Root-Sum-Square, Then Scaled by Geometry

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.

See the relationship before changing it

The figure reads from left to right. The blue card is gdop. The middle card applies the page rule. The green card is position error. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

GDOP changes position error An input card leads through the rule position error = 5.90 m UERE x GDOP to the position error result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. Geometry multiplies the same receiver error without changing the hardware.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 4.

  2. 2

    Name the relationship. position error = 5.90 m UERE x GDOP

  3. 3

    Substitute with units. 5.90 x 4 = 23.60 m

  4. 4

    Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change gdop

Try Predict the direction of position error = 5.90 m UERE x GDOP. Test another gdop, then compare position error.

4
Chapter baseline
Position error

Observe Geometry multiplies the same receiver error without changing the hardware. Reset gdop to 4 and compare position error.

Explain Geometry multiplies the same receiver error without changing the hardware.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only gdop moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only gdop. It is not a field prediction.

TryStart Check derivation with the six range-error terms that sum linearly to 10.8 m.
ObserveRoot-sum-square produces 5.90 m UERE; multiplying by GDOP then expands position error from about 6.3 m to 44 m.
ExplainIndependent errors add by variance rather than magnitude, and the geometry factor multiplies that 5.90 m ranging floor into the final position error.

Ready: use the stated baseline inputs, then compare each displayed result.

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.

Technical boundaries. Biased range errors, multipath, clock correlation, and changing satellite geometry are omitted; independent fixed terms are combined by root-sum-square and one GDOP multiplier.