## 1. Introduction

[2] In modern HF pulsed sounding of the ionosphere using research ionosondes, the phase of the radio echo is measured precisely and frequently; the ‘same echo’ can be identified from one time to the next. Thus, a continuous series of phases can be obtained and analyzed by statistical methods. Since an ionospheric layer is generally in motion (primarily in a horizontal direction), the echo ray path passes through different electron density irregularities at each time, producing random temporal phase variations. Note that such short-period phase variations are distinct from the Doppler imposed by drift or wave motions. One of the most convenient statistical instruments for characterization of short-period variations is the structure function, SF, defined as the square of the difference of a random function, averaged over all pairs of points a certain distance apart. For temporal phase differences at lag τ, this is *D*_{ϕ}(τ) = 〈[ (*t* + τ) − (*t*)]^{2}〉 and for electron density irregularities, δ_{N} () = () at spatial lag , it is *D*_{N} () = 〈[δ_{N} ( + ) − δ_{N} ()]^{2}〉. Among the convenient features of the structure function, and unlike the correlation function or variance, large-scale variations (much larger than ∣∣) only weakly influence the structure function at scale lengths less than ∣∣. Thus, this investigation of small-scale irregularities and short-period phase variations can ignore the behavior of the mid-scale irregularity spectrum (scales exceeding the first Fresnel zone), where good statistical characteristics are difficult to obtain in practice.

[3] A theoretical relation between structure functions of the phase variations and the small-scale ionospheric irregularities was obtained in *Zabotin and Wright* [2001] for a model of frozen irregularities moving with a drift velocity , on the assumption that the irregularities are infinitely stretched along geomagnetic field lines and their two-dimensional (transversal) spectrum is described by a power law Φ(κ_{⊥}) ∝ κ_{⊥}^{−ν}. From the technical viewpoint, this work represented development of earlier results [*Gailit et al.*, 1983; *Denisov and Yerukhimov*, 1966]. In practice, it is found that the SF is log-log-linear in the small-lag regime, and is therefore defined by simply an intercept and slope; we call them structure indices *SIA* and *SIB*. Using the dynasonde ‘B’ ionogram mode (blocks of a few repeated frequency ramps), one can define a ‘rudimentary structure function’ (RSF), containing only a few initial lags, but these are sufficient to determine *SIA* and *SIB*. Thus standard dynasonde ionograms, themselves available in long systematic time-series, become practical sources for irregularity diagnostics [*Zabotin and Wright*, 2001, 2002a, 2002b] in addition to their many other uses.

[4] The ray path traverses many irregularities with various scale lengths, thus integrating their effects. We must not expect a direct relation between specific spatial (*l*) and temporal (τ) scales, such as *l* = *V*τ. In this paper, we conduct a special investigation, based on analysis of theoretical relations from *Zabotin and Wright* [2001], to reveal the range of scales to which the new diagnostic method is most sensitive. In the next section, we reduce the theoretical relations to a form appropriate for further numerical analysis. We find that the phase structure function may be represented as an integral over the irregularity spectrum, and the integrand shows directly the relative contribution of different spatial harmonics. A subsequent section is devoted to detailed numerical investigation of the integrand, and our conclusions are presented in the final section.