Critical satellite geometry and accuracy (GDOP)


Even with four satellites, GPS can run into trouble if the satellite geometry is unfavorable.

Two-solution principle

Normally, solving the GPS equations with four satellites gives two possible receiver positions:
– One realistic solution on or near Earth’s surface
– One “mirror” solution far out in space, which is discarded

Critical case: coplanar satellites

If the four satellites happen to lie (almost) in a single plane with respect to the Earth, these two solutions collapse into one.
This means the system of equations becomes degenerate, and you lose accuracy (or even uniqueness).

Mathematically, this can be checked by examining the unit direction vectors of the satellites:

\[
\hat{s}_i = \frac{s_i}{\| s_i \|}, \quad i = 1,2,3,4
\]

If these four unit vectors all lie in one plane, they lie on a circle on the unit sphere.
This condition can be expressed using a scalar triple product:

\[
(\hat{s}_1 – \hat{s}_4) \cdot \big( (\hat{s}_2 – \hat{s}_4) \times (\hat{s}_3 – \hat{s}_4) \big) = 0
\]

If this determinant is zero, the four vectors are coplanar → critical geometry.

Geometric Dilution of Precision (GDOP)

Even when not exactly coplanar, if satellites are clustered together in the sky, small timing errors can amplify into large position errors.
This effect is called Geometric Dilution of Precision (GDOP):
– Good geometry = satellites spread widely across the sky → low GDOP, high accuracy
– Bad geometry = satellites clustered or nearly coplanar → high GDOP, poor accuracy

Area of uncertainty due to good and poor satellite geometry.
Analysis of DOP and its Preciseness in GNSS Position Estimation – Scientific Figure on ResearchGate. Available from: https://www.researchgate.net/figure/Area-of-uncertainty-due-to-good-and-poor-satellite-geometry_fig2_307910819

✅ In practice, GPS receivers use more than 4 satellites whenever possible and choose the subset with the best geometry to minimize GDOP.