The Error Budget

The understanding and management of errors is indispensable for finding the true geometric range ρ between a satellite and a receiver from either a pseudorange or carrier phase observation.

$$p=\rho +d_{\rho }+c\left(dt-dT\right)+d_{ion}+d_{trop}+{\epsilon }_{m_p}+{\epsilon }_{\mathrm{p \; \; \; \; }\left(\text{pseudorange}\right)}$$

$$\text{Φ}=\rho +d_{\rho }+c\left(dt-dT\right)+N\lambda -d_{ion}+d_{trop}+{\epsilon }_{m_{\text{Φ}}}+{\epsilon }_{\text{Φ }\left(\text{carrier phase}\right)}$$

Both equations include environmental and physical limitations called range biases.

Atmospheric errors are among the biases; two are the ionospheric effect, d_ion, and the tropospheric effect, d_trop.  Other biases, clock errors symbolized by (dt-dT) and receiver noise, ε_ρ and ε_φ, multipath, ε_mρ and ε_mφ, and orbital errors, d_ρ, are unique to satellite surveying methods. As you can see, each of these biases comes from a different source. They are each independent of one another, but they combine to obscure the true geometric range. The objective here is to discuss each of them separately.

Here, we see the formula of the pseudorange error budget on the upper portion. As it indicates, p is the pseudorange, measurement equals ρ, rho, the true range between the GPS receiver and satellite. However, as you can see, there are many more elements, errors, or biases that contaminate the pseudorange— the satellite orbital errors, the ephemeris errors, etc.

The time difference (dt-dT) is between the satellite clock offset, and the receiver clock offset from GPS Time, as was mentioned in the previous lesson. There is also the ionospheric delay, the attenuation, of the signal as it passes through the ionosphere. Distinct from these biases are multipath and receiver noise. These two are distinct because, unlike the others, it is difficult to deal with them in a statistical way. Nevertheless, all are part of the error budget.

Directly below the pseudorange formula, you see the carrier phase formula. It also equals true range symbolized by capital phi,Φ. It includes errors that are very similar to those in the pseudorange formula. But one is included that is obviously different. It is symbolized by the capital N followed by the letter lambda, λ. This is the integer ambiguity. It doesn't occur in the pseudorange measurement.

The management of the errors shown in the formulas, the biases, is indispensable for finding the true range from either a pseudorange or a carrier phase observation. In this lesson, we're going to try to understand the source of these errors so that they can be dealt with.