CN101629850A - Method for inversing land surface temperature from MODIS data - Google Patents

Abstract

The invention relates to a method for inversing land surface temperature from MODIS data, which can be applied in meteorology, environmental monitoring, land management, agricultural condition monitoring, disaster monitoring and other remote sensing application departments. The method comprises four steps: the first step is to calculate the transmittance of 31st and 32nd wave bands of MODIS through water vapor content of atmosphere. The second step is to calculate the emissivity of the 31st and the 32nd wave bands of the MODIS sensor through vegetation index NDVI. The third step is to simplify a radiation transmission equation. The fourth step is to carry out inversion calculation on the MODIS data and obtain the land surface target object temperature and the emissivity distribution situation, so that the method can be used in meteorological forecasting, environmental monitoring, agricultural condition monitoring, disaster monitoring and other departments.

Description

From MODIS data inversion surface temperature method

Technical field

The present invention relates to a kind of method of utilizing the ground level heat radiation information inverting surface temperature that MODIS sensor on the earth observation satellite obtains, this method has overcome in the past the asynchronous and complicated shortcoming that is difficult to practicality of parameter acquiring in the inversion method.Can be applied in remote sensing departments such as meteorology, agricultural, environmental monitoring and damage caused by a drought monitoring.

Background technology

Surface temperature is meant the temperature on top layer, the face of land, and English is skin temperature.The importance of surface temperature in the region resource Environmental Studies has made thermal infrared remote sensing become a key areas of Remote Sensing Study, at present developed the surface temperature remote sensing inversion method of a lot of practicalities, as heat radiation transmission equation method, split window algorithm, single window algorithm and hyperchannel algorithm.Being permitted inversion algorithm is to develop at concrete sensor, Qin et al[2001 for example] develop one at NOAA-AVHRR and split window algorithm [Qin Z.H., Olmo G.D., and Karnieli A., Derivation of split window algorithm and its sensitivity analysis forretrieving land surface temperature from NOAA-advanced very high resolutionradiometer data, J.Geophys.Res., 2001,22,655-22,670.], owing to do not have near-infrared band inverting atmosphere vapour content on the AVHRR sensor, can only obtain atmospheric parameter by meteorological site or other sensors.1999, carried the earth observation satellite successful launch of MODIS remote sensor in 2002, for global and region resource environmental dynamic monitor have been opened up another new approach.MODIS is an intermediate-resolution remote sensing system (as Fig. 1) that has 36 wave bands, can obtain one time the global observation data in per 1～2 day, its flight and sun synchronization, every day, the same area can obtain two scape images round the clock at least, and be free reception, the region resource environmental dynamic monitor of large scale in therefore being very suitable for.Having 8 in 36 wave bands of MODIS is thermal infrared wave band (as table 1), thereby most suitablely analyzes in the face of land of regional scale heat spatial diversity.But, Surface Temperature Retrieval method at the MODIS remotely-sensed data is not a lot of at present, Wan Zhengming etc. in 1996 and 1997 before the MODIS sensor does not also carry satellite Heaven, at this sensor two kinds of inversion algorithms [Wan Z.and Dozier J. has been proposed, A generalized split-window algorithm forretrieving land surface temperature measurement from space, IEEE Trans.Geosci.Remote Sens., 1996,34:892-905.; Wan Z.M., Li Z..L., A Physics-Based Algorithmfor Retrieving land-surface emissivity and temperature from EOS/MODIS data, IEEETrans.Geosci.Remote Sens., 1997,35:980-996.].One of them is to split window algorithm, an inverting surface temperature; The another one algorithm is energy inverting surface temperature and an emissivity simultaneously, the data that this algorithm needs daytime and evening simultaneously, some parameters that need owing to Data Matching reason and this algorithm need other sensor that carries simultaneously or other retrieval products to obtain, and these two algorithms are adopted as the product algorithm by NASA (NASA).But the algorithm more complicated, general scientific research personnel is difficult to realize more complicated.

Domestic many researchers did some researchs at the Surface Temperature Retrieval of MODIS sensor, such as [Mao Kebiao, Qin Zhihao such as Mao Kebiao and Qin Zhihao, execute and build up, the palace roc is split window algorithm research at the MODIS data, Wuhan University's journal (information science version), 2005 (8): 703-708.; Mao Kebiao, Qin Zhihao executes and builds up, with MODIS image and the surface temperature of splitting the window algorithm inverting Shandong Peninsula, China Mining University's journal (natural science edition), 2005 (1): 46-50.].Though the MODIS satellite sensor carries on U.S.'s satellite, China MODIS data ground receiving station is many, such as Ministry of Agriculture's resources remote sensing and Digital Agriculture emphasis chamber, and satellite forecast center of China Meteorological Administration, the geographical grade of the Chinese Academy of Sciences all has receiving station.Though NASA is to whole world issue MODIS temperature product, its precision is more accurate in continental United States, but other some local precision of area is still not enough in the world, need further to improve [WAN, Z., ZHANG, Y., ZHANG, Q.and LI, Z.-L., 2002, Validation of the land-surface temperature products retrieved from TerraModerate Resolution Imaging Spectroradiometer data.Remote Sensing ofEnvironment, 2002,83,163-180.].Need us according to local actual conditions all over the world, further study new algorithm or do further correction, to improve precision.

Table 1MODIS remote sensor technical parameter

[MODIS?Level?1B?Product?User’s?Guide，For?Level?1B?Version?4.2.0(Terra)and?Version?4.2.1(Aqua).]

Summary of the invention

The object of the present invention is to provide a kind of method from remotely-sensed data MODIS inverting surface temperature, to overcome existing Surface Temperature Retrieval method complexity, and be difficult to satisfy the needs that the scientific research personnel personal development uses, and enrich domestic researchist more at thermal infrared remote sensing device development product algorithm, for thermal infrared sensor on 100 satellites of plan emission before China 2020 provides the reference of Surface Temperature Retrieval method, and can also further improve Surface Temperature Retrieval precision in the present Ministry of Agriculture service operation, improve damage caused by a drought monitoring and crops the yield by estimation precision [having drafted one of method of selecting as the Surface Temperature Retrieval method in Ministry of Agriculture's farming feelings monitoring].

For achieving the above object, provided by the inventionly be from remotely-sensed data MODIS inverting surface temperature method step:

The first step by atmosphere vapour content, is calculated the transmitance of MODIS the 31st and 32 wave bands

1-1) utilize MODIS the 19th wave band and the 2nd wave band ratio calculated T, utilize this ratio calculation atmosphere vapour content W[Kaufman Y.J. then, Gao Bo-Cai., Remote Sensing of Water Vapor inthe Near IR from EOS/MODIS, IEEE Trans.Geosci.Remote Sens., 1992,30:871-884.]:

$T=\frac{{\mathrm{\ρ}}_{19}}{{\mathrm{\ρ}}_2}$ (formula 1)

$W={\left(\frac{0.02-\ln T}{0.651}\right)}^2$ (formula 2)

1-2) utilize atmospheric radiation transmission MODTRAN4 analog computation to obtain the relation of atmosphere vapour content and MODIS the 31st and 32 wave band transmitances, with 1-1) in formula below the atmosphere vapour content substitution that calculates, calculate the transmitance (τ of the 31st wave band and 32 wave bands ₃₁, τ ₃₂).

${\mathrm{\τ}}_{31}=2.89798-1.88366e^{\frac{w}{21.22704}}$ (formula 3A)

${\mathrm{\τ}}_{32}=-3.59289+4.60414e^{\frac{-w}{32.70639}}$ (formula 3B)

Second goes on foot, calculates by vegetation index NDVI (Normalized Difference Vegetation Index) emissivity of MODIS sensor the 31st and 32 wave bands.

2-1) read in MODIS the 1st wave band (near-infrared band) and the 2nd wave band (red spectral band), calculate NDVI;

$\mathrm{NDVI}=\frac{B_2-B_1}{B_2+B_1}$ (formula 4)

B wherein ₁Represent the 1st wave band, B ₂Represent second wave band.

2-2) utilize NDVI to judge the type of ground objects of 1 square kilometre of resolution pixel, advance the ASTER wave spectrum storehouse of laboratory (JPL) test to calculate the emissivity (ε of MODIS sensor the 31st wave band and 32 wave bands according to U.S.'s spray vapour then ₃₁, ε ₃₂Represent the 31st and 32 band emission rates respectively).Be calculated as follows:

When NDVI＜0, be water body and snow, ε ₃₁=0.992, ε ₃₂=0.988

When 0＜NDVI＜0.05, be exposed soil, ε ₃₁=0.986, ε ₃₂=0.991

When 0.05＜NDVI＜0.65, be the mixed pixel of vegetation and exposed soil.

Utilize the ratio [KERR of the covering of vegetation in the PV exponential hybrid pixel, Y.H., LAGOUARDE, J.P.and IMBERNON, J., 1992, Accurate land surface temperature retrieval fromAVHRR data with use of an improved split window algorithm.Remote Sensing ofEnvironment, 1992,41,197-209.].Computing formula:

PV=(NDVI-0.05)/0.6 (formula 5)

ε ₃₁＝0.986*(1-PV)+0.972*PV，

ε ₃₂＝0.991*(1-PV)+0.976*PV；

When NDVI＞0.65, be vegetation: ε ₃₁=0.972, ε ₃₂=0.976

The 3rd step was simplified radiation transfer equation

3-1) simplify Planck (Planck) function.In energy-balance equation, each all comprises Planck (Planck) function,

$B_{\mathrm{\λ},T}=\frac{{2\mathrm{\πc}}^{2}h}{{\mathrm{\λ}}^5}{(e^{\frac{\mathrm{ch}}{\mathrm{K\λT}}}-1)}^{-1}$ (formula 6)

B _{λ T}Be the open score radiosity, unit is Wm ^-2μ m ^-1λ is a wavelength, the μ m of unit; H is a Planck's constant (6.6256 * 10-34Js); C is the light velocity (3 * 10 ⁸M/s); K is a Boltzmann constant (1.38 * 10-23J/K); T is absolute temperature (K).Planck (Planck) function is a nonlinear function, for the reduced equation group, need carry out linear simplifiation to Planck (Planck) function.Respectively to the 31st wave band (10.780～11.280 μ m) and 32 wave bands (11.77-12.27 μ m) of MODIS) heat radiation and the variation relation of temperature in the 273K-322K interval calculate, obtain result of calculation as shown in the figure.As can be seen from the figure, caloradiance approaches linear relationship with variation of temperature.Therefore, scatter diagram is set up equation of linear regression, obtains:

To the 31st wave band: B ₃₁(T)=0.13787T ₃₁-31.65677, R ²=0.9971

To the 32nd wave band: B ₃₂(T)=0.11849T ₃₂-26.50036, R ²=0.9978

3-2) simplify the radiation transfer equation group,

For the 31st and 32 wave bands of MODIS, the radiation transfer equation group of simplification can be write as following [Mao Kebiao is at the Surface Temperature Retrieval method research of MODIS data, master thesis, Nanjing University, 2004.5.]:

B ₃₁(T ₃₁)=τ ₃₁(θ) ε ₃₁(θ) B ₃₁(T _s)+[1-τ ₃₁(θ)] [1+ (1-ε ₃₁(θ) τ ₃₁(θ)] B ₃₁(T _a) (formula 7A)

B ₃₂(T ₃₂)=τ ₃₂(θ) ε ₃₂(θ) B ₃₂(T _s)+[1-τ ₃₂(θ)] [1+ (1-ε ₃₂(θ) τ ₃₂(θ)] B ₃₂(T _a) (formula 7B)

τ in the following formula _i(θ) be the transmitances of the 31st and 32 wave bands, ε in angle θ direction _i(θ) be the emissivity of the 31st and 32 wave bands, T in angle θ direction _aBe atmosphere mean effort temperature, T _sBe surface temperature, first B in the left side _i(T _i) be brightness temperature on the star of the 31st and 32 wave bands, first on equation the right is a surface radiation, second on equation the right is an atmospheric effect.3-1) in the reduced equation B of the 31st and 32 wave bands ₃₁(T)=0.13787T ₃₁-31.65677 and B ₃₂(T)=0.11849T ₃₂-26.50036 difference substitution radiation transfer equation groups obtain:

0.13787 ε ₃₁τ ₃₁T _s=0.13787T ₃₁+ 31.65677 ε ₃₁τ ₃₁-(1-τ ₃₁) [1+ (1-ε ₃₁) τ ₃₁] (0.13787T _a-31.65677)-31.65677 (formula 8A)

0.11849 ε ₃₂τ ₃₂T _s=0.11849T ₃₂+ 26.50036 ε ₃₂τ ₃₂-(1-τ ₃₂) [1+ (1-ε ₃₂) τ ₃₂] (0.11849T _a-26.50036)-26.50036 (formula 8B)

For the ease of calculating, the coefficient in the top system of equations is designated as respectively:

A ₃₁＝0.13787*ε ₃₁*τ ₃₁

B ₃₁＝0.13787*T ₃₁+31.65677*τ ₃₁*ε ₃₁-31.65677

C ₃₁＝(1-τ ₃₁)*(1+(1-ε ₃₁)*τ ₃₁)*0.13787

D ₃₁＝(1-τ ₃₁)*(1+(1-ε ₃₁)*τ ₃₁)*31.65677

A ₃₂＝0.11849*ε ₃₂*τ ₃₂

B ₃₂＝0.11849*T ₃₂+26.50036*τ ₃₂*ε ₃₂-26.50036

C ₃₂＝(1-τ ₃₂)*(1+(1-ε ₃₂)*τ ₃₂)*0.11849

D ₃₂＝(1-τ ₃₂)*(1+(1-ε ₃₂)*τ ₃₂)*26.50036

The radiation transfer equation group can be write as:

A ₃₁T _S=B ₃₁-C ₃₁Ta+D ₃₁(formula 9A)

A ₃₂T _S=B ₃₂-C ₃₂Ta+D ₃₂(formula 9B)

The group of solving an equation, the computing formula of top temperature is

T _s=(C ₃₂(B ₃₁+ D ₃₁)-C ₃₁(D ₃₂+ B ₃₂))/(C ₃₂A ₃₁-C ₃₁A ₃₂) (formula 10)

The 4th step inverting surface temperature,

4-1) utilize Planck function (Planck) inverse function to calculate brightness temperature T31 and T32 on the star of the 31st and 32 wave bands, B31 and B32 are respectively brightness on the star of MODIS the 31st and 32 wave bands in the following formula.

T31＝14380/(11.03*ln(2*59500000/(B31*11.03^5)+1))

T32＝14380/(12.02*ln(2*59500000/(B32*12.02^5)+1))

4-2) with the first step and second transmitance and the emissivity that calculate of step, and 4-1) brightness temperature substitution coefficient A respectively on the star that calculates ₃₁, B ₃₁, C ₃₁, D ₃₁, A ₃₂, B ₃₂, C ₃₂, D ₃₂, utilize the surface temperature computing formula to calculate surface temperature.

The invention has the beneficial effects as follows, consider the feature of 1 kilometer resolution of MODIS pixel, the vegetation index that utilizes MODI near infrared and red wave band to calculate calculates the emissivity of thermal infrared wave band 31 and 32, has guaranteed obtaining synchronously of emissivity, and simple and guaranteed certain precision.Utilize near-infrared band inverting atmosphere vapour content, and utilize MODTRAN4 to set up the relation of atmospheric transmittance and steam, thus real-time acquisition atmospheric transmittance parameter.Two key parameters in the Surface Temperature Retrieval have been solved like this.Improved inversion accuracy and computing time, overcome NASA's algorithm complexity in the past, the shortcoming that general scientific research personnel is difficult to realize.Be climate change research, weather forecast, evapotranspiration, agricultural feelings monitoring and disaster monitoring etc. provide effective means and technical support, this method to determine to be adopted by Ministry of Agriculture's farming feelings monitoring system.

Description of drawings

The present invention is further described below in conjunction with drawings and Examples.

Description of drawings

Fig. 1 MODIS remote sensor.

Fig. 2 is a main flow synoptic diagram of the present invention.

Fig. 3 is the atmosphere vapour content remote-sensing inversion result of Bohai Rim.

Fig. 4 is the 31st wave band atmospheric transmittance figure of MODIS.

Fig. 5 is the 32nd wave band atmospheric transmittance figure of MODIS.

Fig. 6 is the NDVI distribution plan.

Fig. 7 is the face of land emissivity distribution plan of MODIS the 31st wave band.

Fig. 8 is the face of land emissivity distribution plan of MODIS the 32nd wave band.

Fig. 9 be inverting and Bohai Rim's surface temperature distribution plan.

Figure 10 contrast verification result.

Embodiment

Here do concrete inverting with a scape MODIS image data of in early August, 2003 China Bohai Rim, this method mainly comprises four steps, as Fig. 2.

The first step by atmosphere vapour content, is calculated the transmitance of MODIS the 31st and 32 wave bands

1-1) utilize MODIS the 19th wave band and the 2nd wave band ratio calculated T, utilize this ratio calculation atmosphere vapour content W then, as Fig. 3:

$T=\frac{{\mathrm{\ρ}}_{19}}{{\mathrm{\ρ}}_2}$

$W={\left(\frac{0.02-\ln T}{0.651}\right)}^2$

1-2) utilize the relation of atmosphere vapour content and MODIS the 31st and 32 wave bands

${\mathrm{\τ}}_{31}=2.89798-1.88366e^{\frac{w}{21.22704}}$

${\mathrm{\τ}}_{32}=-3.59289+4.60414e^{\frac{-w}{32.70639}}$

With 1-1) in formula above the atmosphere vapour content substitution that calculates, calculate the transmitance (τ of the 31st wave band and 32 wave bands ₃₁, τ ₃₂), obtain the transmitance of the 31st and 32 wave bands, as Figure 4 and 5.

Second goes on foot, calculates by vegetation index NDVI (Normalized Difference Vegetation Index) emissivity of MODIS sensor the 31st and 32 wave bands.

2-1) read in MODIS the 1st wave band (near-infrared band) and the 2nd wave band (red spectral band), calculate NDVI;

$\mathrm{NDVI}=\frac{B_2-B_1}{B_2+B_1}$

The result who calculates such as Fig. 6.

2-2) utilize NDVI to judge the type of ground objects of 1 square kilometre of resolution pixel, advance the ASTER wave spectrum storehouse of laboratory (JPL) test to calculate the emissivity (ε of MODIS sensor the 31st wave band and 32 wave bands according to U.S.'s spray vapour then ₃₁, ε ₃₂Represent the 31st and 32 band emission rates respectively).Be calculated as follows:

When NDVI＜0, be water body and snow, ε ₃₁=0.992, ε ₃₂=0.988

When 0＜NDVI＜0.05, be exposed soil, ε ₃₁=0.986, ε ₃₂=0.991

When 0.05＜NDVI＜0.65, be the mixed pixel of vegetation and exposed soil.

PV＝(NDVI-0.05)/0.6

ε ₃₁＝0.986*(1-PV)+0.972*PV，

ε ₃₂＝0.991*(1-PV)+0.976*PV；

When NDVI＞0.65, be vegetation:

ε ₃₁＝0.972，ε ₃₂＝0.976

The emissivity that obtains MODIS the 31st and 32 wave bands after the calculating is respectively as Fig. 7 and Fig. 8.

The 3rd step was set up the radiation transfer equation group of MODIS the 31st wave band and the simplification of 32 wave bands

Brightness temperature on the star of formula calculating MODIS the 31st wave band and 32 wave bands below the utilization of the 4th step

T31＝14380/(11.03*ln(2*59500000/(B31*11.03^5)+1))

T32＝14380/(12.02*ln(2*59500000/(B32*12.02^5)+1))

The first step and second is gone on foot emissivity and the transmitance that calculates, brightness temperature difference substitution coefficient A on the star that calculates ₃₁, B ₃₁, C ₃₁, D ₃₁, A ₃₂, B ₃₂, C ₃₂, D ₃₂,

A ₃₁＝0.13787*ε ₃₁*τ ₃₁

B ₃₁＝0.13787*T ₃₁+31.65677*τ ₃₁*ε ₃₁-31.65677

C ₃₁＝(1-τ ₃₁)*(1+(1-ε ₃₁)*τ ₃₁)*0.13787

D ₃₁＝(1-τ ₃₁)*(1+(1-ε ₃₁)*τ ₃₁)*31.65677

A ₃₂＝0.11849*ε ₃₂*τ ₃₂

B ₃₂＝0.11849*T ₃₂+26.50036*τ ₃₂*ε ₃₂-26.50036

C ₃₂＝(1-τ ₃₂)*(1+(1-ε ₃₂)*τ ₃₂)*0.11849

D ₃₂＝(1-τ ₃₂)*(1+(1-ε ₃₂)*τ ₃₂)*26.50036

Utilize the surface temperature computing formula to calculate surface temperature, as Fig. 9.

T _s＝(C ₃₂(B ₃₁+D ₃₁)-C ₃₁(D ₃₂+B ₃₂))/(C ₃₂A ₃₁-C ₃₁A ₃₂)

In application, evaluation is very important to the method inversion accuracy.Because the pixel resolution of MODIS is 1 kilometer of 1 kilometer *), normally point measurement is measured on the face of land.We are difficult to represent the pixel of a large scale with the mean value of the data of several point measurements, and are to obtain simultaneously when satellite passes by.Here we come our method is estimated with U.S.'s flux netting index certificate.Flux data storehouse (http://public.ornl.gov/ameriflux/datahandler.cfm) is a global microclimate observation grid, mainly is to be used for monitoring carbon dioxide exchange, water vapour, the terrestrial ecosystem ENERGY EXCHANGE BETWEEN SEA of unifying.Also be used to MODIS surface temperature product is verified [Wang, W., S.Liang, Validating MODIS landsurface temperature product, ispmsrs05,17-19, October, Beijing, China, 2005.].Here we have also selected the relatively more smooth and homogeneous website in 3 faces of land (surface temperature FortPeck) are as the actual measured amount data for Brookings, Audubon.The concrete introduction please refer to (http://public.ornl.gov/ameriflux/site-select.cfm).From MODIS inverting surface temperature and flux station data more as shown in figure 10.Mean accuracy approximately is 1.2K, can satisfy our present application demand.

Claims (1)

1, from MODIS data inversion surface temperature method, the steps include:

The first step by atmosphere vapour content, is calculated the transmitance of MODIS the 31st and 32 wave bands

1-1) utilize MODIS the 19th wave band and the 2nd wave band ratio calculated T, utilize this ratio calculation atmosphere vapour content W then.Utilize atmospheric radiation transmission MODTRAN4 analog computation to obtain the relation of atmosphere vapour content and MODIS the 31st and 32 wave band transmitances, formula below the atmosphere vapour content substitution that calculates calculates the transmitance of the 31st wave band and 32 wave bands.

${\mathrm{\τ}}_{31}=2.89798-1.88366e^{\frac{w}{21.22704}}$

${\mathrm{\τ}}_{32}=-3.59289+4.60414e^{\frac{-w}{32.70639}}$

Second goes on foot, calculates by vegetation index NDVI the emissivity of MODIS sensor the 31st and 32 wave bands.

2-1) read in MODIS the 1st wave band (near-infrared band) and the 2nd wave band (red spectral band), calculate NDVI.Utilize NDVI to judge the type of ground objects of 1 square kilometre of resolution pixel, advance the ASTER wave spectrum storehouse of laboratory (JPL) test to calculate the emissivity (ε of MODIS sensor the 31st wave band and 32 wave bands according to U.S.'s spray vapour then ₃₁, ε ₃₂Represent the 31st and 32 band emission rates respectively).Be calculated as follows:

When NDVI＜0, be water body and snow, ε ₃₁=0.992, ε ₃₂=0.988

When 0＜NDVI＜0.05, be exposed soil, ε ₃₁=0.986, ε ₃₂=0.991

When 0.05＜NDVI＜0.65, be the mixed pixel of vegetation and exposed soil.

Utilize the ratio of the covering of vegetation in the PV exponential hybrid pixel.Computing formula:

PV=(NDVI-0.05)/0.6 (formula 5)

ε ₃₁＝0.986*(1-PV)+0.972*PV，

ε ₃₂＝0.991*(1-PV)+0.976*PV；

When NDVI＞0.65, be vegetation: ε ₃₁=0.972, ε ₃₂=0.976

The 3rd step was simplified radiation transfer equation

3-1) simplifying Planck (Planck) Planck (Planck) function is a nonlinear function, for the reduced equation group, need carry out linear simplifiation to Planck (Planck) function.Respectively to the 31st wave band (10.780～11.280 μ m) and 32 wave bands (11.77-12.27 μ m) of MODIS) heat radiation and the variation relation of temperature in the 273K-322K interval calculate, obtain result of calculation as shown in the figure.As can be seen from the figure, caloradiance approaches linear relationship with variation of temperature.Therefore, scatter diagram is set up equation of linear regression, obtains:

To the 31st wave band: B ₃₁(T)=0.13787T ₃₁-31.65677, R ²=0.9971

To the 32nd wave band: B ₃₂(T)=0.11849T ₃₂-26.50036, R ²=0.9978

3-2) simplify the radiation transfer equation group,

For the 31st and 32 wave bands of MODIS, the radiation transfer equation group of simplification can be write as follows:

B ₃₁(T ₃₁)＝τ ₃₁(θ)ε ₃₁(θ)B ₃₁(T _s)+[1-τ ₃₁(θ)][1+(1-ε ₃₁(θ)τ ₃₁(θ)]B ₃₁(T _a)

B ₃₂(T ₃₂)＝τ ₃₂(θ)ε ₃₂(θ)B ₃₂(T _s)+[1-τ ₃₂(θ)][1+(1-ε ₃₂(θ)τ ₃₂(θ)]B ₃₂(T _a)

3-1) in the reduced equation B of the 31st and 32 wave bands ₃₁(T)=0.13787T ₃₁-31.65677 and B ₃₂(T)=0.11849T ₃₂-26.50036 difference substitution radiation transfer equation groups obtain:

0.13787ε ₃₁τ ₃₁T _s＝0.13787T ₃₁+31.65677ε ₃₁τ ₃₁-(1-τ ₃₁)[1+(1-ε ₃₁)τ ₃₁](0.13787T _a-31.65677)-31.65677

0.11849ε ₃₂τ ₃₂T _s＝0.11849T ₃₂+26.50036ε ₃₂τ ₃₂-(1-τ ₃₂)[1+(1-ε ₃₂)τ ₃₂](0.11849T _a-26.50036)-26.50036

For the ease of calculating, the coefficient in the top system of equations is designated as respectively:

A ₃₁＝0.13787*ε ₃₁*τ ₃₁

B ₃₁＝0.13787*T ₃₁+31.65677*τ ₃₁*ε ₃₁-31.65677

C ₃₁＝(1-τ ₃₁)*(1+(1-ε ₃₁)*τ ₃₁)*0.13787

D ₃₁＝(1-τ ₃₁)*(1+(1-ε ₃₁)*τ ₃₁)*31.65677

A ₃₂＝0.11849*ε ₃₂*τ ₃₂

B ₃₂＝0.11849*T ₃₂+26.50036*τ ₃₂*ε ₃₂-26.50036

C ₃₂＝(1-τ ₃₂)*(1+(1-ε ₃₂)*τ ₃₂)*0.11849

D ₃₂＝(1-τ ₃₂)*(1+(1-ε ₃₂)*τ ₃₂)*26.50036

The radiation transfer equation group can be write as:

A ₃₁T _S＝B ₃₁-C ₃₁Ta+D ₃₁

A ₃₂T _S＝B ₃₂-C ₃₂TA+D ₃₂

The group of solving an equation, the computing formula of top temperature is

T _s＝(C ₃₂(B ₃₁+D ₃₁)-C ₃₁(D ₃₂+B ₃₂))/(C ₃₂A ₃₁-C ₃₁A ₃₂)。

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