## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data: Surface Measurements

#### 2.2. Data: In Situ Vertical Profiles

#### 2.3. Data: Climatology

#### 2.4. Methods: Multivariate Empirical Orthogonal Function Reconstruction (mEOF-r)

#### 2.5. Methods: Feed-Forward Neural Networks

#### 2.6. Methods: Long Short-Term Memory Networks

_{i}) is concatenated to the previous cell hidden state (h

_{i}

_{−1}) and then passed through different “gates”, each one aimed at carrying out a specific task to update both the hidden state itself (h

_{i}) and a cell state (C

_{i}), that is directly transmitted to the next cell and basically acts as a network “memory”. The LSTM cell specifically includes a forget gate, an input gate, and an output gate, as depicted in Figure 2a: whose equations thus read:

#### 2.7. Monte-Carlo Dropout

#### 2.8. Code Availability

## 3. Results

## 4. Discussion

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Stukel, M.R.; Aluwihare, L.I.; Barbeau, K.A.; Chekalyuk, A.M.; Goericke, R.; Miller, A.J.; Ohman, M.D.; Ruacho, A.; Song, H.; Stephens, B.M.; et al. Mesoscale ocean fronts enhance carbon export due to gravitational sinking and subduction. Proc. Natl. Acad. Sci. USA
**2017**, 114, 1252–1257. [Google Scholar] [CrossRef] [PubMed] - McWilliams, J.C. A survey of submesoscale currents. Geosci. Lett.
**2019**, 6. [Google Scholar] [CrossRef] - Pilo, G.S.; Oke, P.R.; Coleman, R.; Rykova, T.; Ridgway, K. Patterns of vertical velocity induced by Eddy Distortion in an ocean model. J. Geophys. Res. Oceans
**2018**, 123, 2274–2292. [Google Scholar] [CrossRef] - Carrassi, A.; Bocquet, M.; Bertino, L.; Evensen, G. Data assimilation in the geosciences: An overview of methods, issues, and perspectives. Wiley Interdiscip. Rev. Clim. Chang.
**2018**, 9, 1–50. [Google Scholar] [CrossRef] - Moore, A.M.; Martin, M.J.; Akella, S.; Arango, H.G.; Balmaseda, M.; Bertino, L.; Ciavatta, S.; Cornuelle, B.; Cummings, J.; Frolov, S.; et al. Synthesis of ocean observations using data assimilation for operational, real-time and reanalysis systems: A more complete picture of the state of the ocean. Front. Mar. Sci.
**2019**, 6, 1–6. [Google Scholar] [CrossRef] - Stammer, D.; Balmaseda, M.; Heimbach, P.; Köhl, A.; Weaver, A. Ocean data assimilation in support of climate applications: Status and perspectives. Ann. Rev. Mar. Sci.
**2016**, 8. [Google Scholar] [CrossRef] [PubMed] - Forget, G.; Campin, J.M.; Heimbach, P.; Hill, C.N.; Ponte, R.M.; Wunsch, C. ECCO Version 4: An integrated framework for non-linear inverse modeling and global ocean state estimation. Geosci. Model Dev.
**2015**, 8, 3071–3104. [Google Scholar] [CrossRef] - Rio, M.-H.; Santoleri, R.; Bourdalle-Badie, R.; Griffa, A.; Piterbarg, L.; Taburet, G. Improving the altimeter-derived surface currents using high-resolution sea surface temperature data: A feasability study based on model outputs. J. Atmos. Ocean. Technol.
**2016**, 2769–2784. [Google Scholar] [CrossRef] - Ubelmann, C.; Cornuelle, B.D.; Fu, L.-L. Dynamic mapping of along-track ocean altimetry: Method and performance from observing system simulation experiments. J. Atmos. Ocean. Technol.
**2016**, 33, 1691–1699. [Google Scholar] [CrossRef] - Ciani, D.; Rio, M.; Buongiorno Nardelli, B.; Etienne, H.; Santoleri, R. Improving the altimeter-derived surface currents using sea surface temperature (SST) data: A sensitivity study to SST products. Remote Sens.
**2020**, 12, 1601. [Google Scholar] [CrossRef] - Guinehut, S.; Le Traon, P.Y.; Larnicol, G.; Philipps, S. Combining argo and remote-sensing data to estimate the ocean three-dimensional temperature fields—A first approach based on simulated observations. J. Mar. Syst.
**2004**, 46, 85–98. [Google Scholar] [CrossRef] - Guinehut, S.; Dhomps, A.-L.; Larnicol, G.; Le Traon, P.-Y. High resolution 3-D temperature and salinity fields derived from in situ and satellite observations. Ocean Sci.
**2012**, 8, 845–857. [Google Scholar] [CrossRef] - Uchiyama, Y.; Kanki, R.; Takano, A.; Yamazaki, H.; Miyazawa, Y. Mesoscale reproducibility in regional ocean modelling with a three-dimensional stratification estimate based on aviso-argo data. Atmos. Ocean
**2018**, 56, 212–229. [Google Scholar] [CrossRef] - Hutchinson, K.; Swart, S.; Meijers, A.; Ansorge, I.; Speich, S. Decadal-Scale thermohaline variability in the atlantic sector of the southern ocean. J. Geophys. Res. Oceans
**2016**, 121, 3171–3189. [Google Scholar] [CrossRef] - Meijers, A.J.S.; Bindoff, N.L.; Rintoul, S.R. Estimating the Four-Dimensional structure of the southern ocean using satellite altimetry. J. Atmos. Ocean. Technol.
**2011**, 28, 548–568. [Google Scholar] [CrossRef] - Meinen, C.; Watts, D. Vertical structure and transport on a transect across the North Atlantic current near 42°N: Time Series and Mean. J. Geophys. Res. Oceans
**2000**, 105, 21869–21891. [Google Scholar] [CrossRef] - Buongiorno Nardelli, B.; Guinehut, S.; Verbrugge, N.; Cotroneo, Y.; Zambianchi, E.; Iudicone, D. Southern ocean mixed-layer seasonal and interannual variations from combined satellite and in situ data. J. Geophys. Res. Oceans
**2017**, 122, 10042–10060. [Google Scholar] [CrossRef] - Buongiorno Nardelli, B.; Mulet, S.; Iudicone, D. Three Dimensional Ageostrophic Motion and Water Mass Subduction in the Southern Ocean. J. Geophys. Res. Oceans
**2018**, 23, 1533–1562. [Google Scholar] [CrossRef] - Buongiorno Nardelli, B.; Guinehut, S.; Pascual, A.; Drillet, Y.; Ruiz, S.; Mulet, S. Towards high resolution mapping of 3-d mesoscale dynamics from observations. Ocean Sci.
**2012**, 8, 885–901. [Google Scholar] [CrossRef] - Jeong, Y.; Hwang, J.; Park, J.; Jang, C.J.; Jo, Y.H. Reconstructed 3-D ocean temperature derived from remotely sensed sea surface measurements for mixed layer depth analysis. Remote Sens.
**2019**, 11, 3018. [Google Scholar] [CrossRef] - Takano, A.; Yamazaki, H.; Nagai, T.; Honda, O. A method to estimate three-dimensional thermal structure from satellite altimetry data. J. Atmos. Ocean. Technol.
**2009**, 26, 2655–2664. [Google Scholar] [CrossRef] - Buongiorno Nardelli, B. A multi-year timeseries of observation-based 3D horizontal and vertical quasi-geostrophic global ocean currents. Earth Syst. Sci. Data
**2020**, 12, 1711–1723. [Google Scholar] [CrossRef] - Mulet, S.; Rio, M.-H.; Mignot, A.; Guinehut, S.; Morrow, R. A new estimate of the global 3D geostrophic ocean circulation based on satellite data and in-situ measurements. Deep Sea Res. Part II Top. Stud. Oceanogr.
**2012**, 77–80, 70–81. [Google Scholar] [CrossRef] - Liu, L.E.I.; Xue, H.; Sasaki, H. Reconstructing the ocean interior from high-resolution sea surface information. J. Phys. Oceanogr.
**2019**, 49, 3245–3262. [Google Scholar] [CrossRef] - Lapeyre, G. Surface quasi-geostrophy. Fluids
**2017**, 2, 7. [Google Scholar] [CrossRef] - LaCasce, J.H.; Wang, J. Estimating subsurface velocities from surface fields with idealized stratification. J. Phys. Oceanogr.
**2015**, 45, 2424–2435. [Google Scholar] [CrossRef] - Wang, J.; Flierl, G.; LaCasce, J.H.; McClean, J.L.; Mahadevan, A. Reconstructing the ocean’s interior from surface data. J. Phys. Oceanogr.
**2013**, 43, 1611–1626. [Google Scholar] [CrossRef] - Isern-Fontanet, J.; Hascoët, E. Diagnosis of high-resolution upper ocean dynamics from noisy sea surface temperatures. J. Geophys. Res. Oceans
**2014**, 119, 121–132. [Google Scholar] [CrossRef] - Fresnay, S.; Ponte, A.L.; Le Gentil, S.; Le Sommer, J. Reconstruction of the 3-D dynamics from surface variables in a high-resolution simulation of North Atlantic. J. Geophys. Res. Oceans
**2018**, 123, 1612–1630. [Google Scholar] [CrossRef] - Yan, H.; Wang, H.; Zhang, R.; Chen, J.; Bao, S.; Wang, G. A Dynamical-statistical approach to retrieve the ocean interior structure from surface data: SQG-mEOF-R. J. Geophys. Res. Oceans
**2020**. [Google Scholar] [CrossRef] - Lu, W.; Su, H.; Yang, X.; Yan, X.H. Subsurface temperature estimation from remote sensing data using a clustering-neural network method. Remote Sens. Environ.
**2019**, 229, 213–222. [Google Scholar] [CrossRef] - Bao, S.; Zhang, R.; Wang, H.; Yan, H.; Yu, Y.; Chen, J. Salinity Profile estimation in the pacific ocean from satellite surface salinity observations. J. Atmos. Ocean. Technol.
**2019**, 36, 53–68. [Google Scholar] [CrossRef] - Wu, X.; Yan, X.H.; Jo, Y.H.; Liu, W.T. Estimation of subsurface temperature anomaly in the North Atlantic Using a self-organizing map neural network. J. Atmos. Ocean. Technol.
**2012**, 29, 1675–1688. [Google Scholar] [CrossRef] - Sammartino, M.; Marullo, S.; Santoleri, R.; Scardi, M. Modelling the vertical distribution of phytoplankton biomass in the mediterranean sea from satellite data: A neural network approach. Remote Sens.
**2018**, 10, 1666. [Google Scholar] [CrossRef] - Gueye, M.B.; Niang, A.; Arnault, S.; Thiria, S.; Crépon, M. Neural Approach to inverting complex system: Application to ocean salinity profile estimation from surface parameters. Comput. Geosci.
**2014**, 72, 201–209. [Google Scholar] [CrossRef] - Su, H.; Yang, X.; Lu, W.; Yan, X.-H. Estimating subsurface thermohaline structure of the global ocean using surface remote sensing observations. Remote Sens.
**2019**, 11, 1598. [Google Scholar] [CrossRef] - Su, H.; Wu, X.; Yan, X.H.; Kidwell, A. Estimation of subsurface temperature anomaly in the indian ocean during recent global surface warming hiatus from satellite measurements: A support vector machine approach. Remote Sens. Environ.
**2015**, 160, 63–71. [Google Scholar] [CrossRef] - Bittig, H.C.; Steinhoff, T.; Claustre, H.; Fiedler, B.; Williams, N.L.; Sauzède, R.; Körtzinger, A.; Gattuso, J.P. An alternative to static climatologies: Robust estimation of open ocean CO2 variables and nutrient concentrations from T, S, and O2 data using bayesian neural networks. Front. Mar. Sci.
**2018**, 5, 1–29. [Google Scholar] [CrossRef] - Sauzède, R.; Bittig, H.C.; Claustre, H.; de Fommervault, O.P.; Gattuso, J.P.; Legendre, L.; Johnson, K.S. Estimates of Water-column nutrient concentrations and carbonate system parameters in the global ocean: A novel approach based on neural networks. Front. Mar. Sci.
**2017**, 4, 1–17. [Google Scholar] [CrossRef] - Ballabrera-Poy, J.; Mourre, B.; Garcia-Ladona, E.; Turiel, A.; Font, J. Linear and non-linear T-S models for the Eastern North Atlantic from argo data: Role of surface salinity observations. Deep. Res. Part I Oceanogr. Res. Pap.
**2009**, 56, 1605–1614. [Google Scholar] [CrossRef] - Hochreiter, S.; Schmidhuber, J. Long short-term memory. Neural Comput.
**1997**, 9, 1735–1780. [Google Scholar] [CrossRef] [PubMed] - Gal, Y.; Ghahramani, Z. Dropout As a Bayesian Approximation: Representing Model Uncertainty in Deep Learning. In Proceedings of the 33rd International Conference On Machine Learning, New York, NY, USA, 19–24 June 2016; Volume 48, pp. 1050–1059. [Google Scholar]
- Roberts-Jones, J.; Fiedler, E.K.; Martin, M.J. Daily, Global, high-resolution SST and sea ice reanalysis for 1985–2007 Using the OSTIA System. J. Clim.
**2012**, 25, 6215–6232. [Google Scholar] [CrossRef] - Buongiorno Nardelli, B. ESA-WOC North Atlantic Sea Surface Salinity maps from a multivariate combination of satellite and in situ surface measurements (2010–2018) (Version v1.0), [Data set]. Zenodo
**2020**. [Google Scholar] [CrossRef] - Droghei, R.; Buongiorno Nardelli, B.; Santoleri, R. Combining in-situ and satellite observations to retrieve salinity and density at the ocean surface. J. Atmos. Ocean. Technol.
**2016**, 33, 1211–1223. [Google Scholar] [CrossRef] - Buongiorno Nardelli, B. A Novel approach for the high-resolution interpolation of in situ sea surface salinity. J. Atmos. Ocean. Technol.
**2012**, 29, 867–879. [Google Scholar] [CrossRef] - Droghei, R.; Buongiorno Nardelli, B.; Santoleri, R. A new global sea surface salinity and density dataset from multivariate observations (1993–2016). Front. Mar. Sci.
**2018**, 5, 1–13. [Google Scholar] [CrossRef] - Rio, M.; Mulet, S.; Picot, N. Beyond GOCE for the ocean circulation estimate: Synergetic use of altimetry, gravimetry, and in situ data provides new insight into geostrophic and ekman currents. Geophys. Res. Lett.
**2014**, 41, 8918–8925. [Google Scholar] [CrossRef] - Szekely, T.; Gourrion, J.; Pouliquen, S.; Reverdin, G. The CORA 5.2 dataset for global in situ temperature and salinity measurements: Data description and validation. Ocean Sci.
**2019**, 15, 1601–1614. [Google Scholar] [CrossRef] - Locarnini, R.A.; Mishonov, A.V.; Antonov, J.I.; Boyer, T.P.; Garcia, H.E.; Baranova, O.K.; Zweng, M.M.; Paver, C.R.; Reagan, J.R.; Johnson, D.R.; et al. World Ocean Atlas 2013. Volume 1: Temperature; Levitus, S., Mishonov, A., Eds.; NODC: Silver Spring, MD, USA, 2013; Volume 73, p. 40. [Google Scholar]
- Zweng, M.M.; Reagan, J.R.; Antonov, J.I.; Mishonov, A.V.; Boyer, T.P.; Garcia, H.E.; Baranova, O.K.; Johnson, D.R.; Seidov, D.; Bidlle, M.M. World Ocean Atlas 2013, Volume 2: Salinity; NODC: Silver Spring, MD, USA, 2013; Volume 119, pp. 227–237. [Google Scholar]
- Buongiorno Nardelli, B.; Cavalieri, O.; Rio, M.-H.; Santoleri, R. Subsurface geostrophic velocities inference from altimeter data: Application to the Sicily Channel (Mediterranean Sea). J. Geophys. Res.
**2006**, 111. [Google Scholar] [CrossRef] - Buongiorno Nardelli, B.; Santoleri, R. Methods for the reconstruction of vertical profiles from surface data: Multivariate analyses, residual GEM, and variable temporal signals in the North Pacific Ocean. J. Atmos. Ocean. Technol.
**2005**, 22, 1762–1781. [Google Scholar] [CrossRef] - Buongiorno Nardelli, B. Vortex waves and vertical motion in a mesoscale cyclonic eddy. J. Geophys. Res. Oceans
**2013**, 118, 5609–5624. [Google Scholar] [CrossRef] - Hinton, G.E.; Srivastava, N.; Krizhevsky, A.; Sutskever, I.; Salakhutdinov, R.R. Improving neural networks by preventing co-adaptation of feature detectors. arXiv
**2012**, arXiv:1207.0580. [Google Scholar] - Srivastava, N.; Hinton, G.; Krizhevsky, A.; Sutskeverm, I.; Salakhutdinov, R. Dropout: A simple way to prevent neural networks from overfitting. J. Mach. Learn. Res.
**2014**, 15, 1929–1958. [Google Scholar] - Buongiorno Nardelli, B. Developing a deep Learning network to retrieve ocean hydrographic profiles in the North Atlantic from combined satellite and in situ measurements: Test datasets. (Version v1.0), [Data set]. Zenodo
**2020**. [Google Scholar] [CrossRef] - Efron, B.; Tibshirani, R.J. An Introduction to the Bootstrap; Chapman & Hall/CRC: New York, NY, USA, 1993; Volume 153. [Google Scholar]

**Figure 1.**Examples of the surface daily data taken as input to the reconstruction techniques: OSTIA L4 reprocessed SST (

**a**), SSS L4 developed within ESA-WOC project (

**b**), adjusted ADT L4 derived from DUACS data (

**c**).

**Figure 2.**Diagram showing the elements of a single LSTM cell (

**a**). Stacked LSTM model for the reconstruction of vertical hydrographic profiles (

**b**).

**Figure 3.**RMSD between temperature (

**a**) and salinity (

**b**) climatological and reconstructed profiles, estimated from independent test data. RMSD confidence intervals (one $\sigma $, displayed here as shadowed areas) have been estimated through a Monte Carlo approach—i.e., as the standard deviation of the statistics computed from 1000 resampling with replacement [58].

**Figure 4.**Absolute value of the differences between mEOF-r temperature (

**a**) and salinity (

**b**) and observed test data at 100 m depth, and corresponding LSTM (35–35) differences (

**c**,

**d**). Temperature (

**e**) and salinity (

**f**) predicted LSTM (35–35) reconstruction error at 100 m.

**Figure 5.**Absolute value of the differences between mEOF-r temperature (

**a**) and salinity (

**b**) and observed test data at 500 m depth, and corresponding LSTM (35–35) differences (

**c**,

**d**). Temperature (

**e**) and salinity (

**f**) predicted LSTM (35–35) reconstruction error at 500 m.

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