Why is GPS altitude measurement less accurate than horizontal location measurement?

In the context of GPS, altitude measurement is approximately half as precise as horizontal location measurement.1

This discrepancy arises primarily from the geometry of the satellite constellation. Because all GPS satellites orbit above the Earth, a receiver can only triangulate its position using signals coming from the sky, rather than from all directions around it. This limited viewing geometry makes it mathematically more difficult to resolve the vertical coordinate (altitude) compared to horizontal coordinates (latitude and longitude), which are constrained by the intersection of spheres from a wider range of angles.1

Furthermore, while GPS calculates position based on a mathematical reference ellipsoid, the actual shape of the Earth—the geoid—is irregular due to variations in gravity and mass distribution. These complexities in Earth's figure mean that translating a GPS-derived position into a precise elevation above mean sea level requires accounting for these undulations, which adds another layer of potential error compared to horizontal positioning.23