A stochastic Gauss-Lagrange model for a space-time random field is a mechanism to make a random deformation of a Gaussian field. Typically, the deformation is correlated with the original Gaussian field. In oceanography, Gaussian fields have been used since about 1950 to bescribe the vertical movement of an irregular wave surface. To obtain a more realistic model, one can use a Lagrange model to describe the joint vertical-horizontal motions of individual water particles, which in the simplest case are circles. In a first order Gauss-Lagrange model the horizontal movements are defined as a new Gaussian field, correlated with the vertical movement field. In this way, realistic wave shapes can be produced in a Gaussian framework.
In this paper we investigate the statistical properties of wave characteristics related to asymmetry in the 3D Gauss-Lagrange model, in particular how the vertical and horizontal asymmetry of the waves depend on the two-dimensional spectral density. The studied model can produce front-back asymmetry both of the space waves, i.e. observations of the sea surface at a fixed time, and of the time waves, observed at a fixed measuring station. Slopes are defined as time or space derivatives of the resulting field.
The results are based on a multivariate form of Rice's formula for the expected number of multiple crossings for a vector valued stochastic field. The theory is illustrated by examples, showing how the degree of directional spreading influences the wave asymmetry, important for marine safety.
Keywords: Rice's formula; Random deformation; Wave steepness; Space-time asymmetry
Biography: Professor in Mathematical statistics (em) in Lund, Sweden. Has worked on statistical extreme value theory (book by Leadbetter, Lindgren, Rootzén), and since many years interested in stochastic processes and their role in marine science, meteorology, structural safety, etc.