CL60 FLT CREW FAILED TO SET THE LOCAL ALTIMETER SETTING AS THEY DESCENDED THROUGH TRANSITION LEVEL. THIS RESULTED IN DESCENDING BELOW THE CLEARED ALT. INADVERTENT DESCENT BELOW GS ALSO OCCURRED.

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ON DSCNT; THE ALTIMETER WAS NOT SET TO QNH WHEN DSNDING THROUGH FL180. THE PROB WAS DISCOVERED WHEN APCH ASKED US TO SAY ALT AND GAVE THE CURRENT ALTIMETER SETTING. DURING THE 1.2 HR FLT THE ATIS WAS UPDATED 4 TIMES. THE ALTIMETER SETTING STARTED AT 29.61 AND ENDED AT 29.49. THE LAST 2 CHANGES OCCURRED IN THE FINAL 10 MINS OF THE FLT. THE CREW WAS FOCUSED ON THE LOW CEILING AND VISIBILITY CONDITIONS AT THE TIME AND CAREFULLY BRIEFED THE SETUP AND MISSED APCH PROCS. ONE FIXATION WAS THE DIFFERENCE BTWN PREVAILING VISIBILITY AND SURFACE VISIBILITY. INITIALLY THEY WERE 1/8 SM AND 3/4 SM RESPECTIVELY AND IMPROVED TO 1/4 SM AND 1 - 1/2 SM OF THE LAST ATIS. WHEN THE CTLR ASKED FOR THE ACFT'S ALT; WE HAD BEEN ASSIGNED 3000 FT. WHEN I HEARD THE NEW SETTING OF 29.49 I RESPONDED THAT WE HAD 29.54 (FROM THE PREVIOUS ATIS). BUT AS I ENTERED THE NEW ALTIMETER SETTING I REALIZED THAT WE WERE STILL USING 29.92. AS WE CORRECTED THE PROB WE WERE AT 2800 FT CORRECTING TO 3000 FT. 29.54 HAD BEEN BRIEFED AND SET INTO THE FMS FOR PERFORMANCE CALCULATIONS BUT NEVER ENTERED IN THE ALTIMETERS. WIND; TEMP AND ALTIMETER SETTING ARE ALL ENTERED INTO THE APCH PERFORMANCE SECTION OF THE FMS. OUR CLRNC THROUGH THE TRANSITION ALT WAS 'CROSS ABC AT MAINTAIN ONE ZERO THOUSAND; SPD 250 KTS; ZZZ1 ALTIMETER 29.51. MY TONGUE GOT TIED UP READING THIS BACK AND I LEFT OUT THE ALTIMETER SINCE I HAD IT FROM THE ATIS. (THE LAST 2 ATIS UPDATES WERE STILL TO COME.) ALSO WE HAD BEEN PREVIOUSLY CLRED TO XYZ WHICH IS AFTER ABC; SO I CALLED ATC AND GOT A REVISED CLRNC TO CROSS 10 MI SW OF XYZ AT MAINTAIN 10000 FT AND 250 KTS (NO ALTIMETER THIS TIME). THE FLT HAD ORIGINATED FROM ZZZ2 AT XA30 AFTER DISPATCH CALLED THE PREVIOUS NIGHT. NEITHER CREW MEMBER HAD GOTTEN MORE THAN 3 HRS SLEEP BUT HAD BEEN FULLY RESTED THE DAY BEFORE. I THINK THE EARLY START WAS AFFECTING BOTH OF US. UNFORTUNATELY; THE STORY DOESN'T END HERE. AFTER MAKING WHAT SEEMED ON THE RADIO TO BE A SEMI-GRACEFUL RECOVERY FROM THE ALTIMETER SETTING CONFUSION; WE WERE IN POS AND CLRED TO MAINTAIN 2000 FT UNTIL INTERCEPTING THE APCH INBOUND. WE SWITCHED TO TWR AND WERE IMMEDIATELY CLRED TO LAND. I WAS CLEANING UP THE BEFORE LNDG ITEMS USING THE ELECTRONIC CHKLIST WHEN THE PF SAID WE WERE A LITTLE HIGH AND HE WAS GOING TO DSND TO GET ON THE GLIDE PATH. THIS MADE SENSE BECAUSE WE WERE LEAVING 3000 FT FOR 2000 FT SO I DIDN'T VISUALLY CONFIRM BUT CONTINUED WITH THE FINAL CHKS. WHEN I CLRED THE CHKLIST AND BEGAN MONITORING THE MFD WE WERE 1 DOT BELOW THE FMS GLIDE PATH INDICATION AND FULL SCALE BELOW THE ELECTRONIC GS. I IMMEDIATELY HAD AN UNEASY FEELING ABOUT THIS BECAUSE I THOUGHT WE WERE DSNDING TO INTERCEPT THE GS. WE WERE ACTUALLY DSNDING BELOW THE MINIMUM GS INTERCEPT ALT PRIOR TO THE FAF. WE BOTH REALIZED AND CONFIRMED THIS WITH EACH OTHER AT THE SAME TIME AND BEGAN A SMOOTH CORRECTION THAT RESULTED IN GS INTERCEPT AT THE FAF AS PUBLISHED -- BUT NOT BEFORE SEEING SOMETHING A FEW HUNDRED FT LOWER! THE FINAL APCH SEGMENT; LNDG AND ROLLOUT WERE TEXTBOOK AND THERE WAS RADIO SILENCE BTWN 'CLRED TO LAND' AND 'CONTACT GND.' WE WERE THE ONLY ONES ON FREQ. THIS WHOLE SCENARIO HAS ELEMENTS OF FATIGUE; DISTR; DISORIENTATION; AND POOR SOP COMBINED WITH LOW ALT; LOW IFR CONDITIONS AND RAPIDLY DECREASING BAROMETRIC PRESSURE. IT WAS A RECIPE FOR DISASTER. INITIAL CORRECTIVE ACTION IS TO REVISE THE SOP FOR USE OF THE ELECTRONIC CHKLIST DURING CRITICAL NAV PHASES; AS WELL AS THE QNE/QNH ALTIMETER SETTING PROC. AS A RESULT OF THIS EXPERIENCE; THERE IS AN INCREASED SENSITIVITY TO ALL HUMAN ELEMENTS THAT MAY CONTRIBUTE TO DEVS IN FLT PATH.