Apparently to address this problem, the 2DSA method takes from the concatenated solution of the subgrids only the pattern of active/inactive nonnegativity constraints. Demeler and colleagues construct from the points with nonzero concentration values in the concatenated partial solutions (or a subset thereof) a new grid, conceived to be equal in size to the original low-resolution grids, but now with uneven spacing of the grid points. Again, we can analyze this approach best by comparison with the full, high-resolution grid with the full matrix P, where the unambiguous best-fit nonnegative solution is found exactly with the proven NNLS algorithm (Lawson and Hanson 1974) . The ad hoc exclusion of grid points that did not produce positive concentration values in any of the subgrids is in direct conflict with NNLS. Nothing guarantees that the (s, f r )-values populated in the exact solution will be correctly recognized as populated species (be assigned nonzero values) in the fit with the low-resolution grid of which they are a part. The points populated in the exact solution may therefore simply not be part of the second-stage grid.