Optimal design and re-assessment of offshore structures requires a good understanding of the ocean environment. The motion of the sea surface can be viewed as a three-dimensional, non-linear stochastic process. In order to characterise the wave environment adequately, we need to model its random, non-linear and spread nature. In this paper, we address: the expected shape of a linear wave near a crest or trough, the expected shape of the linear ocean surface at one point, given a crest at a different point, an efficient method to incorporate non-linear effects within linear wave simulations, the distributions of crests and troughs for observed and simulated non-linear waves, and the magnitude of wave non-linearity as a function of wave amplitude. Detailed comparison of theory and full-scale offshore measurements at the Shell Expro Tern platform show good agreement. In particular, the average shape of large waves is consistent with the NewWave theory of Tromans et al. (1991), after allowing for second-order non-linearity. Results validate the application of NewWave for structural design and re-assessment.