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

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, Lecture I : Error Theory and Basics of Adjustment 1.Introduction 2.Quality Estimates for Observations 3.GUM 4.Variance and Covariance Propagation 5.Concept of Parametric Adjustment 6.Further Adjustment Concepts

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

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, D Cartesian Coordinate System Principle: 3D-Cartesian Space Often separated (Due to orientation to normal gravity vector) - Horizontal xy-plane - Vertical direction: z-component Measurements: - Horizontal directions / bearings r 1’ r 2’ … Angles are differences between directions:  = r 2 – r 1 - Slope distances d i - Zenithal angles z i Elevation angles 100gon - z i

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 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: Distance [ m ] Deviation in Height [mm] Deviation for Distance in horizontal plane [mm]

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

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

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

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

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

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, Quality Estimates for Observations Complete Covariance Matrix Meaning: Variance of variable X i Correlation coefficient: n Variables

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, GUM

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, GUM 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, etc Why ?  „Push the button“ does not allow to give realistic estimate!  Repeated observations are not meaningful  Has to include further informations

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, Uncertainty according to GUM Estimates of Type A and B are combined according to the rules of variance propagation: Resulting Uncertainty is u c (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, Variance and Covariance Propagation

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

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

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

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 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, Treatment of redundant observations Coordinate determination of a new point in horizontal plane:  Observations: - 2 distances D1 and D2 - 2 angles a 2 and a 2  Parameter: - Coordinate X N - Coordinate Y N  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, Concept of Adjustment of Observations Functional model: The observations are a function of the coordinates: Stochastic model: variance factor, should be equal to 1 a priori knowledge of the cofactor matrix of the observations Taylor series expansion with approximate coodinates X 0 : Introducing a residual v for the observations:

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 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, 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, Model Quality ( Global Test for Adjustment Model ) 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 Relations between L and X are modeled in proper way, if after the adjustment coincides with the a priori value

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 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 Robust statistics

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, Different Concepts for Adjustment 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.

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, Lecture II : Adjustment of Networks 1.Typical observations and their pre-processing 2.Adjustment Process 3.Quality Measures for Adjustment Results 4.Datum Problem 5.Simulation + Optimization 6.Use of Lasertrackers in Adjustment Process 7.Software Package PANDA

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

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

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 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, Functional relation between observations and coordinates Observations between point P i and P j. X, Y, Z are point coordinates. Parameter o: orientation unknown

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

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, Adjustment Process

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

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

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

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

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

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, Reliability measures 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 Q vv is dominant and determines the fraction of the observation error reflected in the residual. This fraction is called the redundancy number r i : 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 Derived out of Q vv

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 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

Datum Problem Observations Datum definition

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, D-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, Datum Parameters in Geodetic Networks Network Dimension Maximum Datum- defect Datum Parameter Observations with information on datum parameters 1 vertical network 1Translation in zHeight differences: No Absolute heights: Yes 2 horizontal network 4Translation: x, y Orientation: z Scale Distances => scale Azimuth => orientation around z 3 3D-spatial network 7Translation: x, y, z Orientation: x, y, z Scale Distances => scale Azimuth => orientation around z Zenith angles => orientation around x and y

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, 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, 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, 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, Simulated Adjustment and Optimization 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 => Computation of precision and reliability measures is possible, before observations are made !!

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, Simulated network, Example from M. Schlösser Minimum constrain solution, based on 8 outer points N1 to N8 2D-network, 3 points in the middle are observation stands Used standard deviations: directions: 0.3 mgon, distances: 0.2 mm Minimum constraint solution: Datum is based on points N1 to N8

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, Simulated network, Example from M. Schlösser Partial Minimum constraint solution: Datum is based on points N1 to N4 2D-network, 3 points in the middle are observation stands Used standard deviations: directions: 0.3 mgon, distances: 0.2 mm

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, Simulated Example: Check of the Redundancy (Reliability) Colour for each observation indicates the redundancy number r i : r i 0.2 green Remark: Redundancy is depending on datum fixation Here: Minimum constraint solution

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, Combination of Lasertracker with other instruments

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, Characteristics of a Lasertracker  Absolute and relative distances, horizontal and vertical angles/directions  Not automatically oriented to earth gravity field (only by external levels)  Calibration is given by manufacturer, can be controlled by user  Precision of Distance Measurements - Interferometric: Continuous with nm precision : Absolute: mm ppm  Precision of Angles Observations (automatic readings) - Horizontal : 0,6 mgon - Vertical : 0,6 mgon i.e. less precise than total stations with e.g. σ H = σ V = 0.15 mgon

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, Use of Lasertracker with arbitrary orientation : Rotation up to 180°

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, Include for each set-up (position) of lasertracker data a complete set of three orientation angles φ ω κ i.e. use full 3D-rotation matrix for observations To be introduced for each instrumental set up Requires : Approximate values for these rotation angles. (Their determination is without the scope of this tutorial) Treatment of extremly rotated Lasertracker :

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, Rigorous hybrid adjustment is possible: Lasertracker, Total Station and levelling Use lasertracker measurements, i.e. horizontal directions, vertical angles and distances with their adequate covariance-matrix as additional group of observations in adjustment model. L hz v h L vz v z L dist v dist x x L lev v lev x y L az + v az = A * x z L hz, LT v hz, LT x add L v, LT v v, LT x trans L dist, LT v dist, LT..

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, Software Package PANDA

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, All computations are performed with software package PANDA Package for Adjustment of Networks and Deformation Analysis

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, Software Package PANDA Package for Adjustment of Networks and Deformation- Analysis Interactive processing of 1D-, 2D- and 3D-geodetic networks out of all areas of surveying and geodesy Preprocessing of raw total station and leveling data, including plausibility control and computation of approximate coordinates Combined rigorous adjustment of lasertracker. total station, leveling and azimuth observations together with GPS-coordinate sets and existing/given coordinates Deformation analysis for 1D-, 2D- and 3D-networks (e.g. rigorous congruency tests) Interactive simulation and network optimization

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, Independant development for PC under WINDOWS Local cartesian, local spherical, ellipsoidal and global cartesian coordinate systems; different mapping systems Different transformation concepts are included All common datum definitions: Fixed datum, minimum constraint and weak datum; hierarchical concept All common quality measures for precision and reliability Can process large networks (e.g stations) in acceptable time Software Package PANDA Package for Adjustment of Networks and Deformation- Analysis

Niemeier, Wolfgang | Tutorial on Adjustment | IWAA | September 15, Thank you for your attention!