Journal of Marine Science and Engineering

Article The Snell’s Window Image for Remote Sensing of the UpperArticle Sea Layer: Results of Practical Application The Snell’s Window Image for Remote Sensing of the AlexanderUpper A. Sea Molkov Layer * and Lev: Results S. Dolin of Practical Application Institute of Applied Physics of the Russian Academy of Sciences, 46 Uljanova St., 603950 Nizhny Novgorod, AlexanderRussia; [email protected] A. Molkov * and Lev S. Dolin *InstituteCorrespondence: of Applied [email protected]; Physics of the Russian Tel.: Academy +7-831-416-4859 of Sciences, 46 Uljanova St., 603950 Nizhny Novgorod, Russia; [email protected]



 Received:* Correspondence 28 February: a.molkov 2019; Accepted:@inbox.ru 15; Tel.: March +7-831 2019;-416 Published:-4859 19 March 2019

Received: 28 February 2019; Accepted: 15 March 2019; Published: 19 March 2019 Abstract: Estimation of water optical properties can be performed by photo or video registration of rough sea surface from underwater at an angle of total internal reflection in the away from the Abstract: Estimation of water optical properties can be performed by photo or video registration of sun direction at several depths. In this case, the key characteristic of the obtained image will be rough sea surface from underwater at an angle of total internal reflection in the away from the sun the border of the Snell’s window, which is a randomly distorted image of the sky. Its distortion direction at several depths. In this case, the key characteristic of the obtained image will be the changes simultaneously under the action of the sea roughness and light scattering; however, after border of the Snell’s window, which is a randomly distorted image of the sky. Its distortion correctchanges “decoding” simultaneously of this under image, the their action separate of the sea determination roughness and is possible. light scattering This paper; however, presents after the correspondingcorrect “decoding” algorithms of this image for achieving, their separate thesepossibilities determination by is the possible. Snell’s This window paper images. presents These the imagescorresponding were obtained algorithms in waters for achiev with differenting these optical possibilities properties by the and Snell wave’s conditionswindow images under. severalThese typesimages of were illumination. obtained Practical in waters guidelines with different for recording, optical properties processing and and wave analyzing conditions images under of the Snell’sseveral window types of are illumination also formulated.. Practical guidelines for recording, processing and analyzing images of the Snell’s window are also formulated. Keywords: underwater vision; Snell’s window image; inherent optical properties; slope variance; remoteKeywords: sensing underwater vision; Snell’s window image; inherent optical properties; slope variance; remote sensing

1.1. Introduction Introduction TheThe underwater underwater skysky image,image, oror thethe Snell’sSnell’s window, is a contrasting object object frequently frequently observed observed in in photosphotos of of professional professionaland and amateuramateur diversdivers alikealike (Figure1 1).). However,However, thethe useuse ofof such images to to solve solve scientificscientific problems problems has has not not beenbeen noticednoticed untiluntil recently.recently.

(a) (b) (c)

FigureFigure 1.1. TheThe Snell’sSnell’s windowwindow image obtained from oceanographic oceanographic platform platform in in the the Black Black Sea Sea (a ()a )for for a a flatflat seasea surfacesurface fromfrom depthdepth of 0.5 m and for a rough sea sea s surfaceurface from from depth depthss of of (b (b) )1.5 1.5 m m and and (c ()c )5 5m m (attenuation(attenuation coefficientcoefficient waswas aboutabout 0.60.6 mm−11).).

ForFor aa flatflat seasea surface,surface, the Snell’sSnell’s window looks like like a a light light circle circle with with an an angular angular radius radius equal equal to to thethe angleangle ofof totaltotal internalinternal reflectionreflection of 48.7548.75°◦ (Figure 11a).a). An apparent radiance of of the the sea sea surface surface at at anglesangles moremore than than 48.75 48.75°◦ is is determined determined through through the the upwelling upwelling light light reflected reflected by by water water boundary boundary to to the lowerthe lower hemisphere. hemisphere. In fact, In the fact upwelling, the upwelling radianceis radiance too low inis comparison too low in with comparison the downwelling with the one; downwelling one; therefore, the apparent radiance of the surface beyond Snell’s window appears to be comparatively low. Surface waves randomly distort the border of the Snell’s window resulting in J. Mar. Sci. Eng. 2019, 7, 70; doi:10.3390/jmse7030070 www.mdpi.com/journal/jmse

J. Mar. Sci. Eng. 2019, 7, 70; doi:10.3390/jmse7030070 www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2019, 7, 70 2 of 18 therefore, the apparent radiance of the surface beyond Snell’s window appears to be comparatively low. Surface waves randomly distort the border of the Snell’s window resulting in the appearance of lots of dark and light spots near it (Figure1b). Thus, contours of light and dark spots around the Snell’s window border outline the surface patches where the slope exceeds the certain value depending on spot location. This opens up possibilities for retrieval of the slope distribution function or its parameters like slope variance by distortions of the Snell’s window border. The effects of multiple scattering and absorption of light in water also affect the distortion of the Snell’s window border, getting stronger with depth increase (Figure1b,c). The first images of the Snell’s window for estimation of sea roughness parameters were obtained in the Black Sea in 2010 during the photo registration of the underwater solar path at the time of sunset. In this case, the sun solar path was near the border of the of Snell’s window. Its study allowed us to establish that the Snell’s window border is very sensitive to the sea surface roughness and can be used to determine its statistical properties [1]. The first algorithm for estimating the slope variance by value of its border distortion in case of slope less than 20◦ was proposed in [2]. Similar estimation for roughness with breaking waves based on the roughness representation as a sine curve with known characteristics was obtained in [3]. Subsequently, on the basis of a simplified model of the Snell’s window [4], a new algorithm for reconstructing the wind-waves frequency spectra was developed [5]. On the one hand, it was assumed that both algorithms were applicable at optical depths less than one, where the effects of light scattering and absorption can be neglected. On the other hand, it created the background for determining the optical properties of water at greater depths. As a result, in [6], the outcome of the theoretical study in estimating the absorption and scattering coefficient of water through the Snell’s window distortion was presented under the assumption that there was a known parameter, characterizing sea roughness like slope variance. Results of the first testing of the proposed algorithms for images from the Black Sea and empirical regressions between the inherent optical properties (IOP) at 532 nm [7] are presented in [8]. Subsequently, the possibility of determining the absorption coefficient for three spectral channels (420 nm, 530 nm and 650 nm) was demonstrated for fresh water of the Gorky Reservoir, characterized by high concentrations of chlorophyll a and dissolved organic matter [9]. The scope of the present investigation was aimed at estimating both the characteristics of the sea roughness and the optical properties of water for both its types (Case 1 and Case 2), for different illumination and sea roughness conditions and without applying empirical regressions between IOP. In addition, practical recommendations of the Snell’s window image registration, processing and analysis will be made in this work. Additionally, we would like to note that the emphasis on our own research in this article is connected with the fact that there have been no similar results from other research teams. The analysis of modern studies on using Snell’s window image for estimating characteristics of the upper water layer confirms the assertion made. At the present time, a great number of methods, approaches and techniques for studying the sea surface parameters and the optical properties of water have been developed; however, we were interested in the possibility of specifically using underwater images for this purpose.

2. Materials and Methods The work was based on the theoretical model of the accumulated Snell’s window image [8,10], the proposed algorithms for solving inverse problem and the data of field measurements from several expeditions of 2018. For a better understanding of the paper we will briefly consider the theoretical model of the instantaneous image of the Snell’s window, explain the physical principles of its formation and provide a table of abbreviations and symbols used in this paper (Table1). J. Mar. Sci. Eng. 2019, 7, 70 3 of 18

Table 1. Abbreviations and Symbols.

Symbols Meaning

ϑi Angle of incident light ϑ Angle of refracted light Angle of observation of the Snell’s window border for a flat sea surface ϑ Sn (angle of total internal reflection) ϑ− Random angle less than ϑSn ϑ+ Random angle more than ϑSn ϑ Mean angle of distortion of the Snell’s window border Z Depth 4Z Thickness of the water layer ZSecchi Secchi depth x Spatial coordinate of the water surface element t Time ζ Surface elevation η Surface slope 2 σx Slope variance P Surface slope distribution function m Refractive index of water a Absorption coefficient of water b Scattering coefficient of water c Attenuation coefficient of water cλ Attenuation coefficient of water at wavelength λ λ Wavelength τb Optical depth by scattering coefficient γ Angle of light scattering x Phase function dx Variance of a single scattering angle xs Spectrum of the phase function α Sinus of a local incident angle RF Fresnel reflection coefficient for non-polarized light 1 − RF Transmission coefficient Isky Sky luminance distribution Io Radiance of instantaneous Snell’s window image formed by nonscattered light L0 Radiance of accumulated Snell’s window image formed by nonscattered light Radiance of accumulated Snell’s window image obtained with taking into L account multiple scattering and absorption in water A Function describing influence of absorption on the Snell’s window image Radiance of accumulated Snell’s window image in hypothetical water medium [L] a=0 without absorption d Parameter of Snell’s window image describing distortion of its border K Contrast of the Snell’s window image IOP Inherent Optical Properties

2.1. Analytical Model of the Instantaneous Snell’s Window Image Formed by Nonscaterred Light When we constructed the instantaneous Snell’s window image model (see details in [2]), we assumed that it is formed using an ideal optical system placed at depth Z that records the angular radiance distribution of the sea surface (Figure2a). The equations for calculating this radiance were derived in the geometrical optics approximation, mathematical instrument of the radiation transfer theory for turbid water and the boundary conditions for the intensity of light that transfers between mediums with different refraction indexes (air-water). At the same time, we took into account the effects of multiple instances of light scattering and absorption; however, effects of light backscattering were neglected. Optical properties of the water are characterized by absorption coefficient a, scattering coefficient b and phase function x(γ), where γ is a scattering angle. Elevations of sea surface ζ and its q slopes η = −∇ζ/ 1 + (∇ζ)2 were assumed to be stationary and homogeneous Gaussian random 2
fields, which allowed us to use corresponding surface slope distribution function P η, σx with a single J. Mar. Sci. Eng. 2019, 7, 70 4 of 18

theory for turbid water and the boundary conditions for the intensity of light that transfers between mediums with different refraction indexes (air-water). At the same time, we took into account the effects of multiple instances of light scattering and absorption; however, effects of light backscattering were neglected. Optical properties of the water are characterized by absorption coefficient a , scattering coefficient b and phase function x( ) , where  is a scattering angle.

2 J.Elevations Mar. Sci. Eng. of2019 sea, 7surface, 70  and its slopes = − /1 +(   ) were assumed to be stationary4 ofand 18 homogeneous Gaussian random fields, which allowed us to use corresponding surface slope 2 2 distribution function P (, x2) with a single parameter—the slope variance  x (for parameter—the slope variance σx (for one-dimensional case). The influence of surface elevations ζ on theone structure-dimensional of the case). light The field influence at depth ofZ surfaceζ was elevations considered negligible. on the structure Surface of slopes the light were field small at anddepth there Z was nowas shadowing. considered negligible. Surface slopes were small and there was no shadowing.

(a) (b)

FigureFigure 2.2. GraphicalGraphical explanationexplanation ofof ((aa)) thethe Snell’sSnell’s windowwindow forfor aa flatflat seasea surfacesurface andand ((bb)) occurrenceoccurrence ofof spotspot structurestructure nearnear itsits borderborder duedue toto surfacesurface waves.waves.

NeglectingNeglecting the the effects effects of of light light scattering scattering in in water, water, instantaneousinstantaneous radianceradiance ofof the the sea sea surface surface element,element, observableobservable atat angleangle ϑfrom from depth depthZ atZ time at time moment momentt, is determinedt , is determined by the by equation: the equation:

I(, Z , t) =−2 m2 1 R (  , t) I (  , t)−cZ e−/cZ cos /cosϑ , Io(ϑ,oZ, t) = m [1 −RF( Fα( ϑ, t))]I)sky( skyϑi((ϑ i, t))e ) , (1)(1)

where m is the refraction index, c is the attenuation coefficient, Isky is the sky luminance where m is the refraction index, c is the attenuation coefficient, Isky is the sky luminance distribution, ϑdistributioni is the local, incidenti is the angle, local incident angle,

 2 222  q q22 2  q  2− 2q 2 2  11−α(1ϑ−, t)2(−  ,t) m −2 − mα( −ϑ, t)(2  , t) 1 1− −α (ϑ( , t ,) t2)− −m m−2 mm2 −− α(( ϑ ,, tt))2 Rt1 , =+     RF(α(ϑ, t))F (= ( ))  +   (2)(2)  q 2 q22 2 q  2− 2q 2 2   2 −(1−)2( + ,t) +2 − m( − )(2  , t) 1− − ((  ,) t2)+ + m−2 m2 −− ((  , t))2 1 α ϑ, t m α ϑ, t  1 α ϑ, t m m α ϑ, t  isis thethe FresnelFresnel reflectionreflection coefficientcoefficient forfor aa non-polarizednon-polarized lightlight and and

 ,t= sin   , t = m sin  − cos    x  , t α(ϑ, t)( = )sin ϑi(ϑi,(t) =) m(sin( ϑ − cos ϑ · η((x((ϑ)), t)))) (3)(3)

is the sinus of the local incident angle which is a function of observation angle  and surface is the sinus of the local incident angle ϑi which is a function of observation angle ϑ and surface slope η atslope point x at= pointZ tan ϑxZand= tan time momentand timet. moment t . AccordingAccording toto EquationsEquations (1)–(3)(1)–(3) thethe structurestructure ofof thethe Snell’sSnell’s windowwindow imageimage forfor certaincertain reliefrelief ofof thethe seasea surfacesurface isis determineddetermined byby severalseveral factors:factors: thethe naturenature ofof itsits lighting;lighting; radianceradiance attenuationattenuation ofof lightlight penetratingpenetrating intointo thethe waterwater duedue toto itsits reflectionreflection fromfrom seasea surface;surface; changechange ofof lightlight directiondirection whenwhen itit isis penetratingpenetrating throughthrough thethe surface;surface; andand lightlight attenuationattenuation onon thethe wayway fromfrom thethe surfacesurface toto thethe receiver. receiver. InIn the the absence absence of of wind wind roughness, roughness, when when η== 00, a horizontally incident light accordingaccording to to EquationEquation (3)(3) isis refractedrefracted atat anan angle:angle:

◦ ϑ = ϑSn = arcsin(1/m) ≈ 48.75 (4) and the surface radiance in this direction is close to zero according to Equation (5). In a two-dimensional case, we obtain the cone of such rays with angular radius ϑSn. The “intersection” of this cone with a flat sea surface forms the circle border which is a border of the Snell’s window. At the initial depth interval, the apparent radiance of the surface inside the border is determined by the sky luminance distribution Isky and the transmittance dependence 1 − RF. Outside the Snell’s window, the image was J. Mar. Sci. Eng. 2019, 7, 70 5 of 18 formed by backscattering from the water column and light refracted downward by the sea surface, the radiance of which is relatively small. Therefore, the Snell’s window looks like a light circle on a dark background (see Figure1). Surface waves randomly distort the Snell’s window: dark spots appear inside of it, light spots—outside. The mechanism of their appearance is explained in Figure2b. Angles ϑSn, ϑ− and ϑ+ show the directions to the undisturbed Snell’s window border, and to dark and bright spots, respectively. As shown in Figure2b, bright spots outside the undisturbed Snell’s window appear in places of the surface with a positive slope (slope towards the horizon) through which the observer sees fragments of the sky. At the same time, dark spots inside the undisturbed Snell’s window appear on the surface elements with a negative slope, through which the sky is not visible. The number of spots, their sizes and occupied area depends on the intensity of waves. The last one can be estimated on the basis of the accumulated Snell’s window image L0, which is calculated through the instantaneous 2
radiance of the sea surface I and the surface slope distribution function P η, σx :

1 Z  2  2 L0 ϑ, Z, σx = I(ϑ, Z, η)P η, σx dη (5) −1

Obviously, this image depends on the sea surface state, on the basis of which the algorithm to restore the slope variance [2] was developed. In writing Equation (5), the time argument t of function I(ϑ, Z, t) is replaced by surface slope η in view of the fact that η = η(t).

2.2. Analytical Model of the Accumulated Snell’s Window Image Taking into Account Scattering Properties of Water The model of the accumulated Snell’s window image takes into account the effects of multiple scattering and absorption of light in water is quite voluminous and has already been presented in previous work [8,10]. Therefore, in this paper, we present only the general equations that are necessary for the description and implementation of algorithms for the restoration of water optical properties; moreover, we specify the conditions for their derivation and their applicability. According to this model, the radiance of the accumulated image of the Snell’s window L as a function of polar angle ϑ and depth Z can be calculated by the equation:

 2 h  2i L ϑ, Z, a, b, σx = A(ϑ, Z, a) L ϑ, Z, b, σx (6) a=0 where: A(ϑ, Z, a) = exp[−(aZ/ cos ϑ)] (7) is the first function which takes into account the influence of absorption on the Snell’s window image, and:

1 ∞ h i  2
 = 1 R (− 0 − ) 2
R − bZ ( − ( )) ( 0 ) 0 L ϑ, Z, b, σx a=0 π L0 arcsin cos ϑ n x sin ϑ , Z, σx exp cos ϑ 1 xs p cos n x p dp dn x (8) −1 0 is the second function that describes the structure of the image in hypothetical water medium without absorption. Here, xs(p) is the spectrum of phase function x(γ). Equation (8) is obtained by using the Green function in the small-angle diffusion approximation [11]. The proposed model is applicable when the following conditions are met: (1) Light field L should be horizontally uniform in water column with cross size ∆x > ∆x0 ∼ 5Z; (2) Depth Z should be such that the additional light beam after passing through the layer of thickness Z/ cos ϑs remains sufficiently narrow. As we can see from Equations (6)–(8), function L depends on three parameters: absorption and 2 scattering coefficients a and b, and slope variance σx . The first of which is included only in the J. Mar. Sci. Eng. 2019, 7, 70 6 of 18

The proposed model is applicable when the following conditions are met: (1) Light field L should be horizontally uniform in water column with cross

size x   x0 5 Z ; (2) Depth Z should be such that the additional light beam after passing through the layer of

thickness Z / coss remains sufficiently narrow.

J. Mar.As Sci. Eng.we 2019 can , see7, 70 from Equations (6)–(8), function L depends on three parameters: absor6ption of 18 2 and scattering coefficients a and b , and slope variance  x . The first of which is included only in 2 the multiplier A(,,) Z a and two others in multiplier L2,,, Z
 b x . Examples of calculating ( ) ( ) a=0 multiplier A ϑ, Z, a and two others in multiplier L ϑ, Z, b, σx a=0. Examples of calculating these functionsthese functions are shown are shown in Figure in3 ,Fig whereinure 3, whereinthe sky luminance the sky luminance distribution distribution was considered was considered uniform anduniform the water and the surface water was surface flat. Radiancewas flat. Radiance of the surface of the elements, surface elements observable, observable at angles at more angles than more the anglethan the of total angle internal of total reflection internal isreflection considered is considered equal to zero. equal Light to scatteringzero. Light in scattering water was in given water by was the well-knowngiven by the phase well-known function phase [11,12 function]: [11,12]:

−1   x( )=− 2p 2 / dxx − exp1 2 /p d  (9) x(γ) = 2 2/dx γ exp( − 2/)dx γ (9)

where dx is the variance of a single scattering angle. The phase function (9) has been used in a lot of where dx is the variance of a single scattering angle. The phase function (9) has been used in a lot of works on the theory of underwater vision [11]. With appropriate values of parameter it works on the theory of underwater vision [11]. With appropriate values of parameter dx it describes welldescribes the form well of real the seawater form of phase real functionseawater in phase the range function of scattering in the angles range 0.5 of0 < scatteringγ < 300 ÷ angles450. 0.50  30 0  45 0 .

(a) (b) (c)

Figure 3. The structure of functions (a) L ; (b) A ; (c) L from Equation (6) for different depths Figure 3. The structure of functions (a) L;(b) A;(c) [L]a=0 froma=0 Equation (6) for different depths and − − a 1 b −1 1 d −1 2 = 2 nextand parameters:next parameters= 0.06: a m= 0.06, m= 0.24, b m = 0.24, x m= 0.04;, dx σ =x 0.040.;  x = 0 .

Figure3a shows that with increasing depth apparent radiance of the Snell’s window decreases Figure 3a shows that with increasing depth apparent radiance of the Snell’s window decreases and its border blurs. At the same time, according to Figure3c, all curves intersect at angle ϑ = ϑ . and its border blurs. At the same time, according to Figure 3c, all curves intersect at angle = Sn . This means that light scattering in water does not affect the apparent radiance of the Snell’s windowSn This means that light scattering in water does not affect the apparent radiance of the Snell’s window border and, therefore, its radiance is attenuated with depth only due to absorption. In reality, a certain border and, therefore, its radiance is attenuated with depth only due to absorption. In reality, a dependence on the turbidity should be observed, at least due to the backscattering of light in water, certain dependence on the turbidity should be observed, at least due to the backscattering of light in which has not been taken into account. In addition, when deriving Equations (6)–(8), the distortion water, which has not been taken into account. In addition, when deriving Equations (6)–(8), the of the Snell’s window border was implicitly likened to the distortion of luminous half-plane border, distortion of the Snell’s window border was implicitly likened to the distortion of luminous which could also lead to some error in estimating the effect of scattering on the apparent radiance half-plane border, which could also lead to some error in estimating the effect of scattering on the of the Snell’s window border. However, under condition dxbZ << cos ϑSn, this effect should not apparent radiance of the Snell’s window border. However, under condition d bZ  cos , this be significant, which creates the prerequisites for constructing simple algorithmsx for the separate Sn determinationeffect should not of thebe significant, absorption andwhich scattering creates the coefficients prerequisites of water for constructing by radiance attenuationsimple algorithms of the Snell’sfor the window separate border determination and a parameter of the absorption characterizing and its scattering distortion. coefficients of water by radiance attenuation of the Snell’s window border and a parameter characterizing its distortion. 2.3. Algorithms for Solving Inverse Problems 2.3. Algorithms for Solving Inverse Problems If we assume that the attenuation of the apparent radiance of the Snell’s window border is completelyIf we determinedassume thatby the function attenuationA(ϑ, Z of, a the), then apparent the absorption radiance coefficient of the Snell can’s be window expressed border as: is completely determined by function A(,,) Z a , then the absorption coefficient can be expressed as: cos ϑ L(ϑ , Z , a, b, σ2) = Sn Sn 1 x a ln 2 (10) Z2 − Z1 L(ϑSn, Z2, a, b, σx ) through the radiances at two different depths Z1 and Z2 at angle ϑSn coinciding with the direction to the Snell’s window border. Subsequently, we will use the value:

4 Z = Z2 − Z1 (11) J. Mar. Sci. Eng. 2019, 7, 70 7 of 18

to determine thickness of the water layer between horizons Z1 and Z2. After that, with known 2
absorption coefficient and measured radiance distribution L ϑ, Z, a, b, σx , we can find the auxiliary function: 2
h  2i L ϑ, Z, a, b, σx L ϑ, Z, b, σx = (12) a=0 A(ϑ, Z, a) which depends on the scattering coefficient and slope variance. If the last parameter is known, the scattering coefficient of water can be recovered in two ways [8]: (1) By introducing the parameter characterizing the value of the Snell’s window distortion, and determining the scattering characteristics by fitting the theoretical value of this parameter to the experimental one (Algorithm 1); (2) By calculating the parameter characterizing contrast of variations in the apparent radiance of the Snell’s window in a given interval of angles near its border (Algorithm 2). Estimation of the efficiency of each proposed algorithm was carried out on the basis of experiment data, and its results are presented below.

2.4. Experiment In 2018, full-scale measurements were performed in the Black Sea and in the Gorky Reservoir. A Logitech C210 webcam with 30◦ field of view in the underwater box was used to record the Snell’s window images. On its front panel, three replaceable color filters were mounted: a red filter (band 560–2500 nm with no peak), a green filter (band 460–620 nm with peak 530 nm) and a blue filter (band 380–520 nm with peak 420 nm). Images resolution was 1280 × 720 pixels, recording frequency— 15 frames per second. The camera was remotely controlled in online mode. Replacement of the color filter was carried out physically after performing measurements at all depths with the chosen color filter.

2.4.1. The Black Sea Experiment The Snell’s window images were recorded in October of 2018 from the Oceanographic Platform of the Marine Hydrophysical Institute, which is based in 600 m off the seashore (Figure4a). The box with webcam was attached to a triangular rigid metal frame at angle ϑSn, which was lowered from the lower deck of the platform along the stretched cable. The first depth was 0.5 m and the last one 6.5 m; depth increment was 1.0 m. Adjustment and fixation of the camera on the azimuth angle was carried out by lateral stretch marks (Figure4b). Video recording was performed on each horizon for 3 minutes with subsequent transition to the next depth within 1 minute. Therefore, a full measurement cycle on seven horizons for one color filter took about 30 minutes. The first video recording in the green color filter was started at 11:34 AM, then after 50 minutes in red color filter and after a long break for 2 hours in blue color filter. During this time, according to the data from digital anemometer Windsonic, the wind speed decreased from 5–6 m/s at 10:00 AM to 3–5 m/s at 11:30 AM, and at 12:00 PM, the wind completely faded. At the same time, fading swells were observed on the sea surface; their amplitude was about 0.4 m at 11:00 AM and became barely noticeable at 15:00 PM. The swell characteristics were estimated visually by the marks on the supporting piles of the platform at the time of passage. These waves had a noticeable effect on camera oscillation, which was later eliminated during image processing. The sky was clear throughout the day. Simultaneously with the video recording, the vertical profiles of the attenuation coefficient were measured using submersible multichannel optical sonde [13] working at 460 nm, 532 nm and 625 nm; wind speed and direction were measured using the ultrasonic digital anemometer WindSonic and sky radiance distribution was measured using digital camera Nikon D5100 with 10.5 mm f/2.8 G ED DX Fisheye-Nikkor lens, which provided 180◦ field of view. J. Mar. Sci. Eng. 2019, 7, 70 8 of 18 throughout the day. Simultaneously with the video recording, the vertical profiles of the attenuation coefficient were measured using submersible multichannel optical sonde [13] working at 460 nm, 532 nm and 625 nm; wind speed and direction were measured using the ultrasonic digital anemometer WindSonic and sky radiance distribution was measured using digital camera Nikon D5100 with 10.5 mm f/2.8 G ED DX Fisheye-Nikkor lens, which provided 180° field of view.

2.4.2. Gorky Reservoir Experiment A similar work was performed in eutrophic fresh waters of the Gorky Reservoir in the late autumn of 2018 during residual cyanobacteria bloom. The box with webcam was fixed to an aluminum pipe at angle Sn that was lowered from the deck of the anchored vessel (Figure 4c). The first depth was 0.5 m and the last one 2.0 m; depth increment was 0.25 m. Video recording of 60 second was 0.5 m and the last one 2.0 m; depth increment was 0.25 m. Video recording of 60 second J.was Mar. performed Sci. Eng. 2019 on, 7, each 70 horizon; full measurement cycle took under 8 minutes. The maximum depth8 of 18 was comparable with the Secchi depth. Wind and waves were absent; the sky was overcast.

The underwater vision system

The lower deck

(a) (b) (c)

Figure 4.4.( a()a The) The Oceanographic Oceanographic Platform Platform in the in Black the sea; Black (b ) sea the; underwater(b) the underwater vision system vision submerged system fromsubmerged the lower from deck the oflower the platform;deck of the (c) platform the underwater; (c) the visionunderwater system vision submerged system from submerged the scientific from vesselthe scientific in the Gorkyvessel Reservoir.in the Gorky Reservoir.

2.4.2.Concurrent Gorky Reservoir measurements Experiment of the scattering coefficient at 532 nm were performed by a turbidity meterA similarmodel workTurbido was- performed1M [14]. in Determination eutrophic fresh ofwaters the of spectral the Gorky absorption Reservoir in coefficient the late autumn was ofimplemented 2018 during inresidual the laboratory cyanobacteria on bloom. the basis The of box selected with webcam water was samples fixed to using an aluminum a double pipebeam at spectrophotometer and integrating sphere to increase the optical path length [15]. angle ϑSn that was lowered from the deck of the anchored vessel (Figure4c). The first depth was 0.5 m andGeneral the last information one 2.0 m; depthon the increment roughness, was optical 0.25 m.properties Video recording of water ofand 60 secondsky condition was 0.5s mon andthe themeasurement last one 2.0 days m; depth is presented increment in Table was 0.25 2. m. Video recording of 60 second was performed on each horizon; full measurement cycle took under 8 minutes. The maximum depth was comparable with the Table 2. General information on the surface waves, optical properties of water and sky conditions on Secchi depth. Wind and waves were absent; the sky was overcast. the days of experiment according to data of concurrent measurements. Concurrent measurements of the scattering coefficient at 532 nm were performed by a turbidity

−1 −1 −1 2 meter modelBasin Turbido-1Mc460 , m [14 ]. Determinationc532 , m ofc625 the, m spectralZSecchi absorption, m Wind, coefficient m/s  wasx implementedClouds, % in theThe laboratory Black Sea on0.372 the ± basis 0.01 of0. selected361 ± 0.008 water 0.567 samples ± 0.007 using12 ± a0.5 double 0–6 beam ± 0.06 spectrophotometer~0.01 10% and The Gorky integrating sphere to increase- the1.8 optical ± 0.02 path length- [15]. 2.5 ± 0.2 0–2 ± 0.02 0 100% GeneralReservoir information on the roughness, optical properties of water and sky conditions on the measurement days is presented in Table2.

Table 2. General information on the surface waves, optical properties of water and sky conditions on the days of experiment according to data of concurrent measurements.

−1 −1 −1 2 Basin c460, m c532, m c625, m ZSecchi, m Wind, m/s σx Clouds, % The Black Sea 0.372 ± 0.01 0.361 ± 0.008 0.567 ± 0.007 12 ± 0.5 0–6 ± 0.06 ~0.01 10% The Gorky - 1.8 ± 0.02 - 2.5 ± 0.2 0–2 ± 0.02 0 100% Reservoir

3. Results and Discussion

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 J. Mar. Sci. Eng. 2019, 7, 70 9 of 18

3. Results and Discussion

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 J. Mar. Sci. Eng. 2019, 7, 70 9 of 18 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 couldfrequency be obtained spectrum by could section be ofobtained the Snell’s by section window of the in the Snell general’s window direction in the of general the surface direction waves. of Thisthe surface section wave can bes. This used section to recover can be the used optical to recover properties the optical of water properties according of towater [6,8 ,9according]; however, to it[6 is,8,9 a]; suboptimal however, it choice. is a sub Figureoptimal5 demonstrateschoice. Figure the5 demonstrates selection of the selection optimal sectionof the optimal of the Snell’ssection windowof the Snell on’ thes window example on of the its example image obtained of its image at the obtained depth of at 1 mthe with depth Nikon of 1D5100 m with fisheye-Nikkor Nikon D5100 lensfisheye 10.5-Nikkor mm f/2.8 lens G10.5 ED mm DX f/2.8 in the G ED waters DX ofin the waters Black Seaof the on Black a clear Sea day on with a clear small day sea with surface small roughness.sea surface roughness. The direction The of direction the waves of the propagation waves propagation and Sun and position Sun position were set were by azimuth set by azimuth angles ϕanglew ands ϕwS. and The optimalS . The sectionoptimalϕ sectionwill be  the will section be the that section is oriented that is perpendicular oriented perpendicular to the direction to the ofdirection wave propagation, of wave propagation lying within, lying angular within region angular of region 90◦ to 270of 90°◦, relative to 270°, torelative the direction to the direction to the Sun. to Implementationthe Sun. Implementati of the firston of condition the first allows condition to minimize allows the to distortionminimize of the the distortion Snell’s window of the border Snell’s duewindow to the border surface due waves, to the and surface the second waves allows, and tothe avoid second light allows exposure to avoid and sunlight glint. exposure and sun glint.

Figure 5.5. Determination of the optimaloptimal radialradial sectionsection ofof thethe Snell’sSnell’s windowwindow image.image.

InIn practice, practice, it it seems seems that that the the existence existence of still of milder still milder conditions conditions is possible, is possible, namely, namely, in the analysis in the of the section of the Snell’s window in a certain range of angles ±ϑ near direction ϑSn (dotted circles analysis of the section of the Snell’s window in a certain range of angles  near direction Sn in Figure5). Interval ϑ ∈ ϑ − ϑ : ϑ + ϑ is determined by double mean angle ϑ of distortion of (dotted circles in Figure 5). SnInterval Sn  − :  +  is determined by double mean angle  the Snell’s window border under surface wavesSn and Sn light scattering in water. According to estimates fromof distortion [3] maximum of the of ϑ Snellis about’s window 8◦ for winds border of under 10 m/s. surface If we assume waves that and light light scattering scattering in water in water. has theAccording same impact to estimates on the distortionfrom [3] maximum of the Snell’s of  window, is about the 8° totalfor winds distortions of 10 m/s.4ϑ by If waveswe assume and light that scatteringlight scattering will be in about water 30 ◦ has. This the means same that impact optical on receivers the distortion with a of viewing the Snell angle’s window, of 30◦ or more the total can bedistortions used for recording4 by waves of the and Snell’s light window scattering images. will Thebe about simplest 30°.device, This means such asthat a webcam,optical rec satisfieseivers thiswith condition; a viewing therefore, angle of we30° usedor more one incan our be full-scaleused for experiments.recording of the Snell’s window images. The simplestIn addition device, tosuch the as choice a webcam of the, optimal satisfies cross this sectioncondition of; the therefore Snell’s, window,we used thereone in was our a full question-scale regardingexperiments. 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 ϕ ∈ [ϕ1 : ϕ2] marked by shaded area in Figure5. 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 J. Mar. Sci. Eng. 2019, 7, 70 10 of 18

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 J. Mar. Sci. Eng. 2019, 7, 70 10 of 18 radial sections in the sector of azimuth angles  12:   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 ofthe several webcam expeditions, up to 6000 assesspixels thefor realthe SLR possibilities camera. ofThe webcam following implementation results, based ason comprehensible the analysis of opticalvideo recordings receiver. of several expeditions, assess the real possibilities of webcam implementation as comprehensible optical receiver.. 3.2. Image Processing Technique 3.2. Image Processing Technique Figure6 shows instantaneous images of the Snell’s window, registered in three color filters on sevenFigure horizons 6 showsin instantaneous the Black Sea images (on the of the left) Snell and’s the window Gorky, registered Reservoir in (on three the color right), filter bys theon abovementionedseven horizons in webcam. the Black It can Sea be (on seen the that left) the radiance and the ofGorky the images Reservoir in sea (on water the in right) the blue, by andthe greenabovementioned filters decreased webcam in depth. It can approximately be seen that the at theradiance same pace,of the whereas images in the sea radiance water in of the the blue image and in thegreen red filters filter decreased morein depth rapidly. approximately In eutrophic at freshthe same waters, pace, the whereas greatest the decrease radiance was of observed the image in bluein the color red due filter to itsdecreased absorption more by dissolved rapidly. In organic eutrophic matter, fresh average waters, concentrations the greatest of decrease which for was the Gorkyobserved Reservoir in blue arecolor 13 due mg/L. to its Therefore, absorption the by depth dissolved limit oforganic visibility matter, of the average Snell’s concentrations window in blue of colorwhich did for notthe exceed Gorky 1.5 Reservoir m while are the 13 depth mg/L limit. Therefore of visibility, the fordepth green limit color of visibilitywas 2.5 m, of close the Snell to the’s Secchiwindow depth. in blue color did not exceed 1.5 m while the depth limit of visibility for green color was 2.5 m, close to the Secchi depth.

Figure 6. The transform transformationation of the Snell Snell’s’s window images with depth for three three color filter filterss (420 nm, 530 nm and 650 nm) in waters of the Black Sea (on(on the left)left) andand thethe GorkyGorky ReservoirReservoir (on(on thethe right).right).

Images from Figure 66 were were used used to to recoverrecover the the waves waves characteristics characteristics and and the the waterwater opticaloptical properties.properties. The algorithm of their processing isis asas follows:follows: 1. 21 videos, recorded at seven depths in three color filters, were loaded into specially developed software. Each video was divided into frames and an array of frames with dimensions p × p × x y Ni,j was formed, where px = 1280 and py = 720 are resolutions along the Snell’s window image and across it, respectively, Ni,j is the frame number of video according to depth Zi, i = [1 : 7] and each color filter, where j = [1 : 3] is a number of color. J. Mar. Sci. Eng. 2019, 7, 70 11 of 18

2. Correct calculation of the statistical characteristics of the Snell’s window image could be performed only on the basis of data arrays of a uniform dimension. As each video had its own duration the following search procedure of video with minimum length was used:  N = min Ni,j (13)

with the subsequent shortening of the original arrays to dimension px × py × Ni,j, where Ni,j =
Ni,j 1 : N . 3. Then, cycle n = 1 : N was started and ach frame was averaged in cross direction along the transverse coordinate py. As a result, the array px × Ni,j of accumulated radial cross section image of the Snell’s window was formed. 4. For each accumulated radial cross section the search of the Snell’s window border was carried out by its sliding window smoothing with subsequent differentiation by px:

d dLi,j(px) = − Li,j(px), (14) dpx

where index i still corresponds to depth and j—to number of color filter. In Equation (14) variable Z is derived from the number of arguments of initial function L (see Equation (6)) in connection with its introduction to the corresponding index i. Also the viewing angle ϑ is replaced to the pixel coordinate px. Other remaining parameters are omitted for compact representation. The Sn function minimum (14) corresponded to the Snell’s window border pi,j,n. This procedure was necessary, as the webcam was not rigidly fixed underwater and, as a result, was slowly rocking due to current and wind-wave action. The knowledge of the Snell’s window border coordinates made it possible to center each section relative to it. 5. The location of the Snell’s window border in the left or right half of the image determined   4 = Sn − Sn − the distance to the image edge pi,j,n min pi,j,n p1 ; pi,j,n p720 . Minimizing the array of coordinates of the Snell’s window border over all sections allowed to calculate the minimum value ∆p = min 4 pi,j,n, and cut off all sections, reducing them to a single scale. i,j,n h Sn Sn i −∆p + pi,j,n : pi,j,n + ∆p . In other words, all the Snell’s window sections were limited to single scale images with a centered border of the Snell’s window. This allowed us to minimize the influence of camera rocking and ensure the Snell’s window imaging validity from all expeditions. In cases where camera rocking was significant due to long waves, the data was discarded. 6. The final step was the normalization of each section by radiance at first depth and pixel with ∗ coordinate px = 10: ∗ L (p ) = L (p ) L (p ) i,j,n x i,j,n x / i,j,n x i=1. (15)

It is worth noting that we intentionally retreated from the edge of the section by 10 pixels for the purpose of eliminating the influence of image edge effects.

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 px = pSn for each depth into (10) allows to calculate an array of the absorption 0 coefficient for each color filter and the water layer thickness 4Zi0 = Zi0+1 − Zi0 , where i = 1 : 6 (Table3). This table shows that the choice of step 4Zi0 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 J. Mar. Sci. Eng. 2019, 7, 70 12 of 18 the specified parameters. These possibilities are demonstrated in Figure7 on the basis of randomly selected frames from each video recording. Here, the dashed curve marks, the average absorption coefficientJ. Mar. Sci. Eng. and 2019 error, 7, 70 bar define standard deviations. Figure7 shows that the average values12 of of the 18 recovered absorption coefficient can vary more than twofold for optical depths smaller and larger than 1This (in casemeans of thethat Black we have Sea it a is set about of sections 3 m). It isin related an amount to the equal fact that, to the on thetransverse one hand, resolution at small depthsof the aphoto small, i.e. surface, 720 sections area is visible,. Next, summing and its roughness, the sections obviously, together,cannot we will be form considered the accumulated a stationary section and. homogeneousSubstituting radiance Gaussian value random at point process. ppx= On Sn for the each other depth hand, into videos (10) forallows each to depth calculate were an recorded array of sequentially.the absorption This coefficient led to the for fact each that color surface filter waves and the were water different. layer thickness The high spreadZZZi=− of i +1 values i  , where of the recovered absorption coefficient contributes to the “random” sea roughness of each frame. With the i =1: 6 (Table 3). This table shows that the choice of step Zi defines the number of values of the increaseabsorption of coefficient, depth visible which surface can areabe obtained is greatly from expanded, a single pass encompassing in depth. Using many such waves values and for thereby each formingsection and similar each roughness depth, we conditions can estimate for differentthe statistical video characteristics recordings. It leadsof the to absorption obtaining thecoefficient regular averagedepending values on the of specified the absorption parameters. coefficient These with possibilities root-mean-square are demonstrated error (RMSE) in Figure equal 7 on to the 25–50%, basis whichof randomly are in goodselected agreement frames from with each the estimationsvideo recording. according Here, to the the dashed concurrent curve measurements marks, the average data (Tableabsorption4), thus coefficient confirming and the error possibility bar define of the standard solution of deviations the inverse. Figure problem 7 shows by one that frame the under average the statedvalues conditions.of the recovered absorption coefficient can vary more than twofold for optical depths smaller and largerTable 3.thanThe 1 quantity (in case of of recovered the Black absorption Sea it is coefficientabout 3 m). for It a fullis related depth passto the as fact a function that, on of thethe one hand,depth at small step. 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.4Z21 4Z 31This led4Z to41 the 4factZ51 that 4surfaceZ61 waves4Z71 were different. The

high spread of valuesZ1 of the recovered absorption coefficient contributes to the “random” sea roughness of each frame.Z2 Witha21 the increase of depth visible surface area is greatly expanded, encompassing manyZ waves3 anda32 therebya31 forming similar roughness conditions for different video Z a a a recordings. It leads to4 obtaining43 the 42 regular average41 values of the absorption coefficient with Z5 a54 a53 a52 a51 root-mean-square errorZ6 (RMSE)a65 equal ato64 25–50%,a63 which area62 in gooda61 agreement with the estimations according to the concurrentZ7 a measurements76 a75 dataa74 (Table a 4)73, thus confirminga72 a71 the possibility of the solution of the inverse problem by one frame under the stated conditions.

(a) (b) (c)

Figure 7. TheThe results results of of recovery recovery for for the the absorption absorption coefficient coefficient by by a asingle single frame frame of ofeach each color color filter filter as asa function a function of ofwater water layer layer thickness thickness 1– 1–66 m: m: (a ()a 420) 420 nm, nm, (b (b) )530 530 nm nm and and ( (cc)) 650 650 nm. nm. The The error error bar is obtained by averaging the absorptionabsorption coefficientcoefficient valuesvalues forfor thethe samesame thicknessthickness ofof thethe waterwater layer.layer.

Table 4.3. TheThe recoveryquantity resultsof recovered of the absorption coefficientcoefficient offor the a full Black depth Sea frompass as the a Snell’sfunction window of the imagesdepth step in comparison. with related data. The Snell’s Window Image The Snell’s Window Image Sonde Data and Atlas Data Z Z Z Z Z Z 21 31 (1 Frame)41 51 61 71 (150 Frames)

Z1 a , a , a , a , a , a , m−1 a , m−1 a , m−1 420 530 650 420 530 a , m−1 460 532 625 m−1 m−1 m−1 m−1 m−1 650 Z2 a21 0.082 ± 0.132 ± 0.071 ± 0.285 ± 0.099 ± 0.075 ± 0.31 ± 0.12 ± 0.01 0.2 ± 0.02 0.01 Z3 a32 0.045a31 0.031 0.047 0.0091 0.0089 0.0341

Z4 a43 a42 a41

It should be noted thatZ the5 resultsa54 ofa53 concurrent a52 measurementsa51 of the attenuation coefficient by sonde, presented in Table4, were converted into absorption and scattering coefficient on the basis of Z6 a65 a64 a63 a62 a61 regression links [7] between IOP at 532 nm, repeatedly tested for the waters of the Black Sea: Z7 a76 a75 a74 a73 a72 a71

J. Mar. Sci. Eng. 2019, 7, 70 13 of 18

Table 4. The recovery results of the absorption coefficient of the Black Sea from the Snell’s window images in comparison with related data.

The Snell’s Window Image The Snell’s Window Image Sonde Data and Atlas Data (1 Frame) (150 Frames) , a , −1 −1 a625 420 −1 −1 −1 −1 −1 a460 , m a532 , m a530 , m a650 , m , m , m , m m−1 m−1 0.132 ± 0.071 ± 0.285 ± 0.099 ± 0.075 ± 0.31 ± 0.12 ± 0.01 0.082 ± 0.01 0.2 ± 0.02 0.045 0.031 0.047 0.0091 0.0089 0.0341 J. Mar. Sci. Eng. 2019, 7, 70 13 of 18 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: −1 −1 b532 = 0.908c532 − 0.048 ≈ 0.288 m , a532 = c532 − b532 ≈ 0.082 m . (16) −1 −1 bc532=0.908 532 − 0.048  0.288 m , a532=− c 532 b 532 0.082 m . (16) Absorption coefficients for two remaining waves 460 nm and 650 nm are taken from the Atlas of the HydroopticalAbsorption coefficients Characteristics for two of the remaining Black Sea waves [16], based 460 nm on and the results650 nm of are numerous taken from expeditions. the Atlas Overestimatesof the Hydrooptical of the absorption Characteristics coefficient of the at 650 Black nm, Sea obtained [16], frombased the on Snell’s the results window of images, numerous are likelyexpeditions. to be related Overestimates to the fact that of the the color absorption filter has coefficient a wide pass at band 650 nm, in the obtained red spectrum, from where the Snell pure’s waterwindow absorption images, increasesare likely significantly.to be related to the fact that the color filter has a wide pass band in the red spectrum,Having where understood pure water the possibilities absorption ofincreases using a significantly. single frame, let’s consider the accuracy of solving the inverseHaving problem understood by an entire the possibilities video that is of a largeusing array a single of frames. frame, In thislet’s caseconsider we applied the accuracy temporary of accumulationsolving the inverse by frames, problem each by of an which entire was video processed that is a according large array to of the frames. above In mentioned this case we algorithm. applied Thetemporary recovery accumulation of the absorption by frames coefficient, each of is facilitatedwhich was accordingprocessed toaccording the slope to of the straight above linementioned drawn throughalgorithm. all pointsThe recovery of the Snell’s of the windowabsorption borders coefficient radiance is facilitated on a logarithmic according scale to as the a function slope of of straight depth (seeline Equationdrawn through (7)). Figure all points8 presents of the the Snell results’s window obtained borders by processing radiance 150 on frames a logarithmic of each video scale with as a timefunction recording of depth is about(see Equation 10 seconds. (7)). RegressionFigure 8 presents equations the results are listed obtained in captions by process whereingR 2150indicates frames determinationof each video coefficient.with time recording The values is of about the absorption 10 seconds. coefficient Regression are presentedequations inare Table listed4. Asin captions we can 2 see,where using R the indicates short video determination gave strong coefficient. results: the The mean values values of the are closeabsorption to sonde coefficient data with are coefficients presented Rin2 thatTable are 4.over As we 0.94. can Nevertheless, see, using the it isshort necessary video togave carry strong out longer results shootings: the mean in real-casevalues are scenarios close to assonde different data foreignwith coefficients objects can get that in the are webcam’s over 0.94. fieldNevertheless of view:, fish, it is jellyfishnecessary or to mud. carry It willout longer make suchshootings images in unsuitable real-case scenarios for solving as the different inverse foreign problems. 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.

(a) (b) (c)

FigureFigure 8.8. TheThe recoveryrecovery resultsresults ofof thethe absorptionabsorption coefficient coefficient byby 150150 framesframes ofof eacheach colorcolor filterfilter asas aa functionfunction ofof seven depths: depths: (a ()a 420) 420 nm, nm, (b) ( b530) 530 nm nm and and (c) 650 (c) nm 650. Y nm.-axis Y-axis utilizes utilizes the scale the of scale the natural of the naturallogarithm. logarithm.

TheThe repetitionrepetition ofof thethe actionsactions forfor thethe Snell’sSnell’s windowwindow videovideo obtainedobtained inin thethe waterswaters ofof thethe GorkyGorky ReservoirReservoir mademade it it possible possible to to obtain obtain appropriate appropriate estimates estimates of theof the absorption absorption coefficient. coefficient Its.values Its values are presentedare presented in Table in Table5 in comparison 5 in comparison with thewith results the results of related of related data. data.

Table 5. The recovery results of the absorption coefficient of the Gorky Reservoir from the Snell’s window images in comparison with related data. The Snell’s Window Image The Snell’s Window Image Laboratory Results (1 Frame) (150 Frames)

a420, a530, a650, a420, a530, a650, a420, a530, a650, m−1 m−1 m−1 m−1 m−1 m−1 m−1 m−1 m−1 3.610 ± 0.619 ± 0.357 ± 3.987 ± 0.763 ± 0.425 ± 3.626 ± 0.585 ± 0.390 ± 0.07 0.012 0.007 1.19 0.36 0.19 0.43 0.047 0.06 J. Mar. Sci. Eng. 2019, 7, 70 14 of 18

Table 5. The recovery results of the absorption coefficient of the Gorky Reservoir from the Snell’s window images in comparison with related data.

The Snell’s Window Image The Snell’s Window Image Laboratory Results (1 Frame) (150 Frames)

−1 −1 −1 −1 −1 −1 −1 −1 a420 , m a530 , m a650 , m a420 , m a530 , m a650 , m , , m , m m−1 3.610 ± 0.619 ± 0.357 ± 3.987 ± 0.763 ± 0.425 ± 3.626 ± 0.585 ± 0.390 ± 0.07 0.012 0.007 1.19 0.36 0.19 0.43 0.047 0.06

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 theJ. Mar. short Sci. Eng.-wave2019 part, 7, 70 retaining the slope (dotted line in Figure 9). Table data shows that one 14frame of 18 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 The values in the laboratory results section of Table5 are obtained from the spectral absorption probably connected with less sea surface roughness due to calm weather. coefficient in 430–750 nm range (Figure9), 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 3.4. Determination of the Slope Variance and the Scattering Coefficient short-wave part retaining the slope (dotted line in Figure9). Table data shows that one frame usage getsThe a 15–20% task of overestimate, the scattering whereascoefficient 150 recovery frame usage is more increases difficultprecision because it to is 6–8%.based on The an results analysis of ofthe the Gorky Snell Reservoir’s border are distortion, demonstrably which more is affected accurate by than two those reasons of the simultaneously:Black Sea, which sea is probably surface roughnessconnected with and lesslight sea scattering. surface roughness Still, these due two to effects calm weather. can be separated with the help of the two algorithms proposed in Section 2.3.

Figure 9.9.Spectral Spectral absorption absorption coefficient coefficient of the of Gorky the ReservoirGorky Reservoir (black line); (black dotted line) curve; dotted corresponds curve correspondsto the extension to the into extension the short-wave into the partshort and-wave colored part and rectangles colored indicate rectangles the indicate bandwidth the ofbandwidth the used ofcolor the filters.used color filters.

3.4. Determination of the Slope Variance and the Scattering Coefficient 2 Algorithm 1 requires the calculation of the statistical moment d (bx, ) through the following momentsThe 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 2 MM,, 22   and light scattering. Still, these two effects2 can21( beb separated x) ( b with x ) the help of the two algorithms d (bx, ) =− , (17) proposed in Section 2.3. MM2 2 2 00( xx) ( ) 2
Algorithm 1 requires the calculation of the statistical moment d τb, σx through the following moments: 222 ML0 xx = 2
0, , 2
(18)   M( 2 )τ ,σ ( M) 1a=τ0 , σ d τ , σ2 = b x − b x , (17) b x ( 2) 2 2 M0 σx M0(σx )  /3  222 h 2i M1 b,,,  x= L   b  x d  , (19) M( 0 σx ) = L (0, σx ) ,a=0 (18)  /6 a=0 π/3 Z  2  /3h  2i M1 τb, σx 22= L ϑ, τb, σx dϑ, (19) M2 b,  x= 2 L  ,  b ,  x =  d  , (20) ( )  ( ) aa=0 0 π/6 /6

π/3 where b = bZ is the optical depth by the scatteringZ coefficient. Determination of the scattering  2 h  2i M2 τb, σx = 2 L ϑ, τb, σx ϑdϑ, (20) characteristics is performed by fitting the theoretical value of thisa= parameter0 to the experimental one. π/6

where τb = bZ 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 2
one. Figure 10 shows the results of numerical simulation of parameter d τb, σx as functions of the 2 optical depth τb = bZ for three slope variances σx , which were calculated using the Cox and Munk 2 2 2 equation [17] for wind speeds of 0 m/s (σx = 0), 3 m/s (σx ∼ 0.01) and 12 m/s (σx ∼ 0.04) 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 τb 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. J. Mar. Sci. Eng. 2019, 7, 70 15 of 18

2 Figure 10 shows the results of numerical simulation of parameter d (bx, ) as functions of the 2 optical depth b = bZ for three slope variances  x , which were calculated using the Cox and Munk 2 2 2 equation [17] for wind speeds of 0 m/s ( x = 0 ), 3 m/s ( x 0.01) and 12 m/s ( x 0.04 ) 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  b 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. J. Mar. Sci. Eng. 2019, 7, 70 15 of 18 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.

(a) (b)

FigureFigure 10 10.. TheThe comparison comparison of of theoretical theoretical values values of of parameter parameter ddwith with its its empirical empirical values values obtained obtained by by(a )(a 150) 150 frames frames and and (b )(b 1) frame 1 frame of theof the Snell’s Snell window’s window in eachin each color color filter: filter 420: 420 nm nm (blue (blue points), points), 530 530 nm nm(green (green points) points) and and 650 650 nm nm (red (red points). points).

TheThe distribution distribution of of points points resulting resulting from from calculation calculation of of parameter parameter dd byby frames frames in in relation relation to to 2
theoretical curves d τb, σ2x implies the proportionality parameter b fitting between actual depth theoretical curves d (bx, ) implies the proportionality parameter b fitting between actual depth Z, known from field experiments, and optical depth τb = bZ, used in the numerical simulations. Z , known from field experiments, and optical depth  = bZ , used in the numerical simulations. The outcome of this problem based on 150 frames of theb Snell’s window images of the Black Sea Thewaters outcome is represented of this problem in Figure based 10a, whereon 150 the frames points of colorthe Snell matches’s window the color images of the of used the colorBlack filter:Sea waters420 nm is (bluerepresented points), in 530 Figure nm (green10a, where points) the and points 650 color nm(red matches points). the color The change of the used of the color curves filter: for 420one nm set (blue of points points), is not 530 an nm error, (green as one points) might and consider. 650 nm On (red the points). day of theThe first change video of recording the curves in for the one set of points is not an error, as one might consider. On the day of2 the first video recording in the green filter the wind speed has weakened from 4–6 m/s to 3 m/s (σx ∼ 0.01). As a result, a later time 2 greenpoint, filter corresponding the wind speed to greater has weakened depths, “dropped” from 4–6 m/s to3 to m/s 3 m/s wind ( x speed0.01 curve,). As a whereasresult, a later the earlier time point,points corresponding rose above it. Halfto greater an hour depths, later “dropped” the recording to in3 m/s the redwind filter speed was curve, carried whereas out. According the earlier to pointsanemometer rose above data, it. wind Half fadedan hour completely, later the recording while the in swell the red was filter continuing was carried to fade. out For. According this reason, to 2 anemometerall initial points data lay, w onind 3 faded m/s ( σcompletely,x ∼ 0.01) curve while almost the swell evenly, was whereas continuing the laterto fade. points For sunk this reason, beneath allit. initial The recording points lay in on the 3 bluem/s ( filter was carried) curve outalmost a few evenly, hours whereas later, when the later the swell points visibly sunk subsided.beneath 2 it.For The this recording reason, allin the the blue points filter are was close carried to curve outσ ax few= 0 .hours In cases later like, when those the mentioned swell visibly above, subsided. where the wind conditions are changing rapidly, it can be2 argued that without information about the wind For this reason, all the points are close to curve  x = 0 . In cases like those mentioned above, where theor surfacewind conditions waves the are distribution changing rapidly, of the points it can inbe Figureargued 10 that may without have ainformation lot of variation, about depending the wind oron surface the choice waves of the parameter distributionb, which of the could points lead in toFigure overestimates 10 may have or underestimatesa lot of variation, of depending the scattering on thecoefficient. choice of Table parameter6 shows b recovery, which resultscould lead for our to particularoverestimates case. or It underestimates is evident that results of the for scattering 532 nm, coefficient.obtained by Table the Snell’s 6 shows window recovery and results sonde data,for our appear particular very close: case. 12%It is forevident one frame that results and 5% for for 532 150 nm,frames. obtained The scatteringby the Snell coefficients’s window values and sonde for other data wavelengths, appear very (460 close: nm 12% and fo 650r one nm) frame are obtainedand 5% forby 150 subtraction frames. ofThe the scattering reference coefficients data on the values absorption for other coefficient wavelengths according (460 to thenm atlasand 650 [16] nm) from are the obtainedsonde measured by subtraction attenuation of the coefficient. reference data However, on the despite absorption the absence coefficient of the according reliably measuredto the atlas values [16] fromfor these the channels, sonde measured it is apparent attenuation that the coefficient. obtained values However, differ despite by only the30%. absence of the reliably measured values for these channels, it is apparent that the obtained values differ by only 30%. Table 6. The recovery results of the scattering coefficient of the Black Sea from the Snell’s window images in comparison with related data.

The Snell’s Window Image The Snell’s Window Image Sonde Data and Atlas Data (1 Frame) (150 Frames)

b460, b532, b625, b420, b530, b650, b420, b530, b650, m−1 m−1 m−1 m−1 m−1 m−1 m−1 m−1 m−1 0.23 ± 0.288 ± 0.37 ± 0.297 ± 0.256 ± 0.192 ± 0.319 ± 0.302 ± 0.276 ± 0.11 0.006 0.12 0.12 0.081 0.088 0.042 0.029 0.025

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 J. Mar. Sci. Eng. 2019, 7, 70 16 of 18

Snell’s border. Frames for the depth of 1.5 m and three color filters in Figure6 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 d 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 d 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 Table7. 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.

Table 7. The recovery results of the scattering coefficient of the Gorky reservoir from the Snell’s window images in comparison with related data.

The Snell’s Window Image Turbidity Meter The Snell’s Window Image (1 Frame) (150 Frames) −1 −1 −1 −1 −1 −1 −1 −1 −1 b460, m b532, m b625, m b420, m b530, m b650, m b420, m b530, m b650, m - 1.17 ± 0.03 - 2.28 ± 0.93 1.47 ± 0.47 1.69 ± 0.77 2.36 ± 0.17 1.03 ± 0.16 1.31 ± 0.17

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 ϑ1 ≤ ϑ ≤ ϑ2:   Lϑ , τ , σ2
− Lϑ , τ , σ2
2 = 1 b x 2 b x K τb, σx 2 2 . (21) L(ϑ1, τb, σx ) + L(ϑ2, τb, σx ) ◦ ◦ Its estimation for ϑ1 = ϑSn − 5 and ϑ1 = ϑSn + 5 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,J. Mar. Sci. Eng. it is 20 useful19, 7, 70 to use both algorithms as complementary or as a test of each other. 17 of 18

(a) (b)

Figure 11.11. TheThe comparisoncomparison ofof theoreticaltheoretical values values of of parameter parameterK withK with its its empirical empirical values values obtained obtained by (bya) 150(a) 150 frames frames and and (b) one(b) one frame frame of the of Snell’sthe Snell window’s window in each in each color color filter: filter: 420 nm420 (bluenm (blue points), points), 530 nm530 (greennm (green points) points) and and 650 nm650 (rednm (red points). points).

4.4. ConclusionsConclusions The results of of this this study study demonstrate demonstrate the the real real possibility possibility of the of the Snell Snell’s’s window window image image usage usage as a astool a toolfor remote for remote sensing sensing of the of water the water optical optical properties properties and andthe sea the roughness sea roughness characteristics. characteristics. The Thesolution solution of these of problems these problems at the “estimation at the “estimation”” level is possible level isby possible using of bytwo using images, of consecutively two images, consecutivelyregistered at two registered depths at; however, two depths; achievement however, achievement of definite results of definite with results required with accuracy required warrants accuracy field measurements with the greatest possible number of underwater horizons. According to the obtained results using a single frame of the Snell’s window allows solving the inverse problem with an error within 50%, which is quite sufficient for obtaining the primary information about the characteristics of the medium. Processing of short video recordings with the length of under one minute is capable of increasing the accuracy of solving the reverse problem and reducing the error to 10–20%. In our experience, such a data set is not only sufficient to achieve these results, but also to exclude unsuitable frames (frames with moving fish, jellyfish and debris). Nevertheless, even such a simple receiver as a standard webcam with a minimum field of view of 30° can provide data sufficient for a solution to the stated problems. Receivers with a field of view of 60–90° extend the ability to analyze the Snell’s window image by calculating the border distortion in a wider angle range. The information on the border distortion is extremely useful in case of an ambiguous solution of the inverse problems through the usage of the parameters of the accumulated Snell’s window image, for example, in case when some dynamic processes occur in the environment: wind-wave change or turbidity layer drift. Estimation of the total error for the proposed method is mainly related to the performing of a variety of experiments in different water basins under controlled conditions of its optical properties, wave statement and illumination. Comparison of the experimental bulk data with the results of the corresponding calculation by the proposed models will require an improved theoretical model to achieve maximum consistency with real conditions. Performing these actions should improve the accuracy of the estimation of the optical properties by images of the Snell window. Summing up, it can be argued that an underwater optical system based on a simple optical receiver, such as a webcam, can be a part of buoy monitoring systems. This optical component can be used to continuously monitor not only the optical properties of the reservoir, but also the state of the water surface, including the detection of films of surface-active substances.

Author Contributions: Methodology and formal analysis, L.S.D.; software, validation and original draft preparation, A.A.M.

Funding: This research was funded by the state target No. 0035-2019-0006 (Development of Radiophysical Methods for Ocean Research).

Acknowledgments: We are grateful to A.A. Latushkin and D.A. Ryabokon for attenuation coefficient data; G.V. Leshev and A.V. Kupaev for help in experiment performance and equipment setting up; B.V. Konovalov for laboratory analysis and V.V. Pelevin for water sample selection and its transfer to laboratory.

Conflicts of Interest: The authors declare no conflict of interest.

J. Mar. Sci. Eng. 2019, 7, 70 17 of 18 warrants field measurements with the greatest possible number of underwater horizons. According to the obtained results using a single frame of the Snell’s window allows solving the inverse problem with an error within 50%, which is quite sufficient for obtaining the primary information about the characteristics of the medium. Processing of short video recordings with the length of under one minute is capable of increasing the accuracy of solving the reverse problem and reducing the error to 10–20%. In our experience, such a data set is not only sufficient to achieve these results, but also to exclude unsuitable frames (frames with moving fish, jellyfish and debris). Nevertheless, even such a simple receiver as a standard webcam with a minimum field of view of 30◦ can provide data sufficient for a solution to the stated problems. Receivers with a field of view of 60–90◦ extend the ability to analyze the Snell’s window image by calculating the border distortion in a wider angle range. The information on the border distortion is extremely useful in case of an ambiguous solution of the inverse problems through the usage of the parameters of the accumulated Snell’s window image, for example, in case when some dynamic processes occur in the environment: wind-wave change or turbidity layer drift. Estimation of the total error for the proposed method is mainly related to the performing of a variety of experiments in different water basins under controlled conditions of its optical properties, wave statement and illumination. Comparison of the experimental bulk data with the results of the corresponding calculation by the proposed models will require an improved theoretical model to achieve maximum consistency with real conditions. Performing these actions should improve the accuracy of the estimation of the optical properties by images of the Snell window. Summing up, it can be argued that an underwater optical system based on a simple optical receiver, such as a webcam, can be a part of buoy monitoring systems. This optical component can be used to continuously monitor not only the optical properties of the reservoir, but also the state of the water surface, including the detection of films of surface-active substances.

Author Contributions: Methodology and formal analysis, L.S.D.; software, validation and original draft preparation, A.A.M. Funding: This research was funded by the state target No. 0035-2019-0006 (Development of Radiophysical Methods for Ocean Research). Acknowledgments: We are grateful to A.A. Latushkin and D.A. Ryabokon for attenuation coefficient data; G.V. Leshev and A.V. Kupaev for help in experiment performance and equipment setting up; B.V. Konovalov for laboratory analysis and V.V. Pelevin for water sample selection and its transfer to laboratory. Conflicts of Interest: The authors declare no conflict of interest.

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