Root-Sum-Square, Then Scaled by Geometry

Ada re-derives this chapter’s own numbers step by step, at full precision

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Ada ADA · CALCULATION AUDIT

Root-Sum-Square, Then Scaled by Geometry

An autonomous tractor planting rows 50 cm apart and a shipping-container tracker happy with 10 m lean on the very same GPS receiver — what separates them is how six independent range errors combine and how satellite geometry then scales the result. The chapter adds those errors in quadrature to about 6 m rather than the 10.8 m a naive sum would give, then multiplies by geometry so the same receiver spans 6.3 m in the open to 44 m in an urban canyon. This audit re-runs that arithmetic to ask whether root-sum-square, then scaling by geometry, is really what sets the number.

Companion to the chapter GPS Accuracy and Enhancement — every number here comes from that chapter.

Ada: This chapter rests on two multiplicative claims – that six independent range errors combine to about 6 m rather than adding, and that satellite geometry then multiplies that into the real position error. Both deserve a full check. First, the linear sum the chapter warns against: 1.5 + 2.5 + 5.0 + 0.5 + 1.0 + 0.3 = 10.8 m. Independent random errors instead add in quadrature:

  • UERE = sqrt(1.5^2 + 2.5^2 + 5.0^2 + 0.5^2 + 1.0^2 + 0.3^2)
  • = sqrt(2.25 + 6.25 + 25.0 + 0.25 + 1.0 + 0.09) = sqrt(34.84) = 5.90 m

So root-sum-square gives 5.90 m (the chapter’s ~6 m), nearly half the 10.8 m a naive sum would report. Then position error is UERE x GDOP. The knowledge check uses 6 m x 4.0 = 24 m, and the urban-canyon scenario compounds a worse budget with worse geometry: sqrt(46.09) = 6.79 m (6.8 m rounded), then 6.79 x 6.5 = 44 m.

The design-meaningful point is that the same receiver spans 24 m to 44 m of error with no hardware change – geometry is a multiplier, not an addend – so an honest accuracy promise has to track live GDOP and satellite visibility, never just quote the 6 m ranging figure from a clear-sky datasheet.

Every number above is taken from the chapter’s own material and re-derived step by step.