# Sample GeoConv application "723"

In most handheld GPS-devices the datums are defined by ellipsoidal parameters DF, DA and three parameters DX, DY, DZ, which are the parameters of the so called three parameter similarity transformation. Three parameter similarity transformation defines only shifts along X-, Y- and Z-axes (=3 parameters) and is therefore less accurate than corresponding seven parameter similarity transformation, which defines shifts along X-, Y- and Z-axes (=3 parameters), rotation around X-, Y- and Z-axes (=3 parameters) and possible correction of scale (=1 parameter). We could hence say, that seven parameter similarity transformation is "full similarity transformation" and three parameter similarity transformation is "poor mans similarity transformation".

When e.g. national land survey organizations calculate parameters for similarity transformation, they always calculate parameters for a seven parameter similarity transformation and very seldom for a three parameter similarity transformation.

This all leaves the ordinary GPS-user with difficult problems if there aren't any known parameters for the tree parameter similarity transformation and

• Datum is not predefined in GPS or
• Datum is predefined in GPS, but the parameter values are based on old measurements or aren't accurate enough in the area where the user wants to use them.

However, with a couple of applications based on the program GeoConv, the parameters for a three parameter similarity transformation can be calculated.

The first application is based on representative set of points, from which measured coordinates are known in both coordinate systems. Unfortunately such sets of points with coordinate values in both systems are rarely available. Therefore this application is not described here. If you in fact have such information available and want to make the calculations, please contact the author in address eino.uikkanen@iki.fi.

The second solution was built on the idea, that in most cases where the parameters of the three parameter similarity transformation are missing, some good parameters for a seven parameter similarity transformation are known. The basic idea is to reduce the seven parameters to three parameters. That means, that the only input needed is the parameters for a seven parameter similarity transformation. In addition to that, the user has to define a set of (fictitious) points, which very well represent the area in which the best accuracy is desired. This could be even one point in the middle of the area, e.g. in the middle of the country, city, state or region.

The method roughly described is:

• Calculate converted coordinates for a given set of points using a known seven parameter similarity transformation.
• Calculate 3D-cartesian coordinates (X,Y,Z) for both original and converted coordinates.
• Calculate average differences between the cartesian coordinates = average (converted X,Y,Z - original X,Y,Z).
• The result, parameters for the three parameter similarity transformation, are the opposite numbers to calculated averages.

The entire documented application (batch-run), is in text-file 723.bat.txt. If you want to use this run.

• Copy it with another name having extension .bat
• Study the run and modify it to agree with your needs and computer environment
• Make the needed preparations described in the run
• And run it .. Good Luck!

# Test run of "723" and the results

In order to check the functionality of "723" and assess the accuracy of the results I made some test-runs. Here I represent a test, which I made with well known material - 90 points, which Finnish Geodetic Institute (FGI) used to calculate transformation parameters between current Finnish National coordinate system KKJ and ETRS89. FGI has published the coordinate-values of the points and the results of the calculation. Therefore I had in my test all this information available:

• Coordinate-values in current KKJ-system
• Coordinate-values in ETRS89-system based on high precision measurements made by FGI
• Parameters for a seven parameter similarity transformation calculated by FGI and information of the accuracy and residual errors of this transformation

Using this data I calculated parameters for the three parameter similarity transformation for the entire country and separately for the southern and northern part of the country. I made the calculations in two ways. First I calculated parameter-values using "723" and based on the seven parameters calculated by FGI. For the entire country I used four points and for the southern and northern part I used only one point. The results of these calculations I marked "723" in the table below. Then I made calculations based on the measured coordinate values, which of course is the best way if the coordinate values are available. The results of these calculations I marked "EU" in the table below.

I also included in the table the parameters used in most GPS-devices for KKJ-datum. This is marked "Most GPS" in the table below.

The parameters for the seven parameter similarity transformation calculated by FGI are in the first column of the table and marked "FGI seven". These parameters are based on the latest high precision measurements and give at least nearly as good result as a similarity transformation ever can give. Because of the distortion of both scale and orientation in the current KKJ-system, this similarity transformation still leaves residual errors of maximum two meters.

 Parameters Unit Whole country Southern part Northern part FGI seven EU 723 Most GPS EU 723 EU 723 DX Shift / X-axis meter -96.062 -75 -75 -78 -74 -75 -76 -76 DY Shift / Y-axis meter -82.428 -230 -230 -231 -229 -231 -232 -232 DZ Shift / Z-axis meter -121.754 -89 -91 -97 -88 -91 -91 -91 rx Rotation / X-axis second -4.801 ry Rotation / Y-axis second -0.345 rz Rotation / Z-axis second +1.376 m Correction in scale +1.496 Average error meter 0.8 1.6 1.8 1.8 1.4 1.6 0.8 0.8 Maximum error meter 2 3.3 3.9 3.4 2.5 3.8 1.8 1.8

This table shows us that "723" gives almost as good results as calculations based on measured coordinate values, in northern Finland even exactly the same result. This is of course one test-case only and the results could be a lucky coincidence. Anyway, knowing the mathematical base of "723" I can assure, that "723" gives the best results that the available input can theoretically give.

Program GeoConv