Recent experimental and numerical studies of surface gravity waves propagating over a sloping bottom have shown an increase in the probability of extreme waves can be triggered by depth variations in sufficiently shallow waters. This phenomenon is studied here by means of a Boundary Element Method with fast multipole acceleration to solve the fully nonlinear water wave equations. We focus on the case of a random, unidirectional wave field with prescribed statistical properties propagating over a submerged slope, and consider different depth variations, including a step. Validation is provided by comparing with experiments by Trulsen, Zeng & Gramstad (Phys. Fluids, 24, 097101 (2012)). Strongly non-Gaussian statistics are observed in a region localized near the depth transition, beyond which the statistics settle rapidly to the steady statistical state of finitedepth random wave fields. Using a harmonic separation technique, we show that the second-order terms are responsible for the change in the statistical properties near the depth transition. We explore in detail the effects of peak frequency, significant wave height, the inclination of the slope, and the depth of the shallower water side on the kurtosis, skewness and the excess probability of the crest height, including their spatial distributions.

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