AN ACR CREW MADE AN ILLEGAL CIRCLING APCH.

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APCH RWY CLOSED FOR CONSTRUCTION. LNDG RWY REQUIRED CIRCLING APCH. WX AT TIME OF DEP WAS VFR; 4000 FT OVCST; VISIBILITY 7 MI; LIGHT RAIN. AT TOP OF DSCNT; ZSE INFORMED US THE WX HAD DETERIORATED AND ACFT WERE HOLDING. WE CONTACTED DISPATCH TO COORDINATE OUR PLAN DUE TO MARGINAL FUEL. CONFLICTING WX RPTS RECEIVED FROM CTR; DISPATCH AND ATIS. FINALLY DECIDED ON ONE APCH WITH ALTERNATE OF PSC. MISSED THE APCH AT 1000 FT AGL; WHICH IS COMPANY MINIMUMS FOR ALL CIRCLING APCHS AND DIVERTED TO PSC. REFUELED AND WAITED FOR FORECAST IMPROVEMENT. WX AT TIME OF SECOND APCH SEVERAL HRS LATER WAS 300 FT SCATTERED 400 FT BROKEN 1500 FT OVCST VISIBILITY 4 MI LIGHT RAIN AND FOG. SUCCESSFUL APCH AND LNDG. THE NEXT MORNING I REALIZED THAT WE SHOULD NEVER HAVE STARTED THE APCH. THE 400 FT BROKEN LAYER CONSTITUTED A CEILING WHICH WAS BELOW CIRCLING MINIMUMS. I THINK WE WERE 'SUCKED IN' BY THE FREQUENT USE OF ILS APCHS WITH VISIBILITY ONLY MINIMUMS (THIS WAS AN ILS); HIGHER COMPANY MINIMUMS AND THE 1500 FT OVCST LAYER RPTED. I NEED TO REVIEW LEGALITY AS WELL AS SAFETY PROCS IN APCH BRIEFING FROM THE CAPT.