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NEW GEOID FOR THREE ORDNANCE SURVEYS

The purpose of this paper is to demonstrate, using a levelling network, how a least squares adjustment is possible by reparameterising the unknowns into a smaller set of parameters. This technique is based on the well-known result that, if the design matrix X is not of full column rank, the parameters cannot be estimated without imposing a priori constraints (or datum) on the network. However, certain linear cOlnbinations of the parameters are estimable without the imposition of datum constraints. Reparameterisation therefore differs from minimal constraints and inner constraints since the datum is not defined. It also differs from the conventional imposition of side conditions on the residuals. Reparameterisation is a parametric method, and no conditions are imposed on the residuals. Finally, unlike the pseudo-inverse approach, it attempts to reparameterise the parameters into a smaller but estimable set rather than estimate parameters of minimum norm. The method is more appealing if linear combinations of the unknowns (that is, vector g), rather than the unknown parameters themselves, are required. In the levelling example, only height differences that are linear combinations of the absolute heights, rather than the heights themselves, are of interest.