Differences between ellipsoidal height and orthometric altitude (or geoid height) are explained here https://spatialthoughts.com/2019/10/26/convert-between-orthometric-and-ellipsoidal-elevations-using-gdal/
The definition of the projection is the PROJ4 line that can be found on https://spatialreference.org/
examples
- using gdalwarp to convert from UTM33N-WGS84 ellipsoidal height to UTM33N-EGM96 geoid height
gdalwarp -s_srs "+proj=utm +zone=33 +datum=WGS84 +units=m +no_defs" -t_srs "+proj=utm +zone=33 +datum=WGS84 +units=m +no_defs +geoidgrids=egm96_15.gtx" input.tif output.tif
- from Lambert93 planimetric and NGF-IGN69 (with RAF20) to WGS84 ellipsoidal heights
gdalwarp -s_srs "+proj=lcc +lat_0=46.5 +lon_0=3 +lat_1=49 +lat_2=44 +x_0=700000 +y_0=6600000 +ellps=GRS80 +units=m +no_defs +type=crs +geoidgrids=RAF20.gtx" -t_srs "+proj=longlat +datum=WGS84 +no_defs +type=crs" input.tif output.tif
- from altitude to ellipsoidal height use such command (change srs and geoid grid if needed):
gdalwarp -s_srs "+proj=utm +zone=33 +datum=WGS84 +units=m +no_defs +geoidgrids=egm96_15.gtx" -t_srs "+proj=utm +zone=33 +datum=WGS84 +units=m +no_defs" input.tif output.tif
you can choose your input and output spatial reference using the proj4 syntax https://epsg.io/32633 , and note the add of the geoid file with +geoidgrids=egm96_15.gtx in the command.
if you want to use EGM2008 geoid, get the file from https://github.com/OSGeo/proj-datumgrid/blob/master/world/egm08_25.gtx and copy it to the proj directory. In Linux it’s
/usr/share/proj
then you can simply change the geoid file in the command +geoidgrids=egm08_25.gtx