Microwave
remote-sensing systems distinguish between different subjects primarily by the
differences in the signal strength received by the radar [Ulaby, Moore, and
Fung 1986b]. Therefore, the received signal strength is the
most important measurement made by a radar used for remote sensing. To
distinguish among different targets, measurements of angle and distance to a
target are made by recording the arrival times of the received signals. In
addition to this angular dependence, the measured amplitudes depend upon the
statistics of the received signals that result from relative motion between the
sensor and the target. In the case of an imaging radar, the received signals
have a *fading characteristic* because the Doppler frequency shift
associated with each point scatterer on the ground is different, (i.e. the
relative speed between each scatterer and the radar is different).

The fundamental relationship between the radar system parameters, the received signal, and the target is called the

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 written
as

_{
}.
(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.

- Many point scatterers must exist in the unit areas (
_{ }or_{ }) over which_{ },_{ }, and_{ }are nearly constant.

- Many more scatterers must exist in the total illuminated area at any particular instant.

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).

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

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.

_{
}