Radio Science

Estimating ionospheric delay using kriging: 2. Impact on satellite-based augmentation system availability

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Abstract

[1] An augmentation of the Global Positioning System, the Wide Area Augmentation System (WAAS) broadcasts, at each node of an ionospheric grid, an estimate of the vertical ionospheric delay and an integrity bound on the vertical delay error. To date, these quantities have been determined from a planar fit of slant delay measurements, projected to vertical using an obliquity factor specified by the standard thin shell model of the ionosphere. In a future WAAS upgrade (WAAS Follow-On Release 3), however, they will be calculated using an established, geo-statistical estimation technique known as kriging that generally provides higher estimate accuracy than planar fit estimation. This paper analyzes the impact of kriging on system availability. In a preliminary assessment, kriging is found to produce improvements in availability of up to 15%.

1. Introduction

[2] This paper is the second installment of a two-part analysis of satellite-based augmentation systems that use a geo-statistical technique known as kriging [Cressie, 1993; Webster and Oliver, 2001; Blanch, 2002; Wackernagel, 2003] to estimate ionospheric delay. The first installment of this analysis [Sparks et al., 2011] addresses the kriging methodology that is to be used in the next upgrade of the Wide Area Augmentation System (WAAS Follow-On Release 3), the United States' satellite-based augmentation of the Global Positioning System (GPS) for airline navigation. This paper examines kriging's impact on system availability.

[3] WAAS measures the ionospheric slant delay of signals propagating from GPS satellites to multiple, dual frequency receivers distributed across North America in a network of thirty-eight reference stations. From these measurements, WAAS derives a vertical delay estimate at each ionospheric grid point (IGP) in a mask specified by the WAAS Minimum Operational Performance Standards [MOPS WAAS, 2001]. To allow the user to correct for the error due to ionospheric delay, WAAS computes and broadcasts an estimate of the vertical delay at each IGP, i.e., the Ionospheric Grid Delay (IGD), and a safety-critical integrity bound called the Grid Ionospheric Vertical Error (GIVE). Integrity refers to the reliability and trustworthiness of the information broadcast and to the system's ability to warn users promptly when the system should not be used for navigation because of signal corruption or some other error or failure in the system. Derived from inflated and augmented values of the formal estimation error, GIVEs provide protection from the adverse influence of delay estimation error due to ionospheric irregularity, both sampled and undersampled.

[4] To calculate the IGDs and GIVEs, WAAS models the ionosphere as an infinitesimally thin shell at a representative ionospheric altitude. This simple model serves two purposes: (1) to define the ionospheric pierce point (IPP) of each measurement and (2) to convert the slant delay measurement to a vertical delay estimate at the IPP. An IPP is the location where a satellite-to-station raypath or a satellite-to-user-receiver raypath penetrates the thin shell. At regularly spaced intervals in time, WAAS performs delay estimation, converting slant delay measurements in a given epoch to vertical delay estimates and transforming these vertical delay estimates, distributed unevenly in space over the thin shell, into a set of vertical delay estimates distributed at regular spatial intervals on the WAAS ionospheric grid. To infer the ionospheric slant delay (and its integrity bound) associated with a GPS signal detected by the user's receiver, the user must first determine the IPP associated with the signal and then approximate the vertical delay at this IPP using bilinear interpolation of the IGDs (and their integrity bounds) at the nearest IGPs surrounding the IPP. The interpolated vertical delay estimate is then converted, again using the thin shell model, to an estimate of the slant delay from the satellite to the user.

[5] With a slant delay estimate and integrity bound for each GPS signal detected by a user's receiver, the user can determine a correction to his or her position estimate and an integrity bound on that estimate. This integrity bound is used, in turn, to specify the user's Horizontal Protection Level (HPL) and Vertical Protection Level (VPL). The HPL and VPL are the receiver-computed integrity bounds defined by the MOPS WAAS [2001] as, respectively, the radius of a circle in the horizontal plane and the half-length of a segment on the vertical axis perpendicular to this plane, each describing a region whose center coincides with the user's true position and whose breadth is sufficient to provide assurance that the region contains the estimated position. The HPL and VPL define the regions in which the respective time-to-alert requirements can be met. A Horizontal Alert Limit (HAL) and, where applicable, a Vertical Alert Limit (VAL) are associated with each navigation mode (e.g., precision approach, non-precision approach, en route, etc.) supported by WAAS. The MOPS WAAS [2001] defines the HAL and VAL as, respectively, a radius and segment half-length, analogous to the HPL and VPL, each describing regions also centered on the user's position but of such breadth as to meet the requirement to contain the true position within the probability required for a particular navigation mode. When the HPL exceeds the HAL or the VPL exceeds the VAL for a given level of aviation service, that level of service is not available to the user. When the true error in a user's position exceeds the VAL (for equipment aware of the navigation mode) or the computed VPL (for equipment not aware) and WAAS fails to provide notification of the error within the time-to-alert of the applicable phase of flight, WAAS is considered to be broadcasting hazardously misleading information (HMI). For example, the most restrictive integrity requirement on WAAS is that the upper bound on the probability of broadcasting hazardously misleading information be no more than one occurrence in every 10,000,000 runway approaches (resulting in either landings or missed approaches) that use precision approach with vertical guidance, i.e., a probability of broadcasting hazardously misleading information below 10−7.

[6] By this means, WAAS provides vertical guidance down to a minimum height above the runway as determined by the level of the aviation service. The decision height in a precision approach is the height at which a missed approach must be initiated if the required visual reference to continue the approach has not been established. Each level of aviation service specifies a distinct decision height and VAL. For Localizer Performance with Vertical guidance (LPV) service, the decision height is 250 feet and the VAL is 50 m; for LPV200 service, the decision height is 200 feet, and the VAL is 35 m.

[7] The goal of implementing kriging in WAAS is to improve the availability of each level of aviation service. For a user at a given position, the availability of an aviation service can be quantified in terms of the user's computed VPL relative to the VAL specified for the given level of service. Regional availability can thus be quantified by the fraction of the day when the VAL specified for a given level of aviation service exceeds the user's computed VPL.

[8] To appreciate the impact of kriging on availability, it is first necessary to understand how WAAS protects the user from estimation errors due to ionospheric disturbances. A user's computed VPL depends upon the broadcast GIVEs associated with the IGDs used to correct for ionospheric delay in the determination of the user's position. When these GIVEs become sufficiently large, the user's VPL will exceed the VAL for a given level of aviation service, rendering the service locally unavailable. The GIVEs are constructed, in part, from the formal estimation error at each IGP, inflated to account for the statistical uncertainty of this error. When well-sampled ionospheric irregularities are present, the magnitudes of the GIVEs generally increase. Since kriging tends to reduce the estimation error, however, one expects the implementation of kriging to reduce the size of the broadcast GIVEs, and thereby increase the fraction of the total time that the user's VPL is bounded by the VAL.

[9] The impact of kriging on availability is more complicated than this argument would suggest, however, since the GIVEs also provide protection from the influence of undersampled irregularities. The sampling of ionospheric electron density fluctuations is determined by the raypaths that connect satellites to WAAS receivers. When the IPP coverage near an IGP is sparse or highly non-uniform, the region is undersampled, and the corresponding GIVE must be augmented to bound threats comprised of localized electron density gradients that may be present when the ionosphere is disturbed. The WAAS undersampled ionospheric irregularity threat model [Sparks et al., 2001; Altshuler et al., 2001; Paredes et al., 2008] consists of a table of values that govern the amount by which the GIVE is augmented to protect against undersampled threats. To assess fully the impact of kriging on availability, one must also examine its effect on the threat model.

[10] This paper is part of a series that presents a comprehensive account of all aspects of the WAAS GIVE Monitor. The purpose of the GIVE monitor is to calculate the IGDs and GIVEs to be broadcast and to ensure that the vertical delay error bound, computed by the user at an IPP associated with a GPS signal raypath, bounds the actual delay error with a sufficiently high probability for each GPS signal detected by the user's receiver. The GIVE Monitor is designed to handle a single threat, specifically, the threat that local ionospheric behavior does not conform well to the assumed model. The undersampled threat model mitigates this threat in regions where the IPP coverage is poor, and a local irregularity detector mitigates it in regions that are well sampled. The threat model is constructed to respond to a worst-case scenario where irregularities are almost, but not quite, severe enough to trip the irregularity detector identifying the presence of disturbed conditions. The integrity burden for detecting the presence of ionospheric irregularities is shared with a system-wide extreme storm detector that protects the user from highly localized disturbances that originate during extreme storms and have been found to persist many hours into nighttime. Such disturbances are sufficiently localized that they can escape detection by the irregularity detector. In the absence of an extreme storm detector, such disturbances would require an increase in the augmentation of the GIVE provided by the undersampled threat model, thereby reducing WAAS availability under nominal ionospheric conditions.

[11] This paper analyzes how kriging influences system availability. Section 2 discusses how kriging will affect the methodology WAAS uses to protect the user from estimation error due to ionospheric irregularities. Section 3 presents a preliminary assessment of the improvement in aviation service availability that is anticipated in WAAS Follow-On Release 3. Section 4 summarizes our results.

2. Protection From Delay Estimation Error

[12] Airline safety requires that, for each IGP, the broadcast GIVE bound the actual vertical delay estimation error to a high degree of confidence. The GIVE is derived from the formal estimation error variance at the IGP as specified by equation (25) of Sparks et al. [2011]. Even under nominal ionospheric conditions, however, there is a finite statistical probability that the formal delay estimation error significantly underestimates that actual error. Consequently the computation of the GIVE uses an inflated version of the formal error variance. The amount of the inflation depends on the level of ionospheric disturbance: the more disturbed the ionosphere, the greater the inflation.

[13] The inflated formal error variance can only provide protection from the adverse influence of ionospheric irregularities that are well sampled by WAAS. Each GIVE, however, must also provide protection from estimation error due to irregularities that are undersampled. For this reason, the inflated error variance used to define the GIVE is augmented by an additional term that shields the user from the threat to delay accuracy constituted by any undersampled irregularities present.

[14] When the ionosphere becomes sufficiently disturbed, the vertical delay estimate computed at a given IGP can no longer be trusted. To determine when such conditions exist, WAAS uses, at each IGP, an irregularity detector based upon the irregularity metric (see equation (47) of Sparks et al. [2011]). It has been observed, however, that the irregularity detector may fail to detect the presence of dense, highly localized disturbances that linger following the occurrence of an extreme storm. This has motivated the implementation of an extreme storm detector in WAAS.

[15] This section presents the various means WAAS uses to protect the user from delay estimation error. It first discusses how the formal estimation error is inflated prior to constructing the GIVE. The next subsection examines the threat posed by undersampled ionospheric irregularities. The following subsection specifies how the GIVE is calculated. Subsequently, two subsections describe the roles played by the irregularity detector and the extreme storm detector. The final subsection provides a brief overview of how the undersampled threat model is constructed; a more comprehensive account of this threat model will be the subject of a future publication.

2.1. Inflation of Formal Estimation Error

[16] There is some probability that the statistical uncertainty equation image2 as given by equation (25) of Sparks et al. [2011] significantly underestimates the actual error in the estimated vertical delay [Walter et al., 2000]. To allow for this possibility, the GIVE is constructed from an inflated version of the formal error [Blanch, 2003]:

equation image

where we adopt the same variable definitions and notation as in the work of Sparks et al. [2011] (as we do throughout this paper) and introduce

equation image

an inflation factor used to account for ionospheric and statistical uncertainty in the χ2 associated with the delay estimate. (The minimum permitted value of Rirreg2 is 1; any computed value less than this floor value is reset to 1.) The constant χequation image, lower bound2 is defined such that if X is a standard normal random variable and Y is a χ2 random variable with equation imageN − 3 degrees of freedom, then

equation image

where KHMI_GIVE is a scalar constant that defines a confidence interval based upon the standard normal (Gaussian) distribution, and PHMI_R_irreg represents the bound on the probability of broadcasting hazardously misleading information allocated to ionospheric errors. In WAAS, KHMI_GIVE is set to 5.592 (see section 2.3 below), and PHMI_R_irreg is set to 10−10. Note that Rirreg2 is used to scale terms that arise in equation (1) due to the ionospheric covariance only, not the term due to the measurement covariance. Also note that this equation reduces to the formal estimation error variance when Rirreg2 is set to unity.

2.2. The Threat Posed by Undersampled Ionospheric Irregularities

[17] In addition to inflating GIVEs to account for the statistical uncertainty of the formal error variance, WAAS also increases GIVEs to provide protection from delay estimation error due to the undersampling of irregularities in the ionospheric electron density. Such irregularities can increase the delay experienced by a signal propagating toward a user while not affecting the signals observed by WAAS, causing the user's computed position integrity bound to underestimate significantly the true error in the user's position estimate.

[18] A particularly dramatic example of a disturbance illustrating the need for such protection occurred during the Halloween storm of 2003. Beginning at approximately 16:00 UTC on 29 October 2003, a coronal mass ejection struck the earth, initiating a major disruption of the ionosphere that lasted more than forty-eight hours. Under nominal conditions during the night, the electron density in the ionosphere undergoes extensive recombination and diminishes to a very small magnitude. Following the second day of disruption, however, a highly localized remnant of a daytime disturbance persisted for many hours into the early morning.

[19] The colored contours of Figure 1 [from Walter et al., 2004] show the estimated vertical delay over the southeastern United States at 05:00 UTC on 31 October. The disturbance in question hovered over Florida and has hence become known as the Florida event. Circles identify the locations of IPPs associated with delay measurements. Attached to each circle is a line segment pointing to the location of the receiver that recorded the measurement. The length of the segment indicates the magnitude of the elevation angle: longer segments correspond to lower elevation angles. To illustrate how easily this irregularity might have avoided detection, the five measurements of this epoch that did, in fact, sample the irregularity have been removed. When this phenomenon actually occurred, WAAS detected the irregularity and responded appropriately. No misleading information was broadcast. It is evident, however, that the irregularity might have gone undetected had it been located somewhat to the west and slightly to the south.

Figure 1.

A dense, localized irregularity that occurred in the ionosphere over Florida at 05:00 UTC on 31 October 2003. Color contours show the magnitude of the vertical delay. Circles identify the locations of measurement IPPs; the line segment at each circle points to the receiver that recorded the measurement. The five measurements that sampled the irregularity have been removed. (This figure is provided courtesy of Walter et al. [2004].)

2.3. Grid Ionospheric Vertical Error

[20] As noted in the introduction, the top-level integrity requirement for WAAS states that an upper bound on the probability of broadcasting hazardously misleading information is no more than one occurrence in every 10,000,000 runway approaches that use precision approach with vertical guidance, a 10−7 bound on the probability of broadcasting hazardously misleading information. To validate this requirement, WAAS uses a fault tree where each node of the fault tree is associated with a distinct set of threats to system integrity. The fault tree provides a break down of the WAAS integrity requirements allocated to the various system failure modes (both hardware and software). The GIVE Monitor is the integrity monitor that handles the threat posed by ionospheric disturbances. The fault tree allocates to the GIVE Monitor an upper limit of 2.25 × 10−8 per approach on the probability of broadcasting hazardously misleading information due to the missed detection of an ionospheric irregularity threat.

[21] The GIVE at each IGP is specified in terms of a standard normal (Gaussian) distribution that overbounds the tails of the actual distribution of the residual error in vertical delay estimation after the application of the ionospheric correction at the IGP. The overbounding distribution is constructed by requiring that it overbound the tails of the actual error distribution at least to a distance KHMIequation imageGIVE from the origin, where equation imageGIVE is the standard deviation of the overbounding distribution and KHMI = 5.33 defines a confidence interval with a confidence level of (1–10−7), based upon the top-level WAAS integrity requirement. Since the integrity bound on the probability of broadcasting hazardously misleading information due to ionospheric irregularity is smaller than the bound that takes into account all possible sources of hazardously misleading information, the normal distribution based upon equation imageGIVE must be expanded to meet the integrity allocation of the GIVE Monitor. Consequently, the overbounding distribution is broadened by inflating equation imageGIVE by the ratio KHMI_GIVE/KHMI, where KHMI_GIVE = 5.592 is the constant that defines a confidence interval corresponding to a confidence level of (1 − 2.25 × 10−8). By convention, the MOPS WAAS [2001] defines the GIVE to be a confidence bound of this overbounding distribution at a 99.9% confidence level:

equation image

where KGIVE is 3.29, the constant that defines a 99.9% confidence interval. The IGDs and GIVEs actually broadcast by WAAS are quantized to discrete values specified by the MOPS WAAS [2001] for the user receiver ionospheric correction message. The computed GIVE at each IGP is rounded upward to the next larger quantized GIVE value. In contrast, IGD values are rounded to the nearest specified quantized vertical delay value. The maximum GIVE broadcast value (GIVEMAX = 45 m) is safe under all monitored ionospheric conditions; therefore, any computed GIVE value that exceeds GIVEMAX is reset to GIVEMAX.

[22] The overbounding error variance used to define the GIVE at an IGP can be expressed formally as

equation image

where equation imageIGP2 is the inflated value of the estimation error variance that accounts for the presence of well-sampled irregularities, and σundersampled2 is the augmentation of the inflated error variance that protects the user from undersampled irregularities. Here equation imageIGP2 is specified by evaluating equation (1) at the IGP, i.e., by defining w with s = sIGP ≡ [equation image]T and c = cIGP ≡ [equation image]T. Note that the definition of equation imageGIVE2 given by equation (5) differs from its definition in the system: equation (5) omits an enhancement of the formal error variance based upon the maximum error due to antenna bias. This topic will be addressed in more detail in a subsequent publication.

[23] The possible values that σundersampled2 may assume are tabulated in the undersampled threat model and depend only upon metrics that characterize the spatial distribution of IPPs near the IGP. Specifically, σundersampled is tabulated as a function of the fit radius (Rfit) and the relative centroid metric (RCMRcentroid/Rfit) discussed previously (see Figure 4 of Sparks et al. [2011]). This table is constructed from historical WAAS observations on the days during the last solar maximum that exhibited the highest levels of ionospheric disturbance. To identify these data sets, we have assumed that the level of ionospheric disturbance correlates strongly with the level of geomagnetic disturbance as indicated by the Kp and DST indices.

2.4. Irregularity Detector

[24] The irregularity detector at an IGP is said to have tripped when the irregularity metric exceeds a specified threshold value. When the irregularity detector at an IGP has tripped, the GIVE at the IGP is increased to GIVEMAX = 45 m. A 45 m GIVE safely bounds the maximum WAAS ionospheric estimation error ever observed and represents a bound on the maximum delay error possible. When the GIVE is 45 m, vertical guidance is no longer supported by WAAS. (WAAS does, however, continue to support lateral guidance including non-precision approach.)

[25] Since the amount of inflation in equation imageIGP2 is proportional to the value of the goodness-of-fit statistic, the use of an irregularity detector is not required to protect the user from estimation error due to well-sampled irregularities. The principal motivation for introducing the irregularity detector is its impact on the undersampled threat model. Since the values of σundersampled2 tabulated in the threat model depend only upon metrics that characterize the spatial distribution of IPPs near the IGP, the magnitude of σundersampled2 in the GIVE does not depend upon the local level of ionospheric disturbance. The only way to reduce the size of σundersampled2 is to remove threats that have been incorporated into the threat model during its construction. This can be achieved by introducing irregularity detection into both the operational system and the construction of the threat model.

[26] In the absence of an irregularity detector, WAAS would, in effect, always assume that the most threatening undersampled irregularities ever observed are present. By excluding from the undersampled threat model ionospheric threats that cause the irregularity detector to trip, the tabulated values of σundersampled2 become smaller, and hence, the GIVEs become smaller, thereby improving availability under nominal ionospheric conditions. Note that the burden for protecting the user from undersampled ionospheric threats is now shared between the threat model and the irregularity detector; the GIVE at the IGP is increased to its maximum 45 m value under ionospheric conditions that trip the irregularity detector and cause threats to be excluded from the threat model.

2.5. Extreme Storm Detector

[27] To improve system availability further under nominal ionospheric conditions, an additional disturbance detector has been implemented in WAAS, the system-wide extreme storm detector [Sparks et al., 2005]. Extreme storms and their impact on WAAS will be the subject of a subsequent publication. Here we only summarize the purpose of the extreme storm detector. It is designed to protect the user from severe gradient threats that have been observed to occur during and after extreme storms, such as the Florida event observed during the Halloween storm of 2003. In the absence of an extreme storm detector, such a highly localized, dense irregularity might serve as a source of hazardously misleading information.

[28] The extreme storm detector is based upon a metric defined for each epoch as the maximum irregularity metric for all the IGPs in the working set, i.e., the IGPs at which IGDs and GIVEs are available to the user. When this metric exceeds a specified threshold, the GIVEs at all IGPs are increased to 45 m for an extended duration (i.e., several hours).

[29] The extreme storm detector excludes from the undersampled threat model extreme storm events, such as the Florida event, that may not be detected by the local irregularity detector. This serves to decrease the tabulated values of σundersampled, thereby reducing the broadcast GIVEs and improving WAAS availability under nominal ionospheric conditions.

2.6. Construction of the Undersampled Threat Model

[30] This subsection presents an overview describing how the undersampled ionospheric irregularity threat model is constructed. To bound the actual delay estimation error, we require

equation image

where equation imageequation image is the measured vertical delay at the equation imageth IPP, equation imageκ is the corresponding estimated value, equation imageκ2 is the inflated variance of the delay estimate at the IPP, and Kundersampled specifies an upper bound on the square of the residual in terms of the inflated formal error variance (in practice, Kundersampled is set to a value of 5.33). Under quiet ionospheric conditions, this inequality should always be satisfied. Under disturbed conditions, however, it may fail. When it is not satisfied, we define equation imageundersampled, κ2 by requiring

equation image

[31] In constructing an undersampled threat model, the objective is to determine the maximum values of equation imageundersampled, κ2 that could ever be observed, as a function of the IPP distribution metrics, for delay estimates generated at IGPs where the irregularity detector has not tripped. As noted above, threats that cause the irregularity detector to trip are excluded from the threat model since the GIVE at the IGP is increased to its maximum 45 m value under these conditions.

[32] The tabulation of the raw data for the undersampled threat model is performed using the following equation:

equation image

where

equation image

and the maximization is performed over measurements κ and over the time interval T following each fit epoch. The objective of this tabulation is to construct values of σundersampledraw (Rfit, RCM) that reflect the largest errors that could ever be observed due to undersampling in the vicinity of an IGP whose irregularity detector is not in a tripped state. For this reason, vertical delay residuals are computed not only for sets of vertical delays drawn from the entire measurement set in each epoch, but also for sets that systematically exclude some of the measurements in the epoch, a technique known as data deprivation. The dotted lines in Figure 2 of Sparks et al. [2011] show the threat domain associated with each IGP. At any given IGP, only those IPPs are tabulated that lie within the threat domain of the IGP. The time interval T accounts for GIVE computational latency, system broadcast latency, and message latency within the user receiver.

[33] Figure 2 shows the tabulated values of σundersampledraw for the WAAS Release 8/9 undersampled threat model, where planar fits are performed with σdecorrtotal = σdecorrnominal = 0.35 m. Figure 3 shows the corresponding plot when estimation is performed using a kriging model with the parameters σdecorrnominal = 0.3 m, σdecorrtotal = 1.0 m, and ddecorr = 8000 km. The same supertruth data set is used to generate each figure, namely, the twenty-one days from the largest storms of the last solar cycle. To ensure that σundersampled increases monotonically with respect to each IPP distribution metric, the actual threat model used by WAAS is defined as the two-dimensional overbound of the raw data. Figure 4 shows the WAAS Release 8/9 undersampled threat model, and Figure 5 displays the corresponding overbound when kriging is used. Notice that the two threat models closely resemble one another.

Figure 2.

Raw data for the undersampled threat model based upon planar fits using the parameters σdecorrtotal = σdecorrnominal = 0.35 m.

Figure 3.

Raw data for the undersampled threat model based upon kriging using the parameters σdecorrnominal = 0.3 m, σdecorrtotal = 1.0 m, and ddecorr = 8000 km.

Figure 4.

WAAS Release 8/9 undersampled threat model based upon planar fits using the parameters σdecorrtotal = σdecorrnominal = 0.35 m.

Figure 5.

Undersampled threat model based upon a kriging model using the parameters σdecorrnominal = 0.3 m, σdecorrtotal = 1.0 m, and ddecorr = 8000 km.

3. Improvement in WAAS Availability

[34] In this section, we present a qualitative analysis describing how delay estimation based upon kriging influences WAAS availability. In addition we conduct a preliminary quantitative assessment of the expected improvement in WAAS availability that should result from implementation of kriging in the operational system. A more comprehensive analysis of WAAS availability using kriging will appear in a subsequent publication.

[35] A user's computed vertical protection level depends upon the GIVEs broadcast for the IGPs used to interpolate the vertical delay at the user's IPP. Consequently, for a given level of aviation service, two conditions must be satisfied at an IGP before its estimated vertical delay may be included in the computation of the user's position estimate: (1) the computed integrity bound at the IGP must not exceed the limit required by the level of aviation service; and (2) neither the irregularity detector at the IGP nor the system-wide extreme storm detector can be in a tripped state. The level of aviation service can be considered as unavailable at the IGP if either of these conditions is not satisfied.

[36] To understand the impact of kriging on aviation service availability, it is helpful to partition ionospheric disturbances into four classes:

[37] 1. Small, well-sampled irregularities are disturbances that fail to trip the irregularity detector even when well sampled by WAAS. The primary means of protecting the user from estimation error associated with these disturbances is the dependence of the GIVE on the inflated formal error at the IGP. Note that these disturbances need not necessarily be small in any physical sense: if a geomagnetic storm gives rise to a large enhancement in the local electron density that is nevertheless well-modeled by a planar fit, this disturbance is considered operationally to be small.

[38] 2. Large, well-sampled irregularities are disturbances that are well sampled by WAAS and trip the irregularity detector. It is the irregularity detector that prevents the user from incorporating into his or her position calculation the inaccurate vertical delay estimates that such disturbances can cause.

[39] 3. Small, undersampled irregularities are disturbances that are undersampled by WAAS but would not trip the irregularity detector even if they were well sampled. Due to undersampling, the inflation of the formal estimate error may not be sufficient to ensure that the GIVE bounds the true estimate error at an IGP. Thus, the primary means of protecting the user is here provided by the undersampled threat model.

[40] 4. Large, undersampled irregularities are disturbances that fail to trip the irregularity detector but would trip the detector if they were well sampled by WAAS. The user is protected from the adverse influence of such disturbances by both the undersampled threat model and the extreme storm detector.

[41] Let us first examine the effect of kriging on availability when the irregularity detector threshold is held fixed. Kriging's ability to reduce estimation error tends to enhance the availability of a given level of service in the presence of class 1 irregularities: as the magnitude of the inflated formal error diminishes, it less likely that the vertical protection level for the user will exceed the vertical alert limit. Since kriging tends to reduce the size of the χ2 goodness-of-fit statistic associated with the vertical delay estimate at each IGP, kriging will also cause some class 2 irregularities to be reclassified as class 1 irregularities. This serves to enhance availability, as the user will now have access to IGDs whose GIVEs are small enough to permit use of an aviation service that would be rendered unavailable by 45 m GIVEs at these IGPs if estimation were performed with the planar fit model.

[42] Kriging's effect on availability when the irregularity detector threshold is held fixed is more difficult to assess when irregularities are undersampled. If the use of kriging were to result in a reduction in the values of σundersampled that comprise the threat model, the magnitude of the broadcast GIVEs would decrease, and availability would improve. By reducing the magnitude of the residual delay errors, kriging eliminates from the tabulation of the raw threat model data those data points where equation imageundersampled, equation image2 drops below zero (see equation (9)). However, the data points eliminated tend not to be the critical points that determine the threat model, i.e., those exhibiting the maximum equation imageundersampled, κ2 for any particular choice of IPP distribution metric values (Rfit, RCM). Furthermore, since the construction of the threat model tabulates values of equation imageundersampled, κ2 only when the irregularity detector has not tripped, kriging's tendency to reduce the magnitude of the χ2 goodness-of-fit statistic will cause some irregularities to affect the kriging threat model whose influence is absent from the corresponding planar fit threat model. From these general considerations, it is not clear a priori whether the implementation of kriging will produce a less conservative threat model.

[43] Empirically, we have observed that the maximum equation imageundersampled, κ values tabulated in the undersampled threat model are not greatly affected by kriging, at least for the limited set of kriging models that we have examined thus far. Figures 6 and 7 compare the distributions of equation imageundersampled, κ values that result from planar fit estimation and kriging estimation, both using storm data from 30 October 2003. (In these figures, all equation imageundersampled, κ values generated are binned, including those where the irregularity detector has tripped.) The maximum value, mean value and standard deviation for planar fit estimation are 5.48, 1.29, and 0.96, respectively, while the same values for kriging estimation are 4.96, 1.25, and 0.91, respectively. These plots show that the two distributions do not differ significantly. For fixed irregularity detector threshold, we tentatively conclude that kriging does not affect availability through any influence on the threat model. Kriging will cause availability in the presence of class 3 and class 4 irregularities to improve only as a consequence of the reduction in the inflated formal estimate error variance, as in the case of the class 1 irregularities.

Figure 6.

The distribution of equation imageundersampled, κ for data from 30 October 2003, using planar fit estimation with parameters σdecorrtotal = σdecorrnominal = 0.35 m.

Figure 7.

The distribution of equation imageundersampled, κ for data from 30 October 2003, using kriging estimation with parameters σdecorrnominal = 0.3 m, σdecorrtotal = 1.0 m, and ddecorr = 8000 km.

[44] Another factor that influences availability is the threshold at which the irregularity detector trips. Changing the estimation model requires a reassessment of the magnitude of this threshold. The advantage of a high threshold is that the irregularity detector at each IGP trips less often, enhancing availability under disturbed ionospheric conditions. A potential disadvantage of a high threshold, however, is that more threats are included in the threat model. When the trip threshold becomes sufficiently high, these additional threats cause σundersampled values to become unacceptably large, i.e., their inclusion in the GIVE increases the frequency with which the user vertical protection level exceeds the vertical alert limit under nominal ionospheric conditions.

[45] For the WAAS Release 8/9 undersampled threat model based upon planar fits, displayed in Figure 4, the irregularity detector threshold is 2.5 (in dimensionless units). We have conducted threshold studies for kriging models with various sets of parameters. In general, we find that kriging permits use of a higher threshold without significant degradation of the threat model. For the threat model based upon kriging shown in Figure 5, the threshold is 3.0. The greater accuracy of the kriging estimation model permits us to transfer some of the integrity burden for handling ionospheric irregularities from the irregularity detector to the undersampled threat model, simultaneously improving availability.

[46] As discussed above in section 1, the availability of a given level of aviation service can be quantified in terms of the fraction of the day when the vertical alert limit for the service bounds the user's vertical protection levels. We have used the Raytheon Service Volume Model to evaluate the fraction of the service volume for which a given aviation service is available, using both the WAAS Release 8/9 planar fit (with σdecorrtotal = σdecorrnominal = 0.35 m) and the new kriging model (with σdecorrnominal = 0.3 m, σdecorrtotal = 1 m, and ddecorr = 8000 km). The results for Localizer Performance with Vertical guidance (LPV) service and for LPV200 service are presented in Tables 1 and 2, respectively. Under nominal ionospheric conditions (8 July 2009, Dst = −10 nT, Kp = 2-), the improvement in LPV service availability afforded by kriging is modest: there is a slight improvement in the fraction of North America experiencing 100% availability. The improvement is significantly greater for LPV200. The most dramatic benefits of implementing kriging, however, accrue under disturbed ionospheric conditions (22 July 2009, Dst = −78 nT, Kp = 6-), where the improvement is found to range from 12% to 15%. A more extensive analysis of WAAS availability based upon kriging will be presented in a subsequent publication comparing kriging models characterized by different sets of model parameters.

Table 1. The Fraction of North America Experiencing 100% WAAS Availability for LPV Service Over Days of Nominal and Disturbed Ionospheric Conditionsa
Conditions for LPVPlanar FitKriging
  • a

    Vertical alert limit is equal to 50 m; decision height is equal to 250 feet.

Nominal (8 Jul 2009)88.8%92.2%
Moderately disturbed (22 Jul 2009)79.1%91.4%
Table 2. The Fraction of North America Experiencing 100% WAAS Availability for LPV200 Service Over Days of Nominal and Disturbed Ionospheric Conditionsa
Conditions for LPV200Planar FitKriging
  • a

    Vertical alert limit is equal to 35 m; decision height is equal to 200 feet.

Nominal (8 Jul 2009)66.5%75.5%
Moderately disturbed (22 Jul 2009)56.6%71.9%

4. Summary

[47] In a future upgrade of WAAS (WAAS Follow-On Release 3), kriging will be used to estimate the vertical delay and its uncertainty at each ionospheric grid point. In this paper, we have examined how the implementation of kriging will affect the methodology WAAS uses to protect the user from estimation error due to ionospheric irregularities. The undersampled ionospheric irregularity threat model has been found not to change dramatically when it is constructed using kriging. Nevertheless, using kriging to perform delay estimation results in improved WAAS availability for three reasons: (1) kriging reduces the size of the estimation error, thereby decreasing the size of the broadcast GIVEs; (2) kriging generally reduces the magnitude of the χ2 goodness-of-fit statistics associated with vertical delay estimates at IGPs, thereby decreasing the frequency with which the irregularity detectors trip; and (3) kriging permits use of a larger irregularity detector threshold, decreasing yet further the frequency of irregularity detector trips. A preliminary assessment of kriging's impact on system availability has shown an improvement of up to 15%.

Acknowledgments

[48] The research of Lawrence Sparks was performed at the Jet Propulsion Laboratory/California Institute of Technology under contract to the National Aeronautics and Space Administration and the Federal Aviation Administration.

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