Measurement of the electromagnetic field backscattered by a fractal surface for the verification of electromagnetic scattering models

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 48, NO. 4, APRIL 2010 1777 Measu emen o he Elec omagne ic Field Backsca e ed by a F ac al Su ace o he Ve i ica ion o Elec omagne ic Sca e ing Models Giuseppe Ruello, Membe , IEEE, Pablo Blanco-Sánchez, An onio Iodice, Senio Membe , IEEE, Jo di J. Mallo quí, Membe , IEEE, Daniele Riccio, Senio Membe , IEEE, An oni B oque as, Membe , IEEE, and Gio gio F ancesche i, Li e Fellow, IEEE Abs ac —F ac al geome y is widely accep ed as an e icien heo y o he cha ac e iza ion o na u al su aces; he oppo - uni y o desc ibing i egula i y o na u al su aces in e ms o ew ac al pa ame e s makes i s use in di ec and in e se elec omagne ic (EM) sca e ing heo ies highly desi able. In his pape , we p esen an inno a i e p ocedu e o manu ac u ing ac al su aces and o measu ing hei sca e ing p ope ies. A ca dboa d–aluminum ac al su ace was buil as a ep e- sen a ion o a Weies ass–Mandelb o ac al p ocess; he EM ield sca e ed om i was measu ed in an anechoic chambe . A monos a ic ada like con igu a ion was employed. Measu emen esul s we e compa ed o Ki chho app oxima ion and small pe u ba ion me hod closed- o m esul s ha we e analy ically ob ained by employing he ac ional B ownian mo ion o model he su ace shape. Ma ching and disc epancies be ween heo ies and measu emen s a e hen discussed. Finally, ac al and classical su ace models a e compa ed as a as hei use in he EM sca e ing is conce ned. Index Te ms—Elec omagne ic sca e ing by ough su aces, ac als. I. INTRODUCTION ARELIABLE heo y on he elec omagne ic (EM) sca e - ing om na u al su aces equi es an e icien and accu- a e quan i a i e desc ip ion o he na u al su ace shapes. The use o ac al geome y o his pu pose is s ongly sugges ed because i accoun s o he complex beha io o na u e by means o simple models. Elegan me hods o he e alua ion o he EM ield sca e ed om ac al su aces we e ecen ly de eloped wi h encou aging esul s [1], [2]. So a , no sca e ing measu emen campaigns on su aces wi h known ac al pa ame e s we e ca ied ou . Mos o he measu emen campaigns p esen ed in he li e a u e Manusc ip ecei ed Sep embe 5, 2007; e ised June 19, 2008, Decembe 12, 2008, and July 3, 2009. Fi s published Decembe 22, 2009; cu en e sion published Ma ch 24, 2010. This wo k was suppo ed by he Spanish MCYT and EU FEDER unds unde P ojec TEC2008-06764-C02-01. G. Ruello, A. Iodice, D. Riccio, and G. F ancesche i a e wi h he Depa - men o Elec onic and Telecommunica ion Enginee ing, Uni e si y o Naples “Fede ico II,” 80125 Naples, I aly (e-mail: [email p o ec ed]; [email p o ec ed]; [email p o ec ed]; [email p o ec ed]). P. Blanco-Sánchez, J. J. Mallo quí, and A. B oque as a e wi h he Depa - men o Signal Theo y and Communica ions, School o Telecommunica ion Enginee ing, Uni e si a Poli ecnica de Ca alunya, 08034 Ba celona, Spain (e-mail: [email p o ec ed]; [email p o ec ed]; [email p o ec ed]). Colo e sions o one o mo e o he igu es in his pape a e a ailable online a h p://ieeexplo e.ieee.o g. Digi al Objec Iden i ie 10.1109/TGRS.2009.2036007 we e pe o med on a i icially cons uc ed Gaussian su aces [3]–[5] o on na u al su aces desc ibed by means o non ac al pa ame e s like he co ela ion leng h and s anda d de ia ion [1]. Howe e , al eady in he 1970s, i was known ha classical Gaussian pa ame e s es ima ed om samples o na u al su - aces depend on he size o he conside ed su aces [6], [7]; his is conside ed as a majo mo i a ion o concei ing mo e app o- p ia e models o desc ibe na u al su aces. In 1973, Beckmann [6] exp essed he need o a non-Gaussian desc ip ion o na u al su aces o elec omagne ic sca e ing e alua ion pu poses. He poin ed ou ha he p obabili y densi y unc ion (pd ) o a na u al su ace can also be desc ibed by a Gaussian model, bu he pd could be nonc i ical o i s sca e ing p ope ies. In 1988, Wu e al. [8] ag eed ha non-Gaussian p ocesses would be necessa y o na u al su ace desc ip ion. The in oduc ion o he ac al geome y p o ided a powe ul ins umen o com- p ehending and quan i a i ely desc ibing he i egula shapes o na u e. As a ma e o ac , ac al models accoun o he nons a iona y and sel -a ine cha ac e is ics o ac ual p ocesses, so ha in he las decades, se e al esea che s in ensi ely used a ac al ep esen a ion o na u al su aces (e.g., [1], [9]–[17] a e a small bu signi ican pa o he huge li e a u e on he opic). The use o ac als in emo e sensing is highly desi able, also because he na u al su aces can be desc ibed by employing jus ew pa ame e s, and his is e y a ac i e o model-in e sion pu poses [18], [19]. In his pape , we p esen an inno a i e measu emen cam- paign in a con olled en i onmen ca ied ou o es , in monos a ic con igu a ion, he heo ies on he elec omagne ic sca e ing om ac al su aces. In Sec ion II, we in oduce he undamen als o ac al mod- els, and we ecall he main concep s which inspi ed he su ace cons uc ion. In pa icula , we de ine he ac ional B ownian mo ion ( Bm), an e e ywhe e con inuous bu nowhe e di e - en iable p ocess, which is he mos sui able model o desc ibe na u al su aces, and i is cu en ly used in EM sca e ing heo ies. Then, in o de o syn hesize an Bm ac al su ace, we in oduce he Weie s ass–Mandelb o unc ion (WM) whose spec um p o ides a eliable app oxima ion o an Bm p ocess spec um. The alues o he ac al pa ame e s chosen o he nume ical syn hesis o he su ace a e consis en wi h ypical na u al su aces. Then, we p esen he p ocedu e o build he su ace as a supe posi ion o a ca dboa d s uc u e, ep esen ing he low equencies, and w inkled aluminum oils, accoun - ing o he high- equency su ace componen s. Employed 0196-2892/$26.00 © 2009 IEEE Au ho ized licensed use limi ed o: UNIVERSITAT POLITÈCNICA DE CATALUNYA. Downloaded on July 21,2010 a 14:29:17 UTC om IEEE Xplo e. Res ic ions apply. 1778 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 48, NO. 4, APRIL 2010 ma e ials gua an eed o p oduce a cos -e ec i e, po able, and eliable ac al su ace. The co espondence be ween syn he- sized and buil -su ace ac al pa ame e s was checked by means o app op ia e op ical measu emen s [20]. In Sec ion III, we ecall how he Bm su ace desc ip ion can be used in conjunc ion wi h he Ki chho app oxima ion (KA) and he small pe u ba ion me hod (SPM) [1], in o de o p o ide he second-o de s a is ics o he sca e ed ield [21], [22]. The choice o he KA and he SPM is ela ed o he ac ha hese a e he mos used echniques in emo e-sensing applica ions: om SAR da a in e p e a ion [23] o e ie ing algo i hms [18], [19]. Sec ion IV is de o ed o desc ibe he p ocedu e and he implemen a ion o he EM sca e ing measu emen s, pe o med in he anechoic chambe o he Uni e si a Poli ècnica de Ca alunya (UPC). A couple o X-band ho n an ennas was used o illumina e he buil su ace and measu e he ield backsca - e ed om i . The buil su ace was moun ed on a o o , whose mo emen s allowed he acqui ing o a su icien numbe o samples a di e en incidence angles (incidence angles in he ange o 0◦−70◦we e explo ed). Bo h ho izon al- and e ical- pola ized sca e ed ields we e measu ed. The calib a ion p o- cedu e pe o med wi h he use o a ihed al co ne e lec o is also p esen ed. In Sec ion V, we show he ob ained esul s. Calib a ed measu ed da a a e p esen ed as a unc ion o he incidence angle; hey a e hen compa ed wi h he heo e ical esul s in Sec ion III. Ma ching and di e ences a e deeply discussed. In addi ion, a discussion on he main ad an ages o ac als wi h espec o classical models in he EM sca e ing me hods is p esen ed. The aim o Sec ion VI is o summa ize he a ionale, ap- p oach, implemen a ion, and esul s ela ed wi h he p esen - ed wo k. II. SURFACE CONSTRUCTION In his sec ion, we ecall he basic p inciples o he ac al geome y, and we p o ide he a ionale o he syn hesis, he building, and he alida ion o he ac al su ace employed in ou expe imen s. A de ailed desc ip ion o hese opics is p o ided in [20]. A. F ac al Geome y The Bm is widely ecognized as he mos sui able ac al p ocess o model na u al su aces. I allows he desc ibing o he su ace shape in e ms o only wo pa ame e s, he di- mensionless Hu s coe icien ,H, and he inc emen al s anda d de ia ion a uni a y dis ance,s, measu ed in m(1−H).An Bm p ocess ep esen s a s ochas ic su ace z(x, y)i , o e e y x,y, x, and y, he pd o i s inc emen s is Gaussian wi h ze o mean and s anda d de ia ion sτH P z(x, y)−z(x,y)<¯ ζ =1 √2πsτH ¯ ζ  −∞ exp −ζ2 2s2τ2Hdζ (1) whe e τis he dis ance be ween x,yand x,y. I can be demons a ed [13], [24] ha a p ocess sa is ying (1) exis s i 0<H<1, and ha (wi h p obabili y one) any Bm sample su ace has a ac al dimension D=3−H. Fu he - mo e, i we conside he inc emen s a dis ance τ=1m, hei s anda d de ia ion is s. The e o e, he spa ame e ep esen s an index o oughness. The highe i s alue, he oughe he su ace [1]. E alua ion o he Bm powe spec um dese es special ca e due o he nons a iona i y o he su ace. Space- equency and scale- equency app oaches a e equi ed in o de o exp ess he powe spec al densi y o he su ace. Bo h hese app oaches lead o a powe -law spec um [1], [13] W(k)=S0k−α(2) whe e S0and αa e pa ame e s depending on Hand s[1] and k is he spa ial wa enumbe . The Bm is a egula s ochas ic p ocess, whose syn hesis is no a s aigh o wa d ask, and i is usually accomplished in e ms o app op ia e unc ions, as de ailed in he ollowing sec ion. B. Syn hesis The syn hesis o Bm p ocesses can be ob ained ia displaced in e pola ion [25], spec al syn hesis [1], wa ele [25] me hods, and so on. In his pape , we use a WM unc ion, which is a p edic able p ocess ha allows an easy and con ollable modeling o de e minis ic and s ochas ic p ocesses as a unc ion o ew physical pa ame e s. A ma hema ical WM is a supe posi ion o in ini e sinusoidal ones wi h pe iods spaced by an i a ional ac o . Na u al su aces exhibi a ac al beha io in a wide bu ini e ange o scales, so a physical WM [1] can be ob ained by using a 2-D band-limi ed (i.e., wi h a ini e numbe o ones, M)WM unc ion (x, y) (x, y)=B M−1  n=0 Cnν−Hn sin(k0νn(xcos ψn+ysin ψn)+φn) (3) whe e Bis an ampli ude scaling ac o ; νis he i a ional equency scaling ac o ; Cn,ψn, and φna e andom a iables, accoun ing o ampli ude, di ec ion, and phase beha io o each one, espec i ely [1]; and k0is he undamen al one wa enumbe . I was shown ha he WM can be conside ed as a spec al sampled e sion o an Bm p ocess wi h he same ac- al dimension and wi h S0 ela ed o he WM pa ame e s by [1] B2=S0 2πH k−2H 0(νH−ν−H).(4) The syn hesis o a single su ace calls o he se ing o i e pa ame e s: H,B,k0,kM−1, and ν. Fo he su ace conside ed in his pape , hei alues we e chosen acco ding o he ollowing a ionale. 1) The H alue was se a 0.7, because na u al su aces ypically hold H alues anging om 0.6 o 0.9 [24]. 2) The B alue was se a 0.011 m, in acco dance o ypical alues o na u al su aces. This choice co esponds o an s alue o 0.063 m1−H, in acco dance wi h ypical na u al alues [24]. Au ho ized licensed use limi ed o: UNIVERSITAT POLITÈCNICA DE CATALUNYA. Downloaded on July 21,2010 a 14:29:17 UTC om IEEE Xplo e. Res ic ions apply. RUELLO e al.: MEASUREMENT OF THE EM FIELD BACKSCATTERED BY A FRACTAL SURFACE 1779 TABLE I SIMULATED SURFACE PARAMETERS 3) The lowes oughness scale k0 ha con ibu es o he sca e ed ield o ma ion depends on he illumina ed a ea and he anechoic chambe dimensions. In his pape , he undamen al one wa enumbe is o 5.71 m−1. 4) The highes oughness scale kM−1 ha con ibu es o he sca e ed ield o ma ion depends on he inciden wa e- leng h λ. Scales lowe han a ac ion o he wa eleng h do no con ibu e o he ield o ma ion. We ixed λ= 3cm as a e e ence wa eleng h, so ha eliable mea- su emen s could be pe o med a equencies included in he X-band. The e o e, kM−1= 1943.8m−1, ha co e- sponds o a su ace one wa eleng h o 3.2 mm, which is abou λ/10. 5) The ν aluewasse a 0.5e, as a adeo be ween he Bm sampling a e and he memo y cons ain s. Such a choice leads o a numbe o ones o M=20[20]. In Table I, he chosen nume ical alues a e summa ized, and a ep esen a ion o he syn hesized su ace is shown in Fig. 1(a). C. Manu ac u ing The su ace was buil acco ding o a wo-s ep app oach. The la ges spa ial scale oughness ( om me e s ill 0.5 cm) was assembled as supe posi ion o ca dboa d laye s shaped acco ding o he syn hesized su ace-le el cu es wi h a s ep o 0.5 cm (i.e., λ/6) [see Fig. 1(b)]. The choice o ca dboa d allowed an easy manu ac u ing and po abili y o he su ace. Once such a 1.5 m ×1.5 m (i.e., 50 λ×50 λ) mac oscopic s uc u e was buil , we added he mic oscopic oughness by supe posing wo laye s o aluminum oils [see Fig. 1(c)]. The i s laye was glued di ec ly o he ca dboa d wi hou co u- ga ion. The second was manually w inkled and glued on he op. The in ensi y o he co uga ions was no p econ olled bu e alua ed ia lase scansion. Such a p ocedu e allowed he conside ing o he buil su ace as pe ec e lec o . In Fig. 1(d), we show a op iew o he buil su ace. No e ha he aluminum oils co e up he seams, a oiding he possibili y ha hey could bias he esul s wi h hei non ac al s aigh edges. The chosen su ace shape is ci cula , wi h a 1.5-m diame e , in o de o minimize he bo de e ec s [20]. D. Valida ion A high-p ecision lase (wi h esolu ion o 0.7 mm) was employed o p o ide an accu a e analysis o he buil -su ace p ope ies. The measu ed da a we e p ocessed in bo h he spa ial and spec al domains, leading o he esul ha he ac al su ace holds he p esc ibed syn hesized pa ame e s [20] wi hin he ange o scales o in e es o elec omagne ic sca e ing expe imen s [1]. The ob ained measu emen s showed ha he aluminum laye s did no signi ican ly change he su ace ough- ness a he scales o in e es o he elec omagne ic sca e ing. Fig. 1. (a) Syn hesized su ace. (b) Ca dboa d opog aphy. (c) De ail on he aluminum laye s. (d) Top iew o he buil su ace. III. ELECTROMAGNETIC METHODS In his pape , we ocus ou a en ion on me hods ha p o ide simple and e ec i e closed- o m solu ions, because hei use is o in e es in emo e sensing applica ions [1], [18], [19], [26]. In pa icula , we ocus ou analysis on he mos ly used elec o- magne ic me hods, he KA and he SPM. These me hods a e in ensi ely used in emo e sensing applica ions o de eloping Au ho ized licensed use limi ed o: UNIVERSITAT POLITÈCNICA DE CATALUNYA. Downloaded on July 21,2010 a 14:29:17 UTC om IEEE Xplo e. Res ic ions apply. 1780 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 48, NO. 4, APRIL 2010 SAR in e p e a ion ools [23] as well as in e se me hods o e ie ing physical pa ame e s in emo e sensing applica ions [18], [19]. The in oduc ion o he Bm ac al p ocess o su ace p o ile desc ip ion pu poses allowed he de elopmen o imp o ed e sions o hese elec omagne ic sca e ing me hods [1], [2], [21], [22], whose esul s a e ecalled in he ollowing. A. KA Fo mula ion The PO solu ion o he KA leads o exp ess he sca e ed powe densi y Sias a unc ion o he su ace pa ame e s [1], [2] Si∝ ∞  0 J0(ηxyτ)exp−1 2η2 zs2τ2Hτdτ (5) whe e J0is he ze o h-o de Bessel unc ion and η=ki−ks=(ηx,η y,η z), whe e kiand ksa e he inciden and sca e ed wa e ec o s, espec i ely, ηxy =η2 x+η2 y.The su ace oughness, exp essed in e ms o he ac al pa ame e s sand H, as well as he incidence angle a e accoun ed o in he a gumen s o he Bessel and he exponen ial unc ions. A comple e ea ise o he in eg al e alua ion is well beyond he goals o his pape . In li e a u e, closed- o m solu ions can be ob ained ia asymp o ic expansions, as de ailed in [1], [2]. B. SPM Fo mula ion The SPM o mula ion p o ides he ada c oss sec ion as a unc ion o he su ace spec um. I we use he Bm spec um in he exp ession o he i s -o de SPM ada c oss sec ion, we ob ain he ada c oss sec ion as a unc ion o he spec al su ace pa ame e s [1], [2], [22] σo pp =4k4cos4ϑ|βpp|2S0 π(2ksin ϑ)α(6) wi h βpp aking in o accoun he pola iza ion issue, and S0and αa e he Bm powe -law spec um pa ame e s [1], [2]. C. Validi y Limi s The de ini ion o he alidi y limi s o elec omagne ic sca - e ing om ac al su aces is s ill an open p oblem. This is mainly due o he ac ha he Bm is no di e en iable, and i s use in he sca e ing e alua ion equi es a physical-based band- limi ing p ocedu e. Such an ope a ion in luences he de ini ion o he alidi y limi s. So a , he alidi y limi s a e e alua ed ia ela ionships be ween ac al and classical pa ame e s, as epo ed in [1], [2]. Howe e , he equi alence be ween clas- sical and ac al pa ame e s equi es sub le heo e ical issues. To he bes o ou knowledge, no conclusi e esul s exis in he a ailable li e a u e. In a ecen pape [26], an empi ical condi ion was de ined in e ms o he signi ican slope ss( he a io be ween he oo mean squa e (RMS) su ace heigh and he wa eleng h o he dominan spec al peak). Acco ding o [26], o nea -nadi incidence (incidence angles lowe han 30◦), he PO app oach is applicable i ss<0.037 cos3θ.Fo he su ace employed in ou expe imen , he condi ion holds o e e y incidence angle less han 30◦. Despi e he ac ha i does Fig. 2. (a) Top iew o he anechoic chambe . A iden i ies he an enna posi ion, B iden i ies he su ace posi ion. Linea dimensions a e exp essed in cen ime e s. (b) Image o he ansmi ing and ecei ing an ennas. no cons i u e a conclusi e p oo , his esul sugges s ha , a leas o nea -nadi angles, alidi y limi s a e nea ly ul illed. A comple e discussion on his opic equi es a e o mula ion o he alidi y limi s in e ms o ac al pa ame e s, bu i goes beyond he goals o his pape and i is demanded o a u u e discussion. IV. ELECTROMAGNETIC MEASUREMENT SETUP A. Geome y The ac al su ace, buil in acco dance wi h he p ocedu e o Sec ion II, was used o alida e he EM sca e ing me hods p esen ed in Sec ion III. In his sec ion, we p esen he p oce- du e o measu e he EM ield backsca e ed om his su ace. The expe imen s we e pe o med in he anechoic chambe a UPC. A op iew o he measu emen geome y is shown in Fig. 2(a). Two ho n an ennas, one o ansmi ing and one o ecei - ing [see Fig. 2(b)], we e placed in a ixed posi ion [see he A poin in Fig. 2(a)]. The su ace was moun ed in posi ion B [see Fig. 2(a)] a a dis ance o 5.75 m om he an ennas, on a oll- o e azimu h posi ioning sys em ha allows o a ions bo h in Au ho ized licensed use limi ed o: UNIVERSITAT POLITÈCNICA DE CATALUNYA. Downloaded on July 21,2010 a 14:29:17 UTC om IEEE Xplo e. Res ic ions apply. RUELLO e al.: MEASUREMENT OF THE EM FIELD BACKSCATTERED BY A FRACTAL SURFACE 1781 Fig. 3. Measu emen geome y. The o o allows he buil -su ace o a ion a ound he (a) z-and(b)x-axes. he (y,z)plane a ound he x-axis and in he (x, y)plane along he z-axis, in acco dance wi h he geome y o Fig. 3. The spo size o he illumina ing beam co e s he whole su ace. The o a ions a ound he z-axis allowed he acquisi ion o sca e ed ield samples a di e en incidence angles θ[see Figs. 2 and 3(a)]. A a gi en incidence angle, he o a ions a ound he x-axis gua an ee he acquisi ion o many indepen- den sca e ed ield samples. B. Measu emen P ocedu e Fo each su ace (θ,φ)posi ion, he X-band an ennas ac- qui ed 401 samples, uni o mly spaced wi hin he equency ange om 7 o 12 GHz. App op ia e ansmi ing and ecei ing an enna o a ions p o ided he acquisi ion o HH- and VV- pola ized ields. The ield backsca e ed om he ac al su ace was mea- su ed using an HP8510 ne wo k analyze o θ anging om 0◦ o 70◦wi ha2 ◦s ep. Fo each θ alue, 72 independen ield alues we e acqui ed by o a ing he su ace wi h φs eps o 5◦. In ac , he ac al su ace unde analysis is a single ealiza ion o an Bm s ochas ic p ocess. I can be seen as a cell composed o mul iple sca e e s. By changing he illumina ion angle, he ela i e dis ibu ion o he sca e e s changes, so ha wo acqui- si ions a e di e en . The o a ion o a esolu ion elemen by he angle φmo es he sca e ing cen e s. The ange componen o he displacemen causes a sligh ly di e en phase shi o each sca e ing cen e , esul ing in signal deco ela ion. We choose aφs ep o 5◦, in acco dance wi h he esul s ound in he li e a u e conce ning he signal deco ela ion o in e e ome - ic applica ions [27], [28]. In [27], he co ela ion coe icien Fig. 4. Compa ison be ween (do s) da a, (dashed line) KA, and (solid line) SPM o (a) VV and (b) HH pola iza ions. be ween he ield sca e ed by a su ace and he ield sca e ed by he same su ace o a ed by an angle φis compu ed as ρφ=1−2(sin θ)|φ| λ,i 2(sin θ)|φ| λ<1 0,o he wise (7) whe e is he an enna–su ace dis ance. In ac , by using in his exp ession φ=5 ◦, conside ing he alues o λand θbelonging o he in e als de ined ea lie and conside ing =5.75 m (see Fig. 2), we ob ain a null alue o he co ela ion coe icien . C. Calib a ion The calib a ion is pe o med ia a s anda d wo-s ep p oce- du e [29], [30]. Fi s , he powe Pmbacksca e ed by he buil su ace, placed a a dis ance m om he an enna, is measu ed o ob ain he non-calib a ed da a Pm=P G 4π 2 m σm 4π 2 m K. (8) I depends on he an enna gain G, he ansmi ed powe P , and he en i onmen al e ec K. Then, he su ace is eplaced by a pe ec ly conduc ing ihed al co ne e lec o , and he ecei ed powe P i is measu ed a a dis ance i P i =P G 4π 2 i σ i 4π 2 i K. (9) The e o e, he calib a ed ada c oss sec ion σmis ob- ained by he known ihed al ada c oss sec ion σ i as Au ho ized licensed use limi ed o: UNIVERSITAT POLITÈCNICA DE CATALUNYA. Downloaded on July 21,2010 a 14:29:17 UTC om IEEE Xplo e. Res ic ions apply. 1782 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 48, NO. 4, APRIL 2010 Fig. 5. VV pola iza ion. Compa ison be ween (do s) da a, (dashed line) KA, and (solid line) SPM as a unc ion o he equency o incidence angle o (a) 15◦, (b) 30◦,and(c)45 ◦. ollows: σm= 4 m 4 i Pm P i σ i.(10) Based on p e ious measu emen campaigns, he measu e- men expe imen al e o is expec ed o be lowe han 1 dB [31]–[33]. V. R ESULTS In his sec ion, we p esen a compa ison be ween he da a measu ed in acco dance wi h he p ocedu e desc ibed in Sec ion IV and he esul s ob ained by employing he heo e ical me hods ecalled in Sec ion III. The se o acqui ed sca e ing da a allows he in es iga ion o he p ope ies o he backsca - e ed elec omagne ic ield as a unc ion o he incidence angle and o he elec omagne ic equency. A. O e all Compa ison In Fig. 4(a), a compa ison be ween measu ed da a (do s), SPM (solid line), and KA (dashed line) is shown o VV pola iza ion. As s a ed in Sec ion III, bo h heo e ical me hods make use o he Bm p ocess o he su ace desc ip ion. Each do is ep esen a i e o he alue a e aged o e all he φangles and he equency-band measu emen s. The heo e ical cu es a e also a e aged in he conside ed band. No e ha a low incidence angles (up o abou 20◦), KA p edic ions well ma ch Fig. 6. HH pola iza ion. Compa ison be ween (do s) da a, (dashed line) KA, and (solid line) SPM as a unc ion o he equency o incidence angle o (a) 15◦, (b) 30◦,and(c)45 ◦. TABLE II VARIANCE AND CORRELATION LENGTH VALUES AS A FUNCTION OF THE OBSERVED AREA he expe imen al da a; a la ge incidences, we expec ha KA alidi y limi s a e no sa is ied, and he elec omagne ic me hod accu acy dec eases. The SPM seems o be able o ollow he ield beha io o in e media e incidence angles, in acco dance wi h i s alidi y limi s [34]. Simila conclusions can be in e ed by analysis o Fig. 4(b) o HH pola iza ion. Anyway, a comple e discussion on he me hod alidi y calls o a e o mula ion o limi s in e ms o ac al pa ame e s. B. F equency Dependence The ield sca e ed by a su ace is s ongly dependen on he EM inciden wa eleng h; hence, an accu a e s udy on he dependence o he measu ed da a on he equency is equi ed in o de o e i y ha he pe o med a e ages make sense. The e o e, we in es iga ed he EM sca e ed ield o Fig. 4 as a unc ion o he ield equency o ixed incidence angles, (see Figs. 5 and 6, espec i ely) o VV and HH pola iza ions. Au ho ized licensed use limi ed o: UNIVERSITAT POLITÈCNICA DE CATALUNYA. Downloaded on July 21,2010 a 14:29:17 UTC om IEEE Xplo e. Res ic ions apply. RUELLO e al.: MEASUREMENT OF THE EM FIELD BACKSCATTERED BY A FRACTAL SURFACE 1783 Fig. 7. VV pola iza ion. Compa ison be ween (do s) expe imen al esul s and he KA esul s in conjunc ion wi h (solid line) Bm, (long dashed line) Ga-Ga, and (sho dashed line) Ga-Exp o as ollows. (a) σ=1.39 cm; l=17.3cm. (b) σ=1.3cm; l=9.1cm. (c) σ=1.01 cm; l=5.9cm. (d) σ=0.7cm; l=3.3cm. In Figs. 5(a) and 6(a), we show a compa ison be ween measu ed da a (do s), SPM (solid line), and KA (dashed line) as a unc ion o he ield equency, o θ ixed a 15◦.A his incidence angle, he e is a good ma ching be ween measu e- men da a and p edic ion p o ided by bo h heo e ical models, see Fig. 4. Apa om small andom oscilla ions, measu ed da a show a equency beha io simila o hose o he heo- e ical cu es. Such a esul con i ms ha a e aging da a o e he employed equency ange is meaning ul o compa a i e pu poses. In Figs. 5(b) and 6(b), he θ alue is ixed a 30◦. We s ill no e ha he equency dis ibu ion o he acqui ed ield alues is almos linea wi h a law simila o ha o he heo e ical me hod esul s, again jus i ying a e aging da a o e he employed e- quency ange o compa ison pu poses. Fo he highes equen- cies, we can no e a wo s ag eemen be ween he heo e ical models p edic ions and he expe imen al da a esul s, due o he ac ha he buil -su ace oughness is sca cely con olled a he smalles spa ial scales, which a e in ol ed in he sca e ing a he highes equencies. In addi ion, a he highes equencies, he elec omagne ic me hods a e no comple ely adequa e o model he obse ed phenomenon, because hey do no accoun o shadow and mul iple- e lec ions phenomena. The educed equency band whe e his beha io occu s does no impai he compa ison esul s. In Figs. 5(c) and 6(c), he θ alue is ixed a 45◦, in co e- spondence o a signi ican mean di e ence be ween da a and expe imen s, see Fig. 4. Again, he equency dis ibu ion o he acqui ed ield alues is almos linea wi h a law simila o ha o he heo e ical me hod esul s, jus i ying a e aging da a o e he employed equency ange o compa ison pu poses. So a , we limi ed ou a en ion on SPM and KA based on he Bm su ace desc ip ion. Despi e i is widely accep ed ha na u al su aces a e e icien ly desc ibed by ac al models, we ind i use ul o compa e ac al and classical me hods. The esul s o such a compa ison can d i e he choice o he su ace model o be employed in emo e sensing applica ions. C. Gaussian Ve sus F ac al Su ace Models In he ollowing, we compa e he expe imen al da a wi h he esul s o he KA and SPM elec omagne ic me hods ob ained by employing Bm and classical su ace models, i.e., a Gaussian su ace pd , wi h Gaussian (Ga-Ga) o exponen ial (Ga-Exp) co ela ion unc ion. In o de o pe o m he compa ison, i is necessa y o de- e mine he co ela ion leng h and he su ace heigh a iance. The e o e, we es ima ed hese pa ame e s on he la ges possi- ble squa e a ea (90 ×90 cm2)o he buil su ace: The ob ained esul s a e p esen ed in he i s ow o Table II. In Figs. 7(a) and 8(a), he compa ison be ween he measu ed sca e ed ields (do s) and he heo e ical esul s ob ained by using Bm (solid line), Ga-Ga (long dashed line) and Ga-Exp (sho dashed line) su ace models in he KA is p o ided o VV and HH pola iza ions, espec i ely. In Figs. 9(a) and 10(a), he same analysis is shown o he SPM me hod. I is e iden ha he Bm su ace model be e ma ches he da a wi h espec o classical models, a leas o small and mode a e incidence angles. The only excep ion ega ds he HH pola iza ion o he SPM case. The di e ences be ween ac al and classical sca e ing model esul s a e su p isingly wide, and hey lead o hink ha he used classical pa ame e alues a e no ep esen a i e o he conside ed su ace. The e- o e, we e alua ed he classical pa ame e alues land σby conside ing smalle po ions o he buil su ace (see om he second o he ou h ows o Table II), and we compa ed he co esponden e alua ed EM sca e ed ield wi h he mea- su ed da a, as shown in Figs. 7(b)–(d), 8(b)–(d), 9(b)–(d), and Au ho ized licensed use limi ed o: UNIVERSITAT POLITÈCNICA DE CATALUNYA. Downloaded on July 21,2010 a 14:29:17 UTC om IEEE Xplo e. Res ic ions apply. 1784 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 48, NO. 4, APRIL 2010 Fig. 8. HH pola iza ion. Compa ison be ween (do s) expe imen al esul s and he KA esul s in conjunc ion wi h (solid line) Bm, (long dashed line) Ga-Ga, and (sho dashed line) Ga-Exp o as ollows. (a) σ=1.39 cm; l=17.3cm. (b) σ=1.3cm; l=9.1cm. (c) σ=1.01 cm; l=5.9cm. (d) σ=0.7cm; l=3.3cm. Fig. 9. VV pola iza ion. Compa ison be ween (do s) expe imen al esul s and he SPM esul s in conjunc ion wi h (solid line) Bm, (long dashed line) Ga-Ga, and (sho dashed line) Ga-Exp o as ollows. (a) σ=1.39 cm; l=17.3cm. (b) σ=1.3cm; l=9.1cm. (c) σ=1.01 cm; l=5.9cm. (d) σ=0.7cm; l=3.3cm. 10(b)–(d). As expec ed, o classical su ace desc ip ions, land σ alues depend on he dimension o he su ace used o e alua e hem. Figs. 7–10 show ha he a ia ions o he land σ alues s ongly a ec he e alua ion o he sca e ed ield. As a d ama ic consequence, his s ongly educes eliabili y o esul s ob ained by employing classical sca e ing heo ies and hei use o in e se p oblems in emo e sensing applica- ions. No e ha we can ob ain a easonable ag eemen be ween measu emen s and da a by app op ia ely choosing he Gaussian pa ame e s σand l: o ins ance, in he conside ed case s udy, he KA wi h Ga-Ga su ace model wi h σ=1.3cm and l=9.1cm allows he ob aining o a a he good ag eemen Au ho ized licensed use limi ed o: UNIVERSITAT POLITÈCNICA DE CATALUNYA. Downloaded on July 21,2010 a 14:29:17 UTC om IEEE Xplo e. Res ic ions apply. RUELLO e al.: MEASUREMENT OF THE EM FIELD BACKSCATTERED BY A FRACTAL SURFACE 1785 Fig. 10. HH pola iza ion. Compa ison be ween (do s) expe imen al esul s and he SPM esul s in conjunc ion wi h (solid line) Bm, (long dashed line)Ga-Ga, and (sho dashed line)Ga-Exp o as ollows. (a) σ=1.39 cm; l=17.3cm. (b) σ=1.3cm; l=9.1cm. (c) σ=1.01 cm; l=5.9cm. (d) σ=0.7cm; l=3.3cm. wi h measu emen s o angles lowe han 20◦, [see Figs. 7(b) and 8(b)]. Howe e , he σand l alues ha gua an ee his ag eemen a e no ep esen a i e o he su ace. The ob ained esul s lead o he conclusion ha ac al mod- els a e mo e app op ia e han classical ones, o he ollowing easons: 1) he elec omagne ic esul s a e close o expe imen- al da a, and 2) he ac al pa ame e s be e desc ibe he su - aces because hey a e in insic pa ame e s, i.e., hei e alua ion does no depend on he obse e . Bo h esul s sugges he use o ac als o he elec omagne ic sca e ing e alua ion as well as o in e se emo e-sensing p oblems. VI. CONCLUSION In his pape , we p esen ed an inno a i e measu emen p oce- du e o alida ing he heo e ical me hods o e alua ion o he EM ield sca e ed om na u al su aces. To his aim, a ac al su ace wi h assigned pa ame e s was buil as a supe posi ion o ca dboa d and aluminum laye s, as p esen ed in a ecen pape . The cha ac e is ics o such a su ace we e e i ied by measu ing i wi h an op ical high-p ecision ins umen . The su ace was moun ed on a o o in an anechoic chambe , and he EM ield backsca e ed om i was measu ed a di e en incidence angles. P oblems ela ed wi h he acquisi ion o he ield alues we e add essed and sol ed. The compa ison be ween he ob ained calib a ed da a and he heo e ical esul s de i ing om he Bm use in he KA and SPM me hods shows ma ching and disc epancies be ween heo e ical p edic ion and expe imen al esul s. In pa icula , a low incidence angles, he KA appea s o be he mos app op ia e me hod o p edic he sca e ed ield, whe eas a in e media e angles, he SPM be e ma ches he da a, as p edic ed by heo y. In addi ion, p o ided ha he na u al su aces a e well de- sc ibed by ac al laws, we explo ed he possibili y ha classical me hods elying on he Gaussian su ace desc ip ion could be used o he e alua ion o he ield sca e ed om na u al su aces. The e o e, we measu ed he s anda d de ia ion σand he co ela ion leng h lon su ace po ions wi h di e en sizes, and we compa ed he p edic ed EM sca e ed ield wi h da a. The ob ained σand l alues and, as a consequence, he o e- cas ed sca e ed ield alues u n ou o depend on he chosen dimension. I means ha he classical σand lpa ame e s a e no in insic desc ip o s o he su ace. The ob ained expe imen al esul s lead o he conclusion ha he use o ac al su aces in he EM sca e ing heo ies p o ides wo main ad an ages: 1) he esul s a e close o expe imen al da a, and 2) he su ace is e icien ly desc ibed in e ms o only wo in insic pa ame- e s, whose alue does no depend on he obse e . REFERENCES [1] G. F ancesche i and D. Riccio, Sca e ing, Na u al Su aces, and F ac- als. Bu ling on, MA: Academic, 2007. [2] G. F ancesche i, A. Iodice, and D. Riccio, “F ac al models o sca e - ing om na u al su aces,” in Sca e ing, R. Pike and P. Saba ie , Eds. London, U.K.: Academic, 2001, pp. 467–485. [3] T. K. Chan, Y. Kuga, A. Ishima u, and C. T. C. Le, “Expe imen al s udies o bis a ic sca e ing om wo-dimensional conduc ing andom ough su aces,” IEEE T ans. Geosci. Remo e Sens., ol. 34, no. 3, pp. 674–680, May 1996. [4] Y. Oh, K. Sa abandi, and F. T. Ulaby, “An empi ical model and an in e - sion echnique o ada sca e ing om ba e soil su aces,” IEEE T ans. Geosci. 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