to be the area where SST anomalies indicate most clearly ENSO's influence on the global atmosphere. SST anomalies, obtained by subtracting the appropriate monthly averages from the data, are the focus of the majority of previous analyses of the Pacific and other oceans' SSTs. Preliminary data analysis showed that not only Niño 3.4 spatial means but also Niño 3.4 spatial variances varied with month of the year. In this article, we conduct an analysis of the raw SST data and introduce diagnostic plots (here, plots of variability vs. central tendency). These plots show strong negative dependence between the spatial standard deviation and the spatial mean. Outliers are present, so we consider robust regression to obtain intercept and slope estimates for the 12 individual months and for all-months-combined. Based on this mean–standard deviation relationship, we define a variance-stabilizing transformation. On the transformed scale, we describe the Niño 3.4 SST time series with a statistical model that is linear, heteroskedastic, and dynamical.

We describe a method of evaluating systematic errors in measurements of total column dry-air mole fractions of CO2 (XCO2) from space, and we illustrate the method by applying it to the v2.8 Atmospheric CO2 Observations from Space retrievals of the Greenhouse Gases Observing Satellite (ACOS-GOSAT) measurements over land. The approach exploits the lack of large gradients in XCO2 south of 25° S to identify large-scale offsets and other biases in the ACOS-GOSAT data with several retrieval parameters and errors in instrument calibration. We demonstrate the effectiveness of the method by comparing the ACOS-GOSAT data in the Northern Hemisphere with ground truth provided by the Total Carbon Column Observing Network (TCCON). We use the observed correlation between free-tropospheric potential temperature and XCO2 in the Northern Hemisphere to define a dynamically informed coincidence criterion between the ground-based TCCON measurements and the ACOS-GOSAT measurements. We illustrate that this approach provides larger sample sizes, hence giving a more robust comparison than one that simply uses time, latitude and longitude criteria. Our results show that the agreement with the TCCON data improves after accounting for the systematic errors, but that extrapolation to conditions found outside the region south of 25° S may be problematic (e.g., high airmasses, large surface pressure biases, M-gain, measurements made over ocean). A preliminary evaluation of the improved v2.9 ACOS-GOSAT data is also discussed.

As technology progresses, the availability of massive environmental data with global spatial coverage has become quite common. An example of such data is total column ozone (TCO) remotely sensed from a satellite. In their raw form, these data are often spatially (and temporally) dense, but irregular. The problem considered here is one of detecting a large-scale spatial trend at a given time point (actually, in a given time interval). We propose a sequential aggregation method, producing different levels of coarser (spatial) resolution data and, at the same time, preserving both the local information content and the locations of the raw data. In estimating the large-scale trend, we consider different types of smooth spatial trend surfaces on the sphere, all linear combinations of smooth basis functions and satisfying the topological constraints imposed by the sphere. These include trend surfaces based on tensor products of splines, spherical harmonic basis functions, smoothing spherical splines and a new trend surface that we call fixed-rank smoothing (FRS). The FRS trend surfaces can be based on any set of smooth basis functions on the sphere and are estimated via penalized weighted-least-squares regression (ridge regression) using a data-adaptive penalty term. In comparing the various trend surfaces considered, we look at data fidelity, trend-surface consistency when fitted to data at different resolutions, and small- and large-scale spatial behavior of the resulting detrended fine-resolution data. An application to the TCO data reveals that the large-scale spatial trend can be detected and effectively estimated using coarse-resolution data. The FRS trend surfaces are seen to achieve better data fidelity than other trend surfaces considered, and, in terms of trend-surface consistency and small- and large-scale residual behavior, FRS is seen to have as good, and sometimes better, performance. Copyright © 2003 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this article, computation for the purpose of spatial visualization is presented in the context of understanding the variability in global environmental processes. Here, we generate synthetic but realistic global data sets and input them into computational algorithms that have a visualization capability; we call this a simulation-visualization system. Visualization is key here, because the algorithms which we are evaluating must respect the spatial structure of the input. We modify, augment, and integrate four existing component technologies: statistical conditional simulation, Discrete Global Grids (DGGs), Array Set Addressing, and a visualization platform for displaying our results on a globe. The internal representation of the data to be visualized is built around the need for efficient storage and computation as well as the need to move up and downresolutions in a mutually consistent way. In effect, we have constructed a Geographic Information System that is based on a DGG and has desirable data storage, computation, and visualization capabilities. We provide an example of how our simulation-visualization system may be used, by evaluating a computational algorithm called Spatial Statistical Data Fusion that was developed for use on big, remote-sensing data sets., (© The Author(s) 2022, corrected publication 2022.)

Bayesian inference methods are applied within a Bayesian hierarchical modeling framework to the problems of joint state and parameter estimation, and of state forecasting. We explore and demonstrate the ideas in the context of a simple nonlinear marine biogeochemical model. A novel approach is proposed to the formulation of the stochastic process model, in which ecophysiological properties of plankton communities are represented by autoregressive stochastic processes. This approach captures the effects of changes in plankton communities over time, and it allows the incorporation of literature metadata on individual species into prior distributions for process model parameters. The approach is applied to a case study at Ocean Station Papa, using particle Markov chain Monte Carlo computational techniques. The results suggest that, by drawing on objective prior information, it is possible to extract useful information about model state and a subset of parameters, and even to make useful long-term forecasts, based on sparse and noisy observations.

Analyses of ecological data should account for the uncertainty in the process(es) that generated the data. However, accounting for these uncertainties is a difficult task, since ecology is known for its complexity. Measurement and/or process errors are often the only sources of uncertainty modeled when addressing complex ecological problems, yet analyses should also account for uncertainty in sampling design, in model specification, in parameters governing the specified model, and in initial and boundary conditions. Only then can we be confident in the scientific inferences and forecasts made from an analysis. Probability and statistics provide a framework that accounts for multiple sources of uncertainty. Given the complexities of ecological studies, the hierarchical statistical model is an invaluable tool. This approach is not new in ecology, and there are many examples (both Bayesian and non-Bayesian) in the literature illustrating the benefits of this approach. In this article, we provide a baseline for concepts, notation, and methods, from which discussion on hierarchical statistical modeling in ecology can proceed. We have also planted some seeds for discussion and tried to show where the practical difficulties lie. Our thesis is that hierarchical statistical modeling is a powerful way of approaching ecological analysis in the presence of inevitable but quantifiable uncertainties, even if practical issues sometimes require pragmatic compromises.

An infectious disease typically spreads via contact between infected and susceptible individuals. Since the small-scale movements and contacts between people are generally not recorded, available data regarding infectious disease are often aggregations in space and time, yielding small-area counts of the number infected during successive, regular time intervals. In this paper, we develop a spatially descriptive, temporally dynamic hierarchical model to be fitted to such data. Disease counts are viewed as a realization from an underlying multivariate autoregressive process, where the relative risk of infection incorporates the space-time dynamic. We take a Bayesian approach, using Markov chain Monte Carlo to compute posterior estimates of all parameters of interest. We apply the methodology to an influenza epidemic in Scotland during the years 1989-1990., (Copyright 2002 John Wiley & Sons, Ltd.)

Disease incidence or disease mortality rates for small areas are often displayed on maps. Maps of raw rates, disease counts divided by the total population at risk, have been criticized as unreliable due to non-constant variance associated with heterogeneity in base population size. This has led to the use of model-based Bayes or empirical Bayes point estimates for map creation. Because the maps have important epidemiological and political consequences, for example, they are often used to identify small areas with unusually high or low unexplained risk, it is important that the assumptions of the underlying models be scrutinized. We review the use of posterior predictive model checks, which compare features of the observed data to the same features of replicate data generated under the model, for assessing model fitness. One crucial issue is whether extrema are potentially important epidemiological findings or merely evidence of poor model fit. We propose the use of the cross-validation posterior predictive distribution, obtained by reanalyzing the data without a suspect small area, as a method for assessing whether the observed count in the area is consistent with the model. Because it may not be feasible to actually reanalyze the data for each suspect small area in large data sets, two methods for approximating the cross-validation posterior predictive distribution are described., (Copyright 2000 John Wiley & Sons, Ltd.)

Consider a pharmaceutical trial where the consequences of different decisions are expressed on a financial scale. The efficacy of the new drug under consideration has a prior distribution obtained from the underlying biological process, animal experiments, clinical experience, and so forth. Berry and Ho (Biometrics 44, 219-227) show how these components are used to establish an optimal (Bayes) sequential testing procedure, assuming a known constant sample size at each decision point. We show in this article how it is also possible to optimize further, with respect to the sample-size rule. This component of the design, which is missing from most sequential procedures, has the potential to yield considerably larger expected net gains (equivalently, considerably smaller Bayes risks).

This study takes a statistical-modeling point of view to the assessment of health care services and procedures. The emphasis is on small-area prediction of incidence rates from spatially contiguous regions, although suitable modifications can also give doctor-level predictions. The main idea is to recognize the individuality of each region through a spatial Bayesian model for incidence rates. A noise component, because of location error and measurement error, is filtered out using empirical Bayes methods. The resulting smoothed predictors of incidence rates provide an accurate picture of the health care service or procedure under investigation.

Empirical Bayes methods are used to estimate the extent of the undercount at the local level in the 1980 U.S. census. "Grouping of like subareas from areas such as states, counties, and so on into strata is a useful way of reducing the variance of undercount estimators. By modeling the subareas within a stratum to have a common mean and variances inversely proportional to their census counts, and by taking into account sampling of the areas (e.g., by dual-system estimation), empirical Bayes estimators that compromise between the (weighted) stratum average and the sample value can be constructed. The amount of compromise is shown to depend on the relative importance of stratum variance to sampling variance. These estimators are evaluated at the state level (51 states, including Washington, D.C.) and stratified on race/ethnicity (3 strata) using data from the 1980 postenumeration survey (PEP 3-8, for the noninstitutional population).", (excerpt)