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How to compute the Betti numbers of S-D for a surface S and a divisor D?

Let S be a projective non-singular surface and D a Cartier divisor which has a smooth representative. Can the Betti numbers of S-D be represented by the Betti numbers of S and D? In a paper $b_i(S-D)=b_i(S)-b_i(D)$ is used without explanation; however, I cann't prove it and I doubt whether it would hold.