#### 2.1. Comparison between SD and DD under Non-Common Clock Scheme

Following [

20], we simplify the problem and assume that both the SD and DD integer ambiguities have been fixed. All SD observables are equal-weighted (let the weight be 1) and independent. By forming the SD, the satellite clock term is eliminated. For short baseline observables, the influence of atmospheric and ionospheric delays can be ignored. In addition, we only discuss measurements at L1-band and focus on static solutions, the SD measurement equation becomes:

where

$L$ is the carrier phase observable (unit: meter);

${L}_{0}$ is the initial carrier phase observable derived from a priori information;

$i$ denotes the satellite index;

$\Delta $ is the SD operator;

$\delta b$ is the estimated correction of baseline vector;

$A$ is the partial derivative matrix of the SD carrier phase observable respect to baseline vector. The parameter

$u$ includes the receiver clock error difference and UPD difference under the non-common clock scheme, which is the same for all satellites.

$\epsilon $ is the SD observation error.

For single epoch, the covariance sub-matrix of the baseline vector is:

where

${\sigma}^{2}$ is the variance of the observables;

${e}_{n}$ is

$n\times 1$ unit vector;

$n$ is the number of satellites at each epoch;

${I}_{n}$ is

$n\times n$ unit matrix. The baseline vector solution is:

For the DD model, after linearization, the DD measurement equation can be expressed as:

where

D is the DD operator. And the least squares estimation of baseline vector is given by:

The covariance sub-matrix of the baseline vector and the baseline solutions from the DD are exactly the same as those from non-common clock SD (Equations (2) and (3), respectively). Thus the single epoch DD and SD models are equivalent under a non-common clock scheme [

20].

In kinematic mode the single epoch solutions at each epoch are uncorrelated. This case is equivalent to the above single epoch case and the baseline esimations from the DD and SD models are still equivalent. In the static SD mode with multi-epoch joint estimate, the covariance matrix of the baseline vector solution is given by:

where

$i$ is epoch index;

$m$ is the total number of epochs. The baseline vector solution is:

With the same method used above, it can be proved that the static multi-epoch DD and SD models under a non-common clock scheme are also equivalent. Owing to this equivalency, in the following sections we use the non-common clock SD model to represent the DD model for comparison.

#### 2.2. Comparison between Non-Common Clock SD and Common Clock SD without UPD

Under the common clock scheme, the receiver clock error is canceled out by forming SD, thus only UPD is included in parameter

u. [

20] provided the comparison of formal uncertainty expressions between the non-common clock SD and the common clock SD without UPD (UPD = 0). Assume UPD = 0, the measurement equation of the common clock SD is:

The baseline vector solution and its covariance matrix are:

For comparison, we rewrite the Equation (2) of the non-common clock SD model:

where

$M={A}^{T}A,v=\frac{1}{\sqrt{n}}{A}^{T}{e}_{n}$. With the matrix inversion lemma, we have:

It’s easy to find that the first term on the right side of Equation (13) is the same as the baseline solution covariance matrix of the common clock SD model (Equation (11)) and the second term represents the formal uncertainty difference between the non-common clock SD model and common clock SD model without UPD. From the theoretical discussion of [

20], the second term of Equation (13) is always positive, hence the formal uncertainty from the non-common clock SD scheme is always larger than that from the common clock SD scheme without UPD. Geometrical analysis indicates that the difference of formal uncertainty is distributed primarily in the up direction. Here we further evaluate the differences of baseline solutions between the non-common clock SD scheme and the common clock SD scheme without UPD.

Substituting Equations (9) and (11) into Equation (10), we obtain:

Also, substituting Equations (1) and (2) into Equation (3), the estimation of baseline vector is:

where

$\delta b$ is the true value of baseline solution. The second term on the right side of Equation (14) shows that the deviation of the baseline estimation from the common clock SD is only related to the SD observation errors

$\epsilon $. If we regard

${\left({A}^{T}A\right)}^{-1}$ as the weight matrix, the deviation of baseline solution is the weighted average of

$\epsilon $. In contrast for the non-common clock SD the deviation (the second term on the right side of Equation (15)) can be taken as the weighted average of parameter

u (receiver clock error and UPD) and observation errors. Since both the receiver clock error and UPD are satellite independent, it can be proved that:

Then the parameter u can be omitted from Equation (15). Hence, both the estimated solutions (Equations (14) and (15)) are unbiased estimations of the true value, but their dispersions of solutions are different.

For convenience, the baseline vector parameters are expressed in a local geographical coordinate system (east, north and up), the corresponding partial derivative matrix

${A}^{i}$ for satellite

i is also expressed in the same coordinate system:

where

${\rho}^{i}$ is the geometric distance between the

i-th satellite and antenna;

${x}^{i}-x$ are three components of the line from the

i-th satellite to the antenna in the local geographical coordinate system. We assume that the priori position vector of the antenna is close to the true value.

$\theta $ denotes the angle between satellite-antenna vector and the axes of the local geographical coordinate system.

Compared with the true value, the deviations of baseline vector solution in Equations (14) and (15) can be regarded as stochastic variables, and are written respectively as:

Equations (18) and (19) represent the deviations of the baseline vector solution from the true value in the common clock SD without UPD and the non-common clock SD schemes, respectively. However, it is difficult to derive the analytical formula of the inverse of the normal matrix in these two equations. To get the impression of the dispersion differences between the two models intuitively, we discuss their asymptotic case. Assume that the number of satellites is extremely abundant (that is

$n\to \infty $), the occurrences of satellites in the upper hemisphere are uniformly distributed, and the satellite positions in the sky are independent, then the summation matrix elements in Equations (18) and (19) can be expressed using their mathematical expectations, hence:

Then, the statistical result of covariance matrix of the baseline solution from the common clock SD model without UPD is:

While that from the non-common clock SD gives:

Statistically, the formal uncertainties of the horizontal baseline components from the two SD models are equal (corresponding to the yaw angle), and the formal uncertainty of the vertical baseline component (corresponding to the pitch angle) from the common clock SD model without UPD is also equivalent to that of horizontal component, whereas the formal uncertainty of vertical component from the non-common clock SD model is twice as large as that of the horizontal component. These conclusions agree with the results of [

20]. We also give the quantitative asymptotic difference of the vertical formal uncertainty between the two SD schemes, i.e., in the non-common clock SD scheme the vertical uncertainty is twice as large as that in the common clock SD case. Its geometrical cause stems from the uneven satellite distribution in elevation angle. Since the orbital distribution only covers the upper hemisphere, the average value of satellite elevations is not zero, hence the vertical baseline component is correlated with the parameter

u.

We now consider the dispersion of baseline solutions. In statistics, the deviation of baseline solution from the common clock SD model without UPD can be denoted by:

According to the variance formula for the multiplication of two independent stochastic variables:

here

V denotes the variance operator. Similarly, the deviation of baseline solution from non-common clock SD is:

Comparing Equation (25) with Equation (27), statistically, the dispersions of the horizontal component (correspondence to the yaw angle) from both SD models are still equivalent, the dispersion of vertical component from the common clock SD without UPD is also equal to that of the horizontal component, whereas the dispersion of the vertical component from the non-common clock SD is 5.3 times as large as the horizontal component, which is also much more than that of the vertical component from the common clock SD without UPD.

The above discussion has focused on the case of single epoch baseline solutions. For the static multi-epoch case, the baseline solutions for the two SD models are:

where

$i$ is the satellite index. For simplicity, we assume that the number of satellites in each epoch is the same (

$n$ satellites),

j is the index of epoch (

$m$ epochs in all).

Then, covariance matrix from common clock SD model without UPD is:

Whereas that from the non-common clock SD model is:

The dispersion equations can be obtained in a similar way, so their description is omitted. It shows that in multi-epoch static case the proportional relationship of formal uncertainties between horizontal and vertical components is similar to single epoch static case, and the same for dispersion. Since $nm\gg n$, the formal uncertainties and dispersions of the multi-epoch static solutions are smaller than those of single epoch solutions.

Such proportional relationships of formal uncertainties and dispersions can also be extended to the multi-epoch kinematic case. For simplicity, we take two epochs for example. As for the common clock SD model without UPD, both Equations (21) and (25) become a 6 × 6 matrix which consists of two 3 × 3 diagonal sub-matrices like Equations (21) and (25), thus its formal uncertainty and dispersion are the same as those of the single epoch model. For the non-common clock SD, both Equations (22) and (27) become an 8 × 8 matrix which consists of two 4 × 4 diagonal sub-matrices like Equations (22) and (27), and its results are still the same as those of the single epoch model.

Based on the above discussions the following conclusions are reached: both for the single epoch and the multi-epoch solution, both in static and kinematic mode, the formal uncertainty and dispersion of horizontal baseline component from the non-common clock SD scheme (and DD) is equivalent to those from a common clock SD scheme without UPD. For the vertical baseline component, however, the formal uncertainty and dispersion from the common clock SD scheme are still the same as the horizontal baseline components, but those from the non-common SD scheme model are much greater than those from the common clock SD scheme without UPD.

#### 2.3. Common Clock SD with UPD Estimated and Constrained

Although the receiver clock error can be eliminated by the SD under a common clock scheme, the UPD difference still remains in parameter

u. For all the satellites the UPD differences are the same, otherwise the DD ambiguities are no longer integers [

27]. In practice we still need to estimate the parameter

u (or UPD) in a common clock SD scheme. Since the integer part of the parameter

u is included in the integer ambiguities, which are assumed to be fixed already, so the range of parameter

u should be within half a wavelength. If we impose a constraint of a half wavelength on parameters

u, then Equation (2) becomes:

Usually, the measurement error of the SD carrier phase (GPS L1 band) is within 5 mm, and the UPD constraint is 100 mm, so we have $4{\sigma}^{2}/{\lambda}^{2}\approx 1/400\ll n$. Thus in the single epoch solution case, if we estimate the parameter UPD at each epoch with 0.5 wavelength constraint, Equations (32) and (2) will be very close. Similarly, if the constraint on UPD is 0.5 wavelength, the solutions of the common clock SD scheme are very close to those of the non-common clock SD and DD scheme. That means the constraint on u has a very small influence on the covariance matrix.

Next, we discuss the kinematic multi-epoch solution case. For simplicity, we treat UPD as a time invariant parameter with no constraint and suppose the satellite number at each epoch is the same (e.g.,

n). Note that in the previous single epoch solution case we estimate many UPD parameters for each epoch, and in this case we estimate only one UPD parameter for all epochs. For m epochs the statistical result of the covariance matrix is:

Equation (33) reveals an interesting phenomenon. In the first epoch (

m = 1), the covariance of the vertical component is

$12/n$, which is equivalent to Equation (22). As the epoch number increases, the covariance of the vertical component tends to 3/

n, which is equivalent to Equation (21). Similar to Equations (26) and (27), we calculate the statistical dispersion expression (Equation (34)), which indicates that when the observation number

m is large, the dispersion of the vertical component will approach the dispersion of the horizontal components, similar to the common clock SD without UPD case. This multi-epoch case actually represents the kinematic AD case:

Finally, we discuss the multi-epoch static case, which treats both the baseline vector and UPD as time invariant parameters, the solutions are:

The asymptotic covariance matrix is:

The proportional covariance relationship between the vertical and horizontal components is similar to Equation (22) except the n is replaced by nm. For the vertical component dispersion, we can also get the similar proportional relationship between the vertical and horizontal components like Equation (27), except the n is replaced by nm.