Abstract
 Top of page
 Abstract
 1. Introduction
 2. Theoretical Background
 3. Experimental SetUp
 4. Major Technical Issues
 5. Distribution of Sea Surface Elevations
 6. Distribution of Sea Surface Slopes
 7. Distribution of Chords
 8. Calculation and Consistency of the Spectrum
 9. Skewness Function
 10. Conclusion
 Acknowledgments
 References
 Supporting Information
[1] We propose a methodology to extract shortscale statistical characteristics of the sea surface topography by means of stereo image reconstruction. The possibilities and limitations of the technique are discussed and tested on a data set acquired from an oceanographic platform at the Black Sea. A validation is made with simultaneous in situ measurements as well as results from the literature. We show that one and twopoint properties of the shortscale roughness can be well estimated without resorting to an interpolation procedure or an underlying surface model. We obtain the first cumulants of the probability distribution of smallscale elevations and slopes as well as related structure functions. We derive an empirical parametrization for the skewness function that is of primary importance in analytical scattering models from the sea surface.
1. Introduction
 Top of page
 Abstract
 1. Introduction
 2. Theoretical Background
 3. Experimental SetUp
 4. Major Technical Issues
 5. Distribution of Sea Surface Elevations
 6. Distribution of Sea Surface Slopes
 7. Distribution of Chords
 8. Calculation and Consistency of the Spectrum
 9. Skewness Function
 10. Conclusion
 Acknowledgments
 References
 Supporting Information
[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. Twopoint 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 smallscale 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 highresolution 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 phaseresolving 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 brightnessbased spectrum estimates byKosnik and Dulov [2011]); a special kind of brightness field correlation must be adopted in using the subpixel 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 smallscale motions of the sea surface. At least for the first step in learning the smallscale 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 WASSsystem [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 largescale 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 largescale 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 stereobased 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 welldeveloped largescale 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 highquality data sets for three wind speeds (7, 10, 15 m/s) first presented byKosnik 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 setup 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 smallscale 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 smallscale 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 twopoints properties of the smallscale roughness can be characterized in the same manner. We show (section 8) that the autocorrelation function of the smallscale 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.
3. Experimental SetUp
 Top of page
 Abstract
 1. Introduction
 2. Theoretical Background
 3. Experimental SetUp
 4. Major Technical Issues
 5. Distribution of Sea Surface Elevations
 6. Distribution of Sea Surface Slopes
 7. Distribution of Chords
 8. Calculation and Consistency of the Spectrum
 9. Skewness Function
 10. Conclusion
 Acknowledgments
 References
 Supporting Information
[16] The experiment was conducted in October 2009 on the Black Sea platform of Marine Hydrophysical Institute (Crimea, Ukraine). The platform is located 500 meters off the coast by a depth of about 30 m. Stereo images were acquired by means of two synchronized cameras with 72 mm focal length situated at 4.5 m above the sea level with grazing angle of 30°. The cameras were set about 1.5 meter apart (see Figure 1). Stereo system geometry and camera parameters were devised to get the shortest wave extraction and allowed to obtain 16 bit images with 1 mm spatial resolution. The fundamental point here is the small exposure time (1/1000 s) and the small synchronization time of the cameras (better than 0.1 ms). For longer times the shortest waves, which are carried by the orbital velocities of long waves are blurred in the images and become difficult to distinguish. Photographing was made every 10–15 s providing statistically independent realizations of smallscale sea surface topography. Sea surface spatial coordinates were obtained from collected stereo pairs of photographs with the standard approach based on epipolar geometry and pinhole camera model with distortion correction [e.g.,Benetazzo, 2006] and improved homologous point search algorithm [Kosnik and Dulov, 2011]. Simultaneous measurements of temperature, wind speed and direction were recorded by the automatic meteorological station installed on the platform.
[17] Wave height measurements at 10 Hz sampling frequency and 2 mm accuracy were performed with six resistant wave gauges. These gauges were used to record 20 minutes time series during each experimental measurement. Because of some failure of equipment during the experiment, four gauges did not work properly in case 2 and were discarded. Therefore the distribution obtained in this case is more noisy.
[18] Table 1 lists the main environmental parameters for the different experimental conditions. Each series of frames corresponds to a time span of twenty minutes. The full experimental and processing procedure is reported in Kosnik and Dulov [2011].
Table 1. Experimental Conditions  Case Number 

1  2  3 

Day of October, 2009  28  7  19 
Wind speed  7  10  13–17 
Wind direction  North  East  East 
Fetch (km)  0.5  ∼400  ∼400 
Presence of swell  Yes  Yes  No 
Number of frames  76  72  67 
Wave length of spectrum peak (meter)  2.4  32  26 
Wave period of spectrum peak (s)  1.3  4.5  4.1 
Frequency of spectrum peak (Hz)  0.8  0.22  0.24 
Significant wave height (meter)  0.3  0.85  0.9 
4. Major Technical Issues
 Top of page
 Abstract
 1. Introduction
 2. Theoretical Background
 3. Experimental SetUp
 4. Major Technical Issues
 5. Distribution of Sea Surface Elevations
 6. Distribution of Sea Surface Slopes
 7. Distribution of Chords
 8. Calculation and Consistency of the Spectrum
 9. Skewness Function
 10. Conclusion
 Acknowledgments
 References
 Supporting Information
[19] The natural and instrumental limitations impose some restrictions on the recoverable properties of sea surface elevation. Here we discuss the main issues raised by the processing of the current data set. The processing of images of sea surface combines several sequential stages and each of them has a certain limitation and source of error. For example, stereo photography is not applicable at low wind speeds, when the sea surface has very few markers to identify the points for correspondence in stereo pairs. In severe storm conditions the search of corresponding points also faces a difficulty due to the presence of foam that often turns out to be different on the left and right images because of the difference in camera view direction and thus the difference in light reflection from the bubbles.
[20] I1: Limited accuracy—The main errors that arise in stereo analysis include 1) calibration error, i.e. the error of spatial coordinate recovery, 2) the resolution or quantization error, 3) the typical error of corresponding pixel search and 4) the spikes on the recovered surfaces that appear due to the wrong determination of corresponding pixels in stereo pair. Let us consider these factors successively. The error of spatial coordinate recovery is determined by the accuracy of estimation of the epipolar matrix used for triangulation. The respective numerical values are calculated in the calibration procedure [Bouguet, 2004]. However, to evaluate the resulting error in our measurements and to verify the correctness of the calibration results after equipment reinstalling, some test snapshots of subjects with known linear dimensions were made in each experimental run. It was found that a significant error occurs in the direction of the stereo system axis, i.e. in determining the distance to the object. However, linear dimensions of objects at the typical distances from the camera considered in our measurements (∼10 m) are determined with the accuracy better than 1%. Since the systematic error in distance determination contributes only to the low wavenumber trend and does not affect the shortwave statistics, we will consider only the error of measurement of surface displacement relative to its average level, i.e. accept the calibration error ofδS_{calib} = 1%. The parallax error determines the accuracy of coordinate recovery. Benetazzo [2006] suggested the estimation of the maximum error for the 2D model of stereo rig geometry:
[21] Here X axis is parallel to the baseline, Z axis is perpendicular to the baseline and lies in camera optical axes plane, α is the angle between the line of sight of the camera, β is the half view angle of the camera, T is the baseline and N is the number of pixels of the 1D camera. The errors in the world coordinate system are calculated with the use of rotation matrix (er′) = (R)(er). According to these formula, the horizontal coordinate errors are 0.5 mm in baseline direction and 3 mm in perpendicular direction, and the elevation recovery error is 1 mm. On the other hand, we can also estimate the errors for the central pixel considering that it is of order of the spatial pixel size on the surface:
where β_{V} = 11° and β_{H} = 17° are the camera horizontal and vertical angles of view, γ = 30° is the grazing angle, N_{H} = 3888 and N_{V} = 2592 are the numbers of pixels in horizontal and vertical directions, respectively, and L = 10 m is the distance to the surface. We obtain er_{Xc} = 0.8 mm, er_{Yc} = 1.5 mm, er_{hc} = 0.6 mm.
[22] The correctness of the determination of corresponding points on a pair of images is more difficult to estimate. To minimize error in this process we apply several stages to filter out the “false” points in our algorithm [Kosnik and Dulov, 2011]. The difference between stereo reconstruction of sea surface and the classical problem of stereo reconstruction is in the fact that the sea surface does not possess wellselected objects. One can use only the spatial variations of sea surface brightness. This approach is quite suitable for retrieval wavelengths of 1 m or more but for smaller scales significant errors appear due to differences in brightness of images obtained from two different points of observation. To minimize these errors we apply the technique that was first described inKosnik and Dulov [2011]. Briefly, it consists in several stages of image processing before correlation analysis: a) the transform of one image of stereo pair to make it coincide at large scales with the second image; b) removal of low frequencies in Fourier space in order to keep only the small objects at the sea surface that form the image texture; c) application of different approaches to filter out the “false” points at several steps of the algorithm.
[23] Nevertheless the reconstructed surfaces can still contain sporadic unphysical peaks but their number and relative amplitudes are small and their contribution to the short wave statistics is negligible. The typical accuracy of corresponding pixel search can be estimated from correlation coefficient distribution. An example of correlation function is presented in Figure 2. In general the stereo matching algorithm can be significantly improved by minimizing the correlation function width up to the subpixel scales [Wanek and Wu, 2006; Benetazzo, 2006; Bechle and Wu, 2011]. In our case the maximum of correlation function is determined with the accuracy of 1–2 pixel while its halfwidth can be as large as 5–10 pixels. This means that the resulting errors are a few times larger than those predicted by the above formulas. Hence the maximum error on horizontal coordinates is of order of 1 cm and the error on elevation is about several millimeters.
[24] I2: Limited range of scales—Current experimental measurements were aimed at the study of short gravity and capillarygravity waves. A sufficient accuracy for this analysis imposed a limited observation area of about 3 × 4 meters with a pixel footprint size of approximately 1 mm. The window used for the correlation analysis in corresponding pixel search contains several tens of points and the spatial resolution after stereo processing has an order of 1 cm, excluding gaps. These low and highfrequency cutoff limit the accessible wave numbers to an intermediate range, say 5 rad m^{−1} < k < 200 rad m^{−1}. The field of elevations recovered from each stereo pair of images is relative to the mean level over the patch, so that larger waves are only seen through their tilting and hydrodynamical modulation effect but not through their absolute amplitude.
[25] I3: The problem of the mean plane—The mean tilted plane over each image is in principle imposed by the largest waves. However, a systematic bias in the estimation of the mean plane can be introduced by errors in the stereographic reconstruction procedure. Precisely, the orientation between the camera and world coordinate systems is given by successive geometric transformations (rotations and translations) that might be poorly estimated (see Benetazzo [2006] for a complete analysis of the sources of errors).
[26] I4: Gaps in reconstructed data—At the sea surface, smooth areas without notable markers (that is in absence of smallestscale texture) cannot be satisfactorily reconstructed. This results in gaps in reconstructed elevations (seeFigure 3, right). The presence of gaps prevents the application of standard regulargrid algorithms of further data processing. But this difficulty is inherent in smallscale stereo reconstruction [Benetazzo, 2006; Kosnik and Dulov, 2011]. The number of retrieved elevation points for each snapshot varies from 20 387 to 36 235 (in average 29 000) depending on gap locations. For the areas free of gaps, resulting space resolution of stereo reconstruction is of about 1 cm.
[27] I5: Irregular grid—According to the specificity of the stereoreconstruction methodology, the positions of homologous points are strongly associated with irregular ripples pattern and local elevations of sea surface. As a result, reconstructed elevationsz = η(x, y) are found on an irregular horizontal grid. A possible solution is a reinterpolation and projection on a regular grid. However, this implies implicit smoothing of the surface and also causes bias in the statistical estimation.
[28] I6: Gridding effect—The “gridding effect” is the bias introduced while estimating the finite differences on a twodimensional grid. Surface elevation points remain rather close to a regular twodimensional grid, even after the process of identification of homologous points on images of stereo pair. The estimation of the directional slope via two nearest neighbors on a grid is strongly biased by the varying distance between points, according to spatial relative disposition of points: on a diagonal or parallel to the elementary cell.Figure 4 displays an example of the available horizontal coordinates over a frame. It clearly evidences the lacunary as well as gridded nature of the distribution of points.
[29] The aforementioned issues must be addressed and corrected when performing a systematic statistical analysis of the data set.
9. Skewness Function
 Top of page
 Abstract
 1. Introduction
 2. Theoretical Background
 3. Experimental SetUp
 4. Major Technical Issues
 5. Distribution of Sea Surface Elevations
 6. Distribution of Sea Surface Slopes
 7. Distribution of Chords
 8. Calculation and Consistency of the Spectrum
 9. Skewness Function
 10. Conclusion
 Acknowledgments
 References
 Supporting Information
[58] In view of the multiscale and oscillating nature of the sea surface, the structure functions of arbitrary orders have also an oscillating behavior. It was observed in a recent experimental study [Caulliez and Guérin, 2012] that the period of these oscillations is of the order of the dominant wave. Hence positive parametrizations such as (18) and (19)are only expected to hold at small distances compared to the peak wave. In that case the current set of data makes it possible to check the validity of these parametrizations. In view of the limited range of available scales the structure function of the total elevation is not attainable beyond very small lags. However, we can compute the structure function associated to the smallscale process, sayη_{s}, obtained after detrending of the large scale, say η_{L}. The statistical characterization of the smallscale process is relevant in the context of the TwoScale or composite scattering model, which writes the radar cross section as the superposition of two terms. The first term is the contribution of large scales through a Geometrical Optics mechanism. The second term is the field scattered by small rippled on tilted facets. Limiting the skewness function to smallscales is further justified by the fact that the largescale skewness function is in fact not needed at small lags. This can be understood in a TwoScale picture where the sea surface is considered to be the superposition of one large and smallscale (centered) process,η = η_{L} + η_{s}, which we assume to be statistically independent although it is not hydrodynamically true. Assuming minimum scale ∼Lfor the largescale process we may approximateη_{L}(r) ≃ η_{L}(0) + ∇ η_{L}(0) · r in the range r ≪ L and a simple calculation yields the following approximation for the resulting skewness function in this range of lags:
[59] Now the skewness coefficients of slopes are known to be much smaller for long waves than for short wave (in view of the values derived by Cox and Munk [1954]in the “clean” and “slick” case). Since the mss is well distributed across the scales (so that the small and large scale mss are of the same order of magnitude), this means that and so the first term on the righthand side of(31)can be discarded. This shows that in the end only the smallscale skewness function is needed.
[60] To compute the skewness function associated to the smallscale process, the data have been detrended and interpolated on a regular grid with a third order polynomial. Again, to avoid “filling” the gaps in the reconstructed surface with artificially smooth patches, we limited the interpolation to points no further apart than 0.8 cm. The directional dependence of the skewness function is illustrated inFigure 11 for different lags in the three wind cases. At small lags a cos θ azimuthal dependence is clearly identified. This is consistent with the parametrization (18) in Mouche et al. [2007]. However, for large lags and small winds directional properties are different. This can be easily understood since the parametrization based on the limiting behavior of the skewness function at small lags is no longer relevant when the skewness function starts oscillating, that is for lags comparable to the dominant wave (see the discussion above). As to the dependence upon lag, our analysis shows that the relevant parametrization is (18)together with an exponential cutoff:
The validity of the constant factor can be checked by evaluating the renormalized quantity α = S_{3} / (r^{3} cos(θ)) in the limit r 0. As discussed in section 5 it is not possible to determine accurately the structure function at short lags because of the limited resolution. However, the value of α can be extrapolated from larger lags. Figure 12 shows the renormalized skewness function S_{3}(r) / r^{3}in semilog scale for a given fixed direction (θ = π, corresponding to the downwind direction). The linear fit starting at r∼ 15 cm confirms the exponential form of the cutoff. At distance shorter than about 15 cm, the renormalized functionS_{3}(r)/r^{3}exhibits an unphysical blowup due to the lack of accuracy in the determination ofS_{3}. The same was observed for the kurtosis function in section 5. The constant coefficient of the fitting lines coincides with the coefficient αwhile their slopes provide the cutoff distancer_{0}. The obtained numerical values are reported in Table 6. The coefficient α is compared and found consistent with the theoretical factor according to the experimental value of Cox and Munk. The cutoff distancer_{0} = 15 − 20 cm is found quite independent of wind speed.
Table 6. Amplitude Factor in the Skewness Function According to Cox and Munk Experimental Values (Second Column) and Derived From Stereo Data (Third Column)^{a}Wind Speed (ms^{−1})  10^{4} × α, CM  10^{4} × α, Stereo Data  r_{0} (m), Stereo Data 


7  −0.2  −0.6  0.14 
10  −5.4  −6.5  0.12 
14  −13  −7.4  0.22 
[61] Figure 13 shows the overall quality of the presented parametric model (32) for the skewness function (shown here as a function of lag in downwind direction) and a comparison with earlier parametrizations (18) and (19) (just for the case 3, where experimental conditions are comparable). The agreement with the experimental data is excellent at lags in the range of 0.2 m < r < 0.4 m for all wind cases and acceptable at smaller lags, where the accurate determination of the skewness function is problematic. Skewness function at large lags, where r > 0.4 m, can be described by the model (32)only for the high wind conditions. As it was discussed earlier, this can be explained by inapplicability of selected model approach for the bigscale processes close to the dominant wavelength.
10. Conclusion
 Top of page
 Abstract
 1. Introduction
 2. Theoretical Background
 3. Experimental SetUp
 4. Major Technical Issues
 5. Distribution of Sea Surface Elevations
 6. Distribution of Sea Surface Slopes
 7. Distribution of Chords
 8. Calculation and Consistency of the Spectrum
 9. Skewness Function
 10. Conclusion
 Acknowledgments
 References
 Supporting Information
[62] In the present paper we have proposed an extended analysis of the statistical properties of experimental sea surface elevations obtained from stereo imaging technique. This analysis has shown that reconstruction of the topography based on stereo method is an efficient way to derive nontrivial statistical properties of surface short and intermediatewaves (say from 1 cm to 1 m).
[63] Most technical issues, typical for stereo data sets, such as the limited range of scales, the lacunarity of data or the irregular sampling can be partially overcome by appropriate processing of the available points. The proposed technique also allows one to avoid linear interpolation that dramatically corrupts properties of retrieved surfaces. The processing technique imposes that the field of elevation is polynomially detrended, which has the effect of filtering out the large scales. Hence the statistical analysis can only address the smallscale components of the sea surface. The precise cutoff wavelength, which we know is approximatively half the patch size, can be obtained by applying a highpass frequency filter on the reference gauge time records. The results obtained for the detrended images have been shown to be consistent, at least in order of magnitude, with the corresponding gauge measurements as well as other experimental measurements available in the literature. The calculation of the structure functions provides a powerful tool to investigate spectral and statistical properties of the field of elevations. Experimental estimation of the thirdorder structure function, namely the skewness function, is one of the most important and original results of this paper. This function was up to now unavailable in field conditions and its knowledge was limited to theoretical considerations. We have derived an experimental parametrization that confirms a cosθdirectional behavior and an exponential cutoff form (especially for pure windwave seastate).
[64] Due to the lack of precise reference measurements for the smallscale wave field, we could not quantify exactly the accuracy of the retrieval technique. However, it appeared clearly that the obtained accuracy is good enough for the estimation of secondorder statistical quantities, acceptable for thirdorder quantities and insufficient for fourthorder quantities. Therefore, the stereo technique in the present stage should not be thought as a selfcontained universal tool to characterize the surface statistics. Instead, it should be used in conjunction with other well calibrated but sparse reference measurement (such as wave gauges) for crossvalidation and calibration. It then completes the statistical analysis in as much as it provides a snapshot of the threedimensional field and allows for the evaluation of higherorder spatial statistics.
[65] The proposed methodology is a first step before a systematic exploitation of more complete data sets. In particular forthcoming studies should be aimed at the analysis and parametrization of the directional structure functions of arbitrary orders that play an important role in the analytical scattering models for ocean remote sensing.