Intro to Least-Squares Collocation

Introduction to multiple regression equations in datum transformations and their reversibility A. C. Ruffhead Independent researcher, formerly of Defence Geographic Centre, Feltham, UK.

This article was originally published online by Survey Review on 31 October 2016. It is due to appear in the printed journal in 2018. http://dx.doi.org/10.1080/00396265.2016.1244143 DOI: 10.1080/00396265.2016.1244143 This is the accepted manuscript, posted on academic social networks on 31 October 2017, 12 months after Taylor & Francis had published the paper online. ABSTRACT 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. Keywords: multiple regression equations, datum transformations, geodetic datums, reverse transformations

Introduction Multiple Regression Equations (MREs) are one of the most accurate methods for transforming two-dimensional geographical coordinates between geodetic datums. They can be used to model shifts  and  in latitude and longitude respectively from one geodetic datum to another within a defined region, provided there is a sufficiently dense distribution of points that are known in both datums from which the MRE parameters can be derived. The main advantage of MREs over similarity transformations lies in modelling of distortion for better fit in geodetic applications. Non-linear distortions are likely to occur for example when positioning in terms of the older datum is based on inferior surveying methods to that of the newer datum. Multiple regression equations for  and  take the general form (1)  (" )  ai , jU iV j

 i, j

 (" )   bi , jU iV j

(2)

i, j

Both summations are finite and are nth-order polynomials in the sense that n is an upper bound on i+j. The intermediate coordinates U and V are linear functions of  and  in the datum being transformed. The shifts are added to the coordinates being transformed to give approximate values of the coordinates in the other datum. If ellipsoidal heights are known in both datums, a multiple regression equation can be used to model h as well. MREs can also be used to model shifts in Cartesian coordinates (X, Y, Z) if the shift pattern is too non-conformal to be modelled by a 7-parameter similarity transformation. MREs can also be used to model shifts in projected coordinates (x, y). In addition, aside from transformations, MREs can be used to model geoid heights N (DMA, 1987b). In all these cases, the shifts are computed in linear units. In terms of appearance, multiple regression equations are simply polynomial functions of horizontal position (as given by U and V). As a result, some authors (for example, Kutoglu, 2009) prefer to describe them simply as polynomials. The phrase ‘multiple regression’ merely describes the process by which the polynomial functions are i

j

obtained. The selection of variables U V used in the MREs is based on the statistical significance of their contribution to the known datum shifts at control points. The method used is normally the ‘stepwise’ multiple regression procedure. This is described in section 7.2.4.3.3 of DMA (1987a) and by Appelbaum (1982). The full range of multiple regression procedures can be found in Draper and Smith (1966). The nature of polynomials makes it imperative that MREs are only applied within the area of intended use. Large distortions can be realised in very short distances outside of the area of the control points. The outermost control 1

points therefore need to go up to, or even slightly beyond, the area boundary. It is also important that the control points from which the polynomials are derived provide an even coverage of the area for which the MREs are to be applied.

Intermediate coordinates (U and V) For computational convenience, U and V should be within the range -1 to 1, or at least close to that range, to prevent the higher-power terms in U and V being very large in magnitude. For this reason U and V are usually made much smaller in magnitude than would be the case using the values of  and  in degrees. The simplest option is to define U and V to be the values of  and  in radians. See, for example, Gledan and Azzeidani (2014). A logical variation is to select a central point in the region common to both datums; U and V are then defined as relative to that central point and expressed as ‘relative latitude’ in radians and ‘relative longitude’ in radians. See, for example, Soycan (2005). The latitude and longitude of a central point used in this way are often called offsets. A generalisation of these options is to set (3) U  K (in deg   off )

V  K (in deg   off )

(4)

In this case K is a common scaling factor and  off  latitude offset in decimal degrees

 off 

longitude offset in decimal degrees

(5) (6)

If K is the degrees-to-radians conversion factor (/180), then (3) and (4) are equivalent to using centralised coordinates in radians. This choice tends to make U and V too small, causing some coefficients in the polynomial formulae to be very large. A widespread practice, notably in NIMA (2004), is to set K to n/180 where n is a small integer. This is, however, only a convention. There is a broader generalisation that the author would like to introduce for the development of future MREs. That is to define U and V with separate scaling factors: U  K1 (in deg   off ) (7)

V  K 2 (in deg   off )

(8)

This would permit true normalisation of U and V because the scaling factors could be defined to ensure that both intermediate coordinates span the range -1 to 1 in the region of interest. That would provide a computational justification for deriving the coefficients in the polynomial formulae to the same number of decimal places, since all i

j

the terms U V would:  span the range 0 to 1 when i and j are both even;  span the range -1 to 1 in all other cases. It will be shown that another possible use of (7) and (8) is to optimise the reverse transformation arising from an MRE datum transformation.

Alternatives to MREs for distortion modelling Multiple regression equations are not the only means of modelling datum transformations which have distortions that cannot be adequately represented by a first-order conformal model. One alternative is least-squares collocation where shifts are computed from an underlying conformal model and a ‘signal’ derived from deviations at control points (and depending on proximity to those points). It has been used for several datum transformations around the world. See, for example, González-Matesanz et al (2003), You and Hwang (2006), Yun et al (2006) and Kutoglu (2009); the latter source calls the process ‘least square filtering’. One inconvenience is the need to solve a large system of equations whenever shifts are to be computed at new points (although the size of the problem can be reduced by stepwise collocation). A variation on least-squares collocation is the ‘grid-file’ technique implemented by Collier (2002). It is used between Australian datums by Kinneen and Featherstone (2004). A large file of shifts at regularly-spaced points is computed by a one-off application of least-squares collocation. Bi-linear interpolation is used to obtain further shifts in the appropriate grid cells. Some authors mistakenly use the term ‘least-squares collocation’ when they actually mean interpolation from a grid file constructed by least-squares collocation. Merry and Whittal (1998) used ‘Kriging’ to approximate  and  at grid points. The transformation was from Cape Datum to WGS 84 for a test area in South Africa. Kriging, an interpolation technique named after Danie G. Krige, is faster than least-squares collocation but gives very similar results. Interpolation from the grid was done by cubic splines. Another method, described by Featherstone (1997), used conformal models over sub-networks and interpolation in between, so that the conformal-model parameters vary with position. Briggs (1974) developed the method of ‘minimum curvature surfaces’ for geophysical data that can be represented by contour maps. Dewhurst (1990) applied this to a transformation between North American Datums to develop gridded datasets for the shifts, solving a system of difference equations. Computing  and  for an 2

individual point is achieved with bi-linear interpolation based on the 4 surrounding grid points. Dewhurst’s computer program NADCON is now available online for a number of datum transformations; see NGS (2013). NTv2 (National Transformation Version 2) is similar to NADCON although originally designed for Canada. A key difference is that NTv2 allows for sub-grids with a higher density than the main grid. See Junkins and Farley (1995). Greaves (2004) recommends a different kind of grid look-up process. The transformation is for projected grid coordinates from European Terrestrial Reference System 1989 to Ordnance Survey of Great Britain 1936. The method used is a modification of a triangulation with a 6-parameter affine transformation assigned to each triangle. A regular grid smooths out the discontinuities along the edges of the triangles and enables straightforward bilinear interpolation. Another method noted by González-Matesanz et al (2003) was ‘rubber sheeting’. The version highlighted was a Delaunay triangulation of the data points, with some virtual points on the outside. Computing  and  for an individual point is achieved by a linear transformation inside the enclosing triangle. The grid-based alternatives can model distortions more accurately than MREs if the grid is sufficiently dense, but they involve far more data storage. The triangulation-based alternatives can also model distortions more accurately than MREs, but their application is less straightforward because of the need to identify the triangle containing any given point. Multiple regression equations have the merit of being easy to apply.

Appelbaum’s original MREs The use of MREs for datum transformations was first proposed by Appelbaum (1982). The datums considered by Appelbaum in 1982 were European Datum 1950 (ED 50) and World Geodetic System 1972 (WGS 72). The equations covered a ‘limited ED 50 area’ consisting of Denmark, West Germany, Netherlands, Belgium, Luxemburg and France. Appelbaum described his intermediate coordinates as normalised latitude and normalised longitude, which is technically inaccurate. They were defined as follows. (9) U  ( / 60)in deg  2.61

V  ( / 60)in deg  0.24 The first equation was derived from U  3( in rad0.87)

(10) (11)

on the basis that ‘0.87 is the approximate average latitude in radians over the regression area, and 3 is a convenient factor which inhibits large values of equation coefficients’. The equivalent formulae in terms of equations (3) and (4) would be as follows, with K set to /60. U  0.05235988(in deg  49.84732818) (12)

V  0.05235988(in deg  4.58366236)

(13)

Using 53 ‘observation points’, Appelbaum derived expressions for datum shifts from ED 50 to WGS 72. Those for geodetic latitude (arc-seconds), geodetic longitude (arc-seconds) and geodetic height (metres) were as follows.  ()  3.17250  1.96761 U  0.747893 V (14)  0.252615 V 2  4.68674 U 2V 2  ()  5.03830  1.40710 U  1.60471 V

 0.521318 U 2  0.263364 V 2 h(m)  47.1915  35.1158 U  18.2122 V

(15)

(16)  15.8592 U 2  264.165 U 5 Only 33 of the ‘observation points” were Doppler stations. The other 20 points were located to provide more complete area coverage and the reference coordinate differences were obtained by interpolation (with some use made of Doppler stations external to the area).

NIMA’s MREs for transformations to WGS 84 The US Defense Mapping Agency (DMA) derived a series of MREs for transforming a number of local datums to the World Geodetic System 1984 (WGS 84). It should be noted that in 1996 DMA was merged with other bodies into what is now the National Geospatial-Intelligence Agency (NGA). Until 2003 NGA was called the National Imagery and Mapping Agency (NIMA). DMA published a selection of MREs in Appendix D of its Technical Report 8350.2, the third edition of which is NIMA (2004). A more complete set of MREs covering 54 datum transformations around the world can be found in sections 19 and 20 of DMA (1987b). The ‘Appendix D’ selection of MREs to WGS 84 is for ‘seven major continental datums, covering continentalsize land areas with large distortion’. The datums are listed in Table 1, together with the page references of the equations themselves. One of the datums, North American Datum 1927, has been split between USA and Canada for the purpose of datum transformation. For the purpose of this paper, these datums will be designated as NIMA’s ‘Main Eight’.

3

Table 1 Special continental-size local geodetic datums transformed to WGS 84 by MREs Local Geodetic Datum Area of coverage Australian Geodetic 1966 Australian Mainland (excluding Tasmania) Australian Geodetic 1984 Australian Mainland (excluding Tasmania) Campo Inchauspe Argentina (Continental land areas only) Brazil (Continental land areas only) reduced to the area covered Corrego Alegre by the contour charts in Figures 16.28 and 16.29 of DMA (1987b) Western Europe (Austria, Denmark, France, West Germany, The European 1950 Netherlands and Switzerland) Canadian Mainland (Continental contiguous land areas only)  North American 1927 USA Mainland, Alaska & Islands excepted (CONUS)  South American 1969 South America (Continental contiguous land areas only) † in Appendix D of NIMA (2004),

Location of MREs† Page D-5 Page D-6 Page D-7 Page D-8 Page D-9 Page D-10 Page D-11 Page D-12

The areas of coverage are as stated in Annex D of NIMA (2004) except that the area for Corrego Algre has been reduced to the part of Brazil covered by the contour charts in Figures 16.28 and 16.29 of DMA (1987b). It is safe to assume that datum shifts from the local datum to WGS 84 are only available in that sub-region, which is the shaded area in Figure 1. The reason for precise identification of this particular area of coverage will became clear later in this paper.

1 Sub-region of Brazil where there are sufficient known datum shifts from Corrego Alegre to WGS 84 for them to be modelled by MREs

The MREs given in NIMA (2004) are quoted as having a quality of fit of 2.0m. They only cover the shifts  and . The h correction is deemed not applicable because geodetic heights ‘are not available for local geodetic datums’. The MREs in DMA (1987b), however, give formulae for ,  and h although the latter is denoted H. The MREs produced by DMA are 18th-order polynomials. They take the form (1) and (2) with i and j ranging from 0 to 9. The number of terms is generally fewer than 50 but usually more than the 5 used by Appelbaum. Coefficients have been computed to 5 decimal places, rather than the 6 significant figures favoured by Appelbaum. See, for example, equations (19) and (20). The control points from which DMA’s MREs were derived were not solely Doppler stations. Auxiliary points were selected to fill areas of sparse Doppler-station coverage and to provide coverage slightly external to the datum boundary. The process is described in Section 7.2.4.3.3 of DMA (1987a). Latitude and longitude offsets for DMA MREs are roughly central relative to the area of application, but rounded to an exact number of degrees. The DMA MREs use a common scaling factor K for the intermediate coordinates U and V defined by (3) and (4). For the largest areas, K is set to 0.05235988 which is 3/180. Larger integer multiples of /180, such as 0.15707963 (=9/180), were used for smaller areas. This approach to scaling cannot be regarded as normalisation in the true sense of the word. The MREs for transforming North American Datum 1927 (NAD 27) to WGS 84 over Canada cover a longitude range of 83º. As a result, V varies between -2.09 and 2.25, so that its 9th power can be as high as 1487. (To compensate, the coefficient b0,9 in the longitude-shift expansion would need at least 8 decimal places rather than the standard 5.) As an example of the MREs provided by NIMA (2004), the equations for transforming European Datum 1950 to WGS 84 are given in (19) and (20) below. The intermediate coordinates are given in (17) and (18). The area of applicability is Western Europe (continental contiguous land areas). The countries covered are Austria, Denmark, France, West Germany (prior to October 1990), The Netherlands and Sweden.

U  0.05235988 (in deg  52)

(17)

V  0.05235988 (in deg  10)

(18)

4

 ()  2.65261  2.06392U  0.77921V  0.26743U 2  0.10706UV  0.76407U 3  0.95430U 2V  0.17197U 4  1.04974U 4V  0.22899U V  0.05401V  0.78909U 5

2

8

(19)

9

 0.10572U 2V 7  0.05283UV 9  0.02445U 3V 9  ()  4.13447  1.50572U  1.94075V  1.37600U 2  1.98425UV  0.30068V 2  2.31939U 3  1.70401U 4  5.48711UV 3  7.41956U 5  1.61351U 2V 3  5.92923UV 4  1.97974V 5  1.57701U 6  6.52522U 3V 3

(20)

 16.85976U 2V 4  1.79701UV 5  3.08344U 7  14.32516U 6V  4.49096U 4V 4  9.98750U 8V  7.80215U 7V 2  2.26917U 2V 7  0.16438V 9  17.45428U 4V 6  8.25844U 9V 2  5.28734U 8V 3  8.87141U 5V 7  3.48015U 9V 4  0.71041U 4V 9

As this example demonstrates, the polynomial functions differ between the two expansions, although there is some overlap most notably in the low-order terms. The test points for the MRE datum transformations in NIMA (2004) are listed in Table 2, together with the coordinate shifts. They are taken from Sections 19 and 20 of DMA (1987b) rather than from NIMA (2004), because of the greater precision. Table 2 Test points for MRE transformations between WGS 84 and the ‘Main Eight’ local geodetic datums Local Geodetic Datum Local geodetic coordinates WGS 84 coordinates Coordinate shifts Australian Geodetic 1966 -17º0032.776, 144º1137.245 -17º0027.256, 144º1141.167 5.480, 3.922 Australian Geodetic 1984 -20º3800.673, 144º2429.288 -20º3755.170, 144º2433.397 5.503†, 4.109 Campo Inchauspe -29º4745.682, -058º0738.197 -29º4743.729, -058º0740.160 1.953, -1.963 Corrego Alegre -20º2901.019, -054º4713.171 -20º2902.049, -054º4715.272 -1.030, -2.101 European 1950 46º4142.893, 013º5454.088 46º4139.812, 013º5450.602 -3.081, -3.486 North American 1927 (Can) 54º2608.667, -110º1702.410 54º2609.956, -110º1705.566 0.289, -3.156 North American 1927 (US) 34º4708.833, -086º3452.175 34º4709.189, -086º3452.095 0.356, 0.080 South American 1969 -31º5633.954, -065º0618.658 -31º5635.310, -065º0620.816 -1.356, -2.158 † corrected from 5.505 in Table 19.8 of DMA (1987b)

Other examples of MRE transformations In general, MREs have been derived from the older of two geodetic datums to the more recent one. One exception is WGS 84 to Old Egyptian Datum, described in Dawod and Alnaggar (2000). For that transformation, MREs are given for  and  with 7 and 8 terms respectively. Another exception is International Terrestrial Reference System 1994 to ED 50 in Turkey, described in Soycan (2005). The equations are for shifts in Cartesian coordinates (X, Y, Z) and in projected coordinates (x, y) rather than  and . Only second-order polynomials were considered suitable because there was insufficient distortion to justify third-order. Mitsakaki et al (2006) have developed similar second-order polynomials to model (X, Y, Z) and (x, y) for Hellenic Geodetic Reference System 1987 to International Terrestrial Reference System 2000. The area considered was the Gulf of Corinth. Ayer et al (2010) documented third-order MREs for  and  between Ghana War Office Datum and WGS 84. These are among the choice of methods in a computer program ‘GhaTrans’ designed to convert projection coordinates between the Ghana National Grid and the Universal Transverse Mercator. The afore-mentioned second-order and third-order polynomials use all the possible coefficients associated with those functions (6 and 10 respectively). This suggests that the process for deriving the coefficients might not include testing for statistical significance and discarding terms where appropriate. That would be a departure from the regression process advocated by Appelbaum (1982) and DMA (1987a). It is, however, possible that all terms are statistically significant because they are low-order terms capturing moderate distortion. Gledan and Azzeidani (2014) recommends MREs for transforming Europe Libyan Datum 79 to Libyan Geodetic Network 2006. The formulae for  and  are 13th-order polynomials with 6 and 3 terms respectively. The latitude-shift formula is unusual for having terms in 13 and 13 (U and V being  and ). The general section of this source (II part B) should be treated with caution as it contains several copying errors; also, the quoted general form of the MRE does not include the high-order terms used in the transformation. 5

González-Matesanz et al (2003) considers the use of polynomials to model transformations from ED 50 for the Spanish Geodetic Network to European Terrestrial Reference System 1989. They are presented as a means of modelling transformations more accurately than the 7-parameter model rather than a means of fitting distortions, except for the logical expectation ‘that a polynomial is more efficient at absorbing the network behaviour’. The link to MREs is made indirectly through reference to Appelbaum (1982). However, the regression process used to accept/reject terms was ‘progressive elimination’ rather than the more usual stepwise elimination. Both methods are described in Draper and Smith (1966).

Reversing MRE transformations MRE-based transformations are always in terms of coordinates being transformed from Datum 1 to Datum 2. The formulae for applying the shifts  and  for this ‘forward transformation’ are therefore as follows. (21) ( 2)  (1)  

( 2)  (1)  

(22)

There are occasions when it is necessary to apply the reverse transformation. Chapter 8 of DMA (1987a) acknowledges that ‘since many local geodetic systems are currently still in use, an approach is needed for converting GPS-derived WGS 84 coordinates to local geodetic system coordinates’. It is certainly indisputable from (21) and (22) that (23) (1)  ( 2)  

(1)  ( 2)  

(24)

However, neither of these formulae is explicit, since  and  are functions of U and V, which are linear functions of  and  on Datum 1. Appelbaum (1982) fully realised this when he computed U and V from  and  on WGS 72 for the reverse transformation to ED 50. His analysis concluded that none of the differences in the coordinate shifts at his 53 control points between the two directions was greater than 0.02 metres. It must, of course, be noted that a reverse transformation can only be as accurate as the forward transformation. The reverse solution cannot give a better quality of fit to the data from which the original MREs were derived. The accuracy considered in this paper is consistency with the forward transformation. It has been measured by the following process. Consistency measurement algorithm: (i) Select all points at one-degree intersections within the area of coverage. (ii) Transform the points from Datum 1 to Datum 2 using the MREs designed for that process. (iii) Transform the results back by the reverse transformation method under consideration. (iv) Compute the misclosures by comparing the transformed coordinates on Datum 1 with the original coordinates. (These are errors in  and  in arc-seconds.) (v) Convert the errors in  and  to linear units, and from them compute the magnitude of each horizontaldistance error d. (vi) Compute R, the RMS error of the reverse shifts as a horizontal distance, from the root-mean-square of the quantities d.

Reverse shifts from original MREs (Method 1) The solution offered by Appelbaum (1982) is to calculate U and V by the same formula and apply the shift with a change of sign, advice which is echoed in chapter 8 of DMA (1987a). The flaw in this approach is that U and V are supposed to be derived from  and  on Datum 1. So, strictly speaking, the solution is needed to solve the problem. Method 1 (Appelbaum’s method) applies (3) and (4) to the coordinates on Datum 2, so that in deg and in deg are the values-in-degrees of (2) and (2) respectively. The resulting values of U and V are substituted into (1) and (2). The resulting values of  and  are substituted into (23) and (24), which is equivalent to applying the shifts with a change of sign. The RMS horizontal-distance error varies from 1 to 10 millimetres for the reverse transformations from WGS 84 to the datums in Table 1. Full details are given in Table 3. Table 3 Accuracy of reverse MRE transformations from WGS 84 to the ‘Main Eight’ local geodetic datums by method 1 Local Geodetic Datum Intermediate coordinates Points RMS errors in distance (with ,  components) Australian Geodetic 1966 752 3.6 mm (2.1 mm, 2.9 mm) U=0.05235988(+27), V=0.05235988(-134) Australian Geodetic 1984 752 3.8 mm (2.7 mm, 2.6 mm) U=0.05235988(+27), V=0.05235988(-134) Campo Inchauspe 325 10.2 mm (9.4 mm, 3.5 mm) U=0.15707963(+35), V=0.15707963(+64) Corrego Alegre 521 1.1 mm (0.4 mm, 1.0 mm) U=0.05235988(+15), V=0.05235988(+50) European 1950 151 4.3 mm (4.2 mm, 1.1 mm) U=0.05235988(-52), V=0.05235988(-10) North American 1927 (Can) 1311 4.3 mm (3.4 mm, 2.6 mm) U=0.05235988(-60), V=0.05235988(+100) North American 1927 (US) 912 1.6 mm (0.5 mm, 1.5 mm) U=0.05235988(-37), V=0.05235988(+95) South American 1969 1623 2.5 mm (2.0 mm, 1.5 mm) U=0.05235988(+20), V=0.05235988(+60)

The errors in the reverse transformation process are insignificant compared with the accuracy of the MREs as a model of physical data. Nevertheless, it would be reassuring for users of MREs for forward and reverse transformation models to be mutually consistent to the extent that any discrepancies are at sub-millimetre level. 6

Reverse shifts from original MREs with revised offsets (Method 2) Method 2 is the same as Method 1 except that the latitude and longitude offsets are based on the coordinates of the central point on Datum 2. It represents a partial attempt to correct the errors in U and V, in this case by re-centring the intermediate coordinates. One consequence is to ensure that the reverse transformation of the transformed central point is exactly the original central point. (25) U  K (in deg   off  a 0,0 / 3600)

V  K (in deg   off  b0,0 / 3600)

(26)

The resulting values of U and V are substituted into equations (1) and (2).  and  are then substituted into (23) and (24), which is equivalent to applying the shifts with a change of sign. In the case of WGS 84 to ED 50, the constants used in equations (25) and (26) are extracted from equations (17) to (20). The resulting intermediate-coordinate formulae are (27) U  0.05235988 (in deg  51.99926316)

V  0.05235988 (in deg  9.99885154)

(28)

The RMS horizontal-distance error is reduced in all cases, relative to Method 1. The average reduction is 56%, varying from 96% (Australian Geodetic 1966) to 17% (North American Datum over Canada). In general, the larger the central-point shifts (that is, the values of a 0, 0 and b0, 0 ) the greater the improvement in accuracy of Method 2 over Method 1. Full details are given in Table 4. Table 4 Accuracy of reverse MRE transformations from WGS 84 to the ‘Main Eight’ local geodetic datums by method 2 Local Geodetic Datum Intermediate coordinates Points RMS errors in distance (with ,  components) U=0.05235988(+26.9985567), Australian Geodetic 1966 752 0.1 mm (0.1 mm, 0.1 mm) V=0.05235988(-134.00130347) U=0.05235988(+26.99855388), Australian Geodetic 1984 752 0.2 mm (0.2 mm, 0.1 mm) V=0.05235988(-134.00129966) U=0.15707963(+34.99953481), Campo Inchauspe 325 4.8 mm (4.8 mm, 0.6 mm) V=0.15707963(+64.00081421) U=0.05235988(+14.99976579), Corrego Alegre 521 0.9 mm (0.2 mm, 0.9 mm) V=0.05235988(+49.99959430) U=0.05235988(-51.99926316), European 1950 151 0.7 mm (0.6 mm, 0.3 mm) V=0.05235988(-9.99885154) U=0.05235988(-60.00022054), North American 1927 (Can) 1311 3.6 mm (2.9 mm, 2.0 mm) V=0.05235988(+100.00037805) U=0.05235988(-37.00004718), North American 1927 (US) 912 1.2 mm (0.4 mm, 1.1 mm) V=0.05235988(+95.00024566) U=0.05235988(+19.99953471), South American 1969 1623 1.0 mm (0.9 mm, 0.4 mm) V=0.05235988(+59.99950565)

Reverse shifts from original MREs with 4 revised constants (Method 3) Method 3 is the same as Method 2 except that the scaling in each intermediate-coordinate formula is adjusted to allow for the difference between datums. Again, it represents an attempt to make U and V closer to what they should be. (29) U  ( K /  ) (in deg  off a 0,0 / 3600)

V  ( K /  ) (in deg  off b0,0 / 3600)

(30)

where  and  are optimised corrections to scale in latitude and longitude respectively. The resulting values of U and V are substituted into equations (1) and (2).  and  are then substituted into (23) and (24), which is equivalent to applying the shifts with a change of sign. Method 3 was used to obtain the intermediate-coordinate formulae for the reverse transformations from WGS 84 to the datums in Table 1. Those formulae are given in Table 5. The RMS horizontal-distance error relative to Method 1 is reduced in all cases. The average reduction is 80%, varying from 96% (Australian Geodetic 1966) to 38% (Corrego Alegre). The accuracy of Method 3 is summarised in Table 5.

7

Table 5 Accuracy of reverse MRE transformations from WGS 84 to the ‘Main Eight’ local geodetic datums by method 3 Local Geodetic Datum Intermediate coordinates Points RMS errors in distance (with ,  components) U=0.05236014(+26.9985567), Australian Geodetic 1966 752 0.1 mm (0.1 mm, 0.1 mm) V=0.05235996(-134.00130347) U=0.05236017(+26.99855388), Australian Geodetic 1984 752 0.2 mm (0.1 mm, 0.1 mm) V=0.05235960(-134.00129966) U=0.15707464(+34.99953481), Campo Inchauspe 325 1.4 mm (0.9 mm, 0.6 mm) V=0.15707467(+64.00081421) U=0.05235949(+14.99976579), Corrego Alegre 521 0.7 mm (0.1 mm, 0.7 mm) V=0.05235885(+49.99959430) U=0.05235801(-51.99926316), European 1950 151 0.3 mm (0.2 mm, 0.2 mm) V=0.05235897(-9.99885154) U=0.05235999(-60.00022054), North American 1927 (Can) 1311 1.5 mm (0.9 mm, 1.1 mm) V=0.05235741(+100.00037805) U=0.05236052(-37.00004718), North American 1927 (US) 912 0.2 mm (0.1 mm, 0.2 mm) V=0.05235828(+95.00024566) U=0.05235985(+19.99953471), South American 1969 1623 0.6 mm (0.5 mm, 0.4 mm) V=0.05235870(+59.99950565)

The rest of this section is intended for readers seeking computational details on how  and  were optimised for each reverse transformation. A subroutine was written to compute the RMS error of the reverse shifts as a distance, taken over the one-degree intersections of the area of coverage, using Method 3 with any given values for  and . This can be regarded as a function R(,), which is non-linear but calculable. The next stage was to compute initial estimates of  and . A simple method is to make use of the forward latitude shifts at the extremities of the line   off (where V=0) and the forward longitude shifts at the extremities of the line

  off (where

estimates are

U=0). The extremities should be based on the area of coverage. The formulae for those

 N ,( 2 )   S , ( 2 )  N ,(1)  S ,(1)     E , ( 2 ) W ,( 2 ) E ,(1)  W,(1)



(31)

(32)

The resulting value of R(,) is likely to be lower than R(1,1), which is the same as the RMS distance error from Method 2. If it is higher (as in the case of Australian Geodetic 1966), the starting values of  and  should be set to 1. The final stage was to minimise R with respect to  and . This was done numerically using search directions, starting from the initial estimates of  and , until convergence was achieved. For WGS 84 to ED 50, equations (31) and (32) were applied as follows. The points (46N, 010E) and (58N, 010E) on ED 50 were used to estimate , while the points (52N, 004E) and (52N, 010E) on ED 50 were used to estimate . The values obtained were 1.000031114 for  and 1.000026573 for . The resulting provisional intermediate-coordinate formulae were (33) U  0.05235825 (in deg  51.99926316)

V  0.05235849(in deg  9.99885154)

(34)

The resulting value of R was 0.297 mm. Keeping  fixed at 1.000031114, the first search reduced R to 0.270 mm when  was 1.000017571. Keeping  fixed at that value, the second search reduced R to 0.260 mm when  was 1.000035716. Keeping  fixed at 1.000035716, R was minimised when  was 1.000017380, but only reduced by 0.00001 mm. With these values of  and , the operative intermediate-coordinate formulae for WGS 84 to ED 50 by Method 3 are (35) U  0.05235801(in deg  51.99926316)

V  0.05235897(in deg  9.99885154)

(36)

Reverse shifts by predictor-corrector application of the MREs (Method 4) Given that (23) and (24) are implicit equations, the likeliest route to a numerically exact solution is iteration. This seems an obvious approach, although the author has not seen it in any publication. Method 1 is used to obtain an approximate solution ((1) , (1) ) . Using these values for  and , U and V are re-computed from (3) and (4). The datum shifts are re-computed from the MREs and subtracted from the original Datum 2 coordinates to give the Datum 1 coordinates. This approach produced such rapid convergence that further iteration was unnecessary. The RMS horizontaldistance error was below 0.01 millimetres in all cases. Borrowing terminology from the numerical solution of 8

ordinary differential solutions, the two-step process may be classified as a ‘predictor-corrector’ method. The accuracy of Method 4 is summarised in Table 6. Table 6 Accuracy of reverse MRE transformations from WGS 84 to the ‘Main Eight’ local geodetic datums by method 4 Local Geodetic Datum Intermediate coordinates† Points RMS errors in distance Australian Geodetic 1966 752 0.00004 mm U=0.05235988(+27), V=0.05235988(-134) Australian Geodetic 1984 752 0.00015 mm U=0.05235988(+27), V=0.05235988(-134) Campo Inchauspe 325 0.00368 mm U=0.15707963(+35), V=0.15707963(+64) Corrego Alegre 521 0.00005 mm U=0.05235988(+15), V=0.05235988(+50) European 1950 151 0.00014 mm U=0.05235988(-52), V=0.05235988(-10) North American 1927 (Can) 1311 0.00037 mm U=0.05235988(-60), V=0.05235988(+100) North American 1927 (US) 912 0.00005 mm U=0.05235988(-37), V=0.05235988(+95) South American 1969 1623 0.00055 mm U=0.05235988(+20), V=0.05235988(+60) † applied twice: the second time using the approximations to  and  obtained from the prediction run.

Reverse transformations for new MREs The four methods just described for reverse transformations can be applied to any datum transformation with MREs based on equations (1) to (4). They can also be adapted for MREs where (3) and (4) are replaced by truly normalised equations for the reasons discussed earlier. (37) U  K1 (in deg   off )

V  K 2 (in deg   off )

(38)

K1  1 / max in deg   off

(39)

K 2  1 / max in deg   off

(40)

where

K1 in the equation for U and by K 2 in the equation for V. That means for Method 3, the revised scaling constants are K1 /  and K 2 /  . For each of Methods 1 to 4, K is simply replaced by

Practical implementation One obvious characteristic of each reverse transformation is that is only valid for the area that the original MREs are designed for and will be no more accurate than the forward transformation. What is slightly less obvious is the need to use precisely that area of coverage when measuring the accuracy of the reverse transformation relative to the original model. This became apparent when assessing methods 1 to 4 when they were used for transforming WGS 84 back to Corrego Alegre. Applying the consistency measurement algorithm over Brazil as a whole, the misclosures were very large in the north-west, even for Method 4. It has already been noted that the area of coverage for the transformation from Corrego Alegre to WGS 84 is that part of Brazil which is shaded in Figure 1. The MREs would give   600 at the extremities of north-west Brazil compared with a maximum of 1.55 within the shaded region. The problem of large misclosures disappeared when the shaded area was adopted as the area of coverage. It follows that if Method 3 is adopted to find a reverse transformation, the optimisation process to obtain the revised scaling constants should be based on the precise area of coverage. One characteristic common to all of Methods 1 to 4 is that the same MREs are used for the reverse process as for the forward transformation. This means that the reverse transformation cannot be used to verify that the MREs were correctly coded for the forward transformation. To validate the reverse transformations from WGS 84 to local geodetic datums, the test points in Table 2 are recommended. The shifts are, of course, subtracted from the WGS 84 coefficients.

Conclusions Multiple regression equations designed for a geodetic datum transformation do not give exact results when used to compute the reverse transformation, as described in Method 1 which is the process recommended in NIMA (2004). This is because the reverse shifts are functions of the solution. The reverse-transformation accuracy can, however, be improved with little or no extra computation. Revising the offsets in the intermediate coordinates (Method 2) improves the accuracy, on average, by a factor of 2.26. The constant terms on the MREs are merely converted to degrees and added to the offsets. This has already been done for the reverse transformations corresponding to the ‘Main Eight’ NIMA models. The formulae are given in Table 4. Revising all four constants in the intermediate coordinates (Method 3) improves the accuracy, on average, by a factor of 5. The new constants are the offsets as revised in Method 2 and optimised scaling constants. There is no extra computation other than the one-off revision of the constants in the intermediate coordinates. This has already been done for the reverse transformations corresponding to the ‘Main Eight’ NIMA models. The formulae are given in Table 5. Alternatively, the errors can be eliminated altogether for practical purposes by applying the MREs in a predictorcorrector sequence (Method 4), with the original formulae for U and V. This entails twice the computation of the forward-transformation process, but little extra coding. 9

Using a single scale factor on latitude and longitude should be viewed as a tradition rather than a requirement of datum transformation by MREs. Designers of new MREs should feel at liberty to choose separate scale factors that truly normalise the relative coordinates to the range 1. That allows all coefficients to be quoted to the same number of decimal places, and that number would be based the overall precision requirement. The methods for deriving reverse transformations are equally applicable when intermediate coordinates are truly normalised.

Acknowledgements The author wishes to thank the two anonymous reviewers for their valuable comments which helped to improve this paper.

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