Point Accuracy

Points are stored in a Geoida job database as either grid coordinates or as geographic (latitude & longitude) coordinates, depending on job configuration in Job Configuration. When data is imported and converted from another coordinate type into that for which the job is configured, or where conversion from one form to another is made (eg, Inverse or Transformation options or during survey data reduction), some accuracy of coordinate values may be lost in the process due to previous rounding of the original values or precision limitations in the mathematical functions and algorithms used. Converting from one geodetic datum to another has inherent limitations.

Points computed as grid coordinates from the reduction of field observations are subject to the arc-to-chord correction and line-scale factor shortcomings as noted separately below. Latitude and longitude coordinates are computed rigorously using the Vincenty direct non-simplified formulae (References, T. Vincenty, Survey Review, Vol XXIII No 176, April 1975) “which should give complete accuracy over lines of any length, from a few centimetres to nearly 20 000 km”. All computations are made using maximum precision.

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Azimuth, distance, grid bearing and grid convergence

These values (Inverse option or where otherwise derived from point coordinates) are derived primarily from the Vincenty Inverse formulae where the results are of the same order as for the Vincenty Direct formulae, i.e. completely accurate up to 20 000 km, except that the formulae become indeterminate for near-antipodal points (180 degrees difference in longitude). The formulae also become unreliable at distances somewhat less than 1 metre so in these cases inverse values are computed using the Robbins rigorous formulae. Normal-to-geodesic correction is not applied since the correction only becomes significant for azimuth at distances greater than about 1000 km and for distance after about 2500 km (See References, The Australian Geodetic Datum - Technical Manual, Section 3.9 and T. Vincenty, Survey Review, Vol XXIII No 176, April 1975).

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Arc-to-chord correction and line-scale factor

These are computed from 'grid-formulae' which are not rigorous and are stated as being accurate to 0.02 second and 0.1 ppm over any 100 km line in a standard Transverse Mercator zone. (Refer to refer References, The Australian Geodetic Datum - Technical Manual, Sections 5.3.1 and 5.6.3 and compare with 4.1.2).

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Chord-to-arc correction

Chord-to-arc correction is not applied to distance reduction - this correction is used in the most precise geodetic surveys only (Refer References, The Australian Geodetic Datum - Technical Manual, Section 2.3.5).

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Grid-to-Geographic conversion

Redfearn's formulae (Empire Survey Review No 69, 1948) are use to convert between grid projection and geographic coordinates; the stated accuracy is “better than 1mm in any zone of the Map Grid of Australian”. Meridian distance is computed by the series method in which for the limited formula “is correct to less than 0.5mm in latitude 45 degrees”. (See References, The Australian Geodetic Datum - Technical Manual, Section 2.3.5 and The Australian Geodetic Datum - Technical Manual, 'Conversion between Ellipsoidal and Grid Coordinates').

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Geoid-spheroid separation

Computations performed on the spheroid using distance and height measurements which are based on geoidal (orthometric) heights, require the application of the geoid-spheroid separation or undulation (N-value). If a spheroidal height datum is used, the N-value can therefore be disregarded. Values for the geoid-spheroid separation will usually be stated on permanent mark summary sheets obtainable from government geodetic authorities, may also be obtained from government authorities as digital data sets for specific locations (eg, in Australia, AUSGEOID98), or may be scaled from geoid-spheroid separation contour charts (eg, See References, The Australian Geodetic Datum - Technical Manual, Annex E).

The estimated absolute accuracy of the AUSGeoid93 values is 0.510 metres or 2-3 ppm relative; for AUSGeoid98 the accuracy is stated as 0.364 metres. Although based on the global EGM96 geoid model, corrections applied specifically for the Australian region suggest that N-values derived from the AUSGeoid98 model are superior to those that may be derived by GPS means. (Geoscience Australia -

Care should be exercised to ensure that the values for the correct spheroid are used. In Australia, summary sheets may show values for more than one spheroid during the period of transition between the superseded Australian Geodetic Datum (based on the ANS spheroid) and the newer Geocentric Datum of Australia (based on the GRS80 spheroid). The AUSGEOID98 data set provides separation and deflection data for the GRS80 spheroid - this data MUST NOT be used if computing on any other spheroid.

Testing indicates that values for geoid-spheroid separation computed by Geoida by interpolation inside a well-conditioned fixed-point triangle model (using separation values from official sources) are consistent with values derived by bi-cubic interpolation from a grid model and are better than those from a bi-linear interpolation due to the greater density triangle nodes. Extrapolation up to about 1km outside the triangle model is generally consistent (±1mm) with bi-cubic interpolation and is often better than bi-linear interpolation from grid data. However, extrapolation outside a triangle model is not recommended.

Refer to Geoid-Spheroid Separation and the next heading for further discussion.

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Distances and Geoid-spheroid Separation

Where point heights used are spheroidal heights, the value for the geoid-spheroid separation (N) can be omitted or disregarded, since all measured distances will be correctly reduced to the height datum (spheroid) during survey processing options, and the spheroidal distance computed between points in the Inverse option will be correctly converted back to terrain distance. However, if the vertical datum on which point heights are based is related to the geoid (i.e., a spirit-levelled reference such as mean-sea-level) and the value of N is not taken into consideration, then distances reduced to the height datum (geoid/MSL) will not be true spheroidal distances, and 'spheroidal' distances derived by the Inverse option will not be correctly converted back to terrain-height distances.

Omission of geoid-spheroid separation during computations will have variable implications depending on the location. The error will increase as the separation increases and amounts to about one part-per-million for every 6.4 metres of separation; i.e., if the separation is less then 6.4 metres then the error in omitting the correction will be less than 1 ppm. Thus, in situations where heights are based on a geoidal or spirit-levelled datum (i.e. orthometric heights) and the spheroid approximates the geoid, the N-value may be disregarded for normal survey work.

In Australia when the Australian National Spheroid (ANS) which is roughly parallel to the geoid overall is used (AGD66/84 datum), the separation varies from about 1 metre to 23 metres, and the omission of the value of N will thus result in errors up to about 4 ppm.

But if the GRS80 spheroid is used in Australia (GDA94 datum), the geoid-spheroid separation varies from -32 metres at the south-west corner of the continent to +72 metres in the north-east, and ignoring the N-value may result in much larger errors ranging from approximately -5 ppm to +12 ppm because of the increased separation and gradient between the spheroid and the geoid. (See References, Importance of Including the Geoid in Terrestrial Survey Data Reduction to the Geocentric Datum of Australia)

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Gravitational deflections

The amount and direction of gravitational deflection varies from place to place and is related directly to the shape of the geoid. Deflection from the vertical affects horizontal angles and directions, vertical angles, and astronomical azimuths, but may be ignored as insignificant for normal (i.e., non-geodetic) surveys in most areas of generally smaller deflections and non-mountainous terrain. The severity of errors due to deflection increases with distance from the equator and with increase of vertical angle; normally errors of less than 1 second should be expected for horizontal angles and directions, but may be 1 or 2 seconds in some areas and even in the vicinity of perhaps 10 seconds in extreme cases of deflection. Charts or tables of geoid data will give an indication of the size of the errors to be expected. Astronomical azimuths are currently not used in Geoida (v4.50).

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General indications of the accuracy of different transformation methods is given in the relevant sections elsewhere, but because there are many different transformations for different applications, with widely varying accuracies and for many different countries and regions of the Earth, the user is referred to the appropriate custodians of transformation definitions for their particular region and application.

The following indicates estimated accuracies for different transformation methods in Australia for transformations between AGD66, AGD84 and GDA94 datums:

While Geoida may print out transformed coordinates to several decimal places of a metre or seconds of arc, all results will be limited to and within the accuracy that may be stated or expected for the particular methods, even though additional precision may be implied by the results. This additional precision may be necessary for internal computation, but the user should be acquainted with the expected accuracy of the transformation method or the set of parameter values that may be used, and should not infer that the accuracy is better than it really is.

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