  The article investigates the behaviour of the characteristic zero resolution
invariant when transcribed suitably to the case of surfaces in positive
characteristic. By Moh's jumping phenomenon -- or the occurrence of kangaroo
singularities -- one knows that the invariant may increase, thus destroying any
induction.
  We describe in the paper how one can modify in the purely inseparable surface
case the invariant by adding a subtle correction term so as to prohibit its
occasional increases and to establish the induction argument again.
