# 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,code,In the future, there will also be According 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)

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 byare 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

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

Because of

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

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

A general phase-phase linear combination can be formed by

where the combined signal has the frequency and wavelength

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

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

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 combinationsare 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 frequencyto 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)

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

where

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)

where

The related observation equations are the same as Eqs. 6.55 and 6.56, with 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

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

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,is the ionospheric path delay and can be considered a mapping of the zenith delaywhere F is  the mapping function (cf. Sect. 5.1). So one has

wherehave the physical meaning of total electronic contents at the signal path direction and the zenith direction, respectively.is then independent from the zenith angle of the satellite. If the variability of the electronic contents at the zenith direction is stable enough,can be modelled by a step function or a first order polynomial with a reasonably short time intervalby

whereare the beginning and ending time of the GPS survey.can be, e.g., selected by 30 minutes.is the coefficient of the polynomial.

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 vector by

where Eq. 6.65 is used and

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

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

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

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

whereand 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)

and then one gets

Substituting Eq. 6.72 into 6.70 yields

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. Settinginto Eq. 6.61 and omitting the error term, one has

whereis 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

where

The differences of the above observable combinations at the two succeeded epochs andcan be formed:

whereis a time difference operator, for any time function is valid.

Because the time differences of the ionospheric effectsare generally very small, they are called ionospheric residuals. In the case of no cycle slips, i.e., ambiguitiesare constant,equal 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 slipscan lead to a very small combination ofExamples 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 epochscan be formed as

whereis a numerical differentiation operator and

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:

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 bycan be used to predict the future kinematic position by

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

Using the operatorto the un-differenced phase Eq. 6.45 and code Eq. 6.44, one gets

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

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

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 epochs

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.