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 Monitoring of the ionospheric variability is necessary for improving the performance of communication, navigation, and positioning systems. Total electron content (TEC) is an important parameter in analyzing the variability of the ionosphere. Owing to sparse distribution of TEC in a given region, spatial interpolation methods are used for mapping TEC on a dense grid. Typically, TEC maps are produced at time intervals that do not match the variability of the regional ionosphere. To capture the local, small-scale variability, the temporal update period (TUP) in regional TEC monitoring has to be optimized. In this study, the wide sense stationarity (WSS) period is proposed to be used as the optimum TUP of the regional TEC maps. Individual WSS periods of TEC are obtained by the sliding window statistical analysis method. Four types of measures are employed to compute the differences between TEC maps. When WSS periods of the sampling points and the differences of TEC maps are compared with each other, it is observed that for the quiet days of the ionosphere where the general temporal trends dominate, the maximum WSS period in the region can be chosen as the optimum TUP. For the disturbed days of the ionosphere where high temporal and spatial variability is observed, averages or the minimums of WSS periods in a given region have to be chosen as the optimum TUP. Thus, WSS period can be developed into a useful tool in monitoring the ionospheric variability in a given region.
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 The ionosphere affects the operation of various systems such as high frequency (HF), satellite communication, and navigation systems. The performance of these systems can be improved by adjusting the system parameters or correcting the ionospheric error with respect to the varying nature of the ionosphere. Monitoring the ionospheric variability that constitutes an important component of space weather is a complicated task due to the coupling of solar, geomagnetic, and seismic activity. The ionosphere is a temporally and spatially varying medium that is characterized by its electron density distribution. Total electron content (TEC), defined as the total number of free electrons in a tube with cross-sectional area of 1 m2 centered along a raypath, is one of the important parameters used to investigate ionospheric variability. TEC is measured in terms of TECU (1 TECU = 1016 el/m2). TEC can be obtained from the worldwide network of dual frequency global positioning system (GPS) receivers [Manucci et al., 1993; Wilson and Mannucci, 1994; Komjathy, 1997; Komjathy et al., 1998; Manucci et al., 1998; Hernández-Pajares et al., 1999; Schaer, 1999; Arikan et al., 2003, 2004; Nayir et al., 2007]. Owing to sparse and irregular distribution of these receivers, TEC cannot be directly obtained everywhere on the ionosphere. Accurate and robust interpolation methods can be used to estimate TEC for every desired location in a given region [Hernández-Pajares et al., 2002; Opperman et al., 2007; Foster and Evans, 2008; Sayin, 2008; Sayin et al., 2008b; Yilmaz et al., 2009; Arikan et al., 2009]. To better visualize the spatial ionospheric variation, TEC maps can be produced by estimating TEC values on a dense grid.
 The temporal update period (TUP) of consecutive TEC maps (time between consecutive TEC maps) is an important parameter in regional and global monitoring of ionospheric variability. The optimum TUP should be sufficiently small to capture the significant variations in the underlying physics of the ionosphere; however, it should be sufficiently large to allocate computational and storage resources efficiently [Erol and Arikan, 2005; Akdogan et al., 2007; Sayin et al., 2009]. Typically, TEC maps are produced at time intervals that do not necessarily capture the variability of regional ionospheres. For example, the global ionospheric maps (GIM) are produced with a time resolution of 2 hours and a spatial resolution of 2.5° in latitude and 5° in longitude by the analysis centers of the International GNSS Service (IGS), available at ftp://cddisa.gsfc.nasa.gov/gps/products/ionex. Although these maps indicate the general temporal and spatial trends of the ionosphere, regional variations in both time and space cannot be observed with a desired level of sensitivity and accuracy. Furthermore, the Jet Propulsion Laboratory (JPL), http://iono.jpl.nasa.gov, provides GIMs updated every 5 minutes; the Space Weather Application Center-Ionosphere (SWACI), http://swaciweb.dlr.de/, provides TEC maps over Europe for every 5 minutes; the National Oceanic and Atmospheric Administration (NOAA)/Space Weather Prediction Center (SWPC) in the USA, http://www.swpc.noaa.gov, provides TEC maps over the USA for every 15 minutes; and MIT Haystack Observatory, http://madrigal.haystack.mit.edu/madrigal/, provides TEC values over global landmass where GPS receivers are operational for every 5 minutes. Also, there are satellite-based augmentation systems (SBAS) such as the Wide Area Augmentation System (WAAS), http://www.nstb.tc.faa.gov and European Geostationary Navigation Overlay Service (EGNOS), http://www.egnos-pro.esa.int, that provide real-time ionospheric corrections to their users. In the literature, there is no study discussing the optimum temporal update periods or how these 5 minute, 15 minute, or 2 hour temporal update periods are chosen. In this study, we propose a novel technique to choose the optimum TUP adaptively for regional ionosphere monitoring. Owing to the time-varying nature of the ionosphere, standard statistical methods do not provide reliable information. Therefore, the TUP should be estimated by using statistical analysis methods with a regional random field model [Sayin et al., 2008a; Sayin, 2008].
 Investigation of a regional wide sense stationarity (WSS) period can be considered a practical solution in optimizing the TUP [Akdogan et al., 2007; Sayin et al., 2009]. In a WSS period, first- and second-order moments that correspond to the mean and variance are constants [Papoulis and Pillai, 2002]. The regional WSS period of a stochastic process is the time interval in which the mean and variance of the process can be considered as constants. If the WSS period is chosen as the TUP, then significant variations of the regional ionosphere can be captured. In the work of Arikan and Erol , an efficient tool is developed to obtain the statistical characterization of the time variability of HF channel response. The method employs sliding window statistical analysis to estimate the time-varying mean and variance to capture the temporal variability. In the work of Erol and Arikan , the method is applied to the TEC data computed from GPS observables, and the WSS periods of individual GPS stations are estimated manually for both quiet and disturbed days of the ionosphere. In the work of Akdogan et al. , the method is further developed to obtain the WSS periods automatically.
 In this study, the WSS period is proposed to be used as the optimum TUP of the regional TEC maps. For this purpose, GPS-TEC is obtained for a wide selection of European IGS stations for quiet and disturbed days of the ionosphere. The WSS period of TEC for individual IGS stations is estimated for each hour of the chosen days. Also, TEC maps are produced by the ordinary Kriging interpolation algorithm for every 2.5 minute time interval. To quantify the variation from one map to the other, we have employed four different techniques: namely, symmetric Kullback-Leibler Distance (KLD), K; pointwise maximum of the KLD, km; sum of the absolute value of the pointwise differences, A; and the maximum of absolute value of the pointwise differences, Am. The differences between the TEC maps during a day are obtained using all of these methods from 0000 UT to 2400 UT with a step size of 2.5 minutes. The difference levels between maps are compared with the WSS periods of the regional TEC maps. It is observed that WSS periods of TEC for individual GPS stations are a promising and cost-efficient indicator of ionospheric temporal variability for the regional TEC maps for both quiet and disturbed days.
2. Estimation of WSS Period
 In the literature, the variability of the ionosphere is generally investigated using the mean, the standard deviation, and the median values of TEC and layer critical frequencies. Generally, these measured or calculated parameter values are sparse in space and in time. Moreover, hourly, daily, and monthly variations are represented by daily or monthly mean or median values. The long-term statistics of the ionosphere are obtained by relative deviation, relative variability, upper interquartile, lower interquartile, and interquartile difference methods. Although these statistics provide a general trend structure of the ionosphere over daily, monthly, seasonal, yearly, and 11-year solar cycle periods, they are insufficient for ionospheric channel characterization and determination of within-the-hour variability. TEC shows a random time- and space-varying structure, which can be modeled as a stochastic process. Previously, the sliding window analysis method has been used to determine locally stationary time intervals of the stochastic processes to calculate first- and second- order statistics of the HF channel and TEC [Arikan and Erol, 1998; Erol and Arikan, 2005]. The time interval in which the time-varying ionosphere is assumed to be WSS can be taken as the optimum TUP as discussed in Akdogan et al. . Within the WSS interval, the first- and second-order moments of the TEC can be taken as constants [Papoulis and Pillai, 2002].
Arikan and Erol  proposed a statistical analysis method to obtain the statistical characterization of the time variability of an HF channel response. The method employs sliding window statistical analysis to estimate the time-varying first-order and second-order moments of the channel response. In the work of Erol and Arikan , the method is applied manually to the TEC data obtained from one high-latitude and three midlatitude GPS stations, namely, Kiruna (Sweden), Kiev (Ukraine), Ankara (Turkey), and Metzoke Dragot (Israel), for the solar maximum period of 23 to 28 April 2001. The date 23 April 2001 is a negatively disturbed day, 28 April 2001 is a positively disturbed day, and 25 April 2001 is a quiet day of the ionosphere. The quiet and disturbed days of the ionosphere are classified and provided in the web site of the Ionospheric Dispatch Center in Europe (IDCE), http://www.cbk.waw.pl/rwc/idce.html. It is found that for the midlatitude stations, the WSS period is within 15 to 20 minutes during the ionospheric quiet day. During the negatively or positively disturbed days, the WSS can be 3 to 25 minutes. For the high-latitude station, the WSS period is found to be generally shorter than the WSS period of midlatitude stations owing to increased ionospheric activity. The WSS period for the high-latitude GPS station is found to be about 10 minutes for the quiet day and 5 to 7.5 minutes for the disturbed days of the ionosphere. In the work of Akdogan et al. , the method is improved to obtain the WSS periods automatically and is applied to the TEC data obtained from three IGS stations Tromsø (Norway), Ankara (Turkey), and Malindi (Kenya) located in different geomagnetic regions. In daily WSS period analysis, the high-latitude ionosphere has the shortest WSS period and the low-latitude ionosphere has the longest WSS period. In the yearly WSS analysis of 2006, the WSS period of high and midlatitude ionosphere are close to each other, and at low latitude the WSS period is the longest owing to lower variability.
 In this study, the WSS period is investigated to determine whether it is possible to use this parameter as the TUP of the regional TEC maps. The WSS periods are estimated by the method of Akdogan et al.  and compared with the differences of TEC maps. TEC maps are obtained by the Kriging algorithm, which is introduced in the next section.
3. Kriging Interpolation Method
 In geostatistics, environmental sciences, geology, and hydrogeology, random function models are used to model the processes distributed in nature that show spatial and temporal variability [Wackernagel, 1998]. Since TEC is a complex function of many parameters such as solar activity, geomagnetic storms, latitude, longitude, and time, it is necessary to model TEC statistically as a random function. Kriging is a widely used spatial interpolation method that employs this random function model. Kriging estimation at a point is a linear combination of the values at sampled points. The weights are calculated taking into account the correlation or semivariogram of values as a function of the distance between the data points and by minimizing the error variance and satisfying the unbiasedness constraints. In the works of Sayin , Sayin et al. , and Sayin et al. [2008a], several Kriging algorithms, including ordinary Kriging (OK) and univeral Kriging (UK), are compared over the interpolation error values. The OK algorithm is demonstrated to be a robust algorithm owing to its high performance for sparse sampling of the ionosphere. In the works of Sayin  and Sayin et al. [2008a], the OK method is also applied to the TEC values derived from GPS stations for different hours within a day, and it is observed that the OK algorithm can represent the general trends of the TEC variation.
 In this study, the OK algorithm is used as the interpolation method for regional TEC mapping. The experimental semivariogram function is estimated from the residuals, which are found by fitting a linear trend to the TEC values on the sampling points and subtracting the trend from the TEC values. An exponential semivariogram model is fitted to the experimental semivariogram, which is finally used in the OK algorithm as explained in detail by Sayin et al. [2007, 2008a, 2008b] and Sayin . In the next section, the comparison method of WSS periods and the difference of consecutive TEC maps are given.
4. Comparison of WSS Periods and TEC Maps
 In this study, the WSS period for each station is estimated and the TEC maps of the chosen region are produced throughout a day. The differences between TEC map pairs are calculated and compared with the estimated WSS periods.
 The differences between TEC maps are calculated by four different functions having different characteristics. These are symmetric Kullback-Leibler distance (KLD) [Cover and Thomas, 2006], K; pointwise maximum of the KLD, km; sum of the absolute value of the pointwise differences, A; and the maximum of absolute value of the pointwise differences, Am.
 Let z(nt) = [z(1, nt) … z(np, nt)…z(Np, nt)] be a vector whose elements are TEC map values, z(Np, nt), estimated by the OK algorithm at time nt. Np is the total number of points on which the TEC values are estimated, and np is the index of the points.
 The difference function K(nt, nt + τ) is the KLD between two TEC maps estimated at times nt and nt + τ:
The pointwise maximum of the KLD, km(nt, nt + τ), is calculated in equation (3):
where max(.) function gives the maximum argument over the np index. Another difference measure A(nt, nt + τ), which is defined as the sum of the absolute differences of the Kriging maps, is given in equation (4):
The difference function Am(nt, nt + τ), which gives the maximum absolute value of the element-wise subtraction, is given in equation (5):
Since K and A functions are the sum of the measures of distances at all points, they indicate the overall differences throughout the maps. Thus, it makes it difficult to detect small-scale local variations with these two difference measures. However, the Am and km functions are the maxima of the pointwise differences, and they are more sensitive to the small-scale local differences between the TEC maps. By observing both the sum and the maximum differences, global and local differences between the maps can be investigated.
 The results of the comparisons of differences between TEC maps throughout a day and WSS periods are given in the next section.
 The choice of the WSS period of TEC as an optimum TUP is tried over Europe using IGS stations for number of days with different geomagnetic disturbance levels. The 39 IGS stations are denoted by asterisks in Figure 1a, and the convex hull for optimum OK reconstruction is demarcated by Yerevan, Armenia (40.23° N, 44.50° E), Robledo, Spain (40.43° N, 4.25° W), Tromsø, Norway (69.66° N, 18.94° E), and Nicosia, South Cyprus (35.14° N, 33.39° E), as shown in Figure 1b. The complete list of IGS GPS stations is given in the work by Sayin . The vertical TEC values are obtained by IONOLAB-TEC from www.ionolab.org using the Reg-Est algorithm discussed in Arikan et al. , Arikan et al. , Arikan et al. , and Nayir et al. .
 For this study, two geomagnetically disturbed days are chosen as 29 October 2003 and 24 August 2005. Also, 29 April 2006 and 5 May 2006 are chosen as the quiet days. The Ap and Kp indices that show the geomagnetic disturbance level are taken from the web site of the Space Weather Prediction Center, www.swpc.noaa.gov/ftpmenu/indices/old_indices.html. In Table 1, Ap and Kp indices for the chosen days are provided. For the days 29 October 2003 and 24 August 2005, Ap exceeds 30 and Kp exceeds 5, which indicate storm conditions. For the days 29 April 2006 and 5 May 2006, Ap and Kp are smaller than the threshold values and these days are the quiet days of the ionosphere.
Table 1. Ap and Kp Indices of the Days
Kp at UT
29 October 2003
24 August 2005
29 April 2006
5 May 2006
 The error variance of Kriging reconstruction increases rapidly for grid points (indicated with small dots in Figure 1a) that are not encircled by data points. Therefore, to improve the interpolation, we have limited the reconstruction to the convex hull that is demarcated by the data points. The number of points interpolated inside the convex hull encircled by the GPS stations is approximately 1,000. Then the sampling rate (number of points that provide the measurement versus the total number of grid points inside the convex hull) is calculated as × 100 = 4%. An example reconstruction using the ordinary Kriging algorithm is provided in Figure 1b for the date 29 October 2003 at 1200 UT.
 For 0000 UT ≤ nt ≤ 2359 UT, TEC maps are produced with 2.5 minute intervals, since it is observed that the ionosphere does not change significantly within this period. Equations 1, 3, 4, and 5 are applied to the TEC maps where τ varies from 2.5 minutes to 120 minutes, with 2.5 minute steps. K(nt, nt + τ), km(nt, nt + τ), A(nt, nt + τ), and Am(nt, nt + τ) values are compared with wssavg, wssmax, and wssmin in Figures 2 to 345 for 29 October 2003, 24 August 2005, 29 April 2006, and 5 May 2006, respectively. In all figures, subfigures (a), (b), (c), and (d) correspond to difference measures K(nt, τ), A(nt, τ), km(nt, τ) and Am(nt, τ), respectively, and the color bar shows the values of the difference measure in grayscale. Solid lines indicate wssavg, and the triangles and nablas are for wssmin and wssmax, respectively. For all chosen days, wssavg is approximately 10 minutes for every hour from 0000 UT to 2359 UT. Similarly, wssmin does not show significant variance from quiet days to disturbed days, and it is bounded between 5 minutes ≤ wssmin ≤7.5 minutes. Yet, the period of maximum WSS, wssmax can be as long as 35 minutes for quiet days and as short as 10 minutes for disturbed days.
 In Figure 2a, K(nt, nt + τ) indicates the strong disturbance at 1830 UT when the Kp index reaches 9. In Figure 2b, A(nt, nt + τ) is sensitive to the strong disturbance at 0630 UT. Similarly, in Figures 3a and 3b, K(nt, nt + τ) and A(nt, nt + τ) are sensitive to the rising geomagnetic storm indices. When the K(nt, nt + τ) and A(nt, nt + τ) of quiet days of Figure 4 and 5 are compared, it is observed that the K and A measures indicate daily ionization and recombination processes. As the K values indicate, the ionization process during sunrise generates stronger variability in the structure of the ionosphere than the recombination process during sunset. Also, the K and A values of disturbed days are significantly higher in magnitude than those of quiet days.
 From Figures 2 to 5, the km and Am measures provided in (c) and (d) subfigures indicate both general temporal trends and small-scale local variability with very high sensitivity. For example, if 3 to 4 TECU is considered a significant variability between the TEC maps, the time delays corresponding to these Am values can be taken as optimum TUPs. It is observed that for the quiet days given in Figures 4 and 5, wssmax coincides with this value of TUP. Yet, for the disturbed days of Figures 2 and 3, the increased ionospheric variability requires that either wssavg or wssmin be chosen as the optimum TUP. Thus, TEC maps can be updated for time intervals estimated as the WSS periods and can be adjusted adaptively with respect to the variability of the ionosphere.
 Ionospheric variability should be monitored continuously for improving the performance of communication and navigation systems. To capture the local, small-scale variability, the temporal update period is proposed to be used as the optimum TUP of regional TEC maps. As a first step, GPS-TEC for each IGS station in a given region is estimated using IONOLAB-TEC for both quiet and disturbed days of the ionosphere. Then WSS periods of TEC for each station are obtained by the sliding window statistical analysis method. For all chosen days, the average of WSS periods of the IGS stations is approximately 10 minutes. Similarly, minima of the WSS period do not show significant variance from quiet days to disturbed days, and it is bounded between 5 minutes and 7.5 minutes. Yet, the maxima of the WSS period can be as long as 35 minutes for quiet days and as short as 10 minutes for disturbed days. Regional TEC maps using the IGS station coordinates as sample points are constructed by the ordinary Kriging interpolation algorithm with 2.5 minute intervals from 0000 UT to 2359 UT. The symmetric Kullback-Leibler distance, pointwise maximum of the KLD, sum of the absolute value of the pointwise differences, and the maximum of absolute value of the pointwise differences are computed between the TEC maps of varying time delays. When WSS periods of the IGS stations and the differences of TEC maps are compared with each other, for the quiet days of the ionosphere where the general temporal trends dominate, the maximum WSS period in the region can be chosen as the optimum TUP. For the disturbed days of the ionosphere where high temporal and spatial variability is observed, averages or the minima of the WSS periods in a given region should be chosen as optimum TUP. Therefore, to capture and track local and small-scale ionospheric variability, the regional TEC maps should be updated every 20 to 30 minutes for quiet days and every 5 to 10 minutes for disturbed days. According to the changing variability levels during a day, the TUP can be adjusted automatically. Thus, the WSS period can be developed into a useful tool in monitoring the ionospheric variability in a given region. In future studies, the method will be applied for a wider selection of days and the TEC data set will be gathered from a denser network of GPS stations.
 This study is supported by TUBITAK (The Scientific and Technological Research Council of Turkey) grant no. 109E055.