02 / WHY IT WORKS

Why geometry matters

A broad spread of directions helps separate east, north, up and receiver-clock effects. Concentrating satellites can make changes in those unknowns look similar to the measurements. The experiment holds the satellite count fixed, so you can isolate the effect of angular spread.

Read the ellipse as a model

The ellipse shows horizontal uncertainty from the same covariance model as the DOP values. Its long axis is the weakly constrained horizontal direction. We assume independent, equal-variance range errors with σ = 1 m. The axes are one standard deviation along the principal directions; this is not a 95% confidence region or measured scatter.

03 / UNDER THE SURFACE

From intuition to a model

From directions to covariance

For an unweighted local east–north–up model, each row of G contains a unit line of sight and a receiver-clock coefficient. A consistent sign reversal of spatial columns does not change the DOP diagonal. The inverse exists only when the geometry has full rank. Equal independent measurement variance gives position-and-clock covariance P = σ²Q.

Gi=[cos⁡eisin⁡Aicos⁡eicos⁡Aisin⁡ei1]T\mathbf{G}_i=\begin{bmatrix}\cos e_i\sin A_i\\\cos e_i\cos A_i\\\sin e_i\\1\end{bmatrix}^{\mathsf T}Q=(GTG)−1\mathbf{Q}=\left(\mathbf{G}^{\mathsf T}\mathbf{G}\right)^{-1}P=σ2Q\mathbf{P}=\sigma^2\mathbf{Q}

Ai, ei: satellite azimuth and elevation. Q indices are E, N, U, clock; the clock unknown is expressed as a range.

Which DOP should you read?

HDOP concerns horizontal geometry; VDOP concerns vertical geometry; PDOP combines all three spatial directions. GDOP includes the receiver-clock component too. These quantities are dimensionless. Multiplying by an assumed measurement standard deviation yields a model-based RMS scale, not a guaranteed error bound.

HDOP⁡=qEE+qNNVDOP⁡=qUU\begin{aligned}\operatorname{HDOP}&=\sqrt{q_{EE}+q_{NN}}\\[0.4em]\operatorname{VDOP}&=\sqrt{q_{UU}}\end{aligned}PDOP⁡2=HDOP⁡2+VDOP⁡2\operatorname{PDOP}^{2}=\operatorname{HDOP}^{2}+\operatorname{VDOP}^{2}GDOP⁡2=PDOP⁡2+qtt\operatorname{GDOP}^{2}=\operatorname{PDOP}^{2}+q_{tt}

04 / A WORKED EXAMPLE

Holding the number of satellites constant

Begin at 100% angular spread and note PDOP and HDOP. Reduce spread to 25% without removing any of the five satellites. Compare the recomputed values and ellipse dimensions. If PDOP were 2 and the common range-error standard deviation were 3 m, the model would give a 3D RMS scale of 6 m. Real biases or correlated errors violate that simple calculation. Return to 100% to verify the original result is reproduced.

Return to the experiment and try it ↑

05 / PUT IT TO WORK

Try it in GNSS View

  1. Choose your location, constellation selection and elevation mask.
  2. Use the planning chart to compare count and DOP at the same epoch.
  3. Inspect gaps as unavailable solutions; do not interpret them as zero error.
Open the observatory ↗

06 / COMMON QUESTIONS

Is DOP measured in meters?

No. DOP is dimensionless. A measurement-error model is required to turn geometry into an uncertainty scale.

Does adding a satellite always make real positioning better?

Additional valid independent observations help under a fixed ideal model. In practice a biased or poorly weighted observation can degrade a solution.

What does “unavailable” mean?

Too few valid directions, rank deficiency or numerical degeneracy prevents a reliable inverse. The chart must show a gap, not zero.

07 / FURTHER READING

An original GNSS View explanation. These references document the models, terminology and system context.

ESA Navipedia · Positioning Error ↗ESA Navipedia · GNSS Basic Observables ↗