# Ocean Currents Reconstruction from a Combination of
## Data and Methods
### Methods: Rationale of the Optimal Reconstruction  
The ocean surface currents are computed from synthetic altimetry measurements combined with higher resolution synthetic satellite Chl observations, relying on the algorithm firstly developed by [PERSON] [26] and [PERSON] et al. [27]. The principle is to constrain the surface geostrophic flow to follow the evolution of a surface tracer, assuming the tracer can be derived independently and that it is characterized by higher spatial-temporal resolutions than the geostrophic flow, like in our study. With this approach, we attempt to extract useful dynamical information from the surface oceanic tracer evolution to enhance the spatial-temporal content of the geostrophic currents. For clarity, the derivation of the optimal currents is reminded in this section. The sea surface currents are inferred from (3):  
\[\frac{\partial\mathrm{Chl}}{\partial\mathrm{t}}+\mathrm{u}\frac{\partial \mathrm{Chl}}{\partial\mathrm{x}}+\mathrm{v}\frac{\partial\mathrm{Chl}}{ \partial\mathrm{y}}=\mathrm{F} \tag{3}\]  
In (3):  
* (x,y) are the zonal and the meridional directions;
* (u,v) are respectively the zonal and meridional components of the ocean surface flow;
* \(\mathrm{F}\) is the forcing term, representing the Chl source and sinks which, in the present case, include both biological and environmental factors. In particular, \(\mathrm{F}\) includes contributions from the marine currents' vertical advection, horizontal and vertical diffusion as well as the biogeochemical reactions involved in the phytoplankton dynamics [47]. The contributors to the source/sink terms can have different relative magnitudes according to space and time. For example, we expect vertical advection to be dominant under \begin{table}
\begin{tabular}{c c} \hline \hline
**Mission** & **Noise Measurement Error** \\ \hline Jason-3 & 2.4 \\ Sentinel-3A & 2.1 \\ SARAL/AltiKa & 1.75 \\ Cryosat-2 & 2.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mean 1 Hz noise measurement observed for the different altimeters on along-track (L3) DUACS products in the Mediterranean Sea. Noise for raw SLA is indicated in cm rms. Figure 2: Workflow of the OSSE, covering the entire year 2017.  
high wind stress conditions that may trigger upwelling currents or in the presence of mesoscale/submesoscale features generating strong vertical motions [48; 49]. In addition, biogeochemical reactions become dominant during the so-called \"Chl-blooming periods\", i.e., when the near-surface Chl production increases due to an abundance of light and nutrients (e.g., inorganic carbon dioxide, silica, nitrates and phosphorus) involved in the phytoplankton photosynthetic activity [37; 38]. PIT09 [26] derived the expressions of an optimized flow field accounting for the merged contribution of a large-scale, background flow with the dynamical information contained in high-resolution tracers. If the background flow is given by the Synthetic Altimeter Currents (SAC) and the high-resolution tracer by Chl, the zonal and meridional OPtimal Currents (OPC) can be expressed by (4):  
\[\mathrm{u_{OPC}=u_{SAC}-\frac{A(Au_{SAC}+Bv_{SAC}+E)}{A^{2}+B^{2}} =u_{SAC}+u_{CORR}\] \[\mathrm{v_{OPC}=v_{SAC}-\frac{B(Au_{SAC}+Bv_{SAC}+E)}{A^{2}+B^{2}} =v_{SAC}+v_{CORR} \tag{4}\]  
where A = \(\partial_{x}\)Chl, B = \(\partial_{y}\)Chl, E = \(\partial_{t}\)Chl - F. In (4), the subscript \"\(\mathrm{t}\)\" stands for temporal derivative and the subscripts \"x,y\" respectively indicate the spatial derivative in the zonal and meridional directions. Equation (4) expresses the PIT09 method rationale: the geostrophic currents are corrected by means of a factor \((u_{CORR},v_{CORR})\) depending on the spatio-temporal derivatives of the high-resolution tracer observations and on the forcing term (F) regulating the tracer dynamical evolution. Equation (4) is valid when the forcing term F is known perfectly, like in the present study where F is derived from Equation (3).