Phase and amplitude histories model adapted to the spaceborne SAR survey

L.B. Neronskiy, S.G. Likhansky, I.V. Elizavetin and D.V. Sysenko

Abstract: A model of relative movement of a SAR platform and imaged footprint is proposed. The model has been specially adapted to conditions of a spaceborne SAR survey. The main difference of this model from that used before is that it takes into account the SAR platform movement and Earth rotation in addition to commonly used calculations of satellite motion in the disturbed Earth gravitation field. Is shown that SAR motion could be considered as linear, but with velocity derived from orbital motion parameters. The proposed model could be easily applied to analysis of SAR processing algorithms and the influence of the orbit parameter estimation errors on the resulting image quality. Use of the model simplifies the parameter estimation of the SAR sensors intended for planetary investigations.

1 Introduction

The mainstream of spaceborne SAR development results in resolution improvement and a reduction in revisit time. To achieve a few metres azimuth resolution, high accuracy calculations of signal phase history (i.e. signal replica) are needed. For these calculations, one uses the equations of satellite disturbed motion in the gravity power of the Earth and other space bodies, atmosphere resistance power, light pressure power etc. For amplitude history (i.e. weighting function) calculations and azimuth/range ambiguity correction, some simplified models can be used. Also, such simplified motion models can be used in the stage of spacebome SAR design for evaluation of signal para- meter variations [ 11.

In some papers [2, 31 the relative platform-target motion is described by the Taylor expansion of range variation with coefficients derived from platform ephemeris. However, this approach has a disadvantage: it is difficult to analyse variations of the signal amplitudes and phases over focusing time, especially for the spotlight and squint surveillance modes.

To simplify the calculations for signal parameter analy- sis without loss in accuracy, a motion model adapted to spaceborne SAR surveys is presented. It uses Kepler's model of platform motion around the Earth ellipsoid (WGS-84 or another model). The presented compact formulas are based on local representation of a satellite orbit as a circular orbit with the radius equal to the current

0 IEE, 2003 IEE Proceedings online no. 20030443 DOI; 10. 1049/ip-rsn:20030443 Paper first received 22nd October 2002 and in final revised form 19th February 2003 L.B. Neronskiy and S.G. Likhansky are with FSUE 'MNIIP', 34, Kutuzov Ave., Moscow, 121 170, Russia I.V. Elizavetin and D.V. Sysenko are with FSUE "PO mashinostroyenia', 33 Gagarin St., Reutov, Moscow region, 143952, Russia

184

radius of the orbit. The orbit eccentricity influence is taken into account by the platform vertical velocity.

2 Models of platform-target relative motion

2.1 Linear motion model For airborne SAR, linear motion with constant velocity is usually considered. Let us try to compare the airborne motion case with the orbital motion case. Then, the three components of velocity should be considered: air velocity (corresponds to orbital velocity), drift velocity (corres- ponds to Earth rotation), and vertical velocity (could be treated as altitude variation due to orbit ellipsoidal shape). The observation geometry is shown in Fig. 1, where the X-axis lies along the path velocity vector which is the sum of air and drift velocities.

In the orthogonal coordinate system OXYZ the range vector R(t) is equal to the difference of the fixed target vector R, and the time-varying platform vector R,(t):

R(t) = R, - R,(t) (1)

where R, =x0& +yoY, + zoZf represents the target coordinates and xo, yo, zo are unit vectors.

Let us assume, that for the time t = O the platform is located in coordinate system OXYZ where the path velocity vector V, is parallel to axis OX, and X, = 0, Xs(0) = 0.

The current position of the SAR platform is defined by the vector

R,(t) = xo V,t + yoO + Z0(Z, + VHSt) ( 2 )

where V H , is the vertical platform velocity.

presented by Taylor expansion o f t : The slant range variation over the time axis is commonly

R(t) = J y y + Y: + (Z, + VH,t - ZJ2

= R: + V:t2 + 2(H, - Z,) . VHst N

x R, + Cntn (3) n=l

IEE Proc.-Radar Sonar Navig., Vol. 1.50, No. 3, June 2003

Fig. 1 Obsewation geometry of linearly moving airborne SAR

where Ro = R(0) = 2 / [ Y: + (Z, - Z J 2 ] is the initial range for the time t = O , N is the order of expansion which depends on the required calculation accuracy. Let us note, that Ro=R(O) for VHs#O does not correspond to the nearest distance R,, = R(t,,), where R(tn,) = 0.

In the case of horizontal flight, when the platform has zero vertical velocity, the slant range for time t = 0 corres- ponds to its minimal value for all points in the viewing plane. We assume here that the viewing plane passes through the nadir point on the ground and perpendicular to the platform ground trajectory. Radial velocity in the viewing plane is zero for all slant range values even if we take into consideration the surface relief or Earth curvature.

If the platform vertical velocity is non-zero ( V H s # O ) , the initial slant range Ro for the targets in the viewing plane, that is, normal to the ground trajectory,. does not correspond to the nearest slant ranges for which R(tn,) = 0. Thus, during flight with increasing altitude the negative radial velocity appears, resulting in the negative Doppler frequency shift. To compensate this effect, the viewing plane should be squinted towards the flight direction on the pitch angle defined as:

9 vh = arctan (+) (4)

The time of the nearest slant range to the ground target is defined by the equation:

from which, the time correction value is determined as:

The azimuth shift in the image is presented by the following:

Thereby, in the case of a plane Earth surface, the azimuth shift does not depend on the slant range, but it changes proportionally to the deviation of current platform alti- tude Z,(t).

To define the point’s locus where k(f,,)=O the naval term ‘traverse’ is commonly used. However, this term is correct only for horizontal platform motion with VH, = 0, when the nearest slant range lines are located in the plane which is normal to the pass direction. To avoid misunder- standing in further analysis, we will distinguish two

IEE Proc.-Radar Sonar Nuvig., Vol. 150, No. 3, June 2003

geometrical configurations without usage of the term ‘traverse’:

0 a point locus where the target slant ranges are minimal and dR/dt = 0; 0 the plane that passes through platform current position on time t=O and is normal to the platform footprint trajectory.

The signal phase variation over processing time (phase history) is presented as:

where K, = 47112 is a wave number factor. The amplitude variation of the received signal is deter-

mined by antenna pattern G(a,, y) along azimuth a, and viewing angle y . For typical surveillance conditions it could be stated that VH, << V,. It allows us to neglect the vertical velocity and declining of line SS1 in Fig. 1 for purposes of angle estimation. Then, the signal on the SAR receiver output is presented as:

where A is SAR system gain. The azimuth viewing angle is defined as:

a(t> = arctan(g) - a,(t>

Here cr,(t) is the antenna beam positioning angle for the spotlight or squint modes. For stripmap imaging au(t) = 0.

2.2 Orbital model: geocentric coordinate system

2.2. I Modelling problem statement: The com- monly used operation chain for spaceborne radar imaging includes survey planning, orbit and radar parameter esti- mation, onboard instrument control and signal processing using predicted or measured signal parameters. The task of signal parameter analysis considered below differs from the above-mentioned one. Here we suppose that the radar parameters are specified. Then we specify orbit parameters and the platform position for orbit time t = 0, antenna beam angles in azimuth and elevation and the slant range of a control point at t = 0 (or several points). The calculation routine includes calculation of platform footprint geo- graphic coordinates at t = 0, turning the antenna beam plane on course to survey direction and then calculation of target (control point) geographic coordinates. The next step is platform-target distance and viewing angle calcula- tion through trivial modelling of the platform movement and the Earth rotation.

For our task simple orbital geometries can be considered shown in Fig. 2:

inertial on fixed epoch coordinate system OXNYNZN, where axis OXN is pointed at the ascending node, axis OZN to the north, and axis OYN forms a normal right orthogonal system; 0 orbital system OX,Y,Z, that is defined by the turning of system OXNY,Z, around axis OXN where axis OZ, is located in the orbit plane;

platform associated system OX,Y,Z,, where axis OZ, is pointed at zenith, axis OX, lies along the orbital velocity vector and axis OY, forms a right orthogonal system.

2.2.2 Platform coordinates: The initial parameters for the platform motion modelling are: a,, large orbit semi-axis equal to its averaged radius; io, orbit inclination

I85

Fig. 2 Spaceborne SAR viewing geometry

angle; e, , orbit eccentricity; O,, perigee argument, and w p , orbit precession angular velocity.

Current platform coordinates at t = 0 are counted by platform latitude argument 6, value: true anomaly:

8, = 8, - 0, (1 1)

angular orbital velocity:

where ,uo = 3.98602 x 1014 m3/s2 is the Earth gravitation coefficient. current orbital radius:

Polar coordinates of the platform: geographic latitude:

cp, = arcsin(sin &sin 0,)

A& = arctan(tan %,cos io)

A, = AAs + 52,

(14)

(15)

(16)

where 52, is the longitude of ascending node for t = 0. Current platform radius-vector in inertial system OXNY,Z, is defined as:

R, = -xkR,cos 6, - ykR,sin %,cos i, + ziR,sin %,sin io

ascending node longitude:

Greenwich satellite longitude:

(17)

2.2.3 Target coordinates: The location of the target (control point) for platform position at time t = O is determined by given viewing angle $Ja in the orbital plane and range Ro. The target vector coordinates are calculated via parameters of the Earth ellipsoid (WGS-84):

a = 6378 245 km, large semi-axis; e = 0.08181332, eccentricity; w = 7.2921 15 x IOp5 s-’, the Earth rotation angular velocity.

186

Let us take into account the value of orbit precession angular velocity that is equal to up - 2 x lo-’ s-I. Then, the integrated rotation angular target velocity around axis OZNis 0 , = 0 - 0 , - 7 . 2 7 2 ~ 10-5~- l .

Now we can calculate the target coordinates for t = 0. The target azimuth + t is defined by the equation:

where $.y = arcsin(cos i,/cos cp,) is a course angle of the platform, $,, is the angle between the target focusing plane and the vector normal to the orbit plane.

Ground range to the target is defined by central angle f i t :

where

is the Earth radius on the latitude cp, - cp,. The difference between (P, and cpS can be neglected here, or an iteration procedure should be used for target coordinate correction caused by the Earth radii difference, i.e. as a local relief. For these iterations we specify a value of target geocentric angle ,RI, calculate target latitude by (21) and a summary of local Earth radius and relief height R,. Then using (19) we find the target slant range R, for given P I .

The increment of the target longitude to the satellite longitude at t = 0 is:

sin $, A& = arctan cos cp,cot p, - sin cp,cos $,

Target latitude:

sin ppin $, ( sinAA, ) cpI = arccos

Greenwich target current longitude:

JL,(t) = )L.,.,g + AILI5 + w,t

where w, = w - w, - 7.272 x lop5 is a

(22)

residual of the Earth rotation an’gular velocity and orbit precession angular velocity that is equal to wp - 2 x lo-’ s-I.

The target vector in inertial system OXNYNZN at t = 0 is defined as:

(23)

Slant range and viewing angle of the target are calculated as a difference between target and platform vectors by (1). Using the well known equations of Kepler motion [4-61 and taking into account the Earth rotation and orbit precession velocity, we can obtain the time functions of the slant range (phase), speed (Doppler frequency) and viewing angles (signal amplitude).

RI = xiRi,sin A, + y;R,cos @os (P, + ziR,sin (P,

2.3 Orbital movement, sensor coordinates system To derive compact equations which will allow us to calculate platform motion parameters directly, let us consider radar antenna coordinate system OX,Y,Z, (see Fig. 2), which is associated with the platform with axis OZ, pointed at zenith, axis OY, rotated from OY, by angle A$a along the antenna beam direction. Axis OX, forms a right orthogonal system. The viewing geometry for time t = 0 is shown in Fig. 3.

IEE Proc.-Radar Sonar Navig., Vol. 150, No. 3, June 2003

Fig. 3 Viewing geometry for the time t = 0

Let us imagine that the coordinate system OXvYvZ, is 'frozen'. Then the platform motion could be referenced to the ground surface with the negative sign and represented through the rotation velocity vector directed along the Y, axis. The projections of the Earth rotation velocity upon the axis of tied system OX,YvZv could be defined as [7]:

qxv = o,cos Q,cos iocos $, + (cotcos i, - w,)sin $v

= o,,sin A$v wyv = (w,cos io - wo)sin A$, - O,COS W , ~ C O S i,sin A$,

ofzv = cotcos qs = w,sin 0,7sin io = O,,COS A$v

(24)

where

is the platform angular velocity including the Earth rota- tion and orbit precession influence; A$, = t+hv - $0; and $0 = -arcsin(w,cos $,sin io/u,st) is the rotation angle of the target focus plane for zero radial component of the angular velocity (path coordinate system [7, 81).

The range vector variations in the sensor coordinate system OX,YJ, could be represented as:

R(t) = - X!R,{COS ptsin(ostt . cos

-y:R,{cos ptsin(co,y,,,t . sin

+ z:{R, + V,t - R,cos p,cos(o,,t . cos At),,)

+ sin PI . sin o,t)

- sin pt .cos otzvt)

+ sin p,sin(o,,t . sin A$v)} (25)

where

is the platform vertical velocity.

ZEE Pmc.-i?adur Sonar Nuvig., Vol. 150. No. 3, .June 2003

For the important particular side-looking case, where the radar antenna could be oriented in the path coordinate system, the formula is written as:

R(t) = cos P,sin w,,t + sin b, . sin oIzvt)'

+ (R, sin p,cos wfzvt)2

+ (R, + VHst - @os p,cos o,,t)2]1'2 (27)

Line of sight turning:

The current angle Actta between a line of sight in azimuth plane TSTl in Fig. 3 and direction of the antenna pattern maximum is defined by the expression

Aa,a(O = a,(O - %(t>

(29) 1 &(cos P,sin ~ , ~ , t + sin PI . sin o,t) ( R(t)

= arcsin

where a,(t) is the antenna azimuth angle in SAR spotlight or squinted modes of operation, similar to (4).

The target radial velocity is defined as the derivative of the module of target location vector.

3 Exploiting of motion models

3.1 Initial parameters for the simulation Let us illustrate the exploiting of models with parameters as outlined in Table 1. For the linear motion the main parameters are defined by the following equations:

Doppler frequency bandwidth:

(30) 2 v , AFD*y = D

ant

focusing time in the stripmap mode:

finest azimuth resolution in the stripmap mode:

The orbital motion could be approximately considered as linear motion by introducing equivalent linear velocity (LEV) term qe. It must be emphasised that for this velocity the azimuth linear FM rate on the Ro is the same as the linear FM rate of the orbital motion:

VIe = J2c,R, (33)

Table I: Simulation parameters

Parameter Notation Value

Orbit large semi-axis, km Orbit eccentricity Inclination, degrees Node longitude, degrees Perigee argument, degrees Platform latitude argument,

True anomaly, degrees Average flight altitude, km Imaging slant ranges, km

degrees

687 1 0.008 70 0 -90 -90, 45, 0

0, 45, 90 500 580/1180/1780

I87

7600 1

2 7200

2 7000

c (iii)

-200 -150 -100 -50 0 50 100 150 200

e,, deg

Eriations of reference motion velocities against platjbrm Fig. 4 latitude argument '

where C2 is a slant range Taylor expansion factor. Fig. 4 presents some graphics of LEV for three cases of

slant range values 580, 1180 and 1780 km (curves (i), (ii) and (iii)), for platform path velocity (curve (iv)) and platform footprint velocity (curve (v)). Unlike linear motion model, the value of LEV varies depending on slant range and imaged relief elevation. It leads to varia- tions in FM rates. These variations must be taken into account in high-resolution SAR processing algorithms and in the procedures of range migration correction. We note that LEV approximation of orbital motion is a high-order approximation (contrary to the square-law approach by two Taylor expansion terms). Therefore, it does not impact on the phase preserving features of the processing algorithm, so the resulting image could be exploited for future interferometric processing.

3.2 Results of simulation The goal of the simulation is to estimate the provided potential accuracy for phase and amplitude variations over focusing aperture in the cases of

0 the common Kepler motion model versus the model adapted to conditions of spaceborne SAR imagery; 0 a model of linear motion approximation compared with the disturbed motion case; 0 an undisturbed Kepler motion model compared with the disturbed motion case.

In the general case, for SAR motion estimation the numerical integration procedure over a system of disturbed

motion equations is exploited, resulting in the tabulated values of target slant range variations over focusing time.

Using these Tables and the equations listed below, we can derive the values of LEV VI,, time of the nearest range t,,,, radial velocity V,. for the time t = O and also the value of the nearest target slant range Rnr.

R,, = ,/R(0)2 - (37)

where h is the time step of the Tables. The comparison of motion parameters calculated by

disturbed motion equations and by the Kepler motion model is presented in Table 2 and Fig. 5. The calculation of the values tabulated for the disturbed model has been carried out regarding the time t = t,, = 0. As is clear from Table 2, the approximation of the disturbed motion by Kepler motion leads to the variation of the quadratic factor in the Taylor expansion (C,), which defines azimuth FM rate or LEV value. Thus, the remaining errors of the Kepler approximation for the considered conditions amount to 90 mm. If correct data counted by disturbed motion equa- tions are used for LEV approximation, the residual errors do not exceed 22". Also, if correct data are used as precise Kepler model parameters in a local point of orbit, the residual errors do not exceed 20" but their beha- viour over time differs from the remaining errors of the disturbed motion case.

We should also note that the graphs shown in Fig. 5 pertain to the orbits with eccentricity 0.008 for which the variation in altitude is 5 5 4 km. For real spaceborne SAR smaller values of altitude variation (about f 2 0 km for e,, = 0.003) are typical, which leads to the error decreasing to f8.2". In Fig. 6 LEV curves against target slant range are shown. It is clear that curve behaviour depends considerably on the satellite latitude argument or true anomaly value.

3.3 Influence of the motion approximation errors upon the SAR system response As it is shown in the Appendix, the errors of LEV calculation lead to defocusing of the resulting image that

Table 2: LEV values for the Kepler and disturbed motion models

Type of model LEV Vie, m/s

U,= -90" os= -45" U,=O" &=0" 00 = 45" 00 = 90"

Slant range Ro = 580 km Kepler model 7224.225 Disturbed motion 7224.666

Slant range Ro= 1180 km Kepler model 7130.053 Disturbed motion 7130.211

Slant range Ro= 1780 km Kepler model 701 1.502 Disturbed motion 7011.319

7217.559 72 17.853

7 137.237 7 137.81 8

7030.985 7031.877

7184.136 7183.820

7139.943 7139.837

7065.667 7065.628

188 IEE Pvoc-Rrrdar Sonar Nuvig., Vol. 150, No. 3, June 2003

-100 I I

-4 -2 0 2 4 -4 -2 0 2 4 timet, s timet, s

a b

-2 - I 4 0 I I

-4 -2 0 2 4 4 -2 0 2 4 time t, s timet, s

C d

-30 I I I I I 4 -2 0 2 4

time t, s e

Fig. 5 Bold lines: R = 580 km; dotted line: R = 11 80 km, thin lines R = 1780 km a Linear model, Bo = 0" b Kepler approximation, 00 = 0" c Linear model, HO = 45" dKepler approximation, 00 = 45" e Linear model, Ho = 90" f Kepler approximation, 00 = 90"

Remaining errors uf the disturbed motion

7000 400 800 1200 1600

slant range R, km

LEVdependence of the slant range fur the different values Fig. 6 of satellite true anomaly

IEE Proc-Radar Sonar Nuvig., Vol. 150, No. 3, June 2003

time t , s f

causes some deterioration of pulse response: broadening of the response main lobe, especially in the pedestal area, increasing of sidelobe levels, decreasing of the response peak level. The depth of deterioration depends on a non- dimensional parameter:

where AR is a slant range error, 6x is a spatial resolution without errors, A is a SAR wavelength, Ne is the look number.

The slant range errors depend on velocity measurements error in the following way:

For md 5 2 the response deterioration could be regarded as acceptable: response broadening at the -3 dB level not more than 2%, and at the -12 dB level not more than 8%, the increasing of integrated sidelobes level (about 1.3 dB),

189

and the peak decreasing (no more than 0.2dB). As an example, for such an mdf value, let us evaluate the LEV accuracy requirements for the case of L-band SAR with parameters outlined in Table 3 for azimuth resolutions 6 m in stripmap mode and 3 m in spotlight mode.

As it is clear from Table 3, the accuracy of the LEV calculation derived from the Kepler motion model is sufficient for focusing with azimuth resolution 6x = 6 m. For a better resolution of 6x = 3 m the accuracy of Kepler approximation is not sufficient (particularly for the case Bo = 45") and exact calculations of disturbed motion are needed. The exploiting of the linear motion model with the use of LEV derived from disturbed motion conditions is correct. The remaining errors, which are less than 8mm on the edge of the aperture (phase error is 24"), could be neglected. The calculations of azimuth pulse response for Ro = 580 km and T,,,, = 8 s were made using equations given in the Appendix. The convolutions using the exact signal phase history replica and its LEV approximation for worst errors (Fig. 5e) were made. Data collected in Table 5 (see Appendix) show the high accu- racy of the LEV approximation: azimuth resolution practi- cally the same (1.5 m instead of 1.4 m), only near sidelobes are growing (from -21 up to -16.7 dB).

For the calculation of the reference function (signal replica) we need to know the LEV variation law depending on the slant range. This dependence could be calculated using the adapted model proposed here with the appro- priate correction of the LEV variation law over a discrete number of reference slant ranges (3-5 points over the swath). The LEV against range curves depending on different values of the true anomaly are shown in Fig. 6.

3.4 Influence of orbit ellipticity The presence of the vertical component of the satellite velocity vector caused by orbit ellipticity leads to a Doppler frequency shift in the viewing plane, which is oriented perpendicularly to the platform footprint trajec- tory. The line of the nearest ranges drifts relative to this plane. This leads to the appearance of the azimuth errors, which should be compensated. This compensation can be carried out automatically by introducing the Doppler frequency shift into each slant range channel when the reference function (signal replica) is calculated. The rela- tion between the slant range and the radial velocity is shown in Fig. 7. Radial velocity is disposed along the abscissa as an analogue of azimuth shift caused by Doppler frequency shift.

3.5 Angles of antenna beam steering As it has been noted in the Section 2.3, the proposed orbital motion model allows us to take into account the

Table 3: SAR parameters

SAR wavelength i., cm Antenna length Dant, m Azimuth antenna pattern Actant, degrees Slant range Ro, km Weighting window Mainlobe broadening factor k, Looks number Ne Azimuth resolution Sx, m Focusing aperture length La, km Time of signal accumulation T,,,, s Focusing depth A/?, m Acceptable velocity error AV,,, m/s

23.5 12

1.122 1180

Kaiser = 2.9 1.225

1 6 3 28.8 56.6 -3.87 -7.95 304 76 0.92 0.23

1600

E 1200 Y

U-

800

-1 0 0 i o 20 radial velocity V,, m/s

Fig. 7 Relation between the slant range and the radial velocity variation caused by the ellipticity of the orbit for different values of satellite true anomaly

turning of the antenna beam over surface surveillance. In Table 4, the comparison between the signal parameters in the strip-imaging mode for two motion models is shown. The first case is the LEV derived from linear motion conditions and the second one is the Kepler model adapted to the spaceborne radar survey. Orbit parameters are: a, = 683 1 km, e, = 0.008, 00 = 45", Ro = 1180 km. It is seen that if the trajectory curvature is considered, the signal accumulation time and Doppler spectrum width are larger, and the resolution realised in strip mode is better than that predicted for the linear motion model. The differences increase when the orbit to planet radii ratio increases.

In a number of spaceborne SAR systems (SIR-C/ X-SAR, 'Almaz-1', ERS-1/2) the platform is oriented in the path coordinate system. For these cases, if no platform orientation errors occurs the average Doppler frequency in the SAR antenna beam is equal to zero for all slant ranges. Hence, the accurate side looking survey is realised. In such SAR systems as Seasat-A, SIR-A, Radarsat-1, and the designed PALSAR system the platform is oriented in the orbital plane, that leads to squint imaging with additional geometric and radiometric distortions which should be removed during the image processing. Similarly to the case described above, the proposed model could be applied for calculations of signal parameters in highly squinted imaging modes: forward and backward looking survey (see [11>.

3.6 Proposed model usage for SAR parameters analysis Let us give an example of the proposed model usage for SAR parameter analysis. In [9] the way to overcome signal ambiguity constraints by decreasing range swath to azimuth resolution ratio is shown. A linear model with platform speed 7500 m/s for all types of the Earth mapping

Table 4: SAR processing parameters in the stripmap mode

Parameter Motion model Linear Orbital

Predefined LEV, m/s VI, 71 37.273 Signal accumulation time, s T,,,,,, 3.225 3.371 Frequency bandwidth, Hz AFdop 1196.6 1233.2 Azimuth resolution, m SXsm 6 5.74

190 IEE Proc.-Radar Sonar Navig., Vol. 150, No. 3, June 2003

SAR is accepted. The accurate calculations of Radarsat parameters using the proposed model have shown that the LEV value for far ranges falls to 7020m/s. Thus, it is possible to expand the radar survey strip from 800 up to 1000 km for 40 km ground swath and -10 m azimuth resolution. This useful additional ability decreases revisit time and can be used for emergency monitoring.

4 Conclusions

A compact model of orbital motion that simplifies SAR signal analysis and modelling and that provides high calculation accuracy has been presented. It has been shown that SAR-target relative motion in the local area can be considered as linear with velocity defined with the use of the disturbed motion parameters. Based on exact orbital parameters, the LEV model of phase history presen- tation can be used for image synthesis as a first-order approximation. Its deviation of some millimetres from the exact law, together with uncontrolled atmospheric and instrument phase errors, can be the goal of autofocusing techniques.

It has been shown that for elliptical orbits the appearance of vertical velocity leads to drift of the line of the nearest ranges away from the viewing plane. Hence, it would be worth compensating the Doppler shift due to vertical velocity in the phase history of the signal replica. The proposed LEV approximation as a. one-parameter descrip- tion of phase histories in a number of control points could be well used for signal replica interpolation over the range swath.

For antenna steering estimation, the orbital curvature should be taken into account. The convenience of the platform orientation in the path coordinate system is proved, especially for wide swath imaging modes.

5

1

2

3

4

5

6

7

8

9

10

I I

6

References

NERONSKIY, L.B., LIKHANSKY, S., ELIZAVETIN, I.V, and SYSENKO, D.V: ‘Phase history model adapted to the spacebome SAR survey’. Proceedings of EUSAR2002 Conference, Cologne, Germany, June 2002, pp. 557-560 HENRION, S., SAVY, L., and PLANES, J.-G.: ‘Properties of hybrid strip-map/spotlight spacebome SAR processing’. Proceedings of IGARSS’99 HENRION, S., SAVY, L., and PLANES, J.G.: ‘New results for space- bome hybrid strip-map/spotlight SAR high resolution processing’. Proceedings of IGARSS’99 ESCOBAL, P.R.: ‘Methods of orbit determination’ (John Wiley and Sons, N.Y., 1965) ESCOBAL, P.R.: ‘Methods of astrodynamics’ (John Wiley and Sons, N.Y. 1968) CURLANDER, J.C., and McDONOUGH, R.N.: ‘Synthetic aperture radar’ (Wiley and Sons, New York, 1991) NERONSKIY, L., MICHAILOY V, and BRAGIN, I.: ‘Microwave equipment of the Earth and atmosphere remote sensing, synthetic aperture radars’ (SPbGUAP. SPb., 1999), Chap. 2, p. 220 (in Russian) CHANG, C.Y., and CURLANDER, J.C.: ‘Attitude steering for space based synthetic aperture radars’. Proceedings of IGARSS’92, pp. 297-301 FREEMAN, A., JONSON, W.T.K., HUNEYCUTT, B., JORDAN, R., HENSLEY, S., SIQUEIRA, P., and CURLANDER, J.: ‘The ‘Myth’ of the minimum SAR antenna area constraints’, IEEE Trans. Geosci. Remote Sens., 2000, 38, pp. 320-324 NERONSKIY, L.: ‘Estimation of the synthetic aperture radar resolution using transition functions and a correlation function interval of the output signal’, Radiotech. Electron., 1975, 2, pp. 271-279 (in Russian) Radarsat Data Products Specification. RSI-GS-026 3/0 May 8, 2000, www.rsi.ca

Appendix: Influence of the synthetic aperture defocusing errors

6. I Defocusing parameters Let us analyse the influence of defocusing on SAR azimuth pulse response. In terms of offered model defocusing

IEE Proc.-Radar Sonar Nuvig., Vol. 150, No. 3, June 2003

errors are caused by difference between the predicted and the real values of LEV The properties of the azimuth SAR system pulse response are determined by the signal samples digital accumulation over the focusing aperture:

Ns,,,J-’

k,=-Nhvnt/2 h(k) = c - k&XP(-j%(k - k,)J

where N,,, is the number of samples in the aperture, U,(k) and cp,(k) are signal amplitude and phase samples, W(k) is the amplitude weighting function and cpref(k) is the phase reference function (signal replica).

To simplify the further analysis we neglect the range migration effects. Then, for each hologram cell with range value Ro, the function cp,(k) is determined by signal phase history

where Vle is platform linear equivalent velocity and T i s a pulse repetition period. For phase reference function we have:

where VreX is a predicted value of Vie.

acterizing the degree of aperture defocusing Let us introduce a dimensionless parameter mdf char-

where 6xo = Ro/2/2L,,, is azimuth resolution when defo- cusing is absent, A V = Vyef - Vre and AVhxo is a ‘critical’ velocity difference when actual synthetic aperture length differs from predicted one by the value of 6x0:

(44)

As is shown in [lo] for square-law phase history and with expansion to Ne multilook processing the defocusing azimuth resolution is equal to:

6.2 Parameters of pulse response quality We will describe the shape of pulse response by the common set of parameters: 6xe , resolution at the -3 dB level; PSLR, peak values of the sidelobes; and ISLR, integrated level of the sidelobes. As is assumed in the document [ I l l , we will assume that the mainlobe area covers the interval f 1 . 5 . 6 ~ ~ . To this set we include the following parameters: the power losses in the response peak Lm = 20 log(hoe/ho), the width of the pulse response at the - 12 dB level axl2 dB, which is defining the pulse response pedestal, the peak value PSLR*6, and the integrated level ISLR*6 of the far sidelobes out of frames f 6 d x e .

191

0.8 10/ 1.0-

0.8

W In 5 0.6

a, 3 0.4

P

Q

0.2

n-

-

-

-

- o'2L 0 0

a,

s 0.6

a, 9 0.4

P a,

Q

-20 0 20 40 azimuth, m

a

-

-

1.8 -

m C U

.- c 1.6-

P 1.4- ;

r)

0 1 2 3 4 5 6

(ii) / 1.8 -

m C U

.- c 1.6-

P 1.4- ;

r)

1.2 -

0 1 2 3 4 5 6

-40 -20 0 20 40 azimuth, m

b

4 -

3 -

-1

-2

-

-

-3 I 0 1 2 3 4 5 6

defocusing error md,

C

Fig. 8 Pulse response, beam broadening and losses in the output ofSAR system

defocusing error md,

d

a Pulse responses for square window: (i) mdi= 0; (ii) md,= 6 b Pulse responses for Kaiser window, c Beam broadening and losses for Kaiser window at the -3 dB (i) and - 12 dB (ii) levels, for Gaussian weighting (iii) d Response peak power losses (iv), integrated sidelobe levels out of the frame f l . 5 6 ~ ~ (v) and out of frame f66xe (vi) for Kaiser window

= 2.9: (i) mdf = 0; (ii) mdf = 6

6.3 Defocusing errors analysis for various weighting functions Here we outline the comparison of the numerical calcula- tion results of the pulse response shape for different weighting functions: Gaussian, Kaiser, and sinc2. The last function is especially inherent to the SAR antenna pattern. In Fig. 8a the pulse response shape is shown for the case of a square weighting window. The defocusing factor is mdf= 6 that causes the pulse response broadening by 11% at the -3 dB level. As it is seen, the distortions are mainly expressed in the increasing of nearest sidelobes. The level and integrated power of the far sidelobes even decrease slightly.

Curves for beam broadening values at -3dB and -12 dB levels against the defocusing error values are shown in Fig. 8c. In Fig. 8d the graphs of the power losses in the response peak and the integrated sidelobe levels out of the frames f 1 . 5 6 ~ ~ and f66xe , are shown. The similar values are typical for sinc2 weighting due to the SAR antenna pattern in the stripmap single-look imaging mode. Pulse response parameters for square and Kaiser windows, exact focusing and defocusing (mdf= 6 and LEV approximation case) are collected in Table 5. The represented data show that it is reasonable to apply square window processing as the criterion for the autofocusing procedure iterations.

Table 5: Azimuth pulse response parameters depending on defocusing

Errors 6 X , 6x12 dB PSLR PSLR*, ISLR ISLR+6 Lmr dB

Square window (R0=580 km, T,,,,=O.91 s ) m d f = 0 10.0 17.7 -13.3 -27.4 -10.4 -18.1 0 l?ldf= 6 10.1 37.7 -8.9 -27.1 -8.8 -18.3 1.3

Kaiser window (8=2,9, R0=580 km, T,,,= 1.12 s ) mdf= 0 10 18.3 -23.1 -39.4 -21.0 -26.9 0 mdf= 6 14.3 34.3 -16.3 -39.7 -17.1 -27.5 1.9

Kaiser window (8 = 2,9, I?,, = 580 km, T,,,= 8 s ) Exact calculations 1.4 5.5 -23.1 -39.4 -21.0 -26.9 0 LEV approximation 1.5 5.32 -16.7 -38.0 - 19.4 -26.8 0.07

192 IEE Proc-Radar Sonur Nuvig.. Vol. 150, No. 3, June 2003