This paper presents a practical method for computing radio fields in regions of strong focusing, using ray intercept data provided by a standard ray-tracing program. The procedure extends the usefulness of the ray trace by allowing fields to be computed near caustics and cusps where ray density calculations fail. Using a plane wave decomposition of the field components, phase integrals are computed by curvefitting intercepts of rays traced through ionospheric or tropospheric media whose refractive indices vary arbitrarily with altitude. A numerical algorithm is described for performing the plane wave angular spectral integrations. This procedure avoids the complications associated with higher-order asymptotic techniques, allowing a broad range of refractive-index profiles to be analyzed by a single method. It is applied to two sample profiles, and the results agree very closely with higher-order stationary-phase estimates in caustic regions. Moreover, the computer code runs efficiently, despite the presence of highly oscillatory integrands. The method is capable of including the effects of weak collisions, the spherical earth, and azimuthally dependent transmitter configurations. INTRODUCTIONThis paper presents a method for using ray tracing to calculate radio field strengths in strong-focusing regions of the ionosphere or troposphere. Because of its versatility and numerical efficiency, ray tracing is an established technique for predicting radio field strengths in regions without strong focusing. It is desirable to extend the applicability of ray trace results to strong-focusing regions as well. This paper describes a practical numerical method for doing so using state-of-the-art ray-tracing programs [e.g., Jones and Stephenson, 1975].The procedure for computing radio fields in stratified media from ray tracing outside caustic regions uses the well-known correspondence, developed by Booker [ 1939], Budden [ 1961 ], and others, between ray trajectories and the first-order asymptotic approximation to the angular spectral representation of the field components. Budden [1976] and others show how the phase and amplitude of fields are directly related to the path-integrated refractive-index variation and the ray curvature, both easily computed with a ray-tracing program in regions where neighboring rays do not cross. That correspondence solves the radio field problem for stratified media outside caustic areas. 514 Breakdown of the first-order stationary-phase result near caustics has led to the development of higherorder asymptotic formulas that uniformly interpolate the field through caustic regions and can extend field strength calculations based on ray density into strong focusing regions [Maslin, 1976b;Budden, 1976]. Unfortunately, asymptotic methods have two drawbacks that make them inconvenient for use in conjunction with a ray tracing program:1. Different closed-form expressions involving special functions are needed for different degrees of focusing, e.g., Airy functions for caustics and Pearcey's [1946] ...

A method for inverting VLF/LF ionosounder data to obtain ionospheric conductivity profiles is described. The method is applied to altitudes below about 70 km, where the propagation can be assumed to be isotropic. Two cases are evaluated: (1) model ionospheres for which artificial data are generated by calculating reflection coefficients and (2) actual ionospheres for which reflection coefficients are measured by VLF/LF ionosounders during a strong solar proton event. Estimates are given of altitude ranges inside which ionosounder data contain information about conductivity and outside which the inversion breaks down. Calculated profiles are shown to agree well with model profiles within those altitude ranges. Profiles calculated for the solar proton event of September 23, 1978, are similar in magnitude and structure to those which occurred during earlier strong solar proton events.

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