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.

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*

I got hung up on D in question #1.

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

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.