Profile inversion of simple plasmas and nonuniform regions: Three-pole reflection coefficient

Abstract

A mathematical method for the reconstruction of the electron density profile <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N(x)</tex> for an inhomogeneous, stratified, simple plasma is presented. If the reflection coefficient <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r(k)</tex> of the incident probing electromagnetic wave is approximated by a third-order rational approximation, then profiles can be obtained which are similar to profiles obtained from simulated VHF satellite tracking data. The method is based upon the solution of the fundamental integral equation of inverse scattering (Gelfand-Levitan) theory. Using this theory it is possible to obtain an analytical expression for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N(x)</tex> as a function of distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> in the plasma if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r(k)</tex> is a rational function of the wave number <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> . The integral equation is solved by the Laplace transform technique and checked by the differential operator technique. The method is exact once the functional form of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r(k)</tex> is determined. Thus this analysis can supplement information about profiles which are obtained from calculations based on the WKB approximation (which approximation can also be applied to calculate the local wave impedance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W(x)</tex> for propagation in nonuniform regions). The functional characteristics of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N(x)</tex> depend on the pole positions of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r(k)</tex> in the complex <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> plane. By calculating the variations in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N(x)</tex> due to variations in these pole positions, it is possible to set a finite error bound on the profile of the electron density if the error bound in the rational approximation to the reflection coefficient is known.