According to Eq. 2.3.6b, the Doppler modulation rate of a moving target is related to the along-track velocity and cross-track acceleration. The Doppler modulation rates of a moving target and a stationary target differ in the existence of these two motion parameters. The change of the Doppler modulation rate can be expressed as

The Doppler modulation rate directly affects the azimuth focusing performance. In the existence of Df_{dr}, a moving target is smeared since there is the second-order azimuth phase error. The second-order azimuth phase error will lead to the main-lobe broaden, the peak declination of the main-lobe and peak rise of the side-lobes. The second-order azimuth phase error of a moving target in a stationary SAR image can be expressed as

Equation 2.3.12 reaches its largest value at the edge of the synthetic aperture, therefore the maximum value of the second-order azimuth phase error is

By using the typical airborne and spaceborne SAR system parameters as shown in Table 2.2 [5], the numerical second-order phase error can be calculated. Suppose the along-track velocity of a moving target is 10 m/s and the cross-track acceleration is 0.1 m/s^{2}, the second-order azimuth phase error is 149.43 rad in the airborne SAR situation, while the value is 3.013 rad in the spaceborne situation. An airborne SAR has a smaller velocity and nearer range length, and has a higher resolution. Therefore, the defocus of a moving target is more obvious in an airborne SAR image than in a spaceborne SAR image.

Table 2.2 Typical system parameters in airborne and spaceborne SAR

Platform

System parameters

Wavelength

(m)

Range

(km)

Platform

velocity

(m/s)

Synthetic aperture time

(S)

Azimuth

resolution

(m)

Range

resolution

(m)

Airborne

0.032

850

7100

3.4

0.5

1.25

Spaceborne

0.057

30

250

0.64

5

6.25

Fig. 2.5 Peak changes of the mainlobe with motions. a Changes with along-track velocity. b Changes with cross-track velocity

According to the analysis above, airborne SAR moving target imaging is a far more complicate issue than spaceborne SAR. Since the second-order azimuth error also causes the peak declination of the main-lobe, the moving target indication performance will be affected by using CFAR technique. By using the system parameters in Table 2.2, a set of simulations is operated, as shown in Fig. 2.5.

In Fig. 2.5a, the horizontal ordinate denotes the along-track velocity, which is 2, 4, 6, 8, 10 m/s; the vertical ordinate denotes the peak of the mainlobe, which is measured by dB. It can be noted that the peak of the mainlobe decreases with the increase of the along-track velocity. Figure 2.5b illustrates the relationship between the peak of the mainlobe and the cross-track acceleration. The cross-track accelerations are valued as 0.05, 0.1, 0.15, 0.2 and 0.25 m/s^{2}. It can be noted that the peak of the main-lobe also decreases with the increase of the cross-track acceleration. Therefore, it is proved that the change of Doppler modulation rate results in the smear of the image.