Alekseenko, M. A.

Gendrina, I. U.

English

Tomsk State University Repository

МИНИСТЕРСТВО ОБРАЗОВАНИЯ И НАУКИ РОССИЙСКОЙ ФЕДЕРАЦИИ ТОМСКИЙ ЕОСУДАРСТВЕННЫЙ УНИВЕРСИТЕТМ А Т Е Р И А Л ЫV Меадународной молодежной научной конференции«МАТЕМАТИЧЕСКОЕ И ПРОГРАММНОЕ ОБЕСПЕЧЕНИЕ ИНФОРМАЦИОННЫХ, ТЕХНИЧЕСКИХИ ЭКОНОМИЧЕСКИХ СИСТЕМ»Томск, 19-20 мая 2017 г.Под общей редакциейкандидата технических наук И.С. ШмыринаТомскИздательский Дом Томского государственного университета2017ориентированного языка программирования GPSS, получена аналитичеекая модель, раеемотрен иллюетративный пример, который подтверждает работоепоеобноеть веех поетроенных моделей.ЛИТЕРАТУРА1. Барлоу Р. Математическая теория надежности / Р. Барлоу, Ф. Прошан. -  М.: Сов. радио, 1962. -  312 с.2. Барлоу Б. Статическая теория надежности и испытания на безотказность / Б. Барлоу. -  М.: Наука, Глав­ная редакция ф.-м. литературы, 1984. -  327 с.3. Гнеденко Б.В. Математические методы в теории надежности / Б.В. Гнеденко, Ю.К. Беляев, А.Д. Соловьев. -  М.: Наука, Главная редакция физико-математической литературы, 1965. -  524 с.4. Бейтмен Г. Таблицы интегральных преобразований / Г. Бейтмен, А. Эрдейн. -  М.:Наука,1969, Т. 1. -343 с.5. Терпугов А.Ф. Теория случайных процессов / А.Ф. Терпугов. -  Томск: Изд-во ТГУ, 1974.- 121 с.REGRESSION ANALYSIS SIMULATION OF RADIATION TRANSFER IN SYSTEMS VISION THROUGH THE ATMOSPHEREM.A. Alekseenko, I.U. GendrinaTomsk State University masha_af6@mail.ru, igendrina@bk.ruIntroductionSuch method of studies as the system approach has been intensively developed in differ­ent fields of science, industry, and social life in recent decades. System approach is defined to be that which considers any system (object) to be a set of interrelated elements (components), and to have output (purpose), input (resources), connection with environment, and feedback. System approach represents a form of applications in the theory of knowledge and dialectics to studying the processes that occur in nature, community, and thinking. It essentially consists of fulfillment of the requirements of the general system theory; in accordance to this theory, each object in the process of its study should be considered as a large and complex system and, simultaneously, as an element of a more general system. One of the applications of the system approach is the optics [3] and, in particular, the atmospheric optics [4]. The main sys­tem characteristic in these fields is the point spread function (PSF); it is defined as the re­sponse L of linear system to the input signal, representing a point mass 5 (x -X j)8(y-@ )5  located at a certain point( x i , @ )  : i  [ 5  ( x  -  Xi)  5  ( y  -  @ ) ]  =  A ( x ,  у ; Xi, @ ) .An arbitrary object (function) /  ( x , y )  can be considered as a set of point masses. For in­stance:Ч-со Ч-СОf { x , y ) =  j j / ( x i , y i ) 5 ( x - X i , y - y i ) < i X i < i y i  .—CO —COThen, a result of the system impact (image) can be represented in the form:-l-co -i-cog(x,y) = i [ / ( x ,y ) ] =  j j /(xi,yi)A(x,y;Xi,yi)<iXi<iyi .—CO —COObviously, regularities of the image distortion due to impact of any system can be stu­died by analyzing the effect of this system on the point spread function.We will consider a system “underlying system -  atmosphere -  receiving device” which can be regarded as a linear system [3]. These systems are conventionally called the vision systems. Image distortion in the vision systems may be caused by the properties of scattering medium and underlying surface, and by characteristics of receiving optical device. The avail­able images of any objects should be analyzed, and possible distortions of object images should be predicted, by studying the point spread function of this system. One of the methodsfor the PSF determination in this ease is to ealeulate the angular distribution of brightness of surfaee-based point souree, measured with reeeiving deviee at the top of the atmosphere (TOA).The purpose of this work is study dependenee the angular distribution of brightness to radiation at TOA on the geometrieal and optieal observation eonditions, as well as elarify the applieability eonditions of plane and spherieal models.1. Problem statementChange brightness of radiation at point souree in stationary ease is deseribed by the fol­lowing integro-differential transfer equation:((B,grad/(r,(B)) = -a(k ,r) /  (r,(5) + + Ф„ (г,ю) • (1)QHere, X = (f,co) is the point of phase spaee X  = RxQ.  of eoordinates f  gR and diree- tions (d gQ. Фо(г,со) is the distribution density of sourees. is the intensity (bright­ness) at point X = (f,co) .In this paper, we eonsider two models of the atmosphere. The first model of planetary atmosphere represents as plane-parallel layered-homogeneous medium, i.e., all quantities in (1) depends on just one eoordinate, namely, the depth z, while the intensity of the seattered radiation will be a funetion of the eoordinate z and direetion of radiation, eharaeterized by the zenith angle 0 and azimuth cp in the horizontal plane. Values z are varies from z = 0 at the bottom of the atmosphere to z=H  at TOA, where H  is the thiekness of the medium. The souree is on the Earth’s surfaee, and it will be assumed to be the origin of the eoordinates. The reeeiver is on the upper boundary of the medium and has the eoordinates (0 ,0 ,//) . The sys­tem has been the eireular symmetry; therefore, all movements of photons are proeeed in the YOZ [1].Layered-homogeneous plane-parallel model was ehosen as the model of the medium; it eonsists on n homogeneous layers, the geometrieal thiekness, whieh is eharaeterized by eoor­dinate z (fig. 1).Fig. 1. Shematic representation plane-parallel model of the atmosphereThe second model represents the layered-homogeneous spherical atmosphere. In this case, all quantities depend on the distance from Earth’s surface h, while the intensity of scat­tering radiation will be the function of h and direction of radiation й = (0,cp).The Earth’s center to be the origin of coordinates is assumed. The source has the follow­ing coordinates: (0,0,i?„); while the receiver located at TOA and has the coordinates(0,0,7?), where 7?o is the radius of Earth, and R is the outer radius of the atmosphere. This model of the medium atmosphere with the thickness H = R -R ^  is divided into n spherical layers with radii Rj, i = 0,n , R^=R [2] (fig. 2).Fig. 2. Shematic representation spherical model of the atmosphereOptical model of the atmosphere include the following parameters.1. The coefficients of scattering {h,X) and absorption {h,X) . Here, h is the heightabove the Earth’s surface, and X is the wavelength. The coefficients o., and Од piecewise con­stant are assumed , since the atmosphere into homogeneous layers is divided.2. The scattering phase function g[h,[i,X) . Here, p = (й',со) is the cosine of the scatter­ing angle.The scattering phase function is specified by dividing the atmosphere into layers, in which, the scattering phase function constant is assumed. One of the most universal methods for solving equation (1) is the method of imitation simulation, or the Monte Carlo method. This method on the integral transfer equation of the second kind with generalized kernel for the particle collision density is based:/ (x) = 1^ {x \x) / (x ')dx' + i|/(x) .Monte Carlo method is usually used to estimate linear functional of the form (2):^Ф=(/,ф) = |/(^ )ф (^ )й 5 г. (2)If {x^) is a “physicaf’ chain of collisions, then =M^  where ^ • cp(x^).«=0We single out two main algorithms of the Monte Carlo method.1. The algorithm of direct simulation. This method is based on simulation of random tra­jectories of photon passage through the scattering medium. It is noteworthy that the characte­ristics of radiation process, necessary for analysis, are estimated in accordance with their physical meaning. A disadvantage of the direct simulation is that such characteristics as inten­sity, illumination, and others cannot be calculated with a sufficient accuracy. However, the direct simulation can be used to construct some other methods which make it possible to per­form the required calculations.One of these methods is the algorithm of loeal estimate.2. Algorithm of loeal estimate.The algorithm of loeal estimate eonsists of the ealeulation of the following funetional:N(0.) = = j h ( x ' , x ) f { x ' ) d x ' = M Y , Q J i  { x „ , x )  ,e, .r я=о2 7 r f - f, |a* = (co,?*^, A, (?) is the indieator of the region 0j, ф(г*,со*^ is theHere, ?* = r - r\r - rpartiele flux at a preset point of the phase spaee x* = {r* , Q„ is the partiele “weight”.We do not simulate the absorption, but multiply the “weight” by the seattering probabili­ty, namely, by the single seattering albedo. It is noteworthy that the varianee deereases, and the average time of trajeetory simulation on eomputer inereases.This algorithm is used to ealeulate the angular distribution of brightness, whieh, . jUd)represents the following quantity: I(®) = — , where 0  is the value of the eorrespondingto 0 solid angle.2. Initial dataWe will eonsider the proeess of radiative transfer through aerosol-moleeular atmosphere, whieh eomprises layer overeast, by negleeting the refleetion from underlying surfaee. We used the following data:1. Wavelength (mkm) in transparent windows: 0.347; 0.530; 0.694; 0.860; 1,060; 3,390; 10.60.2. Lower boundary of atmosphere above Earth's surfaee, upper limit of the atmosphere H = 30 km above the Earth's surfaee.3. Optieal thiekness for a eloudless atmosphere shown in Table 1 presented.Table 1Optical thickness of the cloudless atonosphereWavelength, mkm Optical thickness0,347 0,2280,530 0,1580,694 0,1240,860 0,0981,060 0,0923,390 0,06710,600 0,0414. Lower boundary of the eloud layer -  1 km above the Earth's surfaee. Upper limit -  2 km above the Earth's surfaee. The optieal model of the eloud layer -  haze H.5. In this work, we eonsidered two models of sourees of radiation: Lambertian, isotropie, and sourees. In ease of isotropie souree, density of the initial areas looks like1p  ( i d )  = —  . For Lambertian this value is defined as p  ( i d )  =  — .27Г 71For the statistieal experiments was used a loeal assessment method in the seheme of eon- strueting eonjugate trajeetories. Caleulations were earried out based software paekage, whieh designed for a eloudless atmosphere. This eomplex was modified in order to be able to eon-duct simulations under the presence of clouds. Numerical simulation results are shown in Ta­ble 2.Table 2The brightness of the scattered radiation to the atmosphere from the cloud layer type “H Haze”Angle of reception, gradWavelengths, mkmX =  0,374 7 = 0,530 X =  0,694 X =  0,860 X =  1,060 X =  3,390 X =  10,604,5 2,24E-05 l,67E-05 l,38E-05 l,13E-05 l,19E-05 5,55E-06 3,51E-0613,5 l,09E-06 7,65E-07 6,60E-07 5,30E-07 4,56E-07 2,09E-07 l,43E-0722,5 2,23E-07 l,83E-07 l,55E-07 l,39E-07 l,12E-07 4,84E-08 3,51E-0831,5 8,44E-08 5,63E-08 5,54E-08 4,73E-08 3,85E-08 l,95E-08 l,32E-0840,5 3,49E-08 2,73E-08 2,81E-08 6,65E-08 4,33E-08 8,70E-09 5,38E-0949,5 l,86E-08 5,15E-08 l,54E-08 l,18E-08 9,30E-09 4,23E-09 4,59E-0958,5 l,30E-08 l,10E-08 l,71E-08 7,75E-09 6,64E-09 3,58E-09 3,57E-0967,5 l,94E-08 l,12E-08 7,95E-09 2,llE-08 l,73E-08 8,16E-08 6,37E-0976,5 l,58E-07 2,13E-08 4,96E-09 3,68E-09 2,90E-09 l,99E-09 l,18E-0985,5 l,20E-08 6,37E-09 l,23E-09 l,03E-09 l,00E-09 5,13E-11 1,16E-113. Statistical analysis of simulation resultRegression analysis was used to establish a functional link between the angular distribu­tion of brightness and optical parameters. This analysis is widely used to restore the characte­ristics of aerosol and clouds, as well as to assess their impact on the climate.For the angular distribution of brightness in the aerosol and molecular cloud and the at­mosphere relative to the wavelength, the regression equation was built. This regression equa­tion was obtained in the form of у = for all wavelengths. The regression coefficients are shown in Tables 4-5. coefficients of determination were also given for each equation.Similarly, the regression equation was built for the angular distribution of brightness in the cloud spherical atmosphere at fixed angles reception. Regression coefficients and the coef­ficient of determination are shown in Table 3.Table 3The coefficients of the regression equation for cloudless atmosphereAngle of reception, grad Coefficient bo Coefficient b\ Coefficient of determination4,5 l,19595E-05 3,25534E-05 0,99363643813,5 2,31401E-07 5,05694E-07 0,97688560822,5 6,98539E-08 l,19799E-07 0,92750353431,5 3,92659E-08 4,0455 lE-08 0,78477320840,5 2,4403E-08 l,51266E-08 0,57006000849,5 l,39106E-08 6,51344E-09 0,48254344858,5 6,82884E-09 3,64745-09 0,60974468367,5 2,74359E-09 2,35355E-09 0,77115510476,5 5,21933E-10 l,23544E-09 0,90722026585,5 3,00774E-11 1,7624E-10 0,929769833Table 4The coefficients of the regression equation for a cloudy atmosphereAngle of reception, grad Coefficient bo Coefficient b\ Coefficient of determination4,5 3,52884E-06 7,12796E-06 0,9854282613,5 l,08652E-07 3,63939E-07 0,99727222,5 3,63765-08 7,53979E-08 0,97307532231,5 l,26022E-08 2,66418E-08 0,97638487658,5 3,68558E-09* 4,34001E-09 0,59850788776,5 -3,34288E-08* 5,0401E-09 0,60609756185,5 -2,37483E-09* 4,50822E-09 0,809969895Statistical evaluation of the regression equations on the significance was conducted by F- test and Student's t-test. With 90% confidence it can be argued that the considered depen­dence are statistically significant.It should be noted that the obtained analytieal data for the eloudless atmosphere are more aeeurate as eompared with the data from the atmosphere eloud layer.This onee again eonfirms that the turbid layered homogeneous environment analysis and the preparation of some of the radiation brightness eharaeteristies of a distorted unlike eharae- teristies in the elear atmosphere.The analysis shows that between the obtained angular distributions of intensity and wa­velength for both eloudless and eloudy for the atmosphere, there is a link that ean be with a good degree of aeeuraey to deseribe hyperbolie regression equation. This faet ean be used for predietion, assessment and analysis of images of objeets observed through the seattering me­dium.ConclusionIn the eourse of paper the following problem was posed and solved:1. Simulation model for the propagation of radiation is eonstrueted, whieh based on the dual transport equation using a loeal assessment method.2. Algorithm of the Monte Carlo method for ealeulation angular distribution of the radia­tion intensity in a eloudless sky is implemented.3. Dependenee angular distribution of brightness on the wavelength, the thiekness of the atmospherie model and radiation sourees is investigated.4. Applying regression analysis to establish funetional relationship between the angular distribution of brightness and parameters of model the atmosphere.REFERENCES1. Krekov G.M. Optical models of atmospheric aerosol / G.M. Krekov, R.F. Rakhimov. -  Tomsk. Publishing House of Tomsk Affiliate of the Siberian Branch of the USSR Academy of Sciences, 1986. -  294 pp.2. The Monte Carlo method in atmospheric optics / G.I. Marchuk [et al]; general editor G.I. Marchuk. -  Novosi­birsk, ’’Nauka”, 1976. -  280 pp.3. Papoulis A. Systems and transforms with applications in optics / A. Papoulis. -  McGraw-Hill Book Company, 1968.-474 pp.4. ZuevV.E. Systems with applications in scattering media / V.E. Zuev, V.V. Belov, V.V. Veretennikov. -  Tomsk. Publishing House “Spectrum” Institute of Atmospheric Optics of the Siberian Branch of the Russian Academy of Sciences, 1997. -  402 pp.

Regression analysis simulation of radiation transfer in systems vision through the atmosphere

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Regression analysis simulation of radiation transfer in systems vision through the atmosphere

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