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/m^{3}) as described in [1],

where r is the density of seawater in kg/m^{3},
*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

r
= r* _{NN}* (

*
S* = *S _{NN}* (

where r* _{NN}* and

where the matrix W* _{ji}*,
the vectors B

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

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 |

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