Optimal design and reassessment of offshore structures requires a good understanding of the ocean environment. The motion of the sea surface can be viewed as a three-dimensional, nonlinear stochastic process in time. In order to characterize the wave environment adequately, we need to model its random, nonlinear, and spread nature. In this paper, we address: • the expected shape of a wave near a crest or trough, • the expected shape of the ocean surface at one point, given a crest at a different point, • an efficient method to incorporate nonlinear effects within linear wave simulations, • the magnitude of wave nonlinearity as a function of wave amplitude. Detailed comparison of theory and full-scale offshore measurements at the Shell Expro Tern platform show good agreement.