## 2 The Unified Forms of Code Biases
### Basic Observation Equations  
According to the uncombined observation model in Refs. [21; 22; 23], the original uncombined observation model with single receiver, single satellite, and single epoch is expressed by using a short general form of pseudorange P and carrier phase L as follows. \[P_{i}^{sys}=\rho^{sys}+c(dt-dT+bias_{P_{i}^{sys}}^{r}-bias_{P_{i}^{sys}}^{s})+T _{w}+I_{i}^{sys}+\varepsilon_{P_{i}^{sys}} \tag{1}\]  
\[L_{i}^{sys}=\rho^{sys}+c(dt-dT+bias_{L_{i}^{sys}}^{r}-bias_{L_{i}^{sys}}^{s})+T _{w}-I_{i}^{sys}-\lambda_{i}^{sys}N_{i}^{sys}+\varepsilon_{L_{i}^{sys}} \tag{2}\]  
where \(c\) is speed of light, \(sys\) represents constellation identification, for instance, GPS or BDS. \(P_{i}^{sys}\) is the pseudorange in frequency i at constellation sys, \(L_{i}^{sys}\) is the carrier phase in frequency i at constellation sys. \(\rho^{sys}\) is the distance between station and satellite at constellation sys, \(dt\) is receiver clock error, \(dT\) is satellite clock error, \(Tw\) is the projection of tropospheric zenith wet path delay, \(I_{i}^{sys}\) is ionospheric delay in \(P_{i}\) at constellation sys. The system frequency ratio \(\gamma_{i}^{sys}=f_{1,sys}^{2}/f_{1,sys}^{2}\), where \(f_{i,sys}\) is the frequency of signal \(i\) at constellation \(sys\). \(N_{i}^{sys}\) is the ambiguity of carrier phase \(L_{i}^{sys}\). \(bias_{P_{IF}^{sys}}^{r}\), \(bias_{P_{IF}^{sys}}^{s}\), \(bias_{L_{i}^{sys}}^{r}\), and \(bias_{L_{i}^{sys}}^{s}\) are hardware delay in pseudorange \(P_{i}^{sys}\) and carrier phase \(L_{i}^{sys}\). \(\varepsilon_{P_{i}^{sys}}\) and \(\varepsilon_{L_{i}^{sys}}\) are observation noise in pseudorange \(P_{i}^{sys}\) and carrier phase \(L_{i}^{sys}\), respectively. Because satellite clock errors degrade the accuracy of PPP and can not be estimated directly, the clock product provided by International GNSS Service (IGS) organization is used to correct satellite clock errors. For legacy GPS, it has been a common practice to define clock offsets in precise ephemeris products with respect to an ionosphere-free dual-frequency combination of conventional reference signals (\(P_{1}^{GPS}\) and \(P_{2}^{GPS}\)) [24]. According to the BDS Interface Control Document (ICD), currently BDS adopts the precise ephemeris (and clock) products referring to the \(P_{1}^{BDS}/P_{2}^{BDS}\) ionosphere-free combination of dual-frequency observations [25; 26]. When the satellite clock products computed by using other combined signals or single signal differing from the conventional reference signal \(P_{1}^{sys}/P_{2}^{sys}\) are employed in GNSS applications, the pseudoranges are well known to be affected by instrumental biases [27; 28]. Therefore, these satellite clock biases, which are inconsistent with satellite clock products, should be corrected by differential code biases (DCBs) for pseudorange-based positioning, timing or ionosphere modeling [29]. The differential code biases (DCBs) provided by GNSS communities particularly for the post-processing PPP applications are as the same physical significance as timing group delays (TGDs) and inter-signal corrections (ISCs) in broadcast ephemeris, but DCBs are more accurate than TGDs and ISCs [30; 31]. The hardware delays can not be estimated, because they can not be separated from the receiver clock and ambiguity respectively. The relationship between hardware delays and DCBs is as follows. \[DCB_{P_{i}P_{j}^{sys}}^{r} =bias_{P_{i}^{sys}}^{r}-bias_{P_{j}^{sys}}^{r},\;\;i\
eq j,\;\;and \;\;i,j=1,2,3 \tag{3}\] \[DCB_{P_{i}P_{j}^{sys}}^{s} =bias_{P_{i}^{sys}}^{s}-bias_{P_{j}^{sys}}^{s},\;\;i\
eq j,\;\;and \;\;i,j=1,2,3 \tag{4}\]  
where \(DCB_{P_{i}P_{j}^{sys}}^{r}\) and \(DCB_{P_{i}P_{j}^{sys}}^{s}\) are difference code biases between \(P_{i}\) and \(P_{j}\) from receiver and satellite. When clock errors from receiver and satellite are based on conventional ionosphere-free model with dual-frequency (CIF2), they can be indicated as follows.