## 1. Introduction

[2] With the ever increasing accuracy of satellite microwave radar sensors for geophysical purposes and the progress of the electromagnetic wave interaction models, there is an increasing need for an accurate description of ocean short waves in various natural conditions. Two-point characteristic functions of the sea surface topography involved in the classical scattering models [see, e.g.,*Voronovich*, 1994] are still based on model considerations rather than direct in situ measurements. In spite of obvious progresses in wave tank measurements such as reported by *Zhang and Cox* [1994]; *Jähne et al.* [2005]; *Zappa et al.* [2008]; and *Caulliez and Guérin* [2012], among others, direct in situ estimation of the small-scale topography of the sea surface is still a challenging issue. Moreover, in view of a statistical characterization it is preferable to rely on direct measurements of the topography rather than resorting to additional a priori assumptions.

[3] The spatial properties of the sea surface are routinely characterized by indirect means such as temporal measurements at a fixed location (gauge, buoys, laser) or remote sensing techniques. Detailed survey of these methods given recently in *Zappa et al.* [2008]shows their shortcomings. Contact measurements by means of wave gauges lead to the variation in time but not in space of the elevations. Radar remote sensing inversion involves a scattering model. Scanning lasers can provide the instantaneous high-resolution field of slopes but are mainly operated in wave tanks. Airborne or spaceborne lidar can be used to measure the slope vector but are restricted to a profile along the track. Recently, promising methods of polarimetric imaging [*Zappa et al.*, 2008] and a phase-resolving spatial reconstruction technique based on a Flash Lidar Camera [*Grilli et al.*, 2011] were proposed but are still at the stage of first publications.

[4] Stereo imaging reconstruction is well suited to assess the statistical characteristics of short waves in natural conditions. Classical utilization of this remote sensing technique does not require an underlying model for the sea surface. This technique has a long history [*Cote et al.*, 1960; *Holthuijsen*, 1983; *Banner et al.*, 1989; *Shemdin et al.*, 1988] and has now been developed to a robust and powerful experimental tool that can be used for regular measurements of oceanic sea state dynamics and sea surface statistical properties. In particular, existing binocular and trinocular systems have been adopted for the observation of the coastal and surf zone [*Wanek and Wu*, 2006; *Bechle and Wu*, 2011; *de Vries et al.*, 2011] as well as for offshore conditions [*Gallego et al.*, 2011a; *Kosnik and Dulov*, 2011; *Gallego et al.*, 2011b; *Fedele et al.*, 2011, 2012; *Benetazzo et al.*, 2012]. In these recent works the classical stereo reconstruction algorithm has been improved using various (both explicit and implicit) additional assumptions on the statistical nature of the sea surface and their brightness distribution. For example, temporal continuity of the surface is commonly implied in processing of continuous video recordings [e.g., *Benetazzo*, 2006]; smoothness of the surface and their brightness field is essential for the application of the variational method introduced in *Gallego et al.* [2011a]; a sea brightness model is needed for the extension of the available range of wavelength (see the brightness-based spectrum estimates by*Kosnik and Dulov* [2011]); a special kind of brightness field correlation must be adopted in using the sub-pixel methods [e.g.,*Wanek and Wu*, 2006; *Bechle and Wu*, 2011; *Benetazzo et al.*, 2012].

[5] However, the processing of stereo data still raises technical issues when it comes to the estimation of key statistical parameters for short waves. First, the aforementioned assumptions may appear to be incorrect for the small-scale motions of the sea surface. At least for the first step in learning the small-scale statistic properties it is interesting to apply only classical stereo reconstruction technique rejecting the additional assumptions. Second, modern devices do not cover the very wide dynamical range that is needed for the observation of surface waves at all scales varying from about 100 m down to 1 mm. Stereo systems aimed at monitoring the long waves (for example, the WASS-system [*Gallego et al.*, 2011a; *Fedele et al.*, 2011; *Benetazzo et al.*, 2012; *Fedele et al.*, 2012]) enables to evaluate the mean level and the mean slope of water because these values are physically determined by the long waves. However, considering short waves only, we cannot obtain precisely these statistical values because long waves are poorly visible in the relatively small scene of observation. For such cases, when the need is to study the fluctuations with respect to a background of large-scale motion, Kolmogorov introduced the structure functions [see, e.g.,*Doob*, 1953; *Monin and Yaglom*, 1999; *Ishimaru*, 1999]. The use of the structure functions minimizes the contribution of large-scale motion and provides us with an approach to learn the statistical properties of short waves at the sea surface without precise knowledge of the mean level and slope. Last, the consideration of the shortest waves in field conditions rises an additional difficulty [*Kosnik and Dulov*, 2011]. The central problem of the stereo reconstruction is to find and localize pairs of corresponding points that are images of the same object in two snapshots of the water surface made from different views. Once the corresponding points are found, the surface topography can be recovered using standard procedures [see, e.g., *Benetazzo*, 2006]. In the laboratory, objects can be artificially introduced on the water surface to facilitate the matching of points [*Tsubaki and Fujita*, 2005]. In field conditions, the only suitable objects are the brightness variations induced by surface waves. In practice, the recovery of the surface topography of waves at a given scale requires the detection of brightness variations due to smaller scale waves. This fact is the principal constraint for the spatial resolution of the stereo-based method. In particular, the shortest waves cannot be recovered because there are no smaller objects on the sea surface. Sharply defined texture (capillary ripples on the sea surface) does not exist everywhere and by all weather conditions. Smooth areas without notable markers cannot be used in processing and this results in gaps in the reconstructed elevation maps [*Benetazzo*, 2006; *Kosnik and Dulov*, 2011]. While this problem does not play a significant role in dealing with well-developed large-scale stereo reconstruction, it requires a special attention in our work.

[6] The aim of this paper is to present a general methodology and some first results on the statistical characterization of ocean short wave fields. The main strength of the technique is that it does not require neither a priory assumptions for the sea surface nor interpolation procedure to compensate for the lack of sampling points. We will use high-quality data sets for three wind speeds (7, 10, 15 m/s) first presented by*Kosnik and Dulov* [2011]. The data were processed to obtain statistically independent sea surface elevation fields using classical stereo reconstruction algorithm with improved method of searching corresponding points [*Kosnik and Dulov*, 2011].

[7] The paper is organized as follows. The theoretical framework of the statistical description of the sea surface is recalled in section 2. The experimental set-up is rapidly described insection 3 and the major technical issues raised by the processing of the acquired data sets are identified and discussed in section 4. We show (section 5) that the distribution of elevations of the small-scale process can be correctly evaluated from the stereo data. The obtained distributions for different sea states can be meaningfully compared with wave gauges measurements after filtering out the large scale components. The retrieval of the sea surface slopes (section 6) requires a similar detrending procedure as well as an extrapolation procedure to compensate for the limited small-scale resolution. The estimated distributions are compared with historical [*Cox and Munk*, 1954] and more recent airborne measurements [*Vandemark et al.*, 2004] by optical means. In addition to the slopes, the distribution of chords at the surface can be derived and successively compared with gauge wire as well as airborne [*Vandemark et al.*, 2004] measurements. The two-points properties of the small-scale roughness can be characterized in the same manner. We show (section 8) that the autocorrelation function of the small-scale process and its Fourier transform are consistent with alternative measurements of the wave spectrum. Our last result (section 9) concerns the derivation of the skewness function that has an important meaning in remote sensing theories and witnesses for the asymmetric nature of waves. We derive it experimentally and provide an original and accurate parametrization of this otherwise unknown function.