3.1. Selection of the Optimal Section of the Snell’s Window Image
The trial registration of the Snell’s window images was used to assess the surface waves characteristics. If the greater surface area was captured, then the chance for full and accurate estimate of surface roughness parameters was higher. However, in the case of swell waves or developed wind waves with the selected direction, the estimate of slope variance or roughness frequency spectrum could be obtained by section of the Snell’s window in the general direction of the surface waves. This section can be used to recover the optical properties of water according to [
6,
8,
9]; however, it is a suboptimal choice.
Figure 5 demonstrates the selection of the optimal section of the Snell’s window on the example of its image obtained at the depth of 1 m with Nikon D5100 fisheye-Nikkor lens 10.5 mm f/2.8 G ED DX in the waters of the Black Sea on a clear day with small sea surface roughness. The direction of the waves propagation and Sun position were set by azimuth angles
and
. The optimal section
will be the section that is oriented perpendicular to the direction of wave propagation, lying within angular region of 90° to 270°, relative to the direction to the Sun. Implementation of the first condition allows to minimize the distortion of the Snell’s window border due to the surface waves, and the second allows to avoid light exposure and sun glint.
In practice, it seems that the existence of still milder conditions is possible, namely, in the analysis of the section of the Snell’s window in a certain range of angles
near direction
(dotted circles in
Figure 5). Interval
is determined by double mean angle
of distortion of the Snell’s window border under surface waves and light scattering in water. According to estimates from [
3] maximum of
is about 8° for winds of 10 m/s. If we assume that light scattering in water has the same impact on the distortion of the Snell’s window, the total distortions
by waves and light scattering will be about 30°. This means that optical receivers with a viewing angle of 30° or more can be used for recording of the Snell’s window images. The simplest device, such as a webcam, satisfies this condition; therefore, we used one in our full-scale experiments.
In addition to the choice of the optimal cross section of the Snell’s window, there was a question regarding the number of sections needed to restore water optical properties or sea waves parameters. From practical experience, reliable results of water optical properties restoration were obtained by processing of at least one minute videos at a frame rate of 15 frames per second. This is processing of 1000 frames. Similar number of sections can be obtained from a single frame by using radial sections in the sector of azimuth angles
marked by shaded area in
Figure 5. The exact number of sections was set by the resolution of the optical receiver: starting from 480 pixels for the webcam up to 6000 pixels for the SLR camera. The following results, based on the analysis of video recordings of several expeditions, assess the real possibilities of webcam implementation as comprehensible optical receiver.
3.3. Determination of Absorption Coefficient
Firstly, let us consider possibilities of a single frame for each depth and color filter. We will assume that for each frame, all preparatory actions from the previous section have been completed. This means that we have a set of sections in an amount equal to the transverse resolution of the photo, i.e., 720 sections. Next, summing the sections together, we will form the accumulated section. Substituting radiance value at point
for each depth into (10) allows to calculate an array of the absorption coefficient for each color filter and the water layer thickness
, where
(
Table 3). This table shows that the choice of step
defines the number of values of the absorption coefficient, which can be obtained from a single pass in depth. Using such values for each section and each depth, we can estimate the statistical characteristics of the absorption coefficient depending on the specified parameters. These possibilities are demonstrated in
Figure 7 on the basis of randomly selected frames from each video recording. Here, the dashed curve marks, the average absorption coefficient and error bar define standard deviations.
Figure 7 shows that the average values of the recovered absorption coefficient can vary more than twofold for optical depths smaller and larger than 1 (in case of the Black Sea it is about 3 m). It is related to the fact that, on the one hand, at small depths a small surface area is visible, and its roughness, obviously, cannot be considered a stationary and homogeneous Gaussian random process. On the other hand, videos for each depth were recorded sequentially. This led to the fact that surface waves were different. The high spread of values of the recovered absorption coefficient contributes to the “random” sea roughness of each frame. With the increase of depth visible surface area is greatly expanded, encompassing many waves and thereby forming similar roughness conditions for different video recordings. It leads to obtaining the regular average values of the absorption coefficient with root-mean-square error (RMSE) equal to 25–50%, which are in good agreement with the estimations according to the concurrent measurements data (
Table 4), thus confirming the possibility of the solution of the inverse problem by one frame under the stated conditions.
It should be noted that the results of concurrent measurements of the attenuation coefficient by sonde, presented in
Table 4, were converted into absorption and scattering coefficient on the basis of regression links [
7] between IOP at 532 nm, repeatedly tested for the waters of the Black Sea:
Absorption coefficients for two remaining waves 460 nm and 650 nm are taken from the Atlas of the Hydrooptical Characteristics of the Black Sea [
16], based on the results of numerous expeditions. Overestimates of the absorption coefficient at 650 nm, obtained from the Snell’s window images, are likely to be related to the fact that the color filter has a wide pass band in the red spectrum, where pure water absorption increases significantly.
Having understood the possibilities of using a single frame, let’s consider the accuracy of solving the inverse problem by an entire video that is a large array of frames. In this case we applied temporary accumulation by frames, each of which was processed according to the above mentioned algorithm. The recovery of the absorption coefficient is facilitated according to the slope of straight line drawn through all points of the Snell’s window borders radiance on a logarithmic scale as a function of depth (see Equation (7)).
Figure 8 presents the results obtained by processing 150 frames of each video with time recording is about 10 seconds. Regression equations are listed in captions where
indicates determination coefficient. The values of the absorption coefficient are presented in
Table 4. As we can see, using the short video gave strong results: the mean values are close to sonde data with coefficients
that are over 0.94. Nevertheless, it is necessary to carry out longer shootings in real-case scenarios as different foreign objects can get in the webcam’s field of view: fish, jellyfish or mud. It will make such images unsuitable for solving the inverse problems.
The repetition of the actions for the Snell’s window video obtained in the waters of the Gorky Reservoir made it possible to obtain appropriate estimates of the absorption coefficient. Its values are presented in
Table 5 in comparison with the results of related data.
The values in the laboratory results section of
Table 5 are obtained from the spectral absorption coefficient in 430–750 nm range (
Figure 9), as the result of the laboratory analysis of the water samples. Estimation of the absorption coefficient at 420 nm was obtained by extending the curve into the short-wave part retaining the slope (dotted line in
Figure 9). Table data shows that one frame usage gets a 15–20% overestimate, whereas 150 frame usage increases precision to 6–8%. The results of the Gorky Reservoir are demonstrably more accurate than those of the Black Sea, which is probably connected with less sea surface roughness due to calm weather.
3.4. Determination of the Slope Variance and the Scattering Coefficient
The task of the scattering coefficient recovery is more difficult because it is based on an analysis of the Snell’s border distortion, which is affected by two reasons simultaneously: sea surface roughness and light scattering. Still, these two effects can be separated with the help of the two algorithms proposed in
Section 2.3.
Algorithm 1 requires the calculation of the statistical moment
through the following moments:
where
is the optical depth by the scattering coefficient. Determination of the scattering characteristics is performed by fitting the theoretical value of this parameter to the experimental one.
Figure 10 shows the results of numerical simulation of parameter
as functions of the optical depth
for three slope variances
, which were calculated using the Cox and Munk equation [
17] for wind speeds of 0 m/s (
), 3 m/s (
) and 12 m/s (
) and luminance distribution for clear sky [
18]. It shows that curves corresponding to different wind-wave conditions are significantly different on the initial interval of the optical depth. As
increases, all curves converge, indicating that the dominant contribution to the borders distortion of the Snell’s window is determined by the light scattering in the water and creating the preconditions for the development of the algorithm for estimating the optical properties of water from greater depths. However, since we do not have data of field measurements for such depths, we will carry out verification of the proposed algorithms only in the initial interval of optical depths.
The distribution of points resulting from calculation of parameter
by frames in relation to theoretical curves
implies the proportionality parameter
fitting between actual depth
, known from field experiments, and optical depth
, used in the numerical simulations. The outcome of this problem based on 150 frames of the Snell’s window images of the Black Sea waters is represented in
Figure 10a, where the points color matches the color of the used color filter: 420 nm (blue points), 530 nm (green points) and 650 nm (red points). The change of the curves for one set of points is not an error, as one might consider. On the day of the first video recording in the green filter the wind speed has weakened from 4–6 m/s to 3 m/s (
). As a result, a later time point, corresponding to greater depths, “dropped” to 3 m/s wind speed curve, whereas the earlier points rose above it. Half an hour later the recording in the red filter was carried out. According to anemometer data, wind faded completely, while the swell was continuing to fade. For this reason, all initial points lay on 3 m/s (
) curve almost evenly, whereas the later points sunk beneath it. The recording in the blue filter was carried out a few hours later, when the swell visibly subsided. For this reason, all the points are close to curve
. In cases like those mentioned above, where the wind conditions are changing rapidly, it can be argued that without information about the wind or surface waves the distribution of the points in
Figure 10 may have a lot of variation, depending on the choice of parameter
, which could lead to overestimates or underestimates of the scattering coefficient.
Table 6 shows recovery results for our particular case. It is evident that results for 532 nm, obtained by the Snell’s window and sonde data, appear very close: 12% for one frame and 5% for 150 frames. The scattering coefficients values for other wavelengths (460 nm and 650 nm) are obtained by subtraction of the reference data on the absorption coefficient according to the atlas [
16] from the sonde measured attenuation coefficient. However, despite the absence of the reliably measured values for these channels, it is apparent that the obtained values differ by only 30%.
A different situation is observed when a single frame of the Snell’s window is used (
Figure 10b). As indicated in
Section 3.2, each frame captures random surface waves. When working at small optical depths the main component of sea roughness is its gravity-capillary part with significant slopes that, in the end, with the averaging over the cross sections of the frame lead to strong distortion of the Snell’s border. Frames for the depth of 1.5 m and three color filters in
Figure 6 can serve as an example of such a situation: in the red filter the border of the Snell’s window was intact, and in the green filter the border of the Snell’s window was significantly distorted. Therefore, recovered values of parameter
are highly variable at small optical depths (see
Figure 10b). However, at greater depths, the visible surface area is greatly expanded, statistical spread is smoothed, variance of parameter
is reduced and the position of experimental points on the plane of
Figure 10b becomes single-valued and close to the results of 150 frames of the Snell’s window (see
Figure 10a).
Similar actions were performed for the data of the Gorky Reservoir and the recovery of the optical properties of water was facilitated by smoothness of water surface. These results are presented in
Table 7. Unfortunately, the correlation between IOP for waters of the Gorky reservoir, similar to [
7], has not yet been developed. Therefore, for comparing the results obtained, we used only data from the turbidity meter at 532 nm. As can be seen, the results obtained by different methods were similar.
The other similar approach to estimating the scattering coefficient (Algorithm 2) was to calculate the contrast of variations in the apparent radiance of the Snell’s window for certain depth in a given angle range
:
Its estimation for
and
leads to the results, similar to those of the previous algorithm (
Figure 11). However, Algorithm 2 is mathematically simpler, and therefore potentially contains less computational errors because of it does not require the computation of auxiliary functions based on the Snell’s window images. Nevertheless, when solving the inverse problem, it is useful to use both algorithms as complementary or as a test of each other.