Nonlinear least-squares method via an isomorphic geometrical setup |
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| Authors: | Georges Blaha Robert P Bessette |
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| Institution: | (1) Nova University Oceanographic Center, 8000 North Ocean Drive, 33004 Dania, Florida, USA;(2) Air Force Geophysics Laboratory, Hanscom Air Force Base, 01731 Bedford, Massachusetts, USA |
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| Abstract: | The resolution of a nonlinear parametric adjustment model is addressed through an isomorphic geometrical setup with tensor
structure and notation, represented by a u-dimensional “model surface” embedded in a flat n-dimensional “observational space”.
Then observations correspond to the observational-space coordinates of the pointQ, theu initial parameters correspond to the model-surface coordinates of the “initial” pointP, and theu adjusted parameters correspond to the model-surface coordinates of the “least-squares” point
. The least-squares criterion results in a minimum-distance property implying that the vector
Q must be orthogonal to the model surface. The geometrical setup leads to the solution of modified normal equations, characterized
by a positive-definite matrix. The latter contains second-order and, optionally, thirdorder partial derivatives of the observables
with respect to the parameters. This approach significantly shortens the convergence process as compared to the standard (linearized)
method. |
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