The two‐dimensional resolution kernel associated with retrieval of ionospheric and atmospheric refractivity profiles by Abelian inversion of radio occultation phase data

Errors in radio occultation experiments associated with departures from sphericity can be analyzed on the basis of straight‐ray tomography in which (1) only one parallel projection of the object is available, (2) diffraction is neglected, and (3) inversion is achieved by assuming the object to be circularly symmetric. The straight‐line tomographic case and the actual curved ray paths in a spherically symmetric atmosphere are related by an analytic transformation. A previously unknown kernel function transforms the two‐dimensional (2‐D) atmospheric refractivity map from the plane of propagation to a 1‐D radial profile along the locus of ray periapsides. This function consists of a discrete, positive singularity plus an extended negative branch; the negative branch rises smoothly from negative infinity in the neighborhood of the positive singularity toward zero. Knowledge of the resolving kernel allows study of horizontal and vertical resolution in a rigorous manner. Spherically symmetric structures are reproduced exactly with the kernel; processing of band‐limited projection data results in a modified kernel of finite vertical resolving power. In general, the horizontal extent of significant contributions, including levels of 10% of the peak, is of the order 2√4.6 rH centered on the locus of periapsides, where r is the radius of the point in consideration and H is the refractivity scale height; the vertical extent of significant contributions extends from approximately one sampling distance below r to a few sampling distances above it. We confirm the expected result that a refractivity structure of radial thickness Δ r will be retrieved properly only if its circumferential extent is approximately 2√2 rΔr or greater centered on the periapsis point. In addition, a positive refractivity structure limited in circumferential extent introduces artifacts of negative refractivity in the retrieved profile. The artifacts first appear at the projected altitude onto the vertical of the structure's circumferential edges and continue downward. The error represented by such negative refractivity artifacts is diminished somewhat by the general exponential increase in refractivity with decreasing altitude of a real atmosphere. Thus the relative error depends on the local scale height and the circumferential extent of the refractivity structure.