[Proj] Transformations with height data to bessel bring unrealistic results

[Mark Johnson]

When transforming positions with DHHN92 (EPSG:5783) height data, to Bessel
based projections are returned.
For my area (Berlin, Germany) the lowest height is just over 24 meters.
Transformations using 34.79 all return  -6.13930772151798 when using
ST_Transform, which performs a pj_transform in the background.
Using -6.13930772151798 to transform back will return the original value.

Between WSG84 and epsg:25833 there are no problems.

I would be interested in learning the reason why this is so.

Here some cs2cs samples with the results:

ETRS89 / UTM zone 33N [epsg:25833]
# +proj=utm +zone=33 +ellps=GRS80 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs

# DHDN / Soldner Berlin
# +proj=cass +lat_0=52.41864827777778 +lon_0=13.62720366666667 +x_0=40000
+y_0=10000 +ellps=bessel +towgs84=598.1,73.7,418.2,0.202,0.045,-2.455,6.7
+units=m +no_defs
echo "389918.0707613 5819698.6197819 34.7900000"  | cs2cs +init=epsg:25833
+to +init=epsg:3068
23183.38 21047.17 -6.14
echo 23183.38 21047.17 -6.139 | cs2cs +init=epsg:3068 +to +init=epsg:25833
389918.07 5819698.62 34.79

# DHDN
# +proj=longlat +ellps=bessel
+towgs84=598.1,73.7,418.2,0.202,0.045,-2.455,6.7 +no_defs
echo "389918.0707613 5819698.6197819 34.7900000"  | cs2cs +init=epsg:25833
+to +init=epsg:4314
13d22'45.996"E 52d31'3.638"N -6.139
echo 13.37770460000026 52.51627039998215 -6.139 | cs2cs +init=epsg:4314 +to
+init=epsg:25833
389796.59 5819544.81 34.80

# DHDN / 3-degree Gauss-Kruger zone 4
# +proj=tmerc +lat_0=0 +lon_0=12 +k=1 +x_0=4500000 +y_0=0 +ellps=bessel
+towgs84=598.1,73.7,418.2,0.202,0.045,-2.455,6.7 +units=m +no_defs
echo "389918.0707613 5819698.6197819 34.7900000"  | cs2cs +init=epsg:25833
+to +init=epsg:31468
4593626.95 5821241.65 -6.14
echo "4593626.95 5821241.65 -6.14"  | cs2cs +init=epsg:31468 +to
+init=epsg:25833
389918.07 5819698.62 34.79

Mark Johnson, Berlin Germany
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