One wavelength () is the distance from one crest to the next crest. The frequency (f) of the wave is the number of crests to travel one wavelength per unit time. The propagation velocity (or phase velocity) of the wave is dependent on the medium through which it is traveling. It is calculated as:

where and are the permeability and permittivity, respectively, of the medium. These are inherent electromagnetic qualities specific to different types of matter. The lowest values these can have are ₀ and ₀ which occur for a vacuum (free space). Thus the maximum velocity of light is within a vacuum and is equal to:

Air is so diffuse that within it the velocity of an EM wave can still be considered to be c. The wavelength and frequency are related to the velocity by the equation:

where v is the velocity in metres per second, f is the frequency in Hertz (cycles per second) and is the wavelength in metres.

The ratio of the electric field intensity E to the magnetic field intensity B at any moment is always equal to the speed of light in a vacuum.

The polarization of the EM wave refers to the direction of the electric field vector. In radar science, the wave is said to be polarized in the direction of its E vector.

Principles of Real Aperture Radar

The SLAR imaging system uses a long, straight antenna mounted on a platform (aircraft or satellite), with its longitudinal axis parallel to the flight path. The antenna emits pulses of electromagnetic energy directed perpendicular to the flight path of the platform and downward to the surface of the Earth, as illustrated in Figure 3. These pulses fall within a narrow area on the ground and are scattered, usually in many directions, including the direction of the antenna. The return echoes arrive at the antenna at different times, depending on the distance from the antenna to the specific scattering object on the ground.

Range Resolution

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 çp 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

Principles of Synthetic Aperture Radar

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 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:

But for v << c, v² / c² approaches 0. Therefore

This frequency is different from the actual frequency emitted from the antenna by the amount

Since the return echo will be shifted by the same amount, the Doppler frequency shift for the target is

This value provides a means of determining exactly where the echo signal came from. For a return signal detected by the antenna at a time corresponding to the slant range R (t = 2R/c) and with a Doppler frequency shift of f_d, the azimuth coordinate must be

Even if another target is at range R and within the beam at the same time, measurement of its Doppler frequency shift f_d still allows us to associate it with an azimuth coordinate x₂. Thus for any target point, we now have two coordinates to localize it--the ground range R_g, relative to the nadir line, and the azimuth distance x₂ relative to broadside.

SAR Azimuth Resolution

where f is the resolution of the Doppler frequency shift--approximately equal to the inverse of the time during which the point target was in the beam, f_d 1 / t_span. This time span can be calculated as

Therefore the azimuth resolution is

which implies that better resolution is obtained with a smaller antenna--not what one would expect since this is opposite to the rule for real aperture radar. This does not mean, of course, that one could simply build an antenna 1 cm long to obtain a resolution of 5 mm. The length of the aperture must be large enough to create the proper interference pattern between the dipoles of the antenna necessary for the desired beamspread at a particular frequency-- _H = / L_a.

This result of _x = L_a / 2 is not strictly true, however, since we assumed that a constant Doppler frequency shift was observed during the entire time t_span. In fact, f_d changes throughout the observation and is only approximately constant for a time much less than t_span. A Fourier analysis of the Doppler waveform results in frequency components at various frequencies so that the target is observed to spread out over multiple resolution cells, covering a distance greater than the resolution x calculated above.

SAR Azimuth Compression

An airborne SAR has an altitude low enough to consider the Earth as a flat surface. The hyperbola will be very flat since the range differences for one target are relatively small. When operating from a satellite, however, the altitude is high enough that the curvature of the Earth becomes a factor, extending the hyperbola. Also, the rotation of the Earth warps it to one side in the range direction. These effects must be corrected before the compression processing can begin.

When two waves are not synchronous with each other, they are said to be out of phase. The phase difference is measured in radians or degrees. For instance, if the wave A's crest occurs at the same time as wave B's trough, the phase difference is (or 180º). Figure 8 illustrates further examples.

The phase difference, in radians, between the transmitted and received signal will be

where Figure 9, shows that the range is

Using the Taylor series to expand this around the position x_c for the slant range R_c gives

when truncated to the second order. Assuming that R_c and R₀ are approximately equal for narrow beam radars, the Doppler frequency shift is

By assuming that the Doppler frequency shift is constant only until the quadratic term adds a value of / 4 to , then the window for observing the waveform is confined to a distance of x_window, where

and so

Therefore the time window is

Therefore, the new azimuth resolution is

A SAR processor which uses this technique is known as unfocused SAR. This technique does not account for the variable rate of phase change, but still manages to produce a resolution much smaller than that for real aperture radars. For example, using the above equation, the resolution for RADARSAT would be

as opposed to the real aperture limit of 3.0 km.

The resolution can get even better, however, by using a focused SAR processor which accounts for the nonlinear phase change. This technique uses all the data collected during the time that the target is within the beam. The quadratic phase is adjusted such that all the return signals due to the target at x₀ (see Figure 9) are added coherently. Any returns from targets not at x₀ will not agree with the adjusted returns so they will cancel. Thus the returns from the target at x₀ will dominate the returns from other targets with the same range but not at x₀. With a few assumptions, this technique results in an azimuth resolution approaching L_a / 2, which is what we calculated before, assuming a constant phase.

Geometric Distortions

Slant vs. Ground Range

Suppose you are imaging objects A and B, separated a distance on the ground, as shown in Figure 11. Because the SAR is viewing from an angle, the two points will look closer together than they actually are. Since angle ACB is approximately 90º, we can say that D_g in slant range presentation is

so that the distance in slant range presentation is always less than in the corresponding ground range projection. As the depression angle gets larger, D_s gets smaller for a given distance D_g. Thus the slant range presentation compresses the terrain features in the near range more than in the far range. This has the effect of greatly stretching the ground range projection near the nadir where approaches 90º.

The ground range presentation is used when one wishes to match the image with known target locations (i.e. geocoding the image) on the ground. EarthView is able to convert an image from slant range to ground range and visa versa, taking into account the curvature of the Earth for satellite imagery.

Relief Displacement

The relief displacement r_g is equal to the distance , however for a small enough value of h, we see that P_gX P'X. Thus we can estimate the relief displacement in ground presentation as

and in slant range presentation as

Thus the relief displacement is proportional to the height of the vertical structure and increases the closer that structure is to the nadir line.

Lay Over

Shadows

in ground range presentation and

in slant range presentation. Thus the length of the shadow is proportional to the height of the vertical structure and increases the further away that structure is from the nadir line or the lower the SAR antenna is.

Image of a Slope

Slopes facing away from the nadir line can be (a) fully imaged, (c) hidden in the radar shadow or (b) strangely imaged as the radar beam just skims along the surface. For case (b), slight irregularities on the slope's surface would have magnified effects. For instance, a bump near the peak would block the radar from the rest of the slope causing a shadow to appear in the image.

All of these slope effects can be found in Figure 15--a subsection of the SAR image 82LWERS1 ( ESA) from the ERS-1 satellite, provided with EarthView.

Surface Interaction with the Radar Beam

Assuming that the EM wave is traveling through a medium that is homogeneous (properties do not vary from point to point), isotropic (properties are the same in all directions from any given point) and non-magnetic, then from Maxwell's equations we can express the complex electric field vector as

where A is the amplitude vector, z is the distance traveled in time t and is a phase offset. The angular frequency is

where f_c is the carrier frequency (the frequency emitted from the antenna) and is the wavelength. The expression for k' is

where k is the wavenumber and _r is the relative permittivity (also called the dielectric constant) of the medium (_r = / ₀ where is the permittivity of the medium and ₀ is the permittivity of free space). It should be noted that _r is a function of the frequency of the EM wave. The relative permeability _r has been assumed to be equal to 1 which is accurate enough for microwave frequencies.

The polarization of the electric field E is the direction of the amplitude vector A at some instant in time. For a linearly polarized wave, the direction of A is fixed in one plane relative to the direction of propagation. An elliptically polarized wave has an electric field consisting of two different components oriented along orthogonal axes such that

resulting in the total amplitude vector A slowly rotating about the z-axis. If A_x = A_y and [₁ - ₂] = /2, then the wave becomes circularly polarized.

An EM wave interacting with a surface is called scattering. There are two types—surface scattering and volume scattering. Surface scattering occurs at the interface between two different homogeneous media such as the atmosphere and the Earth's surface. Volume scattering is the result of interaction with particles within a non-homogeneous medium.

Surface Scattering

where h is the surface roughness, defined as the root-mean-square (rms) height relative to a perfectly smooth surface, is the wavelength of the wave and is the angle of incidence. Peake and Oliver (1971) modified this criterion by defining upper and lower values of h for surfaces of intermediate roughness to obtain a smoothness criterion of

Using the Rayleigh criterion, for instance, when RADARSAT is operating at 5.3 GHz with an incidence angle of 50º, level terrain with h < 9 mm will show up as solid black.

Specular Scattering

where n and n' are the index of refractions of the two mediums, is the angle of incidence and ' is the angle of refraction, shown in Figure 16. However, the index of refraction is equal to

For a non-magnetic medium, _r = 1, and for air, _r 1. Therefore Snell's Law becomes

For the perfect specular reflector, radar returns (backscatter) exist only near vertical incidence (due to a 90º depression angle or the slope of the surface) and the reflected energy is localized to a small angular region around the angle of reflection. Even for non-vertical incidence, however, backscatter can exist for a rough subsurface if the radar is capable of penetrating deep enough. Figure 16 illustrates how it is possible to image a subsurface (i.e. the surface of a different medium below the top surface) that would otherwise be invisible to optical sensors.

Bragg Scattering

where the dominant return is due to the wavelength for which n=1. At steep incidence angles, the backscatter is usually a combination of both specular and Bragg scattering.

The Bragg model is useful for describing the backscatter from the ocean surface. Water has a large dielectric constant making volume scattering impossible (see section 2.5.2). The periodic structure of ocean waves resonate well with radar waves, showing up as periodic bands on the image. When imaging slightly rough terrain with little vegetation, the Bragg model can be used to classify rock types and ages based on the surface roughness.

Volume Scattering

where the relative dielectric constant is given by the complex number _r = ' + i'' and ''/' must be less than 0.1 for the equation to be valid.

Since every type of material has its own characteristic dielectric constant _r, the media are dielectrically variant which results in volume scattering. If the different media are mixed together in a mostly random manner, the backscatter will be in many directions. The fraction of the incident wave that is reflected back to the antenna, then, depends on the shape, density and orientation of the different media as well as on the relative permittivity between them.

Complex Dielectric Constant

Since the dielectric constant depends on the frequency of the EM wave, so does the reflectivity. The higher the frequency (or the lower the wavelength), the lesser the penetration is. If one is imaging a forest, for example, using a high frequency will result in surface scatter from the top of the vegetation canopy while a lower frequency would allow the EM waves to penetrate the canopy, producing volume scattering from the leaves, branches, trunks or ground.

Polarization

Using a filter at the antenna, all polarizations can be screened out except for the desired polarization--either horizontal or vertical. The HH (transmit and receive horizontal) configuration is called the like-polarized return while the HV (transmit horizontal and receive vertical) configuration is called the cross-polarized return. The HV return requires a much higher antenna gain than the HH due to its much smaller energy.

The images produced by these two returns may differ due to the differences between the scattering processes for each. Depolarization, for instance, is usually explained in terms of volume scatter or multiple reflections. At short wavelengths, when the terrain can be considered very rough, like- and cross-polarized images are almost identical. At longer wavelengths, however, when the terrain is considered relatively smooth, noticeable differences stand out. This has resulted in limited geological application in identifying rock types.

Some of the new polarimetric SARs are capable of transmitting and receiving both horizontally and vertically thus allowing HH, HV, VH and VV returns all at the same time. Mathematical analysis of these returns can create a geometric basis which can be used to synthesize the image for any possible transmit/receive polarization for the entire 360º spectrum. Certain objects, especially those man-made, have been found to stand out at a specific transmit/receive polarization, thus making their detection easier in the future.

Speckle

A single resolution cell, or pixel, corresponds to an area of a certain size on the ground. Within that area, there exist many different targets, or scatterers, which provide the surface texture necessary for backscatter. Even though the area may be something on the order of 15 x 15 metres, the wavelength of the EM wave may be only several centimetres--capable of interacting with each individual scatterer. These multiple interactions produce echoes which interfere with each other in either a constructive or destructive manner. Constructive interference results in a strong return signal and a bright pixel in the image. Destructive interference results in a weak return signal and dark pixel in the image.