Investigation into the Accuracy and Practicality of Methods for Transforming Coordinates between Geodetic Datums

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), generalisations 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 nearest-neighbour 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 piecewise-bicubic, 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

Derivation of 9-parameter affine 3D geodetic datum transformations

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 equivalent enlargement 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

Enhancement of inverse-distance-weighting 2D interpolation using accelerated decline

Two-dimensional interpolation – or surface fitting – is an approximation tool with applications in geodetic datum transformations, terrainmodelling 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 othermethods is that data values can be incorporated into the interpolated surface. One disadvantage is the influence of distant points. A simple cut-off limit on distance would affect continuity. This study proposes a transition range of accelerated decline bymeans 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

Partitions of normalised multiple regression equations for datum transformations

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

The role played by invasive species in interactions with endangered and threatened species in the United States: a systematic review

Invasive species are considered to be a leading cause of the decline of threatened species. However, this view has been disputed because much of the evidence base is anecdotal. This systematic review, through an extensive, repeatable search using agreed selection criteria, examined the available scientific evidence on invasive species’ interactions with the 1363 endangered and threatened species protected under the United States Endangered Species Act (ESA). The review found scientific evidence available for 116 endangered or threatened species (8.5% of the ESA list). Of these, 85 species (6.2%) were reported as being negatively impacted by invasive species: 39 located on the continental US and 39 on islands, with seven marine species. The relative percentages of species impacted differed according to location: 4.3% (n = 906) on the continental US, 9.3% (n = 420) on islands. It was found that predation by invasive vertebrates on birds on islands and competition between invasive plants and endangered or threatened plants on the mainland were the main mechanisms of impact. The results of this study contrast markedly with a previous study which found that 49% of imperilled species in the United States were threatened by invasive species. Further research is essential in order to evaluate the impact of invasive species on imperilled species on the ESA list; this would help to reduce the high degree of uncertainty regarding the threat of invasive species due to the lack of empirical information

Partitions of normalised multiple regression equations for datum transformations

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

Partially-conformal variations of the Standard Molodensky datum transformation

Standard Molodensky is a recognised method of transforming coordinates between geodetic datums. Although less accurate than some other methods, it has the merit of being direct. That is to say it can be applied to geodetic coordinates, without involving Cartesian coordinates that give rise to difficulties in computing latitude. This paper considers the use of Standard Molodensky when at least one of the datums is 2D+1D in nature, meaning that that horizontal and vertical positions are obtained by different methods. This was generally the case before 3D positioning by satellites and is a widespread characteristic of local datums that are still used. The 2D+1D property weakens the argument for 3D conformality, and invites the possibility that different translation parameters might be used for horizontal and vertical shifts. The possibility of including a Z-rotation as a 7th parameter is also considered. Besides being ideal for those who favour the simplicity of Standard Molodensky, the variations introduced in this paper offer significant improvements in accuracy such as error reductions of 75%, 69% and 99% in the three selected case studies

Equivalence properties of 3D conformal transformations and their application to reverse transformations

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

Derivation of rigorously-conformal 7-parameter 3D geodetic datum transformations

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

Supplementary data for 'Investigation into the Accuracy and Practicality of Methods for Transforming Coordinates between Geodetic Datums'

Data underpinning Chapters 12 and 13 of the ThesisCoordinates of 4315 points in Great Britain which are common to reference systems ETRS89 and OSGB36Data in Chapter 12 of Thesis Sandi Berk supplied the data in columns B-L and the following technical information: Transverse Mercator projection is used for both CRSs, with the same parameters: the longitude of the central meridian is 15 degrees, the scale factor at the central meridian is 0.9999, the false easting is 500000 m and the false northing is −5000000 m. However, different reference ellipsoids are used: the Bessel 1841 ellipsoid for D48 coordinates (a = 6377397.155 m, b = 6356078.96325 m), and the GRS80 ellipsoid for D96 coordinates (a = 6378137 m, b = 6356752.31414 m). Sandi Berk (of the Surveying and Mapping Authority of the Republic of Slovenia) is contactable via ResearchGate