Airborne SAR processing of highly squinted data using a chirp scaling ...

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 32, NO. 5, SEPTEMBER 1994

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Airbome SAR Processing of Highly Squinted Data Using a Chirp Scaling Approach with Integrated Motion Compensation Albert0 Moreira, Member, IEEE, and Yonghong Huang

Abstract-This paper proposes a new approach for high-resolution airborne SAR data processing, which uses a modified chirp scaling algorithm to accommodate the correction of motion errors, as well as the variations of the Doppler centroid in range and azimuth. By introducing a cubic phase term in the chirp scaling phase, data acquired with a squint angle up to 30" can be processed with no degradation of the impulse response function. The proposed approach is computationally very efficient, since it accommodates the variations of Doppler centroid without using block processing. Furthermore, a motion error extraction algorithm can be incorporated into the proposed approach by means of subaperture processing is azimuth. The new approach, denoted as extended chirp scaling (ECS),is considered to be a generalized algorithm suitable for the high-resolution processing of most airborne SAR systems.

I. INTRODUCTION AR raw data consist of a coherent superposition of the backscattered echoes from the imaged scenario. The backscattered echoes are modulated by the transmitted pulse (Le., chirp signal) and by the natural movement of the sensor (Doppler effect). The image formation process consists of compressing the signal of each scatter, which is time dispersed (modulated) in the along-track (azimuth) and cross-track (range) directions. The first direct step in a time-domain SAR processor consists of the range compression, which is a one-dimensional correlation of each received echo with the complex-conjugated time-inverted replica of the transmitted pulse. Then, azimuth compression is performed. Due to the range migration of each target during the azimuth illumination time and due to the variation of the Doppler modulation as a function of the range distance, the azimuth compression consists of a two-dimensional correlation of the range-compressed signal with a space-variant reference function. The final amplitude image can be calibrated in order to give a reflectivity map of the scenario for a given frequency and polarization. Furthermore, if the phase of the final image is also calibrated, a full polarization matrix can be formed

S

Manuscript received November 8, 1993; revised March 17, 1994. A. Moreira is with the Institute for Radio Frequency Technology, German Aerospace Research Establishment (DLR), D-82230 Oberpfaffenhofen, Germany. Y. Huang is with the Department of Electronic Engineering, Beijing 100083, People's Republic of China. IEEE Log Number 9403635.

that allows a better classification and interpretation of the image. In practical cases, the range and azimuth frequency modulations consist of several hundred points, so that the correlation process is carried out with reduced computational effort in the frequency domain by means of a fast Fourier transform (FFT). This is the basic concept of the hybrid algorithm [24], which performs a one-dimensional frequency-domain correlation for range processing and then generates a two-dimensional time-domain azimuth reference function that is correlated in the Doppler-frequency domain with the azimuth signal. This approach is very accurate with a limitation concerning the update of the azimuth reference function with range. This update is done according to the required depth of focus in azimuth. Otherwise, a new two-dimensional azimuth reference function should be generated for each range position. The update of the azimuth reference function decreases the phase accuracy of the final image, although it has a minor effect in the image intensity. The phase accuracy is very important as far as polarimetric and interferometric applications are concerned. The computational efficiency of the hybrid algorithm is very low if the range migration (especially the range walk) becomes large. In order to increase the efficiency, the range migration can be corrected after range compression by means of an interpolation, so that the azimuth correlation in the frequency domain becomes a one-dimensional process. Due to the fact that the range migration of all targets located at the same range distance is equal, the interpolation can efficiently be performed in the rangetime Doppler-frequency domain. The range-Doppler algorithm [ 5 ] , [ l l ] uses an interpolation kernel (e.g., sinc function) to correct the range migration in this domain. For high squint angles, some complications arise in the range-Doppler algorithm due to the coupling of the range and azimuth signals [ l l ] . Briefly, the range frequency modulation is altered in the range-Doppler domain according to the amount of squint angle. Several methods have been proposed to correct this change [4], [20], which causes a defocusing of the range impulse response function (IRF). The correction of this change has been called secondary range compression (SRC). A more accurate algorithm is the wavenumber algo-

0196-2892/94$04.00 0 1994 IEEE

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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 32, NO. 5 , SEPTEMBER 1994

rithm [18], which is based on wave propagation theory commonly used in seismic processing. This algorithm can also be interpreted as a two-dimensional frequency correlation, whereby a two-dimensional FFT is used to transform the signal from the time domain to the wavenumber domain. The spatial variation of the azimuth signal is corrected by the so-called Stolt interpolation. After adequate phase correction, a two-dimensional IFFT (inverse FFT) is used to form the final image. Several different implementations of this algorithm have been proposed in the last few years [3], [8], [9] and are summarized and analyzed in terms of processing accuracy in [ l]. The range-Doppler and the wavenumber algorithm have mostly been used for efficient SAR processing. However, both algorithms need interpolation for the range cell migration correction (RCMC) or for the Stolt change of variables. In general, the interpolation not only causes phase errors and amplitude artifacts in the SAR image, but also decreases the computational efficiency. The chirp scaling (CS) algorithm has recently been proposed for high-quality SAR processing [6], [19]. This algorithm avoids any interpolation in the SAR processing chain and is suitable for the high-quality processing of several spaceborne SAR systems (e.g., SEASAT, ERS1, RADARSAT). It consists basically of multiplying the SAR data in the range-Doppler domain with a quadratic phase function (chirp scaling) in order to equalize the range cell migration for a reference range, followed by an azimuth and range compression in the wavenumber domain. After transforming the signal back to the rangeDoppler domain, a residual phase correction is carried out. Finally, azimuth IFFT's are performed to generate the focused image. It has been shown in [17] and [22] that the image quality and the phase accuracy of the chirp scaling algorithm is equal or superior to that of the range-Doppler algorithm. For spaceborne SAR systems, the CS algorithm can be used to process data with up to approximately 20" squint angle in L-band without deterioration of the IRF [7]. The SRC is corrected for a reference range in the range-Doppler domain before range compression and it is updated with the azimuth frequency. In order to accommodate higher squint angles, a nonlinear phase term can be introduced into the processing or into the transmitted pulse [7]. In the former case, the nonlinear phase term is added in the wavenumber domain before applying the quadratic chirp scaling phase function. The image quality provided by the chirp scaling algorithm in the case of spaceborne SAR processing is excellent. However, airborne SAR processing requires an update of the Doppler centroid as a function of the range distance and an accurate time-domain motion compensation, which must also be updated with the range distance. The abovementioned requirements cannot be included in the original chirp scaling algorithm, since it is basically a frequency domain focusing approach. Some alternative methods can be implemented in the original chirp scaling algorithm in order to incorporate the Doppler centroid variations (e.g.,

block processing, increase of the azimuth FFT size with adaptive bandpass filtering of the processed image [ 151). In this case, the computational efficiency of the chirp scaling algorithm decreases considerably, and no accurate motion compensation can be performed. The new algorithm, denoted as extended chirp scaling (ECS), has been developed for the processing of airborne data with strong motion errors (e.g., E-SAR system, [lo]) and with variable Doppler centroid in range and/or azimuth. One unique characteristic of the ECS algorithm is the extension of the azimuth spectrum in the range-Doppler domain for accommodating the variations of the Doppler centroid in range. Additionally, short azimuth spectra (subaperture processing) allow the inclusion of the Doppler variation in azimuth as well as the incorporation of the so-called reflectivity displacement method (RDM) in the processing for second-order motion compensation. Unlike other chirp scaling implementations, a deterministic cubic phase term is added by the extended chirp scaling method during the equalization of the range cell migration, so that no deterioration of the geometric resolution is measured up to a 30" squint angle in the C-band mode of the E-SAR system. This paper is organized as follows. The next section shows the mathematical formulation of the extended chirp scaling (ECS) algorithm, including the accurate motion compensation and the highly squinted data processing. Several simulation results are presented in order to validate the proposed approach. Section I11 extends the ECS algorithm to include the update of the Doppler centroid with range. The proposed azimuth spectral extension is extremely accurate, and is based on the fact that the variation of the Doppler centroid as a function of the range distance can be precisely accommodated if the chirp scaling phase function is extended in the azimuth frequency by the amount of the Doppler centroid variation. Section IV describes the azimuth subaperture processing, which allows the Doppler centroid to be updated in each subaperture. In addition, the frequency displacement of the cross-correlation result between adjacent subapertures gives information about the acceleration in line of sight [ 141, which is used for the second-order motion compensation. In Section V, the results of the image processing with strong motion errors are presented and analyzed. Section VI concludes the paper and gives some suggestions for future work. 11. EXTENDED CHIRPSCALINGALGORITHM A . Modeling the Received Signal We consider in this section an airborne-side-looking geometry with constant squint angle. The hyperbolic equation of the range history for a point target is expressed as

~ ( tro> ; = Jri + v 2 ( t - tCl2 (1) where ro is the range to target at the closest approach, u is the aircraft velocity, t is the azimuth time measured in

~~~

~~

1

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MOREIRA AND HUANG. SAR PROCESSING USING CHIRP SCALING WITH MOTION COMPENSATION

the flight direction, and tc is the time at the center of the azimuth illumination path, which is related to the Doppler centroid fdc by the following equation:

where C is a complex constant andf, is the azimuth frequency, which varies within the following range: PRF --

2

where is the radar wavelength. After demodulation, the two-dimensional received SAR signal s(7, t ; ro)of a point target can be written as

2

exp [ - j

*

R(t; ro)

-

T

*

k,

The azimuth antenna pattern a, and the envelope a, of the transmitted pulse are slowly varying functions relative to the signal variations in the azimuth time t and range time (range delay) 7 . In (3), the first exponential term accounts for the range chirp with frequency modulation rate k, and for the range migration. The last term in (3) is the azimuth (Doppler) modulation. In a special case, where the squint angle and azimuth illumination time are small, the range migration in the first exponential term of (3) can be neglected and the received signal is approximated to two one-dimensional functions. In practical cases, however, the range migration leads to a coupling between the range and azimuth modulations, so that the received SAR signal of each target becomes a two-dimensional space-variant function. Due to the large time-bandwidth product of the received SAR signal, the principle of the stationary phase [16] can be used to obtain a signal formulation in the wavenumber domain. In this domain, the form of the azimuth antenna pattern and the envelope of the transmitted pulse remains the same. By performing a series expansion in the range frequency and an inverse Fourier transformation in range, the signal formulation in the rangeDoppler domain is given by [ 161 *

a,

.,(

(7

2

- ";fa; ro

-

ro))

-

*.tu

x

)

2 * v 2 J 1 - [( .fa)l(2 * v)I2

- exp [ - j

?r

The last term in (4) corresponds to the azimuth modulation in the frequency domain. The first exponential term in (4) shows that the frequency modulation of the range chirp is now dependent on the azimuth frequency and the range distance. If this variation is not considered, the range IRF for high squint data processing will be defocused. The modified range-frequency modulation basically consists of two terms:

(3)

S(7,fa; ro) = C

PRF

(5) +fdC%t3fdC+F.

k(f,; ro)

C

where

p

= 4 1 -

(2)2.

(7)

The correction of the term k,, in the range processing is called secondary range compression (SRC). The traditional chirp scaling algorithm assumes one reference range for the SRC and updates it with the azimuth frequency. The missing update of the SRC with range causes a phase error in the range compression for ranges different from the reference range. This phase error is insignificant for a squint angle up to approximately 20" in the case of L-band spaceborne SAR systems [7].

B. Chirp Scaling with Cubic Phase Term The range migration in the range-Doppler domain can be expressed as

where a(f,) is the linear chirp scaling factor given by

In the case of airborne SAR, the scaling factor a( f a ) is not independent on range, so that a linear scaling in range leads to a perfect equalization of the RCMC, if the SRC term can be neglected. The traditional chirp scaling method performs this equalization by means of a quadratic phase term in the range-Doppler domain. Actually, the scaling consists of changing the position of the phase minimum of each chirp signal. No explicit interpolation is carried out. The quadratic phase term of the chirp scaling introduces a frequency offset in the range chirp signal, which can assume values as high as several megahertz for high squint angle. If the frequency offset is high enough so that the signal bandwidth is aliased or shifted outside of the

~

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 32, NO. 5, SEPTEMBER 1994

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processed bandwidth, then the range IRF will deteriorate. In order to minimize this effect, the offset scaling factor due to the squint angle should be removed from the original chirp scaling term a ( f , ) , so that the new scaling factor for the processing is defined by =

4f,>-

(10)

a(fdc).

The scaling factor leads to an equalization of the range migration. However, the range positioning of the targets in the final image will be according to the slant range at the time t, (at the center of the azimuth illumination path). The range positioning can be changed to that of the broadside geometry (range distances according to closest approach) during the slant to ground range transformation. The scaling factor a ’ ( f , ) is now used for the range migration representation in the frequency domain according to (8), i.e., R’(f,; ro) = ro * [l + ~ ’ ( f , ) ] . The quadratic phase function for RCMC can be written as H1(7,fa;

rref) = exp

[

-j

*

&fa;

a

rref)

*

C. Range Compression and Bulk RCMC One basic characteristic of the extended chirp scaling algorithm is that no azimuth compression is performed in the wavenumber domain. Only the range compression and the bulk range cell migration (for the reference range) are performed in this domain. The removal of the azimuth compression from this step is necessary, since second-order motion compensation must be performed after range compression (due to the range update), but before the azimuth compression. The phase-correction term of the ECS algorithm in the wavenumber domain consists of the range compression with the SRC for the reference range and an additional linear phase term with respect to the range frequency, which corresponds to a linear time shift in the range time. Using the wavenumber formulation of the SAR signal, this phase correction can be calculated as

a‘’I :)I.

8 A * ( p 2 - 1) c4 * p 3

(ro - rEf)3 ( a r 2 ~

rEf) = exp - j -

*

a

*

2 R ‘ ( . L ; rref))21

- exp[-j

(.- 2 -

a

(15)

k ( f ; rref)* a ’ ( f )

C

.

When transforming the SAR signal back to the range-

*

k;

R ’ ( f ; rref) C

*

( p 2-

3*c*p3

)31.

(13)

With the additional phase term, up to a 30” squint angle can be processed without deterioriation of the image quality. At the end of this section, a quantitative analysis of the IRF for different squint angles will be presented.

The second term in the above equation corrects a residual phase error that was introduced by the cubic phase term in the scaling function of (13). After the above correction, the SAR signal can be transformed to the signal domain (azimuth and range time) to perform accurate motion compensation.

D. Motion Compensation and Azimuth Compression The motion errors induced by the atmospheric turbulence are a crucial problem in most airborne SAR SYStems. If not corrected, the image quality will considerably degrade [ 141, [23]. The main effects observed are a loss

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MOREIRA AND HUANG: SAR PROCESSING USING CHIRP SCALING WITH MOTION COMPENSATION

of geometric resolution and radiometric accuracy, reduction of image contrast, azimuth ambiguities, and geometric and phase distortions. Once the velocity variations of the aircraft are compensated by means of an on-line variation of PRF, which leads to a constant pixel spacing in azimuth, the aircraft trajectory is corrected to a straight line by applying a time-domain phase-correction function. If the deviations of the aircraft trajectory are greater than one range bin, then the range delay must also be compensated. The first-order motion compensation is defined as being the phase correction for a reference range, and it can be directly carried out with the range uncompressed data (i.e., before the processing starts). The second-order motion compensation includes the update of the phase correction as a function of the range distance. Let &,c(q ro) be the total phase for motion compensation and & , c ( t ;rref)be the first-order motion compensation, then the second-order motion compensation, which is applied after transforming the range-compressed SAR data to the signal domain, is formulated as

* FIRST ORDER MOTION COMPENSATION

* CHIRP SCALING WITH QUADRATIC AND CUBIC PHASE TERMS

* RANGE COMPRESSION * SECONDARY RANGE COMPRESSION * LINEAR RANGE SHIFT FOR BULK RCMC

* PHASE CORRECTION DUE TO

CHIRP SCALING

SECOND ORDER MOTION ERROR CORRECTION

Focussed Image

After performing the second-order motion compensation, only a one-dimensional azimuth compression must be performed. In the case where the squint angle is constant, the SAR signal is transformed again to the range-Doppler domain, and the azimuth modulation is corrected in this domain by a hyperbolic phase modulation:

J

L

(17) Fig. 1 shows the block diagram of the ECS algorithm with the corresponding phase-correction functions. The extra computation consists of the additional phase corrections and of the transformation to the signal domain before azimuth compression (additional azimuth IFFT's and FFT's). If the azimuth time-bandwidth product is low, then the azimuth compression is carried out more efficiently by time-domain correlation approaches (e.g., subaperture processing [ 131).

E. Analysis of the Impulse Response Function The following parameters were used for the simulation of the IRF according to the specifications of the E-SAR system of DLR in the C-band mode [lo]. Radar wavelength: X = 0.0566 m. Sensor velocity: u = 75 m/s. Antenna depression angle: Oi = 37". Squint angle: variable, from -30" to 30" in 5" steps. Chirp frequency modulation rate: k = 2 lOI3 Hz/

-

S.

Fig. 1. Block diagram of the extended chirp scaling (ECS) algorithm for high-precision airborne SAR processing. In this diagram, a constant Doppler centroid value is assumed for processing.

Radar PRF: 1100 Hz. Swath width: 3000 m. Flight altitude: 3000 m. Azimuth and range resolution: 0 . 3

X

2.5 m ( 1 look).

The impulse response functions obtained by processing with traditional chirp scaling and the ECS algorithms for a squint angle of 30" are shown in Fig. 2(a) and (b), respectively. No weighting function was used in the processing and a rectangular pulse envelope and azimuth antenna pattern were adopted, so that a two-dimensional sin (x)/x function is expected for an error-free compression. In both cases, the target is located at the edge of the swath, and the reference range for processing is at the center. This situation corresponds to the worst case for processing as far as the accuracy of the algorithm is concerned. In the chirp scaling algorithm [Fig. 2(a)], the geometric resolution of the range IRF is deteriorated by approximately 12 percent and the peak sidelobe ratio (PSLR) is increased to 7.3 dB (for an error free sin ( x ) / x function it should be 13.2 dB). For the ECS algorithm, the simulation results up to 30" squint angle are almost perfect. The deterioration of the azimuth and range resolution are always lower than 1.7 and 1.4 percent, respectively. The PSLR is not worse than 12.6 dB and the measured phase accuracy is better than 2". For higher squint angles, a more accurate algorithm proposed in [7] should be used, which requires an additional transformation to the wavenumber domain for applying a cubic phase correction before the chirp scaling operation is carried out in the range-Doppler domain.

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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 32, NO. 5 , SEPTEMBER 1994

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range time [ ks ] ----> (a) A

M

IS0

100

range (samples expanded by 4) ---->

-40

-20

0

20

range frequency [ MHz I ---->

40

0

e

(c)

D

M

I 00

II O

range (samples expanded by 4) ----> (b) Fig. 2. Contour plot of the impulse response function (IRF) for 30" squint angle. (a) IRF obtained with the chirp scaling algorithm using a quadratic phase for equalization of the range cell migration. (b) IRF obtained with the extended chirp scaling algorithm using a quadratic and cubic phase term for equalization of the range cell migration. The sensor and processing parameters of the E-SAR system are used for the simulation (2.5 x 0.3 m resolution in range and azimuth with I-look processing).

111. ACCOMMODATION OF DOPPLERCENTROID VARIATIONS WITH RANGE Due to the large variation of the look angle in the airborne SAR geometry, the Doppler centroid can vary several hundred hertz from the near to the far range. The update of the Doppler centroid avoids a loss of signal-tonoise ratio, an azimuth defocusing, and azimuth ambiguities. In the CS algorithm, this variation could be accommodated in an inaccurate and inefficient way by a block processing in the range direction with an overlap between blocks equal to the length of the range reference function. A simple update of the Doppler centroid value in the chirp scaling phase would not work well, since the uncompressed range chirp signals from different range positions are overlapped in the range-Doppler domain. In order to clarify this effect, an illustrative representation of the SAR signal of four point targets in the different domains of the CS algorithm are represented in Fig. 3(a)-(e). The parameters of the E-SAR system mentioned in the previous section were used with an azimuth presuming factor of four, so that the effective PRF is reduced to 275 Hz.

m

0

lOW

500

1%

2000

2500

1

range [ m ] ---a (e) Fig. 3. Contour plot of the signal of four point targets in the different domains of the chirp scaling algorithm. (a) Signal domain (raw data). (b) Range-Doppler domain. (c) Wavenumber domain. (d) Range-Doppler domain. (e) Signal domain (final image). The sensor and processing parameters of the E-SAR system were used for the simulation. A squint angle of 8" was used, so that the PRF ambiguity number is equal to one. For reading the unambiguous azimuth frequency values in (b), (c), and (d), an offset of PRF/4 (275 Hz) must be added.

Due to the variation of the look angle Or from the near to the far range, the Doppler centroid will vary according to [sin Or

-

sin

dd

+ cos 8,

sin OP]. (18)

where

ed

is the drift angle (which is not dependent on the

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MOREIRA AND HUANG: SAR PROCESSING USING CHIRP SCALING WITH MOTION COMPENSATION

.-

m I 0-

range I time [ ps m 1 ---->

L

1035

PRF of 275 Hz, the ambiguous Doppler centroid values in Fig. 3(b) are 30 Hz for the first target in the near range and 150 Hz for the last target in the far range. The azimuth frequency variation of the chirp scaling phase applied in the range-Doppler domain was selected to be centered around the Doppler centroidfdc(rl)of the first target in near range (i.e.,fdc(rl) f PRF/2). Since the bandwidth of the first target in near range is within this variation, the RCMC will be performed accurately and the focusing quality will not deterioriate [see Fig. 3(c) and (d)]. However, for the second, third, and fourth target from the near to the far range, the azimuth frequency variation will not be matched to that of the signal itself, leading to an incorrect RCMC [see Fig. 3(d)], defocused IRF's, and azimuth ambiguities [see Fig. 3(e)]. The solution to this problem is very simple and accurate. After transforming the SAR raw data to the rangeDoppler domain, the azimuth frequency variation is artificially increased by means of an azimuth spectral-length extension, which accommodates the variation of the Doppler centroid with range. The azimuth spectral-length extension should be at least as great as the variation of the Doppler centroid from the near to the far range. In the example of Fig. 4 , the azimuth frequency variation was increased by PRF, i.e., fromfdcl - PRF/2 tof,,, + 3 * PRF/2 [see Fig. 4(b)-(d)]. In the general case, the azimuth spectrum must be extended according to the following limits:

(19) where min [ fd,] and max [ fd,] are the minimum and maximum Doppler centroid values used in the processing. Due to the azimuth spectral-length extension, all of the phase functions (HI new, H 2 , , and H Z 2 ) applied in the rangeDoppler and wavenumber domain are unambiguous, so that the measured quality of the IRF's in Fig. 4(e) are the same as in the case presented in the previous section with constant Doppler centroid. The only limitation of this approach is that the Doppler centroid variation within one range chirp length must be less than the PRF. Otherwise, the azimuth frequencies cannot be represented in an unambigous way after the azimuth spectral extension. range [ m

1 ---->

(e) Fig. 4. Contour plot of the signal of four point targets in the different domains of the extended chirp scaling algorithm. (a) Signal domain (raw data), (b) Range-Doppler domain. (c) Wavenumber domain. (d) Range-Doppler domain. (e) Signal domain (final image). Due to the azimuth spectral extension, the variation of the Doppler centroid can be exactly accommodated by the ECS algorithm. The sensor and processing parameters are the same as for Fig. 3 .

range distance) and 0, is the pitch angle. In the example in Fig. 3 (8' squint angle in middle range, 0" pitch angle), the Doppler centroid varies from approximately 305 to 426 Hz (near to far range, respectively). Due to the

1

I

IV . ACCOMMODATION OF DOPPLER CENTROID VARIATIONS I N AZIMUTH WITH INTEGRATED MOTION COMPENSATION A. Doppler Centroid Variation in Azimuth Due to the motion errors of the aircraft, the antenna must be steered in order to keep the squint angle constant. In the case of the E-SAR system, the antenna is fixed on the fuselage of the aircraft and a wide beam is used in azimuth, so that a constant squint angle can be adopted for the processing. The variations of the squint angle lead to a small-signal loss, which is radiometrically corrected after the processing. Additionally, a deterioration of the

i

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IRF is observed, since the RCMC is a function of the squint angle. Then, for accurate processing, the squint-angle value for the processing must be updated with azimuth. The basic assumption in the following approach is that the Doppler centroid for the processing is constant within the synthetic aperture length but it can vary several hundred hertz along the entire data acquisition interval (data take), which is several minutes long. Since the processed azimuth bandwidth for the E-SAR system is less than onefourth of the total azimuth signal bandwidth, this assumption leads to very accurate results. With the update of the Doppler centroid with azimuth, the signal-to-noise ratio, the ambiguity level and the image resolution are optimized for the entire data take. The proposed approach for accommodating the Doppler centroid variations with azimuth consists of dividing the entire data take in small azimuth subapertures (e.g., 128 points), which are much smaller than one synthetic aperture length. A small overlap between the subapertures is used in order to guarantee a phase continuity between the subapertures. Defining Tsubas the duration of each azimuth subaperture, Tovlas the overlap between the azimuth subapertures, and Tdata -take as the duration of the whole data take we obtain

+

~ ~ (t;7r , d = ~ ( 7t,

i

*


09 1032 ~ > 4 a ' i i~' ~ o . ~ 11.083~ ~ E4

~

~

~

t

>

Altitude . 2205 m Pobrisotion V V

> 11-04-91

Number of Looks

8

*

Peak Sidelobe Rotio --33 dB

37"

*

RF Center Frequency

Wlth Motion Compensotion

/

5 3 GHr

2 0 Compression

(b) Fig. 5 . (Conrinued.)

Multiplication with the function &(7, fa; rref)compensates for a slowly varying azimuth phase, which was introduced by the chirp scaling operation. By means of azimuth IFFT's, the data is transformed from the range-Doppler domain into the range and azimuth time domain (signal domain). At this step, all targets are range compressed and have their azimuth trajectories without any range migration. The second-order motion compensation accounts for the accurate phase correction with range update and compensates the residual azimuth phase error. The azimuth FFT's transform the motion-compensated data into the range-Doppler domain. Multiplication with H3(7, fa; ro) performs the azimuth compression. The final image is obtained after azimuth IFFT's. The above description of the extended chirp scaling algorithm allows the following steps to be included in the processing. Doppler centroid update with range is accommodated by means of an azimuth spectral length extension in the range-Doppler domain.

Doppler centroid update with azimul,. is introduced by means of azimuth subaperture processing. Motion error extraction using the ReJlectivity Displacement Method is incorporated by means of azimuth subaperture processing. Data processing with squint angle up to 30" can be carried out in the C-band without deterioration of the IRF by means of an additional cubic phase term in the chirp scaling operation. Future work includes the image processing of satellite data with the extended chirp scaling algorithm (e.g., SIRC/X-SAR). A feasibility study of dedicated hardware for a real-time SAR processing, which is suitable for airborne and spacebome SAR systems, will be carried out. In addition, the processing of spotlight SAR data and inverse SAR will be considered in connection with the flexibility of the extended chirp scaling algorithm for Doppler centroid update and motion error correction. APPENDIX In the following text, an analysis of the phase errors induced by the chirp scaling operation with quadratic

~

~

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MOREIRA AND HUANG: SAR PROCESSING USING CHIRP SCALING WITH MOTION COMPENSATION

phase function is presented. Then, the anlaysis is extended for the case in which a cubic term is introduced into the original phase function. For airborne SAR imaging, the scaling factor is exactly linear [ 121. The only phase error during the chirp scaling operation occurs due to the assumption of a constant range frequency modulation (no range update). After the chirp scaling operation, the remaining phase error, which is not corrected in the further steps of the processing, is given by

-

(7

2 . r

-C

-

completely: ?r

t cubic $ . = -

- k: -

AkSm(fa)* c 6

By introducing a new term 2 as

-

ro/c, (30) can be rewritten

2

(1

+ a’(&)) .

(24)

Since k,,