EPB electron density slice-plane realizations are formally two-dimensional scalar fields, *N*_{e}(*y*,*z*), where *y* and *z* represent cross-field and altitude, respectively. Assuming that *N*_{e}(*y*,*z*) is statistically homogeneous, the stochastic structure can be characterized by a two-dimensional spectral density function (SDF), which is formally the expectation of the intensity of two-dimensional Fourier decompositions of *N*_{e}(*y*,*z*) realizations.

### Power-law models

Published in situ measurements and remote diagnostics imply an underlying two-component power-power law SDF. The following analytic representation is introduced to guide structure characterization:

$$ \Phi_{N_{e}}(q)=C_{s}\left\{ \begin{array}{c} q^{-p_{1}}\text{ for}~q\leq q_{0} \\ q_{0}^{p_{2}-p_{1}}q^{-p_{2}}\text{ for}~ q>q_{0} \end{array} \right., $$

(1)

where

$$ q=\sqrt{q_{y}^{2}+\beta q_{z}^{2}}, $$

(2)

is the magnitude of the spatial frequency vector [*q*_{y},*q*_{z}] in radians per meter. The *β* coefficient accommodates projection of the radial variation of field-aligned structure. The defining parameters are turbulent strength, *C*_{s}; the break frequency, *q*_{0}; and the spectral indices, *p*_{n} corresponding to subranges of spatial frequencies smaller than (*n*=1) and larger than (*n*=2) *q*_{0}.

In situ measurements are one-dimensional scans. If the structure volume were stochastic in all three dimensions, the measured one-dimensional SDF would be represented by a two-dimensional integration of the three-dimensional SDF. For field-aligned two-dimensional stochastic structures a slice plane containing the one-dimensional scan must be constructed. Configuration-space realizations populate arbitrarily oriented slice planes for extrapolation (Rino et al. 2018). For the EPB analysis, the cross-field orientation of the slice planes were selected for direct structure measurement. One-dimensional SDFs are related to (1) by the integration

$$ \Phi_{N_{e}}^{1}(q)=\int_{-\infty}^{\infty}\Phi_{N_{e}}(q_{y},q_{z}) \frac{dq_{z}}{2\pi }. $$

(3)

For (3) to be well defined, the power-law variation must be specified in more detail. In the EPB realizations, there is a transition from stochastic to trend-like variation at small spatial frequencies. At sufficiently high frequencies, the physics supporting the EPB simulations is incomplete. Furthermore, as already noted, the stochastic structure itself varies with altitude. To capture these details, the following height-dependent one-dimensional SDF *hypothesized* for EPB structure characterization:

$$ \Phi_{N_{e}}\left(q \right) =C_{s}\left\{ \begin{array}{c} q^{-\eta_{1}}\text{for}~q\leq q_{0} \\ q_{0}^{\eta_{2}-\eta_{1}}q^{-\eta_{2}}\text{for}~q>q_{0} \end{array} \right.. $$

(4)

The one-dimensional model captures a broad range of structure characteristics as defined by the turbulent strength *C*_{s}, the spectral indices *η*_{n}, and the break frequency *q*_{0}. For example, if *η*_{1}≃0, *q*_{0} can be interpreted as an outer scale. If *η*_{1}≃*η*_{2}, the SDF is a single power law. Generally, *η*_{1}≤*η*_{2}. However, enhanced low-frequency structure might lead to the opposite ordering, *η*_{1}>*η*_{2}. In all cases, the variation of *C*_{s} provides a measure of overall structure intensity.

Establishing the relation between *η*_{n} and *p*_{n}, which is nominally *η*_{n}=*p*_{n}−1, is beyond the scope of this study. However, ionospheric structure models can be validated by comparing predicted height-dependent one-dimensional structure characteristics with the measured EPB structure.

### Irregularity parameter estimation

Irregularity parameter estimation (IPE) systematically adjusts the defining parameters to minimize a measure of the disparity between an SDF estimate and the theoretical SDF. An IPE procedure for estimating scintillation intensity SDF parameters was introduced by Carrano and Rino (2016). The original IPE procedure was refined to maximize the likelihood that the periodogram was derived from a realization with the theoretical SDF (Carrano et al. 2017). For characterizing the EPB SDFs, the maximum likelihood estimation (MLE) procedure was adapted for power-law SDF estimation as described in Rino and Carrano (2018). Power-law parameter estimation is more challenging than intensity scintillation parameter estimation because of the singular behavior of unmodified power-law SDFs at zero frequency.

The MLE SDF estimate is the average of *M* periodograms, formally

$$ \widehat{\Phi}_{N_{e}}=\frac{1}{M}\sum\limits_{l=1}^{M}\widehat{\Phi }_{n}^{(l)}, $$

(5)

where the periodogram is defined as

$$ \widehat{\Phi }_{n}^{(l)}=\frac{\Delta y}{N}\left\vert \sum_{k=0}^{N-1}N_{e}^{(l)}\left(k\Delta y\right) \exp \{-ink/N\}\right\vert^{2}. $$

(6)

The index *n* corresponds to the spatial frequencies

$$ 2\pi/(N\Delta y)\leq n\Delta q \leq 2\pi /\Delta y, $$

(7)

where *Δ**y* is the *y* sample interval, and *Δ**q*=2*π*/(*N**Δ**y*) is the spatial-frequency resolution. The index *l* identifies the altitude at which the zonal scan is extracted. One can show that

$$ \left\langle \widehat{\Phi}_{N_{e}}\right\rangle={\Phi}_{N_{e}} $$

(8)

MLE exploits the fact that the probability distribution function (PDF) of the periodogram is well approximated by a *χ* distribution with 2 degrees of freedom. The *χ* distribution with 2*M* degrees of freedom follows for the summation.

It is well known that periodogram estimates are contaminated by the sidelobes of end-point discontinuities. Moreover, efficient discrete Fourier transformation (DFT) evaluation requires *N* to be even with as many factors as possible, ideally a power of 2. The Welch method (Welch 1995) uses windowing and segmentation with averaging. Periodogram variants, such as maximum entropy estimates (Fougere 2009), provide additional variants. However, MLE relies on unbiased spectral estimates with *χ* distributions, whereby it is desirable to stay as close to (6) as possible. After some exploration, it was found that using the full 373.6 km *y* extent of the data zero extended to a *nice* FFT number gave the best results. Following (Rino and Carrano 2018), periodograms from two altitudes (*M*=2) were averaged.

Multi-parameter MLE used a MATLAB implementation of the Nelder-Mead simplex algorithm (Olsen and Nelsen 1975). The procedure is surprisingly robust in that fits were made to quasi-deterministic SDFs with no stochastic structure as well SDFs from realizations with fully developed stochastic structure. The two classes are readily distinguished by the reported IPE parameters.