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Leica GPS System 500

Leica GPS System 500. SKI-Pro Adjustment Result Analysis. SKI-Pro Adjustment Result Analysis. Content: SKI-Pro Statistical Tests Adjustment Logfile Examination Exercise: Adjustment Troubleshooting . Identify, remove and correct Bad Vector(s). SKI-Pro Statistical Hypothesis Testing.

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Leica GPS System 500

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  1. Leica GPS System 500 SKI-Pro Adjustment Result Analysis

  2. SKI-Pro Adjustment Result Analysis • Content: • SKI-Pro Statistical Tests • Adjustment Logfile Examination • Exercise: • Adjustment Troubleshooting . • Identify, remove and correct Bad Vector(s).

  3. SKI-Pro Statistical Hypothesis Testing • Statistical testing involves accepting or rejecting a statement using a Prime and Alternate Hypotheses: • The rejection of the Prime hypothesis while in fact the statement is true leads to a Type I decision error having a probability given by . • The acceptance of the Prime hypothesis while in fact the statement is false leads to a Type II decision error having a probability given by  (where (1-) is the power of the test). Statement Decision: accept Prime Decision: reject Prime (reject Alternate) (accept Alternate) True correct decision: Type I error: probability = (1 - ) probability =  False Type II error: correct decision: probability =  probability = (1 - )

  4. SKI-Pro Statistical Test Error Probability Levels Error Probability Levels (Alpha)

  5. SKI-Pro Statistical Test Power Levels Power Levels (1-Beta)

  6. SKI-Pro Variance Ratio F - Test Prime Hypothesis Alternate Hypothesis F - Distribution Power of test F - Test Confidence Level (%) F - Test Error Probability Level (%) (1- ) (1-)   Acceptance Region Rejection Region F - Critical Value

  7. SKI-Pro Individual Blunder Detection W - Test Prime Hypothesis Alternate Hypothesis Tau - Distribution Power of test W -Test Confidence Level (%) W - Test Error Probability Level (%) (1- ) (1-)  /2 /2 Rejection Region Acceptance Region Rejection Region - (W - Critical Value) W - Critical Value

  8. SKI-Pro Group Blunder Detection T - Test Prime Hypothesis Alternate Hypothesis Tau - Distribution Power of test T -Test Confidence Level (%) T - Test Error Probability Level (%) (1- ) (1-)   Acceptance Region Rejection Region T - Critical Value

  9. SKI-Pro Adjustment Logfile Header Information 3D minimally constrained network on WGS'84 ellipsoid STATIONS Number of (partly) known stations 1 Number of unknown stations 3 Total 4 OBSERVATIONS GPS coordinate differences 18 (6 baselines) Known coordinates 3 Total 21 UNKNOWNS Coordinates 12 Total 12 Degrees of freedom 9 ADJUSTMENT Number of iterations 1 Max coord correction in last iteration 0.0000 m # of Baselines x 3 # of Stations x 3 # of Observations - # of Unknowns Solution Convergence

  10. SKI-Pro Adjustment Logfile Statistical Test Values TESTING Alfa (multi dimensional) 0.2876 Alfa 0 (one dimensional) 0.0500 Beta 0.80 Critical value W-test 1.96 Critical value T-test (3 dimensional) 1.89 Critical value F-test 1.20 F-test 0.999 accepted F-Test Error Probability (29%) W-Test Error Probability (5%) (1-Beta) Power Level (80%) Tau Critical Value for W-Outlier Detection Test Tau Critical Value for T-Outlier Detection Test F- Critical Value for Variance Ratio Test Computed Variance Ratio and F-Test Acceptance

  11. SKI-Pro Adjustment Logfile Initial Coordinates Results based on a-posteriori variance factor ELLIPSOID CONSTANTS Ellipsoid WGS'84 Semi major axis 6378137.0000 m Inverse flattening 298.257223563 INPUT APPROXIMATE GPS COORDINATES Station Latitude Longitude Height (m) BABBAGE 45 19 43.55338 N 75 52 01.06364 W 79.6123 BOSSLER 45 23 56.00000 N* 75 55 20.00000 W* 40.0000* known DUNROBIN 45 24 25.91648 N 76 03 04.32882 W 53.4054 PIER51 45 23 07.36787 N 75 56 35.80863 W 33.0777 Computed Variance Factor 2applied to Adjustment Error Estimates 3-D Cartesian Reference System Initial WGS-84 Coordinate Values

  12. SKI-Pro Raw Observations & Initial Error Estimates INPUT OBSERVATIONS Station Target St ih Tg ih Reading DX BABBAGE BOSSLER -5557.2502 m DY 4350.4324 m DZ 5447.6065 m DX BABBAGE DUNROBIN -15511.8947 m DY 2534.1525 m DZ 6105.6145 m INPUT STANDARD DEVIATIONS OF OBSERVATIONS Station Target Sd abs Sd rel Sd tot DX BABBAGE BOSSLER 0.0044 m DY -0.0389 0.0086 cor m DZ 0.3145 -0.6161 0.0077 cor cor m DX BABBAGE DUNROBIN 0.0036 m DY -0.0409 0.0127 cor m DZ -0.0771 0.3398 0.0142 cor cor m GPS Raw Vector Components GPS Vector Covariance Lower Triangular Part of GPS Vector Covariance Matrix

  13. SKI-Pro Adjusted Coordinates Network Constraining Status COORDINATES (MINIMALLY CONSTRAINED NETWORK) Station Coordinate Corr Sd BABBAGE Latitude 45 19 43.55338 N 0.0000 0.0035 m Longitude 75 52 01.06364 W 0.0000 0.0029 m Height 79.6123 -0.0000 0.0071 m BOSSLER Latitude 45 23 56.00000 N* 0.0000 fixed m Longitude 75 55 20.00000 W* 0.0000 fixed m Height 40.0000* 0.0000 fixed m DUNROBIN Latitude 45 24 25.91648 N 0.0000 0.0036 m Longitude 76 03 04.32882 W 0.0000 0.0023 m Height 53.4054 -0.0000 0.0063 m PIER51 Latitude 45 23 07.36787 N -0.0000 0.0014 m Longitude 75 56 35.80863 W 0.0000 0.0015 m Height 33.0777 -0.0000 0.0032 m Correction from initial to adjusted coordinate values Adjusted Coordinates Coordinate Standard Deviations (1-Sigma precision estimates) Coordinates held fixed

  14. SKI-Pro External Reliability on Coordinates External Reliability EXTERNAL RELIABILITY Station Ext Rel Station Target BABBAGE Latitude -0.0097 m DY BABBAGE BOSSLER Longitude -0.0060 m DX BABBAGE DUNROBIN Height -0.0086 m DZ BABBAGE BOSSLER BOSSLER Latitude -0.0000 m DY BOSSLER PIER51 Longitude 0.0000 m DX BOSSLER PIER51 Height 0.0000 m DZ BOSSLER PIER51 DUNROBIN Latitude 0.0094 m DZ BOSSLER DUNROBIN Longitude 0.0078 m DX BOSSLER PIER51 Height 0.0112 m DY DUNROBIN PIER51 PIER51 Latitude 0.0130 m DY BOSSLER PIER51 Longitude 0.0095 m DX BOSSLER PIER51 Height -0.0125 m DY BOSSLER PIER51

  15. SKI-Pro Absolute & Relative Confidence Regions Point Confidence Regions ABSOLUTE STANDARD ELLIPSES Station A B A/B Phi BABBAGE 0.0037 0.0026 m 1.4 29 deg BOSSLER 0.0000 0.0000 m 0.0 90 deg DUNROBIN 0.0037 0.0021 m 1.7 14 deg PIER51 0.0016 0.0013 m 1.2 48 deg RELATIVE STANDARD ELLIPSES Station Station A B A/B Psi Sd Hgt BABBAGE BOSSLER 0.0037 0.0026 m 1.4 58 deg 0.0071 m BABBAGE DUNROBIN 0.0047 0.0028 m 1.7 73 deg 0.0083 m BABBAGE PIER51 0.0037 0.0026 m 1.4 69 deg 0.0072 m DUNROBIN PIER51 0.0036 0.0019 m 2.0 85 deg 0.0060 m BOSSLER DUNROBIN 0.0037 0.0021 m 1.7 -80 deg 0.0063 m Semi Minor Axis Axis Ratio Error Ellipse Orientation from North Semi Major Axis Confidence Regions between Points Standard Deviation in Height Difference

  16. SKI-Pro Residuals & Adjusted Observations ADJUSTED OBSERVATIONS Station Target Adj obs Resid Resid(ENH) Sd DX BABBAGE BOSSLER -5557.2484 -0.0018 -0.0040 0.0028 m DY 4350.4414 -0.0090 -0.0034 0.0059 m DZ 5447.6030 0.0035 0.0083 0.0054 m DX BABBAGE DUNROBIN -15511.8957 0.0010 0.0030 0.0027 m DY 2534.1441 0.0084 0.0041 0.0069 m DZ 6105.6167 -0.0022 -0.0071 0.0067 m DX BABBAGE PIER51 -6898.1738 -0.0013 0.0004 0.0027 m DY 2917.5259 0.0068 0.0022 0.0059 m DZ 4388.3274 -0.0039 -0.0077 0.0054 m DX DUNROBIN PIER51 8613.7219 -0.0000 -0.0001 0.0016 m DY 383.3818 -0.0002 -0.0071 0.0050 m DZ -1717.2892 -0.0100 -0.0070 0.0050 m DX BOSSLER DUNROBIN -9954.6473 -0.0068 -0.0070 0.0021 m DY -1816.2973 -0.0018 -0.0078 0.0052 m DZ 658.0137 -0.0110 -0.0077 0.0051 m Residuals in Geographic Units Standard Deviations of Adjusted Observations Residuals in Cartesian Units

  17. SKI-Pro Blunder Detection & Internal Reliability Blunder Detection W & T Tests TEST OF OBSERVATIONS Station Target MDB Red BNR W-test T-test DX BABBAGE BOSSLER 0.0154 m 59 2.5 -0.53 0.77 DY 0.0267 m 52 2.9 -1.24 DZ 0.0237 m 46 3.2 -0.11 DX BABBAGE DUNROBIN 0.0153 m 44 3.2 0.48 0.28 DY 0.0388 m 72 1.7 0.79 DZ 0.0416 m 79 1.4 0.12 DX DUNROBIN PIER51 0.0116 m 20 5.8 -0.37 1.51 DY 0.0255 m 40 3.2 -0.83 DZ 0.0251 m 55 2.5 -2.02** DX BOSSLER DUNROBIN 0.0144 m 80 1.6 -0.91 2.15** DY 0.0268 m 67 2.4 -1.42 DZ 0.0247 m 51 3.0 -2.05** Redundancy Number in % Bias to Noise Ratio Standardized Residuals Minimum Detectable Bias Vector Standardized Residuals Flagged Measurements as “possible” Blunders

  18. SKI-Pro Estimated Errors from W & T Test Failures ESTIMATED ERRORS FOR OBSERVATIONS WITH REJECTED W-TESTS (max 10) Record Station Target W-test Fact Est err 5 DZ BOSSLER DUNROBIN -2.05 1.0 -0.0181 m 4 DZ DUNROBIN PIER51 -2.02 1.0 -0.0181 m ESTIMATED ERRORS FOR OBSERVATIONS WITH REJECTED T-TESTS (max 10) Record Station Target T-test Fact Est err 5 DX BOSSLER DUNROBIN 2.15 1.1 -0.0077 m DY -0.0035 m DZ -0.0196 m Estimated Errors on Rejected Standardized Residuals from Blunder Detection W-Test Estimated Errors on Rejected Vector Residuals from 3-D Blunder Detection T-Test

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