A power law power spectral density model of total electron content structure in the polar region



[1] Measurements by the early warning radar at Thule, Greenland, together with GPS measurements from the Air Force Research Laboratory (AFRL) Ionospheric Measurement System (IMS) receiver, have been analyzed to develop a model for the structure of the total electron content (TEC) of the polar ionosphere. For the model the TEC measurements are related to the Wide Band Model (WBMOD), a climatological model of small-scale ionization structure. The TEC data agree very well with the predictions of a simple extension of WBMOD to larger scales, where a slightly steeper spectral slope (p ≈ 3) is used for the TEC structure (compared to p = 2.7 for the small-scale structure of WBMOD's polar region model). The benefit of this approach is that the TEC model subsumes the climatology of WBMOD, which is built upon two solar cycles of ionospheric measurements.

1. Introduction

[2] Data measured by the early warning radar (EWR) at Thule, Greenland, have been analyzed to develop a model for the structure of the total electron content (TEC) of the polar ionosphere. Two-frequency returns at 2 Hz pulse repetition frequency were obtained from satellites of opportunity and used to determine the ionospheric TEC along the radar line-of-sight. These “slant” TECs were then converted to equivalent vertical TEC. The data set used to develop the model includes measurements for 60 passes spanning October 2000 through February 2001 (7 different days, all times of day, and 11 satellites including 3 with highly spherical radar cross section). Additionally, the data set was augmented with GPS TEC measurements from the Air Force Research Laboratory (AFRL) Ionospheric Measurement System (IMS) receiver at Thule. Electronic Systems Center (ESC) provided data acquired by AFRL from the AN/GMQ-35 Ionospheric Measuring System (IMS) unit at Thule, Greenland. IMS is currently owned and operated by the Air Force Weather Agency (AFWA) and maintained by Detachment 11, Space and Missile Systems Center (SMC Det 11), supported by AFRL. The GPS measurements span October 2000 through March 2001, with 1 day of passes for each month (3 days in January). We will show that these data agree very well with the predictions of a simple extension of WBMOD to larger scales. WBMOD contains a climatological model of small-scale ionization structure [Secan, 1993]. The TEC structure model is taken as this extrapolation of WBMOD to larger scales. The benefit of this approach is that the TEC model subsumes the climatology of WBMOD, which is built upon two solar cycles of ionospheric measurements.

[3] The high-latitude ionosphere is characterized by structured ionization. Medium- to large-scale ionization enhancements originating in the auroral and subauroral regions are convected through the auroral zone and polar caps. The polar plasma patches are typically 500–1000 km in size and may travel at speeds in excess of 1 km/s. Smaller-scale structures are produced as plasma instabilities (e.g., equation image gradient drift instability) break up these larger-scale structures during transport [Tsunoda, 1988; Kelley et al., 1982]. This small-scale structure is generally geomagnetic field aligned and is responsible for producing scintillation of radar signals. These amplitude and phase fluctuations are the result of distortions imposed on the wave as it propagates through the irregularities. Additionally, Sun-aligned plasma arcs have been discovered, which sweep across the polar region with plasma density enhancements an order of magnitude larger than the background [Chang et al., 1998]. These arcs, which sometimes stretch across the entire polar cap, are typically 100–200 km across and move transversely over the polar cap at speeds up to 300 m/s. The large-scale structures (polar plasma patches and arcs) cause variations in the TEC of radar lines-of-sight to targets of interest, causing variable ionospheric-induced delay. Scintillation and TEC variations can degrade UHF radar detection and acquisition, tracking, and coherent processing.

[4] Figure 1 displays an incoherent-scatter radar measurement of a polar plasma patch as it detaches from the northern edge of the subauroral trough (measured by the Chatanika radar in Alaska). The contours represent electron density (as critical frequency in MHz) obtained by incoherent processing of radar returns. The shaded area represents the presence of meter-scale irregularities filling the large-scale polar plasma patch. In Figure 1 the tick-mark separations are 100 km in both dimensions, so one can see that this patch exceeds 400 km in horizontal length and represents a substantial localized enhancement in electron density. If not properly accounted for, radar range determinations for lines-of-sight traversing this patch will be substantially in error. Furthermore, as the radar line-of-sight passes into (or out of) this enhancement, the effective time variability of TEC will result in an artificial Doppler contribution which must be accounted for to obtain accurate trajectory estimates.

Figure 1.

Chatanika radar image of a polar plasma patch [from Tsunoda, 1988]. Contours are labeled by plasma frequency in MHz.

[5] Figure 2 shows UV images of a discrete auroral arc spanning the auroral oval (theta aurora, measured by the DE-1 imager) as it transits the polar cap. The images are about 1 hour apart with the arc moving from the dusk side toward the center of the polar cap. Like the polar plasma patches, these features are typically associated with localized order-of-magnitude electron density enhancements. The effects on UHF radar range and Doppler determination are the same as with plasma patches.

Figure 2.

Hourly UV images of a discrete auroral arc [from Chang et al., 1998].

[6] Models exist for the small-scale ionization structure that develops from these larger structures. Currently the best such model is WBMOD [Secan, 1993]. We seek a corresponding model for the morphology of the polar ionospheric TEC structure. Additional kinds of data will ultimately be required to build a complete TEC model (e.g., higher data rate “swath” data from beacon satellites to determine TEC gradients and spatially diverse multireceiver data to determine spatial joint statistics).

2. Analysis of EWR and IMS GPS Data

[7] Previous analyses of Thule EWR TEC measurements were performed as part of the Ionospheric Range Delay Study [Knepp and Brown, 1988], and expressions were derived for the expected accuracy of TEC determinations obtained from EWR two-frequency measurements. TEC can be measured by the EWR but with relatively low accuracy. Given the measured range difference Δr = r1r2 between returns from the same target at two different frequencies f1 and f2, the TEC is obtained as

equation image

where re is the classical electron radius and c is the speed of light. For a single pulse pair of the EWR the uncertainty in TEC is σTEC = 5 TEC units for 25 dB SNR (one TEC unit is 1016 electrons per square meter). Twenty pulse pairs must be averaged to obtain σTEC = 1 TEC unit. Typical TECs are 10–40 TEC units or more.

[8] TEC is a line-integrated electron density. We must distinguish between “slant TEC” and “vertical TEC.” For slant TEC the integration is performed along the radar line-of-sight to the target. For vertical TEC the line is assumed to be in the local vertical direction. Assuming that the dominant contribution of the ionosphere occurs near its peak height and ignoring horizontal gradients (obviously a weak assumption), one may convert slant TEC to an equivalent vertical TEC (here denoted TECV) using the geometric formula

equation image

where θ is the elevation angle of the radar line-of-sight at the radar, Re is Earth's radius, and hmax is the height of the peak of the ionospheric electron density. In the figures that follow, we take hmax = 300 km.

[9] Figure 3 shows a typical example of the TECs derived from the EWR data. The TEC is given in TEC units (TECu). Four quantities are displayed. The scatter points are the individual two-frequency pulse pair slant TEC values (which, owing to the inherent inaccuracy of the individual measurements, sometimes yield negative TEC values). The upper line is the 20-point running average slant TEC. The lower line is the equivalent vertical TEC, again using a 20-point running average. The stem points are equation image, the standard deviation of the vertical TEC obtained over the 20-point running average. Localized TEC enhancements are evident in Figure 3, and these are presumably associated with polar plasma patch ionization enhancements.

Figure 3.

Derived TECs from EWR data. The scatter points are pulse-by-pulse derived slant TEC. The upper curve is the 20-point smoothed slant TEC. The lower curve is the equivalent vertical TEC derived from the smoothed slant TEC. The stem points represent the standard deviation of the TEC over each smoothing interval.

[10] The power spectral density (PSD) of the smoothed vertical TECs from the EWR data can be obtained from the Fourier transform of the TEC time histories. That is, given the TEC spectrum

equation image

we obtain the TEC PSD as

equation image

The divisor TTEC is included as a normalization factor such that the integral of the PSD over all frequencies yields the TEC variance.

[11] We chose to employ the fast Fourier transform (FFT) to obtain the TEC spectra. This means that the data must be resampled to provide equally time-spaced values for the FFT. This was done by interpolating the data of each pass with a cubic spline. When using the spline as a replacement for the original data, it must be borne in mind that the original data were sampled at a maximum rate of 2 Hz. Thus the spectrum obtained from the interpolating spline must only be considered valid for frequencies below 2 Hz; larger frequency components are simply an artifact of the spline interpolation. It is also important that the time series satisfy the assumed periodicity of the FFT; and thus a window function must be employed. We chose a simple sine-squared window function. We therefore obtain the TEC spectra as

equation image

where N is the order of the FFT and Tdur is the duration of the pass. The window function is given by

equation image

[12] The PSDs of all 60 passes of the October 2000 to February 2001 EWR data are shown in Figure 4. Note that for frequencies below 1 Hz the PSDs have a consistent power law behavior. Linear fits of log10 (PSDTEC) with log10(f) for frequencies below 1 Hz yield a median slope of −p, where p = 2.68 over the 60 passes. The curve in Figure 4 is a power law model for the TEC PSD using this spectral slope parameter. We use a two-component power law in order to truncate the PSD for frequencies above the data-sampling rate. Our TEC fit has the functional form

equation image

where σTEC = 2.52 TECu and TTEC = 1496 s. TTEC is a large timescale that we will later show is determined by a TEC scale size parameter LTEC and an effective scan velocity. Again, the second power law regime is constructed to provide a steep taper for frequencies above the data-sampling rate.

Figure 4.

Power spectral densities derived from EWR data for 60 passes and median power law model (overlying curve).

[13] It is interesting to note that the spectral slope obtained here is consistent with the spectral slope obtained from phase scintillation data derived from satellite beacon signals. Since TEC is a line-integrated quantity derived from differential phase measurements, its spectral slope parameter p is directly comparable to the phase spectral slope used to describe small-scale irregularity structure. WBMOD [Secan, 1993], which is based on fits to phase spectra obtained from satellite beacon data, uses a spectral slope in the polar/auroral region of pauroral = 2.7. The consistency of the EWR-derived TEC spectral slope with that of WBMOD's small-scale spectral slope will next be shown to extend to the IMS GPS data set.

[14] We have performed a similar analysis using the AFRL IMS GPS-derived TEC data. As with the EWR data, we fit the spectrum from each GPS pass in the data set with a power law PSD. Cumulative distributions of the spectral slopes and TEC standard deviation are shown in Figure 5. Note in Figure 5 that at the 50th percentile level, the value of the spectral slope is very near WBMOD's high-latitude value of pauroral = 2.7. The distribution of spectral slope values is also in agreement with the database of small-scale spectral slopes used in the development of WBMOD (J. A. Secan, private communication, 2001).

Figure 5.

Cumulative distribution in percent of σTEC (a) and spectral slope (b) from AFRL IMS data.

[15] The values of the vertical TEC standard deviation from the IMS data (Figure 5) are also in reasonable agreement with the predictions of WBMOD. Using the relationship (derived in the next section) of σTEC to WBMOD's CkL strength parameter (spectral strength at a 1 km scale size; see Secan [1993]), we find that a typical WBMOD-predicted value of CkL = 0.5 × 1034 (typical polar value for 90th percentile probability of occurrence at high sunspot number) and using a value of LTEC = 500 km (see the next section) yields σTEC = 5 TECu (or log(σTEC) = 0.7), which lies at the 90th percentile level of Figure 5.

[16] It is gratifying that the EWR and IMS data are fully in agreement with WBMOD when predominantly small spectral scales are considered. The TEC model we seek, however, is meant to apply to intermediate scale structure; that is, to spatial scales that exceed the outer scale size Lo of the small-scale ionization structure. The outer scale size, which is a parameter in the power law PSD description of small-scale ionization structure used in WBMOD, is of the order Lo ∼ 10 km. It should not be assumed that the spectral slope at small scales is the same at intermediate scales. (If it were, then the TEC model could be obtained by simply extending the outer scale size to a larger value.) As we develop the TEC structure model in the next section, we will show that a fit of the IMS data using only those scales that exceed the outer scale size produces a slightly steeper spectral slope, and the scale size Lo represents the place in the spectrum where the break to steeper slope occurs.

3. TEC Structure Model

[17] The results of the last section provide an argument in favor of the notion that a TEC structure model can be obtained as a suitable extrapolation of WBMOD to larger scale sizes. In this section we develop this approach.

[18] Since our goal is to extend the small-scale ionization structure model of WBMOD to larger scales for our TEC model, we begin by examining the phase spectral model of small-scale ionization structure. WBMOD is built upon the formulation of Rino [1979]. In Rino's formulation, the phase PSD of transionospheric signals is given by

equation image

where TRino is a strength parameter and fo is the frequency scale associated with the outer scale size of the electron density fluctuations. The strength parameter TRino is given by

equation image

Here λ is the field wavelength, re is the classical electron radius, L is the ionospheric layer thickness, Cs is a strength parameter for the electron density fluctuations (related to WBMOD's CkL), and veff is an effective scan velocity (the velocity of the line-of-sight through the generally moving plasma). G is a geometrical enhancement factor that accounts for the geomagnetic field-aligned character of small-scale ionization structure. In the following we will take G = 1 because, at least for the present, we do not want the TEC model to depend on geomagnetic field line geometry.

[19] The TEC frequency spectrum is obtained from equations (8) and (9) by dividing away the (λre)2 and setting G = 1. We also choose to define a timescale by TTEC = 1/fo, and we recast the strength parameter in terms of the standard deviation of the TEC fluctuations to obtain

equation image

The standard deviation of the TEC fluctuations σTEC is given by

equation image

[20] Equations (10) and (11) define the TEC frequency/time spectral model. This can be readily converted to a spatial spectral model by using the relationship veff = (2πf)/k and by defining the TEC scale size parameter,

equation image

In order to roughly model the EWR and IMS data presented in the preceding section, we take LTEC = 500 km (although the model is fairly insensitive to values ranging from 100 to 1000 km). Remembering that for a change of variables in a PSD we must multiply by the determinant of the Jacobian of the transformation, which in this case is simply veff/2π, we obtain

equation image

where we may rewrite the TEC variance of equation (11) as

equation image

The strength parameter CsL is related to the WBMOD strength parameter CkL by

equation image

[21] As we mentioned at the end of the last section, the TEC model is meant to describe only those structures that exceed the outer scale size of the small-scale electron density fluctuations, Lo. We have fit the IMS data using only these larger scales, i.e., with the data restricted to frequencies below 0.016 Hz. This corresponds to timescales exceeding 63 s, which corresponds to the line-of-sight traversing 2πLo = 63 km with an effective velocity of veff = 1 km/s. This is a typical value for the plasma drift speed in the polar region (note that the line-of-sight motion through the plasma due to the GPS satellite speed is negligible).

[22] Figure 6 shows the cumulative distribution functions for the intermediate-scale fits of the IMS data using equation (10). Comparison of Figures 5 and 6 shows that for the intermediate-scale structure, the spectral slopes are somewhat larger at the 50th percentile level than for the small-scale case. For the present we shall follow WBMOD in using a fixed spectral slope for the TEC structure model, which we take to be the 50th percentile value p = 2.94.

Figure 6.

Cumulative distribution in percent of σTEC (a) and spectral slope (b) from AFRL IMS intermediate-scale data.

[23] Equations (10)–(15) with LTEC = 500 km and p = 2.94 constitute the TEC structure model. A proviso for the use of this model is that it is meant to describe intermediate scale structures, and thus it applies only to scales larger than the outer scale size parameter Lo of the small-scale ionization structure formulation (Lo ∼ 10 km). One way to require this formally would be to incorporate a second power law regime (as we did in equation (7)) that is designed to make the PSD drop steeply for scale sizes smaller than Lo. For simplicity, we have chosen not to do this, and hence it behooves the user of this model to be aware of its regime of applicability.

4. TEC Realizations

[24] The TEC PSD model can be used to generate realizations of TEC for use in simulations. The procedure to generate the realizations is as follows. Given two sequences of Gaussian random numbers, g1k and g2k (with zero mean and unit variance), where 1 ≤ kN (N is the order of the FFT), form the complex random numbers

equation image

A realization of the TEC spectrum is then generated by

equation image

where Tdur is the desired time duration of the realization and fk = (k − 1)/Tdur. The realization is then given by

equation image

[25] The quantity TEC0 is a user-added offset supplied to ensure that the TEC never drops below a specified minimum value. Since the FFT of equation (18) produces a complex result, the realization actually contains two independent real realizations given by Re (TECk) and Im (TECk). The times for the TEC realizations are given by tk = (k − 1)T/N.

[26] Figure 7 shows two realizations generated as the real and imaginary parts of equation (18), where an offset of TEC0 = 5 TECu was used. Here we truncated the spectrum at a timescale of 63 s so as to avoid the generation of small-scale structure below the spatial scale of 10 km. Comparison of the structure of these realizations with the IMS GPS data sample of Figure 8 (which has been converted to equivalent vertical TEC) shows that the model has successfully captured the behavior of the data.

Figure 7.

Realizations generated as real and imaginary parts of a single complex realization using the TEC model.

Figure 8.

IMS GPS equivalent vertical TEC from 14 October 2000.

5. Summary

[27] A model for the intermediate-scale structure of the ionospheric TEC of the polar region has been developed as an extension of WBMOD. The model is based on a power law PSD with a strength parameter related to the CkL strength parameter of WBMOD. As such, the TEC model subsumes the climatology of WBMOD, which was formulated using an extensive data set spanning two solar cycles.

[28] Two-frequency data from both the early warning radar and AFRL Ionospheric Measurement System GPS receivers at Thule, Greenland, were used in the TEC model development. These data spanned a 6-month period during the winter of 2000–2001, a time near solar maximum. The fact that these data were found to be in excellent agreement with the predictions of WBMOD for the geophysical conditions of this time period is taken as an indication that the TEC model may be reasonably extended to other geophysical/solar conditions using the climatology of WBMOD. It is, however, desirable to revisit this assumption as more data becomes available over the course of the next solar cycle.

[29] The TEC model of this paper is limited in a number of respects. The limited nature of the data available to us did not allow for any study of the joint statistics of TEC structure for spatially separated regions, and thus no accounting is given for the well-known large-scale structure of TEC in the polar region and the associated convective flow. Rather, the simple PSD approach of the current model generates realizations with TEC structure that are homogeneous on a large scale. A related limitation of the PSD approach is that, when generating random realizations of TEC structure from the model, the power law nature of the PSD is manifested as structure with a range of scale sizes. However, it is possible for smooth intermediate-scale size structures with steep gradients to manifest the same PSD power law behavior, and such structures are in fact observed at times in the polar region. The method of statistical signal generation applied in section 4 to generate random realizations from the TEC model cannot generate such smooth, steep-sided structures, but will always generate structure-filled TEC enhancements.

[30] This TEC model applies only to scale sizes that are larger than the outer scale size of the small-scale ionospheric irregularity structure, a size of about 10 km. For scales smaller than this the full geomagnetic field-aligned formulation of small-scale ionization structure should be used, as in the work of Rino [1979].


[31] The author would like to thank Jim Secan and Ed Fremouw of NorthWest Research Associates for their helpful discussions concerning this work.