(3) Error related to the receivers: Receivers may be subject to varying degrees of
multipath effects and electromagnetic interference. This part of error also includes
the noise of the receiver itself. Multipath error does not show normal distribution
but sinusoidal motion with a period of several minutes as the satellite moves [7].
The above error can be classified into deviation and noise according to the characteristics of magnitude and speed of changing. The deviation varies slowly in a certain
period of time. For example, the magnitude of ionospheric delay generally remains
unchanged in a few seconds or even minutes. The noise changes quickly and is difficult
to measure. But its mean, variance, correlation function and other statistical indicators
in a certain period of time can be calculated.
3 Characteristic Analysis and Experimental Study of BDS
Positioning Error
The data characteristics of positioning error are the guarantee of choosing a reasonable
distribution model and improving the accuracy of BDS integrity risk evaluation. BDS
has two positioning methods: Single-point positioning and differential positioning. The
error of single-point positioning can reach to tens of meters without any compensation
which is difficult to meet the positioning performance requirements of many occasions.
Differential positioning uses the spatial and temporal correlation of satellite clock error,
ephemeris error and atmospheric delay error to eliminate the measurement error of user
stations and improve the positioning performance by calculating the error correction of
reference stations with known positions. Formulas (1) and (2) are pseudo-range
observer expressions q
ðiÞ
u and q
ðiÞ
uc for single-point positioning and differential positioning, respectively.
q
ðiÞ
u ¼ R
ðiÞ
u þ cðDt u À tÞ þ g
ðiÞ
u þ I
ðiÞ
u þ T
ðiÞ
u þ e
ðiÞ
u
ð1Þ
q
ðiÞ
uc ¼ R
ðiÞ
u þ cDt ur þ g
ðiÞ
ur þ I
ðiÞ
ur þ T
ðiÞ
ur þ e
ðiÞ
ur
ð2Þ
Among them, superscript i represents satellite number, subscript u, r represent user
receiver and reference receiver, respectively. R
ðiÞ
u is the real distance from satellite i to
the user receiver. Dt u ; t; g
ðiÞ
u ; I
ðiÞ
u ; T
ðiÞ
u ; e
ðiÞ
u represent clock error of the user receiver,
satellite clock error, ephemeris error, ionospheric error, tropospheric error, noise,
respectively, q
ðiÞ
uc ; g
ðiÞ
ur ; I
ðiÞ
ur ; T
ðiÞ
ur ; e
ðiÞ
ur represent the corresponding correction residues and
satellite clock error can be eliminated completely. g
ðiÞ
ur ; I
ðiÞ
ur ; T
ðiÞ
ur are nearly to zero under
short baseline. c is the speed of light. Differential positioning could eliminate the
amount of common error.
Because the ionospheric delay error, tropospheric delay error and multipath effect
error show non-zero mean and asymmetric non-Gaussian characteristics, the superposition of the error makes the positioning error projected to the position domain shows
non-Gaussian characteristics.
38
H. Li et al.
multipath effects and electromagnetic interference. This part of error also includes
the noise of the receiver itself. Multipath error does not show normal distribution
but sinusoidal motion with a period of several minutes as the satellite moves [7].
The above error can be classified into deviation and noise according to the characteristics of magnitude and speed of changing. The deviation varies slowly in a certain
period of time. For example, the magnitude of ionospheric delay generally remains
unchanged in a few seconds or even minutes. The noise changes quickly and is difficult
to measure. But its mean, variance, correlation function and other statistical indicators
in a certain period of time can be calculated.
3 Characteristic Analysis and Experimental Study of BDS
Positioning Error
The data characteristics of positioning error are the guarantee of choosing a reasonable
distribution model and improving the accuracy of BDS integrity risk evaluation. BDS
has two positioning methods: Single-point positioning and differential positioning. The
error of single-point positioning can reach to tens of meters without any compensation
which is difficult to meet the positioning performance requirements of many occasions.
Differential positioning uses the spatial and temporal correlation of satellite clock error,
ephemeris error and atmospheric delay error to eliminate the measurement error of user
stations and improve the positioning performance by calculating the error correction of
reference stations with known positions. Formulas (1) and (2) are pseudo-range
observer expressions q
ðiÞ
u and q
ðiÞ
uc for single-point positioning and differential positioning, respectively.
q
ðiÞ
u ¼ R
ðiÞ
u þ cðDt u À tÞ þ g
ðiÞ
u þ I
ðiÞ
u þ T
ðiÞ
u þ e
ðiÞ
u
ð1Þ
q
ðiÞ
uc ¼ R
ðiÞ
u þ cDt ur þ g
ðiÞ
ur þ I
ðiÞ
ur þ T
ðiÞ
ur þ e
ðiÞ
ur
ð2Þ
Among them, superscript i represents satellite number, subscript u, r represent user
receiver and reference receiver, respectively. R
ðiÞ
u is the real distance from satellite i to
the user receiver. Dt u ; t; g
ðiÞ
u ; I
ðiÞ
u ; T
ðiÞ
u ; e
ðiÞ
u represent clock error of the user receiver,
satellite clock error, ephemeris error, ionospheric error, tropospheric error, noise,
respectively, q
ðiÞ
uc ; g
ðiÞ
ur ; I
ðiÞ
ur ; T
ðiÞ
ur ; e
ðiÞ
ur represent the corresponding correction residues and
satellite clock error can be eliminated completely. g
ðiÞ
ur ; I
ðiÞ
ur ; T
ðiÞ
ur are nearly to zero under
short baseline. c is the speed of light. Differential positioning could eliminate the
amount of common error.
Because the ionospheric delay error, tropospheric delay error and multipath effect
error show non-zero mean and asymmetric non-Gaussian characteristics, the superposition of the error makes the positioning error projected to the position domain shows
non-Gaussian characteristics.
38
H. Li et al.