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Tutorial Series · Part 11 of 27

Satellite Geometry and DOP

Show why satellite layout affects position uncertainty in GNSS.

By

  • gnss
  • dop
  • uncertainty
Series Context

Navigation from First Principles

This article is part of an ordered course. Use the previous and next links to stay in sequence.

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Two GNSS receivers can have the same measurement noise but different position accuracy because the satellites are arranged differently in the sky. GPS.gov lists satellite geometry as one of the factors affecting user accuracy.1 Navipedia’s DOP discussion connects satellite geometry to the covariance of the position estimate.2

Why geometry matters

If all visible satellites are bunched in one part of the sky, the receiver has poor geometric leverage. If satellites surround the receiver across the sky, the position estimate is usually better conditioned.

A simple model

For a linearised GNSS position update:

Δy=GΔx+ϵ\Delta \mathbf{y}=\mathbf{G}\Delta \mathbf{x}+\boldsymbol{\epsilon}

Assuming independent equal-variance measurement noise:

E[ϵϵT]=σ2IE[\boldsymbol{\epsilon}\boldsymbol{\epsilon}^T]=\sigma^2\mathbf{I}

The state covariance is:

PΔx=σ2(GTG)−1\mathbf{P}_{\Delta x}=\sigma^2(\mathbf{G}^T\mathbf{G})^{-1}

The matrix G\mathbf{G} is built from the receiver-to-satellite geometry. The measurement noise σ\sigma matters, but the geometry matrix can amplify or reduce how that noise appears in position.

A generic DOP-like value can be shown as:

DOP=trace⁡((GTG)−1)DOP = \sqrt{\operatorname{trace}((\mathbf{G}^T\mathbf{G})^{-1})}

Interactive demo

The demo below links a draggable skyplot to a horizontal uncertainty ellipse. It uses a reduced teaching approximation, but it still captures the main idea: geometry changes how measurement noise maps into position uncertainty.

Interactive Demo

Skyplot DOP Playground

Drag the satellites to spread them around the sky. The left plot shows geometry in the skyplot, while the right plot turns that geometry into a horizontal uncertainty ellipse.

N E S W

Skyplot: centre is overhead, outer ring is the horizon.

Ellipse: bigger and longer means weaker horizontal geometry.

Controls

Try clustering the satellites on one side, then use “Spread geometry” to see the DOP drop.

Readouts

Satellites drawn4
Approximate DOP0.00
Horizontal uncertainty0.0 m
Geometry verdictwell spread

What to watch

  • Clustering satellites worsens the displayed DOP and stretches the ellipse.
  • Spreading satellites around the sky improves the geometry.
  • Increasing measurement noise inflates uncertainty even when geometry is unchanged.
  • DOP is about layout, not the whole error budget.

Try it

Place four satellites close together in the north-east sky. Note the elongated uncertainty ellipse. Then drag them around the sky. The ellipse should shrink and become more balanced.

Limits of DOP

Good DOP does not guarantee good accuracy. Multipath, non-line-of-sight reception, atmospheric errors and receiver quality can still degrade the solution. Geometry is important, but it is only one part of the accuracy story.1

Footnotes

  1. GPS.gov, “GPS Accuracy”. https://www.gps.gov/gps-accuracy - Official GPS accuracy FAQ covering satellite geometry, signal blockage, atmospheric conditions, receiver design, multipath, jamming and smartphone accuracy. Accessed 2026-06-11. ↩ ↩2

  2. ESA Navipedia, “Positioning Error / Dilution of Precision”. https://gssc.esa.int/navipedia/index.php?title=Positioning_Error - DOP and covariance relationship for satellite geometry. Accessed 2026-06-11. ↩