Polarimetric Interferometry

Transcription

1 Matin Hellmann 1, Shane R. Cloude 1 Geman Aeospace Cente (DLR), VO-ST, Linde Höhe, D Köln, Gemany, matin.hellmann@dl.de Shane R Cloude, scloude@ieee.og Abstact This lectue pesents the ole of Polaimety in SAR Intefeomety. A geneal fomulation fo vecto wave intefeomety is pesented that includes conventional scala intefeomety pesented in the espective fome lectue as a special case. Based on this fomulation, the coheence optimization poblem can be solved to obtain the optimum scatteing mechanisms that lead to the best phase estimates. Compaison with conventional single-polaization estimates illustates the significant pocessing gains that ae possible if thee is access to full polaimetic intefeometic data. A compaison with conventional single-polaization pesented in fome lectues illustates the significant pocessing gains that ae possible if access to full polaimetic intefeometic data is possible. The stong polaization dependence of the coheence will be addessed and the analytical solution fo optimum polaization states that maximize the intefeometic coheence will be deived and applied to expeimental data. These impoved intefeogammes allow an impovement of the accuacy of deived DEM poducts. The intoduction of a new coheent decomposition theoem fo intefeometic applications based on the Singula value spectum of a 3 x 3 complex matix allows the decomposition of polaimetic intefeometic poblems into a set of coheent scatteing mechanisms. As a consequence, it is possible to geneate intefeogams elated to cetain independent scatteing mechanisms and extact the height diffeences between them. The limitation of this technique is the existence of independent scatteing mechanisms located at diffeent height positions. To explain the physical oigin of these mechanisms, a coheent electomagnetic scatteing model will be established which, additionally, can be used to establish the suitability of the decomposition algoithm fo solving the poblem of estimating the location of the effective scatteing cente, which is a citical point in the physical intepetation of intefeogams. Howeve, these intoduction of Polaimety in intefeometic pocessing equies that fully coheent polaimetic data must be collected in ode to sepaate the scatteing mechanisms.. In this connection polaimetic Diffeential Intefeomety will be consideed also. The phase diffeence between the optimum intefeogams obtained by the application of the algoithm on the SIR-C data tuned out to be stongly coelated with the actual foest height. This was a majo esult indicating the potential of the coheent combination of polaimety and intefeomety. Intoduction Polaimetic SAR Intefeomety was a fist step in the abatement of the scatteing ambiguity poblem in the height diection. By combining intefeometic and polaimetic techniques, it enables the sepaation of diffeent scatteing mechanisms within a esolution cell and at the same time, the estimation of the associated heights. Both, ada polaimety and ada intefeomety ae phase sensitive techniques. The use of polaimetic SAR data has been widely addessed in the last decade. The tight elation between natual media physical popeties and thei polaimetic featues leads to highly desciptive esults that can be intepeted by analyzing undelying scatteing mechanisms. Intefeometic data on the othe hand povide infomation concening the coheence of the scatteing mechanisms and can be used to etieve obseved media stuctues and complexity. The complementay aspect of polaimetic Hellmann, M.; Cloude, S.R. (007) Polaimetic Intefeomety. In Rada Polaimety and Intefeomety (pp ). Educational Notes RTO-EN-SET-081bis, Pape 8. Neuilly-su-Seine, Fance: RTO. Available fom: RTO-EN-SET-081bis 8-1

2 Repot Documentation Page Fom Appoved OMB No Public epoting buden fo the collection of infomation is estimated to aveage 1 hou pe esponse, including the time fo eviewing instuctions, seaching existing data souces, gatheing and maintaining the data needed, and completing and eviewing the collection of infomation. Send comments egading this buden estimate o any othe aspect of this collection of infomation, including suggestions fo educing this buden, to Washington Headquates Sevices, Diectoate fo Infomation Opeations and Repots, 115 Jeffeson Davis Highway, Suite 104, Alington VA Respondents should be awae that notwithstanding any othe povision of law, no peson shall be subject to a penalty fo failing to comply with a collection of infomation if it does not display a cuently valid OMB contol numbe. 1. REPORT DATE 01 FEB TITLE AND SUBTITLE Polaimetic Intefeomety. REPORT TYPE N/A 3. DATES COVERED - 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Geman Aeospace Cente (DLR), VO-ST, Linde Höhe, D Köln, Gemany 8. PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR S ACRONYM(S) 1. DISTRIBUTION/AVAILABILITY STATEMENT Appoved fo public elease, distibution unlimited 13. SUPPLEMENTARY NOTES See also ADM , The oiginal document contains colo images. 14. ABSTRACT 15. SUBJECT TERMS 11. SPONSOR/MONITOR S REPORT NUMBER(S) 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT UU a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified 18. NUMBER OF PAGES 14 19a. NAME OF RESPONSIBLE PERSON Standad Fom 98 (Rev. 8-98) Pescibed by ANSI Std Z39-18

3 and intefeometic infomation leads to a combination of both appoaches. In Polaimetic- Intefeometic Synthetic Apetue Rada (POL-INSAR) Imaging it is possible to ecove textual and spatial popeties simultaneously. This includes the extaction of Digital Elevation Maps (DEM) fom eithe fully Polaimetic (scatteing matix) o intefeometic SAR image data takes with the additional benefit of obtaining co-egisteed thee-dimensional POL-IN-DEM infomation. Ove the next few yeas seveal fee-flying emote sensing satellites will be deployed in obit, poviding the intenational scientific, commecial and militay communities with a wealth of new data. Many of these will cay advanced multi-channel imaging adas designed to combine vaious levels of polaisation divesity with ada intefeomety. Polaimetic intefeomety has poved to be a valuable tool fo many applications. One is emote sensing whee it has been shown in seveal ecent publications that by using intefeogams in multiple polaisation channels, estimation of vegetation height, undelying gound topogaphy and mean extinction is possible [11,14,3]. Polaimetic Synthetic Apetue Rada intefeomety (POLInSAR) can also be used to enhance the detection of militay tagets hidden beneath foliage. The key idea is to note that fo andom volume scatteing the intefeometic coheence is invaiant to changes in wave polaisation. On the othe hand, in the pesence of a taget the coheence changes with polaisation. It can be shown that unde geneal symmety constaints this change is linea in the complex coheence plane. These obsevations can be used to devise a filte to suppess the etuns fom foliage clutte while maintaining the signal fom hidden tagets. Rada Polaimety As shown in a pevious lectue, an impotant extension to single-channel SAR emote sensing is the utilisation of polaised waves. A polaimetic SAR system measues the electic field, backscatteed by the scene, including its polaisation state. The inteaction of the tansmitted wave with a scatteing object tansfoms its polaisation. One special chaacteistic of SAR polaimety is that it allows a discimination of diffeent types of scatteing mechanisms. This becomes possible because the obseved polaimetic signatues depend stongly on the actual scatteing pocess. In compaison to conventional single-channel SAR, the inclusion of SAR polaimety consequently can lead to a significant impovement in the quality of data analysis. Cetain polaimetic scatteing models even povide a diect physical intepetation of the scatteing pocess, allowing an estimation of physical gound paametes like soil moistue and suface oughness [11], as well as unsupevised classification methods with automatic identification of diffeent scattee chaacteistics and taget types [4,5]. SAR polaimety additionally offes some limited capability fo sepaating multiple scatteing mechanisms occuing inside the same esolution cell and can be deemed as a fist step in esolving the ambiguous scatteing poblem in SAR, as mentioned above. With polaimetic decomposition techniques a eceived signal can be split into a sum of thee scatteing contibutions with othogonal polaimetic signatues. This can be used fo extacting the coesponding taget types in the image, even in the case that they ae occuing supeimposed. Also, if a signal is distubed by undesied othogonal contibutions, in this way the elevant components can be extacted. 8 - RTO-EN-SET-081bis

4 The Phase In ada polaimety [1] we analyze the shape of the tansmit (and eceive) polaisation ellipse, as shown schematically in Figue 1a, fo the puposes of impoved infomation extaction. Figue 1a shows the spatial helix esulting fom a combination of hoizontal (H, in geen) and vetical (V, in blue) tansmitted components. y $ x $ E, ( z t) $y E, ( z t) 0 0 x $ z 0 z $ Figue 1a : The Polaisation Ellipse and Spatial helix decomposed into othogonal components x (hoizontal H) and y (vetical V) [Pottie, lectues on polaimety] By contolling the elative amplitudes we can otate the polaisation fom H though 45 degees to V. Howeve, by adjusting the elative timing (phase) of the blue and geen components we can also adjust the shape of the ellipse as shown in Figue 1b. It is this combined amplitude and phase dimension that leads to inceased infomation content in emote sensing applications, since the level of scatteing we obseve fom natual teain depends on the shape of this ellipse. [1,,3,4,5] Figue 1b : The Polaisation Ellipse RTO-EN-SET-081bis 8-3

5 To epesent this combined amplitude and phase contol mathematically, we descibe the wave using a pai of complex numbes, e x and e y as shown in equation 1. The phase diffeence i.e. φ = ag(e x e y ) then contols the shape of the ellipse, with linea polaisations defined by φ = 0. Note that the ellipse is actually a dynamic quantity, being the time locus of the helix in a fixed spatial plane. Consequently the locus can move clockwise o counte-clockwise (when viewed in the z diection), coesponding to what ae temed left and ight-handed polaisations espectively. The set of all possible left and ight handed ellipses can then be conveniently mapped onto the nothen and southen hemisphees of the Poincaé sphee. Figue 1c [1,6] $z CR ELLIP. POLARISATION LEFT HAND LINEAR POLARISATION -45 H $x 0 V +45 y $ ELLIP. POLARISATION RIGHT HAND CL Figue 1c : The Poincaé sphee Futhemoe, ou ability to extact quantitative infomation ests on the stability of this phase o the obustness of the spatial helix to small time and spatial shifts. A geneic way to define the stability of this helix is to use the wave coheency matix [J], taken at a point in space with position vecto. and fomed as an aveage of all possible complex poducts between e x and e y as shown in equation 1, [6] e x x ˆ + e y y ˆ E = e x J e y []= E.E T = e ()e x x() e x ()e y () ()e x () e y ()e y () e y (1) 8-4 RTO-EN-SET-081bis

6 As its name suggests, this matix allows us to calculate not only the wave intensity (fom the diagonal components) but also the coheence, which is a measue of the phase stability of the wave, as defined in equation [7]. ex () e y () () e (). e () e () ~ γ,0 ~ xy = γ xy 1 () e x x y y A key benefit of employing atios such as equation is that absolute amplitude tems cancel, so emoving some of the stuctual dependence in scatteing fom andom media. It is this obsevation that shifts inteest in polaimety towads the study of atios as potentially moe obust indicatos of physical stuctue (see examples in table I). In active micowave sensing we assume ~ γ xy =1 fo the tansmitted wave and hence the tansmitted spatial helix is vey stable. Howeve, when the wave is scatteed o eflected fom natual media, its phase and amplitude will in geneal be modified (as shown schematically in Figue a). This pocess again must be descibed by a set of complex numbes, this time by a set of fou, being the elements of the coheent scatteing matix [S] defined as shown in equation 3. This matix chaacteises all possible phase and amplitude changes due to copola (diagonal elements) and coss-pola (off diagonal) scatteing. In pactice, fo the common case of backscatte, the ecipocity theoem fo electomagnetic waves educes this set to thee complex numbes, as the coss-polaisation tems ae equal S HV = S VH. Note that while this is widely tue, thee ae a few special but impotant cases whee it beaks down, as fo example in low fequency adio wave popagation though the ionosphee, whee the eaths magnetic field lines beak this ecipocity symmety and as a esult the coss polaisation tems ae no longe equal. This obsevation can be used to calibate the effects of Faaday otation due to tansionospheic popagation, an impotant issue fo the deployment of low fequency space-bone adas [8]. E s = e ik S S HH HV T [ S]. E [ S] = k = S [ C] = k. k VH S S VV ecipocity SHV = SVH S S HH VV HV (3) One key idea in polaimety is that if we know all fou of these [S] matix elements then we can calculate the phase stability of the scatteed signal fo abitay incident ellipse, using a 3 x3 covaiance matix [C] as shown in equation 3. In this way we don t have to actually change the shape of the tansmit ellipse (which would call fo contol of the antenna and micowave electonics) but can simulate the same effect off-line in the pocessing stages. Fo this eason thee has been a lot of inteest in the development of micowave switching systems that ae capable of measuing all fou elements of [S] (the simplest is to switch each tansmit pulse between X and Y othogonal polaisations with simultaneous eception of the X and Y components). Note that one impotant step is to calibate system distotion effects due to cosstalk (which causes poblems with estimation of the off-diagonal elements of [S]) and channel imbalance due to phase and amplitude distotions of the ada system itself. The development of obust calibation pocedues has been a key enabling step in the quantitative exploitation of this technology [9]. Such systems ae called quadpol as they measue 4 complex numbes fo each pixel in the image and allow the use to exploe the whole Poincaé sphee. Thee ae cuently seveal matue aibone quadpol ada sensos with such a capability, but significantly thee will soon be a new geneation of fee-flying satellite adas opeating in this mode. The Euopean Teasa-X/L, Japanese ALOS-PALSAR and Canadian Radasat- ae impotant examples. The main question then becomes, how can we find the best polaisation combination to deive infomation RTO-EN-SET-081bis 8-5

7 poducts exploiting the scatteing of waves fom sufaces and vegetation? To answe this we must look moe caefully at equation and the whole issue of coheence. Coheence and Entopy To calculate polaimetic coheence, we fist choose a pai of polaisations x and y, then measue the (complex) components of the signal in these two channels and estimate the coheence by aveaging. Howeve, even fo a fixed wave, the coheence obtained with this method will depend on the choice of ou efeence pai x and y (e.g. choosing x=y will give a coheence of 1, while less obvious but moe impotant is the idea that fo evey wave we can choose an othogonal pai x and y so that the coheence is zeo). This goes against the idea that the spatial helix is somehow independent of the coodinates we use to epesent it, and that consequently we should be able to descibe its stability in coodinate invaiant tems. One way to do this is to descibe the helix stability using a genealised coheence o entopy (anothe popula way is to use the degee of polaisation [6]). The wave entopy is fomally defined fom the atio of eigenvalues of [J] (see equation 4) and has a value of 0 when the helix is pefectly stable and 1 when it becomes noise like [,4,6]. ([ ]) 0 H w = p i log p i 1, p i = λ i J i=1 λ (4) By extension, we can also descibe the loss of helix stability afte scatteing by the entopy of the 3 x3 covaiance matix [C] in equation 3, as defined in equation 5 [,4] ([ ]) 3 0 H s = p i log 3 p i 1, p i = λ i C i=1 λ (5) It is impotant to ealize that this scatteing entopy is chaacteistic of the scatteing medium itself. Fo example, fo low fequency volume scatteing fom a cloud of ellipsoidal paticles of dielectic constant ε and axial atio m, the nomalized eigenvalues of [C] can be evaluated explicitly as shown in equation 6 [,4,16,17] m = > 1 1 polate paticles spheical paticles < 1 oblate paticles R = mε + m + ε + 1 λ1 = R + 6R + 7 λ = ( R 1) λ3 = ( R 1) (6) Fo spheical paticles (R =1) this leads to zeo entopy but fo a cloud of wet dipoles (m and ε lage) the entopy ises to Hence a measuement of entopy elates to infomation about composition of the volume. Impotantly, we can estimate scatteing entopy numeically on a pixel-by-pixel basis fom quadpol ada imaging data. Figue b shows an example of the entopy o phase stability of a mixed scene, being the Obepfaffenhofen aea as collected by the DLR L-Band ESAR system. We note that ove non-vegetated sufaces (left bottom cone in figue b) the entopy is low and hence the scatteed wave helix is vey stable fo all types of tansmit polaisation. This can be exploited fo quantitative moistue and oughness estimation of non-vegetated land sufaces by choosing appopiate obust atios of scatteing elements as shown fo example in table I. [10,11,1,13,14,15] 8-6 RTO-EN-SET-081bis

8 The uban aeas (uppe ight cone) in figue show modeate entopy, but the wost case aises fo vegetation (lowe ight cone). Hee we see high entopy due to volume scatteing by the andom components of the vegetation cove (as in equation 6). These obsevations ae independent of the actual scene consideed and hence have been suggested by seveal authos as suitable fo obust unsupevised classification of land cove [3,4,5,13]. While useful fo classification and limited composition studies, such high entopy fo vegetation cove esticts ou ability to fully exploit polaisation fo quantitative paamete estimation. Yet vegetation cove is of pime impotance in emote sensing applications. Somehow, in ode to poceed, we have to find a way to educe the entopy. Impotantly this can be achieved by combining polaimety with intefeomety, to fom the new topic of imaging polaimetic intefeomety o POLInSAR as we now show. Figue a: Depolaisation and Entopy: Figue b : Entopy image L-Band Obepfaffenhofen, Gemany DLR ESAR Data RTO-EN-SET-081bis 8-7

9 Contolling Entopy : Volume Decoelation in Rada Intefeomety Rada intefeomety employs spatial sepaation by a baseline vecto b of multiple sensos (fo singlepass) o a single senso at multiple times (fo epeat-pass) [18]. It then uses phase diffeence as a poxy fo elevation, enabling detemination of scattee height, hence leading to poducts such as high esolution digital elevation model (DEM) geneation. Again howeve, the accuacy of this pocess is govened by phase stability o coheence. In this case we can define a coheency matix as shown in equation 7 [] J x = p x p x ()p x () p x ()p x ( + b) ()p x ( + b) p x ( + b)p x ( + b) (7) whee x coesponds to a single selected polaisation channel. The pesence of vegetation is now modelled as a finite bounded vetical andom distibution of scattees with a spatial weighting to account fo the fact that scattees deepe in the volume will have a smalle influence due to wave extinction. With this model, the coheence of vegetation can be expessed as shown in equation 8 [19,] γ v e iφ ( z o ) = p x p x ()p x ( + b) ()p x (). p x + b = e iφ ( z o) p 1 e phv 1 p e p 1 h v 1 p 1 = σ cosθ ( )p x + b ( ) = eiφ ( zo ) σ 1 e iφ (z o ) p = p 1 + ik z, cosθ o (e σ 1 h v / cosθ o 1) e ik z z' e k z = 4πΔθ λsinθ h v 0 σ 1 z' cosθ o dz' 4πB n λrsinθ (8) whee B n is the nomal component of the baseline to the line of sight. Thee ae two key featues of this model: Coheence (and theefoe entopy) can now be contolled by selecting the baseline B n. The intefeometic coheence is independent of x i.e. of polaisation The fist means that, unlike in polaimety, we can now design the senso to contol the obseved entopy of vegetation scatteing (contast equations 6 and 8). Howeve, the second seems to indicate that we do not need polaisation divesity, as equation 8 does not change with x. Why then do we need to conside POLInSAR? The answe to this appaent contadiction is hidden in equation 8 itself. We see that the coheence is a function of seveal paametes, the unknown height of the vegetation, the unknown wave extinction and the unknown gound topogaphic phase. It follows that one channel of intefeomety by itself cannot be used fo unambiguous paamete etieval. The situation is futhe complicated by the fact that fo micowaves the extinction can be elatively small and hence thee can be penetation of vegetation ight down to the undelying suface. This equies us to conside combined suface and volume scatteing, so focing us to modify equation 8 to at least a two-laye model as shown in equation 9 [19,0,1,]. γ x = e iφ z γ o v + μ x 1+ μ x (9) ( ) whee μ x is the atio of suface-to-volume scatteing, which changes with fequency, vegetation density and suface conditions. Howeve, it is now that polaisation divesity helps, as fom figue b we see that suface scatteing has low entopy and hence we can contol its influence in 9 by changing 8-8 RTO-EN-SET-081bis

10 x at the same time as leaving the volume coheence unchanged. Consequently by using POLInSAR we can incease the numbe of obsevations faste than the numbe of unknowns and hence achieve paamete estimation with a coheence o entopy unde ou contol. This is one eason why thee is such an inteest in developing POLInSAR sensos fo vegetation mapping [,3,4]. Seveal futhe examples can be found as pat of the poceedings of a ecent ESA funded wokshop focussing on this topic ( Figue 3 shows an example POLInSAR poduct, obtained using the L-band aibone E-SAR senso opeated by DLR in Gemany. Hee we show a ada-deived quantitative tee height estimation ovelaid on a ada-deived DEM. It uses the model of equation 9 with polaisation divesity ove x to isolate the height h v and φ(z 0 ) dependence and povide a map of tee height ove the mountainous teain. Quantitative compaisons with in-situ measuements indicate an accuacy of height estimation aound 10%. [1,,4]. While tee height is itself a useful poduct, it can also povide the basis fo vaious impotant seconday poducts. Fo example, in Figue 4 we show a foest biomass map deived using the height data in Figue 3 coupled to allometic equations deived fom foesty tables fo this egion [4]. In the uppe Figue we also show a conventional SAR image of the scene, which displays Figue 3: Tee Height and Topogaphy Estimated using L-Band DLR E-SAR Polaimetic Intefeometic Data none of the impotant foest stuctual infomation seen in the height/biomass poducts. This nicely illustates the potential infomation gain obtained by using POLInSAR sensos fo vegetation applications. RTO-EN-SET-081bis 8-9

11 Polaimetic Intefeomety Figue 4 : HH RCS image (uppe) and ada deived tee height/biomass map fo the same scene (lowe) (see efeence [4]) Conclusions and Futue Developments In this teatment we have developed as a theme the impotance of multi-channel phase in ada emote sensing and used it to suppot the idea of combining polaisation divesity with intefeomety in futue ada sensos. Key to success is the genealised coheence o entopy and key to obustness the development of physical models fo the inteaction of polaised waves with natual sufaces. We have concentated on one impotant example, namely tee height and biomass estimation, but thee ae many othe application aeas whee this technology is being consideed. Table I povides a selective suvey of diffeent geo-physical paametes and examples of the types of algoithms cuently being developed. We can see that polaimetic and/o intefeometic phase appeas in evey aea. This table 8-10 RTO-EN-SET-081bis

12 povides a snapshot in time, each aea is ongoing in eseach and development and exciting futue technology innovations such as bistatic ada and satellite ada constellations will equie paallel impovements in ou undestanding of the inteaction of polaised waves with natual media in ode to fully exploit the scientific and commecial potential of ada in emote sensing. By using moe than polaimetic data sets polaimetic intefeomety can be extend to moe complex appoaches. One of possible appoache is diffeential polaimetic intefeomety. By using 3 o moe tempoal sepaated data sets it could be possible to enhance the potential of conventional diffeential intefeomety by using the polaimetic infomation in ode to analyse the changes in scatteing pocesses ove time. POL-IN-SAR imaging, when applied to Repeat-Pass Image Ovelay Intefeomety, povides diffeential backgound validation and measuement, stess assessment, and envionmental stess-change monitoing capabilities. Anothe appoach cuently unde investigation is polaimetic SAR tomogaphy, which is the extension of conventional two-dimensional SAR imaging pinciple to thee dimensions. A eal theedimensional imaging of a scene is achieved by the fomation of an additional synthetic apetue in elevation by a coheent cobination of images acquied fom seveal paallel flight tacks. It can be seen as a diect appoach to esolve the SAR scatteing ambiguity poblems. The intoduction of tomogaphic SAR offes the possibility of a diect localisation and identification of all scatteing contibutions in a volume. This geatly extends the potential of SAR, paticulaly fo the analysis of volume stuctues like fo example foests as shown in Figue 5. Figue 5 Tomogaphic slice geneated fom 13 paallel flight tacks with a mutual distance of 0m. Senso: DLR ESAR in L-band. Scene: Onbepfaffenhofen, Gemany. (Reigbe[5]) Acknowledgements Thank is due to Pofesso Eic Pottie of the Univesity of Rennes, Fance, and D. Andeas Reigbe fom the Technical Univesity of Belin fo thei valuable assistance in composing this tutoial. RTO-EN-SET-081bis 8-11

13 Poduct Rada Paamete Polaimetic and Intefeometic Measuement Paametes Bae Suface Roughness Bae Suface Moistue 0 R(ks) 1 s=ms oughness k=wavenumbe=π/λ 0 M( θ,ε ) 1 θ = angle of incidence ε dielectic constant tanω tanβ = sinφ cosφ tanγ R = M = S HH S VV S HH S VV S HH S VV S HH + S VV 4 S HV + 4 S HV + 4 S HV Suface Re (S Slope φ = tan 1 HH S VV ).S HV 4 S HV S HH + S VV tanγ = ange slope φ + π π if φ π 4 β = tanω = azimuth slope 4 4 φ + π π 4 if φ > π π 4 φ = ada look angle 4 Souce Ref. 10,11,15 10,11,15 1,14 Tue Gound Topogaphy z o = z ef + ˆ φ k z k z = 4πΔθ λsinθ 4πB n λrsinθ ˆ φ = ag( γ HH VV γ HV ( 1 L)) A = γ HV 1, B = Re(( γ HH VV γ HV ). C = γ HH VV γ HV AL + BL + C = 0 ˆ L γ HV ), 1,,3 Vegetation Componen t Stuctue Vegetation Height and Extinction > 1 polate paticles m = 1 spheical paticles <1 oblate paticles mε + P = m + ε + 1 h v = top height in m σ = mean extinction (m -1 ) θ = angle of incidence ˆ φ = gound topogaphic phase (see above) V = 4 S HV S HH + S VV a = (1 V ),b = (1+ 3V ),c = (1 7V ) ap + bp + c = 0 ˆ P min L 1 = γ HV e i ˆ φ p e p1hv h v,σ p 1 e ph v p = σ whee cosθ p 1 = p + ik z 1 1 4,16,17 0,1, Table I : Examples of Geophysical Paamete Estimation using Rada Polaimety and Intefeomety 8-1 RTO-EN-SET-081bis

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