Temperature and salinity fields are performed by using the objective analysis method (Bretherton, 1975). The procedure used is briefly outlined below:

We consider the problem of calculating the gridded estimate  of a scalar variable  , with referring to the coordinates of the grid , (j=1, n). We use measurements  at a limited number of data points . Thus, the data and the field are defined by:

   , j=1,m

 , i=1,n

where the are zero-mean measurement errors. represent both the instrumental errors and the subgrid variability which are not represented in the estimate but exist in the data. are not correlated with the measurements and with the fields .

A background field (climatlogy or mean) is removed from the original data prior to the analysis, leading to the anomalies:

Thus, the problem consists in estimating from . The solution is:  

The matrix of errors is : 

The matrix is the covariance between the field estimated (at grid point) and the data, is the covariance matrix between all pairs of data and R the covariance matrix of the errors.

In general, the optimal estimate at a data point does not precisely correspond to the measurement there. This discrepancy or residual is given by: