Radio Science Bulletin 325 - June 2008 - URSI

screen (which is implemented in a simple Fourier-transformprocedure), and repeating the process through all the screensand then to the receiver. This procedure is now usuallycalled the Fourier split-step method, but I referred to it at thetime as the Phase-screen/Diffraction Method (PDM). Mycontribution to the theory was the realization that the Phasescreen/DiffractionMethod solution approach, when appliedto the parabolic wave equation for the mutual coherencefunction (as opposed to the parabolic wave equation for thewave itself), could be solved analytically for the generalcase of nonuniform background ionization and varyingscreen velocities, in the strong-scatter regime [17]. Iimplemented this solution on primary rays computed usingHaselgrove’s Equations in various ionosphere models,including ionosphere models derived from ionograms andincoherent-scatter radar measurements taken during thecourse of the SRI channel-probe campaigns. These resultsclearly showed that the unexpected scattering-functionshapes, measured by the channel probe, were due tononuniform plasma motion of the polar ionosphere alongthe extended length of the HF ray paths of the channelprobelink.Examples of Phase-screen/Diffraction Methodderivedscattering functions are shown in Figure 6. Forthese cases, Haselgrove’s Equations were used to propagatea 12 MHz ray from the southern tip of Greenland to a pointapproximately 1464 km to the north on a one-hop F-regionlow-ray propagation mode in a generic, but strong, daytimeionospheremodel. The traced ray determined theencountered electron density along the path, as well as theangles of the ray direction with respect to the geomagneticfield. At this high-latitude location and for a northerlydirected and highly oblique path like this, the geomagneticfield lines crossed the ray path almost orthogonally,vertically. Naturally occurring small-scale ionizationirregularity structure tends to be elongated along thegeomagnetic field lines, with scale sizes that are reasonablyparameterized by a power spectral density (PSD) with anouter scale size of L ⊥ = 10 km orthogonal to the geomagneticfield lines and L || = 100 km along them, and a so-calledfreezing scale of l f = 1km, where is assumed that thepower spectral density breaks to a steeper spectral slope,indicating declining presence of relevant structure. (Detailsabout the power spectral density parameterization can befound in [17]. For the interested reader, in the examples ofFigure 6, using the notation of [17], I used a two-componentpower spectral density, with an inner spectral slope ofn = 1.9 , breaking to the slightly steeper spectral slope ofn′ = 2 at the freezing scale. The strength of the powerspectral density of the irregularity structure was taken to be20% of the ambient electron density at each point along theray, that is, σ Ne Ne= 0.2 .) To generate the Phasescreen/DiffractionMethod solution for the scatteringfunction, I collapsed the ionization to 11 or 12 equidistantphase screens along the ray path, and applied the averageaspect angle of the ray to the geomagnetic field lines overeach collapsed interval.In Figure 6a, all 12 phase screens were movingtogether in the positive (easterly) direction, resulting in ascattering function that exhibited the parabolic delay-Doppler correlation that had been anticipated by the simpleFigure 6a. Phase-screen/Diffraction Methodscattering functions for uniform plasma motion.Figure 6b. Phase-screen/Diffraction Methodscattering functions for a dominantly positivelymoving plasma with a negative velocity sheer.42TheRadio Science Bulletin No 325 (June 2008)