Pseudorange: The Distance That Is Not Quite a Distance

The basic GNSS observable is signal travel time multiplied by the speed of light, giving an apparent range.¹ It is called pseudorange because it contains more than pure geometry: receiver clock error, satellite clock error, atmospheric delays, instrumental delays, multipath and noise can all appear in the measurement model.²

Why timing matters

The speed of light is about:

$$c \approx 299,792,458\,m/s$$

A receiver-clock error of only $1\,\mu s$ corresponds to roughly:

$$299,792,458 \times 10^{-6} \approx 300\,m$$

GNSS positioning is not just a geometry problem. It is also a timing problem.

A simple measurement model

A simplified pseudorange model is:

$$P_i = \rho_i + c(\delta t_r - \delta t_i^s) + T_i + I_i + M_i + \epsilon_i$$

where:

• $\rho_i$ is geometric range.
• $\delta t_r$ is receiver clock offset.
• $\delta t_i^s$ is satellite clock offset.
• $T_i$ is tropospheric delay.
• $I_i$ is ionospheric delay.
• $M_i$ is multipath.
• $\epsilon_i$ is measurement noise.

For a first demo, collapse all non-clock errors into noise and solve:

$$P_i = \lVert \mathbf{r}-\mathbf{s}_i\rVert + c\delta t_r$$

Unknowns in 3D are:

$$[x, y, z, c\delta t_r]^T$$

This is why the beginner rule is “four satellites for 3D position plus receiver clock”.³

Interactive demo

The embedded visualiser makes the clock term explicit by drawing the true geometric ranges and the biased pseudoranges together.

What to watch

• Increasing clock bias expands or contracts all pseudorange circles together.
• Ignoring clock bias gives a visibly wrong position estimate.
• Solving for clock bias pulls the estimate back toward the true receiver when enough satellites exist.
• Even a very small timing offset creates a large metre-scale range error.

Try it

Set the clock bias to $1\,\mu s$. The range bias should be about $300\,m$. Then set it to $10\,ns$. Even that is about $3\,m$, which is large for precise navigation.

Limits of this model

Real receivers do not simply solve one clean pseudorange equation. They use satellite navigation messages, clock corrections, atmosphere models, code tracking, carrier tracking and quality checks. This post isolates the receiver clock effect because it is the most important beginner insight.

Footnotes

1. ESA Navipedia, “GNSS Basic Observables”. https://gssc.esa.int/navipedia/index.php/GNSS_Basic_Observables - Reference for pseudorange and carrier-phase observables. Accessed 2026-06-11. ↩

2. ESA Navipedia, “GNSS Measurements Modelling”. https://gssc.esa.int/navipedia/index.php/GNSS_Measurements_Modelling - Measurement equations including geometric range, clock terms, troposphere, ionosphere, multipath and noise. Accessed 2026-06-11. ↩

3. ESA Navipedia, “An intuitive approach to the GNSS positioning”. https://gssc.esa.int/navipedia/index.php/An_intuitive_approach_to_the_GNSS_positioning - Beginner-friendly explanation of pseudorange geometry, clock offset, 3D positioning and DOP. Accessed 2026-06-11. ↩