1. Introduction
Global navigation satellite system (GNSS)-based safety-critical navigation systems are supporting world-wide aircraft operations while meeting the strict aviation requirements for various phases of flight from the en-route to the precision approach [
1,
2,
3]. Safety-critical navigation systems guarantee that the aircraft navigation error does not exceed the pre-defined alert limit to an extremely high probability, which is defined by the system integrity requirement [
1]. Currently, there are two standardized systems from the International Civil Aviation Organization (ICAO) Standards and Recommended Practices (SARPS) [
1]: the ground based augmentation system (GBAS) [
4] and the satellite based augmentation system (SBAS) [
5]. Both of these systems achieve the aircraft precision approach with vertical guidance meeting the integrity requirement. Both systems rely upon monitoring GNSS signals by the precisely known positions of ground receivers, and broadcast range error corrections to users by exploiting the differential GNSS (DGNSS) technique. More importantly, integrity parameters accompany the corrections, providing information about the uncertainty of the corrections, as the corrections generated by ground systems cannot entirely capture the range error that an aircraft is experiencing.
The ionosphere is the most challenging error source when quantifying the uncertainty of the corrections due to its variable and unpredictable delay error characteristics in GNSS signals [
6]. In GBAS and SBAS, which exploit ground receivers to correct the range error, a spatially decorrelated ionosphere results in residual ionospheric delay errors after applying the corrections. The ionospheric spatial decorrelation is usually quantified by a parameter of spatial gradient, which is defined as a difference in total electron content (TEC) or ionosphere delay error (which is proportional to TEC) per unit distance of the ionosphere pierce point (IPP). It was estimated that the ionosphere spatial gradient ranges from less than 0.01 TEC unit (TECU)/km under a quiet ionospheric condition to as large as 2.54 TECU/km observed during the storm event on 23 October 2013 [
7,
8].
The degree of ionospheric spatial decorrelation experienced by aircraft varies depending on ionospheric conditions. Without a method to distinguish different ionospheric spatial decorrelation conditions, the system should assume the worst ionospheric spatial decorrelation models when protecting user position error. Aviation communities have conducted extensive analyses and development for the past 30 years on an integrity algorithm against the residual ionospheric error uncertainty to protect aircraft navigation safety. GBAS determines a conservative standard deviation, denoted as
vertical ionospheric gradient (vig), of the ionospheric spatial gradient using historical datasets, and broadcasts it to a user within an airport using Very High Frequency (VHF) data broadcast (VDB) to bound correction uncertainty against residual ionospheric error under nominal ionospheric conditions [
7,
9,
10]. Users always assume the pre-defined
vig when computing the nominal user position error bound. Furthermore, even though the
vig parameter overbounds various nominal ionospheric conditions from quiet to active events during most operational times, there is a possibility that the
vig parameter does not overbound the actual error due to the excessive ionospheric spatial gradient [
11]. In addition to the
vig parameter, GBAS applies an upper bound of the spatial gradient threat model which was constructed using the worst-case spatial gradients observed from several severe ionospheric storm events to simulate the system availability against the potential threat conditions [
8,
9,
12,
13].
Unlike GBAS which monitors GNSS signals by ground receivers installed at each airport, the SBAS system monitors GNSS signals via the network of ground stations over the coverage area. The ionospheric range error and uncertainty information are then computed at the SBAS master station for each ionosphere grid point (IGP) and they are broadcast on the SBAS signal via a geostationary orbiting (GEO) satellite. As in GBAS, no detailed knowledge about the current ionospheric spatial decorrelation condition causes the SBAS ionospheric correction uncertainty or grid ionospheric vertical errors (GIVE) to be conservative by assuming the worst-case condition. This assumption was made in several terms used to determine GIVE. These include a sigma that specifies the nominal ionospheric decorrelation, an inflation factor that the nominal decorrelation sigma must be multiplied by to protect against all the errors for all ionospheric conditions, and an undersampled uncertainty term that represents a worst-case threat that the system would encounter in an undersampled geometry [
14]. Of special note, the conservatism on the undersampled uncertainty term comes from an assumption that the worst-storm ever seen is about to influence its delay computation at all times [
14,
15].
As conservative as the parameters are, the reviewed methods still yield the systems operating in the conterminous United States (CONUS) region sufficient for meeting the current requirements which are Category (CAT)-I service for GBAS and localizer precision with vertical guidance (LPV) 200 for SBAS which guide aircraft down to 200 ft of the ground [
16]. Nevertheless, both systems are actively working on improving system availability. GBAS is working on developing auto-land capability (defined by CAT-II/III service requirements) to guide an aircraft to the runway surface in all weather and visibility conditions [
17], and SBAS is expanding its coverage for LPV-200 service [
15]. Therefore, the model parameters must not be too conservative for system availability, but must safely bound true ionospheric behavior in order to assure integrity. We, therefore, require a method that will accurately describe the ionospheric behavior without being overly pessimistic.
In response to this, this study investigates the feasibility of utilizing space weather information for GNSS-based safety-critical navigation systems based on the well-known fact that the ionosphere is coupled to the space weather activity [
18,
19,
20,
21]. Space weather (or geomagnetic) indices are provided by many organizations including the Space Weather Prediction Center (SWPC) from the National Oceanic and Atmospheric Administration (NOAA) [
22] and the National Aeronautics and Space Administration (NASA) Space Weather Laboratory [
23]. In addition, space weather prediction technologies have been significantly advanced and matured in recent years [
22,
24]. If the level of ionospheric spatial decorrelation is understood with reliable and measurable space weather indices in real-time or in advance, it can be utilized to describe the ionospheric spatial decorrelation without being overly conservative.
Previously, there were studies for the relationship between space weather and ionosphere. Mendillo et al. [
19] found that ionospheric TEC enhancements coincided with increases of space weather activity. Balan and Rao [
18] showed a correlation between space weather intensity with ionospheric activity strength by examining both low- and mid-latitude TEC and peak electron density data. Based on the relationship, GBAS and SBAS communities have utilized space weather indices to help target days on which anomalous or nominal behavior was likely to have occurred when developing the ionospheric models in the studies mentioned above [
7,
8,
25]. However, due to a lack of information on the relationship between space weather indices and the ionospheric spatial gradient, which the safety-critical navigation systems specifically require, space weather indices could not be used to a broader extent for reducing the conservatism on the ionospheric spatial decorrelation models.
This study focuses on the relationship between the ionospheric spatial decorrelation and space weather indices which is an essential understanding for GNSS-based safety-critical systems. In our previous study, we have shown that the worst-case spatial gradient threat points observed from several ionospheric storm days are highly correlated with the disturbance storm time (Dst) index [
26]. However, the study was limited to only an extreme ionospheric storm condition, and the relationship was derived using only several tens of spatial gradient threat points due to limited data observed during storm days. This paper uses all pairs of stations from dense GNSS networks in the conterminous United States (CONUS) region that provide an IPP separation distance less than 100 km to obtain spatial gradient measurements under both ionospherically quiet and active conditions. The statistics are then compared with space weather indices, disturbance storm time (Dst) index, and the interplanetary magnetic field (IMF) Bz, for the correlation study to the ionosphere. In addition, a case study is addressed discussing an example application of the developed relationship to GBAS which potentially reduces conservatism without modifying a core system architecture.
This paper is organized as follows: The dataset used for the correlation study is described in
Section 2. Methods and algorithms for estimating TEC gradients in the CONUS region are discussed in
Section 3.
Section 4 presents the Gaussian probability density function (pdf) overbounding technique which is a key procedure for ensuring system integrity when modeling the navigation error.
Section 5 shows the results of correlation analysis between statistics of TEC gradients and space weather indices. The discussion and conclusion are given in
Section 6.
3. TEC Gradient Estimation in the CONUS Region
Precise TEC estimates were generated using LTIAM software developed to monitor ionospheric events continuously [
25]. The software utilizes GNSS dual frequency carrier (
) and code (
) measurements to accurately estimate TEC based on the fact that the carrier measurement provides a range with much lower noise level (
) compared with the code measurements,
, but contains unknown integer ambiguity numbers (
,
) not included in the code measurement. The LTIAM software computes the slant TEC based on the following steps:
First, the slant TEC estimates using carrier and code measurements are computed for each station
and satellite
pair, respectively, as shown in Equations (2) and (3), where
is a constant equal to 40.3 m
3·s
−2,
is the combination of the carrier frequency of the signals (Hz) as
, and
is the wavelength of the GNSS carrier signal.
The two TEC estimates include common terms which are a true TEC that we want to extract ultimately (), and the inter-frequency hardware biases on the receiver () and satellite () where is the speed of light. The integer ambiguity numbers, as well as remaining noise, are additionally contained in . Utilizing the characteristics that the remaining error included in the is negligible, and the included in the is assumed to follow a zero-mean Gaussian distribution, the last term which includes the integer ambiguities in Equation (2) is estimated utilizing Equation (3) as shown in the next paragraph.
Second, a leveling is performed by computing the level parameter,
, in order to remove the integer ambiguity numbers as shown in Equation (2).
is computed for each continuous arc by averaging the difference between the
and
over the epoch by applying a satellite elevation (
) dependent weighting as in Equation (4) where
is the number of data points within a continuous arc.
The leveled carrier-derived TEC measurement (
) is then computed by adding
to the
as in (5).
Third, the inter-frequency bias terms (
, and
) are estimated. LTIAM applies estimates of
provided by the International GNSS Service (IGS) to remove the satellite hardware bias, and search for
which minimizes the cumulative vertical TEC standard deviation to the mean on a given day [
35]. After applying the hardware biases,
, which is used for the gradient estimation, remains.
Finally,
and
are used to compute a vertical TEC gradient based on the station-pair method [
7] between stations
and
observing satellite
. The slant TEC is converted to an equivalent vertical TEC (
) using a geometric mapping function. In this study, we applied the mapping function derived by approximating the ionosphere which originally stretches from a height of about 50 km to more than 1000 km above the surface of the Earth [
36] with a thin-shell model. The thin-shell model treats the entire ionosphere to be a shell of finite thickness containing the condensed TEC located at 350 km [
5]. The mapping function or an obliquity factor,
, which is a function of satellite elevation and ionospheric model height is shown in Equation (6) where
is the radius of the Earth,
is the height of the ionospheric shell, and
is the elevation angle of the line of sight between a station
and a satellite
. The difference of the
between stations
and
becomes a differential vertical TEC (
) as shown in Equation (7). The vertical TEC gradient (
) is then computed by dividing
with the distance
between ionospheric pierce points (IPP) (points where the line-of-sight (LOS) from two stations and the thin-shell ionospheric model intersect) as in Equation (8).
Based on the reviewed algorithm, differential TEC as a function of IPP separation distance derived using data on 26 July 2004 is shown in
Figure 2b. The numbers of station pairs used for statistical analysis in 2004 and 2010 are also shown in
Figure 2a along with the differential TEC, revealing a significant increase in the number of CORS stations. In
Figure 2b, horizontal axes divide the IPP separation distance, and vertical axes divide differential vertical TEC into bins, the color of which corresponds to the number of performed measurements. Herein, TEC
V gradients were estimated using station pairs with IPP distances of less than 100 km, mimicking the situation when an aircraft receives DGNSS corrections at an airport. The histogram above shows a linear and smooth dependence of differential TEC on IPP separation for the whole range of distances. For the correlation study, the bin from 80 km to 100 km was used to derive the reliable statistics considering the number of data points as well as the estimate uncertainty due to the remaining bias (e.g., the receiver inter frequency bias calibration error or the carrier-phase leveling error). The same amount of bias divided by a shorter distance would magnify the bias effect on vertical gradient estimates, and this effect is shown in
Section 4.
4. Gaussian PDF Overbounding Technique
GNSS-based safety-critical navigation systems including GBAS and SBAS commonly model the ionospheric errors as a Gaussian distribution because of its simplicity for defining the error distribution with only two parameters (its mean and standard deviation) [
37]. In this context, a standard deviation determined without thoroughly considering non-Gaussian tails may threaten the users. Thus, the Gaussian pdf overbounding technique, which has been utilized for the certification of GNSS-based aviation systems to ensure safety from navigation errors, is vital for handling errors to protect users even from low-probability threats. This method overbounds distributions reliably and conservatively by including non-Gaussian tails to achieve the required level of integrity. In this study, it is observed that the TEC gradient distribution has thick Gaussian tails so that the use of this method is required due to the subsequent application of analysis results in safety-critical applications.
Figure 3a shows a pdf for normalized vertical TEC gradients in log scale. The normalized vertical TEC gradients are computed by removing their means and dividing them by their standard deviations, which were computed separately for each bin as illustrated in
Figure 3b. Bin sizes for 2004 and 2010 were chosen as 20 km and 10 km, respectively, considering the number of data points required for reliable statistics. Normalized vertical TEC gradients displayed in
Figure 3a show that the distribution has non-Gaussian (thick) tails. Considering that the ionospheric noise is modeled as a Gaussian distribution that is commonly used in GNSS-based aviation systems, the non-Gaussian tails must be appropriately overbounded by a Gaussian distribution with an inflated standard deviation. Thus, the nominal sigma (1σ) of a zero-mean Gaussian distribution (the dashed curve) was inflated to cover the non-Gaussian tails of the actual distribution. The inflation factor (f) bounding the distribution to the level of a probability of 1 × 10
−5 is determined to take into account worst-case TEC gradients, and the inflated distribution is also shown in the same figure (solid line). The ‘σ
VTG overbound’ is then computed as |μ
VTG| + f∙σ
VTG by applying an inflation factor, where μ
VTG is a mean and σ
VTG is a standard deviation of a vertical TEC gradient for each bin as shown in
Figure 3b. As mentioned in
Section 3, the σ
VTG overbound increases as the distance decreases because the remaining biases are divided by the short distances and magnify the effect of biases on the estimates. In this study, σ
VTG overbound from the last bin (the bin of 80–100 km in 2004, and the bin of 90–100 km in 2010), which contains the largest number of data points with the longest IPP separation distances, is used for the correlation study.
6. Discussion and Conclusions
This study examined the correlations of vertical TEC gradients with indices describing the intensity of space weather activity. Correlations obtained under active ionospheric conditions for both Dst and IMF Bz were higher than those determined under quiet conditions. Space weather events in Dataset 2 were much less pronounced (worst observed Dst of approximately −27 nT during our processed nine days) than those in Dataset 1 (worst observed Dst of approximately −170 nT during our processed nine days). The increase of the
VTG_overbound values was observed roughly one day after that of space weather indices, whereas the ionosphere was instantly impacted by the space weather activity in Dataset 1. The weaker correlation observed under quiet conditions was partially due to the delayed response of the TEC gradient statistics to space weather activity, as supported by previous studies on the relationship between time delay (between space weather activity and ionospheric storm occurrence) and the intensity of ionospheric activities at low- and mid-latitude stations [
18]. It observed that time delays were inversely related to storm intensity based on data for more than 60 geomagnetic storms.
Another interesting observation is the decreased tendency of the Dst index in Dataset 2 to exhibit a time delay, as compared to that of IMF Bz, i.e., the time delay was clearly observed in the case of IMF Bz, being relatively unpronounced in the case of Dst. Prior studies have revealed that IMF Bz exhibits a faster response than Dst [
33,
34], providing an explanation of the delays observed for these space weather activity indices.
These correlations would be applicable to safety-critical navigation systems. As a case study,
Figure 7 shows an example application of the developed relationship to the local area augmentation system (LAAS), which is one of the GBAS operating in the CONUS region. Currently, LAAS is applying the worst-case
VTG_overbound of 0.025 TECU/km to an aircraft regardless of ionospheric conditions [
7]. This could pose too much conservatism for the majority of the operation time. The solid red line in
Figure 7 is the currently broadcast upper bound of a standard deviation of the vertical TEC gradients. We plotted a total of 18
VTG_overbound points from both Dataset 1 and Dataset 2 as a function of Dst index. Triangle marks indicate the points observed from quiet ionospheric conditions, and circle marks represent the points observed from active ionospheric conditions. As discussed, the dependency or correlation of
VTG_overbound in active ionospheric conditions is higher with the Dst index than those of quiet ionospheric conditions. Based on the observed points, we discussed the potential possibility of reducing the conservatism with the adaptive ionospheric model, instead of using the worst-case upper bound model. The adaptive ionospheric model can be utilized to determine corresponding
VTG_overbound values by applying a real-time or a predicted Dst index while reducing conservatism. Even though we found that the correlation might be utilized to improve the availability of a safety-critical navigation system, it should be noted that great care must be taken in the practical application of these kinds of ionospheric models. Most importantly, safety verification should be conducted from factors such as the number of datasets used to develop a model or uncertainty of a real-time or predicted space weather index.
This study provides an understanding of the correlation between ionospheric spatial decorrelation and space weather activities. The results, combined with dramatically improving space weather technologies, could potentially provide the opportunity to foresee ionospheric impact under both nominal and ionospheric active conditions to broad applications which utilize DGNSS techniques. Further investigations are required to make this concept fully applicable to aviation systems. For instance, we limited the temporal resolution of TEC gradient statistics to ‘one day’ for the correlation study to ensure a sufficient amount of data. The increased number of station-satellite pairs obtained from multi-constellation GNSS satellites should enable the collection of a more substantial amount of data to reduce the time resolution and to improve the accuracy of analysis in terms of temporal correlation.