Effect of the relativistic Doppler shift on geolocation of SAR data. Jezek, Carande, Olmsted

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1 Eect o the reltiitic Doppler hit on geoloction o SAR t Jezek, Crne, Olmte Abtrct We compre the eect o clicl n reltiitic Doppler requency hit on SAR geoloction etimte in the long trck or zimuth irection. For impliie geometry, we clculte tht there my be much 10 m bi between long trck poition etimte uing the clicl n reltiitic Doppler eqution. We conclue tht reltiitic Doppler correction i importnt or uture ytem tht trie to pproch 1-m reolution n highly ccurte geoloction. Introuction Spceborne ynthetic perture rr (SAR) rely on requency chirpe ignl to obtin reolution in the cro trck (rnge irection). Mtch iltering the chirp llow long pule with low erge power to be compree into ery hort pule. SAR rely on the Doppler hit to ocu the bem in the long trck (light) irection. Eentilly, the rition in the Doppler requency i equilent to chirp in the zimuth irection. Een more imply, the ierence between the Doppler hite receie ignl n the requency o the trnmitte ignl cn be ue to etimte the long trck poition o the trget. Conequently, etimting the proper Doppler requency eqution in SAR proceing lgorithm i eentil to reuce error in long trck poition error. Commercilly ilble SAR proceor ue the clicl eqution or etimting the Doppler requency. In thi pper, we exmine the conequence o uing the clicl eqution rther thn the reltiitic ormultion or pceborne SAR ytem. Similr to the reult o Olmte (1993), we preict tht there i meurble ierence in zimuth poition etimte by the clicl n reltiitic ecription Clicl Doppler Shit The clicl Doppler requency eqution re gien in tnr text, which emontrte tht the Doppler requency i ierent or the ce o ttionry receier n moing trnmitter n the ce o moing receier n ttionry trnmitter. Firt conier the trnmitter mounte on tellite it moe towr trget or receier on ttionry erth. The moing trnmitter ire ignl. The Doppler requency obere by ttionry receier/trget own rnge will be 1 1 Here i the operting requency o the ource, i the propgtion elocity o the we, n i the pee o the ource.

2 The ttionry receier/trget now ctter ignl with requency bck to the ource. The ignl obere by the moing ource i (1 + ) (1 + ) (1 ) Now elute the ierence between the trnmitter requency n the obere Doppler requency. (1 + ) (1 ) 3 Curlner n McDonough (1991, p. 17) gie thi reult prt o the impliie ecription o SAR methoology Curlner n McDonough reult. They go on to how tht the ierence between the Doppler hite ignl n the trnmitte ignl i proportionl to the long trck (zimuthl) poition o the trget. Reltiitic Doppler Shit Stellite borne ytem trel t bout 7 km/ec. We eek to inetigte whether the clicl Doppler hit i uiciently ccurte to be ue or SAR t geoloction. The reltiitic Doppler hit i gien by γ (1 β coα) 4 Vt β c γ (1 β 1/ ) Here Vt i the pee o the tellite, c i the pee o light, lph i the ngle between the tellite light irection n the line connecting the tellite to the trget. It i the me whether or not the ource or trget i moing. Coniering moing tellite n ttionry trget, the requency obere t the trget will be gien by eqution 3. There i non-zero Doppler een t the bro ie ngle becue o the time iltion o the trnmitter clock obere on the urce. Now conier the tellite. The tellite ee the trget moing towr it. The trget

3 ctterer rition bck to the tellite t requency gien by (4). At the tellite, the trget clock now pper low n o the tellite etect requency 5 γ ( 1 β coα ) γ (1 β coα ) When lph i 90 egree, the tellite elocity i orthogonl to the line connecting the tellite poition n the trget. In the clicl ce, thi poition woul correpon to zero Doppler. But or the reltiitic ce, there i non-zero Doppler lue 6 γ Note tht in the o broie irection, the expreion (5) reuce to the clicl reult. Strt by neglecting econ orer term n then expning the enomintor Tylor erie. 1 β coα ( β coα) (1 + β coα) 7 Subtrcting rom eqution 7 yiel eqution 3. Eqution 6 oe not he clicl counterprt n i the reult o time iltion eect which re unccounte or in the clicl ce. The clicl n reltiitic Doppler hit or typicl tellite prmeter re compre in igure 1. Eect on SAR Azimuth Loction It i intereting to expn eqution 5 keeping more o the higher orer term. Thi i one ollow β coα + 6β co α... (1 β coα) 8 Then, (1 + β coα + 6β co α)(1 β ) 3 4 (1 + β coα β (1 6 co α) β coα 6β coα) 9 Neglecting higher orer term in either β or in the prouct o co α n β

4 (1 β + β coα) 10 An coα (1 β ) 11 β Switching the ngle to be eine per Curlner igure 1.8 inθ R( x x R (1 β ) β (1 β )) R( β (1 β )) λ V t 1 Note gin tht when β i mll, the reult reuce to the clicl ce. Alo note tht when β i inclue, the Doppler hit eine by Curlner n McDonough i moiie by n itionl term. Conier the broie ce gin. Suppoe we o not ue the proper bie Doppler n imply tke Then x will be in error by n mount. x e R( (1 β )) R( β ) λ λ V V t t RV t c 13 The error i bout 10 m or rnge o 1000 km. Thi i hl the lue the Olmte, 1993, etimte. Concluion Thi imple clcultion ume lt, ttionry erth n neglect iue ocite with multi-look proceing. Neerthele, our imple clcultion oe ugget meurble eect tht my he conequence or SAR geoloction or other ytem tht rely on Doppler bem hrpening technique. We beliee it my be ueul to ttempt urther experiment irt through SAR imultor tht ccount or erth rottion eect n ole the complete et o rnge, Doppler n erth moel eqution. Thee coul be ollowe up with creully eigne experimentl progrm, the lter requiring creul ement o ll geoloction error. Reerence

5 Curlner, J., n R. McDonough, Synthetic Aperture Rr Sytem n Signl Proceing. Wiley n Son, New York, 647 p. Olmte, C Alk SAR Fcility Scientiic Uer Guie. Appenix B, ASF Publiction ASF-SD-003, 53 p. Acknowlegment Thi reerch w upporte by grnt rom the NASA Polr Ocen n Ice Sheet Progrm n the NASA Pthiner Progrm. We thnk John Bolzn or ueul comment n icuion. Figure 1. Comprion between reltiitic n clicl etimte o the ierence between the trnmitter requency n the Doppler requency. The x-xi i the ngle between the tellite elocity ector n the line connecting the tellite to the trget (90 egree correpon to the clicl zero-oppler ce). Stellite pee i 7 km/ec. Trnmitter requency i 5 GHz.