SRV-03

Position dilution of precision

PDOP ≈ √(σE²+σN²+σU²)/σUERE.

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GNSSGPS ICDIGS

Governing equation

\mathrm{PDOP}=\dfrac{\sqrt{\sigma_E^2+\sigma_N^2+\sigma_U^2}}{\sigma_{\mathrm{UERE}}}

$\sigma_{\mathrm{UERE}} \in \left[0.2, 8\right]$

where

\sigma_E: East RMS (m)
\sigma_N: North RMS (m)
\sigma_U: Up RMS (m)
\sigma_{\mathrm{UERE}}: UERE (m)
PDOP: PDOP (—)

The figure, part by part

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①

Governing law — red

\mathrm{PDOP}=\dfrac{\sqrt{\sigma_E^2+\sigma_N^2+\sigma_U^2}}{\sigma_{\mathrm{UERE}}}

$\sigma_{\mathrm{UERE}} \in \left[0.2, 8\right]$

$\sigma_{\mathrm{UERE}} = 0.2$ $PDOP = 14.5$

Along the curve

②

Domain of the variable — blue

\sigma_{\mathrm{UERE}}_{\min} = 0.2,\quad \sigma_{\mathrm{UERE}}_{\max} = 8

$\sigma_{\mathrm{UERE}} \in \left[0.2, 8\right]$

$\sigma_{\mathrm{UERE}} = 0.2$ $\sigma_{\mathrm{UERE}} = 0.2$

Along the curve

③

Response curve — green

\mathrm{PDOP}=\dfrac{\sqrt{\sigma_E^2+\sigma_N^2+\sigma_U^2}}{\sigma_{\mathrm{UERE}}}

$\sigma_{\mathrm{UERE}} \in \left[0.2, 8\right]$

$\sigma_{\mathrm{UERE}} = 0.2$ $PDOP = 14.5$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}PDOP}{\mathrm{d}\sigma_{\mathrm{UERE}}}

$\sigma_{\mathrm{UERE}} \in \left[0.2, 8\right]$

$\sigma_{\mathrm{UERE}} = 0.249$ $PDOP^{\prime} = -48.64$

Along the curve

Lecture brief

Historical brief

From chain and theodolite through Bowditch’s compass-rule (1802) to GPS PDOP, surveying is the propagation of small errors across a network. These sheets close traverses, reduce stadia and state map scale. This sheet (SRV-03 — Position dilution of precision) is the form associated with GPS ICD · IGS. Working symbols: $\sigma_E$ , $\sigma_N$ , $\sigma_U$ , $\sigma_{\mathrm{UERE}}$ $\rightarrow$ $PDOP$ . PDOP maps user-equivalent range error onto 3-D position through the geometry matrix (HᵀH)⁻¹.

Purpose

Purpose: compute

PDOP

from

\sigma_E

\sigma_N

\sigma_U

\sigma_{\mathrm{UERE}}

in Surveying via

\mathrm{PDOP}=\dfrac{\sqrt{\sigma_E^2+\sigma_N^2+\sigma_U^2}}{\sigma_{\mathrm{UERE}}}

PDOP ≈ √(σE²+σN²+σU²)/σUERE. Use it when a real surveying question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\sigma_E = 1.200\,\mathrm{m}

\sigma_N = 1.100\,\mathrm{m}

\sigma_U = 2.400\,\mathrm{m}

\sigma_{\mathrm{UERE}} = 1.500\,\mathrm{m}

, the governing relation

\mathrm{PDOP}=\dfrac{\sqrt{\sigma_E^2+\sigma_N^2+\sigma_U^2}}{\sigma_{\mathrm{UERE}}}

yields

PDOP = 1.933\,\mathrm{—}

. Enter component RMS errors and UERE. Pedagogical ratio. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

East RMS \sigma_E1.2 mNorth RMS \sigma_N1.1 mUp RMS \sigma_U2.4 mUERE \sigma_{\mathrm{UERE}}1.5 m

Outputs

PDOP PDOP1.933 —

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Narration of this film

Enter component RMS errors and UERE. Pedagogical ratio.

PDOP maps user-equivalent range error onto 3-D position through the geometry matrix (HᵀH)⁻¹.

Reading speed

Watch on YouTube