1. INTRODUCTION
The term DOP has been well designed and widely used (Parkinson Reference Parkinson and Spilker1996) and the DOP value is often used as a major index to gauge the contribution of satellites. Various types of DOPs were defined such as the geometric (GDOP), positional (PDOP), horizontal (HDOP), vertical (VDOP) and time (TDOP) (Langley Reference Langley1999). Relationships among the vertical error of navigation, VDOP and the pseudo-range error in GPS were discussed by Leva (Reference Leva1994) and a closed-form expression for DOP was given by Doong (Reference Doong2009). A particular type of DOP called an ambiguity DOP (ADOP) was designed by Teunissen (Reference Teunissen and Odijk1997) and its corresponding properties were also discussed. The ADOP in a closed-form expression for a hierarchy of multi-frequency single-baseline GNSS models was given (Odijk and Teunissen Reference Odijk and Teunissen2008).
The commonly used DOP in PNT relies mainly on the geometry of measurements. It is useful when the least squares estimation and only sole satellite constellations are used for the solution. The observation geometric strength represented by the DOP is usually determined by the design matrix of the model parameters such as the parameters of position and time offset (Teunissen and Kleusberg Reference Teunissen and Kleusberg1999, p 449–451). Generally, the stronger the observation geometry, the less the DOP value and the more precise the PNT solutions. When additional observations/satellites are used in the solution, the increase of the number of observations can always reduce the GDOP value (Yarlagadda et al. Reference Yarlagadda, Ali, Al-Dhahire and Hershey2000), regardless of whether the geometric structure is enhanced by the additional observations/satellites. The precision of positioning can then be improved.
The Chinese Compass/BeiDou and European Galileo systems have experienced quick developments in recent years and nine Compass satellites have been successfully launched (as of July 2011). Additional phase and code measurements from these new multi-GNSS signals have become available and will contribute to the improvement of the phase bias estimation or ambiguity resolutions (Feng Reference Feng2008; Feng and Li Reference Feng and Li2008; Feng and Rizos Reference Feng and Rizos2009). In addition, Different combinations of measurements may also cause changes in the DOP values, for example, non-differential measurements and differential measurements may have different DOP values (Park and Kim Reference Park and Kim2000).
Some problems exist when the commonly used DOP definition is applied to measure the possible precision of PNT solutions. These include:
1. It is well known that when multiple frequencies and multiple constellations of GNSS are used in PNT, the number of measurements is increased, which is an advantage. Some additional parameters however may also need to be introduced for the interoperability of the different systems. The DOP values calculated using a fixed functional model may not be applicable anymore;
2. Usually, multiple GNSS constellations may increase the noise floor and may introduce additional outliers in the measurements. DOP computation based on the whole design matrix may not be suitable because the outlying measurements could significantly impact the precision which cannot be reflected by usual DOP;
3. If different stochastic models are used for different satellite constellations, the DOP value from the equal-weight-based approach is not suitable anymore. Instead the weighted least squares estimation (Sairo et al. Reference Sairo, Akopian and Takala2003) or the iterative weighted total least-squares adjustment (Shen and Li Reference Shen, Li and Chen2010) may be applied;
4. When a priori information of the model parameters is employed in the parameter estimation, the DOP calculated based on the design matrix of measurement equations does not work;
5. Especially when non-least squares estimation, such as robust estimation is used (Hampel et al. Reference Hampel, Ronchetti, Rousseeuw and Stahel1986; Yang Reference Yang1991), the DOP based on least squares principle does not work either;
6. DOP is defined as the square-root of the trace of the normal matrix which neither reflects the correlations among the parameters or the ill-posed problems, nor reflects the systematic error influences. In real time GNSS positioning, the carrier phase ambiguity resolution is often ill-posed or the geometry is nearly collinear as in the case of poor satellite geometry, which may result in unreliable or unsuccessful ambiguity resolution and hence unreliable positioning if no care is taken to mitigate this situation (Li et al. Reference Li, Shen and Feng2010). In this case, the regularisation or ridge regression method (Xu et al. Reference Xu, Fukuda and Liu2006; Xu Reference Xu2009) is recommended. In addition, calculating GDOP using the ridge regression method was also proposed for improving the accuracy and performance of positioning (Kelly Reference Kelly1990).
To sum up, the DOP values should reflect not only the geometrical strength, but also the other important information related to the PNT. In fact, the precision of a GNSS PNT solution relies on the following four main factors; the magnitude of errors, the geometric strength of measurements, the number of model parameters and the principle used for the parameter estimation. In order to reasonably evaluate and analyse the error influence or DOP of satellite positioning, a suitable DOP statistics that takes into account the above four factors needs to be constructed.
The influence function (IF) defined in statistics (Hampel et al. Reference Hampel, Ronchetti, Rousseeuw and Stahel1986; Yang Reference Yang1991, Reference Yang1997, Reference Yang, Song and Xu2002) is another important criterion in the quality assessment of measurements and the parameter estimates. Both DOP and IF can be used as the criteria for evaluating an observations’ contribution to the parameter estimates, thus they should be intrinsically related. To establish the relationship between the DOP and IF, a new generalised DOP (G-DOP) based on different backgrounds such as the principles of a robust estimation and the Bayesian estimation need to be developed. If the robust principle is employed, the first type of G-DOP, expressed as G-DOPI in this paper, is proposed. If additional parameters for the interoperability of different navigation systems are considered in the functional model, then the second type of G-DOP, expressed as G-DOPII, is defined. If a priori weights on the model parameters are considered, the third G-DOP expressed as G-DOPIII is recommended. It is obvious that more information can be included in the calculation of G-DOP values if some particular analysis or computation is needed.
All the aforementioned three types of generalised DOPs are defined by the square root of the variance covariance matrix (VCM) of the estimated parameters, which is the same as that of the ones that have been used. The only differences are in the forms of expression, which are related to the estimation principle, the parameter characteristic as well as whether the a priori information is used.
2. INFLUENCE FUNCTION OF SIGNAL-IN-SPACE ERROR
The influence of the signal-in-space error (SISE) can be expressed by an IF. The IF is related to the geometric structure of observations, which in turn is related to the DOP value. Assuming that the pseudo-range measurement of the receiver to the satellite is L i, the measurement equation is (Langley Reference Langley1999);
where ρ i is the geometric range between the antenna phase centres of both receiver and satellite, dt u and dt i denote the clock offsets of the receiver and satellite respectively, dI i and dT i are the ionospheric and tropospheric delays respectively, e i is the sum of the measurement noise and un-modelled observation errors such as multipath effects and c denotes the speed of light in vacuum. Equation (1) can be written in the following error equation form:
where V denotes the residual vector, A is the design matrix, 
denotes the unknown parameter vector which is generally the vector of corrections to the model parameters’ approximate values and L denotes the observed-minus-computed observations.
Assume that the contaminated distribution function of L i is G i, which is composed of two terms (Hampel et al. Reference Hampel, Ronchetti, Rousseeuw and Stahel1986):
where δl i is the contaminated symmetric distribution and ε is the contaminated ratio with 0⩽ε<1 and F i is the normal distribution function of L i.
Assuming that the measurements are independent, a general risk function based on the maximum likelihood principle (M-estimation) can be expressed as:
where ρ(·) is a convex function, which is continuous and differentiable with the scalar derivative as 
, is the estimated parameter vector. The solution to the risk function (4) must meet the following requirements:
Based on the estimation principle (4) and the estimation equation (5), an IF is defined by Hampel (Hampel et al. Reference Hampel, Ronchetti, Rousseeuw and Stahel1986, Yang Reference Yang1991):
![$${\bf IF}(L_i ;{\bf \hat X},F_i ) = \mathop {{\rm lim}}\limits_{\varepsilon \to 0} \displaystyle{{{\bf \hat X}[(1 - \varepsilon )F_i + \varepsilon \delta l_i ] - {\bf \hat X}(F_i )} \over \varepsilon} $$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151022042007115-0798:S0373463311000415_eqn6.gif?pub-status=live){kind=small}
where ![${\bf \hat X}[(1 - \varepsilon )F_i + \varepsilon \delta l_i ]$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151022042007115-0798:S0373463311000415_eqnU4.gif?pub-status=live){kind=small}
and 
are the estimated vectors based on the contaminated and normal distributions respectively.
Considering the error equation (2) and the M-estimation equation (5), we have:
where ai is the ith row vector of A, p i is the weight of L i and F i represents the marginal distribution of L i.
Substituting (7) into (6), we obtain the following derivative with respect to ε (at ε=0):
where 
. Given the following equations (Hampel et al. Reference Hampel, Ronchetti, Rousseeuw and Stahel1986; Yang Reference Yang1991, Reference Yang1997):
where E denotes expectation, we obtain:
By substituting equations (9)–(11) into Equation (8), with:
and Z the diagonal matrix with elements:
The IF can also be expressed in the equivalent weight matrix 
which has the diagonal elements of 
(Yang Reference Yang1991, Reference Yang1997):
In the least squares estimation, ρ(e i)=e i2, ψ(e i)=ρ′(e i)=2e i, ψ′(e i)=2 and
Hence;
Then the IF becomes;
where IFLS represents the influence function under the LS estimation.
According to the IF expressed in Equation (12), the effects of the SISE on the model parameters mainly depend on ψ. Usually, ψ is a descending function which decreases with the increase of the error, or, it is a re-descending function which decreases to zero when the error is larger than the given threshold value. In other words, the influence of SISE on the model parameters decreases with the increase of the magnitude of the SISE. On the other hand, the influence of SISE e i is related to the observation geometric structure: the stronger the geometric structure, the smaller the value of M− 1 and the less the effects of the SISE.
3. THE FIRST TYPE OF G-DOP: CONSIDERING A ROBUST RISK FUNCTION
Usually, the DOP is defined based on the cofactor matrix of the parameter estimates. If the principle of maximum likelihood estimation is applied, the integrated IF of the SISE of the measurements L 1, L 2,…L n can be expressed as:
![$$\mathop {{\bf IF}}\limits_{m \times 1} ({\bf L};{\bf \hat X}) = \left[ {\matrix{ {IF(L_1, L_2, \cdots L_n, \hat X_1 )} \cr {IF(L_1, L_2, \cdots L_n, \hat X_2 )} \cr \vdots \cr {IF(L_1, L_2, \cdots L_n, \hat X_m )} \cr}} \right] = - {\bf M}^{ - 1} {\bf A}^{\rm T} {\bf P}\left[ {\psi (e)} \right]$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151022042007115-0798:S0373463311000415_eqn18.gif?pub-status=live)
where [ψ(e)] is the vector composed of the elements of ψ(e i). The posterior covariance matrix of the model parameters is then defined as (Yang Reference Yang1997):
Based on Equation (12), the approximate estimate of 
can be expressed as (Yang Reference Yang1997):
can also be written as the following expression if the equivalent weight matrix is used:
with the elements of the equivalent weight matrix calculated by the residual as:
where v i denotes the residuals of L i.
If Huber's ψ function (Huber Reference Huber1981) is applied to Equation (22) then:
where m 0 denotes the number of 
that have zero value.
In the LS estimation; E([ψ(e)][ψ(e)]T)=E(eeT)=Σe=P−1σ 02 and M=ATPA, thus:
where 
is the LS estimate of variance scale.
Similar to the usual GDOP definition, we define:
![$${\rm G {\mbox {\hbox -}} DOP}_{\rm I} = [{\rm tr}({\bf M}^{ - 1} )]^{{\textstyle{1 \over 2}}} $$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151022042007115-0798:S0373463311000415_eqn26.gif?pub-status=live){kind=small}
as the first type of G-DOP.
If the equivalent weight matrix is used, Equation (26) can be also simplified as:
![$${\rm G {\mbox {\hbox -}} DOP}_{\rm I} = [{\rm tr}({\bf A}^{\rm T} {\bf \bar PA})^{ - 1} ]^{{\textstyle{1 \over 2}}} $$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151022042007115-0798:S0373463311000415_eqn27.gif?pub-status=live){kind=small}
From the analysis of the G-DOPI expression, it is found that:
(1) The first type of G-DOP, as the commonly defined/used GDOP, reflects the observation geometric configuration. The stronger the observation geometry, the smaller the G-DOPI value.
(2) The combined effects of measurement errors and the risk function are reflected in M− 1. If there exist outliers, the derivative of ψ, ψ′(e i) will be reduced whereas M− 1 will be increased. It means that when outliers exist in some observations, the contribution of those observations will be removed so that they will not affect the precision of the parameter estimates. This is more reasonable because when |e i| increases, ψ(e i) decreases and the corresponding
of L i will also decrease. In this case, G-DOPI should be increased. The usual DOP in the LS estimation will not help us to estimate the actual and reasonable variation of the precision, when there exist outliers which do not make any contribution to the model parameter estimates.(3) Both DOP and IF are related to the inverse of the normal matrix M− 1. Another expression of G-DOP can be easily obtained from (26);
(28)
![$${\rm G {\mbox {\hbox -}} DOP}_{\rm I} = \left\{ {{\rm tr[}E({\bf IF} \cdot {\bf IF}^{\rm T} ]} \right\}^{{\textstyle{1 \over 2}}} /\sigma _0 $$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151022042007115-0798:S0373463311000415_eqn28.gif?pub-status=live){kind=small}
where σ 0 is the standard deviation.
It should be noted that the IF is the integrated influence vector that contains the effects of the errors e 1, e 2,…, e n. Furthermore, Equation (28) can be expressed in the form of IF components:
![$${\rm G {\mbox \hbox{-}} DOP}_{\rm I} = \left\{ \sum\limits_{\,j = 1}^m {E[IF^2 ({\bf L},\hat X_i )]}\right\} ^{{\textstyle{1 \over 2}}} \bigg/\sigma _0} $$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151022042007115-0798:S0373463311000415_eqn29.gif?pub-status=live)
where 
is a scalar quantity and it can be expressed as:
![$$\eqalign{IF({\bf L},\hat X_i ) =\left[ {M_{\,j1}^{ - 1}, M_{\,j2}^{ - 1}, \cdots, M_{\,jm}^{ - 1}} \right]{\rm} \left[ {\matrix{ {\sum\limits_{i = 1}^n {a_{i1} p_i \psi (e_i )}} \cr {\sum\limits_{i = 1}^n {a_{i2} p_i \psi (e_i )}} \cr \vdots \cr {\sum\limits_{i = 1}^n {a_{im} p_i \psi (e_i )}} \cr}} \right]= \sum\limits_{k = 1}^m {M_{\,jk}^{ - 1}} \cdot \sum\limits_{i = 1}^n {a_{ik} p_i \psi (e_i )}} $$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151022042007115-0798:S0373463311000415_eqn30.gif?pub-status=live)
In the LS estimation:
Equation (31) is the same as that of the usual DOP.
It should be pointed out that if the outliers are detected and identified by the DIA method (Teunissen Reference Teunissen1990), the G-DOPI is nearly equivalent to the commonly used DOP.
4. THE SECOND TYPE OF G-DOP: CONSIDERING ADDITIONAL MODEL PARAMETERS
The issues of compatibility and interoperability need to be addressed when multi-GNSS constellations are used. One feasible approach is to add extra model parameters to the functional model for compensating the systematic offsets/errors such as the coordinate system offset, the systematic time offsets and the satellite orbit errors etc. In this case, the observation equation matrix becomes:
where Δi is the random error vector, Sj denotes the systematic error vector that only includes the additional model parameters, ai is the ith row of the design matrix and bi is the coefficient vector of the additional model parameters. For a single navigation system, Sj may be the receiver time offset parameter dt (Milbert Reference Milbert2008, Reference Milbert2009) and Sj can be merged into the parameter vector X. If each satellite system has its own systematic parameters Sj, then they can be estimated under the support of multi-GNSS. At least the effects of Sj can be compensated by estimating their values.
Equation (32) can be written in the form of the following error equation:
where the subscript “J” (J=1, 2, …, N), represents different satellite systems, such as GPS, GLONASS, COMPASS etc. The parameter estimates and the IF from a robust estimator could be;
![$$\left[ {\matrix{ {{\bf \hat X}} \cr {{\bf \hat S}} \cr}} \right] = \left[ {\matrix{ {{\bf A}^{\rm T} {\bf \bar PA}} & {{\bf A}^{\rm T} {\bf \bar PB}} \cr {{\bf B}^{\rm T} {\bf \bar PA}} & {{\bf B}^{\rm T} {\bf \bar PB}} \cr}} \right]^{ - 1} \left[ {\matrix{ {{\bf A}^{\rm T} {\bf \bar PL}} \cr {{\bf B}^{\rm T} {\bf \bar PL}} \cr}} \right]$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151022042007115-0798:S0373463311000415_eqn34.gif?pub-status=live){kind=small}
![$$\left[ {\matrix{ {IF(\Delta _J, {\bf \hat X})} \cr {IF(\Delta _J, {\bf \hat S})} \cr}} \right] = \left[ {\matrix{ {{\bf A}^{\rm T} {\bf \bar PA}} & {{\bf A}^{\rm T} {\bf \bar PB}} \cr {{\bf B}^{\rm T} {\bf \bar PA}} & {{\bf B}^{\rm T} {\bf \bar PB}} \cr}} \right]^{ - 1} \left[ {\matrix{ {{\bf A}_J^{\rm T} {\bf \bar P}_J \Delta _J} \cr {{\bf B}_J^{\rm T} {\bf \bar P}_J \Delta _J} \cr}} \right]\vskip10pt $$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151022042007115-0798:S0373463311000415_eqn35.gif?pub-status=live){kind=small}
From the estimator (34), it is easy to obtain the effects of the systematic error vector SJ on the common model parameter vector by:
Similarly, the IF of the random errors Δ1,…, ΔN on the model parameters can be expressed as:
In multi-GNSS, the influence of the systematic error of a single constellation on the estimates of PNT can be analysed using the total influence function (TIF) by:
Since the additional model parameters will degrade the precision, so the DOP values will be enlarged. The second type of G-DOP (G-DOPII) that takes into account the additional model parameters can be defined as:
![$${\rm G {\mbox \hbox{-}} DOP}_{{\rm II}} = \left\{ {{\rm tr}\left[ {\matrix{ {{\bf A}^{\rm T} {\bf \bar PA}} & {{\bf A}^{\rm T} {\bf \bar PB}} \cr {{\bf B}^{\rm T} {\bf \bar PA}} & {{\bf B}^{\rm T} {\bf \bar PB}} \cr}} \right]^{ - 1}} \right\}^{{\textstyle{1 \over 2}}} $$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151022042007115-0798:S0373463311000415_eqn39.gif?pub-status=live)
Using the matrix equivalent equation (Koch Reference Koch1987), the G-DOPII can be obtained as:
![$$\eqalign{{\rm G {\mbox \hbox{-}} DOP}_{{\rm II}}\! =\! \{ {\rm tr}[{\bf N}_{\bf X}^{ - 1}\! +\! {\bf N}_{\bf X}^{ - 1} {\bf N}_{{\bf XS}} ({\bf N}_{\bf S}\! -\! {\bf N}_{{\bf SX}} {\bf N}_{\bf X}^{ - 1} {\bf N}_{{\bf XS}} )^{ - 1}{\bf N}_{{\bf SX}} {\bf N}_{\bf X}^{ - 1}\! +\! ({\bf N}_{\bf S}\! -\! {\bf N}_{{\bf SX}} {\bf N}_{\bf X}^{ - 1} {\bf N}_{{\bf XS}} )^{ - 1} ]\} ^{{\textstyle{1 \over 2}}}} $$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151022042007115-0798:S0373463311000415_eqn40.gif?pub-status=live){kind=small}
where 
, 
and 
.
From the analysis of the G-DOPII and TIF, the following findings can be obtained:
1. Generally, multi-GNSS will improve the G-DOP, especially when multi-GNSS meet the requirements of compatibility and interoperability, no additional parameters need to be introduced to the functional model for compensating the systematic error effects. The G-DOPII can then be changed into:
(41)where G-DOPR denotes the GDOP based on a robust estimation. It is obvious that the more satellites available, the smaller the G-DOPR value.2. If the requirement of interoperability is not met, additional parameters for individual constellations need to be added to compensate for their systematic errors in the integrated navigation or positioning solutions. The G-DOP becomes larger and the level of improvement will be reduced. From Equation (39), we can obtain:
which means that G-DOPII⩾GDOP. Therefore, the compatibility and interoperability of multi-GNSS is important for improving the precision of multi-GNSS positioning.3. If SJ (J=1,…, N) have different signs, meaning that they have random characteristics, then
will have the nature of cumulative decay. As an extreme case: when N approaches infinity, will be close to zero. This means that when multi-GNSS is available the systematic errors among different systems in the integrated positioning may be cancelled out or compensated.4. A large SJ will result in a descending ψ(eJ) since eJ includes both ΔJ and SJ. Therefore the G-DOPII reflects the impacts of the systematic errors as well.
![$${\rm G {\mbox \hbox{-}} DOP}_{\rm R} = \left\{ {{\rm tr}\left[ {\sum\limits_{J = 1}^N {{\bf A}_J^{\rm T} {\bf \bar P}_J {\bf A}_J}} \right]^{ - 1}} \right\}^{{\textstyle{1 \over 2}}} $$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151022042007115-0798:S0373463311000415_eqn41.gif?pub-status=live)
![$$\eqalign{\{& {\rm tr}[{\bf N}_{\bf X}^{ - 1} +{\bf N}_{\bf X}^{ - 1} {\bf N}_{{\bf XS}} ({\bf N}_{\bf S} - {\bf N}_{{\bf SX}} {\bf N}_{\bf X}^{ - 1} {\bf N}_{{\bf XS}} )^{ - 1} {\bf N}_{{\bf SX}} {\bf N}_{\bf X}^{ - 1} + ({\bf N}_{\bf S} - {\bf N}_{{\bf SX}} {\bf N}_{\bf X}^{ - 1} {\bf N}_{{\bf XS}} )^{ - 1} ]\} ^{{\textstyle{1 \over 2}}} \cr&\geqslant [{\rm tr}({\bf N}_{\bf X}^{ - 1} )]^{{\textstyle{1 \over 2}}}} $$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151022042007115-0798:S0373463311000415_eqnU21.gif?pub-status=live)
will have the nature of cumulative decay. As an extreme case: when N approaches infinity, 
will be close to zero. This means that when multi-GNSS is available the systematic errors among different systems in the integrated positioning may be cancelled out or compensated.It should be mentioned that if no additional parameters are to be estimated in the data processing, the G-DOPII will be equal to the commonly used GDOP.
5. THE THIRD TYPE OF G-DOP: CONSIDERING THE A PRIORI WEIGHTS OF SOME PARAMETERS
Different types of systematic errors and offsets in multi-GNSS may exist and in many cases these errors are treated in different ways, depending on their characteristics. For example, some of the systematic errors are treated as constants, e.g. the coordinate system errors are usually expressed as; ΔX0=[Δx 0 Δy 0 Δz 0]T. Some other errors are time-varying, which may be modelled by a polynomial function or an autoregressive process. For example, the linear polynomial function Δt=dt 0+(t−t 0)dt 1 is commonly used for systematic time errors. These systematic errors, or the values (or even their approximate values) of the additional model parameters at the first observation epoch, expressed as Δx 0, Δy 0, Δz 0, Δt 0 and Δt 1, are normally unknown. After the first epoch's data processing, a set of approximate values along with the variance-covariance matrix of the parameter estimates can be obtained. The variance-covariance matrix is usually called an a priori weight matrix (for the next epoch's data processing). Thus, the DOP value should reflect the a priori information of the additional model parameters.
If ΔX0 and Δt are merged into the additional model parameter vector S, with a priori estimates of 
and 
, then the error equation can be expressed as:
where 
denotes the estimated S vector.
To conveniently express the estimator under the Bayesian rule (Koch Reference Koch1990), the weight matrix 
for parameter is used. The Bayesian estimator of model parameters including the additional parameters can be then expressed as:
![$$\left[ {\matrix{ {{\bf \hat X}} \cr {{\bf \hat S}} \cr}} \right] = \left[ {\matrix{ {{\bf A}^{\rm T} {\bf PA}} & {{\bf A}^{\rm T} {\bf PB}} \cr {{\bf B}^{\rm T} {\bf PA}} & {{\bf B}^{\rm T} {\bf PB} + {\bf P}_{{\bf \bar S}}} \cr}} \right]^{ - 1} \left[ {\matrix{ {{\bf A}^{\rm T} {\bf \bar PL}} \cr {{\bf B}^{\rm T} {\bf \bar PL} + {\bf P}_{{\bf \bar S}} {\bf \bar S}} \cr}} \right]$$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151022042007115-0798:S0373463311000415_eqn43.gif?pub-status=live){kind=small}
Therefore, the third type of G-DOP (G-DOPIII) which takes into account the a priori information of some model parameters is defined as:
![$${\rm G {\mbox \hbox{-}} DOP}_{{\rm III}} = \left\{ {{\rm tr}\left[ {\matrix{ {{\bf A}^{\rm T} {\bf PA}} & {{\bf A}^{\rm T} {\bf P{\rm B}}} \cr {{\bf B}^{\rm T} {\bf PA}} & {{\bf B}^{\rm T} {\bf PB} + {\bf P}_{{\bf \bar S}}} \cr}} \right]^{ - 1}} \right\}^{{\textstyle{1 \over 2}}} $$](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151022042007115-0798:S0373463311000415_eqn44.gif?pub-status=live)
In positioning and navigation with multi-GNSS, the G-DOPIII will be improved even in the case where additional systematic parameters need to be introduced for addressing the interoperability problem of different systems if the a priori information is used in the estimation process.
It should be pointed out that G-DOPIII is not new and that it depends on the data processing procedure. If the a priori information is employed in the data processing the G-DOPIII is valid. In the case where there is no a priori information available, the usual DOP is still used. G-DOPIII should be used if some estimated information can be obtained during the data processing.
6. SIMULATED EXAMPLES
A simulation was carried out in this research. The time period of the simulation is 24 hours starting from 0000 hours March 28, 2010, in GPS time. The sampling interval is 300 s. The mask elevation angle is 5 degrees. For GPS and GLONASS satellites, their orbit parameters are obtained from broadcast ephemerides. In the time period, 30 GPS and 21 GLONASS satellites were involved. Circular orbits for Galileo and Compass were used. 27 Galileo satellites and 27 Compass MEO satellites distributed evenly in three orbital planes were simulated respectively. The simulated Keplerian orbit parameters for these satellites are listed in Table 1.
Table 1. Orbital Parameters of Compass and Galileo Satellites.
In Table 1, a denotes the major semi-axis, i is the inclination angle, e is the orbit eccentricity, ω is the argument of perigee and M 0 is the mean anomaly.
The five Compass GEO satellites are located at 58·75°E, 80°E, 110·5°E, 140°E and 160°E respectively. The inclinations of the three IGSO (Inclined Geo-Synchronous Orbit) satellites are all 55° and their right ascension of the ascending nodes are Ω=0°, 120° and 240° respectively. The cross node is at 118°E.
Table 2. The first type of G-DOP values with different proportion of outliers.
Table 3. Average G-DOP of 24 hours of schemes 1, 2 and 3.
(Note that when the number of visible GLONASS satellites is less than four, the corresponding G-DOP is not included in the calculation of the average values).
Table 4. Average G-DOP for periods of 10-epochs for schemes 1, 4 and 5.
At the simulated station (34·8°N, 113·7°E, 110·4 m), the number of satellites in view is shown in Figure 1.
Figure 1. Numbers of visible satellites from the four GNSS and the sum of all the visible satellites.
The flowing three scenarios are used for the simulation:
Case 1: 5%, 10% and 20% outliers were simulated in the pseudo-range measurements. The GDOPs in the two cases: with and without the outliers were compared, as shown in Table 2.
Case 2: In the integrated positioning system of the four GNSS constellations, some additional parameters are added for compensating the systematic errors. Three parameters for systematic time errors and six parameters for systematic time and coordinate errors are used. The G-DOP values with and without additional parameters were calculated and compared in Figure 2 and Table 3.
Figure 2. The second type of G-DOP changes with observation epochs.
Case 3: The a priori information of the additional parameters for the compensation of time and coordinate errors and offsets were used in the positioning solution. The third type G-DOP was calculated and compared with the cases of with and without the additional parameters but not using the a priori information. The results are shown in Figure 3 and Table 4 respectively.
Figure 3. The third type of G-DOP changes with observation epochs.
In addition, the following five schemes were designed for cases 2 and 3:
Scheme 1: Only four parameters were considered in the positioning model: three for position and one for receiver clock. This means that the systematic errors of both time and coordinate systems were not included in the parameter vector.
Scheme 2: Seven parameters were considered in the positioning model in which three additional parameters for coordinate systematic errors were included in addition to the three position and one receiver clock parameters.
Scheme 3: Ten model parameters were included: three position and one receiver clock parameters, three parameters for coordinate systematic errors and three parameters for time systematic errors.
Scheme 4: Similar to scheme 2 but an a priori variance-covariance matrix of the systematic error parameters is used.
Scheme 5: Similar to scheme 3 but an a priori variance-covariance matrix of the six systematic parameters is considered.
From the analysis of the above results, it is concluded that:
• From Table 2, it is found that the first type of G-DOP increases with the increase of the number of outliers. This type of G-DOP reflects the outlier impact on the observation geometry based on the robust estimation. It is reasonable that when there are outliers in the measurements the geometry strength should be weakened.
• From Figure 2 and Table 3, it is found that when interoperability problems exist among the GNSS, systematic error parameters should be included in the positioning model. This will result in an increase in the value of the second type of G-DOP. The second type of G-DOP increases with the increase in the number of the systematic error parameters. It is reasonable that when the number of model parameters increases the strength of the corresponding observation geometry will be weakened.
• From Figure 3, it is found that if the a priori information of the systematic error parameters is continuously used in the estimation process, the value of the third type of G-DOP will be getting smaller and smaller with the increase in the number of observation epochs. Therefore, the a priori information of the systematic error parameters can strengthen the observation structure and weaken the adverse effects of the interoperability parameters on the integrated positioning of multi-GNSS and this can be reflected by the third type of G-DOP. From Table 4, it is concluded that the fewer the number of systematic parameters, the quicker the convergence of the third type of G-DOP.
7. CONCLUSIONS AND REMARKS
The observation geometry strength is related not only to the design matrix of the observation equation but also to the health of observations, i.e. related to the precision and reliability of the measurements. The first type of G-DOP can reflect the contributions of the outlying measurements to the DOP based on the robust estimation principle. The outlying measurements will not change the original DOP, however it does change the first type of G-DOP.
The observation geometry strength is also related to the number of model parameters. The higher the number of the model parameters to be estimated, the weaker the observation strength. The second type of G-DOP can reflect the effects of the additional systematic error parameters on the positioning precision. It shows that if the interoperability of multi GNSS is not fulfilled, the increased number of satellites will not significantly reduce the DOP value because additional model parameters should be introduced and estimated, which weakens the contribution of the increased number of measurements.
If the a priori information of the model parameters is used in the positioning process, the precision of the position will benefit from both a priori information of the parameters and new measurements. The third type of G-DOP can reflect both the contribution of the a priori information of model parameters and new measurements to the DOP. From this point of view, if the systematic errors change smoothly with time or position, the a priori estimates of the systematic parameters and their corresponding variance-covariance matrix should be taken into account in the actual GNSS positioning to reduce the DOP values.
ACKNOWLEDGEMENTS
This project is sponsored by The Natural Science Foundation of China (41020144004, 40774001).
