N-dimensional B-spline surface estimated by lofting for locally improving IRI
N-dimensional surfaces are defined by the tensor product of B-spline basis functions. To estimate the unknown control points of these B-spline surfaces, the lofting method also called skinning method by cross-sectional curve fits is applied. It is shown by an analytical proof and numerically confirmed by the example of a four-dimensional surface that the results of the lofting method agree with the ones of the simultaneous estimation of the unknown control points. The numerical complexity for estimating vn control points by the lofting method is O(vn+1) while it results in O(v3n) for the simultaneous estimation. It is also shown that a B-spline surface estimated by a simultaneous estimation can be extended to higher dimensions by the lofting method, thus saving computer time.
An application of this method is the local improvement of the International Reference Ionosphere (IRI), e.g. by the slant total electron content (STEC) obtained by dual-frequency observations of the Global Navigation Satellite System (GNSS). Three-dimensional B-spline surfaces at different time epochs have to be determined by the simultaneous estimation of the control points for this improvement. A four-dimensional representation in space and time of the electron density of the ionosphere is desirable. It can be obtained by the lofting method. This takes less computer time than determining the four-dimensional surface solely by a simultaneous estimation.
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