6

BASICS OF RADAR IMAGING

In this chapter, Section 6.1 surveys several popular imaging radars and possible applications. Section 6.2 covers the geometry of stripmap SAR, which consists of broadside SAR and squint SAR. Section 6.3 describes the relationship between Doppler frequency and radar image processing, where both broadside SAR and squint SAR are covered. A key problem of radar image processing is range migration, which is described in Section 6.4. Radar image distortion and resolution are covered in Sections 6.5 and 6.6. Most hand drawings in this chapter are for illustration purposes and are not to scale.

6.1 BACKGROUND

Imaging radar has been used to measure the motion of the earth's surface to help us better understand earthquakes and volcanoes. It can be used to study the movements and changing size of glaciers and ice floes to understand long-term climate variability. The development of highly detailed and accurate elevation maps is based on imaging radar. Imaging radar is also used to find oil or other natural resources, to study land cover and land use change, to assess the health of crops and forests, and to plan for urban development.

In early 1950, scientist Carl Wiley found that the use of Doppler frequency analysis could improve radar image resolution of side-looking radar. This new finding led to the development of the SAR (synthetic aperture radar) technique. The SAR technique derives from the along-track (or azimuth) processing of the signal data by synthesizing an aperture that is longer than the actual physical antenna to yield a higher resolution. Depending on the system's configuration, SAR has been used to acquire data in three major modes: (1) stripmap SAR, which images a long strip terrain during the full transit distance; (2) scan SAR, which scans the ground in wider swaths by varying the elevation angle of the antenna beam along the flight path; and (3) spotlight SAR, which images a scene with finer resolution at multiple viewing angles during a single pass.

FIGURE 6.1 Configurations of (a) a stripmap SAR and (b) a scan SAR.

Figure 6.1 displays two configurations of the SAR system: the stripmap SAR and the scan SAR. Figure 6.1a shows that with fixed antenna angle, the radar scans through the ground with a fixed width of strip along the flight path. Figure 6.1b shows that the antenna varies the angle along the flight path and scans through a wider swath on the ground.

Figure 6.2a shows the spotlight SAR, which achieves finer resolution on the target by rotating the antenna and aiming at a target along the flight path. In addition to the stripmap, scan, and spotlight modes, other techniques are also used for radar image processing. Interferometric SAR (InSAR) has been widely used in many applications by applying multiple radar antennas and/or different acquisition geometries. InSAR may also achieve the same purpose by scanning the same ground with a single antenna but at different flight paths. Figure 6.2b displays an example of InSAR, where A1 and A2 are two radars separated by distance D. The inverse SAR (ISAR) is an imaging system that deals with moving targets while the radar is stationary.

FIGURE 6.2 Imaging radar for (a) a spotlight SAR and (b) an interferometric SAR.

The following chapters focus only on discussion of the stripmap SAR, which consists of the broadside-mode SAR and the squint-mode SAR.

6.2 GEOMETRY OF IMAGING RADAR

Figure 6.3 shows an imaging radar that operates on a flying vehicle, which could be a satellite or an airplane. The radar antenna is oriented parallel to the flight direction; that is, it is looking sideward to the ground. The radar is moving along the flight path above the earth with height H at velocity . The radar antenna, which is assumed to be a phased array, has dimensions of length L and width W. The ground surface area from which the radar pulse is reflected is called the footprint. Swath is the ground surface area covered by the consecutive radar pulses. The radar transmits short pulses with duration T_p and repeats at period PRI = T = 1/f_PRF to the ground, where f_PRF is the pulse repetition frequency. The 3-dB beamwidth along the track is θ_H = λ/L, while across the track it is θ_V = λ/W and the wavelength of the transmitted signal is λ. The pulse is directed at some angle off nadir (directly below the radar) called the look angle or incident angle θ_ι. The distance from radar antenna to center of footprint is represented by R₀.

FIGURE 6.3 Geometry of stripmap imaging radar.

FIGURE 6.4 Geometry of (a) a broadside SAR and (b) a squint SAR.

Imaging radar is extremely sensitive to the ground surface roughness of the area being imaged. It does not detect the visible color of the surface, but detects the moisture and electrical properties of the surface. Ground targets are illuminated numerous times by the radar beams. The time interval during which the target is illuminated depends on the beamwidth of the radar antenna and the speed of the flying vehicle, namely, ΔT = R₀θ_H/V. The returned signals from ground targets at the same slant range will arrive at the same time. These equal-slant-range targets are separable only in Doppler frequency, because of the radial velocity difference of the radar relative to the target.

Figure 6.4a shows the geometry of a broadside-mode side-looking aperture radar system, where the center of the radar beams is perpendicular to the flight path. The radar moves at speed V along the flight path, which is parallel to the y axis. The three radar positions A, B, and C are the locations where the radar transmits a signal and receives an echo from the ground targets at three pulse periods. The length of the center beam between radar and ground target is R₀. The radar–target distance is called slant range; however, when the target is under the center beam of the radar and the center beam is perpendicular to the flight path, the distance is called range. The radar beam covers the ground with an elliptical shaped area. The two axes of the elliptical shape are determined by R₀θ_H = R₀λ/L and R₀θ_V = R₀λ/W, respectively, where θ_H and θ_V are the radar beamwidths and L and W are the antenna sizes. The consecutive radar beams cover the ground and form a rectangular area, which can be processed to form a rectangular-shape radar image.

Figure 6.4b shows the geometry of a forward-looking aperture radar system with nonzero squint angle, where the center of the radar beam forms an angle θ_q with the axis perpendicular to the flight path. All other parameters are the same as in Fig. 6.4a, except that the radar beams cover the ground with a rhombus area, which can be processed to form a rhombus-shape radar image.

6.3 DOPPLER FREQUENCY AND RADAR IMAGE PROCESSING

The radar image is formed by processing the 2D raw data collected by range radar. Each row of 2D raw data is formed by the target echo returned from every radar pulse, which is transmitted at the rate of f_PRF. The 2D radar image data are represented in complex numbers and normally can be processed separately by processing range data first, followed by azimuth data. Various algorithms are available to process the 2D raw data to generate a radar image, and they are discussed in later chapters.

Figure 6.5 shows the configuration of imaging radar and the corresponding radar transmitter and receiver signal. Figure 6.5a shows a radar pulse that covers an elliptical-shape ground area. The two axes of the elliptical shape are R₀λ/L, and R₀λ/W, respectively, where L and W are the length and width of radar antenna. The two slant ranges from the radar to the closest target and farthest target along the x axis are represented as R_N and R_F, respectively. R₀ is the distance from the radar center beam to the target. The radar moves at speed V in the y direction at x = 0 and height z = H. Figure 6.5b shows the relationship between the transmitted pulse and the returned echoes from all targets within the elliptical shaped area in one pulse period. All targets with equal distance from the radar will superimpose on each other and share the same data from the range viewpoint. Delay time t and slant range r are related by t = 2r/c, where c is the speed of light; therefore, t and r are used interchangeably for convenience. Notice that the parameter x in Fig. 6.5b refers to the slant range, while in Fig. 6.5a it refers to the ground range.

FIGURE 6.5 (a) Imaging radar and (b) radar pulse and received echo.

The transmitter signal from a pulsed LFM radar can be written as follows, from Eq. (5.8):

Here a(t) is the amplitude of the signal, Rect (t/T_p) is a rectangular function with |t| ≤ (T_p/2), and T_p is the pulse duration time. The symbol f_c is the carrier frequency, and α is the LFM pulse chirp rate.

The echo signal from a single target, with distance R from the radar and target reflection coefficient σ, can be represented as

The baseband signal s_b(t) can be obtained by removing the carrier frequency followed by a lowpass filter through the quadrature demodulation process. The demodulated baseband signal can be represented as:

Let the single target and the imaging radar be located at (X₀, 0, 0) and (0, u, H) respectively, where X₀ and H are constant but the radar position u is a variable. The slant range R can be represented as follows:

The slant range R(u) is a function of both target location and radar position; therefore the 2D received signal can be represented as follows:

The data collected along the flight path, or y axis, are discrete, with sampling frequency equal to f_PRF. The sample spacing along the y axis can be computed as Δy = V/f_PRF.

FIGURE 6.6 (a) Single channel radar range data; (b) M × N radar imaging data array.

Figure 6.6 displays the formation of raw data for the radar image. Figure 6.6a represents a single channel of slant range data, which corresponds to a given radar position u = u₀ from an echo signal as shown in Fig. 6.5b. This is a 1D data digitized with N samples starting from ground location x₀ to x_N−1. Figure 6.6b shows an M × N radar image data array, which corresponds to M radar positions of u = u_i for i = 1, 2,..., M. At each position u_i, the radar receives a returned signal with N samples. A more detailed analysis of radar image data is addressed later.

The principle of using radar to determine target reflectivity and range distance has been addressed in Chapters 4 and 5. The following sections discuss the key parameters of Doppler frequency, which serves as the basis to synthesize a LFM signal along the azimuth direction. Broadside SAR is addressed first, followed by squint SAR later.

6.3.1 Broadside SAR

In a broadside SAR system, the radar beam is perpendicular to its direction of movement. Figure 6.7 displays a broadside SAR system with three radar positions, A, B, C, together with a ground point target located at (D, 0, 0). At radar position A, the radar beam begins to illuminate the target. At position B the center of the radar beam is on the target. At position C the radar beam ends the illumination on the target. Position B is chosen to be at (0, 0, H), with beamwidth θ_H = λ/L. The distance between positions A and C is Ls, which is the synthetic aperture length, and Ls = R₀θ_H = R₀λ/L.

A simplified version of Fig. 6.7 is shown in Fig. 6.8a, where the plane covered by points A, B, C and the point target on the ground forms a new x′–u coordinate system. The distance D from the target to the y axis in Fig. 6.7 is replaced by the slant range D′ in Fig. 6.8, where D′ = (D² + H²)^1/2 = R₀, and R₀ is the closest distance between target and radar. Radar positions A and C are the two ends of radar's 3-dB beamwidth, and the slant range r between position A (or C) and the point target equals

Figure 6.8b shows the relationship between the slant range r and the point target. Here radar position B is chosen to be the origin of the x′–u coordinate system, and θ is the angle between the slant range r and the x′ axis. In practical applications |u| R₀, the slant range r can be computed as

FIGURE 6.7 Configuration of a broadside SAR system.

FIGURE 6.8 A simplified broadside SAR system.

or

Equation (6.5b) expresses a parabolic function, with the vertex located at r = R₀ and u = 0. Figure 6.9 shows the slant range r versus the radar movement along the flight path. The origin of the r–u coordinate is set at r = R₀. The plot is based on Eq. (6.5b) with the assumption that both slant range r and radar position u are normalized with respect to R₀, and the transmitted signal is a pulsed signal. For illustration purposes, we set θ_H = 0.0349 (2°) and Ls = 0.0349R₀. Notice that the x′ axis in Fig. 6.8 is a reference axis connecting point B and the ground target, which is perpendicular to the flight path. The r axis of Fig. 6.9 is a slant range axis connecting the radar and the ground target. For a radar pulse with finite duration, the returned echo signal will last for the same time interval. Figure 6.9a shows a pulse duration equivalent to R₀/200. Figure 6.9b shows the range-compressed version of Fig. 6.9a. Only range-compressed signals are considered in the following discussion,

FIGURE 6.9 Echo signal from the point target before (a) and after (b) range compression.

FIGURE 6.10 Broadside SAR with multiple targets.

Figure 6.10 displays the geometry of two cases of broadside SAR with three targets on the ground. The corresponding slant ranges between radar and targets are shown as r₁, r₂, and r₃, respectively.

Figure 6.10a shows the three targets, labeled as 1, 2, and 3, and located at (u₁, R₀), (u₂, R₀), and (u₃, R₀), respectively, where R₀ is the closest distance between targets and flight path. Figure 6.10b displays the three targets aligned along the x′ axis with the same cross-range u₀, and located at (u₀, R₀₁), (u₀, R₀₂), and (u₀, R₀₃), respectively. Each target corresponds to one of the three targets to flight path distances R₀₁, R₀₂, and R₀₃, respectively. The trajectories of echo signals in the u–r or azimuth–slant range domain corresponding to target 1–3 of Fig. 6.10 are shown in Fig. 6.11, where θ_H = 10° is assumed.

FIGURE 6.11 Slant range R(u) versus radar position u for three targets at equal (a) and different (b) ranges.

Figure 6.11a displays the trajectories of echo signals due to three targets, which share the same distance R₀ but are located at three different positions, u₁, u₂, and u₃, along the u axis. For simplicity, we have made u₃ = 0. The peaks of the three curves appear at u₁, u₂, and u₃ where the radar has the closest distance to the ground targets, respectively. Since the three targets have the same closest distance R₀ from the radar, the parabolic trajectories appear to have the same curvature for all three targets. In addition, the synthesized aperture lengths Ls1 = Ls2 = Ls3.

Figure 6.11b shows the trajectories of echo signals due to three targets, which line up in the direction perpendicular to the flight path (u axis) but have different distances R_0i from the radar. The parabolic trajectories appear to have a different curvature for different targets. The greater the distances R_0i is, the flatter the parabolic trajectory becomes and the wider the synthetic aperture length is. The peaks of the three parabolic trajectories are all aligned along the r axis with u = u₀. For simplicity, we have made u₀ = 0.

In general, the ground targets are randomly distributed; therefore the trajectories of echo signals will be a combination of those in both Figs. 6.11a and 6.11b.

In the SAR system described above, the radar moves along the flight path, or u axis. The Doppler frequency therefore occurs and can be expressed in terms of the radar position u, the slow time s (along the u axis), and the slant range r. Discussion of the Doppler frequency in terms of u or s will lead to synthesis of pulsed LFM along the azimuth direction, while the representation of Doppler frequency in terms of slant range r will serve as the basis for “range cell migration correction.”

Figure 6.12 shows a broadside SAR with radar moving at velocity = VÛ, where Û is the unit vector along the u axis. The radar position B is set to be the origin of the x′–u coordinate, and has the shortest range R₀ from the target. This is also the position where the target is under the center beam of radar. The radar positions A and C are two endpoints of the 3-dB beamwidth. The angle between radar position u and the x′ axis is represented as θ_u. At radar position A or C, the angle θ_u equals θ_H/2, where θ_H is the radar's 3-dB beamwidth and the synthetic aperture length Ls = R₀θ_H.

FIGURE 6.12 Broadside SAR with single point target.

At any radar position = uÛ, the radial velocity of radar is defined as The symbol is the unit vector from radar position to the target. The dot product of Û and is defined as Û · = sinθ_u. At position B the radar's radial velocity

Given the radar position u, where |u| ≤ Ls/2, the relationship between V_r and u can be expressed as follows:

Here sgn (u) denotes “sign of u.” The value of V_r is positive when the radar is located at the left side of the origin (or u is negative) and becomes negative when the radar is at the right side of the origin (or u is positive).

Normally R₀ |u|; therefore Eq. (6.6a) becomes

As discussed in Chapter 4, the relationship between the radial velocity and the Doppler frequency is f_D = 2V_r/λ. The Doppler frequency f_D is positive for inward velocity and becomes negative for outward velocity . Therefore, at any radar position u, the Doppler frequency can be expressed as

The change rate of f_D with respect to the radar position u is

Given the parameters V and λ, Eq. (6.7b) states that the change rate of f_Du is always negative and its magnitude decreases when R₀ increases. The subscript u used in f_Du and indicates that f_Du and are associated with variable u. Here θ_u 0 for small beamwidth.

Equation (6.7a) states that the Doppler frequency f_Du = 0 when u equals zero. The maximum value of Doppler frequency occurs at both ends of the 3-dB radar beamwidth, which implies u_m = ±Ls/2; that is

where Ls = R₀λ/L.

The bandwidth of the Doppler frequency therefore becomes

Since the slow time s = u/V, the Doppler frequency f_D and associated parameters can also be represented in terms of s as follows, from Eq. (6.7a):

The total time duration during which the Doppler frequency is observable along the flight path can be computed as

From Eqs. (6.8a) and (6.8b), it is clear that for a single target with a small radar beamwidth, the following conditions apply:

1. The Doppler frequency f_Ds is linearly propositional to the slow time s.
2. The rate of Doppler frequency change is independent of the slow time s.

These two characteristics of Doppler frequency associated with slow time s fit the characteristics of the linear frequency modulation waveform discussed in Chapter 5. In other words, a new signal along the flight path can be derived, and it can be considered as a synthesized pulsed linear frequency-modulated (LFM) signal in terms of slow time s. This LFM signal has the following characteristics:

Here the subscript s is dropped from f_D and related parameters for convenience.

Derivation of the synthesized LFM waveform is based on the 3-dB beamwidth of the radar antenna. Therefore it is amplitude-modulated by the antenna's 3-dB radiation pattern. Figure 6.13 shows a typical radiation pattern of antenna array and a pulsed LFM waveform along the azimuth direction.

Figure 6.14 displays the 3-dB beamwidth of the antenna radiation pattern shown in Fig. 6.13a. This radiation pattern serves as a built-in windowing function on the synthesized LFM waveform. By multiplying the data of Fig. 6.14a by those of Fig. 6.13b, one obtains the synthesized and symmetric LFM waveform along the azimuth axis, which is shown in Fig. 6.14b. The synthesized LFM signal has a duration time T_a = Ls/V, and its maximum frequency f_Dm = ±(V/L) occurs at two ends of the synthetic aperture length Ls.

FIGURE 6.13 (a) Radiation pattern from a typical antenna array; (b) real part of a LFM signal.

FIGURE 6.14 (a) 3-dB beamwidth of a radiation pattern from a typical antenna array; (b) real part of amplitude-weighted LFM signal.

FIGURE 6.15 Doppler frequency and multiple targets

Figure 6.15 displays the relationship between the Doppler frequency and the radar displacement u under multiple target situations. Figure 6.15a shows the geometry of the SAR system with three targets located at (u₁, R₀), (u₂, R₀), and (u₃, R₀₃), respectively. The slant ranges r₁, r₂, and r₃ are the respective target–radar distances. The origin of the u–x′ coordinate system is chosen with radar position u₃ = 0 and x′ = 0. The radar moves at velocity = VÛ along the u axis, and the x′ axis is perpendicular to the u axis.

Figure 6.15b plots the Doppler frequency f_D against the radar position along the u axis or azimuth direction. Equations (6.7a), (6.7b), and (6.7c) are used for this illustration. Targets 1 and 2 have identical slopes, namely, −2V/(λR₀). Their frequency bandwidth and the synthetic aperture length are also the same: B_D1 = B_D2 = (2VLs/λR₀) = (2V/L) and Ls1 = Ls2 = (R₀λ/L). However, the zero-frequency positions of targets 1 and 2 are different and located at u₁ and u₂, respectively. Target 3, has a shorter closest distance to radar R₀₃, and its aperture duration time is shorter than those of targets 1 and 2. The frequency bandwidth of target 3 is identical to those of targets 1 and 2. The zero frequency occurs at u₃ = 0. The center frequency f_Dc is zero for all three targets.

Since the slow time s = u/V, Fig. 6.15b can also be considered as f_D versus slow time s by dividing u with V. All parameters discussed above will apply, except the slope of f_D becomes −(2V²/λR₀).

The preceding descriptions are based on a Doppler frequency related to the radar position u, or slow time s, along the azimuth direction. The correlation between Doppler frequency and slant range r is discussed next.

Consider the point target case and use Fig. 6.12 for illustration. Given the radar position u and the slant range r, Eq. (6.6a) can be rewritten as follows:

Equation (6.7) then becomes

Let f_DU and f_DL represent the upper and lower bounds of the Doppler frequency spectrum. These two frequencies occur at both ends of the radar 3-dB beamwidth where a target is under illumination. For a forward-looking SAR, the Doppler frequency f_DU occurs at the position where the radar beam begins to illuminate the target, while f_DL occurs at the position where the radar beam ends the illumination on the target. Their values are f_DU = V/L and f_DL = −V/L in the broadside case.

The bandwidth of Doppler frequency for broadside SAR can then be computed as

FIGURE 6.16 Doppler frequency versus slant range for single target.

Figure 6.16 displays the correlation between the Doppler frequency and the slant range r. The plotted is based on the single-target model shown in Fig. 6.12 and with the help of Eq. (6.10b). The following parameters are used for the plot: f_c = 2.0 GHz, V = 300 m/s, R₀ = 10,000 m, L = 1 m, and θ_H = 0.15 radian or 8.59°. As can be seen, both ends of the curve have values V/L = 300 Hz and − V/L = − 300 Hz, respectively. The center of the curve is located at range R₀ = 10,000 m, where the Doppler frequency f_Dc = 0.

The plot in Fig. 6.17 is similar to that of Fig. 6.16 except with multiple targets. The plot reflects the three-target model shown in Fig. 6.15a. The left curve represents target 3, and the right one represents targets 1 and 2. The maximum slant range occurs at both ends of the curve, which correspond to the two ends of the Doppler frequency. Both curves have the same Doppler frequency center f_Dc = 0. The start and end of the Doppler frequencies are f_DU = V/L = 300 Hz and f_DL = − V/L = −300 Hz, respectively, and are the same for both curves. The bandwidth B_D = 2V/L = 600 Hz are also identical for both curves. The slant ranges R₀₁ = 10,000 m and R₀₂ = 10,025 m are used to plot the drawing.

Up to this point, we have shown that f_D, , f_DU, f_DL, and B_D can be represented in terms of spatial position u, slow time s, and slant range r. Although f_D and are in different forms when expressed in terms of u, s, or r, they all share the same results of f_DU, f_DL, and B_D.

FIGURE 6.17 Doppler frequency versus slant range for multiple targets.

The following are part of the technical specifications of the ERS-1/2 satellite systems. They will be used to compute the parameters related to Doppler frequency discussed in this section.

Antenna size (L × W): 10 × 1m

Velocity (V): 7125 m/s

Sampling rate (f_s): 18.96 MHz

LFM chirp rate 4.1778 × 10¹¹ Hz/s

Radar wavelength (λ): 0.0566 m

Pulse duration time (T): 37μs

f_PRF: 1680 Hz

The bandwidth and resolution of range imaging can be computed as

The related parameters of Doppler frequency can be computed as

f_DU: V/L = 712 Hz

f_DL: −V/L = −712 Hz

B_D: 2V/L = 1424 Hz

Δu: V/f_PRF = 4.24 m

Notice that the cross-range Doppler frequency bandwidth of 1424 Hz is much less than that of the range (time-domain) LFM signal, which has a bandwidth of 15.46 MHz.

6.3.2 SAR with Squint Angle

For the broadside SAR systems, the radar beam is perpendicular to the flight path. For satellite SAR systems, the earth's rotation induces an effective squint angle that varies by about ±3° for each orbit. Some applications require the radar beam to look forward or aft on the ground targets. A nonzero squint angle SAR system is discussed next.

Figure 6.18 shows the geometry of a typical forward-looking radar with squint angle θ_q and a point target located at (0, R₀) in a u–x′ coordinate system. A small-squint SAR system is considered as θ_q < 0.5θ_H, while a low-squint SAR system is considered as 0.5θ_H < θ_q < 10°. The three radar positions of u₁, u₂, and u₃ correspond to the start, center, and end of the radar beam illuminated on the target. Narrow beamwidth θ_H is assumed in our discussion.

Figure 6.18a shows a small-squint SAR; Fig. 6.18b, a low-squint SAR. The slant ranges R₁, R₂, and R₃ correspond to the three radar positions u₁, u₂, and u₃, respectively. The three shaded areas in Fig. 6.18 represent the 3-dB radar beamwidth corresponding to positions u₁, u₂, and u₃, respectively. The slant range R₂, corresponding to the center beam of radar at position u₂, is also represented as R_c. The distance between u₁ and u₃ equals the synthetic aperture length Ls. The closest distance between the radar and the target is R₀ = R_c cos θ_q. The radar beamwidth is shown as θ_H, and the squint angle θ_q is the angle between the radar center beam at position u₂ and the x′ axis. The radar moves from left to right at speed V along the u axis. If the radar moves from right to left, the same geometry becomes a backward-looking radar. Forward looking radar is the major configuration discussed in the following chapters.

FIGURE 6.18 Geometry of a forward-looking radar system with nonzero squint angle.

For a squint angle that satisfies θ_q < θ_H/2, as shown in Fig. 6.18a, the radar moves from u₁ to u₂, then from u₂ to u = 0. The slant range r decreases from R₁ to R₂ = R_c and then to R₀. When the radar moves from u = 0 to u₃, the slant range r increases from R₀ to R₃. The radar positions u₁, u₂, and u₃, can be represented as follows:

The corresponding slant range R₁, R₂ (=R_c), and R₃ can be represented as

In the case of θ_q > θ_H/2, as shown in Fig. 6.18b, Eq. (6.11b) remains the same, but u₃ in Eq. (6.11a) changes to become

In the broadside case, we have shown that the Doppler frequency f_D and the related parameters can be expressed in terms of the radar position u, slow time s, or slant range r. Those equations used in the broadside case are now extended to the squint angle case by factoring in Eq. (6.11). Since u = sV, the radar position u will be dropped in the following discussion.

The small-squint-angle SAR and the low-squint-angle SAR are considered separately in the following sections to derive the Doppler frequency f_D and other associated parameters.

6.3.2.1 SAR with a Small Squint Angle.

Figure 6.18a serves as a small-squint-angle SAR that satisfies θ_q ≤ 0.5θ_H. The Doppler frequency and related parameters f_D, , and B_D can be derived as follows:

1. When radar moves from s₁ to s₃, from Eq. (6.7), the Doppler frequency and the change rate of f_D with respect to the slow time s can be derived as

   

   From Eq. (6.10), the Doppler frequency and the change rate or slope of f_D with respect to slant range r can be derived as follows:

   

   Although f_Ds and f_Dr appear to differ, one can observe that f_Ds = f_Dr = f_D, but Notice that is dependent on sgn (u), but is not.

2. When s = 0, r = R₀; therefore

   

   and

   

3. The cenroid of the Doppler frequency f_Dc occurs when s = s₂, r = r₂ = R_c, and θ_u = θ_q:

   

   

   Here, sgn (u₂) = −1 is applied, as can be seen from Fig. 6.18a. Therefore, the center of the Doppler frequency f_Dc = f_Dcs = f_Dcr.

4. The upper bound of the Doppler frequency occurs at s = s₁ and r = R₁, and the lower bound of the Doppler frequency occurs at s = s₃ and r = R₃; therefore

   

   and

   

   Here, sgn (u₁) = − 1 and sgn (u₃) = 1 are applied as can be seen from Fig. 6.18a. Therefore the upper bound and the lower bound of f_D satisfy the relation f_DU = f_DUs = f_DUr and f_DL = f_DLs = f_DLr.

5. The bandwidth of the Doppler frequency is then

   

   For a small beamwidth θ_H, cos(0.5θ_H) 1, sin(0.5θ_H) 0.5θ_H, and θ_H = λ/L; therefore

   

   In the case where the small radar beamwidth is used and the squint angle is zero, the bandwidth B_D of Eq. (6.15b) becomes identical to that of the broadside case as shown in Eq. (6.10e).

The synthetic aperture length Ls can be computed as

Ls = R₁ sin(0.5θ_H + θ_q) + R₃ sin(0.5θ_H − θ_q).

The pulse duration time is computed as follows:

The correlation between Doppler frequency and slow time s and slant range r is shown in Fig. 6.19. Both plots are based on the small-squint-angle SAR with a relatively large radar beamwidth under a single-target situation for illustration purposes. The following parameters are used to generate the drawings: V = 300 m/s, L × W = 1 × 1 m, f_c = 2.0 GHz, R₀ = 10,000 m, and θ_q = 3° and θ_H = λ/L = 0.15 radian or 8.59°.

Figure 6.19a shows the Doppler frequency distribution in terms of the slow time s, which decreases linearly from slow times s₁ to s₃. The linear relation reflects the linear FM characteristics under the squint-mode radar in terms of slow time s. Figure 6.19b, based on Eq. (6.12c), illustrates Doppler Frequency distribution in terms of slant range r and appears to be a hyperbolic format. The two plots shown in Fig. 6.19 share the same parameters of the Doppler frequency spectrum, namely, the same f_DU, f_DL, f_Dc, and B_D values.

FIGURE 6.19 Small θ_q Doppler frequency versus slow time s (a) and slant range r (b).

6.3.2.2 SAR with a Low Squint Angle.

A low-squint-angle SAR that satisfies 0.5θ_H < θ_q <10° is shown in Fig. 6.18b. For such a configuration, the Doppler frequency and related parameters f_D, , and B_D can be derived as follows:

1. At any radar position u, where u₁ < u < u₃ and R₁ < r < R₃, one obtains

   

   

   Equations (6.16a) and (6.16b) show that f_Ds = f_Dr = f_D, but Both f_D and are positive because the value u, shown in Fig. 6.18b, is always negative.

2. The centroid of the Doppler frequency, which occurs at s = s₂ and r = R₂ = R_c, can be computed as follows:

   

   Therefore f_Dcs = f_Dcr = f_Dc.

3. The upper bound of the Doppler frequency occurs at s = s₁ and r = R₁ and the lower bound, at s = s₃ and r = R₃; therefore

   

   and

   

   Therefore f_DUu = f_DUr = f_DU and f_DLu = f_DLr = f_DL.

4. The bandwidth of the Doppler frequency can be computed as follows:

   

   Comparing Eqs. (6.15a) with (6.17a), one can see that the Doppler frequency bandwidth B_D is identical for both small- and low-squint-angle SAR radars.

For small radar beamwidth θ_H, the following approximations can be derived:

Again, the bandwidth B_D shown in Eq. (6.17b) is identical to that of Eq. (6.15b).

The synthetic aperture length, or equivalently the beamwidth, can be computed as

Ls = R₁ sin(θ_q + 0.5θ_H) − R₃ sin(θ_q − 0.5θ_H)

and the pulse duration time, as T_a = Ls/V.

Figure 6.20 displays two drawings of the Doppler frequency based on the low-squint-angle SAR with a relatively small radar beamwidth. Figure 6.20a is plotted in terms of slow time s and Fig. 6.20b, in terms of slant range r. A single target is assumed, and the following parameters are used to generate the drawings: V = 300 m/s, L × W = 1 × 1 m, f_c = 6 GHz, R₀ = 10,000 m, and θ_q = 9° and θ_H = λ/L = 0.05 radian or 2.865°.

Figure 6.20a, based on Eq. (6.16a), shows that the Doppler frequency linearly decreases from slow times s₁ to s₃. Figure 6.20b, based on Eq. (6.16b), also appears to be in linear format. This is because the slant range R₀ is a large value relative to the movement of radar. Unlike the f_D shown in Fig. 6.16, where both positive and negative f_D exist, only positive f_D exist for both plots of Fig. 6.20. The Doppler centroid f_Dc is located in the middle of the bandwidth of f_D. Again, the two plots shown in Fig. 6.20 share the same parameters of f_D, namely, f_DU, f_DL, and B_D.

FIGURE 6.20 Low θ_q Doppler frequency versus slow time s (a) and slant range r (b).

For comparison purposes, three drawings of the Doppler frequency are displayed in Fig. 6.21. The drawings are in terms of the slant range r and squint angles θ_q. Figure 6.21a corresponds to the broadside SAR; Fig. 6.21b, a small-squint-angle SAR; Fig. 6.21c, a large-squint-angle SAR. The parameters used to generate the drawings are listed below:

Radar carrier frequency f_c = 2 GHz; (wavelength λ = 0.15 m)

Antenna array size L × W = 1 × 1 m

Radar beamwidth θ_H = 8.6°

Small squint angle θ_q = 3°

Large squint angle θ_q = 15°

Shortest target–radar distance, R₀ = 10,000 m

Radar direction of movement (orientation) = from left to right

Radar speed (velocity) V = 300 m/s

FIGURE 6.21 Comparison of Doppler frequencies for different SAR systems.

The key Doppler frequency–related parameters for the three cases shown in Fig. 6.21 can be computed and are listed below:

Although the three cases from Fig. 6.21 share the same radar beamwidth θ_H = 8.6°, the frequency bandwidth is different in all three cases. Since the Doppler frequency bandwidth is squint-angle-dependent, the larger the squint angle θ_q, the smaller the Doppler frequency bandwidth B_D becomes.

Figure 6.22 displays the relationship of the Doppler frequency of a forward-looking squint SAR with respect to radar positions u. Figure 6.22a shows the geometry of a squint radar with three ground targets located at (u₁, R₀₁), (u₂, R₀₂), and (u₃, R₀₃), respectively, with R₀₁ = R₀₂ = R₀ and u₃ = 0. The slant ranges between radar and targets 1, 2, and 3 are r₁, r₂, and r₃, respectively.

Figure 6.22b shows that the Doppler frequency f_D varies linearly along the u axis. The two lines on the left, which correspond to targets 1 and 2, have the same slope. The third line has a steeper slope and is related to target 3. The centers of all three lines are identical, and serve as the centroid of Doppler frequency f_Dc. The intersections of the three lines with the u axis are u_a, u_b, and u_c, which are the radar positions when the targets are under the center beam of the radar. Because of the nonzero squint angle, u_a, u_b, and u_c are different from u₁, u₂, and u₃. The Doppler frequency bandwidths of all three lines are identical. Targets 1 and 2 have the same synthetic aperture length, while target 3 has a shorter one.

FIGURE 6.22 (a) Multiple-target squint SAR system; (b) plot of Doppler frequency f_D versus radar displacement u.

6.4 RANGE MIGRATION AND CURVATURE

To help the reader better understand range migration and curvature, Fig. 6.23 is presented as a simplified version of Fig. 6.18. Notice that radar position u is at the left side of the point of origin and is a negative value. Leting θ = π/2 − θ_q, one can obtain the following equation:

Therefore

FIGURE 6.23 A simplified single-target squint SAR system.

Ignoring the higher-order items following the square bracket, Eq. (6.18) becomes

The range migration ΔR is defined as the difference between R(u) and R_c, that is, ΔR = R(u)–R_c. The linear part of Eq. (6.19) is called range walk, and the quadratic part of Eq. (6.19) is called range curvature. Because |u − u_c|≤ Ls/2, the maximum range migration can be obtained from Eq. (6.19) as follows:

Given a range resolution ΔR_r, in general, there will be no range migration correction required if ΔR_max is less than ΔR_r/4. Therefore, the criterion for determining whether range migration correction is required, is to check whether

FIGURE 6.24 Single-target trajectory in squint SAR system.

For the broadside mode, θ_q = 0 and R_c = R₀, this criterion becomes

Figure 6.24 is a graphical representation of a point target range migration under the squint SAR. Figure 6.24a shows slant range R(u) plotted against radar position u. The two slant range points R₁ and R₃ are the start- and endpoints where the target is under the radar's illumination. R₂ is the point at which the center beam of the radar is on the target, which is normally represented as R₂ = R_c. The corresponding radar positions are u₁, u₂, and u₃. The difference between R₁ and R₃ is the total range migration. The distance between the curve R(u) and the line linking R₁ and R₃ is the range curvature. The vertical dashed line represents the range cell lines; the horizontal dashed lines, the radar positions or azimuth cell lines. The intersection points of these two lines are the radar image data points. R(u) represents a single-target trajectory seen by the radar through the synthetic aperture length. It migrates from range cell R₁ to range cell R₂, then stops at range cell R₃. The total range migration in this example is 5 range cells, which will cause distortion of the image if not corrected.

Figure 6.24b shows the corresponding range migration in the Doppler frequency domain, namely, the target trajectory R(f_D) plotted against the slant range r. It illustrates the Doppler frequency migration through different range cells. The vertical dashed lines denote the range cell lines; the horizontal dashed lines, the Doppler frequency bin lines. R(f_D) migrates from f_DU at range cell R₁ to f_Dc at range cell R_c, then stops at f_DL at range cell R₃. The total Doppler frequency migration in this example is 5 range cells.

In general, the range migration is relatively small for airborne-based platforms; yet for satellite-based platforms, the range migration is severe. The requirement for range migration correction depends on the choice of radar platforms, the methods used to process the radar image, and the waveform of radar (frequency bandwidth and beamwidth of the transmitted signal).

Imaging radar records the phase and polarization of the reflected signal. The radar image consists of pixels, which represent the radar backscatter from the ground targets. Darker areas represent low backscatter, while bright areas represent high backscatter. The radar images come with certain geometric distortions, such as slant range to ground distortion, image layover, and shadowing. This is discussed in the next section.

6.5 GEOMETRIC DISTORTIONS OF THE RADAR IMAGE

Because the image radar measures the distance to features in slant range rather than the true horizontal distance along the ground, this results in a varying image scale. Consider the example shown in Fig. 6.25, where the radar maps the three features with different height and shapes on the ground. The three features, described and labeled as a, b, c,..., j, are mapped and appear as b′, a′, c′,..., j′, to the radar image plane. On the basis of Fig. 6.25, the geometric distortions of radar image are described below.

6.5.1 Layover

Layover occurs when the radar beam reaches the top of a tall feature (point b in Fig. 6.25) before it reaches the base (point a in Fig. 6.25). The return signal from the top of the feature will be received before the signal from the bottom will. As a result, the top of the feature is displaced toward the radar from its true position on the ground, and “lays over” the base of the feature (b' to a').

FIGURE 6.25 Geometric distortions of radar image.

6.5.2 Foreshortening

Foreshortening occurs when the radar beam reaches the base of a tall feature tilted toward the radar (e.g., a mountain) before it reaches the top. Because the radar measures distance in slant range, the slope (d to e) will appear compressed and the length of the slope will be represented incorrectly (d′ to e′).

6.5.3 Shadowing

The shadowing effect occurs when the radar beam cannot reach part of a tall feature (h to i). In the image plane, h to i will appear as dark. In addition, any target between i and j will also be shown as dark.

In addition to the distortions mentioned above other distortions can occur in the radar image, and they are described below.

6.5.4 Slant-to-Ground Range Distortion

The slant range is the distance seen by radar to the point target and is represented by R_s. The ground range distance is the horizontal distance along the ground corresponding to the point target measured in the slant range and is represented as R_g. The relationship between R_s and R_g is R_g sinθ = R_s, where θ is the radar incident angle. Features of the radar image displayed in slant range must be converted into ground range to reflect their real-world positions relative to one another.

6.5.5 Speckle

Speckle appears as a grainy “salt and pepper” texture in an image. This is caused by random interference from the multiple scattering returns that occur within each resolution cell and produce random bright and dark areas in the radar image. In order to clearly identify objects within an image, it may be necessary to clean up the image through speckle reduction. Speckle reduction can be achieved in either of two ways:

1. By multilook processing—multiple measurements of an object's (pixels) backscatter from different locations
2. By spatial filtering—calculating the average (often median) value of adjacent pixels to average out speckle

6.6 RADAR IMAGE RESOLUTION

The resolution of radar image consists of both range and angular resolutions. A resolution cell in the slant range is formed by ΔR_s by ΔR_a, while on the ground it is formed by ΔR_g by ΔR_a. Figure 6.26 displays the resolution cell in both slant range and on the ground.

FIGURE 6.26 The resolution cell of a side-looking radar.

The term resolution cell refers to an area of the image, while a pixel (picture element) corresponds to the location of a digital sample in an image. Normally, at least 2 pixels per range resolution are needed. This is similarly true for angular resolution.

The range resolution of imaging radar is determined by the ability of the radar to distinguish between two point targets on the ground in the range direction. For real aperture radar (RAR), it is dictated by the time duration of the radar pulse T_p, and the angle of incidence θ, such that two targets on the ground can be distinguished only if they are separated by more than one pulsewidth.

For pulse-based RAR, the resolution in slant range depends on the transmitted pulse duration T_p or on the frequency bandwidth B of the pulse:

The factor originates from the two-way travel of the radar signal. The corresponding ground range resolution is

or

Along the flight track or azimuth direction, RAR resolution corresponds to the size of the antenna footprint on the ground. From Fig. 6.26, the footprint on the ground is R_sθ_H, where R_s is the slant range from the radar to the center of the footprint and θ_H is the 3-dB beamwidth with θ_H = λ/L. The RAR angular resolution in the azimuth direction is then

6.6.1 Example of Real Aperture Radar (RAR) Resolution: ERS-1/2-Imaging Radars

Major specifications of ERS-1 and ERS-2 radars are:

Pulse duration time T_p: 0.0371 ms

Average angle of incidence θ: 20°

Signal wavelength λ: 0.056 m

Mean range to a target on the earth R_s: 850 km

Radar moving speed V: 7km/s

Antenna size L × W: 10 × 1 m

Without considering the LFM signal for radar pulse, the ERS-1/2 radars yield the following results:

Now consider linear frequency modulation. The ERS-1/2 satellite radars utilize the LFM signal with frequency bandwidth at 15.46 MHz and compressed pulse duration T_p = 1/B. The slant and ground range resolutions can be computed as

This is much better than the pure pulse case of ΔR_sr = 5.56 km and ΔR_gr = 16 km.

Equation (6.24) shows that the azimuth resolution can be improved if the length of the antenna L increases. A synthetic aperture radar with a small physical antenna achieves a high resolution by synthesizing a long length of antenna. A long length of synthesized aperture corresponds to wider bandwidth of Doppler frequency, which in turn improves the azimuth resolution.

Computation of the Doppler frequency bandwidth requires computing the synthetic aperture length Ls and the upper and lower bounds of the Doppler frequency f_DU and f_DL.

The synthetic aperture length Ls equals the product of a 3-dB beamwidth multiplied by slant range R_s:

Consider the broadside SAR case. The upper bound of the Doppler frequency f_DU can be obtained as follows:

Similarly, the lower bound of the Doppler frequency f_DL can be obtained as

The bandwidth of the Doppler frequency is

The along-track spatial resolution is the product of the synthesized LFM pulse duration T_a and the relative velocity of the flying vehicle; that is

where T_a = 1/B_D. Therefore

For an antenna length equal to 10 m, the azimuth resolution becomes 5 m, which is much smaller than the RAR case of 4.76 km.

The cross-range resolution for SAR is independent of the slant range R_s. Equation (6.26b) states that an arbitrarily high resolution can be obtained by using a shorter antenna. However, smaller radar antennas output less power, which reduce the target signal-to-noise power ratio. Therefore, a tradeoff between the antenna length and cross-range resolution must be made in designing the SAR antenna.

The criteria used to check whether range migration correction is required will now be applied on ERS-1/2 radars.

Given the wavelength λ = 0.056 m, the satellite–earth distance R₀ = 850 km, the antenna length L = 10 m, and—assuming that the squint angle θ_q = 0°—the maximum range migration amount can be computed as follows, from Eq. (6.20):

From Eq. (6.22), no range migration correction is required if

However, the slant range resolution ΔR_sr/4 = 2.425 m, from Eq. (6.25), is less than (R₀λ²/8L²) = 3.333 m. Therefore, range migration correction is required.