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A MANUAL FOR GEODETIC

POSITION COMPUTATIONS

IN THE MARITIME

PROVINCES

D. B. THOMSON

E. J. KRAKIWSKY

J. R. ADAMS

February 1978

TECHNICAL REPORT

NO. 217

TECHNICAL REPORT

NO. 52

PREFACE

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PREFACE

This "manual" is the second of three being written to cover the correct and practical use of the geodetic information of the redefined Maritime Geodetic Network. While the first manual dealt with a single terrain point, this involves two points and the observations between them. The third manual will centre on terrestrial networks (many terrain"points and observations amongst them). This manual was written as' ~ guide to the use and interpretation of geodetic information for two terrain points. It is to serve mainly as a surveyors handbook for Geodetic Position Computations in the three-dimensional, ellipsoidal, and conformal mapping plane environments in the maritime provinces. No derivations or extensive explanations of the mathematical formulae are given. The equations required to solve the position and associated error transformation problems are stated, the notation used is explained, and a numerical example is presented. A reader desiring extensive background information as to the relevance of this manual, and a detailed explanation of the origins of the mathematical formulae, is referred to the reference material. It should be noted that the material presented in this manual has been rigorously developed. Approximations made, and their affects are clearly indicated. Further approximations, for whatever reasons, are left to the professional judgement of the surveyor •

• This "f.tanual" was written in partial fulfillment of a contract (U.N.B. Contract No. 132 730) with the Land Registration and Information Service, Surveys and Mapping Branch, Summcrside, P.E.I.

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TABLE OF CONTENTS

Preface • • • • • • • • • • • • • • • Acknowledgements • • • • • • • • • • Table of Contents • • • • • • • • • • List of Tables and Figures

1. Introduction

1. Computations in Three Dimensions 2.1 Notation • 2.2 General Concepts

2.2.1 The Direct Problem .2. 2. 2 The Inverse Problem Error Propagation • • • • • 2.3.1 Error Propagation in the Direct Problem 2.3.2 Error Propagation in the Inverse Problem 2.4 New Brunswick Numerical Example 2.4. 2.4. Direct Problem • Inverse Problem 2.5 Prince Edward Island Numerical Example • 2.5. 2.5. Direct Problem Inverse Problem 2.6 Nova Scotia Numerical Example • 2.6. 2.6. Direct Problem • Inverse Problem

1. Computations on the Ellipsoid • 3.1 Notation • 3.2 Reduction Formulae • 3.2. 3.2.33.2.

Introduction Reduction of (^) Horizontal• • • • (^) Directions• • • • Reduction of Horizontal Angles

iii

i ii iii vi 1 5 5 5 6 12 14 14 19 23 23 27 28 28 33 34 34 39 41 41 43 43 45 45

3.2o

Reduction of Zenith Distances o o o o o o • • • Reduction of Astronomic Azimuths • • o Reduction of Spatial Distances o o Magnitude of Corrections o • • o • o • o • • • • • Error Propagation ~hrough Reduction Formula • • • •

1. 3 "Reduction" of Computed Geodetic Quantities to the Terrain...................... 3-4 Puissant's ~ormula o •• o • o ••••••••

Direct Problem • Inverse Problem The Gauss Mid-Latitude Formula • 3.5. 3.5. Direct Problem • • • • • • • • • • • • Inverse Problem • • • • • • 3.6 Error Propagation Through Position Computations. 1 •

3.6.i· Direct Problem Error Propagation • 3.6.2 Inverse Problem Error Propagation Introduction to Numerical Examples • • • • 3.7. 3.7. Use of Computed Geodetic Azimuth Ellipsoid Direct Problem Flow Chart •

....^ <^ ••••• -~^ •^ ....•••^ •

3.8 New Brunswick Numerical Example

3.8.1 Direct Problem •........ 3.8.2 Inverse Problem •....... Prince Edward Island Numerical Example 3.9.1 Direct Problem • (^).... 3.9.2 Inverse Problem (^)..... Nova Scotia Numerical Example • 3.10. 3.10. Direct Problem........ •.. •. •. • •... Inverse Problem • • • (^)........... ... Computations on a Conformal Mapping Plane

4o2^ Notation Reduction^ •^ of•^ Observations•^ •^ •^ •^ •^ •^ •^ • •^ • •^ o •^ • •^ • • • 4.2.1 Reduction of Horizontal Directions (from the

Ellipsoid to the Mapping Plane) • • • • • • • • • • • • 98 iv

Figure 1- Fiqure (^) l- Fiqure 2- Fiqure 3- Fiqure (^) 3- Figure 3- Fiqure 3- Fiqure (^) 3- Fiqure (^) 4~ Fiqure 4- Fiqure (^) 4- Fiqure 4- Figure 4- Fiqure A- Fiqure A-

Table 3-

LIST OF TABLES AND FIGURES

Distance Reduction • • • • • • • • • • • Local Geodetic System Position Vector Local Astronomic Observations • • • • Spatial Distance Reduction • Gravimetric Correction • Skew Normal Correction • Normal Section to Geodesic Correction • Ellipsoid Direct Problem Flow Chart • • Reduction of Horizontal Directions •. Reduction of Horizontal Angles Reduction of Azimuth • •

.... ..

Page 3 8 10 47 so 51 52 74 99 •• 101 ••• 102 Siqn of (T-t) Correction Stereographic • • • • • lOS Mapping Plane Direct Problem Flow Chart ••••• 124 Rotation Matrices • Reflection Matrices

Effect of Geoidal Height on Distance Reduction • 53

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1. INTRODUCTION

In "A Manual for Geodetic Coordinate Transformations in the Maritime Provinces". [Krakiwsky et al, 1977], it was shown that a terrain point i could be described mathematically by any one of three different sets of coordinates (three-dimensional (X.,l. y (^) i' z.),ellipsoidall. ( ~.l. , A.l. ) ,. conformal mapping pl~ne (Xl.. , Y.))l. and their associated accuracies (variance-covariance matrices). Furthermore, it was shown, by rigorous coordinate and variance-covariance matrix transformations, that the coordinates and associated accuracies in all three systems were equivalent. In this second handbook, we introduce a second point j, and treat two different problems involving i and j simultaneously in each of the three-dimensional, ellipsoidal, and conformal mapping plane environments. One of the problems - the so-called inverse problem - involves the computation of the azimuths, distance , and associated accuracies between the two points. A rigorous procedure for each of the three environments is given, and it is shown, via appropriate "reductions", that the solutions are equivalent. The other problem - the so-called direct problem - involves^. the computation of the coordinates and associated variance-covariance matrix of the second point j using observations made from i to j. Again, solutions are given for the three-dimensional, ellipsoidal, and conformal mapping plane environments and, using appropriate "reduction~" of observed and computed data and coordinate transformations, it is shown that the solutions are equivalent.

1

3 \ \ \ \ \ \ \ \ \

I. (^) l.J..

Fiqure 1- Distance Reduction

I I I I I I I

j

GEOID

CONFORMAL MAPPING

PLANE

4

processes required for position computations in the maritime provinces. In closing, the reader should take special note of the fact that the measurement reduction processes are reversible; that is, one may compute a distance on a conformal mapping plane and "reduce" it up to the terrain. This is an important point for surveyors who are often faced with the need for terrain values for computed distances and azimuths. This inverse reduction process is covered in Chapters 3 and 4.

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referred to the reference ellipsoid. Computations of geodetic positions in three dimensions, for whic~ formulae are given in this chapter, are based on three dimensional Euclidean geometry and employ vector and matrix a~gebra. Since distances, zenith angles and azimuths of lines are actually observed in three dimensional space, they require no "reduction" to some surface and need only be corrected for refraction effects and instrumental corrections such as heights of instrument above the actual terrain point or zero error for electromagnetic distance measurements. Readers not familiar with tl;le AliTerage Terrestrial, Geodetic, Local Geodetic and L.ocal Astronomic coordinate systems are referred to, for example, Krakiwsky and Wells [1971]. It should be mentioned here that we present only 9ne method for solving the direct and inverse problems in the 3-D environment. There are other methods and the interested reader is referred to, for excurple, Krakiwsky ani Thomson {1974]. If the reader is unfamiliar with rotation matrices please review Appendix I before continuing. 2.2.1 The Direct Problem The direct problem may be stated as; given the coordinates (Xi, Yi, Zi )G of point i, the terrestrial spatial distance r^ l.J..^ ,^ .astronomic^ azimut: -A (^) 4J.• , and zenith. angle Z (^) l.J.. from.i to a second point j, compute the coordinates (X., J Y.,J Z.)JG of point j. We note here that if we are given (ljl.,1. A..,1. h.1. )G of point i a coordinate transformation [Krakiwsky et al.,l977] yields

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To solve the direct problem we must know the relationship between the Local Geodetic coordinate system and Geodetic system, and between the Local Astronomic system, where we observe, and the Local Geodetic system in which we compute. If we know these relationships the observed quantities of azimuth, zenith angle and distance can be used to determine.the coordinates of a secpnd point. The relationship between the Local Geodetic and Geodetic system is examined-first. From Figure 2-1 we can rotate the vector (r^ -+ ij) LG from the I.ocal Geodetic to the Geodetic system using [Krakiwsky and Wells, 1971]. (2-1)

We can then obtain the (rj)G^ -+ using

(2-2)

Expanding (2-1) and substituting into (2-2) yields

(2-3)

(2-4)

-Continuing and referring to Figure 2.2, the position vector (rij)LA can be written as a function of the observables, namely

·ex.> l. "l^ l.J_._^. sin^ z^ l.J..^ cos^ A^ l.J.. (r^ +^ ij) LA =^ (Y.)l. =^ +rij^ sin^ z^ ..^ ~in^ A^ ••^ (2-6) l.J l.J (Z.)l. (^) LA +rij^ cos z (^) l.J..

The relationship betwe~n the_vector (~ij)LA and (~ij)LG is

given by [Krakiwsky and Wells, 1971].

(2-7) ti and ~i are the meridian and priiDe vertical co~nents of the deflection of the vertical at the terrain point i. llazl.J .. is the difference between the geodetic and astronomic azimuths of the terrain point i and is given by [Krakiwsky and Wells, 1971]

Now (2-7) is expanded with (2-6) substituted in it. We are going to assume that the deflection components and Aaz are all less than 30~0 of arc in the maritimes which allows us to write with better than .01 m accuracy that

r 1 J. (sin z (^) l.J.. cos Al.J.. + !Jaz.l.J .. sin z (^) l.J.. sin Al.J.. + E;. l. (^) cos z (^) l.J.. ) (r^ + 1 .J.)LG • r .. (-flaz .. sin z .. cos A.. + sin z .. sin A.. + n. cos z .. ) l.J l.J l.J l.J l.J l.J l.. l.J r 1 j (-E;.l. sin z (^) l.J.. cos Al.J.. - 1"1.l. sin z (^) l.J.. sin A 1 .J. + cos Zl.J .. ) (2-9)

y LA

ZLA

A ...,J.J .. .......

___^ .......^ ........^ ....... ~I^ / "'

l

Figure 2- Local Astronomic Observations