## 1. Introduction

[2] The development of fully consistent inversions of sea surface short wave characteristics via the ever increasing capabilities (radiometric precision, spatial resolution) of remote sensing measurements has considerably advanced. Yet, difficulties remain, mostly associated to stringent requirements to have adequate understandings and means to describe very precisely the sea surface statistical properties in relation to surface wave dynamics. The simplest linear superposition and Gaussian models remain in common use. Such models provide insight and are often accurate enough for many practical purposes. Yet, common visual inspections of natural ocean surface waves often reveal geometrical asymmetries. Namely, when the steepness of a wave locally increases, its crest becomes sharper and its trough flatter. Harmonic phase couplings occur, and an ocean surface wave field can become rapidly a non-Gaussian random process. For remote sensing applications and model developments, the statistical description of random nonlinear gravity waves is then certainly not straightforward, but must be taken into account to improve uses and interpretation of measurements, e.g., to correct for the sea state bias in altimetry, to explain the upwind/downwind asymmetry of the radar cross section or to interpret the role of fast scatterer in Doppler spectra.

[3] As usually described, nonlinear surface gravity waves are generally prescribed in the context of the potential flow of an ideal fluid. For small wave steepness, the resulting nonlinear evolution equations can first been solved by means of a perturbation expansion [*Tick*, 1959]. This approach consists in finding iteratively a perturbative solution of the equations of motion for both the surface elevation and the velocity potential, by matching the boundary conditions at the bottom and at the free surface [*Hasselmann*, 1962; *Longuet-Higgins*, 1963; *Weber and Barrick*, 1977]. Following an other approach, *Zakharov* [1968] showed that the wave height and velocity potential evaluated on the free surface are canonically conjugate variables. This helps to uniquely formulate the water wave equations as a Hamiltonian system. For water waves, the Hamiltonian is the total energy *E* of the fluid. The Hamiltonian approach is based on operators expansions technique [*Zakharov*, 1968; *Creamer et al.*, 1989; *Watson and West*, 1975; *West et al.*, 1987; *Fructus et al.*, 2005], albeit using truncated Hamiltonian. We refer to *Elfouhaily* [2000] for a comparison and discussion between the two approaches. For two-dimensional water waves, where the free surface evolves as a function of one variable in space, effective methods have been improved and include conformal mapping variables [*Zakharov et al.*, 2002; *Ruban*, 2005; *Chalikov and Sheinin*, 2005]. A recent review on numerical methods for irrotational waves can be found in the paper by *Dias and Bridges* [2006]. For the three-dimensional problem, one loses the possibility to employ complex analysis, except to still consider a quasi-planar approximation, i.e., very long crested waves. Consequently, for the general problem, the perturbative technique has the advantage of simplicity, but remains essentially a low-frequency expansion and produces some nonphysical effects at higher frequencies, such as the divergence of the second-order spectrum. The Hamiltonian approach will be capable of handling stronger nonlinearities but is more tedious, remains essentially numerical and does not provide explicit statistical formulas. Finally, a Lagrangian description of surface wave motion may be more appropriate to describe steep waves [*Chalikov and Sheinin*, 2005]. In such a context, the Gerstner wave [*Gerstner*, 1809] is a first well-known exact solution for rotational waves in deep water, and *Stokes* [1847] derived a second-order Lagrangian approximation for irrotational waves leading to a well-known and observed net mass transport, the Stokes drift phenomenon, in the direction of the wave propagation.

[4] The aim of this paper is to build on this latter simplified phase perturbation methodology to propose a simple, versatile model, that can reproduce the lowest-order nonlinearity of the perturbative expansion but does not suffer from its related shortcomings. This analytical model is certainly not properly new, as it is widely used by the computer graphics community [*Fournier and Reeves*, 1986; J. Tessendorf, unpublished data, 2004] to produce real-time realistic looking sea surfaces. The terminology choppy wave model (henceforth abbreviated to “CWM”) originates from the visual effect imposed by the transformation compared to linear waves. In addition to gravity waves nonlinear interaction, the model can incorporate further physical features such as the horizontal skewness induced by wind action over the waves, an effect that we will not consider in this paper and which will be left for subsequent work.

[5] On the mathematical level, the model identifies completely with the perturbative expansion in Lagrangian coordinates as proposed four decades ago by *Pierson* [1962, 1961]. In the case of a single wave, it coincides with the Gerstner solution and is consistent with the Stokes expansion [*Stokes*, 1880] at third order in slope. Our present contribution is to provide a complete, nontrivial statistical study of this model and a comparison with the classical approaches. As understood, the CWM does not claim to compete with Hamiltonian-based methods and is in fact limited to the lowest-order nonlinearity. Its main strength is to provide a good compromise between simplicity, stability and accuracy. More precisely, it is (1) numerically efficient, as time evolving sample surfaces can be generated by FFT; (2) analytically tractable, as it provides explicit formulas for the first- and second-order point statistics; and (3) robust to the frequency regime, as it is found to be equivalent to the canonical approach [*Creamer et al.*, 1989] at low frequencies while remaining stable at higher frequencies.

[6] In the following we have studied the two- and three-dimensional case pertaining to long-crested or truly two-dimensional sea surfaces, which from now on we will refer to as the 2-D and 3-D case. Since the methodology remains the same in both instances, we have chosen to give a complete exposure of the technique in the 2-D case which is considerably simpler. All the analytical results of the 2-D case (section 2) have their counterpart in the 3-D case (section 3). In the subsequent study, the emphasis will be put on the spatial properties of a “frozen” surface, even through some temporal properties will also be discussed. Using a phase perturbation in the Fourier domain, the nonlinear local transformation simply consists in shifting the horizontal surface coordinates. Starting with a linear, reference surface, assumed to be a second-order Gaussian stationary process in space and time with given power spectrum, the complete first- and second-order properties of the resulting, non-Gaussian, random process is derived and related to the statistics of the reference surface. In particular, the resulting spectrum, which we refer to as dressed, has been related to the reference spectrum, termed undressed, in a way which is found to be very similar to *Weber and Barrick*'s [1977] and *Creamer et al.*'s [1989], but corrects the former and extends the latter to the 3-D case. As well, the sea surface slope statistical description is modified to exhibit a non-Gaussian behavior with a measurable peakedness effect, i.e., an excess of zero and steep slopes. A comparison with recent airborne laser measurements which allows to discriminate the slope statistics of gravity waves from smaller, short gravity, and capillary waves, is presented in section 4. As found, the CWM brings the excess kurtosis of omnidirectional slopes significantly closer to the data.