Spline smoothing of two-dimensional data series with precision estimations applied to satellite navigation
Noise two-dimensional data series are a common occurrence in the geodetic field. As a results of this, numerous algorithms have been formulated to separate the systematic components from the noise within the observed data series. These algorithms, however, are not applicable to all types of data series. In this thesis, the performance of the piecewise cubic function, as a means of smoothing such data series, is investigated. Cubic splines have, in the past, been used a smoothing algorithms. However, they proved ineffective in dealing with two-dimensional data series as they were developed with a weighting scheme for only one of the two variables. Hence they cannot accept a fully populated covariance matrix associated with each observed data point. The spline algorithm developed in this thesis uses both parameterized cubic splines and the method of least-squares to formulate a weighting scheme which allows the incorporation of the two-dimensional covariance matrices of the observations. The resulting spline approximation technique is then used to smooth the navigation data sets of the three ice camps of the Lomonosov Ridge Experiment (LOREX) in the vicinity of the North Pole in 1979. The Navy Navigation Satellite System (Transit) was used as the primary positioning system. Error models evaluating the accuracy of the position fixes using Transit satellites at high latitudes are developed. Smoothed positions and velocities for the three ice camps at one hour intervals are computed for the duration of the expedition. To evaluate the performance of the spline algorithm, the smoothed data series produced by the spline algorithm (DSPLIN) and the real-time smoothing technique (SMOBS) used during LOREX are compared with those generated by the precise dynamic package (GEODOP), i.e. DSPLN versus GEODOP and SMOBS versus GEODOP. The smoothed data series produced by the precise dynamic technique (GEODOP) are hence used a reference standard. In the comparisons between the smoothed data series of positions and velocities for the three ice camps, a reduction of about 56 and 47 percent in the root mean square of the differences in position and velocity respectively, is achieved by DSPLIN over SMOBS. In the same comparisons, the maximum discrepancy between individual smoother positions is reduced by about one-half (i.e. from 3758 m to 1564 m). The computed standardised position differences between the smoothed positions produced by the spline algorithm and the precise dynamic technique shoes that the precision estimates computed by the spline algorithm are consistent with accuracy only over certain periods of time in the LOREX data spans.