The stochastic Lagrange wave model is a realistic alternative to the Gaussian linear wave model, which has been successfully used in ocean engineering for more than half a century. In this paper we present the slope distributions and other characteristic distributions at level crossings for asymmetric Lagrange time waves, i.e. what can be observed at a fixed measuring station, thereby extending results previously given for space waves. The distributions are given as expectations in a multivariate normal distribution, and they have to be evaluated by simulation or numerical integration. Interesting characteristic variables are the slope in time, the slope in space, and the vertical particle velocity when the waves are observed close to instances when the water level crosses a predetermined level. The theory has been made possible by recent generalizations of Rice's formula for the expected number of marked crossings in random fields.

In many applications, such as remote sensing or wave slamming on ships and offshore structures, it is important to have a good model for wave slope. Today, most models are based on the assumption that the sea surface is well described by a Gaussian random field. However, since the Gaussian model does not capture several important features of real ocean waves, e.g. the asymmetry of crests and troughs, it may lead to unconservative safety estimates. An alternative is to use a stochastic Lagrangian wave model. Few studies have been carried out on the Lagrangian model; in particular, very little is known about its probabilistic properties. Therefore, in this paper we derive expressions for the level-crossing intensity of the Lagrangian sea surface, which has the interpretation of wave intensity, as well as the distribution of the wave slope at an arbitrary crossing. These results are then compared to the corresponding intensity and distribution of slope for the Gaussian model.

Gaussian wave models have been successfully used since the early 1950s to describe the development of random sea waves, particularly as input to dynamic simulation of the safety of ships and offshore structures. A drawback of the Gaussian model is that it produces stochastically symmetric waves, which is an unrealistic feature and can lead to unconservative safety estimates. The Gaussian model describes the height of the sea surface at each point as a function of time and space. The Lagrange wave model describes the horizontal and vertical movements of individual water particles as functions of time and original location. This model is physically based, and a stochastic version has recently been advocated as a realistic model for asymmetric water waves. Since the stochastic Lagrange model treats both the vertical and the horizontal movements as Gaussian processes, it can be analysed using methods from the Gaussian theory. In this paper we present an analysis of the stochastic properties of the first-order stochastic Lagrange waves model, both as functions of time and as functions of space. A Slepian model for the description of the random shape of individual waves is also presented and analysed.