Polarization Invariants and Retrieval of Surface. Parameters Using Polarization Measurements in. Remote Sensing Applications

Transcription

1 Thi i a rerint coy of an article ublihed in Alied Otic (Shetoaloff Yu. K. Polarization Invariant and Retrieval of Surface Parameter Uing Polarization Meaurement in Remote Sening Alication, Alied Otic, 011, Vol. 50, I. 36, DOI: /AO ). The final verion of the article can be found at the following URL on the OSA webite: htt:// Polarization Invariant and Retrieval of Surface Parameter Uing Polarization Meaurement in Remote Sening Alication Yuri K. Shetoaloff 1,* * Correonding author: he169@yahoo.ca Uing olarization meaurement in remote ening and otical tudie allow retrieving more information. We conider relationhi between the reflection coefficient of lane and rough urface for linearly olarized wave. Certain olarization roertie of reflected wave and olarization invariant, in articular at incident angle of forty five degree, allow finding amlitude and hae characteritic of reflected wave. Baed on thi tudy, we introduce method for finding dielectric ermittivity, temerature and geometrical characteritic of oberved urface. Exerimental reult rove that thee method can be ued for different ractical uroe in technological and remote ening alication, in a broad range of electromagnetic ectrum. OCIS code: , , ,

2 1. Introduction The vector nature of electromagnetic field and wave i rereented by olarization arameter. Polarization henomena are cloely related to coherence of electromagnetic wave. Incoherent electromagnetic emiion i non-olarized. On the other hand, it i known that if we receive uch a non-olarized ignal uing a narrow bandwidth filter or a olarization filter for light, then the outut will be a ignal that ha ome degree of olarization. So, olarization characteritic deend on the ignal arameter, a well a ignal interaction with the receiving and emitting device and roagation media. In thi tudy, we aume that characteritic of emitting and receiving device are matched, that i, the receiving device obtain the whole ectrum of reflected or emitted ignal, o that the ignal olarization roertie are not ditorted by the receiver. Polarization roertie of electromagnetic ignal are often decribed by arameter that can be conidered, from a mathematical erective, to be indeendent [1]. On the other hand, the ignal olarization roertie are defined by characteritic of emitting device and medium that interact with thi ignal. In thi ene, the ignal olarization arameter are deendent, although not necearily in a direct way. For intance, reflection coefficient on arallel and erendicular olarization both deend on the dielectric ermittivity of reflecting medium. Once exoed, uch inherent relationhi can enhance our undertanding of the nature of electromagnetic ignal and their interaction with media. In articular, they allow u to extract more information from olarization meaurement, and increae the

3 accuracy of their interretation. However, in many intance, the relationhi of olarization arameter and characteritic of tudied object i not traightforward. Thi i why the interretation of olarization meaurement can be ambiguou. Given thi, the exitence of ecial obervation condition, when olarization arameter and characteritic of tudied object have imle hyical, and accordingly mathematical, relationhi, can be beneficial for ractical alication. Such olarization invariant are dicued in thi article. Firt, we reent the reult of a theoretical tudy of olarization roertie of linearly olarized electromagnetic wave reflected from flat and rough urface. Baed on thi tudy, we introduce method for determining characteritic of reflected and refracted ignal, and method for finding dielectric ermittivity, temerature and geometric characteritic of oberved object. We alo ugget a calibration method for determining the noie level. Then, in a earate ection, we reent the reult of exerimental tudie of the rooed method and dicu their ractical alication.. Relationhi of reflection coefficient and emiivitie of linearly olarized wave. Retrieving a medium arameter Let u conider a flat urface. In our tudy, we ue the direct relationhi between the reflection coefficient on arallel olarization and the reflection coefficient on erendicular olarization, at the ame arbitrary incident angle. In thi, we follow the aroache uggeted in [,3]. Let u conider the comlex reflection coefficient R 3

4 and R, which correond, reectively, to arallel and erendicular olarization. The magnetic ermittivitie of both media are aumed to be equal to one. ε in α coα R =. (1) ε in α + coα ε coα ε in α R =. () ε coα + ε in α Here, α i the incidence angle (angle between the normal to the urface and the direction of obervation) and ε i the dimenionle dielectric ermittivity, which i comlex-valued. Therefore, in general, the reflection coefficient incororate hae information. Formula () define the Frenel reflection coefficient on arallel olarization, while formula (1) define the Frenel reflection coefficient on erendicular olarization, taken with the ooite ign. The reaon for the ign change i thi. At normal angle of incidence α = 0, reflection coefficient have to be equal, becaue the obervation condition for the reflected wave are identical in thi cae, and meaurement on erendicular and arallel olarization are inditinguihable. Therefore, the comlex value of reective reflection coefficient hould be equal. Thi i oible only if the reflection coefficient are reented in the form of (1) and (). Indeed, in thi cae, we have for the normal angle of incidence R = = ( ε 1 )/( ε + 1) R. If we would like to ue incident wave a the reference oint for the hae of reflection coefficient, then we would obtain R =, which i incorrect, becaue, a we found out, obervation at normal angle R of incidence on both olarization are identical, and conequently reflection coefficient hould be the ame. The aforementioned conideration i by no mean a 4

5 rincial one, becaue, eentially, we are talking about what reference oint i to be taken to calculate the hae hift between the comlex reflection coefficient. So, if one decide to ue (1) with the ooite ign, that i in a traditional form that comute the hae hift relative to incident wave, then all reult that we obtain in thi article remain valid if one ubtitute R by ( R ). Note that (1) and () alo rereent the general cae of reflection from the flat boundary between medium with arbitrary dielectric ermittivitie ε 1 and ε, where the index correond to reflecting medium. In thi cae, we hould aume in (1) and () that ε = ε / ε 1. So, all reult that we obtain below are alo valid for the cae when the incident wave come from a medium whoe dielectric ermittivity differ from one. Certainly, the dielectric ermittivity of both media can be comlex. There i an intereting reflection roerty of a boundary between media that have the ame dielectric ermittivity but different magnetic ermittivity. In thi cae, the refection coefficient on both olarization are the ame and do not deend on obervation angle. In articular, thi mean that the reflection coefficient of a urface with large irregularitie (whoe linear ize i ubtantially greater than the wavelength) and a flat urface are the ame. Thi effect might have a ractical value. More detail are in the Aendix. find that Uing the value of dielectric ermittivity from (1) and ubtituting it into (), we R + R coα R =. (3) 1+ R coα 5

6 (The derivation i in the Aendix, formula (A1) (A3).) Note that ince the equation (1) (3) are comlex-valued, (3) include the relationhi between the abolute value and hae of reflection coefficient. Let u aume that R = R ex( iϕ ), and R = R ex( iϕ ). Subtituting thee value into (3) and equating the real and imaginary art, we obtain the following ytem of equation. R inϕ + R R coα in( ϕ + ϕ ) = R in(ϕ ) + R coα inϕ. (4) R coϕ + R R coα co( ϕ + ϕ ) = R co(ϕ ) + R coα coϕ. (5) Uing (4), (5), we can find hae ϕ and ϕ through the abolute value of reflection coefficient. In order to find the olution, we may quare both left and right ide of (4) and (5) and um u the aroriate ide of thee equation. After tranformation, we obtain the following. 4 R + R co α R (1 + R co α ) coϕ =. (6) R coα ( R R ) Now, uing (4) or (5), we can find the value of ϕ by olving the aroriate equation with reect to either in ϕ or tan( ϕ / ), deending on the choen tranformation. In ractice, it i uually eaier to meaure the aroriate emiivitie 1 R χ = and 1 R χ = a ooed to comlex reflection coefficient with hae hift. Thi way, we meaure characteritic relevant to energy, while the hae information i detroyed. (Here and below we ue the definition of emiivity 6

7 a the ratio of the radiation emitted by a urface to the radiation emitted by a erfect blackbody at the ame temerature from [4]. However, one hould undertand that radiation i emitted from the urface layer that ha finite thickne.) Uing the rooed aroach, we can retore the hae of reflection coefficient when we meaure only the energy characteritic of ignal. In the cae of 0 α = 45, the hae value in (6) are undefined. However, it follow from (3) that the following relationhi hold true regardle of the abolute value of reflection coefficient and, conequently, the dielectric ermittivity. R = R. (7) ϕ = ϕ. (8) Firt, thi effect wa dicovered in [5]. In [6], it wa obtained baed on different conideration a a limit of certain exreion. Phyically, the relationhi (7) and (8) are due to the geometry of interaction of electrical diole (ocillating electrical charge) near the medium urface with vector electromagnetic field of incident wave. In order to undertand the effect, the analogy with Brewter angle can be ued. Radiation diagram for the Brewter angle and when the angle of incidence i 0 45 are hown in Fig. 1. 7

8 Brewter angle 45 degree 90 degree 90 degree 0 (45 α rfr ) Fig. 1. Radiation diagram (circle) for the Brewter angle (left) and (right) angle Theoretically, the reflected wave at arallel olarization i abent at Brewter angle, becaue electrical charge of the medium ocillate erendicular to the direction of the refracted wave, which i alo erendicular to the direction of the reflected wave. The radiation diagram of thee ocillating charge i uch that there i no emiion in the direction of the reflected wave at arallel olarization. However, the reflected wave i actually the reult of interaction of radiation of electrical charge of the media, that are excited by the incident electromagnetic wave, with the incident electromagnetic wave. Thi roce haen in a layer of the medium with finite thickne. Conequently, in ractice, the reflected ignal with arallel olarization never become exactly zero at Brewter angle. In work [7], the author ay: Studie of the light reflected at Brewter angle howed that there are mall 8

9 deviation from the value redicted by Frenel formula. It turn out that there i no uch incident angle when the intenity of the reflected wave at the arallel olarization become zero and the electrical vector of the reflected wave ocillate erendicular to the lane of incidence. The author exlain that thi effect occur becaue reflection and refraction haen not on a mathematical urface boundary, but in a thin tranactional layer between different media. In cae of incident angle of 0 45, the direction of incident and reflected wave are erendicular, a it i hown in Fig. 1. The direction of the refracted wave (let u denote it by angle α rfr counted from the normal direction) differ from erendicular to the reflected wave by (45 0 ). So, unlike in the cae of Brewter angle, the α rfr radiation diagram of diole i oriented in uch a way that the emitted ignal i not zero at arallel olarization in the direction of reflected wave. At the ame time, the exitence of imle relationhi (7) mean that there i a certain hyical connection between the arallel olarized radiation emitted by diole at angle (45 0 ) relative to the axi of ymmetry of their radiation diagram, and the radiation emitted by diole that are excited by the electric field of a erendicularly olarized wave. The exlanation of the detail of thi effect require further tudy. What would be the ue of thi effect? Electrical charge on the urface hould not radiate in the ame way a electrical charge that are deeer in the medium. The orientation of their radiation attern might be different. So, actually, we may benefit from thi effect at leat in two way. The detailed tudy of certain electrodynamic ymmetry, which (7) and (8) reent, hould give u a better undertanding of the α rfr 9

10 nature of electromagnetic wave, becaue Frenel formula follow from Maxwell equation. On the other hand, the exerimentally meaured deviation from (7) and (8) would allow u to evaluate effect caued by the finite thickne of the boundary layer, in which the rocee of reflection and refraction actually take lace, and o allow u to better undertand the roertie of thi layer. Another ueful conequence of (1) and () i that we can exre the comlexvalued dielectric ermittivity through the comlex-valued reflection coefficient (derivation i in Aendix, formula (A4)) a follow. (1 + R)(1 + R ) ε =. (9) (1 R )(1 R ) We can validate (9) a follow. If we aume that α = 0 in (1) and (), then we can find ε = 1+ R ) /(1 R ) and ε = 1+ R ) /(1 R ) reectively. At normal angle ( ( of incidence, reflection coefficient in (9) are equal, and o formula (9) roduce the ame value, which confirm it correctne. We can alo find the dielectric ermittivity earately from (1) or () when we know the obervation angle and the aroriate reflection coefficient. However, in the cae of (9), we only have to meaure the reflection coefficient, and we do not need to meaure the obervation angle. Thi i an advantageou feature, becaue accurately meauring obervation angle often reent a challenge, eecially in remote ening alication. Note that we can ue everal meaurement at different angle with (9). Then, if the meaurement error are not correlated, their average value will roduce more accurate reult. In fact, in many intance, everal uch meaurement, aided by ome additional information, let u ay about urface 10

11 roughne, can allow the dicovery of ytematic error a well, thu alo reducing the total error. An intereting thing about (9) i that by combining (4) (6) and (9) we can find the comlex-valued dielectric ermittivity without knowing the hae tructure of reflection coefficient, only their abolute value. For intance, we can meaure emiivite 1 R χ = and 1 R χ = (which do not contain hae information) and find the hae of reflection coefficient ϕ and ϕ uing (4) (6). Then, (9) can be rewritten a follow. (1 + 1 χ (coϕ + iinϕ ))(1 + 1 χ (coϕ + iinϕ )) ε =. (10) (1 1 χ (coϕ + iinϕ ))(1 1 χ (coϕ + iinϕ )) 3. Polarization invariant 0 At obervation angle of 45, equation (3) can be viewed a a olarization invariant. For comlex reflection coefficient, we can rewrite it a follow. R =1. (11) R If we ue emiivitie for the reective olarization, then (11) can be reented in the following form. χ = 1. (1) χ χ The general form of olarization invariant (11) for an arbitrary angle of incidence i a follow. 11

12 R (1 + R coα ) = R + R coα 1. (13) An equation that i imilar to (13), but written in term of emiivitie, can be derived from (5) by erforming the aroriate tranformation. Uing (6), we obtain the following. co ϕ from χ (1 + coα coϕ 1 χ ) χ χ = 1+ co α (1 χ ) + coα coϕ 1 χ. (14) Tranforming (14), we find that the olarization invariant for an arbitrary angle of incidence i a follow. χ χ + coα coϕ 1 χ ( χ χ ) χ co α (1 χ ) χ = 1. (15) If the imaginary art of the dielectric ermittivity i mall and can be neglected, then we can aume ϕ = Finding temerature uing olarization meaurement The form of olarization invariant (1) and (15) are more aroriate when we conider electromagnetic radiation emitted by the body itelf, for intance, infrared or microwave radiation [8]. In the cae of olarization meaurement, we can define the reective emiivitie a follow: T χ =, T χ = T, where T and T are the T brightne temerature on erendicular and arallel olarization; T i the thermodynamic temerature of a remote object of interet. (Thi follow from the reviouly mentioned definition of emiivity from [4].) Subtituting thee value 1

13 into (1), we can find the thermodynamic temerature of a remote object by uing meaurement on two olarization a follow. T T T T =. (16) The hyical imortance of (16) i that it tie olarization meaurement of electromagnetic emiion in any range of electromagnetic ectrum (to which Frenel formula and Maxwell equation can be alied), to thermodynamic temerature. Thi i a conequence of the continuity of the ectrum of electromagnetic emiion emitted by bodie whoe temerature i not zero (Planck ectrum). For intance, it i oible to find the temerature of the Earth urface uing atellite or airborne meaurement of the Earth microwave radiation. Uing (15), we can derive a formula imilar to (16) for an arbitrary obervation angle. In order to do that, we hould multily the numerator and denominator in (15) by the temerature T. Let u alo denote x = 1 χ. Then, x =1 χ and χ = 1 x. Subtituting thee value into (15) and doing aroriate tranformation, we obtain the following equation. T T (1 x ) + xcoα coϕ ( T T ) x T co α T = 1. (17) Tranforming (17), we obtain the following quadratic equation. x ( T co α T ) coα coϕ ( T T ) x + ( T T ) = 0. (18) Generally, it ha two root. Poitive ign and a value of le than 1 i a good indicator of which olution i the right one. If both root atify to the condition, then the 13

14 choice hould be baed on inut data. Once we know the correct olution x, the temerature can be found a T = T /(1 x ). If ϕ = 0, then olution of (18) become coα ( T 0.5 =. (19) 1, T co α T x T ) ± in α ( T T T ) 0 For α = 45, (19) tranform to (16), which erve a a validation of (19). In ractical alication, the value of ϕ may be unknown. However, in many intance, the imaginary art of the dielectric ermittivity i very mall. So, for uch obervation, we can aume ϕ = Polarization invariant a an unambiguou criterion for urface roughne Recall that the olarization invariant (11), (13), (15) have been derived for a flat urface. When urface roughne change, the value of R R in (11) will change / accordingly. We can then define a quantitative characteritic of the urface roughne S a follow. R S =. (0) R Note that S, in general, can be comlex-valued. The arameter S (let u call it the roughne coefficient) i enitive to all tye of urface irregularitie. It i eecially enitive when the linear ize of roughne i greater than the wavelength. To undertand the influence of urface irregularitie on the roughne coefficient, we ued model uggeted in [8-10] for comuting reflection coefficient. Here, the rough urface wa modeled by a combination of different geometrical form at 14

15 different obervation angle. In the range of dielectric ermittivity from.5 to 70, an increae in the average loe of the urface irregularitie lead to a monotonic decreae of reflection coefficient on erendicular olarization and monotonic increae of reflection coefficient on arallel olarization, when comared to a medium with the ame dielectric ermittivity but a flat urface. So, in thi cae, the roughne coefficient (0) monotonically decreae tarting from a value of one, which correond to a flat urface. Intead of reflection coefficient, we can alo ue emiivitie to introduce a imilar roughne coefficient. For intance, for the angle o α = 45, the roughne coefficient S χ i a follow (note that it ha a maximum value of one for a flat urface). S χ χ = χ χ. (1) In thi cae, S i real-valued, not comlex-valued a i the cae with (0). Note that χ when the urface i flat, both (0) and (1) are real. The imaginary art of S in (0) can aear only for non-flat urface. follow. The aroriate roughne coefficient for an arbitrary angle of obervation i a S χ = χ + coα coϕ χ 1 χ ( χ χ ) χ co α (1 χ ) χ. () Fig. illutrate the deendence () grahically for two value of dielectric ermittivity. 15

16 1 roughne coefficient loe, degree Diel. Perm.=3 Diel. Perm=30 Fig.. Deendence of roughne coefficient on the average loe of a rough urface 0 modeled by an aembly of cone, at an obervation angle of Uing olarization invariant for calibration In ractical alication, received ignal are often accomanied by ome background radiation, which can come from external ource uch a relic background radiation or cloud emiion in microwave meaurement, noie of electronic receiving device, or white noie in otical meaurement. The good thing i that calibration rocedure, to comenate for noie, can be built into the device itelf baed on the olarization invariant. Let u aume that there i no external noie, and the device, or internal, noie level i N. Let T r and T r be non-calibrated meaurement of brightne temerature on erendicular and arallel olarization reectively, o that the 16

17 calibrated brightne temerature are T = ( T N) and T = ( T N), and the thermodynamic temerature of emitting object i T. For intance, in the cae of 0 obervation angle of 45, we can ubtitute thee value into (16) and obtain a quadratic equation with reect to N. Solving thi equation, we obtain the following two root. (Which of the two root ha a ractical meaning i determined by the r r device ecific.) N { T ) [ 4 ( )] } 1/ r S T ± SχT + S T Tr T = χ 0.5 ( χ r. (3) The advantage of thi method i that it doe not require u to know the value of the dielectric ermittivity and emiivity of the oberved object in order to etimate the noie level. Certainly, we can ue the ame aroach to etimate the value of external noie. If the nature of external and internal noie i the ame, then external noie can be included into the overall noie defined by (3). If the internal or external noie deend on the olarization, we till can ue thi aroach. However, we have to add a econd meaurement at a different angle. Thi way, we obtain two mathematically indeendent equation, and accordingly can find two unknown variable, which are the noie for each olarization channel. For intance, when hae hift ϕ = 0, the aroriate econd equation, baed on (19), i a follow. ( 1 ( T N ) / T ) 0.5 = 0.5 coα ( T T N + N ) ± in α[( T N )( T N ) ( T N ) ]. (4) = ( T N )co α ( T N ) Here, N rereent the value of noie in the aroriate olarization channel. 7. Exerimental obervation and dicuion 17

18 Temerature determination Prooed method for remote temerature determination on the bai of olarization meaurement were teted exerimentally. The meaurement were erformed outdoor. In the cae of 3.4 cm wavelength, the antenna rojector zone wa ued for obervation. For the 3.4 cm and 1.5 cm wavelength, an adjutment T wa b made for the background radiation coming from the ky. The value of radiation from the ky Tky wa meaured by the ame radiometer; it could be u to 15 K in cloudy weather. Then, uing the value of dielectric ermittivitie from [11,1] or the one found in our reviou exerimental tudie, we evaluated the background radiation a follow: T b = T ky R, where R correond to the reflection coefficient (1) or () for the aroriate olarization at the angle of obervation we ued. Finally, the adjutment T b wa ubtracted from the meaured brightne temerature. Meaurement at. cm wavelength wa done from a tower at a ditance at which the antenna radiation diagram wa fully formed. In all cae, the receiver error wa about 0.7 K or le. Table 1 reent the average abolute error in our exeriment on temerature determination by microwave radiometer for different urface and different wavelength. 18

19 Table 1. Accuracy of temerature determination ( 0 C ) oberved in exeriment. Angle, degree Surface (wavelength, cm) freh water (3.4) alt water (3.4) dry and (3.4) wet and (3.4) dry freh now (.) dene now (.) lain wood urface (1.5) We can evaluate the accuracy δ T of temerature determination by differentiating (16). Auming that the meaurement error of brightne temerature are accordingly δ T and δ T, we obtain the following. T δt + T ( TδT T δt ) δ T =. (5) (T T ) In many intance, we can aume that δt δt. Denoting δ Tbt = δt = δt, we can rewrite (5) a follow. T ( T T ) + T δ T δt. (6) bt (T T ) 19

20 We can ee from (6) that when the brightne temerature are cloe, that i T T, which wa the cae with freh dry now and dry and, we have δt δt. Given thi, bt it i not clear why the error in cae of freh now i o large, even we take into account the influence of mall urface roughne (when the linear ize of urface roughne i le than the wavelength). The mot likely reaon i that the now layer wa not electrohyically uniform. The thickne of the electromagnetic kin-layer wa everal ten of wavelength, becaue of the low now denity and low now conductivity. So, in fact, the now we were dealing with wa a multilayer medium, comoed of everal layer that fell at different time and were ubjected to change under the influence of temerature variation, unlight and wind. Thee factor could have caued the large oberved error. Otherwie, the etimate baed on (16) and (18) correlated with exerimental obervation relatively well. In general, the higher the dimenionle dielectric ermittivity, the more accurate temerature meaurement are. Excet for the cae of dry now, the main ource of error are background radiation and influence of mall cale urface roughne. Both factor can otentially be taken into account baed on a riori information about the urface and obervation condition. The reence of ytematic error can alo indicate the exitence of ome contant factor, e.g. calibration error. Infrared meaurement are widely ued for temerature determination. In ome intance, when the oberved urface are mooth and the obervation angle i not normal, the accuracy of temerature meaurement can otentially be increaed by uing olarized infrared meaurement. 0

21 Meaurement of roughne coefficient We tudied exerimentally the oibility of uing the introduced roughne coefficient for ractical evaluation of average loe of rough urface. Obervation were done with 3.4 cm radiometer in the antenna rojector zone. We ued and to model a rough urface coniting of an aembly of cone, whoe linear length exceeded the radiometer wavelength by everal time. The and comoition wa ilica; it had a tyical aturated andy color, average ize grain of the order of tenth of a millimeter, with the value of the dielectric ermittivity of dry and of For the exeriment, the and wa moderately moitened. We reeated the exeriment four time for different value of dielectric ermittivity (adding water and mixing the and). The dielectric ermittivity ε of wet and wa in the range , which wa found uing the meaured emiivitie of and with a flat urface. The moiture content of wet and wa not required ince we meaured the dielectric ermittivity directly. We ued formula (9) and (10) for calculation of dielectric ermittivity both for the loe and nadir obervation. The imaginary art of dielectric ermittivity wa very mall, and we conidered the dielectric ermittivity to be real-valued. In order to verify the accuracy, we alo ued the average value of emiivity found from nadir obervation on two orthogonal olarization. A deendence of the exerimentally determined roughne coefficient on the average loe veru the theoretical curve i hown in Fig. 3. We can ee that exerimental reult cloely follow the theoretical curve. 1

22 1.0 roughne coefficient loe, degree theoretical curve exeriment Fig. 3. Roughne coefficient of the rough and urface meaured by 3.4 cm 0 radiometer at 45 obervation angle veru theoretical curve. Dimenionle dielectric ermittivity i 7.4; roughne i modeled by and cone. Error bar correond to one tandard deviation. The meaurement error wa etimated by differentiating (1). S δ χ χ δχ + χ ( χ δχ χ δχ ) =. (7) (χ χ ) where δχ and δχ are the meaurement error of the aroriate emiivitie. Exerimental reult in Fig. 3 are within the range of oible error defined by (7). There i a clear indication of ytematic error in Fig. 3, which wa atifactorily exlained by the cooerative influence of mall cale roughne and by the ytematic meaurement error of angle of obervation. (The influence of mall cale

23 roughne doe not deend much on the loe of large cale roughne, a we found from reviou exeriment. The ame reult wa reorted in theoretical and exerimental tudie of urface with mall cale roughne at different angle of obervation and different wavelength [8,9,13].) Interretation of atellite obervation We analyzed data from atellite Komo-1151 acquired by a olarization radiometer at 3.4 cm wavelength (two channel on arallel and erendicular 0 olarization, and one nadir channel). The angle of obervation wa about 60. The time of obervation wa February, 1980, 8 AM Greenwich time. The atellite oition and orientation data were available, o that the location of antenna ot and obervation angle were calculated accurately. The linear ize of an antenna ot i on the order of ten kilometer. Other available nadir channel of radiometer at 0.8, 1.35 and 8.5 cm were ued to ae additional information, uch a temerature variation, atmohere arameter, and alo to verify value of dielectric ermittivity. The dielectric ermittivity can be evaluated uing both nadir and oblique angle of obervation. We ued both aroache for comarion uroe. Below, for imlicity, we dicu nadir meaurement. The error were etimated uing differential of formula for evaluation of aroriate arameter, imilar to (5), (7). Baed on available data, we found that the relative error of evaluation of dielectric ermittivity wa about δε / ε 10δχ, where δχ i the relative error of valuation of emiivity at nadir obervation. Thi n n 3

24 inaccuracy wa roduced by error of temerature determination (3 K), radiometer noie (1 K), calibration error (1 K). The error are indeendent, o we ued their quare average, which in our cae wa δχ , o that δε / ε n The error of determination of teene of the urface deend on the accuracy of valuation of emiivity, error of valuation of dielectric ermittivity and imerfection of our rough urface model. Taking into account thee factor, we found that for the average value of teene of 10, 0, 30 the error were 0 0 accordingly.6, 1.8, 1 0. The antenna ot trajectory went aroximately over 6-th meridian of wetern longitude and between 16 and 9 degree of northern latitude, which included Sahara deert and the Atla Mountain foothill in Northern Africa. The reult of comuting the average loe from exerimental data, in articular from the roughne coefficient, are reented in Fig. 4. Note that the roughne arameter that i baed on olarization meaurement i enitive to all urface irregularitie whoe linear cale i greater than everal wavelength. So, large cale geomorhologic and geograhic characteritic, which are of the order of a hundred meter and more (let u call them relief unevenne ), are alo included into thi integral urface roughne defined by olarization meaurement. The roughne coefficient allow the introduction of a new characteritic of earth urface, namely the roughne of mall cale urface irregularitie (let u call them urface roughne ), which are not aeed by geograhic method. Thi i becaue, in many intance, the contribution of different cale relief unevenne and 4

25 urface roughne into the integral roughne coefficient can be earated baed on reliminary information and additional meaurement. Thi further increae the information caabilitie of olarization meaurement with regard to evaluation of urface geometry. Fig. 5 reent the elevation rofile for the ame trajectory in 90-meter reolution [14], averaged over the antenna ot uing value of elevation in equally ditanced 30 oint, while Fig. 6 how the average loe over the antenna ot. We ued the loe data with one quare kilometer reolution [15], o that the average loe wa calculated over the larger area correonding to the antenna ot. We ued data with dicretization of one degree of loe, o that even if the actual value of loe were not zero, but le than one degree, they were aumed to be equal to zero. Similarly, all loe between one and two degree were aumed to be equal to one degree, and o on. So, the calculated value have bia toward maller than actual value of average loe. Thi i why the calculated average value of the loe i zero through a large art of the trajectory. Within the antenna ot, there were imultaneouly area with different loe, o that the roduced average value of the loe could have a fractional value. The average error wa calculated baed on the aumtion that the loe value are uniformly ditributed within each interval of one degree. Generally, terrain that i rougher in a geological ene i alo rougher on a maller cale. For intance, a mountain area ha tone on the urface, while flat valley uually do not. Similarly, higher elevation correlate with rougher terrain. Thee relationhi are clearly oberved in the reented figure. 5

26 We can ee that our evaluation of the average urface loe correlate well both with elevation data and change of average loe. Beide thee comarion, we alo did a viual analyi of detailed otical image with reolution of a few ten of meter (available on ublic webite Google Ma ). Qualitatively, the reult are in very good agreement with otical data, both for the urface roughne and dielectric ermittivity. Average teene, a an integral characteritic, alo correlate well with ecific of certain geograhical object, uch a lateau, ridge of and dune, alt an, foothill, etc; ee comment below. Although not reented in the article, we alo ued information from other ource for comarion, uch a geomorhologic data [16], weather condition, etc, which can be tranlated into value of dimenionle dielectric ermittivity and average loe. Thi rior geological and geomorhologic information alo correond well to the reult we obtained from interretation of atellite meaurement. 6

27 4 Average teene, degree Exeriment Northern latitude, degree Fig. 4. Change of average roughne of the urface along the oberved trajectory baed on exerimental data. Elevation, m Northern latitude, deg. Fig. 5. Elevation rofile. 7

28 Sloe, deg Northern latitude, deg Fig. 6. Geograhical loe. In Fig. 4, we identify the following geograhical area. 1. Flat area, outh-eat of Nema.. Plateau, and dune. 3. Jafene (and, alt an). 4. Erg Chache (large ridge of and dune). 5. Gbat El Eglab (lateau). 6. Plateau, Atla Mountain foothill. For determining the average loe from the value of the roughne coefficient, we ued tatitical model of rough urface, including one with variation in the loe value [9,17]. In thee model, the reence of loe variation lead to curve which are oitioned lightly lower than the curve for a rough urface with a contant loe. Therefore, the dicued reult confirm validity of the theoretical tudy. Overall, at obervation angle of more than thirty degree, the roughne coefficient unambiguouly correlate with geometry of urface irregularitie, defined a the 8

29 average loe of urface roughne. Thi roerty of the roughne coefficient can be ued in different alication. For intance, the roughne coefficient can be ued to determine geometrical characteritic of urface in remote ening and radar obervation or for recognition uroe. It can alo be ued to meaure the urface roughne of item in roduction environment by uing, for intance, laer radiation. In the lat cae, the meaurement ytem can be calibrated by amle item with known roughne or by tatitical evaluation, imilar to aroache decribed in [8]. Another advantageou feature of introduced roughne coefficient i that it deend le on the dielectric ermittivity than the geometry of urface irregularitie. Combined with the fact that the dielectric ermittivity can be accurately etimated uing many method, it mean that the roughne coefficient can rovide an accurate characterization of urface geometrical characteritic. On the other hand, if we know the roughne characteritic, then we can interret olarization meaurement more accurately. For intance, if we know the urface roughne coefficient in term of emiivitie, then we can rewrite (16) a follow. T T = S (T χ T ) (8) Thi way, we can more accurately determine the temerature of a rough urface. 9

30 Dielectric ermettivity Northern latitude, degree Exeriment Geological data Fig. 7. Change of dielectric ermittivity along the oberved trajectory. Fig. 7 how the change of dielectric ermittivity along the trajectory of antenna ot. We found the following zone, hown in Fig. 7, for which we can identify the tye of mineral, rock and oil, and the aroriate value of dielectric ermittivity. 1. Sandtone, and atche (fluiol).. Sand, oil (aliol). 3. Sand dune (and from rock formation), alt an (arenool). 4. Sand dune ridge (Erg Chache) (aliol). 5. Salt an, and, rock formation (aliol). 30

31 6. Ridge of and dune on lateau El Glab. Granite and re-cambrian era metamorhic rock, uch a hale, ilt (rotruion of a continental latform). Aliool, letool (thin, hallow oil on hard rock). 7. Rock formation on lateau, with ridge of and. 8. Atla Mountain foothill. Rock formation (granite, metamorhic rock), atche of and. We then ued otical image in order to identify the mineral in articular area and to determine their urface coverage. Value of dielectric ermittivity for articular mineral at comarable frequencie are available from different ource, in articular [11,1]. We alo comared thee value of dielectric ermittivity with data reviouly collected from other ource, including unublihed reort, and our direct meaurement of dielectric ermittivity in the range of cm in the laboratory etting, and found the reult of thi comarion atifactory. The deendence between the emiivity and the dielectric ermittivity of emitting urface i non-linear. We ued the following aroach to calculate dielectric ermittivity on the bai of geological data and otical obervation. The integral emiivity χ T of the urface that i comoed of everal tye of mineral, each having emiivity χ i and the urface coverage weight W i, i defined a i N = = χ T W i χi. Uing (1) at α = 0 and the relationhi i= 1 χt ( α = 0) = 1 ε ε T T 1, we

32 can tranform thi equation to ε ε T T i = E, where = N ε i 1 E = W i = ε + 1 i 1 i. So, we can find the effective dielectric ermittivity ε T for a urface that include everal emitting object with different dielectric ermittivitie a ε T 1+ = 1 E E. Thu obtained reult were in very good agreement with the value of dielectric ermittivity found from atellite meaurement on the bai of (9). Although additional reearch i needed, the uggeted aroach demontrate high otential for ractical alication. Concluion The reult of thi tudy howed that the combined meaurement of electromagnetic radiation uing orthogonal linear olarization allow retrieving more information about the characteritic of reflected and emitted ignal, uch a ignal amlitude and hae tructure, and alo allow finding a medium arameter (dielectric ermittivity, temerature, average urface loe). We alo conidered certain relationhi between reflection coefficient that hold true for all angle of obervation, which become eecially imle when the 0 angle of obervation i 45. Similar olarization invariant exit for emiivitie. It turn out that uch introduced olarization invariant have a ractical value. They can be ued for evaluation of geometry, temerature and dielectric ermittivity of oberved urface, calibration of receiving and emitting device and evaluation of 3

33 external noie. Exerimental obervation confirmed the validity and ractical value of rooed method. On a fundamental hyical level, there i a certain geometrical ymmetry in the interaction of electrical diole, located on and near the medium urface, with linearly olarized electromagnetic wave incident at an angle of forty five degree. Thi effect may erve a one of the connection oint between geometrical and hyical otic and theory of electromagnetim, which would alo allow taking into account finer effect, uch a, for intance, the aforementioned finite thickne of medium layer in which the refraction and reflection rocee take lace. Acknowledgement The author greatly areciate the effort of Alexander Shetoaloff, whoe dedicated and enthuiatic hel in retrieving geograhical, climate, oil and mineral related information wa crucial for thi roject. The author thank A. Kyriako for the dicuion and advice. Aendix Let u find ε in α and ε from (1). coα(1 + R ) 1 R ε in α =. (A1) 1+ R 1+ R + R coα ε = in α + co α = 1 R. (A) (1 R ) 33

34 Subtituting thee value into () and doing aroriate tranformation, we obtain the following. R 1+ R + R coα coα(1 + R ) coα (1 R ) 1 R = = 1+ R + R coα coα(1 + R ) coα + (1 R ) 1 R coα(1 + R = coα(1 + R + R + R coα ) coα(1 R coα ) + coα(1 R ) R + R coα = ) 1+ R coα. (A3) If we ubtitute R co α from (A3) into (A) and erform aroriate tranformation, we obtain the following. ( R ) R 1+ R + 1+ R co (1 ) (1 )(1 ) + R α R + R + R ε = = =. (A4) (1 R ) (1 R ) (1 R )(1 R ) Below, we conider the cae when magnetic ermittivitie of media differ from one. Let u denote μ = μ 1 / μ. Indexe 1 and indicate accordingly the medium from which the incident wave come, and the reflecting medium. Uing general formula for reflection coefficient from [7], we can rewrite (1) and () a follow. μ ε in α coα R =. (A5) μ ε in α + coα με coα ε in α R =. (A6) με coα + ε in α Here, ε = ε / ε 1, the ame a in (1) and (). Doing tranformation of (A5) and (A6) imilar to (A1) (A3), we obtain. R (1 R ) = 1. (A7) μ (1 R ) + c (1 + R ) + 1 R 34

35 0 When μ = 1, (A7) tranform to (A3). When α = 45, (A7) tranform to equation: R = 4(1 R 1 μ (1 R ) (1 ) (1 + + R + R ). (A8) ) If we aume μ = 1, then (A8) tranform to (7), that i R =. R When ε = 1, that i ε 1 = ε R, then, a it follow from (A5) and (A6), = = ( μ 1) /( μ + 1). In other word, the reflection coefficient are the ame on R both olarization and do not deend on the obervation angle. Creating different ubtance with the ame dielectric ermittivity i oible; for intance, uing mixture or olution. In thi cae, the reflection and refraction roertie of the boundary between uch ubtance will deend only on the magnetic ermittivity, which in ome cae can be ueful. Reference 1. D. K. Kalluri, Electromagnetic of time varying comlex media: frequency and olarization tranformer (Boca Raton : Taylor & Franci, nd ed., 010).. R. M. A. Azzam, Direct relation between Frenel' interface reflection coefficient for the arallel and erendicular olarization, J. Ot. Soc. Am., 69, (1979). 3. V.V. Bogorodky, A.I. Kozlov, Yu. K. Shetoalov, Determination of roughne rank and dielectric ermittivity of the earth urface by microwave meaurement, Journal of Technical Phyic. 54, No. 1, (1984). 4. McGraw-Hill Dictionary of Phyic (McGraw-Hill Profeional Publihing, nd edition, 1996). 35

36 5. R. M. A. Azzam, On the reflection of light at 45 angle of incidence, J. Mod. Ot., 6, 113 (1979). 6. Yu. K. Shetoalov, On the relationhi of Frenel reflection coefficient at obervation angle of forty five degree, Journal of Technical Phyic, 53, No. 1, 144 (1983). 7. A. N. Matveev, Otika (Otic) (Vyhaya Shkola, Mokva, 1985). 8. Yu. K. Shetoalov, Statitical roceing of aive microwave data. IEEE Tranaction on Geocience and Remote Sening, 31, (1993). 9. Yu. K. Shetoalov, Multile incoherent wave cattering on tatitically rough urface with large tee roughne, Radiotekhnika, No 4, (1989). 10. P. P. Bobrov, T. A. Belyaeva, Yu. K. Shetoalov, I. M Shchetkin, Peculiaritie of microwave radiation from eriodically uneven ground, Journal of Communication Technology & Electronic, 45, (000). 11. D. J. Daniel, Surface-enetrating radar--iee Radar, Sonar, Navigation and Avionic, Serie 6 (London, The Intitute of Electrical Engineer, 1996). 1. J. L. Davi, A. P. Annan, Ground-enetrating radar for high-reolution maing of oil and rock tratigrahy, Geohyical Proecting, 37 (1989). 13. S. M. Rytov, Yu. A. Kravtov, V. I. Tatarkii, Princile of tatitical radiohyic (Sringer-Verlag, Berlin, New York, 1987). 14. A. Jarvi, H. I. Reuter, A. Nelon, E. Guevara, Hole-filled eamle SRTM Data, V4, International Centre for Troical Agriculture (CIAT) (008). htt://rtm.ci.cgiar.org. 15. Global GIS Databae: Digital Atla of Africa, U.S. Geological Survey (001). 36

37 16. R. J. Huggett, Fundamental of geomorhology (Routledge, 007). 17. Yu. K. Shetoalov, Microwave olarization roertie of the rough urface with large tee roughne. Part, Proceeding of Higher Educational Intitution. Radiohyic, 8, (1985). 37