Batch Least Squares Differential Corrections

Batch Least Squares Differential Corrections (LS) methods have the following characteristics:

Batch Least Squares Differential Corrections
Property Manifestation in LS
Input Tracking measurements with tracking platform locations, and an a priori orbit estimate
Output Refined orbit estimate
A priori orbit estimate Required
Output error magnitudes Small when compared to IOD outputs
Methods A sequence of linear LS corrections where sequence convergence is defined as a function of tracking measurement residual RMS (Root Mean Square). Each linear LS correction is characterized by a minimization of the sum of squares of tracking measurement residuals (see below for equations). LS methods produce orbit error covariance matrices that are optimistic (i.e. orbit element error variances are typically too small by at least an order of magnitude).
Measurement editing during calculations Often require inspection and manual measurement by humans. Therefore, LS algorithms require elaborate software mechanisms for measurement editing.
Sources First derived by Gauss in 1795 (for orbit determination!), then independently by Legendre

Operationally, LS may be the only method used, or it may be used to initialize optimal orbit determination (OOD).

In the LS methods used here, the sum of the squares is represented by the following equations:

Minimization is characterized by: