We show that the variable cohomology of a regular quadric bundle arising from a linear system of quadrics can be identified with the intersection cohomology of a double covering. As a consequence we show that the middle cohomology of a general smooth complete intersection of four quadrics in an odd-dimensional projective space is isomorphic to the cohomology \(H^3(\tilde Z_0)\) of a resolution of singularities of a double solid.