PARTIAL LOSS OF ACFT CTL AND LNDG MINIMUMS VIOLATED WHEN CFI AND STUDENT PLT ARE UNABLE TO MAINTAIN ALT IN A C172 WHEN PICKING UP HVY ICING S OF APA; CO.

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AIRPLANE BEGAN TO ACCUMULATE LIGHT RIME ICE IN CRUISE AT 10000 FT ON VICTOR 81 SBOUND FROM BJC VOR. WX WAS FORECAST TO IMPROVE THE FARTHER S WE WENT. ABOUT 30 DME S OF BJC ICING BECAME INCREASINGLY HEAVIER; UNTIL AT APPROX HOHUM INTXN WE WERE HAVING DIFFICULTY MAINTAINING ALT. DEN DEP RADIOED HE NEEDED US TO MAINTAIN 10000 FT. I TOLD HIM WE WERE PICKING UP ICE AND REQUESTED VECTORS ILS RWY 35R AT APA. HE INFORMED ME THE GS WAS OTS. I REPLIED I KNEW THAT. HE THEN OFFERED ME VECTORS TO DEN WHERE THE WX WAS HIGHER. I REPLIED WE'D PROBABLY HAVE TO STICK WITH APA. HE VECTORED US TO THE E; AT WHICH TIME I ASKED FOR VECTORS TO GET US THERE AS QUICKLY AS POSSIBLE. HE INFORMED ME WE WERE ABOUT 2 MI FROM THE APCH; AND GAVE US AN INTERCEPT TO THE LOC AND OVER TO CENTENNIAL TWR. WE CHKED IN WITH TWR; SHE CLRED US TO LAND. ICING WAS MODERATE AT THAT POINT. WE HAD FULL THROTTLE AT 70 KIAS AND DSNDING 400 FPM. WE WERE UNABLE TO MAINTAIN APCH MINIMUMS; AND AT ONE POINT TWR SAID 'YOU PROBABLY KNOW THIS; BUT I'M GETTING AN ALT ALERT.' I SAID WE WERE DOING THE BEST WE COULD. WE (MY STUDENT AND I) BRIEFED THE APCH AND KNEW WE WERE GOING TO HAVE TO FIND THE RWY REGARDLESS OF THE WX. THOUGH THE GS WAS OTS; IT APPEARED TO BE WORKING AS WE CROSSED CASSE INTXN. IT SHOWED FULL DEFLECTION TO THE TOP. WE SAW THE APCH LIGHTS ABOUT 400 FT AGL; ALMOST 500 FT BELOW THE LOC APCH MINIMUMS. WE LANDED WITHOUT INCIDENT (WITH 2 INCHES OF ICE). CAUSES: WE TOOK OFF INTO FORECAST ICING CONDITIONS; BUT WE HAD JUST COME THROUGH AND WE HAD RPTED TRACE RIME. I THOUGHT IF WE COULD GET UP HIGH ENOUGH (10000 FT) WE COULD FLY OVER THE ICING LAYER. BUT THE CONDITIONS HAD CHANGED. ONCE AT 10000 FT; I WAS RELUCTANT TO DSND BECAUSE I ASSUMED (CORRECTLY!) THE ICING WAS WORSE BELOW US. PREVENTION: I WILL NOT TAKE OFF INTO POSSIBLE ICING CONDITIONS; EVEN IF I HAVE JUST COME THROUGH AND I THINK I KNOW HOW THE CONDITIONS ARE. SUPPLEMENTAL INFO FROM ACN 545946: FROM OUR PREVIOUS FLT; WE KNEW THERE WAS A TEMP INVERSION AND DECIDED IF WE GOT TO THE WARM AIR AT 10000 FT THERE WOULD BE NO PROB AND IF WE PICKED UP ICE IN THE CLB WE COULD RETURN TO BJC WHERE THERE WAS A 1500 FT CEILING (VFR CONDITIONS). DURING THE CLB WE PICKED UP A TRACE OF ICE AND WERE THANKFUL TO BE AT 10000 FT AND AT THE TOPS OF THE CLOUDS WITH OCCASIONAL BLUE SKY. THEN THE CLOUD GOT DARKER AND WE STARTED PICKING UP A LITTLE MORE ICE UNTIL I COULD NO LONGER HOLD ALT AT 80 KIAS. MY INSTRUCTOR ASKED TO DIVERT TO APA. ATIS RPTED CEILING 500 FT AND VISIBILITY 2 MI. OUR DECISION TO TAKE OFF FROM BJC IN THE FIRST PLACE WAS FLAWED; BUT FELT SAFE BECAUSE WE HAD JUST FLOWN INTO THERE. POOR DECISION MAKING LED TO THIS PROB. I AM GRATEFUL THAT MY FIRST TIME IN IMC WAS WITH A COMPETENT INSTRUCTOR WHO KEPT HIS HEAD AND GOT US OUT OF A BAD SIT. DEN APCH AND APA TWR WERE EXTREMELY HELPFUL.