Doppler Modulation Rate
According to Eq. 2.3.6b, the Doppler modulation rate of a moving target is related to the alongtrack velocity and crosstrack 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 secondorder azimuth phase error. The secondorder azimuth phase error will lead to the mainlobe broaden, the peak declination of the mainlobe and peak rise of the sidelobes. The secondorder 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 secondorder azimuth phase error is
By using the typical airborne and spaceborne SAR system parameters as shown in Table 2.2 [5], the numerical secondorder phase error can be calculated. Suppose the alongtrack velocity of a moving target is 10 m/s and the crosstrack acceleration is 0.1 m/s^{2}, the secondorder 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 alongtrack velocity. b Changes with crosstrack velocity
According to the analysis above, airborne SAR moving target imaging is a far more complicate issue than spaceborne SAR. Since the secondorder azimuth error also causes the peak declination of the mainlobe, 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 alongtrack 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 alongtrack velocity. Figure 2.5b illustrates the relationship between the peak of the mainlobe and the crosstrack acceleration. The crosstrack 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 mainlobe also decreases with the increase of the crosstrack acceleration. Therefore, it is proved that the change of Doppler modulation rate results in the smear of the image.