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Network Least Squares Adjustments Demystified

By Bruce Carlson and Dean Goodman

Carlson Software

Course Outline Measurement Errors

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

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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 +- 0.01’ and

3PPM.

Types of Errors

There are three classifications of errors:Blunders

Systematic Errors

Random Errors

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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.

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

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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

+- 0.01 feet accuracy. You can only measure from the point to the height

of instrument to a +- 0.01’ accuracy.

Random errors can be adjusted out of the traverse.

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.

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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.

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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,

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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.

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.

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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: For additional precision To help remove instrument errors To mitigate curvature and refraction errors To improve vertical closures.

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.

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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.

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.

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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.

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.

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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.

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).

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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.

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.

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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

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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

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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

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Preprocessing

Select the preprocessing settings Curvature and Refraction corrections

Point Substitution feature

Measurement tolerances

Closure computation

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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

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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

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Output Options

Output rounding options

Units

Output coordinate file

Optional coordinate file for ground coordinates

Sample Traverse Network

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

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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

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Non-contiguous control pointsSee SEP1.PRJ and SEP2.PRJ

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Traverse Networks

See N_DRUIDH.PRJ and DEANPROP.PRJ

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Traverse with Triangulation

See NORTHLN6.PRJ

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GPS Vectors

See “CC&V Control.prj”

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Traverse combined with GPS vectors

See M07052.PRJ

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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.

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What is Relative Positional Precision?

2011 ALTA Standards

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Kentucky Survey Standards

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

KAR 18:150 Standards of Practice

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KAR 18:150 Standards of Practice

KAR 18:150 Standards of Practice

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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.

Sample ALTA Survey Project

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

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Thank you for your attention