1 / 63

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.

macaria
Télécharger la présentation

Tutorial: Error Theory and Adjustment of Networks

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related