Topography Reconstruction by Interferometric SAR Look Vector's Orthogonal Decomposition

Transcription

1 005 WSEAS Int. Conf. on REMOTE SENSING, Venice, Italy, Novembe -4, 005 (pp89-94) Topogaphy Reconstuction by Intefeometic SAR Look Vecto's Othogonal Decomposition S. REDADAA 1 and M. BENSLAMA 1 Laboatoy of Automatic and Infomatics LAIG Univesity of Guelma B.P 401 Guelma 4000 ALGERIA Laboatoy of Electomagnetism and telecommunications LET Univesity of Constantine, Constantine 5000 ALGERIA edasdz@yahoo.f, Abstact: - Topogaphy econstuction by intefeometic synthetic apetue ada (InSAR) efes to a method used to detemine taget's thee- dimensional (3D) position. This position is obtained by pincipal measuements and the imaging geomety of system. By coheently combining signals fom the conventional and additional SAR antennas, the intefeometic phase diffeence between the eceived signals can be fomed fo each imaged point. In this scenaio, the phase diffeence is essentially elated to the geometic path length diffeence to the image point, which depends on the topogaphy. The ange equation, the Dopple equation and the intefeometic phase equation of InSAR povide the elation between the fundamental measuements and the point s position on topogaphy. So the point s position may be obtained by solving thee intefeometic equations. The InSAR equations contain the unknown point s position in a highly nonlinea manne and ae difficult to be solved diectly. Fo this poblem, eseaches have esoted to the othogonal decomposition of the look vecto of ada and have deived seveal econstuction algoithms. In this pape, we analyze the topogaphy econstuction algoithm based on Madsen's othogonal decomposition (MOD). We show that the plane wave model has been intoduced in the deivation of this algoithm which causes a significant econstuction eo especially fo aibone case. Key-Wods: Look vecto's othogonal decomposition, Topogaphy econstuction, InSAR. 1 Intoduction Intefeometic ada has been poposed and successfully demonstated as a topogaphic mapping technique by Gaham [1], Zebke and Goldstein [], and Gabiel and Goldstein [3]. A ada intefeomete is fomed by elating the signals fom two spatially sepaated antennas; the sepaation of the two antennas is called the baseline. The spatial extent of the baseline is one of the majo pefomance dives in an intefeometic ada system: if the baseline is too shot the sensitivity to signal phase diffeences will be so small to be undetectable, while if the baseline is too long additional noise due to spatial decoelation coupts the signal. Two distinct implementation appoaches have been developed fo topogaphic ada intefeometes; they diffe in how the intefeometic baseline is fomed. In the fist case the baseline is fomed by two physical antennas which illuminate a given aea on the gound simultaneously: this is the usual appoach fo aicaft implementations whee the physical mounting stuctues may be spaced fo sufficient baseline. This appoach is used by [] fo the NASA CV-990 ada, and is also used in the TOPSAR topogaphic mapping ada mounted on the NASA DC-8 aicaft [4]. The second type of implementation utilizes a single satellite antenna in a nealy-exact epeating obit. Then it foms an intefeomete baseline by elating ada signals on the epeat passes ove the same site. Topogaphic maps using this technique have been demonstated by Goldstein et al [] and Gabiel et al [3], [5]. Although, intefeometic topogaphic mapping is possible using each appoach, a significant advantage accues fom using the single-pass aicaft implementation ove the epeat-obit spacecaft method. Specifically in the fist appoach, we avoid poblems associated with tempoal decoelation of the suface, that is, the esult that changes on the wavelength scale of the suface lead to additional decoelation noise in the intefeogam. If the change on the suface is lage, as could happen if, pecipitation occued, the phases of the eceived signals may be wholly unelated. On the othe hand, spacebone platfoms povide views of inaccessible egions of the Eath. The Acoss-tack intefeometic technique exploits the phase diffeences of at least two complex-valued SAR images acquied fom slightly diffeent positions

2 005 WSEAS Int. Conf. on REMOTE SENSING, Venice, Italy, Novembe -4, 005 (pp89-94) and/o at diffeent times to extact height infomation of topogaphy. It can measue the slant ange diffeence at the ode of sub-wavelength, thus it has the ability to extact accuately topogaphy height infomation [6]-[8]. The aibone InSAR imaging geomety is depicted in Fig.1. Fig.1 Aibone InSAR imaging geomety. The coodinate system denoted by XYZ $$$ is the ECR (Eath Centeed Rotating) coodinate system with the oigin located at the eath's gavitational cente, with Y $ paallel to the nominal tack A i, the slant ange vecto fom antenna A i to taget P is given by i, whee the subscipt i efes to the antenna numbe o tack numbe. The angle θ, β espectively epesents the look angle and squint angle of antenna A. The taget location 1 vecto is given by vecto P, the baseline vecto is defined as b = A A 1. The slant ange i, the Dopple fequency f D and the intefeometic phase ϕ ae fundamental measuements of InSAR, the elation among them can be descibed by thee basic intefeometic equations : the equation of ange sphee, the Dopple equation and the phase equation [9]. These equations ae espectively given by: v v P A i = i (1) f D =< $ i, v > () πq ϕ = ) (3) ( 1 whee Q = 1 fo the standad mode InSAR system, Q = fo the ping-pong mode and $ i is the unit vecto in the line-of-sight diection of i. Topogaphy econstuction by InSAR efes to detemine taget's 3D location using pincipal measuements and the imaging geomety of system; it may be achieved by solving the equation set (1)-(3). Howeve, the equation set is difficult to be solved diectly, that due to its highly nonlinea chaacteistics. To solve this poblem, seveal econstuction algoithms have been poposed and they ae geneally divided into two classes: numeical algoithms [10] and analytical algoithms [11]-[1]. In numeical algoithms, the lineaization of the nonlinea equation set may be achieved by Taylo expanding InSAR equations (1)-(3) to the fist ode and a linea equations system is obtained. Then the unknown position coodinates may be easily achieved by solving the new equation set. The main dawback of numeical algoithms is thei inaccuacy due to using only the constant and the linea tem of Taylo seies. Howeve, analytical algoithms may give moe accuate solution than numeical ones because they detemine the position coodinates by solving the equation set fomed by (1)-(3) without intoducing any appoximation. In analytical algoithms, two eseach tends ae esoted to solve the highly nonlinea equation set. Thus, analytical algoithms fall into two classes: the diect analytical method [11] and the look vecto's othogonal decomposition method [13]. In this pape, we will focus on the study of topogaphy econstuction based on the look vecto's othogonal decomposition. Accoding to (1), the taget location can be expessed as P = A1 + 1 = A $ 1 (4) whee the antenna vecto A 1 may be povided by navigation equipments and the slant ange 1 may be detemined accoding to the pinciple of ada anging. Theefoe, the detemination of the taget location is educed to the detemination of the unit vecto. Then, look vecto's othogonal decomposition method is intoduced just fom this idea. The look vecto is pojected onto an othogonal basis that is defined as local antenna coodinate system, and then tansfomed into the geneal coodinate system. Thus, substituting the unit vecto in the geneal coodinate system into (4) may achieve topogaphy econstuction. Accoding to the idea of the othogonal decomposition, Madsen has intoduced a local coodinate system defined on the aicaft and has given a method fo the othogonal decomposition of the look vecto [13]. Howeve, the look vectos fo the two acquisitions have been assumed to be paallel, and thus the ange diffeence appoximately equals to the pojection of the baseline vecto on the look diection. That is to say, a plane wave model of the electomagnetic wave font has been intoduced. The simplified model may be valid only fo spacebone InSAR geometies with small swaths and small atios of the baseline to the slant ange [14]. Howeve, the aibone InSAR has a elatively lage baseline-ange

3 005 WSEAS Int. Conf. on REMOTE SENSING, Venice, Italy, Novembe -4, 005 (pp89-94) atio than the spacebone one. Thus, the plane wave model will intoduce a significant height econstuction eo. Plane wave model Accoding to (3), the intefeometic phase of InSAR is popotional to the ange diffeence and may be expessed by the baseline vecto and ange vecto as [9] ( ) < ˆ 1 π Q π Q b ˆ, 1 b> b ϕ = = (5) 1 1 The unit vectos of 1 and b espectively denoted by $1 b π and $b. x =, γ = Q 1, κ = < > 1 1 $, b $. The Taylo expanding of (5) at x = 0 is given by 3 γκ γ γ κ 3 ϕ x + x + x +... (6) γ + 1 ( γ + 1) if b << 1, the following elation can be obtained only by using the constant and linea tem of Taylo seies γκ πq ϕ pw x = < $ 1, b > (7) whee ϕ pw is the plane wave model of the intefeometic phase [], [9], [13], [15]-[16]. As discussed above, the taget position is detemined by the intesection of thee sufaces descibed by (1)-(3) [9]. Fig. depicts the ange sphee, the Dopple cone and the phase cone elevant to antenna A 1. The ange sphee is centeed at antenna A 1 and has a adius equal to slant ange 1. The Dopple cone has a geneating axis along the velocity vecto and the cone angle is popotional to the Dopple fequency. Fom (5), the phase suface is a hypeboloid with focuses located at A i and a symmetical axis along the baseline vecto. Thus, the taget position is given by the intesection locus by the thee sufaces. Unde the plane wave model, the intefeometic phase hypeboloid educes to a phase cone, and the taget position is detemined by the intesection locus of ange sphee, the Dopple cone and the phase cone, illustated by P in Fig.. Accoding to this intepetation, the plane wave model may intoduce an unavoidable econstuction eo. We note that a conventional SAR system esolves tagets in the ange diection by measuing the time it takes a ada pulse to popagate to the taget and etun to ada. The along-tack location is detemined fom the Dopple fequency shift that esults wheeve the elative velocity between the ada and taget is not zeo. A taget in the ada image could be located anywhee on the intesection locus, which is a cicle in the plane fomed by the ada line of sight to the taget and vecto pointing fom the aicaft to nadi. Fig. Geometic intepetation of topogaphy 3D econstuction by InSAR. Physically, the so called plane wave model is that when b << 1 is valid. The signals scatteed by the same point P on the suface of topogaphy aive at the antennas in a paallel manne, thus, the wave font may be appoximated as a plane. So, the ange diffeence fom antennas to point P appoximately equals the pojection of the baseline vecto onto the look diection, whee Δ = A1C =< $ 1, b > as illustated in Fig.3. Howeve, the wave font is actually spheical, the ange diffeence is AB 1. Thus, an intefeometic phase analytic eo will be unavoidably intoduced. Fom (5) and (6), this eo can be expessed as πq πq Δϕ = ϕ ϕ pw = ( 1 ) = 1. 1 (8) b < ˆ 1,bˆ > b b < > 1 ˆ,bˆ A 1 b B C A 1 Fig.3 The plane wave model. As illustated in Fig.4, fo height measuement by InSAR, the univesal geomety may be the twodimensional (D) InSAR geomety, unde which the baseline component lies in the plane of the look vecto and the nadi diection, nomal to the flight diection. Accoding to this geomety, we have < ˆ ˆ1, b >= sin( θ α ) (9) P

4 005 WSEAS Int. Conf. on REMOTE SENSING, Venice, Italy, Novembe -4, 005 (pp89-94) whee α is the angle the baseline makes with espect to a efeence hoizontal plane. Fom (8), we obtain the height econstuction eo intoduced by the plane wave model 1 ϕ ΔP z = 1 + sin( θ ) Δϕ (10) πqb cos( θ α ) πq1 $Z (0, 0) A 1 A1 α A $Y θ 1 b δ A 1 P P 3 Madsen's othogonal decomposition In 3D InSAR imaging geomety, the baseline compises a component along tack and component acoss tack. These two components coespond to two diffeent types of applications: the diffeential intefeomety and the topogaphy measuement. On this decomposition basis, Madsen has intoduced an othogonal decomposition that defines a local coodinate system. The MMC (Madsen Moving Coodinate) system vnw $$$ is illustated in Fig.5, with $X paallel to the antenna velocity vecto v. Its thee othogonal bases ae v v$ = v b << b, v$ >, v$ > n$ = (11) b << b, v$ >, v$ > w$ = n$ v$ Fig.4 D InSAR geomety. As mentioned above, the plane wave model was oiginally applied to the spacebone InSAR system with a small swath [17]. The model allows simplified estimation of the baseline based on tie points (points of known elevation on the gound) distibuted in ange and in azimuth if thee is divegence of the baseline. Howeve, one should be awae that significant systematic econstuction eos ae intoduced. The plane wave model is inaccuate fo satellite geometies with lage swaths, and will intoduce a systematic height econstuction eo. Fo cases with paallel tacks, the ERS esults show that the plane wave model causes systematic height eos on the ode of 10 s of mete. Moeove, unde D InSAR, the systematic eos incease in cases with lage tack divegences. Although these eos may be mitigated though tie point baseline tweaking [14], [18], they can't be eliminated completely. Theefoe, the model must be caefully teated to get an accuate topogaphy econstuction by spacebone InSAR. Fo aibone InSAR geometies, the phase eo and the height econstuction eo intoduced by the plane wave model ae appoximately of the same ode as the accuacies of the phase measuement and the height econstuction by InSAR [19]. On the othe hand, even fo the aibone InSAR with dual antennas constucted on the igid platfom, the attitude of the aicaft may be dynamic changed. So, tack divegences also exist in aibone InSAR as that in spacebone InSAR, which in tun inceases the height econstuction eo intoduced by the plane wave model. Fig.5 MMC and ECR systems Then, the unit look vecto $ 1 may be expessed by the othogonal basis as $ $ $ 1 = μ 1v + η1n+ ζ 1 $w, μ $, v$ 1 =< 1 > (1) η $, n$ 1 =< 1 > ζ $, w$ 1 =< 1 > Unde the plane wave model, the intefeometic phase now eads [9] πq ϕ = < $ 1, b > (13) Substituting () and (13) into (1) deives the unit vecto in the local system f D v ϕ b ˆ v f D 1 vnw = (14) πqbn bn v ± 1 μ1 η1 whee bv =< b, v $ >, b n =< b, n$ >, b = b v v $ + b n n $. The sign of ζ 1 is given by "+" fo the left looking of ada and "-" fo the ight.

5 005 WSEAS Int. Conf. on REMOTE SENSING, Venice, Italy, Novembe -4, 005 (pp89-94) 4 Topogaphy econstuction algoithm The tansfomation fom the local coodinate system to the geneal one can be achieved by equation (15) Γ = 0 cosα sinα (15) 0 sinα cosα Theoetically, afte tansfomation by (15), we have fulfilled the tansfomation. Howeve, in aibone InSAR, the vaiation of aicaft attitude (depicted by yaw, pitch and oll) causes the unit look vecto in the geneal coodinates to be tansfomed into a new coodinate. Thus, in ode to achieve the topogaphy 3D econstuction, the unit look vecto must be tansfomed back into the geneal coodinate system fom the new one again. The tansfomation can be fulfilled by Eule otation matices coesponding to thee attitude angles. The above tansfomation yields the unit look vecto in the geneal coodinate system including the attitude of the aicaft. $ 1 = YPR$ 1vnw (16) whee R = 0 cosθ sinθ, 0 sinθ cosθ cosθ p 0 sinθ p P = 0 1 0, (17) sinθ p 0 cosθ p cosθ y sinθ y 0 Y = sinθ cos 0 y θ y whee Y, P, R ae Eule otation matices and θ, θ p, θ y ae the attitude angles espectively fo oll, pitch and yaw. Finally, we calculate the ( x, y, z) coodinates of the tagets by using (14) - (17) with (1) and (4). 5 Conclusion In this pape, the topogaphy econstuction algoithm based on Madsen's othogonal decomposition has analyzed. Madsen has given an othogonal decomposition of the look vecto unde the plane wave model of the electomagnetic wave font. Howeve, InSAR systems equie extemely accuate knowledge of the baseline length and oientation angle-millimete o bette knowledge fo the baseline length and 10's of ac second fo the baseline oientation angle. These equiements necessitate an extemely igid and contolled baseline, a pecise baseline metology system and igoous calibation pocedues. Also, we note that the phase accuacy equiements fo intefeometic systems typically ange fom This imposes athe stict monitoing of phase changes, which ae not elated to the imaging geomety in ode to poduce accuate topogaphic maps. Fom ou analysis, we may finally conclude that the plane wave model intoduce a significant econstuction eo especially fo the aibone InSAR, and must be discaded to accuately econstuct the topogaphy. Refeences: [1] L.C. Gaham, Synthetic apetue ada intefeomety fo topogaphic mapping, Poceedings IEEE, vol. 6, 1974, pp [] H.A. Zebke and R.M Goldstein, Topogaphic mapping fom intefeometic synthetic apetue ada obsevations, Jounal of Geophysical Reseach, vol. 91, No.B5, 1986, pp [3] A.K. Gabiel and R.M. Goldstein, Cossed obit intefeomety: Theoy and expeimental esults fom SIR-B, Intenational Jounal of Remote Sensing, vol. 9, No.8, 1988, pp [4] H.A. Zebke, S. N. Madsen, J. Matin, et al., The TOPSAR intefeometic ada topogaphic mapping instument, IEEE Tansactions on Geoscience and Remote Sensing, vol. 30, No.5, 199, pp [5] A.K. Gabiel, R.M. Goldstein, and H.A Zebke, Mapping small elevation changes ove lage aeas: diffeential ada intefeomety, Jounal of Geophysical Reseach, vol.94, No.B7, 1989, pp [6] K. A. Câmaa de Macedo and R. Scheibe, Pecise topogaphy and apetue dependent motion compensation fo aibone SAR, IEEE Tansactions on Geoscience and Remote Sensing Lettes, vol., No., 005, pp [7] S. Redadaa, A. Boualleg, D. Benatia, and M. Benslama, Influence of the InSAR paametes on the height esolution, 1 st Intenational Confeence on Telecomputing and Infomation Technology ICTIT'004, Amman, Jodan, 004, pp [8] S. Redadaa and M. Benslama, Application of the InSAR technique in the topogaphy field, accepted in 10 th Intenational Symposium on Micowave and Optical Technology ISMOT005, August -5, Fukuoka, Japan, 005. [9] P.A. Rosen, F.K. Li, S. Hensley, S.N. Madsen, I.R. Joughin, E.Rodiguez and R.M. Goldstein, Synthetic apetue ada intefeomety, Poc. IEEE, Vol.88, No.3, 000, pp [10] K.H. Gutjah, A new InSAR geolocation algoithm, Euopean Confeence on Synthetic Apetue Rada, EUSAR 000, Munich, Gemany, 000, pp [11] W. Goblisch, The exact solution of the imaging equations fo coss tack intefeometes,

6 005 WSEAS Int. Conf. on REMOTE SENSING, Venice, Italy, Novembe -4, 005 (pp89-94) IEEE Intenational Geoscience and Remote Sensing Symposium, IGARSS 97, Singapoe, 1997, pp [1] A.J. Wilkinson, Techniques fo 3D suface econstuction using ada intefeomety, Ph.D Thesis, Univesity of London, England, [13] S.N. Madsen, H.A. Zebke, and J. Matin, Topogaphic mapping using ada intefeomety: pocessing techniques, IEEE Tansactions on Geoscience and Remote Sensing, vol.31, No.1, 1993, pp [14] D. Small, P. Pasquali, and S. Füglistale, A compaison of phase to height convesion methods fo SAR intefeomety, IEEE Intenational Geoscience and Remote Sensing Symposium, IGARSS 96, Nebaska, USA, 1996, pp [15] E. Rodiguez and J. Matin, Theoy and design of intefeometic synthetic apetue adas, IEE Poceedings-F, vol. 139, No., 199, pp [16] J.0. Hagbeg and L.M.H. Ulande, On the optimization of intefeometic SAR fo topogaphic mapping, IEEE Tansactions on Geoscience and Remote Sensing, vol. 31, No.1, 1993, pp [17] F.K. Li, and R. M. Goldstein, Studies of multibaseline spacebone intefeometic synthetic apetue adas, IEEE Tansactions on Geoscience and Remote Sensing, vol. 8, No.1, 1990, pp [18] D. Small, C. Weme, and D. Nüesch, Baseline modelling fo ERS-1 SAR intefeomety, IEEE Intenational Geoscience and Remote Sensing Symposium, IGARSS 93, Tokyo, Japan, 1993, pp [19] V. Mstik, J.G. VanBlaicum, G. Cadillo, and M. Fennel, Teain height measuement accuacy of intefeometic synthetic apetue adas, IEEE Tansactions on Geoscience and Remote Sensing, vol. 34, No. 1, 1996, pp.19-8.