  Bathymetric changes have been experimentally shown to affect the occurrence
of rogue waves. We recently derived a non-homogeneous correction to the
spectral analysis, allowing to describe the evolution of the rogue wave
probability over a shoal. Here, we extend this work to the evolution of the
excess kurtosis of the surface elevation, that plays a central role in
estimating rare event probabilities. Furthermore, we provide an upper bound to
the excess kurtosis. In intermediate and deep water regimes, a shoal does not
affect wave steepness nor bandwidth significantly, so that the vertical
asymmetry between crests and troughs, the excess kurtosis, and the exceedance
probability of wave height stay rather constant. In contrast, in shallower
water, a sharp increase in wave steepness increases the vertical asymmetry,
resulting in a growth of both the tail of the exceedance probability and the
excess kurtosis.
