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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.

   figure621
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:

equation640

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:

equation653

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:

  equation671

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:

equation690

equation699

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:

  equation716

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:

  equation739

That is, by the inverse transform,

equation745

or

  equation752

Substituting Equation 3.9 into Equation 3.6 yields

equation767

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

equation794

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

equation813

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

  equation820

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):

  equation832

and

  equation841

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

  equation848

  equation857

Similary Equation 3.15 becomes (using 3.17)

  equation866

Hence using Equation 3.16

equation880

and Equation 3.17 we get

equation893

Using some trigonometric identities this can be written as

equation901

and since tex2html_wrap_inline4832 is small, it follows that

equation907

But since tex2html_wrap_inline4834 we arrive at the property that

  equation909

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

equation915

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

equation927

equation929

equation931

equation934

to yield

equation936

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:

equation947

equation953

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

equation958

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