and the Doppler frequency rate f R , can be related to the coefficients of this polynomial. The relationships are:
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1 Algrithm fr Estimating R and R - (David Sandwell, SIO, August 4, 2006) Azimith cmpressin invlves the alignment f successive eches t be fcused n a pint target Let s be the slw time alng the satellite track If the trajectry f the satellite was a straight line then the range t the pint target wuld vary as a hyperbla Hwever, because the aperture (< 30 km) is shrt cmpared with the nminal range (~850 km), a parablic apprximatin is cmmnly used R(s) = + R R (s " s ) + 2 (s " s ) 2 + where is the clsest apprach f the spacecraft t the target and s is the time f clsest apprach We shwed abve that the critical parameters fr fcusing the SAR image, the Dppler centrid f DC and the Dppler frequency rate f R, can be related t the cefficients f this plynmial The relatinships are: f DC = "2 R # and f R = 2 R " where λ is the wavelength f the radar In additin, if ne assumes a linear trajectry f the spacacraft V relative t the target then the Dppler centrid and Dppler rate can be apprximately related t the velcity and clsest range as f DC = "2V # ( x " s V ) and f R = 2V 2 " We will nt cnsider the Dppler centrid further because it is accurately estimated frm the raw signal data [Madsen, 1989] Fr C-band SARs such as ERS-1 and Envisat, the abve frmula fr the f R prvides adequate fcus Hwever, fr L-band SARs such as ALOS, the aperture is much lnger s ther factrs must be cnsidered such as the curvature and ellipticity f the rbit as well as the rtatin rate f the Earth Curlander and McDnugh [1991] discuss the estimatin f the Dppler rate and there are tw main appraches The Autfcus apprach uses the crudely-fcussed imagery t imprve n the estimate f the Dppler rate The rbit apprach uses the mre precise gemetry f the elliptical rbit abut a rtating elliptical Earth t prvide a mre exact estimate f R Here we cnsider a new, and mre direct apprach, t estimating R Cnsider the fllwing three vectrs which frm a triangle;
2 r " the vectr psitin f the satellite in the Earth-fixed crdinate system; r R e " the vectr psitin f a pint scatterer n the Earth and smewhere in the SAR scene; r " the line-f-sight vectr between the satellite and the pint scatterer The three vectrs frm a triangle such that R r e = R r s + R r The scalar range, which is a functin f slw time, is given by R(s) = r (s) " r R e # + R R (s " s ) + 2 (s " s ) 2 + Measurements f scalar range versus slw time can be used t estimate the cefficients ff the parablic apprximatin The algrithm is t: (1) Use the precise rbit t calculate the psitin vectr f the satellite and cmpute a time series R(s) ver the length f the aperture (2) Perfrm a least-squares parablic fit t this time series t estimate and (3) Cmpute the effective speed as V e 2 = The nly remaining step is t select the psitin f the pint target R r e = R r s + R r Actually this can be any pint in the image s the selectin criteria is that the pint lie at the prper radius f the surface f the Earth R e = R r r e and the vectr is perpendicular t the velcity vectr f the satellite V r If we prescribe the length f the -vectr then the angle between the satellite psitin vectr and the line-f-sight vectr is given by the law f csines as shwn in the diagram belw θ R e
3 q 3 q 2 This diagram has the satellite at the gray dt with the velcity vectr pinting int the page The csine f the lk angle is cs" = R 2 s + R 2 2 ( # R e ) 2 Next cnsider a lcal c-rdinate system where the q 2 -axis is aligned with the velcity vectr f the satellite as shwn in the fllwing diagram V q 1 In this primed c-rdinate system, the q 1 -axis is parallel t R r s, the q 2 axis is parallel t V r, and the q 3 axis is perpendicular t bth and is given by their crss prduct q 3 = r r V " r V The fllwing diagram has the velcity vectr f the satellite ging int the page and aligned with the q 2 crdinate The q 1 vectr is the radial vectr frm the center f the earth The line-f-sight right-lk vectr is in the q 1 - q 3 plane as shwn in the diagram q 1 q 3 θ
4 The line f sight vectr is R r = ("cs# q ˆ 1 0 q ˆ 2 "sin# q ˆ 3 ) Example with ALOS L-Band Orbit and SAR Data Nw that we have an algrithm fr finding a pint n the surface f the Earth that is within the SAcene, we can cmpute the time-evlutin f the range t that pint as the satellite rbits abve the rtating Earth We cnsider data frm a descending rbit ver Kga Japan where three radar reflectrs have been deplyed (FBS343_RSP058_ ) The image is a fine-beam, single plarizatin having a nminal lk angle f 343 degrees A Hermite plynmial interplatin was used t calculate the x-y-z psitin f the satellite frm 28 psitin and velcity vectrs spaced at 60-secnd intervals Thus the entire arc is 28 minutes r a quarter n an rbit The accuracy f the Hermite interplatr was checked by mitting a central pint and perfrming an interplatin using 6 surrunding pints The accuracy f the interplatin was fund t be better than 02 mm suggesting that the 60-s interval prvides an accurate representatin f the rbital arc Next we cmpute the range t the grund pint as a functin f time befre and after the perpendicular LOS vectr r We used a befre/after time interval f 3 secnds which is abut twice the length f the synthetic aperture fr ALOS A secnd-rder plynmial was fit t the range versus time functin and the three cefficients prvide estimates f, R, and R /2 The range versus time, as well as the residual f the fit, are shwn belw One can learn a great deal frm this exercise
5 1) The first thing learned is that the parablic apprximatin t a hyperbla used in the SAR prcessr has a maximum errr f abut 1 cm at a time ffset f 3 secnds This crrespnds t a small fractin f the 23-cm wavelength Nte that the actual aperture length fr ALOS is nly +/- 15 secnds s this apprximatin is justified 2) The range at zer ffset has an errr f -024 m This is due t apprximating the shape f the earth as a sphere having a lcal radius given by the WGS84 ellipsid frmula In ther wrds there is a small errr in the LOS vectr because it intersects the surface f the Earth at a latitude that is slightly different frm the spacecraft latitude s the earth radii will differ slightly 3) The Dppler shift when the satellite is perpendicular t the target can be calculated frm the range rate cefficient This value seems t small because fr this pass the yaw angle was 3 degrees which wuld prduce a Dppler shift f 375 m/s I dn't yet understand this
6 4) Finally the range acceleratin can be used t calculate the effective speed f the satellite V e 2 = Fr this example we arrive at a speed f 7174 m/s By trial and errr we fund the ptimal fcus parameter crrespnds t a speed between 7173 and 7183 s this new estimate is accurate The simple Cartesian grund speed apprximatin prvided an estimate f 7208 m/s which prvided pr fcus
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