## 1. Introduction

[2] The problem of real-height electron density inversion from ionosonde data has a long history, starting with *Appleton* [1930]. It has always been formulated essentially as an applied mathematical task and has therefore required a set of significant physical idealizations. A main simplification inherent to all “profile inversion” methods (until now) is the assumption of a horizontally plane-stratified ionosphere, thus reducing the problem to only one dimension. All plasma density surfaces are then horizontal planes, and only a vertical gradient is assumed. The wave vector of a sounding signal propagating in this medium is always vertical. Lower frequencies (and signals of both characteristic polarizations) pass through exactly the same plasma densities up to (and down from) their reflection heights. The radio propagation direction relative to the geomagnetic field, an important parameter of the inversion calculations, is simply assumed to be the local dip angle.

[3] The actual properties of the real ionosphere differ significantly from this idealistic picture. Practically all processes in the ionosphere create horizontal gradients of electron density. Specific sources of large-scale horizontal gradients and tilts include the solar terminator (near sunrise and sunset), acoustic gravity waves and traveling ionospheric disturbances, storm time density enhancements, the equatorial anomaly, plasma bubbles, particle precipitation, and the general latitudinal/longitudinal dependence of the ionosphere. Midscale (∼10–100 km) and small-scale (∼0.1–2 km) irregularities add additional complexity to the ionosphere, leading to multibeam reflections and “spreading” of echoes over large intervals of group range. In combination, these effects cause the sounding signal to propagate along noncanonical raypaths that only by chance are “vertical.” Thus actual data are usually in more or less explicit disagreement with the usual assumptions. Conventional inversion methods are consequently unreliable: They often cannot cope with internal contradictions of the input data and either fail numerically or produce distorted and always inaccurate results. Indeed, the inaccuracies are unknown and are nonquantifiable by these methods. Even relatively small tilts result in the use of an incorrect value for the propagation angle between the wave vector and magnetic field, with serious consequences for certain aspects of the inversion problem [*Wright*, 1990].

[4] Two methods of ionogram inversion widely used at present, POLAN [*Titheridge*, 1985] and NhPC (part of the Digisonde data processing system ARTIST) [*Reinisch and Huang*, 1983], were developed more than 20 years ago, and little new attention has since been given to the problem. However, new opportunities arise from the computing power of modern PCs, and new requirements are now placed on the quality of ionospheric profile information. The latter include the need for realistic error assessments in modern assimilative modeling [*Schunk et al.*, 2004; *Wang et al.*, 2004; *Khattatov et al.*, 2004]. New opportunities and new requirements are weighty reasons for reconsidering the problem of electron density inversion from ionograms.

[5] Meanwhile, modern digital ionosondes (i.e., dynasondes) have advanced their capabilities to measure accurately both the group time of propagation and the directions of arrival for each ionogram echo. Three-dimensional distributions of apparent echolocations are presented in Figure 1.

[6] Each echolocation datum in these plots represents the end point of a “group path vector” [*Paul et al.*, 1974] aligned with the direction of arrival of the echo as measured at the dynasonde location, of length equivalent to half the group time of propagation, which is determined with remarkable precision by the stationary phase method. Actual positions of the reflection points (where the reflection conditions are met) are not measured directly and will differ because of refraction in the inhomogeneous magnetized ionospheric plasma. In the “forward problem,” these effects can be calculated by ray tracing if the plasma distribution is known in three dimensions. In the inversion problem, ray tracing becomes a component in a procedure to reconcile the spatial electron density distribution with the measured echo ranges and their angular positions. This is the basic strategy of the new inversion scheme NeXtYZ.

[7] Three-dimensional (3-D) ionogram inversion does not aspire to the spatial coverage of radio tomography. The characteristic feature of the “all-sky view” of the ionosonde is its extraordinary sensitivity to local spatial structure, providing the perspective from a single ground location. The 3-D ionogram inversion problem may be formulated as the recovery of parameters of a parameterized model that describes locally both the vertical and horizontal gradients of ionospheric plasma density. The “wedge-stratified ionosphere” (WSI) model is the appropriate substitution for the former “plane-stratified ionosphere” model.