Sect_11.1.3

11.1.3 Transforming GPS Surveying Results: Coordinate Systems & Datums

TRANSFORMATION MODELS AND PROCEDURES

There are a number of ways of defining the relationship between one reference system and another. (In subsequent discussions the reference system coordinate sets will simply be referred to as "networks", and although implying they are distinct collections of points, they are in fact often the same physical points but for which two or more sets of coordinates are available.) The choice of the most appropriate network transformation model is influenced by such factors as:

Whether the model is to be applied to a small area, or over a large region.
Whether one (or both) networks have significant distortions.
Whether the networks are three-dimensional in nature, 2-D or even 1-D.
The accuracy required.
Whether the transformation parameters are available, or must be determined.

The most general of the transformations is the affine transformation. An affine transformation transforms straight lines to straight lines and parallel lines remain parallel. Generally the size, shape, position, and orientation of lines in a network will change. The scale factor depends on the orientation of the line but not on its position within the net. Hence the lengths of all lines in a certain direction are multiplied by the same scalar. Alternatively it is possible to define a projection transformation where the scale factor is also a function of position.

A transformation in which the scale factor is the same in all directions is called a similarity transformation, and is by far the most widely used of the transformation models. A similarity transformation preserves shape, so angles will not change, but the lengths of lines and the position of points may change. An orthogonal transformation is a similarity transformation in which the scale factor is unity. In this case the angles and distances within the network will not change, but the positions of points do change on transformation. The 3-D similarity transformation model relating coordinates of points in the X_BY_BZ_B network to coordinates in the X_AY_AZ_A network is:

(11.1-1)

where s is the scale factor and R is a 3x3 orthogonal rotation matrix (eqn (11.1-4)). Note that there are seven parameters: three rotation angles, three translation components and one scale factor. The translation terms T_x, T_y, T_z are the coordinates of the origin of the X_AY_AZ_A net in the frame of the X_BY_BZ_B net.

The seven parameter 3-D similarity transformation model.

It is presumptuous to assume that similarity transformations, rather than affine or projection transformations, correctly describe the differences between any two datums, or coordinate sets, but there's no denying their popularity. Why are similarity transformations so popular? Some of the reasons are:

A comparatively small number of parameters are involved, the minimum necessary to account for differences between 3-D reference systems: between their two origins, and the differences in the orientation and lengths of the respective coordinate axes.
It is a simple model that can be implemented easily in software (there are several computational procedures to chose from).
If the seven parameters must be determined from common stations in the two networks, it is preferable to estimate a small number of parameters as there are generally only a limited number of common points.
The model is adequate for relating two networks which are homogeneous (that is, no local distortion in scale or orientation).

There are, however, circumstances where even the seven parameter similarity transformation model is too elaborate, and a number of simpler models may be adequate:

The simplest procedure is not to use an algorithmic model at all. An example would be a contour map of the differences between the coordinate sets for the common points of two networks. Such a procedure is adequate for converting the coordinates of the non-common points from one reference system to the another, but no mathematical relationship is explicitly established between the reference systems.
Another option is to merely use a subset of the similarity transformation parameters. Because geodetic networks normally cover only small portions of the earth's surface, some transformation parameters are highly correlated. Often the rotation parameters are insignificant and/or poorly determined. Hence a transformation model based on, say, a simple three parameter block shift (no scale or rotation parameters) can be used.
If the networks are relatively small in extent, of the order of less than say 100km width and breadth, a local 2-D transformation may be defined. By converting the Cartesian or ellipsoidal coordinates to topocentric quantities, the three axes are oriented (east, north, and normal to the horizontal plane) in such a way as to solve the original 3-D transformation problem through separate horizontal and vertical procedures. For example, the horizontal transformation may involve two block shift parameters (along the east, north axes), a rotation about the normal axis, and a scale factor.

The two-dimensional transformation is described in HOFMANN-WELLENHOF et al (1998).

The similarity transformation model may be considered a suitable compromise between elaborate models such as the affine or projection transformations on the one hand, and crude limited parameter models on the other. When used wisely, and with an appreciation of its shortcomings, the similarity transformation is ideal for relating 3-D GPS networks to other GPS or terrestrial networks. In particular, using a similarity transformation on a large network may distort local scale and orientation. Therefore an important consideration is the magnitude of local distortions in scale and orientation. In this regard it should be noted that:

Similarity transformations will tend to smooth out local distortions.
It may be more appropriate to divide a region into smaller zones, each with their own set of transformation parameters.
The results of the transformation may be unreliable outside the area spanned by the common points used to derive the transformation parameters.

The issue of network distortions, and the common strategy of determining "tailored" transformation parameters for small areas, requires that a distinction be made between:

Universal or global model parameters, used, for example, to relate two homogeneous satellite datums and derived from globally distributed network data. The transformation parameters in the Table in section 1.2.3 are examples of such global models.
National parameters, intended for use on a datum-wide basis. They are often derived by the national geodetic authority and typically relate a satellite datum to the national geodetic datum, for example WGS84 and AGD84 (section 11.2.6).
Local parameters are determined for any given area, for example, by a surveyor so that he/she may relate his/her survey work to the surrounding network. This is often the case for GPS surveys.

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