An accurate quantification of the role of the ocean as source/sink of
greenhouse gases (GHGs) requires to access the high-resolution of the GHG
air–sea flux at the interface. In this paper we present a novel method to
reconstruct maps of surface ocean partial pressure of CO

The ocean can be thought of as a complex system in which a large number of
different processes (e.g., physical, chemical, biological, atmosphere–ocean
interactions) interact with each other at different spatial and temporal
scales

The most commonly used methods to estimate air–sea CO

Another new avenue for inferring air–sea GHG fluxes is through inverse modeling
applied to vertical column densities (VCDs) extracted from satellite
spectrometers, i.e., Greenhouse gases Observing SATellite (GOSAT) and SCanning
Imaging Absorption SpectroMeter for Atmospheric CHartographY (SCIAMACHY), at
low spatial resolution

In this regard, the last few years have seen the appearance of interesting
new developments on multiscale processing techniques for complex signals
coming from Earth observations

These advances open a wide field of theoretical and experimental research and
their use in the analysis of complex data coming from satellite imagery has
been proven innovative and efficient, showing a particular ability to perform
fusion of satellite data acquired at different spatial resolutions

Unlike the Lagrangian approach to reconstruct tracer maps at high resolution

The eastern boundary upwelling systems (EBUSs) and oxygen minimum zones
(OMZs) are likely to contribute significantly to the gas exchange between the
ocean and the atmosphere

This paper is organized as follows: Sect.

The input data combines air–sea CO

It is known that the evolution of a concentration,

We obtain the partial pressure of ocean CO

The raw data of CarbonTracker fluxes of CO

Oceanic

We use here the high-resolution satellite ocean data for chlorophyll

Estimated fluxes from CarbonTracker data. Shown are the results on the Benguela upwelling system on 23 March 2006. Left are the CarbonTracker fluxes, right are our results.

In this study we use Chl

Snapshots of both Chl

Snapshot of Chl

Snapshot of SST fields corresponding to 21 September 2006 regridded
at

We use SST derived from OSTIA and MODIS products. OSTIA (Operational SST and
Sea Ice Analysis system) is a new analysis of SST that uses satellite data
provided by the GHRSST (Group for High Resolution SST) project, together with
in situ observations, to determine the SST with a global coverage and without
gaps in data. The data sets are produced daily and at spatial resolution of

Among the available data in SOCAT version 2

2000, one cruise: ANT-18-1

2004, one cruise: 0404SFC-PRT

2005, five cruises: QUIMA2005-0804, QUIMA2005-0821, QUIMA2005-0922, QUIMA2005-1202, QUIMA2005-1220

2006, nine cruises: GALATHEA, QUIMA2006-0326, QUIMA2006-0426, QUIMA2006-0514, QUIMA2006-0803, QUIMA2006-0821, QUIMA2006-0921, QUIMA2006-1013, QUIMA2006-1124

2008, seven VOS cruises: QUIMA2008-1, QUIMA2008-2, QUIMA2008-3, QUIMA2008-4, QUIMA2008-5, QUIMA2008-6, QUIMA2008-7

2010, one cruise: ANT27-1

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

The idea behind the methodology hinges on the fundamental discovery of a
simple functional dependency between the transitions – those being measured
by the dimensionless values of the singularity exponents computed within the
framework of the microcanonical multifractal formalism – of the respective
physical variables under study: SST, ocean color and oceanic partial pressure
(

In the ocean, the turbulence causes the formation of unsteady eddies on many scales
which interact with each other

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 (

We will not review here the details of the computation of the singularity
exponents

Some examples of the singularity exponents of Chl

Another important idea implemented in the methodology is the coupling of the
physical information contained in SST and OC images with the ocean

Therefore, in our methodology, the local connection between different tracer
concentrations (SST and Chl

Once we have introduced these coefficients in the linear combination on
satellite data, we obtain a proxy for singularity exponents of

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

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

After selecting a given area of study, compute the singularity exponents of SST, Chl

Using Eq. (

Obtain the regression coefficients

Calculate the singularity exponents of available satellite SST, Chl

Use coefficients obtained in step (iii) and apply Eq. (

Using Eq. (

Use Eq. (

The methodology has been successfully applied to dual ROMS simulation data at
two resolutions, obtaining a mean absolute error of

Since the key element for the application of our inferring algorithm relies
on the ability to obtain the singularity exponents and their quality, the
success of our methodology applied to satellite data depends on the quality
and the properties of the input data. In order to assess such properties, we
perform a statistical analysis of the different data sets. First, we analyze
the Chl

Further information can be obtained by computing statistical quantities such
as standard deviation, skewness and kurtosis. Table

We have repeated the same analysis for SST data sets. The PDFs of the SST
values for OSTIA and MODIS products are shown in Fig.

Values of the standard deviation, skewness and kurtosis for the different products.

If turbulence is dominated by coherent structures localized in space and
time, then PDFs are non-Gaussian, and the kurtosis will be higher than 3. To
analyze this feature we turn to the statistical analysis of the singularity
exponents, which, as explained before, have the ability to unveil the cascade
structures given by the tracer gradients. In Fig.

Finally, we obtain the singularity spectra from the empirical distributions
of singularity exponents shown in Fig.

We now apply the methodology to infer ocean

Henceforward we use the following notation for the three different sources of

Values of the standard deviation, skewness and kurtosis of the singularity exponents for the different products.

For the inference we use the following three combinations of Chl

Figure

Maps of

What is new in the reconstructed

Longitudinal profiles of

Since the underlying aim of this work is to develop a methodology to infer
super-resolution

Similar results are found when one compares the spatial distribution of the

Spatial distribution of the time averages of

Number of valid points in the

Comparison of the probability distribution functions of
CarbonTracker and inferred

First, we compute the number of valid points in the

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

Furthermore, to examine the transition fronts for the different products, we
compute the singularity spectra for the three product combinations (see
Fig.

Next, we perform a validation analysis of the results of our super-resolution

An example of the qualitative comparison of values of

Values of

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

Mean difference, absolute error and relative error of

The number of valid intersections is the largest with the OSTIA-GLOBCOLOUR
combination (Table

In order to quantitatively study the difference between

where

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

We summarize in Table

Finally, if we only compare the statistical errors at the common valid
intersections between the

Mean difference, absolute error and relative error of

In this work we have presented a method to infer high-resolution CO

We are aware that further investigations could improve the algorithm. The
multiple linear regression coefficients could be derived differentiating the
seasons (i.e., coefficients would vary as a function of calendar month)
considering the marked seasonal cycle in the Benguela upwelling system.
Additionally, future work will focus on the extension of the computations to
larger areas in order to infer global high-resolution CO

This work was supported by the ESA Support To Science Element Grant
no. 400014715/11/I-NB OceanFlux-Upwelling Theme. The Surface Ocean
CO