It is known that the evolution of a concentration, c, in the atmosphere is given by the advection–reaction–diffusion equation: ∂c∂t=-u∇c+1ρ∇(ρTd∇c)+1ρg+F, with the wind field u, the density of the air ρ, the turbulent diffusivity tensor Td, the chemical reaction rate g and the net flux at the air–sea interface F . Using optimal control and inverse problem modeling, a map of F can be derived using Earth observation data . It would be ideal if we could use data of atmospheric CO2 concentrations from space measured by satellite sensors such as SCIAMACHY (SCanning Imaging Absorption SpectroMeter for Atmospheric CHartographY) aboard ENVISAT (Environmental Satellite), in orbit since 2002, and GOSAT (Greenhouse gases Observing SATellite), in orbit since January 2009, to derive the air–sea flux. However SCIAMACHY and GOSAT sampling is not dense enough, with very suboptimal sampling of the Benguela upwelling system. This led us to use data of CO2 fluxes from CarbonTracker (http://www.esrl.noaa.gov/gmd/ccgg/carbontracker/) at spatial resolution of 1∘ × 1∘ (∼ 100 km × ∼ 100 km) . CarbonTracker system assimilates and integrates a diversity of atmospheric CO2 data into a computation of surface CO2 fluxes, using a state-of-the-art atmospheric transport model and an ensemble Kalman filter.

We obtain the partial pressure of ocean CO2 by using the equation of the net flux in the air–sea interface: F=αK(pCO2air-pCO2ocean), where α is the gas solubility, which depends on sea surface temperature (SST) and sea surface salinity (SSS), and K, the gas transfer velocity, is a function of wind, salinity, temperature, and sea state, which can be obtained from satellite data. To estimate the gas transfer velocity we use well-accepted relationships for the transfer velocity in air–sea gas exchange from wind speed, the parameterization developed by . The CO2 gas solubility is derived according to . Input data for SST are derived from OSTIA (Operational SST and Sea Ice Analysis system) product, SSS are derived from LEGOS (Laboratoire d'Etudes en Géophysique et Océanographie Spatiales) product compiled by and winds from Cross-Calibrated Multi-Platform Ocean surface winds from JPL (Jet Propulsion Laboratory) PO.DAAC (Physical Oceanography Distributed Active Archive Center, http://podaac.jpl.nasa.gov/). We assume pCO2air to be constant in the domain of study, and it is derived from the Globalview-CO2 product of the Cooperative Atmospheric Data Integration Project coordinated by Carbon Cycle Greenhouse Gases Group (www.esrl.noaa.gov/gmd/ccgg/globalview/). We use values taken at the closest sea-level station to the Benguela, located at Ascension Island (7.97∘ S and 14.40∘ W) as our reference atmospheric CO2. The partial pressure of CO2 is determined from its mole fraction (xCO2), using the following equation : pCO2 = xCO2×p, where p is the total pressure of the mixture. We assume this pressure to be close to 1 atm for the conversion following ORNL/CDIAC 105 report (program developed for CO2 systems calculation).

The raw data of CarbonTracker fluxes of CO2 in the area of interest are strongly binned and exhibit strong gradients across those bins. This turns out to be suboptimal for our super-resolution approach. developed an optimal control approach to invert interfacial fluxes using a simplified inverse problem of atmospheric transport. The inverse problem is solved using the Galerkin finite element method and the dual weighted residual (DWR) method for goal-oriented mesh optimization. An adaptation of this approach has been applied to the CarbonTracker data set. However, the estimations are expensive and computing results for all the time frames of interest was infeasible. Therefore, an anisotropic diffusion-based approach has been applied to the raw fluxes of the CarbonTracker data set. The diffusion is steered by the direction of the low-altitude wind field. The results thus retain the structure of the CarbonTracker fluxes very well while suppressing artifacts. Examples of this process are shown in Fig. . Results are comparable to the physically more accurate approach of .

Oceanic pCO2 is a complex signal depending, at any spatial resolution, on sea surface temperature, salinity, chlorophyll concentration, dissolved inorganic carbon, alkalinity and nutrient concentrations. Both the biological pump, with chlorophyll a as a proxy, and the physical pump, driven by the temperature and salinity (e.g., solubility, water mass), govern the evolution of pCO2 in the surface ocean.

We use here the high-resolution satellite ocean data for chlorophyll a, as a proxy for the biological carbon pump and for SST, as a proxy for the thermodynamical pump, (see Sect.  for more details on the connection of these oceanic variables).

Among the available data in SOCAT version 2 (Surface Ocean CO2 Atlas, http://www.socat.info) over the 2000–2010 period in our region of interest, we find the following cruises with pCO2 measurements:

2000, one cruise: ANT-18-1

The small number of cruises found in 1 decade (24 cruises) shows that the scarcity of cruises in the Benguela region is a fact. This indeed demonstrates the crucial need of developing a robust method to infer high-resolution pCO2 from space. Moreover for some of these cruises, for instance, the track of GALATHEA cruise is too close to the coast and is out of the original CarbonTracker domain. Due to this restriction we only document the offshore conditions of this upwelling system. Owing to the relatively large number of cruises during 2005, 2006 and 2008 (a total of 20 cruises, representing 83 % of all available cruise data from 2000 through 2010), in this validation, we focus the analysis on the set of QUIMA-cruises during 2005 (QUIMA2005), 2006 (QUIMA2006) and 2008 (QUIMA2008) and we present the global analysis using all available cruises during these 3 years. analyzed this data to study the sea surface pCO2, fCO2 and CO2 air–sea fluxes in the offshore Benguela upwelling system between 2005 and 2006 (for each month from July 2005 up to November 2006) and extended the study including cruises data from 2007 to 2008. The QUIMA line crosses the region between 5 and 35∘ S, with all the cruises following the same track.

In the ocean, the turbulence causes the formation of unsteady eddies on many scales which interact with each other . Most of the kinetic energy of the turbulent motion is contained in the large-scale structures. The energy cascades from the large-scale structures to smaller-scale structures by an inertial and essentially inviscid mechanism. This process continues, creating smaller and smaller structures which produces a hierarchy of eddies. Moreover, the ocean is a system displaying scale-invariant behavior, that is, the correlations of variables do not change when we zoom in or we zoom out the system, and can be represented by power laws in particular, with the scaling exponents h.

It can be shown that the scaling exponents are the values taken on by localized singularity exponents, which can be computed at high precision in the acquired data using the microcanonical multifractal formalism. Hence, within that framework, the multifractal hierarchy of turbulence, defined by a continuum of sets (Fh) indexed by scaling exponents (h), is obtained as the level sets of the geometrically localized singularity exponents.

We will not review here the details of the computation of the singularity exponents h(x), leaving the reader to consult references for an effective description of an algorithm able to compute the h(x) at every point x in a signal's domain.

Some examples of the singularity exponents of Chl a and SST images for the different products described in Sect.  are shown in Figs. c, d, c, d, respectively. As compared to the corresponding images of Chl a and SST shown in Figs. a, b, a, b, one can see the ability of the singularity exponents to unveil the cascade structures arisen by tracer-gradient variances hidden in satellite images.

Another important idea implemented in the methodology is the coupling of the physical information contained in SST and OC images with the ocean pCO2. For instance, it is known that marine primary production is a key process in oceanic carbon cycling, and variations in the concentration of phytoplankton biomass can be related to variations in carbon concentrations. Surface temperature is also related to gas solubility in the ocean, and areas with high temperatures are more suitable for releasing CO2 to the atmosphere. We have studied the relationship of SST and Chl a variables with pCO2 using the outputs of a coupled Regional Ocean Modeling System (ROMS) with the BIOgeochemical model of the Eastern Boundary Upwelling System (BIOEBUS) . The ROMS includes several levels of nesting and composed grids, which makes it an ideal model for the basis of our methodology in working in two spatial resolutions. BIOEBUS has been developed for the Benguela to simulate the first trophic levels of the Benguela ecosystem functioning and also to include a more detailed description of the complete nitrogen cycle, including denitrification and anammox processes as well as the oxygen cycle and the carbonate system. This model coupled to ROMS has been also shown to be skillful in simulating many aspects of the biogeochemical environment in the Peru upwelling system . When one compares SST and Chl a with pCO2 one finds undetermined functional dependency. However, when comparing their corresponding singularity exponents one obtains a clear simpler dependency. This is due to the fact that SST, Chl a and pCO2 are variables of different dimensions while singularity exponents are dimensionless quantities. These results show that there is a good correlation between the turbulent transitions given by the singularity exponents and that singularity exponents are good candidates for a multi-resolution analysis performed on the three signals, SST, Chl a and pCO2. Furthermore, the log-histograms and singularity spectrum show that singularity exponents of pCO2 images possess a multifractal character. Therefore, such signals are expected to feature cascading, multiscale and other characteristic properties found in turbulent signals as described in and . Consequently the use of nonlinear and multiscale signal processing techniques is justified to assess the properties of the pCO2 signal along the scales.

Therefore, in our methodology, the local connection between different tracer concentrations (SST and Chl a) with pCO2, is performed in order to obtain a proxy for pCO2 at high resolution by using the following linear combination of multiple linear regressions: S(pCO2)(x)=a(x)S(SST)(x)+b(x)S(Chla)(x)+c(x)S(pCO2LR)(x)+d(x), where S(pCO2)(x) refers to the singularity exponent of pCO2 at x, S(SST)(x) to singularity exponent of SST at x, S(Chla)(x) to singularity exponent of Chl a signal at x. In order to propagate the pCO2 signal itself along the scales in the multi-resolution analysis we introduce S(pCO2LR) to refer to the singularity exponent from pCO2 at low resolution interpolated on the high-resolution grid. a(x), b(x) and c(x) are the regression coefficients associated with singularity exponents, and d(x) is the error associated with the multiple-linear regression. These regression coefficients are estimated using simulated data from the ROMS-BIOEBUS model developed for the Benguela upwelling system and described above.

Once we have introduced these coefficients in the linear combination on satellite data, we obtain a proxy for singularity exponents of pCO2 at high resolution and we can perform the multi-resolution analysis to infer the information across the scales.

Among the functionals that are most commonly used for analyzing the scaling properties of multifractal systems, wavelets occupy a prominent position. Wavelets projections are integral transforms that separate the relevant details of a signal at different scale levels, and since they are scale-tunable, they are appropriate for analyzing the multiscale behavior of cascade processes and for representing them. However, as shown in , and , not all multi-resolution analyses are equivalent; the most interesting are those which are optimal for inferring information along scales, in particular, in a context where information is to be propagated along the scales from low resolution to high resolution.

The effective determination of an optimal wavelet for a given category of turbulent signals is, in general, a very difficult open problem. This difficulty can be contoured by considering multi-resolution analysis performed on the signal of the singularity exponents h(x) themselves. Indeed, since the most singular manifold (the set Fh associated with the lowest singularity exponents) is associated with the highest frequencies in a turbulent signal, and since the multifractal hierarchy Fh converges to this set, it is physically evident that the multifractal hierarchy corresponds to a description of the detail spaces of a multi-resolution analysis performed on a turbulent signal. Consequently, designating the approximation and detail spaces computed on the S(pCO2)(x) signal as Vj and Wj, respectively, and their corresponding orthogonal projections from space L2(R2) as Aj and Pj, the reconstruction formula, Aj-1pCO2=AjpCO2+Pjh, consists of reconstructing a signal across the scales using the detail spaces of the singularity exponents and hence re-generating a physical variable according to its cascade decomposition. From these ideas, which are described more fully in , we can deduce the following algorithm for reconstructing a super-resolution pCO2 signal from available high-resolution SST, Chl a, and low-resolution pCO2: (i)

After selecting a given area of study, compute the singularity exponents of SST, Chl a and pCO2 at low and high resolution from ROMS-BIOEBUS output. This is done once and then they can be used for every computation performed over the same area.

The methodology has been successfully applied to dual ROMS simulation data at two resolutions, obtaining a mean absolute error of pCO2 reconstructed values with respect to ROMS simulated high-resolution pCO2 equal to 3.2 µatm (0.89 % of relative error) (V. Garçon, personal communication, 2014).

We now apply the methodology to infer ocean pCO2 maps at super resolution from pCO2 at low resolution derived from CarbonTracker data (see Sect. ) in the offshore area of the Benguela region.

Henceforward we use the following notation for the three different sources of pCO2: we refer to the values of ocean pCO2 derived from CarbonTracker as pCO2Ctrack, values of inferred pCO2 at higher resolution from pCO2 at low resolution together with computation of the cascade onto SST and chlorophyll a concentrations as pCO2infer, and finally pCO2insitu refers to the values of the in situ measurements of pCO2.

For the inference we use the following three combinations of Chl a and SST products described in Sect. : MERIS-OSTIA, GLOBCOLOUR-OSTIA, GLOBCOLOUR-MODIS. We do not include the MERIS-MODIS combination in the analysis due to the fact that the use of such satellite data results in a too drastic reduction of the coverage of the resulting pCO2infer field, but using merged products offers wider coverage instead. The inferred pCO2 obtained from two merged products for Chl a, GLOBCOLOUR GSM and GLOBCOLOUR AVW is very similar, with a slight improvement when GSM is used. Thus for the sake of clarity, we only show figures for GLOBCOLOUR-GSM and some statistical results making comparisons with AVW. Therefore from now on we use GLOBCOLOUR to refer to the Chl a obtained with the GSM merged method.

Figure d shows one example of pCO2infer field corresponding to 22 March 2006, when we use SST data from OSTIA (Fig. a), ocean color from GLOBCOLOUR (Fig. b) at high resolution and pCO2 at low resolution (Fig. c) derived from CarbonTracker air–sea flux of CO2 (Fig. e) and using the Eq. (). The air–sea flux of CO2 at super resolution (Fig. f) is obtained from the pCO2infer field and a constant value of atmospheric pCO2 equal to 385.6 µatm. On this day the images of the pCO2infer and fluxes of CO2 combine to give good coverage and clear identification of small-scale structures and gradients, as described below. Note that the air–sea CO2 flux from CarbonTracker presents a large land mask close to the coast and consequently, we study the offshore area of the Benguela upwelling. Comparing the figures one can see that values of pCO2 and CO2 flux over the domain (from 4.5∘ E to coast (taking out the mask of the CarbonTracker domain and from 20.5 to 35∘ S) vary between 360 and 380 µatm and between -4 × 10-8 and 0.5 × 10-8 mol C m-2 s-1, respectively. The resultant flux of CO2 is positive (towards the atmosphere) in the region 25–28∘ S and from 7∘ E eastward to the coast and is negative (into the ocean) south of 30∘ S and east of 6∘ E. Thus, we see that in the southern part offshore the Benguela area there is a strong CO2 sink and the northern part behaves as a weak CO2 source.

What is new in the reconstructed pCO2 is, for instance, that the cascade of information across the scales enhances gradients in the field of pCO2. It is striking that the high-resolution map provides the position of the north–south dipole “front” located at 30∘ S (i.e., -1.5 × 10-8 isoline in green) which could not be inferred accurately from the low-resolution map. The low-resolution map provides an estimate of the location of the “front” that is ∼ 1.5∘ north of the location inferred from the high-resolution map. Moreover one can see small structures in the pCO2infer field at 33–35∘ S, 9–12∘ E in the pCO2infer field (Fig. d). The small spatial scale variability is captured in the super-resolution pCO2 field and not in pCO2Ctrack as shown in the longitudinal profile of the images plotted in Fig.  at latitude 33.5∘ S (see Fig. ). The same high spatial variability given by the small-scale structures of the SST and OC images can be seen in their corresponding longitudinal profiles displayed in Fig. a and b. It is worth noting the change in the shape of the profiles between the pCO2infer and pCO2Ctrack and fluxes of CO2 at large scale, from 5.5–10.5∘ E, showing that the method not only introduces small-scale features but also modifies the large-scale spatial variability.

Since the underlying aim of this work is to develop a methodology to infer super-resolution pCO2 from space using remote observations, we perform a validation study of the different data used in the inferring computations. This provides us an evaluation where satellite products are more suitable for our methodology and thus gives confidence to our method as well as a better understanding of its limitations. The evaluation analysis is addressed taking into account two main concerns: one related to the number of valid points yielded in the pCO2infer field, and another with regard to the degradation of the information contained in the transition fronts. A valid point is a pixel where we have simultaneous Chl a, SST and pCO2 values from CarbonTracker, from which we can obtain a value of pCO2infer, in other words without missing information. One example comparing the reconstructed pCO2 field obtained from the above-mentioned product combinations is plotted in Fig. . The general pattern is quite similar in all of them with some differences in the details of the small scales and in the missing points due to cloudiness (white patches). This example clearly shows how different coverage of the pCO2 can be in the field depending on the product combination.

Similar results are found when one compares the spatial distribution of the pCO2infer values to the time averages over 2006 and 2008 for the three product combinations (Fig. ). The same pattern with an area of higher pCO2 between 24 and 30∘ S and lower pCO2 values outside this region is produced with the three combinations. The most noticeable differences are located in the most northern region and in the southeastern region off Benguela. This can be quantified by computing the standard deviation of the reconstructed pCO2 values among the different combination of data sets. Figure d shows the spatial distribution of the time average over 2006 and 2008 of the standard deviation computed in each pixel among the pCO2infer values obtained from the three product combinations. The larger values of the dispersion (not greater than 5 µatm) are found in the area between the latitudes of 20∘ S and 23∘ S – and in the southern region, in particular, between the latitudes of 31.5∘ S and 35.5∘ S – and between the longitudes of 11∘ E and 13.5∘ E. The low value of the dispersion indicates that the method is robust when different data sets are used in the inference.

First, we compute the number of valid points in the pCO2infer field for each product combination. Table  summarizes the total number of valid points for each product combination for both years 2006 and 2008. As expected, the number of valid points is found to be the highest for the combination of merged products OSTIA SST and GLOBCOLOUR-GSM, with NGO=27 313 043 points, followed by the combination MODIS SST and GLOBCOLOUR Chl with NMG=20 397 047 points and finally by the OSTIA SST and MERIS Chl combination with NOM=98 00 776 points. Looking at the different proportions, we find that the number of valid points is 2.78 times larger when using the merged products OSTIA and GLOBCOLOUR-GSM than using OSTIA and MERIS, 1.33 times larger than using MODIS and GLOBCOLOUR-GSM and 1.08 times larger using OSTIA SST and GSM Chl a than using MODIS SST and GSM Chl a. Furthermore, if we know that the total number of pixels in the domain taking out the points of the CarbonTracker mask and for the 2 years is Np=55 711 378, one can estimate the loss of valid points for each combination, LPx. LPx is computed by taking the relative difference between the number of total available pixels in the domain Np and the number of points in the inferred pCO2 field obtained for each product combination, Nx, and dividing it by the total number of pixels Np, LPx=Np-NxNp100 %. Here the subscript x refers to the product combination (e.g., LPx = LPOM, LPOG and LPMG for the loss of valid points with the OSTIA-MERIS, OSTIA-GLOBCOLOUR and MODIS-GLOBCOLOUR product combinations, respectively). The loss of valid points due to cloudiness in the ocean color and SST images is less severe for the OSTIA-GLOBCOLOUR combination, with a loss of 51 %, and is more affected by the cloudiness the OSTIA-MERIS combination with a loss of 82 %.

Next we explore the quality of the information contained in the transition fronts, in particular, in the non-merged products such as MERIS OC and MODIS SST as compared to the merged products: GLOBCOLOUR OC and OSTIA SST. The PDFs of pCO2 values from CarbonTracker and pCO2infer values for the three combinations of OC and SST products, i.e., MERIS-OSTIA, GLOBCOLOUR-OSTIA, MODIS-GLOBCOLOUR (see Fig. ) show that there is a good correspondence of all pCO2infer values with those from pCO2Ctrack. Indeed the histograms show also a better agreement between merged products and CarbonTracker: the peak of the PDF for pCO2infer is closer to the CarbonTracker peak in the case of OSTIA and GLOBCOLOUR than when using MERIS and MODIS products.

Furthermore, to examine the transition fronts for the different products, we compute the singularity spectra for the three product combinations (see Fig. ). At low values of h (singularity exponent), related to the most singular manifolds, the shape of singularity spectrum for inferred data from merged products better matches a binomial cascade, with an improved description of the dimension of the sharpest transition fronts. We know from the theory that tracers advected by the flow in the turbulent regime, as happens in the ocean, shows multifractal behavior with a characteristic singularity spectrum D(h), which is similar, for some types of turbulence, to D(h) for the binomial multiplicative process.

Next, we perform a validation analysis of the results of our super-resolution pCO2 algorithm with field observations of oceanic pCO2. We perform the validation using pCO2 ocean data from in situ measurements (pCO2insitu) taken in the Benguela region (see Sect. ). We decided to carry out the validation directly on pCO2 rather than on the air–sea CO2 flux since the field measurements provide oceanic pCO2 data.

An example of the qualitative comparison of values of pCO2Ctrack, pCO2infer for all the product combinations and pCO2insitu at the intersections of the QUIMA cruise during 4–7 July 2008, as a function of the longitudinal coordinate of the intersections, is shown in Fig. . While there are visible differences between various pCO2 values, the values of pCO2infer approximate better pCO2insitu values than those of pCO2Ctrack. The small-scale patterns are well reproduced in the inferred pCO2 field. Values of pCO2infer exhibit gradients and small-scale fluctuations, likely induced by the presence of fronts, which can be also detected in the profile of the in situ measurements of pCO2. Most days pCO2infer and pCO2Ctrack values overestimate pCO2insitu values. On some days, pCO2infer values follow the same trend, with the same small-scale fluctuations as pCO2insitu.

First, we analyze the number of valid intersections for each product combination. A valid intersection is a placement in space and time common to the inferred, CarbonTracker and in situ pCO2, without missing values. Among the 20 available cruises in the Benguela during 2005, 2006 and 2008, we find that the total number of in situ measurements in the Benguela region under study is Ninsitu=17 355 and within the CarbonTracker domain this number is reduced to NCtrack=8377 measurements. To estimate the loss of valid intersections due to the land mask of the CarbonTracker, we compute the relative difference of the number of intersections between the cruise trajectories and the CarbonTracker domain with respect to the number of the in situ measurements, LCrack=Ninsitu-NCtrackNinsitu100 % = 52 %, showing that half of the measurements fall within the coastal region of the Benguela (land masked by CarbonTracker).

The number of valid intersections is the largest with the OSTIA-GLOBCOLOUR combination (Table ). To quantify the loss of valid intersections between the in situ measurements and points in the pCO2infer field, likely due to cloudiness, we compute the relative difference between the number of measurements in the CarbonTracker domain and the valid points in the inferred pCO2 field with respect to the number of intersections measurements of each cruise and the pCO2Ctrack field, Linfer=NCtrack-NinferNCtrack100 %. We repeat such a computation for the three product combinations. The percentage of losses of intersections in inferred field Linfer becomes twice as large than in the case of the OSTIA-SST and MERIS Chl combination, and even higher than with the CarbonTracker domain mask.

In order to quantitatively study the difference between pCO2Ctrack and pCO2insitu values as well as the difference between pCO2infer and pCO2insitu measurements, we compute the following statistical quantities:

Mean difference (MD): average of all the intersections of the difference between pCO2Ctrack and pCO2insitu values as well as the difference between pCO2infer and pCO2insitu values at the same intersection, i,MDCtrack=1N∑i=1N(pCO2Ctrack(i)-pCO2insitu(i)),MDinfer=1N∑i=1N(pCO2infer(i)-pCO2insitu(i)),

where N is the number of intersections.

We started the statistical validation by analyzing each QUIMA cruise separately (not shown) and we found that in most of the cruises, the absolute error for inferred pCO2 is relatively small (less than 15 µatm) except on 21 August 2006 and 17 May 2008, with an error of 44 and 30 µatm, respectively. Then we address the global validation using all available cruises during these years.

We summarize in Table  the results of the computations of the errors given by Eq. (–) by taking averages of all valid intersections found during 2005, 2006 and 2008. The absolute error, AE is smaller in the three cases of pCO2infer (17.77, 16.47 and 16.62 µatm for OSTIA-MERIS, OSTIA-GLOBCOLOUR and MODIS-GLOBCOLOUR combinations, respectively) than for pCO2Ctrack (21.34, 22.08 and 22.07 µatm, respectively), showing that the estimated pCO2 field at super resolution using our algorithm is improving the pCO2 field obtained from CarbonTracker. The smallest AE is for the combination of SST and Chl provided by merged products. The values of pCO2Ctrack are, on average, larger than pCO2insitu (MDCtrack=2.97, 8.83 and 14.93 µatm) while the differences between pCO2infer and pCO2insitu values compensate each other (MDinfer=0.15, 3.42 and 8.42 µatm). In all cases the MDCtrack and MDinfer are positive, meaning that the pCO2 values are overestimated. Finally, comparing the relative error of pCO2Ctrack and pCO2infer with respect to pCO2insitu, we found that the relative error is low in all cases, smaller for pCO2infer than for pCO2Ctrack.

Finally, if we only compare the statistical errors at the common valid intersections between the pCO2infer using the three product combinations with pCO2Ctrack and with the in situ measurements (see Table ), we obtain 458 mutual intersections. We obtain similar results when taking into account all the intersections. The absolute error is smaller in the case of pCO2infer (17.65 µatm) than with pCO2Ctrack (20.24  µatm), indicating that our algorithm improves the estimation of ocean pCO2. The smallest AE is again for the combination with merged products. MD is positive showing that most of the time pCO2infer and pCO2Ctrack values are overestimated (see Fig. ). Again the relative error is small, less than 0.06, for all the product combinations.