# Motion Error Compensation in Stationary SAR Imaging

## Motion Error Analysis in Airborne SAR

In airborne SAR imaging, the platform is assumed to be moving along a straight trajectory with a constant velocity. However, in practical applications, these conditions cannot be satisfied, and the motion error compensation must be induced in the imaging process. The motion error of the stationary target imaging is the same as that of moving target imaging, so the motion error compensation is necessary in GMTIm algorithms.

The geometry of airborne SAR with motion error is illustrated in Fig. 6.1. It can be noted that the actual trajectory of the platform is a non-ideal trajectory. Suppose

**Fig. 6.1 ****Geometry of an airborne SAR with the motion error**

**Fig. 6.2 ****Geometry of the range difference**

the coordinate of sample A is (X*(t _{a}), Y(t_{a}), Z(t_{a})),* the coordinate of Target P is

*(X*the instantaneous slant range from A to P is

_{p}, Y_{p}, Z_{p}),

After Taylor expansion and neglect the higher order phase of *t _{a},* it yields

where the nearest range of Target P *R _{0} =* y^Yp^+Z

^{2}, sin b = R

^{p}, cos b = R

^{p}. The

last two terms in Eq. 6.2.2 is the range difference between the actual trajectory and the ideal trajectory, as shown in Fig. 6.2 Suppose the range difference is *AR(t _{a}), *i.e., A

*R(t*— Y

_{a}) =*(t*sin b — Z

_{a})*(t*cos b, substitute A

_{a})*R(t*into Eq. 6.2.2, it yields

_{a})

It can be noted from Eq. 6.2.3 that the motion error of the platform is separated into two parts: the inconstant velocity of the platform and the inconstant slant range. Both errors will cause the error in the instantaneous slant range, and in turn lead to the azimuth smear [1]. Two kinds of errors and the compensation methods are introduced below.