Alternative transformation from Cartesian to geodetic coordinates by least squares for GPS georeferencing applications

The inverse transformation of coordinates, from Cartesian to curvilinear geodetic, or symbolically (x,y,z)→(λ,φ,h) has been extensively researched in the geodetic literature. However, published formulations require that the application must be deterministically implemented point-by-point individually. Recently, and thanks to GPS technology, scientists have made available thousands of determinations of the coordinates (x,y,z) at a single point perhaps characterized by different observational circumstances such as date, length of occupation time, distance and geometric distribution of reference stations, etc. In this paper a least squares (LS) solution is introduced to determine a unique set of geodetic coordinates, with accompanying accuracy predictions all based on the given sets of individual (x,y,z) GPS-obtained values and their variance–covariance matrices. The (x,y,z) coordinates are used as pseudo-observations with their attached stochastic information in the LS process to simultaneously compute a unique set of (λ,φ,h) curvilinear geodetic coordinates from different observing scenarios.