In the graphic arts industry, there is a need to convert colorimetric readings taken on one backing (usually white) to values that would have been measured with a different backing (usually black). We describe and compare different models for such a conversion. Starting from published models using linear scaling, we developed a new nonlinear model for a strongly scattering substrate. Another new model was derived from the Clapper-Yule model, which includes effects of internal reflectances. All these models are applicable in both the spectral and the tristimulus domain. For calibration, we used measurements of the bare substrate on both backings. We intentionally make only use of the measured spectral or XYZ values, and do not require knowledge of the nominal CMYK values. This is particularly useful for arbitrary patches measured with a stand-alone measurement device. The test data sets consisted of a large set of test prints, originating from digital or conventional printing processes, and covering typical ranges of mass per area. Both new models outperformed linear regression models and the spectral versions always yielded better results than their corresponding versions in tristimulus space.