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I have a shapefile whose Geometry is of POINTZM type.

The XY coordinate System is NAD_1983_CORS96_StatePlane_North_Carolina_FIPS_3200_Ft_US and the Z coordinate system is North_American_Vertical_Datum_1988

I want to transform the data, so that the data is projected in a Geographic (Lat/Long) projection, with the WGS84 horizontal datum and the EGM96 vertical datum.

I can't figure out how I can do this using the Project tool in ArcToolbox, of ArcGIS for Desktop 10.3.

How do I do this?

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Project tool has Vertical parameter since 10.4. The link also mentions that you need to install additional package with coordinate system data.

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This is apparently not possible with ArcGIS for Desktop, see Esri's article FAQ: Are vertical datum transformations supported in ArcGIS?

If you go to the ArcGIS idea page referenced in the article's comments, there is a mention of this being possible in Pro and Runtime, programmatically. To be verified with Esri, I couldn't find any confirmation of this.

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