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

A mathematical method for the reconstruction of the electron density profile
N(x)
for an inhomogeneous, stratified, simple plasma is presented. If the reflection coefficient
r(k)
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
N(x)
as a function of distance
x
in the plasma if
r(k)
is a rational function of the wave number
k
. 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
r(k)
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
W(x)
for propagation in nonuniform regions). The functional characteristics of
N(x)
depend on the pole positions of
r(k)
in the complex
k
plane. By calculating the variations in
N(x)
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.