**2.5 Singular value decomposition (SVD) analysis of the cross-covariance**

A more sophisticated method to analyze the relationship between any two satellite data sets is the SVD analysis of the cross-covariance matrix between the two data sets with the same

Using SVD Analysis of Combined Altimetry and Ocean Color

oscillation on phytoplankton production in open waters.

Satellite Data for Assessing Basin Scale Physical-Biological Coupling in the Mediterranean Sea 129

Gabes and the Nile River delta display positive correlations. The Adriatic basin is essentially controlled by the local winter climatic conditions, rather than the nutrient inputs from deeper layers or land sources (Santoleri et al., 2003). The Gulf of Gabes signal may be an artifact produced by direct bottom reflection in areas of shallow clear waters (Jaquet et al., 1999). The other regions with positive correlations are located over the continental shelf and receive important terrestrial inputs that may override the control of seasonal thermocline

Fig. 2. Correlation coefficient between SLA and CHL seasonal cycles. The correlation in the

Although the large-scale patterns in oceanic areas observed in Figure 2 imply that SLA and related thermocline variations have key relevance on seasonal CHL, other physical processes also modulate the biological response in the Mediterranean. The correlation between SLA and CHL anomalies at inter-annual time scale is mostly negative in oceanic areas (Figure 3), suggesting the prevalence of the coupling between SLA and CHL typical of temperate areas, as observed for the seasonal cycles. However, areas showing positive correlations increase with respect to Figure 2. Water discharges from Rhone, Ebro and Nile rivers and from the Black Sea cause these positive correlations in their influence areas. Interestingly, positive correlations are also found in the southern part of the Western basin. This area is characterized by an intense mesoscale activity produced byAlgerian eddies detached from the African coast and propagating to the north (Millot and Taupier-Letage, 2005). Vertical transfer of nutrients through eddy pumping is a dominant process modulating the biological activity in this region (Arnone and La Violette, 1986; Taupier-

The correlation between SLA and CHL anomalies at intra-annual time scales is shown in Figure 4. Correlations are not statistically significant in most of the Mediterranean Sea. Indeed, the values of the significant correlations are notably lower than the correlations between seasonal cycles and anomalies at inter-annual time scales, suggesting that direct correlation is not adequate to analyze the relationships between SLA and CHL anomalies at

red and blue areas is statistically significant at 95% level or more.

Letage et al., 2003).

intra-annual time scales.

data length in time, but not necessarily the same spatial domain (Bretherton et al., 1992). The SVD analysis isolates covarying (coupled) spatial patterns of variability that tend to be linear related to one another. The SVD analysis is a generalization of empirical orthogonal function (EOF) analysis. Rather than extracting the modes that explain the greatest variance in a single data set, as in EOFs, the SVD analysis finds the covarying modes that explain as much as possible of the covariance between the two data sets.

Consider two data sets **s**(**x**,*t*) and **c**(**y**,*t*), consisting of SLA anomaly values at *Ns* grid points and CHL anomaly values at *Nc* grid points (possibly different), both for the same *T* observation times. The data time series **s**(*t*) and **c**(*t*) at each of the grid points can be expanded in terms of a set of *N* < min(*Ns*,*Nc*) vectors or patterns

$$\mathbf{s}(t) \equiv \sum\_{k=1}^{N} a\_k(t)\mathbf{p}\_k \tag{2}$$

$$\mathbf{c}(t) \equiv \sum\_{k=1}^{N} b\_k(t)\mathbf{q}\_k \tag{3}$$

The time series *ak*(*t*) and *bk*(*t*) are the expansion coefficients and the vectors **p***k* and **q***k* are the corresponding spatial patterns. The SVD spatial patterns are othonormal. Each pair of coefficients and patterns together (for a given *k*) make up a mode. The coefficients and patterns are chosen so that the first mode maximizes 1 1 *a tb t* () () , the cross-covariance of the expansion coefficients, where the brackets denote the time average over the T observation

times. Successive pairs explain the maximum squared temporal covariance subject to orthogonality of the spatial patterns among themselves. The SVD modes are the eigensolutions of the cross-covariance matrix between the two time series.

### **3. Correlations between SLA and CHL**

Ocean phytoplankton growth mainly depends on the availability of light and nutrients. Whereas light is rarely limiting in surface waters of the Mediterranean Sea (exceptions are some areas affected by river discharges), nutrient availability generally regulates phytoplankton growth. Since nutrient concentrations are higher in the deep ocean, physical processes that favor the supply nutrients from deeper layers into the surface euphotic zone will stimulate phytoplankton growth. Stratification and mixed layer depth changes are important factors regulating deep nutrient-rich waters supply to the upper ocean layer. SLA is indicative of changes in the thermocline depth because SLA primarily reflects the first baroclinic mode, which is related to the main thermocline (Stammer, 1997; Wunsch, 1996).

Figure 2 shows the correlation between the seasonal cycles of SLA and CHL at each grid point with shaded colors. Correlations that are not statistically significant at the 95% level are shaded white. Negative correlations are observed in most of the Mediterranean Sea, with highest (absolute) values in the northern part of the Western basin and lowest values between the Ionian and Levantine basins. Also, higher correlations are generally observed in oceanic water, off from the shelf. Inverse correlations in the seasonal cycles of SLA and CHL are typical of temperate regions where summer stratification inhibits the vertical flux of nutrients and winter mixing supplies nutrient-rich subsurface waters fueling phytoplankton growth. A few areas such as the Adriatic and Aegean basins, the entrance of the Gulf of

data length in time, but not necessarily the same spatial domain (Bretherton et al., 1992). The SVD analysis isolates covarying (coupled) spatial patterns of variability that tend to be linear related to one another. The SVD analysis is a generalization of empirical orthogonal function (EOF) analysis. Rather than extracting the modes that explain the greatest variance in a single data set, as in EOFs, the SVD analysis finds the covarying modes that explain as

Consider two data sets **s**(**x**,*t*) and **c**(**y**,*t*), consisting of SLA anomaly values at *Ns* grid points and CHL anomaly values at *Nc* grid points (possibly different), both for the same *T* observation times. The data time series **s**(*t*) and **c**(*t*) at each of the grid points can be

> 1 () () *N*

1 () () *N*

The time series *ak*(*t*) and *bk*(*t*) are the expansion coefficients and the vectors **p***k* and **q***k* are the corresponding spatial patterns. The SVD spatial patterns are othonormal. Each pair of coefficients and patterns together (for a given *k*) make up a mode. The coefficients and patterns are chosen so that the first mode maximizes 1 1 *a tb t* () () , the cross-covariance of the expansion coefficients, where the brackets denote the time average over the T observation times. Successive pairs explain the maximum squared temporal covariance subject to orthogonality of the spatial patterns among themselves. The SVD modes are the

Ocean phytoplankton growth mainly depends on the availability of light and nutrients. Whereas light is rarely limiting in surface waters of the Mediterranean Sea (exceptions are some areas affected by river discharges), nutrient availability generally regulates phytoplankton growth. Since nutrient concentrations are higher in the deep ocean, physical processes that favor the supply nutrients from deeper layers into the surface euphotic zone will stimulate phytoplankton growth. Stratification and mixed layer depth changes are important factors regulating deep nutrient-rich waters supply to the upper ocean layer. SLA is indicative of changes in the thermocline depth because SLA primarily reflects the first baroclinic mode, which is related to the main thermocline (Stammer, 1997; Wunsch, 1996). Figure 2 shows the correlation between the seasonal cycles of SLA and CHL at each grid point with shaded colors. Correlations that are not statistically significant at the 95% level are shaded white. Negative correlations are observed in most of the Mediterranean Sea, with highest (absolute) values in the northern part of the Western basin and lowest values between the Ionian and Levantine basins. Also, higher correlations are generally observed in oceanic water, off from the shelf. Inverse correlations in the seasonal cycles of SLA and CHL are typical of temperate regions where summer stratification inhibits the vertical flux of nutrients and winter mixing supplies nutrient-rich subsurface waters fueling phytoplankton growth. A few areas such as the Adriatic and Aegean basins, the entrance of the Gulf of

*k t bt* 

eigensolutions of the cross-covariance matrix between the two time series.

**3. Correlations between SLA and CHL** 

*k t at* 

*k k*

*k k*

**s p** (2)

**c q** (3)

much as possible of the covariance between the two data sets.

expanded in terms of a set of *N* < min(*Ns*,*Nc*) vectors or patterns

Gabes and the Nile River delta display positive correlations. The Adriatic basin is essentially controlled by the local winter climatic conditions, rather than the nutrient inputs from deeper layers or land sources (Santoleri et al., 2003). The Gulf of Gabes signal may be an artifact produced by direct bottom reflection in areas of shallow clear waters (Jaquet et al., 1999). The other regions with positive correlations are located over the continental shelf and receive important terrestrial inputs that may override the control of seasonal thermocline oscillation on phytoplankton production in open waters.

Fig. 2. Correlation coefficient between SLA and CHL seasonal cycles. The correlation in the red and blue areas is statistically significant at 95% level or more.

Although the large-scale patterns in oceanic areas observed in Figure 2 imply that SLA and related thermocline variations have key relevance on seasonal CHL, other physical processes also modulate the biological response in the Mediterranean. The correlation between SLA and CHL anomalies at inter-annual time scale is mostly negative in oceanic areas (Figure 3), suggesting the prevalence of the coupling between SLA and CHL typical of temperate areas, as observed for the seasonal cycles. However, areas showing positive correlations increase with respect to Figure 2. Water discharges from Rhone, Ebro and Nile rivers and from the Black Sea cause these positive correlations in their influence areas. Interestingly, positive correlations are also found in the southern part of the Western basin. This area is characterized by an intense mesoscale activity produced byAlgerian eddies detached from the African coast and propagating to the north (Millot and Taupier-Letage, 2005). Vertical transfer of nutrients through eddy pumping is a dominant process modulating the biological activity in this region (Arnone and La Violette, 1986; Taupier-Letage et al., 2003).

The correlation between SLA and CHL anomalies at intra-annual time scales is shown in Figure 4. Correlations are not statistically significant in most of the Mediterranean Sea. Indeed, the values of the significant correlations are notably lower than the correlations between seasonal cycles and anomalies at inter-annual time scales, suggesting that direct correlation is not adequate to analyze the relationships between SLA and CHL anomalies at intra-annual time scales.

Using SVD Analysis of Combined Altimetry and Ocean Color

inputs.

Satellite Data for Assessing Basin Scale Physical-Biological Coupling in the Mediterranean Sea 131

areas of the Mediterranean Sea present the negative covariations (i.e. SLA increases and CHL decreases, or vice versa) typical of temperate areas, particularly in the Tyrrhenian, Ionian and Aegean basins. In the Levantine basin and in the southern part of the Western basin, areas with positive covariations are observed. These regions are characterized by high levels of mesoscale eddy variability (Pujol & Larnicol, 2005). Mesoscale eddies have important biological and biogeochemical consequences, driving vertical motions of water and lifting subsurface nutrients into the surface euphotic layer (McGillicuddy et al., 1998; Oschlies & Garçon, 1998). Positive covariations are also observed in the Adriatic basin, which is driven by the local winter climatic conditions (Santoleri et al., 2003), and in coastal areas such as Rhone, Ebro, Po and Nile river deltas, suggesting the influence of riverine

Fig. 5. First spatial patterns of (a) SLA and (b) CHL anomalies at inter-annual time scales. The patterns are scaled to represent the amplitude of SLA and CHL anomalies associated

with 1 standard deviation of the first expansion coefficients.

Fig. 3. Correlation coefficient between SLA and CHL anomalies at inter-annual time scales. The correlation in the red and blue areas is statistically significant at 95% level or more.

Fig. 4. Correlation coefficient between SLA and CHL anomalies at intra-annual time scales. The correlation in the red and blue areas is statistically significant at 95% level or more.
