Data Combinations (GPS Observation Equations and Equivalence Properties)

Data combinations are methods of combining GPS data measured with the same receiver at the same station. Usually, observables are the code pseudoranges, carrier phases and Doppler at working frequencies such as C/A code,tmp2A-1080code,tmp2A-1081In the future, there will also be tmp2A-1082According to the observation equations of the observables, a suitable combination can be advantageous for understanding and solving GPS problems.


For convenience, code, phase and Doppler observables are simplified and rewritten as (cf. Eqs. 6.1-6.3)

tmp2A-1086

Where j is the index of frequency f, the means of the other symbols are the same as the notes of Eqs. 6.1-6.3. Equation 6.47 is an approximation for code. A general code-code combination can be formed bytmp2A-1087tmp2A-1088are arbitrary constants. However, in order to make such a combination that still has the sense of a code survey, a standardised combination has to be formed by

tmp2A-1091

The newly-formed code R can then be interpreted as a weight-averaged code survey oftmp2A-1092The mathematical model of the observable Eq. 6.44 isgenerally still valid for R. Denoting the standard deviation of code observabletmp2A-1093 the newly-formed code observation R has the standard deviation of

tmp2A-1096

Because of


tmp2A-1097

(cf., e.g., Wang et al. 1979; Bronstein and Semendjajew 1987), one has the property of

tmp2A-1098

where m is the maximum index. Therefore in our case, one hastmp2A-1099for combinations of two or three code observables.

A general phase-phase linear combination can be formed by

tmp2A-1100

where the combined signal has the frequency and wavelength

tmp2A-1101

means the measured distance (with ambiguity!) and can be presented alternatively as

tmp2A-1103

Mathematical model of Eq. 6.45 is generally still valid for the newly-formedtmp2A-1104 Denoting the standard deviation of phase observabletmp2A-1105the newly- formed observation has a variance of

tmp2A-1110

with m = 2 or 3 for combinations of two or three phases.

That is, the data combination will degrade the quality of the original data. Linear combinationstmp2A-1111are called wide-lane and x-lane combinations with wavelengths of about 86.2 cm and 15.5 cm. They reduce the first order ionospheric effects on frequencytmp2A-1112to 40% and 20%, respectively.is called a narrow-lane combination.

Ionosphere-Free Combinations

Due to Eqs. 6.44-6.47, phase-phase and code-code ionosphere-free combinations can be formed by (cf. Sect. 5.1)

tmp2A-1117

The related observation equations can be formed from Eqs. 6.44 and 6.45 as

tmp2A-1118

where

tmp2A-1119

denote the residuals after the combination of code and phase, respectively.

The advantages of such ionosphere-free combinations are that the ionospheric effects have disappeared from the observation Eqs. 6.55 and 6.56 and the other terms of the equations have remained the same. However, the combined ambiguity is not an integer anymore, and the combined observables have higher standard deviations. Equations 6.55 and 6.56 are indeed first order ionosphere-free combinations.

Second order ionosphere-free combinations can be formed by (see Sect. 5.1.2 for details)

tmp2A-1121

where

tmp2A-1122

The related observation equations are the same as Eqs. 6.55 and 6.56, withtmp2A-1123 given above.

Geometry-Free Combinations

Due to Eqs. 6.44-6.46, code-code, phase-phase and phase-code geometry-free combinations can be formed by

tmp2A-1125

For an ionospheric model of the second order, one has approximately

tmpD-1_thumb

The geometry-free code-code and phase-phase combinations cancel out all other terms in the observation equations except the ionospheric term and the ambiguity parameters. Recalling the discussions of Sect. 5.1,tmpD-2_thumbis the ionospheric path delay and can be considered a mapping of the zenith delaytmpD-3_thumbwhere F is  the mapping function (cf. Sect. 5.1). So one has

tmpD-6_thumb

wheretmpD-7_thumbhave the physical meaning of total electronic contents at the signal path direction and the zenith direction, respectively.tmpD-8_thumbis then independent from the zenith angle of the satellite. If the variability of the electronic contents at the zenith direction is stable enough,tmpD-9_thumbcan be modelled by a step function or a first order polynomial with a reasonably short time intervaltmpD-10_thumbby

tmpD-15_thumb

wheretmpD-16_thumbare the beginning and ending time of the GPS survey.tmpD-17_thumbcan be, e.g., selected by 30 minutes.is the coefficient of the polynomial.tmpD-18_thumb

Geometry-free combinations of Eqs. 6.60, 6.61 and 6.63 (only for j = 1) can be considered a linear transformation of the original observable vectortmpD-19_thumb by

tmpD-24_thumb

where Eq. 6.65 is used and

tmpD-25_thumb

Equation 6.68 is called an ambiguity-ionospheric equation. For any viewed GPS satellite, Eq. 6.68 is solvable. If the variance vector of the observable vector is

tmpD-26_thumb

then the covariance matrix of the original observable vector is (cf. Sect. 6.2)

tmpD-27_thumb

and the covariance matrix of the transformed observable vector (left side of Eq. 6.68) is (cf. Sect. 6.4)

tmpD-28_thumb

Taking all the data measured at a station into account, the ambiguity and the ionospheric parameters (as a step function of the polynomial) can be solved by using Eq. 6.68 with the weight of Eq. 6.69. Taking the data station by station into account, all ambiguity and ionospheric parameters can be determined. The different weights of the code and phase measurements are considered exactly here. Due to the physical property of the ionosphere, all solved ionospheric parameters shall have the same sign. Even though the observation Eq. 6.68 is already a linear equation system, an initialisation is still helpful to avoid numbers from ambiguities that are too big. The broadcasting ionospheric model can be used for initialisation of the related ionospheric parameters.

A geometry-free combination of Eq. 6.62 can be used as a quality check of the Dop-pler data.

Standard Phase-Code Combination

Traditionally, phase and code combinations are used to compute the wide-lane ambiguity (cf. Sjoeberg 1999; Hofmann-Wellenhof et al. 1997). The formulas can be derived as follows. Dividing Xj into Eq. 6.63 and forming the difference for j = 1 and j = 2, one gets

tmpD-29_thumb

wheretmpD-30_thumband they are called wide-lane observable and am biguity; c is the velocity of light and A1 is the ionospheric parameter. The error term is omitted here. Equation 6.60 can be rewritten as (by omitting the error term)

tmpD-32_thumb

and then one gets

tmpD-33_thumb

Substituting Eq. 6.72 into 6.70 yields

tmpD-34_thumb

Equation 6.73 is the most popular formula for computing wide-lane ambiguities using phase and code observables. The un-differenced ambiguity N1 can be derived as follows. SettingtmpD-35_thumbinto Eq. 6.61 and omitting the error term, one has

tmpD-37_thumb

wheretmpD-38_thumbis the wide-lane frequency.

Compared with the adjustment method derived in Sect. 6.5.2, it is obvious that the quality differences of the phase and code data are not considered by using Eqs. 6.73 and 6.74 for determining the ambiguity parameters. Therefore, the method proposed in Sect. 6.5.2 is suggested for use.

Ionospheric Residuals

Considering the GPS observables as a time series, the geometry-free combinations of Eqs. 6.60-6.64 can be rewritten as

tmpD-40_thumb

where

tmpD-41_thumb

The differences of the above observable combinations at the two succeeded epochstmpD-42_thumb andtmpD-43_thumbcan be formed:

tmpD-48_thumb

wheretmpD-49_thumbis a time difference operator, for any time functiontmpD-50_thumb is valid.

Because the time differences of the ionospheric effectstmpD-51_thumbare generally very small, they are called ionospheric residuals. In the case of no cycle slips, i.e., ambiguitiestmpD-52_thumbare constant,tmpD-53_thumbequal zero. Equations 6.79-6.81 are called ionospheric residual combinations. The first combination of Eq. 6.79 can be used for a consistency check of two code measurements. Equations 6.80 and 6.81 can be used for a cycle slip check. Equation 6.81 is a phase-code combination, due to the lower accuracy of the code measurements; it can be used only to check for big cycle slips. Equation 6.80 is a phase-phase combination, and therefore it has higher sensibility related to the cycle slips. However, two special cycle slipstmpD-54_thumbcan lead to a very small combination oftmpD-55_thumbExamples of the combinations can be found, e.g., in (Hofmann-Wellenhof et al. 1997). That is, even the ionospheric residual of Eq. 6.80 is very small; it may not guarantee that there are no cycle slips.

Differential Doppler and Doppler Integration Differential Doppler

The numerical differentiation of the original observables given in Eqs. 6.44 and 6.45 at the two succeeded epochstmpD-56_thumbcan be formed as

tmpD-65_thumb

wheretmpD-66_thumbis a numerical differentiation operator andtmpD-67_thumb

The left-hand side of Eq. 6.83 is called differential Doppler. Ionospheric residuals are negligible and omitted here. The third terms of Eqs. 6.82 and 6.83 on the right-hand side are small residual errors. For convenience of comparison, the Doppler observable model of Eq. 6.46 is copied below:

tmpD-70_thumb

It is obvious that Eqs. 6.83 and 6.84 are nearly the same. The only difference is that in Doppler Eq. 6.84 the observed Doppler is an instantaneous one and its model is presented by theoretical differentiation, whereas the term on the left-hand side of Eq. 6.83 is the numerically differenced Doppler (formed by phases) and its model is presented by numerical differentiation. Doppler measurement measures the instantaneous motion of the GPS antenna, whereas differential Doppler describes a kind of average velocity of the antenna during the two succeeded epochs. The velocity solution of Eq. 6.83 (denoted bytmpD-71_thumbcan be used to predict the future kinematic position by

tmpD-73_thumb

In other words, differential Doppler can be used as the system equation of a Kalman filter for kinematic positioning. The Kalman filter will be discussed in the next t. A Kalman filter using differential Doppler will be discussed in Sect. 9.8.

Doppler Integration

Integrating the instantaneous Doppler Eq. 6.84, one has

tmpD-74_thumb

Using the operatortmpD-75_thumbto the un-differenced phase Eq. 6.45 and code Eq. 6.44, one gets

tmpD-77_thumb

where the same symbols are used for the error terms (later too). Differencing the first equation of Eq. 6.86 with the integrated Doppler leads to

tmpD-78_thumb

That is, the integrated Doppler can be used for cycle slip detection. Such a cycle slip detection method is very reasonable. Phase is measured by keeping track of the partial phase and accumulating the integer count. If any loss of lock of the signal happens during the time, the integer accumulating will be wrong, i.e., cycle slip happens. Therefore, an external instantaneous Doppler integration can be used as an alternative method of cycle slip detection. The integration can be made first by fitting the Doppler with a suitable order polynomial, and then integrating that within the time interval.

Code Smoothing

Comparing the two formulas of Eq. 6.86, one has

tmpD-79_thumb

Equation 6.88 can be used for smoothing the code survey by phase if there are no cycle slips.

Differential Phases

The first formula of Eq. 6.86 is the numerical difference of the phases at the two succeeded epochstmpD-80_thumb

tmpD-82_thumb

All other terms on the right-hand side are of low variation ones except the ambiguity term. Any cycle slips will lead to a sudden jump of the time difference of the phases. Therefore, the time differenced phase can be used as an alternative method of cycle slip detection.

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