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Coherent Scatter Radar and the Scatter Theorem


Several types of scatter radar techniques are used for upper atmosphere research. Coherent scatter radar is a volume scattering technique whereby the radar detects energy scattered from within a medium when there are regular spatial variations of the refractive index due to irregularities. This is the analogue of Bragg scattering of X-rays from crystals. The ionosphere contains irregularities of many different sizes and in general a small fraction of the incident energy is scattered at each boundary. Signals backscattered from plasma wave irregularities with a spacing of half the radar wavelength will reinforce by constructive interference in the direction back to the radar and can produce a signal strong enough to be detected by the receiver. Thus, a radar of wavelength tex2html_wrap_inline4734 will effectively select from the scattering medium the spatial component of wavelength tex2html_wrap_inline4736 along the radar beam (a more detailed derivation of this property follows). The term ``coherent'' applies to the constructive interference possible when there is a scattering structure with an organized spatial content at half the radar wavelength.

The scatter theorem describes the signals that can be measured by a coherent scatter radar. Figure 3.1 (left) shows the geometry used in the derivation of the scatter theorem for a bistatic radar system.

Figure 3.1: Geometry of scatter theorem analysis

The origin of the coordinate system lies within the scattering region under observation, tex2html_wrap_inline4738 points to a position within this region, tex2html_wrap_inline4740 and tex2html_wrap_inline4742 are the positions of the transmitter and the receiver, respectively. They are fixed in space and can be considered constant in this derivation. We introduce further the vectors tex2html_wrap_inline4744 and tex2html_wrap_inline4746 . Considered is a volume element dV at the position tex2html_wrap_inline4738 . The electric field produced by the transmitter is given by:


where tex2html_wrap_inline4752 is the electric field at the transmitter, tex2html_wrap_inline4754 is the transmitter antenna voltage gain and tex2html_wrap_inline4586 is the angular frequency of the wave.gif The electric field at the receiver tex2html_wrap_inline4758 from the volume element dV is then given by:


where tex2html_wrap_inline4762 is the amplitude scattering cross section of electrons for signals incident along tex2html_wrap_inline4764 and scattered along tex2html_wrap_inline4766 and tex2html_wrap_inline4768 is the electron density at tex2html_wrap_inline4738 and time t. To simplify this equation it is assumed that for a narrow beam, the voltage gain function for the transmitting and receiving antennas can be set to constants tex2html_wrap_inline4754 and tex2html_wrap_inline4776 respectively. A similar simplification is done with the scattering cross section tex2html_wrap_inline4762 which is set to a constant, since all spatial information is contained in the electron density term D. The differential voltage produced at the receiver input by volume element dV of the scatterer is then:


Assuming tex2html_wrap_inline4784 and tex2html_wrap_inline4786 , performing a Taylor expansion on tex2html_wrap_inline4788 and tex2html_wrap_inline4790 and neglecting quadratic terms, one obtains the following approximations for the transmitter (t) and receiver (r) distances:



This approximation can now be used to rewrite the amplitude term tex2html_wrap_inline4792 , but not the phase term. Using the unit vectors tex2html_wrap_inline4794 and tex2html_wrap_inline4796 , equation 3.3 becomes:


We express the electron density tex2html_wrap_inline4798 through its Fourier component tex2html_wrap_inline4800 . Thus if the transform is denoted by tex2html_wrap_inline4802 the definition of N is:


That is, by the inverse transform,




Substituting Equation 3.9 into Equation 3.6 yields


In these steps we also used the fact that tex2html_wrap_inline4804 . If we integrate over the volume where backscatter occurs, the tex2html_wrap_inline4738 -dependent exponential term can be approximated by a delta function tex2html_wrap_inline4808 , where


This means we select the component of N that corresponds to tex2html_wrap_inline4812 . After the integration over tex2html_wrap_inline4814 , taking the delta function into account, the final result can be written as follows


where C denotes some constant.

This shows that the radar is sensitive to the Fourier components of the electron density fluctuation with wave vector tex2html_wrap_inline4812 , corresponding to the wavelength


where tex2html_wrap_inline4582 is the angle between tex2html_wrap_inline4822 and tex2html_wrap_inline4824 . This important property can be derived by considering the following equation derived from the conservation of momentum and energy in two dimensions (see Figure 3.1, right):




Rewriting Equation 3.14 in terms of the wavelength tex2html_wrap_inline4734 for the x and y components gives



Similary Equation 3.15 becomes (using 3.17)


Hence using Equation 3.16


and Equation 3.17 we get


Using some trigonometric identities this can be written as


and since tex2html_wrap_inline4832 is small, it follows that


But since tex2html_wrap_inline4834 we arrive at the property that


which means that the angle tex2html_wrap_inline4660 which tex2html_wrap_inline4812 makes with the negative x-axis is along the bisector of the transmitter and receiver directions. Using this fact we can write Equation 3.14 as


where tex2html_wrap_inline4840 is the unit vector pointing along the incident direction and tex2html_wrap_inline4842 is perpendicular to tex2html_wrap_inline4840 . This can be simplified using the following trigonometric identities





to yield


By comparing the left and right side of this equation it follows that the direction tex2html_wrap_inline4846 and wavelength tex2html_wrap_inline4848 is described by:



We can convert tex2html_wrap_inline4660 to the angle tex2html_wrap_inline4582 that was used earlier using


which yields Equation 3.13 again. For a monostatic backscatter radar, with transmitting and receiving antennas at the same location, tex2html_wrap_inline4854 and tex2html_wrap_inline4856 .

next up previous
Next: Introduction to SuperDARN Up: The SuperDARN HF radar Previous: The SuperDARN HF radar

Andreas Schiffler
Wed Oct 9 10:05:17 CST 1996