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The adjustment of systems of highly non-linear, redundant equations   , deriving from observations of certain geophysical processes and geodetic data cannot be based on conventional least-squares techniques, and is based on various numerical inversion techniques. Still these techniques lead to solutions trapped in local minima, to correlated estimates and to solution with poor error control. To overcome these problems, we propose an alternative numerical-topological approach inspired by lighthouse beacon navigation, usually used in 2-D, low-accuracy applications. In our approach, an  m -dimensional grid  G  of points around the real solution (an  m -dimensional vector) is at first specified. Then, for each equation    an uncertainty    is assigned to the corresponding measurement   , and the sets    of the grid points which satisfy the condition    are detected. This process is repeated for all equations, and the common section  A  of the sets of grid points    is defined. From this set of grid points, which define a space including the real solution, we compute its center of weight, which corresponds to an estimate of the solution, and its variance–covariance matrix. An optimal solution can be obtained through optimization of the uncertainty    in each observation. The efficiency of the overall process was assessed in comparison with conventional least squares adjustment.

Adjustment of highly non-linear redundant systems of equations using a numerical, topology-based approach