A method to reconstruct missing data in sea surface temperature data using a neural network is presented. Satellite observations working in the optical and infrared bands are affected by clouds, which obscure part of the ocean underneath. In this paper, a neural network with the structure of a convolutional auto-encoder is developed to reconstruct the missing data based on the available cloud-free pixels in satellite images. Contrary to standard image reconstruction with neural networks, this application requires a method to handle missing data (or data with variable accuracy) in the training phase. The present work shows a consistent approach which uses the satellite data and its expected error variance as input and provides the reconstructed field along with its expected error variance as output. The neural network is trained by maximizing the likelihood of the observed value. The approach, called DINCAE (Data INterpolating Convolutional Auto-Encoder), is applied to a 25-year time series of Advanced Very High Resolution Radiometer (AVHRR) sea surface temperature data and compared to DINEOF (Data INterpolating Empirical Orthogonal Functions), a commonly used method to reconstruct missing data based on an EOF (empirical orthogonal function) decomposition. The reconstruction error of both approaches is computed using cross-validation and in situ observations from the World Ocean Database. DINCAE results have lower error while showing higher variability than the DINEOF reconstruction.

DINCAE (Data INterpolating Convolutional Auto-Encoder) is a neural network used to reconstruct missing data (e.g., obscured by clouds or gaps between tracks) in satellite data. Contrary to standard image reconstruction (in-painting) with neural networks, this application requires a method to handle missing data (or data with variable accuracy) already in the training phase. Instead of using a standard L2 (or L1) cost function, the neural network (U-Net type of network) is optimized by minimizing the negative log likelihood assuming a Gaussian distribution (characterized by a mean and a variance). As a consequence, the neural network also provides an expected error variance of the reconstructed field (per pixel and per time instance). In this updated version DINCAE 2.0, the code was rewritten in Julia and a new type of skip connection has been implemented which showed superior performance with respect to the previous version. The method has also been extended to handle multivariate data (an example will be shown with sea surface temperature, chlorophyll concentration and wind fields). The improvement of this network is demonstrated for the Adriatic Sea. Convolutional networks work usually with gridded data as input. This is however a limitation for some data types used in oceanography and in Earth sciences in general, where observations are often irregularly sampled. The first layer of the neural network and the cost function have been modified so that unstructured data can also be used as inputs to obtain gridded fields as output. To demonstrate this, the neural network is applied to along-track altimetry data in the Mediterranean Sea. Results from a 20-year reconstruction are presented and validated. Hyperparameters are determined using Bayesian optimization and minimizing the error relative to a development dataset.

A tool for multidimensional variational analysis (divand) is presented. It allows the interpolation and analysis of observations on curvilinear orthogonal grids in an arbitrary high dimensional space by minimizing a cost function. This cost function penalizes the deviation from the observations, the deviation from a first guess and abruptly varying fields based on a given correlation length (potentially varying in space and time). Additional constraints can be added to this cost function such as an advection constraint which forces the analysed field to align with the ocean current. The method decouples naturally disconnected areas based on topography and topology. This is useful in oceanography where disconnected water masses often have different physical properties. Individual elements of the a priori and a posteriori error covariance matrix can also be computed, in particular expected error variances of the analysis. A multidimensional approach (as opposed to stacking two-dimensional analysis) has the benefit of providing a smooth analysis in all dimensions, although the computational cost is increased. Primal (problem solved in the grid space) and dual formulations (problem solved in the observational space) are implemented using either direct solvers (based on Cholesky factorization) or iterative solvers (conjugate gradient method). In most applications the primal formulation with the direct solver is the fastest, especially if an a posteriori error estimate is needed. However, for correlated observation errors the dual formulation with an iterative solver is more efficient. The method is tested by using pseudo-observations from a global model. The distribution of the observations is based on the position of the Argo floats. The benefit of the three-dimensional analysis (longitude, latitude and time) compared to two-dimensional analysis (longitude and latitude) and the role of the advection constraint are highlighted. The tool divand is free software, and is distributed under the terms of the General Public Licence (GPL) (http://modb.oce.ulg.ac.be/mediawiki/index.php/divand).

HF radar measurements are used to optimize surface wind forcing and baroclinic open boundary condition forcing in order to constrain model coastal surface currents. This method is applied to a northwestern Mediterranean (NWM) regional primitive equation model configuration. A new radar data set, provided by two radars deployed in the Toulon area (France), is used. To our knowledge, this is the first time that radar measurements of the NWM Sea are assimilated into a circulation model. Special attention has been paid to the improvement of the model coastal current in terms of speed and position. The data assimilation method uses an ensemble Kalman smoother to optimize forcing in order to improve the model trajectory. Twin experiments are initially performed to evaluate the method skills. Real measurements are then fed into the circulation model and significant improvements to the modeled surface currents, when compared to observations, are obtained.