Various types of a positioning apparatus which estimate the position utilizing the satellite signals from the satellites are disclosed.  There are two kinds of the positioning methods for the positioning apparatus, one is a point positioning and the other is a differential positioning.  A basic estimation of the point positioning is to derive a nonlinear simultaneous equation comprising a receiver's position (error) in three dimensions and a clock error of the receiver as the unknown values from a code pseudorange.  And applying a Newton method, an extended Kalmann Filter to this nonlinear samultaneous equation, the point positioning could be estimated.  The various types of a method to eliminate an influence of an ionosphere and a troposphere are disclosed.  One method set a delay of the ionosphere and a delay of the troposphere to “0” at initial condition.  And the other method estimates these delays utilizing a phase difference such as a single phase difference. 
The Geodetic Society of Japan, [Shinteiban GPS—Jinko-eisei-niyoru seimitsu-sokutei system—(in Japanese), A new edition GPS—A precise positioning system—], JAPAN ASSOCIATION OF SURVEYORS, Nov. 15, 1989, p. 121-140