CN105488805B

CN105488805B - Multifrequency dual polarization forest land accumulated snow passive microwave mixed pixel decomposition method

Info

Publication number: CN105488805B
Application number: CN201510933687.9A
Authority: CN
Inventors: 顾玲嘉; 任瑞治
Original assignee: Jilin University
Current assignee: Jilin University
Priority date: 2015-12-15
Filing date: 2015-12-15
Publication date: 2017-11-17
Anticipated expiration: 2035-12-15
Also published as: CN105488805A

Abstract

The invention discloses a kind of multifrequency dual polarization forest land accumulated snow passive microwave mixed pixel decomposition method, belong to the technical field of remote sensing image processing, carry out using passive microwave data the problem of error is larger during snow depth inverting to forest land accumulated snow for prior art.Acquisition of the present invention first to forest land underlying surface land classification data and reclassify, then according to establishing multifrequency dual polarization forest land accumulated snow passive microwave Pixel Unmixing Models on the basis of classification results, it is finally based on the less qualitative solving equations of dynamic window data selection strategy, classification data and error information corresponding to each underlying surface after being decomposed, during the present invention consider microwave pixel spatial coherence, propose the input of 8 neighborhood window datas and 4 neighborhood window datas input the scheme of two kinds of input datas, and preferably propose four kinds of solutions according to both the above input scheme and to solving result.

Description

Multifrequency dual polarization forest land accumulated snow passive microwave mixed pixel decomposition method

Technical field

The invention belongs to the technical field of remote sensing image processing, and in particular to one kind is based on multifrequency dual polarization forest land accumulated snow quilt Dynamic Microwave Hybrid picture element decomposing method.

Background technology

Accumulated snow is one of most active natural cause of earth's surface, and the freshwater resources that land has 3/4 on the earth are deposited in the form of ice and snow Seasonal Snow Cover in the area on, earth's surface surface 1/3 be present, Eurasia and North America area in the winter time at least 80% earth's surface Covered by accumulated snow.Accumulated snow is an important factor for determining radiation balance, not only with strong climatic effect, and to energy and Water Cycle process has extremely important influence., can be rapid because passive microwave remote sensing has very high temporal resolution Covering the whole world, therefore, in monitoring Global and the accumulated snow change in time and space of continental-scale, effect is especially prominent for it.It can not only Accumulated snow is observed round-the-clockly, can also penetrate most of snow cover so as to detect the information of snow depth and water equivalent of snow.Wherein, product Snow depth degree is one of the Main Factors for reflecting accumulated snow total amount, and snow disaster, snow melt flood warning, monitoring and assessment it is important because Son.

It is very big, it is necessary to right that domestic and international accumulated snow research shows that underlay surface properties influence on the inversion result of snow depth, water equivalent of snow The main underlying surface type for influenceing the bright temperature value of accumulated snow passive microwave is analyzed, and carries out passive microwave mixed pixel on this basis Decompose.Foster etc. is (referring to J.L.Foster, D.K.Hall, and A.T.C.Chang, (1984) " An overview of passive microwave snow research and results,”Rev.Geophys.Space Phys.22(2), Forest cover degree parameter 195-208) is introduced, improves the inversion accuracy of forest area water equivalent of snow to a certain extent.Tait (ginsengs See A.B.Tait (1998) " Estimation of snow water equivalent using passive microwave radiation Data,”Remote Sensing of Environment,64(3):286-291) utilize SSM/I sensors Passive microwave bright temperature data, with reference to the U.S. and Russian ground snow depth observation data, according to underlay surface properties (whether have snow slush, Whether with the presence of deep frost layer development, whether have complexity alpine terrain and whether have forest), underlying surface is divided into 16 classes. In a kind of, combined and return instead using two difference on the frequencies of 19GHz and 36GHz, 19GHz and 85GHz and a variety of polarization modes Drill, its result shows to work as presence of the underlying surface without snow slush and deep frost layer, and without forest cover, flat country inversion error is most Small, coefficient correlation is up to 0.75；And snow slush, deep frost layer develop without forestland distribution complicated landform underlying surface efficiency of inverse process Worst, coefficient correlation only 0.22, it is very big that this also illustrates that underlay surface properties influence on the inversion result of water equivalent of snow.Derkesen etc. (referring to C.Derksen, A.Walker, and B.Goodison (2005) " Evaluation of passive microwave snow water equivalent retrievals across the boreal forest/tundra of western Canada, " Remote Sensing of Environment, 96 (3-4), 315-327) research Canada West forest district During snowpack, the snow depth inversion algorithm sensitive to different earth's surface cover types is developed, has mainly considered bare area, coniferous forest, fallen leaves Four kinds of different underlying surfaces such as woods and incomplete wood, its weight are all types of coverages in pixel.For efficiency of inverse process not Good Canadian tundra area, Derkesen is (referring to C.Derksen, and A.Walker (2003) " Combining SMMR and SSM/I data for time series analysis of central North American snow water Journal of Hydrometeorology, 4 (2), 304-316) built again using 2002/03-2006/07 winter data The snow depth inversion method for relying only on the bright temperature of 37V polarization is found.In recent years, domestic scholars are to the accumulated snow snow depth inverting suitable for the country Algorithm is also explored and developed.Che et al. is (referring to T.Che, X.Li, R.Jin, R.Armstrong, and T.J.Zhang (2008).“Snow depth derived from passive microwave remote-sensing data in China, " Annals of Glaciology, 49 (1), 145-154) based on Chang algorithms, number is observed by weather station According to have modified SMMR (1980 and the observation snow depth of meteorological site in 1981 data), the SMM/I of regional respectively (2003 Weather station snow depth observation data) coefficient, SMMR and SMM/I snow depth inverting regression algorithm is obtained, the root mean square of inverting snow depth misses Difference is respectively 6.22cm and 5.99cm.Jiang etc. is (referring to L.M.Jiang, P.Wang, L.X.Zhang, H.Yang, and J.Y.Yang(2014).“Improvement of snow depth retrieval for FY3B-MWRI in China,” Science China Earth Sciences, 44 (3), 531-547) using 2002-2009 national meteorological site ground Area avenges parameter estimator data and corresponding time, the AMSRE L2A data in space, with reference to the underground properties of regional, Regional is divided into four kinds of forest, farmland, meadow and bare area underlying surface types, these four different underlying surface types of first inverting Snow depth, then in conjunction with the snow depth in weight calculation mixed pixel, it is applied to the snow depth product of FY3B/MWRI business.(the ginseng such as Gu See L.J.Gu, R.Z.Ren, K.Zhao and X.F.Li (2014) " Snow depth and snow cover retrieval from FengYun3B microwave radiation imagery based on a snow passive microwave Unmixing method in Northeast China, " J.Appl.Remote Sens.8 (1), 084682) it is empty based on 1km Between the Chinese soil of resolution ratio utilize data, with reference to passive microwave antenna gain function, propose accumulated snow passive microwave mixed pixel Decomposition model, the main underlying surface type of Northeast Area of China is divided into five kinds of meadow, farmland, bare area, forest and water body underlays Noodles type, it is applied to the TMI data that No. three B stars of wind and cloud carry.Due to the space point of microwave radiance transfer data Resolution is relatively low (>=10km), and mixed pixel is prevalent in passive microwave remote sensing data, and its presence is the inverting of snow depth later stage The main reason for precision is difficult to reach requirement.In most observation areas, by difference in passive microwave antenna footprint Earth's surface medium composition, the bright temperature of underlying surface that microwave radiometer obtains is that the mixed pixel for representing some scale integrates bright temperature, And the underlying surface microwave radiation characteristics of different type underlying surface have larger difference, wherein forest is in the heterogeneity in pixel Influence maximum.

The content of the invention

Had a great influence to solve the heterogeneity in the pixel of forest land, and cause its using snow depth inversion method when error compared with The problem of big, the present invention observes Underlying Surface Data in pixel according to forest land passive microwave, with reference to passive microwave antenna gain function, Provide a kind of multifrequency dual polarization forest land accumulated snow passive microwave mixed pixel decomposition method.

The technical solution adopted in the present invention is specific as follows：

A kind of multifrequency dual polarization forest land accumulated snow passive microwave mixed pixel decomposition method, the application conditions of this method are winters Underlying surface land classification data in the passive microwave remote sensing data and resolution spectrum remotely-sensed data of forest land sector of observation, method Including following process：1) acquisition of forest land underlying surface land classification data and reclassify, 2) establish multifrequency dual polarization forest land product The dynamic Microwave Hybrid pixel analysis model of snow cover, 3) the less qualitative solving equations based on dynamic window data selection strategy.

1) acquisition of forest land underlying surface land classification data and reclassify

The land cover classification product in resolution spectrum remotely-sensed data is obtained, under in resolution spectrum remotely-sensed data Land classification data in pad face are reclassified, specific as follows：

A) data that classification defined in former grouped data is belonged to aciculignosa type are uniformly newly defined as coniferous forest class Type B；

B) will be evergreen broad-leaved vegetation, broad leaved and deciduous broad leaved vegetation and annual broad-leaved vegetation pattern defined in former grouped data Data are uniformly newly defined as broad-leaf forest type N；

C) other grouped datas S₁,S₂,…S_nKeep former type constant；

2) multifrequency dual polarization forest land accumulated snow passive microwave Pixel Unmixing Models are established

From website download multifrequency dual polarization passive microwave remote sensing data, the data preferably have been subjected to demarcation, atmospheric correction, Reason correction and standardization pretreatment.

First, passive microwave remote sensing data is the bright temperature value of mixing that antenna for radiometer measures ground on satellite, and on satellite The bright temperature value of mixing that antenna for radiometer measures ground is the integration of the actual bright gentle antenna gain function of earth's surface：

In formula (1),It is the bright temperature value of mixing of antenna for radiometer measurement,It is actual bright in ground location (x, y) Temperature value, lower footnote F are passive microwave observation frequency ranges,It is antenna gain function, integration covers the elliptic region of whole footmark, day The formula of line gain function is defined as：

A and b represents footmark corresponding to different frequency range in passive microwave data.

The true bright temperature value in groundIt may be considered the linear superposition of n kinds underlying surface classification value in its footmark：

Wherein T_iRepresent different underlying surface classification value, LⁱRepresent i-th of underlying surface grouped data.

The frequency range of microwave radiance transfer used in the present invention be K frequency (18.7GHz) and Ka frequency (36.5GHz), K frequency with Ka frequencies are the conventional effective frequency ranges sentenced and know accumulated snow, and each frequency range includes horizontal polarization and vertical polarization two ways.

The present invention measures for the Snow Thickness in forest land, therefore only considers the main underlying surface composition of forest, such as：Broad-leaved Woods, coniferous forest and other underlying surfaces composition that may be included, then the forest land accumulated snow passive microwave at ground location (x, y) place is in K frequencies It can be expressed as with the radiation value of Ka frequencies：

T represents bright temperature value in formula (4)；L represents underlying surface grouped data, if the underlying surface be present in (x, y) position, then L (x, Y)=1；Otherwise, L (x, y)=0.Upper footnote C, B and S₁,S₂,…S_nUnderlying surface land type, the broad-leaved of the position are represented respectively Woods underlying surface land type, coniferous forest underlying surface land type and other underlying surface land types that may be present；Lower footnote K and Ka represents passive microwave K and Ka frequency range respectively, wherein lower footnote H and V represent horizontal polarization and vertical polarization respectively；ε represents residual Residual quantity；L^B(x,y)、L^N(x, y) andRepresent respectively ground location (x, y) place broad-leaf forest, coniferous forest and The distribution of other underlying surfaces, and have：

The bright temperature of the bright temperature value of mixing that microwave radiance transfer radiometer observes in different frequency range (F) and polarization mode (V, H) ValueThe convolution of the bright temperature of the actual underlying surface type of passive microwave antenna and ground can be regarded as, formula (4) is substituted into formula (1) Then have：

It is antenna gain function to make αWith underlying surface grouped data L convolution, then have：

C=B, N, S (7)

Formula (6) is deformed into using formula (7)：

Formula (8) is multifrequency dual polarization forest land accumulated snow passive microwave Pixel Unmixing Models.

3) the less qualitative solving equations based on dynamic window data selection strategy

α values are calculated according in formula (8), belongs to linear and owes to determine row solving equations problem, theoretical above formula (8) should have Numerous solution.Therefore, following multipoint observation constraint solving method should be used, when such as observation station number being p, formula (8) is：

E represents residual matrix in formula (9).

Under normal circumstances due to the spatial coherence of microwave pixel, there is identical interior all similar underlying surfaces around observation station Bright values, therefore assume that interior all similar underlying surfaces have identical bright values around observation station here, the condition can be with As the constraint of formula (8), formula (8) is write as：

Generally consider to solve by least square method in this case：

Wherein,

With

In addition, the passive microwave classification for solving to obtain for certain frequency range same position, should also meet following condition：

Formula (11) can be solved by nonnegative least.For solving mixed problem, a common most young waiter in a wineshop or an inn is converted into Multiplication its meet bright temperature solution on the occasion of condition.

T=(A^tA)^-1A^tY (17)

Resolution error is：

E=T- (A^tA)^-1A^tY (18)

Chosen for p value in formula (10), using dynamic window data selection strategy：

First, it is contemplated that the spatial coherence of microwave pixel, with position (x to be decomposed₀,y₀) place tested F-band Centered on passive microwave mixed pixel, the passive microwave mixed pixel of 8 neighborhood windows and the neighborhood window of surrounding 4 around it are chosen Passive microwave mixed pixel is solved as equation group input data.

Correspondingly the scheme of two kinds of input datas is：

(A) 8 neighborhood window datas input：According to Fig. 1, by the multifrequency dual polarization quilt containing 9 pixels including current pixel Dynamic microwave data (mixes bright temperature value) it is changed into a n dimensional vector nSubstitution formula (12).If calculate Obtained result meets formula (15) and formula (16), obtains and rationally solves, component corresponding to each underlying surface is bright after being decomposed Warm data T₁, obtain error information E₁；

(B) 4 neighborhood window datas input：According to Fig. 2, by containing 5 multifrequency dual polarization passive microwaves including current pixel Data (mix bright temperature value) it is changed into a n dimensional vector nSubstitute into formula (12).If it is calculated As a result meet formula (15) and formula (16), obtain and rationally solve, classification data corresponding to each underlying surface after being decomposed T₂, obtain error information E₂。

But according to the above method classification data not necessarily optimal solution, therefore according to (A), (B) scheme Solving result, consider the following four situation being likely to occur, determine the classification finally exported：

Situation one：(A) scheme of selection, if solution is unreasonable, uses (B) scheme；If (B) Scheme Solving is still unreasonable, Illustrating the mixed pixel of input can not decompose, and retain the bright temperature value of preimage member；

Situation two：(A) scheme of selection, if solution is unreasonable, uses (B) scheme；If (B) Scheme Solving is reasonable, output Classification value T₂And error E₂, final classification value is T₂, error information be E₂；

Situation three：(A) scheme of selection, if solving reasonable, output classification value T₁With error matrix E₁；Continue to use (B) Scheme, if (B) Scheme Solving is unreasonable, final classification value is T₁, error be E₁；

Situation four：(A) scheme of selection, if solving reasonable, output classification value T₁With error matrix E₁；Continue to adopt (B) side Case, if (B) Scheme Solving is reasonable, export classification value T₂With error matrix E₂.Respectively to error information E₁And E₂Matrix Calculating E is designated as with result₁' and E₂', compare E₁' and E₂' size.If E₁' ＞ E₂', then final classification value is T₂, error information It is E₂；If E₁' ＜ E₂', then final classification value is T₁, error information be E₁。

Beneficial effects of the present invention：

In terms of forest snow remote sensing, microwave remote sensing can play it and penetrate the ability of forest canopy, effectively detect Accumulated snow information below Forest Canopy, this point are that optical remote sensing can not be reached.Utilize the K frequencies of passive microwave radiometer (18.7GHz) and Ka frequency (36.5GHz) microwave radiation bright temperature data carry out mathematic interpolation, can effectively obtain sylvan life accumulated snow letter Breath, it is thus identified that the validity that microwave remote sensing detects for sylvan life accumulated snow.But due to passive microwave data spatial resolution compared with Low, inhomogeneity can all cause the inverting of snow depth to complicate in microwave pixel, and the particularly presence of vegetation can reduce microwave to product The sensitiveness of snow.Emphasis of the present invention carries out snow depth inverting research for winter sector of observation for the region of Type of Forest Land, proposes more Frequency dual polarization forest land accumulated snow passive microwave mixed pixel decomposition method, it is divided into pin according to the main Types included in microwave pixel Ye Lin, broad-leaf forest and other underlying surfaces, obtain the passive microwave classification on each underlying surface respectively, use it for coniferous forest, The snow depth inverting of broad-leaf forest underlying surface, the accumulated snow of more accurate different woodland type under Global Scale can be handled out in time Supplemental characteristic, solve the problems, such as that current forest land precision of snow parameter inversion is relatively low.

Brief description of the drawings

Fig. 1 is 8 neighborhood window input data positions selection of the invention.

Fig. 2 is 4 neighborhood window input data positions selection of the invention.

Fig. 3 is the forest land sector of observation passive microwave radiometer K frequency horizontal polarization data of the embodiment of the present invention 1.

Fig. 4 is the forest land sector of observation passive microwave radiometer Ka frequency horizontal polarization data of the embodiment of the present invention 1.

Fig. 5 is the forest land sector of observation passive microwave radiometer K frequency horizontal polarization classification data of the embodiment of the present invention 1.

Fig. 6 is the forest land sector of observation passive microwave radiometer Ka frequency horizontal polarization classification data of the embodiment of the present invention 1.

Fig. 7 is that the original passive microwave data of the embodiment of the present invention 1 and passive microwave classification data carry out snow depth inverting Comparative result.

Fig. 8 is the inventive method flow chart of steps.

Embodiment

Technical solution of the present invention is further illustrated in the form of specific embodiment below.

Embodiment 1

The present invention utilizes FY-3B MWRI passive microwave remote sensing datas and MODIS land cover classification data products, with reference to Multifrequency dual polarization forest land accumulated snow passive microwave mixed pixel decomposition method, realizes in January, 2012 Heilongjiang Province of China Yichun Effective decomposition of forest land accumulated snow passive microwave mixed pixel.

Comprise the following steps that：

(1) acquisition of forest land grouped data and reclassify

The land classification data that the present invention uses are the Land cover types productions of MODIS intermediate-resolution spectral remote sensing data Product.The Land cover types product of intermediate-resolution MODIS spectral remote sensing data is downloaded in MODIS official websites.MODIS is ground on land Study carefully the Year Land cover classification product MCD12Q1 (spaces point of group development MODIS Aqua and Terra satellite data synthesis Resolution 500m), NPP (net primary productivity) land cover classification data set in MCD12Q1 sort products is selected, the time is 2012.

Underlying surface is divided into 9 classes by NPP (net primary productivity) land cover classification data sets, including：Water, evergreen needle are planted Quilt, evergreen broad-leaved vegetation, aciculignosa of falling leaves, broad leaved and deciduous broad leaved vegetation, annual broad-leaved vegetation, annual herb vegetation, no plant The soil and city of quilt.

Obtaining its data according to sector of observation mainly includes：Evergreen aciculignosa, evergreen broad-leaved vegetation, fallen leaves needle are planted Quilt, broad leaved and deciduous broad leaved vegetation, annual broad-leaved vegetation, annual herb vegetation and city, therefore grouped data product is divided again Class is specific as follows：

(a) by be defined in former grouped data evergreen aciculignosa and fallen leaves aciculignosa type be uniformly newly defined as Coniferous forest type N；

(b) it will defined in former grouped data be the class of evergreen broad-leaved vegetation, broad leaved and deciduous broad leaved vegetation and annual broad-leaved vegetation Type is uniformly newly defined as broad-leaf forest type B；

(c) annual herb vegetation pattern is defined as Q；

(d) classes of cities is defined as U.

Using above-mentioned sorting technique, can obtain research area's broad-leaf forest type, broad-leaf forest type, annual herb vegetation and Four kinds of underlying surface grouped data L of classes of cities^N、L^B、L^QAnd L^U。

(2) multifrequency dual polarization forest land accumulated snow passive microwave Pixel Unmixing Models are established

The TMI (MWRI) loaded on No. three B stars of wind and cloud is first, China satellite-borne microwave remote sensing instrument, scanning Mode is conical scanning, and its design frequency is 10.65~150GHz, and wherein 150GHz is pilot passageway.Each frequency has vertical Directly round-the-clock, round-the-clock surface temperature, the soil water can be provided with two kinds of different polarization modes of level, the remotely sensed image of these frequencies Point, the abundant information such as Flood and drought, snow depth, typhoon structure, atmospheric water content.It is distant that MWRI passive microwaves are downloaded from website Feel data, its spatial resolution is 10km.The product has been subjected to demarcation, atmospheric correction, geographical correction and standardization pretreatment.

The bright temperature that antenna for radiometer measures ground on satellite is the integration of the actual bright gentle antenna gain function of earth's surface：

Footmark corresponding to a and b expression passive microwave data Mid Frequencies, is defined as a=5km and b=5km.

For the true bright temperature in groundIt may be considered the linear superposition of 4 kinds of underlying surface classification values in its footmark：

Wherein T_iRepresent different underlying surface (component) bright temperature, LⁱRepresent i-th of underlying surface grouped data.

Microwave radiance transfer K frequencies (18.7GHz) and Ka frequencies (36.5GHz) are the conventional effective frequency ranges sentenced and know accumulated snow, are considered To being mainly made up of broad-leaf forest, coniferous forest, annual herb vegetation and city underlying surface for sector of observation, ground location (x, y) The forest land accumulated snow passive microwave at place can be expressed as in the radiation value of K frequencies and Ka frequencies：

Wherein T represents bright temperature value；L represents underlying surface grouped data, if the underlying surface be present in (x, y) position, then L (x, y) =1；Otherwise, L (x, y)=0.Upper footnote C, B, N, Q and U represent respectively observation position underlying surface type, broad-leaf forest, coniferous forest, Annual herb vegetation and city underlying surface.Lower footnote K and Ka represent passive microwave K and Ka frequency range respectively, wherein lower footnote H and V represents horizontal polarization and vertical polarization respectively.ε represents residual error amount.L^B、L^N、L^QAnd L^UIt is wealthy that ground location (x, y) place is represented respectively Ye Lin, the distribution of coniferous forest, annual herb vegetation and city underlying surface, and have：

L^B(x,y)+L^N(x,y)+L^Q(x,y)+L^U(x, y)=1 (5 ')

The bright temperature of forest land accumulated snow that microwave radiance transfer radiometer observes in different frequency range (F) and polarization mode (V, H) ValueThe convolution of the bright temperature of the actual underlying surface type of passive microwave antenna and ground can be regarded as, (4) substitution (1) is had：

It is antenna gain function to make αWith underlying surface grouped data L convolution, have：

C=B, N, Q, U (7)

Then (6) formula is changed into：

Formula (8) includes 4 equations, it is known that 4 forest lands observed mix bright temperature value and 4 different underlying surface classification The α values that are calculated are, it is necessary to obtain 16 unknown quantitys, i.e. 4 bright temperature of broad-leaf forest, 4 bright temperature of coniferous forest, 4 annual herbs The bright temperature in bright gentle 4 cities of vegetation, this belongs to linear and owes to determine row solving equations problem.

(3) the less qualitative solving equations based on dynamic window data selection strategy

In theory, formula (7) should have numerous solution.Accordingly, it is considered to using following multipoint observation constraint solving method:

Wherein p is the number of observation station, and E represents residual matrix.It is assumed here that around the observation station it is interior it is all it is similar under There are identical bright values in pad face, and the condition can be as the constraint of formula (8).Formula (8) as determined linear system can be write Into：

Consider least square method：

Wherein

With

In theory, formula (11) can be solved by nonnegative least.For solving mixed problem, it is converted into common Least square method its meet bright temperature solution on the occasion of condition.

T=(A^tA)^-1A^tY (17)

Resolution error is：

E=T- (A^tA)^-1A^tY (18)

First, it is contemplated that the spatial coherence of microwave pixel, with position (x to be decomposed₀,y₀) place F-band it is passive micro- Ripple mixed pixel T_F ^MCentered on, passive microwave mixed pixel (as shown in Figure 1) and the surrounding 4 for choosing 8 neighborhood windows around it are adjacent The passive microwave mixed pixel of domain window s (as shown in Figure 2) to be solved as equation group input data, and F takes K and Ka frequencies respectively Section microwave data.

Correspondingly the scheme of two kinds of input datas is：

(a) 8 neighborhood window datas input：According to Fig. 1, by the multifrequency passive microwave number containing 9 pixels including current pixel According to being changed into a n dimensional vector nSubstitute into formula (12).If the result being calculated meet formula (15) and Formula (16), obtain and rationally solve, classification data T corresponding to each underlying surface after being decomposed₁, obtain error information E₁；

(b) 4 neighborhood window datas input：According to Fig. 2, will turn containing 5 multifrequency passive microwave data including current pixel It is changed into a n dimensional vector nSubstitute into formula (12).If the result being calculated meets formula (15) and formula (16), obtain and rationally solve, classification data T corresponding to each underlying surface after being decomposed₂, obtain error information E₂；

Dynamic window data selection strategy is designed, solves less qualitative equation group.According to (a), the solving result of (b) scheme, Consider following four situation, determine the classification finally exported：

Situation one：(a) scheme of selection, if solution is unreasonable, uses (b) scheme；If Scheme Solving is still unreasonable (b), Illustrating the mixed pixel of input can not decompose, and retain the bright temperature value of preimage member；

Situation two：(a) scheme of selection, if solution is unreasonable, uses (b) scheme；If Scheme Solving is reasonable (b), output Classification value T₂With error information E₂, final classification value is T₂, error information be E₂；

Situation three：(a) scheme of selection, if solving reasonable, output classification value T₁With error information E₁；Continue to use (b) Scheme, if (b) Scheme Solving is unreasonable, final classification value is T₁, error information be E₁；

Situation four：(a) scheme of selection, if solving reasonable, output classification value T₁With error information E₁；Continue to use (b) Scheme, if (b) Scheme Solving is reasonable, export classification value T₂With error information E₂.Component corresponding to each underlying surface is calculated respectively Error E₁Summation is designated as E₁', error E₂Summation be designated as E₂', compare E₁' and E₂' size.If E₁' ＞ E₂', then final component Bright temperature value is T₂, error information be E₂；If E₁' ＜ E₂', then final classification value is T₁, error information be E₁。

Compliance test result：

According to Chang microwave snow depth inversion method, it is assumed that accumulated snow is homogeneous, individual layer dry snow, snow density 0.3g/ cm³, snow particle diameter is 0.3mm, and snow depth inversion formula is as follows：

SD=C × (T_KH-T_KaH) (19)

Wherein SD is snow depth (cm)；C is regression coefficient, generally takes 1.59；T_KHAnd T_KaHWei not passive microwave K and Ka The horizontal polarization data of frequency range.By passive microwave radiometer K and the Ka frequency range of Heilongjiang Province of China Yichun forest land sector of observation Horizontal polarization data (as shown in Figure 3 and Figure 4), utilize accumulated snow passive microwave mixing picture in multifrequency dual polarization forest land proposed by the present invention First decomposition method, the horizontal polarization classification data (as shown in Figure 5 and Figure 6) after K and Ka frequency ranges are decomposed can be obtained.

Using the classification after original passive microwave K and Ka frequency range horizontal polarization data and corresponding Decomposition of Mixed Pixels Data, pixel and Chang is carried out according to the affiliated classification data of its actual underlying surface land classification where test point position The result (as shown in Figure 7) of snow depth inverting.As seen from Figure 7, in Chang snow depths inversion method, using the classification after decomposition Data carry out the result that snow depth inverting obtains, relative to the result of initial data snow depth inverting, closer to the snow in actual measurement forest land Deep data.Using accumulated snow passive microwave mixed pixel decomposition method in multifrequency dual polarization forest land proposed by the present invention, pin can be obtained Ye Lin, broad-leaf forest underlying surface snow depth, effectively improve the precision of later stage snow depth inverting.

Claims

1. a kind of multifrequency dual polarization forest land accumulated snow passive microwave mixed pixel decomposition method, the application conditions of this method are winter woodss Underlying surface land classification data in the passive microwave remote sensing data and resolution spectrum remotely-sensed data of ground sector of observation, specific bag Include following process：

1) acquisition of forest land underlying surface land classification data and reclassify；

2) multifrequency dual polarization forest land accumulated snow passive microwave Pixel Unmixing Models are established；

3) the less qualitative solving equations based on dynamic window data selection strategy；

The acquisition of the forest land underlying surface land classification data and the step of reclassify：Obtain in resolution spectrum remotely-sensed data Land cover classification product, the underlying surface land classification data in resolution spectrum remotely-sensed data are reclassified, have Body is as follows：

A) data that classification defined in former grouped data is belonged to aciculignosa type are uniformly newly defined as coniferous forest type N；

B) it will defined in former grouped data be the data of evergreen broad-leaved vegetation, broad leaved and deciduous broad leaved vegetation and annual broad-leaved vegetation pattern Uniformly it is newly defined as broad-leaf forest type B；

C) other grouped datas S₁,S₂,…S_nKeep former type constant；

Described the step of establishing multifrequency dual polarization forest land accumulated snow passive microwave Pixel Unmixing Models：The multifrequency dual polarization woods Area snow cover moves Microwave Hybrid pixel analysis model：

<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <mi>M</mi> </msubsup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <mi>M</mi> </msubsup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <mi>M</mi> </msubsup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <mi>M</mi> </msubsup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <msup> <mi>&alpha;</mi> <mi>B</mi> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> </mrow> <mo>)</mo> </mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <mi>B</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <mi>B</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <mi>B</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <mi>B</mi> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <msup> <mi>&alpha;</mi> <mi>N</mi> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> </mrow> <mo>)</mo> </mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <mi>N</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <mi>N</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <mi>N</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <mi>N</mi> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <msup> <mi>&alpha;</mi> <msub> <mi>S</mi> <mn>1</mn> </msub> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> </mrow> <mo>)</mo> </mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <msup> <mi>&alpha;</mi> <msub> <mi>S</mi> <mn>2</mn> </msub> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> </mrow> <mo>)</mo> </mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <msub> <mi>S</mi> <mn>2</mn> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <msub> <mi>S</mi> <mn>2</mn> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <msub> <mi>S</mi> <mn>2</mn> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <msub> <mi>S</mi> <mn>2</mn> </msub> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mn>...</mn> <mo>+</mo> <msup> <mi>&alpha;</mi> <msub> <mi>S</mi> <mi>n</mi> </msub> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> </mrow> <mo>)</mo> </mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mi>&epsiv;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>

T in formula (8)^MThe bright temperature value of forest land accumulated snow that passive microwave radiometer observes in different frequency range and polarization mode is represented, under Footnote K and Ka represent passive microwave K and Ka frequency range respectively, and lower footnote H and V represent horizontal polarization and vertical polarization respectively；Two kinds Polarization mode；ε represents residual error amount；

Wherein,

<mrow> <msup> <mi>&alpha;</mi> <mi>C</mi> </msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mo>&Integral;</mo> <mo>&Integral;</mo> <mover> <mi>G</mi> <mo>&OverBar;</mo> </mover> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <msup> <mi>L</mi> <mi>C</mi> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mi>d</mi> <mi>x</mi> <mi>d</mi> <mi>y</mi> <mo>,</mo> <mi>C</mi> <mo>=</mo> <mi>B</mi> <mo>,</mo> <mi>N</mi> <mo>,</mo> <msub> <mi>S</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>S</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>...</mo> <msub> <mi>S</mi> <mi>n</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>

In formula (7), L (x, y) represents the underlying surface land classification data of (x, y) position, upper footnote C, B, N and S₁,S₂,…S_nPoint Underlying surface type, broad-leaf forest underlying surface land type, coniferous forest underlying surface land type and other underlays of the position are not represented Face land type；If there is the underlying surface in (x, y) position, then L (x, y)=1, otherwise, L (x, y)=0；It is antenna gain letter Number, and have

<mrow> <mover> <mi>G</mi> <mo>&OverBar;</mo> </mover> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mi>l</mi> <mi>n</mi> <mn>2</mn> </mrow> <mi>&pi;</mi> </mfrac> <mo>&CenterDot;</mo> <mi>exp</mi> <mo>&lsqb;</mo> <mo>-</mo> <mi>l</mi> <mi>n</mi> <mn>2</mn> <mo>&CenterDot;</mo> <mrow> <mo>(</mo> <mfrac> <msup> <mi>x</mi> <mn>2</mn> </msup> <msup> <mi>a</mi> <mn>2</mn> </msup> </mfrac> <mo>+</mo> <mfrac> <msup> <mi>y</mi> <mn>2</mn> </msup> <msup> <mi>b</mi> <mn>2</mn> </msup> </mfrac> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>

In formula (2), a and b represent the footmark in passive microwave remote sensing data corresponding to K and Ka frequency ranges respectively；

The step of less qualitative solving equations based on dynamic window data selection strategy：Asked using multipoint observation constraint Solution, when observation station quantity is p, it is assumed that interior all similar underlying surfaces have identical bright values around observation station, then formula (8) turns It is changed to

<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <mi>M</mi> </msubsup> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <mi>M</mi> </msubsup> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <mi>M</mi> </msubsup> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <mi>M</mi> </msubsup> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <mi>M</mi> </msubsup> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <mi>M</mi> </msubsup> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <mi>M</mi> </msubsup> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <mi>M</mi> </msubsup> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <mi>B</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <mi>N</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> </msubsup> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <mi>B</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <mi>N</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> </msubsup> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <mi>B</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <mi>N</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> </msubsup> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <mi>B</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <mi>N</mi> </msubsup> </mtd> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> </msubsup> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>&alpha;</mi> <mn>1</mn> <mi>B</mi> </msubsup> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msubsup> <mi>&alpha;</mi> <mi>p</mi> <mi>B</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>&alpha;</mi> <mn>1</mn> <mi>N</mi> </msubsup> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msubsup> <mi>&alpha;</mi> <mi>p</mi> <mi>N</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>&alpha;</mi> <mn>1</mn> <mi>S</mi> </msubsup> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msubsup> <mi>&alpha;</mi> <mi>p</mi> <mi>S</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>&alpha;</mi> <mn>1</mn> <msub> <mi>S</mi> <mi>n</mi> </msub> </msubsup> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msubsup> <mi>&alpha;</mi> <mi>p</mi> <msub> <mi>S</mi> <mi>n</mi> </msub> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mi>E</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>

P is the number of observation station in formula (10), and E represents residual matrix；

Then, it is solved using nonnegative least,

T=(A^tA)^-1A^tY (17)

Resolution error is：

E=T- (A^tA)^-1A^tY (18)

In formula (17) and formula (18),

<mrow> <mi>A</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>&alpha;</mi> <mn>1</mn> <mi>B</mi> </msubsup> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msubsup> <mi>&alpha;</mi> <mi>p</mi> <mi>B</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>&alpha;</mi> <mn>1</mn> <mi>N</mi> </msubsup> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msubsup> <mi>&alpha;</mi> <mi>p</mi> <mi>N</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>&alpha;</mi> <mn>1</mn> <msub> <mi>S</mi> <mn>1</mn> </msub> </msubsup> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msubsup> <mi>&alpha;</mi> <mi>p</mi> <msub> <mi>S</mi> <mn>1</mn> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>&alpha;</mi> <mn>1</mn> <msub> <mi>S</mi> <mi>n</mi> </msub> </msubsup> </mtd> <mtd> <mo>...</mo> </mtd> <mtd> <msubsup> <mi>&alpha;</mi> <mi>p</mi> <msub> <mi>S</mi> <mi>n</mi> </msub> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>.</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow>

Constraints：

First, the multifrequency dual polarization passive microwave remote sensing data for being tested area is obtained, according to the spatial coherence of microwave pixel, with Position (x to be decomposed₀,y₀) place passive microwave mixed pixel centered on, the passive microwave for choosing 8 neighborhood windows around it mixes The passive microwave mixed pixel for closing 4 neighborhood windows of pixel and/or surrounding is solved as equation group input data, and corresponding two The scheme of kind of input data is：

(I) 8 neighborhood window data inputs：By containing position (x to be decomposed₀,y₀) 9 pixels including pixel the bright temperature value of mixingSubstitute into formula (12)；If the result being calculated meets formula (15) and formula (16), obtain and rationally solve, divided Classification data T corresponding to each underlying surface after solution₁, obtain error information E₁；

(II) 4 neighborhood window data inputs：By containing position (x to be decomposed₀,y₀) 5 pixels including pixel the bright temperature value of mixingSubstitute into formula (12)；If the result being calculated meets formula (15) and formula (16), obtain and rationally solve, divided Classification data T corresponding to each underlying surface after solution₂, obtain error information E₂。

2. a kind of multifrequency dual polarization forest land accumulated snow passive microwave mixed pixel decomposition method according to claim 1, it is special Sign is, obtains the method for optimal result according to (I), the solving result of (II) two schemes after step 3), is specifically divided into four Kind situation：

Situation one：(I) scheme of selection, if solution is unreasonable, uses (II) scheme；If (II) Scheme Solving is still unreasonable, say The mixed pixel of bright input can not decompose, and retain the bright temperature value of preimage member；

Situation two：(I) scheme of selection, if solution is unreasonable, uses (II) scheme；If (II) Scheme Solving is reasonable, output group Divide bright temperature value T₂With error information E₂, final classification value is T₂, error information be E₂；

Situation three：(I) scheme of selection, if solving reasonable, output classification value T₁With error information E₁；Continue using (II) side Case, if (II) Scheme Solving is unreasonable, final classification value is T₁, error information be E₁；

Situation four：(I) scheme of selection, if solving reasonable, output classification value T₁With error information E₁；Continue using (II) side Case, if (II) Scheme Solving is reasonable, export classification value T₂With error information E₂；Respectively to E₁And E₂Matrix Calculating and be designated as E₁' and E₂', compare E₁' and E₂' size；If E₁' ＞ E₂', then final classification value is T₂, error information be E₂；If E₁' ＜ E₂', then final classification value is T₁, error information be E₁。

3. a kind of multifrequency dual polarization forest land accumulated snow passive microwave mixed pixel decomposition method according to claim 1, it is special Sign is that the passive microwave remote sensing data is by demarcation, atmospheric correction, geographical correction and standardization pretreatment.

4. a kind of method for establishing multifrequency dual polarization forest land accumulated snow passive microwave Pixel Unmixing Models, is comprised the following steps that：

First, passive microwave remote sensing data is that antenna for radiometer measures the bright temperature value of mixing on ground on satellite, and is radiated on satellite The bright temperature value of mixing that meter antenna measures ground is the integration of the actual bright gentle antenna gain function of earth's surface：

<mrow> <msubsup> <mi>T</mi> <mi>F</mi> <mi>M</mi> </msubsup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mo>&Integral;</mo> <mo>&Integral;</mo> <mover> <mi>G</mi> <mo>&OverBar;</mo> </mover> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <msubsup> <mi>T</mi> <mi>F</mi> <mi>C</mi> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mi>d</mi> <mi>x</mi> <mi>d</mi> <mi>y</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>

In formula (1),It is the bright temperature value of mixing of antenna for radiometer measurement,Be in the actual bright temperature value of ground location (x, y), Lower footnote F is passive microwave observation frequency range,It is antenna gain function, the defined formula of antenna gain function is：

In formula (2), a and b represent K frequencies and footmark corresponding to Ka frequencies in passive microwave data respectively；

The true bright temperature value in groundIt is the linear superposition of n kinds underlying surface classification value in its footmark：

<mrow> <msubsup> <mi>T</mi> <mi>F</mi> <mi>C</mi> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msup> <mi>L</mi> <mi>i</mi> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <msub> <mi>T</mi> <mi>i</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>

Wherein T_iRepresent different underlying surface classification value, LⁱRepresent i-th of underlying surface grouped data；

Secondly because the frequency range of used passive microwave is K frequencies and Ka frequencies, each frequency range includes horizontal polarization and vertical pole Change two ways；Then the forest land accumulated snow passive microwave at ground location (x, y) place can be expressed as in the radiation value of K frequencies and Ka frequencies：

<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <mi>C</mi> </msubsup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <mi>C</mi> </msubsup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <mi>C</mi> </msubsup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <mi>C</mi> </msubsup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <msup> <mi>L</mi> <mi>B</mi> </msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <mi>B</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <mi>B</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <mi>B</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <mi>B</mi> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <msup> <mi>L</mi> <mi>N</mi> </msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <mi>N</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <mi>N</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <mi>N</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <mi>N</mi> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <msup> <mi>L</mi> <msub> <mi>S</mi> <mn>1</mn> </msub> </msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <msup> <mi>L</mi> <msub> <mi>S</mi> <mn>2</mn> </msub> </msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <msub> <mi>S</mi> <mn>2</mn> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <msub> <mi>S</mi> <mn>2</mn> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <msub> <mi>S</mi> <mn>2</mn> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <msub> <mi>S</mi> <mn>2</mn> </msub> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mn>...</mn> <mo>+</mo> <msup> <mi>L</mi> <msub> <mi>S</mi> <mi>n</mi> </msub> </msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mi>&epsiv;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>

T represents bright temperature value in formula (4)；L represents underlying surface grouped data, if the underlying surface be present in (x, y) position, then L (x, y)= 1；Otherwise, L (x, y)=0；Upper footnote C, B, N and S₁,S₂,…S_nUnderlying surface land type, the broad-leaf forest of the position are represented respectively Underlying surface land type, coniferous forest underlying surface land type and other underlying surface land types that may be present；Lower footnote K and Ka Passive microwave K and Ka frequency range is represented respectively, wherein lower footnote H and V represent horizontal polarization and vertical polarization respectively；ε represents residual error Amount；L^B(x,y)、L^N(x, y) andRepresent respectively ground location (x, y) place broad-leaf forest, coniferous forest and its The distribution of its underlying surface, and have：

The bright temperature value of mixing of antenna for radiometer measurementAntenna gain function can be regarded asWith the actual underlying surface type in ground The convolution of bright temperature, formula (4), which is substituted into formula (1), then to be had：

<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <mi>M</mi> </msubsup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <mi>M</mi> </msubsup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <mi>M</mi> </msubsup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <mi>M</mi> </msubsup> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mo>&Integral;</mo> <mo>&Integral;</mo> <mover> <mi>G</mi> <mo>&OverBar;</mo> </mover> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> <mrow> <mo>(</mo> <msup> <mi>L</mi> <mi>B</mi> </msup> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <mi>B</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <mi>B</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <mi>B</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <mi>B</mi> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <msup> <mi>L</mi> <mi>N</mi> </msup> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <mi>N</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <mi>N</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <mi>N</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <mi>N</mi> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <msup> <mi>L</mi> <msub> <mi>S</mi> <mn>1</mn> </msub> </msup> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mn>...</mn> <mo>+</mo> <msup> <mi>L</mi> <msub> <mi>S</mi> <mi>n</mi> </msub> </msup> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> <mo>)</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>H</mi> </mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>V</mi> </mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>H</mi> </mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>T</mi> <mrow> <mi>K</mi> <mi>a</mi> <mi>V</mi> </mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>)</mo> </mrow> <mi>d</mi> <mi>x</mi> <mi>d</mi> <mi>y</mi> <mo>+</mo> <mi>&epsiv;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msup> <mi>&alpha;</mi> <mi>C</mi> </msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mo>&Integral;</mo> <mo>&Integral;</mo> <mover> <mi>G</mi> <mo>&OverBar;</mo> </mover> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <msup> <mi>L</mi> <mi>C</mi> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mi>d</mi> <mi>x</mi> <mi>d</mi> <mi>y</mi> <mo>,</mo> <mi>C</mi> <mo>=</mo> <mi>B</mi> <mo>,</mo> <mi>N</mi> <mo>,</mo> <mi>S</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>

Formula (6) is deformed into using formula (7)：