The constellation of Earth-observing satellites has now produced atmospheric greenhouse gas concentration estimates covering a period of several years. Their global coverage is providing additional information on the global carbon cycle. These products can be combined with complex inversion systems to infer the magnitude of carbon sources and sinks around the globe. Multiple factors, including the atmospheric transport model and satellite product aggregation method, can impact such flux estimates. Analysis of variance (ANOVA) is a well-established statistical framework for estimating common signals while partitioning variability across factors in the analysis of experiments. Functional ANOVA extends this approach with a statistical model that incorporates spatiotemporal correlation for each ANOVA component. The approach is illustrated on inversion experiments with different satellite retrieval aggregation methods and identifies consistent significant patterns in flux increments that span large spatial scales. Functional ANOVA identifies these patterns while accounting for the uncertainty at small spatial scales that is attributed to differences in the aggregation method. Functional ANOVA is also applied to a recent flux model intercomparison project (MIP), and the relative magnitudes of inversion system effects and data source (satellite versus in situ) are similar but exhibit slightly different importance for fluxes over different continents. In all examples, the unexplained residual variability is locally sizable in magnitude but with limited spatial and temporal correlation. These common behaviors across flux inversion experiments demonstrate the diagnostic capability for functional ANOVA to simultaneously distinguish the spatiotemporal coherence of carbon cycle processes and algorithmic factors.

Many of the key processes in the global carbon cycle have undergone substantial change in recent decades, yet their impacts remain challenging to estimate. This is due in large part to the sparsity of direct observations of carbon fluxes. In particular, a lack of global coverage requires alternative approaches for understanding the global carbon cycle. Fluxes can be inferred indirectly with atmospheric transport models in combination with information on atmospheric carbon dioxide concentration. Regular global

Some of these sources of uncertainty, such as the data source or inversion system, can be represented as discrete instances of multiple categorical factors, and partitioning their relative contributions to the range of solutions can guide priorities for future research in carbon cycle science. In this work we are particularly interested in flux estimates derived from different inversion systems, such as those investigated in model intercomparison projects

Given a set of flux maps obtained under different scenarios, or combinations of these factors of interest, our goal is to find common features among the scenarios and to identify systematic ways or regions in which fluxes from different scenarios differ. Analysis of variance (ANOVA) is a statistical modeling framework that facilitates the estimation of the common and factor-specific effects. It further characterizes the magnitude of the differences within factors relative to the inherent variability within a scenario. Statistical model assumptions dictate the estimation of this within-scenario variability and will be an additional focus of our investigation. The ANOVA methodology has been extended to functional data, such as time series and spatial fields, where it can provide a coherent depiction of space–time patterns and anomalies due to various factors

The ANOVA approach can be particularly useful for analyzing output from a collection of Earth system models or assimilation systems with a common quantity of interest and similar experimental setup. This setup is often formalized as a MIP, an enterprise becoming commonplace among multi-component process models in Earth system modeling

Functional ANOVA extends the classical approach to settings with quantities of interest that are functions of known inputs such as space and/or time. The statistical model is typically extended with a specification for the relationships among the ANOVA components across space and time. For applications involving spatial fields, individual ANOVA effects are typically assumed to be spatially correlated, and this structure can be estimated from the data available. This strategy has been applied to output from regional climate models

In this paper, we implement the functional ANOVA methodology for multiple flux inversion solutions in order to identify meaningful, spatially coherent, carbon cycle signals and partition variability among various solutions in the multi-model ensemble. We illustrate the approach for a recent MIP effort

In subsequent sections, we employ the functional ANOVA methodology for multiple collections of flux inversions using in situ data and products from OCO-2. For the satellite data, the inversion systems use retrievals of XCO

An ongoing NASA effort under the Making Earth System Data Records for Use in Research Environments (MEaSUREs) program aims to provide inversion-ready data products that use OCO-2 and GOSAT retrievals. The effort produces spatially aggregated and gap-filled estimates of XCO

Other spatial aggregation approaches have been devised for OCO-2 inversions. The resulting data products are all structured like Level 3 products and have similar spatial resolution but differ in the underlying methodology, particularly in handling spatial correlation in the Level 2 XCO

The ANOVA methodology is particularly convenient for analyzing the quantitative outcomes of experiments or trials under various discrete combinations of one or more factors of interest. The approach formulates a statistical model that is outlined in the next section. The parameters of the model include an overall mean response and additive effects for each combination of factors. Replication within factor combinations allows the decomposition of variability between (mean differences) and within combinations (noise, unexplained variability). Our demonstration of the functional ANOVA for the fused

The changing carbon cycle of the middle and high latitudes plays a critical role in the global climate system. These land areas are major carbon sinks during JJA. The multiple inversions from OCO-2 reported in

Monthly flux estimates from CMS-Flux for combinations of year and aggregation method. The top two rows depict fluxes for JJA 2015 from the two aggregation methods, and the bottom two rows depict fluxes for JJA 2016. The Control case uses the super-obs aggregation approach, and the Fused case uses the local kriging aggregation approach

Our second demonstration of the functional ANOVA approach comes from a multi-institution flux MIP using data products from OCO-2, which provides global estimates of XCO

Multiple flux inversion teams applied a common inversion protocol to their individual inversion systems as part of the OCO-2 Version 9 Model Intercomparison Project

The V9 MIP flux experiment suite includes estimates from 10 inversion systems

As

Flux inversion systems in this study; see Tables 1–2 of

Monthly flux estimates from four inversion systems for June–July–August (JJA) 2016 over North America, with columns for the different inversions. Panel

Monthly flux estimates from four inversion systems for JJA 2016 over Africa. Panel

ANOVA is a statistical method with a long history connected to designed experiments. In such experiments, one or more factors can be controlled at levels selected by the experimenters, and ANOVA provides a framework for estimating the factors' impact on response variables of interest. The method relies on replication within combinations of factors, or treatments, in order to estimate a mean response for each combination of factors, along with a partitioning of variability between and within treatments. The treatment means are typically re-parameterized into an overall mean and individual effects for each level of the factors, as well as interaction effects.

The classic implementation of ANOVA considers a univariate response, such as an integrated or average carbon flux over a region of interest. This is frequently extended to a multivariate response with MANOVA (multivariate ANOVA), and the decomposition of variance is accompanied by estimation of the correlation structure among the multivariate responses

In the current work, we invoke a two-way functional ANOVA in the context of carbon flux fields over land. In the two-way model, there are two experimental factors examined, generically termed factor A and factor B. In the fused CO

The response for the functional ANOVA models in Eqs. (

Summary of flux inversion results used in functional ANOVA examples. Region numbers indicate TransCom regions used in each example

The effects model (

In classic univariate ANOVA, the effects model parameters are estimated by assembling a series of contrast effects of reduced dimension to ensure identifiability. For factor A, there are

Since the quantities of interest are spatial fields, the ANOVA effects are functions of location. The estimation can account for this structure and exploit potential spatial correlation if a suitable spatial statistical model is incorporated in a hierarchical fashion. To that end, a Gaussian process (GP) is assumed for each spatial field. In this study, the flux inversion results are reported on a spatial grid. For each ANOVA component, the collection of

Analogous GP assumptions are made for the remaining ANOVA model components. When

Functional ANOVA for geophysical model output has often used different time points to yield multiple pseudo-replicates

The spatiotemporal covariance for the error process is

Bayesian inference is commonly used for hierarchical spatiotemporal statistical models and has been implemented in previous work on functional ANOVA incorporating spatial dependence

The posterior distribution (Eq.

The functional ANOVA model and MCMC algorithm for the carbon flux examples are broadly similar to previous demonstrations with climate model output

The MCMC algorithm outlined in the previous section yields a large collection of random draws from the posterior distribution of the spatiotemporal covariance parameters and ANOVA components. The posterior samples can be summarized for individual parameters, as well as for arbitrary functions of them. For example, while the MCMC samples the contrast effects

It is also worth emphasizing that the inference can be broadly partitioned into two categories of quantities. The first category includes the covariance parameters

The CMS-Flux inversion results over Eurasia for JJA 2015 and 2016 using the super-obs and data fusion aggregation methods were incorporated into the first functional ANOVA implementation. Table

Functional ANOVA spatiotemporal covariance parameter estimates for the CMS-Flux inversions in the fused CO

The contrast in spatial coherence among the ANOVA components is also evident in the location-specific posterior means of the ANOVA components, which are summarized in Fig.

Posterior means for functional ANOVA model components for the records of fused CO

For this example, there is particular interest in the impact of the aggregation method on the estimated fluxes from an inversion system. Figure

Posterior probabilities for the records of fused CO

The functional ANOVA inference was carried out separately for flux fields from the OCO-2 V9 flux MIP over North America and Africa for JJA 2016. In both cases the two ANOVA factors are the flux inversion system with four levels (modeling groups) and the data source with two levels (IS and LNLG). These two regions represent distinct scenarios for the methodology for a number of reasons. The carbon cycles of the temperate and boreal land regions, and transitions therein, of North America differ from the tropical and subtropical areas of Africa. In addition, data availability for the two regions is markedly different. As shown in Fig. 1 of

Table

The two regions differ slightly in the relative variability of the model, data source, and interaction effects (

Functional ANOVA spatiotemporal covariance parameter estimates for North America and Africa for JJA 2016. The posterior median is the value listed first in each cell, and the values in parentheses are the lower and upper endpoints of 95 % credible intervals. Data and standard deviations have units of gC m

Noting again that the analysis is carried out on the inversions' deviations from their respective priors, Fig.

The MCMC procedure provides samples from the full joint posterior distribution, and the samples can be summarized in various ways to describe the uncertainty for quantities of interest. Figure

Posterior mean for the functional ANOVA overall mean flux increment

Posterior mean for the functional ANOVA main effect for flux model (

Posterior credible intervals for the functional ANOVA overall mean

Figure

Posterior mean for the functional ANOVA overall mean flux increment

Posterior mean for the functional ANOVA main effect for flux model (

Posterior credible intervals for the functional ANOVA data source effect

Flux inversions produce estimates of the land–ocean–atmosphere exchange of carbon as spatiotemporal fields, providing critical information on the global climate system. These estimates can be variable due to the combinations of factors, such as the atmospheric transport model and the input data source(s), used in the inversions. Recent application of classical analysis of variance (ANOVA) to spatially aggregated fluxes provided a statistical model for partitioning variability among these factors while estimating the common signals from the various flux inversions

The CMS-Flux inversion system was used in a set of inversion experiments to investigate the impact of satellite retrieval aggregation on flux inferences, contrasting a super-obs method with data fusion for aggregating fine-scale OCO-2 retrievals. Overall the aggregation method effect estimated via functional ANOVA was small in magnitude and in its extent of spatial dependence, particularly relative to the overall mean flux increment. Estimated flux differences across years exhibit substantial spatial coherence relative to the aggregation method and interaction effects. At the same time, the interannual variability is of a similar order of magnitude as the aggregation method effect, so detecting year-to-year differences is more challenging with a limited number of pseudo-replicates.

The functional ANOVA was also implemented for a subset of inversions from the OCO-2 flux MIP over both North America and Africa for JJA 2016. The functional ANOVA identified local consensus in flux increments for both continents in the presence of variability across inversion systems and atmospheric

The four inversion systems represented in the MIP functional ANOVA use the same inverse method and have similar spatial resolution in their flux solutions. This subset was selected to illustrate the ANOVA, including the Vecchia approximation for GPs, for a factor with more than two levels, where a more complex set of contrasts is employed to preserve the sum-to-zero constraints. This demonstration indicates that the extension to more than two levels per factor is attainable methodologically and computationally. The estimation could be extended to the full collection of inversion systems in the OCO-2 MIP. This extension would modestly increase the computational burden of the MCMC, but the intensive operations on the GP precision matrices would still be executed just once per ANOVA component per MCMC iteration, as noted in the Supplement (Sect. S.1.3.1). MCMC convergence could be somewhat more challenging with more levels per factor.

ANOVA methods and the resulting estimates can be used to devise potentially unequal weights for combining flux estimates into a consensus flux estimate

The current implementation of functional ANOVA for carbon flux estimates has extended related applications to climate models

Since the functional ANOVA model is applied to flux increments, or the difference between posterior and prior fluxes, some of the functional ANOVA results can be challenging to interpret, particularly for the MIP experiments. For the OCO-2 MIP, each modeling group selected its own prior flux and strategy for gridding results to the common output resolution. These choices, along with transport model impacts, all likely contribute to the spatial patterns in model effects inferred through the functional ANOVA. These characteristics could be more explicitly controlled in the design of future multi-model flux inversion studies. For example, multi-model experiments that use a common prior flux across inversion systems would help diagnose the impact of other sources of variability, such as transport model effects.

As the satellite CO

The OCO-2 V9 MIP surface gridded fluxes used in the examples are available from

The supplement related to this article is available online at:

JH: project administration, conceptualization, methodology, software, formal analysis, data curation, writing (original draft and review and editing), visualization. MK: conceptualization, methodology, software, writing (original draft and review and editing). HN: conceptualization, methodology, software, data curation, writing (review and editing). VY: project administration, conceptualization, methodology, supervision, writing (review and editing). JL: conceptualization, methodology, formal analysis, data curation, writing (review and editing).

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The authors thank David Baker, Amy Braverman, Noel Cressie, Susan Kulawik, and the OCO-2 flux team for helpful discussions. The authors further appreciate the thoughtful comments from two reviewers on this paper.

This research has been supported by the National Aeronautics and Space Administration (NASA) through the Making Earth System Data Records for Use in Research Environments (MEaSUREs) program. This research was performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA.

This paper was edited by Hans Verbeeck and reviewed by Julia Marshall and three anonymous referees.