With the increase in the number of statistical estimates of surface ocean CO2, the Surface Ocean CO2 Mapping (SOCOM) community collated 14 of these methods in an intercomparison of “gap-filling” methods (Rödenbeck et al., 2015). The intercomparison gives an overview of the SOCOM landscape, with regression and statistical interpolation approaches making up eight and four of the 14 methods, respectively (Rödenbeck et al., 2015). Two model-based approaches were also compared.

While SOCOM intercomparison did not seek to identify an optimal mapping method, it assessed members according to how well they represented interannual variability (IAV) relative to climatological surface ocean pCO2 increasing at the rate of atmospheric CO2 concentrations (Riav). Two methods, the Jena-MLS (mixed-layer scheme) and MPI-SOMFFN (self-organising map feed-forward neural network), achieved lower Riav scores compared to other members of the comparison. MPI-SOMFFN is a global implementation of a two-step clustering-regression approach and has been widely adopted in the literature (Landschützer et al., 2015, 2016, 2018; Ritter et al., 2017). The elegance of the clustering-regression approach, particularly the clustering step, is that it reduces the problem into smaller parts with more coherent variability and reduces the computational size of the problem per cluster – a beneficial attribute when using regression methods that do not scale well to big datasets.

The SOCOM intercomparison found that the gap-filling methods were in agreement in regions with a large number of seasonally resolving persistent measurements, but the different methods did not agree in regions where data were sparse (e.g. the Southern Ocean). Similarly, Ritter et al. (2017) found little agreement in the Southern Ocean on seasonal timescales, yet on decadal timescales, there was agreement on the direction of trends between gap-filling methods.

The assessment of gap-filling methods is largely limited by the distribution of the observational coverage, which is particularly true for the Southern Hemisphere where data are sparse (Rödenbeck et al., 2015; Bakker et al., 2016). The standard use of root mean squared error (RMSE) and bias as measures of uncertainty gives larger weighting to observation-heavy regions or periods compared with data-sparse regions and periods, potentially leading to underestimates of uncertainty (Lebehot et al., 2019). Note that the term “error” refers here to the error introduced by the gap-filling method relative to the observations. The Riav score improves on the standard implementation of RMSE and bias by weighting the uncertainties annually, thus giving a less temporally biased estimate of uncertainty.

Previous studies have compared their methods' estimates to independent datasets, where measurements of pCO2 are not included in the SOCAT datasets (Landschützer et al., 2013, 2014; Jones et al., 2015; Denvil-Sommer et al., 2019). These data serve as good validation data, particularly with the inclusion of derivations of pCO2 from autonomous platforms in the Southern Ocean, a historically undersampled area especially during winter (Boutin and Merlivat, 2013; Gray et al., 2018).

One of the concluding statements in the SOCOM intercomparison is that pseudo or synthetic data (deterministic model output) experiments should be used to test and compare methods. Gregor et al. (2017) did just this, but their study was limited to the Southern Ocean, and the synthetic data did not fully capture the variability represented by observations, in part due to coarse synthetic data resolution (5 d mean and 1/2∘ spatially). The authors found that the ensemble average performed slightly better than ensemble members, in agreement with ensemble averaging approaches previously used in ocean CO2 studies (Khatiwala et al., 2013). On the other hand, Lebehot et al. (2019) investigated the performance of an interpolation method in the North Atlantic using an ensemble of model outputs. Their approach offered a unique way of assessing a gap-filling method at places and times where no observations were made.

The main aim of this study is to present and evaluate a new machine-learning approach to estimate surface ocean pCO2. We propose the use of an ensemble average, where we hypothesise that the “whole is greater than the sum of its parts” as the strengths of the ensemble members are often complementary in such a way to overcome the weaknesses (Khatiwala et al., 2013; Gregor et al., 2017). Further, we aim to evaluate the method for a selection of existing gap-filling methods. From this comparison, we aim not only to gain a sense of our method's performance but also the state of gap-filling based estimates; i.e. where would we be able to improve in future work?

The experimental design, outlined below, is summarised in Fig. 1:

In the first step (denoted as “K-means clustering” in Fig. 1), we generate climatological biomes using the oceanic CO2 biomes by Fay and McKinley (2014), and a selection of features variables (five combinations) and number of clusters (a range of 11 to 25 clusters, stepping by two) resulting in a total of 41 clustering configurations.

Standard machine-learning implementation requires a training and a predictive dataset. The training dataset consists of a target variable that is being predicted (in this case, pCO2) and one or more feature variables that have samples that correspond with target samples, e.g. sea surface temperature (SST), Chl a and mixed-layer depth (MLD) co-located in space and time, where feature variables may directly or indirectly influence the target variable. Features variables are used to predict once a machine-learning model has been trained and must thus be available for the full prediction domain.

Here, we use surface ocean pCO2 calculated from the SOCAT v5 monthly gridded fCO2 (fugacity of CO2) product (hereinafter SOCAT v5, as shown in Fig. 2) as the target variable (Sabine et al., 2013; Bakker et al., 2016). SOCAT v5 is a quality-controlled dataset that contains observations of surface ocean fCO2, which is converted to pCO2 with 1pCO2=fCO2⋅exp⁡Patmsurf⋅B+2⋅δR⋅T-1, where Patmsurf is the atmospheric pressure at the surface of the ocean, T is the SST in ∘K, B and δ are virial coefficients, and R is the gas constant (Dickson et al., 2007). We used ERA-Interim Patmsurf (Dee et al., 2011) and National Oceanic and Atmospheric Administration (NOAA) daily optimally interpolated SST version 2 (dOISSTv2) that uses only Advanced Very High Resolution Radiometer (AVHRR; Reynolds et al., 2007; Banzon et al., 2016) data.

An important consideration in the use of the SOCAT database is that in situ measurements (i.e. ship measurements) are not collected at the surface. The in situ temperatures that coincide with pCO2 in the SOCAT database are thus different from surface temperature products used to estimate pCO2 and calculate fluxes (Goddijn-Murphy et al., 2015; Bakker et al., 2016). The discrepancy in in situ and remotely sensed temperature results in a theoretical difference between pCO2 measured at the ship intake depth and the surface due to warming or cooling (Takahashi et al., 1993). Goddijn-Murphy et al. (2015) suggest that a correction for the theoretical difference in pCO2 should be made using the empirical relationship between pCO2 and temperature (Takahashi et al., 1993). While this merits further coordinated consideration by the marine CO2 observation community, we do not apply such a temperature correction in this study, as we aim to be consistent with the earlier pCO2 estimates from the SOCOM intercomparison (Rödenbeck et al., 2015). However, we do present the potential impact of this discrepancy in Sect. S2.4.

Feature variables in both the training and predictive datasets are globally gridded products, including satellite observations, in situ measurements and reanalysis products (Table 1; see Sect. S1 for details). All feature variables are gridded to a monthly frequency onto a global 1∘×1∘ resolution grid. Thereafter, data processing steps are applied as shown in Table 1 and described in detail in the Supplement (Sect. S1), with the final output being a complete dataset ranging from 1982 to 2016. Note that the clustering and regression steps use different subsets of the feature variables, as indicated in Table 1.

In this paragraph, we briefly describe the data processing steps shown in Table 1; detailed product descriptions and in-depth processing steps are in Sect. S1. We derive an additional SST feature, SST′, by subtracting the annual mean of SST from each respective year, leaving the annual mean anomalies (Reynolds et al., 2007; Banzon et al., 2016). We use the log10 transformation of the Globcolour Chl a global product (Maritorena et al., 2010). Cloud gaps and the period before the start of the product (1982–1997) are filled with the climatology (1998–2016), and high-latitude winter regions (where there is no climatology for Chl a) are filled with low-concentration random noise to be consistent with regions of low-concentration Chl a (Gregor et al., 2017). We derive an additional Chl a feature, Chl a′, using the same procedure as described for the SST annual mean anomalies. We use a log10 transformation of MLD from Argo float density profiles (Holte et al., 2017) to create a monthly climatology, thus imposing the assumption that there is no interannual variability. Wind speed is calculated from 6-hourly data using the equation in Table 1 before taking the monthly average. Atmospheric pCO2 is calculated with pCO2=xCO2atm×Patm, where xCO2atm is the mole fraction of atmospheric CO2 (from ObsPack v3 by Masarie et al., 2014) and Patm is the reanalysed mean sea-level pressure (from ERA-interim 2; Dee et al., 2011) – further details for the procedure are in Sect. S1. The climatology of eddy kinetic energy (EKEclim) is calculated from u and v surface current components (integrated for depth < 15 m) from the Globcurrent product (Rio et al., 2014), where u′ is calculated as u¯-u and similarly with v (Table 1).

The seasonal and interannual variability of global surface ocean pCO2 is complex due to interactions of various driver variables acting on the surface ocean at different space scales and timescales (Lenton et al., 2012; Landschützer et al., 2015; Gregor et al., 2018). Machine-learning algorithms applied globally struggle to represent the pCO2 accurately unless spatial coordinates are included as feature variables (Gregor et al., 2017). This is due to the fact that pCO2 may respond inconsistently to observable feature variables in different regions as it is not possible to observe all feature variables that drive pCO2. A common practice to avoid the inclusion of coordinates is to separate the ocean into regions where processes that drive pCO2 are coherent and then apply individual regressions to each region – five of the eight regression methods in Rödenbeck et al. (2015) apply this approach. We adopt two such approaches to develop regions of internal coherence with respect to CO2 variability, namely regions defined by biogeochemical properties and clusters defined by a clustering algorithm.

Our first “clustering” approach uses the oceanic CO2 biomes by Fay and McKinley (2014) that divide the ocean into 17 biomes. Fay and McKinley (2014) define their biomes by establishing thresholds for SST, Chl a, sea-ice extent and maximum MLD. Unclassified regions from the original biomes are manually assigned based on their geographical extent, resulting in six additional regions (Fig. 3). We maintain these as separate regions from the original Fay and McKinley (2014) biomes. Their study originally did not classify these regions in the core biomes because the physical and biogeochemical properties were not accounted for by the set thresholds from their study. This would suggest that drivers of CO2 in these regions could be quite different from the adjacent open-ocean biomes. Note that we may refer to the modified Fay and McKinley (2014) ocean CO2 biomes as “CO2 biomes” or as “BIO23” from here on (Fig. 3). For later analyses, we group certain biomes together, as shown by the brackets above the colour bar in Fig. 3.

We also use K-means clustering, which groups data based on Euclidean distances. More specifically, we implement mini-batch K means from Python's Scikit-learn package (Sculley, 2010; Pedregosa et al., 2011), which is described in the Supplement (Sect. S2.2; Fig. S2). We apply clustering with various feature combinations and the number of clusters (shown by orange hexagons in Fig. 1). We tested a range of 11 to 25 clusters (stepping by two). The performance of each clustering configuration is not tested with a clustering metric; instead, we test the performance based on the test scores of the regressions in the next step as a more complete indicator of performance. We find optimal results with respect to RMSE and biases with 21 and 23 clusters. We selected 21 clusters (Fig. S2). Each method of defining regional coherence with respect to pCO2 variability has its methodological weaknesses so in this study, we adopted the approach of incorporating both K means and CO2 biomes into the ensemble average (Fig. 1). Although this likely weakens the geophysical meaning of the ensemble domains, we show that it strengthens the overall performance of the ensemble average.

Here, we describe the underlying machine-learning principles of regression. The co-located data (i.e. SOCAT v5) are split into training and test subsets with a roughly 80 : 20 split. The test subset is isolated from the training process to attain a reliable estimate of uncertainty. We make the split between training and test subsets based on a random subset of years in the time series (1982–2016): 1984, 1990, 1995, 2000, 2005, 2010 and 2014. We avoid using a shuffled train-test split (completely random), as this leads to artificially low uncertainties in machine-learning algorithms that are prone to overfitting (see the experiment in Sect. S2.1), where the models can reproduce the shuffled test data better, as these data are adjacent to samples of the same ship track.

We further reduce the possibility of overfitting by tuning the hyperparameters for each model to be more generalised, i.e. able to fit the data that the model has not been exposed to. The search for the optimal hyperparameters is achieved with grid-search cross validation, where a portion of the training subset is iteratively kept separate from the training process for a certain set of hyperparameters (Hastie et al., 2009). The hyperparameters that result in the best score from the grid search are used for the fit with the full training subset (see Sect. S2.3 for more details). We use a variation of K-fold cross validation called “group K fold” in Scikit-learn (Pedregosa et al., 2011). Rather than having arbitrary splits for each fold, a given grouping variable is used to split the data – in this case, years. Using years as the grouping variable reduces bias towards the second half of the time series where data are less sparse.

The train-test split and cross validation are applied identically to each of the four machine-learning algorithms for each clustering configuration. We use the following machine-learning algorithms: ERT – Geurts et al. (2006); GBM – Friedman (2001); SVR – Drucker et al. (1997); and FFNs. The details of these methods and how they were tuned are explained in the Supplement (Sect. S2.3). The first two methods, ERT and GBM, are new to this application. SVR has been implemented as a single global domain by Zeng et al. (2017), and FFN is used by several different methods, some of which are in the SOCOM intercomparison (Landschützer et al., 2014; Zeng et al., 2014; Sasse et al., 2013).

Regression performance is tested using RMSE primarily but also bias (Eqs. 3 and 4 below) and Riav (Eq. 5), with only the models from the best averaged clustering configuration used for the rest of the study.

Standard practice in machine learning is to set aside a test subset of the data, as described in Sect. 2.4. We use this standard approach in the second step of our experiment (regression comparison) as an estimate of the performance for each of the machine-learning models (164 in total). However, this grouped train-test split gives a bias and RMSE estimate limited to the random test years of test subset (see Sect. 2.4). To overcome this limitation, we iteratively apply the train-test split method with multiple selections of years. The splits in the test fold are based on a subset of years spaced 5 years apart. We then refactor the five test-fold estimates into a complete test estimate (with the same structure as the original SOCAT v5), thus giving a complete estimate of bias and RMSE (Fig. 1, step 3). This robust test-estimate method ensures that correct biases and RMSE scores are reported even if methods are prone to overfitting (see Sect. S2.1 and Fig. S1). We limit this procedure to only the CO2 biome and best clustered regressions as it has 5 times the computational cost of a single train-test split.

For method validation, we use observation data that are not used in SOCAT (Fig. 4 and Table 2) as they are either (1) included in the Lamont–Doherty Earth Observatory (LDEO) database but not in SOCAT; (2) not measured with an infrared analyser; or (3) derived from two other variables in the marine carbonate system, where these include dissolved inorganic carbon (DIC), pH and total alkalinity (TA) – where the Southern Ocean Carbon and Climate Observation and Modeling (SOCCOM) floats use empirically calculated TA.

The uncertainty of pCO2 that is calculated from DIC and TA is dependent on the accuracy of these two measurements, as well as the derivation of pCO2 with dissociation constants, for which we use the CBSys package in Python (Hain et al., 2015). CBSys implements the constants from Lueker et al. (2000) that reports an uncertainty of 1.9 % standard deviation of the calculated pCO2 where DIC and TA uncertainties are 2.0 and 4.0 µmol kg-1, respectively. The measurements in GLODAP v2 are slightly larger than this at 4 and 6 µmol kg-1, which would result in an error larger than 1.9 % – this is 12 µatm for a 400 µatm estimate at a hypothetical 3 % error. However, this error may be larger, as reported in Table 2, where Bockmon and Dickson (2015) showed that the uncertainty for DIC and TA is likely closer to ±10 µmol kg-1. While this potentially large error range may seem concerning, we argue that the inclusion of these data in data-sparse regions is more valuable than their omission. Additionally, GLODAP v2 data have been adjusted on a per-profile basis to minimise the biases through the comparison of deep slow-changing ocean properties (Olsen et al., 2016). Williams et al. (2017) estimated the error for pCO2 calculated empirically to be 2.7 %, where TA was calculated empirically with the locally interpolated alkalinity regression (LIAR) algorithm (Carter et al., 2016). Note that the datasets in Table 2 likely suffer from biases unaccounted for due to temperature mismatches as discussed in Sect. 2.2 (Goddijn-Murphy et al., 2015). It is important to note that each of the validation datasets are compared independently of each other, thus avoiding the complications of accounting for the biases between datasets. All pCO2 data are then gridded to the same time and space resolution as the feature variables (monthly × 1∘) using xarray and pandas packages in Python (McKinney, 2010; Hoyer and Hamman, 2017).

Bulk sea–air CO2 flux (FCO2) is calculated with 2FCO2=kw⋅K0⋅pCO2sea-pCO2atm, where K0 is the solubility of CO2 in seawater (Weiss, 1974) and kw is the gas-transfer velocity calculated from wind speed using formulation by Nightingale et al. (2000), as this parameterisation was the closest match to in situ observations of CO2 fluxes (Goddijn-Murphy et al., 2016). The ERA-interim v2 wind product is used to calculate kw. pCO2sea is from the gap-filling methods, and pCO2atm is atmospheric pCO2. All ancillary variables required in these calculations are the same as those listed in Table 1, except for pCO2atm, which is the CarboScope atmospheric pCO2 product from Rödenbeck et al. (2014). One of the problems with the bulk estimates of sea–air CO2 fluxes is that models of gas exchange in the surface layer of the water column are simplified, but there are approaches, such as the rapid equilibrium model, that account for more complex temperature gradients in the upper layer of the surface ocean (Wanninkhof et al., 2009; Woolf et al., 2016). However, for the sake of consistency with past studies, we use the bulk approximation of sea–air fluxes (Eq. 2), where kw is scaled to 16 cm h-1 as in the SOCOM intercomparison (Rödenbeck et al., 2015).

The results from the regression comparisons (step 2 in Fig. 1) are depicted in Fig. 5a–c, which plot the matrix of the (a) average bias, (b) RMSE and (c) Riav for each combination of the experimental number of clusters and clustering features.

Results show that the configuration that includes EKEclim (column E in Fig. 5a–c) as a clustering feature has the lowest average RMSE and absolute bias for nearly all clustering configurations, regardless of the number of clusters (rows in Fig. 5a, b). The increased dynamics associated with high-EKE regions might change the way pCO2 behaves compared to low-EKE regions (Boutin and Merlivat, 2013; Monteiro et al., 2015; du Plessis et al., 2017, 2019). The optimal number of clusters within this configuration is either 21 or 23, based on the smallest bias and RMSE scores (as indicated by the black box in Fig. 5), while we do not weight Riav strongly in this assessment as a Riav score of less than 0.3 is in the top-performing category in the SOCOM intercomparison (Rödenbeck et al., 2015). While the individual regression methods' bias and RMSE scores (Figs. S5 and S6, respectively) do not match the distributions exactly, the two selected clustering configurations (black boxes in Fig. 5) score consistently low for both metrics (with the exception of ERT – discussed in greater detail further on). We are motivated to select only one clustering configuration for the sake of simplicity. Furthermore, we select the configuration with 21 clusters (rather than 23), as fewer clusters further reduce the possible complexity at little cost. The selected clustering configuration with 21 clusters has the following features: SST, log10(MLDclim), pCO2clim, log10(Chl aclim) and log10(EKEclim), and is hereinafter abbreviated as K21E (see Fig. S2 for the distribution of the climatology for these clusters).

Comparatively, the Fay and McKinley (2014) CO2 biomes have an average RMSE score of 18.98 µatm (Table 3) but have a lower mean Riav (0.26) and smaller bias (0.03 µatm) than the K21E configuration. Given that the CO2 biomes perform well and provide an alternate clustering approach, we include the regression estimates. The eight machine-learning models from K21E and BIO23 (four each) were used to create an ensemble average by averaging pCO2 estimates (CSIR-ML8, Council for Scientific and Industrial Research – Machine Learning ensemble with Eight members).

All regression methods have lower RMSE scores for K21E than for BIO23, but Riav and bias do not indicate that either of the two clustering approaches is preferable (Table 3). Comparing the RMSE scores of the individual regression methods, we see that the model scores are ranked the same in each cluster from first to last: SVR, ERT, GBM and FFN. However, it is important to note that this ranking does not apply to bias or Riav, where ERT has low RMSE but the largest bias and Riav in each clustering approach. CSIR-ML8 only slightly betters its members, with RMSE and bias scores of 17.25 and 0.04 µatm, respectively. However, the ensemble average Riav (0.25) is only just less than the average of the ensemble members' average (0.26).

Here, we study the change in the bias and RMSE for all selected methods (i.e. K21E, BIO23 and CSIR-ML8; Table 3) across 1982–2016 (Fig. 6). Most notable is that bias scores for all models have the same interannual tendencies, with a positive bias at the beginning of the time series (1982 to 1993) that is strongest before 1990, strongly influencing the mean bias (Table 4). Secondly, the biases for K21E (solid lines) are, on average, smaller than for BIO23 (dashed lines), as shown for the annually averaged results in Table 4 (0.73 and 2.24 µatm, respectively). These biases are larger than those reported in Table 3 (with averages of absolute biases of 0.48 and 0.41 µatm for K21E and BIO23, respectively), but this is likely since selected test years (black triangles in Fig. 6b) fall on years of low bias. While FFN has the largest RMSE (18.93 and 20.24 µatm for K21E and BIO23), it has a smaller bias compared to other regression methods (0.04 and 1.60 µatm, respectively), motivating including FFN regressions in the ensemble average (Table 4). Conversely, the ERT approach has a significant positive bias likely due to the method's resilience to outliers, where sparse measurements could be treated as outliers (2.08 and 3.88 µatm for K21E and BIO23, respectively, with p>0.95 for both values; Table 4; Gregor et al., 2017). A second ensemble average without ERT regressions, thus with six members (CSIR-MLR6 version 2019a; hereafter called CSIR-ML6), has lower biases compared to CSIR-ML8 (0.98 and 1.48 µatm, respectively; Table 4).

Similar to the biases, RMSEs for all models (Fig. 6b) have similar interannual tendencies and variability, with a sharp peak in the year 2000 (> 20 µatm, where the mean RMSE is 18.61 µatm). The increased RMSE scores are likely due to the spatial distribution of sampling density (see Fig. S7); e.g. an increase in sampling in the high latitudes during spring and summer, a region and period of high variability and biogeochemical complexity, would increase the weight of these data in the final RMSE calculation, thus resulting in larger RMSE scores. The increase in the number of samples from 2002 to 2016 results in a sharp decrease in RMSE (< 19 µatm for the majority of this period). Both ensemble averages perform slightly better than all other methods for the majority of the time series with RMSE scores of 17.16 and 17.25 µatm for CSIR-ML6 and CSIR-ML8, respectively (see Table S1 comparisons of ensemble averages with different members).

The Riav scores for the robust errors (Table 4) are lower than train-test results with a single split reported in Table 3, likely due to an increase of standard deviation for the IAV benchmark (Eq. 5). The lowest score is held by CSIR-ML6 (0.20) and is lower (better) than the average for its members (0.21). These Riav estimates compare well to the Jena-MLS and SOM-FFN, which both scored < 0.3 (Rödenbeck et al., 2015).

The spatial distribution of the bias and RMSE is now studied for CSIR-ML6 (Fig. 7a and b, respectively), particularly focusing on the regional patterns emerging from the data. CSIR-ML6 clearly represents the subtropical regions (NH-ST and SH-ST) with relatively low biases and RMSE scores (|bias| < 5 µatm and RMSE < 10 µatm). The equatorial regions (EQU), especially the eastern Pacific, contrasts this with large uncertainties in both bias and RMSE (> |10 µatm| and 30 µatm, respectively). The high-latitude oceans (NH-HL and SH-HL) have considerable uncertainties due to the large interannual variability of surface ocean pCO2 caused by the formation and retreat of sea ice (around Antarctica; Ishii et al., 1998; Bakker et al., 2008) and phytoplankton spring blooms (Atlantic sector of the Southern Ocean, North Pacific and Arctic Atlantic; Thomalla et al., 2011; Lenton et al., 2013; Gregor et al., 2018). There are two bands of overestimates on the southern and northern boundaries of the North Atlantic Gyre, where the latter coincides with the Gulf Stream. Regression approaches may be prone to a positive bias in the North Atlantic, as this was also shown by Landschützer et al. (2013, 2014).

In summary, the robust test estimates show that there is a positive bias in pCO2 predictions before 1990 for all models, but it is largest for ERT, and excluding these models from the ensemble results in better pCO2 predictions. The spatial evaluation of the performance metrics for CSIR-ML6 shows that regions with specific oceanic features (e.g. western boundary currents) mostly have positive biases. However, it is important to note that these uncertainty assessments are limited as the characteristics and biases of the dataset are intrinsic to the models. Validation with independent data is thus a more reliable estimate of the performance of these methods.

Here, we validate the accuracy of pCO2 estimates from CSIR-ML6 with independent data (that are not in SOCAT v5, as described in Table 2). To further study the behaviour of our ensemble average estimates relative to previous studies, we compare the results from four independent methods of the SOCOM intercomparison project against the independent data calculated over individual data points (Rödenbeck et al., 2015). Those four independent methods are the Jena mixed-layer scheme (Jena-MLS version oc_v1.6; Rödenbeck et al., 2014); JMA-MLR, updated on 2 December 2018 (Iida et al. 2015); MPI-SOMFFN v2016 (Landschützer et al., 2017); and University of East Anglia – Statistical Interpolation (UEA-SI version 1.0; Jones et al., 2015). pCO2 estimates by the Jena-MLS were resampled to monthly temporal resolution and interpolated to a 1∘ grid using Python's xarray package. Note that these datasets will also suffer from the same temperature biases discussed in Sect. S2.4.

The performance of each gap-filling method is represented with a Taylor diagram for each independent validation dataset (Fig. 8; Taylor, 2001). The most important characteristic learnt from these plots is that the gap-filling methods are tightly bunched for nearly all validation datasets, indicating a similar RMSE, correlation and standard deviation relative to the reference datasets. Poor estimates in Fig. 8a–d may indicate that the training data for gap-filling methods is the limiting factor. Secondly, the gap-filling methods almost always underestimate the standard deviation of the validation datasets, being below the black arched line for all but the station HOT (Fig. 8e).

All methods fail to represent the standard deviation of the two global validation datasets, LDEO and GLODAP v2 (Fig. 8a, b), with centred RMSE scores greater than 35 µatm. However, calculating RMSE annually results in scores of ∼ 27 µatm for LDEO and ∼ 35 µatm for GLODAP v2, much lower than shown in Fig. 8a–b, due to high RMSE scores (> 40 µatm) for a small subset of years (Sect. S3.4 and Fig. S7). Estimates of the Southern Ocean datasets (Fig. 8c, d), SOCCOM and CARIOCA, have lower RMSE scores (∼ 16 and ∼ 23 µatm, respectively) relative to LDEO and GLODAP v2. However, for standard deviation scores of similar magnitude and low correlation coefficients, the datasets are not well constrained (Table 5). The SOCCOM dataset also has the largest average absolute bias for estimates, with gap-filling methods underestimated by at least 11 µatm (Table 5). This large bias may be because SOCCOM floats have a proportionately large number of winter samples – suggesting that our knowledge of Southern Ocean winter fluxes is largely underestimated (Williams et al., 2017). In contrast, all methods estimate the two time series stations, HOT and BATS (Fig. 8e, f and Table 5), relatively well with correlation scores > 0.8 and low average bias ∼ 4.5 µatm.

Despite all scores being closely grouped (Fig. 8), Table 5 shows that the CSIR-ML6 method scores significantly lower RMSE scores (using a two-tailed Z test with p<0.05) for all but one of the datasets (SOCCOM). However, bunching of the RMSE scores (Fig. 8) is beneficial with regard to achieving low p values. No single method dominates the biases, with JMA-MLR and MPI-SOMFFN each scoring the lowest bias on two occasions. To summarise, all gap-filling methods underperform when validated against independent observational products. Tight bunching of gap-filling method scores per validation dataset shows that training data may limit all methods in the same manner.

In this section, we assess the regional implications of the differences in gap-filling methods' estimates (within CSIR-ML6 and the four independent methods described in Sect. 3.3) of the sea–air CO2 flux (FCO2) over the period 1990 to 2016. FCO2 was calculated using the same gas-transfer velocity and solubility for each gap-filling method (Sect. 2.7). Differences in FCO2 are thus driven by variations in pCO2 from each gap-filling method.

The average FCO2 for 1990–2016 by CSIR-ML6 (Fig. 9a) contextualises the regional distribution of fluxes: strong outgassing in the equatorial Pacific, strong sink in the midlatitudes, a moderate uptake for the most part of the subtropics and weak source in the majority of the Southern Ocean (in agreement with, e.g. Takahashi et al., 2009). The global annual time series for FCO2 as simulated by CSIR-ML6 (Fig. 10a) indicates a strengthening for 2000 to 2016 (as for the other methods). To give spatial context to this strengthening, we display the differences in FCO2 between 2016 and 2000 (Fig. 9b), since those are the two years where the difference in global FCO2 is greatest for CSIR-ML6 (Fig. 10a). Note that Fig. 9b serves as a snapshot for the change in FCO2 between those two years, whose interpretation cannot be linked to an overall anthropogenically forced change as the comparison between the two years could reflect interannual, decadal or multi-decadal variability. The differences in FCO2 between 2016 and 2000 are negative in the high latitudes and moderately positive in the subtropics, indicating a respective increase and decrease in the CO2 ocean uptake between the two years. The eastern equatorial Pacific is the only region that shows a considerable increase in FCO2 (> 10 g C m-2 yr-1) between the two specific years.

The annual change in FCO2 is also studied for the different regions. The Southern Hemisphere high-latitude (SH-HL) region is the strongest contributor to the trend (Fig. S10b), where there is a steady increase in the uptake of CO2 since the 2000s for all methods (Landschützer et al., 2015; Gregor et al., 2018). On average, the Northern Hemisphere high latitudes (NH-HL) are a weaker sink relative to the SH-HL, because the SH-HL is more than double the area of the NH-HL (Fig. S10c). The equatorial (EQU) region is the only persistent source of CO2 to the atmosphere (also seen in Fig. 9a). The subtropical regions (Fig. 10c, e) contribute to global flux on similar orders of magnitude; however, there is a large divergence between gap-filling methods in the SH-HL.

We use the average interquartile range between the 1-year rolling mean estimates (IQRIA) as a measure of agreement or divergence between gap-filling methods, where large values indicate a divergence (Sect. 2.8.2). We also show the IQRIA scaled to the range of the regional interannual variability (max–min) as a percentage (relative IQRIA), which shows if the trend for a particular region is agreed on by all methods (the smaller the percentage, the better the agreement across methods). The disagreement between methods in the SH-ST is substantial (Fig. 10e), with diverging FCO2 throughout the period with an IQRIA of 0.11 Pg C yr-1 and a large relative IQRIA of 28 %. Similarly, the IQRIA for the SH-HL region (Fig. 10f) is 0.08 Pg C yr-1, but the relative IQRIA is lower at 14 %, indicating that all methods agree on the observed strong trend. Compared to the Southern Hemisphere, the Northern Hemisphere regions are both relatively well constrained, with IQRIA estimates of 0.04 and 0.05 Pg C yr-1 for the NH-ST and NH-HL regions, respectively (Fig. 10c, d). However, a larger relative IQRIA of 20 % suggests that the interannual FCO2 estimates in the NH-ST region are potentially not resolving the trend, or more likely that there is a weak trend with a small difference between the minimum and maximum interannual estimates of FCO2. The EQU region (Fig. 10b) has an IQRIA and relative score at 0.03 Pg C yr-1 and 14 %.

The CSIR-ML8 method is not included in the IQRIA calculations but is included in Fig. 10 to show the impact of the ERT models' positive bias in pCO2 on FCO2 (Fig. 6a). The biases are positive at the beginning and negative end of the time series, with the average absolute difference between the CSIR methods being 0.08 Pg C yr-1. The positive biases have the strongest impact on the SH-ST that occupies 36 % total area (Fig. S10c), with only 11 % of the total observations in SOCAT, suggesting that this method is sensitive to imbalanced datasets.

In order to better understand the regional distribution of the uncertainties in FCO2, we assess the level of agreement between independent gap-filling methods in their interannual surface ocean pCO2 estimates (Fig. 11). We use pCO2 for this representation as no spatial integration occurs – only time averaging.

The interannual estimates of IQRIA (Fig. 11a) show the disagreement between methods is relatively small in the majority of the ocean (< 6 µatm), with the exceptions being the Southern Ocean, South Atlantic, southeastern Pacific and eastern equatorial Pacific with differences of > 10 µatm, where these regions coincide with regions of low sampling density (Fig. 2). The IQRIA scaled to the maximum–minimum range of interannual pCO2 suggests that the NH-ST trend is relatively well constrained (< 10 %), which is in conflict with the IQRIA for FCO2 in Fig. 10c (where the relative IQRIA is 20 %). The disagreement may stem from the magnifying impact that wind speed has on FCO2; i.e. small differences in pCO2 may become large when fluxes are calculated. The same principle may apply to the EQU in Fig. 11b, where relative IQRIA is large (> 10 %) for pCO2, but low wind speeds result in a low relative IQRIA for FCO2 (7 % in Fig. 10b). The largest relative IQRIA scores occur in the SH-ST (> 10 % in Fig. 11c) where data are sparse, specifically the South Atlantic and southeastern Pacific (Fig. 2a). The relative IQRIA scores suggest that the gap-filling methods agree on pCO2 in the SH-HL east of the Greenwich meridian (> 0∘ E).

In summary, we show that there is an agreement between gap-filling methods in the Northern Hemisphere for interannual pCO2, but the methods show considerable disagreement in the Southern Hemisphere, particularly in the subtropics. Disagreements in the equatorial and Southern Hemisphere high-latitude regions are large (> 10 %) and should be treated with caution when considering trends in these regions.

In their study, Khatiwala et al. (2013) stated that “our comparison of different methods suggests, that multiple approaches, each with its own strengths and weaknesses, remain necessary to quantify the ocean sink of anthropogenic CO₂”. In our study, we embrace this philosophy by creating an ensemble average of two-step machine-learning models that estimate global surface ocean pCO2. We show robustly that the CSIR-ML6 method reproduces the available data with greater accuracy than previous methods, albeit in an incremental way. Our method is methodologically consistent with regard to feature variables. Though there is variability in the clustering and regression, we create the ensemble average with a good understanding of each model's biases (Figs. 6 and S8). The argument that ensemble averages reduce transparency is also somewhat diminished by the fact that little additional information that can be gained from highly non-linear models, with the exception of basic diagnostics such as feature-variable importance (see Fig. S11) from decision-tree-based approaches (Pedregosa et al., 2011; Castelvecchi, 2016). Our results thus show that there is, in fact, a benefit in creating an ensemble average of models (Table 5), and if carefully implemented, it is an additional tool that can be used to reduce the uncertainties in gap-filling estimates of pCO2.

It could be argued that an exhaustive search for the optimal configuration (Fig. 5) for CSIR-ML6 may result in poorly trained individual models. However, we think that the merit of introducing and assessing regression algorithms new to the application (for gradient-boosting machines and extremely randomised trees) outweighs the marginal loss in potential performance for individual methods. Moreover, lessons learnt from our study can be used to improve on future iterations. It also makes the case for ensemble averages stronger, as the CSIR-ML6 performs well relative to other gap-filling methods.

In the search for the optimal clustering configuration (Fig. 5a, b), we show that including EKE (along with SST) as a clustering feature variable leads to an improvement in bias and RMSE for nearly all numbers of clusters, albeit a small improvement. Increased intraseasonal variability of pCO2 appears to be associated with regions of high EKE compared to low EKE regions (Monteiro et al., 2015; du Plessis et al., 2017, 2019). Moreover, the importance of EKE as a part of the clustering constraints also shows that more thought should be given to how we sample pCO2 in high-EKE regions and at what resolution regression methods are run at.

Our findings suggest the following about the individual regression methods: the SVR and GBM algorithms produce good estimates with lower RMSE scores and biases, the FFN approach has larger RMSE scores yet low biases than the other methods, and the ERT approach has low RMSE scores but large biases in the estimates (Fig. 6a, b; Table 4). We do not include the ERT approach in the ensemble average (CSIR-ML6) due to the large time-evolving biases, suggesting that ERT (with our tuning) is not suitable for estimating surface ocean pCO2. The bias in ERT may be due to its sensitivity to imbalanced datasets (Crone and Finlay, 2012), where the data in SOCAT v5 are sparse before 2000. Returning to the above quote by Khatiwala et al. (2013), we thus find that the weaknesses of ERT outweigh its strengths.

While we see that the improvements in the performance of gap-filling methods are relatively stagnant (relative to the training and validation data), the differences between the methods' estimates of pCO2 and FCO2 vary significantly in some regions, particularly in regions where data are sparse, such as in the Southern Hemisphere oceans (Fig. 2). We also find that training the gap-filling methods with limited training data exposes the intrinsic biases of the algorithms, or in the words of Ritter et al. (2017): “the difference [between gap-filling methods] is a result of how the spatial and seasonal heterogeneity and the sparseness of the data is dealt with”. Conversely, as the number of training data increases, the biases are reduced, and the methods converge.

The Northern Hemisphere subtropical regions are a good example of a region where the gap-filling methods converge (Fig. 11b), as also shown by the low RMSE scores and high correlation for the two mooring stations, HOT and BATS (Fig. 8e, f). One of the reasons that the methods predict the variability well in the subtropics (Fig. 8e, f) is that these regions are less biogeochemically complex and driven primarily by seasonal changes in SST (Bates, 2001; Dore et al., 2009). This strong SST-driven seasonality in the subtropics is shown by the high seasonal cycle reproducibility (Fig. 12).

The gap-filling methods' divergences also serve as a metric to inform where there are not enough data to constrain the pCO2 or FCO2 estimates; i.e. the divergences inform us where estimates should be treated with caution. The IQRIA, when scaled to the range of interannual variability (Fig. 11b), should be taken into account when analysing interannual trends of ΔpCO2 (Fig. 13). For instance, significant trend estimates in ΔpCO2 for CSIR-ML6 (p<0.05) are negative for the majority of the global ocean, even in regions where method estimates are too disparate to resolve interannual variability (relative IQRIA>15 %; dotted regions in Fig. 13). However, the relative IQRIA is not without its limits, as there may be regions where methods are in agreement but share the same biases, thus reporting false confidence in the estimates. Regions of false confidence would most likely occur in data-sparse areas but could only truly be identified with better data coverage in these regions.

In our study, we show that all gap-filling methods suffer from the same uncertainties where there are data to test and validate the estimates (Fig. 8), and result in divergences between estimates when there are insufficient data to constrain the methods (Fig. 11b). From these points, it may seem that we may have in fact “hit the wall” in terms of better resolving surface ocean pCO2. In this section, we discuss how we might overcome this proverbial wall: first, by addressing the existing uncertainty and biases, and then discussing how we could improve on estimates in data-poor regions.