Proximity sensor for height from snow covered ground
Some of the options mentioned in your question would work for the purpose, with varying precision and suitability:


Ultrasonic: Probably your best bet: Narrow aperture piezoelectric ultrasonic sensors (single-unit send/receive) are reliable in ranges from a few inches to a hundred feet or more. Cost goes up disproportionately with range, though. Precision is fair-to-high, and it works with most surfaces, including reflective and transparent ones i.e. packed snow or sheet ice. Scan frequency is fairly high.


Senix TSPC-21S series has a range precisely matching your requirement, 50 feet range / fractional inch interpolated precision, but costs around $500-800. Some Chinese clones do exist, at around a tenth the price or less, but no idea on precision or reliability. We had ordered one, it arrived badly damaged, seller was unresponsive. Maybe give it a try if you don't need it waterproof.






Laser: Pulsed time-of-flight principle, similar to ultrasonic sensors. Scanner type, since reflector type will not work with snow. Range up to 40 feet, precision extremely high. 


SensoPart FT 90 ILA might suit the requirement if 30 feet range is acceptable. It claims 2 mm precision, which is meaningless for uneven snow surfaces.







The following would not be recommended:


Optical: These too use lasers, but operate on a stereoscopic triangulation principle. Range is limited, typically to 1 meter. Precision is very high, scan frequency is high.
Radar: Insufficient precision and insufficient minimum range. Integrated RADAR devices are usually for 10 feet minimum to several miles maximum range, weight is massive as range increases, and precision is in tens of feet.
Infrared: Insufficient maximum range for your purposes. Conventional IR reflectometry works up to a couple of feet at the most, and is more popular in sub-foot ranges. However, the Laser device listed above is also infrared, in that it uses an infrared scanning laser.


I hope this helps