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Prediction of global ionospheric VTEC maps using an adaptive autoregressive model
© The Author(s) 2018
- Received: 7 February 2017
- Accepted: 12 December 2017
- Published: 2 February 2018
In this contribution, an adaptive autoregressive model is proposed and developed to predict global ionospheric vertical total electron content maps (VTEC). Specifically, the spherical harmonic (SH) coefficients are predicted based on the autoregressive model, and the order of the autoregressive model is determined adaptively using the F-test method. To test our method, final CODE and IGS global ionospheric map (GIM) products, as well as altimeter TEC data during low and mid-to-high solar activity period collected by JASON, are used to evaluate the precision of our forecasting products. Results indicate that the predicted products derived from the model proposed in this paper have good consistency with the final GIMs in low solar activity, where the annual mean of the root-mean-square value is approximately 1.5 TECU. However, the performance of predicted vertical TEC in periods of mid-to-high solar activity has less accuracy than that during low solar activity periods, especially in the equatorial ionization anomaly region and the Southern Hemisphere. Additionally, in comparison with forecasting products, the final IGS GIMs have the best consistency with altimeter TEC data. Future work is needed to investigate the performance of forecasting products using the proposed method in an operational environment, rather than using the SH coefficients from the final CODE products, to understand the real-time applicability of the method.
The ionosphere plays an important role in the dynamics of space weather of solar–terrestrial space. The ionosphere is important in matters of national defense, aviation security, economic development, and human life. Ionosphere monitoring using dual-frequency measurements from the Global Navigation Satellite System (GNSS) has been a topic for several decades (Komjathy 1997; De Franceschi and Zolesi 1998; Mannucci et al. 1998; Hernández-Pajares et al. 1999; Schaer 1999). GNSS provides an opportunity for long-term monitoring of the ionosphere with high accuracy and temporal and spatial resolution at relatively low cost, either in a regional context or on a global scale. The Ionosphere Associate Analysis Centers (IAACs) of the International GNSS Service (IGS) (Dow et al. 2009) have been providing reliable global ionospheric maps (GIMs) since 1998 (Hernández-Pajares et al. 2009). The IGS final vertical total electron content (TEC) maps are used for the scientific analysis of the ionosphere and practical applications. However, the IGS final GIM product is released with a time delay of approximately 2 weeks, limiting their application in real-time scenarios, including real-time precise positioning (Shi et al. 2012; Rovira-Garcia et al. 2015) and space missions, such as the Soil Moisture and Ocean Salinity (SMOS) from the European Space Agency (ESA) (Silvestrin et al. 2001; García-Rigo et al. 2011). Rapid GIM products, e.g., UQRG (ftp://newg1.upc.es/upc_ionex) and WHUD (ftp://pub.ionosphere.cn) provided by the Technical University of Catalonia (UPC) and Wuhan University (WHU), respectively, are available with one-day latency. Meanwhile, real-time GIMs produced by IAACs will be available in the near future. However, the accuracy of real-time ionospheric products on a global scale might be limited by data availability, as the public real-time data stream is currently more concentrated in certain regions, i.e., North America, Europe, and Australia. In addition, the applications might be limited by a latency of a few seconds needed to get the real-time products of ionosphere. Thus, short-term predictions of global ionospheric vertical TEC (VTEC) maps are important for technological applications. Since real-time satellite orbits and clocks are available, the limiting factor in high-accuracy positioning is the ionospheric delay (Rovira-Garcia et al. 2015). Short-term predictions could be used to generate real-time global VTEC maps (Orús Pérez et al. 2010). Along with many other applications, such as automobiles, road mapping, and location-based services, short-term predictions could be used to achieve sub-meter accuracy for mass-market single-frequency receivers (García-Rigo et al. 2011).
To meet the needs presented by the study of ionospheric physics and application in GNSS positioning, a few ionospheric models have been constructed, i.e., the Klobuchar model (Klobuchar 1987), the International Reference Ionosphere (IRI) model (Rawer et al. 1978; Bilitza and Reinisch 2008; Bilitza et al. 2011), and the NeQuick model (Radicella and Leitinger 2001; Nava et al. 2008). Many scholars have investigated the accuracy of these models for different regions of the world during periods of different solar and geomagnetic activities (Abdu et al. 1996; Araujo-Pradere et al. 2003; Bertoni et al. 2006; Lee and Reinisch 2006; Bhuyan and Borah 2007; Mosert et al. 2007; Adewale et al. 2011; Nigussie et al. 2013; Okoh et al. 2013; Olwendo et al. 2013; Wichaipanich et al. 2013; Wang 2016). The annual mean of the root-mean-square (RMS) of the differences between IGS GIMs and IRI predictions in 2014 was approximately 10 total electron content units (TECU, 1016 el/m2) (Wang et al. 2016). Thus, these empirical models are suitable for use in the scientific study of ionosphere behavior, which can provide predictions of the ionosphere, but they might not be appropriate for other applications that require high accuracy. Additionally, other models are built to represent the majority of the variations and the temporal–spatial distribution of the global or regional ionospheric TEC. For instance, global models are constructed by using empirical orthogonal function analysis to reproduce the major variations in TEC and the ionospheric climatology (Ercha et al. 2012; Wan et al. 2012; Mukhtarov et al. 2013). Also, regional models are studied over many countries and regions by using different methods to capture more details of the ionosphere (Bouya et al. 2010; Habarulema 2010; Chen et al. 2015; Fuller-Rowell et al. 2016; Huang et al. 2017).
It is possible to obtain better VTEC maps by forecasting in the short term than those derived from empirical models. Several methods have been developed for ionospheric forecasting in recent years, such as the autocorrelation method (Muhtarov and Kutiev 1999), the autoregressive moving average (ARMA) method (Krankowski et al. 2005), a method based on neural networks (Tulunay et al. 2006), and an autoregressive model for predicting VTEC values (Karthik et al. 2012). However, many predictions are investigated at a certain location or over a regional area. In terms of global ionospheric VTEC maps forecasting, the Center for Orbit Determination in Europe (CODE), an IAAC, has provided predicted ionospheric products (one- and two-day-ahead VTEC maps, named C1PG and C2PG, respectively) for public access since 2008, via the FTP server of the Crustal Dynamics Data Information System (CDDIS, ftp://cddis.gsfc.nasa.gov/). Shortly afterward, the European Space Agency (ESA) and the Technical University of Catalonia (UPC) released their two-day-ahead VTEC maps through the FTP server of CDDIS, as well. Least-squares collocation (LSC) is used by CODE to extrapolate the coefficients of spherical harmonics (SH) for predicting VTEC maps (Schaer 1999). UPC VTEC forecasting is based on the discrete cosine transform (DCT) technique (García-Rigo et al. 2011). Moreover, a linear regression module is used to forecast the DCT coefficients and predict VTEC maps. Unlike the two methods above, the development of adaptive autoregressive modeling (AARM) for the prediction of global ionospheric VTEC maps will be presented in this manuscript. The first section of the manuscript is devoted to a detailed explanation of the AARM for ionospheric forecasting. The performance of AARM forecasting is investigated through a comparison between VTEC map predictions and IGS final products. Additionally, a comparison between VTEC predictions and external independent JASON data is conducted. Finally, conclusions are summarized in the last section.
Technical description of AARM
Forecasting global VTEC maps using AARM
Test and reference data
Additionally, an independent source of VTEC measurements observed by a dual-frequency altimeter instrument on the JASON 2 satellite is used to validate the VTEC values of GNSS-derived VTEC maps over oceans, where a few GNSS receivers exist. JASON data have previously been used to validate final IGS ionospheric products (Hernández-Pajares et al. 2009). However, there are two aspects that should be taken into account. First, the GNSS-derived VTEC includes the plasmaspheric electron content contribution, which is typically 10% during the daytime and as high as 60% at night (Yizengaw et al. 2008), in contrast to JASON VTEC, covering heights from the bottom of ionosphere to the JASON orbit at an altitude of approximately 1300 km. Unless otherwise indicated, JASON VTEC is hereafter referred to as J2TEC. The amount of GNSS experimental data available over the oceans and in southern latitudes is much less than that in northern latitudes. GNSS-derived VTEC maps over these areas might have lower accuracy. To a certain extent, the comparison of VTEC between J2TEC and GNSS-derived VTEC might allow an estimation of the relative accuracy of the predicted and final ionospheric VTEC maps over the oceans and southern latitudes.
Comparison of predicted VTEC maps
Minimum, maximum, and mean values of bias and RMS values of the differences between forecasting VTEC values and final CODE VTEC values in 2009 and 2015
Latitudinal and longitudinal behavior
As shown in Fig. 8a, the bias in 2009 all over the world is approximately zero TECU. In Fig. 8b, the geographic distribution of bias is more uneven than that in Fig. 8a. Additionally, the bias is mostly positive in the Northern Hemisphere and negative in the middle and high latitudes of the Southern Hemisphere. Additionally, the range of bias values between forecasting products and CODG in 2015 is much larger than those in 2009. This result may be due to the higher solar activity in 2015. Additionally, it should be noted that the biases between C1PG and CODG are obviously different in the Northern Hemisphere and Southern Hemisphere. C1PG overestimates VTEC values by more than 2 TECU in the Northern Hemisphere, in contrast to the underestimation of approximately 2 TECU that occurs in the Southern Hemisphere, except in some areas of both low latitude and low longitude. This phenomenon is also shown in Fig. 5, which shows the latitudinal distribution of the bias. Figure 8 indicates that C1PG-predicted VTEC maps exhibit apparent systematic error both in 2009 and 2015.
Figure 9 presents the differences between forecasting VTEC maps and CODG in terms of RMS values in 2009 and 2015, respectively. All sub-figures show that RMS values in the EIA area are higher than those in other areas, particularly in 2015. During the solar minimum, both C1PG and ARPG have good consistency with the final CODG in terms of RMS values, which are less than 3 TECU, while in 2015, the highest RMS values are approximately 8 TECU in the EIA area. Furthermore, Fig. 9 shows that the forecast performance of ARPG is better than that of C1PG in the EIA area and the Southern Hemisphere, especially in 2015.
Bias and RMS values of the differences between forecasting VTEC values and final CODE VTEC values in 2009 and 2015, in TECU
Validation with JASON data
As shown in Fig. 11, the absolute values of bias in low latitudes are smaller than those in mid-high latitudes. Both forecasting VTEC maps and final GIMs are smaller than JASON VTEC values, especially in mid-high latitudes. Other studies also obtained the same results, in which JASON VTEC values are larger than the VTEC values derived from GNSS measurements (Mandrake et al. 2005; Hernández-Pajares et al. 2009; García-Rigo et al. 2011). However, the possible reasons for this fact are not clear yet and should be investigated in the future. Additionally, Fig. 11 depicts that ARPG has almost the same performance as CODG. Meanwhile in 2015, GIMs are larger than JASON VTEC values in low and mid-low latitudes and smaller than those in other latitudes. ARPG also has a similar performance to CODG in 2015. Moreover, RMS values of the differences between forecasting VTEC maps, final GIMs, and J2TEC in 2009 show a similar trend, as presented in Fig. 12. Compared to J2TEC, the final GIMs (including CODG and IGSG) show better consistency than the forecasting VTEC maps, and this behavior is more obvious in the mid-high latitudes of the Southern Hemisphere in 2009, as well as in both the low latitudes of the Northern Hemisphere and the mid-high latitudes of the Southern Hemisphere in 2015. Therefore, the performance of forecasting products is not as good in mid-high solar activity as they are in low solar activity, particularly in EIA areas and the Southern Hemisphere.
Overall, the final IGS GIMs have the best consistency with JASON data among the forecasting VTEC maps and final GIMs. Since there are few GNSS measurements over ocean areas and the Southern Hemisphere, VTEC maps provided by IAACs might have low accuracy over these areas. From the bias presented in Fig. 11, the forecasting ARPG VTEC maps perform similarly as the final CODE GIMs. With respect to bias, it seems that C1PG has good performance in the Northern Hemisphere, sometimes even better than IGSG. According to the results in “Latitudinal behavior” and “Latitudinal and longitudinal behavior” sections, which indicate that C1PG has apparent systematic error both in 2009 and 2015, C1PG provides overestimated VTEC values in the northern latitudes and underestimated VTEC values in the southern latitudes. Therefore, there is reason to believe that it is just a coincidence that the performance of C1PG is better than that of CODG in mid- and high latitudes both in 2009 and 2015, especially even better than IGSG in the mid- and high latitudes in 2015. From the RMS values presented in Fig. 12, ARPG has similar performance to CODG in 2009, but not as good in 2015, especially in the EIA area and the Southern Hemisphere.
From the comparative results presented above, the forecasting VTEC maps derived by the proposed adaptive AR model perform better than CODE’s forecasting product when comparing with final CODE GIMs. The performance of C1PG is not as good as that of ARPG, especially during periods of mid-to-high solar activity. There might be two reasons for this problem. On the one hand, it might be inappropriate to extrapolate the SH coefficients by using the least-squares collocation method. The periodic features of SH coefficients should be well known before using the LSC method. So far, the periodic features of VTEC are known quite well, e.g., the periods of 11, 1, 1/2 year, 14.77 days, and 1 solar day. SH coefficients have no obvious physical significance because they are estimated mathematically by the least-squares method. Therefore, although VTEC is represented by the linear combination of SH coefficients, the periodic features of SH coefficients might not be the same as those of VTEC values. ARPG is derived from the extrapolated SH coefficients, calculated by using the proposed adaptive AR model. The AARM method does not involve the specific periodic features of SH coefficients. Rather, the AARM method reflects the features of SH coefficients to describe the relationship between current and historical SH coefficients, while C1PG is derived from the extrapolated SH coefficients considered by CODE to have the same periodic features of VTEC values. Additionally, the VTEC could present a more complex variation during periods of mid-to-high solar activity. It is possible that C1PG has even larger errors in 2015, as shown in Figs. 8 and 9. On the other hand, the forecasting ionospheric products C1PG are generated in an operational environment, in contrast to ARPG, estimated by postprocessing. Thus, the performance of ARPG might worsen in an operational environment of providing daily forecasting products.
An improved algorithm is proposed for the short-term prediction of global ionospheric VTEC maps. A time series of SH coefficients from the previous 30 days is constructed and used to perform autoregressive modeling to predict SH coefficients. Global ionospheric forecasting VTEC maps are generated from the predicted SH coefficients. Comparisons and validations are conducted to assess the performance of the forecasting VTEC maps. Results show that the forecasting ionospheric products from CODE (C1PG) have an apparent systematic error of greater than 1 TECU, especially in areas with mid-to-high solar activity. ARPG performs similarly as the final CODE GIMs (CODG), such that no apparent systematic error is detected. ARPG shows better consistency with CODG than C1PG, especially in southern mid- and high latitudes. Additionally, independent VTEC data from JASON are collected to evaluate the performance of forecasting products over ocean areas. A good agreement between the final IGS GIMs and JASON data is found, and the RMS of their difference is approximately 3 TECU and greater than 6 TECU in low solar activity and mid-to-high solar activity, respectively.
However, the methods for forecasting products from CODE, ESA, and UPC, as well as ARPG, presented in this manuscript are actually based on mathematical fitting without consideration of the physical processes. “Comparison of predicted VTEC maps” section shows that the performance of forecasting ionospheric VTEC maps was limited during geomagnetic storms on DOY 77, 174, and 355 in 2015. The main reason might be that the AARM model could not adapt to the sudden change in geomagnetic activity. On the other hand, the accuracy of the final VTEC maps might be lower during geomagnetic storms. Additionally, limited data availability and inhomogeneous distribution of GNSS stations could also impact the performance of forecasting. Therefore, a key priority for the prediction of global ionospheric VTEC maps in the future should be importing physical information in terms of solar and geomagnetic activities, such as the solar flux and Dst.
In general, the final CODE products of SH coefficients have been used to forecast VTEC maps in this study, while in an operational environment, the forecasting should require SH coefficients of the recent days in a short period of time. WHU has planned to generate ultra-rapid ionospheric products with a latency of 2 h. By then, forecasting VTEC maps using ultra-rapid SH coefficients might be a valid approach to understand the real-time applicability of the proposed method. Like the final combined IGS GIMs that partially filter systematic errors with the combination process (Orús et al. 2005; García-Rigo et al. 2011), combined IGS predicted products could also have improved performance in the future.
CW proposed the research topic, developed the algorithm, and wrote the main manuscript text; CS and SX designed the experiment and checked the final results; XL and FL have contributed to writing and improving the paper. All authors read and approved the final manuscript.
The authors appreciate CODE for providing SH coefficients and final ionospheric products. Also, the authors would like to thank IGS for final GIMs and NASA/CNES for JASON data. The authors would like to thank the anonymous reviewers and the editor for their comments and useful suggestions, which have been very helpful in revising the manuscript. This study has been funded by the National Key Research and Development Program of China (No. 2016YFB0501802), the Fundamental Research Funds for the Central Universities (No. 2042016kf0061) and the National Natural Science Foundation of China (No. 41274049), as well as by the China Postdoctoral Science Foundation (No. 2016M600613).
The authors declare that they have no competing interests.
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