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4. Interpolation onto a regular grid 
4.1 Optimal interpolation method 
The optimal interpolation method was used to compute climatological property distributions 
of the selected standard levels on a regular grid. The methodology we use in this study has 
been described previously in the oceanography literature, where it is variously referred to as 
Gauss-Markov, optimal or statistical interpolation, or objective analysis (Gandin, 1963; 
Bretherton et al., 1976). The technique is commonly used to interpolate irregularly sampled, 
noisy data onto regular grids for subsequent analysis. 
The method employs a location-dependent background or first-guess field G. An analysed 
grid-point value F 0 is the first guess evaluated at the grid point plus an interpolated analysis 
error. The latter is an interpolation to the grid-points of the differences between observation 
values and values of the background field at the observing points. Thus an analysed grid- 
point value F 0 is: 
F 0 = 27 Wj(F r Gj) +G a 
The weights w-, are those weights which minimise the ensemble average of the squared 
difference between the analysis value and the true value of the field signal. For any specific 
set of observing sites (i,j) and grid-point (o) locations the first-guess-minus-observation 
differences and the grid-point first-guess error are considered to be stochastic variables with 
a joint statistical distribution for which the covariances are known or can be computed or 
modelled. 
The minimisation gives a set of linear equations for optimal weights w\ 
(A +AI) W = p (1) 
where A is the signal correlation matrix with elements A ts =p(Ax id ), I is the identity matrix, and 
A,=c n 2 /c f 2 , an AXjj represent the spatial separation between points i and j, p,=p(A x i0 ). 
The method requires knowledge of variances of signal and noise and of the spatial 
autocorrelation function p for increment fields. The signal is defined as variability with scales 
larger than the smallest scales of interest. The noise is variability with smaller scales, plus 
random instrumental errors. 
An advantage of the optimal interpolation method is that it also returns an estimate of the 
uncertainty (error variance). The relative error s'depends on the observation locations, and 
on the levels of signal and noise variance: 
So' = (1 -II Poi A~ 1 ijPoj), (2) 
Another important advantage over empirical distance-weighting schemes is that the optimal 
interpolation method takes into account relative separations among the observing sites but 
not only the individual separations between the grid-point and the observing sites. The 
presence of the correlations among input increments in the weight determination algorithm 
controls for redundancy of information from sources whose increments are statistically 
related. 
4.2. Data reduction 
The optimal interpolation requires inversion of the covariance matrix, which becomes 
impractical for a large number of observational points. Usually, only the data points closest to