A new formulation for interpreting radio occultation phase data is achieved by treating the radio occultation method as a straight-ray tomography experiment in which only one parallel projection of the object is available. This approach yields the inherent two-dimensional, geometrical optics (GO), resolution kernel directly. The kernel has a strong positive peak in the vertical direction at its lowest point and a decaying negative branch significant over a few kilometers above the positive peak; along the ray the kernel is constant. The three-dimensional resolution kernel incorporating the full-wave nature of radio waves resembles the two-dimensional result in the vertical direction smoothed on the order of a Fresnel zone. In the across-propagation horizontal direction it has the shape of the Fresnel zone; along the ray its shape is unchanged from GO. The curving of rays due to the spherically symmetric background is included in the kernel by using a set of transformations which convert the problem into an equivalent problem where curved rays are represented as straight rays. The curved-ray kernel exhibits the shortcoming of the radio occultation method in the event of critical refraction. For perfect data for a spherical atmosphere the inverse problem is solved exactly with the kernel formulation.; Large-scale departures from sphericity are modeled as locally spherical structures with center of symmetry offset from the center of mass. A first-order perturbation analysis yields analytic expressions for the errors in radius, refractivity, pressure, and temperature for general conditions. The resulting errors depend on the orientation and magnitude of the horizontal gradient at ray periapsis, and on the trajectory. For the atmosphere of Mars fractional errors in refractivity, temperature, and pressure can be as large as a few percent for altitudes above 30 km. For Earth's troposphere the errors are highly sensitive to horizontal gradients because of the large bending; typically, the error magnitude remains less than two percent because the horizontal gradients are mild. Comparison of results from the analysis with those from simulated occultations by the atmospheres of Mars and Earth and subsequent Abel inversion shows good agreement.
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