Here is a picture of an idealized transmitted radar signal. The electronic strength of the transmitted signal is shown on the y-axis, and the distance from the transmitter is shown on the x-axis. The signal is seen to oscillate, or exactly repeat itself over and over again along the x-axis.
If you were walking away from the transmitter, you would walk through many cycles of the repeating pattern. You would walk through a single cycle of the pattern when it repeated itself just once. A single cycle of the wave is indicated by the green line. The distance walked through a single cycle is called the wavelength, and is 2 cm in the example in the picture, represented by the blue line.
The phase of the wave is the total number of cycles of the wave at any given distance from the transmitter, including the fractional part. Therefore, the phase at any given distance from the transmitter is given by the that distance divided by the wave length. In an equation,
phase (in cycles) = distance from transmitter / wavelength (1)
For example, at the first peak of the wave (0.5 cm on the x-axis), the phase is 1/4 cycle. At the 1-cm mark, the phase is 1/2 cycle. At the 3-cm mark on the x-axis, the phase of the wave is 1.5 cycles. Frequently, one cycle of phase is said to correspond to 360 degrees, but we will stick to phase measured in cycles to describe the interferometry concept as it applies to SRTM.
The amplitude of the wave is the peak signal strength, which is 3 in the picture. In an actual transmitted signal, the strength (and the amplitude) would continually decrease as you went farther along on the x-axis, but we can safely ignore this aspect of the wave in discussing SRTM interferometry.
*NOTE: The illustrations below are not to scale.
As shown in Figure 2, when a radar signal is transmitted from the Shuttle and hits a target on the Earth, part of the signal is reflected back toward the Shuttle. A receiver on the Shuttle measures the strength of the reflected wave, and that strength, when plotted versus distance from the target, would look much like Figure 1. (Again, we will ignore the actual decrease in amplitude as you proceed away from the target).
If the Shuttle had two receivers separated by a fairly big distance (60 m in the case of SRTM), then you would have the components of an interferometer. The two receivers are said to be at the ends of the "interferometric baseline." An interferometer measures the difference in phase between two signals received at the ends of a baseline, as shown in Figure 3. The interferometer accomplishes the phase differencing by comparing or "interfering" the signals at the two ends of the baseline by a signal-processing technique called "complex cross correlation." This phase difference is called the "interferometric phase." The interferometric phase is determined by effectively subtracting the measured phase at each end of the baseline. Because each received phase depends on the distance between the receiver and the target (as in Figure 1), the interferometric phase is a measurement of the DIFFERENCE between the distances from each receiver to the target.
In order to see how radar interferometry is sensitive to topography, which is the height of the target, Figure 4 shows two different targets at two different heights. It can be seen that the differential distance of each of these targets between the ends of the baseline depends on the height of the target. For the higher target (target 2), the differential distance is greater than for the lower one (target 1). The interferometric phase for target 2 is therefore larger than that for target 1. Note that the differential distance gets larger as the incidence angle (theta_1 < theta_2) in the picture to the target gets larger. The interferometric phase can be related to the incidence angle by the very-nearly-exact formula
interferometric phase = B sin(theta) /wave length (2)
where B is the baseline length. For a given target height, theta can be calculated and the interferometric phase calculated using (2). Eq. (2) can also be used to start with the measured interferometric phase and solved
(2) for theta and then find the topographic height. Conceptually, this is how SRTM interferometry determines the topographic height.
[Text written by Robert Treuhaft, JPL] |