Your grid currently lacks projection information. Hence the line
Coordinate System is ''
in the output. With proper projection information, the output would start (for example) like:
Coordinate System is:
PROJCS["UTM Zone 4, Northern Hemisphere",
It is important to understand the difference between assigning a projection and reprojecting. Using the
a_srs switch of
gdal_translate will assign a projection where it is wrong or missing (as it is here).
gdalwarp will reproject from one system to another. So
gdal_translate is the tool you want to use here. You will only use
gdalwarp after determining the existing projection, if you want it in a different one.
With vector data it is sometimes easier to eyeball a map and figure out what projection it is using, but this is pretty difficult to do with raster data. You have two options for determining the projection. The first is to contact the original source of your data to find out what it is. The second option is to try assigning likely candidates, and see if your data align with other known data (another raster or vector layer with known coordinate reference system). Since you have a list of about 20 candidates in Austria (link, entire list duplicated below), this is at least feasible. Obviously, you have already determined that it is not EPSG:32633. This is not surprising, as EPSG:32633 corresponds to UTM zone 33N. Since UTM North zones have their vertical origin at the equator, your raster lower extent of 249995 would be ~250 km from the equator.
Because false eastings and false northings are used in order to avoid having negative coordinates, the negative minimum X value of your grid (-144178) leads me to believe that the correct projection from the list you linked to is one of the ones with a false easting of 0, giving 9 possibilities. If you knew whether this data were in the West, Central, or East of Austria, you could whittle the list further from 9 to 3 possibilities.
You can either (a) use
gdal_translate first, then use
gdalwarp, or (b) specify the missing CRS (as will as your desired output CRS) as part of
gdalwarp. For example, if you determine that the existing projection is Austria Gauss-Kruger West EPSG:31254, you would do one of the following:
Assign projection, then reproject
gdal_translate -a_srs:31254 DHM_10meter.asc temp_file.asc
gdalwarp -t_srs EPSG:32633 temp_file.asc output_name.asc
Reproject while specifying missing CRS
gdalwarp s_srs EPSG:31254 -t_srs EPSG:32633 DHM_10meter.asc output_name.asc
Let us know how it goes.
Table of possible projections
EPSG code Projection name Base GeogCRS Map Projection Longitude Orig Latitude Orig Prime Meridian false easting false northing Projection Method
3416 ETRS89 / Austria Lambert ETRS89 Austria Lambert 13d 20m 47d 30m Greenwich 400000 400000 Lambert Conic Conformal (2SP)
31251 MGI (Ferro) / Austria GK West Zone MGI (Ferro) Austria Gauss-Kruger West Zone 28d 0d Ferro 0 -5000000 Transverse Mercator
31252 MGI (Ferro) / Austria GK Central Zone MGI (Ferro) Austria Gauss-Kruger Central Zone 31d 0d Ferro 0 -5000000 Transverse Mercator
31253 MGI (Ferro) / Austria GK East Zone MGI (Ferro) Austria Gauss-Kruger East Zone 34d 0d Ferro 0 -5000000 Transverse Mercator
31254 MGI / Austria GK West MGI Austria Gauss-Kruger West 10d 20m 0d Greenwich 0 -5000000 Transverse Mercator
31255 MGI / Austria GK Central MGI Austria Gauss-Kruger Central 13d 20m 0d Greenwich 0 -5000000 Transverse Mercator
31256 MGI / Austria GK East MGI Austria Gauss-Kruger East 16d 20m 0d Greenwich 0 -5000000 Transverse Mercator
31257 MGI / Austria GK M28 MGI Austria Gauss-Kruger M28 10d 20m 0d Greenwich 150000 -5000000 Transverse Mercator
31258 MGI / Austria GK M31 MGI Austria Gauss-Kruger M31 13d 20m 0d Greenwich 450000 -5000000 Transverse Mercator
31259 MGI / Austria GK M34 MGI Austria Gauss-Kruger M34 16d 20m 0d Greenwich 750000 -5000000 Transverse Mercator
31281 MGI (Ferro) / Austria West Zone MGI (Ferro) Austria West Zone 28d 0d Ferro 0 0 Transverse Mercator
31282 MGI (Ferro) / Austria Central Zone MGI (Ferro) Austria Central Zone 31d 0d Ferro 0 0 Transverse Mercator
31283 MGI (Ferro) / Austria East Zone MGI (Ferro) Austria East Zone 34d 0d Ferro 0 0 Transverse Mercator
31284 MGI / Austria M28 MGI Austria M28 10d 20m 0d Greenwich 150000 0 Transverse Mercator
31285 MGI / Austria M31 MGI Austria M31 13d 20m 0d Greenwich 450000 0 Transverse Mercator
31286 MGI / Austria M34 MGI Austria M34 16d 20m 0d Greenwich 750000 0 Transverse Mercator
31287 MGI / Austria Lambert MGI Austria Lambert 13d 20m 47d 30m Greenwich 400000 400000 Lambert Conic Conformal (2SP)
31288 MGI (Ferro) / M28 MGI (Ferro) Austria zone M28 28d 0d Ferro 150000 0 Transverse Mercator
31289 MGI (Ferro) / M31 MGI (Ferro) Austria zone M31 31d 0d Ferro 450000 0 Transverse Mercator
31290 MGI (Ferro) / M34 MGI (Ferro) Austria zone M34 34d 0d Ferro 750000 0 Transverse Mercator