451-200 Survey Networks Theory, Design and Testing

1. 451-200 Survey Networks Theory, Design and Testing Allison Kealy akealy@unimelb.edu.au Department of Geomatics The University of Melbourne Victoria 3010

2. Introduction • Survey network adjustment is also known as • Variation of coordinates • Least squares adjustment • Least squares estimation • Survey adjustment • Use routinely for survey computations. Survey Networks: Theory, Design and Testing

3. Advantages Networks adjustment is widely adopted due to • Consistent treatment of redundant measurements • Rigorous processing of measurement variability • Ability to statistically test and analyse the results Survey Networks: Theory, Design and Testing

4. Implementations • Many commercial and proprietary network adjustment packages are available • SkiPro • CompNET • Star*Net • TDVC, DNA • Wide variation in ease of use, sophistication and available features Survey Networks: Theory, Design and Testing

5. Non-Network Adjustment • Coordinate geometry computations • Also known as “COGO” packages • Simple 2D or 3D geometry computations for radiations, intersections etc • Traverse adjustment • Known as Bowditch or traverse rules • Valid method of distributing errors • Not statistically rigorous Survey Networks: Theory, Design and Testing

6. Input Data • Survey measurments • Horizontal angles • Vertical angles • Distances (slope and horizontal) • Level differences • GPS positions and baselines • Azimuths/bearings • Measurement precisions Survey Networks: Theory, Design and Testing

7. Input Data (continued) • Fixed and adjustable coordinate indicators • Known coordinates of unknown stations • Approximate coordinates of unknown stations • Auxiliary data such as • Coordinate system and datum • Atmospheric refraction • Default values for precisions etc Survey Networks: Theory, Design and Testing

8. Algorithm – Functional Model • Describe the geometric relationship between measurements and stations • Very well understood for conventional measurements • GPS knowledge well established • Sets the response of station positions to different measurement types Survey Networks: Theory, Design and Testing

9. Algorithm – Stochastic Model • Models the statistical properties of the measurements • Assumes a Gaussian or normal distribution function of random error • Effectively a “weighting” of the “importance” of different measurements based on precision data • Precision levels are often not well estimated Survey Networks: Theory, Design and Testing

10. Results Output • Adjusted coordinates for all stations • Precision of all coordinates • Error ellipses for all stations • Adjusted measurements • Measurement residuals • Differences between the measured and adjusted values for any measurment Survey Networks: Theory, Design and Testing

11. Statistical Testing Information • Unit weight precision • Also known as sigma zero (s0) • Squared quantity known as estimate of the variance factor or unit weight variance • Indicates overall or global quality of the solution • t statistics for each measurement • Indicates local quality of individual measurements Survey Networks: Theory, Design and Testing

12. Reliability Indicators • Reliability is a measure of the susceptibility to error • Global and local values can be computed • Indicated by either • Redundancy numbers • Reliability factors • Generally only useful for internal comparisons of measurements Survey Networks: Theory, Design and Testing

13. Network Analysis • Analysis of the results of survey networks is essential • Assessment of station coordinate precisions against specifications is often first priority • Networks may also be tested for accuracy if suitable independent checks are available • Testing of networks for gross errors and other factors is mandatory Survey Networks: Theory, Design and Testing

14. Network Testing • The estimate of the variance factor is used as a global test of the entire survey network • Individual measurements are locally tested against the student t distribution • Both test distributions are independent of the number of redundancies in the network • The confidence of the testing improves with higher redundancy numbers Survey Networks: Theory, Design and Testing

15. Network Testing (continued) • Global and local test values are influenced by • Blunders or gross errors e.g. reading or transcription errors • Systematic errors, e.g. calibration errors or anomalous refraction • Precision errors, e.g. under or over estimation of the repeatability of an instrument or the influence of environmental factors Survey Networks: Theory, Design and Testing

16. Network Testing (continued) • An initial global test is required to determine the likelihood of errors in individual measurements • Local errors are tested, de-activating the measurements with the worst t statistic and re-processing the adjustment • Measurements are deactivated until all local tests are acceptable or the point of “diminishing returns” is reached • If the global test still fails then systematic or precision errors are investigated Survey Networks: Theory, Design and Testing

17. Network Design • Networks must be designed to suit • The survey problem • Specifications for precision and accuracy • Expectations for reliability • Limitations on physical access • Restrictions placed o time and/or cost • Availability of equipments • Availability of staff Survey Networks: Theory, Design and Testing

18. Network Design (continued) • Network design is part experience and part science • Experience comes from practiced knowledge of network types, error propagation and geometry • Scientific analysis comes from the interpretation of error ellipses and other indicators of network quality Survey Networks: Theory, Design and Testing

19. Network Design (continued) • Basic network types comprise • Level networks • Resection • Intersection • Control traverse • Control networks • The choice of type is primarily based on the survey problem, specifications for precision/accuracy and available equipments Survey Networks: Theory, Design and Testing

20. Level Network • Measurement data is level differences only • All horizontal angles must be fixed • At least one station height must be fixed to set the vertical datum • Level differences are typically set s proportional to the square root of the run length Survey Networks: Theory, Design and Testing

21. Resection • Measurement data is horizontal angles only • All coordinates of the resection targets must be held fixed • The height of the instrument station must be held fixed • Horizontal angle precisions are set from the standard deviations of the means of the multiple rounds of observations Survey Networks: Theory, Design and Testing

22. Control Traverse • Measurement data is horizontal and vertical angles, distances and perhaps level differences • At least one known control station and one reference object are needed • Precision data may be estimated from experience or adopted from instrument specifications Survey Networks: Theory, Design and Testing

23. Control Networks • All measurement data types • At least one control station and one reference object needed • Precision data may be estimated from experience, adopted from the instrument specifications or computed • High numerical and geometric redundancies leading to very high reliabilities Survey Networks: Theory, Design and Testing

24. Steps in Survey Design • Using available information lay out possible positions of stations • Check line of sights • Do field recce and adjust positions of stations • Determine approximate coordinates • Compute values of observations from coordinates • Compute standard deviation of measurements Survey Networks: Theory, Design and Testing

25. Steps in Survey Design • Perform least square adjustment, to compute observational redundancy numbers, standard deviations of coordinates and error ellipses • Inspect the solution for weak areas based on redundancy numbers and ellipse shapes • Evaluate cost of survey • Write specification Survey Networks: Theory, Design and Testing

26. Conclusions • Any survey work involves a component of network design and almost invariably requires testing • Efficient and appropriate network design is a learned skill, supplemented by experience • Network testing is essential to determine the quality of the survey • http://www.geom.unimelb.edu.au/kealyal/200/Teaching/net_design_test.html Survey Networks: Theory, Design and Testing

27. Survey Network Configurations • Station coordinates can be fixed, constrained or free • Good approximations for the free stations are necessary for convergence • There must be sufficient measurements to geometrically define all the free coordinates Survey Networks: Theory, Design and Testing

28. Survey Network Configurations • Assuming we have sufficient station coordinates and measurements to define the datum, orientation and scale, station coordinates are defined by the measurements as follows: Survey Networks: Theory, Design and Testing

29. Survey Network Configurations • Strength or weakness of the determination depends on the geometry of the relationship between the stations and the measurements • Every station can be tested for the minimum numerical requirement to define all the coordinates of the station Survey Networks: Theory, Design and Testing

30. Externally Constrained Networks • Assume survey networks are externally constrained • Externally constrained networks contain sufficient fixed or constrained station coordinates to define the datum, orientation and scale of the networks • Datum • Locates network relative of coordinate system origin • three coordinates fixed, one in each dimension • Orientation • Fix the orientation of the network relative to the coordinate system • Use bearings or planimetric coordinate of another stations Survey Networks: Theory, Design and Testing

31. Externally Constrained Networks • Scale • Use distances to fix the scale of the network relative to the coordinate system • Fix planimetric coordinates of another station • Minimal Constraints Survey Networks: Theory, Design and Testing

32. Free Networks • Free or internally constrained • All stations open to adjustment • Based on initial coordinates of stations • Datum, scale and orientation arbitrary Survey Networks: Theory, Design and Testing

33. Testing of Adjustments • Factors affecting adjustments • Mathematical model • Stochastic model • Gross errors • Confidence intervals • Redundant Measurements Survey Networks: Theory, Design and Testing