1. Introduction
The Mediterranean Sea is a hot spot for climate change and plays a major role in the climate variability over Europe. This has led to efforts being made to better characterize its surface currents and to develop numerical models of its circulation. However, the dynamics in most of the Mediterranean are characterized by a Rossby radius of deformation of the order of 10–15 km, which requires spatial resolutions higher than the available observations of surface velocities (~100 km). Besides, current observations of sea surface temperature (SST) do have the appropriate spatial resolutions. However, the lack of a direct connection between SST and surface currents has proven to be an obstacle difficult to overcome for their quantitative estimation from SST satellite observations.
The diagnosis of surface velocities from SST images is based on the exploitation of the properties of upper-ocean dynamics. One of such properties is that small-scale patterns tend to be advected by larger-scale velocities in a turbulent flow. Then, mesoscale currents (~20 km) can be inferred, in principle, by tracking submesoscale patterns (1–10 km) in consecutive cloud-free SST images (e.g., Emery et al. 1986; Bowen et al. 2002). Another possibility consists of the exploitation of heat conservation and the inversion of the temperature equation to retrieve ocean currents using a sequence of SST images (e.g., Kelly 1989; Ostrovskii and Piterbarg 1995; Chen et al. 2008). A third approach, which has emerged during the last years, is based on the property that SST and surface currents have a similar complex phase in the upper ocean under the appropriate conditions (e.g., Isern-Fontanet et al. 2006a; LaCasce and Mahadevan 2006; Isern-Fontanet et al. 2008; Isern-Fontanet and Hascoët 2014). Indeed, Lapeyre and Klein (2006) put forward that potential vorticity (PV) in the upper ocean is correlated to the sea surface buoyancy (SSB), which locks the phase between SSB and the streamfunction. Contrary to the previous approaches, this property allows diagnosing velocities from a single SST field.
Indeed, Lapeyre and Klein (2006) and LaCasce and Mahadevan (2006) proposed that the mesoscale and submesoscale dynamics of the upper-ocean layers could be modeled using an effective version of the surface quasigeostrophic (SQG) equations. This implies that the surface streamfunction can be retrieved directly from SSB by convolving it with a kernel proportional to k−1, where k is the modulus of the two-dimensional wavevector. This kernel is usually known as the transfer function in signal processing theory. Other solutions can be found depending on the geometrical conditions (finite depth vs semi-infinite ocean; Tulloch and Smith 2006), the stratification (constant vs exponential; LaCasce 2012), or the relative contribution of the interior PV to respect the surface boundary condition (Lapeyre and Klein 2006; Klein et al. 2010). All of these solutions are also characterized by the phase locking between the streamfunction and SSB. It is worth mentioning that although the relevant dynamical variable in the PV inversion problem is the SSB, the field that can be remotely observed is the SST. Therefore, some additional assumptions regarding the alignment between SST and buoyancy gradients (Klein et al. 1998; Ferrari and Paparella 2003) have to be made if the objective is to retrieve ocean dynamics from SST. Because these solutions imply that SSB (or SST) shares the phase with the streamfunction, it is necessary to verify this condition before applying these transfer functions and identify under which conditions the assumption of zero phase shift holds.
The validity of the SQG approach implies an energy spectrum with slopes of k−5/3 (Blumen 1978) and a spectrum of k−11/3 for sea surface height (SSH). This has motivated the study of its validity through the analysis of altimetric observations of SSH. Le Traon et al. (2008) showed that spectral slopes in the mesoscale band in high energy areas are significantly different from a k−5 law, predicted by quasigeostrophic (QG) turbulence, but they very closely follow the k−11/3 slope, which indicates that the surface QG is a much better dynamical framework than the QG turbulence theory to describe the ocean surface dynamics. However, results in low energy areas revealed shallower slopes (Xu and Fu 2011). Sasaki and Klein (2012) further investigated such areas using numerical simulations and showed that even in low energy areas SSH had spectra of the type k−4, close to SQG predictions, and suggested that instrumental noise could hide such slopes. Furthermore, they showed that the scales for which the k−4 slope could be seen depend on the energy level. In particular, for low energy areas with sea level variance on the order of 25 cm2, the k−4 slope can be observed at scales shorter than 150 km. These studies have been complemented by the analysis of high-frequency radar just off the U.S. West Coast, which found spectral slopes equivalent to a SSH k−4 slope for scales smaller than 100 km (Kim et al. 2011). Nevertheless, this does not necessarily imply that velocities can be diagnosed from SST. On one side, Armi and Flament (1985) pointed out that different dynamical regimes could have similar spectral slopes and stressed the importance of the complex phases. On the other side, Klein and Hua (1990) showed that the emergence and evolution of the SST mesoscale variability depends on the mixed layer (ML) dynamics, which can introduce a phase shift between SST and currents.
The Mediterranean Sea is a semienclosed basin with an excess of evaporation mainly compensated by the entrance of the relatively fresher waters from the Atlantic Ocean through the Strait of Gibraltar. Satellite images of SST have been widely used to investigate the circulation patterns in the Mediterranean Sea and track coherent vortices (e.g., Puillat et al. 2002; Hamad et al. 2005). However, the difficulty to quantitatively estimate surface velocities from SST patterns has limited its use to qualitative analysis. Consequently, quantitative studies have been based on altimetric (e.g., Larnicol et al. 2002; Pujol and Larnicol 2005; Isern-Fontanet et al. 2006c) and in situ measurements (e.g., Millot 1999; Malanotte-Rizzoli et al. 1999; Poulain et al. 2012, and references therein). The picture that emerges from both satellite and in situ observations reveals that incoming fresh waters circulate anticlockwise along the coast in the western Mediterranean Sea and the Levantine Basin (Millot and Taupier-Letage 2005; Poulain et al. 2012). The circulation in the Ionian Sea is still under debate, and different circulation schemata have been proposed (e.g., Millot and Taupier-Letage 2005; Pinardi et al. 2005; Poulain et al. 2012). The major discrepancies concern the existence of a jet that crosses the Ionian Sea, which is supported by the observations provided by surface drifters (Millot and Gerin 2010; Poulain et al. 2012), although some researchers consider it an artifact (Millot and Gerin 2010). In addition to these patterns, the instability of the inflow and local wind action often generate coherent vortices in several parts of the basin that tend to propagate following rather well-defined patterns in some areas of the Mediterranean Sea (Isern-Fontanet et al. 2006b). Although the use of SST has proven to be in good agreement with in situ measurements (e.g., Taupier-Letage et al. 2003; Isern-Fontanet et al. 2004) and very useful to get insight about ocean circulation in the area, there are some potential difficulties. By the end of summer, surface resident waters are warmer than incoming Atlantic water, while in winter the Atlantic water is warmer than the resident Mediterranean waters. This has two main implications for the diagnosis of surface velocities from SST: some features may not be visible during certain periods of the year, and the alignment between salinity and temperature gradients may vary (Taupier-Letage 2008).
The main objective of this study is to characterize the transfer function between SSB and SSH and better determine the conditions under which SSB shares the complex phase with SSH in the Mediterranean Sea. Because SSB cannot be determined from direct observations, the study has been extended to include the analysis of SST to clearly identify the capability to reconstruct surface velocities from satellite observations of SST. The paper is organized as follows. Section 2 defines the theoretical framework used in this study. Section 3 describes the numerical simulations used for this study and the characteristics of the areas analyzed. Section 4 describes our results and section 5 discusses them.
2. Theoretical framework
a. The transfer function formalism
b. The transfer function at the ocean surface
The transfer function Fb(k) is a real function that controls the amplitude of the resulting streamfunction and only depends on the modulus of the wavevector.
3. Numerical simulations
a. The Mediterranean Forecasting System
In this study, we used the nowcasts provided by the Mediterranean Forecasting System (MFS) running at the Istituto Nazionale di Geofisica e Vulcanologia (INGV) (http://gnoo.bo.ingv.it/mfs/). The model is an implementation of the Nucleus for European Modelling of the Ocean–Océan Parallélisé (NEMO–OPA), version 3.2 (Madec et al. 1998; Madec 2008) that covers the entire Mediterranean Sea and part of the Atlantic Ocean with a spatial resolution of 
The system assimilates sea level anomaly (SLA), sea surface temperature, in situ temperature profiles from Voluntary Observing Ship (VOS) expendable bathythermographs (XBTs), in situ temperature and salinity profiles from Argo floats, and in situ temperature and salinity profiles from CTD using the three-dimensional variational data assimilation (3DVAR) scheme (Dobricic and Pinardi 2008). The background error correlation matrix is estimated from the temporal variability of parameters in a historical model simulation. Background error correlation matrices vary seasonally and in 13 regions of the Mediterranean that have different physical characteristics (Dobricic et al. 2007). The mean dynamic topography used for the assimilation of SLA has been computed by Dobricic (2005). Satellite SST data are used for the correction of surface heat fluxes.
b. Regions under study
This study has been focused on four regions of the Mediterranean Sea representative of its circulation: the Algerian Basin, the Ionian Sea, the Levantine Basin, and the Gulf of Lions (Fig. 1). The dynamics and environmental conditions present during the period analyzed were characterized by the sea level variance (SLV), the root-mean-square (RMS) of wind stress, and the RMS of the mixed layer depth (MLD; see Table 1). Here, the MLD was estimated using the classical difference criterion with a reference depth of 10 m, as in de Boyer Montégut et al. (2004) and D’Ortenzio et al. (2005). Notice that the SLV values shown in Table 1 correspond to low energy areas.
(from top to bottom) RMS of the MLD, RMS of wind stress, and SLV. Squares indicate the location and extension of the four areas analyzed: 1) Algerian Basin, 2) Ionian Sea, 3) Levantine Basin, and 4) Gulf of Lions.
Citation: Journal of Physical Oceanography 44, 5; 10.1175/JPO-D-13-0186.1
Spatially averaged (indicated by the angle brackets) SLV 
The Algerian Basin is a highly energetic area characterized by the presence of eddies with sizes ranging from 30 to 150 km generated by the instability of the fresh waters of Atlantic origin that flow along the Algerian coast. These eddies tend to follow well-defined patterns characterized by a recirculation loop approximately enclosed within the box used in our analysis (Isern-Fontanet et al. 2006b). The presence of such eddies produced the largest spatially averaged SLV (Table 1). On the contrary, the Ionian Sea had the lowest SLV of all four boxes, mainly concentrated in the southern part of the box. Its dynamics are characterized by a lower concentration of intense eddies (Isern-Fontanet et al. 2006b) and the presence of a jet crossing the box from east to west (e.g., Poulain et al. 2012). The Levantine Basin had dynamical characteristics that resemble those of the Algerian Basin. It is characterized by the presence of energetic eddies generated in part by the instability of the waters that circulate counterclockwise along the coast. However, the length of the time series here analyzed was not long enough to produce more homogeneous patterns of SLV. Indeed, the box in this area was dominated by the presence of a highly energetic eddy in its southwestern corner, which is evident in Fig. 1. Finally, the Gulf of Lions, which is an area of deep-water formation, was characterized by the strongest wind forcing, very deep MLD, and quite active dynamics, mainly in its southwestern part due to the contribution of Algerian eddies. In addition to these eddies, this area is characterized by the presence of smaller vortices (~50 km) generated by the instability of the Liguro–Provençal Current that follows the continental shelf (e.g., Rubio et al. 2005).
PDF of the alignment between temperature and salinity gradients. Dotted line corresponds to the Algerian Basin, dashed line to the Ionian Sea, dashed–dotted line to the Levantine Basin, and solid line to the Gulf of Lions.
Citation: Journal of Physical Oceanography 44, 5; 10.1175/JPO-D-13-0186.1
4. Results
a. Phase shift between SSB and SSH
(left) Time series for the mean MLD 
Citation: Journal of Physical Oceanography 44, 5; 10.1175/JPO-D-13-0186.1
Spatial correlations between the vorticity diagnosed from SSH and the vorticity diagnosed from the streamfunctions derived from SSB (black line) and SST (gray line) using the spectral amplitudes of SSH.
Citation: Journal of Physical Oceanography 44, 5; 10.1175/JPO-D-13-0186.1
The difficulty in reaching correlations higher than 0.9 still indicated the existence of a phase shift between SSB and SSH. To estimate it, daily phase shifts associated with spatial correlations larger or equal to 0.9 were averaged (Fig. 5). Results revealed that even for high spatial correlations, there was a phase shift between SSB and SSH of the order of 
(left) Spectral dependence of the mean phase shift between SSB and SSH. (right) Mean transfer function modulus. The mean has been taken over those days with correlations of vorticity larger or equal than 0.9. Dotted line corresponds to the Algerian Basin, dashed line to the Ionian Sea, dashed–dotted line to the Levantine Basin, and solid line to the Gulf of Lions.
Citation: Journal of Physical Oceanography 44, 5; 10.1175/JPO-D-13-0186.1
b. The transfer function between SSB and SSH
In addition to the phase shift, the characteristics and temporal evolution of the transfer function were investigated. The transfer function was computed from daily SSB and SSH fields using Eq. (12). Results shown in Fig. 3 revealed that it was characterized by negative slopes particularly for wavelengths shorter than 100 km, and a clear seasonal cycle characterized by steeper slopes for wavelengths shorter than 100 km during summer. A qualitative correspondence between the patterns observed in the transfer function and those observed for the temporal evolution of the phase shift could be found. In general, all boxes exhibited similar features with the exception of the Gulf of Lions, which had less marked patterns (not shown).
The transfer function associated with the smallest phase shifts was estimated by averaging the observations for those days with vorticity correlations larger or equal to 0.9 as before (Fig. 5). From results, two different bands in most basins could be clearly identified: one characterized by an SQG-like slope, that is, Fb(k) ~ k−1, between 20 and 100 km (see Table 2); and another characterized by a plateau, that is, Fb(k) ~ C, for wavelengths larger than 100 km. Nevertheless, the situation was slightly different in the Gulf of Lions, where the Fb(k) ~ k−1 extended along the whole range of wavelengths analyzed.
Spectral slopes between 20 and 100 km of the transfer function shown in Fig. 5 and the associated uncertainties in the four boxes analyzed.
c. Dependence on environmental conditions
The strong seasonal dependence of the phase shift and transfer function pushed us to investigate which parameters better characterized such evolution. We focused on the spatial averages of the MLD 
Scatterplot of the correlations between the vorticity diagnosed from SSB and model’s vorticity for the band between 20 and 300 km and different environmental parameters: ML depth 
Citation: Journal of Physical Oceanography 44, 5; 10.1175/JPO-D-13-0186.1
The averaged phase shift and transfer function were explored using the MLD as a criterion to classify the daily results. In particular, the mean transfer function was obtained under different MLD ranges, as is shown in Fig. 7. For MLD deeper than 70 m, the results were similar to those obtained selecting correlations larger or equal to 0.9 (Fig. 5), except for the Gulf of Lions, which exhibited larger phase shifts and more of a departure from the mean transfer functions than that observed for the other boxes. The reduction of the MLD threshold value used to select which days were averaged lead to an increase of the phase shift and a change of the transfer function for wavelengths shorter than 100 km from Fb(k) ~ k−1 to Fb(k) ~ k−2. The plateau for wavelengths longer than 100 km did not change.
(left) Spectral dependence of the mean phase shift between surface buoyancy and SSH. (right) Mean transfer function modulus. (from top to bottom) Observations for MLD shallower than 20 m, observations for MLD between 20 and 70 m, and results for MLD deeper than 70 m. Dotted line corresponds to the Algerian Basin, dashed line to the Ionian Sea, dashed–dotted line to the Levantine Basin, and solid line to the Gulf of Lions.
Citation: Journal of Physical Oceanography 44, 5; 10.1175/JPO-D-13-0186.1
Mean spectra of SSB (dotted line), SST (dashed line), and SSH (solid line) for different MLD in the Levantine Basin. The spectra of SSH have been multiplied by k2 for comparison with SST and SSB.
Citation: Journal of Physical Oceanography 44, 5; 10.1175/JPO-D-13-0186.1
d. Flow reconstruction from SST
The capability to diagnose surface dynamics from SST was investigated, focusing on the reconstruction capabilities for the surface streamfunction, surface velocity 
Correlation between reconstructed fields from SST using the optimal transfer function and fields provided by the model obtained by binning the MLD and taking the median in each bin. Solid line corresponds to wavelengths longer than 100 km and dashed line to wavelengths between 20 and 100 km. (from top to bottom) Streamfunction ψ, vorticity ζ, zonal velocity u, and meridional velocity υ.
Citation: Journal of Physical Oceanography 44, 5; 10.1175/JPO-D-13-0186.1
5. Discussion
It has been shown that the transfer function between the streamfunction and the surface buoyancy does not follow the classical SQG solution for large scales, with the exception of the Gulf of Lions, and the existence of a phase shift between the streamfunction and surface buoyancy has been evidenced. Indeed, results have shown that the amplitude of the mean transfer function for both, surface density and SST, has an SQG-like response for wavelengths shorter than 100 km and deep ML in all the regions analyzed. On the contrary, it has almost flat amplitude for longer wavelengths in most boxes. The qualitative comparison of these results with the analytical solution obtained assuming a finite depth and a constant stratification [Eqs. (6) and (8)] suggests a dominance of the interior solution at large scales and of the surface solution at smaller scales, in agreement with theoretical studies (Lapeyre and Klein 2006; Lapeyre 2009). The above results were observed mainly during winter, when a deep ML can be found. On the contrary, observations for shallow ML reveal a different response characterized by steeper slopes in the amplitude of the transfer function at short wavelengths, that is, Fb(k) ~ k−2.
A recent study by Callies and Ferrari (2013) showed that the interior QG solution dominates the flow at scales larger than 20 km in highly energetic areas, such as the Gulf Stream (
Scatterplot of the correlations for vorticity diagnosed from SST using the SQG model 
Citation: Journal of Physical Oceanography 44, 5; 10.1175/JPO-D-13-0186.1
Indeed, current observations of SSH and SST have synergetic characteristics. On one side, along-track altimetric measurements of SSHs are very well suited to quantify across-track currents and have been widely used to estimate turbulence spectra in the ocean (e.g., Le Traon et al. 1990; Stammer 1997; Xu and Fu 2011), but their sampling geometry and the number of available platforms strongly constrain the reconstruction of two-dimensional streamfunctions (Pascual et al. 2006). On the other side, SST measurements provided by infrared radiometers are able to locate ocean structures with small phase shifts, if the environmental conditions are appropriate. This suggests to derive the transfer function between simultaneous SST and SSH measurements to obtain its best estimation. Furthermore, along-track SSH measurements could then be used to quantify if the reconstruction is good enough to be used for scientific or operational applications by correlating across-track velocities derived from SSH with those diagnosed from the combination of SSH and SST. Our results have shown that the different response in the transfer function originates from a different slope of SSH (Fig. 8). Such a result should be verified with satellite observations of SSH. However, the main difficulty is the low signal-to-noise ratio of available altimetric measurements in the Mediterranean Sea, which could hide such behavior.
Another approach to exploit the complementary behavior between SST and SSH is to use the information provided by each field to reconstruct the flow at different scales. Indeed, our results reveal that surface velocities can be reconstructed from SST fields in the band between 20 and 100 km with correlations on the order of 0.8, if the ML is deep enough (Fig. 9). Furthermore, the transfer function for this band is close to the SQG predictions (Fig. 7). This suggests that the spatial resolution of SSH observations could be improved by merging the low-resolution altimetric measurements (λ > 100 km) with the high-resolution (20 < λ < 100 km) streamfunctions derived from SST using the SQG model as a first approach or incorporating the along-track SSH spectrum. This idea is similar to the underlying approach proposed by Gaultier et al. (2013). In particular, their procedure is based on extracting high-resolution information from the SST gradients and finite size Lyapunov exponents computed from altimetry (e.g., d’Ovidio et al. 2009) to correct the low-resolution altimetric maps. The underlying concept to retrieve the high-resolution information about the velocity field has some similarities with the technique proposed by Turiel et al. (2005) and Isern-Fontanet et al. (2007). Nevertheless, the method to reconstruct the velocities from this information was different.
As it has been outlined in the introduction, there are alternative methods to derive surface currents from SST. Among different techniques, the maximum cross correlation (MCC) method is by far the most widely used (Emery et al. 1986; Bowen et al. 2002; Barton 2002; Notarstefano et al. 2008). The basic idea consists of computing spatial correlations over windows between consecutive images to estimate their displacement. In general, the MCC method acts poorly in regions of uniform SST and presents some difficulties to estimating the velocity along the front of the scalar field (e.g., Zavialov et al. 2002). Consequently, the resulting velocity fields tend to be sparse, which requires merging of the estimated velocities with other measurements such as altimetry to retrieve the two-dimensional velocity field (Wilkin et al. 2002). Other approaches used to estimate velocities from SST are the constrained optical flow methods to solve the heat conservation equation (e.g., Kelly 1989; Vigan et al. 2000) or variational filter and interpolation techniques (Afanasyev et al. 2002). All these techniques rely on the availability of cloud-free very high-resolution (~1 km) sequences of images over short enough time periods. This latter restriction is imposed by the lack of absolute conservation of SST. These requirements impose strong restrictions on their applicability and geographically and seasonally limit the regions over which velocities can be estimated. On the contrary, our approach based on the combination of SST and SSH is able to provide dense velocity fields from an SST snapshot.
The diagnosis of subsurface fields requires separating the interior and surface contributions due to their different vertical variations. Besides, the empirical determination of the transfer function would be only able to provide the total contribution implying the need to find a criterion to separate both contributions. A possible solution could be the method of Wang et al. (2013), who propose to first determine the surface contribution using surface buoyancy (or SST) through the inversion of the problem given by Eqs. (1) and (2) with Q = 0 and bs ≠ 0 and then find the amplitude of the barotropic and first baroclinic mode by matching to the surface streamfunction (or SSH) resulting from the subtraction of the surface component.
The underlying assumption needed to reconstruct the flow from SST is that 
Mean spatial correlation between surface buoyancy anomaly and PV anomaly in the Levantine Basin for MLD deeper than 70 m (solid line) and MLD shallower than 20 m (dashed line).
Citation: Journal of Physical Oceanography 44, 5; 10.1175/JPO-D-13-0186.1
Some studies have investigated and characterized the phase shift between SST and SSH and the mechanisms associated with it. Hausmann and Czaja (2012) investigated the phase shift between SST and SSH associated with mesoscale eddies in the global ocean. They showed that SST anomalies tend to be westward shifted with respect to the eddy core for both cyclonic and anticylonic eddies and poleward (equatorward) for anticyclonic (cyclonic) eddies, recalling how vortices propagate in the ocean (Cushman-Roisin and Beckers 2011). A qualitative exploration of our simulations also revealed a systematic shift associated with eddies, although our analysis did not allow us to determine any preferred direction. Notice that the propagation paths of Mediterranean vortices form complex patterns due to the interaction between them and the constrains imposed by the topography (Isern-Fontanet et al. 2006b). Besides, the strong dependence on the MLD of the observed phase shifts suggests that it can play a nonnegligible role. Indeed, Klein and Hua (1990) investigated the role of the ML dynamics in the emergence of SST patterns and identified two different regimes. One regime is characterized by the ML deepening that generated the spatial variability of the SST that was a linear combination of subsurface temperatures and relative vorticity depending on the strength of the ML deepening. The other regime, which can be found after the wind stops, is characterized by the advection of the SST by the flow. This process generates fronts and energetic smaller scales.
Our results have shown that it is possible to reconstruct surface currents from SST observations in the Mediterranean Sea. However, the need to minimize the phase shift between SST and SSH implies that the reconstruction is constrained to winter, when less cloud-free images are available. This would force the use of the geostationary satellite to minimize the impact of clouds. Besides, our results have shown that the reconstruction in the Ionian Basin was not possible during most of the analyzed period. Because the performance of the reconstruction from SSB was similar to the results in the other areas, it points to the lower capability of SST to identify SSB patterns.
6. Conclusions
The conditions under which the SST can be used to diagnose surface velocities in the Mediterranean Sea have been identified. It has been shown that the reconstruction of upper-ocean dynamics from SST works better for larger scales than shorter scales and for ML deeper than approximately 70 m. In addition, results have confirmed that the transfer function has a SQG-like behavior for wavelengths shorter than 100 km. However, in most of the studied areas the transfer function differs from the SQG for longer wavelengths and was close to a constant. This implies that SST at these wavelengths can be considered a proxy of the streamfunction in some areas of the Mediterranean Sea. On the other side, our results have pointed to a new approach to improve the estimation of surface currents from satellite observations through the combination of simultaneous SST and SSH measurements.
Acknowledgments
This study is a contribution to the MED3D Project funded by the Spanish R+D+i Plan (CTM2009-11020) and to the MICROVELS Project funded by the Fundación Ramón Areces (CIVP16A1819). JIF is funded through a Ramon y Cajal contract.
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In our notation, a phase shift with no imaginary part has been absorbed into the definition of the transfer function, that is, 