VELOCITY ESTIMATION OF MOVING TARGETS USING SAR

Junfeng Wang and Xingzhao Liu

Shanghai Jiaotong University, China

ABSTRACT A new scheme is presented for the velocity estimation of moving targets in synthetic aperture radar (SAR) imaging. First, the moving target is imaged using a range-Doppler algorithm, where clutter lock, range correction and focus filtering are automatically done. Then, the parameters obtained in this algorithm are used to estimate the velocity of the moving target. The Doppler centroid is used to estimate the velocity of the moving target in range. The out-of-focus coefficient is used to estimate the velocity of the moving target in azimuth. Especially, the ambiguity of the Doppler centroid is resolved according to the range-correction coefficient and the out-of-focus coefficient. This scheme is accurate and computationally efficient.

Index Terms— Synthetic aperture radar (SAR), moving targets, velocity estimation

1. INTRODUCTION The velocity estimation of moving targets is an issue of interest in SAR imaging. This can be achieved using single-sensor SAR or multiple-sensor SAR. In this paper, we only treat single-sensor SAR. Moreover, we only treat airborne broadside-looking SAR though the addressed ideas and methods can be extended into other cases. Generally, the velocity of the moving target in the imaging plane is decomposed into the component in range and the component in azimuth. They are estimated from the Doppler centroid and the Doppler rate, respectively.

The Doppler centroid estimated by clutter lock is limited within the baseband. It may be ambiguous, i.e., the true Doppler centroid may be this estimate plus a multiple of the pulse repetition frequency (PRF). This happens when the velocity of the moving target in range is so large that the absolute value of the Doppler centroid attains or exceeds half the PRF. In such cases, the estimate of the Doppler centroid cannot be used to calculate the velocity of the moving target in range directly.

The ambiguity of the Doppler centroid is solved by finding an unambiguous Doppler centroid and then calculating the ambiguity number. (The unambiguous Doppler centroid is not as accurate as the Doppler centroid estimated by clutter lock, and thus is only used to calculate

the ambiguity number.) Typical methods include the multiple-look method [1, 2], the wavelength-diversity method [3, 4], and the sharpest-projection method [5]. In the multiple-look method, the displacement between two looks in range is used to estimate the unambiguous Doppler centroid. This method may not work when the velocity of the moving target is large such that the two looks are seriously blurred and therefore cannot be registered. In the wavelength-diversity method, the unambiguous Doppler centroid is estimated from the skew of the two-dimensional spectrum. This method is not very robust against the background clutter. The sharpest-projection method is based on the principle that when the unambiguous Doppler centroid is correctly estimated, in the range-Doppler domain, the norm of the signal has the sharpest projection on the range axis. This method is computationally expensive.

In this paper, a new scheme is presented for the velocity estimation of moving targets in SAR imaging. First, the moving target is imaged using an automatic range-Doppler algorithm. In this algorithm, clutter lock is carried out using the nominal-spectrum method [6], range correction is carried out using the method in [7], and the focus filtering is carried out using the sharpest-image method in [8]. Then, the parameters obtained in this algorithm are used to estimate the velocity of the moving target. The Doppler centroid is used to estimate the velocity of the moving target in range. The out-of-focus coefficient is used to estimate the velocity of the moving target in azimuth. Especially, the ambiguity of the Doppler centroid is resolved according to the range-correction coefficient and the out-of-focus coefficient. This scheme is accurate and computationally efficient.

2. VELOCITY ESTIMATION 2.1. Signal from a Scatterer The signal from a scatterer is written as

rjts

4exp)( , (1)

where t is slow time, is the scattering coefficient of the scatterer, r is the range of the scatterer to the radar, and is the carrier wavelength. r is written as

20

20 VttvxtvRr xR

. (2) (2) is derived according to the geometry of SAR imaging in

340978-1-4577-1005-6/11/$26.00 ©2011 IEEE IGARSS 2011

the imaging plane (figure 1). The radar moves at velocity V in the x-axis, and is situated at (0, 0) at t=0. The scatterer moves at velocity vx in the x-axis and velocity vR in the R-axis, and is situated at (x0, R0) at t=0. Let tc be the time when the radar bore-sight is directed to the scatterer, i.e.,

xc vV

xt 0 , (3)

and Rc be the range of the scatterer to the radar at tc, i.e., cRc tvRR 0 . (4)

Then, approximating r by its second-order Taylor series at tc, we obtain

2c

2

2tt

RvV

ttvRrc

xcRc

. (5)

The substitution of (5) into (1) yields )(ts

2c

2

24exp tt

RvV

ttvRjc

xcRc

. (6)

Figure 1. Geometry of SAR Imaging in Imaging Plane.

2.2. Estimation of Velocity in Range The instantaneous Doppler frequency of the scatterer is defined as the derivative of the phase of s(t), i.e.,

c

24tt

RvV

vc

xR

. (7)

The Doppler centroid is found by letting t=tc in (7), i.e.,

Rc v4 . (8)

From (8), we obtain the velocity in range

cRv4

. (9)

Therefore, the velocity in range can be estimated from the Doppler centroid. In fact, (9) is written as

cR kMT

v2

, (10)

where M is the number of the Doppler samples, T is the pulse repetition interval, and kc is the Doppler centroid with the Doppler interval as the unit.

The Doppler centroid can be estimated by clutter lock.

However, the Doppler centroid estimated by clutter lock is limited within the baseband. It may be ambiguous, that is, the true Doppler centroid may be the estimate plus a multiple of the PRF. This happens when the absolute value of the velocity in range is so large that the Doppler centroid attains or exceeds half the PRF. In such cases, the estimate of the Doppler centroid cannot be used to calculate the velocity in range directly. In order to solve the ambiguity of the Doppler centroid, an unambiguous Doppler centroid needs to be found to calculate the ambiguity number. Only used to calculate the ambiguity number, the unambiguous Doppler centroid does not have to be as accurate as the Doppler centroid estimated by clutter lock. So, an accurate, unambiguous Doppler centroid can be determined by

Mk

Mkk rc round0

. (11)

where k0 is the Doppler centroid estimated by clutter lock, kr is a rough but unambiguous Doppler centroid, and round( ) carries out the rounding operation and is used to calculate the ambiguity number. In order to solve the ambiguity of the Doppler centroid, kr needs to be found.

Now consider how to find kr. From (7), we obtain

cx

cc

vV

Rtt 24

. (12)

The substitution of (12) into (5) yields

cx

cRc

vV

RvRr 24

222

2

32 c

x

c

vV

R . (13)

(13) gives the relation of range to instantaneous Doppler frequency. Approximately, it is also taken as the relation of range to spectral Doppler frequency. So, in range correction [7], the shift made to the Doppler slice in range should be

222

2

2 324 cx

cc

x

cR

vV

R

vV

Rvr . (14)

Extending (14) to the fundamental interval and discretizing r and , we obtain

MkM

kMkkM

Mkkkk

Mn

i

i

i

i

i

i

k

2 ,)(2

20 ,)(2

0

2

10

0

2

10

, (15)

21 4 x

cR

vVdT

Rv , (16)

22

2

2 32 x

c

vVdT

R , (17)

where k is the index of , d is the sampling interval in range, and nk is the shift made to the Doppler slice in range when d is taken as the unit. 1 and 2 are called the range-correction coefficients and can be estimated in range correction. The

x

R

Radar

Scatterer vx

vR

V

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Fourier transform of s(t) is approximately

c

x

c RjjvV

RS

44

exp2

)( 2

2

28 c

x

cc

vV

Rjtj . (18)

So, the phase response of the focus filter should be 2

28)( c

x

c

vV

R . (19)

Extending (19) to the fundamental interval and discretizing , we obtain

MkM

kM

Mkk

Mkk

Mkk

k

2 ,

20 ,

)(

0

20

0

20

, (20)

22 )(2 x

c

vVTR , (21)

where (k) is the phase response of the focus filter in terms of k over the fundamental interval. is called the out-of-focus coefficient, and can be estimated in focus filtering [8]. Letting (16) over (21), we obtain

12Td

vR. (22)

Substituting (22) into (10), we obtain an estimate of kc, 14dM

kr. (23)

Once kr is found, kc is calculated using (11). Here, kr is found from 1 and . It can also be found from 1 and 2 theoretically. This, however, is not a good method because the estimate of 2 may not be accurate enough.

Alternately, we can find the velocity in range from the Doppler centroid estimated by clutter lock and then solve the ambiguity of the velocity in range. 2.3. Estimation of Velocity in Azimuth From (21), we obtain the velocity in azimuth

21 c

x

RT

Vv . (24)

Therefore, the velocity in azimuth can be estimated from the out-of-focus coefficient. In the estimation, Rc is taken as the range of the central range bin.

3. FURTHER DISCUSSION 3.1. Finer Imaging A right-located, well-focused image can be generated using the automatic range-Doppler algorithm. However, since some approximations are assumed, such images may not be

satisfactory in particular applications. Actually, once the velocity is estimated, the moving target can be reimaged using a finer algorithm. Taking the moving target as the frame of reference for motion analysis, we can set up another imaging geometry. Then, a finer algorithm can be used to reimage the moving target. 3.2. Target Isolation The aforementioned scheme for automatic imaging and velocity estimation applies to a single moving target. It may not work when there exists a stationary background or other moving targets. In such a case, automatic imaging and velocity estimation can be carried out in the following way [9, 10]. First, the stationary background and the moving targets are imaged with the assumption that the radar moves regularly and no target moves. Then, the complex image of each moving target is isolated from the stationary background and other moving targets. Finally, each moving target is reprocessed with the aforementioned scheme for automatic imaging and velocity estimation. Target isolation may be difficult in some applications. Further investigation needs to be made on this topic.

4. RESULTS Multiple sets of simulated signals are used to evaluate our scheme. The radar is carried on a platform moving at a velocity of 250 m/s and transmits pulses at a period of 0.002 s. The beam is directed broadside and has an angle of 0.02 rad in azimuth. The pulses have a wavelength of 3 cm and a bandwidth of 200 MHz. The sampling frequency is 300 MHz. 512 echoes with 256 range bins each are recorded. When the slow time is 0, the platform is situated at (0, 0), and the center of the target is situated at (0, 10 km).

Velocity in Range (m/s) Velocity in Azimuth (m/s) Truth Estimate Truth Estimate

9 9.00146 9 9.00594 6 5.99854 6 5.98984 3 2.99927 3 2.96872

0 0.00000 0 0.00000 3 2.99927 3 3.01740 6 5.99854 6 5.97656 9 9.00146 9 9.01865

Table 1. Truths and Estimates of Velocities in Different Simulations.

Table 1 shows the truths and the estimates of the

velocities in different simulations. It can be seen that this scheme has high estimation accuracy. When the velocity in range is 9 m/s, 6 m/s, 6 m/s or 9 m/s, the Doppler centroid is 1200 rad/s, 800 rad/s, 800 rad/s or 1200 rad/s, respectively. Since the absolute value of the Doppler

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centroid is larger than 500 rad/s, a half of the PRF, the Doppler centroid estimated by clutter lock is ambiguous. It is around 200 rad/s, 200 rad/s, 200 rad/s or 200 rad/s, respectively. If the ambiguity of the Doppler centroid were not solved, the estimate of the velocity in range would be around 1.5 m/s, 1.5 m/s, 1.5 m/s or 1.5 m/s, respectively. In fact, as we see from table 1, the estimate of the velocity in range is around 9 m/s, 6 m/s, 6 m/s or 9 m/s, respectively. This indicates the effectiveness of this scheme in solving the ambiguity of the Doppler centroid.

SNR (dB) Estimate of Velocity in Range (m/s)

Estimate of Velocity in Azimuth (m/s)

20 5.99854 5.97656 10 5.99487 5.97656 0 5.98755 6.02498 10 5.96191 5.97656 20 5.88501 5.92811

Table 2. Estimates of Velocities under Different SNRs (Truths: 6 m/s).

Figure 2. Image when Velocities in Range

and Azimuth are 6 m/s, and SNR is 20 dB.

The program is developed in MinGW C on a Dell Precision T3400 workstation with a 3 GHz Intel Core 2 Duo processor and a 3 GB memory. For the above 7 sets of signals, the computation times are 2.000 s, 1.703 s, 1.703 s, 1.422 s, 1.703 s, 1.890 s and 2.015 s, respectively. It can be seen that this scheme is also computationally efficient. In fact, the times are mainly those for the sharpest-image autofocus. In its implementation, the phase response of the focus filter is estimated from all the range bins. If it is estimated from part of the range bins, the computational efficiency will be further improved.

The estimates of the velocities under different signal-to-noise ratios (SNR) are shown in table 2. Here, Gaussian noises with different intensities are added to the signals when the velocities in range and azimuth are both 6 m/s. It can be seen that with the noises, the errors increase. Nevertheless, the estimates are accurate. This means that this scheme is robust against noise. Figure 2 shows the image when the velocities in range and azimuth are both 6 m/s, and

the SNR is 20 dB.

5. CONCLUSION In SAR imaging, the velocity of moving targets can be effectively estimated using the scheme presented in this paper. First, the moving target is imaged using a range-Doppler algorithm, where clutter lock, range correction and focus filtering are automatically done. Then, the parameters obtained in this algorithm are used to estimate the velocity of the moving target. The Doppler centroid is used to estimate the velocity of the moving target in range. The out-of-focus coefficient is used to estimate the velocity of the moving target in azimuth. Especially, the ambiguity of the Doppler centroid is resolved according to the range-correction coefficient and the out-of-focus coefficient. This scheme is accurate and computationally efficient.

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