CHAPTER 1

SYNTHETIC APERTURE RADAR (SAR) IMAGING BASICS

The word radar is an acronym for radio detection and ranging. A radar measures the distance, or range, to an object by transmitting an electromagnetic signal to and receiving an echo reflected from the object. Since electromagnetic waves propagate at the speed of light, one only has to measure the time it takes the radar signal to propagate to the object and back to calculate the range to the object. The total distance traveled by the signal is twice the distance between the radar and the object, since the signal travels from the radar to the object and then back from the object to the radar after reflection. Therefore, once we measured the propagation time (t), we can easily calculate the range (R) as

(1-1) Numbered Display Equation

where c is the speed of light in a vacuum. The factor ½ accounts for the fact that the radar signal actually traveled twice the distance measured: first from the radar to the object and then from the object to the radar. If the electric property of the propagation medium is different from that of the vacuum, the actual propagation velocity has to be estimated for advanced radar techniques, such as synthetic aperture radar (SAR) interferometry.

Radars provide their own signals to detect the presence of objects. Therefore, radars are known as active, remote-sensing instruments. Because radars provide their own signal, they can operate during day or night. In addition, radar signals typically penetrate clouds and rain, which means that radar images can be acquired not only during day or night but also under (almost) all weather conditions. For these reasons, radars are often referred to as all-weather instruments. Imaging, remote-sensing radars, such as SAR, produce high-resolution (from submeter to a few tens of meters) images of surfaces. The geophysical information can be derived from these high-resolution images by using proper postprocessing techniques.

This book focuses on a specific class of implementation of synthetic aperture radar with particular emphasis on the use of polarization to infer the geophysical properties of the scene. As mentioned above, SAR is a way to achieve high-resolution images using radio waves. We shall first describe the basics of radar imaging. This shall be followed by a description of the synthetic aperture principle. Finally, we shall discuss some advanced SAR implementations, such as SAR polarimetry and polarimetric SAR interferometry.

1.1 BASIC PRINCIPLES OF RADAR IMAGING

Imaging radars generate surface images that are at first glance very similar to the more familiar images produced by instruments that operate in the visible or infrared parts of the electromagnetic spectrum. However, the principle behind the image generation is fundamentally different in the two cases. Visible and infrared sensors use a lens or mirror system to project the radiation from the scene on a two-dimensional array of detectors, which could be an electronic array or, in earlier remote-sensing instruments, a film using chemical processes. The two-dimensionality can also be achieved by using scanning systems or by moving a single line array of detectors. This imaging approach—an approach with which we are all familiar from taking photographs with a camera—conserves the relative angular relationships between objects in the scene and their images in the focal plane, as shown in Figure 1-1. Because of this conservation of angular relationships, the resolution of the images depends on how far away the camera is from the scene it is imaging. The closer the camera, the higher the resolution and the smaller the details that can be recognized in the images. As the camera moves further away from the scene, the resolution degrades and only larger objects can be discerned in the image.

Figure 1-1 Passive imaging systems conserve the angular relationships between objects in the scene and their images in the focal plane of the instrument

Imaging radars use a quite different mechanism to generate images, with the result that the image characteristics are also quite different from those of visible and infrared images. There are two different mechanisms by which radars can be used to produce images; the two types of radars are broadly classified as real aperture and synthetic aperture radars. We shall discuss the differences between these two types in more detail later in this chapter.

Radar images are typically acquired in strips as the satellite or aircraft carrying the radar system moves along its flight path. These strips are often referred to as swaths or tracks. To separate objects in the cross-track direction and the along-track direction within a radar image, two different methods must be implemented. The cross-track direction, also known as the range direction in radar imaging, is the direction perpendicular to the direction in which the imaging platform is moving. In this direction, radar echoes are separated using the time delay between the echoes that are backscattered from the different surface elements. This is true for both real aperture and synthetic aperture radar imagers. The along-track direction, also known as the azimuth direction, is the direction parallel to the movement of the imaging platform. The angular size (in the case of the real aperture radar) or the Doppler history (in the case of the synthetic aperture radar) is used to separate surface pixels in the along-track dimension in the radar images. As we will see later, only the azimuth imaging mechanism of real aperture radars is similar to that of regular cameras. Using the time delay and Doppler history results, SAR images have resolutions that are independent of how far away the radar is from the scene it is imaging. This fundamental advantage enables high-resolution, spaceborne SAR without requiring an extremely large antenna.

Another difference between images acquired by cameras operating in the visible and near-infrared part of the electromagnetic spectrum and radar images is the way in which they are acquired. Cameras typically look straight down, or at least have no fundamental limitation that prevents them from taking pictures looking straight down from the spacecraft or aircraft. Not so for imaging radars. To avoid so-called ambiguities, which we will discuss in more detail later, the imaging radar sensor has to use an antenna that illuminates the surface to one side of the flight track. Usually, the antenna has a fan beam that illuminates a highly elongated, elliptically shaped area on the surface, as shown in Figure 1-2. The illuminated area across track generally defines the image swath.

Figure 1-2 Imaging geometry for a side-looking radar system

Within the illumination beam, the radar sensor transmits a very short effective pulse of electromagnetic energy. Echoes from surface points farther away along the cross-track coordinate will be received at proportionally later times (see Figure 1-2). Thus, by dividing the receive time in increments of equal time bins, the surface can be subdivided into a series of range bins. The width in the along-track direction of each range bin is equal to the antenna footprint along the track xa. As the platform moves, the sets of range bins are covered sequentially, thereby allowing strip mapping of the surface line by line. This is comparable to strip mapping with a so-called push broom imaging system using a line array in the visible and infrared part of the electromagnetic spectrum. The brightness associated with each image pixel in the radar image is proportional to the echo power contained within the corresponding time bin. As we will see later, the real difference between real aperture radars and synthetic aperture radars lies in the way in which the azimuth resolution is achieved.

This is also a good time to point out that there are two different meanings for the term range in radar imaging. The first is the so-called slant range and refers to the range along the radar line of sight, as shown in Figure 1-3. Slant ranges are measured along the line connecting the radar and the object being imaged, often called the target or the scatterer. The second use of the term range is for the ground range, which refers to the range along a smooth surface (the ground) to the scatterer. The ground range is measured from the so-called nadir track, which represents the line described by the position directly underneath the radar imaging platform. One has to be careful to take topography into account when resampling radar images from slant range to ground range. This will be discussed in more detail in Section 1.6.2.

Figure 1-3 Definition of some common radar imaging angles

Before looking at radar resolutions, let us define a few more terms commonly encountered in radar imaging. The look angle is defined as the angle between the vertical direction and the radar beam at the radar platform. The incidence angle is defined as the angle between the vertical direction and the radar wave propagation vector at the surface (as shown in Figure 1-3). When surface curvature effects are neglected, the look angle is equal to the incidence angle at the surface when the surface is flat. In the case of spaceborne systems, surface curvature must be taken into account; this leads to an incidence angle that is always larger than the look angle for flat surfaces. It is quite common in the literature to find authors using the terms look angle and incidence angle interchangeably. This is only correct for low-flying aircraft, and only when there is no topography present in the scene. As we will see next, if topography is present (i.e., if the surface is not flat), the local incidence angle might vary in the radar image from pixel to pixel.

Consider the simple case of a single hill illuminated by a radar system, as is shown in Figure 1-4. Also shown in the figure are the local normal to the surface for several positions on the hill. Relative to a flat surface, it is clear that for points on the hill facing the radar, the local normal tilts more toward the radar; therefore, the local incidence angle will be smaller than for a point at the same ground range but on a flat surface.

Figure 1-4 Topographic variations in the image will cause the local incidence angle to be different from that expected for a flat surface with no relief

A term commonly encountered in the military literature is depression angle. This is the angle between the radar beam and the horizontal at the radar platform. The depression angle is, therefore, related to the look angle in that one is equal to 90° minus the other. A small look angle is equivalent to a large depression angle, and vice versa. Similarly, one often finds the term grazing angle describing the angle between the horizontal at the surface and the incident wave in the military literature. The grazing angle is, therefore, related to the incidence angle in the same way that the depression angle is related to the look angle. In this text, we shall use look angle and incidence angle to describe the imaging geometry.

1.2 RADAR RESOLUTION

The resolution of an image is defined as that separation between the two closest features that can still be resolved in the final image. First, consider two point targets that are separated in the slant range direction by xr. Because the radar waves propagate at the speed of light, the corresponding echoes will be separated by a time difference Δt equal to:

(1.2-1) Numbered Display Equation

where c is the speed of light and the factor 2 is included to account for the signal round trip propagation, as was previously described. Radar waves are usually not transmitted continuously; instead, a radar usually transmits short bursts of energy known as radar pulses. The two features can be discriminated if the leading edge of the pulse returned from the second object is received later than the trailing edge of the pulse received from the first feature, as shown in Figure 1-5.

Figure 1-5 If the radar echoes from two point targets are separated in time by more than or equal to the length of the radar pulse, it is possible to recognize the echoes as those from two different scatterers, as shown in the top two panels. If the time difference between the echoes is less than the radar pulse length, it is not possible to recognize two distinct scatters, as in the case of the bottom panel

Therefore, the smallest separable time difference in the radar receiver is equal to the effective time length τ of the pulse. Thus, the slant range resolution of a radar is:

(1.2-2) Numbered Display Equation

Now let us consider the case of two objects separated by a distance xg on the ground. The corresponding echoes will be separated by a time difference Δt equal to:

(1.2-3) Numbered Display Equation

The angle θ in Eq. (1.2-3) is the local incidence angle. (This should actually be called the incident angle, or angle of incidence. Since incidence angle is used almost universally in the literature, we shall continue to use that term to avoid confusion.). As in the case of the slant range discussed above, the two features can be discriminated if the leading edge of the pulse returned from the second object is received later than the trailing edge of the pulse received from the first feature. Therefore, the ground range resolution of the radar is given by

(1.2-4) Numbered Display Equation

In other words, the ground range resolution is equal to half the footprint of the radar pulse on the surface. Note that Eq. (1.2-4) implies that the ground range resolution is different for different incidence angles. This also means that the local slopes in the images will affect the ground range resolution. For slopes facing the radar, the ground range resolution will be poorer than that for slopes facing away from the radar.

Occasionally, the effective pulse length is described in terms of the system bandwidth B. As we will show in the next section, to a good approximation,

(1.2-5) Numbered Display Equation

A pulsed radar determines the range by measuring the round trip time by transmitting a pulse signal. In designing the signal pattern for a radar sensor, there is usually a strong requirement to have as much energy as possible in each pulse in order to enhance the signal-to-noise ratio. This can be done by increasing the transmitted peak power or by using a longer pulse. However, the peak power is usually strongly limited by the available power sources, particularly in the case of spaceborne sensors. On the other hand, an increased pulse length leads to a worse range resolution [see Eq. (1.2-4)]. This dilemma is usually resolved by using modulated pulses, which have the property of a wide bandwidth even when the pulse is very long. After so-called pulse-compression, a short effective pulse length is generated, increasing the resolution. One way to modulate the pulse is to vary the radar signal frequency linearly while the pulse is being transmitted. This waveform is known as chirp. Although other ways of modulating pulses are occasionally used, the chirp modulation is by far the most common; we shall therefore use this scheme to illustrate how signal modulation can be used to enhance radar range resolution.

In a chirp, the signal frequency within the pulse is linearly changed as a function of time. If the frequency is linearly changed from f0 to f0 + Δf, the effective bandwidth would be equal to:

(1.2-6) Numbered Display Equation

which is independent of the pulse length τp. Thus, a pulse with long duration (i.e., high energy) and wide bandwidth (i.e., high range resolution) can be constructed. The instantaneous frequency for such a signal is given by:

(1.2-7) Numbered Display Equation

The corresponding signal amplitude is:

(1.2-8)

where inline (x) means the real part of x. Note that the instantaneous frequency is the derivative of the instantaneous phase. A pulse signal such as shown in Eq. (1.2-8) has a physical pulse length τp and a bandwidth B. The product τpB is known as the time bandwidth product of the radar system. In typical radar systems time bandwidth products of several hundreds are used.

At first glance, it might seem that using a pulse of the form shown in Eq. (1.2-8) cannot be used to separate targets that are closer than the projected physical length of the pulse (as shown in the previous section). It is indeed true that the echoes from two neighboring targets that are separated in the range direction by much less than the physical length of the signal pulse will overlap in time. If the modulated pulse and, therefore, the echoes have a constant frequency, it will not be possible to resolve the two targets. However, if the frequency is modulated as described in Eq. (1.2-7), the echoes from the two targets will have different frequencies at any instant of time and, therefore, can be separated by frequency filtering.

In actual radar systems, a matched filter is used to compress the returns from the different targets. Consider an example in which we transmit a signal of the form described in Eq. (1.2-8). The signal received from a single point scatterer at a range R is a scaled replica of the transmitted signal delayed by a time t = 2R/c. The output of the matched filter for such a point scatterer is mathematically described as the convolution of the returned signal with a replica of the transmitted signal. Being careful about the limits of the integration, one finds that for large time-bandwidth products,

(1.2-9)

where

and

This compressed pulse has a half power width of approximately 1/B, and its peak position occurs at time 2R/c. Therefore, the achievable range resolution using a modulated pulse of the kind described in Eq. (1.2-8) is a function of the chirp bandwidth and not the physical pulse length. In typical spaceborne and airborne SAR systems, physical pulse lengths of several tens of microseconds are used, while bandwidths of several tens of megahertz are no longer uncommon for spaceborne systems, and several hundreds of megahertz are common in airborne systems.

So far, we have seen the first major difference between radar imaging and that used in passive imaging systems. The cross-track resolution in the radar case is independent of the distance between the scene and the radar instrument and is a function of the system bandwidth. Before looking at the imaging mechanisms in the along-track direction, we will examine the general expression for the amount of reflected power that the radar receiver would measure. This is described through the so-called radar equation, which we will examine in the next section.

1.3 RADAR EQUATION

One of the key factors that determines the quality of the radar imagery is the corresponding signal-to-noise ratio (SNR), commonly called SNR. This is the equivalent of the brightness of a scene being photographed with a camera versus the sensitivity of the film or detector. Here, we consider the effect of thermal noise on the sensitivity of radar imaging systems. The derivation of the radar equation is graphically shown in Figure 1-6.

Figure 1-6 Schematic of the derivation of the radar equation

In addition to the target echo, the received signal also contains noise, which results from the fact that all objects at temperatures higher than absolute zero emit radiation across the whole electromagnetic spectrum. The noise component that is within the spectral bandwidth B of the sensor is passed through with the signal. The receiver electronics also generates noise that contaminates the signal. The thermal noise power is given by:

(1.3-1) Numbered Display Equation

where k is Boltzmann's constant (k = 1.6 × 10−23 W/K/Hz) and T is the total equivalent noise temperature in kelvin. The resulting SNR is then:

(1.3-2) Numbered Display Equation

Note that the noise bandwidth is usually larger than the transmit bandwidth, because of the hardware limitation.

One common way of characterizing an imaging radar sensor is to determine the surface backscatter cross section σN, which gives an SNR = 1. This is called the noise equivalent backscatter cross section. It defines the weakest surface return that can be detected and, therefore, identifies the range of surface units that can be imaged.

1.4 REAL APERTURE RADAR

The real aperture imaging radar sensor also uses an antenna that illuminates the surface to one side of the flight track. As mentioned before, the antenna usually has a fan beam that illuminates a highly elongated elliptical shaped area on the surface (see Figure 1-2). As shown in Figure 1-2, the illuminated area across track defines the image swath. For an antenna of width W operating at a wavelength λ, the beam angular width in the range plane is given by:

(1.4-1) Numbered Display Equation

and the resulting surface footprint or swath S is given by

(1.4-2) Numbered Display Equation

where h is the sensor height above the surface, θ is the angle from the center of the illumination beam to the vertical (the look angle at the center of the swath), and θr is assumed to be very small. Note that Eq. (1.4-2) ignores the curvature of the Earth. For spaceborne radars, this effect should not be ignored. If the antenna beam width is large, one needs to use the law of cosines to solve for the swath width.

A real aperture radar relies on the resolution afforded by the antenna beam in the along-track direction for imaging. This means that the resolution of a real aperture radar in the along-track direction is determined by the size of the antenna as well as the range to the scene. Assuming an antenna length of L, the antenna beam width in the along-track direction is

(1.4-3) Numbered Display Equation

At a distance R from the antenna, this means that the antenna beam width illuminates an area with the along-track dimension equal to

(1.4-4) Numbered Display Equation

To illustrate, for h = 800 km, λ = 23 cm, L = 12 m, and θ = 20°, then xa = 16 km. Even if λ is as short as 2 cm and h as low as 200 km, xa will still be equal to about 360 m, which is considered to be a relatively poor resolution, even for remote sensing. This has led to very limited use of the real-aperture technique for surface imaging, especially from space. A real aperture radar uses the same imaging mechanism as a passive optical system for the along-track direction. However, because of the small value of λ (∼1 μm) in the case of optical systems, resolutions of a few meters can be achieved from orbital altitudes with an aperture only a few tens of centimeters in size. From aircraft altitudes, however, reasonable azimuth resolutions can be achieved if higher radar frequencies (typically X-band or higher) are used. For this reason, real aperture radars are not commonly used in spaceborne remote sensing (except in the case of scatterometers and altimeters that do not need high-resolution data).

In terms of the radar equation, the area responsible for reflecting the power back to the radar is given by the physical size of the antenna illumination in the along-track direction and the projection of the pulse on the ground in the cross-track direction. This is shown in Figure 1-2 for the pulses in the radar swath. The along-track dimension of the antenna pattern is given by Eq. (1.4-4). If the pulse has a length τp in time, and the signal is incident on the ground at an angle θi, the projected length of the pulse on the ground is

(1.4-5) Numbered Display Equation

Therefore, the radar equation in the case of a real aperture radar becomes

(1.4-6) Numbered Display Equation

The normalized backscattering cross section (σ0) is defined as

(1.4-7) Numbered Display Equation

where A0 is the illuminated surface area, Er is the reflected electric field, and Ei is the incident electric field.

This shows that when a real