合成孔徑雷達原理（二）

Range Resolution

Range is the direction perpendicular to flight path of the aircraft. The vertical beamwidth , shown in Figure 3, is determined by the wavelength () and the antenna height W_a such that _v =  / W_a. Referring to Figure 4, we see that the width of the ground swath W_g, also called the range extent, is simply calculated as:

and so

where  is the incidence angle of the beam at the midpoint of the swath (equal to the look angle ) and R_m is the slant range (distance) from the antenna to the midpoint of the swath.

Ground resolution is defined as the minimum distance on the ground at which two object points can be imaged separately, shown in Figure 4 as the distance R_g. Two objects can be distinguished if the leading edge of the pulse echo from the more distant object arrives at the antenna later than the trailing edge of the pulse echo from the nearer object. Thus if their distances to the antenna are different by at least half of the pulse length (the distance the pulse travels in the time _p during which the transmitter is turned on) then the objects are resolvable. Therefore the ground range resolution is

where R_g is the slant range resolution,  is the incidence angle, L_p is the pulse length (in units of distance), _p is the pulse duration (in units of time) and c is the speed of light. Note that as you approach the nadir line directly below the aircraft,  approaches zero making sin() approach zero so that the ground range resolution becomes extremely poor. This makes sense since the objects near the nadir line are all virtually the same distance from the antenna, making them impossible to distinguish.

To achieve a ground range resolution small enough to be useful, the pulse duration ?p would have to be too short to generate enough energy to produce a sufficient echo signal to noise ratio (SNR). To get around this, a technique known as pulse compression is often used together with a type of processing of the returned signal known as matched filtering (which we will not go into here, though a good description can be found in Curlander, see SAR References) to produce both high resolution and a high SNR. The ground range resolution possible with these techniques is

where B_R is the frequency bandwidth of the transmitted radar pulse. The resolution can be further improved by increasing the frequency bandwidth.

Azimuth Resolution

Azimuth is the direction parallel to the flight path of the aircraft. Figure 3 shows that the angular spread of the radar beam in the azimuth direction is equal to _H =  / L_a where  is the wavelength of the transmitted beam and L_a is the length of the radar antenna in the azimuth direction. This spread is due to interference of the waves emitted from and received by the dipoles of the antenna, which causes the angular spread to decrease as the aperture length increases (a longer antenna produces a tighter beam, much as a longer barrel of a gun produces less dispersion). Two objects on the ground and with the same slant range R can only be imaged separately if they are not both within the radar beam at the same time. Thus the azimuth resolution is

Principles of Synthetic Aperture Radar

Because of the way in which microwaves interact with the atmosphere and the ground, only a select few frequency bands are useful for imaging. These are shown in Figure 5. The wavelength affects the penetration depth and also the size of a target necessary to return the radar. This shall be discussed later.

For a specific frequency f (or wavelength ) and slant range R, the azimuth resolution is entirely dependent on the aperture length L_a. However, for the altitudes at which satellite or airborne imaging sensors operate (in the 1-10 GHz region), engineering difficulties make it is impossible to achieve values of L_a /  greater than several hundred. For example, the new Canadian satellite RADARSAT, launched in 1995, orbits at 792 km and operates at a frequency of 5.3 GHz (C-Band). The real aperture limit to the azimuth resolution would be

Through the use of synthetic aperture technology, however, RADARSAT can achieve a resolution as low as 9 m.

It was in 1951 that Carl Wiley first realized that the Doppler spread of the echo signal could be used to synthesize a much longer aperture (thus the name synthetic aperture radar) to greatly improve the resolution of a side-looking radar.

The Doppler Effect

When a detector moves with respect to a source of waves, a shift in frequency is observed known as the Doppler effect. When a source emitting waves with frequency f recedes from the observer at a velocity v, the observed frequency is

When the source approaches the observer, the observed frequency is

where v is always positive. Thus a receding source exhibits a lower frequency and an approaching source exhibits a higher frequency than the actual frequency emitted.

When two target points on the ground are separated in the azimuth direction, they are at slightly different angles from the antenna with respect to the line of flight. Because of this, they have slightly different speeds at any given moment relative to the antenna. Therefore the signal echoed from each target will have its frequency shifted a different amount from the original. To calculate the Doppler frequency shift for a specific target, refer to Figure 6 as we first calculate dR/dt--the relative speed with which the target and antenna approach each other.

To find the R-directional component of V_rel (the velocity with which the target approaches the sensor), we can project V_rel (parallel to x) onto a line parallel to R to get

Now, we can calculate the frequency observed by the ground target: