<p>In contrast to the numerous interpolation methods for grid point information, for information of a&#xa0;grid of cells (e.g., from Earth observation data or modelling results), there is only pycnophylactic interpolation which does not disturb the original information. Averaging the new information surface over the original grid cells yields their original values (volume-/mass-preserving property). However, pycnophylactic interpolation is a&#xa0;complex iteration and rarely used. I thus derive a&#xa0;new interpolation approach for grid cell information: integral-differential interpolation. It exploits the fact that the original grid cell values imply an integral calculus, which also determines an abstract integral field. The integral values between the supporting points (defined by the original grid) are approximated using classical interpolation methods. The new higher-resolution grid or a&#xa0;smooth surface is derived from the integral field by difference/differential calculus. The theory of the new approach is simpler and more transparent than that of pycnophylactic interpolation. Therefore, the basis for error quantification could be also derived. Another advantage is the wide range of possibilities: various classical mathematical interpolation methods can be applied to the integral field and the partial linear spline interpolation results can be applied directly to the original cell information. The new approach is applied to a&#xa0;demonstration example, namely the fields of the maximum wind gusts (Germany region) of the storms Kyrill 2007 and Sabine 2020.</p>