The c i,j provide an estimate of the size-and-shape distribution with cðs; f r Þ % c i;j ðDsDf r Þ. Signal offsets from systematic time-invariant [b(r n )] and radial-invariant [b(t m )] noise contributions are indicated in Eq. (3), but their simultaneous optimization with the method of separation of linear and nonlinear parameters (Ruhe and Wedin 1980) poses no significant further complications (Schuck and Demeler 1999) and therefore they will be dropped from further consideration in order to make the notation more transparent in the following. 2 We can introduce a new index l that lexicographically orders all data points (a total of L = N 9 M), and a single index k that enumerates all size-and-shape grid points (s i , f r,j ) from 1 to K = S 9 F, which allows us to write (3) as a simple sum 1 We also analyzed the data with ULTRASCAN II version 9.9 to confirm our results as far as possible. Unfortunately, the current lack of a manual section for the use of the 2DSA analysis and the excessive computational times involved prevented us from a direct comparative analysis of the same data with the full 2DSA model as described by Brookes et al. (2009) . Further, a detailed comparison does not seem possible due to seemingly unavoidable data truncation steps when loading data in ULTRASCAN II, and due to our inability to write the entire calculated distribution into a text file.