GNSS Observations

\rho_{S,f} = r_S + c\left( \Delta t_s - \Delta t_R \right) + T_S
ρS,f=rS+c(Δts−ΔtR)+TS\rho_{S,f} = r_S + c\left( \Delta t_s - \Delta t_R \right) + T_S
+ b^\rho_{S,f} + b^\rho_{R,f} + I_{S,f} + M^\rho_{S,f} + \epsilon^\rho_{S,f}
+bS,fρ+bR,fρ+IS,f+MS,fρ+ϵS,fρ+ b^\rho_{S,f} + b^\rho_{R,f} + I_{S,f} + M^\rho_{S,f} + \epsilon^\rho_{S,f}
\phi_{S,f} = r_S + c\left( \Delta t_s - \Delta t_R \right) + T_s
ϕS,f=rS+c(Δts−ΔtR)+Ts \phi_{S,f} = r_S + c\left( \Delta t_s - \Delta t_R \right) + T_s
+ b^\phi_{S,f} + b^\phi_{R,f} - I_{S,f} + \lambda_f N_{S,f} + M^\phi_{S,f} + \epsilon^\phi_{S,f}
+bS,fϕ+bR,fϕ−IS,f+λfNS,f+MS,fϕ+ϵS,fϕ+ b^\phi_{S,f} + b^\phi_{R,f} - I_{S,f} + \lambda_f N_{S,f} + M^\phi_{S,f} + \epsilon^\phi_{S,f}

HARDWARE

BIAS

IONOSPHERE DELAY

CARRIER

AMBIGUITY

MULTIPATH EFFECTS

FREQUENCY INDEPENDENT EFFECTS

Bias and Error Assumptions

b_{S,f}^\phi \approx b_{S,f}^\rho
bS,fϕ≈bS,fρb_{S,f}^\phi \approx b_{S,f}^\rho
b_{R,f}^\phi \approx b_{R,f}^\rho
bR,fϕ≈bR,fρb_{R,f}^\phi \approx b_{R,f}^\rho

assume no code-carrier bias

\Delta b_{S,f_i,f_j}, \ \Delta b_{R, f_i, f_j}
ΔbS,fi,fj, ΔbR,fi,fj\Delta b_{S,f_i,f_j}, \ \Delta b_{R, f_i, f_j}
  • receiver code-carrier bias usually compensated for by manufacturer

since we will use geometry free combinations, we only care about inter-frequency biases

(IFB)

E\left[ M_{f_i} M_{f_j} \right] \approx 0
E[MfiMfj]≈0E\left[ M_{f_i} M_{f_j} \right] \approx 0
M^\phi \ll M^\rho
Mϕ≪MρM^\phi \ll M^\rho

multipath uncorrelated across different signals

carrier pseudorange noise / multipath is small compared to code

not true

\epsilon^\phi \ll \epsilon^\rho
ϵϕ≪ϵρ\epsilon^\phi \ll \epsilon^\rho
\approx \text{constant}
≈constant\approx \text{constant}

over 1 day

Geometry-Free Combinations

\phi_{S,f_i} - \phi_{S,f_j} = \Delta b_{S,f_i,f_j} + \Delta b_{R,f_i,f_j} - \left(I_{S,f_i} - I_{S,f_j} \right)
ϕS,fi−ϕS,fj=ΔbS,fi,fj+ΔbR,fi,fj−(IS,fi−IS,fj)\phi_{S,f_i} - \phi_{S,f_j} = \Delta b_{S,f_i,f_j} + \Delta b_{R,f_i,f_j} - \left(I_{S,f_i} - I_{S,f_j} \right)
+ \lambda_{f_i}N_{S,f_i} - \lambda_{f_j}N_{S,f_j} + M^\phi_{S,f_i} - M^\phi_{S,f_j} + \epsilon^\phi_{S,f_i} - \epsilon^\phi_{S,f_j}
+λfiNS,fi−λfjNS,fj+MS,fiϕ−MS,fjϕ+ϵS,fiϕ−ϵS,fjϕ+ \lambda_{f_i}N_{S,f_i} - \lambda_{f_j}N_{S,f_j} + M^\phi_{S,f_i} - M^\phi_{S,f_j} + \epsilon^\phi_{S,f_i} - \epsilon^\phi_{S,f_j}
\rho_{S,f} - \phi_{S,f} \approx 2I_{S,f} - \lambda_fN_{S,f} + M_{S,f}^\rho + \epsilon_{S,f}^{\rho}
ρS,f−ϕS,f≈2IS,f−λfNS,f+MS,fρ+ϵS,fρ\rho_{S,f} - \phi_{S,f} \approx 2I_{S,f} - \lambda_fN_{S,f} + M_{S,f}^\rho + \epsilon_{S,f}^{\rho}
\Delta ADR_{f_i,f_j}
ΔADRfi,fj\Delta ADR_{f_i,f_j}
CMC_f
CMCfCMC_f

Geometry-Free Combinations

We can remove satellite IFB using estimates from IGS

\Delta b_{S,f_i,f_j}
ΔbS,fi,fj\Delta b_{S,f_i,f_j}

We express ionosphere delays in terms of TEC

\Delta ADR_{f_i,f_j} \approx \Delta b_{R,f_i,f_j} - \frac{40.3\times 10^{16}}{\alpha_{f_i,f_j}}TEC
ΔADRfi,fj≈ΔbR,fi,fj−40.3×1016αfi,fjTEC\Delta ADR_{f_i,f_j} \approx \Delta b_{R,f_i,f_j} - \frac{40.3\times 10^{16}}{\alpha_{f_i,f_j}}TEC
+ \lambda_{f_i} N_{S,f_i} - \lambda_{f_j} N_{S,f_j} + \cdots
+λfiNS,fi−λfjNS,fj+⋯+ \lambda_{f_i} N_{S,f_i} - \lambda_{f_j} N_{S,f_j} + \cdots
CMC_f \approx \frac{40.3\times 10^{16}}{f^2}TEC - \lambda_f N_{S,f} + \cdots
CMCf≈40.3×1016f2TEC−λfNS,f+⋯CMC_f \approx \frac{40.3\times 10^{16}}{f^2}TEC - \lambda_f N_{S,f} + \cdots

multipath / noise / unmodeled errors

ILS TEC Estimation

Use iterative least-squares to solve large sparse system for 1 day of data

Example of 2-Frequency TEC From Velox LEO Satellite

before algorithm

after algorithm

2016-01-24