Abstract
Let the configuration or first-order design matrix A of a geodetic net be given and let the weight or second-order design matrix P of m observations be unknown. For some predetermined choice of the dispersion matrix Σ of the n unknown net coordinates, the unique minimum Euclidean norm solution for the weight matrix P is P = (A′)+Σ−1A+. A′ indicates the transpose and A+ the pseudoinverse of the (in general) rectangular matrix A. For a general first-order design matrix A and a given Σ, there does not exist a solution for a diagonal positive definite weight matrix P. Our solution for P belongs to the first category of an optimal design of C. R. Rao. The designs P = I and P = (A′)+Σ−1A+ yield the same best linear unbiased estimator for the unknowns, but the variance covariance matrix for these designs is different. Examples, especially the structure by G. I. Taylor and T. Karman for the homogeneous and isotropic geodetic net, illustrate theoretical results.
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