rus
Issue #8/2020
V. P. Budak, A. V. Grimailo
Statistical Lens Development When Taking into Account Wave Correlation of Randomly Uneven Surface
Statistical Lens Development When Taking into Account Wave Correlation of Randomly Uneven Surface
DOI: 10.22184/1993-7296.FRos.2020.14.8.708.716
The article is devoted to modelling a randomly uneven surface, taking into account the correlation of slopes. The facts of observation of the underwater relief from space are presented, and an explanation is proposed based on the assumption that a statistical lens can form on the water surface. A spectral representation method is described, which allows one to construct random surfaces with a given correlation function. This method is implemented in the MATLAB software environment. To check the program, the relief of the CD-ROM was reproduced, obtained using an atomic force microscope. As a next step, a disturbed water surface is simulated, which acts on the transmitted rays like a real optical lens. The results obtained confirm the possibility of the formation of giant lenses on the surface of the seas and oceans, thus helping to distinguish the bottom topography from space.
The article is devoted to modelling a randomly uneven surface, taking into account the correlation of slopes. The facts of observation of the underwater relief from space are presented, and an explanation is proposed based on the assumption that a statistical lens can form on the water surface. A spectral representation method is described, which allows one to construct random surfaces with a given correlation function. This method is implemented in the MATLAB software environment. To check the program, the relief of the CD-ROM was reproduced, obtained using an atomic force microscope. As a next step, a disturbed water surface is simulated, which acts on the transmitted rays like a real optical lens. The results obtained confirm the possibility of the formation of giant lenses on the surface of the seas and oceans, thus helping to distinguish the bottom topography from space.
Теги: randomly uneven surface slope correlation statistical lens корреляция уклонов случайно неровная поверхность статистическая линза
Statistical Lens Development When Taking Into Account Wave Correlation of Randomly Uneven Surface
V. P. Budak, A. V. Grimailo
National Research Univarsity ”Moscow Power Engineering Institute’’, Moscow, Russia
The article is devoted to modelling a randomly uneven surface, taking into account the correlation of slopes. The facts of observation of the underwater relief from space are presented, and an explanation is proposed based on the assumption that a statistical lens can form on the water surface. A spectral representation method is described, which allows one to construct random surfaces with a given correlation function. This method is implemented in the MATLAB software environment. To check the program, the relief of the CD-ROM was reproduced, obtained using an atomic force microscope. As a next step, a disturbed water surface is simulated, which acts on the transmitted rays like a real optical lens. The results obtained confirm the possibility of the formation of giant lenses on the surface of the seas and oceans, thus helping to distinguish the bottom topography from space.
Keywords: randomly uneven surface, slope correlation, statistical lens
Received on: 24.09.2020
Accepted on: 06.10.2020
INTRODUCTION
In August 1965, on the Gemini 5 mission, the American astronaut L. G. Cooper was able to observe the underwater relief through the water column. In the final report [1] on the mission, he reported that the topography of the coastal bottom and the structure of underwater currents near the California Peninsula were discernible from space orbit.
Since that time, a large amount of similar evidence has been accumulated. Thus, in June 1970, A. G. Nikolaev and V. I. Sevastiyanov, aboard the Soyuz 9 spacecraft, were the first Soviet cosmonauts to encounter this phenomenon [2]. They saw how the South American continent descends into the ocean, and how the continuation of the Amazon River is visible from space in the ocean. After that, several more similar cases were recorded [2]:
in August 1974 G. V. Sarafanov and L. S. Demin observed at depths of hundreds of meters the bottom of the Mozambique Strait, which separates the island of Madagascar from the continent of Africa;
in June 1975, P. I. Klimuk and V. I. Sevastiyanov, when flying over the Atlantic Ocean from New Foundland Island to the Canary Islands, clearly saw the ocean currents and the ocean floor in the shallow area;
in June 1978, members of the main crew of the second expedition of the Salyut‑6 orbital station V. V. Kovalenok and A S. Ivanchenkov observed the underwater topography of the Pacific Ocean floor at depths of up to four hundred meters.
As an illustrative example of this phenomenon, one can cite a satellite image [3] of the northeastern part of the Caspian Sea (Fig. 1). In this shallow water zone with depths of up to five to eight meters, the bottom topography is well reflected.
Currently, this effect is not yet fully studied, and among the versions about the causes of its occurrence, it is impossible to single out the only true one. According to the authors, the most probable of them seems to be the version associated with the formation of the so-called statistical (or stochastic) lens, a state in which the agitated surface of water affects the light rays passing through it like a real optical lens.
The study of this phenomenon is all the more important because it can play a significant role in solving many other problems associated with radiation transfer. One of the most important, in the context of this discussion, is the task of building a model of a reflective surface.
When creating it, it is important to take into account that light is reflected not only from the edges of the surface of the material, but also from the bulk of the material itself: the light penetrates into the subsurface layers of the substance, where it is scattered on particles, and then exits back into the surrounding space. Therefore, the most appropriate way of constructing a model of a reflecting surface is its interpretation in the form of a scattering layer bounded from below by a diffuse substrate, and from above by a randomly uneven Fresnel boundary [4].
There are many papers that touch on the issue of constructing a randomly uneven surface (RUS), including taking into account the correlation of slopes (e. g., [5, 6]). Despite the abundance of such works, so far no detailed study has been carried out on how taking into account the correlation affects the transmission of radiation through the RUS. The aim of this work is to answer this question.
PLOTTING A CORRELATED RANDOMLY UNEVEN SURFACE
We used the spectral representation method to simulate a randomly uneven surface. The main idea of this method [7] is to obtain a numerical model of a random process x(t) with a correlation function B(x) by approximating the stochastic integral
where ξ(dλ), η(dλ) are real orthogonal stochastic measures on the semiaxis [0,+∞); ν(dλ) = 2μ(dλ); ξ(dλ) = 2 Re ζ(dλ); η(dλ) = –2 Im ζ(dλ); ζ(dλ) is the stochastic spectral measure of the process x(t).
In the case under consideration, it was necessary to construct a numerical model of a correlated randomly uneven surface, which can be considered as a homogeneous isotropic random field on a plane.
We denote as Ψ(x) a random real field with the correlation function B(x), for which the expansion is given in the general case
, (1)
where p(λ) is the spectral density; . Further, for a simpler form of formulas, we will assume that the mathematical expectation , and the variance .
In the case of a finite spectrum, that is, when
,
the required field Ψ(x) can be constructed [8] by the formula
, (2)
where (ξk, ηk) are collectively independent standard Gaussian random variables; k = 1, 2, …, N. This representation can be used to construct a field with a continuous spectrum based on some quadrature formula
with non-negative coefficients, the sum of which is equal to one.
We split the space into parts D1, ..., DN and let random points λ1, ..., λΝ be distributed in these parts according to the densities
, .
Then, by virtue of (1), we have:
, .
When solving practical problems, the studied properties of the solution are often determined with the required accuracy by the correlation function of the initial field [8]. In this case, we can assume N = 1, D1 = Rn.
Let us further consider a specific example of modelling a randomly uneven surface with the correlation function B(r) = exp(–k0 r), where k0 is a constant and r2 = x2 + y2. Taking into account the assumed isotropy of a randomly uneven surface (that is, the presence of axial symmetry), we pass from the cosine Fourier transform to the Hankel transform. In this case, the spectral expansion of the correlation function B(r) takes the following form
,
where J0 is the Bessel function of the first kind.
Carrying out standard operations and using [9, 10], one can obtain a formula that will take its final form if we introduce averaging over realizations and provide for the case when DΨ(x) ≠ 1. With these transformations, the expression for modelling Ψ(x, y) is written as
(4)
where σ is the standard deviation.
Software implementation of the model and analysis of results
The model proposed in the previous section for constructing a correlated randomly uneven surface was implemented in the Mathworks MATLAB software environment. However, in order to make this model more “flexible”, the final appearance of the surface ΨΣ(x, y) is constructed as an overlay (superposition) of two surfaces Ψ1 and Ψ2 obtained by formula (4) with different parameters of
ΨΣ(x, y) = Ψ(x, y | ω1) + Ψ(x, y | ω2),
where ω is a point in the phase space Φ = {M, σ, k0}.
Such a modification makes it possible, by selecting certain parameters, to construct surfaces of various types that retain their statistical properties. For example, in this way you can build a smoothly polished mirror surface or an undulating ocean surface with fine ripples.
To check the correct operation of the program, obtained after the implementation of the model in MATLAB, and to evaluate its capabilities, we reproduced the relief of a real surface using an atomic force scanning microscope. A fragment of a CD ROM was used as a sample for the experiment. Below is the obtained surface relief of the sample (Fig. 2).
To obtain a surface ΨΣ(x, y) of a similar type, the following parameters ω1 and ω2 were selected:
ω1 = {M = 1, σ = 2, k0 = 1,09},
ω1 = {M = 100, σ = 0,6, k0 = 1}.
One of the implemented surfaces obtained with such parameters can have the form shown in Fig. 3.
The next step is the selection of such parameters of the RUS, at which it would be possible to observe the appearance of a statistical lens. As a result of numerous reproduction of random surfaces with different values, we were able to capture this effect.
Implementation of a random surface with parameters
ω1 = {M = 500, σ = 11,25, k0 = 0,025},
ω1 = {M = 100, σ = 0,2, k0 = 3}
might look like shown in Fig. 4. This option reproduces a relatively weakly agitated water surface with fine ripples (it is known that it is possible to record the underwater relief on images from space in rather rare cases; high transparency of water and calm weather are essential conditions [11]).
If an annular source of parallel rays is placed above the centre of the depression, then tracing their course after refraction by this surface (refractive index of water n = 1.33), one can see that most of the rays converge at some distance from the RUS (Fig. 5). Having placed a screen at the place where the rays converge, we will see the image of the luminous ring (Fig. 6), although to some extent it is blurred in comparison with the original one.
Special attention should be paid to the problem of finding the intersection points of rays with a randomly uneven surface. The task itself is quite complex. A simple enumeration of the parameters of the surface elements and determination of their belonging to the points of intersection of the ray with the corresponding planes can give an answer with a significant error due to the accumulation of machine error, and the solution takes significant computational time.
Although in the considered example, due to its simplicity, it was possible to restrict oneself to a trivial approach, the authors decided to use a more universal method in terms of the computation time in order to further use this model in solving more complex problems.
To determine the coordinates of the intersection points, the original RUS is approximated by the surface , for which the system of equations with respect to ξ is then solved
This system is reduced to a nonlinear equation with one unknown, since the approximating surface is specified explicitly as . However, the resulting equation in the general case can have a large number of roots, and therefore the estimate of the initial value comes to the fore.
In the case under consideration, this problem is solved by dividing the domain of definition of the function into a set of parts and multiplying the values of the function on the corresponding boundaries of adjacent segments. After that, the segment closest to zero on the right is selected, on which the function changes its sign. On this segment, the value of is refined using the bisection method.
This method can be vectorized and thus search for the coordinates of intersection points for all rays at once, which is its great advantage. The disadvantages include the likelihood of missing roots. However, such an approach by its nature corresponds to the problem being solved, and the partition of the domain of definition of the function into sufficiently small parts allows us to consider the probability of missing a root as negligible.
Conclusion
The results presented in this work show that large-scale water surfaces (such as the sea or the ocean) can indeed form, under certain conditions, giant lenses through which one can observe from space objects located deep under water.
In addition, the example with the water surface and the surface of the CD-ROM speaks of the versatility of the model, due to its ability to form surfaces of various origins of irregularities and their different types.
As the next stages in the development of the model, we can consider the inclusion of scattering in the volume of the substance under the surface. In the future, this will pave the way for the development of a universal model for the reflection of light from a real surface.
Also promising is the development and use of the method of modelling a randomly uneven surface, proposed in [12]. Its essence consists in modelling not the RUS itself, but only the points of intersection of the rays with it. This approach will significantly reduce the computation time compared to standard methods.
REFERENCES
Gemini Program Mission Report. Gemini V. Huston, 1965.
Lazarev A. I., Sevast’janov V. I. Licom k licu lica ne uvidat’. Nauka i zhizn’. 1987; 9: 27–32. (In Russ.) ISSN 0028–1263.
Deshifrirovanie mnogozonal’nykh aehrokosmicheskikh snimkov. Metodika i rezul’taty. – M.; Berlin: Nauka; Akademi-ferlag, 1982. 83 p. (In Russ.)
Budak V. P., Grimailo A. V. The model of reflective surface based on the scattering layer with diffuse substrate and randomly rough Fresnel boundary. CEUR Workshop Proc. 2019; 2485: 198–201. DOI: 10.30987/graphicon‑2019-2-198-201.
Gartley M. G., Brown S. D., Schott J. R. Micro-scale surface and contaminate modeling for polarimetric signature prediction. Polariz. Meas. Anal. Remote Sens. VIII. 2008; 6972: 697213. DOI: 10.1117/12.801904.
Gartley M. G., Schott J. R., Brown S. D. Micro-scale modeling of contaminant effects on surface optical properties. Imaging Spectrom. XIII. 2008; 7086: 70860H. DOI: 10.1117/12.796428.
Prigarin S. M. Modeli sluchajjnykh processov i polejj v metodakh Monte-Karlo. Palmarium Academic Publishing, 2014. 160 p. (In Russ.) ISBN 978-3-659-98980-3.
Mikhajjlov G. A. Chislennoe postroenie sluchajjnogo polja s zadannojj spektral’nojj plotnost’ju. Dokl. AN SSSR. 1978; 238(4): 793–795. (In Russ.)
Ermakov S. M., Mikhajjlov G. A. Kurs statisticheskogo modelirovanija. – M.: Nauka, 1976. 320 p.
Korn G., Korn T. Spravochnik po matematike (dlja nauchnykh rabotnikov i inzhenerov). – M.: Nauka. 1974. 832 p. (In Russ.)
Pokazeev K. V., Chaplina T. O., Chashechkin Ju. D. Optika okeana: Uchebnoe posobie. – M.: MAKS Press. 2010. 216 p. (In Russ.) ISBN 978-5-317-03439-9.
Kargin B. A., Rakimgulov K. B. A weighting Monte Carlo method for modelling the optical radiation field in the ocean-atmosphere system. Russ. J. Numer. Anal. Math. Model. 1992. 7(3): 221–240.
The authors declare the distribution of the contribution of each to the mutual work: Budak V. P. – research administration and discussion of results,
Grimailo – formulation and implementation of the algorithm.
AUTHORS
Budak Vladimir P., e-mail: BudakVP@gmail.com; Dr. of Tech. Sc., Prof., Lighting Engineering Department, National Research University “Moscow Power Engineering Institute“, Moscow, Russia.
ORCID iD: 0000-0003-4750-0160
ResearcherID: G‑4515–2014
Scopus ID: 10142738100
Grimailo Anton V., Master Graduate of the Lighting Engineering Department, e-mail: GrimailoAV@gmail.com; National Research University “Moscow Power Engineering Institute“, Moscow, Russia.
ORCID iD: 0000-0002-1253-7687
Scopus ID: 57211639687
V. P. Budak, A. V. Grimailo
National Research Univarsity ”Moscow Power Engineering Institute’’, Moscow, Russia
The article is devoted to modelling a randomly uneven surface, taking into account the correlation of slopes. The facts of observation of the underwater relief from space are presented, and an explanation is proposed based on the assumption that a statistical lens can form on the water surface. A spectral representation method is described, which allows one to construct random surfaces with a given correlation function. This method is implemented in the MATLAB software environment. To check the program, the relief of the CD-ROM was reproduced, obtained using an atomic force microscope. As a next step, a disturbed water surface is simulated, which acts on the transmitted rays like a real optical lens. The results obtained confirm the possibility of the formation of giant lenses on the surface of the seas and oceans, thus helping to distinguish the bottom topography from space.
Keywords: randomly uneven surface, slope correlation, statistical lens
Received on: 24.09.2020
Accepted on: 06.10.2020
INTRODUCTION
In August 1965, on the Gemini 5 mission, the American astronaut L. G. Cooper was able to observe the underwater relief through the water column. In the final report [1] on the mission, he reported that the topography of the coastal bottom and the structure of underwater currents near the California Peninsula were discernible from space orbit.
Since that time, a large amount of similar evidence has been accumulated. Thus, in June 1970, A. G. Nikolaev and V. I. Sevastiyanov, aboard the Soyuz 9 spacecraft, were the first Soviet cosmonauts to encounter this phenomenon [2]. They saw how the South American continent descends into the ocean, and how the continuation of the Amazon River is visible from space in the ocean. After that, several more similar cases were recorded [2]:
in August 1974 G. V. Sarafanov and L. S. Demin observed at depths of hundreds of meters the bottom of the Mozambique Strait, which separates the island of Madagascar from the continent of Africa;
in June 1975, P. I. Klimuk and V. I. Sevastiyanov, when flying over the Atlantic Ocean from New Foundland Island to the Canary Islands, clearly saw the ocean currents and the ocean floor in the shallow area;
in June 1978, members of the main crew of the second expedition of the Salyut‑6 orbital station V. V. Kovalenok and A S. Ivanchenkov observed the underwater topography of the Pacific Ocean floor at depths of up to four hundred meters.
As an illustrative example of this phenomenon, one can cite a satellite image [3] of the northeastern part of the Caspian Sea (Fig. 1). In this shallow water zone with depths of up to five to eight meters, the bottom topography is well reflected.
Currently, this effect is not yet fully studied, and among the versions about the causes of its occurrence, it is impossible to single out the only true one. According to the authors, the most probable of them seems to be the version associated with the formation of the so-called statistical (or stochastic) lens, a state in which the agitated surface of water affects the light rays passing through it like a real optical lens.
The study of this phenomenon is all the more important because it can play a significant role in solving many other problems associated with radiation transfer. One of the most important, in the context of this discussion, is the task of building a model of a reflective surface.
When creating it, it is important to take into account that light is reflected not only from the edges of the surface of the material, but also from the bulk of the material itself: the light penetrates into the subsurface layers of the substance, where it is scattered on particles, and then exits back into the surrounding space. Therefore, the most appropriate way of constructing a model of a reflecting surface is its interpretation in the form of a scattering layer bounded from below by a diffuse substrate, and from above by a randomly uneven Fresnel boundary [4].
There are many papers that touch on the issue of constructing a randomly uneven surface (RUS), including taking into account the correlation of slopes (e. g., [5, 6]). Despite the abundance of such works, so far no detailed study has been carried out on how taking into account the correlation affects the transmission of radiation through the RUS. The aim of this work is to answer this question.
PLOTTING A CORRELATED RANDOMLY UNEVEN SURFACE
We used the spectral representation method to simulate a randomly uneven surface. The main idea of this method [7] is to obtain a numerical model of a random process x(t) with a correlation function B(x) by approximating the stochastic integral
where ξ(dλ), η(dλ) are real orthogonal stochastic measures on the semiaxis [0,+∞); ν(dλ) = 2μ(dλ); ξ(dλ) = 2 Re ζ(dλ); η(dλ) = –2 Im ζ(dλ); ζ(dλ) is the stochastic spectral measure of the process x(t).
In the case under consideration, it was necessary to construct a numerical model of a correlated randomly uneven surface, which can be considered as a homogeneous isotropic random field on a plane.
We denote as Ψ(x) a random real field with the correlation function B(x), for which the expansion is given in the general case
, (1)
where p(λ) is the spectral density; . Further, for a simpler form of formulas, we will assume that the mathematical expectation , and the variance .
In the case of a finite spectrum, that is, when
,
the required field Ψ(x) can be constructed [8] by the formula
, (2)
where (ξk, ηk) are collectively independent standard Gaussian random variables; k = 1, 2, …, N. This representation can be used to construct a field with a continuous spectrum based on some quadrature formula
with non-negative coefficients, the sum of which is equal to one.
We split the space into parts D1, ..., DN and let random points λ1, ..., λΝ be distributed in these parts according to the densities
, .
Then, by virtue of (1), we have:
, .
When solving practical problems, the studied properties of the solution are often determined with the required accuracy by the correlation function of the initial field [8]. In this case, we can assume N = 1, D1 = Rn.
Let us further consider a specific example of modelling a randomly uneven surface with the correlation function B(r) = exp(–k0 r), where k0 is a constant and r2 = x2 + y2. Taking into account the assumed isotropy of a randomly uneven surface (that is, the presence of axial symmetry), we pass from the cosine Fourier transform to the Hankel transform. In this case, the spectral expansion of the correlation function B(r) takes the following form
,
where J0 is the Bessel function of the first kind.
Carrying out standard operations and using [9, 10], one can obtain a formula that will take its final form if we introduce averaging over realizations and provide for the case when DΨ(x) ≠ 1. With these transformations, the expression for modelling Ψ(x, y) is written as
(4)
where σ is the standard deviation.
Software implementation of the model and analysis of results
The model proposed in the previous section for constructing a correlated randomly uneven surface was implemented in the Mathworks MATLAB software environment. However, in order to make this model more “flexible”, the final appearance of the surface ΨΣ(x, y) is constructed as an overlay (superposition) of two surfaces Ψ1 and Ψ2 obtained by formula (4) with different parameters of
ΨΣ(x, y) = Ψ(x, y | ω1) + Ψ(x, y | ω2),
where ω is a point in the phase space Φ = {M, σ, k0}.
Such a modification makes it possible, by selecting certain parameters, to construct surfaces of various types that retain their statistical properties. For example, in this way you can build a smoothly polished mirror surface or an undulating ocean surface with fine ripples.
To check the correct operation of the program, obtained after the implementation of the model in MATLAB, and to evaluate its capabilities, we reproduced the relief of a real surface using an atomic force scanning microscope. A fragment of a CD ROM was used as a sample for the experiment. Below is the obtained surface relief of the sample (Fig. 2).
To obtain a surface ΨΣ(x, y) of a similar type, the following parameters ω1 and ω2 were selected:
ω1 = {M = 1, σ = 2, k0 = 1,09},
ω1 = {M = 100, σ = 0,6, k0 = 1}.
One of the implemented surfaces obtained with such parameters can have the form shown in Fig. 3.
The next step is the selection of such parameters of the RUS, at which it would be possible to observe the appearance of a statistical lens. As a result of numerous reproduction of random surfaces with different values, we were able to capture this effect.
Implementation of a random surface with parameters
ω1 = {M = 500, σ = 11,25, k0 = 0,025},
ω1 = {M = 100, σ = 0,2, k0 = 3}
might look like shown in Fig. 4. This option reproduces a relatively weakly agitated water surface with fine ripples (it is known that it is possible to record the underwater relief on images from space in rather rare cases; high transparency of water and calm weather are essential conditions [11]).
If an annular source of parallel rays is placed above the centre of the depression, then tracing their course after refraction by this surface (refractive index of water n = 1.33), one can see that most of the rays converge at some distance from the RUS (Fig. 5). Having placed a screen at the place where the rays converge, we will see the image of the luminous ring (Fig. 6), although to some extent it is blurred in comparison with the original one.
Special attention should be paid to the problem of finding the intersection points of rays with a randomly uneven surface. The task itself is quite complex. A simple enumeration of the parameters of the surface elements and determination of their belonging to the points of intersection of the ray with the corresponding planes can give an answer with a significant error due to the accumulation of machine error, and the solution takes significant computational time.
Although in the considered example, due to its simplicity, it was possible to restrict oneself to a trivial approach, the authors decided to use a more universal method in terms of the computation time in order to further use this model in solving more complex problems.
To determine the coordinates of the intersection points, the original RUS is approximated by the surface , for which the system of equations with respect to ξ is then solved
This system is reduced to a nonlinear equation with one unknown, since the approximating surface is specified explicitly as . However, the resulting equation in the general case can have a large number of roots, and therefore the estimate of the initial value comes to the fore.
In the case under consideration, this problem is solved by dividing the domain of definition of the function into a set of parts and multiplying the values of the function on the corresponding boundaries of adjacent segments. After that, the segment closest to zero on the right is selected, on which the function changes its sign. On this segment, the value of is refined using the bisection method.
This method can be vectorized and thus search for the coordinates of intersection points for all rays at once, which is its great advantage. The disadvantages include the likelihood of missing roots. However, such an approach by its nature corresponds to the problem being solved, and the partition of the domain of definition of the function into sufficiently small parts allows us to consider the probability of missing a root as negligible.
Conclusion
The results presented in this work show that large-scale water surfaces (such as the sea or the ocean) can indeed form, under certain conditions, giant lenses through which one can observe from space objects located deep under water.
In addition, the example with the water surface and the surface of the CD-ROM speaks of the versatility of the model, due to its ability to form surfaces of various origins of irregularities and their different types.
As the next stages in the development of the model, we can consider the inclusion of scattering in the volume of the substance under the surface. In the future, this will pave the way for the development of a universal model for the reflection of light from a real surface.
Also promising is the development and use of the method of modelling a randomly uneven surface, proposed in [12]. Its essence consists in modelling not the RUS itself, but only the points of intersection of the rays with it. This approach will significantly reduce the computation time compared to standard methods.
REFERENCES
Gemini Program Mission Report. Gemini V. Huston, 1965.
Lazarev A. I., Sevast’janov V. I. Licom k licu lica ne uvidat’. Nauka i zhizn’. 1987; 9: 27–32. (In Russ.) ISSN 0028–1263.
Deshifrirovanie mnogozonal’nykh aehrokosmicheskikh snimkov. Metodika i rezul’taty. – M.; Berlin: Nauka; Akademi-ferlag, 1982. 83 p. (In Russ.)
Budak V. P., Grimailo A. V. The model of reflective surface based on the scattering layer with diffuse substrate and randomly rough Fresnel boundary. CEUR Workshop Proc. 2019; 2485: 198–201. DOI: 10.30987/graphicon‑2019-2-198-201.
Gartley M. G., Brown S. D., Schott J. R. Micro-scale surface and contaminate modeling for polarimetric signature prediction. Polariz. Meas. Anal. Remote Sens. VIII. 2008; 6972: 697213. DOI: 10.1117/12.801904.
Gartley M. G., Schott J. R., Brown S. D. Micro-scale modeling of contaminant effects on surface optical properties. Imaging Spectrom. XIII. 2008; 7086: 70860H. DOI: 10.1117/12.796428.
Prigarin S. M. Modeli sluchajjnykh processov i polejj v metodakh Monte-Karlo. Palmarium Academic Publishing, 2014. 160 p. (In Russ.) ISBN 978-3-659-98980-3.
Mikhajjlov G. A. Chislennoe postroenie sluchajjnogo polja s zadannojj spektral’nojj plotnost’ju. Dokl. AN SSSR. 1978; 238(4): 793–795. (In Russ.)
Ermakov S. M., Mikhajjlov G. A. Kurs statisticheskogo modelirovanija. – M.: Nauka, 1976. 320 p.
Korn G., Korn T. Spravochnik po matematike (dlja nauchnykh rabotnikov i inzhenerov). – M.: Nauka. 1974. 832 p. (In Russ.)
Pokazeev K. V., Chaplina T. O., Chashechkin Ju. D. Optika okeana: Uchebnoe posobie. – M.: MAKS Press. 2010. 216 p. (In Russ.) ISBN 978-5-317-03439-9.
Kargin B. A., Rakimgulov K. B. A weighting Monte Carlo method for modelling the optical radiation field in the ocean-atmosphere system. Russ. J. Numer. Anal. Math. Model. 1992. 7(3): 221–240.
The authors declare the distribution of the contribution of each to the mutual work: Budak V. P. – research administration and discussion of results,
Grimailo – formulation and implementation of the algorithm.
AUTHORS
Budak Vladimir P., e-mail: BudakVP@gmail.com; Dr. of Tech. Sc., Prof., Lighting Engineering Department, National Research University “Moscow Power Engineering Institute“, Moscow, Russia.
ORCID iD: 0000-0003-4750-0160
ResearcherID: G‑4515–2014
Scopus ID: 10142738100
Grimailo Anton V., Master Graduate of the Lighting Engineering Department, e-mail: GrimailoAV@gmail.com; National Research University “Moscow Power Engineering Institute“, Moscow, Russia.
ORCID iD: 0000-0002-1253-7687
Scopus ID: 57211639687
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