Calculation of Helmert 7 transformation parameters

The task is intended to calculate the datum - parameters of spheroid's orientation of defined coordinate system relative to the spheroid WGS84. In GIS Panorama there are allowed two types of Datum - based on transformations of Helmert and Molodensky. In this task parameters are calculated dX, dY, dZ, Rx, Ry, Rz, M of Helmert transformation:

Xgc wgs84 = | 1 Rz-Ry | * |Xgc | |dX |

Ygc wgs84 = (1 + M) * |-Rz 1 Rx | * |Ygc | + |dY |

Zgc wgs84 = | Ry-Rx 1 | * |Zgc | |dZ |

In the GIS Panorama for all standard types of maps the Datum is defined. The task of calculating datum arises when the use of a standard datum leads to unacceptable errors. This is true for the coordinate system of 1942 year, and the coordinate system, the basic geodetic network of which is based on the points of the state geodetic network in the coordinate system of 1942. This is explained by the fact that at development of the state geodetic network the errors have accumulated which values reach several metres.

In local territories these errors can be eliminated with use of the calculated Datum, that actually is the analog of using an additional transformation «shift, scale, turn» at recalculations into other coordinate systems.

The input data for calculations are the geodetic coordinates of points at WGS84 and geodetic or rectangular coordinates of defined coordinate system.  To obtain reliable parameters it is necessary to use the points measured with the maximum precision (centimetric and more precise) and located on considerable distance from each other (minimum of ten kilometres).

 It should be noted that in the above formula the geocentric coordinates should be calculated by the heights measured from a surface of spheroid (geodetic heights). To receive spheroidal heights on WGS84 it is possible by methods of space geodesy, and heights of the majority of national coordinate systems are normal, i.e. are measured from level surface - the geoid. The surface of geoid is located unevenly concerning spheroid as it depends on the location of the gravitational mass in the body of the Earth. The most precise in open access at present is the model of geoid EGM2008 (the model of geoid EGM2008 contains heights matrix of geoid concerning a surface of spheroid WGS84 and is accessible to downloading in a matrix format of the GIS Panorama by the address <http://gistoolkit.ru/download/geoid/egm8_1.zip>).
 

For cases when heights in defined coordinate system are counted relative to the geoid, 3 variants of Datum calculation are possible:

1.Instead of spheroidal one to give onto the task's input the normal heights. In this case the linear part of displacement of a geoid relative to the spheroid will go into a Datum that will allow to recalculate directly the coordinates from defined coordinate system with normal heights on WGS84. This way is admissible at a slight nonlinear component of displacement of a geoid relative to the spheroid within processed area.

Currently the GIS Panorama does not use geoid model at recalculations of coordinates, i.e. actually all heights are considered as the spheroidal. Therefore for the further use of the calculated Datum in GIS Panorama the datum should be calculated only in this way.

2.Before Datum calculation to recalculate heights from spheroid WGS84 onto geoid with use of EGM2008 (the mode is used «Recalculate heights of points from spheroid on the geoid by model of geoid EGM2008»). In this case the Datum will be calculated by normal heights instead of spheroidal heights, that is not absolutely correct from the mathematical point of view, but allows to get a higher precision of coordinates recalculation by the calculated Datum.

Calculated in this way datum can be used for recalculation of coordinates from WGS84 into the necessary coordinate system, but in this case heights of all processed data should be preliminaryly  be converted to the geoid by EGM2008 with use of the third-party software.

3.There are geoinformation systems in whom it is possible to use geoid model at the description of coordinate system. And often the geoid model can be specified relative to another Datum and even another spheroid. But if the geoid model is on another spheroid or a Datum, then inside the GIS at height recalculation the calculated Datum is not used, i.e. the chain of recalculation of coordinates looks like this:

BL-> (Datum)-> BwgsLwgs

Hgeoid-> (dHegm)-> Hwgs

In this case accuracy of recalculation of heights depends only on accuracy of the geoid model.

If to specify at the description of coordinate system that the geoid model is defined on the same spheroid, then the chain of coordinates recalculation  looks like this:

BL-> ( Datum)-> BwgsLwgsHwgs

Hgeoid-> (dHegm)-> Hsph

In this case the linear part of shift on height will be eliminated at recalculation of coordinates inside the GIS by the calculated Datum.

Thus, for obtaining the maximum accuracy, the geoid model should be defined on an spheroid of defined coordinate system.

EGM2008 geoid model is defined relative to WGS84, but because the size of a cell of a model's matrix in times more than real displacement of spheroids in space, then it is possible to specify at the description of coordinate system that model of geoid EGM2008 is  defined on an spheroid of defined coordinate system.

For use of the calculated Datum in GIS, which allow to use geoid model at the description of the coordinate system, at Datum calculation the mode should be set «Recalculate heights of points from a geoid to the spheroid on model of geoid EGM2008».

Then the calculated Datum should be described according to the rules adopted in the GIS. For example, for ErdasImagine into the file spheroid.tab you must make changes for the necessary spheroid:

"NewCS" dxdydzrxryrzm

"NewCS Height" SURFACE BASEDATUM = "NewCS"

         {

                 RASTER RESAMPLE = "Bilinear"        

                 HEIGHT = "egm2008ww1mgh.bin"        

         } DESCRIPTION = "Calculated"

Preparation of input data

Coordinates of points should be preliminarily written down in two text files - catalogs of coordinates. In one catalogue geodetic coordinates of points on WGS84 are specified, in the second one there are specified the geodetic or rectangular coordinates of points in defined coordinate system, i.e. on an spheroid, the orientation of which is to be determined in relation to WGS84. The order of recording the points in catalogs does not matter, because at loading the catalog of defined coordinate system the points automatically are sorted in the order of their arrangement in catalog WGS84.

The string with the coordinates of the point in the catalog coordinates is entered in the following format:

Point_Number Latitude Longitude Elevation Weight_in_plan Weight_by_elevation

The fields must be separated by one or more spaces. As the point number it is permissible to use any text characters with no spaces.  The weights can be omitted, in this case they will be equal 1. Latitude and longitude can be entered in degrees or in degrees, minutes, seconds, separated by a space. For rectangular coordinates instead of latitude the coordinate Х in meters is entered, instead of a longitude - Y is entered.

If in the catalog of points coordinates of defined coordinate system there are high-rise points with unknown plane coordinates their names should begin with the same symbols (prefix) for their identification at loading. Such points should not have plane coordinates (X and Y) and point weight in the plan.

Main parameters

Before loading the catalogs it is necessary to specify a format of recording the coordinates in catalogues (fractions of degrees, degrees, minutes, seconds or metres) and a prefix of high-rise points. Other parametres can be entered after loading of catalogues.

The catalog of points on WGS84 must be opened at first, because opening of this catalogue nulls the list of loaded points. At opening the catalogue in defined coordinate system the coordinates of points are filled, that previously were loaded from the catalogue WGS84.  

Coordinates of points can be recalculated on model of geoid EGM2008 before Datum calculation in accordance with the above recommendations. To do this, select one of the modes «Recalculate heights of points from spheroid onto the geoid by model of geoid EGM2008» or « Recalculate heights of points from a geoid onto spheroid by model of geoid EGM2008». Before use the geoid model should be copied into the directory of the GIS Panorama.

If coordinates of points in the catalogue are entered in meters, then you must enter the parameters of the projection in the standard dialog of input of projection parameters.

Additional parameters

Datum calculation is carried out in two steps. At first the Datum is calculated at the direct solving the system of the normal equations of Helmert transformation.

Then it is specified by consecutive iterations. In the field the Maximum number of iterations is indicated the maximum quantity of iterations of refining the calculated parametres. If to enter into this field a zero, refinement will not be carried out. In this case the parameters calculated at the direct solution will be result.

After calculating the Datum a calculation of residual discrepancies on points with use of the calculated Datum is carried out. These discrepancies are shown in the table of coordinates. For search of blunders of discrepancies exceeding the tolerance specified in the field Show by red divergence more … MSE, are shown in the table by red color.

If the geoid model EGM2008 is used, then the height between cells of a matrix of a geoid model can be obtained by bilinear or bicubic interpolation. Bicubic interpolation gives better results, but is carried out significantly slowly. It is also necessary to take into account that the selected interpolation method should be the same what is used in third-party GIS at interpolation of model of geoid EGM2008.

Not all coefficients can be calculated, but only those which are specified in group Calculated coefficients. Not calculated coefficients should be entered into the appropriate fields. Iterations of parameters refinement will be executed as long as the amendment to all the calculated coefficients will not be less than the tolerance entered in the field refine up to.

Order of processing

Calculation of parameters is performed by pressing the button Calculate. After calculation of parameters in group of Results of calculation there are shown:

- calculated coefficients from the direct and refined solution;

- amendments in the last iteration;

- number of refinement iterations;

- mean-square error (MSE) on reference points (with the enabled check-box);

- mean-square error on control points (with the disabled check-box).

In the table the residual discrepancies on points are shown. Residual discrepancies for convenience of the analysis are always displayed in meters. If the projection parameters of defined coordinate system are set, then discrepancies are calculated in the specified projection. If geodetic coordinates were used, then the discrepancies are calculated in topocentric coordinate system on the specified spheroid with the centre in a defined point.

According to results of the discrepancies analysis, it is possible to exclude some points from calculations, having removed the check mark in point number. Such points do not participate in calculation of coefficients, but by them the discrepancies and MSE on control points are calculated.

To change influence of separate points onto results of solution it is possible, changing weight of points in the plan or on height. The point weight can be edited in the table during the task performance, and will not be stored in the catalog of coordinates at its completion.

At each calculation of datum the report file calcdatm.txt is created in the folder from which the catalogue WGS84 is loaded. In it intermediate results of calculations are printed. To see the report of last solution it is possible by pressing the button Report...

After calculation the parameters can be saved to XML for their subsequent use in the GIS Panorama at creation of digital maps. By pressing the button Save to XML the dialog is called Add reference system.

By default the new reference system is created in preset in the GIS Panorama list “LCS of Subjects RF.xml”. It is possible to change a list file, by clicking the button to the right of a field File XML and having chosen it in standard dialog of file open. If to enter a new name in dialog of a file choice, the new list will be created. For identification of reference system it is necessary to enter the name and optional comment. After button pressing Save to XML the reference system is stored into the selected list and dialog is closed.

тпIf in the basic dialog the parameters of defined coordinate system were entered, then in XML the record will be created with the chosen parametres of a projection, spheroid and the calculated datum. If projection parametres were not entered, then a record will be created with type of a map "Latitude/longitude", the chosen spheroid and the calculated Datum.