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Tutorial: Error Theory and Adjustment of Networks

Wolfgang Niemeier Hamburg, Sept. 15, 2010. Institut für Geodäsie und Photogrammetrie. Tutorial: Error Theory and Adjustment of Networks. Introduction Quality Estimates for Observations GUM Variance and Covariance Propagation Concept of Parametric Adjustment Further Adjustment Concepts.

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Tutorial: Error Theory and Adjustment of Networks

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  1. Wolfgang NiemeierHamburg, Sept. 15, 2010 Institut für Geodäsie und Photogrammetrie Tutorial: Error Theory and Adjustment of Networks

  2. Introduction Quality Estimates for Observations GUM Variance and Covariance Propagation Concept of Parametric Adjustment Further Adjustment Concepts Lecture I : Error Theory and Basics of Adjustment Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 2

  3. Typical Measuring Instruments Levelling: Lasertracker Total Station Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 3

  4. Principle: 3D-Cartesian Space Often separated (Due to orientation to normal gravity vector) - Horizontal xy-plane - Vertical direction: z-component Measurements: - Horizontal directions / bearings r1’ r2’ … Angles are differences between directions: = r2 – r1 - Slope distances di - Zenithal angles ziElevation angles 100gon - zi 3D Cartesian Coordinate System Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 4

  5. Validity of 3D-Cartesian Coordinate System • A local cartesian coordinate system neglects the curvature of the earth.This effect can be computed for deviation in heights and for distances: Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 5

  6. Quality Estimates for Observations Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 6

  7. Example: Distance measurements • Basic statistics: • Random variable • Measurements • Processing of repeated observations: • Arithmetic mean • Empirical residuals: • Empirical variance (standard deviation *2 ) Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 7

  8. Classical Concept: Normal Distribution Reasoning Gauß: „Guess“ (1801) Cramer: Central limit theorem (1947) Hagen: Sum of elementary errors (1837) Normal distribution 2 parameters: μ meanσ2 variance Density function: Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 8

  9. Normal distribution is still valid !?! But other concepts exist: • modulated normal distribution mixture of 2 or more normal distributions • random noise, white, colored spectral decomposition of error budget • robust estimates estimates, if several gross (systematic) errors are in the data • extended uncertainty estimates GUM Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 9

  10. Classical statistical treatment Property of observation: Expection of mean: Sum of residuals Confidence regionfor μ mostly at 95% confidence level Quantile of normal distribution Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 10

  11. Correlation/Covariance between observations Two variables with series of n observations: Mean valueand (true) errors Covariance matrix: Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 11

  12. Quality Estimates for Observations n Variables • Complete Covariance Matrix Meaning: Variance of variable Xi Correlation coefficient: Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 12

  13. GUM Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 13

  14. Guide to the Expression of Uncertainty in Measurement (GUM) ISO/BIPM Publication,1995 Idea: „New concept, to attach realistic estimate for variability (variance) to a measuring quantity. Two Catagories: Type A : Classical statistical components, e.g. variance estimates Type B : Further influence factors, to account for centering errors, systematic effects, atmospheric effects, etcWhy ? „Push the button“ does not allow to give realistic estimate! Repeated observations are not meaningful Has to include further informations GUM Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 14

  15. Uncertainty according to GUM Estimates of Type A and B are combined according to the rules of variance propagation: Resulting Uncertainty is uc (similar to standard deviation ?) Extended uncertainty Problem (and main criticism): - Need estimates for each influence factor of Type B- Type B effects have to be modelled as stochastic variables Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 15

  16. Variance and Covariance Propagation Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 16

  17. Variance and Covariance Propagation • For derived variables: General function: For uncorrelated observations: Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 17

  18. (n,1) random vector X with covariance matrix Task: Determine covariance matrix for (m,1) vector YA and b are non-stochastic quantities, i.e. constants.Introducing the given functionFinal formula: Variance-Covariance Progagation (Linear Function) SYY Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 18

  19. Concept of Parametric Adjustment or: Adjustment of Observations Gauss - Markov - Model Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 19

  20. Treatment of redundant observations • Linear Regression: Processing strategies: • Use line between each 2 points • Use all points to determine the parameter of the line • Transformation: • Using just 2 identical points • Using all available identical points: Overdetermined transformation (adjustment approach) Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 20

  21. Treatment of redundant observations • Coordinate determination of a new point in horizontal plane: • Observations: • 2 distances D1 and D2 • 2 angles a2 and a2 • Parameter: • Coordinate XN • Coordinate YN • Type of processing: • Use two angles • Use two distances • Use one distance and one angle • Use all information : Adjustment Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 21

  22. Concept of Adjustment of Observations Stochastic model: • Functional model:The observations are a function of the coordinates: Taylor series expansion with approximate coodinates X0: variance factor, should be equal to 1 a priori knowledge of thecofactor matrix of theobservations Introducing a residual v for the observations: Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 22

  23. Solution Algorithm for Gauss-Markov Model Least-squares-method optimizes the weighted sum of quadratic residuals: With the functional model The quadratic sum of residuals is a function of the parameters x To find the minimum of the sum we set the gradient to zero: Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 23

  24. Solution Algorithm for Gauss-Markov Model • Estimated parameters: • Cofactor matrix for parameters: • Estimation for variance factor: „a posteriori“ Final results, to be used in practise: • Final coordinates: • Final covariance matrix: Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 24

  25. Model Quality ( Global Test for Adjustment Model ) Relations between L and X are modeled in proper way, if after the adjustment coincides with the a priori value • Statistical Test: • Reasons for test failure • Gross and systematic errors in observations • Functional relationship are not correct • Calibration parameters not considered • Weighting between groups of observation not correct: need: variance-component estimation Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 25

  26. Robust statistics • If is not valid, especially if several gross errors are expected in data, e.g. 2 – 10 % (break even point ?) • Robust statistics(Hampel and Huber (since 1965), Rouseeuw (1987): Basic Idea: Restrict influence of outlying observations on results : • Various theoretical concepts, algorithms,... Easy applicable in practise: • Reweighting:Observations with large normalised residual: Increase variance, i.e. reduce weight of observation in adjustment model • Common strategy: - Best result is least-squares-solution - If errors are indicated, use reweighting Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 26

  27. Conditional Adjustment Only observations are adjusted, example: levelling loops Coordinates are computed with adjusted quantities in separate step Parametric Adjustment (Gauss-Markov-Model) The observations are considered as function of the parameters L=f(X) For each observation its relation to coordinates has to be expressed. Examples: Normal network adjustments Combined Adjustment of Observations and Parameters Only implict formulation is possible: f(L,X)=0 Basis are functional relations between various observations and various parameters in each equation. Different Concepts for Adjustment Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 27

  28. Typical observations and their pre-processing Adjustment Process Quality Measures for Adjustment Results Datum Problem Simulation + Optimization Use of Lasertrackers in Adjustment Process Software Package PANDA Lecture II : Adjustment of Networks Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 28

  29. Typical observations and their pre-processing Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 29

  30. Typical Measuring Instruments Levelling: Lasertracker Total Station Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 30

  31. Typical observations and their pre-processing • Typical observations in local 3D networks • Leveling • Height differences • Total Station • Distances • Directions/Bearings • Zenith angels • Laser tracker (horizontation ?, see last chapter) • Distances • Directions/Bearings • Zenith angels Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 31

  32. Functional relation between observations and coordinates • Observations between point Pi and Pj. X, Y, Z are point coordinates. Parameter o: orientation unknown Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 32

  33. Additional Parameters: • Calibration is not perfect, add: • Additional constant for distances • Scale factor for distances • Refraction unknown for zenith anglesto account for atmospheric effects, e.g. in tunnels • Deviations of the verticalaffects directions and zenith angles;important to account for gravity field variations Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 33

  34. Adjustment Process Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 34

  35. Functional Model : Relations between Observations and Coordinates with: • Linearise functional relations:=> Observation Equations Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 35

  36. Stochastic model: Summarize knowledge on variability of observations • Normal assumption:For each type of observation just one variance estimate exists: => Derived a priori covariance function (often as diagonal matrix) Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 36

  37. Adjustment computations Solution, according to derivation in Lecture I: Estimated parameters: Covariance matrix of parameters: Variance after the adjustment: But: Is this result final or are there outliers, systematic effects, etc ? Is the achieved quality sufficient ? Is the datum of the computation chosen properly ? Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 37

  38. Quality Measures Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 38

  39. Precision of coordinates: Derived out of Relative measures: • for distances (derived quantity) • relative confidence ellipses between points • Absolute measures: • variances for coordinates - confidence ellipses for points General probability relation for confidence ellipses: For a single point Pj : As relative measures between point Pi and Pj: Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 39

  40. Precision of coordinates: Derived out of= Qxx Cofactor matrix Qxx as basic information for precision values: Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 40

  41. Reliability Concept Example: Determination of point N, Points P1 and P2 are known. Only distances measured: no control 2 distances and 1 angle measured Controlability ! 2 distances and 2 angle measured: Separability !(Errors are detectable) Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 41

  42. Reliability measures Derived out of Qvv • The residuals are important to assess the results, because the true errors are unknown. The residuals are a linear function of the observations: For uncorrelated observations the diagonal of Qvv is dominant and determines the fraction of the observation error reflected in the residual. This fraction is called the redundancy number ri: A redundancy number of zero indicate an observation without control!r < 0.2 bad control; 0.2 < r < 0.8 ok; 0.8 < r well controlled Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 42

  43. Check of gross errors • The residual v follows the normal distribution: The standardized residual w is defined as: with Residuals with w > y are detected as gross errors! Quantile of normal distribution Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 43

  44. Datum Problem Observations Datum definition

  45. 2D-network with angle-observations, only • - Can be positioned and oriented arbitrarily in coordinate frame- Scale is not defined, i.e. absolute size is not fixed Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 45

  46. Datum Parameters in Geodetic Networks Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 46

  47. Different models for datum fixation • Zero Variance computational base. Select number of coordinates according to free datum parameters and set them fix (no variance) • Minimum constraint solution Position network geometry on set of all approximate parameters Gives best (smallest) variances for points • Weak datum Introduce „given coordinates“, i.e. use coordinate values for selected points as observations and attach variance information to them the network. Network geometry will be effected. • Hierachical adjustment Fix coordinates of several (higher order ?) points. If number of free datum parameters is exceeded, the network geometry will be effected / detroyed. Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 47

  48. Zero Variance Computational Base Plane network, datum parameters 4: Approach: Select given coordinates of two points to define the datum Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 48

  49. Minimum Constraint Solution Plane network, datum parameters 4: Approach: Position (inner) network geometry on all approximate coordinate (Similar to a Helmert-transformation on all given points) Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 49

  50. Objective: Evaluate and optimize network design. Quality check, before observations are taken:Is expected precision and reliability sufficient ? Type of instrument and number of observations:Is the selected instrumentation adequate ? Configuration of network points:Is the strenght of the geometry of the network sufficient ? Precision estimates: Out of and Reliability estimates: Out of Simulated Adjustment and Optimization =>Computation of precision and reliability measures is possible, before observations are made !! Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 2010 50

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