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Propagation of Refraction Errors in Trigonometric Height Traversing and Geodetic Levelling

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Propagation of Refraction Errors in Trigonometric Height Traversing and Geodetic Levelling PROPAGATION OF REFRACTION ERRORS IN TRIGONOMETRIC HEIGHT TRAVERSING AND GEODETIC LEVELLING G. A. KHARAGHANI November 1987 TECHNICAL REPORT NO. 132 PREFACE In order to make our extensive series of technical reports more readily available, we have scanned the old master copies and produced electronic versions in Portable Document Format. The quality of the images varies depending on the quality of the originals. The images have not been converted to searchable text. PROPAGATION OF REFRACTION ERRORS IN TRIGONOMETRIC HEIGHT TRAVERSING AND GEODETIC LEVELLING Gholam A. Kharaghani Department of Surveying Engineering University of New Brunswick P.O. Box 4400 Fredericton, N.B. Canada E3B5A3 November 1987 PREFACE This report is an unaltered printing of the author's M.Sc. thesis of the same title, which was submitted to this Department in August 1987. The thesis supervisor was Professor Adam Chrzanowski. Any comments communicated to the author or to Dr. Chrzanowski will be greatly appreciated. ABSTRACT The use of trigonometric height traversing as an alternative to geodetic levelling has recently been given considerable attention. A replacement for geodetic levelling is sought to reduce the cost and to reduce the uncertainty due to the refraction and other systematic errors. As in geodetic levelling, the atmospheric refraction can be the main source of error in the trigonometric method. This thesis investigates the propagation of refraction errors in trigonometric height traversing. Three new models for the temperature profile up to 4 m above the ground are propos.ed and compared with the widely accepted Kukkamaki 's temperature model. The results have shown that the new models give better precision of fit and are easier to utilize. A computer simulation of the influence of refraction. in trigonometric height traversing suggests that the accumulation of the refraction effect becomes randomized to a large extent over long traverses. It is concluded that the accumulation of the refraction effect in short-range trigonometric height traversing is within the limits of Canadian specifications for the first order levelling. - ii - TABLE OF CONTENTS ABSTRACT . • ii LIST OF TABLES . .. vi LIST OF FIGURES • . • • • viii ACKNOWLEDGEMENTS . • xi Chapter 1. INTRODUCTION . 1 2. A REVIEW OF METHODS FOR THE DETERMINATION OF THE REFRACTION CORRECTION • • • • • • • • • • • 6 Determination of the Vertical Refraction Angle by the Meteorological Approach • • • • • • 7 Refractive index of air • • • • • . • • . • 7 Angle of refraction error • • • • • • • • • • . 9 Determination of the Vertical Refraction Angle Using Lasers of Different Wavelengths • . 13 Angle of Refraction Derived from the Variance of the Angle-of-Arrival Fluctuations • 15 Determination of the Vertical Refraction Correction Using the Reflection Method • . 16 Comments on the Discussed Methods ••••••.• 17 3. REFRACTION CORRECTION IN GEODETIC LEVELLING USING THE METEOROLOGICAL METHOD • • • • • • • • • • • 19 Refraction Correction Based on Direct Measurement of Temperature Gradient • • • • • • . • . 20 Kukkamaki's equation for geodetic levelling refraction correction • • • • • • • . 22 Refraction Correction Formulated in Terms of Sensible Heat Flux . • • • • • • • • • • • . 27 Review of the meteorological parameters • • • . 27 Thermal stability parameter •••••••••. 30 Profile of mean potential temperature gradients 31 The Angus-Leppan equation for refraction correction • . • . • • • • • • • • . 35 Investigation by Holdahl • • • • • • • • • 36 Comments on the Meteorological Methods • • • • . • 38 - iii - 4. REFRACTION CORRECTION IN TRIGONOMETRIC HEIGHT TRAVERSING • • • • • • • • • • • • • • • • 39 Reciprocal Trigonometric Height Traversing •••• 41 Formulae of reciprocal trigonometric height traversing • • • • • • • • • • • • • • . 42 Achievable accuracy using reciprocal trigonometric height traversing • • . 44 Precision of refraction corrections in reciprocal method • • • • • • • • • . 47 Proposed method for the calculation of refraction correction • • • • • • • 49 Refraction in Leap-Frog Trigonometric Height Traversing • • • • • • • • • • • • • • • • . 52 Leap-Frog Trigonometric Height Traversing Formulae • • • • • • • • • • • • . • • 53 Achievable Accuracy Using Leap-Frog Trigonometric Height Traversing ••• 55 Precision of refraction correction in leap- frog method • • • • • • • • • • • • . • 56 5. TEST SURVEYS AT UNB •••••• . 59 Background of Trigonometric Height Traversing at UNB • • • • • • • • • • • • • • • • • • . • 59 Description of the Test Areas and Scope of the Tests . 61 South-Gym test lines • • • • • • • • • . • 61 Head-Hall test line • • • • • • • • • • . 64 Description of the Field Equipment • • • • • . 66 Temperature gradient • • • • • • • • • • • • . 66 Trigonometric height traversing •••••••• 67 Investigation of Temperature Models as Function of Height . 67 Choice of models • o • • • • • • o o • • • • • 67 Temperature gradient measurement • o • • o • • 69 Determination of the coefficient of temperature models . 7 4 Comparison and field verification of the results • o • • • o o o • • • • • . 83 Computed Versus Measured Refraction Effect . 93 Tests on 20 June 1985 • • 0 • • • • • • • . 93 Tests on 19 July 1985 o • • • • • o • • • . 95 Tests on 23 and 24 July 1985 • . • • • 100 Tests on 29 July 1985 and estimation of standard deviation of vertical angle measurements • • • • • • • • • • • o 0 112 Comments on South-Gym test surveys • • • o 0 115 Tests on 06 August 1985 • • • • • • • . • 116 6o SIMULATIONS OF REFRACTION ERROR IN TRIGONOMETRIC HEIGHT TRAVERSING • • • • • • • • o • • • • • 120 - iv - Simulation Along a Geodetic Levelling Line on Vancouver Island • • • • • • • • • • • • • 120 Computation of the refraction error in geodetic levelling • • • • • • • • • • • • • • • 122 Refraction error in trigonometric height traversing • • • • • • • • • • • • • . 124 Results of simulations • • • • • • • • • • • 125 Simulation of the Refraction Error Using other Values of Temperature Gradient Measurements 127 Simulation on the Test Lines at UNB • • • • 1.31 7. CONCLUSIONS AND RECOMMENDATIONS • . 145 Conclusions . 145 Recommendations . 149 REFERENCES . 151 - v - LIST OF TABLES Table ~ 5 .1. The Time Averaged Temperatures on Gravel Line 71 5.2. The Time Averaged Temperatures on Grass Line . 72 5.3. The Time Averaged Temperatures on Asphalt Line . 73 5.4. Mean standard deviations . 75 5.5. Curve fitting and coefficient of refraction computations (Kukkamaki's model, i1 in Table 5.4, over asphalt) ••••••••••••••• 77 5.6. Curve fitting and coefficient of refraction computations (model, i4 in Table 5.4, over asphalt) •••••••••••••.. 78 5.7. Curve fitting with test of the significance of coefficient (Kukkamaki's model, i1 in Table 5. 4) . 80 5.8. Curve fitting with the significance of coefficient test (model i3 in Table 5.4). • .•.•.•• 81 5.9. Refraction effect [mm] computed using the seven models versus the measured value (BM1-BM2). 86 5.10. Refraction effect [mm] computed using the seven models versus the measured value (BM2-BM3). • 87 5.11. Refraction effect [mm] computed using the seven models versus the measured value (BM3-BM1). 88 5.12. Correlation Coefficients Matrices • 89 5.13. Preliminary test measurements using UNB trigonometric method at South-Gym area from BMl to BM2 • • • • • • • • • • • • • • • • • • • • 93 5.14. Discrepancies between the results obtained using trigonometric height traversing and geodetic levelling for BMl to BM2 •••••••••••. 96 - vi - 5.15. Discrepancies between the results obtained using trigonometric height traversing and geodetic levelling for BM2 to BM3 •••••••••••• 97 5.16. Discrepancies between the results obtained using trigonometric height traversing and geodetic levelling for BM3 to BMl ••...•••••.. 98 5.17. Computed refraction using measured temperature gradient . • • • • • • • • • • • • • • • • . 99 5.18. t-test on the significance of the correlation coefficients • • • . • . • • • . • • • • • 111 5.19. Computed refraction effect versus the measured value for Head-Hall test line • 119 6.1. Average 6t, b and H along the levelling routes • 128 6.2. Average 6t, b and H in Fredericton, N.B •• 129 6.3. Average 6t, b and H along levelling routes in United States (after Holdahl [1982]) • • • • 130 - vii - LIST OF FIGURES Figure 1.1. Methods of trigonometric height traversing • . 5 2 .1. Vertical Refraction Angle . • 10 2.2. Principle of refraction by reflection . 17 3 .1. Refraction effect in a geodetic levelling set- up . •. • 23 3.2. Profile of mean potential temperature e • • 32 4.1. Ellipsoidal section for reciprocal trigonometric height traversing • • • • • • • • • • • • • • . 42 4.2. Standard deviation of refraction correction in reciprocal height traversing as a function of distance •••.••.••••••••••.•. 49 4.3. Ellipsoidal section for leap-Frog trigonometric height traversing • . • • • • • • • • • • • • . 54 4.4. Standard deviation of refraction correction in leap-frog height traversing as a function of distance. • • • • • . • • • • • • • • • • • 58 5 .1. Plan and profiles of South-Gym test lines . 63 5.2. Plan and profile of Head-Hall test line . 65 5.3. Refraction Coefficient Contours . 79 5.4. Test of the significance of coefficient for models in Table 5.4 .••.••.•••••••.•• 82 5.5. Refraction effect computed using the seven models versus the measured value for BM1-BM2 ••••• 90 5.6. Refraction effect computed using the seven models versus the measured value for BM2-BM3 ••... 91 5.7. Refraction effect computed using the seven models versus the measured
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