A study of the equatorial anomaly ionosphere using tomographic images

Authors


Abstract

[1] Tomographic techniques are applied to relative total electron content (TEC) data obtained using radio signals transmitted by the polar orbiting Navy Navigation Satellite System (NNSS) and received at a chain of six stations located at approximately 121°E longitude. Images reconstructed for each satellite passage provide a picture of electron density over a 25° latitude range and to an altitude of about 1000 km at a longitude of 121°E. Over 350 ionospheric images of the northern equatorial anomaly region have been reconstructed for October/November 1994. The daytime equatorial anomaly crest is a prominent feature of the reconstructed images. The rms difference between foF2 computed from tomographic images and foF2 measured by two ionosondes is found to be less than 12%. Causes for the differences are discussed. We show how the crest develops on average days in October and November and discuss the nature of seasonal variations. The fully developed anomaly core is aligned approximately along geomagnetic field lines, resulting in the existence of strong and directional gradients in the anomaly region. As a consequence, the crest latitude and the maximum vertical TEC (VTEC) latitude are noncoincident. The tilt of the anomaly core also causes the slant TEC integrated through the crest to be highly directional. In fact, the use of spherical stratification in predicting slant TEC based on the knowledge of the vertical TEC may give an error as large as 30 to 50% in some directions. Implications for GPS ranging measurements are discussed.

1. Introduction

[2] About fifteen years ago, a new experimental tool was proposed and shown to be capable of reliably imaging the ionospheric density structures. By applying the tomographic technique to satellite radio beacon data recorded at a chain of stations, the feasibility of reconstructing ionospheric structures was demonstrated. Since then, several receiver chains have been established specifically to take advantage of this new experimental tool. Some interesting results have been obtained. Readers interested in the historical account, details of the tomographic technique, and the early results should consult the reviews by Kunitsyn and Tereshchenko [1992] and Leitinger [1999]. One such chain is the low-latitude ionospheric tomographic network (LITN). The LITN consisted of six stations nearly aligned along the 121 ± 1°E meridian, ranging from 14.6°N to 31.3°N in geographic latitude or 3.3°N to 19.7°N in geomagnetic latitude. The specifics of each LITN station, data processing techniques and early results have already been published in a sequence of papers [Yeh et al., 1994; Huang et al., 1997, 1999; Franke et al., 1999; Andreeva et al., 2000, 2001; Yeh et al., 2001] and will not be repeated here. In this paper we report some new results based on tomographic images reconstructed from the LITN database for October/November 1994, a solar minimum year.

[3] In section 2, we first compare the foF2 obtained from tomographic reconstructions with the foF2 observed by two ionosondes. The root-mean-square dispersion between these two sets of foF2 data is generally less than 12%. The experimental results are given in section 3, which is organized into three parts: the development of the equatorial anomaly, the degree that the latitude of the maximum VTEC is noncoincident with the latitude of the anomaly crest, and some specific properties of the slant TEC resulting from strong gradients near the anomaly crest. Finally, the paper is discussed and then concluded in section 4.

2. Comparison With Ionosonde Data

[4] In processing the LITN data, an algorithm making use of the phase difference is used [Kunitsyn et al., 1994, 1995; Andreeva et al., 2001]. The advantages of phase difference radio tomography have been amply discussed in these referenced papers. They demonstrate that the phase difference method is better in imaging the horizontal structures than either the total phase or the relative phase method. In implementing the data inversion, a priori knowledge about any other ancillary ionospheric data can be incorporated into the algorithm. But, in this work, even though independent data from two ionosondes is available, we do not use the ionosonde data in the tomographic inversion. Thus, the ionospheric images so reconstructed are completely independent of the ionosonde data. In the literature, several comparison studies between the tomographically reconstructed ionospheric images and other ionospheric measurements have been reported [Kersley et al., 1993; Raymund et al., 1993; Foster et al., 1994]. Generally, they yield good agreement. However, all the earlier comparisons were carried out for a few sample cases only. In contrast, the comparisons carried out in this work are done on statistical basis and encompass hundreds of cases.

[5] Two ionosondes are situated long the LITN receiver chain. One ionosonde is situated at Lunping (25.0°N, 121.2°E), which is only 7.5 km away from Chungli, a tomography station in the middle of LITN chain. Another ionosonde is situated at Manila (14.7°N, 121.1°E), which is at the southern end of the LITN chain. The simplest quantity to be compared is foF2, which is experimentally measured, routinely scaled, and tabulated at these two ionosonde stations. The hourly tabulations have been obtained from the National Geophysical Data Center for this comparison. Using tomographic images, the peak densities and hence foF2 can be calculated at the same latitude as Lunping and Manila. These two sets of foF2 values are compared in Figures 1 and 2 for the months of October and November 1994, respectively. On the plot, a 45°-line is drawn to show the line of exact agreement between tomographic foF2 and ionosonde foF2. Thus, a point away from this line indicates a discrepancy. Ideally, in making the comparisons, the two sets of foF2 should be obtained at the same time and at the same location. This is not possible. The passage of an NNSS satellite is dictated by the satellite orbit and usually does not occur on the hour. The hourly ionosonde foF2 data must therefore be interpolated for the fractional hours of the satellite passage time. Additionally, the satellite almost never passes overhead of either of the two ionosonde stations. It commonly passes several hundred kilometers to the east or to the west of the station. Again, some interpolation scheme is needed. This is done by assuming that the longitude dependence can be interpreted as a time-dependence. Such simplified interpolations have improved the agreement between the two sets of foF2 data, especially on occasions of high foF2, but the ionosphere varies in a complex way and is known to depend on time as well as the longitude. The imperfect interpolation schemes therefore become sources of error in the comparison study. In spite of these imperfect conditions, the experimental data points seem to follow the 45° line fairly closely as shown in all four plots in Figures 1 and 2. A closer examination of these figures reveals two interesting features:

  1. There is a slight indication that the scatter of data points from the 45°-line at Lunping is larger than those at Manila for both months. To show it quantitatively, we define the dispersion as the percent root-mean-square deviation of the corresponding tomographic foF2 relative to the ionosonde foF2. The calculated dispersion for October 1994 is 11.5% at Lunping versus 8.8% at Manila. Correspondingly, the dispersion for November 1994 is 10.3% at Lunping and 7.9% at Manila.
  2. During times of high density, especially when foF2 exceeds 13 MHz, the experimental points clearly show a saturation trend. That is, the ionosonde foF2 is consistently larger than the corresponding tomographic foF2. Furthermore, the high electron density points are more numerous in October than those in November at either Lunping or Manila, suggestive of seasonal dependence.
Figure 1.

A comparison of foF2 computed tomographically versus foF2 reported by the ionosonde stations for October 1994. The upper panel refers to the Lunping station, the lower panel the Manila station. In each case, the latitude selected from the tomographic image corresponds to that of the ionosonde station.

Figure 2.

Same as Figure 1, except for November 1994.

[6] Both of these interesting features support the idea that strong spatial gradients present in the anomaly region may be responsible for producing disagreement between tomographic foF2 and ionosonde foF2. This point is discussed and further amplified in section 4 later. In view of the nonideal conditions discussed above, the comparisons, in our view, yield adequate agreement between tomographic foF2 and ionosonde foF2 for all four sets of data carried out in this study.

3. Experimental Results

[7] In this section, we present some results based on the more than 350 reconstructed images for the months of October and November 1994. We show first the hourly contour plots of the averaged Nmax as a function of latitude and local time for each month. Next, we show some plots of the crest latitude versus the latitude of maximum VTEC. Lastly, we show some specific properties of slant TEC near the anomaly core that result from the tilted nature of ionization contours in this region.

[8] Corresponding to each satellite pass, an ionospheric image is tomographically reconstructed at the longitude of 121°E over a latitude range from about 12°N to 30°N. From these images, the peak density values Nmax can be scaled for all the latitudes in the range. At a given latitude, the average Nmax for a given hour is computed by averaging all values of Nmax in a one-hour window centered about that hour. Based on these data, contours or images of Nmax on the local time versus latitude plane can be plotted. The results for both October and November months are depicted in two panels of Figure 3. Such contours, if plotted, will show very little structure after local midnight except for continued decrease of ionization density until about 08:00 LT. Similarly, very little structure is observed after 18:00 LT except for continued decrease of electron density to midnight. For these reasons the contours shown in Figure 3 are confined to 10:00 to 18:00 LT only. For the month of October, the upper panel shows that the anomaly crest already exists at around 18.5°N at 10:00 LT. The crest then moves northward as it intensifies. On the average, the crest is most intense and reaches its northern most point at 22.5°N at 13:00 LT. Thereafter, the crest moves southward to 20°N and then returns to northward motion, reaching 22°N at 16:00 LT. The crest finally recedes southward as it weakens until 18:00 LT or shortly afterwards. Near the anomaly region, this north-south undulation of the crest in the afternoon is responsible for producing a double humped diurnal variation of Nmax at a given latitude, as can be seen in Figure 3. In gross features, the observed behavior is similar for the month of November as depicted in the lower panel of Figure 3, except for some significant departures enumerated in the following. When compared with October, the morning development of the anomaly crest in November is later by about one hour and, once triggered, the crest develops faster. The motion of the anomaly crest is faster in October than in November, resulting in a higher crest latitude of 22.5°N in October versus 20.0°N in November, both at 13:00 LT. Consistent with well known seasonal variations, the ionization density in November is generally slightly lower than that in October. In the afternoon, the anomaly crest largely disappears shortly after 17:00 LT in November while the crest still lingers even after 18:00 LT in October. Many of these behaviors have been reported previously [Yeh et al., 2001], except in the current work the months October and November are separated and the averaging is carried out over one hour instead of two hours so that more details are revealed here.

Figure 3.

Contours of average Nmax for the month of October 1994 (upper panel) and November 1994 (lower panel). The average Nmax is computed by taking the mean of Nmax values in a one-hour window centered on the hour at a given latitude.

[9] The original investigations of the equatorial anomaly were carried out by using foF2 or Nmax, as was done in its early discovery [Appleton, 1946; Liang, 1947] and later works as summarized and reviewed by several authors [e.g., Rao, 1963; Rastogi, 1966; Raghavarao et al., 1988]. When the geo-stationary satellite ATS 6 was moved to 34°E longitude in 1975, eight stations were set up to measure VTEC from 8.5°N geographic latitude (or 0.5°S geomagnetic latitude) to 30.3°N geographic latitude (or 24.5°N geomagnetic latitude). The unprecedented close spacings of these stations provided a unique opportunity to study latitudinal structures across the anomaly region in the Indian sector. Detailed results have been published [Rastogi and Klobuchar, 1990]. They show that, by and large, the latitudinal VTEC behavior mimics the latitudinal Nmax behavior studied earlier [Rastogi, 1966; Raghavarao et al., 1988], but there also exist some important and detailed differences between the two quantities. For example, the daily variation of Nmax at the anomaly crest exhibits a double hump as is also observed in our data shown in Figure 3, while that of VTEC reveals only one hump [Huang et al., 1987]. The latitude of VTEC maximum is slightly different from the latitude of the crest, how different is hard to quantify for lack of spatial resolution. The crest-to-trough ratio in Nmax is much higher than that in VTEC. In an earlier publication [Andreeva et al., 2000], the ionospheric images for three daytime examples were given. These reconstructed images had a spatial resolution in the neighborhood of 50 km. One prominent feature appearing in all three images is the tilted anomaly core. One consequence of the tilted anomaly core is that the latitude of VTEC maximum does not generally coincide with the latitude of the crest. As a matter of fact, for the three examples published previously [Andreeva et al., 2000], the VTEC maximum occurs at a latitude lower than that of the crest by 2.5° to 4.5°. This results from the thickened ionosphere on the equatorward side of the crest. Thus, it is of interest to carry out a statistical study to show the difference between these two latitudes. This is done in Figure 4. For this figure, the data points in one day are binned into six four-hour intervals, each bin contains four hours as indicated in the plot. The regression line through the experimental points is drawn as a solid line, while a dashed line shows the condition of exact agreement between the latitude of the VTEC maximum and that of the crest. After midnight but before 08:00 LT, the maximum in VTEC and the maximum in Nmax may occur on individual days. When they do occur, they are not repeated from one day to the next and do not show any systematic trend in gradients. On averaging, the maxima that occur on individual days get smeared out except for the continued decay into early morning hours. This is why the solid line and the dashed line nearly coincide in the two time intervals 00–04 LT and 04–08 LT. Beginning at the next time interval, from 08–12 LT, the regression line shows a systematic departure from the exact agreement line. The noncoincidence of the solid line from the dashed line indicates a departure of about 1° in latitude. This departure increases in the next two time intervals and becomes about 1.8° in the time interval 16–20 LT. In the last time interval, 20–24 LT, the departure shrinks to very small values as shown by the frame on the right lower corner of Figure 4. The fact that, in the daytime, the VTEC maximum occurs at a latitude lower than the crest latitude suggests that the ionosphere thickens toward the equator on a statistical basis. A mean value of about 1° to 1.8° lower is somewhat smaller than the three sample cases reported earlier [Andreeva et al., 2000].

Figure 4.

Latitude of VTEC maximum (horizontal axis as denoted by LTmax) versus latitude of the crest (vertical axis as denoted by LNmax) in six four-hour intervals for October/November 1994. The solid line refers to the regression line and the dashed line corresponds to an exact agreement between the latitude of VTEC maximum and the latitude of the crest.

[10] The tilted nature of the anomaly core suggests that TEC values integrated through this region may be very directional. This is investigated here. In our computations, we adopt a geometry depicted in Figure 5. For this latitudinal cross section at a longitude of 121°E, two lines are drawn through the anomaly crest: one radially showing the local vertical and another slanted at a zenith angle χ. The angle is reckoned positive when the slant line is tilted northward as shown in Figure 5 and negative when the line is tilted southward. The slant TEC is then computed through the reconstructed ionosphere for both positive and negative χ angles. Superposed on the geometry plot of Figure 5 is the image of the ionospheric density for September 8, 1994, one of the three days for which the image was published in the work of Andreeva et al. [2000]. Along the ground surface, the six stations of LITN chain are marked. The computed slant TEC for these three examples are shown in the top panel of Figure 6. As expected, each of the three curves rises very quickly from the minimum near χ = 0. If the ionosphere is horizontally stratified, such rise should follow the sec χ behavior exactly. For a spherically stratified ionosphere, the behavior generally depends on the vertical ionization distribution and hence can become very complicated. If however χ is small, sec χ dependence is still approximately correct up to about 68.5° at which a 10% error is expected. However, the equatorial ionosphere is not spherically stratified. The tilt of the anomaly core shown in Figure 5 is very obvious. Thus, the curves in the top panel of Figure 6 depart from the simple sec χ behavior even for small χ. In order to show this departure, a ratio rT is computed. The numerator of rT is the slant TEC through the crest for a given χ, i.e., it is just the value computed in the top panel of Figure 6. The denominator of rT is the slant TEC computed through a spherically symmetric ionosphere whose height distribution is identical to that at χ = 0. When so defined, rT is unity when χ = 0 as seen at the bottom panel of Figure 6. Computations for rT have been carried out for these three reconstructed images. When χ departs from 0, the ratio rT also departs from unity; rT decreases as χ increases and rT increases initially as χ decreases to a maximum value at χmax after which rT decreases. For the geometry shown in Figure 5, when Icrest − χ = 90° (where Icrest is the dip angle at the crest) the slant line is exactly parallel to the geomagnetic line. It is known that the location of the crest depends on the equatorial dynamo. As the strength of the dynamo field varies daily the latitude, and consequently the magnetic dip, of the crest will change accordingly. Thus the test for parallelism must be carried out on a case by case basis. For three examples depicted in Figure 6 the computed results are displayed in Table 1. They show that maximum TEC is obtained when Icrest − χmax is within 3° to 4° of 90°. Thus the alignment of the crest along geomagnetic field lines is fairly precise. At the angle, where rT is maximum, its value varies from 1.29 to 1.36 for the three examples (see bottom panel of Figure 6). This means that if one uses the spherically symmetric ionosphere model in predicting the slant TEC from a known VTEC value, one will have committed an error of 29 to 36%. The computations in Figure 6 were carried out for a reference point located at the anomaly crest. Similar computations can also be carried out using a reference point at the height of peak density but shifted ±2° north or south of the crest. The ratios in these two cases are presented in Figure 7. From the curves presented, the deviations from the prediction using spherically symmetric ionosphere can vary, depending on the angle χ, from −30% to +45% for a point north of the crest and from −20% to +27% for a point south of the crest. Similar computations for a point displaced by ±3° and ±4° in latitude from the crest have also been carried out. The results depend critically on the ionization distribution and do not show any systemic behaviors. These curves will not be presented. Suffice it to say that some errors can exceed ±50%.

Figure 5.

The geometry used in computing the slant TEC in a latitudinal cross section at 121°E. The zenith angle χ is reckoned positive when the line is slanted northward as shown and negative when the line is slanted southward. Superposed on the geometry is the image of the ionosphere reconstructed for 13:50 LT, September 8, 1994 [Andreeva et al., 2000]. Marked on the Earth surface are latitudes of six tomographic stations, from south to north: Manila, Baoguio, Kaoshiung, Chungli, Wenzhou and Shanghai.

Figure 6.

The upper panel depicts the slant TEC as a function of the zenith angle χ for three examples published by Andreeva et al. [2000]. The lower panel shows the ratio rT of two slant TECs computed for these three examples. The numerator of rT is the slant TEC integrated through the crest of a given reconstructed image as a function of χ for each example. The denominator of rT is the slant TEC through a spherically stratified ionosphere with a distribution identical to the reconstructed ionosphere at χ = 0 for that example.

Figure 7.

The upper panel depicts the ratio rT as a function of χ through a point of Nmax that is 2° north of the crest. The lower panel is similar to the upper panel, except that it is at a point of Nmax 2° south of the crest.

Table 1. Degree of Alignment of Three Fully Developed Equatorial Anomalies With Respect to the Local Geomagnetic Field Linesa
DateχmaxIcrestIcrest − χmax
  • a

    Exact alignment is achieved when Icrest − χmax is 90°.

8 Sept. 1994−63.5°30.4°93.9°
3 Oct. 1994−61.5°32.8°93.3°
5 Oct. 1994−60.0°33.2°93.2°

4. Discussion and Conclusion

[11] In this paper, we have presented our study of the behavior of the equatorial anomaly ionosphere gleaned from reconstructed tomographic images. Because of the high spatial resolution we are able to report some details that have apparently not been seen before. Here we wish to discuss some of these results.

[12] To gain confidence on the tomographic technique, a comparison study has been carried out in section 2. Using tomographic images, the foF2 values at the same latitude as the two ionosondes are computed and compared. Ideally, such comparisons for the two sets of data should be done at the same time and same geographic location. However, the time of satellite passage and its track are determined by orbital mechanics and are not under our control. Even after some interpolations, the situation is still not quite ideal. It is known that the probing ray of a supposedly vertical ionosonde is generally not reflected from directly overhead. Even in a horizontally stratified ionosphere, a vertical ordinary ray will bend toward the pole and, at the point of reflection, the ordinary ray becomes perpendicular to the local geomagnetic field [Forsgren, 1951; Yeh and Liu, 1972]. Thus, generally the reflection point departs from exactly overhead. The deviation distance is zero only at the equator and, for a station of Lunping's latitude, it can be about 10 km. Of course, Lunping is very nearly under the equatorial anomaly crest where the gradients are very strong. The ionosonde ray in such an inhomogeneous ionosphere will deviate fairly substantially before it is reflected back. The recorded ionosonde foF2 may therefore not be the true overhead value. These observations may help explain two aspects of our results. The first aspect has to do with data points falling below the 45°-line on occasions of large foF2 (larger than 13 MHz) revealed in Figures 1 and 2. Occasions of high foF2 refer to full development of the anomaly and hence high gradients. It may explain why ionosonde foF2 on these occasions is on the average higher than the true overhead value. The second aspect refers to higher dispersion at Lunping than at Manila in each of two months. It should be noted that Lunping (or Chungli) is in the middle of the tomographic chain. The reconstructed images are expected to be more reliable at latitudes near the middle of the chain (Chungli) than at latitudes near the end of the chain (Manila), contrary to results depicted in Figures 1 and 2. Again, Chungli is near the anomaly crest where gradients are very strong and therefore is situated in an environment for errors in both vertical sounding and tomographic imagery. More investigations of course are necessary to positively identify the sources of the observed discrepancy. In a different vein, these findings already suggest the need to exercise caution. In the ionospheric tomography literature, there have been suggestions of using all sorts of ancillary ionospheric data to aide tomographic reconstructions [Leitinger, 1999]. In doing so, one must ascertain whether such ancillary data are accurate and compatible with the tomographic data. When the error of the ancillary data is too large or there exist compatibility problems, the introduction of such data into the inversion algorithm may only degrade the reconstructed images, thus achieving an undesirable result.

[13] The daily development of the anomaly near Lunping has been reported by Huang et al. [1987] using TEC observations. In their experiment, the recorded dispersive Doppler of NNSS satellites yields the relative TEC whose initial value is provided by the Faraday rotation data on the beacon signal transmitted by the geostationary satellite ETS-2. The slant TEC is then converted to VTEC as a function of latitude for local time of the satellite passage. The interpolation of successive NNSS passages provides a contour plot of VTEC in the local time-latitude plane for that day. Several plots are given by Huang et al. [1987], one for a normal day and several for magnetically disturbed days. On January 11, 1986, their contours reveal that the anomaly crest begins to form at 09:00 LT. In the next few hours, the crest develops and intensifies as it moves to higher latitudes. It becomes fully developed at around 12:00 LT while at its northern most latitude of 22°N. Shortly afterwards, the crest begins to weaken as it recedes equatorward until its disappearance at around 16:00 LT. Such gross features are also observed in our data. However, in comparing their results with ours, one must be cautioned to note that in converting the slant TEC to VTEC near the anomaly crest region, the effects examined in Figures 6 and 7 are operative. One must therefore be cautious in data interpretation.

[14] The development of the crest using the tomographic images is depicted in Figure 3. Comparing the October (upper panel of Figure 3) contours with the November (lower panel) contours, the anomaly crest develops earlier by about 1–2 hours, moves northward with a larger speed and ends at a higher latitude (22.5° versus 20°) at 13:00 LT, is slightly more intense (1.6 MHz versus 1.5 MHz), and lingers longer into the evening hours. These possibly can all be explained in terms of seasonal behavior by carrying out a model study similar to that done by Anderson [1981] to earlier data [Rao, 1963; Rastogi, 1966]. When compared with our own results published earlier [Yeh et al., 2001], the current results show more details because the data are averaged over one hour for October and November separately instead of averaging over two hours by combining both October and November months together. Additionally, the treatment of data is different. Here, we average Nmax in one-hour intervals at a given latitude, while in the earlier paper [Yeh et al., 2001], we average, in two-hour intervals, the latitude for those days with the observed anomaly crest. In the daytime, the two data treatments yield roughly similar results because the crest is a common daily feature. However, after local sunset the results differ because the appearance of crest is no longer so common. When averaging Nmax, the crest that may appear for some days is smeared in the averaging process and thus disappears in the averaged data. This is why our Figure 2 when extended into midnight only shows a continuous decay in electron density. On the other hand, on the days with crests at night, the location of the crest given by Figure 2 of Yeh et al. [2001] still shows their possible movement.

[15] In this paper, we have also investigated the possible existence of a systematic difference between the latitude of maximum Nmax (i.e., the crest) and the latitude of maximum VTEC. We find that there is no systematic difference before 08:00 LT. In the 08:00–12:00 LT interval, the latitude of VTEC maximum is systematically displaced to the south of the crest latitude by 1°. This displacement increases in the next two four-hour intervals to 1.8° in the 16:00–20:00 LT interval, then collapses to nearly zero in the 20:00–24:00 LT interval. Now, VTEC (denoted by V in the following equation) is a product of Nmax and slab thickness τ, i.e.,

equation image

Differentiating both sides of (1) with respect to the latitude, L, and noting that at a latitude where V is a maximum, we have dV/dL = 0, we obtain

equation image

Since Nmax and τ are both positive quantities, (2) can be satisfied only when dNmax/dL and dτ/dL have opposite signs or when both gradients are equal to zero. That is, if dNmax/dL is positive (i.e., the crest appears at higher latitude than VTEC maximum as observed in Figure 4), then dτ/dL must be negative (i.e., the ionosphere thins out with the increasing latitude). Thus, the noncoincidence of the crest latitude and the maximum VTEC latitude results from the latitude dependence of the ionospheric thickness.

[16] One of the prominent features in a fully developed anomaly crest is its tilt with a rough alignment along the geomagnetic field. As a consequence, the slant TEC through the crest is highly aspect sensitive. To quantitatively show this effect, computations have been carried out as depicted in the upper panel of Figure 6 for three examples. To separate the obliquity factor a ratio rT is computed. The numerator of this ratio is the slant TEC through the crest, while the denominator is the slant TEC through the same point by assuming a spherically symmetric ionosphere. When so defined, rT is unity for an overhead ray, i.e., χ = 0, and also unity for all values of χ if the ionosphere is truly spherically stratified. Thus, the departure of rT from unity indicates the degree of horizontal inhomogeneity. Computational results depicted in Table 1 show that rT maximizes with a slant angular deviation of only 3° to 4° from exact alignment with the geomagnetic field. Such a small departure from being exactly parallel to the magnetic field lines shows the the dominating influence of the equatorial fountain. Along the magnetic field lines rT peaks at about 1.30 to 1.36 for the three examples. These results have applications in GPS ranging [Klobuchar et al., 2001]. At GPS frequencies, every slant TEC increment of 1018 electron/m2 produces an excess time delay of 52.5 nanoseconds or an excess range of 15.75 m. If one uses a spherical stratification hypothesis in predicting slant TEC from VTEC, a 30% to 36% error will be made. This corresponds to an error of 2.77–5.29 × 1017 electrons/m2 in slant TEC or 14.3–27.8 nanoseconds in excess time delay or 4.3–8.35 m in excess range. Depending on applications, such errors can be important. Computations carried out at a point away from the crest show even larger errors are possible.

Acknowledgments

[17] The material is based upon work supported by the U.S. National Science Foundation under grant ATM 00-03418 and by the Russian Foundation for Fundamental Investigation under grant 01-05-64806. The ionosonde data were obtained from the National Geophysical Data Center, NOAA.

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