spherical cap harmonic analysis of the arctic ionospheric tec for one solar cycle

Spherical cap harmonic analysis of the Arcticionospheric TEC for one solar cycleJingbin Liu1, Ruizhi Chen2, Jiachun An3, Zemin Wang3, and Juha Hyyppa1

1Department of Remote Sensing and Photogrammetry, Finnish Geodetic Institute, Masala, Finland, 2Conrad BlucherInstitute for Surveying and Science, Texas A&M University Corpus Christi, Corpus Christi, Texas, USA, 3Chinese AntarcticCenter of Surveying and Mapping, Wuhan University, Wuhan, China

Abstract Precise knowledge of the Arctic ionosphere total electron content (TEC) and its variations hasscientific relevance due to the unique characteristics of the polar ionosphere. Understanding the ArcticTEC is also important for precise positioning and navigation in the Arctic. This study utilized the sphericalcap harmonic analysis (SCHA) method to map the Arctic TEC for the most recent solar cycle from 2000 to2013 and analyzed the distributions and variations of the Arctic TEC at different temporal and spatialscales. Even with different ionosphere conditions during the solar cycle, the results showed that theexisting International Global Navigation Satellite Systems Service stations are sufficient for mapping theArctic TEC. The SCHA method provides adequate accuracy and resolution to analyze the spatiotemporaldistributions and variations of the Arctic TEC under different ionosphere conditions and to track ionizationpatches in this polar region (e.g., the ionization event of 26 September 2011). The results derived from theSCHA model were compared to direct observations using the Super Dual Auroral Radar Network radar. TheSCHA method is able to predict the TEC in the long and short terms. This paper presented a long-termprediction with a relative uncertainty of 75% for a latency of one solar cycle and a short-term predictionwith errors of ±2.2 total electron content units (TECUs, 1 TECU=1016 elm�2), ±3.8 TECU, and ±4.8 TECU fora latency of 1, 2, and 3 days, respectively. The SCHA is an effective method for mapping, predicting, andanalyzing the Arctic TEC.

1. Introduction

Human activities such as resource utilization, scientific research, air and marine traffic, and resourceexploration are currently increasing in the Arctic region. The sovereignty of the Arctic is a politically importanttopic because natural resources are becoming exhausted, and new resources are being sought in the Arcticregions. In addition, with receding polar ice, the Northwest and Northeast Passages are now considered aspossible new routes that will increase transportation efficiency. Oil exploration is also increasing in Arcticregions. Finally, search and rescue operations are likely to increase due to increasing transportation throughthe area. Thus, reliable Global Navigation Satellite Systems (GNSS) positioning and navigation solutions in theArctic are becoming ever more important.

Ionospheric disturbances can be severe at high latitudes due to the proximity to the polar cap, which isconnected to the solar wind via open magnetic field lines. Thus, the Arctic is a highly variable environment,both spatially and temporally. Ionospheric models used for satellite-based augmentation systems (SBASs)often suffer from a large grid size and an insufficient number of measurements. The ionospheric plasmaexhibits dispersive effects on radio signals that lead to several difficulties, including phase advance, codedelay, phase and amplitude scintillation, and a loss of lock for GNSS signals [Jakowski et al., 2005]. These ef-fects are directly connected to the total electron content (TEC) and gradients of the TEC within the iono-sphere. In addition, the Arctic regions are located within or close to the polar cap, which is connected to thehighly variable solar wind by open magnetic field lines. This effect, in connection with changes in solar illu-mination, leads to changes over temporal (e.g., diurnal, seasonal, and solar cycles) and spatial scales (small,medium, and large scales). In Arctic regions, these ionospheric effects are particularly important due to thelow elevation of the GPS (Global Positioning System) and Galileo orbits, resulting in a longer signal paththrough the ionosphere. Aquino et al. [2009] discovered that the scintillation index S4 increases for the GNSSsatellites at low elevations observed at high latitudes.

LIU ET AL. ©2013. American Geophysical Union. All Rights Reserved. 601

PUBLICATIONSJournal of Geophysical Research: Space Physics

RESEARCH ARTICLE10.1002/2013JA019501

Special Section:The Causes and Consequencesof the Extended SolarMinimum between SolarCycles 23 and 24

Key Points:• SCHA has adequate accuracy forthe mapping and prediction of theArctic TEC

• SCHA is able to track TEC distribution,variation, and ionization patches

• GNSS and SCHA provide a promisingapproach for analyzing the Arctic TEC

Correspondence to:J. Liu,[email protected]

Citation:Liu, J., R. Chen, J. An, Z. Wang, and J.Hyyppa (2014), Spherical cap harmonicanalysis of the Arctic ionospheric TEC forone solar cycle, J. Geophys. Res. SpacePhysics, 119, 601–619, doi:10.1002/2013JA019501.

Received 30 SEP 2013Accepted 19 DEC 2013Accepted article online 28 DEC 2013Published online 30 JAN 2014

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http://dx.doi.org/10.1002/2013JA019501

http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2169-9402/specialsection/SOLARMIN1




Unlike the equatorial region, where the ionosphere is affected more directly by solar activity, the ionosphereof the Arctic region is also driven by electron precipitation and irregularities in electron density [Skone, 1998].Therefore, the ionospheric activity of the Arctic region differs substantially from that of the equatorial region;for example, electron precipitation leads to the auroral light at night. As a consequence, in the Arctic region,the ionosphere exhibits much greater irregularity compared to other regions, and the total electron contentof the ionosphere is even more difficult to model for the following two reasons:

1. The ionospheric TEC variation in the Arctic is much more complicated than in other regions, and com-monly used models, such as the Klobuchar model, Global Ionosphere Model (GIM), and InternationalReference Ionosphere (IRI) model, are estimated using data that are observed at low and middle latitudesrather than Arctic latitudes [He et al., 2011].

2. The conventional ionosphere models based on geodetic coordinates have different spatial resolutions inthe east-west and north-south directions, especially in the area close to the pole.

The IRI model is the internationally recognized and recommended standard for the specification of plasmaparameters in Earth’s ionosphere [Bilitza et al., 2011]. This model describes monthly averages of the electrondensity, electron temperature, ion temperature, ion composition, and several additional parameters in thealtitude range from 60 to 1500 km. IRI is a data-based empirical model based on most of the available andreliable sources of ionospheric plasma data. These data sources include the worldwide network ofionosondes that has monitored ionospheric electron densities at and below the F peak for more than half acentury and the powerful incoherent scatter radars that measure plasma densities, temperatures, and ve-locities throughout the entire ionosphere; unfortunately, the latter data are available only at a few selectedlocations (approximately eight in operation as of 2011) [Bilitza et al., 2011]. Due to the complicated structureof the polar ionosphere, the nonuniform distribution of ionosonde stations, and the paucity of data frompolar sectors (both north and south), the inclusion of high-latitude characteristics in IRI modeling remains anongoing, high-priority topic for IRI researchers [Szuszczewicz et al., 1993; Bilitza, 1995; Zhang andPaxton, 2008].

Ionospheric parameters deduced from GNSS measurements represent a promising new resource for im-proving the IRI model and are an excellent candidate for data assimilation into the IRI model [Hernández-Pajares et al., 2009; Bilitza et al., 2011]. Of greatest interest are electron density profiles deduced via tomog-raphy and occultation, and integral measurements (TEC) are widely used for space weather applications[Garner et al., 2008]. The ongoing validation of these techniques has been successful [Bilitza et al., 2011].Komjathy et al. [1998] were among the first to assimilate global TEC maps deduced from GPS measurementsinto the IRI model to update the monthly average model based on the daily and hourly ionospheric condi-tions monitored by GPS. Hernandez-Pajares et al. [2002] directly used individual slant TEC measurements toadjust IRI model parameters. More recently, Schmidt et al. [2008] and Zeilhofer et al. [2009] developed a 4-Drepresentation of the ionospheric electron density. Other IRI-GPS assimilation schemes have been developed,such as the work of Fuller-Rowell et al. [2006], who used IRI-deduced empirical orthogonal functions to rep-resent the TEC over the continental U.S. (US-TEC), and Angling et al. [2009], who used IRI and GPS data in theirelectron density assimilative model.

To map and predict the regional TEC with comparable accuracy, Liu et al. [2008a, 2011] presented a regionalmodel based on the spherical cap harmonic analysis (SCHA) method. This approach is used to model the TECvalues distributed over a spherical cap using the spherical cap harmonic functions [Haines, 1988]. Unlike theconventional approaches in which the future TEC is predicted by extrapolation from an existing model, thespherical cap harmonic analysis approach utilizes models of the past to predict the future models; the futureTEC is then predicted via the predicted models. This method has been applied to modeling the ionosphericTEC observed by the Australian Regional GPS Network (ARGN), and the obtained results confirm the useful-ness of SCHA for near-real-time regional TEC mapping as well as the potential for its application to themodeling of other ionospheric parameters [Zahra et al., 2010]. Liu et al. [2010] presented promising resultsobtained by mapping the Arctic TEC using the existing International GNSS Services (IGSs) data and products.Natural Resources Canada subsequently used this method to develop the GPS TEC mapping service overCanada [Ghoddousi-Fard et al., 2011].

The present study presents the mapping and prediction of ionospheric TECs of the Arctic region using GNSSdata and products and analyzes the distribution and variations of the Arctic ionosphere for the solar cycles

Journal of Geophysical Research: Space Physics 10.1002/2013JA019501


from 2000 to 2013. The TEC observables derived from the GNSS data were mapped using the spherical capharmonic analysis model. The ionospheric TEC variations were then analyzed using different spatiotemporalscales in the time and frequency domains. Because ionosphere refraction is one of major error sources inGNSS positioning, the knowledge of ionosphere variations will enable the development of precise position-ing services in the Arctic region. As a detailed study case, a geomagnetic storm that occurred on 26September 2011 was analyzed using the spherical cap harmonic model to map the ionospheric TEC over theArctic region for this particular day, and various ionization patches were identified via SCHA-derived TECmapping. The result was then compared to the findings of a paper published in Science [Zhang et al., 2013]that presented the directly observed results of the evolution of the ionization patches, including the forma-tion, polar cap entry, transpolar evolution, polar cap exit, and sunward return flow using the Super DualAuroral Radar Network (SuperDARN).

2. Regional TEC Mapping Using GPS Data

This section outlines the traditional regional TECmodels and the SCHAmodel. The results of thesemodels arecompared in the following sections. This section also introduces the data processing methods used for all ofthe models in this study.

2.1. Deriving TEC From GPS Measurements

As the ionosphere is a dispersive medium for radio signals, ionospheric delays of radio signals are a functionof the radio frequencies and ionospheric TEC. The slant TEC corresponds to the total number of electronsalong a satellite-receiver path and is estimated for each satellite/receiver pair using the geometry-free linearcombination of the measurements of a dual-frequency GPS receiver as follows [Bergeot et al., 2010]:

TECk ¼ �ν� P2 � P1ð Þ þ ν� Br þ Bsð Þ (1)

where ν ¼ �f 21f22= 40:28 f 21 � f 22

� �� ¼ �9:52437 by taking the frequencies of GPS carriers L1 = 1.57542GHzand L2 = 1.22760GHz. TECk is the TEC observation derived from pseudorange measurements at epoch k,and it has the unit of TECU (1 TECU= 1016 electronsm�2). Pi i ¼ 1; 2ð Þ are pseudorange of the correspond-ing frequency. Br and Bs are the receiver and satellite interfrequency hardware delays, respectively, that areincluded in the IGS ionosphere products.

In the work of ionospheric TEC mapping, an idealized single-layer model assumes that all free electrons arecontained in a shell of infinitesimal thickness. The idealized layer typically has an altitude (H) of between 350and 450 km. A mapping function is then used to convert the derived slant TEC to a vertical TEC. In this study,the height of the assumed single layer was selected as H= 428.8 km, which is close to the optimal approxi-mation of the Chapman profile mapping function that has less than 1% of TEC mapping error [Schaer, 1999].The mapping function is the triangular function [Sardon et al., 1994].

As shown in equation (1), the noise of GPS pseudoranges has an amplified effect on the accuracy of the TECestimate. In this study, the method of carrier phase smoothing was used to improve the accuracy of the TECestimate. When cycle slips are not present or can be recovered and the interfrequency hardware delay biasesB are considered stable over a period of a few days [Schaer, 1999], the accuracy of ionospheric TEC estimatecan be improved using a recursive smoothing process over time as follows:

TeECk ¼ K � 1K

� ν� λ2φ2 � λ1φ1ð Þ � 1K

∑K

k¼1ν� P2 � P1ð Þð Þ

� 1K

∑K�1

k¼1ν� λ2φ2 � λ1φ1ð Þð Þ þ ν� Br þ Bsð Þ

(2)

where TeECk is the smoothed TEC measurement, Pi; and ϕi i ¼ 1; 2ð Þ are pseudorange and carrier phasemeasurements of two frequencies, respectively, and K is the smoothing length.

We assume that the pseudorange and carrier phasemeasurements are uncorrelated; thus, the variance of thesmoothed TEC can be estimated as follows:

σ2TeECK

¼ K � 1K

σ2TECϕ þ 1

Kσ2TECp þ σ2B (3)

whereσ2TECp andσ2TECϕ are the variance of TECmeasurements derived from the pseudorange and carrier phase

measurements, respectively, and σ2B is the variance of the hardware delay biases.



The variance of the smoothed TEC is used to determine the weight of smoothed TEC measurements in theleast squares solution. In this study, the variances of the pseudorange and carrier phase TEC measurementsare given as σTECp ¼ 3 TECU and σTECϕ ¼ 0:1 TECU, which corresponds to an accuracy of 0.3m for P code

pseudorange measurements and 0.05 cycles for carrier phase measurements. To avoid the overweighting ofsmoothed TEC measurements, a limit of K≤ 112 is applied to the smoothing process. Consequently, the noiselevel of the smoothed TEC is σ

TeECK≥0:3 TECU.

2.2. Traditional Regional TEC Models

For single-frequency receivers, one alternative for mitigating the effect of ionosphere delay is to calculate theTEC value via a TEC model. Some models have a global coverage, while others can cover a specific region asthese models are derived from GPS data collected in the corresponding region. Global models such as theKlobuchar model and the GIM have limited coverage and degraded accuracy in the Arctic region. RegionalTEC models usually have better mapping accuracy because of the greater station density in the regional GPSnetworks. Regional TEC models mainly have two categories: one is grid-based models such as the SBASmodel, while the other is mathematical function based, e.g., the polynomial model [Schaer, 1999; Liu et al.,2008b; Komjathy, 1997], the triangle series model [Georgiadou, 1994; Georgiadou and Kleusberg, 1988], andthe low-degree spherical functionmodel [Wilson et al., 1995]. These three types of basic functionmodels havemany variants. The characteristics of these three models are investigated and compared by Liu et al. [2008b]with observations from China. In this study, these models are processed with observation in the Arctic, andtheir mapping performances in the Arctic region are compared with that of the spherical cap harmonicanalysis model. The mathematical functions of these three models can be found in Liu et al. [2010, 2011], andthe following parameter configurations were used for the Arctic region in the present study.

For the polynomial model, the latitude and longitude of central point of the region are 75° and 0°, respec-tively, with the degree of 7 and the order of 8. Thus, there are 56 model parameters to be estimated. Thetriangular series model takes the value of TEC as a series function of different effect factors on the ionosphere,such as local time and latitude. The number of model parameters is 15. The low-degree spherical functionmodel has a similar function forma as the global spherical harmonic model [Schaer, 1999;Wilson et al., 1995].In this study, the degree is 4, and there are 25 model parameters to be estimated. Different with the case ofthe whole globe, the spherical functions are not the solution of the Laplace’s equation over a partial region ofthe globe, and they are not hence the “harmonic functions.”

2.3. The Spherical Cap Harmonic Analysis Model

The spherical cap harmonic analysis model consists of a set of spherical cap harmonic functions derived bysolving a Laplace’s equation on a specific spherical cap. A spherical cap has a colatitude range of ([0,θ0]), θ0π. When θ0 = π, the spherical cap harmonic function becomes the traditional spherical harmonic function[Schaer, 1999]. The spherical cap harmonic analysis model of regional TEC is expressed as follows:

Ev θc; ; λcð Þ ¼ ∑K max

k¼0∑

min k;Mð Þ

m¼0ePmnk mð Þ cosθcð Þ eCm

k cos mλcð Þ þ eSmk sin mλcð Þh i

(4)

where (θc,λc) is the spherical cap coordinate of the ionosphere pierce point (IPP);Ev(βc,λc) is the vertical TEC atthe IPP (θc,λc);KMAX and M are the maximum degree and order of the series, respectively;eP cosθð Þ is the nor-malized associated Legendre function;and eCm

k and eSmk are normalized spherical cap harmonic coefficients.

In the case of the spherical cap (θ0≠ π), the boundary conditions of Laplace’s equations are met by real de-grees nk(m) rather than integer degrees as in the case of the globe. The real degrees nk(m) are solved by aniterative bisection solution, and k is the index of the real degrees (0≤ k≤ KMAX). Additional mathematicalrepresentations of the spherical cap harmonic function and the real degrees solution can be found in Haines[1985, 1988], Li [1993], and Liu et al. [2008a]. In the present study, the geographical North Pole is the sphericalcap pole of the interested area, and the half angle is 30° (θ0 = 30∘), the maximum degree is 8 (KMAX = 8), andthe maximum order is 6 (M=6). The number of model parameters is 75 in total.

3. Mapping the Arctic TEC Using GPS Data

This study compares the three traditional TEC models and the SCHA model for mapping the Arctic TEC usingthe same GPS measurements and relevant products. These measurements and products include (1) GPS



measurements from IGS tracking stations, (2) IGS precise orbit data in sp3 format, and (3) Global IonosphereMapping products and differential code bias (DCB) products of receivers and satellites provided by the Centerfor Orbit Determination in Europe (CODE).

The IGS tracking stations located above the 55° latitude were used in this study, and their geographical lo-cations are shown in Figure 1. Among these stations, 44 IGS stations, marked as “Mapping stations” inFigure 1, were used for TEC mapping, and other seven stations, marked as “Verification stations,” were usedfor verifying the estimated SCHAmodels of TECmapping. The observations of these stations are independentbecause they use different receivers, although some of stations have close locations. The measurement dataset was provided in receiver-independent exchange format by IGS central bureau via ftp access. The samplerate of the GPSmeasurements is 30 s, and the elevation cutoff threshold is 20° in the data processing. Becausethe ionospheric TEC is typically related to the local time, this study divides 24 h of one calendar day into 12sessions with a session length of 2 h. TEC measurements in each session are represented by a set of modelparameters. Therefore, there are 12 sets of model parameters for each calendar day.

The sp3 satellite orbit products are used to calculate the precise positions of satellites and further calculatethe positions of the ionosphere pierce points and elevations of GPS signal paths. The Spline interpolationmethod is used to interpolate the satellite positions at the observation epochs.

The values of the interfrequency hardware biases in equation (1) are retrieved from the DCB products pro-vided by CODE and are removed from TEC measurements. In addition, the GIM products are used as a ref-erence for comparison. The GIM products can be interpolated for calculating the TEC of anytime at anylocation using the interpolation method presented in Schaer et al. [1998].

4. Mapping Performance of the Four Models in Temporal and Spatial Scales

As noted by Liu et al. [2008a, 2010], these traditional models were originally proposed for lower and middlelatitude regions, and they have an optimal performance for specific conditions. For example, the polynomialmodel is suitable for a small area, and it has been adopted in Bernese GPS analysis software [Dach et al., 2007].The function expressions of these models are based on two dimensional geographic coordinates (latitude

Figure 1. The geographical locations of the IGS tracking stations in the Arctic area. Land is indicated by brown, and sea/ocean is representedby blue. The yellow points indicate the locations of the 44 IGS stations used for estimating the SCHAmodels, while the magenta points showthe seven IGS stations used for verifying the estimated models.



and longitude), and they do not consider the noncoherence of spatial scale in the east-west and north-southdirections, which becomes particularly large in the Arctic. This study presents comparatively the mappingperformance of these models as well as the spherical cap harmonic analysis model for the Arctic region.

Figure 2 shows the annual mapping root-mean-square (RMS) errors (top) and the relative errors (bottom) ofthe four models in the Arctic region during the solar cycles from 2000 to 2013. The mapping error is definedas the difference between the model estimate and the TEC measurement, while the relative error is definedas the ratio in percentage between RMS error and mean TEC for a time period. Figure 3 shows the estimated

Figure 2. Annually ionospheric TEC mapping performance of the four models at the Arctic region during the most recent solar cycle (2000–2013).

Figure 3. Estimated (top) daily average ionospheric TEC, (middle) solar activity indices, and (bottom) geomagnetic A index during the most recent solar cycle (2000 to 2013).



daily average ionospheric TEC; solar activity indices, including the number of sunspots and radio flux; andgeomagnetic A index during the period from 2000 to 2013. Figures 2 and 3 show that the polynomial modelhas the worst mapping accuracy, and the triangle series model has significantly larger mapping errors thanthe spherical function model and spherical cap harmonic analysis model, which have comparable perfor-mance. When the ionosphere undergoes active conditions, which are indicated by solar and geomagneticindices in Figure 3, all of these models have degraded mapping accuracy than that for the calm ionosphereconditions. During the whole solar cycle, the spherical cap harmonic analysis model has mapping RMS errorsof less than 5 TECU at most of times, and the mapping error is less than 2 TECU under calm ionosphereconditions during 2006 to 2010.

The spatial distribution of the mapping residuals is another important measure of mapping performance.Figures 4 and 5 show the spatial distribution of mapping residuals of the spherical cap harmonic analysismodel under the most active (December 2002) and calm (December 2008) ionosphere conditions, respec-tively. These 2 days are close to the southern (winter) solstice of year 2002 and 2008 in the Arctic area, andthey have the maximum and minimum ionospheric activity levels separately. Note that Figures 4 and 5 usedifferent color scales. At calm ionosphere conditions, the spherical cap harmonic analysis model has mappingerrors of less than 3 TECU for the entire area. At active ionosphere conditions, mapping errors are less than5 TECU for most of the areas, and they are less than 10 TECU for the whole area. For all ionosphere conditions,the larger errors typically occur at 12 to 14 h local time, which is marked by the magenta lines. During theperiod of 12 to 14 h local time, the ionosphere TEC typically has the largest amplitude and variation during1 day [Schaer, 1999; Liu and Gao, 2004]. Lower latitude areas have larger mapping errors because there ismore solar radiation than in higher latitude areas in December. The mapping errors of the SCHA model showno “boundary effect,” which is significant for polynomial models [Liu et al., 2011].

5. Arctic TEC Spatial Distributions and the Variations With Respect to Local Time

The previous section has showed that the spherical cap harmonic analysis model has an adequate mappingaccuracy for the entire Arctic region under different ionosphere conditions. From this section, the Arcticionosphere TEC values estimated by the SCHA model are analyzed in the temporal and frequency domains.

Figures 6 and 7 show the ionosphere TEC mapping distribution of spherical cap harmonic analysis modeledfor the Arctic region in 1 day under the most active (December 2002) and calm (December 2008) ionosphereconditions, respectively. These days have the maximum and minimum ionospheric activity levels separately.Under active ionosphere conditions (Figure 6), the overall ionosphere TEC level is generally 6 times higherthan that during calm ionosphere conditions (Figure 7). For all ionosphere conditions, the ionospheric TECspatial distribution over the entire arctic region is strongly correlated with the solar radiation, which is relatedto the local time and latitude. In December, more northern latitude areas receive less solar radiation. This isknown as the polar night phenomenon. The series in Figures 6 and 7 show clearly the diurnal variation of theionosphere TEC over the entire study region. At a given time, ionosphere TEC becomes higher when the localtime (longitude) approaches 12 to 14 h and the latitude becomes lower. At a specific location, the ionosphereTEC becomes higher when the local time is closer to 12 to 14 h. In the north polar area, where the polar nightoccurs in December, the ionosphere TEC remains relatively low throughout the entire day. The spatial dis-tribution of the ionosphere TEC shows high correlations with the time of day.

6. Time Series Analysis

In the spherical cap harmonic analysis model, the coefficient eC0;0 represents the average regional TEC, andthe accuracy was evaluated using the data of an Australian network (ARGN) by Zahra et al. [2010] and aChinese network by Liu et al. [2011]. Figure 8 shows the time series of the regional TEC average with 2 hresolution at four seasonal days and the immediately prior and subsequent days—spring equinox (days79–81), summer solstice (days 173–175), autumnal equinox (days 265–267), and winter solstice (days355–357)—under active (2002) and calm (2008) ionosphere conditions. The regional TEC average levels atthe four seasonal days are related to the solar radiation of the Arctic. At the spring equinox in 2002, theionosphere level is higher than the value at the summer solstice, which is most likely caused by the highersolar activity level, as shown in Figure 3.



The results in Figure 8 also show that the regional TEC average of the Arctic region has a lower am-plitude of diurnal variation than does the middle latitude region [Liu et al., 2011]. Within the Arctic, thistrend is also apparent in Figures 9–12, which show the ionosphere TEC values of different seasons atfour locations of different latitudes (60°N, 70°N, 80°N, and 90°N) and longitudes (0°E, 90°E, 180°E, and90°W). In all seasons, at a given longitude, the diurnal variation amplitude of the ionospheric TEC is lowerat higher latitude where the location is closer to the North Pole. For different seasons, the diurnal variationamplitudes of the ionospheric TEC are higher in March and September than in June and Decemberbecause part of the Arctic region experiences the polar day and polar night in June and December,

Figure 4. Spatial distribution of mapping residuals of 12 2 h sections under active ionosphere conditions (December 2002). The color bar has a unit of TECU. The magenta lines in eachsubplot indicate the longitudes where local times are 12 and 14 h at the middle of the 2 h section of the corresponding subplot.



respectively. The results of other time points show similar results, and this trend is maintained undervarious ionosphere conditions.

7. Frequency Domain Analysis and Ionosphere Prediction

Spherical cap harmonic analysis model has a prediction capability for future TEC because the estimatedmodel coefficients have coherent spectrum parameters in frequency domain with ionosphere TEC. This is asignificant advantage of the spherical cap harmonic analysis model comparing to the conventional regionalmodels. In this study, we present the spectrum analysis of the time series of the estimated model coefficients.Based on the frequency domain spectrum parameters, the prediction in short and long terms is operatedwith the method of least squares collocation.

The fast Fourier transform technique is used to analyze the frequency spectrum of the time series of theestimatedmodel coefficients. Figure 13 shows the various period components of the time series of themodelcoefficient C0,0 which represents the regional TEC average. The primary components in the frequency do-main include a component of 11.22 years with an amplitude of 2.2 TECU, an annual component with an

Figure 5. Spatial distribution of mapping residuals of 12 2 h sections under calm ionosphere conditions (December 2008). The color bar hasa unit of TECU. The magenta lines in each subplot indicate the longitudes where local times are 12 and 14 h at the middle of the 2 h sectionof the corresponding subplot.



amplitude of 2.58 TECU, a semiannual component with an amplitude of 1.21 TECU, a monthly 27 days com-ponent with an amplitude of 0.98 TECU, a diurnal component with an amplitude of 2.63 TECU, and a semi-diurnal component with an amplitude of 0.78 TECU. These components are related to the well-knowngeophysics and ionosphere phenomena. The 11.2 year component is typically related to solar activity cycles.

Figure 6. Spatial distribution of the Arctic ionosphere TEC of 12 2 h sections under active ionosphere conditions (22 December 2002). The color bar has a unit of TECU. The magenta lines ineach subplot indicate the longitudes where local times are 12 and 14 h at the middle of the 2 h section of the corresponding subplot.



For the annual component, it is associated with solar ionization flux, which generally exhibits annual varia-tions of 6–7% on a global scale and exhibits a maximum in December and minimum in June due to changesin the F2 layer electron density and the distance between the Sun and Earth. The daily component corre-sponds to the Earth’s rotation and the period of the hour angle of the Sun. All of these components indicate

Figure 7. Spatial distribution of the Arctic ionosphere TEC of 12 2 h sections under calm ionosphere conditions (22 December 2008). The color bar has a unit of TECU. The magenta lines ineach subplot indicate the longitudes where local times are 12 and 14 h at the middle of the 2 h section of the corresponding subplot.



that the solar radiation has strong impact on regional ionosphere activity. The semiannual component isrelated to the common fact that the time series of Arctic average TEC values has two local peaks during a1 year period, as shown in Figure 15. The first peak is observed commonly in the middle of May, and thesecond appears around autumnal equinox. These TEC observations in the Arctic are slightly different fromthat in the midlatitude (e.g., China) and global areas [Liu et al., 2011; Schaer, 1999], where the two peaksusually occur in April and October. These semiannual variations are caused by the temperature of theneutrosphere and the thickness of the neutral components O and N2 in the ionosphere [Schaer, 1999]. Itshould be noted that there are more spectral peaks in Figure 13, such as the 5.6 year and 9 year periods withamplitudes of over 1 TECU. These spectral peaks are not related to any well-known physics processes. Dataset of a longer span and other instruments are needed to confirm and interpret these spectral peaks.

Other model coefficients have significant frequency domain spectrum components of short periods,typically including semidiurnal, diurnal, monthly, semiannual, and annual components with different

Figure 8. The time series of the C0,0 coefficient with 2 h resolution at spring equinox, summer solstice, autumnal equinox, and winter sol-stice under (top) active ionosphere condition (2002) and (bottom) calm ionosphere condition (2008).

Figure 9. Ionosphere TEC values at four locations at different latitudes along the longitude of 0°E for 4months in 2002: March, June,September, and December.



amplitudes [Liu et al., 2008a]. Figure 14 shows the frequency spectrum of the coefficients C1,0 and S1,1 ofwhich the most significant periodical component is the diurnal component.

Based on the spectral analysis, the coefficients of the spherical cap harmonic analysis model can be predictedusing the least squares collocation method [Schaer, 1999]. Therefore, a regional TEC model can be predicted,and TEC values at a given time and location can be forecasted with the predicted model. TEC prediction canbe conducted over the short and long terms.

The method of least squares collocation represents a time series with three components: trend function,signal, and noise. Trend function is the deterministic component of the time series, while signal is the sto-chastic component. The trend function for each coefficient has the form of harmonic function, which uses theperiodical components obtained from the spectral analysis. Schaer [1999] and Liu et al. [2011] have presentedthe mathematical models for estimating the deterministic component, stochastic component, and noise aswell as the corresponding covariance.

Long-term TEC prediction can be performed only for the deterministic component because the autocorre-lation function of the stochastic component of TEC time series drops quickly within 5–6 days [Liu et al., 2008a,

Figure 10. Ionosphere TEC values at four locations at different latitudes along longitude of 90°E for 4months in 2002: March, June,September, and December.




2011]. Figure 15 shows the observed regional average TEC and its trend function of the solar cycles from 2000to 2013 and the predicted TEC for the coming 11 years from 2013 to 2024. The trend function shows thevariations of the Arctic TEC in large temporal scales and the relation between seasonal variations and thesolar calendar, e.g., the equinox days and the solstice days. Figure 15 shows that the Arctic ionospheric TEChas reached its latest peak at the summer of 2013, and the subsequent minimum and maximum ionosphereTEC conditions will occur in the winter of 2018 and the summer of 2024, respectively. For the latency of a solarcycle, the uncertainty of the long-term TEC prediction is 5.86 TECU with the 95% confidence level, whichcorresponds roughly to 75% of the maximum TEC trend.

For short-term prediction, the stochastic signal component of each coefficient can also be predicted usingleast squares collocation. The prediction of the model can achieve a better accuracy by adding the predictedtrend function to the predicted signal components for each coefficient. Figure 16 illustrates the time series ofthree estimated coefficients—C0,0, C1,0, and S1,1—for 33 days in 2013 and their respective trend functionsover the same period and the predicted coefficients for the last 3 days using the data of the previous 30 days.When all of the coefficients are predicted in the same way, TEC values at a given time and location can bepredicted by using the predicted model. Figure 17 shows the predicted results at two given locations for the


Figure 13. Frequency spectra of the coefficient C0,0 in the (top) long and (middle and bottom) short terms.



short-term scenario that predicts the SCHA models of 1, 2, and 3 days ahead by using the measurements ofthe previous 30 days. The prediction accuracies are ±2.2 TECU, ±3.8 TECU, and ±4.8 TECU, respectively.

8. Study Case of Tracking Ionization Patches

Ionization patches are common in Arctic ionosphere, and their movement and associated density gradientshave variably negative effects on high-frequency radio communications and satellite navigation and commu-nication. Their formation and dynamics are poorly understood, particularly under disturbed space weatherconditions. By referring to the result of direct observations using the SuperDARN radar network, we study thepossibility of tracking ionization patches using the SCHAmodel, which is useful toward automatic identificationand tracking of ionization patches in future.

The TECmapping of the SCHAmodel can represent ionospheric TEC variations related to local time and latitudes.As a study case, this section utilizes the SCHAmodel to track the ionization patches that occurred over the Arcticduring a geomagnetic storm, which occurred on 26 September 2011. These ionization patches were directlyobserved by Zhang et al. using SuperDARN and GPS data. The related results were presented recently in a paperin Science [Zhang et al., 2013]. Because the SuperDARN network is sparse, and parts of areas have no TEC dataavailable as pointed it out in Zhang et al. [2013], the observation is not continuous in the spatial domain. The TEC

Figure 14. Frequency spectra of coefficients (top) C1,0 and (bottom) S1,1.

Figure 15. The time series of the observed regional average TEC from 2000 to 2013 and its trend function of the past solar cycle and thepredicted regional average TEC for the next 11 years from 2013 to 2024. The deterministic components at four seasonal days in everyyear are indicated with specific symbols.



mapping of the SCHA model has full spatial coverage in the study region, and it can be used to track the evo-lution of the ionization patches. These results are comparable with the observations of the SuperDARN radar.

By combining the knowledge of ionosphere dynamics showed in Zhang et al. [2013], Figure 18 revealed theformation and evolution of the patches using the SCHAmappingmodel. For the reason of comparison, the timetags for each subplot are referred to that presented in Zhang et al. [2013]. Figure 18a showed the initial con-dition of TEC distribution. In Figure 18b, a local TEC enhancement (ringed in black), compared to the initialcondition of Figure 18a, indicated a large patch, which formed in westward of the cusp region near noon andcrossed through the throat. This patch was recognized as an ionization patch in Zhang et al. [2013] by combingTEC observables and measurements of other instruments of the SuperDARN network, such as solar wind dy-namic pressure and interplanetary magnetic field. The interval covered by Figures 18b and 18c represent thegrowth phase of the ionization patches. The patch shown in Figures 18b and 18c did not directly move acrossthe polar cap; rather, it was “stored” westward of the cusp region, where it grew in size. Figure 18e showed thespatially continuous TEC mapping, which confirmed the uncertainty raised by the missing observations [Zhanget al., 2013]. In Figures 18f–18i, the patch is seen exiting the nightside auroral oval and moving sunward.

Figure 16. The time series of the estimated coefficients (top) C0,0 (middle) C1,0 and (bottom) S1,1 for 33 days (days 170 to 203) in 2013(in red lines); their respective trend functions for the same period (in blue lines); and the predicted coefficients for the last 3 days (in green lines)using the data of the previous 30days, respectively.

Figure 17. Comparison of TEC values calculated by the estimatedmodel and the predictedmodel for 3 days (days 200 to 203) in 2013 at twogiven locations: (top: 60°N, 24°E) and (bottom: 87.5°N, 24°E).



9. Conclusions and Outlook

This manuscript utilizes the spherical cap harmonic analysis method tomap the Arctic ionosphere TEC during thewhole period of the past solar cycle from 2000 to 2013 and analyzes the Arctic TEC over large temporal andspatial scales. The promising results show that the existing IGS stations are sufficient formapping arctic TEC usingthe SCHA method with a mapping accuracy of 5 TECU for solar active periods and 2TECU for solar calm periods.

Compared to the traditional ionosphere TEC models, the SCHA model has the best mapping accuracy andreliability for the entire Arctic area under the various ionosphere conditions during a period of more than onesolar cycle. The consistent mapping accuracy and reliability provides a sound basis for further analyzing theArctic TEC and its temporal and spatial variations using the SCHA model. The study case of 26 September2011 shows that the SCHA method presents adequate temporal and spatial resolutions to track some ioni-zation patches in the polar region, and it is a potentially effective observation resource to complement theSuperDARN radar network, of which the observation is sparse and insufficient for some areas.

Figure 18. Arctic ionospheric TEC mapping during an evolution of the ionosphere patches. The time tags of each subplot are the same as presented in Zhang et al. [2013]. The black circlesand ellipses highlight the ionization patches, the evolution of which is followed in this figure.



The frequency spectrum analysis shows that the coefficients of the SCHA model have period parametersconsistent with the ionosphere TEC, which provides the SCHA method with a prediction capability for futureTEC. In the long term, this method predicts the deterministic component of the regional average TEC with arelative precision of more than 75% for the latency of a solar cycle of 11.2 years. For the short-term intervals,all of the model coefficients are predicted for the both deterministic and stochastic components. The pre-dicted models are then used to predict the TEC value of a specific time and location. The prediction accu-racies are ±2.2 TECU, ±3.8 TECU, and ±4.8 TECU for the cases of 1, 2, and 3 days in advance, respectively.

Precise knowledge about Arctic ionosphere TEC and its variations has scientific relevance because the polarionosphere is unique in physics. This knowledge is also important for developing precise positioning andnavigation solutions for the Arctic, which are required by increasing human activities in this area, because theionosphere is one of the main error sources in GNSS positioning. The SCHA method is essential to develop areal-time TEC mapping and prediction service for the entire Arctic region and to improve the current GNSSpositioning performance and reliability. For example, the broadcast ionosphere models of the current GNSSconstellations, which are commonly used in single-frequency receivers, will be evaluated and improved forthe Arctic region.

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spherical cap harmonic analysis of the arctic ionospheric tec for one solar cycle

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