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Ken Brown (of Factorization Lemma fame) published a paper relating Cech hypercohomology and and sheaf cohomology, extending a result of Verdier (that is relegated to an appendix in, I believe, SGA 4. Brown calls it "Verdier's Hypercovering theorem"). There is an evident way to compare Cech cohomology and Cech hypercohomology (as discussed in the paper), and so I believe that it should answer your question:

Click!

Edit: If the link does not work, the paper is Abstract Homotopy Theory and Generalized Sheaf Cohomology by K.S. Brown

Ken Brown (of Factorization Lemma fame) published a paper relating Cech hypercohomology and sheaf cohomology, extending a result of Verdier (that is relegated to an appendix in, I believe, SGA 4. Brown calls it "Verdier's Hypercovering theorem"). There is an evident way to compare Cech cohomology and Cech hypercohomology (as discussed in the paper), and so I believe that it should answer your question:

Click!

Edit: If the link does not work, the paper is Abstract Homotopy Theory and Generalized Sheaf Cohomology by K.S. Brown

Ken Brown (of Brown Representability fame) published a paper relating Cech hypercohomology and and sheaf cohomology, extending a result of Verdier (that is relegated to an appendix in, I believe, SGA 4. Brown calls it "Verdier's Hypercovering theorem"). There is an evident way to compare Cech cohomology and Cech hypercohomology (as discussed in the paper), and so I believe that it should answer your question:

Click!

Edit: If the link does not work, the paper is Abstract Homotopy Theory and Generalized Sheaf Cohomology by K.S. Brown

Ken Brown (of Factorization Lemma fame) published a paper relating Cech hypercohomology and and sheaf cohomology, extending a result of Verdier (that is relegated to an appendix in, I believe, SGA 4. Brown calls it "Verdier's Hypercovering theorem"). There is an evident way to compare Cech cohomology and Cech hypercohomology (as discussed in the paper), and so I believe that it should answer your question:

Click!

Edit: If the link does not work, the paper is Abstract Homotopy Theory and Generalized Sheaf Cohomology by K.S. Brown

Ken Brown (of "Brown Representability" fame) published a paper relating Cech hypercohomology and and sheaf cohomology, extending a result of Verdier (that is relegated to an appendix in, I believe, SGA 4. Brown calls it "Verdier's Hypercovering theorem"). There is an evident way to compare Cech cohomology and Cech hypercohomology (as discussed in the paper), and so I believe that it should answer your question:

Click!

Ken Brown (of Brown Representability fame) published a paper relating Cech hypercohomology and and sheaf cohomology, extending a result of Verdier (that is relegated to an appendix in, I believe, SGA 4. Brown calls it "Verdier's Hypercovering theorem"). There is an evident way to compare Cech cohomology and Cech hypercohomology (as discussed in the paper), and so I believe that it should answer your question:

Click!

Edit: If the link does not work, the paper is Abstract Homotopy Theory and Generalized Sheaf Cohomology by K.S. Brown