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I have some geotiff file and I need to change the position: 500 meters to east and 500 m to south. Resolution is 12,5 meters.

I've tried using GeoTIFF Tools GUI, and I've created .gtf file, but I don't know what I should write in.

Also I've tried with GeoTiffExaminer, but after I wrote in X vector X and Y vector Y values nothing changes.

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You can use gdal_translate http://www.gdal.org/gdal_translate.html with the "assign upper-left lower-right" option

-a_ullr ulx uly lrx lry:
    Assign/override the georeferenced bounds of the output file. This assigns georeferenced bounds to the output file, ignoring what would have been derived from the source file. So this does not cause reprojection to the specified SRS. 

Gdal_translate will make a copy of the image. If you prefer to update in place, use gdal_edit.py http://www.gdal.org/gdal_edit.html with the same option

-a_ullr ulx uly lrx lry:

    Assign/override the georeferenced bounds of the dataset.
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