For the purpose of this simplification, the following assumptions are made:

- The difference in total signal travel time between pseudoranges and carrier phases will be neglected. Hence, the pseudorange clock terms will be assumed to be identical to the corresponding carder phase clock terms.

- The differences between the frequency dependent pseudorange and carrier phase receiver eccentrities will be neglected. Similarly, the differences between the frequency dependent pseudorange and carder phase satellite eccentricities will be neglected. Hence, the geometric range from receiver i to satellite k is assumed to be independent of the frequency used and the same for both pseudoranges and carder phases. This geometric range will be denoted as \(\rho _i^k\).

- It follows from the structure of the above observation equations and the fact that we only consider the single-channel case, that not all parameters on the fight-hand side of these equations are separably estimable. A number of these parameters will therefore be lumped together into one single parameter. For each channel, the receiver-satellite range , the clock terms \(d{t_i}\)and\(d{t_k}\), and the tropospheric delay \(T_i^k\)will be lumped together in one single parameter\(s_i^k\):

\(s_i^k = \rho _i^k + c\left( {d{t_i} - d{t^k}} \right) + T_i^k\)

\(d_i^k = c\left[ {{d_i}\left( t \right) + {d^k}\left( {t - \tau _i^k} \right)} \right] + dm_i^k\) and \(\delta _i^k = c\left[ {{\delta _i}\left( t \right) + {\delta ^k}\left( {t - \tau _i^k} \right)} \right] + \delta m_i^k\)

- The instrumental delays of both receiver and satellite, and the multipath delay will be lumped together.

- The carrier phase observation equation the non-zero intial phases will be lumped together with the carrier phase ambiguity term:

- The ionosphere causes the integrated carrier phase count to decrease (that is, the apparent phase velocity is greater\({L_1}\) than the velocity of light!), but causes the pseudo-range to appear longer than the geometric range. In the remainder of these notes only the group delay term dion will be used in the pseudo-range and phase observation equations. Note that the time delay is proportional to the inverse of the frequency squared. That is, higher frequencies are less affected by the ionosphere, and hence the ionospheric time delay for L1 observations (1575.42MHz) is less than for L2 observations (1227.60MHz).

- Carrier phase ambiguities are assumed to be constant during the entire observational time span.

- Instrumental delays are assumed negligible and temporarily ignored and \({a_2} = {\lambda _2}{M_2}\)

- All remaining parameters on the right-hand side of the above four equations, except the carrier phase ambiguities, are assumed to change with time. However, the functional dependency on time will be assumed unknown.

- Unmodelled errors \(e_{1,2}^k\) and\(\varepsilon _{1,2}^{\rm{k}}\) will be treated as noise. error propagation, needed to obtain variances and co-variances of the derived observables

- Simplifying the notation, "i" denoting the receiver and the upper index "k" will be removed and \({a_1} = {\lambda _1}{M_1}\) and \({a_2} = {\lambda _2}{M_2}\)

\(M_i^k = \left[ {{\phi _0}\left( {{t_0}} \right) - {\phi ^k}\left( {{t_0}} \right)} \right] + N_i^k\)

- Based on the first-order expression for the ionospheric range delay, \(I = 40.3TEC/{f^2}\), the ionospheric range delay on L 2 will be expressed in terms of the ionospheric range delay on as

\(I_{i,2}^k = qI_{i,1}^kwhereq = f_1^2/f_2^2\)

\({P_1} = s + {I_1};{{\rm{\Phi }}_1} = s - {I_1} + {a_1}\)

\({P_2} = s + q{I_1};{{\rm{\Phi }}_2} = s - q{I_1} + {a_2}\)

\({P_2} = s + q{I_1};{{\rm{\Phi }}_2} = s - q{I_1} + {a_2}\)