Adjustment of Level Nets Introduction In this chapter

Adjustment of Level Nets

Introduction • In this chapter we will deal with differential leveling only • In SUR 2101 you learned how to close and adjust a level loop – that is actually a least squares solution • In complex situations where an interconnected level net is involved, we need least squares • Method of observation equations is most common

Leveling Observation Equation The observed elevation difference between stations I and J The ΔElev may be simply a Backsight minus Foresight, or the result of many Backsights and Foresights along a leveling path

Unweighted Example

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Compute Residuals

Weighted Level Adjustment • Recall that weights are inversely proportional to line length • Weights are also inversely proportional to number of setups • Weights can also be estimated a priori based on standard deviations of readings, propagated as error of a sum

Weighted Level Example Same as previous example, but consider line lengths for weighting

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Reference Standard Deviation Standard deviation of an observation of unit weight

Unweighted Example – S 0

Weighted Example – S 0

Another Weighted Adjustment From: To: ΔE (m) σ (m) A B 10. 509 0. 006 B C 5. 360 0. 004 C D -8. 523 0. 005 D A -7. 348 0. 003 B D -3. 167 0. 004 A C 15. 881 0. 012 Benchmark A has an elevation of 437. 596 m. What are the most probable values for B, C, D?

Observation Equations +B -B = A + 10. 509 + v 1 = 448. 105 + v 1 +C -C = 5. 360 + v 2 +D = -8. 523 + v 3 -D = -A – 7. 348 + v 4 = -444. 944 + v 4 -B +D = -3. 167 + v 5 +C = A + 15. 881 + v 6 = 453. 477 + v 6 See spreadsheet for solution.

Units for Standard Deviation of Unit Weight Unweighted (all unit weight), first example: feet Weighted by line length, second example: Weighted by standard deviation, third example: Unitless

S 0 – Weights by A-Priori σ • When weights are based on a priori standard deviations, the computed reference variance should be the input value (typically 1) • We can do a Chi-square test to see if it is significantly different from 1 • If a Chi-square test fails, it may be due to one or more blunders or incorrect a priori standard deviations • Blunders can cause the reference variance to be much greater than 1

Chi-Square Test for Example H 0: σ 2 = 1 H a : σ2 ≠ 1 (two-tail test at 0. 05 significance) Degrees of freedom = 6 -3 = 3 χ20. 975, 3 = 0. 216 0. 652 = 0. 42 > 0. 216, therefore do not reject the null hypothesis In other words, we have no statistical evidence that the a priori standard deviations were incorrect.

Reference Variance Test • We can make the reference variance equal 1 by adjusting the a priori standard deviations • Usually, if the computed value is < 1, nothing further is done • If the value fails the Chi-Square test by being much greater than 1, the first thing to do is look for blunders • Large residuals often indicate blunders