CN107748733A - Suitable for the Radiance transfer calculation method of cloud micro-properties consecutive variations - Google Patents

Suitable for the Radiance transfer calculation method of cloud micro-properties consecutive variations Download PDF

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CN107748733A
CN107748733A CN201710831636.4A CN201710831636A CN107748733A CN 107748733 A CN107748733 A CN 107748733A CN 201710831636 A CN201710831636 A CN 201710831636A CN 107748733 A CN107748733 A CN 107748733A
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张峰
颜佳任
吴琨
石怡宁
杨全
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Nanjing University of Information Science and Technology
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Abstract

The present invention relates to the Radiance transfer calculation method suitable for cloud micro-properties consecutive variations, based on two-stream approximation scheme, this method comprises the following steps：Step 1）By the radiation flux of cloud micro-properties consecutive variations be expressed as constant term and disturbance term and；Step 2）The first radiation flux being calculated according to traditional second-rate radiation transfer equation, and the first radiation flux is set as the constant term；Step 3）The form for being parameterized dissymmetry factor g and single scattering albedo ω by perturbation method substitutes into formation heterogeneity in traditional second-rate radiation transmission Strength Equation and causes disturbance term equation group, and by causing disturbance term solving equations to obtain two micro-disturbance factors the heterogeneityWith

Description

Suitable for the Radiance transfer calculation method of cloud micro-properties consecutive variations
Technical field
The invention belongs to cloud radiation transmission field, more particularly to a kind of radiation suitable for cloud micro-properties consecutive variations Transmit computational methods.
Background technology
Cloud covers 50% or so of ground sphere area, is the important fitter of earth atmosphere system radiation budget, and cloud passes in radiation Play the part of highly important effect in defeated, cloud about reflects 50W/m2Emittance.In climatic model, to cloud radiation transmission The accurate calculating of process will largely influence the result of climatic simulation and prediction.IPCC third times are reported[3]Point out current One of key restriction factors of climatic model levels are exactly cloud radiation parameter scheme, and this is also that pattern is probabilistic most Main cause.Therefore the accurate radiation transmission parameterized procedure for calculating climatic model medium cloud, to accurate simulation climate system medium cloud- Radiation interaction, improved mode forecast have very important meaning.Current numerical model can't resolve Cloud microphysical The non-uniformity problem of matter in vertical direction.The change of cloud in vertical direction is typically very fast, and cloud cluster may be Just withered away in hundreds of meters, or optical property change is violent.But in present mode, it is equal that pattern is generally divided into several Even mode layer, the optical property in each mode layer is fixed, and this obviously have ignored the change of optical property in layer knot.When So, if the layer knot quantity in pattern is enough, vertical resolution is fine enough, can ignore vertical direction heterogeneity and bring Radiance transfer calculation error.But the number of plies of existing climatic model is typically between 30-100[4], this is obviously also not enough to neglect Non-uniformity problem slightly in vertical direction, increasing the vertical rate respectively of pattern can greatly increase the operation time of pattern again, transport Efficiency is calculated to substantially reduce.The problem of cloud micro-properties consecutive variations, DSMC (Monte Carlo method) energy The radiation effect of this consecutive variations is directly simulated, and the complicated road radiation transmission process in arbitrary medium can be handled.It is this Method can obtain very high computational accuracy, but need substantial amounts of distinct photons experiment, and it is very time-consuming undoubtedly so to do, Therefore it is not directly applicable in pattern.
The content of the invention
Present invention aims at when existing engineering project encounters problems in the process of implementation, there is provided it is micro- that one kind is applied to cloud The Radiance transfer calculation method of physical characteristic consecutive variations, is specifically realized by following technical scheme：
The Radiance transfer calculation method suitable for cloud micro-properties consecutive variations, based on two-stream approximation scheme, bag Include following steps：
Step 1) by the radiation flux of cloud micro-properties consecutive variations be expressed as constant term and disturbance term and；
The first radiation flux that step 2) is calculated according to traditional second-rate radiation transfer equation, and set the first radiation Flux is the constant term；
The form that step 3) is parameterized dissymmetry factor g and single scattering albedo ω by perturbation method substitutes into traditional two Flow and the disturbance term equation group as caused by heterogeneity is formed in radiation transmission Strength Equation, and by causing to the heterogeneity Disturbance term solving equations obtain two micro-disturbance factor εωAnd εgThe second radiation flux represented；
Step 4) carries out the summation completion cloud micro-properties with the second radiation flux to the first radiation flux and continuously become The solution of the radiation flux of change.
The further design of the Radiance transfer calculation method suitable for cloud micro-properties consecutive variations is, when this When method is based on solar shortwave radiation, the form of expression of the radiation flux of the cloud micro-properties consecutive variations in the step 1 Such as formula (1)：
Wherein, Fi ±(i=1,2) is disturbance term,For constant term.
The further design of the Radiance transfer calculation method suitable for cloud micro-properties consecutive variations is, described In step 3), the equation group F of the micro-disturbance item of effect caused by heterogeneityi ±(i=1,2) such as formula (2)
Wherein, FsRepresent incident solar flux, τ is optical thickness, τ0Represent cloud top to cloud base optical thickness,
Represent respectively It is that optical thickness isWhen ω, g corresponding to be worth,μ0For the cosine value of sun altitude；
Disturbance term F caused by obtaining vertical heterogeneous stratification is solved according to formula (2)i +And Fi -Expression formula such as formula (3)：
Wherein,
K meets
In addition, Xi、αi、Pi ±、And It is excessive parameter, wherein, AiAnd BiDetermined by boundary condition, AiAnd BiIt is expressed as：
Wherein,Represent diffusion reflectivity；
AiAnd BiCorresponding boundary condition is：
The further design of the Radiance transfer calculation method suitable for cloud micro-properties consecutive variations is, described In step 3), traditional second-rate radiation transfer equation such as formula (4)：
Corresponding boundary condition is in formula (4)
The further design of the Radiance transfer calculation method suitable for cloud micro-properties consecutive variations is, when this When method is based on earth's surface long-wave radiation, the form of expression of the radiation flux of the cloud micro-properties consecutive variations in the step 1 Such as formula (5)：
Wherein,For disturbance term,For constant term.
The further design of the Radiance transfer calculation method suitable for cloud micro-properties consecutive variations is, described In step 3), traditional second-rate radiation transfer equation such as formula (6)：
In formula (6), B (τ)=a1+b1τ, a1And b1It is constant,
The boundary condition of formula (6) is：I-(τ=0)=0, I+(τ=τ0)=(1- εs)I-(τ=τ0)+εsπBs(Ts), TsTable Show surface temperature, BsRepresent Planck blackbody transmitting；
Tried to achieve according to formula (6)Such as formula (7)：
Wherein,
Wherein, K and H is amount, the ε determined by boundary conditionsEarth's surface emissivity, a1And b1Represent constant.
The further design of the Radiance transfer calculation method suitable for cloud micro-properties consecutive variations is, described In step 3), the equation group of the micro-disturbance item of effect caused by heterogeneityEquation group such as formula (8)：
Wherein, ω (τ) represents individual layer scattering albedo, and τ is optical thickness, τ0Represent cloud top to cloud base optical thickness,
Optical thickness is respectivelyWhen ω, g corresponding to be worth, μ0For the cosine value of sun altitude,
Disturbance term caused by obtaining vertical heterogeneous stratification is solved according to formula (8)WithExpression formula such as formula (9)：
Wherein,
Wherein, R represents Reflectivity for Growing Season,
AiAnd BiCorresponding boundary condition is：
Advantages of the present invention is as follows：
(1) set forth herein the Radiance transfer calculation method suitable for cloud micro-properties consecutive variations can effectively solve gas The vertical problem of non-uniform of time pattern medium cloud internal opticses property, so as to improve accuracy of the radiation calculation.No matter for shortwave or Long wave, this method have higher computational accuracy when calculating cloud heterogeneous, than traditional second-rate radiation transmission algorithm, especially Transmitted for shortwave radiation, what computational accuracy improved becomes apparent, and this can be used to explain that the absorption errors of air medium cloud reach 7% Absorption it is abnormal.
(2) this method is as applied in climatic model, it is only necessary to which in the Radiation Module of climatic model plus one parameterizes Program, it is simple to operate and computational efficiency can't be influenceed.
(3) because the amplitude of cloud micro-properties change is bigger, then the error being calculated as homogeneous atmosphere is higher, Raising of this method to computational accuracy is obvious, the calculating effect for increasing faster cloud can be become apparent from this way.
(4) this method is not only applicable to the radiation calculating of cloud micro-properties consecutive variations, and it is micro- also to significantly improve snow The radiation of physical characteristic consecutive variations calculates.The method is adapted to all continuous comprising dissymmetry factor and single scattering albedo in fact The Radiance transfer calculation of the inhomogeneous medium of change.
Embodiment
Technical scheme is further illustrated below.
The Radiance transfer calculation method suitable for cloud micro-properties consecutive variations of the present embodiment, based on two-stream approximation side Case, it is characterised in that this method comprises the following steps：
Step 1) by the radiation flux of cloud micro-properties consecutive variations be expressed as constant term and disturbance term and；
The first radiation flux that step 2) is calculated according to traditional second-rate radiation transfer equation, and set the first radiation Flux is the constant term；
The form that step 3) is parameterized dissymmetry factor g and single scattering albedo ω by perturbation method substitutes into traditional two Heterogeneity, which is formed, in stream radiation transmission Strength Equation causes disturbance term equation group, and by causing disturbance to the heterogeneity Item solving equations obtain two micro-disturbance factor εωAnd εgThe second radiation flux represented；
Step 4) carries out the summation completion cloud micro-properties with the second radiation flux to the first radiation flux and continuously become The solution of the radiation flux of change.
The Radiance transfer calculation method suitable for cloud micro-properties consecutive variations of the present embodiment is directed to the ripple of cloud radiation The long method of pushing over each provided for the second-rate radiation transmission of long wave and shortwave parametrization equation is as follows：
(1) it is applied to the second-rate radiation transfer equation of shortwave of cloud micro-properties consecutive variations
The solar radiation transmission fundamental equation of mean square parallactic angle can be written as：
In above formula, I (τ, μ) represents intensity of solar radiation, and μ is the cosine value of sun altitude, and τ is optical thickness, ω (τ) Represent individual layer scattering albedo, FsIncident solar flux is represented, P (τ, μ, μ ') represents phase function, for Eddington approximation, Phase function can be expressed as P (τ, μ, μ ')=1+3g (τ) μ μ ', (- 1≤μ≤1), wherein, g (τ) is dissymmetry factor.For dissipating Air is penetrated, solar flux density up and down is as follows：
The dissymmetry factor g (τ) and individual layer scattering albedo ω (τ) of consideration be all with optical thickness consecutive variations, This is more conform with the situation of true medium.Dissymmetry factor and individual layer scattering albedo is given below with optical thickness consecutive variations Form：
Above formula is all based on individual layer air, τ0It is the optical thickness of whole individual layer air,WithIt is optical thicknessWhen value, in a practical situation, εωAnd εgIt is to be much smaller thanWithSmall parameter.
Theoretical according to micro-disturbance, radiosity can be by micro-disturbance factor εωAnd εgExpand to following form:
Then can be expressed as by Eddington approximation, radiation intensitySo Combine equation (1.1-1.2) afterwards, we can obtain：
Wherein, γ1(τ)=1/4 [7- ω (τ) (4+3g (τ))], γ2(τ)=- 1/4 [1- ω (τ) (4-3g (τ))], γ3 [2-3g (τ) μ of (τ)=1/40]；Rdir(Rdif) be cloud base portion either ground direct projection (diffusion) albedo.(1.3) are substituted into (1.5) and ignore more than second order and second order in a small amount, we can obtain：
Wherein,
By above formula and (1.4) formula In (1.5) formula of substitution, we can respectively obtain F+ and F- expression formula：
Wherein,By above equation, two equation groups can be written as respectivelyWithObtained equation groupFor：
(1.8) formula is the radiation transfer equation of the conforming layer knot of standard, and its solution is:
Wherein,
According to (1.7) formula, represent heterogeneity and draw The equation group F of the micro-disturbance item of the effect riseni ±(i=1,2) it is：
In above formulaWe allowIt can obtain：
Wherein,
Formula is asked optical thickness It is secondary to lead, it can obtain：
Above formula
(1.12) formula is solved, can be obtained:
Finally, disturbance term F caused by vertical heterogeneous stratification can be obtainedi +And Fi -Expression formula:
Wherein,
AiAnd BiDetermined by boundary condition,
(2) it is applied to the long wave two-stream approximation radiation transfer equation of cloud micro-properties consecutive variations
The fundamental equation that long-wave radiation is transmitted in scattering atmosphere can be written as：
Wherein B (τ) is Planck function, and phase function can be by Legnedre polynomial PlExpansion, independently of azimuthal phase letter Number can be written asAccording to two-stream approximation, radiation intensity up and down is I (τ ,+μ1)=I+ (τ) and I (τ ,-μ1)=I-(τ), then according to Gaussian integration methodMoreover, the phase function under the limitation of two-stream approximation P (τ, μ, μ ')=1+3g (τ) μ μ '=1 ± g (τ) are written as, therefore the integration of equation (2.1) can replace and beSo can be by equation (2.1) by up and down Intensity be expressed as form：
In above formula
Analysis above, it is as traditional if single scattering albedo and dissymmetry factor are not with optical thickness change The two-stream approximation algorithm of infrared radiation transmissions.Consider dissymmetry factor and single scattering albedo be with optical thickness exponentially Version, shaped like
τ0It is the optical thickness of whole individual layer air,WithIt is that optical thickness isWhen corresponding value, in real medium In, εωAnd εgIt is to be much smaller thanWithSmall parameter.(2.3) formula is substituted into γ1(τ), γ2(τ), γ3In (τ), then ignore two More than rank and second order a small amount of, we can obtain following formula
In above formulaAll it is known quantity, Theoretical according to micro-disturbance, radiation intensity can be separated into often It is several, εωItem and εg, its radiation intensity up and down can be written as
(2.4-2.5) is substituted into (2.2) formula, it can be deduced that following form
Equation (2.6) is deployed, because to be far smaller than single order a small amount of for magnitude more than second order and second order a small amount of, therefore here Do not consider, so equation (2.6) is segmented intoWithThree parts, whereinWithAll it is magnitude ratioSmall one-level It is a small amount of, therefore merge into hereinObtained equation groupIt is as follows：
Formula (2.7c) is boundary condition, and the radiation intensity for representing top down is 0, and bottom up radiation intensity is bottom The downward radiation intensity in portion multiply reflectivity and earth's surface Planck blackbody emissivity and, εsIt is earth's surface emissivity, π Bs(Ts)=σ Ts 4 It is the flux density of Planck blackbody transmitting, then we are carried out according to linear function form to the Planck function of a level Parametrization, B (τ)=a1+b1τ, wherein a1And b1It is constant.(2.7) are solved with can to obtain downward radiation upwards strong Degree：
Wherein,
ObtainEquation group it is as follows：
In order to try to achieve the analytic solutions of above formula, it will be assumed thatTherefore we can obtain：
Wherein,
In order that formula seems simpler It is single, introduce hereFormula (2.10) carries out derivation to optical thickness again, can obtain：
Wherein,
For formula (2.11), we are easy to solution and draw MiAnd NiSolution：
Wherein, According toWe can be readily available：
Wherein,
AiAnd BiIt is to be determined by boundary condition, (2.9c) is substituted into equation (2.13) to obtain
The calculating of radiosity is according to two-stream approximation, Ke YiyouObtain.
The Radiance transfer calculation method suitable for cloud micro-properties consecutive variations that the present embodiment proposes can be solved effectively The vertical problem of non-uniform of climatic model medium cloud internal opticses property, so as to improve accuracy of the radiation calculation.No matter for shortwave also It is long wave, new second-rate radiation transport approach has higher when calculating cloud heterogeneous than traditional second-rate radiation transmission algorithm Computational accuracy, especially for shortwave radiation transmit, computational accuracy improve becomes apparent, this can be used to reduce air medium cloud Up to 7% absorption errors.This method is as applied in climatic model, it is only necessary to adds one in the Radiation Module of climatic model The program of parametrization, it is simple to operate and computational efficiency can't be influenceed.Because the amplitude of cloud micro-properties change is bigger, then The error being calculated as homogeneous atmosphere is higher, and at this moment raising of the new parameter method to computational accuracy is obvious, so This method can become apparent to the calculating effect for increasing faster cloud.This method is not only applicable to cloud micro-properties consecutive variations Radiation calculate, can also significantly improve snow microphysical property consecutive variations radiation calculate.The method is adapted to all to include in fact The Radiance transfer calculation of the inhomogeneous medium of dissymmetry factor and single scattering albedo consecutive variations.
The foregoing is only a preferred embodiment of the present invention, but protection scope of the present invention be not limited thereto, Any one skilled in the art the invention discloses technical scope in, the change or replacement that can readily occur in, It should all be included within the scope of the present invention.Therefore, protection scope of the present invention should be with scope of the claims It is defined.

Claims (7)

1. a kind of Radiance transfer calculation method suitable for cloud micro-properties consecutive variations, based on two-stream approximation scheme, it is special Sign is that this method comprises the following steps：
Step 1) by the radiation flux of cloud micro-properties consecutive variations be expressed as constant term and disturbance term and；
The first radiation flux that step 2) is calculated according to traditional second-rate radiation transfer equation, and set the first radiation flux For the constant term；
The form that step 3) is parameterized dissymmetry factor g and single scattering albedo ω by perturbation method substitutes into traditional second-rate spoke Formation disturbance term equation group as caused by heterogeneity in intensity transmission equation is penetrated, and by being disturbed caused by the heterogeneity Dynamic item solving equations obtain two micro-disturbance factor εωAnd εgThe second radiation flux represented；
Step 4) carries out summation to the first radiation flux and the second radiation flux and completes the cloud micro-properties consecutive variations The solution of radiation flux.
2. the Radiance transfer calculation method according to claim 1 suitable for cloud micro-properties consecutive variations, its feature It is when this method is based on solar shortwave radiation, the radiation flux of the cloud micro-properties consecutive variations in the step 1 The form of expression such as formula (1)：
<mrow> <mtable> <mtr> <mtd> <mrow> <msup> <mi>F</mi> <mo>+</mo> </msup> <mo>=</mo> <msubsup> <mi>F</mi> <mn>0</mn> <mo>+</mo> </msubsup> <mo>+</mo> <msubsup> <mi>F</mi> <mn>1</mn> <mo>+</mo> </msubsup> <msub> <mi>&amp;epsiv;</mi> <mi>&amp;omega;</mi> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mn>2</mn> <mo>+</mo> </msubsup> <msub> <mi>&amp;epsiv;</mi> <mi>g</mi> </msub> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mi>F</mi> <mo>-</mo> </msup> <mo>=</mo> <msubsup> <mi>F</mi> <mn>0</mn> <mo>-</mo> </msubsup> <mo>+</mo> <msubsup> <mi>F</mi> <mn>1</mn> <mo>-</mo> </msubsup> <msub> <mi>&amp;epsiv;</mi> <mi>&amp;omega;</mi> </msub> <mo>+</mo> <msubsup> <mi>F</mi> <mn>2</mn> <mo>-</mo> </msubsup> <msub> <mi>&amp;epsiv;</mi> <mi>g</mi> </msub> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
Wherein, Fi ±(i=1,2) is disturbance term,For constant term.
3. the Radiance transfer calculation method according to claim 2 suitable for cloud micro-properties consecutive variations, its feature It is in the step 3), the equation group F of the micro-disturbance item of effect caused by heterogeneityi ±(i=1,2) such as formula (2)
<mrow> <mfrac> <mrow> <msubsup> <mi>dF</mi> <mi>i</mi> <mo>+</mo> </msubsup> </mrow> <mrow> <mi>d</mi> <mi>&amp;tau;</mi> </mrow> </mfrac> <mo>=</mo> <mrow> <mo>(</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>1</mn> <mn>0</mn> </msubsup> <msubsup> <mi>F</mi> <mi>i</mi> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>2</mn> <mn>0</mn> </msubsup> <msubsup> <mi>F</mi> <mi>i</mi> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mi>&amp;tau;</mi> </mrow> </msup> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>1</mn> <mi>i</mi> </msubsup> <msubsup> <mi>F</mi> <mn>0</mn> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>2</mn> <mi>i</mi> </msubsup> <msubsup> <mi>F</mi> <mn>0</mn> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>3</mn> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mi>&amp;tau;</mi> </mrow> </msup> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <msub> <mi>F</mi> <mi>s</mi> </msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>&amp;tau;</mi> <mo>/</mo> <msub> <mi>&amp;mu;</mi> <mn>0</mn> </msub> </mrow> </msup> <mo>,</mo> </mrow>
<mrow> <mfrac> <mrow> <msubsup> <mi>dF</mi> <mi>i</mi> <mo>-</mo> </msubsup> </mrow> <mrow> <mi>d</mi> <mi>&amp;tau;</mi> </mrow> </mfrac> <mo>=</mo> <mrow> <mo>(</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>2</mn> <mn>0</mn> </msubsup> <msubsup> <mi>F</mi> <mi>i</mi> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>1</mn> <mn>0</mn> </msubsup> <msubsup> <mi>F</mi> <mi>i</mi> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mi>&amp;tau;</mi> </mrow> </msup> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>2</mn> <mi>i</mi> </msubsup> <msubsup> <mi>F</mi> <mn>0</mn> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>1</mn> <mi>i</mi> </msubsup> <msubsup> <mi>F</mi> <mn>0</mn> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>4</mn> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mi>&amp;tau;</mi> </mrow> </msup> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <msub> <mi>F</mi> <mi>s</mi> </msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>&amp;tau;</mi> <mo>/</mo> <msub> <mi>&amp;mu;</mi> <mn>0</mn> </msub> </mrow> </msup> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
Wherein, FsRepresent incident solar flux, τ is optical thickness, τ0Represent cloud top to cloud base optical thickness,
Represent it is that optical thickness is respectivelyWhen ω, g corresponding to be worth,μ0For the cosine value of sun altitude；
Disturbance term F caused by obtaining vertical heterogeneous stratification is solved according to formula (2)i +And Fi -Expression formula such as formula (3)：
Wherein,
<mrow> <msub> <mi>X</mi> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <msubsup> <mi>&amp;psi;</mi> <mi>i</mi> <mo>+</mo> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;psi;</mi> <mi>i</mi> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>1</mn> <mn>0</mn> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>2</mn> <mn>0</mn> </msubsup> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>-</mo> <mfrac> <msubsup> <mi>&amp;eta;</mi> <mrow> <mn>4</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mrow> <mn>4</mn> <mi>k</mi> <mrow> <mo>(</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>1</mn> <mn>0</mn> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>2</mn> <mn>0</mn> </msubsup> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> <msub> <mi>Y</mi> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <msubsup> <mi>&amp;zeta;</mi> <mi>i</mi> <mo>+</mo> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;zeta;</mi> <mi>i</mi> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>1</mn> <mn>0</mn> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>2</mn> <mn>0</mn> </msubsup> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>+</mo> <mfrac> <msubsup> <mi>&amp;mu;</mi> <mrow> <mn>5</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mrow> <mn>4</mn> <mi>k</mi> <mrow> <mo>(</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>1</mn> <mn>0</mn> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>2</mn> <mn>0</mn> </msubsup> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> </mrow>
K meetsIn addition, Xi、αi、Pi +、Pi -、 With AndIt is excessive parameter, wherein, AiAnd BiDetermined by boundary condition, AiAnd BiIt is expressed as：
Wherein, RdifRepresent diffusion reflectivity；
AiAnd BiCorresponding boundary condition is：Fi -(τ=0)=0, Fi +(τ=τ0)=RdifFi -(τ=τ0)。
4. the Radiance transfer calculation method according to claim 3 suitable for cloud micro-properties consecutive variations, its feature It is in the step 3), traditional second-rate radiation transfer equation such as formula (4)：
<mrow> <mfrac> <mrow> <msubsup> <mi>dF</mi> <mn>0</mn> <mo>+</mo> </msubsup> </mrow> <mrow> <mi>d</mi> <mi>&amp;tau;</mi> </mrow> </mfrac> <mo>=</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>1</mn> <mn>0</mn> </msubsup> <msubsup> <mi>F</mi> <mn>0</mn> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>2</mn> <mn>0</mn> </msubsup> <msubsup> <mi>F</mi> <mn>0</mn> <mo>-</mo> </msubsup> <mo>-</mo> <mover> <mi>&amp;omega;</mi> <mo>^</mo> </mover> <msubsup> <mi>&amp;gamma;</mi> <mn>3</mn> <mn>0</mn> </msubsup> <msub> <mi>F</mi> <mi>s</mi> </msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>&amp;tau;</mi> <mo>/</mo> <msub> <mi>&amp;mu;</mi> <mn>0</mn> </msub> </mrow> </msup> <mo>,</mo> </mrow>
<mrow> <mfrac> <mrow> <msubsup> <mi>dF</mi> <mn>0</mn> <mo>-</mo> </msubsup> </mrow> <mrow> <mi>d</mi> <mi>&amp;tau;</mi> </mrow> </mfrac> <mo>=</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>2</mn> <mn>0</mn> </msubsup> <msubsup> <mi>F</mi> <mn>0</mn> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>1</mn> <mn>0</mn> </msubsup> <msubsup> <mi>F</mi> <mn>0</mn> <mo>-</mo> </msubsup> <mo>+</mo> <mover> <mi>&amp;omega;</mi> <mo>^</mo> </mover> <msubsup> <mi>&amp;gamma;</mi> <mn>3</mn> <mn>0</mn> </msubsup> <msub> <mi>F</mi> <mi>s</mi> </msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>&amp;tau;</mi> <mo>/</mo> <msub> <mi>&amp;mu;</mi> <mn>0</mn> </msub> </mrow> </msup> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
Corresponding boundary condition is in formula (4)
5. the Radiance transfer calculation method according to claim 1 suitable for cloud micro-properties consecutive variations, its feature It is when this method is based on earth's surface long-wave radiation, the radiation flux of the cloud micro-properties consecutive variations in the step 1 The form of expression such as formula (5)：
<mrow> <mtable> <mtr> <mtd> <mrow> <msup> <mi>I</mi> <mo>+</mo> </msup> <mo>=</mo> <msubsup> <mi>I</mi> <mn>0</mn> <mo>+</mo> </msubsup> <mo>+</mo> <msubsup> <mi>I</mi> <mn>1</mn> <mo>+</mo> </msubsup> <msub> <mi>&amp;epsiv;</mi> <mi>&amp;omega;</mi> </msub> <mo>+</mo> <msubsup> <mi>I</mi> <mn>2</mn> <mo>+</mo> </msubsup> <msub> <mi>&amp;epsiv;</mi> <mi>g</mi> </msub> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mi>I</mi> <mo>-</mo> </msup> <mo>=</mo> <msubsup> <mi>I</mi> <mn>0</mn> <mo>-</mo> </msubsup> <mo>+</mo> <msubsup> <mi>I</mi> <mn>1</mn> <mo>-</mo> </msubsup> <msub> <mi>&amp;epsiv;</mi> <mi>&amp;omega;</mi> </msub> <mo>+</mo> <msubsup> <mi>I</mi> <mn>2</mn> <mo>-</mo> </msubsup> <msub> <mi>&amp;epsiv;</mi> <mi>g</mi> </msub> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
Wherein,For disturbance term,For constant term.
6. the Radiance transfer calculation method according to claim 5 suitable for cloud micro-properties consecutive variations, its feature It is in the step 3), traditional second-rate radiation transfer equation such as formula (6)：
<mrow> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <msubsup> <mi>dI</mi> <mn>0</mn> <mo>+</mo> </msubsup> </mrow> <mrow> <mi>d</mi> <mi>&amp;tau;</mi> </mrow> </mfrac> <mo>=</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>1</mn> <mn>0</mn> </msubsup> <msubsup> <mi>I</mi> <mn>0</mn> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>2</mn> <mn>0</mn> </msubsup> <msubsup> <mi>I</mi> <mn>0</mn> <mo>-</mo> </msubsup> <mo>-</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>3</mn> <mn>0</mn> </msubsup> <mi>B</mi> <mrow> <mo>(</mo> <mi>&amp;tau;</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <msubsup> <mi>dI</mi> <mn>0</mn> <mo>-</mo> </msubsup> </mrow> <mrow> <mi>d</mi> <mi>&amp;tau;</mi> </mrow> </mfrac> <mo>=</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>2</mn> <mn>0</mn> </msubsup> <msubsup> <mi>I</mi> <mn>0</mn> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>1</mn> <mn>0</mn> </msubsup> <msubsup> <mi>I</mi> <mn>0</mn> <mo>-</mo> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>3</mn> <mn>0</mn> </msubsup> <mi>B</mi> <mrow> <mo>(</mo> <mi>&amp;tau;</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
In formula (6), B (τ)=a1+b1τ, a1And b1It is constant,
The boundary condition of formula (6) is：I-(τ=0)=0, I+(τ=τ0)=(1- εs)I-(τ=τ0)+εsπBs(Ts),
TsRepresent surface temperature, BsRepresent Planck blackbody transmitting；
Tried to achieve according to formula (6)Such as formula (7)：
<mrow> <msubsup> <mi>I</mi> <mn>0</mn> <mo>+</mo> </msubsup> <mo>=</mo> <msup> <mi>Ke</mi> <mrow> <mo>-</mo> <mi>k</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>-</mo> <mi>&amp;tau;</mi> <mo>)</mo> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>Hxe</mi> <mrow> <mo>-</mo> <mi>k</mi> <mi>&amp;tau;</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>G</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>G</mi> <mn>1</mn> </msub> <mi>&amp;tau;</mi> <mo>,</mo> </mrow>
<mrow> <msubsup> <mi>I</mi> <mn>0</mn> <mo>-</mo> </msubsup> <mo>=</mo> <msup> <mi>Kxe</mi> <mrow> <mo>-</mo> <mi>k</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>-</mo> <mi>&amp;tau;</mi> <mo>)</mo> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>He</mi> <mrow> <mo>-</mo> <mi>k</mi> <mi>&amp;tau;</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>G</mi> <mn>3</mn> </msub> <mo>+</mo> <msub> <mi>G</mi> <mn>1</mn> </msub> <mi>&amp;tau;</mi> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
Wherein,
<mrow> <msub> <mi>G</mi> <mn>3</mn> </msub> <mo>=</mo> <mfrac> <msubsup> <mi>&amp;gamma;</mi> <mn>3</mn> <mn>0</mn> </msubsup> <mrow> <msubsup> <mi>&amp;gamma;</mi> <mn>1</mn> <mn>0</mn> </msubsup> <mo>-</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>2</mn> <mn>0</mn> </msubsup> </mrow> </mfrac> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>-</mo> <mfrac> <mrow> <msub> <mi>b</mi> <mn>1</mn> </msub> <msubsup> <mi>&amp;gamma;</mi> <mn>3</mn> <mn>0</mn> </msubsup> </mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mfrac> <mo>,</mo> <mi>H</mi> <mo>=</mo> <mfrac> <mrow> <msup> <mi>xe</mi> <mrow> <mo>-</mo> <msub> <mi>k&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <msub> <mi>G</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>G</mi> <mn>3</mn> </msub> <mi>R</mi> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>R</mi> <mo>)</mo> </mrow> <msub> <mi>G</mi> <mn>1</mn> </msub> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>R</mi> <mo>)</mo> </mrow> <msubsup> <mi>&amp;sigma;T</mi> <mi>s</mi> <mn>4</mn> </msubsup> <mo>&amp;rsqb;</mo> <mo>-</mo> <msub> <mi>G</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>x</mi> <mi>R</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>x</mi> <mi>R</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>-</mo> <mi>R</mi> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mn>2</mn> <msub> <mi>k&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> <mo>&amp;rsqb;</mo> </mrow> </mfrac> <mo>,</mo> </mrow>
<mrow> <mi>K</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>x</mi> <mi>R</mi> <mo>)</mo> </mrow> </mfrac> <mo>&amp;lsqb;</mo> <msub> <mi>&amp;epsiv;</mi> <mi>s</mi> </msub> <msubsup> <mi>&amp;sigma;T</mi> <mi>s</mi> <mn>4</mn> </msubsup> <mo>-</mo> <mrow> <mo>(</mo> <mi>x</mi> <mo>-</mo> <mi>R</mi> <mo>)</mo> </mrow> <msup> <mi>He</mi> <mrow> <mo>-</mo> <msub> <mi>k&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> <mo>-</mo> <mrow> <mo>(</mo> <msub> <mi>G</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>G</mi> <mn>3</mn> </msub> <mi>R</mi> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>R</mi> <mo>)</mo> </mrow> <msub> <mi>G</mi> <mn>1</mn> </msub> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>&amp;rsqb;</mo> <mo>,</mo> <mi>R</mi> <mo>=</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mi>s</mi> </msub> <mo>,</mo> </mrow>
Wherein, K and H is amount, the ε determined by boundary conditionsEarth's surface emissivity, a1、b1Represent constant.
7. the Radiance transfer calculation method according to claim 6 suitable for cloud micro-properties consecutive variations, its feature It is in the step 3), the equation group of the micro-disturbance item of effect caused by heterogeneityEquation group such as formula (8)：
<mrow> <mfrac> <mrow> <msubsup> <mi>dI</mi> <mi>i</mi> <mo>+</mo> </msubsup> </mrow> <mrow> <mi>d</mi> <mi>&amp;tau;</mi> </mrow> </mfrac> <mo>=</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>1</mn> <mn>0</mn> </msubsup> <msubsup> <mi>I</mi> <mi>i</mi> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>2</mn> <mn>0</mn> </msubsup> <msubsup> <mi>I</mi> <mi>i</mi> <mo>-</mo> </msubsup> <mo>+</mo> <mrow> <mo>(</mo> <msubsup> <mi>I</mi> <mn>0</mn> <mo>+</mo> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mn>1</mn> <mi>i</mi> </msubsup> <mo>-</mo> <msubsup> <mi>I</mi> <mn>0</mn> <mo>-</mo> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mn>2</mn> <mi>i</mi> </msubsup> <mo>-</mo> <mi>B</mi> <mo>(</mo> <mi>&amp;tau;</mi> <mo>)</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>3</mn> <mi>i</mi> </msubsup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mi>&amp;tau;</mi> </mrow> </msup> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <mfrac> <mrow> <msubsup> <mi>dI</mi> <mi>i</mi> <mo>-</mo> </msubsup> </mrow> <mrow> <mi>d</mi> <mi>&amp;tau;</mi> </mrow> </mfrac> <mo>=</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>2</mn> <mn>0</mn> </msubsup> <msubsup> <mi>I</mi> <mi>i</mi> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>1</mn> <mn>0</mn> </msubsup> <msubsup> <mi>I</mi> <mi>i</mi> <mo>-</mo> </msubsup> <mo>+</mo> <mrow> <mo>(</mo> <msubsup> <mi>I</mi> <mn>0</mn> <mo>+</mo> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mn>2</mn> <mi>i</mi> </msubsup> <mo>-</mo> <msubsup> <mi>I</mi> <mn>0</mn> <mo>-</mo> </msubsup> <msubsup> <mi>&amp;gamma;</mi> <mn>1</mn> <mi>i</mi> </msubsup> <mo>+</mo> <mi>B</mi> <mo>(</mo> <mi>&amp;tau;</mi> <mo>)</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>3</mn> <mi>i</mi> </msubsup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mi>&amp;tau;</mi> </mrow> </msup> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
Wherein, ω (τ) represents individual layer scattering albedo, and τ is optical thickness, τ0Represent cloud top to cloud base optical thickness, Optical thickness is respectivelyWhen ω, g corresponding to be worth, μ0For the cosine value of sun altitude,
Disturbance term caused by obtaining vertical heterogeneous stratification is solved according to formula (8)WithExpression formula such as formula (9)：
<mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>I</mi> <mi>i</mi> <mo>+</mo> </msubsup> <mo>=</mo> <msubsup> <mi>D</mi> <mrow> <mn>1</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>k</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>-</mo> <mi>&amp;tau;</mi> <mo>)</mo> </mrow> </mrow> </msup> <mo>+</mo> <msubsup> <mi>D</mi> <mrow> <mn>2</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>k</mi> <mi>&amp;tau;</mi> </mrow> </msup> <mo>+</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>1</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>k&amp;tau;</mi> <mn>0</mn> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mi>&amp;tau;</mi> </mrow> </msup> <mo>+</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>2</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mi>&amp;tau;</mi> </mrow> </msup> <mo>+</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>3</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mi>&amp;tau;</mi> </mrow> </msup> <mo>+</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>4</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <msup> <mi>&amp;tau;e</mi> <mrow> <mo>-</mo> <mi>k</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>-</mo> <mi>&amp;tau;</mi> <mo>)</mo> </mrow> </mrow> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>5</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <msup> <mi>&amp;tau;e</mi> <mrow> <mo>-</mo> <mi>k</mi> <mi>&amp;tau;</mi> </mrow> </msup> <mo>+</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>6</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msubsup> <mi>P</mi> <mrow> <mn>7</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mi>&amp;tau;</mi> <mo>+</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>7</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mrow> <mo>(</mo> <mi>&amp;tau;</mi> <mo>+</mo> <msub> <mi>&amp;sigma;</mi> <mrow> <mn>8</mn> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mi>&amp;tau;</mi> </mrow> </msup> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>I</mi> <mi>i</mi> <mo>-</mo> </msubsup> <mo>=</mo> <msubsup> <mi>D</mi> <mrow> <mn>1</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>k</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>-</mo> <mi>&amp;tau;</mi> <mo>)</mo> </mrow> </mrow> </msup> <mo>+</mo> <msubsup> <mi>D</mi> <mrow> <mn>2</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>k</mi> <mi>&amp;tau;</mi> </mrow> </msup> <mo>+</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>1</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>k&amp;tau;</mi> <mn>0</mn> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mi>&amp;tau;</mi> </mrow> </msup> <mo>+</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>2</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mi>&amp;tau;</mi> </mrow> </msup> <mo>+</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>3</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mi>&amp;tau;</mi> </mrow> </msup> <mo>+</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>4</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <msup> <mi>&amp;tau;e</mi> <mrow> <mo>-</mo> <mi>k</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>-</mo> <mi>&amp;tau;</mi> <mo>)</mo> </mrow> </mrow> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>5</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <msup> <mi>&amp;tau;e</mi> <mrow> <mo>-</mo> <mi>k</mi> <mi>&amp;tau;</mi> </mrow> </msup> <mo>+</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>6</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msubsup> <mi>P</mi> <mrow> <mn>7</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mi>&amp;tau;</mi> <mo>+</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>7</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mrow> <mo>(</mo> <mi>&amp;tau;</mi> <mo>+</mo> <msub> <mi>&amp;sigma;</mi> <mrow> <mn>8</mn> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mi>&amp;tau;</mi> </mrow> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced>
Wherein,
<mrow> <msub> <mi>Y</mi> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mo>(</mo> <msubsup> <mi>P</mi> <mrow> <mn>5</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>&amp;chi;</mi> <mrow> <mn>5</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>1</mn> <mn>0</mn> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>2</mn> <mn>0</mn> </msubsup> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>1</mn> <mi>i</mi> </mrow> <mo>&amp;PlusMinus;</mo> </msubsup> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <msubsup> <mi>P</mi> <mrow> <mn>1</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>&amp;PlusMinus;</mo> <msubsup> <mi>P</mi> <mrow> <mn>1</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>2</mn> <mi>i</mi> </mrow> <mo>&amp;PlusMinus;</mo> </msubsup> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <msubsup> <mi>P</mi> <mrow> <mn>2</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>&amp;PlusMinus;</mo> <msubsup> <mi>P</mi> <mrow> <mn>2</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>3</mn> <mi>i</mi> </mrow> <mo>&amp;PlusMinus;</mo> </msubsup> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <msubsup> <mi>P</mi> <mrow> <mn>3</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>&amp;PlusMinus;</mo> <msubsup> <mi>P</mi> <mrow> <mn>3</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
<mrow> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>4</mn> <mi>i</mi> </mrow> <mo>&amp;PlusMinus;</mo> </msubsup> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <msubsup> <mi>P</mi> <mrow> <mn>4</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>&amp;PlusMinus;</mo> <msubsup> <mi>P</mi> <mrow> <mn>4</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>5</mn> <mi>i</mi> </mrow> <mo>&amp;PlusMinus;</mo> </msubsup> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <msubsup> <mi>P</mi> <mrow> <mn>5</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>+</mo> <msubsup> <mi>P</mi> <mrow> <mn>5</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>6</mn> <mi>i</mi> </mrow> <mo>&amp;PlusMinus;</mo> </msubsup> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <msubsup> <mi>P</mi> <mrow> <mn>6</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>&amp;PlusMinus;</mo> <msubsup> <mi>P</mi> <mrow> <mn>6</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
<mrow> <msubsup> <mi>&amp;chi;</mi> <mrow> <mn>3</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>G</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>G</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>1</mn> <mi>i</mi> </msubsup> <mo>-</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>2</mn> <mi>i</mi> </msubsup> <mo>)</mo> </mrow> <mo>-</mo> <mn>2</mn> <msub> <mi>a</mi> <mn>1</mn> </msub> <msubsup> <mi>&amp;gamma;</mi> <mn>3</mn> <mi>i</mi> </msubsup> <mo>,</mo> <msubsup> <mi>&amp;chi;</mi> <mrow> <mn>6</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>=</mo> <mo>-</mo> <mrow> <mo>(</mo> <msub> <mi>G</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>G</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>1</mn> <mi>i</mi> </msubsup> <mo>-</mo> <msubsup> <mi>&amp;gamma;</mi> <mn>2</mn> <mi>i</mi> </msubsup> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msub> <mi>a</mi> <mn>1</mn> </msub> <msubsup> <mi>&amp;gamma;</mi> <mn>3</mn> <mn>10</mn> </msubsup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mrow>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mo>(</mo> <msup> <mi>&amp;alpha;</mi> <mo>-</mo> </msup> <mo>-</mo> <msup> <mi>R&amp;alpha;</mi> <mo>+</mo> </msup> <mo>)</mo> <msub> <mi>d</mi> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>k&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> <mo>+</mo> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mi>R</mi> <mo>)</mo> <msub> <mi>X</mi> <mi>i</mi> </msub> <mo>+</mo> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mi>R</mi> <mo>)</mo> <msub> <mi>Y</mi> <mi>i</mi> </msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>k&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> <mo>+</mo> <mo>(</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>1</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>R&amp;sigma;</mi> <mrow> <mn>1</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>)</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> <mo>+</mo> <mo>(</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>2</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>R&amp;sigma;</mi> <mrow> <mn>2</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>)</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> <mo>+</mo> <mo>(</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>3</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>R&amp;sigma;</mi> <mrow> <mn>3</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>)</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <msup> <mi>&amp;alpha;</mi> <mo>+</mo> </msup> <mo>-</mo> <msup> <mi>R&amp;alpha;</mi> <mo>-</mo> </msup> <mo>)</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mfrac> <mrow> <mo>(</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>4</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>R&amp;sigma;</mi> <mrow> <mn>4</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>)</mo> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>+</mo> <mo>(</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>5</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>R&amp;sigma;</mi> <mrow> <mn>5</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>)</mo> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>k&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> <mo>+</mo> <mo>(</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>6</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>R&amp;sigma;</mi> <mrow> <mn>6</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>)</mo> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>R</mi> <mo>)</mo> <msubsup> <mi>P</mi> <mrow> <mn>7</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>+</mo> <mo>(</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>7</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>R&amp;sigma;</mi> <mrow> <mn>7</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>)</mo> <mo>(</mo> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>+</mo> <msub> <mi>&amp;sigma;</mi> <mrow> <mn>8</mn> <mi>i</mi> </mrow> </msub> <mo>)</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <msup> <mi>&amp;alpha;</mi> <mo>+</mo> </msup> <mo>-</mo> <msup> <mi>R&amp;alpha;</mi> <mo>-</mo> </msup> <mo>)</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>B</mi> <mi>i</mi> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mo>(</mo> <msup> <mi>&amp;alpha;</mi> <mo>+</mo> </msup> <mo>-</mo> <msup> <mi>R&amp;alpha;</mi> <mo>-</mo> </msup> <mo>)</mo> <mo>(</mo> <mo>-</mo> <msub> <mi>X</mi> <mi>i</mi> </msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>k&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> <mo>-</mo> <msub> <mi>Y</mi> <mi>i</mi> </msub> <mo>+</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>1</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>k&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> <mo>+</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>2</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>3</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>6</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>7</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <msub> <mi>&amp;sigma;</mi> <mrow> <mn>8</mn> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow> <msup> <mi>&amp;alpha;</mi> <mo>+</mo> </msup> <mrow> <mo>(</mo> <msup> <mi>&amp;alpha;</mi> <mo>+</mo> </msup> <mo>-</mo> <msup> <mi>R&amp;alpha;</mi> <mo>-</mo> </msup> <mo>)</mo> </mrow> <mo>-</mo> <msup> <mi>&amp;alpha;</mi> <mo>-</mo> </msup> <mrow> <mo>(</mo> <msup> <mi>&amp;alpha;</mi> <mo>-</mo> </msup> <mo>-</mo> <msup> <mi>R&amp;alpha;</mi> <mo>-</mo> </msup> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mn>2</mn> <msub> <mi>k&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mfrac> <mrow> <msup> <mi>&amp;alpha;</mi> <mo>-</mo> </msup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>k&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mi>R</mi> <mo>)</mo> </mrow> <msub> <mi>X</mi> <mi>i</mi> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mi>R</mi> <mo>)</mo> </mrow> <msub> <mi>Y</mi> <mi>i</mi> </msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>k&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> <mo>+</mo> <mrow> <mo>(</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>1</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>R&amp;sigma;</mi> <mrow> <mn>1</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> <mo>+</mo> <mrow> <mo>(</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>2</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>R&amp;sigma;</mi> <mrow> <mn>2</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> <mo>+</mo> <mrow> <mo>(</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>3</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>R&amp;sigma;</mi> <mrow> <mn>3</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> <mo>&amp;rsqb;</mo> </mrow> <mrow> <msup> <mi>&amp;alpha;</mi> <mo>+</mo> </msup> <mrow> <mo>(</mo> <msup> <mi>&amp;alpha;</mi> <mo>+</mo> </msup> <mo>-</mo> <msup> <mi>R&amp;alpha;</mi> <mo>-</mo> </msup> <mo>)</mo> </mrow> <mo>-</mo> <msup> <mi>&amp;alpha;</mi> <mo>-</mo> </msup> <mrow> <mo>(</mo> <msup> <mi>&amp;alpha;</mi> <mo>-</mo> </msup> <mo>-</mo> <msup> <mi>R&amp;alpha;</mi> <mo>+</mo> </msup> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mn>2</mn> <msub> <mi>k&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msup> <mi>&amp;alpha;</mi> <mo>-</mo> </msup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>k&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> <mfrac> <mrow> <mo>(</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>4</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>R&amp;sigma;</mi> <mrow> <mn>4</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>)</mo> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>+</mo> <mo>(</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>5</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>R&amp;sigma;</mi> <mrow> <mn>5</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>)</mo> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>k&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> <mo>+</mo> <mo>(</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>6</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>R&amp;sigma;</mi> <mrow> <mn>6</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>)</mo> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>R</mi> <mo>)</mo> <msubsup> <mi>P</mi> <mrow> <mn>7</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>+</mo> <mo>(</mo> <msubsup> <mi>&amp;sigma;</mi> <mrow> <mn>7</mn> <mi>i</mi> </mrow> <mo>+</mo> </msubsup> <mo>-</mo> <msubsup> <mi>R&amp;sigma;</mi> <mrow> <mn>7</mn> <mi>i</mi> </mrow> <mo>-</mo> </msubsup> <mo>)</mo> <mo>(</mo> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>+</mo> <msub> <mi>&amp;sigma;</mi> <mrow> <mn>8</mn> <mi>i</mi> </mrow> </msub> <mo>)</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> </mrow> <mrow> <msup> <mi>&amp;alpha;</mi> <mo>+</mo> </msup> <mrow> <mo>(</mo> <msup> <mi>&amp;alpha;</mi> <mo>+</mo> </msup> <mo>-</mo> <msup> <mi>R&amp;alpha;</mi> <mo>-</mo> </msup> <mo>)</mo> </mrow> <mo>-</mo> <msup> <mi>&amp;alpha;</mi> <mo>-</mo> </msup> <mrow> <mo>(</mo> <msup> <mi>&amp;alpha;</mi> <mo>-</mo> </msup> <mo>-</mo> <msup> <mi>R&amp;alpha;</mi> <mo>+</mo> </msup> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mn>2</mn> <msub> <mi>k&amp;tau;</mi> <mn>0</mn> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced>
Wherein, R represents Reflectivity for Growing Season,
AiAnd BiCorresponding boundary condition is：
<mrow> <msubsup> <mi>I</mi> <mi>i</mi> <mo>-</mo> </msubsup> <mrow> <mo>(</mo> <mi>&amp;tau;</mi> <mo>=</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msubsup> <mi>I</mi> <mi>i</mi> <mo>+</mo> </msubsup> <mrow> <mo>(</mo> <mi>&amp;tau;</mi> <mo>=</mo> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mi>s</mi> </msub> <mo>)</mo> </mrow> <msubsup> <mi>I</mi> <mi>i</mi> <mo>-</mo> </msubsup> <mrow> <mo>(</mo> <mi>&amp;tau;</mi> <mo>=</mo> <msub> <mi>&amp;tau;</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>.</mo> </mrow>
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