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How would you adjust this leveling data?  

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GeeOddMike
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In a “recent” posting the use of least squares adjustment as “data doctoring” prompted me to generate an example of its utility and rigor.

How would you derive heights for the unknown points in the level network shown below? What are the heights of the unknowns? How accurate are the new heights with respect to the known heights? Optional: Prove your answer is be best possible.

BTW, the sample data was taken from the text “Linear Algebra, Geodesy and GPS” by Gilbert Strang and Kai Borre. 

E7398A58 DD94 4D7A BEB2 45575CB924E5

 

 

  

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John Hamilton
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Using Star*Net, I weighted the data at 0.006 mm/km (way higher than the normal 0.0015 m/km that I usually use). 

MicroSurvey STAR*NET-PRO Version 9,2,4,226
Run Date: Thu Oct 10 2019 15:45:01

Summary of Files Used and Option Settings
=========================================

Project Folder and Data Files

Project Name TEST LEVELING
Project Folder C:\PROJECTS
Data File List 1. test leveling.dat

Project Option Settings

STAR*NET Run Mode : Adjust with Error Propagation
Type of Adjustment : Lev
Project Units : Meters
Input/Output Coordinate Order : North-East
Create Coordinate File : Yes

Instrument Standard Error Settings

Project Default Instrument
Differential Levels : 0.006000 Meters / Km

Summary of Unadjusted Input Observations
========================================

Number of Entered Stations (Meters) = 3

Fixed Stations Elev Description
A 10.0210
B 10.3210
C 11.0020

Number of Differential Level Observations (Meters) = 5

From To Elev Diff StdErr Length
A E 0.7320 0.0059 970
A F 1.9780 0.0060 1002
B E 0.4200 0.0062 1070
C F 0.9880 0.0063 1110
E F 1.2580 0.0057 890

Adjustment Statistical Summary
==============================

Number of Stations = 5

Number of Observations = 5
Number of Unknowns = 2
Number of Redundant Obs = 3

Observation Count Sum Squares Error
of StdRes Factor
Level Data 5 4.636 1.243

Total 5 4.636 1.243

The Chi-Square Test at 5.00% Level Passed
Lower/Upper Bounds (0.268/1.765)

Adjusted Elevations and Error Propagation (Meters)
==================================================

Station Elev StdDev 95% Description
A 10.0210 0.000000 0.000000
B 10.3210 0.000000 0.000000
C 11.0020 0.000000 0.000000
E 10.7445 0.003671 0.007195
F 11.9976 0.003711 0.007274

Adjusted Observations and Residuals
===================================

Adjusted Differential Level Observations (Meters)

From To Elev Diff Residual StdErr StdRes File:Line
A E 0.7235 -0.0085 0.0059 1.4 1:5
C F 0.9956 0.0076 0.0063 1.2 1:6
E F 1.2531 -0.0049 0.0057 0.9 1:8
B E 0.4235 0.0035 0.0062 0.6 1:7
A F 1.9766 -0.0014 0.0060 0.2 1:4

Elapsed Time = 00:00:00

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Bill93
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Posted by: @john-hamilton

Using Star*Net, I weighted the data at 0.006 mm/km

Project Default Instrument
Differential Levels : 0.006000 Meters / Km

Being pedantic here, I am assuming you mean 0.006 meters per sqrt(km).  But neither you nor Star*Net mentioned the square root, so I'm not 100% sure how this is treated.

 

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GeeOddMike
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To Bill93 and MightyMoe,

Sorry for the mislabeling. The point labeled “F” should have been labeled “D.” 

I had hoped there would be some non-LSQ user responses showing how they approach this situation. 

 

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MathTeacher
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@john-hamilton

Shouldn't the distance from A to F be 1020 instead of 1002?

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John Hamilton
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MathTeacher
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@john-hamilton

How does it affect the output?

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MightyMoe
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I got hung up on D in question #1.

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Bill93
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I guess Point F should be labeled Point D or vice versa, as I can't find D either.

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Jim Frame
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Posted by: @bill93

I am assuming you mean 0.006 meters per sqrt(km).  But neither you nor Star*Net mentioned the square root, so I'm not 100% sure how this is treated.

Standard error of height per unit distance.  Star*Net also allows this error to be expressed as height per turn.

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