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

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1 RADIO SCIENCE, VOL. 39,, doi: /2002rs002818, 2004 A ower law ower sectral density model of total electron content structure in the olar region L. J. Nickisch Mission Research Cororation, Monterey, California, USA Received 11 November 2002; revised 24 Setember 2003; acceted 16 October 2003; ublished 24 January [1] Measurements by the early warning radar at Thule, Greenland, together with GPS measurements from the Air Force Research Laboratory (AFRL) Ionosheric Measurement System (IMS) receiver, have been analyzed to develo a model for the structure of the total electron content (TEC) of the olar ionoshere. 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 redictions of a simle extension of WBMOD to larger scales, where a slightly steeer sectral sloe ( 3) is used for the TEC structure (comared to = 2.7 for the small-scale structure of WBMOD s olar region model). The benefit of this aroach is that the TEC model subsumes the climatology of WBMOD, which is built uon two solar cycles of ionosheric measurements. INDEX TERMS: 2439 Ionoshere: Ionosheric irregularities; 2475 Ionoshere: Polar ca ionoshere; 6929 Radio Science: Ionosheric hysics (2409); 6934 Radio Science: Ionosheric roagation (2487); 6964 Radio Science: Radio wave roagation; KEYWORDS: total electron content, ionoshere, olar Citation: Nickisch, L. J. (2004), A ower law ower sectral density model of total electron content structure in the olar region, Radio Sci., 39,, doi: /2002rs Introduction [2] Data measured by the early warning radar (EWR) at Thule, Greenland, have been analyzed to develo a model for the structure of the total electron content (TEC) of the olar ionoshere. Two-frequency returns at 2 Hz ulse reetition frequency were obtained from satellites of oortunity and used to determine the ionosheric TEC along the radar line-of-sight. These slant TECs were then converted to equivalent vertical TEC. The data set used to develo the model includes measurements for 60 asses sanning October 2000 through February 2001 (7 different days, all times of day, and 11 satellites including 3 with highly sherical radar cross section). Additionally, the data set was augmented with GPS TEC measurements from the Air Force Research Laboratory (AFRL) Ionosheric Measurement System (IMS) receiver at Thule. Electronic Systems Center (ESC) rovided data acquired by AFRL from the AN/ GMQ-35 Ionosheric Measuring System (IMS) unit at Thule, Greenland. IMS is currently owned and oerated by the Air Force Weather Agency (AFWA) and maintained by Detachment 11, Sace and Missile Systems Coyright 2004 by the American Geohysical Union /04/2002RS Center (SMC Det 11), suorted by AFRL. The GPS measurements san October 2000 through March 2001, with 1 day of asses for each month (3 days in January). We will show that these data agree very well with the redictions of a simle 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 extraolation of WBMOD to larger scales. The benefit of this aroach is that the TEC model subsumes the climatology of WBMOD, which is built uon two solar cycles of ionosheric measurements. [3] The high-latitude ionoshere 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 olar cas. The olar lasma atches are tyically km in size and may travel at seeds in excess of 1 km/s. Smaller-scale structures are roduced as lasma instabilities (e.g., ~E ~B gradient drift instability) break u these larger-scale structures during transort [Tsunoda, 1988; Kelley et al., 1982]. This small-scale structure is generally geomagnetic field aligned and is resonsible for roducing scintillation of radar signals. These amlitude and hase fluctuations are the result of distortions imosed on the wave as it roagates through 1of8

2 Figure 1. Chatanika radar image of a olar lasma atch [from Tsunoda, 1988]. Contours are labeled by lasma frequency in MHz. the irregularities. Additionally, Sun-aligned lasma arcs have been discovered, which swee across the olar region with lasma density enhancements an order of magnitude larger than the background [Chang et al., 1998]. These arcs, which sometimes stretch across the entire olar ca, are tyically km across and move transversely over the olar ca at seeds u to 300 m/s. The large-scale structures (olar lasma atches and arcs) cause variations in the TEC of radar lines-of-sight to targets of interest, causing variable ionosheric-induced delay. Scintillation and TEC variations can degrade UHF radar detection and acquisition, tracking, and coherent rocessing. [4] Figure 1 dislays an incoherent-scatter radar measurement of a olar lasma atch as it detaches from the northern edge of the subauroral trough (measured by the Chatanika radar in Alaska). The contours reresent electron density (as critical frequency in MHz) obtained by incoherent rocessing of radar returns. The shaded area reresents the resence of meter-scale irregularities filling the large-scale olar lasma atch. In Figure 1 the tick-mark searations are 100 km in both dimensions, so one can see that this atch exceeds 400 km in horizontal length and reresents a substantial localized enhancement in electron density. If not roerly accounted for, radar range determinations for lines-of-sight traversing this atch will be substantially in error. Furthermore, as the radar line-of-sight asses into (or out of) this enhancement, the effective time variability of TEC will result in an artificial Doler contribution which must be accounted for to obtain accurate trajectory estimates. [5] Figure 2 shows UV images of a discrete auroral arc sanning the auroral oval (theta aurora, measured by the DE-1 imager) as it transits the olar ca. The images are about 1 hour aart with the arc moving from the dusk side toward the center of the olar ca. Like the olar lasma atches, these features are tyically associated with localized order-of-magnitude electron density enhancements. The effects on UHF radar range and Doler determination are the same as with lasma atches. [6] Models exist for the small-scale ionization structure that develos from these larger structures. Currently the best such model is WBMOD [Secan, 1993]. We seek a corresonding model for the morhology of the olar ionosheric TEC structure. Additional kinds of data will ultimately be required to build a comlete TEC model (e.g., higher data rate swath data from beacon satellites to determine TEC gradients and satially diverse multireceiver data to determine satial joint statistics). 2. Analysis of EWR and IMS GPS Data [7] Previous analyses of Thule EWR TEC measurements were erformed as art of the Ionosheric Range Delay Study [Kne and Brown, 1988], and exressions were derived for the exected 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 Dr = r 1 r 2 between returns from the same target at two different frequencies f 1 and f 2, the TEC is obtained as TEC ¼ 2Dr ; ð1þ r e c 2 1 f f 2 2 where r e is the classical electron radius and c is the seed of light. For a single ulse air of the EWR the uncertainty in TEC is s TEC = 5 TEC units for 25 db SNR (one TEC unit is electrons er square meter). Twenty ulse airs must be averaged to obtain s TEC = 1 TEC unit. Tyical TECs are 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 erformed 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 ionoshere occurs near its eak height and ignoring horizontal gradients (obviously a weak assumtion), one may convert slant 2of8 Figure 2. Hourly UV images of a discrete auroral arc [from Chang et al., 1998].

3 the Fourier transform of the TEC time histories. That is, given the TEC sectrum F TEC ðf Þ¼ Z 1 1 TECðtÞe i2ft dt; ð3þ we obtain the TEC PSD as PSD TEC ð f Þ¼ Fð f ÞF*ð f Þ T TEC : ð4þ Figure 3. Derived TECs from EWR data. The scatter oints are ulse-by-ulse derived slant TEC. The uer curve is the 20-oint smoothed slant TEC. The lower curve is the equivalent vertical TEC derived from the smoothed slant TEC. The stem oints reresent the standard deviation of the TEC over each smoothing interval. TEC to an equivalent vertical TEC (here denoted TEC V ) using the geometric formula sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TEC V ¼ TEC 1 R e cos q 2 ; ð2þ R e þ h max The divisor T TEC is included as a normalization factor such that the integral of the PSD over all frequencies yields the TEC variance. [11] We chose to emloy the fast Fourier transform (FFT) to obtain the TEC sectra. This means that the data must be resamled to rovide equally time-saced values for the FFT. This was done by interolating the data of each ass with a cubic sline. When using the sline as a relacement for the original data, it must be borne in mind that the original data were samled at a maximum rate of 2 Hz. Thus the sectrum obtained from the interolating sline must only be considered valid for frequencies below 2 Hz; larger frequency comonents are simly an artifact of the sline interolation. It is also imortant that the time series satisfy the assumed eriodicity of the FFT; and thus a window function must be emloyed. We chose a simle sine-squared window function. We therefore obtain the TEC sectra as where q is the elevation angle of the radar line-of-sight at the radar, R e is Earth s radius, and h max is the height of the eak of the ionosheric electron density. In the figures that follow, we take h max = 300 km. [9] Figure 3 shows a tyical examle of the TECs derived from the EWR data. The TEC is given in TEC units (TECu). Four quantities are dislayed. The scatter oints are the individual two-frequency ulse air slant TEC values (which, owing to the inherent inaccuracy of the individual measurements, sometimes yield negative TEC values). The uer line is the 20-oint running average slant TEC. The lower line is the equivalent vertical TEC, again using a 20-oint running average. The stem oints are s TECV, the standard deviation of the vertical TEC obtained over the 20-oint running average. Localized TEC enhancements are evident in Figure 3, and these are resumably associated with olar lasma atch ionization enhancements. [10] The ower sectral density (PSD) of the smoothed vertical TECs from the EWR data can be obtained from 3of8 F TEC ðf Þ¼ T dur FFT½TECðtÞW ðtþš; N ð5þ where N is the order of the FFT and T dur is the duration of the ass. The window function is given by W ðtþ ¼sin 2 t T dur : ð6þ [12] The PSDs of all 60 asses 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 ower law behavior. Linear fits of log 10 (PSD TEC ) with log 10 (f) for frequencies below 1 Hz yield a median sloe of, where = 2.68 over the 60 asses. The curve in Figure 4 is a ower law model for the TEC PSD using this sectral sloe arameter. We use a twocomonent ower law in order to truncate the PSD for

4 Figure 4. Power sectral densities derived from EWR data for 60 asses and median ower law model (overlying curve). frequencies above the data-samling rate. Our TEC fit has the functional form PSD Fit ðf Þ¼2 ffiffiffi Gð 2 Þ Gð Þ s 2 TEC T TEC 1 þ TTEC 2 f h i 2 =2 5 ; ð7þ 1 þð0:75f Þ 8 where s TEC = 2.52 TECu and T TEC = 1496 s. T TEC is a large timescale that we will later show is determined by a TEC scale size arameter L TEC and an effective scan velocity. Again, the second ower law regime is constructed to rovide a stee taer for frequencies above the data-samling rate. [13] It is interesting to note that the sectral sloe obtained here is consistent with the sectral sloe obtained from hase scintillation data derived from satellite beacon signals. Since TEC is a line-integrated quantity derived from differential hase measurements, its sectral sloe arameter is directly comarable to the hase sectral sloe used to describe small-scale irregularity structure. WBMOD [Secan, 1993], which is based on fits to hase sectra obtained from satellite beacon data, uses a sectral sloe in the olar/auroral region of auroral = 2.7. The consistency of the EWRderived TEC sectral sloe with that of WBMOD s small-scale sectral sloe will next be shown to extend to the IMS GPS data set. [14] We have erformed a similar analysis using the AFRL IMS GPS-derived TEC data. As with the EWR data, we fit the sectrum from each GPS ass in the data set with a ower law PSD. Cumulative distributions of 4of8 the sectral sloes and TEC standard deviation are shown in Figure 5. Note in Figure 5 that at the 50th ercentile level, the value of the sectral sloe is very near WBMOD s high-latitude value of auroral = 2.7. The distribution of sectral sloe values is also in agreement with the database of small-scale sectral sloes used in the develoment of WBMOD (J. A. Secan, rivate communication, 2001). [15] The values of the vertical TEC standard deviation from the IMS data (Figure 5) are also in reasonable agreement with the redictions of WBMOD. Using the relationshi (derived in the next section) of s TEC to WBMOD s C k L strength arameter (sectral strength at a 1 km scale size; see Secan [1993]), we find that a tyical WBMOD-redicted value of C k L = (tyical olar value for 90th ercentile robability of occurrence at high sunsot number) and using a value of L TEC = 500 km (see the next section) yields s TEC = 5 TECu (or log(s TEC ) = 0.7), which lies at the 90th ercentile level of Figure 5. [16] It is gratifying that the EWR and IMS data are fully in agreement with WBMOD when redominantly small sectral scales are considered. The TEC model we seek, however, is meant to aly to intermediate scale structure; that is, to satial scales that exceed the outer scale size L o of the small-scale ionization structure. The outer scale size, which is a arameter in the ower law PSD descrition of small-scale ionization structure used in WBMOD, is of the order L o 10 km. It should not be assumed that the sectral sloe at small scales is the same at intermediate scales. (If it were, then the TEC model could be obtained by simly extending the outer scale size to a larger value.) As we develo 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 roduces a slightly steeer sectral sloe, and the scale size L o reresents the lace in the sectrum where the break to steeer sloe occurs. 3. TEC Structure Model [17] The results of the last section rovide an argument in favor of the notion that a TEC structure model can be obtained as a suitable extraolation of WBMOD to larger scale sizes. In this section we develo this aroach. [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 hase sectral model of small-scale ionization structure. WBMOD is built uon the formulation of Rino [1979]. In Rino s formulation, the hase PSD of transionosheric signals is given by PSD f ðf Þ¼ T Rino ½ f 2 o þ f 2 Š =2 ; ð8þ

5 Figure 5. data. Cumulative distribution in ercent of s TEC (a) and sectral sloe (b) from AFRL IMS where T Rino is a strength arameter and f o is the frequency scale associated with the outer scale size of the electron density fluctuations. The strength arameter T Rino is given by ffiffiffi T Rino ¼ðlr e Þ 2 Gð2 Þ ð2þ þ1 Gð 2 þ 1 2 Þ v 1 eff GC s L: ð9þ 5of8 Here l is the field wavelength, r e is the classical electron radius, L is the ionosheric layer thickness, C s is a strength arameter for the electron density fluctuations (related to WBMOD s C k L), and v eff is an effective scan velocity (the velocity of the line-of-sight through the generally moving lasma). G is a geometrical enhancement factor that accounts for the geomagnetic fieldaligned character of small-scale ionization structure. In the following we will take G = 1 because, at least for the resent, we do not want the TEC model to deend on geomagnetic field line geometry. [19] The TEC frequency sectrum is obtained from equations (8) and (9) by dividing away the (lr e ) 2 and setting G = 1. We also choose to define a timescale by T TEC = 1/f o, and we recast the strength arameter in terms of the standard deviation of the TEC fluctuations to obtain PSD TEC ðf Þ¼2 ffiffiffi Gð 2 Þ s 2 Gð TEC T TEC Þ ½1 þ TTEC 2 f 2 Š : =2 ð10þ

6 Figure 6. Cumulative distribution in ercent of s TEC (a) and sectral sloe (b) from AFRL IMS intermediate-scale data. The standard deviation of the TEC fluctuations s TEC is given by relationshi v eff = (2f )/k and by defining the TEC scale size arameter, s 2 TEC ¼ 1 Gð2 1 2 Þ 2ð2Þ þ1 Gð 2 þ 1 2 Þ v 1 eff T 1 TEC C sl: ð11þ L TEC ¼ v eff T TEC : ð12þ 2 [20] Equations (10) and (11) define the TEC frequency/time sectral model. This can be readily converted to a satial sectral model by using the In order to roughly model the EWR and IMS data resented in the receding section, we take L TEC = 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 6of8

7 Figure 7. Realizations generated as real and imaginary arts of a single comlex realization using the TEC model. multily by the determinant of the Jacobian of the transformation, which in this case is simly v eff /2, we obtain PSD TEC ðkþ ¼2 ffiffiffi Gð 2 Þ s 2 Gð TEC L TEC Þ ½1 þ L 2 TEC k2 Š ; =2 ð13þ where we may rewrite the TEC variance of equation (11) as s 2 TEC ¼ Gð Þ Gð 2 þ 1 2 Þ L 1 TEC C sl: ð14þ The strength arameter C s L is related to the WBMOD strength arameter C k L by C s L ¼ 2 þ1 C k L: ð15þ 1000 [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, L o. We have fit the IMS data using only these larger scales, i.e., with the data restricted to frequencies below Hz. This corresonds to timescales exceeding 63 s, which corresonds to the line-of-sight traversing 2L o = 63 km with an effective velocity of v eff = 1 km/s. This is a tyical value for the lasma drift seed in the olar region (note that the line-of-sight motion through the lasma due to the GPS satellite seed is negligible). [22] Figure 6 shows the cumulative distribution functions for the intermediate-scale fits of the IMS data using equation (10). Comarison of Figures 5 and 6 shows that for the intermediate-scale structure, the sectral sloes are somewhat larger at the 50th ercentile level than for the small-scale case. For the resent we shall follow WBMOD in using a fixed sectral sloe for the TEC structure model, which we take to be the 50th ercentile value = of8 [23] Equations (10) (15) with L TEC = 500 km and = 2.94 constitute the TEC structure model. A roviso for the use of this model is that it is meant to describe intermediate scale structures, and thus it alies only to scales larger than the outer scale size arameter L o of the small-scale ionization structure formulation (L o 10 km). One way to require this formally would be to incororate a second ower law regime (as we did in equation (7)) that is designed to make the PSD dro steely for scale sizes smaller than L o. For simlicity, we have chosen not to do this, and hence it behooves the user of this model to be aware of its regime of alicability. 4. TEC Realizations [24] The TEC PSD model can be used to generate realizations of TEC for use in simulations. The rocedure to generate the realizations is as follows. Given two sequences of Gaussian random numbers, g 1k and g 2k (with zero mean and unit variance), where 1 k N (N is the order of the FFT), form the comlex random numbers r k ¼ðg 1k þ ig 2k Þ: ð16þ A realization of the TEC sectrum is then generated by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F k ¼ r k PSD TEC ðf k Þ T dur ; ð17þ 2 where T dur is the desired time duration of the realization and f k =(k 1)/T dur. The realization is then given by TEC k ¼ 2 T dur FFT 1 ðf k ÞþTEC 0 : ð18þ [25] The quantity TEC 0 is a user-added offset sulied to ensure that the TEC never dros below a secified minimum value. Since the FFT of equation (18) roduces a comlex result, the realization actually contains two indeendent real realizations given by Re (TEC k ) and Im Figure 8. IMS GPS equivalent vertical TEC from 14 October 2000.

8 (TEC k ). The times for the TEC realizations are given by t k =(k 1)T/N. [26] Figure 7 shows two realizations generated as the real and imaginary arts of equation (18), where an offset of TEC 0 = 5 TECu was used. Here we truncated the sectrum at a timescale of 63 s so as to avoid the generation of small-scale structure below the satial scale of 10 km. Comarison of the structure of these realizations with the IMS GPS data samle of Figure 8 (which has been converted to equivalent vertical TEC) shows that the model has successfully catured the behavior of the data. 5. Summary [27] A model for the intermediate-scale structure of the ionosheric TEC of the olar region has been develoed as an extension of WBMOD. The model is based on a ower law PSD with a strength arameter related to the C k L strength arameter of WBMOD. As such, the TEC model subsumes the climatology of WBMOD, which was formulated using an extensive data set sanning two solar cycles. [28] Two-frequency data from both the early warning radar and AFRL Ionosheric Measurement System GPS receivers at Thule, Greenland, were used in the TEC model develoment. These data sanned a 6-month eriod during the winter of , a time near solar maximum. The fact that these data were found to be in excellent agreement with the redictions of WBMOD for the geohysical conditions of this time eriod is taken as an indication that the TEC model may be reasonably extended to other geohysical/solar conditions using the climatology of WBMOD. It is, however, desirable to revisit this assumtion as more data becomes available over the course of the next solar cycle. [29] The TEC model of this aer is limited in a number of resects. The limited nature of the data available to us did not allow for any study of the joint statistics of TEC structure for satially searated regions, and thus no accounting is given for the well-known large-scale structure of TEC in the olar region and the associated convective flow. Rather, the simle PSD aroach of the current model generates realizations with TEC structure that are homogeneous on a large scale. A related limitation of the PSD aroach is that, when generating random realizations of TEC structure from the model, the ower law nature of the PSD is manifested as structure with a range of scale sizes. However, it is ossible for smooth intermediate-scale size structures with stee gradients to manifest the same PSD ower law behavior, and such structures are in fact observed at times in the olar region. The method of statistical signal generation alied in section 4 to generate random realizations from the TEC model cannot generate such smooth, stee-sided structures, but will always generate structure-filled TEC enhancements. [30] This TEC model alies only to scale sizes that are larger than the outer scale size of the small-scale ionosheric irregularity structure, a size of about 10 km. For scales smaller than this the full geomagnetic fieldaligned formulation of small-scale ionization structure should be used, as in the work of Rino [1979]. [31] Acknowledgments. The author would like to thank Jim Secan and Ed Fremouw of NorthWest Research Associates for their helful discussions concerning this work. References Chang, S. W., W. J. Burke, N. C. Maynard, and J. D. Scudder (1998), The re-formation of the olar ca during quiet times: The theta aurora, Eos Trans. AGU, 79, 269. Kelley, M. C., J. F. Vickrey, C. W. Carlson, and R. Torbert (1982), On the origin and satial extent of high-latitude F region irregularities, J. Geohys. Res., 87(A6), Kne, D. L., and W. A. Brown (1988), Ionosheric range delay study, art 2, Bmews site I, Final Tech. Re., Contract F C-0113, Raytheon, Waltham, Mass. Rino, C. L. (1979), A ower law hase screen model for ionosheric scintillation: 1. Weak scatter, Radio Sci., 14(6), Secan, J. A. (1993), WBMOD ionosheric scintillation model An abbreviated user s guide, Re. NWRA-CR-93-R098, NorthWest Res. Assoc., Bellevue, Wash. Tsunoda, R. T. (1988), High-latitude F region irregularities: A review and synthesis, Rev. Geohys., 26(4), L. J. Nickisch, Mission Research Cororation, 10 Ragsdale Dr., Suite 201, Monterey, CA , USA. (nickisch@ mrcmry.com) 8of8