The authors consider surfaces which are defined as the graphs of real-valued functions defined on a domain \(\Omega\subset\mathbb{R}^2\) and deal, in particular, with \(C^1\) cubic splines defined on triangulations obtained from convex quadrangulations by drawing both diagonals in each quadrilateral. For such spline surfaces a compression scheme is developed, which does not require the construction of wavelets. The key to the presented method is to work with \(C^1\) cubic spline spaces which can be parametrized locally using the well-known FVS-macro-elements. The algorithms are based on constructing hierarchical bases for certain nested sequences of such spline spaces.