Stability of the Solutions for the Positions of Earth Surface Stations Determined by Doppler-Satellite Short-arc Method

With problems in geodesy and surveying,if the position of stations can be determined many,many times and adjusted by the sequential method,the results can be proved to be the same as with Kalman filter.In fact,with such problems sequential adjustment and Kalman filter can be proved to be one the same thing.Since in the short-arc Doppler satellite positioning computations,the positions of the satellites and ground points are all considered known only approximately,and corrections to these unknowns are sought,thus they are in nature same as the so-called "Free network adjustment",the problem can be solved by first making the sum of the squares of the observed residuals minimum and then make the sum of the squares of the corrections to the unknowns(corrections to its ground points) minimum.Since the minimum requirements are made one after another,the auther suggests this method may be called the method of separate minimization.This method has been discussed by many geodetists of the world.With Doppler satellite positioning adjustments,we can solve this problem by requiring the sum of the residuals of observations and the sum of squares of the corrections to the approximate initial Positions of the stations minimum at the same time.This method,the auther calls the method of sum minimization.Viewed from the theory of Kalman filter,the method of semishort-arc sum-minimization is uniformally completely controlable.And it is proaved in this paper it is also uniformally completely observable.Then from the theory of Kalman filter,the solutions in this case,the position of the ground points are uniformally asymtotic stable.That is,however large the error of the approximate initial points may be,when the number of times of observing the NNSS passes are large enough,the erronous initial points will converge to the theoretical best estimate,i.e.best point positions.An obvious reason for this,in the case of single point determination using Doppler satellites,many,as much as 50 or 100 of satellite passes are observed.The random errors would be eliminated mostly,and the result will converge to the theoretical best points.