Geodesy

The Black-Allan Line Revisited
A paper (26 pages) on the straight line section of the Victorian-New South Wales border discussing some historical aspects of the border survey (1870-72) and the possible types of lines (geodesic, normal section, curve of alignment, etc.) that could have been candidates for the straight line from Cape Howe to the nearest source of the Murray River. A Power Point presentation is also provided.
Black-Allan Line Revisited.pdf
Black-Allan Line Revisited.ppt

Computation of Meridian Distance on the ellipsoid
A set of lecture notes (30 pages) including Matlab functions. Most of the material is contained in Geometric Geodesy (Part A)
Meridian Distance.pdf

Curve of Alignment on an ellipsoid
Notes detailing the equation of the curve of alignment and a technique for computing the position of a point on a curve of alignment (20 pages). These notes are contained in Geometric Geodesy (Part B).
Curve of Alignment.pdf

Eccentricity of the Normal Ellipsoid
The eccentricity of the normal ellipsoid is a function of the defining constants that are usually; equatorial radius, geocentric gravitational constant, dynamical form factor and angular velocity. These notes (4 pages) show how the eccentricity is calculated from these constants using an iterative sequence. Maxima code is given and the value of eccentricity-squared for the GRS80 ellipsoid is evaluated to 45 decimal digits.
Eccentricity of the Normal Ellipsoid.pdf

Electronic Distance Measurement
These notes (17 pages) describe the principles of short range distance measurement with a reflector using an infrared or visible laser electromagnetic wave and phase measurement.  Digital phase measurement of a modulated wave is outlined and the propagation of waves through the atmosphere is discussed.  Formula for refractive index are given and atmospheric corrections (first- and second-velocity corrections) and path length corrections developed.
Notes on Electronic Distance Measurement.pdf

Geometric Geodesy (Part A)
A set of lecture notes that are an introduction to ellipsoidal geometry related to geodesy written by R.E. Deakin and M.N. Hunter (v + 140 pages)
Geometric Geodesy A(2013).pdf

Geometric Geodesy (Part B)
A set of lecture notes on ellipsoidal geometry related to geodesy written by R.E. Deakin and M.N. Hunter (iii + 208 pages). These notes are mainly concerned with the computation of distance and direction between points on a reference ellipsoid and are a collection of papers written by the authors and includes a paper written by C.F.F. Karney and R.E. Deakin that is a translation of F.W. Bessel’s (1826) original paper on the topic of geodesic computation.
Geometric Geodesy B(2010).pdf

Geodesics on an Ellipsoid – Bessel’s Method
Technical paper by R.E. Deakin and M.N. Hunter on F.W. Bessel’s method of computing geodesics (66 pages). This paper also includes the modifications to Bessel’s equations by Rainsford and Vincenty as well as Matlab functions for the forward and reverse cases. This paper is contained in Geometric Geodesy (Part B).
Geodesics – Bessel method.pdf

Geodesics on an Ellipsoid – Pittman’s Method
Paper by R.E. Deakin and M.N. Hunter presented at the Spatial Sciences Institute Biennial International Conference (SSC2007), Hobart, Tasmania, Australia, 14-18 May, 2007 (19 pages). This paper is contained in Geometric Geodesy (Part B).
Geodesics – Pittman method.pdf

The calculation of longitude and latitude from geodetic measurements – F.W.Bessel
A translation of Bessel’s original paper in Astronomische Nachrichten (Astronomical Notes) 4(86), 241-254, 1826 by Charles Karney and Rod Deakin (11 pages)
Bessel_arxiv_0908.1824v1.pdf

Great Elliptic Arc on an ellipsoid
Notes detailing the equation of the great elliptic arc on an ellipsoid and a technique for computing the position of a point on the arc (19 pages). These notes are contained in Geometric Geodesy (Part B).
Great Elliptic Arc.pdf

Great Elliptic Arc Distance
These notes (43 pages) provide a detailed derivation of series formula for (i) Meridian distance M as a function of latitude φ and the ellipsoid constant eccentricity-squared e2, and φ and the ellipsoid constant third flattening n; (ii) Rectifying latitude µ as functions of φ ,e2 and φ ,n; and (iii) latitude φ as functions of µ ,e2 and µ ,n These series can then be used in solving the direct and inverse problems on the ellipsoid (using great elliptic arcs) and are easily obtained using the Computer Algebra System Maxima. In addition, a detailed derivation of the equation for the great elliptic arc on an ellipsoid is provided as well as defining the azimuth and the vertex of a great elliptic. And to assist in the solution of the direct and inverse problems the auxiliary sphere is introduced and equations developed.
Great Elliptic Arc Distance.pdf

Local Geodetic Horizon Coordinates
Notes on transforming X,Y,Z Cartesian coordinates to Local Horizon coordinates E,N,U (5 pages).
LOCAL HORIZON COORDINATES.pdf

Loxodrome on an ellipsoid
Notes detailing the equation of the loxodrome on an ellipsoid and a technique for computing the azimuth and distance of a loxodrome (20 pages). These notes are contained in Geometric Geodesy (Part B).
Loxodrome on Ellipsoid.pdf

Normal Section Curves on the Ellipsoid
Notes on normal section curves (53 pages) including proof that normal sections are arcs of ellipses, Cartesian and polar equations of normal section curves, azimuth of normal sections and computation of normal section arc length using Romberg integration. These notes are contained in Geometric Geodesy (Part B).
Normal Section.pdf

Reference Values for the GRS80 and WGS84 Ellipsoids
Two Matlab functions refval.m and refval2.m are given (with their help messages and output) that compute the geometric and physical constants of the GRS80 and WGS84 ellipsoids respectively given their defining parameters.
MATLAB function refval.pdf
MATLAB function refval2.pdf

Relationships between Astronomic and geodetic coordinates
Notes explaining the connection between observations (horizontal direction and zenith distance observations) related to the earth’s equipotential surfaces and the ellipsoid (8 pages)
Laplace.pdf

Skew-Normal Correction
Notes on the correction to geodetic direction due to height of stations above the ellipsoid (8 pages).
SKEW NORMAL CORRECTION.pdf

The Gravity Field of the Earth
Notes on the Earth’s Gravity Field (hand-written 59 pages) beginning with explanations of Gravitation, Gravitational Potential and Gravity, then Gravitational Potential V, Rotational Potential R and Gravity Potential W = V + R. A solution for the Earth’s gravitational potential V is developed by solving Laplace’s equation as a spherical harmonic series containing Associated Legendre Functions. Working forms of the gravitational potential V are developed and a worked example of the calculation of V for Melbourne, Australia is given using the fully-normalized coefficients for the geopotential model EGM96.
The Gravity Field of the Earth.pdf

The Normal Gravity Field
Notes on the Normal Gravity Field (hand-written 38 pages). The source of the normal gravity field is a model earth which bets fits the actual shape of the Earth. An ellipsoid (an ellipse of revolution) is assumed for the model earth and this ellipsoid is said to have the same mass M of the earth, but with homogenous density; the same angular velocity ; and the surface of this ellipsoid is said to be a level surface (an equipotential surface) of its own gravity field. This geocentric equipotential ellipsoid is the basis of the Geodetic Reference System 1980 (GRS80).
The Normal Gravity Field.pdf

The Geoid
Technical paper published in The Australian Surveyor, Vol. 41, No. 4, 1996, pp. 294-305.
The_Geoid.pdf

Transforming Cartesian coordinates X,Y,Z to Geographical coordinates φ,λ,h
A paper by G.P. Gerdan and R.E. Deakin published in The Australian Surveyor, Vol. 44, No. 1, pp. 55-63, June 1999.
Transforming Cartesian Coordinates.pdf

Traverse Computation on the Ellipsoid and on the UTM projection
Notes describing (in detail) the methods and formula used in reducing traverses of measured directions and distances on the earth’s surface to: (i) a set of quasi-measurements on the reference ellipsoid and then (ii) a set of plane directions and distances on the UTM projection.
Trav_Comp_V2.1.pdf

Traverse Computations: Ellipsoid versus the UTM projection
Paper presented at the Victoria/South Australia Survey Conference, Water Wine Wind, Mount Gambier, SA, 15-17 April, 2005 (11 pages)
Mt Gambier.pdf

Traverse Computation on the UTM projection for Surveys of Limited Extent
Paper discussing various aspects of the transverse Mercator projection and computational methods that can be adopted if the survey area is small, say less than 25 square km (20 pages)
Cadastral Trav Comp.pdf

Useful Derivatives for Geodetic Transformations
This paper (19 pages – unfinished) gives a detailed derivation of derivatives that may be useful in analyzing the precision of geodetic coordinate transformations. For example, networks of GPS measurements are processed in X,Y,Z Cartesian coordinates but locations of network stations are usually transformed to geodetic coordinates (latitude, longitude and height) related to the reference ellipsoid. And may be transformed to E,N grid coordinates on a map projection. If the precisions of coordinates are known in one system, then precisions in the transformed system can be evaluated by propagation of variances expressed mathematically as a sequence of matrix operations. One of the matrices involved is the Jacobian matrix of first-order partial derivatives, and this paper gives the derivation of various partial derivatives as well as examples of their application.
Useful Derivatives V2 01Dec2014.pdf

Satellite Orbits
Technical paper (51 pages) providing a brief historical account of Kepler’s laws and how these are an outcome of Newton’s laws of motion and universal gravitation. The equation of motion for the N-body problem is derived and then simplified to the two-body problem (earth-satellite).
Satellite Orbits.pdf