Gravity and Navigation: Is Gravity Just 9.81?

Introductory physics often uses $g = 9.81\,m/s^2$. Navigation needs more care. Earth’s gravity varies with latitude, height, and local mass distribution. WGS 84 includes an ellipsoidal normal Earth gravitational model, and NGA’s Earth Gravitational Model material discusses gravity anomalies, geoid heights, deflections of the vertical, and gravity disturbances used in WGS 84 support and navigation systems.¹²

Why gravity modelling matters

An INS subtracts or adds a gravity model when converting accelerometer specific force into acceleration. If the gravity model is wrong, the error can look like a small acceleration bias. Small acceleration errors integrate into velocity and position errors.

Gravity in the navigation equations

Specific force relationship:

$$\mathbf{a}_n = \mathbf{C}_{b}^{n}\mathbf{f}_b + \mathbf{g}_n$$

If the gravity model has an error:

$$\delta \mathbf{g} = \mathbf{g}_{true} - \mathbf{g}_{model}$$

then the navigation acceleration has an error:

$$\delta \mathbf{a} \approx \delta \mathbf{g}$$

Position error from a constant acceleration error is approximately:

$$\delta p(t) = \frac{1}{2}\delta g\,t^2$$

Vertical datums add another important idea. GNSS gives ellipsoidal height, while many practical heights are orthometric heights related to the geoid. NOAA’s geoid service describes converting GNSS-derived ellipsoidal height to orthometric height using geoid height.³

A common relationship is:

$$H \approx h - N$$

where $H$ is orthometric height, $h$ is ellipsoidal height, and $N$ is geoid undulation.

Interactive demo: gravity model playground

The demo below uses a synthetic anomaly map and a simplified normal-gravity equation. The aim is to make gravity-model mismatch and height-datum confusion visible without implying that the browser is running a full geopotential model.

What to watch

• Switching from a constant gravity value to a latitude-aware model changes the implied acceleration error.
• Tiny gravity mismatches integrate into meaningful position error over time.
• Accelerometer bias and gravity-model error play similarly in the simplest teaching model.
• The geoid and ellipsoid diagram explains why GNSS height is not automatically “height above sea level”.

Try this

Set a high latitude, raise the height, and keep the constant 9.81 model selected. Then extend the duration. The induced position error stays conceptually small in acceleration terms, but integration makes it accumulate.

What this leaves out

Real gravity modelling is a specialised topic involving geopotential models, local gravity surveys, terrain effects and frame conventions. This post is a conceptual bridge, not a geodesy course.

Footnotes

1. National Geospatial-Intelligence Agency, “World Geodetic System 1984 (WGS 84)”. https://earth-info.nga.mil/GandG/wgs84/ - Official reference for WGS 84 as a 3D coordinate reference frame for latitude, longitude, and height in navigation, positioning, and targeting. Accessed 2026-06-11. ↩

2. NGA Office of Geomatics, “Earth Gravitational Model (EGM)”. https://earth-info.nga.mil/GandG/wgs84/gravitymod - Gravity anomalies, geoid heights, deflections of the vertical, and gravity disturbances used in WGS 84 support and navigation systems. Accessed 2026-06-11. ↩

3. NOAA NGS, “The Geoid Height Service”. https://www.ngs.noaa.gov/web_services/geoid.shtml - Geoid height service for converting GNSS-derived ellipsoidal height to orthometric height. Accessed 2026-06-11. ↩