[精品論文]REGULARIZATION MODIFICATION INVERSION OF RADIATIVE TRANSFER EQUATIONS FOR RETRIEVE IOP OF WATER.doc

精品論文regularization modification inversion of radiative transfer equations for retrieve iop of water5zheng ye, wei ran, gu yanfeng(school of electronic and information. harbin institue of techonlogy, harbin 150001)abstract: as a physical-based model, radiative transfer equations (rte) has proved to be an outstanding tool to predict light fields distribution in certain water body. moreover, the inversion technique provide us a possibility to retrieve water properties fromtheir spectrum based on rte.10however, existing inverse algorithms to calculate iops based on rte are lack of stability due to the ill-posed nature of inverse problem. in this paper, we propose a modification inverse method to solve rte.a regularization term is forced to gauss-newton iterations; therefore, the sensitivity to model noise is reduced, name-ly stability is improved. the experiment results show that our algorithm can acquire comparable retrieve outputs when noise is added to input spectrum.15keywords: signal procssing; regularization; rte; retrieve; stability0introductionwater is an important source in our planet, therefore many techniques are used to survey its physical and chemical properties. among them, remote sensing (rs) that can acquire image of20researching area in wide space scale has developed to study certain water body such as interior lake or near shore seawater. as a kind of rs image, hyperspectral will provide us not only spatial but also spectral information of waters, that make the possibility of quantity analysis of water. nowadays, three kindsof hyperspectral based water parameters analysis are used: experience-based retrieving methods, semi-empirical approach and physical based approaches.25compare to the others, physical based methods are more reliable and approximate to actual physical process during the imaging of hyperspectral. roughly speaking, such methods are solve inverse problem described by physical transfer model. many physical based models are proposed, for example, single-scattering, quasi-single scattering, two-stream models, etc. those models are naturally mathematical description of photon interact with corpuscle in water. rte (radiative30transfer equation) model, proposed by c.d. mobley, is most classical one.based on above models, several inverse approaches are proposed. in1-3look-up table methods with interpolation is proposed; in4 spectral decomposition algorithms is proposed by oyamaa et al. in5 6 , algorithms based on an mathematical reformation of bio-opticalmodel, matrix-inversion, is proposed to linearization original bio-optical. in7 simplex concept is35introduced by peter. gege to get optimal parameters of rte model; in8-10, the established model is inversed by newton-gauss methods, that is typical descent algorithm to deal with non-linear curve fitting problems. the processing to acquire model parameters is typical inverse problems in mathematical, which is seldom posed-ness. that means, one of the following conditions:the existence, the uniqueness or the stability will not satisfied. the first two difficulties can be solved40by enlarging or reducing solution spaces, for example, in2, the author raise the accuracy ofinversion by add some boundaries to variable, namely, limit the dynamic range of parameters. however, the last one, stability that is more difficulty to achieve,is usually the reason lead toill-posedness ofinversion. gauss-newton methods are adopted, such method as we can see infoundations: research fund for the doctorial program of higher education of china under the grant(no.20092302110033)brief author introduction:zhang ye(1960-),male,professor. major research area is remote sensing image processing technique,such as hyperspectral classficiation,target detection,quantity inversion. e-mail: - 13 -section 3, willbe very sensitivity to disturbance- little error in the input result large changes in45the output due to ill-condition matrix existed in their iteration.in this paper, we proposed a modification version of gauss-newton methods to retrieve water color factors (wcf) via an physical-based model. first, rte model is represented to parameterization the relationship between apparent optical properties (aops) and inherent optical properties (iops), and then the inverse algorithm to retrieve iops form aopsis50proposed. it can be considered as an robust version of newton-gauss algorithms, which is a classical trust-region search methods. the proposed method improve their stability by add a regularization term to the original algorithms, the experiment shows that tolerance to error of proposed algorithms are improved and the sensitivity to disturbance is reduced.1the basic theory of radiative transfer equations55any water bodies, including seawater, lake water, river or other nature waters, can be seen as a 3-d medium that consisting of different particles. in macroscopic scale, the energy of lights or electromagnetic incident on this 3-d cubewill be reflected, scattering or attenuation by body and the resulting remote sensing images will generated because of the properties diversity in different spatial locations; in microscopic scale, the processing in above can be interpreted by the60interaction between photon and corresponding particle. usually, three substances can affect the distribution of lights fields under waterobviously, that is color dissolved organic matter (cdom), phytoplankton (ph) or chlorophyll-a (chl-a) and total suspend matter (tsm). in this section ,we will describe how dose those matter will affect the ultimate signal received by airborne or space borne remote sensing senor.651.1 the optical properties of waterthe manner that certain water body reflect lights are determined by their optical properties. iop (inherent optical properties) is defined as properties just impact by water components, compare to aop (apparent optical properties) of water.the iop of water are mainly constituted by absorption coefficients, volume scattering70functions and scattering coefficients. the absorption coefficients a is used to measure the reduction by energy conversion (electromagneticto other forms).volume scattering functions ( , ) is defined as follows: ( ; ) limis ( ; )(1)here， i ( , ) = i ( g; t; )v 0 ei ( ) vxis energy intensity:s s x75i ( g;t; ) qt(w sr 1 nm1 )(2)iop.e i is incident irradiance; note that , althoughi s is defined by irradiance, an aop, it is also aintegrate eq.(1)to solid angle() , we can get scattering coefficients.b ( ) = 2 ( , ) sin d(3)80the aop of waters are optical properties that involve to energy incident on water, such asradiance, irradiance, remote sensing reflectance etc. the radiance is defined as follows:l x ( g; t; ) qta(w m 2 sr 1 nm1 )(4)here, q is total energy from solid angle , incident on a region with area a at time t in wavelength .85the upwelling and downwelling irradiance is the integration of radiance at upper-semisphere andunder semisphere respectively:e ( g; t; ) = l ( g; t; ) cos d ( )(5)u x xue ( g; t; ) = l ( g; t; ) cos d ( )(6)d x x dthe remote sensing reflectance is the ratio between upwelling radiancel ande ( g; t; ) , that90is :rrs= luedud x(7)95100105this aop is the most important one to signal received by sensor, because it is the surface reflectance of water body.1.2 the establishment ofradiative transfer equationsaccording to c.d. mobleys theory11, all scattering and absorption effect in water can be modeled by one of the follow six transfer processing:1. esd: elastic scattering, namely scattered wavelength is same to incident one, but the light beam energy decrease;2. isd: inelastic scattering, namely scattered wavelength is different to incident one,moreover, the light beam energy decrease;3. ta: true absorption. all incident light beam energy is transformed to other forms;4. esi: elastic scattering, namely scattered wavelength is same to incident one, but the light beam energy increase;5. isi: inelastic scattering, namely scattered wavelength is different to incident one, moreover, thelight beam energy increase;6. te: true emission. all energy with other forms is transformed to light beam;based on the 6 modeling processing, and definition(1)-(6), we can get the general formation of radiative transfer equation:1 = = l + = l = c = l + _ e + _ i + _ s(8)110in eq.(8):v t n2 n2 n2 is solid angle;xv = v ( g; t; )gis the speed of light in position x , at time tand wavelength ;xl = l ( g; t; )is the radiance (eq.(4)of light in positiongx , at time t and wavelength ;115n = n ( g; t; )wavelength ;gxis index of refraction, it is also a function of position x , at time tandthe last three terms in right side of eq.(8) is corresponding to esd+esi, isd+isi, terespectively:( )_gl ()e x; t; ; gx( gx; t; ; )( )(9)in2 ( g; t; ) gx; t; ; d l ()_ ( x; t; ) gxx; t; ; ( g)( )(10) n2 ( g; t; ) i x; t; ; d d 0 xx( g )s ( g, ) s ()_120s x; ; gn2 ( g; t; )(11)in eq.(9)(11),x ( g; ; ) ( x; ; ) is elastic volume scattering function as definition in eq.(1);is inelastic volume scattering functions, defined as:g( g )ir ( x; ) i x; ; eg; v (12)i ( xg) s s0 ( x, ) is source function of emitter and( )is distribution functions.we can see froml n 2125eq.(8) and their terms expansion from eq.(9)(11) thatn 2is used instead of l , that is theresult oflaw in radiativetransfer, derived from snells theory.130eq.(8) just give us a general formation of rte, however, the actual water have many properties that is benefit to simplify eq.(8). first, most waters can be considered as homogeny in same depth, and the index of refraction is constant, moreover, the light fields are independence oftime, therefore ,we can get a standard formulation of eq.(8):()dl ( z; )g = c ( z, ) l x; ; dz(13)where = cos ( ) ;+l ( z; ) ( z; ; ) d () + s ( z; )1351.3 the correlation between iop and aoprsfrom rte models (eq.(8) or eq.(13), we can get an explicit formation of iops(scattering coefficients, absorptions coefficients) and aops (remote sensing reflectance), named bio-optical models:rrs(0 ) = fbbba + b (14)140where, r (0 ) is reflectance just beneath water surface and a is absorption coefficient,back-scattering coefficient, defined as:b ( ) = 2 2 ( ; ) sin dbb is(15)b0rsf is a isotropic factor; in10, c.d. mobley develop an analytical expression ofrsfunction of r (0 ) and iops in shallow waters.r asu ,w k r = r(0 ) 1 exp k + z rsrsd cos b vu , b r k (16)+ b exp k + z v dcos bwhere,rb isbottom reflectance,k d is downwelling diffuse attenuation,k u ,w andku , b are145upwelling diffuse attenuation due to water and bottom ,z b is depth of water column, v isviewing angle. in8, the author proved that the following analytical equations can quantity unknown parameters in eq.(14) and(16).the isotropic factors, f isf = p (1 + p x + p x 2 + p x3 ) 1 + p1 s 12345 cos (17)6 7(1 + p u ) 1 + p1 cos v 150here,x = bba + bb(18)pi , i = 1, 2 6 is undetermined coefficients can be derived by lsq;and downwelling diffuse attenuation, k dis regressed by a function with form:k d = a + bb0 cos (19)155 0 is parameters to be determined.the two upwelling diffuse attenuation terms,sk u ,w and ku , b is fitted by following expressions,with two unknown parametersk i (i = 1, 2 ) :s = ( a + b ) (1 + x )1,w 1 + 1 (20)u ,wb2,w cos s k = ( a + b ) (1 + x)1, b 1 + 1 (21)u , bb2 , b cos 160165thus, based on the eq.(16) and eq.(17)(21), we can get a final equations to predict aopsdistribution according to a set of iops, that is also the direct model which is need to inversed.2regularization numerical solution of rtein last section, we give a formulation that connect the remote sensing reflectance can be accepted by sensor and some iops, such as specific absorption, specific scattering absorption, etc. in this section, we will propose our algorithm to inverse them, it can be seen as a modification of gauss-newton methods, a classical descent methods to deal with non-linear optimization problems.2.1 gauss-newtons methodsubstitute eq.（14）,（17） （21） to eq.（16）, we can get a final expression aboutrrs and170a, bb , and with bio-optical model, we can know the correlation between concentration of water component. according to bio-optical model, the iops of certain water body, can written as linear combination of substance concentration and corresponding specific inherent optical properties,siop.175a ( ) = aw ( ) + ach ( )chl a + atsm ( )tsm +acdom ( )cdom bb ( ) = bb,w ( ) + bb,ch ( )chl a + bb,tsm ( )tsm +bb,cdom ( )cdom (22)(23)i ii denote concentrations, anda , b represent siops. after bio-optical model is adopted, the180rte model will be formulated finally, that is a non-linear expressions, therefore, the inversion of them is a typical non-linear optimization problem, in 8-10, authors use gauss-newton (gn) algorithms to calculate concentrations. gn algorithms is classical approaches to inverse non-linearleastsquareproblem,thetargetfunctionofgnpossessesfollowingtform: f (x ) = f1 ( x ) , f2 (x ) , fm ( x ) m , andfi ( x ) = yi m ( xi , ti )(24)m ( xi , ti ) is model function with known shape ( ti is variable) but undetermined parametersyi is value acquired by measurement. the minimization of eq. (24) is identical to:x = arg min f (x )xxi ,185= min f 2 ( x ) = min f t ( x ) f ( x )(25)here:i x i =1 xmf (x ) =f 2 ( x ) = f t ( x ) f ( x )(26) ii=1gn algorithm use taylors expansion to deal withf ( x ) , and get:f (x + h ) = f (x ) + jt ( x ) h(27)190be:j is jacobianmatrix of target function, and h is steps. insert eq（.27）to（26）,f (x + h ) willandf ( x + h ) = f ( x ) + ht jt f + 1 ht jt jh (28)2f ( x + h ) = l (h ) = jt f + jt jh(29)195let eq.(29) equal to zero, we can get:( t )tj j hgn = j f (30)moreover, we can prove that ifj t j is positive define or j is full rank, thenhgn is a descentdirections.based on the necessary conditions for local minimizer,hgnthat satisfy eq.(30) willbe stationary point ofl (h )and ifj t j is positive, it will be a minimizer, therefore, we can we200can get the finalx at certain stopping rules by solving eq. (30)iteratively when given an initialvalue x 02.2 the improvement of gn1it is straightly to see that matrix inversion will be involved when solving eq.(30). however,we can see in the following paragraph thatthe inverse operation toj t j maybe lead to instability205if j t j has small singular value.to analysis the influence of disturbance, we add a noise term , that is satisfying = ,to eq.(30):( t )tand letjt j = k andj j hgn = j f + (31)jt f = d , then, we can implement singular value decomposition (svd)210to k , and get:k 1h= hture +ns 1 (ut )v(32)gngn i i iii=1here,si is singular value of matrix k , andui , v i are vectors ofu, v which is satisfying:215eq.(33)(37)is called svd of k .k = udiag ( s ) vtui u j = ijvi v j = i