Application of neural networks for efficient calculation of sea water density or salinity from the UNESCO equation of state

We consider two related problems which arise in oceanic modeling and data assimilation. First, in most ocean models, the UNESCO International Equation of State for Seawater (UES) is evaluated at each point of a three-dimensional grid for each time step. The UES gives the expression for the density of seawater as a function of the temperature, the salinity, and the pressure. Mathematically, this function is represented by a ratio of multidimensional polynomials which contain more than 40 parameters. For high resolution models, the solution of this equation consumes a significant part of the overall computation time. Second, in data assimilation systems, the assimilation of temperature into ocean circulation models which employ the full equation of state, without making corresponding adjustments to the salinity, can lead to unstable stratification. To eliminate such problems, salinity also needs to be adjusted. This requires the inversion of the UES to obtain salinity as a function of temperature, density and depth (or pressure). Unfortunately, it is not a simple matter to extract salinity, give temperature and density, from the UES since this represents what is essentially an inverse problem. Here we propose solutions for both problems using neural networks (NNs) - a technique which is well suited for nonlinear modeling and inversion. We have developed an approach for obtaining simpler and faster local parameterizations for seawater density, using neural networks which are excellent devices for approximating multidimensional functions. The neural network (NN) approach has also allowed us to invert the density to obtain salinity as a function of temperature, density and depth. Two NNs were trained which represent density and salinity. All of NNs weights and biases are different for density and for salinity fields. In each case they are determined during the process of training the NN. The NN is 2-3 times faster than USE for estimating the density. Creating density lookup tables for each model level may make the calculations of the sea water density even faster. However, for inversion of UES for salinity, the NN provides a fast and robust solution for which there may be no better alternative. The NN which estimates of salinity have an RMS error of 0.1 psu. In terms of the related error in density, this accuracy corresponds to an RMS error of 0.08 kg m-3. Two particular applications of NNs have been considered here: (1) accelerating numerical estimation of a complicated multi-variate function by modeling this function using a simple NN, and (2) inversion of a multi-variate function using NNs. There are many similar problems (subsystems) in atmospheric and oceanic numerical models, these subsystems also may be accelerated and/or inverted through the application such a NN approach.