12.2.1.2 Weighted least-squares fitting  12.2.1 Surface fitting  12.2.1 Surface fitting

12.2.1.1 ORDINARY LEAST SQUARES FITTING

The general problem of fitting functions to data located in a latitude $(\lambda)$ - longitude $(\phi)$ coordinate system can be expressed as follows. Let $S(\lambda_j,\phi_j)$ be an observation at location j and suppose that we can represent the observations at m locations by

$$S(\lambda_j,\phi_j)=\sum_i^n f_i(\lambda_j,\phi_j) \Theta_i+ r_j \qquad for \qquad j = 1,2,...,m; m > n$$ (12.3)

 

where f_i, i = 1,2,...,n are known functions (usually polynomials), and r_j is the error of approximation. The parameter to be estimated is $\Theta_i$. The matrix equivalent of the above expression is

$$S = F \Theta + r$$ (12.4)

 

where S is column vector of length m, F is an $m \timesn$ matrix, $\Theta$ is a column vector of length n, and r is a column vector of length m. Our analysis will be $S^a = F \Theta$. We find $\Theta$ by requiring that $(r^Tr) = \sum_j^m r_j^2$ be a minimum. This is equivalent to the requirement that

$$\sum_j^m [S^a(\lambda_j,\phi_j) - S(\lambda_j,\phi_j]^2 =\su......n f_i(\lambda_j,\phi_j) \Theta_i -S(\lambda_j,\phi_j)\Bigr]^2$$ (12.5)

 

be a minimum. The solution is

$$\Theta = (F^T F)^{-1} F^T S$$ (12.6)

 

where T means matrix transpose and -1 means matrix inverse. (F^T F)^-1 F^T is called a pseudo-inverse because F is not square.

   
 12.2.1.2 Weighted least-squares fitting  12.2.1 Surface fitting  12.2.1 Surface fitting 

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