1. Network Least Squares Adjustments Demystified
By Bruce Carlson and Dean Goodman
Carlson Software

2. Course Outline Measurement Errors Adjustments
Types of Errors
Error propagation
Adjustments
Averaging
Traditional Adjustments
Least Squares Adjustment
Advantages of LSA
Creating a Project in SurvNET
Sample traverse project in Carlson SurvNET
Processing unique datasets
Traverse networks
Traverses with triangulation/resection data
GPS Vectors
Combination of Traverse/GPS data
ALTA Surveys
Certifications
Relative error ellipses

3. Surveyors make measurements
One aspect of a land surveyor’s job is to make measurements.
There are no perfect measurements, all measurements have errors.
Errors are inherent in the instrumentation we use.
A 5” instrument measures angles to +- 5”.
An EDM may measures distances to ’ and 3PPM.

4. There are three classifications of errors:
Types of Errors
There are three classifications of errors:
Blunders
Systematic Errors
Random Errors

5. Blunders Blunders are mistakes and cannot be adjusted.
Blunders MUST be removed from the data set prior to adjustment.
Examples of Blunders:
Failure to level the instrument.
Failure to put the instrument or target over the point.
Assigning the wrong point number to a point.
Instrumentation out of adjustment.
Recording the wrong measurement.

6. Systematic Errors
Systematic Errors are predictable errors and should be removed from the dataset prior to adjustment.
Examples of systematic errors:
Atmospheric corrections – temperature, and pressure.
Curvature and Refraction corrections.
Wrong prism constant

7. Random Errors
Random errors are errors inherent in the instrumentation used to make the measurement.
A theodolite may measure to a +- five second precision.
An EDM may measure to 0.01 feet plus 3 PPM
You can only set your instrument over a point to a feet accuracy.
You can only measure from the point to the height of instrument to a ’ accuracy.
Random errors can be adjusted out of the traverse.

8. Error Propagation Random errors propagate throughout the traverse.
Errors can accumulate, cancel or decrease.
Prior to a least squares adjustments, each measurement is assigned an initial standard error based on the equipment specifications.
Error propagation can then be predicted using statistical, mathematical models.

9. Error Propagation
As you traverse further from known control points, the less you know your position.
The next slide shows the error ellipses of a traverse with one control point.

11. Error Propagation
Error ellipses increase as you get further from a known point.
The next slide shows the error ellipses of the same traverse with two known points,

13. Error Propagation
Additional control points and redundant measurements add strength to the traverse and make the statistics more meaningful.
GPS can be used to add additional control.
Traverse densification and measurements to points from different locations can add redundancy to the traverse.
Triangulation can also be useful in adding redundancy.

14. Adjustments Why do we adjust traverses?
All traverses have errors; they do not close exactly on the terminal point.
If we do not adjust the traverse, all the error is placed in the last leg of the traverse which is not a valid assumption.
Error adjustments are important to future work on a project. Placing all the error in one measurement can prove problematic for both project design and layout.

15. Averaging Averaging is a type of adjustment.
We average SETS of angles measured in both direct and reverse faces:
For additional precision
(Sm = S / √N) Standard deviation of the mean equals the standard deviation of the single measurement divided by the square root of the number of measurements.
As a check against blunders
To help remove instrument errors
We average distances and zenith angles measured in both faces and in both directions:
To mitigate curvature and refraction errors
To improve vertical closures.

16. Traditional Adjustments
Prior to the advent of high powered computers, when only calculators or hand calculations were used, there were three popular adjustments:
Transit Rule
Crandall’s Rule
Compass Rule
Typically angles were balanced prior to adjusting the traverse with these methods.

17. Crandall’s Rule
Crandall’ Rule is a “special case” least squares adjustment.
Crandall’s Rule assumes there is NO error in the angles – angles/directions are assigned an infinite weight.
Therefore, adjustments are made only to distances.
This adjustment was typically used to match bearings with previous surveys but under certain conditions can give unexpected results.
The assumption that angles contain no error is not a valid assumption.

18. Transit Rule
Transit Rule adjusts both angles and distances but makes the assumption that angles are measured with a higher precision than distances.
This assumption may have been valid in the past when using a 10” theodolite for angles while pulling a chain for distances, but it is not valid for today’s measuring equipment.

19. Compass Rule
The Compass Rule assumes both angles and distances are measured with equal precision.
Of the traditional adjustments, this assumption is most valid for today’s measuring equipment.
The Compass Rule remains a very popular form of adjustment by surveyors but has distinct disadvantages when compared to Least Squares Adjustments.

20. What is Least Squares?
Least squares is a statistical method used to compute a best-fit solution for a mathematical model when there are excess measurements of certain variables making up the mathematical model.
Least squares requires a mathematical model, a system of equations. It requires redundant measurements of one or more variables (the known variables). Lastly it requires variables that are unknown that are being solved for.
The least squares criteria is reached when the sum of the squares of the residuals have been minimized.
Least Squares Adjustment applies the least possible amount of correction when adjusting the measurements which arguably makes it the best adjustment.

21. Advantages of LSA
It allows the simultaneous adjustment of a network of traverses. Traditional adjustments can only adjust one traverse at a time.
It allows the combined adjustment of traverse data, GPS data and level data – 1D, 2D or 3D adjustments.
It allows complete control of the adjustment process. Measurements are weighted based on the equipment used and the number of measurements made.

22. Advantages of LSA
It allows processing data of different precision – weights can be applied to individual measurements or groups of measurements.
It allows flexible control. The control points can be anywhere in the traverse, you don’t need to start on a known point. Control points do not have to be contiguous and they can be side-shots.
It can handle resection data (measurements from an unknown point to known points), triangulation (angle-only measurements) and trilateration (distance-only measurements).

23. Advantages of LSA
Automatic reduction to State Plane Coordinate systems if desired.
Extensive data analysis provides more information for evaluation of traverse networks.
Enhanced blunder detection tools.
Allows flexible field procedures; the data does not have to be in any specific order.
Provides tools for ALTA or State survey certifications.

24. Common misconceptions of LSA
Least Squares Adjustments are just too complicated.
Reports are intuitive and easily understood
Flexibility of LSA makes processing of difficult datasets easy.
Least Squares Adjustments are only necessary for very precise surveys.
LSA can and should be used for any type survey.
LSA can be used for simple loop traverses as well as complex traverse networks.

25. Creating a Project in SurvNET
Choose a coordinate system
Select data files to be processed
Enter pre-processing settings
Enter preliminary standard errors
Select adjustment and output options
Define control points (known points)
Process data
Analyze report

27. Coordinate System Select the adjustment model
Choose the type coordinate system for your project
Local (assumed)
NAD83 State Plane
NAD27 State Plane
UTM
User defined

29. Input Data Files Select the data files to be processed
Multiple files of each type can be selected
Traverse data files
GPS Vector data files
Differential/Trig Level files

31. Preprocessing Select the preprocessing settings
Curvature and Refraction corrections
Point Substitution feature
Measurement tolerances
Closure computation

33. Standard Errors
Select preliminary standard errors for measurement weighting
Standard errors should reflect the realistic error expectations for the equipment used
Distance errors
Angle errors
Setup errors
Control errors
Leveling and GPS errors

35. Adjustment Settings
Number of iterations to perform prior to showing convergence error
Confidence Interval for statistical analysis
ALTA certifications
Select connections for relative error ellipse calculations

37. Output Options Output rounding options Units Output coordinate file
Optional coordinate file for ground coordinates

38. Sample Traverse Network
Sample Least Squares Adjustment project presented using Carlson SurvNET software
See project PARK.PRJ

40. Processing Unique Datasets
Least Squares adjustments can process data that the traditional methods of adjustment cannot
Traverses with non-contiguous control points
Networks of Traverses
Triangulation
Total Station Traverse with GPS vectors

41. Non-contiguous control points
See SEP1.PRJ and SEP2.PRJ

44. See N_DRUIDH.PRJ and DEANPROP.PRJ
Traverse Networks
See N_DRUIDH.PRJ and DEANPROP.PRJ

47. Traverse with Triangulation
See NORTHLN6.PRJ

50. GPS Vectors
See “CC&V Control.prj”

52. Traverse combined with GPS vectors
See M07052.PRJ

54. ALTA and Kentucky Survey Standards
ALTA Title Surveys require that the surveyor make the following Certification:
The relative positional precision between any two property corners shown on the survey shall be within 0.07 of a foot plus 50 parts per million at a 95% confidence level
This certification requires the statistical model produced by a least squares adjustment.

55. What is Relative Positional Precision?

56. 2011 ALTA Standards

57. Kentucky Survey Standards
The Kentucky Survey Standards as stated in KAR 18:150 are similar to the ALTA standards we just discussed.

58. KAR 18:150 Standards of Practice

59. KAR 18:150 Standards of Practice

60. KAR 18:150 Standards of Practice

61. Relative Error Ellipses
The determination of the Relative Positional Precision between any two points requires the calculation of the relative error ellipse between the points
Relative error ellipses are confidence regions established for the coordinate difference between any two points.
This is not the same thing as an error ellipse. Error ellipses are the measure of the positional error of a point. It is defined by an ellipse which is defined by a semi-major axis and semi-minor axis., and the direction of the maximum error.

62. Sample ALTA Survey Project
Sample ALTA Survey project presented using Carlson SurvNET
See project ALTA.PRJ

64. Thank you for your attention