Are rogue waves really unexpected?

Abstract: An unexpected wave is defined by Gemmrich & Garrett (2008) as a wave that is much taller than a set of neighboring waves. Their definition of "unexpected" refers to a wave that is not anticipated by a casual observer. Clearly, unexpected waves defined in this way are predictable in a statistical sense. They can occur relatively often with a small or moderate crest height, but large unexpected waves that are rogue are rare. Here, this concept is elaborated and statistically described based on a third-order nonlinear model. In particular, the conditional return period of an unexpected wave whose crest exceeds a given threshold is developed. This definition leads to greater return periods or on average less frequent occurrences of unexpected waves than those implied by the conventional return periods not conditioned on a reference threshold. Ultimately, it appears that a rogue wave that is also unexpected would have a lower occurrence frequency than that of a usual rogue wave. As specific applications, the Andrea and WACSIS rogue wave events are examined in detail. Both waves appeared without warning and their crests were nearly $2$-times larger than the surrounding $O(10)$ wave crests, and thus unexpected. The two crest heights are nearly the same as the threshold~$h_{0.3\cdot10^{6}}\sim1.6H_{s}$ exceeded on average once every~$0.3\cdot 10^{6}$ waves, where $H_s$ is the significant wave height. In contrast, the Andrea and WACSIS events, as both rogue and unexpected, would occur slightly less often and on average once every~$3\cdot10^{6}$ and~$0.6\cdot10^6$ waves respectively.