Theoretical statistical relationships between the intensities of an image of the sea surface and its slopes: a result comparison of rect and Gaussian glitter functions

Abstract. The reflection of the sunlight over the sea surface is called glitter pattern. In previous works where the one-dimensional case was analyzed, the glitter function was mathematically described by a rect function. This rect function has proven to be a very good representation of the glitter pattern. A Gaussian glitter function is used like a first approximation to the rect function. The statistical relationship between the variance and the correlation function of the intensities of the image, the glitter pattern, and the variance of the sea surface slopes are obtained and analyzed. The analytical solutions for these relationships are given by different equations; however, the graphic representations are very similar.


Introduction
To measure the wave motion, the use of radar images and optical processing of aerial photographs has been used. In the last century, Barber 1 showed that the periodicity and directionality of waves can be estimated from the optical diffraction patterns of an image of the sea surface. Cox and Munk 2-4 studied the distribution of intensity or glitter patterns in aerial photographs of the sea. One of their conclusions was that for constant and moderate wind speed, the probability density function of the slopes is Gaussian as a first approximation. This could be taken as an indication that in certain circumstances the ocean surface could be modeled as a Gaussian random process.
Álvarez-Borrego 5-7 derived the equations which describe the glitter pattern in one and two dimensions. With the glitter function, we can find the variance and the correlation function of the sea surface slopes from the statistics of the intensities in the image. Álvarez-Borrego 8 considered the problem of retrieving spatial information of the statistical properties of random rough sea surfaces from images via remote sensing. He obtained expressions relating the variance and the correlation function in the image and the surface heights. He considered the detector located at an arbitrary height above the sea level.
Zhang and Wang 9 carried out comparison studies with measurements from moderate resolution imaging spectroradiometer for several popular sun glitter models.
Baxter 10 studied the ocean wave slope and height retrieval using imagery collected from a polarimetric camera system mounted on an airborne platforms, enabling measurements over large areas and in regions devoid of wave buoys developing a technique to calculate significant wave height. Kay et al. 11 demonstrated that it is feasible to generate high-resolution sea surface models and use them in radiative transfer modeling of the ocean surface. Incorporating correct statistics of surface elevation as well as slope gave estimates of reflected radiance.
Recently, an improved model to obtain some statistical properties of the sea surface slopes via remote sensing using variable reflection angle has been considered. 12 The statistical parameters of the sea surface slopes were obtained from the statistics of the glitter pattern considering the detector located at any heights over the mean of the sea level.
In the present article, we derivate new relationships of the variance and the correlation function of the intensities in the image of the sea surface and sea surface slopes for different incidence angles by considering a Gaussian glitter function. We also compare these results with the obtained previously by Álvarez-Borrego and Martín-Atienza. 12 In other words, illumination changes in the glitter pattern can be modeled using a Gaussian glitter function and the comparison with previous result will give us an idea about the behavior of these theoretical relationships. In our work, we assume that all the sea waves are moving along a single direction (in the open ocean the wind waves can be assumed to be mono-directional, however, this is not true for shallow environments because of the varied topography 13 ) and vertical plane containing this direction includes both the observer and the sun. 14 However, for any angle of reflection of the sunlight, the glint points on one transect taken from a two-dimensional glitter image does not contain the same number of glitter as a one-dimensional (1-D) profile because of the x and y components. The material of this work is organized as follows: in Sec. 2, the statistical properties of sea surface slopes are described; in Sec. 2.1, the statistical properties using a rect glitter function are obtained; in Secs. 2.1.1 and 2.1.2, the relationships among the variances and the correlation function of the intensities in the image and sea surface slopes considering a Gaussian probability density function are calculated, respectively; in Sec. 2.2, the statistical properties using a Gaussian glitter function like the mean, variance, and the correlation function of the image are obtained; in Sec. 3, the results and discussions are presented and in Sec. 4, the conclusions are given.

Statistical Properties of Sea Surface Slopes
This section describes the geometric model used and the different theoretical relationships between the variances and the correlation functions using different glitter function.

1-D Geometric Model
We propose to use the same 1-D geometry as in Álvarez-Borrego and Martin-Atienza, 12 in which θ d is a variable angle subtended by the optical system of the detector with the normal at one point of the surface (Fig. 1). This model does not impose a restriction on the sensor field of view. In addition, we neglect the effect of shadowing, i.e., when some reflecting points are obscured by roughness of the sea surface.
The geometric model, considering θ d as a variable, is shown in Fig. 1. The surface ζðxÞ is illuminated by a uniform incoherent source S of limited angular extent, with wavelengthλ and an apparent diameter β. Its image is formed in D by an aberration free optical system. The incidence angle θ s is defined as the angle between the incidence direction and the normal to the mean surface. From Fig. 1, we can observe that, ðθ d Þ i corresponds to the angle subtended by the optical system of the detector with the normal to the point i of the surface, and is given by where H are the height of the detector, and Δx is the interval between surface points. Here, α i is the slope angle at each i point in the surface and is subtended between the normal to the mean surface and the normal to the surface in the i point, i.e., In this realistic physical situation, the angle ðθ d Þ i is changing with respect to each point in the surface. It is worth noticing that by using a variable ðθ d Þ i the sensor field of view is not restricted.
According to Álvarez-Borrego and Martín-Atienza, 12 the glitter function can be expressed as By combining Eqs. (2) and (4)- (6), we obtained which is the slope angle interval, where a bright spot is received by the sensor.

2.1.1
Relationship between the variance of the intensities in the image and surface slopes considering the Gaussian probability density function for the slopes and using the rect glitter function The mean of the image μ I may be written as 12 where pðM i Þ is the Gaussian probability density function in one dimension.
Þβ∕4 the resulting expression is The variance of the intensities in the image is defined by Álvarez-Borrego and Martín-Atienza: 12 Fig. 1 Geometry of the model to get some statistical properties of marine surface.
which is the required relation between the variance of the intensities in the image, σ 2 I , and the variance of the surface slopes, σ 2 M .

2.1.2
Relationship between the correlation function of the intensities in the image and surface slopes considering the Gaussian probability density function for the slopes and the rect glitter function The relationship between the correlation function of the surface slopes, C M ðτÞ, and the correlation function of the glitter pattern, C I ðτÞ, is given by where τ is the lag and BðM 1i Þ and BðM 2j Þ are the glitter function in two different slopes, and and after an analytical integration we obtain and Equation (13) is the required relation between the correlation function of the intensities in the image, C I ðτÞ, and the correlation function of the surface slopes, C M ðτÞ.

Relationship between the Variance and the Correlation Functions of the Intensities in the Image with Sea Surface Slopes Considering the Gaussian Probability Density Function and Using the Gaussian Glitter Function
In this article, we define the Gaussian glitter function as and considering Eqs. (4)- (7), and the same definition for the mean and the variance of the intensities of the image, we find and Þβ∕8 gives information of the width of the Gaussian glitter function.
In addition, the relationship between the correlation function of the surface slopes, C M ðτÞ, and the correlation function of the glitter pattern, C I ðτÞ, is given by where and Therefore, the result of the first integral is Although we obtain an analytical relationship for the first integral, for the second integral, the process must be numeric. In order to avoid computer memory problems, the 16,000 data-point profile can be divided into 16 consecutive intervals. The value of ðθ d Þ i varies point to point in the profile. For each interval and for each θ s value, the relationship between the correlation functions C I ðτÞ and C M ðτÞ is calculated. Then, to compare the results using the rect and the Gaussian glitter function we used a value of σ M ¼ 0.2121. It is necessary to normalize the correlation functions, by using the corresponding variances. A theoretical variance σ 2 I can be calculated from Eqs. (10) and (16) (for both rect and Gaussian glitter function cases), respectively. It is also possible to estimate the variance by averaging the theoretical variances calculated for each of the 16 intervals. In the rect glitter function case, these results can be seen in Álvarez-Borrego and Martín-Atienza. 12 We compare both variances in Table 1 for the Gaussian glitter function case for different values of H.

Results and Discussions
The next figures show the comparison of Eqs. (10) and (13), where the rect function and the Gaussian function are used like glitter functions.
In Fig. 2, the variance of the surface slope is represented against the variance of the intensities of the image. Values of σ 2 M from 0 to 0.04 correspond linearly to a wind velocity in the range from 0 to 12 to 14 m∕s, according to Cox and Munk. 15 The shape of the curves for the rect and the Gaussian cases is very similar. Their differences correspond to the changes in the values of the variance of the intensities. The profile of the source in the Gaussian case is less intense than for the rect glitter function. This figure shows how these relationships change for different heights (100, 500, 1000, and 5000 m). It can be seen that when H increases, the line corresponding to a value of incidence angle of 50 deg descends and even cross with the others lines.
When H increases, the lines with larger θ s go down until the arrangement of the curves changes. If the camera is fixed at H ¼ 100 m, it will receive more reflected light at large θ s because of the geometry of the reflection. When H increases, the camera sensor will receive less reflected light for large incidence angles.
In Fig. 3, the shape of the curves is also very similar for the rect and Gaussian cases (excepting for H ¼ 100 m). This figure shows these normalized relationships using the two glitter functions) for 100, 500, 1000, and 5000 m. The correlation functions have a behavior similar to the variance curves when the sensor height changes.

Conclusions
The behavior of the curves for different H is the same, notwithstanding the use of the rect or the Gaussian glitter function. Both glitter functions are easy to use in the calculations of these relationships. The differences in the values of the variance of the intensities of the image in both cases are proportional to the profile of the source used. Both glitter functions represent well the physics of the problem, which can act as a function to modulate the intensity of luminescence on brightness pattern, which is consistent with different  actual physical situation such as the presence of clouds or aerosols between the sensor and sensing area of sea surface.