Nonlinear Parametric Least-Squares Adjustment.

Abstract

Abstract : A parametric adjustment model expresses n observables in terms of u parameters, where the structure linking the two groups is in general nonlinear. This model can be expanded in the Taylor series based on an initial point P. The standards procedure includes only the linear terms, constituting a known linearized model at P which is resolved upon applying the least-squares (L.S.) criterion . If necessary, the solution is iterated, i.e., the initial point in a given step is revised using the L.S. result from the previous step. In the present study, the resolution of a nonlinear adjustment model is addressed through an isomorphic geometrical setup with tensor structure and notations, represented by a u-dimensional model surface embedded in an n-dimensional observational space . In this geometrical context, the observations correspond to the observational-s pace coordinates of the point called Q, and the adjusted parameters correspond to the model-surface coordinates of the point called P. The L.S. criterion implies that the vector PQ must be orthogonal to the model surface. Keywords: Least squares method, Nonlinear model, Variance covariances, Weights, Tensor analysis, Observational space, Model surface, Coordinate system, Covariant components, Contravariant components, Metric tensor, Associated metric tensor.