Prepared by Indrajith D. Wijayratne Associate Professor of surveying Michigan Technological University Houghton, MI 49931

Copyright © 2002 by Indrajith Wijayratne

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SURVEY ADJUSTMENTS Two basic methods • Direct adjustments - Adjustment of measurements • Indirect adjustments - Computation of parameters directly by an adjustment procedure

An adjustment is performed when • There are redundant measurements • Only random errors are present in the measurements

Adjustments can be • Simple, e.g. compass rule for traverses • Rigorous, e.g. Least Squares A rigorous adjustment can • Accommodate different precision of measurements • Yield most probable values for parameters

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CONCEPT OF WEIGHTS IN MEASUREMENTS Weight is the worth or reliability of one measurement relative to a standard or another measurement Measurements with higher precision (smaller std. deviation) should be assigned higher weights Measurements with higher weights should receive smaller corrections after an adjustment Weights are relative, and therefore, determined by comparing with another measurement Weights are inversely proportional to some precision index of measurements Weight of an uncorrelated measurement is inversely proportional to square of std. deviation (variance) Since weights are relative, any error estimate, similar to std. deviation, could be used, e.g. E95 Weight of a mean value computed from repeated measurements is proportional to the number of repetitions Weight assigned to an elevation difference, determined by differential leveling, is inversely proportional to the number of setups or the length of the level line

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Simple Adjustments Using Weights • weighted mean • adjustment of angles in a triangle or a traverse • adjustment of level lines Examples:

Self-study Problems 1. Problem 9.3 on page 167 of text. 2. The three interior angles of a triangle ABC have been measured as follows; A = 40° 12' 13" B= 52° 46' 02" (a) (b)

C = 87° 01' 41"

Adjust the angles assuming that they were measured with equal precision Adjust the angles assuming the following estimated standard deviations for each of the angles A = 0.8"

B = 0.5"

C = 0.3"

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ADJUSTMENT BY LEAST SQUARES (Chapter 10) Random errors behave as normally distributed variables Smaller errors have higher probabilities Most (statistically) probable value of a quantity is the one, which makes residuals small Can be achieved by making the sum of squares of residuals the minimum If the measurements were not made with the same precision then the sum of squares of weighted residuals should be minimum This results in an (statistically) equitable distribution of errors among measurements Residuals (errors) can only be determined if there are redundant measurements Simple mean is the least squares estimate of the measured quantity, if the measurements were repeated with the same precision If the measurements with different precision are used, then weighted mean is the least squares estimate

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Benefits of Least Squares Adjustments • Can be applied to any measurement problem in science or engineering • Offers unique values for parameters i.e. does not depend on the P.O.B. • Offers (statistically) best estimates for parameters based on available measurements • Any number/type of measurements, any network configuration can be adjusted simultaneously • A rigorous error analysis is possible • Offers estimates of uncertainties of adjusted values

Some disadvantages • Precision estimates of measurements are needed • The method is computation-intensive, and therefore, a high speed computer is needed • A large redundancy (large degree of freedom) is necessary to get a meaningful adjustment • A knowledge of statistics is necessary to do a good analysis of results

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Two basic methods • Observation equations (parametric) method • Condition equations method

Observation Equation Method There exists a functional relationship between every measurement and certain parameters Every measurement (observation) can be related to certain number of parameters by an equation The number of equations that can be written, therefore, is exactly equal to the number of measurements used in the adjustment A solution to these equations is obtained by minimizing the sum of the (weighted) squares of residuals This ‘least squares’ solution results in direct determination of parameters Condition Equation Method Condition equation is a functional relationship among measurements Certain geometric conditions or constraints exist among measurements, e.g. sum of 3 angles of a plane triangle is 180° Again, a solution is obtained by minimizing weighted squares of residuals

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The solution only offers the corrections (residuals) to measurements Parameters have to be computed using corrected measurements

The ‘observation equation’ method is preferred due to • Easy adaptation for a computer solution • Availability of error estimates of parameters directly as a by-product

In both methods, equations can be either linear or non-linear Non- linear equations have to be linearized before least squares principle can be applied Approximate values for parameters are needed to obtain a solution for non-linear problems Examples of level net adjustments will follow

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Steps needed in a least squares adjustment • Identify points (coordinates) or benchmarks (elevations) that need to be adjusted • Identify coordinates, elevations, azimuths or any other quantities that will be held fixed • Ensure that there are redundant measurements • Organize measurements in some order • Assign weights to different observations based on some precision estimate • Form observation equations • Form normal equations by minimizing sum of weighted squares of residuals • Solve normal equations • Analyze results

Even though the formation of observation equations and normal equations and the solution of normal equations can be done using software, the surveyor should have a thorough understanding of the process in order to do a thorough post adjustment analysis

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Least Squares Adjustment of Leveling Networks (Chapter 11)

Leveling network is formed when several individual level lines are connected together using common benchmarks

This type of leveling networks are used when high order benchmarks are to be established by conventional leveling

Network usually has several known benchmarks that should be preferably of higher order of accuracy than that needed for new benchmarks

For the purpose of least squares adjustment, the elevation difference between any two benchmarks, existing or new, is considered a single measurement

Least squares adjustment is performed by holding the elevations of all known benchmarks fixed

If a weighted least squares solution is sought, weights can be applied to known benchmark elevations as well based on their order of accuracy

EDM Calibration (sec 10.12 p.202) Calibration baselines are established for the benefit of surveyors to calibrate their EDM instruments

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Usually established by NGS on the request of surveyors Typical baseline consists of three segments whose lengths have been measured and adjusted to a very high degree of accuracy When the calibration of an EDM is needed, different combination of measurements are made on the calibration baseline Three distance measurements are possible from each of the four monuments to make a total of twelve measurements In order to get reliable results, these measurements are adjusted by a Least Squares method and a statistical analysis is performed Following are possible errors in an EDM measurement • Tribrach errors (optical plummet, level bubble both in EDM and prism) • Temperature, pressure, humidity • Frequency drift • Change of system prism constant. i.e. a combined EDM and prism constant • Random measurement errors Least Squares adjustment of measured distances of the baseline is made to determine any scaling factor needed for distances measured with this instrument and the correct system constant All tribrachs used should be well adjusted and the measured distances should be corrected for change in temperature, pressure, humidity, i.e. meteorology correction

Example:

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Least Squares Adjustment of Horizontal Control Networks • • • •

Traverse Networks Trilateration Networks Triangulation Networks GPS Networks

Type of Horizontal Control Networks • Two dimensional e.g. traverse •

Three dimensional e.g. GPS Baselines

Typical measurements • • • •

Distances Horizontal angles Azimuths or Directions Zenith or Vertical angles

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Parameters in 2-D surveys can be • Latitudes and Longitudes • Plane coordinates ( State Plane or Project)

Observation equations for distances, azimuths and angles are nonlinear Least squares method needs linear observation equations Non-linear observation equations are linearized using Taylor series expansion The series is truncated by dropping all the terms after the first order terms All remaining terms are linear and this results in a linear functional relationship between observations and parameters Note here that the variables (parameters) in the new equations are the differential changes to the original variables that is the result of Taylor series expansion The effect of the truncated terms can be made negligible by making these differential changes approach zero This is achieved by successive iterations of solutions To evaluate the coefficients of individual terms of the series, a set of values for variables (coordinates) is needed

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These are computed using measurements The least squares solution provides a set of corrections to be added to the initial values of coordinates Initial coordinates are updated by applying these corrections, and the coefficients of Taylor series are re-computed by using the updated coordinates for the next iteration Note that the updated coordinates are closer to the expected values, and therefore, the effect of terms dropped becomes still smaller This procedure is repeated (iterations) until corrections fall below a pre-set limit In software solutions, it is important to terminate iterations if the solution does not converge to a pre-set limit, and therefore, a maximum number of iterations may be specified Probable causes for the solution to diverge are errors in measurements or input, improper mathematical and stochastic model, etc. that result in ill-conditioned matrices

What does Least Squares process do? • Determines smallest possible corrections to measurements based on their estimated precision • Provides best estimates of parameters along with their uncertainties

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Requirements: • Redundant measurements • Some controls (constraints) • Precision estimates for measurements

Guidelines for Projects that Require Least Squares Adjustments Field: • Choose equipment/methods of measurements based on on accuracy requirements of the project •

Use well adjusted instruments and procedures to eliminate systematic errors in measurements

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Have internal checks to detect blunders in measurements

• Make many additional measurements to have a large degree of freedom •

Apply necessary corrections for remaining systematic errors

• Assess the precision of measurements accurately • Assess the quality of control which will be used • Avoid weak geometry in nets

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Office: • Correct measurements for all remaining systematic effects • Make a list of all field measurements and organize them in a manner suitable for input into the software • Make a visual check of measurements to ensure that there are no obvious blunders • Assign ID labels to all points • Check the accuracy of control point coordinates including field checks and recent recovery notes in the case of national control points • Make a pre-adjustment analysis of measurements if software provides it • Adjust and analyze results

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