Geographic Transformations

This transformation is actually an approximation of the Seven Parameter transformation, except using position vector rather than coordinate frame rotation.

The approximation is arrived at by making three assumptions:

1. The sine of a small angle is equal to the angle (in radians) itself;
2. The multiplication of two sine terms is zero; and
3. The cosine of a small angle is one.

This approximation is valid only for small angles.

In all other aspects, this transformation is the same as the Seven Parameter transformation. In processing new data projects, use the Seven Parameter transformation in lieu of Bursa/Wolf. The Bursa/Wolf approximation is provided to reproduce numbers/calculations that were originally accomplished using the approximation.

This transformation will produce the same results as the Seven Parameter transformation, with all three rotation parameters and the scale parameter set to zero.

As with the Seven Parameter Transformation, this transformation proceeds in three phases. First, the geographic coordinates are converted to three-dimensional Cartesian, geocentric coordinates using the ellipsoid of the original datum. Second, the three translation parameters, Delta X, Delta Y, and Delta Z, are used to translate the geocentric coordinates. Third, the resulting geocentric coordinates are converted back to geographic form using the target ellipsoid.

As in all other cases for the translation parameters, the geocentric parameters must be given in units of meters.

This transformation method supports a priority-ordered list of grid files in arbitrary formats. It is specific to geodetic transformation definitions, and may not be used for datums. Each grid file entry includes the grid format, direction of the grid, and the path to the grid. The first grid that provides coverage of the input point is used for conversion.

Grid Formats:

Grid direction must be ‘Fwd’ (Forward) or ‘Inv’ (Inverse/Reverse).

Transformation Definition Example Snippet:

[…]

METHOD GRID_INTERP \

GRID_FILE “NADCON,Fwd,.\GridData\Nadcon\arhpgn.l?s” \

GRID_FILE “NADCON,Fwd,.\GridData\Nadcon\alhpgn.l?s”

In addition to the Seven Parameter transformation parameters, Molodensky-Badekas allows a rotation origin point to be specified. The additional parameters are:

Rotation Origin X: The X component of the point (in the source Cartesian coordinate reference system) about which the rotation will be performed.

Rotation Origin Y: The Y component of the point (in the source Cartesian coordinate reference system) about which the rotation will be performed.

Rotation Origin Z: The Z component of the point (in the source Cartesian coordinate reference system) about which the rotation will be performed.

This transformation is based on the series of Multiple Regression developments published by the US Defense Mapping Agency (NIMA) in Technical Report 8350.2-B, December 1987. Essentially, these are formulas developed from applying linear regression techniques to a varying number of points where the source and target ellipsoid coordinates are rather precisely known.

These regression formulas are based on normalized input coordinates. It is assumed that the normalized coordinates define the useful range of the datum transformation. In theory, therefore, a geographic coordinate that produces a normalized coordinate greater than 1.0 or less than -1.0 would normally be considered outside the useful range of the transformation. In this implementation of the regression technique, a geographic coordinate is considered outside the useful range of a transformation if the absolute value of either normalized coordinate exceeds 1.4.

In the event that a coordinate is given that is outside of the useful range of the multiple regression formula as described above, a fallback technique is used to calculate a datum shift. In this case, the fallback technique is the Molodensky, Six Parameter , or Seven Parameter transformation, depending upon how many parameters have been defined in the base definition. That is, when defining the datum definition, temporarily set the technique specification to Seven Parameter and set the desired fallback parameters. Then, the technique can be set back to the Multiple Regression selection and the parameters values will be preserved.

Currently, the parameters to such a transformation consist of a preprocessed transformation definition file. These files contain all of the coefficients of the multiple regression formula in a compact form. This form also facilitates the actual testing of each parameter file individually, as the DMA-provided test case is included in the file. Currently, no provisions are made for users to implement their own multiple regression parameter files.

This transformation is a rigorous implementation of the standard three-dimensional transformation with coordinate frame rotation. The seven provided parameters must indicate the transformation to convert source datum coordinates to target datum coordinates. For many typical GIS applications, you can simply change the sign of each of the seven parameters to effect an inverse. However, this technique is not exact. For precise results, a rigorous inversion is necessary in order to determine the appropriate parameters.

Essentially, this transformation proceeds in three phases. First, the geographic coordinates are converted to three-dimensional Cartesian, geocentric coordinates using the ellipsoid of the original datum. Second, the three-dimensional transformation defined by the seven parameters is applied producing a modified set of geocentric Cartesian coordinates. Third, the resulting geocentric coordinates are converted back to geographic form using the target ellipsoid.

The seven parameters are:

Delta X: The amount the intermediary geocentric X coordinate is translated. This value must be given in meters and the direction of the translation is given by the sign of the value.

Delta Y: The amount the intermediary geocentric Y coordinate is translated. This value must be given in meters and the direction of the translation is given by the sign of the value.

Delta Z: The amount the intermediary geocentric Z coordinate is translated. This value must be given in meters and the direction of the translation is given by the sign of the value.

X Rotation: The amount of rotation about the X axis which is applied to the intermediary geocentric coordinates. This value is given in seconds of arc, and the direction of the rotation is indicated by the sign of the value.

Y Rotation: The amount of rotation about the Y axis which is applied to the intermediary geocentric coordinates. This value is given in seconds of arc, and the direction of the rotation is indicated by the sign of the value.

Z Rotation: The amount of rotation about the Z axis which is applied to the intermediary geocentric coordinates. This value is given in seconds of arc, and the direction of the rotation is indicated by the sign of the value.

Scale: A scale factor that is applied to the intermediary geocentric coordinates. The value is given as a value in parts per million and is the difference of the actual scale factor and unity. For example, a value for the scale parameter of -2.5 produces an actual scale factor of 0.9999985. That is, the actual scale factor used is arrived at by multiplying the parameter value by 1.0x10-06 and adding the result (algebraically) to 1.0.