Math Bridge: GPS error and geometry

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Math BridgeUX DesignStruggle-friendly runway

Why does satellite geometry turn 6 metres into 39?

Combine independent errors, then expose the separate geometry multiplier.

UX Uma, the guideUX Uma guides
The one targetCalculate UERE and show why clustered satellites amplify it.
The chapter caseSix error sources; 6 m rounded UERE; GDOP 1.5, 4.0, 6.5.
What it buys youAn accuracy claim that carries both noise and sky geometry.

A field team faces an unresolved physical question: Why does satellite geometry turn 6 metres into 39? They must answer it before changing separation on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is separation. The middle card applies this page's relationship. The green card is variance sum. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Separation changes variance sum An input card leads through the page relationship to the variance sum result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Clustered directions cannot separate position changes well, so the same range noise stretches into a larger location region.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for separation is 10.

  2. 2

    Name the relationship. 1.5²+2.5²+5.0²+0.5²+1.0²+0.3² = 34.8 m² UERE = √34.8 = 5.90 m ≈ 6 m 6x1.5 = 9.0 m; 6x4.0 = 24.0 m; 6x6.5 = 39.0 m

  3. 3

    Substitute the chapter fixture. Set separation to 10. The page ledger gives variance sum as 34.840.

  4. 4

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

Predict, then change separation

Try Predict the direction of variance sum. Move one control, calculate, then check your prediction.

10
Chapter baseline
Variance sum

Observe Clustered directions cannot separate position changes well, so the same range noise stretches into a larger location region. Reset the control to 10 and compare variance sum.

Explain Only separation moves here. The other chapter fixtures remain fixed.

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 separation moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only separation. It replays the chapter fixture; it is not a field prediction.

1. Separate error size from error direction

Independent random errors can point positive or negative, so they partly cancel. Their variances always add. Geometry is different: it describes how the satellite directions turn the remaining range error into a position error.

UX Uma: First combine the range evidence; only then multiply by geometry.

2. Name every algebra move

1

Square each errorThis removes sign and creates variance.

2

Add variancesΣσ².

3

Undo the squareUERE=√Σσ².

4

Apply geometryσposition=GDOP×UERE.

3. Reproduce the chapter table

1.5²+2.5²+5.0²+0.5²+1.0²+0.3² = 34.8 m²
UERE = √34.8 = 5.90 m ≈ 6 m
6×1.5 = 9.0 m; 6×4.0 = 24.0 m; 6×6.5 = 39.0 m

The ionosphere contributes 25/34.8 = 71.8% of variance. Linear addition gives 10.8 m, 1.83× the root-sum-square result.

4. Try the satellite separation

TryMove two toy satellite directions apart and watch their geometry multiplier.

Separation
Variance sum
UERE
Linear sum
Linear/RSS
Ionosphere variance
Good geometry
Quiz geometry
Canyon geometry
Toy DOP
Toy position error

ObserveAt 10°, toy DOP is 8.14; at 90°, it falls to 1.41.

ExplainClustered directions cannot separate position changes well, so the same range noise stretches into a larger location region.

Technical boundaries.

The slider is a two-direction, flat-world picture of a real matrix problem.

Receiver
Real GPS solves 3-D position plus clock bias
Errors
Root-sum-square assumes independence
Confidence
One sigma is not a guaranteed maximum

Use receiver covariance, satellite health, multipath tests, and the product's confidence rule.

5. Match accuracy to consequence

Use a wide, uncertain zone when the error supports only a hint. Require corrections, better geometry, fusion, or a guarded fallback before position can trigger a costly or safety-sensitive action.

6. Record the accuracy evidence

Store the error-source assumptions, satellite set, GDOP, covariance or confidence, age, environment, enhancement service, outage behavior, and field truth used to validate the claim.

7. Check yourself

Why not add the six errors directly?
Answer: Independent random errors combine by variance, so square, add, then take the square root.
What does GDOP=6.5 do to 6 m?
Answer: It multiplies the range error to about 39 m position error.
Does the two-satellite slider model real GPS?
Answer: No. It only makes direction clustering visible; real receivers solve a larger 3-D system.
Honesty boundary.

The six errors and GDOP examples are the chapter's values; the two-direction DOP is a deliberately bounded teaching model.

5.90 m
Exact chapter RSS
6 m
Chapter rounded UERE
√2/sin(φ)
2-D geometry illustration

Correct, not complete: this ledger does not qualify a receiver, correction service, or location-triggered action.

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