12.2.1.4 ADVANTAGES AND DISADVANTAGES  12.2.1 Surface fitting  12.2.1.2 WEIGHTED LEAST SQUARES

12.2.1.3 OTHER METHODS OF SURFACE FITTING

The system of equations

$$\Theta = (F^T F)^{-1} F^T S \qquad or \qquad \Theta = (F^TB F)^{-1} F^T B S$$ (12.9)

 

is often numerically ill-conditioned (not stable numerically).

Moreover, some of the higher order polynomial components may account for very little of the observed variability in S. These problems can be avoided through the use of orthogonal polynomials $g_i(\lambda_j,\phi_j)$ (Dixon et al. 1972), where $g_ig_j = \delta_{ij}$, instead of the functions $f_i(\lambda_j,\phi_j)$ in the representation (12.1), such that

$${rl}S(\lambda_j,\phi_j) =\sum_i^ng_i(\lambda_j,\phi_j)\beta_i + r_j$$

$$= G \beta + r$$ (12.10)

 

Ordering the columns of G according to the contribution of each component vector to the total variance of S gives control over the amount of detail to be retained in the analysis. After ordering, as terms are successively dropped from the end of the summation, the analysis becomes more and more smooth.

• Splines: Although polynomials of sufficiently high degree can be made to pass through all the data points, the resulting surfaces are not the smoothest possible. Since most meteorologists prefer the smoothest possible analysis consistent with the data, the use of splines has become more popular, starting with Fritsch (1971).
• Cross-validation technique (Wahba and Wendelbeger, 1980): A much more sophisticated form of surface fitting, involving splines. This method allows one to control the smoothing and filtering properties of the analysis while also imposing dynamical constraints.

   
 12.2.1.4 ADVANTAGES AND DISADVANTAGES  12.2.1 Surface fitting  12.2.1.2 WEIGHTED LEAST SQUARES