ERRORS AND ADJUSTMENTS

+υυ

y = f (x)

y

Error size

xΔυ

y

Example

(a) (d)(c)(b)

• Which set of shots is more– Accurate?– Precise?

Precision ExampleObservation Pacing,

pTaping,t

EDM,e

1 571 567.17 567.1332 563 567.08 567.1243 566 567.12 567.1294 588 567.38 567.1655 557 567.01 567.114

• Which observation of a distance is– More precise?– More accurate?

e

567.0 567.1 567.2 567.3 567.4

t t t t teee e

Mean of e Mean of t

Advantages of Least Squares• Errors adjusted according to

laws of probability• Easy to perform with today’s

computers• Provides a single solution to a

set of observations• Forces observations to satisfy

geometric closures• Can perform presurvey

planning

Example (cont.)• Construct the 95% confidence interval• From the F0.025 distribution table (D.4) we

find

– F0.025,24,30 = 2.21 and F0.025,30,24 = 2.14• Construct the 95% confidence interval

2122

2122

2.25 1 2.25 2.210.49 2.14 0.49

2.14 10.15

Population versus Sample• Below are random samples having 10 values each.

32.2 30.0 24.2 18.9 17.2 22.4 21.3 21.3 26.4 24.5

28.0 21.2 18.9 33.2 30.2 26.5 25.2 29.0 21.8 26.3

33.9 21.3 21.3 25.2 18.9 19.6 28.5 36.0 27.1 30.6

24.2 24.4 28.5 25.3 32.2 19.6 32.9 21.3 24.0 26.5

μ = 23.84σ2 = 22.05

μ = 26.03 σ2 = 19.39

μ = 26.24σ2 = 36.29

μ = 25.89σ2 = 18.41

Basics of Error Propagation• Control, distances, directions, and angles

all contain random errors• So what are the errors in the computed

– Latitudes?– Departures?– Coordinates?– Areas?– And so on?

• If we can answer this, we can answer the question, “did our observations meet acceptable closures?”