We consider two related problems which arise in oceanic modeling and data assimilation. (1) In most ocean models, the UNESCO International Equation of State for Seawater (UES) [1] is evaluated at each point of a three-dimensional grid for each time step. For high-resolution models, the solution of this equation consumes a significant part of the overall computation time. (2) 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 problems in the assimilation process. To perform such an adjustment, we must invert 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, given temperature and density, from the UES since this represents what is essentially an inverse problem. Alternately, we could use a simpler and more readily invertible version of the equation of state for seawater such as that given by [2], but then we sacrifice accuracy, particularly for extreme values. Here we propose solutions for both problems using neural networks (NNs) - a technique which well-suited for nonlinear modeling and inversion.

The UES gives the following expression for the density r (kg/m³) as described in [1],

                      (1)

where r is the density of seawater in kg/m³, T is the temperature in C, S is the salinity in psu, P is the pressure, and K(T,S,P) is the secant bulk modulus [1].

The empirically based UES equation (1) is given over a three-dimensional domain for D = { -2 < T < 40C, 0 < S < 40 psu, and 0 < P < 10000 decibars}. This domain represents all possible combinations of T, S, and Z which are encountered globally. Mathematically, the functions (T,S,0) and K(T,S,P) are represented by multidimensional polynomials and, as a result, the density (1) is the ratio of two, three-dimensional polynomials which contain more than 40 parameters.

Due to the limitations of the UES described above, we have developed an alternate approach for obtaining simpler and faster local parameterizations for (T,S,Z), using neural networks which are excellent devices for approximating multidimensional functions [3]. The neural network (NN) approach has also allowed us to invert the density to obtain salinity as a function of temperature, density and depth, S(T,r,Z).

We have developed regional NN representations for density and salinity for a three-dimensional subdomain, D_S, where D_S = { -2 < T < 35C, 5 < S < 38 psu, and 0 < Z < 5700 m}. In this three-dimensional volume, 4,000 points (T_i, S_i, Z_i) were generated. The UES was used to estimate the density of seawater, r_i, for each point. This simulated data set (r_i; T_i, S_i, Z_i) was used as a proxy for experimental data in order to train the NNs to extract density and salinity. Two expressions were obtained:

                                                                    r = r_NN (T,S,Z)                           (2a)

                                                                    S = S_NN (T,,Z)                             (2b)

where r_NN and S_NN are the NN representations which can be expressed as follows and f = r or S,

                (3)

where the matrix  W_ji, the vectors B_j, w_j, and b collectively represent the weights and biases of the NN, and a and b are scaling parameters. All of these coefficients are different for than they are for S, and r in each case they are determined during the process of training the NN.

The NN for estimating is 2-3 times faster than USE. 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 (2b) provides a fast and robust solution which there is no better alternative to the best of our knowledge. To evaluate the errors in using the NN approach to estimate salinity, we used the UES to estimate the density of seawater, r_i, at 16,000 points (T_i, S_i, Z_i). Then the NN for S_NN (2b) was applied to calculate a new salinity, s_i, using the corresponding values (T_i, r_i, Z_i). Then the differences (S_i - s_i) were utilized to estimate the accuracy of the NN-derived salinities (first line in Table). To further evaluate the quality of the NN-derived salinities, the UES was applied again, this time to the triad (T_i, s_i, Z_i) to recalculate the density of seawater, r'_i. If the NN-obtained values for salinity were perfect, then the density, r'_i, would be equal to r_i. The differences between these two values, (r_i - r'_i), were then used to further estimate the accuracy of the salinity-trained NN to retrieve salinities in terms of the density (second line in Table). Table shows that the NN estimates of salinity (5b) 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.
 

Units	Min error	Max error	Mean error	RMS error
psu	-0.33	0.85	0.00	0.10
Kg m-3	-0.27	0.71	0.00	0.08

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 in meteorology and oceanography which also may benefit through the application of NNs.
 

REFERENCES

[1] Fofonoff, N.P., and R.C. Millard Jr., "Algorithms for computation of fundamental properties of seawater", UNESCO technical paper in marine science 44, UNESCO, 1983

[2] Mamayev, O.I., 1975: Temperature - Salinity Analysis of the World Ocean Waters. Elsevier Scientific, Amsterdam

[3] Chen, T., and H. Chen, "Approximation capability to functions of several variables, nonlinear functionals, and operators by radial basis function neural networks," Neural Networks, vol. 6, pp.904-910, 1995