    Figure 2.1 Geometry of the radar equation [Ulaby, Moore, and Fung 1986b, Fig. 7.1]

The fundamental relationship between the radar system parameters, the received signal, and the target is called the radar equation. Consider the scattering from an isolated scatterer shown in Fig 2.1. When a power is transmitted by an antenna, with gain in the direction of the target, the power per unit solid angle in the direction of the target is [Ulaby, Moore, and Fung 1986b]. At the scatterer, , (2.1)

where is the power density at the scatterer. The spreading loss is the reduction in power density associated with the spreading of power over the surface of a sphere with radius centered at the antenna.

The total power intercepted by the scatterer is the product of the power density and the effective receiving area of the scatterer: . (2.2)

The actual value of depends on the effectiveness of the scatterer as a receiving antenna.
If the scatterer is a dielectric (not a perfect conductor or insulator), some of the power received is absorbed ( ), while the remainder is reradiated
( ). The total reradiated power is given by . (2.3)

The effective receiving area of the scatterer is a function of its orientation relative to the incident beam. The scatterer's reradiation pattern may not be the same as its effective receiving pattern. Also, the scatterer gain in the direction of the receiving antenna may be different from the gain of the transmitting antenna. Therefore, new quantities are necessary to describe the path between the scatterer and the receiving antenna: , . (2.4)

Here, is the total reradiated power, is the gain of the scatterer in the direction of the receiver, and is the spreading factor for the reradiation. The power entering the receiver is , (2.5a)

where is the effective aperture of the receiving antenna, not its actual area.
Combining relations in (2.1) through (2.5) yields  . (2.5b)

The factors relating to the scatterer are contained in the square brackets. They are difficult to measure individually, but fortunately their relative contributions are not required to determine the magnitude of the received radar signal. Typically they are grouped into a single term known as the radar scattering cross section , . (2.6)

Since the fractional power ( ) and gain are both dimensionless, has units of area. The cross section is a function of the propagation directions of the waves incident upon and reradiated by the scatterer, as well as scatterer shape and dielectric properties. Writing the radar equation in terms of gives . (2.7)

Equation 2.7 applies to the general case of the transmitting antenna located separately from the receiving antenna (a bistatic radar). However, a single antenna is often used to transmit and receive (a monostatic radar). In this case, the transmitter and receiver distances are the same, as well as the antenna gains and effective apertures. Therefore, . (2.8)

The effective area of an antenna can be related to its gain by , (2.9)

and the radar equation may be rewritten as . (2.10)

In environmental remote sensing applications, most targets occupy an area. Writing the radar equation for an area target is based on the assumption that the observed area may be considered as a collection of many point scatterers, where no individual target dominates the scattered signal. The amplitude and phase of the electric field received from each scatterer are added together in a phasor summation to determine the total received electric field at any instant in time. If the received power from each scatterer is averaged over the number of scatterers , the phase contributions cancel. The average received power may be writtenas . (2.11)

Now, Equation 2.10 can be rewritten for an area target, .(2.12)

An average scattering cross section per unit area is defined similarly to the average received power. The scattering cross section for an individual target is normalized by the corresponding area on the ground . For a monostatic radar, the average value of this quotient is defined to be the radar backscattering coefficient , which is given by , (2.13)

where the brackets represent a statistical average. If the target area is comprised of unit areas , such that , , and are nearly constant and each contains enough scatterers that a reasonable average may be obtained, Equation 2.12 may be rewritten by replacing with : . (2.14)

The radar equation can also be expressed as an integral if the limit for infinitesimal area is taken. In general, , , and the product may vary over the entire illuminated area, so they appear inside the integral , (2.15)

Two conditions must be satisfied whenever Equation 2.14 or 2.15 is used.

1. Many point scatterers must exist in the unit areas ( or ) over which , , and are nearly constant.
2. Many more scatterers must exist in the total illuminated area at any particular instant.

### 2.1.1.2 Imaging geometry and spatial resolution

In order to properly interpret radar data, it is necessary to account for the geometry of the acquisition. Radars used for remote sensing of terrain are almost always side-looking systems, while most optical instruments are nadir-looking. This difference exists because optical instruments are able to distinguish among targets based upon their angular distance from the sensor. However, a radar can only distinguish the returns from various targets based upon the arrival time of the received signals. A nadir-looking radar could not distinguish between two scatterers a and b (see Figure 2.2) that are equal distances from the sensor because a single incident wave front illuminates both points at the same instant, so the scattered returns from both points arrive at the receiving antenna simultaneously. This leads to an ambiguity in range for any right/left-symmetric, equidistant points. If the radar illumination is restricted to one side of the platform, the wave front illuminates the same two points at different times. Their scattered returns arrive at the sensor separated in time and are thereby distinguishable from each other. Figure 2.2 Rationale for side-looking radars: (a) shows ambiguity associated with a nadir-looking radar, while (b) shows how a side-looking geometry resolves the ambiguity

The imaging zone of a remote-sensing radar typically consists of a horizontally thin swath extending from the side of the platform and orthogonal to the direction of motion of the platform. This is because radars carried on airborne or spaceborne platforms for remote sensing typically rely on platform motion to achieve area coverage, (i.e. they are not gimbaled). Figure 2.3 illustrates the imaging geometry for a side-looking airborne radar (SLAR). Figure 2.3 SLAR imaging geometry

Using the geometry in Figure 2.3, the SLAR resolution can be expressed in terms of the system characteristics. Most remote sensing radars transmit energy in discrete pulses. For any pulse radar to resolve two separate targets in range, the targets must be separated by at least half of the pulse length, which is equal to in line-of-sight distance (slant range) or in ground distance (ground range). Here is the speed of light, is the pulse duration, and is the incidence angle [Ulaby, Moore, and Fung 1986b]. That minimum separation allows the return signal from the nearer target to be received one full pulse duration before the return signal from the more distant target is received. This minimum separation distance can be expressed in terms of the slant-range resolution as (2.16)

The size of the resolution cell on the ground, the ground-range resolution , can then be defined by projecting the slant-range resolution onto the ground plane: . (2.17)

For real-aperture SLAR systems, the azimuth (along-track) resolution is simply the arc length corresponding to the slant-range distance from the sensor . If is written in terms of the radar's elevation above the target and the incidence angle , the azimuth resolution can be written as , (2.18)

where is the horizontal antenna beam width, is the wavelength of the incident wave, and is the antenna length (see Figure 2.3). These resolutions result in conflicting constraints on SLAR systems. For a given altitude the ground-range resolution improves for larger incidence angles, while the azimuth resolution improves for smaller incidence angles. This conflict is usually avoided with the use of a Synthetic Aperture Radar (SAR), which removes the dependence of the azimuth resolution on the incidence angle.

The side-looking configuration of radar systems creates several distortions, two of which are elevation distortion and range distortion. Both distortions arise because SLAR systems measure distances along the radar's line of sight, (i.e. the slant range) [Ulaby, Moore, and Fung 1986b]. Elevation distortions occur when structures rise steeply above the mean ground level, (e.g. mountain cliffs). The tops of these structures can be closer to the radar than the bottom, and thus appear closer to the flight line in the radar image. This effect is sometimes called layover. Range distortions occur because the slant-range distance measured by the radar is not linearly related to distance measured along the ground (ground-range distance). As a result, features in the near range of the radar image are compressed in the range direction, relative to features in the far range of the image.

When analyzing radar data, it is desirable to use a parameter that has been normalized so that returns from similar terrains are approximately equal even if different radar systems are used to observe them. For this reason, is usually the parameter that corresponds to pixel brightness in radar images, and is therefore the parameter of interest. 