*3.2. Fusion Method*

As discussed in Turiel et al. [10], ocean scalars as chlorophyll and temperature (denoted respectively by *c* and *θ*) are multifractal in the microcanonical sense. This implies that a singularity exponent can be assigned at each point and that they are arranged in a particular way, which in turn implies that they are related to a property of the flow (its circulation) rather than to any specificity of the associated scalar.

Assuming that the singularity lines of Chl-*a* and SST coincide as it was shown in Nieves et al. [13], it follows that their gradients must be related by a smooth 2 × 2 matrix *ρ* [17] :

$$
\nabla \mathcal{C}(\vec{x}) \, \, = \, \rho(\vec{x}) \, \nabla \theta(\vec{x}) , \, \tag{2}
$$

Although a large number of *ρ*(*x*) verifying Equation (2) exist, the indetermination is considerably reduced by imposing that the matrix *ρ*(*x*) must be smooth: a non-smooth *ρ*(*x*) would be a source of singularities, and hence the singularity lines of *c* and *θ* would differ. If we further assume that the gradients of *c* and *θ* are aligned, then Equation (2) leads to:

$$a(\vec{x}) \approx a(\vec{x})\,\theta(\vec{x}) + b(\vec{x}) \tag{3}$$

with smooth functions *a* and *b* (i.e., they have small gradients as compared to those of *c* and *θ*).

Let us now suppose that the function *c* is contaminated by some source of noise *n*, so in fact our data sets consists of *θ*(*x*) (assumed noiseless) and *c* (*x*) = *c*(*x*) + *n*(*x*). We can construct a filtered version of *c*, denoted by *c <sup>f</sup>* , as:

$$\mathcal{L}\_f(\vec{x}) = \mathfrak{d}(\vec{x})\,\theta(\vec{x}) + \hat{\mathfrak{d}}(\vec{x}) \tag{4}$$

where *a*ˆ(*x*) and ˆ *b*(*x*) are estimates of the actual parameters *a*(*x*) and *b*(*x*) in Equation (3). These estimates are obtained from the values of *c* and *θ* by performing scale-invariant linear regressions weighted around each point. The weight of a specific point *x* is defined as *wx*(*x* ) = |*x* − *x* | <sup>−</sup>2.

The total weight of a point *x*, *N*(*x*), is defined as follows:

$$N(\vec{x}) \equiv \sum\_{\vec{x}' \neq \vec{x}' \in \text{so}} \frac{1}{|\vec{x}' - \vec{x}|^2} \tag{5}$$

The weighted regression uses all available data of each field, using the function *N*(*x*) that decay as a power-law with the distance (Equation (5)); thus it is scale-invariant, as it does not introduce any preferred scale [16,17].

With *a*ˆ(*x*) and ˆ *b*(*x*), an estimation of the chlorophyll *c <sup>f</sup>* may be obtained applying Equation (4) as soon as the value of *θ* at that location *x* is available. This algorithm was shown in Umbert et al. [17] to lead to a large increase in the quality of SMOS SSS maps using OSTIA SST maps as a template; besides, the method leads to significant restoration of the structure of singularity exponents.

#### **4. Results**

#### *4.1. Scalar Synergy*

Figure 1 shows examples of daily maps of MODIS Chl-*a* concentration (top panel) and SST (middle panel) in 1 January 2006. The values of Chl-*a* are derived from measurements in the visible part of the spectrum, which may be affected by artifacts like aerosols, sun glint and high turbidity of the water. Several flags are introduced to characterize Chl-*a* data quality and a result, maps of Chl-*a* usually suffer from a larger incompleteness than those of SST, which are derived from infrared measurements in the 4 μm range. An example of the application of the fusion algorithm in the global ocean for 1 January 2006 is shown in the bottom panel of Figure 1. As stated before, the algorithm allows estimating the local Chl-*a*-SST regression by taking into account all possible couples of data (SST,Chl-*a*) weighted using function *N*(*x*) in Equation (5). Fused Chl-*a* maps integrate the relation between structures present in the SST and Chl-*a* specific structures. In fused daily products we recognize some expected Chl-*a* global patterns, near the ocean surface, where availability of sunlight is not limiting, phytoplankton growth depends on temperature and nutrient levels. High chlorophyll concentrations are found in nutrient-rich, cold polar waters and where ocean currents cause upwelling, which brings nutrient-rich deep-cold water to the surface.

We will focus in the region of the Gulf of California, a narrow sea between mainland Mexico and the Baja California peninsula, where high primary productivity levels are found as a result of an efficient nutrient transport of waters from under a shallow pycnocline into the euphotic zone [21]. Figure 2 shows the input variables of our algorithm (top panel: Chl-*a*, middle panel: SST) and the output fused Chl-*a* (bottom panel). The corresponding singularity exponents are shown in the right column. Rich singularity structures can be recognized in the original chlorophyll concentration maps associated with the fronts mainly caused by the horizontal transport from high primary production areas onto less productive ones. The SST field exhibits also the richness of patterns associated with the circulation in that area. Where both images are not affected by data gaps, the singularity structure (although not the magnitude) is similar between them. Places where the correspondence of both singularity images fails may identify places at which intrinsic dynamics of the variable competes with flow advection.

Once the data fusion is applied, the L4 Chl-*a* is extrapolated to all the pixels where SST was available. The structure of Chl-*a* is well represented (compared to the original image), although a smoothing of the original variable degrades the dynamic range of the variable. The singularity exponents of the fused Chl-*a* data reveal that most of the structures present in the SST map are re-integrated in the chlorophyll map at the same time that Chl-*a* specific magnitude and structures are maintained; for instance, the strong gradient associated with the biological activity present in the Gulf of California, appear delineated in the fused Chl-*a*. This implies that the fusion algorithm can partially integrate SST structures without destroying Chl-*a* ones.

**Figure 1.** (**top panel**) MODIS Aqua Level 3 Chl-*a* (mg/m3), (**middle panel**) MODIS Aqua Level 3 SST ( ◦C) and (**bottom panel**) Level 4 Chl-*a* concentration (mg/m3) extended to the areas where MODIS SST was available, for 1 January 2006.

**Figure 2.** (**top row**) MODIS Chl-*a* (mg/m3), (**middle row**) MODIS SST (◦C) and (**bottom row**) fused Chl-*a* (mg/m3), for 1 January 2006 (**left column**) and associated singularity exponents (**right column**).

#### *4.2. Interpretation of Auxiliary Parameters*

As shown in Equation (4), the functions *a*ˆ(*x*) and ˆ *b*(*x*) provide information about the local functional dependence between SST and Chl-*a*. The local slope, *a*ˆ(*x*), will be negative at those places where SST decreases as Chl-*a* increases in the neighborhood, and the converse. Considering that cold waters tend to have more nutrients than warm waters, phytoplankton is more abundant where surface waters are cold. So, as we move from one given point to another with colder water, Chl-*a* would usually increase and thus the slope *a*ˆ(*x*) will be negative. However, the relationship changes from point to point and it should be expected to change from one image to the next one. Figure 3 shows the seasonal average of the slope and intercept (considering winter as January-February-March, spring as April-May-June, summer as July-August-Septemebr and fall as October-November-December).

The coherent patterns of the auxiliary parameters of the method delineate areas with different relation between Chl-*a* and SST. In the same spirit, Longhurst [22] introduced the concept of ocean biogeochemical provinces, characterized by their particular physical and biological behavior. Longhurst definition is based on the mixed layer depth lying close to the ocean-atmosphere interface. Specific provinces have common characteristics and can generally be classified as four general biomes: the coastal, polar, westerly and trade winds biomes. A visual comparison between local functions of the fusion method and the Longhurst definition is shown in Figure 3.

As expected, negative slopes are present in the upwelling areas associated with the easternmost currents of the great anticyclonic gyres, corresponding to the Benguela and Canary currents in the Atlantic Ocean and the Peru and California currents in the Pacific Ocean. Notice that this negative slope is present all year long. These oceanic areas, as well as the upwelling zones and continental margins, are rich in Chl-*a* as a result of the proximity to areas where the resurgence of nutrients take place and the local circulation is favorable to nutrient accumulation. A band of cool, chlorophyll-rich water is also apparent all along the equator; the strongest signal at the Atlantic Ocean and the open waters of the Pacific Ocean also leads to negative values of the local intercept ˆ *b*(*x*).

**Figure 3.** (**left column**) Seasonal mean local slope estimation *a*ˆ(*x*) for spring, summer, fall and winter. (**right column**) Local intercept estimation ˆ *b*(*x*) for spring, summer, fall and winter. These are the mean estimations used in the derivation of the fused maps as the one presented in Figure 1. (**bottom row**) Chl-*a* mean concentration for year 2006 and SST mean for the same year.

Negative values of *a*ˆ(*x*) are also found in areas where Chl-*a* concentration decreases as SST increases; this situation, which is typically found in the (oligotrophic) subtropical gyres, intensifies in the Atlantic Ocean during the northern hemisphere winter and spring. The Pacific Ocean exhibits an intensified negative pattern in the Northern subtropical gyre during boreal spring and a negative pattern in the southern hemisphere during austral spring. In both cases, such intensification in the oligotrophic subtropical gyres is driven by the seasonal cycle of sea surface temperature.

Subpolar gyres are also characterized by high Chl-*a* concentrations linked to nutrient accumulation during winter, when the mixing layer reaches the deep ocean followed by the stratification of the water column during spring. During the dark winter months, the local slope between SST and Chl-*a* is positive. However, when sunlight returns and nutrients are trapped near the surface during spring and summer, the phytoplankton flourishes in high concentration, seen as negative values of *a*ˆ(*x*).

The local regression coefficient (not shown) has small values along the Equatorial Pacific, and in the Southern and Indian Ocean indicating that horizontal advection of Chl-*a* cannot be locally explained by SST variability in these regions only. Therefore, either additional variables should be taken into account, or a more sophisticated relation between Chl-*a* and SST should be used. For example, in ocean regions of High Nutrient Low Chlorophyll (HNLC) as the equatorial Pacific and the Southern Ocean, low Chl-*a* concentrations are due to a stoichiometric imbalance of iron which have no link to SST. Another possible cause for the low values of the local regression coefficient in some regions is the lack of enough points to provide a quality reconstruction. For instance, 85% of the data points are missing in the Equatorial Pacific due to the large cloudiness.
