Patent Publication Number: US-7586440-B2

Title: Independent positioning device and independent positioning method

Description:
This application is the U.S. national phase of international application PCT/JP2005/015404 filed 25 Aug. 2005, which designated the U.S. and claimed priority of JP 2004-244808 filed 25 Aug. 2004, the entire contents of each of which are hereby incorporated by reference. 
     TECHNICAL FIELD 
     The invention relates to point positioning apparatus and point positioning method configured to estimate a receiver&#39;s position by utilizing the satellite signals transmitted from the satellites. 
     BACKGROUND ART 
     Various types of a point positioning apparatus and method which estimate the point position utilizing the satellite signals of positioning from the satellites are disclosed. A basic positioning estimation of these apparatus is to introduce a nonlinear simultaneous equation comprising of a code pseudorange, a receiver&#39;s position in three dimensions, and a clock error of a receiver. And this estimation applied a Newton method, an extended Kalmann Filter to the nonlinear simultaneous equation. Applying to this estimation, various types of a method to eliminate an influence of an ionosphere and a troposphere are disclosed, too. One method set the delay of the ionosphere and the delay of the troposphere to “0” at initial condition. And the other method estimated these delay utilizing a phase difference, including single phase difference. 
     The Geodetic Society of Japan, [Shinteiban GPS—jinkoueisei niyoru seimitsusokuisisutemu—(in Japanese), A new edition GPS—A precise positioning system—], JAPAN ASSOCIATION OF SURVEYORS, Nov. 15, 1989, p. 121-140 
     DISCLOSURE OF THE INVENTION 
     A conventional point positioning apparatus and method utilizing the code pseudorange estimate the position by ignoring an actual delay of the ionosphere and an actual delay of the troposphere. The conventional point positioning apparatus and method had not estimated the point position until an estimation result had a satisfactory accuracy. Consequently the estimation result of the positioning varied widely, so we could not estimate the position with a high accuracy. Furthermore in the case of using the single phase difference with the code pseudorange, the delay data of the ionosphere, and the delay data of the troposphere, estimation result could have the high accuracy. But this estimation process was complicated, and the estimation result varied more widely under the influence of a threshpassed of noise after a estimation process with the single phase difference. Despite a complication of the estimation process, the estimation result could not be so high accuracy. And the complication of the estimation process made a processing speed very slow. 
     A feature of the invention is to provide a point positioning apparatus and method that estimates the receiver&#39;s position with a high accuracy and without the complication of the processing. 
     A point positioning apparatus of the invention estimates the distances between the receiver and the satellites respectively and estimates the receiver&#39;s position by utilizing these distances. A point positioning apparatus includes a satellite data observing device, an ionosphere data getting device, a troposphere data getting device, and a position estimating device. The satellite data observing device observes the orbits and the errors of the respective satellites by use of the navigation messages contained in the satellite signals or the values have been estimated off-line. The ionosphere data getting device gets the delay data of the ionosphere. The troposphere data getting device gets the delay data of the troposphere. The position estimating device estimates the receiver&#39;s position at least by means of applying a parameter estimation algorithm to a regression equation. The regression equation is comprised of the explanatory variables and the objective variables. The explanatory variables are an unknown values including of the receiver&#39;s position that calculated by a linear approximation utilizing previous estimation results and the orbits data of the satellites, an integer ambiguity, a clock error of the receiver, the clock errors of the satellites, the delays of the ionosphere, and the delays of the troposphere. The objective variables are the observable values including the carrier phases, the code pseudoranges, the clock errors of the satellites included in the navigation message, the delay data of the ionosphere, and the delay data of the troposphere. 
     A point positioning method of the invention estimates the distances between the receiver and the satellites respectively and estimates the receiver&#39;s position by utilizing these distances. A point positioning method includes a satellite data observing process, an ionosphere data getting process, a troposphere data getting process, and a position estimating process. The satellite data observing process is an observation of the orbits and the errors of the respective satellites by use of the navigation messages contained in the satellite signals or the values have been estimated off-line. The ionosphere data getting process is an acquisition of the delay data of the ionosphere. The troposphere data getting process is an acquisition of the delay data of the troposphere. The position estimating process is an estimation of the receiver&#39;s position at least by means of applying a parameter estimation algorithm to a regression equation. The regression equation is comprised of the explanatory variables and the objective variables. The explanatory variables are an unknown values including of the receiver&#39;s position that calculated by a linear approximation utilizing previous estimation results and the orbits data of the satellites, an integer ambiguity, a clock error of the receiver, the clock errors of the satellites, the delays of the ionosphere, and the delays of the troposphere. The objective variables are the observable values including of the carrier phase, the code pseudorange, the clock errors of the satellites included in the navigation message, the delay data of the ionosphere, and the delay data of the troposphere. 
     The point positioning apparatus and the positioning method set the carrier phases of every satellites and every carrier waves, the code pseudoranges of every satellites and every PN codes, the orbit data of every satellites, the clock error data of the every satellites, the delay data of the ionosphere, and the delay data of the troposphere as observable values. And the point positioning apparatus and point positioning method set the position of receiver in three dimensions, the clock error of the receiver, and the integer ambiguity of every carrier wave and every satellite as unknown value. The regression equation is composed of the observable values as objective variables and the unknown values as the explanatory variables. At this point, the receiver&#39;s position is linear estimated by the previous estimation results of the receiver&#39;s position and the orbit data of the every satellite. Applying the parameter estimation algorithm such as least-square method to a regression equation, the unknown values including at least the receiver&#39;s position could be estimated. 
     In an aspect of the invention, the point positioning apparatus and the point positioning method estimate the receiver&#39;s position by fixing the integer ambiguity with an integer ambiguity estimation method. 
     In this aspect, applying the integer ambiguity estimation method such as LAMBDA method to the integer ambiguity of the carrier phases, the integer ambiguity could be fixed as an integer value. By applying this fixed integer ambiguity to the regression equation, the amount of the unknown values could decrease and accuracy of the estimation result could improve. 
     In another aspect of the invention, the point positioning apparatus and the point positioning method memorize the carrier phases of every satellites, the code pseudoranges of every satellites, the clock error of the receiver, the orbit data of every satellites, the clock errors of every satellites, the delay data of the ionosphere, the delay date of the troposphere over the number of epochs. 
     In this aspect, getting the data over the number of epochs, the amount of the observable values and the unknown values would increase. But the amount of the observable values is more than one of the unknown values, so the unknown values of the regression equation could be estimated easier. 
     In another aspect of the invention, the point positioning apparatus and the point positioning method applies a state estimation algorithm to the regression equation. 
     In this aspect, applying the state estimation algorithm such as the Kalmann Filter and a nonlinear Filter to the regression equation described above, a position could be estimated while the receiver is moving. 
     In another aspect of the invention, the point positioning apparatus and the point positioning method add the orbit data of every satellites as the objective variable and the errors of the satellite&#39;s orbits as the explanatory variable into the regression equation. By using this regression equation the positioning apparatus and the positioning method estimates the receiver&#39;s position. 
     In this aspect of the invention, applying the error of the satellite&#39;s orbit which is related to a satellite&#39;s position as unknown value to the explanatory variable. 
     According to this invention, the estimation accuracy of the receiver&#39;s position and the integer ambiguity is high by utilizing the regression equation which comprising the delay data of the ionosphere, the delay data of the troposphere, the orbit data of the satellites, the clock error of the satellites, the carrier phases, the code pseudoranges as observational value, and the receiver&#39;s position; the integer ambiguity as unknown value. Accordingly, utilizing the easy estimation such as only one regression equation, the point position of the receiver is estimated with high accuracy. 
     In another aspect of the invention, applying the integer ambiguity estimation method such as LAMBDA method, estimating of the point position of the receiver has higher accuracy. 
     In another aspect of the invention, utilizing the observational variables over the number of epochs, the required number of the observational satellites is reduced and estimating the point position of the receiver has high accuracy. 
     In another aspect of the invention, applying the state estimation algorithm such as the Kalmann Filter, estimating the point position of the receiver which is moving has high accuracy. 
     In another aspect of the invention, utilizing the estimation equation comprising of the orbit data of the satellites as the observable value and the error of the satellite&#39;s orbit, the distance between the receiver and the satellites could be estimated with high accuracy. Accordingly, the receiver&#39;s position could be estimated with very high accuracy. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram illustrating an embodiment of the point positioning apparatus. 
         FIG. 2  is a flowchart illustrating an estimation process of a positioning system composed of the point positioning apparatus and a GPS receiver. 
         FIG. 3  is a scatter diagram of the receiver&#39;s position with an embodiment of the point positioning method. 
         FIG. 4  is a diagram of the receiver&#39;s ellipsoid height with an embodiment of the point positioning method. 
     
    
    
     EXPLANATION OF REFERENCE NUMBERS 
     
         
           10  point positioning apparatus 
           11  a navigation message analysis device 
           12  a satellite information processing device 
           13  a point position operating device 
           20  a GPS antenna 
           30  a GPS receiver 
       
    
     THE BEST MODE FOR CARRYING OUT THE INVENTION 
     With reference to the accompanying drawings, an embodiment of the invention which is a point positioning apparatus will be described below. The following explanation of an embodiment explains GPS (Global Positioning System), but can apply to other all GNSS (Global Navigation Satellite System)  FIG. 1  is a block diagram illustrating an embodiment of the point positioning apparatus.  FIG. 2  is a flowchart illustrating an estimation process of a positioning system composed of the point positioning apparatus and a GPS receiver. 
     As shown  FIG. 1 , the point positioning apparatus connects to the GPS receiver  30 , and includes a navigation message analysis device  11 , a satellite information processing device  12 , and a point position operating device  13 . 
     The GPS receiver  30  connects to a antenna, and gets a L1 carrier phases, a L2 carrier phases, a pseudorange based on the C/A code, a pseudorange based on the P(Y) code, and a navigation message on L1 carrier by known method from the GPS satellite (S 1 , S 2 ). The GPS receiver  30  outputs the navigation massage to the navigation massage analysis device  11 . The GPS receiver  30  outputs the carrier phase and the code pseudorange (the pseudorange based on the C/A code and the pseudorange based on the P(Y) code) to the point position operating device  13 . 
     The navigation message analysis device  11  gets a delay data of the ionosphere, the clock errors of the respective satellites, and the ephemeris data of the respective satellites by analyzing the navigation message inputted from the GPS receiver  30 . The navigation message analysis device  11  gets a data of a troposphere by using the mathematical model. And the navigation message analysis device  11  outputs the each data to the satellite information processing device  12 . 
     In reference to the ephemeris data, the satellite information processing device  12  chooses the GPS satellites utilized for estimating a receiver&#39;s position. The satellite information processing device  12  outputs to the chosen data. These chosen data have the ephemeris data, the clock delay of the satellite, the delay data of the ionosphere, and the delay data of the troposphere concerned with the chosen GPS satellites respectively (S 3 ). 
     The point position operating device  13  formulates the observed positioning data consisting of the L1 carrier phase, the L2 carrier phase, the pseudorange based on the C/A code, the pseudorange based on the P(Y) code, the ephemeris data, a clock error of the GPS satellite, the delay data of the ionosphere, the delay data of the troposphere by a liner regression equation to described below. Applying a least-square method that is a kind of a parameter estimation algorithm to this linear regression equation, the point position operating device  13  estimates the receiver&#39;s position, an integer ambiguity NL 1  of the L1 carrier wave, an integer ambiguity NL 2  of the L2 carrier wave (S 4 ). This estimation processing should be repeated until an error of the estimation result drops blow a predetermined threshpassed. When the error of the estimation result drops below the predetermined threshpassed, the estimation result of the receiver&#39;s position is outputted as an estimated receiver&#39;s position. 
     The navigation message analysis device  11 , the satellite information processing device  12 , and the point position operating device  13  are comprised of an arithmetic logical unit like a microprocessor which operates the algorithm described below. And these devices are comprised of the arithmetic logical units respectively or are comprised of the only arithmetic logical unit. 
     Next described below is a minute detail relating to an estimation algorithm which estimates the integer ambiguity of L1 carrier wave, the integer ambiguity of L2 carrier wave, and the receiver&#39;s position. 
     Generally an observation equation of a carrier phase φ pL, u  is described by an expression (1), and an observation equation of a code pseudorange ρ pc,u  is described by an equation (2), where u is receiver, p is a satellite number (observable), L is a kind of a carrier wave, c is a kind of code. Meanwhile, an error of a multipath is too small to be ignored. 
     
       
         
           
             
               
                 
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     where λL is a wavelength of L carrier wave, r pu (t,t−τ pu ) is the geometric distance between the receiver u at the time t and the satellite p at the time t−τ pu . Thus τ pu  denotes the travel time from the satellite p to the receiver u. And where δI pu (t) is the delay of the ionosphere to the L1 carrier wave, δT pu (t) is the delay of the troposphere to the L1 and L2 carrier wave. δt u (t) is a clock error of receiver u in rue time t, δt p (t−τ pu ) is a clock error of GPS satellite p by time (t−τ pu ). N pL,u  is the integer ambiguity concerned with a distance between the receiver u and the GPS satellite p, εp L,u (t), e pL,u (t) are the observation noise respectively. 
     Therefore the observation equation of the L1 carrier phase φ pL1,u  is described by an expression (3), and the observation equation of L2 carrier phase φ pL2,u  is described by an expression (4). 
     
       
         
           
             
               
                 
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     where fL 1  is a central frequency of the L1 carrier wave, fL 2  is a central frequency of the L2 carrier wave. 
     Furthermore, the L1 carrier phase φ pL1,u  and the L2 carrier phase φ pL2,u  are described by an expression (3′), (4′) respectively. 
     
       
         
           
             
               
                 
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                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                   , 
                                   u 
                                 
                                 p 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     3 
                     ′ 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           Φ 
                           
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                             , 
                             u 
                           
                           p 
                         
                         ≡ 
                           
                         ⁢ 
                         
                           
                             λ 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                           
                           ⁢ 
                           
                             
                               φ 
                               
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                                 , 
                                 u 
                               
                               p 
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               r 
                               u 
                               p 
                             
                             ⁡ 
                             
                               ( 
                               
                                 t 
                                 , 
                                 
                                   t 
                                   - 
                                   
                                     τ 
                                     u 
                                     p 
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               
                                 f 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                                 2 
                               
                               
                                 f 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                                 2 
                               
                             
                             ⁢ 
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 I 
                                 u 
                                 p 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           + 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 T 
                                 u 
                                 p 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             c 
                             ⁡ 
                             
                               [ 
                               
                                 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       t 
                                       u 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                 
                                 - 
                                 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       t 
                                       p 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         t 
                                         - 
                                         
                                           τ 
                                           u 
                                           p 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           + 
                           
                             
                               λ 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                             ⁢ 
                             
                               N 
                               
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                                 , 
                                 u 
                               
                               p 
                             
                           
                           + 
                           
                             
                               λ 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                             ⁢ 
                             
                               
                                 ɛ 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                   , 
                                   u 
                                 
                                 p 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     4 
                     ′ 
                   
                   ) 
                 
               
             
           
         
       
     
     The observation equation of the C/A code pseudorange ρ pCA,u  is described by an expression (5), and the observation equation of P code ρ pP,u  is described by an expression (6). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     4 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       ρ 
                       
                         CA 
                         , 
                         u 
                       
                       p 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         r 
                         u 
                         p 
                       
                       ⁡ 
                       
                         ( 
                         
                           t 
                           , 
                           
                             t 
                             - 
                             
                               τ 
                               u 
                               p 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           I 
                           u 
                           p 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     + 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           T 
                           u 
                           p 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     + 
                     
                       c 
                       ⁡ 
                       
                         [ 
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 t 
                                 u 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           - 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 t 
                                 p 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   t 
                                   - 
                                   
                                     τ 
                                     u 
                                     p 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         e 
                         
                           CA 
                           , 
                           u 
                         
                         p 
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ρ 
                       
                         P 
                         , 
                         u 
                       
                       p 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         r 
                         u 
                         p 
                       
                       ⁡ 
                       
                         ( 
                         
                           t 
                           , 
                           
                             t 
                             - 
                             
                               τ 
                               u 
                               p 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         
                           f 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                           2 
                         
                         
                           f 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                           2 
                         
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           I 
                           u 
                           p 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     + 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           T 
                           u 
                           p 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     + 
                     
                       c 
                       ⁡ 
                       
                         [ 
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 t 
                                 u 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           - 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 t 
                                 p 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   t 
                                   - 
                                   
                                     τ 
                                     u 
                                     p 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         e 
                         
                           P 
                           , 
                           u 
                         
                         p 
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     At this point the geometric distance r pu (t,t−τ pu ) is described by an expression (7). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     5 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         r 
                         u 
                         p 
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ≡ 
                     
                       
                         r 
                         u 
                         p 
                       
                       ⁡ 
                       
                         ( 
                         
                           t 
                           , 
                           
                             t 
                             - 
                             
                               τ 
                               u 
                               p 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       x 
                                       u 
                                     
                                     ⁢ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       x 
                                       p 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         t 
                                         - 
                                         
                                           τ 
                                           u 
                                           p 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                           
                         
                       
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       y 
                                       u 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       y 
                                       p 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         t 
                                         - 
                                         
                                           τ 
                                           u 
                                           p 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       z 
                                       u 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       z 
                                       p 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         t 
                                         - 
                                         
                                           τ 
                                           u 
                                           p 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Next, we apply a Linear Taylor series expansion to an unknown receiver&#39;s position u(t) with a priori estimation of the receiver&#39;s position u (j) (t), and apply a linear approximation to the geometric distance r pu (t), then a following equation (an expression (9)) should be derived mathematically. At this point u(t) and u (j) (t) are defined by an following definitional equation respectively. u(t)≡[x u (t), y u (t), z u (t)] T , u (j) (t)≡[x u   (j) (t), y u   (j) (t), z u   (j) (t)] T . 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     6 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         r 
                         u 
                         p 
                       
                       ≅ 
                       
                         
                           r 
                           
                             u 
                             ⁡ 
                             
                               ( 
                               j 
                               ) 
                             
                           
                           p 
                         
                         + 
                         
                           
                             
                               [ 
                               
                                 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     r 
                                     u 
                                     p 
                                   
                                 
                                 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   u 
                                 
                               
                               ] 
                             
                             
                               u 
                               = 
                               
                                 u 
                                 ⁡ 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                             T 
                           
                           ⁢ 
                           
                             ( 
                             
                               u 
                               - 
                               
                                 u 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           p 
                         
                       
                     
                     = 
                     1 
                   
                   , 
                   2 
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   
                     
                       n 
                       s 
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     where 
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         g 
                         
                           p 
                           , 
                           
                             ( 
                             j 
                             ) 
                           
                         
                       
                       ≡ 
                       
                         
                           
                             [ 
                             
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   r 
                                   
                                     u 
                                     
                                       ( 
                                       j 
                                       ) 
                                     
                                   
                                   p 
                                 
                               
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   u 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                           T 
                         
                         ⁢ 
                         p 
                       
                     
                     = 
                     1 
                   
                   , 
                   2 
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   
                     n 
                     s 
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     By this expression (9), the expression (3′), (4′), (5), (6) are re-described to the following expression (10), (11), (12), (13) respectively. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     7 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     Φ 
                     
                       
                         L 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                       , 
                       u 
                     
                     
                       p 
                       , 
                       
                         ( 
                         j 
                         ) 
                       
                     
                   
                   = 
                   
                     
                       r 
                       
                         u 
                         
                           ( 
                           j 
                           ) 
                         
                       
                       p 
                     
                     + 
                     
                       
                         g 
                         
                           p 
                           , 
                           
                             ( 
                             j 
                             ) 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           u 
                           - 
                           
                             u 
                             
                               ( 
                               j 
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         I 
                         u 
                         p 
                       
                     
                     + 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         T 
                         u 
                         p 
                       
                     
                     + 
                     
                       c 
                       ⁡ 
                       
                         ( 
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               u 
                             
                           
                           - 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               p 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         λ 
                         
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                       
                       ⁢ 
                       
                         N 
                         
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                           , 
                           u 
                         
                         p 
                       
                     
                     + 
                     
                       
                         λ 
                         
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                       
                       ⁢ 
                       
                         ɛ 
                         
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                           , 
                           u 
                         
                         p 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
             
               
                 
                   
                     Φ 
                     
                       
                         L 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                       , 
                       u 
                     
                     
                       p 
                       , 
                       
                         ( 
                         j 
                         ) 
                       
                     
                   
                   = 
                   
                     
                       r 
                       
                         u 
                         
                           ( 
                           j 
                           ) 
                         
                       
                       p 
                     
                     + 
                     
                       
                         g 
                         
                           p 
                           , 
                           
                             ( 
                             j 
                             ) 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           u 
                           - 
                           
                             u 
                             
                               ( 
                               j 
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         
                           f 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                           2 
                         
                         
                           f 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                           2 
                         
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         I 
                         u 
                         p 
                       
                     
                     + 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         T 
                         u 
                         p 
                       
                     
                     + 
                     
                       c 
                       ⁡ 
                       
                         ( 
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               u 
                             
                           
                           - 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               p 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         λ 
                         
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                       ⁢ 
                       
                         N 
                         
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                           , 
                           u 
                         
                         p 
                       
                     
                     + 
                     
                       
                         λ 
                         
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                       ⁢ 
                       
                         ɛ 
                         
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                           , 
                           u 
                         
                         p 
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   
                     ρ 
                     
                       CA 
                       , 
                       u 
                     
                     
                       p 
                       , 
                       
                         ( 
                         j 
                         ) 
                       
                     
                   
                   = 
                   
                     
                       r 
                       
                         u 
                         
                           ( 
                           j 
                           ) 
                         
                       
                       p 
                     
                     + 
                     
                       
                         g 
                         
                           p 
                           , 
                           
                             ( 
                             j 
                             ) 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           u 
                           - 
                           
                             u 
                             
                               ( 
                               j 
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         I 
                         u 
                         p 
                       
                     
                     + 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         T 
                         u 
                         p 
                       
                     
                     + 
                     
                       c 
                       ⁡ 
                       
                         ( 
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               u 
                             
                           
                           - 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               p 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       e 
                       
                         CA 
                         , 
                         u 
                       
                       p 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   
                     ρ 
                     
                       P 
                       , 
                       u 
                     
                     
                       p 
                       , 
                       
                         ( 
                         j 
                         ) 
                       
                     
                   
                   = 
                   
                     
                       r 
                       
                         u 
                         
                           ( 
                           j 
                           ) 
                         
                       
                       p 
                     
                     + 
                     
                       
                         g 
                         
                           p 
                           , 
                           
                             ( 
                             j 
                             ) 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           u 
                           - 
                           
                             u 
                             
                               ( 
                               j 
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         
                           f 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                           2 
                         
                         
                           f 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                           2 
                         
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         I 
                         u 
                         p 
                       
                     
                     + 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         T 
                         u 
                         p 
                       
                     
                     + 
                     
                       c 
                       ⁡ 
                       
                         ( 
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               u 
                             
                           
                           - 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               p 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       e 
                       
                         P 
                         , 
                         u 
                       
                       p 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     At this point, a L1 carrier phase Φe p,(j)   L1,u , a L2 carrier phase Φe p,(j)   L2,u , a C/A code pseudorange ρe p,(j)   CA,u , a P code pseudorange ρe p,(j)   P,u  are defined by the following expression (14), (15), (16), (17) respectively. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     8 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     Φ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       e 
                       
                         
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                         , 
                         u 
                       
                       
                         p 
                         , 
                         
                           ( 
                           j 
                           ) 
                         
                       
                     
                   
                   ≡ 
                   
                     
                       Φ 
                       
                         
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                         , 
                         u 
                       
                       
                         p 
                         , 
                         
                           ( 
                           j 
                           ) 
                         
                       
                     
                     - 
                     
                       ( 
                       
                         
                           r 
                           
                             u 
                             
                               ( 
                               j 
                               ) 
                             
                           
                           p 
                         
                         - 
                         
                           
                             g 
                             
                               p 
                               , 
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                           
                           ⁢ 
                           
                             u 
                             
                               ( 
                               j 
                               ) 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
             
               
                 
                   
                     Φ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       e 
                       
                         
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                         , 
                         u 
                       
                       
                         p 
                         , 
                         
                           ( 
                           j 
                           ) 
                         
                       
                     
                   
                   ≡ 
                   
                     
                       Φ 
                       
                         
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                         , 
                         u 
                       
                       
                         p 
                         , 
                         
                           ( 
                           j 
                           ) 
                         
                       
                     
                     - 
                     
                       ( 
                       
                         
                           r 
                           
                             u 
                             
                               ( 
                               j 
                               ) 
                             
                           
                           p 
                         
                         - 
                         
                           
                             g 
                             
                               p 
                               , 
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                           
                           ⁢ 
                           
                             u 
                             
                               ( 
                               j 
                               ) 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       e 
                       
                         CA 
                         , 
                         u 
                       
                       
                         p 
                         , 
                         
                           ( 
                           j 
                           ) 
                         
                       
                     
                   
                   ≡ 
                   
                     
                       ρ 
                       
                         CA 
                         , 
                         u 
                       
                       
                         p 
                         , 
                         
                           ( 
                           j 
                           ) 
                         
                       
                     
                     - 
                     
                       r 
                       
                         u 
                         
                           ( 
                           j 
                           ) 
                         
                       
                       p 
                     
                     + 
                     
                       
                         g 
                         
                           p 
                           , 
                           
                             ( 
                             j 
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         u 
                         
                           ( 
                           j 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
             
               
                 
                   
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       e 
                       
                         P 
                         , 
                         u 
                       
                       
                         p 
                         , 
                         
                           ( 
                           j 
                           ) 
                         
                       
                     
                   
                   ≡ 
                   
                     
                       ρ 
                       
                         P 
                         , 
                         u 
                       
                       
                         p 
                         , 
                         
                           ( 
                           j 
                           ) 
                         
                       
                     
                     - 
                     
                       r 
                       
                         u 
                         
                           ( 
                           j 
                           ) 
                         
                       
                       p 
                     
                     + 
                     
                       
                         g 
                         
                           p 
                           , 
                           
                             ( 
                             j 
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         u 
                         
                           ( 
                           j 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     By these expressions, the expression (10), (11), (12), (13) are re-described to the following expression (18), (19), (20), (21) respectively. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     9 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     Φ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       e 
                       
                         
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                         , 
                         u 
                       
                       
                         p 
                         , 
                         
                           ( 
                           j 
                           ) 
                         
                       
                     
                   
                   ≅ 
                   
                     
                       
                         g 
                         
                           p 
                           , 
                           
                             ( 
                             j 
                             ) 
                           
                         
                       
                       ⁢ 
                       u 
                     
                     - 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         I 
                         u 
                         p 
                       
                     
                     + 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         T 
                         u 
                         p 
                       
                     
                     + 
                     
                       c 
                       ⁡ 
                       
                         ( 
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               u 
                             
                           
                           - 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               p 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         λ 
                         
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                       
                       ⁢ 
                       
                         N 
                         
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                           , 
                           u 
                         
                         p 
                       
                     
                     + 
                     
                       
                         λ 
                         
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                       
                       ⁢ 
                       
                         ɛ 
                         
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                           , 
                           u 
                         
                         p 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
             
               
                 
                   
                     Φ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       e 
                       
                         
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                         , 
                         u 
                       
                       
                         p 
                         , 
                         
                           ( 
                           j 
                           ) 
                         
                       
                     
                   
                   ≅ 
                   
                     
                       
                         g 
                         
                           p 
                           , 
                           
                             ( 
                             j 
                             ) 
                           
                         
                       
                       ⁢ 
                       u 
                     
                     - 
                     
                       
                         
                           f 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                           2 
                         
                         
                           f 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                           2 
                         
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         I 
                         u 
                         p 
                       
                     
                     + 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         T 
                         u 
                         p 
                       
                     
                     + 
                     
                       c 
                       ⁡ 
                       
                         ( 
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               u 
                             
                           
                           - 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               p 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         λ 
                         
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                       ⁢ 
                       
                         N 
                         
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                           , 
                           u 
                         
                         p 
                       
                     
                     + 
                     
                       
                         λ 
                         
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                       ⁢ 
                       
                         ɛ 
                         
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                           , 
                           u 
                         
                         p 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
             
               
                 
                   
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       e 
                       
                         CA 
                         , 
                         u 
                       
                       
                         p 
                         , 
                         
                           ( 
                           j 
                           ) 
                         
                       
                     
                   
                   ≅ 
                   
                     
                       
                         g 
                         
                           p 
                           , 
                           
                             ( 
                             j 
                             ) 
                           
                         
                       
                       ⁢ 
                       u 
                     
                     + 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         I 
                         u 
                         p 
                       
                     
                     + 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         T 
                         u 
                         p 
                       
                     
                     + 
                     
                       c 
                       ⁢ 
                       
                         ⌊ 
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               u 
                             
                           
                           - 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               p 
                             
                           
                         
                         ⌋ 
                       
                     
                     + 
                     
                       e 
                       
                         CA 
                         , 
                         u 
                       
                       p 
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       e 
                       
                         P 
                         , 
                         u 
                       
                       
                         p 
                         , 
                         
                           ( 
                           j 
                           ) 
                         
                       
                     
                   
                   ≅ 
                   
                     
                       
                         g 
                         
                           p 
                           , 
                           
                             ( 
                             j 
                             ) 
                           
                         
                       
                       ⁢ 
                       u 
                     
                     + 
                     
                       
                         
                           f 
                           1 
                           2 
                         
                         
                           f 
                           2 
                           2 
                         
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         I 
                         u 
                         p 
                       
                     
                     + 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         T 
                         u 
                         p 
                       
                     
                     + 
                     
                       c 
                       ⁡ 
                       
                         [ 
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               u 
                             
                           
                           - 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               p 
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       e 
                       
                         P 
                         , 
                         u 
                       
                       p 
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     This equation denotes the approximate linear regression equation that is comprised of some explanatory variables and some objective variables. The objective variables are comprised of the carrier phases and the code pseudoranges. The explanatory variables are comprised of the receiver&#39;s position, the delay of the ionosphere, the delay of the troposphere, the integer ambiguity, and the measurement error. 
     Define a matrix G (j)  as an expression (22) here. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     10 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     G 
                     
                       ( 
                       j 
                       ) 
                     
                   
                   ≡ 
                   
                     [ 
                     
                       
                         
                           
                             g 
                             
                               1 
                               , 
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             g 
                             
                               2 
                               , 
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             g 
                             
                               
                                 n 
                                 s 
                               
                               , 
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     And a matrix G (j)   u  is defined as an expression (23). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     11 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       G 
                       u 
                       
                         ( 
                         j 
                         ) 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   r 
                                   
                                     u 
                                     
                                       ( 
                                       j 
                                       ) 
                                     
                                   
                                   1 
                                 
                               
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   x 
                                   u 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   r 
                                   
                                     u 
                                     
                                       ( 
                                       j 
                                       ) 
                                     
                                   
                                   
                                     
                                         
                                       * 
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                               
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   y 
                                   u 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   r 
                                   
                                     u 
                                     
                                       ( 
                                       j 
                                       ) 
                                     
                                   
                                   
                                     
                                         
                                       * 
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                               
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   z 
                                   u 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   r 
                                   
                                     u 
                                     
                                       ( 
                                       j 
                                       ) 
                                     
                                   
                                   2 
                                 
                               
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   x 
                                   u 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   r 
                                   
                                     u 
                                     
                                       ( 
                                       j 
                                       ) 
                                     
                                   
                                   2 
                                 
                               
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   y 
                                   u 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   r 
                                   
                                     u 
                                     
                                       ( 
                                       j 
                                       ) 
                                     
                                   
                                   2 
                                 
                               
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   z 
                                   u 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   r 
                                   
                                     u 
                                     
                                       ( 
                                       j 
                                       ) 
                                     
                                   
                                   
                                     n 
                                     s 
                                   
                                 
                               
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   x 
                                   u 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   r 
                                   
                                     u 
                                     
                                       ( 
                                       j 
                                       ) 
                                     
                                   
                                   
                                     n 
                                     s 
                                   
                                 
                               
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   y 
                                   u 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   r 
                                   
                                     u 
                                     
                                       ( 
                                       j 
                                       ) 
                                     
                                   
                                   
                                     n 
                                     s 
                                   
                                 
                               
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   z 
                                   u 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             r 
                             
                               u 
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                             p 
                           
                         
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             x 
                             u 
                             
                               ( 
                               j 
                               ) 
                             
                           
                         
                       
                       = 
                       
                         
                           ( 
                           
                             
                               x 
                               u 
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                             - 
                             
                               x 
                               p 
                             
                           
                           ) 
                         
                         
                           r 
                           
                             u 
                             
                               ( 
                               j 
                               ) 
                             
                           
                           p 
                         
                       
                     
                     , 
                     
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             r 
                             
                               u 
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                             p 
                           
                         
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             y 
                             u 
                             
                               ( 
                               j 
                               ) 
                             
                           
                         
                       
                       = 
                       
                         
                           ( 
                           
                             
                               y 
                               u 
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                             - 
                             
                               y 
                               p 
                             
                           
                           ) 
                         
                         
                           r 
                           
                             u 
                             
                               ( 
                               j 
                               ) 
                             
                           
                           p 
                         
                       
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             r 
                             
                               u 
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                             p 
                           
                         
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             z 
                             u 
                             
                               ( 
                               j 
                               ) 
                             
                           
                         
                       
                       = 
                       
                         
                           
                             ( 
                             
                               
                                 z 
                                 u 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                               - 
                               
                                 z 
                                 p 
                               
                             
                             ) 
                           
                           
                             r 
                             
                               u 
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                             p 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               p 
                               = 
                               1 
                             
                             , 
                             2 
                             , 
                             … 
                             ⁢ 
                             
                                 
                             
                             , 
                             
                               n 
                               s 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     Additionally define the L1 carrier phase Φe p,(j)   L1,u , the L2 carrier phase Φe p,(j)   L2,u , the C/A code pseudorange ρe p,(j)   CA,u , the P code pseudorange ρe p,(j)   P,u  as an expression (24) in matrix. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     12 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       Φ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         e 
                         
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                           , 
                           u 
                         
                         
                           ( 
                           j 
                           ) 
                         
                       
                     
                     ≡ 
                     
                       [ 
                       
                         
                           
                             
                               Φ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                   , 
                                   u 
                                 
                                 
                                   1 
                                   , 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               Φ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                   , 
                                   u 
                                 
                                 
                                   
                                     n 
                                     s 
                                   
                                   , 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
                       Φ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         e 
                         
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                           , 
                           u 
                         
                         
                           ( 
                           j 
                           ) 
                         
                       
                     
                     ≡ 
                     
                       [ 
                       
                         
                           
                             
                               Φ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                   , 
                                   u 
                                 
                                 
                                   1 
                                   , 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               Φ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                   , 
                                   u 
                                 
                                 
                                   
                                     n 
                                     s 
                                   
                                   , 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       ρ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         e 
                         
                           CA 
                           , 
                           u 
                         
                         
                           ( 
                           j 
                           ) 
                         
                       
                     
                     ≡ 
                     
                       [ 
                       
                         
                           
                             
                               ρ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   CA 
                                   , 
                                   u 
                                 
                                 
                                   1 
                                   , 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               ρ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   CA 
                                   , 
                                   u 
                                 
                                 
                                   
                                     n 
                                     s 
                                   
                                   , 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
                       ρ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         e 
                         
                           P 
                           , 
                           u 
                         
                         
                           ( 
                           j 
                           ) 
                         
                       
                     
                     ≡ 
                     
                       [ 
                       
                         
                           
                             
                               ρ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   P 
                                   , 
                                   u 
                                 
                                 
                                   1 
                                   , 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               ρ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   P 
                                   , 
                                   u 
                                 
                                 
                                   
                                     n 
                                     s 
                                   
                                   , 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     By these expressions (22), (23), (24), the expression (18), (19), (20), (21) are re-described to the following expression (25) in a vector-matrix. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     13 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             Φ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               e 
                               
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                                 , 
                                 u 
                               
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             Φ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               e 
                               
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                                 , 
                                 u 
                               
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             ρ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               e 
                               
                                 CA 
                                 , 
                                 u 
                               
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             ρ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               e 
                               
                                 P 
                                 , 
                                 u 
                               
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 G 
                                 u 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                             
                               1 
                             
                             
                               
                                 - 
                                 I 
                               
                             
                             
                               
                                 - 
                                 I 
                               
                             
                             
                               I 
                             
                             
                               I 
                             
                             
                               0 
                             
                           
                           
                             
                               
                                 G 
                                 u 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                             
                               1 
                             
                             
                               
                                 - 
                                 I 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     
                                       f 
                                       
                                         L 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         1 
                                       
                                       2 
                                     
                                     
                                       f 
                                       
                                         L 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         2 
                                       
                                       2 
                                     
                                   
                                 
                                 ⁢ 
                                 I 
                               
                             
                             
                               I 
                             
                             
                               0 
                             
                             
                               I 
                             
                           
                           
                             
                               
                                 G 
                                 u 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                             
                               1 
                             
                             
                               
                                 - 
                                 I 
                               
                             
                             
                               I 
                             
                             
                               I 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               
                                 G 
                                 u 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                             
                               1 
                             
                             
                               
                                 - 
                                 I 
                               
                             
                             
                               
                                 
                                   
                                     f 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
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                                     2 
                                   
                                   
                                     f 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                     
                                     2 
                                   
                                 
                                 ⁢ 
                                 I 
                               
                             
                             
                               I 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                         
                         ] 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               u 
                             
                           
                           
                             
                               
                                 c 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   t 
                                   u 
                                 
                               
                             
                           
                           
                             
                               
                                 c 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   t 
                                   p 
                                 
                               
                             
                           
                           
                             
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   I 
                                   u 
                                 
                               
                             
                           
                           
                             
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   T 
                                   u 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   λ 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   N 
                                   
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                     , 
                                     u 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   λ 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   N 
                                   
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                     
                                     , 
                                     u 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       [ 
                       
                         
                           
                             
                               
                                 λ 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 ɛ 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                   , 
                                   u 
                                 
                                 p 
                               
                             
                           
                         
                         
                           
                             
                               
                                 λ 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 ɛ 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                   , 
                                   u 
                                 
                                 p 
                               
                             
                           
                         
                         
                           
                             
                               e 
                               
                                 CA 
                                 , 
                                 u 
                               
                               p 
                             
                           
                         
                         
                           
                             
                               e 
                               
                                 P 
                                 , 
                                 u 
                               
                               p 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     or, 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     14 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     y 
                     u 
                     
                       ( 
                       j 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         H 
                         
                           ( 
                           j 
                           ) 
                         
                       
                       ⁢ 
                       
                         θ 
                         u 
                       
                     
                     + 
                     
                       υ 
                       u 
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
             
               
                 
                   
                     y 
                     u 
                     
                       ( 
                       j 
                       ) 
                     
                   
                   ≡ 
                   
                     
                       [ 
                       
                         
                           
                             
                               Φ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                   , 
                                   u 
                                 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               Φ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                   , 
                                   u 
                                 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               ρ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   CA 
                                   , 
                                   u 
                                 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               ρ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   P 
                                   , 
                                   u 
                                 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                     T 
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
             
               
                 
                   
                     H 
                     u 
                     
                       ( 
                       j 
                       ) 
                     
                   
                   ≡ 
                   
                     [ 
                     
                       
                         
                           
                             G 
                             u 
                             
                               ( 
                               j 
                               ) 
                             
                           
                         
                         
                           1 
                         
                         
                           
                             - 
                             I 
                           
                         
                         
                           
                             - 
                             I 
                           
                         
                         
                           I 
                         
                         
                           I 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             G 
                             u 
                             
                               ( 
                               j 
                               ) 
                             
                           
                         
                         
                           1 
                         
                         
                           
                             - 
                             I 
                           
                         
                         
                           
                             
                               - 
                               
                                 
                                   f 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                   2 
                                 
                                 
                                   f 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                   2 
                                 
                               
                             
                             ⁢ 
                             I 
                           
                         
                         
                           I 
                         
                         
                           0 
                         
                         
                           I 
                         
                       
                       
                         
                           
                             G 
                             u 
                             
                               ( 
                               j 
                               ) 
                             
                           
                         
                         
                           1 
                         
                         
                           
                             - 
                             I 
                           
                         
                         
                           I 
                         
                         
                           I 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             G 
                             u 
                             
                               ( 
                               j 
                               ) 
                             
                           
                         
                         
                           1 
                         
                         
                           
                             - 
                             I 
                           
                         
                         
                           
                             
                               
                                 f 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                                 2 
                               
                               
                                 f 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                                 2 
                               
                             
                             ⁢ 
                             I 
                           
                         
                         
                           I 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
             
               
                 
                   
                     θ 
                     u 
                   
                   ≡ 
                   
                     
                       [ 
                       
                         
                           
                             u 
                           
                           
                             
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 t 
                                 u 
                               
                             
                           
                           
                             
                               c 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 t 
                                 p 
                               
                             
                           
                           
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 I 
                                 u 
                               
                             
                           
                           
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 T 
                                 u 
                               
                             
                           
                           
                             
                               
                                 λ 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 N 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                   , 
                                   u 
                                 
                               
                             
                           
                           
                             
                               
                                 λ 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 N 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                   , 
                                   u 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                     T 
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
             
               
                 
                   
                     υ 
                     u 
                   
                   ≡ 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 λ 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 ɛ 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                   , 
                                   u 
                                 
                                 p 
                               
                             
                           
                           
                             
                               
                                 λ 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 ɛ 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                   , 
                                   u 
                                 
                                 p 
                               
                             
                           
                           
                             
                               e 
                               
                                 CA 
                                 , 
                                 u 
                               
                               p 
                             
                           
                           
                             
                               e 
                               
                                 P 
                                 , 
                                 u 
                               
                               p 
                             
                           
                         
                       
                       ] 
                     
                     T 
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
     And now, described above, the navigation message includes the clock errors of each GPS satellite, so the navigation message analysis device  11  gets the clock errors of each GPS satellite from the navigation message. An observable value of these clock errors of the GPS satellites δte p  is described by a following expression, and could be added to the regression equation described above.
 
δ te   p =└0 0 1 n     s   0 0 0 0 ┘θ u   +e   δt     p     [Expression 15]
 
     Additionally, the delay of the ionosphere and the delay of the troposphere could be calculated from some kind of the GPS satellite information on the navigation message, applying one mathematical model calculates the delay data of the ionosphere and another mathematical model calculates the delay data of the troposphere. An observable value of the delay data of the ionosphere δIe u  and an observable of the delay data of the troposphere δTe u  are described by following expressions respectively, and could be added to the regression equation described above.
 
δ Ie   u =└0 0 1 n     s    0 0 0 0 ┘θ u   +e   δIu  
 
δ Te   u =└0 0 1 n     s    0 0 0 0 ┘θ u   +e   δTu   [Expression 16]
 
     As a result, the objective variable of this linear regression equation, which is comprised of the carrier phase, the code pseudorange, the clock errors of the GPS satellites, the delay data of the ionosphere, and the delay data of the troposphere, is the observable value. And the explanatory variable of this linear regression equation, which includes the integer ambiguity and the receiver&#39;s position, is the unknown value. 
     Namely, the regression equation described by the expression (26) re-described by the following expressions (31), (32), (33), (34). 
                   [     Expression   ⁢           ⁢   17     ]                             y     u   ,   E       (   j   )       =         H   E     (   j   )       ⁢     θ   u       +     υ     u   ,   E                 (   31   )                 y     u   ,   E       (   j   )       ≡       [           Φ   ⁢           ⁢     e       L   ⁢           ⁢   1     ,   u       (   j   )               Φ   ⁢           ⁢     e       L   ⁢           ⁢   2     ,   u       (   j   )               ρ   ⁢           ⁢     e     CA   ,   u       (   j   )               ρ   ⁢           ⁢     e     P   ,   u       (   j   )               δ   ⁢           ⁢   t   ⁢           ⁢     e   p             δ   ⁢           ⁢   I   ⁢           ⁢     e   u             δ   ⁢           ⁢   T   ⁢           ⁢     e   u             ]     T             (   32   )                 H   E     (   j   )       ≡     [           G   u     (   j   )           1         -   I           -   I         I       I       0             G   u     (   j   )           1         -   I             -       f     L   ⁢           ⁢   1     2       f     L   ⁢           ⁢   2     2         ⁢   I         I       0       I             G   u     (   j   )           1         -   I         I       I       0       0             G   u     (   j   )           1         -   I               f     L   ⁢           ⁢   1     2       f     L   ⁢           ⁢   2     2       ⁢   I         I       0       0           0       0       I       0       0       0       0           0       0       0       I       0       0       0           0       0       0       0       I       0       0         ]             (   33   )                 υ     u   ,   E       ≡       [             λ     L   ⁢           ⁢   1       ⁢     ɛ       L   ⁢           ⁢   1     ,   u     p               λ     L   ⁢           ⁢   2       ⁢     ɛ       L   ⁢           ⁢   2     ,   u     p             e     CA   ,   u     p           e     P   ,   u     p           e     δ   ⁢           ⁢     t   p               e     δ   ⁢           ⁢     I   u               e     δ   ⁢           ⁢   Tu             ]     T             (   34   )               
where y (i)   u,E  is a vector-matrix of the objective variable, H (j)   E  is a calculating matrix, θ u  is a vector-matrix of the explanatory variable, and ν u,E  is a vector-matrix of the error.
 
     At this point, a covariance matrix of the error R based on the vector-matrix of the error ν u,E  is defined by a following expression (35). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     18 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     R 
                     ≡ 
                     
                       Cov 
                       ⁢ 
                       
                           
                       
                       [ 
                       
                           
                       
                       ⁢ 
                       
                         υ 
                         
                           u 
                           , 
                           E 
                         
                       
                       ] 
                     
                   
                   = 
                   
                       
                     
                       
                         
                           
                             [ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     
                                       σ 
                                       
                                         Φ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         1 
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                       I 
                                       
                                         n 
                                         s 
                                       
                                     
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     
                                       σ 
                                       
                                         Φ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         2 
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                       I 
                                       
                                         n 
                                         s 
                                       
                                     
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     
                                       σ 
                                       CA 
                                       2 
                                     
                                     ⁢ 
                                     
                                       I 
                                       
                                         n 
                                         s 
                                       
                                     
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     
                                       σ 
                                       P 
                                       2 
                                     
                                     ⁢ 
                                     
                                       I 
                                       
                                         n 
                                         s 
                                       
                                     
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     
                                       σ 
                                       
                                         δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           t 
                                           p 
                                         
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                       I 
                                       
                                         n 
                                         s 
                                       
                                     
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     
                                       σ 
                                       
                                         δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           I 
                                           u 
                                         
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                       I 
                                       
                                         n 
                                         s 
                                       
                                     
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     
                                       σ 
                                       
                                         δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           T 
                                           u 
                                         
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                       I 
                                       
                                         n 
                                         s 
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             
 
                           
                           ⁢ 
                           where 
                           ⁢ 
                           
                             
 
                           
                           ⁢ 
                           
                             σ 
                             
                               Φ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                             2 
                           
                         
                         ≡ 
                         
                           
                             λ 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                             2 
                           
                           ⁢ 
                           
                             σ 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                             2 
                           
                         
                       
                       , 
                       
                         
                           σ 
                           
                             Φ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                           2 
                         
                         ≡ 
                         
                           
                             λ 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                             2 
                           
                           ⁢ 
                           
                             σ 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
     Therefore, the estimated explanatory variable θe (j+1)  which is a estimation of the explanatory variable θ (j+1)  is defined by a following expression (36).
 
θ e   (j+1)   ={[H   E   (j) ] T   R   −1   [H   E   (j) ]} −1   [H   E   (j) ] T   R   −1   y   u,E   (j)   (36)[Expression 19]
 
     Additionally a variance of the estimated explanatory variable θe (j+1)  is defined by an following expression (37).
 
 Cov[θe   (j+1)   ]≡E [(θ e   (j+1) −θ)(θ e   (j+1) −θ) T   ]=[[H   E   (j) ] T   R   −1   [H   E   (j) ]] −1   (37)[Expression 20]
 
     As a reference to the calculated result of this expression (36), and applying a least-square method, the linear regression equation described above is calculated repeatedly. And the unknown values composing the explanatory variable θ converge, then the estimated values is determined by the converged values. At this point, a convergence condition of the unknown values composing the explanatory variable θ is what the variation (error) caused in the repeated calculation of estimating the unknown values drops blow the predetermined threshpassed. And then under this condition the estimation values composing the explanatory variable θ are determined. Utilizing the estimation operation described above, the integer ambiguity of L1 carrier wave N L1 , the integer ambiguity of L2 carrier wave N L2 , and the receiver&#39;s position could be estimated. 
     And according to the present embodiment, the integer ambiguity of L1 carrier wave NL 1 , the integer ambiguity of L2 carrier wave, and the receiver&#39;s position could be estimated without the estimation operation utilizing the single difference. As a consequence, we could estimate the point position of the receiver by using easier algorithm than the conventional one, and make an operation speed of the point positioning faster. 
     At this time using the estimation operation described above, the unknown values are the components of the explanatory variables θ u , and specifically the regression equation described by an expression (31) has some unknown parameters. The number of the unknown parameters is 3 in the receiver&#39;s position u, 1 in the clock error of the receiver δt u , n s  (equal to the number of the observable GPS satellite) in the clock errors of the GPS satellites δt p , n s  in the delay of the ionosphere δI, n s  in the delay of the troposphere δT, n s  in the integer ambiguity of the L1 carrier wave, and n s  in the integer ambiguity of the L2 carrier wave, an amount of the unknown values is 4+5*n s . 
     On the other hand, the observable values is composed of the L1 carrier phase Φe (j)   L1,u , the L2 carrier phase Φe (j)   L2,u , the C/A code pseudorange ρe (j)   CA,u , the P(Y) code pseudorange ρe (j)   P,u , the clock error data of the GPS satellite δte p , the delay data of the ionosphere δIe, and the delay data of the troposphere δTe. The number of these observable values are n s  that is a number of the observable GPS satellites, and an amount of the observable values is 7*n s . 
     In consequence, a required condition to solve the unknown values of this regression equation is the number of the unknown values 4+5*n s  is equal or less than the number of the observable values 7*n s  as the following expression.
 
4+5 *n   s &lt;=7 *n   s  
 
     This means n s &gt;=2. 
     More specifically, it could be only 2 satellite signals from the respective GPS satellites to estimate the point position of the receiver. However, if we apply the least-square method to this regression equation described above, the inverse matrix of the H (i)   E  in the expression (33) must exist. 
     But when the number of the observable GPS satellites n s  is 2, this inverse matrix must be a single matrix. And when the number of the observable GPS satellites n s  is 3, it has a large possibility that this inverse matrix would be a single matrix. Under this condition we could not have the inverse matrix. So we could not solve the unknown values of the regression equation described above. 
     This embodiment of invention described above use P code to execute the estimation operation, but P(Y) code is a secret code, so observing the P code pseudorange is very difficult in fact. Under this condition the observable value derived from the P(Y) code (include n s  parameter) decreases, a required condition to solve the unknown values of this regression equation is following expression.
 
4+5 *n   s &lt;=6 *n   s  
 
     This means n s &gt;=4. So it could be at least 4 satellite signals from the respective GPS satellites to estimate the point position of the receiver. 
     Following section describes the result of a simulation which is operated by this embodiment which is operated by this embodiment of the point positioning method. This result of the simulation is operated without utilizing the P code for the estimation equation. 
       FIG. 3  is a scatter diagram of the receiver&#39;s position with an embodiment of the point positioning method. The receiver&#39;s position is plotted in two dimensions, a longitude and a latitude.  FIG. 4  is a diagram of the receiver&#39;s ellipsoid height with an embodiment of the point positioning method. 
     A circuit plot shows an estimation result using this embodiment in FIGS.  3 , 4 . And an area configured by a continuous line  101  contains all circle plots in  FIG. 4 . A quadrangle plot shows a consuetudinary estimation result only using the code pseudorange in FIGS.  3 , 4 . And an area configured by a dotted line  102  contains almost quadrangle plots in  FIG. 4 . A triangle plot shows a consuetudinary estimation result using the code pseudorange, the delay data of the ionosphere, the delay data of the troposphere, and the clock error of the GPS satellite in FIGS.  3 , 4 . And an area configured by dash line  103  contains almost triangle plots in  FIG. 4 . An asterisk plot in  FIG. 3  and a continuous line in  FIG. 4  show a relative positioning result. 
     And a table 1 shows an average and a standard deviation of the receiver&#39;s position (the longitude, the latitude, and the ellipsoid height) estimated by the point positioning method described above. A table 2 shows an average and a standard deviation of the receiver&#39;s position (the longitude, the latitude, and the ellipsoid height) estimated by the consuetudinary point positioning method only using the code pseudorange. A table 3 shows an average and a standard deviation of the receiver&#39;s position (the longitude, the latitude, and the ellipsoid height) estimated by the consuetudinary point positioning method using the code pseudorange, the delay data of the ionosphere, the delay data of the troposphere, and the clock error of the GPS satellite. 
     In this simulation, an initial value of the receiver&#39;s position is given the coordinate value described in RINEX DATA “APPROXPOSITIONXYZ”. In addition, the delay data of the ionosphere is calculated applying what is called broadcasting model (Klobuchar model) to the data related to the delay of the ionosphere included in the navigation message. 
     The delay data of the troposphere is calculated applying a GPS satellite&#39;s elevation angle figured from the navigation message to a following equation. 
     
       
         
           
             [ 
             
               Expression 
               ⁢ 
               
                   
               
               ⁢ 
               21 
             
             ] 
           
         
       
       
         
           
             
               δ 
               ⁢ 
               
                   
               
               ⁢ 
               T 
               ⁢ 
               
                   
               
               ⁢ 
               e 
             
             = 
             
               
                 2.47 
                 
                   
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ξ 
                   
                   + 
                   0.0121 
                 
               
               ⁡ 
               
                 [ 
                 m 
                 ] 
               
             
           
         
       
     
     where ξ is an elevation angle. 
     The respective variances (σ) of the observation error are configured by 1.5 m at the code pseudorange, λ/10+1.5/10 m at the carrier phase, 3.6 m at the clock error of GPS satellite, 7.0 m at the delay of the ionosphere, and 0.7 m at the delay of the troposphere. On this simulation which estimates the receiver&#39;s position applying the least-square method, a calculation of the receiver&#39;s position should be operated repeatedly until a norm of a shift of the estimated receiver&#39;s position bellows 1*10 −3  m. 
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 Latitude 
                 Longitude 
                 Ellipsoid Height 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 Average 
                 34.98188223 
                 [°] 
                 135.9641106 
                 [°] 
                 224.93518 
                 [°] 
               
               
                 Standard 
                 0.195322465 
                 [m] 
                 0.362454829 
                 [m] 
                 0.526664731 
                 [m] 
               
               
                 Deviation 
               
               
                 Differential 
                 34.981876 
                 [°] 
                 135.964098 
                 [°] 
                 224.177 
                 [°] 
               
               
                 Positioning 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE 2 
               
               
                   
                   
               
               
                   
                 Latitude 
                 Logitude 
                 Ellipsoid Height 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 Average 
                 34.98188284 
                 [°] 
                 135.9641387 
                 [°] 
                 245.547332 
                 [°] 
               
               
                 Standard 
                 1.419451503 
                 [m] 
                 1.719274211 
                 [m] 
                 3.161014416 
                 [m] 
               
               
                 Deviation 
               
               
                 Differential 
                 34.981876 
                 [°] 
                 135.964098 
                 [°] 
                 224.177 
                 [°] 
               
               
                 Positioning 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE 3 
               
               
                   
                   
               
               
                   
                 Latitude 
                 Logitude 
                 Ellipsoid Height 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 Average 
                 34.98186963 
                 [°] 
                 135.9641115 
                 [°] 
                 236.1081403 
                 [°] 
               
               
                 Standard 
                 1.347937297 
                 [m] 
                 1.572959303 
                 [m] 
                 3.184914299 
                 [m] 
               
               
                 Deviation 
               
               
                 Differential 
                 34.981876 
                 [°] 
                 135.964098 
                 [°] 
                 224.177 
                 [°] 
               
               
                 Positioning 
               
               
                   
               
            
           
         
       
     
     According to these results, the receiver&#39;s position estimated by the method of point positioning described above could be estimated preciously. 
     As described above, we could estimate the point position of the receiver by using easier algorithm than the consuetudinary one, and the point position estimated by using this embodiment could be more precise than the consuetudinary one. 
     Thus applying the least-square method to the estimation operation, a LMBDA method using the variance estimated from an expression (37) should be applied. 
     Accordingly, the L1 integer ambiguity and the L2 integer ambiguity are fixed, and the receiver&#39;s position could be estimated more precisely. 
     And on this embodiment described above there is the point positioning method utilizing the observed value at only one epoch. But the point positioning method utilizing the observed values for some epochs could be considered. Utilizing this method the point positioning apparatus has a memory memorized the carrier phases, the code pseudoranges, the ephemeris data, the clock errors of the GPS satellites, the delay of the ionosphere, and the delay data of the troposphere for each GPS satellite for some epochs. 
     Utilizing the observed values for some epochs, for example utilizing only C/A code, the number of the observed values for K epochs is a following expression.
 
6 *n   s   *K= 6 K*n   s  
 
     On the other hand at a kinematic point positioning the number of the observing epoch is plural, so the number of the unknown values (unknown parameters) except for the L1 integer ambiguity and the L2 integer ambiguity is increased in proportion to the number of the observing epochs. And the number of the unknown values is a following expression.
 
4+5 *n   s +( K− 1)(4+3 n   s )=4 K +(3 K+ 2)* n   s  
 
     Therefore, the requirement to resolve the regression equation applying the least-square method is a following equation.
 
6 K*n   s &gt;=4 K +(3 K+ 2)* n   s  
 
(3 K− 2)* n   s &gt;=4 K  
 
     Namely, n s &gt;=4K/(3K−2) 
     And in the case where the number of the epochs increases from 2 to infinity, calculating an integer available for us, the number n s  of the GPS satellites is a following equation.
 
n s &gt;=2
 
     Thus increasing the number of the satellites, we could operate the point positioning with observing at least 2 GPS satellites. 
     Furthermore applying the Kalmann Filter to the kinetic point positioning as described above, we could estimate the receiver&#39;s position more precisely. Next an estimation algorithm of the point positioning applied the Kalmann Filter is described below. 
     At first the expression (31) is re-described to a following expression (38) with using an observation time t corresponding to the order of epoch.
 
 y   u,E   (j) ( t )= H   E   (j) ( t )θ u ( t )+ν u,E ( t )  (38)[Expression 22]
 
     t=1, 2, 3, . . . 
     This expression (38) re-described to a following vector-matrix expression (39). 
     
       
         
           
             
               
                 
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                           1 
                         
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                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
     At this point the clock error of the GPS satellite multiplied the speed of light cδt p , the delay of the ionosphere δI u , and the delay of the troposphere δT u  are defined by an expression (40a), an expression (40b), an expression (40c) respectively.
 
 cδt   p ( t )= cδte   p ( t )− e   δt     p   ( t )  (40a)
 
δ I   u ( t )=δ Ie   u ( t )− e   δI     u   ( t )  (40b)
 
δ T   u ( t )=δ Te   u ( t )− e   δT     u   ( t )  (40c)[Expression 24]
 
     These expression (40a-40c) are applied to the regression equation (39), furthermore, the observational vectors y (j)   L1,u (t), y (j)   L2,u (t), y (j)   CA,u (t), and y (j)   P,u (t) are defined by an following expression (41-44) respectively. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     25 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     y 
                     
                       
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                     + 
                     
                       c 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       t 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           e 
                           p 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     + 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       I 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           e 
                           u 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     - 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       T 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           e 
                           u 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       y 
                       
                         
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                         , 
                         u 
                       
                       
                         ( 
                         j 
                         ) 
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       Φ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           e 
                           
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                             , 
                             u 
                           
                           
                             ( 
                             j 
                             ) 
                           
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     + 
                     
                       c 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       t 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           e 
                           p 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     + 
                     
                       
                         
                           f 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                           2 
                         
                         
                           f 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                           2 
                         
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       I 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           e 
                           u 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     - 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       T 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           e 
                           u 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       y 
                       
                         CA 
                         , 
                         u 
                       
                       
                         ( 
                         j 
                         ) 
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       ρ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           e 
                           
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                             , 
                             u 
                           
                           
                             ( 
                             j 
                             ) 
                           
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     + 
                     
                       c 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       t 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           e 
                           p 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     + 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       I 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           e 
                           u 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     - 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       T 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           e 
                           u 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       y 
                       
                         P 
                         , 
                         u 
                       
                       
                         ( 
                         j 
                         ) 
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       ρ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           e 
                           
                             P 
                             , 
                             u 
                           
                           
                             ( 
                             j 
                             ) 
                           
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     + 
                     
                       c 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       t 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           e 
                           p 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     + 
                     
                       
                         
                           f 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                           2 
                         
                         
                           f 
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                           2 
                         
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       I 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           e 
                           u 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     - 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       T 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           e 
                           u 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
     A new regression equation is re-described to a following expression (45). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     26 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               y 
                               
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                                 , 
                                 u 
                               
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               y 
                               
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                                 , 
                                 u 
                               
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               y 
                               
                                 CA 
                                 , 
                                 u 
                               
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               y 
                               
                                 P 
                                 , 
                                 u 
                               
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   G 
                                   u 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             
                               1 
                             
                             
                               I 
                             
                             
                               0 
                             
                           
                           
                             
                               
                                 
                                   G 
                                   u 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             
                               1 
                             
                             
                               0 
                             
                             
                               I 
                             
                           
                           
                             
                               
                                 
                                   G 
                                   u 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               
                                 
                                   G 
                                   u 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                         
                         ] 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 u 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 c 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   t 
                                   u 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   λ 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   N 
                                   
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                     , 
                                     u 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   λ 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   N 
                                   
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                     
                                     , 
                                     u 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                         
                       
                         [ 
                         
                           
                             
                               
                                 
                                   
                                     e 
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         t 
                                         p 
                                       
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     e 
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         I 
                                         u 
                                       
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     e 
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         T 
                                         u 
                                       
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     λ 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                     ɛ 
                                     
                                       
                                         L 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         1 
                                       
                                       , 
                                       u 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     e 
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         t 
                                         P 
                                       
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       f 
                                       
                                         L 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         1 
                                       
                                       2 
                                     
                                     
                                       f 
                                       
                                         L 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         2 
                                       
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       e 
                                       
                                         δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           I 
                                           u 
                                         
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     e 
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         T 
                                         u 
                                       
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     λ 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                     ɛ 
                                     
                                       
                                         L 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         2 
                                       
                                       , 
                                       u 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     e 
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         t 
                                         P 
                                       
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     e 
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         I 
                                         u 
                                       
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     e 
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         T 
                                         u 
                                       
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                 + 
                                 
                                   e 
                                   
                                     CA 
                                     , 
                                     u 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     e 
                                     
                                       δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         t 
                                         P 
                                       
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     
                                       f 
                                       
                                         L 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         1 
                                       
                                       2 
                                     
                                     
                                       f 
                                       
                                         L 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         2 
                                       
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       e 
                                       
                                         δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           I 
                                           u 
                                         
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     
                                       e 
                                       
                                         δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           T 
                                           u 
                                         
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       P 
                                       , 
                                       u 
                                     
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
     At this point the Markov process model of the velocity of an unknown receiver&#39;s position u(t) and a clock error of the receiver cδt u (t) could be assumed, and a new state vector η u(t)  is defined by an following expression (46). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     27 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       η 
                       u 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             u 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             v 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             c 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               u 
                             
                           
                         
                       
                       
                         
                           
                             N 
                             
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                               , 
                               u 
                             
                           
                         
                       
                       
                         
                           
                             N 
                             
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                               , 
                               u 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   46 
                   ) 
                 
               
             
           
         
       
     
     And for this state vector η u(t)  a new state equation described by a following expression (47) is configured.
 
η u ( t+ 1)= Aη   u ( t )+ Bw ( t )  (47)[Expression 28]
 
     Additionally the observation equation described by the expression (45) re-describes to a following expression (48). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     29 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               y 
                               
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                                 , 
                                 u 
                               
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               y 
                               
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                                 , 
                                 u 
                               
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               y 
                               
                                 CA 
                                 , 
                                 u 
                               
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               y 
                               
                                 P 
                                 , 
                                 u 
                               
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 G 
                                 u 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           
                             0 
                           
                           
                             1 
                           
                           
                             
                               
                                 λ 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                               
                               ⁢ 
                               I 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             
                               
                                 G 
                                 u 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                           
                             
                               
                                 λ 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                               
                               ⁢ 
                               I 
                             
                           
                         
                         
                           
                             
                               
                                 G 
                                 u 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             
                               
                                 G 
                                 u 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           
                             0 
                           
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                       
                       ] 
                     
                     ⁢ 
                     
                         
                       
                         
                           [ 
                           
                             
                               
                                 
                                   u 
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   v 
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   c 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     t 
                                     u 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     λ 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                     N 
                                     
                                       
                                         L 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         1 
                                       
                                       , 
                                       u 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     λ 
                                     
                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                     N 
                                     
                                       
                                         L 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         2 
                                       
                                       , 
                                       u 
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                         + 
                         
                             
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       
                                         e 
                                         
                                           δ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             t 
                                             P 
                                           
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     + 
                                     
                                       
                                         e 
                                         
                                           δ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             I 
                                             u 
                                           
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     - 
                                     
                                       
                                         e 
                                         
                                           δ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             T 
                                             u 
                                           
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     + 
                                     
                                       
                                         λ 
                                         
                                           L 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           1 
                                         
                                       
                                       ⁢ 
                                       
                                         ɛ 
                                         
                                           
                                             L 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             1 
                                           
                                           , 
                                           u 
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         e 
                                         
                                           δ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             t 
                                             P 
                                           
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     + 
                                     
                                       
                                         
                                           f 
                                           
                                             L 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             1 
                                           
                                           2 
                                         
                                         
                                           f 
                                           
                                             L 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             2 
                                           
                                           2 
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           e 
                                           
                                             δ 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               I 
                                               u 
                                             
                                           
                                         
                                         ⁡ 
                                         
                                           ( 
                                           t 
                                           ) 
                                         
                                       
                                     
                                     - 
                                     
                                       
                                         e 
                                         
                                           δ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             T 
                                             u 
                                           
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     + 
                                     
                                       
                                         λ 
                                         
                                           L 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           2 
                                         
                                       
                                       ⁢ 
                                       
                                         ɛ 
                                         
                                           
                                             L 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             2 
                                           
                                           , 
                                           u 
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         e 
                                         
                                           δ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             t 
                                             P 
                                           
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     - 
                                     
                                       
                                         e 
                                         
                                           δ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             I 
                                             u 
                                           
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     - 
                                     
                                       
                                         e 
                                         
                                           δ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             T 
                                             u 
                                           
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     + 
                                     
                                       e 
                                       
                                         CA 
                                         , 
                                         u 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         e 
                                         
                                           δ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             t 
                                             P 
                                           
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         t 
                                         ) 
                                       
                                     
                                     - 
                                     
                                       
                                         
                                           f 
                                           
                                             L 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             1 
                                           
                                           2 
                                         
                                         
                                           f 
                                           
                                             L 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             2 
                                           
                                           2 
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           e 
                                           
                                             δ 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               I 
                                               u 
                                             
                                           
                                         
                                         ⁡ 
                                         
                                           ( 
                                           t 
                                           ) 
                                         
                                       
                                     
                                     - 
                                     
                                       
                                         
                                           e 
                                           
                                             δ 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               T 
                                               u 
                                             
                                           
                                         
                                         ⁡ 
                                         
                                           ( 
                                           t 
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         e 
                                         
                                           P 
                                           , 
                                           u 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   48 
                   ) 
                 
               
             
           
         
       
     
     This observation equation re-describes to an following vector-matrix expression (49).
 
 y   u,R   (j) ( t )= C   u   (j) ( t )η u ( t )+ν u,R ( t )  (49)[Expression 30]
 
     t=1, 2, . . . 
     Namely, it means these expression (47), (48), (49) constitute the Kalmann Filter. At this point the covariance matrix νu,R(t) could be configured as the expression (35). And utilizing the state equation described by the expression (47) and the observation equation described by the expression (48), even if the receiver moves, the estimation of the receiver&#39;s position could be operated. 
     According to the embodiment described above, even if the receiver moves, we could estimate the receiver&#39;s position precisely with estimating the movement of the receiver by the Kalmann Filter. 
     Meanwhile, in the embodiment described above, a GPS satellite&#39;s position is not estimated. But utilizing a following method (algorithm), the GPS satellite&#39;s position could be estimated. 
     At first the distance from the receiver to the GPS satellite is defined. And when both the GPS satellite&#39;s position and the receiver&#39;s position would be estimated, the linear approximation of the receiver&#39;s position could be described below. 
     
       
         
           
             [ 
             
               Expression 
               ⁢ 
               
                   
               
               ⁢ 
               31 
             
             ] 
           
         
       
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 r 
                                 u 
                                 p 
                               
                             
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 x 
                                 u 
                               
                             
                           
                           = 
                             
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   x 
                                   u 
                                 
                                 - 
                                 
                                   x 
                                   p 
                                 
                               
                               ) 
                             
                             
                               r 
                               u 
                               P 
                             
                           
                         
                         , 
                       
                     
                   
                   
                     
                       
                         
                           
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 r 
                                 u 
                                 p 
                               
                             
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 y 
                                 u 
                               
                             
                           
                           = 
                             
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   y 
                                   u 
                                 
                                 - 
                                 
                                   y 
                                   p 
                                 
                               
                               ) 
                             
                             
                               r 
                               u 
                               P 
                             
                           
                         
                         , 
                       
                     
                   
                   
                     
                       
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               r 
                               u 
                               p 
                             
                           
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               z 
                               u 
                             
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 z 
                                 u 
                               
                               - 
                               
                                 z 
                                 p 
                               
                             
                             ) 
                           
                           
                             r 
                             u 
                             p 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   ( 
                   
                     
                       p 
                       = 
                       
                         1 
                         , 
                         
                             
                         
                         ⁢ 
                         2 
                       
                     
                     , 
                     … 
                     ⁢ 
                     
                         
                     
                     , 
                     
                       n 
                       s 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     The linear approximation of the GPS satellite&#39;s position could be described 
     
       
         
           
             [ 
             
               Expression 
               ⁢ 
               
                   
               
               ⁢ 
               32 
             
             ] 
           
         
       
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 r 
                                 u 
                                 p 
                               
                             
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 x 
                                 p 
                               
                             
                           
                           = 
                             
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   x 
                                   u 
                                 
                                 - 
                                 
                                   x 
                                   p 
                                 
                               
                               ) 
                             
                             
                               r 
                               u 
                               P 
                             
                           
                         
                         , 
                       
                     
                   
                   
                     
                       
                         
                           
                             
                               
                                   
                               
                               ⁢ 
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   r 
                                   u 
                                   p 
                                 
                               
                             
                             
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 y 
                                 p 
                               
                             
                           
                           = 
                             
                           ⁢ 
                           
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     y 
                                     u 
                                   
                                   - 
                                   
                                     y 
                                     p 
                                   
                                 
                                 ) 
                               
                               
                                 r 
                                 u 
                                 P 
                               
                             
                           
                         
                         , 
                       
                     
                   
                   
                     
                       
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               r 
                               u 
                               p 
                             
                           
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               z 
                               p 
                             
                           
                         
                         = 
                           
                         ⁢ 
                         
                           - 
                           
                             
                               ( 
                               
                                 
                                   z 
                                   u 
                                 
                                 - 
                                 
                                   z 
                                   p 
                                 
                               
                               ) 
                             
                             
                               r 
                               u 
                               p 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   ( 
                   
                     
                       p 
                       = 
                       1 
                     
                     , 
                     2 
                     , 
                     … 
                     ⁢ 
                     
                         
                     
                     , 
                     
                       n 
                       s 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     And when each satellite&#39;s position is defined as sp≡[x p , y p , z p ] T , following equation is effected. 
     
       
         
           
             [ 
             
               Expression 
               ⁢ 
               
                   
               
               ⁢ 
               33 
             
             ] 
           
         
       
       
         
           
             
               
                 δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   r 
                   u 
                   p 
                 
               
               
                 δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 u 
               
             
             = 
             
               - 
               
                 
                   δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     r 
                     u 
                     p 
                   
                 
                 
                   δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     s 
                     p 
                   
                 
               
             
           
         
       
     
     Thus, an estimation value of the receiver&#39;s position u and the GPS satellite&#39;s position s p  are approximated as a following expression (50) by the linear Taylor series expansion using the observation value of the receiver&#39;s position u (j)  and the GPS satellite&#39;s position se p . 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     34 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           r 
                           u 
                           p 
                         
                         ≅ 
                         
                           
                             r 
                             
                               u 
                               ⁡ 
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                             p 
                           
                           + 
                           
                             
                               
                                 [ 
                                 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       r 
                                       u 
                                       p 
                                     
                                   
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     u 
                                   
                                 
                                 ] 
                               
                               
                                 
                                   u 
                                   - 
                                   
                                     u 
                                     ⁡ 
                                     
                                       ( 
                                       j 
                                       ) 
                                     
                                   
                                 
                                 , 
                                 
                                   
                                     s 
                                     p 
                                   
                                   - 
                                   
                                     se 
                                     p 
                                   
                                 
                               
                               T 
                             
                             ⁡ 
                             
                               [ 
                               
                                 u 
                                 - 
                                 
                                   s 
                                   p 
                                 
                                 - 
                                 
                                   ( 
                                   
                                     
                                       u 
                                       
                                         ( 
                                         j 
                                         ) 
                                       
                                     
                                     - 
                                     
                                       se 
                                       p 
                                     
                                   
                                   ) 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         ( 
                         
                           
                             p 
                             = 
                             
                               1 
                               , 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                           
                           , 
                           … 
                           ⁢ 
                           
                               
                           
                           , 
                           
                             n 
                             s 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   50 
                   ) 
                 
               
             
           
         
       
     
     Consequently, the expressions (10), (11), (12), (13) are reconfigured to the following expressions (51), (52), (53), (54) respectively. 
     
       
         
           
             Expression 
             ⁢ 
             
                 
             
             ⁢ 
             35 
           
         
       
       
         
           
             
               
                 
                   
                     
                       
                         
                           Φ 
                           
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                             , 
                             u 
                           
                           
                             p 
                             , 
                             
                               ( 
                               j 
                               ) 
                             
                           
                         
                         = 
                         
                           
                             r 
                             
                               u 
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                             p 
                           
                           + 
                           
                             
                               g 
                               
                                 p 
                                 , 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                               ⌊ 
                               
                                 u 
                                 - 
                                 
                                   s 
                                   p 
                                 
                                 - 
                                 
                                   ( 
                                   
                                     
                                       u 
                                       
                                         ( 
                                         j 
                                         ) 
                                       
                                     
                                     - 
                                     
                                       se 
                                       p 
                                     
                                   
                                   ) 
                                 
                               
                               ⌋ 
                             
                           
                           - 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               1 
                               u 
                               p 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             T 
                             u 
                             p 
                           
                         
                         + 
                         
                           c 
                           ⁡ 
                           
                             ( 
                             
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   t 
                                   u 
                                 
                               
                               - 
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   t 
                                   p 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             λ 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                           ⁢ 
                           
                             N 
                             
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                               , 
                               u 
                             
                             p 
                           
                         
                         + 
                         
                           
                             λ 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                           ⁢ 
                           
                             ɛ 
                             
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                               , 
                               u 
                             
                             p 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   51 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           Φ 
                           
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                             , 
                             u 
                           
                           
                             p 
                             , 
                             
                               ( 
                               j 
                               ) 
                             
                           
                         
                         = 
                         
                           
                             r 
                             
                               u 
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                             p 
                           
                           + 
                           
                             
                               g 
                               
                                 p 
                                 , 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                               ⌊ 
                               
                                 u 
                                 - 
                                 
                                   s 
                                   p 
                                 
                                 - 
                                 
                                   ( 
                                   
                                     
                                       u 
                                       
                                         ( 
                                         j 
                                         ) 
                                       
                                     
                                     - 
                                     
                                       se 
                                       p 
                                     
                                   
                                   ) 
                                 
                               
                               ⌋ 
                             
                           
                           - 
                           
                             
                               
                                 f 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                 
                                 2 
                               
                               
                                 f 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                                 2 
                               
                             
                             ⁢ 
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               1 
                               u 
                               p 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             T 
                             u 
                             p 
                           
                         
                         + 
                         
                           c 
                           ⁡ 
                           
                             ( 
                             
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   t 
                                   u 
                                 
                               
                               - 
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   t 
                                   p 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             λ 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                           
                           ⁢ 
                           
                             N 
                             
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                               , 
                               u 
                             
                             p 
                           
                         
                         + 
                         
                           
                             λ 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                           
                           ⁢ 
                           
                             ɛ 
                             
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                               , 
                               u 
                             
                             p 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   52 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           ρ 
                           
                             CA 
                             , 
                             u 
                           
                           
                             p 
                             , 
                             
                               ( 
                               j 
                               ) 
                             
                           
                         
                         = 
                         
                           
                             r 
                             
                               u 
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                             p 
                           
                           + 
                           
                             
                               g 
                               
                                 p 
                                 , 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                               ⌊ 
                               
                                 u 
                                 - 
                                 
                                   s 
                                   p 
                                 
                                 - 
                                 
                                   ( 
                                   
                                     
                                       u 
                                       
                                         ( 
                                         j 
                                         ) 
                                       
                                     
                                     - 
                                     
                                       se 
                                       p 
                                     
                                   
                                   ) 
                                 
                               
                               ⌋ 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             I 
                             u 
                             p 
                           
                         
                         + 
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             T 
                             u 
                             p 
                           
                         
                         + 
                         
                           c 
                           ⁡ 
                           
                             ( 
                             
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   t 
                                   u 
                                 
                               
                               - 
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   t 
                                   p 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           e 
                           
                             CA 
                             , 
                             u 
                           
                           p 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   53 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           ρ 
                           
                             P 
                             , 
                             u 
                           
                           
                             p 
                             ⁡ 
                             
                               ( 
                               j 
                               ) 
                             
                           
                         
                         = 
                         
                           
                             r 
                             
                               u 
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                             p 
                           
                           + 
                           
                             
                               g 
                               
                                 p 
                                 , 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                               ⌊ 
                               
                                 u 
                                 - 
                                 
                                   s 
                                   p 
                                 
                                 - 
                                 
                                   ( 
                                   
                                     
                                       u 
                                       
                                         ( 
                                         j 
                                         ) 
                                       
                                     
                                     - 
                                     
                                       se 
                                       p 
                                     
                                   
                                   ) 
                                 
                               
                               ⌋ 
                             
                           
                           + 
                           
                             
                               
                                 f 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                                 2 
                               
                               
                                 f 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                                 2 
                               
                             
                             ⁢ 
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               I 
                               u 
                               p 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             T 
                             u 
                             p 
                           
                         
                         + 
                         
                           c 
                           ⁡ 
                           
                             ( 
                             
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   t 
                                   u 
                                 
                               
                               - 
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   t 
                                   p 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           e 
                           
                             P 
                             , 
                             u 
                           
                           p 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   54 
                   ) 
                 
               
             
           
         
       
     
     At this point the observational L1 carrier phase Φe p(j)   L1,u , the observational L2 carrier phase Φe p,(j)   L2,u , the observational C/A code pseudorange ρe p,(j)   CA,u , and the observational P code pseudorange ρe p,(j)   P,u  are redefined to a following expressions (55), (56), (57), (58) respectively. 
     
       
         
           
             
               [ 
               
                 Expression 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 36 
               
               ] 
             
             ⁢ 
             
               
                 
                   
                     
                       
                         
                           
                             Φ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               e 
                               
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                                 , 
                                 u 
                               
                               
                                 p 
                                 , 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                           
                           ≡ 
                           
                             
                               Φ 
                               
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                                 , 
                                 u 
                               
                               
                                 p 
                                 , 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                             - 
                             
                               ⌊ 
                               
                                 
                                   r 
                                   
                                     u 
                                     
                                       ( 
                                       j 
                                       ) 
                                     
                                   
                                   p 
                                 
                                 - 
                                 
                                   
                                     g 
                                     
                                       p 
                                       , 
                                       
                                         ( 
                                         j 
                                         ) 
                                       
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         u 
                                         
                                           ( 
                                           j 
                                           ) 
                                         
                                       
                                       - 
                                       
                                         se 
                                         p 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ⌋ 
                             
                           
                         
                       
                       
                         
                           ( 
                           55 
                           ) 
                         
                       
                     
                     
                       
                         
                           
                             Φ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               e 
                               
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                                 , 
                                 u 
                               
                               
                                 p 
                                 , 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                           
                           ≡ 
                           
                             
                               Φ 
                               
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                                 , 
                                 u 
                               
                               
                                 p 
                                 , 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                             - 
                             
                               ⌊ 
                               
                                 
                                   r 
                                   
                                     u 
                                     
                                       ( 
                                       j 
                                       ) 
                                     
                                   
                                   p 
                                 
                                 - 
                                 
                                   
                                     g 
                                     
                                       p 
                                       , 
                                       
                                         ( 
                                         j 
                                         ) 
                                       
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         u 
                                         
                                           ( 
                                           j 
                                           ) 
                                         
                                       
                                       - 
                                       
                                         se 
                                         p 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ⌋ 
                             
                           
                         
                       
                       
                         
                           ( 
                           56 
                           ) 
                         
                       
                     
                     
                       
                         
                           
                             ρ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               e 
                               
                                 CA 
                                 , 
                                 u 
                               
                               
                                 p 
                                 , 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                           
                           ≡ 
                           
                             
                               ρ 
                               
                                 CA 
                                 , 
                                 u 
                               
                               
                                 p 
                                 , 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                             - 
                             
                               ⌊ 
                               
                                 
                                   r 
                                   
                                     u 
                                     
                                       ( 
                                       j 
                                       ) 
                                     
                                   
                                   p 
                                 
                                 - 
                                 
                                   
                                     g 
                                     
                                       p 
                                       , 
                                       
                                         ( 
                                         j 
                                         ) 
                                       
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         u 
                                         
                                           ( 
                                           j 
                                           ) 
                                         
                                       
                                       - 
                                       
                                         se 
                                         p 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ⌋ 
                             
                           
                         
                       
                       
                         
                           ( 
                           57 
                           ) 
                         
                       
                     
                     
                       
                         
                           
                             ρ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               e 
                               
                                 P 
                                 , 
                                 u 
                               
                               
                                 p 
                                 , 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                           
                           ≡ 
                           
                             
                               ρ 
                               
                                 P 
                                 , 
                                 u 
                               
                               
                                 p 
                                 , 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                             - 
                             
                               ⌊ 
                               
                                 
                                   r 
                                   
                                     u 
                                     
                                       ( 
                                       j 
                                       ) 
                                     
                                   
                                   p 
                                 
                                 - 
                                 
                                   
                                     g 
                                     
                                       p 
                                       , 
                                       
                                         ( 
                                         j 
                                         ) 
                                       
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         u 
                                         
                                           ( 
                                           j 
                                           ) 
                                         
                                       
                                       - 
                                       
                                         se 
                                         p 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ⌋ 
                             
                           
                         
                       
                       
                         
                           ( 
                           58 
                           ) 
                         
                       
                     
                   
                 
                 
                   
                       
                   
                 
               
             
           
         
       
     
     Thus the expressions (51), (52), (53), (54) are re-described to an expression (59), (60), (61), (62) respectively. 
     
       
         
           
             [ 
             
               Expression 
               ⁢ 
               
                   
               
               ⁢ 
               37 
             
             ] 
           
         
       
       
         
           
             
               
                 
                   
                     
                       
                         
                           Φ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             e 
                             
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                               , 
                               u 
                             
                             
                               p 
                               , 
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                           
                         
                         ≅ 
                         
                           
                             
                               g 
                               
                                 p 
                                 , 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                             ⁢ 
                             u 
                           
                           - 
                           
                             
                               g 
                               
                                 p 
                                 , 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                               s 
                               p 
                             
                           
                           - 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               I 
                               u 
                               p 
                             
                           
                           + 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               T 
                               u 
                               p 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                         
                           c 
                           ⁡ 
                           
                             ( 
                             
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   t 
                                   u 
                                 
                               
                               - 
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   t 
                                   p 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             λ 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                           ⁢ 
                           
                             N 
                             
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                               , 
                               u 
                             
                             p 
                           
                         
                         + 
                         
                           
                             λ 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                           ⁢ 
                           
                             ɛ 
                             
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                               , 
                               u 
                             
                             p 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   59 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           Φ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             e 
                             
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                               , 
                               u 
                             
                             
                               p 
                               , 
                               
                                 ( 
                                 j 
                                 ) 
                               
                             
                           
                         
                         ≅ 
                         
                           
                             
                               g 
                               
                                 p 
                                 , 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                             ⁢ 
                             u 
                           
                           - 
                           
                             
                               g 
                               
                                 p 
                                 , 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                             ⁢ 
                             
                               s 
                               p 
                             
                           
                           - 
                           
                             
                               
                                 f 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                                 2 
                               
                               
                                 f 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                                 2 
                               
                             
                             ⁢ 
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               I 
                               u 
                               p 
                             
                           
                           + 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               T 
                               u 
                               p 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                         
                           c 
                           ⁢ 
                           
                             ( 
                             
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   t 
                                   u 
                                 
                               
                               - 
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   t 
                                   p 
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             λ 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                           
                           ⁢ 
                           
                             N 
                             
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                               , 
                               u 
                             
                             p 
                           
                         
                         + 
                         
                           
                             λ 
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                           
                           ⁢ 
                           
                             ɛ 
                             
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                               , 
                               u 
                             
                             p 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   60 
                   ) 
                 
               
             
             
               
                 
                   
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       e 
                       
                         CA 
                         , 
                         u 
                       
                       
                         p 
                         , 
                         
                           ( 
                           j 
                           ) 
                         
                       
                     
                   
                   ≅ 
                   
                     
                       
                         g 
                         
                           p 
                           , 
                           
                             ( 
                             j 
                             ) 
                           
                         
                       
                       ⁢ 
                       u 
                     
                     - 
                     
                       
                         g 
                         
                           p 
                           , 
                           
                             ( 
                             j 
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         s 
                         p 
                       
                     
                     + 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         I 
                         u 
                         p 
                       
                     
                     + 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         T 
                         u 
                         p 
                       
                     
                     + 
                     
                       c 
                       ⁢ 
                       
                         ⌊ 
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               u 
                             
                           
                           - 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               p 
                             
                           
                         
                         ⌋ 
                       
                     
                     + 
                     
                       e 
                       
                         CA 
                         , 
                         u 
                       
                       p 
                     
                   
                 
               
               
                 
                   ( 
                   61 
                   ) 
                 
               
             
             
               
                 
                   
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       e 
                       
                         P 
                         , 
                         u 
                       
                       
                         p 
                         , 
                         
                           ( 
                           j 
                           ) 
                         
                       
                     
                   
                   ≅ 
                   
                     
                       
                         g 
                         
                           p 
                           , 
                           
                             ( 
                             j 
                             ) 
                           
                         
                       
                       ⁢ 
                       u 
                     
                     - 
                     
                       
                         g 
                         
                           p 
                           , 
                           
                             ( 
                             j 
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         s 
                         p 
                       
                     
                     + 
                     
                       
                         
                           f 
                           1 
                           2 
                         
                         
                           f 
                           2 
                           2 
                         
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         I 
                         u 
                         P 
                       
                     
                     + 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         T 
                         u 
                         p 
                       
                     
                     + 
                     
                       c 
                       ⁡ 
                       
                         [ 
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               u 
                             
                           
                           - 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               t 
                               p 
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       e 
                       
                         P 
                         , 
                         u 
                       
                       p 
                     
                   
                 
               
               
                 
                   ( 
                   62 
                   ) 
                 
               
             
           
         
       
     
     The observational L1 carrier phase Φe p,(j)   L1,u , the observational L2 carrier phase Φe p,(j)   L2,u , the observational C/A code pseudorange ρe p,(j)   CA,u , and the observational P code pseudorange ρe p,(j)   P,u  are redefined to a following expressions respectively. 
     
       
         
           
             [ 
             
               Expression 
               ⁢ 
               
                   
               
               ⁢ 
               138 
             
             ] 
           
         
       
       
         
           
             
               
                 
                   
                     
                       Φ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         e 
                         
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                           , 
                           u 
                         
                         
                           s 
                           , 
                           
                             ( 
                             j 
                             ) 
                           
                         
                       
                     
                     ≡ 
                     
                       [ 
                       
                         
                           
                             
                               Φ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                   , 
                                   u 
                                 
                                 
                                   1 
                                   , 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               Φ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                   , 
                                   u 
                                 
                                 
                                   
                                     n 
                                     s 
                                   
                                   , 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
                       Φ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         e 
                         
                           
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                           , 
                           u 
                         
                         
                           s 
                           , 
                           
                             ( 
                             j 
                             ) 
                           
                         
                       
                     
                     ≡ 
                     
                       [ 
                       
                         
                           
                             
                               Φ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                   , 
                                   u 
                                 
                                 
                                   1 
                                   , 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               Φ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                   , 
                                   u 
                                 
                                 
                                   
                                     n 
                                     s 
                                   
                                   , 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       ρ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         e 
                         
                           CA 
                           , 
                           u 
                         
                         
                           s 
                           , 
                           
                             ( 
                             j 
                             ) 
                           
                         
                       
                     
                     ≡ 
                     
                       [ 
                       
                         
                           
                             
                               ρ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   CA 
                                   , 
                                   u 
                                 
                                 
                                   1 
                                   , 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               ρ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   CA 
                                   , 
                                   u 
                                 
                                 
                                   
                                     n 
                                     s 
                                   
                                   , 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
                       ρ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         e 
                         
                           P 
                           , 
                           u 
                         
                         
                           s 
                           , 
                           
                             ( 
                             j 
                             ) 
                           
                         
                       
                     
                     ≡ 
                     
                       [ 
                       
                         
                           
                             
                               ρ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   P 
                                   , 
                                   u 
                                 
                                 
                                   1 
                                   , 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               ρ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   P 
                                   , 
                                   u 
                                 
                                 
                                   
                                     n 
                                     s 
                                   
                                   , 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     Consequently, the expressions (59), (60), (61), (62) are re-described to the vector-matrix as a following expression (63). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     39 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               Φ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                   , 
                                   u 
                                 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               Φ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                   , 
                                   u 
                                 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               ρ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   CA 
                                   , 
                                   u 
                                 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               ρ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 e 
                                 
                                   P 
                                   , 
                                   u 
                                 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   G 
                                   u 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                               
                                 1 
                               
                               
                                 
                                   - 
                                   
                                     G 
                                     
                                       D 
                                       , 
                                       u 
                                     
                                     
                                       ( 
                                       j 
                                       ) 
                                     
                                   
                                 
                               
                               
                                 
                                   - 
                                   I 
                                 
                               
                               
                                 
                                   - 
                                   I 
                                 
                               
                               
                                 I 
                               
                               
                                 I 
                               
                               
                                 0 
                               
                             
                             
                               
                                 
                                   G 
                                   u 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                               
                                 1 
                               
                               
                                 
                                   - 
                                   
                                     G 
                                     
                                       D 
                                       , 
                                       u 
                                     
                                     
                                       ( 
                                       j 
                                       ) 
                                     
                                   
                                 
                               
                               
                                 
                                   - 
                                   I 
                                 
                               
                               
                                 
                                   
                                     - 
                                     
                                       
                                         f 
                                         
                                           L 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           1 
                                         
                                         2 
                                       
                                       
                                         f 
                                         
                                           L 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           2 
                                         
                                         2 
                                       
                                     
                                   
                                   ⁢ 
                                   I 
                                 
                               
                               
                                 I 
                               
                               
                                 0 
                               
                               
                                 I 
                               
                             
                             
                               
                                 
                                   G 
                                   u 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                               
                                 1 
                               
                               
                                 
                                   - 
                                   
                                     G 
                                     
                                       D 
                                       , 
                                       u 
                                     
                                     
                                       ( 
                                       j 
                                       ) 
                                     
                                   
                                 
                               
                               
                                 
                                   - 
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                                 I 
                               
                               
                                 I 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 
                                   G 
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                                     ( 
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                                       , 
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                                       ( 
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                                       f 
                                       
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                                         ⁢ 
                                         
                                             
                                         
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                                         ⁢ 
                                         
                                             
                                         
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                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         [ 
                         
                             
                         
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                                     ⁢ 
                                     
                                         
                                     
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                                       ⁢ 
                                       
                                           
                                       
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                                     ⁢ 
                                     
                                         
                                     
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                                       ⁢ 
                                       
                                           
                                       
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                                     , 
                                     u 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       + 
                       
                         
 
                       
                       ⁢ 
                       
                         [ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               
                                 
                                   λ 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
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                                 ⁢ 
                                 
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                                       ⁢ 
                                       
                                           
                                       
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                                     u 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
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                                     ⁢ 
                                     
                                         
                                     
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                                       ⁢ 
                                       
                                           
                                       
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                                       2 
                                     
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                                     u 
                                   
                                 
                               
                             
                           
                           
                             
                               
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                                   , 
                                   u 
                                 
                               
                             
                           
                           
                             
                               
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                                   , 
                                   u 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   where 
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       G 
                       
                         D 
                         , 
                         u 
                       
                       
                         ( 
                         j 
                         ) 
                       
                     
                     ≡ 
                     
                       
                         [ 
                         
                           
                             
                               
                                 g 
                                 
                                   1 
                                   , 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                             
                               0 
                             
                             
                               … 
                             
                             
                               
                                   
                               
                             
                             
                               0 
                             
                           
                           
                             
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                                 g 
                                 
                                   2 
                                   , 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                             
                               0 
                             
                             
                               … 
                             
                             
                               ⋮ 
                             
                           
                           
                             
                               … 
                             
                             
                               … 
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               … 
                             
                             
                               0 
                             
                             
                               
                                 g 
                                 
                                   
                                     n 
                                     s 
                                   
                                   , 
                                   
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             n 
                             s 
                           
                           × 
                           3 
                           ⁢ 
                           
                             n 
                             s 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           matrix 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   63 
                   ) 
                 
               
             
           
         
       
     
     An observation vector sk of the satellite with a precise orbit is defined to an following expression.
 
 sk=s+e   s   , s =[( s   1 ) T ,( s   1 ) T ,Λ,( s   1 ) T   ]T , 3n s ×1  [Expression 40]
 
     Applying this expression to the expression (63), the expression (64) is derived. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     41 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     [ 
                     
                       
                         
                           
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                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 e 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                   , 
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                                   ( 
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                                   ) 
                                 
                               
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                                 ( 
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                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
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                             ⁢ 
                             
                                 
                             
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                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
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                                   ( 
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                               ⁡ 
                               
                                 ( 
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                             ⁢ 
                             
                                 
                             
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                                   ( 
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                                 ( 
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                             ⁢ 
                             
                                 
                             
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                                   ( 
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                               ⁡ 
                               
                                 ( 
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                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             sk 
                             ⁡ 
                             
                               ( 
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                             ⁢ 
                             
                                 
                             
                             ⁢ 
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                                 s 
                               
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                                 ( 
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                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 Ie 
                                 u 
                               
                               ⁡ 
                               
                                 ( 
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                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 Te 
                                 u 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
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                     ] 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   G 
                                   u 
                                   
                                     ( 
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                         ] 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 u 
                                 ⁡ 
                                 
                                   ( 
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                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 c 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     δt 
                                     u 
                                   
                                   ⁡ 
                                   
                                     ( 
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                                 ⁡ 
                                 
                                   ( 
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                                   ) 
                                 
                               
                             
                           
                           
                             
                               
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                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
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                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
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                                     s 
                                   
                                   ⁡ 
                                   
                                     ( 
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                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
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                                     u 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     t 
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
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                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
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                                     u 
                                   
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                                     ( 
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                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
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                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
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                                 ⁢ 
                                 
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                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
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                                     ⁢ 
                                     
                                         
                                     
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                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                     
                                     , 
                                     u 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       [ 
                       
                         
                           
                             
                               
                                 λ 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 
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                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                     
                                     , 
                                     u 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
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                                   ⁢ 
                                   
                                       
                                   
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                                       L 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                     
                                     , 
                                     u 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 e 
                                 
                                   CA 
                                   , 
                                   u 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 e 
                                 
                                   P 
                                   , 
                                   u 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 e 
                                 s 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 e 
                                 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     t 
                                     s 
                                   
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 e 
                                 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     I 
                                     u 
                                   
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 e 
                                 
                                   δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     T 
                                     u 
                                   
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   64 
                   ) 
                 
               
             
           
         
       
     
     Utilizing the regression equation described above, we could estimate the GPS satellite&#39;s position precisely. And estimating the GPS satellite position precisely, we could estimate the receiver&#39;s position more precisely. 
     Furthermore, utilizing a following algorithm, we could apply the Kalmann Filter to the regression equation, too. 
     Namely, the state vector s, the clock errors of the GPS cδt p , the delay of the ionosphere δI u , the delay of the troposphere δT u  are defined the following expressions (65a), (65b), (65c), (65d) respectively.
 
 s ( t )= sk ( t )− e   s ( t )  (65a)
 
 cδt   p ( t )= cδte   p ( t )− e   δt     p   ( t )  (65b)
 
δ I   u ( t )=δ Ie   u ( t )− e   δI     u   ( t )  (65c)
 
δ T   u ( t )=δ Te   u ( t )− e   δT     u   ( t )  (65d)[Expression 42]
 
     These expressions are applied to the regression equation (64). Furthermore the observational vectors y (j)   L1,u (t), y (j)   L2,u (t), y (j)   CA,u (t), and y (j)   P,u (t) are defined by a following expressions (66), (67), (68), (69) respectively. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     43 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       y 
                       
                         
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                         , 
                         u 
                       
                       
                         ( 
                         j 
                         ) 
                       
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       Φ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           e 
                           
                             
                               L 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                             , 
                             u 
                           
                           
                             ( 
                             j 
                             ) 
                           
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     + 
                     
                       
                         
                           G 
                           
                             D 
                             , 
                             u 
                           
                           
                             ( 
                             j 
                             ) 
                           
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                       ⁢ 
                       
                         sk 
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     + 
                     
                       c 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           te 
                           p 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     + 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           Ie 
                           u 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     - 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           Te 
                           u 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   66 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             y 
                             
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                               , 
                               u 
                             
                             
                               ( 
                               j 
                               ) 
                             
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                         = 
                         
                           
                             Φ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 e 
                                 
                                   
                                     L 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                   , 
                                   u 
                                 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               
                                 G 
                                 
                                   D 
                                   , 
                                   u 
                                 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             ⁢ 
                             
                               sk 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                           + 
                           
                             c 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               te 
                               p 
                             
                             ⁢ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             
                               f 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                               2 
                             
                             
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     Consequently a new regression equation is described as an expression (70). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Expression 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     44 
                   
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                   ( 
                   70 
                   ) 
                 
               
             
           
         
       
     
     And defining the state equation to this observational equation (a new regression equation), the Kalmann Filter could be configured. 
     Meanwhile, in this embodiment described above, the delay of the ionosphere and the delay of the troposphere are estimated at the receiver. But the base station whose position is known and fixed exist, the delay of the ionosphere and the delay of the troposphere could be estimated at the base station using the method described above. And giving the delay of the ionosphere and the delay of the troposphere estimated at base station, and more the error of the satellite orbit and the clock error of the GPS satellite estimated at base station to the receiver, at the receiver we could estimate the receiver&#39;s position more precisely. 
     Additionally, with the embodiment described above, the least-square method is applied to the linear regression equation, but the other estimation algorithm of the parameter could be applied. 
     And with the embodiment described above, the linear Taylor series expansion is applied to the linear approximation of the receiver&#39;s position, but the other expression of the linear approximation could be applied. 
     And with the embodiment described above, the LAMBDA method is applied to the linear regression equation, but the other estimation algorithm to fix an integer ambiguity could be applied. 
     And with the embodiment described above, the Kalmann Filter is applied to the linear regression equation, but the other estimation algorithm of condition could be applied.