INSTRUCTOR AND STUDENT LAND AT UNCONTROLLED ARPT WITH SNOW COVERED RWY THAT IS DEEPER THAN SUSPECTED.

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HAVING PREVIOUSLY FLOWN WITH MY STUDENT WHO IS PVT PLT RATED; AND SATISFIED WITH HIS ABILITY TO ADEQUATELY PREFLT THE AIRPLANE AND BRIEF HIMSELF WITH ALL INFO PERTAINING TO THE FLT; I OPTED TO OBTAIN WX AND OTHER INFO RELATED TO THE FLT FOR HIM; AS HE WAS RUNNING LATE. AFTER DETERMINING THAT ZZZ WOULD MEET THE REQUIREMENT; I BRIEFED MYSELF ONLINE FOR WX ALONG THE RTE AND NOTAMS; AND OTHER INFO PERTAINING TO THE FLT. THE WX WAS PERFECT VFR; AND I DID NOT SEE ANY NOTAMS FOR THE ARPT. THE STUDENT COMPLETED THE PREFLT OF THE ACFT; AND I ALSO DID A THOROUGH WALKAROUND OF THE ACFT. THE FLT PROCEEDED WITHOUT INCIDENT. UPON ARR AT THE ARPT; WE FLEW OVER THE FIELD AT ZZZ APPROX 1500 FT AGL (6500 FT MSL) TO INSPECT THE WINDSOCK; TFC; AND RWYS. THE RWY APPEARED TO BE COVERED WITH A LIGHT SMATTERING OF SNOW; AND THERE WERE MULTIPLE SETS OF TRACKS INDICATING THAT ACFT HAD BEEN TAKING OFF AND LNDG AT THE ARPT. IT APPEARED NO MORE THAN 1 INCH OF SNOW ON THE RWY. HAVING NOT OBSERVED ANYTHING THAT WOULD INDICATE THE ARPT WAS NOT IN FULL OP; WE DECIDED TO ENTER A L DOWNWIND; AND MADE APPROPRIATE CALLS ON THE CTAF. MY STUDENT COMPLETED THE APPLICABLE CHKLISTS; AND ESTABLISHED A STABLE APCH. WE TOUCHED DOWN AT APPROX XA50 AND THE ACFT SANK INTO THE SNOW ON THE RWY AND CAME TO A STOP IN APPROX 160 FT. WE SHUT DOWN THE ACFT; AND EXITED THE ACFT TO INSPECT IT. THERE WAS ABOUT 14 INCHES OF SNOW ON THE RWY. THERE WAS NO DAMAGE TO THE ACFT; OR INJURY TO THE PLTS. THERE WAS NO DAMAGE TO THE RWY. I IMMEDIATELY CONTACTED THE OPS MGR AT THE FLT SCHOOL TO INFORM HIM OF THE SITUATION. HE INSTRUCTED ME TO CONTACT ANY PERSONS NECESSARY TO ENSURE THE ARPT WOULD BE NOTAMED CLOSED DUE TO ACFT ON THE RWY. I CALLED FSS AND INFORMED THEM OF THE ACFT ON THE RWY; REQUESTING THE CLOSURE OF THE FIELD. THE CHIEF PLT PICKED UP BOTH ME AND MY STUDENT AND DROVE US BACK. WE DISCUSSED THE FLT AS I HAVE OUTLINED IT HERE. THE EVENT COULD HAVE BEEN AVOIDED HAD I CALLED FSS FOR A BRIEFING INSTEAD OF RELYING ON THE INFO I USED ONLINE TO LOCATE NOTAMS. ALSO; I LEARNED THAT YOU CANNOT JUDGE THE DEPTH OF SNOW ON THE RWY FROM 1500 FT AGL; AND WHEN IN DOUBT; DON'T LAND.