Papers by Andrew Ruffhead
Reports on Geodesy and Geoinformatics, 2023
Two-dimensional interpolationor surface fittingis an approximation tool with applications in geod... more Two-dimensional interpolationor surface fittingis an approximation tool with applications in geodetic datum transformations, terrain modelling and geoid determination. It can also be applied to many other forms of geographic point data, including rainfall, chemical concentrations and noise levels. The problem of fitting of a smooth continuous interpolant to a bivariate function is particularly difficult if the dataset of control points is scattered irregularly. A typical approach is a weighted sum of data values where the sum of the weights is always unity. Weighting by inverse distance to a power is one approach, although a power greater than 1 is needed to ensure smooth results. One advantage over other methods is that data values can be incorporated into the interpolated surface. One disadvantage is the influence of distant points. A simple cutoff limit on distance would affect continuity. This study proposes a transition range of accelerated decline by means of an adjoining polynomial. This preserves smoothness and continuity in the interpolating surface. Case studies indicate accuracy advantages over standard versions of inverse-distance weighting.
Survey Review, 2023
This paper proposes a new method of deriving 9-parameter affine 3D datum transformations by ordin... more This paper proposes a new method of deriving 9-parameter affine 3D datum transformations by ordinary least-squares. Unlike previous methods, it covers all versions of the transformation. Initially, an 'average' scale factor is computed by distance analysis. Removing the scaling effect, the 'RIGOPT' subroutine is applied to optimise the rigid transformation that consists of 3 translations and 3 rotations. Using an equivalentenlargement hypothesis, the number of scale factors is increased to 3 by a short series of single-search-direction optimisations. The minimisation of residuals is verified by enclosing-interval analysis. The case studies cover datasets in Western Australia, Great Britain and Sweden.
Bulletin of Geodetic Sciences, 2022
Multiple regression equations (MREs) provide an empirical direct method of transforming coordinat... more Multiple regression equations (MREs) provide an empirical direct method of transforming coordinates between geodetic datums. Since they offer a means of modelling distortions, they are capable of a more accurate fit to datum-shift datasets than more basic direct methods. MRE models of datum shifts traditionally consist of polynomials based on relative latitude and longitude. However, the limited availability of low-power terms often leads to high-power terms being included, and these are a potential cause of instability. This paper introduces three variations based on simple partitions and 2 or 4 smoothly conjoined polynomials. The new types are North/South, East/West and Four-Quadrant. They increase the availability of low-order terms, enabling distortions to be modelled with fewer side effects. Case studies in Great Britain, Slovenia and Western Australia provide examples of partitioned MREs that are more accurate than conventional MREs with the same number of terms.
Survey Review, 2021
Seven-parameter conformal coordinate transformations, also known as Helmert transformations, can ... more Seven-parameter conformal coordinate transformations, also known as Helmert transformations, can be constructed in more than one way. Two possible orderings of the rotations are in common use, giving rise to Helmert versions 1 and 2. It is demonstrated how the rotation parameters of either version can be converted into the rotation parameters of the other. This is useful when software is designed for the other version. It also enables computation of the same-formula inverse transformation by changing the sign of the equivalent 'other version' parameters. These results were primarily intended for conformal transformations between geodetic datums. They can, however, be extended to coordinate transformations in disciplines such as photogrammetry where rotations sometimes exceed 90 degrees.
Survey Review, 2021
This paper proposes a new method of deriving rigorously-conformal 7-parameter 3D coordinate trans... more This paper proposes a new method of deriving rigorously-conformal 7-parameter 3D coordinate transformations between geodetic datums. The problem of linearisation is reduced by distance analysis which provides an estimate of scale-change. The resulting 6-parameter transformation is linearised to enable an initial least-squares estimate of the rotation parameters. The 6-parameter transformation is then re-linearised to obtain a least-squares estimate of the corrections to the rotations. The validity of the scale-change estimate can be tested and is verified in almost all cases. The exception is transformations covering very small areas where short distances maximise the impact of measurement errors in the control data. Even there, the method can be adapted to optimise the transformation. The method can also be used to obtain pseudo-optimal conformal transformations that provide a closest fit to published Bursa-Wolf transformations.
Error Analysis of Numerical Extrapolation Processors, 1976
Polynomial extrapolation is discussed in a general context. Truncation and rounding errors a... more Polynomial extrapolation is discussed in a general context. Truncation and rounding errors are considered both for their precise form and as to how they may best be estimated. Neville’s process is described with provision for interpolation points placed an increasing distance from the point of interest (“outward parameters”).
Neville extrapolation is applied to numerical differentiation. Optimal sequences of interpolation points are then selected with the objective of minimising the relative contribution of rounding errors to the total error. Equally-spaced outward parameters are shown to come near to minimising it, whatever the length of the iteration. The use of optimal sequences to achieve a specified error tolerance is investigated.
Rational extrapolation processes are considered with emphasis on the general approach of Larkin. Characteristics of the actual interpolation functions are discussed. Error expansions are investigated with the objective of discovering why extrapolation appears to be more efficient than polynomial extrapolation. Possible reasons for this, based on zero summations of error terms, are discussed.
Survey Review, 2018
This paper provides an introduction to multiple regression equations as a method of performing ge... more This paper provides an introduction to multiple regression equations as a method of performing geodetic datum transformations. The formulae are particularly useful when there are non-linear distortions that need to be built into the transformation model. However, the equations take the form of a one-way transformation, usually a local geodetic datum to a global datum. The standard procedure for applying the equations to obtain the reverse transformation only gives approximate results relative to the original model. This paper quantifies the problem and describes three methods for computing the reverse transformation (or inverse transformation) more accurately.
Survey Review, 2016
For three-parameter datum transformations to be applied rigorously, geodetic coordinates on the f... more For three-parameter datum transformations to be applied rigorously, geodetic coordinates on the first ellipsoid need to be converted to Cartesian coordinates before application of the shifts, then converted to geodetic coordinates on the second ellipsoid. The Standard Molodensky method of datum transformation is more direct but is inexact. It also fails to reproduce the original coordinates when applied forward and back. However, this paper shows a pattern of proportionality between the misclosures and the errors in the forward approximations. This gives rise to a new method of computing the transformations, best described as " Standard Molodensky in two stages with applied misclosures " (SMITSWAM). The method is shown to be more than 1600 times more accurate than Standard Molodensky, coming close to the accuracy of the rigorous approach. SMITSWAM is also shown to be around 48% faster than the traditional form of the rigorous method which uses iteration.
Survey Review, Oct 1, 1998
Inverse projection algorithms are sometimes inconsistent with the projections, in the sense that ... more Inverse projection algorithms are sometimes inconsistent with the projections, in the sense that the application of one after the other produces departures from the original coordinates. Eliminating the misclosure by error modelling is one option, but this only produces a correction on a specific grid rather than to the generic inverse projection algorithm. This paper describes an iterative approach to eliminating misclosures, using approximate partial derivatives. The Syrian Stereographic Projection is given as an example of the problem and how it can be solved.
Survey Review, Jan 1, 1994
This paper considers the problem of calculating the vertical gravitational attraction at an arbit... more This paper considers the problem of calculating the vertical gravitational attraction at an arbitrary point of a rectangular mass with uniform density. A rigorous approach is needed for a large mass which is very close to that point. Treating the mass as an infinite summation of point masses requires the computation of a triple integral. However, the problem can be solved by treating the mass as a summation of cylindrical strips. A subroutine and an algorithm are proposed.
Survey Review, Apr 1, 1987
This paper provides an introduction to least-squares collocation, a process already well establis... more This paper provides an introduction to least-squares collocation, a process already well established in statistical geodesy for prediction, filtering and modelling. Variance/covariance propagation laws are shown to have hitherto-unnoticed applications including prediction and filtering formulae usually obtained by least-squares criteria. An example from coordinate transformations is given in which collocation is shown to be more accurate than traditional least-squares modelling.
Computer Physics Communications, 1985
In the production of geographic data covering regions of the world, a typical project requirement... more In the production of geographic data covering regions of the world, a typical project requirement is a set of matrices totalling 100 million heights interpolated from contour data. An investigation on a sample area using the DAP shows a 10 to 1 advantage in speed, even when based on a method originally designed for a serial computer. The use of the DAP to manipulate and re-arrange data with inconvenient structures is also demonstrated.
BIT, 1980
The first derivative of a real-valued function may be approximated at a certain point by the deri... more The first derivative of a real-valued function may be approximated at a certain point by the derivative of a polynomial collocating with the function at this point and a number of other distinct points. The particular points which minimise the magnification of any rounding errors in the function values for any fixed point of truncation error are shown to be closely related to the turning points of a related Chebyshev polynomial.
BIT, 1975
The relationships between various numerical methods for obtaining polynomial approximations to th... more The relationships between various numerical methods for obtaining polynomial approximations to the first derivative of a known function are investigated, and their computational advantages discussed. Optimal sequences of interpolation points are then selected with the objective of minimising the relative contribution of rounding errors to the total error, and geometric sequences, though non-optimal in this sense, are considered for computational reasons.
Teaching Documents by Andrew Ruffhead
UEL ACE Surveying Working Paper No 01/2020, 2020
This paper provides an introduction to methods of performing coordinate transformations between g... more This paper provides an introduction to methods of performing coordinate transformations between geodetic datums. The emphasis is on the types of transformation which can be expressed directly by formulae. The paper aims to be comprehensive. To that end it takes account of the fact that some methods have slightly different forms, something which is seldom acknowledged. Each method is classified according to whether it is conformal, near-conformal or non-conformal.
This paper is primarily concerned with the application of transformations for which “optimal” parameters are already known. Methods for optimising parameters are not covered here, partly because they constitute a major topic in their own right. A typical approach is minimisation of the sum of the squares of residual shifts, although there are many variations. It is assumed that the derivation methods are statistically based and are designed to produce transformation models that fit actual data to a particular level of accuracy.
Published parameters for a datum transformation are primarily designed for a one-way transformation of coordinates, usually from a local geodetic datum to a global datum. Where reverse transformations are needed, there is a general tendency to recommend the original formulae with negated parameters, giving results that are only approximate relative to the forward transformation. This paper recommends mathematically-exact reverse formulae where they exist and computationally-convergent reverse processes where they do not.
Thesis Chapters by Andrew Ruffhead
None (PhD Thesis), 2021
This thesis is a study of methods of transforming coordinates between geodetic datums, the method... more This thesis is a study of methods of transforming coordinates between geodetic datums, the methods being generally known as datum transformations. Direct methods are described and categorised as conformal, near-conformal and non-conformal. New variations on all three types are included in the direct methods: SMITSWAM (which avoids changes of coordinate-type), partially-conformal variants of Standard & Abridged Molodensky, and normalised generalisations of multiple regression equations (5 types). Reverse transformations are extensively covered, as are methods of derivation. In both cases, new algorithms are included. Direct methods, with the exception of multiple regression equations, do not capture distortions in datum transformations. The thesis therefore includes a review of composite methods which extract a trend model and apply a surface-fitting technique (SFT) to the residuals. Sometimes the SFT is used as a gridding method, producing regularly-spaced data that can be interpolated as a final stage of the composite process. The SFTs selected for detailed study include new variations on inverse-distance-to-a-power weighting and nearestneighbour interpolation. These are called HIPFEAD and LIVONN respectively. In both cases, the variations are shown to have advantages in terms of accuracy of fit. Least-squares collocation and radial basis functions are shown to produce reusable vectors-described here as "revamped signals"that enable interpolation without gridding. Where the composite methods are used for gridding, it is shown that geodetic coordinates can be used, avoiding the need for projected grid coordinates. The interpolation options applied are piecewise-bilinear and piecewisebicubic, the latter being an algorithm (believed to be new) that uses up to 12 "grid" points. Case studies were considered using 6 datasets, two for Great Britain, one each for Western Australia, Ghana, Sweden and Slovenia. These showed beneficial properties of the new methods, both in the direct and composite categories. They also enabled comparisons of transformation methods generally.
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Papers by Andrew Ruffhead
Neville extrapolation is applied to numerical differentiation. Optimal sequences of interpolation points are then selected with the objective of minimising the relative contribution of rounding errors to the total error. Equally-spaced outward parameters are shown to come near to minimising it, whatever the length of the iteration. The use of optimal sequences to achieve a specified error tolerance is investigated.
Rational extrapolation processes are considered with emphasis on the general approach of Larkin. Characteristics of the actual interpolation functions are discussed. Error expansions are investigated with the objective of discovering why extrapolation appears to be more efficient than polynomial extrapolation. Possible reasons for this, based on zero summations of error terms, are discussed.
Teaching Documents by Andrew Ruffhead
This paper is primarily concerned with the application of transformations for which “optimal” parameters are already known. Methods for optimising parameters are not covered here, partly because they constitute a major topic in their own right. A typical approach is minimisation of the sum of the squares of residual shifts, although there are many variations. It is assumed that the derivation methods are statistically based and are designed to produce transformation models that fit actual data to a particular level of accuracy.
Published parameters for a datum transformation are primarily designed for a one-way transformation of coordinates, usually from a local geodetic datum to a global datum. Where reverse transformations are needed, there is a general tendency to recommend the original formulae with negated parameters, giving results that are only approximate relative to the forward transformation. This paper recommends mathematically-exact reverse formulae where they exist and computationally-convergent reverse processes where they do not.
Thesis Chapters by Andrew Ruffhead