**2. Mathematical Model of DGNSS/DBA Combined Positioning**

*2.1. DGNSS Positioning Observation Equation*

GNSS receivers can simultaneously receive observation data such as code pseudorange, carrier phase, Doppler shift, and SNR. The raw observation values contain the receiver geometric position parameters, clock error as well as hardware delays parameters, and various error corrections, such as tropospheric errors and ionospheric errors. Therefore, the raw code pseudorange observation equation between satellite *s* and receiver *i* can be expressed as:

$$P\_i^s = \rho\_i^s + c(dt\_i - dt^s) + I\_i^s + T\_i^s + M\_i^s + \varepsilon\_i^s \tag{1}$$

where *P<sup>s</sup> <sup>i</sup>* denotes the code pseudorange observation between satellite *s* and receiver *i* (Unit: m); *ρ<sup>s</sup> <sup>i</sup>* = - (*X<sup>s</sup>* − *X*) <sup>2</sup> + (*Y<sup>s</sup>* <sup>−</sup> *<sup>Y</sup>*) <sup>2</sup> + (*Z<sup>s</sup>* <sup>−</sup> *<sup>Z</sup>*) <sup>2</sup> denotes the geometric distance between satellite *s* and receiver *i* at the moment of signal transmission, (*X<sup>s</sup>* ,*Y<sup>s</sup>* , *Zs*) and (*X*,*Y*, *Z*) are the satellite *s* and unknown receiver *i* antenna center position. *c* is the speed of light in vacuum; *dti* denotes the receiver clock error; *dt<sup>s</sup>* denotes the satellite clock error; *Is <sup>i</sup>* is the ionospheric error; *<sup>T</sup><sup>s</sup> <sup>i</sup>* is the tropospheric error; *<sup>M</sup><sup>s</sup> <sup>i</sup>* is the multipath delay error; *<sup>ε</sup><sup>s</sup> i* contains the code pseudorange measurement noise and other uncorrected errors.

For the short and medium baselines, the receiver clock error and satellite clock error are eliminated in the double-difference observation equation, the ionospheric and tropospheric errors can be neglected, and the DGNSS observation equation can be simplified and expressed as [34]:

$$
\nabla \Delta P\_{i,j}^{s\_1 s\_k} = \nabla \Delta \rho\_{i,j}^{s\_1 s\_k} + \nabla \Delta \varepsilon\_{i,j}^{s\_1 s\_k} \tag{2}
$$

where ∇Δ denotes the double-difference operator; *i* and *j* denotes the reference station receiver and mobile receiver; *s*<sup>1</sup> and *sk* denote the reference and nonreference satellite, respectively.
