  Recent analyses of wetting in the semi-infinite two dimensional Ising model,
extended to include both a surface coupling enhancement and a surface field,
have shown that the wetting transition may be effectively first-order and that
surprisingly the surface susceptibility develops a divergence described by an
anomalous exponent with value $\gamma_{11}^{\rm eff}=\frac{3}{2}$. We reproduce
these results using an interfacial Hamiltonian model making connection with
previous studies of two dimensional wetting and show that they follow from the
simple crossover scaling of the singular contribution to the surface
free-energy which describes the change from apparent first-order to continuous
(critical) wetting due to interfacial tunnelling. The crossover scaling
functions are calculated explicitly within both the strong-fluctuation and
intermediate-fluctuation regimes and determine uniquely and more generally the
value of $\gamma_{11}^{\rm eff}$ which is non-universal for the latter regime.
The location and the rounding of a line of pseudo pre-wetting transitions
occurring above the wetting temperature and off bulk coexistence, together with
the crossover scaling of the parallel correlation length, is also discussed in
detail.
