NUMERICAL MODELING OF MICROWAVE EMISSION FROM IRREGULAR LAYER LUIS MANUEL CAMACHO-VELAZQUEZ. Presented to the Faculty of the Graduate School of

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1 NUMERICAL MODELING OF MICROWAVE EMISSION FROM IRREGULAR LAYER by LUIS MANUEL CAMACHO-VELAZQUEZ Preeted to the Faculty of the Graduate School of The Uverty of Tea at Arlgto Partal Fulfllmet of the Requremet for the Degree of DOCTOR OF PHILOSOPHY THE UNIVERSITY OF TEXAS AT ARLINGTON December 0

2 Copyrght by Lu Mauel Camacho-Velazquez 0 All Rght Reerved

3 ACKNOWLEDGEMENTS I would le to epre my cere apprecato to my upervg profeor Dr. Sabu Tuata for h gudace atace ad tmulatg dcuo throughout my Doctor of Phloophy degree tude. My ucce th Ph.D. program largely due to h tme ecouragemet ad practcal ght for ot oly the academc actvte but alo peroal ue. I would alo le to tha Dr. Mgyu Lu for h advce regardg the umercal aaly apect my reearch. I would alo le to tha the member of my dertato commttee Dr. Joatha Bredow Dr. Tucay Atou ad Dr. Laureao Hoyo for ther uggeto ad for allocatg ther precou tme partcpatg my dertato defee eamato. Alo pecal tha to the Hgh Performace Computg ytem at the Uverty of Tea at Arlgto for provdg computatoal reource for the umercal aaly. I am deeply thaful to the Uverdad Autóoma de Nuevo Leó Meco for the upport to carry out graduate tude at UTA. Epecally to Joé A. Gozález Trevño Joé L. Catllo Ocaña Cátulo E. Vela Vllarreal Rogelo G. Garza Rvera Eteba Báez Vllarreal ad Eugeo López Guerrero. Furthermore I would le to hare the atfacto of th degree wth my fred Joé Mrele Jr. Humberto Ochoa ad Suma K. Guala for ther dcuo help ad geeroty. Fally I would le to epre my love ad apprecato to my wfe Fa ad our chldre Etefaa Xmea ad Lu for ther patece udertadg ad ucodtoal upport. I alo tha my paret Joé Delfo Camacho Nava from whom I foud out that learg a lfetme proce ad Luva Velázquez de Camacho for her prayer ad love. Lat

4 but ot leat I wat to tha my brother ter ad famly--law for ther ecouragemet. I dedcate th wor to all of them. November v

5 ABSTRACT NUMERICAL MODELING OF MICROWAVE EMISSION FROM IRREGULAR LAYER Lu Mauel Camacho-Velazquez PhD. The Uverty of Tea at Arlgto 0 Supervg Profeor: Sabu Tuata A umercal approach to model the mcrowave emo from a rregular layer developed. The phae matr of the layer compoet a well a the emo charactertc are etmated ug a umercal approach amely the Fte-Dfferece Tme-Doma (FDTD) method. The phae matr the tegrated to a layer model that accout for the catterg betwee the layer ad groud ug the radatve trafer theory. The effect of terface roughe o emo are corporated to the model through the urface phae matrce whch are computed ug the Itegral Equato Model (IEM). The valdty of the phae matrce computato teted agat theoretcal model. Smlarly the umercal model valdated by comparg t reult wth feld meauremet. For th purpoe the compoet of the rregular layer are choe to be cor tal repreeted by vertcal delectrc cylder. The bottom half-pace bottom choe to be ol wth rough urface. The mcrowave emo model predcto are the compared to thoe feld meauremet howg a good agreemet. v

6 TABLE OF CONTENTS ACKNOWLEDGEMENTS... ABSTRACT... v LIST OF ILLUSTRATIONS... LIST OF TABLES... v Chapter Page. INTRODUCTION.... Remote eg.... Mcrowave frequecy rage for emo from ol Clafcato of approache to model emo ad catterg from vegetato caope Curret wor Approach ued tudy RADIATIVE TRANSFER FORMULATION Stoe parameter.... Phae fucto ad phae matr Phae matr of rough urface Radatve trafer equato Scatterg from a homogeeou layer wth rregular boudare Emo model for a homogeeou layer Formulato of urface catterg matrce ug the Itegral Equato method... 9 v

7 3. FINITE DIFFERENCE TIME DOMAIN (FDTD) METHOD Itroducto to the Fte Dfferece Tme Doma (FDTD) method Mawell equato three dmeo The Yee algorthm Fte dfferece ad otato Fte dfferece epreo for Mawell equato three dmeo NUMERICAL COMPUTATION OF EMISSION FROM FINITE-SIZE OBJECTS Emo theory Valdato of FDTD reult by comparo wth aalytcal reult ug Me ere oluto Valdato of FDTD reult by comparo wth method of momet (MoM) reult Cro ecto computato ug FDTD ad Me ere oluto Computato of emvty from fte ze obect Itroducto Emo from fte-ze obect Numercal modelg approach Numercal computato of emvty from lole obect Numercal computato of emvty of loy delectrc obect Reult ad aaly MODELING OF MICROWAVE EMISSION FROM IRREGULAR LAYER Itroducto v

8 5. Cotructo of phae matr for a crcular cylder Cotructo of phae matr of cor tal repreeted by cylder Valdato of phae matrce for eergy coervato Verfcato of computato of bacward ad forward phae matrce Emo catterg mecham Determato of effectve permttvty of layer Valdato of model by comparo wth feld meauremet CONCLUSIONS AND RECOMMENDATIONS Summary ad cocluo Future wor APPENDIX A. LIST OF ABBREVIATIONS REFERENCES... 0 BIOGRAPHICAL INFORMATION v

9 LIST OF ILLUSTRATIONS Fgure Page. Geometry for pecfc tety ad the chage tety I Scatterg from a homogeeou layer above a homogeeou half pace Poto of feld compoet. The E r -compoet are the mddle of edge ad the H r -compoet are the ceter of the face Leapfrog tme tegrato Starcaed phere cubcal Yee cell The brghte temperature of a obect related to t phycal temperature by t emvty Cofgurato for mulato of plae wave mpgg o a delectrc phere Comparo of cattered feld value obtaed by ug FDTD ad Me ere oluto. (a) E ad E (b) H ad H (ormalzed to θ φ θ H r c ) Geometry ued to compute catterg from a cylder Fgure reproduced from [47] Geeral catterg patter of a cylder Fgure reproduced from [47] Normalzed btatc catterg coeffcet v. catterg agle for fte delectrc cylder computed wth FDTD ad MoM. The parameter of the cylder are L =0.0λ0 a = 0.4λ0 o ε c = 8 6 ad cdet agle θ = 0 (top) o ad θ = 80 (bottom) Comparo of aborpto cro ecto of a 4 - cm dameter phere ug aalytcal oluto ad FDTD mulato. The properte φ

10 of the phere are ε r = 4. 0 ad σ = 0.S / m. The Yee cell = 0. 00m ( λ ) Comparo of aborpto cro ecto of a 4 - cm dameter phere ug aalytcal oluto ad FDTD mulato. The properte of the phere are ε r = 4. 0 ad σ = 0.S / m. The Yee cell = m ( λ ) Comparo of aborpto cro ecto of a 4 - cm dameter phere ug aalytcal oluto ad FDTD mulato. The properte of the phere are ε r =0. 0 ad σ = 0.5S / m The Yee cell = m ( λ ) Comparo of aborpto cro ecto of a 4 - cm dameter phere ug aalytcal oluto ad FDTD mulato. The properte of the phere ε r = 4. 0 ad σ = 0.5S / m. The Yee cell = 0. 00m ( λ ) Comparo of aborpto cro ecto of a 4 - cm dameter phere ug aalytcal oluto ad FDTD mulato. The properte of the phere are ε r =0. 0 ad σ = 0.S / m. The Yee cell = m ( λ ) Comparo of aborpto cro ecto of a 4 - cm dameter phere ug aalytcal oluto ad FDTD mulato. The properte of the phere are ε r =0. 0 ad σ = 0.S / m. The Yee cell = m ( λ ) Vrtual eclog urface ued for computato of cro ecto. For aborpto cro ecto the eclog urface geerated the FDTD total feld rego whle for the catterg cro ecto the eclog urface geerated the cattered feld rego Vrtual eclog urface ued for computato of cro ecto ug the Gaua quadrature techque. For aborpto cro ecto the eclog urface ue the FDTD total feld value whle for the catterg cro ecto the eclog urface ue the cattered feld by ubtractg the cdet feld from the total feld Emvty of lole cylder wth repect to cdet agle whe varyg the cell ze from m to 0.005m. The dmeo of the cylder are: dameter = 0.05m ad heght = 0.05m. The delectrc properte are ε = 4. 0 ad σ = 0S / m. r 0 0

11 The frequecy of aaly 5 GHz. The polarzato vertcal Emvty of lole cylder wth repect to cdet agle whe varyg the cell ze from from m to 0.000m. The dmeo of the cylder are: dameter = 0.04m ad heght = 0.04m. The delectrc properte are ε r = 4. 0 ad σ = 0S / m. The frequecy of aaly 6 GHz. The polarzato vertcal Emvty of lole cylder wth repect to cdet agle whe varyg the cell ze from from m to 0.000m. The dmeo of the cylder are: dameter = 0.04m ad heght = 0.04m. The delectrc properte are ε r = 4. 0 ad σ = 0S / m. The frequecy of aaly 5 GHz. The polarzato vertcal Emvty of PEC cube wth repect to cdet agle whe varyg the cell ze from from m to 0.000m. The dmeo of the cube are: de = 0.04 m. The frequecy of aaly 6 GHz. The polarzato vertcal Emvty of PEC cube wth repect to cdet agle whe varyg the cell ze from from m to 0.000m. The dmeo of the cube are: de = 0.04m. The frequecy of aaly 5 GHz. The polarzato vertcal Emvty at 6 GHz for a phere ( r = λ0 ) wth ε r = 4. 0 ad σ = 0.0.5S / m Geometry ued for FDTD mulato of plae wave mpgg o ladme-le obect ( d =.4λ0 ad h = 0.8λ0 ) Emvty at 6 GHz for a crcular cylder wth ε r = 4. 0 ad σ = S / m for vertcal polarzato. Dmeo of the cylder are d =.4λ0(cm) ad h = 0.8λ0 (4cm) Emvty at 6 GHz for a crcular cylder wth ε r = 4. 0 ad σ = S / m for horzotal polarzato. Dmeo of the cylder are d =.4λ0(cm) ad h = 0.8λ0 (4cm) Geometry ued for FDTD mulato of plae wave mpgg o cylder ( d =.λ0(6cm) ad h = 3.6λ0(8cm ) )... 67

12 4.5 Emvty at 6 GHz for a crcular cylder wth ε r = 4. 0 ad σ = S / m for vertcal polarzato. Dmeo of the cylder are d =.λ0(6cm) ad h = 3.6λ0(8cm ) Emvty at 6 GHz for a crcular cylder wth ε r = 4. 0 ad σ = S / m for horzotal polarzato. Dmeo of the cylder are d =.λ0(6cm) ad h = 3.6λ0 (8 cm) Geometry ued for FDTD mulato of plae wave mpgg o cylder ( d =.6λ0(8cm) ad h =.6λ0 (8cm) ) Emvty at 6 GHz for a crcular cylder wth ε r = 4. 0 ad σ = S / m for vertcal polarzato. Dmeo of the cylder are d =.6λ0(8cm) ad h =.6λ0 (8cm) Geometry ued for FDTD mulato of plae wave mpgg o cylder ( d =.0λ0 (0 cm) ad h =.λ0 (6cm) ) Emvty at 6 GHz for a crcular cylder wth ε r = 4. 0 ad σ = S / m for vertcal polarzato. Dmeo of the cylder are d =.0λ0 (0 cm) ad h =.λ0 (6cm) Geometry ued for FDTD mulato of plae wave mpgg o cylder ( d =.λ0(6cm) ad h =.8λ0 (9cm) ) Emvty at 6 GHz for a crcular cylder wth ε r = 4. 0 ad σ = S / m for vertcal polarzato. Dmeo of the cylder are d =.λ0(6cm) ad h =.8λ0 (9cm) Geometry ued for FDTD mulato of plae wave mpgg o cylder ( d =.8λ0 (4 cm) ad h = 0.4λ0(cm) ) Emvty at 6 GHz for a crcular cylder wth ε r = 4. 0 ad σ = S / m for vertcal polarzato. Dmeo of the cylder are d =.8λ0 (4 cm) ad

13 h = 0.4λ0(cm) Geometry of the obect for computato of the phae matr Computatoal tructure of the phae matr for the frt two Stoe parameter Cofgurato for modelg of emo from ol the preece of vertcal cyldrcal vegetato compoet. Delectrc cylder are ued to model the vegetato compoet Geometry ued for FDTD mulato of plae wave mpgg o cylder Cor tal repreeted by delectrc cylder (a) Aborpto ad (b) catterg cro ecto umercally computed for a delectrc cylder repreetg a cor tal Eergy coervato tet reult for umercally-computed * * phae matrce S ad T Emo from urface wth a cover of a layer wth phae matrce characterzed a thoe of a phere. The phae matrce were obtaed ug Me-formulato feld FDTD-computed feld (phae matr ormalzed by aalytcal catterg cro ecto FDTD) ad FDTD-computed feld (phae matr ormalzed by catterg cro ecto ug method decrbed chapter 4 FDTD) (a) V-polarzato ad (b) H-polarzato Geeral geometry for modelg of emo from ol the preece of homogeeou layer Emo catterg mecham for upwellg ad dowwellg emo from the layer ad upwellg emo from the bottom half-pace Effectve reflectvty matr from the medum - medum 3 terface Comparo of model-predcted mcrowave emo ad feld meauremet (ARC) from ol wth a cover of cor tal

14 LIST OF TABLES Table Page 3. Locato of the dfferet feld the Yee cell Parameter of obect ued FDTD mulato of plae wave mpgg o delectrc phere Parameter of cdet plae wave ad cattered feld Parameter ued for comparo de computato of aborpto cro ecto of a 4-cm-dameter phere ug aalytcal oluto ad FDTD mulato Parameter ued for computato of emo from groud covered wth a homogeeou layer wth phae matrce mlar to thoe of a phere Parameter ued for model-predcted emo v

15 CHAPTER INTRODUCTION. Remote eg The term mcrowave remote eg ecompae the phyc of radowave propagato ad teracto wth materal meda cludg urface catterg volume catterg ad emo the mcrowave rego. Seor ued mcrowave remote eg are dvded to two group accordg to ther mode of operato: actve eor are thoe that provde ther ow ource of llumato ad therefore cota a tramtter ad a recever whle pave eor are mply recever that meaure the radato emaatg from the cee uder obervato. I pave mcrowave remote eg radato meaured (emo reflecto polarzato). From thee meauremet parameter uch a temperature radato ol moture ea urface alty ad ea urface wd are ferred. Actve eor are alo etve to uch parameter but actve trumet are typcally much larger requre larger atea ad ue more power. Some applcato of mcrowave to remote eg uch a catterg ad emo from agrcultural ol ow ad atmophere are gve []. Model have bee developed to characterze the emo from ol to terpret the oberved relato betwee radometer obervato ad ol parameter []. I fact eteve reearch ha bee carred out to etmate the emo from bare urface [3-5]. However the emo from ol may cae affected by a vegetato layer whch atteuate the ol emo ad add t ow cotrbuto to the lad urface emo. I order to accout for t

16 model have bee developed that corporate the effect of vegetato o ol mcrowave emo [6]. The problem of electromagetc catterg by vegetato ha receved much atteto recet year partcularly wth the deploymet of ar ad atellte-bor Sythetc Aperture Radar (SAR) trumet. It therefore epecally mportat to have depedable model to characterze the electromagetc behavor of vegetato. Numerou reearch wor have bee publhed to documet the mert of the propoed model ad may more have reported the reult of epermetal obervato; however may ureolved queto rema about the ature of the wave-target teracto proce both the caopy volume ad betwee the caopy elemet ad the uderlyg ol urface. For the mot part the applcablty of a gve model to a gve target type evaluated by comparg the model-calculated catterg coeffcet or brghte temperature to the correpodg meaured value. The model baed o certa aumpto ad a fucto of a et of electromagetc ad phycal parameter of the target from whch may are dffcult to meaure ad are therefore elected uch that the ft betwee the model ad obervato are optmzed. I geeral catterg cot of both urface catterg due to the boudary dcotute ad volume catterg due to homogeete the medum. Hece the modelg of catterg requre a approprate combato of urface-catterg ad volumecatterg method. I volume catterg two geeral approache have bee ued: the wave approach ad the tety approach alo referred to a radatve trafer approach [7 8]. The wave approach rele o Mawell equato ad ca be rgorouly formulated. However to obta ueful practcal reult ome appromato mut be made. Whe the volumecatterg problem formulated term of the averaged power or tety baed o the radatve trafer prcple [8-0] trog delectrc fluctuato ad certa type of multple catterg ca be cluded.

17 Surface catterg from a rough urface ha bee tuded etevely [3 4]. The problem of catterg from rough urface ha bee tuded ug low frequecy (Small Perturbato Method -- SPM) ad hgh frequecy (Krchhoff) appromato. For rough urface wth dfferet cale of roughe two-cale model combg the two aforemetoed appromato were ued. However a rough urface realty may have a cotuou cale of roughe ad may ot follow a two-cale model. A the catterg elemet of rough urface preet a comple geometry ad are radomly dtrbuted ther electromagetc catterg volve comple teracto. The Itegral Equato Method (IEM) [] combe the two appromato together ad ca be ued to model urface wth arbtrary roughe cale.. Mcrowave frequecy rage for emo from ol It ha bee foud that the mcrowave frequecy rage that wor bet for ol moture motorg - GHz (L-bad) becaue at th frequecy rage (a) the atmophere effect traparet (b) vegetato em-traparet hece allowg obervato of the urface ad (c) the mcrowave meauremet trogly depedet o ol moture. Addtoally becaue of the log electromagetc wavelegth of the cdet wave ( cm for a.4 GHz wave); the urface roughe effect are mall. I prcple vegetato effect are alo mall but they caot be eglected f the obervato agle large ad/or the vegetato cover thc or dee. Hece mot of the meauremet a well a model predcto are carred out that frequecy rage; however ucertate related to urface roughe ad the aborpto catterg ad emo by vegetato mut be reolved before ol moture retreval algorthm ca be appled wth ow ad acceptable accuracy ug atellte obervato. 3

18 .3 Clafcato of approache to model emo ad catterg from vegetato caope I the mcrowave rego mot vegetato caope are codered homogeeou meda ad ome cae aotropc alo. Geerally effort to model the radar baccatterg ad radometrc emo from vegetato follow oe of two bac approache. I the frt approach the caopy treated a a cotuou radom medum wth a average delectrc cotat ε ad a fluctuatg compoet ε ( y z) a f whch the geeral cae may be formulated a a teor. Whe modeled ug the ecod approach the caopy treated a the um of dcrete elemet each characterzed by a catterg phae fucto that relate the drectoal dtrbuto of eergy cattered by the elemet to the eergy cdet upo t. Model to compute the emo are maly clafed the followg group: (a) mple model where vegetato treated a a purely aborbg layer ad corporated to a atteuato parameter; (b) parametrc model whch rely o vegetato parameter; ad (c) phycal model..4 Curret wor To get a better udertadg of mcrowave emo from vegetato meauremet at.7 ad 5. GHz for row crop have bee performed ug a metallc cree uder the vegetato layer o that the emo from the ol ot codered []. A model baed o radatve trafer theory wa ued at hgh-frequece (0 ad 37 GHz) to evaluate the effect of gle-plat cottuet o the total emo from crop. The effect were evaluated by mea of meauremet carred out o plat atural codto ad after equetal cut of frut leave ad tem. The aalyzed crop had vertcal tem [3]. 4

19 I a program cotg of umerou feld ad arcraft epermet t wa determed that a -cm wavelegth radometer wa the bet gle eor for ol moture reearch [4]. Furthermore order to ae the effect of ma-made vegetato caope a feldapproach two-dmeoal model wa developed to epla wave propagato through uch caope. The model wa foud to provde ecellet agreemet wth the epermetal reult although a 3D model ha to be developed [5]. Ug the Ulaby ad El-Raye model a tudy of the vegetato effect o the mcrowave emo from ol wa performed to etmate the aborpto lo ad optcal depth of plat caope for frequece ragg from to 40 GHz [6]. A tudy to quatfy the etvte of ol moture retreved ug a L-bad glepolarzato algorthm to three lad urface parameter for cor ad oybea te wa carred out [7]. May retreval approache are baed o the τ - ω model [8]. The τ - ω model ha bee foud to be accurate at L-bad. It rele o two vegetato parameter: the optcal depth (τ ) whch parameterze the atteuato effect; ad the gle-catterg albedo (ω ) whch parameterze the catterg effect wth the caopy. Hece a characterzato of the depedece of vegetato model parameter o crop tructure cdece agle ad polarzato at L-bad for everal crop type ug the τ - ω model wa carred out [9]. Several baccatter ad emo model of cor caope for ol moture retreval geeral aume a uform dtrbuto of leave ad tal ad hece treat the caopy a a uform layer. The model parameter are optmzed to yeld good agreemet wth epermetal data whe data retrcted to a partcular obervato agle ad polarzato. However azmuthal agle depedece of emo by the caopy ot codered by thee model. Therefore a mulato tudy wa performed at L-bad to aalyze the effect of tructural 5

20 feature of vegetato caope o brghte temperature. The mulato were performed o qua-perodc cor caope cotg of oly tal. It wa how that at low agle of cdece the effect the caopy row tructure reduced ad the caopy loo mlar a uform caopy. At hgher agle of cdece the effect of caopy tructure o emo oberved through the azmuthal depedece of the caopy emo. I order to mplfy the aaly the uderlyg groud aumed to be a perfect coducto (PEC) plae to elmate emo by the groud [0]. Wth repect to phycal model a mcrowave radometry model for a vegetato caopy wa developed by Karam []. The model baed o a teratve oluto of the radatve trafer equato. I the model fte-legth delectrc cylder were ued to repreet ome vegetato compoet. The catterg ampltude teor for fte legth cylder were calculated through etmatg the er feld wth t correpodg de fte cylder havg mlar radal properte. I th regard a valdato of both urfacead volume-curret model for electromagetc catterg from fte-legth delectrc cylder wa carred out []. Thee approache requre the legth of the cylder to be much larger tha t dameter. A dcrete model to evaluate vegetato effect pave mcrowave ol moture retreval wa gve [3]. I th model vegetato caopy coted of dcrete catter uch a leave tem brache wth varou ze ad oretato. A Matr Doublg algorthm wa ued for the vegetato layer o the multple catterg wth the caopy could be accouted for. The urface catterg wa modeled by the IEM algorthm. Prelmary reult howed that the emo from ol domat; the vegetato maly act a a aborpto layer. 6

21 A comparo betwee theory ad epermet of radometrc meauremet over bare feld ad covered wth gra oybea cor ad alfalfa made wth.4- ad 5-GHz mcrowave radometer wa carred out [4]. The meaured reult were compared wth radatve trafer theory treatg the vegetated feld a a two-layer radom medum. It wa foud that the preece of a vegetato cover geerally gve a hgher brghte temperature tha that epected from a bare ol. A maor compoet of the emo from vegetato-covered ol the emo comg from the urface. A ey factor that affect the urface emo the urface roughe. For eample ug lad urface proce/radobrghte (LSP/R) model for prare gralad the L-bad brghte predcted appeared to be lower tha epected. A crucal reao for the uderetmato wa that the urface wa aumed to be mooth. I [5] the urface roughe effect were cluded by mea of the IEM model. Furthermore aother parameter to be cluded mulato the vegetato volumetrc fracto. A aaly about the effect of the varato of th parameter addto to urface roughe wa gve [6]. A more compreheve aaly of other model for vegetato ha bee decrbed [3]..5 Approach ued tudy All pheomea volvg electromagetc feld ca be mathematcally decrbed by applyg Mawell' equato. Ufortuately olvg thee equato aalytcally ofte very dffcult ad certa appromato have to be troduced to obta a practcal oluto. Oe opto to crcumvet th ue the ue of other aaly techque. Some of thee techque are the umercal method whch are ued to perform a mulato of electromagetc wave propagato dfferet meda [7]. The obtaed reult ug umercal method are accurate to wth the tolerace that ca be arbtrarly preet ad the mulated obect ca be of ay 7

22 hape ad ze lmted oly by the avalable computatoal reource. For thoe reao umercal method repreet very attractve tool for modelg ad aaly of varou electromagetc problem. For th purpoe th reearch the Fte-Dfferece Tme- Doma Method (FDTD) ued whch olve Mawell equato [8 9]. The FDTD method ha prove to be a effectve techque of calculatg the teracto of electromagetc wave wth bode of dfferet materal ad comple hece realtc geometre [30 3]. Numercal method dd ot ga much mportace utl recetly becaue of the legthy computatoal tme ad eormou amout of memory requred to get oluto; but ow wth the curret powerful computatoal reource umercal method are becomg the prmary method to oberve electromagetc wave propagato catterg ad peetrato. I th tudy the vegetato layer repreeted a a homogeeou layer o top of a homogeeou half pace repreetg the ol. O top of the homogeeou layer aother homogeeou half pace preet repreetg ar. Therefore a ey compoet of th reearch to characterze the catterg ad emo properte of the homogeeou layer. I [3] a 3D-FDTD algorthm wa developed ad ued to compute the emvty of fte-ze cyldrcal-hape obect though t ca be ued for ay arbtrary-hape obect. A mlar approach alo ued th reearch to atta ot oly the emo properte of the layer compoet uder tudy but alo the catterg charactertc. A drect method ued to aalyze a pave emo problem by mulatg a actve catterg problem whch a umercal method ued to mulate the teracto of a plae wave ad the obect. Cotrary to the lmtato wth repect to the geometry of the body other approache [33] th tudy a obect of arbtrary hape could be aalyzed. The catterg properte of the obect are alo obtaed ad ued to geerate the phae matr whch relate the cattered tete to the cdet tete. The phae matr 8

23 the tegrated to a radatve trafer model volvg the homogeeou layer ad the homogeou half pace metoed above. The orgazato of th dertato a follow: Chapter offer a decrpto of the radatve trafer theory. The cotructo of urface catterg matrce ug the Itegral Equato Method alo gve. Radatve trafer equato do ot have cloed-form oluto; a teratve proce ued to get the oluto whch provde ome ght to the mechac of catterg. Hece radatve trafer theory et up a teratve tegral equato to compute the emvty from the layer. Th proce of ettg up the equato for teratve oluto gve alo Chapter. A decrpto of the umercal techque to be ued amely the FDTD method gve Chapter 3. A valdato of the accuracy of the computatoal approach ued to perform the mulato gve Chapter 4. Th valdato carred out by comparg the reult obtaed ug the FDTD approach wth aalytcal oluto a well a publhed reult. Alo the umercal computato of emvty from fte-ze obect gve Chapter 4. Chapter 5 gve the umercal approach to geerate the phae matr ad the approach ued to tegrate the phae matr to a layer model. The model the valdated by comparg t predcto wth feld meauremet. The cotrbuto of th wor ad uggeto for future wor related to th dertato are dcued Chapter 6. 9

24 CHAPTER RADIATIVE TRANSFER FORMULATION The bac developmet ad formulato of radatve trafer theory provded by Chadraehar [34]. Several reearcher have cotrbuted to th theory [5 35]. For eae of referece a bref dcuo of th model gve here. The clacal dervato of the radatve trafer theory baed o the amout of radat eergy traported acro the medum epreed term of pecfc tety I υ. The pecfc tety defed term of amout of power dp flowg alog the rˆ drecto wth the old agle d Ω through a elemetary area da a frequecy terval ( υ υ dυ ) a follow: dp = I ( rˆ)(co α) dadω dυ (.) υ where α the agle betwee the outward ormal â to da ad the ut vector rˆ a how Fgure.. Fgure. Geometry for pecfc tety ad the chage tety I. 0

25 The dmeo of I υ Wm r Hz. It coveet to coder the tety I at frequecy υ therefore I ( rˆ) = I ( rˆ ) dυ. Hece the power at a gle frequecy ca be wrtte υ a dp = I( rˆ)(co α ) dadω (.) The equato of trafer gover the varato of tete a medum that aborb emt ad catter radato. Wth the medum coder a cyldrcal volume of ut cro ecto ad legth dl a how Fgure.. Eergy balace requre that the chage tety I propagatg through the cyldrcal volume alog the dtace dl due to aborpto lo catterg lo thermal emo ad catterg the drecto of propagato.e. di = κ I dl κ I dl κ J dl κ J dl (.3) a a a where κ a ad κ are the volume aborpto ad volume catterg coeffcet repectvely. The frt two term o the rght had de of (.3) repreet repectvely aborpto lo ad lo due to catterg away from the drecto of propagato ad J a ad J are the emo ource fucto ad the catterg ource fucto repreetg the eergy cattered to the drecto of propagato. Equato (.3) the radatve trafer equato where ππ J gve by J ( θ φ ) = ( θ φ ; θ φ) ( θ φ)θ dθ dφ 4π P I (.4) 0 0 where P ( θ φ ; θ φ) the phae fucto accoutg for catterg wth the medum.

26 . Stoe parameter Lght get polarzed o catterg; therefore eact treatmet of polarzed radato requred. To decrbe a polarzed beam geerally four parameter hould be pecfed whch wll gve the tety the degree of polarzato the plae of polarzato ad the ellptcty of the radato at each pot ad ay gve drecto [34]. However t dffcult to clude uch dvere quatte a tety a rato a agle ad a pure umber ay ymmetrcal way formulatg the equato of trafer. I radatve trafer theory the Stoe parameter are ued to decrbe the partally polarzed electromagetc wave. Coder a ellptcally polarzed moochromatc plae wave r r r E = ( E vˆ E hˆ) ep( ω t) v h (.5) propagatg through a dfferetal old agle d Ω a medum wth trc mpedace η. The vertcal ad horzotal polarzato tate are repreeted by vˆ ad ĥ ut vector ad vertcally ad horzotally polarzed cdet feld compoet are deoted by E v ad E h repectvely. The modfed Stoe parameter I v I h U ad V the dmeo of tety are defed a v E v I dω = η (.6) h E h I dω = η (.7) ( * v E ) η ( * v E ) η U dω = Re h (.8) V d = Ω Im h (.9) I (.6) to (.9) the ymbol deote a tme average. Thee four Stoe parameter have the ame dmeo ad hece are more coveet to ue tha ampltude ad phae

27 whch have dfferet dmeo. It ha bee how that the ampltude phae ad polarzato tate of ay ellptcally polarzed wave ca be completely characterzed by thee parameter [8]. Due to the addtvty of the average power of depedet wave the Stoe parameter are addtve oly for coheret wave.. Phae fucto ad phae matr Scatterg the teracto betwee electromagetc wave ad the medum. Th teracto ca be characterzed term of t catterg properte.e. catterg cro ecto or t thermal emo pectrum. Durg catterg the medum tae eergy from the cdet wave ad reradate depedg upo the rato of catter ze to the wavelegth of the cdet wave. The phae fucto gve the rate at whch the eergy cattered per ut old agle a gve drecto to the average eergy all drecto [34]. I addto to the agular dtrbuto of the cattered radato at mcrowave frequece the catterg ad emo from atural urface are largely determed by the urface roughe ad the homogeeou profle of the delectrc cotat temperature ad the volume catterg property. I the remote eg of ol moture at mcrowave frequece the o-uform moture ad temperature profle the ear-urface rego ad the rough urface play a domat role. The effect of uburface volume catterg play a ecodary role becaue of hgh aborpto due to ol moture. Phae matr a quatty that relate the correpodg Stoe vector of the cattered wave ad the cdet wave. Sce the charactertc of the urface catterg ad the volume catterg are dfferet the phae matrce defed for thee two cae wll alo be dfferet. For urface catterg the cdet wave ecte urface curret o the urface that reradate the cattered wave. So the quatte ued are geerally ormalzed by the llumated area. 3

28 Whle for volume catterg the ource of re-radato are volume curret ad the the quatte are ormalzed by the volume of the partcle... Phae matr of rough urface To relate the cattered tety to the cdet tety coder a plae wave llumatg a rough urface area A [3]. The relato betwee the vertcally ad horzotally v h polarzed cattered feld compoet E E ad thoe of the cdet feld compoet E E v h E E v h e = R R S S vv hv S S vh hh E E v h (. 0) where S ( p q = vor h ) the catterg ampltude meter R the dtace from the ceter pq of the llumated area to the pot of obervato ad the wave umber. The catterg 0 coeffcet σ pq defed a σ 0 pq < E p > = (.) A E /(4π R ) q The matr equato relate the cattered tete I to the cdet tete I through the dmeole quatty referred to a phae matr P I = PI dω 4π (.) The compoet of I are the Stoe parameter defed by (.6) to (.9) for the cdet plae wave. The compoet of the cattered tety 4 I are alo Stoe parameter but are defed for phercal wave. To um up all poble cdet tete from all

29 drecto cotrbutg to.e. I alog a gve drecto (.) tegrated over all old agle I = PI dω 4π 4π (.3) Here I I are colum vector whoe compoet are the Stoe parameter. The detaled cotet of the phae matr are ummarzed below 0 ( Acoθ ) = σ coθ P = 4π M (.4) where the Stoe matr M ha bee provded by Ihmaru [8] Svv Shv Re( SvvS Im( SvvS * hv * hv ) ) S S vh hh Re( S Im( S vh vh S S * hh * hh ) ) Re( S Im( S Re( S Im( S vv vv S S vv hv * hh * hh S S * vh * hh S S ) ) vh vh S S * hv * hv ) ) Im( S Re( S Im( S Im( S vv hv * vvshh * vvshh S S S S * vh ) * hh ) * vhshv * vhshv ) ) The theory for cotructo of the phae matr for a homogeeou medum ha bee developed for ome tme already [5 7 36]. Several factor are codered to geerate the phae matr le ze hape oretato volumetrc dety etc. The approach ued to geerate the phae matr th tudy wll be gve Chapter 5..3 Radatve trafer equato The radatve trafer equato for partally polarzed wave a dcrete homogeeou medum hould clude both catterg ad aborbg charactertc the dfferetal equato ad ca be wrtte a [5] or di dl κ e = κ ei Pe I dω κ aj 4π 5 4π a (.5)

30 di dl κ = κ ei P I dω κ aj 4π 4π a (.6) Mot of the tude to date have dealt wth the pecal cae of phercal partcle. For thee partcle or for o-phercal partcle wth radom oretato κ e ad κ reduce to calar [37]. If th o t coveet to covert the depedet varable (.5) or (.6) to optcal thce τ. The (.5) ad (.6) reduce repectvely to ad di = I dτ 4π di ω = I dτ 4π 4π 4π P I dω ( ω) e J a P I dω ( ω) J a (.7) (.8) where τ = κ e dl ad ω the albedo whch the rato of κ to κ e. It may be helpful to ote that P = ω P. The radatve trafer equato formulated o the ba of eergy balace. e Thu the phae chage of the cattered wave ad t cro-correlato term are gored the oluto of the trafer equato. Although the radatve trafer approach caot accout for phae effect multple catterg calculato t doe clude phae effect the phae fucto calculato ad t accout for hgher order multple catterg more effectvely tha the wave approach. It aumed that there o correlato betwee feld; hece addto of tete codered tead of addto of feld. I the radatve trafer formulato the phae matr characterze the couplg of tete ay drecto due to catterg ad the etcto matr decrbe the atteuato of pecfc tety due to aborpto ad catterg. 6

31 7.4 Scatterg from a homogeeou layer wth rregular boudare For bouded meda catterg or reflecto may occur at the boudary. Both cdet ad cattered tete are eeded the boudary codto. Therefore t eceary to plt the tety matr to upward I ad dowward I compoet ad rewrte the trafer equato a two equato. Coder the problem of a plae wave ar cdet upo a homogeeou layer above the groud urface. The geometry of the catterg problem depcted Fgure.. The homogeeou layer aumed to have uch charactertc that the upward tety I ad the dowward tety I atfy the radatve trafer equato. Fgure. Scatterg from a homogeeou layer above a homogeeou half pace. Upo re-wrtg (.8) term of thee tete the equato ca be epreed a φ µ φ µ φ φ µ µ κ π φ µ κ φ µ µ π d d z I P z I z I dz d e ) ( ) ( 4 ) ( ) ( 0 0 = φ µ φ µ φ φ µ µ κ π π d d z I P ) ( ) ( (.9) φ µ φ µ φ φ µ µ κ π φ µ κ φ µ µ π d d z I P z I z I dz d e ) ( ) ( 4 ) ( ) ( 0 0 =

32 π κ P ( µ µ φ φ) I ( z µ φ) dµ dφ (.0) 4π 0 0 where µ = coθ ; µ = coθ ; I I are colum vector cotag the four modfed Stoe parameter ad P the phae matr. The eplct form of the phae matrce for the IEM model gve et ecto. The trafer equato gve by (.9) ad (.0) ca be olved by ug a umercal techque..5 Emo model for a homogeeou layer Coder the emo problem of a homogeeou rregular layer above a homogeeou half pace (Fgure.). The coheret ource term repreeted by the emtted tety e I due to the layer temperature profle T. The coheret ource term (.7) I e cluded. e I gve by I e K = ( ω) J a = ε rt (.) λ where K the Boltzma cotat λ the operatg wavelegth the free pace ad ε r the relatve permttvty of the layer. The tety vector cot of a vertcally ad a horzotally polarzed compoet.e. oly the frt two Stoe parameter. Alo the layer temperature profle T aumed depedet of azmuthal agle. The frt pot above the coequece of the coheret ature of atural emo whch caue the thrd ad the fourth Stoe parameter to vah. Th becaue cro polarzato of vertcal ad horzotal emo compoet zero. The ecod pot allow to 8

33 replace the elemet of the phae matr by ther zeroth-order Fourer compoet both the trafer equato ad the aocated boudary codto..6 Formulato of urface catterg matrce ug the Itegral Equato method A approprate decrpto of the urface requred for all urface modelg. Two prcpal parameter characterze the urface: the tadard devato of heght ad correlato legth whch decrbe the degree of vertcal ad horzotal roughe. The problem of catterg from rough urface ha bee tuded ug low frequecy (Small Perturbato Method SPM) ad hgh frequecy (Krchhoff) appromato. For rough urface wth dfferet cale of roughe two-cale model combg the above two appromato were ued. But real rough urface may have a cotuou cale of roughe ad may ot follow a two-cale model. O the other had a the catterg elemet of rough urface preet a comple geometry ad are radomly dtrbuted ther electromagetc catterg volve comple teracto. The Itegral Equato Method (IEM) combe the two appromato together ad ca be ued to model urface wth arbtrary roughe cale. At low frequece IEM reduce to the frt-order SPM ad at hgh frequece; t reduce to the Krchhoff term [3]. The IEM model wa frt developed to decrbe electromagetc wave catterg for a radomly rough perfectly coductg urface [38] ad later for a radomly rough delectrc urface []. The frt complete vero of the IEM model wa developed ad propoed by Fug [] baed o a more rgorou oluto of a par of tegral equato goverg the urface curret. I th model a mplfyg aumpto wa made o the pectral form of the Gree fucto by gorg the phae term. I partcular t wa argued that the abolute value of the dfferece the urface heght at two urface pot appearg the phae of the 9

34 Gree fucto ca be gored [39]. Th argumet wa arrved at by otg that a) f the two pot are cloe together the dfferece heght hould be mall for a cotuou urface ad b) f the two urface pot are far apart there hould be a eglgble amout of correlato betwee the two pot ad hece wll ot cotrbute gfcatly to the cattered power. I IEM for a gve cdet plae wave defed ug E r ad H r the tagetal E r ad H r feld o the terface betwee two meda are foud by olvg Electrc Itegral Feld Equato (EFIE) ad Magetc Itegral Feld Equato (MFIE). The ug the Stratto-Chu Radato Itegral the far zoe feld ug the tagetal feld at the delectrc terface foud. Oce epreo for the cattered E r ad H r feld have bee derved the average cattered power a gve drecto foud by averagg the feld quatte over varou realzato of the rough urface where the realzato themelve are pced from varou probablty dtrbuto of the urface. Mot of the appromato made the dervato are related to mag the feld quatte a depedet of the local coordate a poble. The gle baccatterg coeffcet from the IEM model are gve by wth 0 W ( 0) σ pp = σ I pp (.)! ep[ zσ ] = ( ) f ep[ ] { [ F ( 0) F ( 0)]} I pp = z pp zσ z pp pp (.3) where pp deote the polarzato tate ad f vv R co v = (.4) ( θ ) f R h hh = (.5) co( θ ) 0

35 F hh F vv ( 0) Fvv ( 0) ( θ )( Rv ) = co( θ ) ( 0) Fhh ( 0) ( θ )( Rh ) = co( θ ) µ Tε ε ε µ Tε µ T µ T ( θ ) ( ) ( ) ε co θ co θ ( θ ) ( ) ( ) µ T co θ co θ (.6) (.7) I the above equato θ the agle of cdece ad vertcally polarzed Freel reflecto coeffcet; ε ad R h ad R v are the horzotally µ T are the relatve permttvty ad permeablty of the urface; σ the tadard devato of the urface heght; the wave umber; = co( θ ) ; ( θ ) z = ; ad ow urface correlato fucto whch ca be calculated by W the Fourer traform of the th power of a W 0 ( ξ ) J ( Kξ ) ( K) = ρ ξdξ 0 (.8) where ρ the urface correlato ad J 0 the Beel fucto to the zeroth order. More detal about dervato are gve [3 36]. I the above epreo f pp the Krchoff feld coeffcet ad F pp the complemetary feld coeffcet. The epreo for the catterg coeffcet volve a ere ad roughly the umber of term requred for covergece gve by [3] = ( σ ) 5.7( σ ) 9.488( σ ).47( σ ) (.9) 3 4 where ( σ ) the ormalzed tadard devato of the urface roughe. The cotructo of btatc gle-catter catterg ad tramo phae coeffcet phae matr alo gve detal [36].

36 CHAPTER 3 FINITE-DIFFERENCE TIME-DOMAIN METHOD All pheomea volvg electromagetc feld ca be mathematcally decrbed by applyg Mawell' equato. Ufortuately olvg thee equato aalytcally ofte very dffcult ad certa appromato have to be troduced to obta a practcal oluto. Soluto to thee equato have a log htory ad a large varety of method have bee developed. The mot atfactory oluto the mathematcally eact oe. Aalytcal method ued for obtag uch oluto are: eparato of varable ere epao coformal mappg ad tegral method. However thee method ca be ued oly the aaly of very pecfc feld problem. I may practcal tuato the complety of the problem mply too large for t to be olved aalytcally. I th cae other aaly techque eed to be codered uch a umercal graphcal ad epermetal method. Uually epermetal oluto are very epeve ad graphcal oe are ot accurate eough. O the other had umercal method are relatvely epeve ad the obtaed reult are accurate to wth the tolerace that ca be arbtrarly preet. For thoe reao umercal method repreet very attractve tool for modelg ad aaly of varou electromagetc problem. The Fte Dfferece Tme-Doma (FDTD) method a wdely ued computatoal techque for provdg accurate full-wave aaly of electromagetc pheomea.

37 3. Itroducto to the Fte Dfferece Tme-Doma (FDTD) method The Fte-Dfferece Tme-Doma Method (FDTD) a Mawell equato olver [8 9]. It ha prove to be a effectve techque of calculatg the teracto of electromagetc wave wth bode of dfferet materal ad comple geometre [3]. Alo FDTD codered to be oe of the mot popular ad robut mea to mplemet egeerg electromagetc model [30]. I th chapter the foudato of FDTD electromagetc feld aaly the algorthm troduced by Kae Yee [9] gve. Yee choe a geometry for patally amplg the electrc ad magetc feld vector compoet whch robutly repreet both the dfferetal ad tegral form of Mawell equato. The decrpto gve here follow cloely the oe gve [40]. 3. Mawell equato three dmeo Aume a rego of pace that ha o electrc or magetc curret ource but may have materal that aborb electrc or magetc feld eergy. I th tuato the tmedepedet Mawell equato are gve dfferetal ad tegral form by Faraday law: Ampere law: t A r B r r = E M t r r B da = L r r E dl r D r r = H J t A r r M da (3.) (3.) (3.3) 3

38 Gau law for the electrc feld: t A r r D da = L r r H dl A r r J da (3.4) Gau law for the magetc feld: A D r = 0 (3.5) D da = 0 (3.6) A B r = 0 (3.7) B da = 0 (3.8) I lear otropc odperve materal to relate D r to E r ad B r to H r the followg epreo ca be ued: r r r r r r D = ε E = ε E B = µ H = µ H (3.9) rε 0 rµ 0 The ource J r ad M r ca act a depedet ource of E r - ad H r -feld eergy J r ource ad ource M r. Materal wth otropc odperve electrc ad magetc loe that atteuate E r - ad H r -feld va covero to heat eergy ca be volved. Th yeld r r r r r r J = J σ E M = M ρ H (3.0) ource ource where σ ad ρ are electrc coductvty (Seme/meter) ad the equvalet magetc lo (ohm/meter) repectvely. By ubttutg (3.9) ad (3.0) to (3.) ad (3.3) the Mawell curl equato lear otropc odperve loy materal are gve by: r H r = E t µ µ r r ( M ρ H ) ource (3.) 4

39 5 ( ) E J H t E ource r r r r σ ε ε = (3.) The vector compoet of the curl operator of (3.) ad (3.) Cartea coordate are repreeted by the followg ytem of coupled calar equato: ( ) = ource z y H M y E z E t H ρ µ (3.3) ( ) = y ource z y H M z E E t H y ρ µ (3.4) ( ) = z ource y z H M E y E t H z ρ µ (3.5) ( ) = ource y z E J z H y H t E σ ε (3.6) ( ) = y ource z y E J H z H t E y σ ε (3.7) ( ) = z ource y z E J y H H t E z σ ε (3.8) The ytem of coupled partal dfferetal equato of (3.3) to (3.8) form the ba of FDTD umercal algorthm for electromagetc wave teracto wth geeral threedmeoal obect. The FDTD algorthm eed ot eplctly eforce the Gau law relato dcatg zero free electrc ad magetc charge (3.5) to (3.8); becaue thee relato are theoretcally a drect coequece of the curl equato. Neverthele the FDTD pace lattce mut be tructured o that the Gau law relato are mplct the poto of the E r ad H r compoet ad the umercal pace-dervatve operato upo thee compoet that model the acto of the curl operator.

40 3.3 The Yee algorthm Yee orgated a et of fte-dfferece equato for the tme-depedet Mawell curl equato ytem of (3.3) ad (3.8) for the lole materal cae ρ = 0 ad σ = 0 [9]. The Yee algorthm olve for both electrc ad magetc feld tme ad pace ug the coupled Mawell curl equato tead of olvg for the electrc feld aloe (or the magetc feld aloe) wth a wave equato. z E y E E E y H z E z E z H E z H y E z H y H E y y E E E y H z Fgure 3. Poto of feld compoet. The E r -compoet are the mddle of edge ad the H r -compoet are the ceter of the face. The Yee algorthm ceter t E r ad H r compoet three-dmeoal pace o that every E r compoet urrouded by four crculatg H r compoet ad every H r compoet urrouded by four crculatg E r compoet (Fg. 3.). 6

41 Th provde a mple pcture of three-dmeoal pace beg flled by a terled array of Faraday law ad Ampere law cotour. It poble to detfy Yee E r compoet aocated wth dplacemet curret flu lg H r loop a well a H r compoet aocated wth magetc flu lg E r loop. Actually the Yee algorthm multaeouly mulate the potwe dfferetal form ad the macrocopc tegral form of Mawell equato. Aother feature of the Yee pace lattce are : a) the fte-dfferece epreo for the pace dervatve ued the curl operator are cetral-dfferece ature ad ecod-order accurate; b) cotuty of tagetal E r ad H r aturally mataed acro a terface of dmlar materal f the terface parallel to oe of the lattce coordate ae. For th cae there o eed to pecally eforce feld boudary codto at the terface. At the begg of the problem the materal permttvty ad permeablty at each feld compoet locato pecfed. Th yeld a tepped or tarcae appromato of the urface ad teral geometry of the tructure wth a pace reoluto et by the ze of the lattce ut cell; ad c) the locato of the E r ad H r compoet the Yee pace lattce ad the cetral- dfferece operato o thee compoet mplctly eforce the two Gau law relato. Coequetly the Yee meh dvergece-free wth repect to t E r ad H r feld the abece of free electrc ad magetc charge. The Yee algorthm alo ceter t E r ad H r compoet tme what termed a a leapfrog arragemet (Fg. 3.). All of the E r computato the modeled pace are completed ad tored memory for a partcular tme pot ug prevouly tored H r data. Afterward all of the H r computato the pace are completed ad tored memory ug the E r data ut computed. The cycle tart aga wth the recomputato of the E r compoet baed o the recetly obtaed H r. Th proce carre o utl tme-teppg cocluded. 7

42 H -/ H / E - E t Fgure 3. Leapfrog tme tegrato. 3.4 Fte dfferece ad otato I [9] Yee troduced the otato how table 3. for pace pot ad fucto of pace ad tme a uform rectagular lattce. Table 3. Locato of the dfferet feld the Yee cell Elemet ( ) E ( ) E y ( ) E z ( ) E z ( ) H y ( ) H z Locato = y = y z = z = y = y z = z = y = y z = z = y = y z = z = y = y z = z = y = y z = z 8

43 The lattce pace cremet the y ad z coordate drecto are deoted by y ad z repectvely; ad ad are teger. Hece ay fucto u of pace ad tme evaluated at a dcrete pot the grd ad at a dcrete pot tme deoted a u ( y z t) = u (3.9) where teger. t the tme cremet aumed uform over the obervato terval ad a Yee made ue of cetered fte-dfferece epreo for the pace ad tme dervatve that are both mply programmed ad ecod-order accurate the pace ad tme cremet. A epreo for the frt partal pace dervatve of u the -drecto evaluated at the fed tme t = t gve by: u u u [ ] ( y z t) = O ( ) (3.0) A ± cremet the ubcrpt ( -coordate) of u deote a pace fte- dfferece over ( ) ±. A ecod-order accurate cetral dfferecg ued. Data for the cetral dfferece are tae to the rght ad left of the obervato pot by oly / rather tha a full. Th otato choe to terleave E r ad H r compoet the pace lattce at terval of /. The dfferece of two adacet E r compoet eparated by ad located ± ( ) o ether de of a H r compoet would be ued to provde a umercal appromato for r E / to permt teppg the H r compoet tme. For the ae of completee t hould be added that a umercal appromato aalogou to (3.0) for u / y 9

44 or u / z ca be wrtte mply by cremetg the or ubcrpt of u by ± ( / ) y or ( / ) z ± repectvely. pot ( ) The epreo for the frt tme partal dervatve of u evaluated at the fed pace follow by aalogy: u t u [ ] ( y z t) = O ( t) u t (3.) Now the ± cremet the upercrpt (tme coordate) of u deotg a tme fte- dfferece over ± ( / ) t. Th otato wa choe to terleave the E r ad H r compoet tme at terval of ( / ) t for purpoe of mplemetg a leapfrog algorthm. 3.5 Fte dfferece epreo for Mawell equato three dmeo The dea ad otato how above are appled to acheve a umercal appromato of the Mawell curl equato three dmeo gve by (3.3) to (3.8). The E feldcompoet gve by (3.6): E t H = ε y z H y z ( J σ E ) ource Coder a typcal ubttuto of cetral dfferece for the tme ad pace dervatve (3.6) for eample at E. Itally 30

45 3 = ource y y z z E J z H H y H H t E E σ ε (3.) All feld quatte the rght had de are evaluated at tme-tep cludg the electrc feld E appearg due to the materal coductvty σ. Sce E value at tme-tep are ot aumed to be tored the computer memory (oly the prevou value of E at tmetep are aumed to be memory) ome way to etmate uch term eeded. Oe way ug what called a em-mplct appromato: = E E E (3.3) The E value at tme-tep are aumed to be mply the arthmetc average of the tored value of E at tme-tep ad the yet-to-be-computed ew value of E at tmetep. By ubttuto of (3.3) to (3.) after multplyg both de by t the followg obtaed

46 3 = t E E J z H H y H H t E E ource y y z z σ ε (3.4) The term E ad E appear o both de of (3.4). Collectg all term of thee two type ad olatg E o the left-had de yeld = ource y y z z J z H H y H H t E t E t ε ε σ ε σ (3.5) Both de are dvded by t ε σ. Hece the dered eplct tme-teppg relato for E :

47 33 = ource y y z z J z H H y H H t t E t t E ε σ ε ε σ ε σ (3.6) The em-mplct aumpto of (3.3) yeld umercally table ad accurate reult for value of σ from zero to fty [3]. The term of th type troduced o the rght-had de of (3.4) ca be grouped wth a le term o the left-had de ad the olved eplctly. Smlarly fte-dfferece epreo baed o Yee algorthm for the y E ad z E feld compoet gve by Mawell equato (3.7) ad (3.8) repectvely ca be derved. The tme-teppg epreo for the y E ad z E are the = ource z z y y J H H z H H t t E t t E ε σ ε ε σ ε σ (3.7)

48 34 = ource y y z z J y H H H H t t E t t E ε σ ε ε σ ε σ (3.8) Lewe fte-dfferece equato for (3.3) to (3.5) to tme-tep H y H ad z H ca be derved ug a em-mplct procedure aalogou to (3.3). A magetc lo term repreeted by H ρ o the rght-had de of each equato. Th yeld three equato havg a form mlar to that of the E r equato. Hece the followg tme-teppg epreo for the H compoet located at the upper rght corer of the ut cell ca be obtaed: = 3 3 ource z z y y M y E E z E E t t H t t H µ ρ µ µ ρ µ ρ (3.9) Smlarly the tme-teppg epreo for the y H ad z H compoet located at the upper frot corer ad the rght frot corer of the ut cell repectvely ca be repreeted a:

49 35 = 3 ource z z y y M z E E E E t t H t t H µ ρ µ µ ρ µ ρ (3.30) ad = 3 ource y y z z M E E y E E t t H t t H µ ρ µ µ ρ µ ρ (3.3) repectvely. I cocluo wth the ytem of fte-dfferece epreo gve by (3.6)-(3.3) the ew value of a electromagetc feld vector compoet at ay lattce pot deped oly o t prevou value the prevou value of the compoet of the other feld vector at adacet pot ad the ow electrc ad magetc curret ource.

50 Fgure 3.3 Starcaed phere cubcal Yee cell. For llutrato purpoe of a dcretzed obect Fgure 3.3 a phere appromated ug cubcal Yee cell wth the tarcag of the phere urface how clearly. 36

51 CHAPTER 4 NUMERICAL COMPUTATION OF EMISSION FROM FINITE-SIZE OBJECTS 4. Emo theory It well ow that all matter radate electromagetc eergy [4]. The radato a coequece of the teracto betwee the atom ad the molecule the materal. A materal alo may aborb ad/or reflect eergy cdet upo t. Whe thermodyamc equlbrum wth t evromet a materal aborb ad radate eergy at the ame rate. A blacbody defed a a deal materal that aborb all of the cdet radato reflectg oe. Sce eergy aborbed by a materal would creae t temperature f o eergy were emtted a perfect aborber alo a perfect emtter. The blacbody radato pectrum gve by Plac radato law whch wa formulated the ba of the quatum theory of matter. Th pectrum ued a a referece agat whch the radato pectra of real bode at the ame phycal temperature are compared. The pectral polarzato ad agular varato of the radato emtted aborbed ad cattered by a medum are govered by the geometrcal cofgurato of the urface ad teror of the medum ad by the patal dtrbuto of t delectrc ad temperature. Radometry the meauremet of electromagetc radato. A mcrowave radometer a hghly etve recever capable of meaurg low level of mcrowave radato. Whe a cee uch a terra oberved by a mcrowave radometer the radato receved by the atea partly due to elf-emo by the cee ad partly due to reflected radato orgatg from the urroudg uch a the atmophere. Through proper choce of the 37

52 radometer parameter (wavelegth polarzato ad vewg agle) t ometme poble to etablh ueful relato betwee magtude of the eergy receved by the radometer ad pecfc terretral or atmopherc parameter of teret. For eample obervato of bare ol urface have how that the radometrc repoe of the 0-30 cm wavelegth rage flueced trogly by the ol moture cotet [4]. Such depedece ca be ued over large area to remotely map ol moture cotet a mportat phycal parameter may hydrologcal agrcultural ad meteorologcal applcato. The cece of mcrowave radometry alo referred a pave mcrowave remote eg cotrat to radar whch ow a actve mcrowave remote eg. Numerou vetgato have bee coducted to evaluate the ue of pave mcrowave eor for meteorologcal hydrologcal oceaographc ad mltary applcato. Fgure 4. The brghte temperature of a obect related to t phycal temperature by t emvty. Every obect wth a phycal temperature above abolute zero radate eergy. The amout of eergy radated uually repreeted by a equvalet temperature T B better ow a brghte temperature (Fg. 4.) ad t defed a B ( φ p) e( θ φ p) T P T θ = (4.) where: 38

53 T P : Phycal temperature ( θ φ p) e : Emvty θ φ : Elevato ad azmuth agle p : Polarzato (vh) Hece whe aalyzg the radato from a fte-ze obect a parameter to coder the emvty whch the rato of eergy radated by a partcular materal to eergy radated by a blac body. Th chapter preet a method to umercally compute the emvty from a fte ze obect. I [43] the gfcace of the equato obtaed for the geeralzed Krchhoff law for a fte-ze aotropc medum the fact that t relate the thermal emo from the orgal obect to the aborpto cro ecto of the complemetary obect therefore relatg a pave emo problem to a actve catterg problem. Hece although the purpoe to model the emo from a obect a umercal method ca be ued to mulate the teracto of a plae wave ad the obect ad hece fd the aborpto ad catterg properte of the obect. Before the emvty of a fte-ze obect wa computed everal tet were coducted order to etablh the accuracy of the FDTD approach to compute the cattered electromagetc feld. The tet coted of comparg the FDTD reult wth aalytcal reult a well a publhed reult ug a dfferet umercal method. 4. Valdato of FDTD reult by comparo wth aalytcal reult ug Me ere oluto Oe of the tet wa the comparo of cattered feld of a delectrc phere. A phere ued becaue the cattered feld from a phere due to a plae wave cdece have aalytcal 39

54 oluto. The oluto are uually called Me ere oluto. Therefore a plae wave mpgg o a phere wa mulated the FDTD approach a how Fg. 4.. The parameter of the phere are gve table 4. where λ 0 the wavelegth of the cdet wave free pace. Table 4. how the parameter of the cdet plae wave a well a the cattered feld. The term rage refer to the dtace from the ceter of the phere at whch the cattered feld value were computed. θ z θ y φ φ E H Fgure 4. Cofgurato for mulato of plae wave mpgg o a delectrc phere. Table 4. Parameter of obect ued FDTD mulato of plae wave mpgg o delectrc phere. Parameter Relatve permttvty 3.5 Coductvty 0.09 S/m Value Radu mm (0.67 λ ) 0 40

55 Table 4. Parameter of cdet plae wave ad cattered feld. Polarzato Parameter Icdet feld elevato agle Icdet feld azmuth agle Vertcal o θ = o φ = 0.0 Value Scattered feld azmuth agle Frequecy o φ = GHz Rage mm (0.809λ 0) from ceter of phere Fgure 4.3 how the value of both E r ad H r feld computed wth FDTD approach ad Me ere oluto. The value of the compoet of H r are ormalzed by H r c. A good agreemet ca be oberved. 4

56 FDTD Me.8 FDTD Me E θ (V/m) H θ (0-4 A/m) θ (deg) θ (deg) FDTD Me FDTD Me 3 0. E φ (0-4 V/m).5.5 H φ (A/m) θ (deg) (a) (b) Fgure 4.3 Comparo of cattered feld value obtaed by ug FDTD ad Me ere oluto. (a) E θ ad E φ (b) H θ ad H φ (ormalzed to θ (deg) H r c ). 4.3 Valdato of FDTD reult by comparo wth method of momet (MoM) reult Although a eact aalytcal oluto for the catterg from fte-legth cylder doe ot et umercal method ca deed provde accurate reult. The tudy of the catterg behavor of a delectrc cylder relevat th tudy becaue everal vegetato compoet ca be repreeted by them. I mcrowave remote eg a vegetato caopy ca be codered a a multlayered medum above a half pace repreetg the groud. Each layer ca be modeled a a eemble of dvdual delectrc obect of dfferet type ze ad 4

57 oretato [3]. Amog the mot commo compoet a vegetated medum are cyldrcal tructure uch a tem brache tru eedle ad tal [44 45]. The cylder are geerally crcular homogeeou ad loy. Whle the umercal treatmet of the problem of catterg from a fte cylder ha geerally bee lmted to the method of momet [46] the FDTD method utable for problem volvg homogeeou cylder. I th ecto t carred out a comparo of the ormalzed btatc coeffcet of fte-ze cylder ug two umercal method: the FDTD approach ad the method of momet (MoM). For th purpoe the reult publhed [47] are ued. For the comparo coder a ftely-log crcular delectrc cylder of radu a ad legth L wth permttvty ε = εcε0 ad permeablty µ = µ 0 located free pace. Here ε 0 ad µ 0 are the free pace permttvty ad permeablty repectvely ad ε c = ε ε the (comple) relatve permttvty of the cylder. A rectagular coordate ytem ( y z ) defed wth t org the ceter of the cylder ad the z-a cocdet wth the cylder a a llutrated Fgure 4.4. Aumg a agular frequecy ω = π f ( f beg the frequecy) ad tme depedece of the form t e ω for all electromagetc quatte the cylder codered to be llumated by a uform plae wave gve by r E ( ) ˆ ( ) ˆ 0 r r = q e q = h v. r (4.) where 0 = ω ε 0µ 0 the free pace waveumber ad ˆ = ˆ θ coφ yˆ θ φ zˆ coθ (4.3) 43

58 the propagato vector of the cdet wave from the drecto ( θ ˆ or ˆ are deoted by ymbol wth hat o them. A cattered wave codered the drecto ( θ φ φ ). Ut vector uch a ) wth propagato vector ˆ = θ coφ ˆ θ φ yˆ coθ zˆ (4.4) Fgure 4.4. Geometry ued to compute catterg from a cylder. Fgure reproduced from [47]. The relatve delectrc cotat ε c of vegetato tructure greatly deped o ther water cotet. Tree tru ad brache are uually drer ad therefore have a lower ε c tha cor tal. For comparo purpoe oly oe value for the delectrc cotat ε c ued. The value choe [47] ε = 8 6 ad cotet wth groud data meauremet ad Ulaby emprcal model [48]. c 44

59 I radar remote eg partcularly whe coderg catterg from a layer of delectrc obect the teret le the ma catterg lobe. Th a coequece of the large umber of catterer uch vegetato meda. Sce the cotrbuto from the gle catterer are added all together the relatve weght of the catterg from the de lobe become eglgble. Therefore whe comparg catterg coeffcet ug dfferet umercal method t codered acceptable f the two method agree de the ma catterg lobe ad doe ot produce hgh catterg elewhere. I the followg ecto the value of the catterg coeffcet (catterg cro ecto per ut area) computed by both umercal method are compared. I partcular for two cdet agle θ ) the btatc catterg coeffcet over a rage of catterg agle ( φ ( φ θ ) ug both method determed. I agreemet wth the reult of [47] the llutrato of reult lmted to the cae of catterg the ame plae φ = φ of the cdet wave. Fgure 4.5 Geeral catterg patter of a cylder. Fgure reproduced from [47]. 45

60 A cylder wth dmeo choe cotetly wth the groud data [49] ad [50] codered. The geeral hape of the catterg patter of a delectrc cylder llutrated Fgure 4.5. The ampltude of the catterg ha t mamum a cocal rego alo referred to a the catterg coe. I a ecto φ = φ of the catterg patter there wll be two relatve mama correpodg to the ma catterg coe oe the forward drecto the other oe the pecular drecto wth repect to the cylder de. I the followg the ormalzed btatc catterg coeffcet σ ( π a ) are plotted a a fucto of the catterg agle for a fed cdet agle both hh- ad vv-polarzato. The catterg agle θ the plot rage betwee 0 ad π therefore oly the pecular lobe at θ = π wll be obervable. θ The cae eamed a cylder of legth L =0.0λ0 ad radu a = 0.4λ0 wth ε = 8 6. For a wavelegth λ 0 = 60 cm (or f = 500 MHz P-bad) th correpod to a c tree tru 6 m log wth a dameter of 4.8cm. The ormalzed btatc catterg coeffcet σ ( π a ) plotted Fgure 4.6 a a fucto of the catterg agle θ for a cdet agle o θ = 0 both hh- ad vv-polarzato. The old curve repreet the FDTD oluto whle the dahed curve obtaed ug the method of momet. A very good agreemet ca be oberved betwee the two oluto ecept for vv-polarzato at agle θ far from the pecular catterg lobe located at appromately o θ = 60 ad for h-h polarzato at repectvely. Whe treatg remote eg problem oly the catterg wth o θ = 00 0 db of the pea value relevat therefore we are cocered wth achevg a good appromato oly the rego urroudg the mama. 46

61 For ome of the dfferece the plot a poble caue would be the fact that our approach 60 evaluato pot for θ were ued whle [47] 0 pot were ued. A mprovemet the FDTD reult epected by reducg the ze of the Yee cell; hece the dcretzed model of the cylder would approach more the curvature of the cylder. Wth the prevou reult t wa how that the FDTD approach ued th tudy provde accurate value for cattered feld the far-zoe for cylder ze ad delectrc cotat mlar to thoe ued to repreet vegetato elemet. hh polarzato v-v polarzato 0 FDTD MoM 0 FDTD MoM 0 0 σ hh /(π a ) [dbm] 0-0 σ vv /(π a ) [dbm] catterg agle θ [deg] catterg agle θ [deg] 0 FDTD MoM 0 FDTD MoM 0 0 σ hh /(π a ) [dbm] 0-0 σ vv /(π a ) [dbm] catterg agle θ [deg] catterg agle θ [deg] Fgure 4.6 Normalzed btatc catterg coeffcet v. catterg agle for fte delectrc cylder computed wth FDTD ad MoM. The parameter of the cylder are L =0.0λ0 o o a = 0.4λ ε = 8 6 ad cdet agle θ = 0 (top) ad θ = 80 (bottom). 0 c 47

62 4.4 Cro ecto computato ug FDTD ad Me ere oluto For the aaly coducted th tudy two parameter of teret are the aborpto ad catterg cro ecto. Hece a fal tet to be coducted to ae the accuracy of the FDTD approach the comparo of aborpto cro ecto computed ug FDTD method ad Me ere oluto. The obect uder tudy wa alo a phere. The frequecy of aaly wa the rage GHz (C-bad). The phere had a phycal dameter of 4 cm ( λ 0 ). The relatve permttvty value are 4. 0 ad The coductvty value are 0. ad 0. 5 S / m. The Yee cell ze wa choe to be = y = z = 0. 00m ( λ 0 ) m ( λ 0 ) ad m ( λ 0 ) for ε r = 4. 0 whle for ε r = 0. 0 the cell ze wa m ( λ 0 )ad m ( λ 0 ). Table 4.3 how the parameter ued aborpto cro ecto reult comparo. Table 4.3 Parameter ued for comparo de computato of aborpto cro ecto of a 4-cm-dameter phere ug aalytcal oluto ad FDTD mulato. Relatve permttvty Coductvty Yee cell ze ε = 4.0 σ = 0.S / m = m r ε = 4.0 σ = 0.S / m = m r ε =0.0 σ = 0.5 S / m = m r ε = 4.0 σ = 0.5 S / m = m r ε =0.0 σ = 0.S / m = m r ε =0.0 σ = 0.S / m = m r Fgure 4.7 to 4. how the comparo of aborpto cro ecto of a 4 - cm - dameter phere computed ug aalytcal oluto ad the FDTD approach. 48

63 6 5 Aalytcal FDTD Aborpto Cro Secto (m ) Frequecy (GHz) Fgure 4.7 Comparo of aborpto cro ecto of a 4 - cm dameter phere ug aalytcal oluto ad FDTD mulato. The properte of the phere are ε r = 4. 0 ad σ = 0.S / m. The Yee cell = 0. 00m ( λ ) Aalytcal FDTD Aborpto Cro Secto (m ) Frequecy (GHz) Fgure 4.8 Comparo of aborpto cro ecto of a 4 - cm dameter phere ug aalytcal oluto ad FDTD mulato. The properte of the phere are ε r = 4. 0 ad σ = 0.S / m. The Yee cell = m ( λ ). 0 49

64 6 5 Aalytcal FDTD Aborpto Cro Secto (m ) Frequecy (GHz) Fgure 4.9 Comparo of aborpto cro ecto of a 4 - cm dameter phere ug aalytcal oluto ad FDTD mulato. The properte of the phere are ε r =0. 0 ad σ = 0.5S / m The Yee cell = m ( λ ) Aalytcal FDTD Aborpto Cro Secto (m ) Frequecy (GHz) Fgure 4.0 Comparo of aborpto cro ecto of a 4 - cm dameter phere ug aalytcal oluto ad FDTD mulato. The properte of the phere ε = 4. 0 ad σ = 0.5S / m. The Yee cell = 0. 00m ( λ ). 0 r 50

65 6 5 Aalytcal FDTD Aborpto Cro Secto (m ) Frequecy (GHz) Fgure 4. Comparo of aborpto cro ecto of a 4 - cm dameter phere ug aalytcal oluto ad FDTD mulato. The properte of the phere are ε =0. 0 ad σ = 0.S / m. The Yee cell = m ( λ ). 0 r 6 5 Aalytcal FDTD Aborpto Cro Secto (m ) Frequecy (GHz) Fgure 4. Comparo of aborpto cro ecto of a 4 - cm dameter phere ug aalytcal oluto ad FDTD mulato. The properte of the phere are ε =0. 0 ad σ = 0.S / m. The Yee cell = m ( λ ). r 0 5

66 Fgure 4.7 to 4. how both that a) the FDTD approach ca be ued to compute the requred feld to obta the aborpto cro ecto ad b) the tegrato approach ued to compute the aborpto cro ecto atfactory. I et ecto more formato wll be gve about the tegrato approach. I the rage of aaly how th ecto ce the aborpto cro ecto drectly proportoal to the coductvty the target a the coductvty hgher the relatve umercal error the FDTD reult wth repect to the Me ere oluto decreae. Alo a epected a the cell ze reduced the umercal dpero reduced accordgly; hece decreag the error. 4.5 Computato of emvty from fte ze obect 4.5. Itroducto All ubtace at a fte abolute temperature radate electromagetc eergy. The empha etg emo modelg for pave remote eg ha bee o emo from eteve-area target. Th ecto preet the computato of the emvty of a fte-ze obect. I order to tegrate the effect of fte-ze obect to a eteve area target model t mperatve to have a good udertadg of the emo properte of fte-ze obect. Sce the purpoe to ue the reult of th tudy pave remote eg model uch a thoe decrbed [3 5] t eeded to characterze both the emo behavor of the obect ad t catterg properte. To date mot of the modelg tudy o obect for actve eg. Wth repect to the emvty tude remote eg modelg reported lterature are maly for radom urface. Th ecto focue o emo modelg for fteze obect wth arbtrary hape. I [33 5] aalytcal epreo are reported to fd the emo from a body. A geeral epreo gve for a obect wth arbtrary hape. 5

67 However to obta practcal reult aumpto were made to mae the formulato mathematcally tractable; appromato ug caocal obect have bee ued. I [ ] the equato to compute the flu of eergy emtted to a partcular drecto of pace ad wth a partcular polarzato proportoal to the aborptvty. The complety of the aalytcal relato betwee the obect Stoe parameter ad t aborpto properte determed by the geometry ad delectrc properte of the obect. The gfcace of thoe equato to compute the emo for a fte-ze aotropc medum the fact that t relate the emo from the obect to the aborpto cro ecto of the obect therefore relatg a pave emo problem to a actve catterg problem. Coequetly to model the emo from a obect a umercal method ca be ued to mulate the teracto of a terrogatg wave ad the obect amely a actve remote eg modelg ad hece fd the cro ecto of the obect. Addtoally umercal method allow for modelg of emo from a obect of arbtrary hape ad delectrc value. The FDTD approach ha bee how adequate to perform th ta [40]. I th tudy a 3D-FDTD algorthm wa developed ad ued to mulate a plae wave mpgg o a obect from everal agle ad polarzato. Both the aborpto ad the catterg cro ecto of a obect are obtaed to compute the emvty of a obect Emo from fte-ze obect The charactertc a a ource of a fte-ze obect gve by t brghte temperature emtted a pecfc drecto. Whe th emtted by a blacbody the brghte temperature equal to the phycal temperature T of the body. The rato betwee the brghte temperature of the body ad that oe of a blacbody the emvty e. I th ecto a approach to compute the emvty from a obect preeted. A mlar approach gve [5] however t vald for a eteded urface. Coder that a wave wth 53

68 polarzato tate α cdet o the obect the drecto î. The the rato of the total cattered power to the um of cattered ad aborbed power the obect amely the albedo ω computed a ca Q ω = (4.5) ab ca Q Q where ca Q ad ab Q are the catterg ad aborpto cro ecto repectvely. The aborpto cro ecto defed a the power dpated the obect uder plae wave cdece dvded by the power dety of the cdet wave [53]. Th ca be calculated by tegratg Poytg vector of the total feld over a cloed urface cotag the obect. Smlarly the catterg cro ecto formulato the total power cattered by the obect ued. Accordg to [5] the followg epreo ca be ued to obta both the aborpto ad catterg cro ecto: Q ab = S0 r r Re E H S * r da (4.6) Q ca = S0 r r Re E H S * r da (4.7) where E r H r ad E r H r are the total ad cattered feld repectvely. The fractoal power aborbed by the obect therefore a ( ˆ ) ( ˆ ) =. (4.8) α ω α 54

69 drecto ( î ) the by The quatty a ( ˆ ) α the aborptvty. The brghte temperature T α emtted the wth polarzato tate α from a obect ept at the temperature T gve ( ˆ ) = e ( ˆ )T Tα α. (4.9) Accordg to Krchhoff law f the obect thermal equlbrum the aborpto mut be equal to the emo ad therefore the emvty equal to the aborptvty: e ( ˆ ) a ( ˆ ). (4.0) α = α Numercal modelg approach Accordg to Eq. (4.5) both the aborpto ad catterg cro ecto of the obect uder aaly are eeded. They wll be computed ug the FDTD approach. A plae wave cdet at varou agle mulated ad both the total ad cattered electromagetc feld tagetal to a vrtual urface eclog the obect (Fg. 4.3 ad 4.4) are ampled at everal pot to compute the Poytg Vector at thoe pot. The reultg Poytg vector value are the tegrated over the eclog urface to compute the power ether leavg or eterg the urface. 55

70 Eclog urface Samplg pot Fgure 4.3 Vrtual eclog urface ued for computato of cro ecto. For aborpto cro ecto the eclog urface geerated the FDTD total feld rego whle for the catterg cro ecto the eclog urface geerated the cattered feld rego. Fgure 4.4 Vrtual eclog urface ued for computato of cro ecto ug the Gaua quadrature techque. For aborpto cro ecto the eclog urface ue the FDTD total feld value whle for the catterg cro ecto the eclog urface ue the cattered feld by ubtractg the cdet feld from the total feld. The amplg of the requred electromagetc feld wa performed two way: (a) ug a vrtual eclog cube (Fg. 4.3) ad (b) ug amplg pot accordg to the Gaua quadrature techque (Fg. 4.4). The former approach mplfe the amplg ce 56

71 oly the tagetal compoet at each of the face are recorded. Sce the urface wa a cube the requred computatoal pace could be reduced. The later approach requre both all rectagular compoet to be recorded ad more computatoal memory; however t provde a more prece tegrato. I the frt approach the eclog area formed by dfferetal patche. The value of the E r ad H r feld at the ceter of the patche are requred. However due to the ature of the FDTD algorthm the feld are computed at hfted poto wth repect to each other the grd. To crcumvet th problem oce the patche formg the eclog urface were defed the value of the E r ad H r feld at the ceter of them were obtaed by performg a tr-lear terpolato ad the ampled. The terpolated value for E r ad H r were aumed to be cotat the complete patch area. To compute the aborpto cro ecto the equato Q ab = S0 r r Re E H S * r da wa ued. Due to the FDTD dcretzato the umerator wa replaced by the followg appromato: P = N = v * { E H } Re v a. (4.) where N the total umber of patche ad a = beg the legth of each patch de equal to the legth of the FDTD cell. The cdet power dety wa computed aalytcally a: 57

72 S r Ec =. (4.) η where E v c ad η are the cdet feld ad the trc mpedace of free pace repectvely Numercal computato of emvty of lole obect Before computg ad aalyzg the emvty from everal target the emvty from lole target computed. The purpoe to ae the method to compute the emvty wth a a pror ow reult. Sce the lo zero the the aborptvty ad hece the emvty mut be zero too. The frt aaly wa coducted o a cylder wth dameter ad heght equal to λ 0. Although the emvty wa reduced by reducg the cell ze hece by creag the umber of pot per wavelegth for the mulato the emvty wth repect to cdet agle howed cotecy due to reoace. Therefore t wa cocluded that cylder wth dmeo the ame a the wavelegth hould be avoded due to reoace problem. For a 5 5 GHz cdet wave the relatve dmeo of the cylder ( d = λ 0 ad 6 5 λ 6 h = 0 ) wth repect to 0 λ wa maller therefore the reoace wa avoded. Fgure 4.5 how that the emvty wth repect to cdet agle approache to a zero value a the umber of pot per wavelegth creaed. Lewe the tred of the curve repreetg the emvty value ted to be flat. The pea at o 45 due to umercal dpero. 58

73 = m = m = m 0.05 Emvty Icdet agle θ (deg) Fgure 4.5 Emvty of lole cylder wth repect to cdet agle whe varyg the cell ze from m to 0.005m. The dmeo of the cylder are: dameter = 0.05m ad heght = 0.05m. The delectrc properte are ε r = 4. 0 ad σ = 0S / m. The frequecy of aaly 5 GHz. The polarzato vertcal. Fgure 4.6 ad 4.7 how the emvty of a cylder wth dmeo d = 0.8λ0 ad h = 0.8λ for 6 GHz ad 5 GHz cdet wave repectvely. A prevou cae the 0 emvty how both a tedecy to approach the correct value a the umber of pot per wavelegth creaed; ad a pea at o 45 due to umercal dpero. 59

74 = m = m = m 0.05 Emvty Icdet agle θ (deg) Fgure 4.6 Emvty of lole cylder wth repect to cdet agle whe varyg the cell ze from from m to 0.000m. The dmeo of the cylder are: dameter = 0.04m ad heght = 0.04m. The delectrc properte are ε r = 4. 0 ad σ = 0S / m. The frequecy of aaly 6 GHz. The polarzato vertcal. 60

75 = m = m = m 0.05 Emvty Icdet agle θ (deg) Fgure 4.7 Emvty of lole cylder wth repect to cdet agle whe varyg the cell ze from from m to 0.000m. The dmeo of the cylder are: dameter = 0.04m ad heght = 0.04m. The delectrc properte are ε r = 4. 0 ad σ = 0S / m. The frequecy of aaly 5 GHz. The polarzato vertcal. Addtoally the performace of the FDTD tegrato approach wa aeed by computg the emvty of a perfect-electrc-coductor (PEC) obect. I cae of a PEC obect all eergy cattered. Therefore the aborptvty ad hece emvty zero. For mplcty to repreet the coductg obect the computatoal pace a perfect electrc coductor (PEC) cube wa ued. Th way the PEC model wa appled by ettg the electrc feld taget to the obect urface to be zero. 6

76 = m = m = m 0.05 Emvty Icdet agle θ (deg) Fgure 4.8 Emvty of PEC cube wth repect to cdet agle whe varyg the cell ze from from m to 0.000m. The dmeo of the cube are: de = 0.04 m. The frequecy of aaly 6 GHz. The polarzato vertcal. Fgure 4.8 ad 4.9 how that for cdet frequece of 6 GHz ad 5 GHz a epected the emvty ted to approach zero a the umber of pot per wavelegth creaed. The effect of the umercal dpero alo vble. However cotrary to prevou cae the cdet wave doe ot travel through the PEC obect therefore the effect le at o 45 whe compared wth o 0 ad o 90 cdece. 6

77 = m = m = m 0.05 Emvty Icdet agle θ (deg) Fgure 4.9 Emvty of PEC cube wth repect to cdet agle whe varyg the cell ze from from m to 0.000m. The dmeo of the cube are: de = 0.04m. The frequecy of aaly 5 GHz. The polarzato vertcal Numercal computato of emvty of loy delectrc obect The FDTD approach preeted ca be ued to obta the emvty of obect wth arbtrary hape. I th ecto the emvty calculato are preeted for everal fte-ze obect. A relatve permttvty ε = 4. 0 ad value for coductvty σ ragg from 0. to r 0.5S m were ued. The frequecy of aaly wa the md-rage of C bad ( 6 GHz ). The emvty geerally a fucto of both θ ad ϕ agle defed a how Fgure 4.. However ce the obect preeted th ecto have ymmetry aroud the a of rotato th cae z-a the emvty value are depedet of ϕ hece all reult are how oly a a fucto of θ. Both the vertcal ad horzotal polarzato are codered the FDTD mulato. The cdet plae wave wa a modulated Gaua pule wth 63

78 cetral frequecy at mm wa ued. 6 GHz. To repreet the FDTD pace a regular grd wth a cell ze of Reult ad aaly Fgure 4.0 how that the emvty from a phercal obect due to t ymmetry oly a fucto of delectrc properte. For the permttvty value uder aaly the emvty hgher for hgh coductvty (lo) hece a hgher cotrat. The reult obtaed from the phere are a good referece to aalyze ad udertad the reult from other geometrcal hape. Although mall the FDTD-computed emvty howed ome varato wth repect to agle of obervato. Th wa a coequece of the umercal dpero due to the dcretzato of the computatoal pace. Th varato reduce a the ze of the grd cell reduced. Fgure 4.0 Emvty at 6 GHz for a phere ( r = λ0 ) wth ε = 4. 0 ad σ = 0.0.5S / m. r 64

79 d z θ h y φ Fgure 4. Geometry ued for FDTD mulato of plae wave mpgg o ladme-le obect ( d =.4λ0 ad h = 0.8λ0 ) σ = 0.00 S/m σ = 0.5 S/m σ = 0.50 S/m σ = 0.00 S/m σ = S/m σ = S/m Emvty Icdet agle θ (deg) Fgure 4. Emvty at 6 GHz for a crcular cylder wth ε r = 4. 0 ad σ = S / m for vertcal polarzato. Dmeo of the cylder are d =.4λ0(cm ) ad h = 0.8λ0 (4cm). Fgure 4. how the emvty from the ladme-le obect for vertcal polarzato. For the permttvty aalyzed t ca be oberved that for low coductvty the 65

80 emvty how more varato wth repect to the obervato agle havg a pea value at about 50. A the coductvty creaed the emvty ted to be more uform wth repect to the obervato agle. Fgure 4.3 how the emvty from the ladme-le obect for horzotal polarzato. Smlarly to prevou cae t ca be oberved that for low coductvty the emvty how more varato wth repect to the obervato agle havg alo a pea value at about 50. A the coductvty creaed the emvty ted to be more uform wth repect to the obervato agle σ = 0.00 S/m σ = S/m σ = S/m 0.6 Emvty Icdet agle θ (deg) Fgure 4.3 Emvty at 6 GHz for a crcular cylder wth ε r = 4. 0 ad σ = S / m for horzotal polarzato. Dmeo of the cylder are d =.4λ0(cm ) ad h = 0.8λ0 (4cm). Regardg the emvty of the cylder repreeted Fg. 4.4 t ca be oberved that t ha alo more varato wth repect to the obervato agle for low coductvty havg a mmum at about 30 for both vertcal ad horzotal polarzato (Fg. 4.5 ad 4.6). 66

81 However t how a hgher emvty for horzotal polarzato. Smlarly a the ladmele obect the emvty ted to be more uform depedetly of the obervato agle whe the coductvty tae a hgher value. d z θ h y φ Fgure 4.4 Geometry ued for FDTD mulato of plae wave mpgg o cylder ( d =.λ0(6cm) ad h = 3.6λ0(8cm ) ) σ = 0.00 S/m σ = 0.5 S/m σ = 0.50 S/m σ = 0.00 S/m σ = S/m σ = S/m Emvty Icdet agle θ (deg) Fgure 4.5 Emvty at 6 GHz for a crcular cylder wth ε r = 4. 0 ad σ = S / m for vertcal polarzato. Dmeo of the cylder are d =.λ0(6cm) ad h = 3.6λ0(8cm ). 67

82 σ = 0. S/m σ = 0.3 S/m σ = 0.5 S/m 0.6 Emvty Icdet agle θ (deg) Fgure 4.6 Emvty at 6 GHz for a crcular cylder wth ε r = 4. 0 ad σ = S / m for horzotal polarzato. Dmeo of the cylder are d =.λ0(6cm) ad h = 3.6λ0 (8 cm). I order to ga more ght the behavor of emvty of fte ze obect wth repect to dmeo ad delectrc properte more obect are teted. I followg le the obect a well a the emvty plot are how. 68

83 d z θ h y φ Fgure 4.7 Geometry ued for FDTD mulato of plae wave mpgg o cylder ( d =.6λ0(8cm) ad h =.6λ0 (8cm) ) σ = 0.00 S/m σ = 0.5 S/m σ = 0.50 S/m σ = 0.00 S/m σ = S/m σ = S/m Emvty Icdet agle θ (deg) Fgure 4.8 Emvty at 6 GHz for a crcular cylder wth ε r = 4. 0 ad σ = S / m for vertcal polarzato. Dmeo of the cylder are d =.6λ0(8cm) ad h =.6λ0 (8cm). 69

84 z θ d h y φ Fgure 4.9 Geometry ued for FDTD mulato of plae wave mpgg o cylder ( d =.0λ0 (0 cm) ad h =.λ0(6 cm) ) σ = 0.00 S/m σ = 0.5 S/m σ = 0.50 S/m σ = 0.00 S/m σ = S/m σ = S/m Emvty Icdet agle θ (deg) Fgure 4.30 Emvty at 6 GHz for a crcular cylder wth ε r = 4. 0 ad σ = S / m for vertcal polarzato. Dmeo of the cylder are d =.0λ0 (0 cm) ad h =.λ0(6 cm). 70

85 z θ d h y φ Fgure 4.3 Geometry ued for FDTD mulato of plae wave mpgg o cylder ( d =.λ0(6cm) ad h =.8λ0 (9cm) ) σ = 0.00 S/m σ = 0.5 S/m σ = 0.50 S/m σ = 0.00 S/m σ = S/m σ = S/m Emvty Icdet agle θ (deg) Fgure 4.3 Emvty at 6 GHz for a crcular cylder wth ε r = 4. 0 ad σ = S / m for vertcal polarzato. Dmeo of the cylder are d =.λ0(6cm) ad h =.8λ0 (9cm). 7

86 z θ d h y φ Fgure 4.33 Geometry ued for FDTD mulato of plae wave mpgg o cylder ( d =.8λ0 (4 cm) ad h = 0.4λ0(cm) ). Emvty σ = 0.00 S/m σ = 0.5 S/m σ = 0.50 S/m σ = 0.00 S/m σ = S/m σ = S/m Icdet agle θ (deg) Fgure 4.34 Emvty at 6 GHz for a crcular cylder wth ε r = 4. 0 ad σ = S / m for vertcal polarzato. Dmeo of the cylder are d =.8λ0 (4 cm) ad h = 0.4λ0(cm). 7

87 The plot how that the cremet the emvty wth repect to the cremet coductvty ot lear. Alo for relatvely low value of coductvty the emvty how otceable varato wth repect to the agle of vew. However a the coductvty creae the value of the emvty ted to approach a pecfc value depedetly of the vew agle. For the cae aalyzed th chapter the value were aroud 0.5. I order to ga a ght the behavor of the catterg ad aborpto cro ecto wth repect to the parameter ued th tudy to etmate the emvty a qualtatve aaly preeted by Ihmaru could be ued [8]. The aaly epla how the catterg ad aborpto cro ecto are related to the geometrc cro ecto wavelegth ad delectrc cotat of a obect. It tate that f the ze of a partcle much greater tha a wavelegth the total cro ecto twce the geometrc cro ecto tot Q approache geo Q of the partcle a the ze creae. Addtoally the total aborbed power whe the partcle very large caot be greater tha the product of the cdet power dety ad the geometrcal cro ecto gve by S geo Q ad thu the aborpto cro ecto ab Q approache a cotat omewhat le tha the geometrc cro ecto. Ug the method developed Chapter 4 of th tudy the emvty would alo be cotat. The aaly alo tate that f the ze of the partcle much maller tha a wavelegth the catterg cro ecto ca Q verely proportoal to the fourth power of the wavelegth ad proportoal to the quare of the volume of the partcle; hece the catterg cro ecto mall. For th codto the total cro ecto would be motly due to the aborpto ad hece the emvty would be hgh. Although ome udertadg ca be gaed from that aaly thee two oppote codto certaly do ot repreet the cae preeted th tudy ce the dmeo of the cylder uder tudy ad the cdet wavelegth are ot too dfferet. The the value of 73

88 the rato ze/wavelegth of the obect uder aaly would be the mddle rage of thoe etreme cae requrg a more eteve aaly. I cocluo the aaly of emvty urface ad fte ze obect optc rego ha bee tuded for a log tme; however the emvty of obect the mcrowave rego a ogog reearch [ ]. Hece more tude are eeded order to etablh a relatohp betwee emvty ad obect properte uch a dmeo delectrc a well a frequecy ad polarzato of terrogatg wave. However at th pot due to both teve ad eteve computatoal reource requred t left a a topc for future tudy. I ummary th ecto a 3D-FDTD algorthm wa developed to compute emvty from fte-ze obect. Some tedece of the emvty plot wth repect to the properte of both the obect ad the terrogatg wave ca be oberved. The method to compute the aborpto ad catterg cro ecto of fte ze obect wll be tegrated to a pave remote eg model gve Chapter 5. 74

89 CHAPTER 5 MODELING OF MICROWAVE EMISSION FROM IRREGULAR LAYER 5. Itroducto The umercal modelg of mcrowave emo from a rregular layer coverg a homogeeou half pace gve; cludg the catterg mecham affectg the emo. The catterg behavor of the layer characterzed by the umercally-geerated phae matrce. The approach to geerate the phae matrce alo gve here. I order to valdate the phae matrce both a eergy coervato tet ad comparo wth aalytcal reult are carred out. The layer alo characterzed by t effectve permttvty whch etmated by a lear model. Fally the umercal model valdated by comparg model-predcted reult wth feld meauremet. 5. Cotructo of phae matr for a crcular cylder I order to ue the radatve trafer formulato for the homogeeou layer the gle catterg phae matr for the catterer eeded. The phae matr a quatty that relate the Stoe vector of the cattered wave to that of the cdet wave. The value of the cattered feld r H r eeded for cotructo of the phae matr E are computed at a cotat dtace R from the ceter of the obect ad at agle ( θ ϕ ). Fgure 5. how the geometry of the obect for computato of the phae matr. Sce both vertcal ad horzotal cdece polarzato are eeded the FDTD mulato eecuted for the ame polar agle but varyg the cdece polarzato 75

90 accordgly. The umber of both the cdet ad cattered agle choe accordg to the Gaua quadrature techque. At each mulato the mpgg feld wll have a cdece agle ( θ φ ). The feld wll be computed ad recorded at ( φ ) at a dtace R. θ d z θ R E r H r h y φ Fgure 5. Geometry of the obect for computato of the phae matr. I the ame mulato the cattered feld at dfferet frequece ca be computed; however curretly our approach th requre that the delectrc properte be cotat whch ot the cae for the delectrc properte of vegetato compoet. Therefore the FDTD mulato wll be doe for gle frequecy. Th alo allow reducg the computato tme by electg the parameter of the terrogatg wave uch a way that the tme-doma durato of the cdet pule hort. For th purpoe a MATLAB program wa develop to geerate the coordate of the cattered-feld poto. Sce the obect uder tudy how ymmetry aroud oe a ( z ) the cdet agle wll vary oly θ. Thee coordate ( y z ) put fle read by the FDTD program ad the cattered feld are obtaed. Due to the hft the E r ad H r feld poto the Yee cell a trlear terpolato wa performed durg the mulato to obta both E r ad H r feld at the ame pot. The output of the program are the cattered feld 76

91 r H r. I the approach E E r E r y ad E r z are ormalzed by E r c ad H r H r y H r z by H r = E r η where η the charactertc mpedace of the hot medum repectvely. c c Sce the rectagular compoet of the E r ad H r feld are computed the FDTD program a coordate traformato eeded to fd the phercal compoet eeded for computato of the phae matr. For ae of completee the R compoet alo how the epreo (5.); however oly the tagetal compoet of the cattered feld the drecto of propagato are eeded ce thoe are the compoet eeded to compute the power dety. The coordate traformato gve by AR θ coφ θ φ coθ A = Aθ coθ coφ coθ φ θ Ay (5.) A φ coφ 0 A φ z where A repreet ether E or H. For emo problem the emtted tete ad phae matrce are depedet of azmuthal agle. Therefore oly the zeroth-order Fourer compoet of the tete ad the phae matrce epaded wth repect to the azmuthal agle are eeded. The ze of the tety vector ad phae matrce ued the radatve trafer formulato are depedet upo the umber of cdece ad catterg agle choe a well a the umber of Stoe parameter computed. However atural emo oly the frt two Stoe parameter the vertcal ad horzotal polarzato are coupled together. Hece f N polar agle are choe accordace wth the Gaua quadrature techque (or ay other tegrato techque ued for th purpoe) the tety vector wll be ad the phae matrce wll have a dmeo of 77 N -colum vector N N for each Fourer compoet. Fgure 5. how the computatoal tructure of the phae matr for the frt two Stoe Parameter.

92 Icdet agle N N Scattered agle N VV VH v-cat N HV HH h-cat v-c h-c Fgure 5. Computatoal tructure of the phae matr for the frt two Stoe parameter. Followg the approach how [3] the phae matrce for the frt two Stoe parameter are computed. However [3] the catterer are repreeted by phere allowg to compute the requred cattered feld aalytcally. Sce o aalytcal oluto et for the cattered feld from fte-ze cylder th tudy the cattered feld are olved ug a umercal approach. Wth the cattered feld compoet ow all the elemet of the phae matr ca be computed. For may applcato oly le ad cro polarzato are of teret; for uch cae oly the frt four elemet of the phae matr are eeded a P = κ ( θ φ) E 4π r η 0 0 * ( Eθ ' Hφ ' ) v Re * ( Eφ ' Hθ ' ) cdece vcdece ( E θ ' r φ ' H ( E H * φ ' ) hcdece * θ ' ) hcdece (5.) where κ ( θ φ) the effectve volume-catterg coeffcet ( Np / m ) ad o the catter umber dety. A etmato of κ ( θ φ) gve by κ ( θ φ) = 0 Q ( θ φ) (5.3) Hece 78

93 P 4π r η = Q ( θ φ) E 0 * ( Eθ ' Hφ ' ) v Re * ( Eφ ' Hθ ' ) cdece vcdece ( E θ ' r φ ' H ( E H * φ ' ) hcdece * θ ' ) hcdece (5.4) where Q (θ ) the catterg cro ecto of the gle compoet. Aother mportat parameter for characterzg a homogeeou medum t aborpto lo repreeted by the volume-aborpto coeffcet κ ( θ φ) ( θ φ ) Q ( θ φ ) a gve by κ = (5.5) a 0 a where Q ( θφ) the aborpto cro ecto. The ut of κ ( θ φ ) a a are alo Np / m. The total cro ecto alo ow a the etcto cro ecto for a partcle e ( θ φ ) Q ( θ φ ) Q ( θ φ ) Q = (5.6) a Mot of the tude have dealt wth the pecal cae of phercal partcle. For thee partcle or for ophercal partcle wth radom oretato κ e ad κ reduce to calar. However for th tudy oe of thoe codto are met becaue κ e ad κ are agle depedet hece they are treated dfferetly. Both parameter the catterg ad aborpto cro ecto are computed umercally ug the approach gve Chapter 4. The requred feld are to be computed at a average dtace from the ceter of the catterer. Although the approach propoed here vald for ay arbtrary-hape obect the aaly wll be focued o cyldrcal-hape obect. I order to valdate the umercal model by comparg t reult wth feld meauremet the compoet of the rregular layer were choe to be cor tal repreeted by vertcal delectrc cylder. The bottom half-pace bottom wa choe to be ol wth rough urface. 79

94 Fgure 5.3 how a geeral cofgurato for modelg emo from ol the preece of vegetato compoet. The vegetato cover modfe the emo from ol by catterg the ol emo ad addg t ow emo to the brghte temperature read by a radometer. The vegetato compoet are both modeled a delectrc cylder ad aumed to be all vertcal. The cylder are codered to be homogeeou crcular loy ad of the ame legth ad dameter. The groud uderlyg the lab repreeted by a homogeeou loy delectrc halfpace ad the terface betwee the lab ad groud tae to be rough wth the tadard devato of heght σ ad correlato legth L decrbg the degree of vertcal ad horzotal roughe. ε 0 µ 0 ε r µ r σ Sol Rough urface Fgure 5.3 Cofgurato for modelg of emo from ol the preece of vertcal cyldrcal vegetato compoet. Delectrc cylder are ued to model the vegetato compoet. 80

95 θ d z θ h y φ φ E H Fgure 5.4 Geometry ued for FDTD mulato of plae wave mpgg o cylder. For emo from ol mot of the data avalable are baed o a aaly doe at L - bad. Therefore order to compare predcto of the model wth feld meauremet the aaly doe at a frequecy of.4ghz at vertcal ad horzotal polarzato. Fgure 5.4 llutrate the geometry for the cylder the lab ued for the FDTD mulato. 5.3 Cotructo of phae matr of cor tal repreeted by cylder A aaly correpodg to cor tal repreeted by vertcal cylder wa coducted (Fgure 5.5). The aaly wa performed at L bad. The delectrc properte of cor tal were obtaed from a tudy reported [56]; whch meauremet of vegetato compoet at ad GHz bad a a fucto of moture cotet ad mcrowave frequecy were performed. The materal meaured cluded the leave ad tal of cor ad wheat. Delectrc meauremet alo were made of the lqud cluded the vegetato materal after t wa etracted from the vegetato by mechacal mea. 8

96 Fgure 5.5 Cor tal repreeted by delectrc cylder. For the aaly carred out th tudy the volumetrc moture cotet m v wa codered to be 0. 5 order to eep the value of ε relatvely low o that the computer memory requremet wa ot hgh. The delectrc cotat for cor tal would the be ε = ε ε = Smlarly to eep computer memory requremet maageable the heght of the cor tal wa et to be.5λ 0 ( 30 cm ) ad t dameter wa 0.5λ 0 ( 3 cm ). A more detaled report about tal dmeo ca be foud [57]. The rage at whch the cattered feld wa computed obtaed from a average dtace amog catterer tag a value of cor feld dety from [58]. Wth the method how Chapter 4 ad the delectrc properte tae from [57] ad dmeo decrbed above both the aborpto ad catterg cro ecto of the cylder repreetg a cor tal were computed (Fgure 5.6). 8

97 V-pol H-pol Aborpto Cro Secto (m ) Icdet agle θ (Deg) (a) V-pol H-pol Scatterg Cro Secto (m ) Icdet agle θ (Deg) (b) Fgure 5.6 (a) Aborpto ad (b) catterg cro ecto umercally computed for a delectrc cylder repreetg a cor tal. 5.4 Valdato of phae matrce for eergy coervato Before tegratg the umercally-computed phae matr for a fte-ze obect to the layer model t eed to be valdated for eergy coervato whch requre that for each cdet agle ( θ φ ) 83

98 π π p 4π 0 0 P pq θ dθ d ( φ φ ) = (5.7) where P deote the elemet of the phae matr P( θ θ ) pq ad the ummato over p clude the elemet of the frt two Stoe parameter ( I v I h ) [5]. o Aumg φ = 0 equato (5.7) ca be alo epreed a pqdµ dφ = p π 4π 0 P (5.8) where µ the drectoal coe ( co θ ) for the cattered feld..5.4 V-pol H-pol.3. Eergy coervato Icdet agle θ (Deg) Fgure 5.7 Eergy coervato tet reult for umercally-computed * phae matrce S. 84

99 Fgure 5.7 how the eergy coervato tet reult for the umercally-computed phae matr * S wth repect to the cdet agle θ. Reult how that the eergy coerved at each cdet agle. The ame reult were obtaed whe tetg the phae matr * T. 5.5 Verfcato of computato of bacward ad forward phae matrce I order to verfy the preco of umercally-computed phae matrce a comparo wth aalytcally-computed phae matrce wa carred out. For th purpoe a cofgurato a how Fgure 5.0 ued. The phae matrce decrbg the catterg ad aborpto behavor of the homogeeou layer are aumed to have a behavor le the oe from a phere. The parameter ued for the comparo are gve Table 5.. Table 5. Parameter ued for computato of emo from groud covered wth a homogeeou layer wth phae matrce mlar to thoe of a phere Parameter Value Scatterer relatve permttvty ε r = Effectve permttvty of layer ε rl = Permttvty of top half-pace ε ra = Scatterer volume fracto v = 30% Scatterer radu 3 r = 0 m Slab thce d = 0. 00m Bacgroud relatve permttvty ε rb = Permttvty of bottom half pace ε = f rg Layer temperature o T L = 93 K Bottom half pace temperature o T B = 93 K Frequecy f = 35GHz Parameter of urface (groud) σ = 0.0 σ = L =.00 L = m Parameter of top boudary σ = 0.50 σ = m m L =.50 L = m -3-3

100 The reult how Fgure 5.8 are obtaed ug phae matrce computed wth (a) feld computed aalytcally wth Me oluto; (b) FDTD-computed feld wth the phae matrce ormalzed by cotat aalytcal cro ecto (FDTD); ad (c) FDTD-computed feld wth phae matrce ormalzed by the correpodg catterg cro ecto ug Q ca S = 0 r r Re E H S * r da where E r ad H r are the cattered electrc ad magetc feld repectvely; dety of the cdet wave ad S 0 the vrtual urface eclog the obect. S r the power 86

101 60 55 Me FDTD FDTD Brghte Temperature ( o K) θ (Deg) (a) Me FDTD FDTD Brghte Temperature ( o K) θ (Deg) (b) Fgure 5.8 Emo from urface wth a cover of a layer wth phae matrce characterzed a thoe of a phere. The phae matrce were obtaed ug Me-formulato feld FDTDcomputed feld (phae matr ormalzed by aalytcal catterg cro ecto FDTD) ad FDTD-computed feld (phae matr ormalzed by catterg cro ecto ug method decrbed chapter 4 FDTD) (a) V-polarzato ad (b) H-polarzato. 87

102 Addtoally although the catterg cro ecto for a phere depedet of agle ad polarzato due to FDTD-related umercal dpero the umercally-computed cro ecto deped lghtly o cdet agle. Hece order to aure eergy coervato the ca c thrd cae the phae matr ormalzed by Q ( pol) θ. Fgure 5.8 how the model-predcted brghte temperature for both vertcal ad horzotal polarzato. A very good agreemet betwee the reult obtaed by ug the aalytcally-computed ad the umercally-computed phae matrce ca be oberved. 5.6 Emo catterg mecham The geeral cofgurato of a layer betwee two half pace gve Fgure 5.9. The total emo to a top homogeeou half pace (medum ) compoed of cotrbuto from both the emo by a homogeeou layer (medum ) ad the emo from a homogeeou half-pace (medum 3). Becaue of the homogeete the layer ad the dcotuty at the terface betwee medum ad meda ad 3 repectvely; the emo wll further udergo multple catterg procee; whch are decrbed th ecto. Sce the cluo volume fracto very low the layer ad the hot medum delectrc properte are the ame a thoe of the medum the effectve permttvty of medum very cloe to the permttvty of medum ; hece the reflectvty at the top boudary eglected. I order to mplfy the aaly the emo from the homogeeou layer decompoed t upwellg ad dowwellg compoet whch are referred to a u d repectvely. The emo from the bottom half-pace wll be referred to a u g. u u ad 88

103 Addtoally the volume dety of the layer codered to be low; hece the dtace betwee vegetato compoet aumed to be large o that oly the frt order catterg codered. ε 0 µ 0 d S T S* T* ε r µ r σ Sol ε rg µ rg σ g Rough urface Fgure 5.9 Geeral geometry for modelg of emo from ol the preece of homogeeou layer. Q u u Q ~ QT R3u d Q T u g Medum u u u d T T Medum ~ R 3 u g Medum 3 Fgure 5.0 Emo catterg mecham for upwellg ad dowwellg emo from the layer ad upwellg emo from the bottom half-pace. The multple catterg mecham geometry of the mcrowave emo problem for a homogeeou layer (medum ) wth rregular boudare how Fgure 5.0. The meda ad 3 are aumed to be homogeeou half pace ad the layer bacward ad forward volume catterg phae matrce are the umercally computed gle catterg phae matr 89

104 of the layer compoet S ad ~ T repectvely. The urface phae matr R 3 repreet the effectve reflectvty matr at the terface betwee meda ad 3; t computato gve below. Uder local thermodyamc equlbrum the upwellg ad dowwellg emtted tete epreed a brghte temperature from a elemetary layer of optcal thce τ at level τ are gve by the Krchhoff-Plac law [3] r ( - ω( θ φ) ) T( τ ) τ u ( τ; θ φ) = u ( τ ; θ φ) = U ( θ φ) ε (5.9) u d where T ( τ ) the temperature profle of the layer; U the dagoal matr of the coe of the cattered agle ω the gle-catterg albedo ε r the relatve permttvty of the layer ad the layer relatve permeablty µ r aumed to be. Both the gle-catterg albedo ad the optcal thce have agle depedecy. Equato (5.9) ued to determe the homogeeou layer upward ad dowward emo ource amely uu ad u d repectvely. The emo from lower homogeeou half-pace u g ca be determed ug ( θ ) u g = T g *ε g (5.0) where g T ad ( θ ) ε g are the temperature ad emvty of the groud repectvely. Fgure 5. how the catterg proce at the terface betwee homogeeou pace (medum ) ad homogeeou half-pace (medum 3) due to a ut cdet tety. S the medum- bacward catterg phae matr for upward cdece. The urface phae matr R 3 to accout for the catterg at the lower boudary obtaed by ug the Itegral Equato Method []. 90

105 Followg thoe catterg procee the correpodg effectve reflectvty phae ~ matr R 3 at the boudary obtaed the followg way: ~ R = R R S R R S R S R R S R S R S R L (5.a) = ( S R ) 3 ~ R = R (5.b) 3 3 ( I S ) ~ 3 = R 3 R3 R (5.c) 3 3 where I repreet the detty matr. Medum R 3 R3S R R 3 3S R3S R3 R 3S R3S R3S R3 Medum S S S R 3 R R R Medum 3 Fgure 5. Effectve reflectvty matr from the medum - medum 3 terface. compoet a: The effectve reflectvty phae matr ca be epreed term of t Fourer ~ m m * m 3 3 m = R I fm S R3 R where π fm = π m = 0 m (5.) However for emo oly the zeroth order of the Fourer compoet ued hece * 0 ~ = R3 I fm S R3 R (5.3) 9