SRV-24

Distance error propagation

σD = √(σE² + σN²) for D from a ΔE, ΔN pair, first-order.

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ErrorsVariance propagation

Governing equation

\sigma_D=\sqrt{\sigma_E^2+\sigma_N^2}

$\sigma_E \in \left[0.5, 50\right]$

where

\sigma_E: Easting σ (mm)
\sigma_N: Northing σ (mm)
\sigma_D: Distance σ (mm)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\sigma_D=\sqrt{\sigma_E^2+\sigma_N^2}

$\sigma_E \in \left[0.5, 50\right]$

$\sigma_E = 0.5$ $\sigma_D = 6.021$

Along the curve

②

Domain of the variable — blue

\sigma_E_{\min} = 0.5,\quad \sigma_E_{\max} = 50

$\sigma_E \in \left[0.5, 50\right]$

$\sigma_E = 0.5$ $\sigma_E = 0.5$

Along the curve

③

Response curve — green

\sigma_D=\sqrt{\sigma_E^2+\sigma_N^2}

$\sigma_E \in \left[0.5, 50\right]$

$\sigma_E = 0.5$ $\sigma_D = 6.021$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\sigma_D}{\mathrm{d}\sigma_E}

$\sigma_E \in \left[0.5, 50\right]$

$\sigma_E = 0.811$ $\sigma_D^{\prime} = 0.134$

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-24 — Distance error propagation) is the form associated with Variance propagation. Working symbols: $\sigma_E$ , $\sigma_N$ $\rightarrow$ $\sigma_D$ . σf² = (∂f/∂x)² σx² + (∂f/∂y)² σy². For D = √(ΔE²+ΔN²) this collapses to the RSS of the components when they are equal-weight.

Purpose

Purpose: compute

\sigma_D

from

\sigma_E

\sigma_N

in Surveying via

\sigma_D=\sqrt{\sigma_E^2+\sigma_N^2}

σD = √(σE² + σN²) for D from a ΔE, ΔN pair, first-order. 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 = 8.000\,\mathrm{mm}

\sigma_N = 6.000\,\mathrm{mm}

, the governing relation

\sigma_D=\sqrt{\sigma_E^2+\sigma_N^2}

yields

\sigma_D = 10.00\,\mathrm{mm}

. An error ellipse, a distance, a σD. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Easting σ \sigma_E8 mmNorthing σ \sigma_N6 mm

Outputs

Distance σ \sigma_D10.00 mm

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Narration of this film

An error ellipse, a distance, a σD.

σf² = (∂f/∂x)² σx² + (∂f/∂y)² σy². For D = √(ΔE²+ΔN²) this collapses to the RSS of the components when they are equal-weight.

Reading speed

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