Abstract
 Top of page
 Abstract
 1. Introduction
 2. Bounding the Probability of Transmitting HMI
 3. Review of WAAS Ionospheric Algorithms
 4. The Probability of Sudden Ionospheric Decorrelation
 5. Method of Determining an Upper Bound on P_{D}(t,T_{ns})
 6. Algorithm for Calculating σ_{decorr}
 7. Data Processing
 8. Results
 9. Conclusions
 Acknowledgments
 References
 Supporting Information
[1] In the absence of selective availability, the ionosphere represents the largest source of positioning error for singlefrequency users of the Global Positioning System (GPS). In differential GPS systems such as the Wide Area Augmentation System (WAAS), vertical ionospheric delays are modeled at regularly spaced intervals in geographic latitude and longitude. The broadcast bound on the error at each of these points is designated the grid ionospheric vertical error (GIVE). Higher performance standards planned for future implementations of WAAS require a reduction in the magnitude of the GIVE broadcast under nominal conditions. Achieving this reduction depends upon a better understanding of the decorrelation of ionospheric delay, in both space and time. In this paper we focus on temporal decorrelation. We report a methodology for assessing the impact on WAAS posed by a sudden increase in the level of ionospheric disturbance. The methodology is based upon forming an estimate of the probability P_{D} that a WAAS user will confront a sudden increase in the level of ionospheric disturbance following a period of relative calm. We have determined a limiting upper bound of P_{D} to be 4 × 10^{−7}, which lies well within the margin needed to meet WAAS integrity requirements.
1. Introduction
 Top of page
 Abstract
 1. Introduction
 2. Bounding the Probability of Transmitting HMI
 3. Review of WAAS Ionospheric Algorithms
 4. The Probability of Sudden Ionospheric Decorrelation
 5. Method of Determining an Upper Bound on P_{D}(t,T_{ns})
 6. Algorithm for Calculating σ_{decorr}
 7. Data Processing
 8. Results
 9. Conclusions
 Acknowledgments
 References
 Supporting Information
[2] The Wide Area Augmentation System (WAAS) is designed to provide reliable differential GPS corrections for aircraft navigation [RTCA, 1996]. In the absence of selective availability, the largest source of positioning error is the carrier phase advance and pseudorange group delay caused by the ionosphere. Since WAAS user measurements generally do not coincide with reference station measurements, it is necessary to rely on ionospheric correlation to infer the state of the ionosphere in regions sampled by the user. In WAAS, vertical ionospheric delays are modeled at regularly spaced intervals in geographic latitude and longitude, i.e., at ionospheric grid points (IGPs). The broadcast bound on the error at each of these points is designated the grid ionospheric vertical error (GIVE). A critical integrity requirement of WAAS is that the broadcast GIVE at each IGP bounds the residual error with a very high degree of confidence. Specifically, the probability of broadcasting hazardously misleading information (HMI) must not exceed 10^{−7} per approach at any point in the service volume under the worst foreseeable conditions. Here hazardously misleading information refers to a broadcast GIVE that fails to bound the actual error, and foreseeable conditions include any event that is regular, common, or strongly correlated with observed parameters (by definition, unforeseeable conditions must be extremely rare, not adhering to any predictable pattern).
[3] The threat posed by the ionosphere manifests itself in three ways: (1) instantaneous residual errors due to mismodeling of the ionosphere at the IGPs; (2) residual errors that arise when interpolating IGP delays to a user position; and (3) residual errors that grow over the life span of the broadcast message.
[4] The broadcast GIVE must protect the user from each of these threats. The rate, in both space and time, at which neighboring measurements of ionospheric delay become decorrelated is a critical component in the calculation of the WAAS GIVE. Irregularities in the ionosphere represent a threat to the accuracy of the confidence bounds describing the integrity of the broadcast corrections [Hansen et al., 2000a, 2000b; Lejeune et al., 2001].
[5] Under nominal quiettime conditions, a planar fit of slant delay measurements projected to vertical provides estimates of the local vertical delay that are of sufficient accuracy for WAAS operation. When the ionosphere is disturbed, the residual error associated with the planar fit increases, indicating that delay estimates based on this fit are less reliable. Consequently, the confidence bounds must be increased or the fit declared unusable. As long as the fit residuals accurately reflect the degree of disturbance of the ionosphere, the integrity of the corrections should remain high. Since fits are performed at finite intervals, however, it is possible that significant growth in the degree of disturbance could occur between fit evaluations. In this case the fit residuals no longer accurately reflect the true ionospheric state as encountered by the user.
[6] The WAAS GIVE is based, in part, on the uncertainty in the vertical ionospheric delay as modeled by the planar fit. In the first phase of WAAS implementation, this uncertainty is conservatively assumed to be a constant (35 cm) independent of both measurement elevation angle and distance from the IGP. Subsequent implementations of WAAS that have higher performance requirements will demand a reduction in the magnitude of the GIVE broadcast under nominal conditions. Achieving this reduction will require a better understanding of the decorrelation of ionospheric delay, in both space and time.
[7] In this paper we focus on the temporal decorrelation of the ionospheric delay. The objective is to establish a methodology for assessing the risk to the WAAS user of sudden increases in the level of ionospheric disturbance. Our means of risk assessment relies on defining P_{D}, the probability that a WAAS user will sample a region of the ionosphere during the onset of a significant disturbance. An upper bound on P_{HMI}, the probability of broadcasting HMI, is shown to depend linearly upon P_{D}. We have determined a limiting upper bound on P_{D} of 4 × 10^{−7}, which falls well within the margin needed to meet WAAS integrity requirements.
[8] In section 2 we derive an upper bound on P_{HMI} and examine its dependence on P_{D}. In section 3 we review the WAAS model for ionospheric delay and related algorithms. We provide in section 4 a precise definition of P_{D}, upon which a quantitative evaluation can be based. Section 5 describes our method for estimating an upper bound on P_{D}. Section 6 presents an iterative method for calculating σ_{decorr}, the standard deviation of the local vertical total electron content of the ionosphere relative to a planar approximation. In section 7 we discuss the manner in which observational data have been processed. Section 8 reports the results of our analysis, and conclusions are presented in section 9.
2. Bounding the Probability of Transmitting HMI
 Top of page
 Abstract
 1. Introduction
 2. Bounding the Probability of Transmitting HMI
 3. Review of WAAS Ionospheric Algorithms
 4. The Probability of Sudden Ionospheric Decorrelation
 5. Method of Determining an Upper Bound on P_{D}(t,T_{ns})
 6. Algorithm for Calculating σ_{decorr}
 7. Data Processing
 8. Results
 9. Conclusions
 Acknowledgments
 References
 Supporting Information
[9] Under disturbed ionospheric conditions, the magnitude of the errors in the vertical delays estimated by WAAS tends to increase. Consequently, WAAS uses an irregularity detector to monitor the level of ionospheric disturbance. When the detector at a given IGP trips, the GIVE at that IGP is set to an appropriately large number or the IGP is designated as not monitored. The transmission of HMI can occur when the irregularity detector does not trip and the broadcast GIVE fails to bound the actual error with a sufficiently high probability. In this section we will derive an upper bound on the probability of transmitting HMI.
[10] Let us characterize the level of ionospheric disturbance by some scalar parameter σ, where σ ≤ 1 corresponds to the nominal (or undisturbed) ionosphere and σ > 1 indicates the presence of an irregularity. Then, the probability of transmitting HMI as a function of σ can be represented formally as
where P_{MD}(σ) is the probability of missed detection, i.e., the probability that the irregularity detector will fail to trip, and P_{GIVE}(σ) is the probability that the broadcast GIVE fails to bound the actual error. If we postulate that at any given instant σ can be chosen at random from some a priori distribution with density f(σ) normalized such that ∫_{0}^{∞}f(σ)dσ = 1, then the P_{HMI} can be written as an integral over σ:
By dividing the domain of integration into two regions separated by σ = 1, an upper bound on P_{HMI} may be written as follows:
where
[11] The WAAS GIVE algorithm is constructed to ensure that when the ionosphere is actually in a nominal or undisturbed state, the broadcast GIVE will bound the actual delay error with a high degree of certainty, that is, P_{GIVE}(σ) is negligibly small when σ ≤ 1. We can therefore anticipate that P_{HMI}^{max} will satisfy the integrity requirement of 10^{−7} per approach provided the second term on the righthand side of equation (3) is sufficiently small.
[12] Demonstrating that the probability of WAAS transmitting HMI is less than the integrity limit of 10^{−7} is actually a much more complex task than the simple discussion presented here might suggest, and a detailed discussion of it is far beyond the scope of this paper. There are other possible sources of HMI, and thus integrity analysis demands that P_{HMI}^{max} be, in fact, nearly an order of magnitude smaller than 10^{−7}. An independent analysis of
has indicated that P_{D} must be less than 4 × 10^{−5} in order that the upper bound on P_{HMI}^{max} be sufficiently small to ensure WAAS integrity (T. R. Schempp, personal communication, 2003).
4. The Probability of Sudden Ionospheric Decorrelation
 Top of page
 Abstract
 1. Introduction
 2. Bounding the Probability of Transmitting HMI
 3. Review of WAAS Ionospheric Algorithms
 4. The Probability of Sudden Ionospheric Decorrelation
 5. Method of Determining an Upper Bound on P_{D}(t,T_{ns})
 6. Algorithm for Calculating σ_{decorr}
 7. Data Processing
 8. Results
 9. Conclusions
 Acknowledgments
 References
 Supporting Information
[17] Our goal is to assess the risk to the WAAS user posed by a sudden increase in the level of ionospheric disturbance. We will need to address two distinct aspects of this problem: the degree of disturbance and the timescale of its onset. Let P_{D} be the probability that a WAAS user will sample a region of the ionosphere experiencing a sudden, significant growth in perturbation following a period of relative calm. To quantify the period of relative calm, we define a nonstorm duration as a period of time T_{ns} during which the local ionospheric storm detector has not tripped. As a measure the magnitude of a disturbance, we define the decorrelation ratio to be
where σ_{decorr} is a representative standard deviation of the local vertical total electron content with respect to a plane, and σ_{crit} is a critical bound required for user safety, that is, the decorrelation ratio must be less than 1 to a very high degree of confidence. In current WAAS algorithms, σ_{crit} = R_{irreg}σ_{decorr}^{nom}, where R_{irreg} is the irregularity inflation factor [Walter et al., 2000] and σ_{decorr}^{nom} is defined in the previous section. If time t = 0 corresponds to the conclusion of a nonstorm duration of length T_{ns}, then we define P_{D} = P_{D}(t,T_{ns}) to be the probability at time t > 0 that a user samples the ionosphere in a region where the local ionospheric decorrelation ratio exceeds 1.
[18] We anticipate that P_{D}(t,T_{ns}) will tend to be an increasing function of t, becoming flat when t is sufficiently large. In other words, the ionosphere can become progressively more disturbed with the passage of time following a nonstorm duration, increasing the likelihood that the decorrelation ratio will exceed unity. This remains true until the time is sufficiently great that the state of the ionosphere is statistically uncorrelated with the prior nonstorm duration, at which point the probability of the decorrelation ratio exceeding unity becomes nearly constant. Rather than approximate P_{D}(t,T_{ns}) directly, we will seek an upper bound on P_{D}(t,T_{ns}) for t ≤ t_{sample}, where t_{sample} is a sample period of interest. The nonstorm duration period and sample period currently of interest in WAAS are, respectively, T_{ns} = 900 s (15 min is the length of time that a WAAS message may be used) and t_{sample} = 85 s (the time it takes for WAAS to perform a planar fit at each active IGP; currently there are 170 IGPs and two planar fits are performed per second).
5. Method of Determining an Upper Bound on P_{D}(t,T_{ns})
 Top of page
 Abstract
 1. Introduction
 2. Bounding the Probability of Transmitting HMI
 3. Review of WAAS Ionospheric Algorithms
 4. The Probability of Sudden Ionospheric Decorrelation
 5. Method of Determining an Upper Bound on P_{D}(t,T_{ns})
 6. Algorithm for Calculating σ_{decorr}
 7. Data Processing
 8. Results
 9. Conclusions
 Acknowledgments
 References
 Supporting Information
[19] For the decorrelation ratio to exceed unity, the ionosphere must be sufficiently disturbed. In ionospheric science, the geomagnetic index Kp is often used to approximate the degree of disturbance of the ionosphere, since ionospheric disturbances are known to be highly correlated with perturbations of the Earth's magnetic field. The initial strategy for determining an upper bound on P_{D}(t,T_{ns}) has been to assume P_{D}(t,T_{ns}) to be of the form
where P_{i}^{K} is the probability that the value of the Kp index lies within the range i − Δ ≤ Kp ≤ i + Δ, where Δ = 0.3 (see Table 1), and P_{D}(t,T_{ns}∣K_{i}) is the conditional probability that a user, at time t following a nonstorm duration of length T_{ns}, samples the ionosphere in a region where the local ionospheric decorrelation ratio exceeds 1, given that the Kp index lies within the range i − Δ ≤ Kp ≤ i + Δ. Note that P_{i}^{K} = 1. An upper bound on P_{D}(t,T_{ns}) can then be calculated from a determination of upper bounds on the P_{D}(t,T_{ns}∣K_{i}). The expectation is that for low values of i (i.e., for low values of Kp), upper bounds on the P_{D}(t,T_{ns}∣K_{i}) will be small, thereby ensuring that the bound on P_{D}(t,T_{ns}) will be small.
Table 1. The Probability That the Value of the K_{p} Index Lies Within the Range i − Δ ≤ K_{P} ≤ i + Δ (Where Δ = 0.3) as Tabulated From K_{p} Data Over the Time Period 1932–2000^{a}i  P_{i}^{K} 


0  0.0852 
1  0.2497 
2  0.2556 
3  0.2089 
4  0.1199 
5  0.0525 
6  0.0184 
7  0.0066 
8  0.0026 
9  0.0006 
[20] Let _{D}(t,T_{ns}K_{i}) be an estimate of P_{D}(t,T_{ns}∣K_{i}) determined purely from data collected on days in which ionospheric storms have occurred. On such days the decorrelation ratio is more likely to attain values greater than 1, ensuring that P_{D}(t,T_{ns}∣K_{i}) ≤ _{D}(t,T_{ns}∣K_{i}). An upper bound on P_{D}(t,T_{ns}∣K_{i}) may then be determined as follows:
where t′ ≈ t_{sample}. Since P_{D}(t,T_{ns}∣K_{i}) is roughly an increasing function of time, we can safely assume that P_{D}(t,T_{ns}∣K_{i}) ≤ _{D}(t′,T_{ns}∣K_{i}) for t ≤ t′. Note that P_{D}^{storm}(t′,T_{ns}) is likely to be a conservative upper bound, i.e., the true value of P_{D}(t,T_{ns}) may be considerably less than this bound, depending upon the data sets used to calculate P_{D}^{storm}(t′,T_{ns}).
6. Algorithm for Calculating σ_{decorr}
 Top of page
 Abstract
 1. Introduction
 2. Bounding the Probability of Transmitting HMI
 3. Review of WAAS Ionospheric Algorithms
 4. The Probability of Sudden Ionospheric Decorrelation
 5. Method of Determining an Upper Bound on P_{D}(t,T_{ns})
 6. Algorithm for Calculating σ_{decorr}
 7. Data Processing
 8. Results
 9. Conclusions
 Acknowledgments
 References
 Supporting Information
[21] The value of σ_{decorr} in the neighborhood of a given ionospheric grid point (IGP) can be defined in terms of the χ^{2} associated with the planar fit at that IGP. Let us define σ_{decorr} to be the value of σ such that the χ^{2} per degree of freedom is unity. Since the planar ionospheric model has three fit parameters, the χ^{2} per degree of freedom is χ^{2}(σ^{2})/(N − 3), where N is the total number of points in the fit. Thus, to obtain an estimate of σ_{decorr}, we solve
using a NewtonRaphson method iteration:
It can be shown that
where I is the identity matrix.
[22] The initial guess for σ_{decorr} is set according to the equation
where
(σ_{bias}^{satellite})^{2} is the variance of the hardware bias for each satellite and (σ_{bias}^{receiver})^{2} is the variance of the hardware bias for each receiver (the latter two variances are assumed to be constant for all satellites and receivers, respectively). This choice of the initial guess is based upon a suggestion by E. Altschuler (personal communication, 2002); we approximate σ_{decorr} by replacing each σ_{IPP,i}^{2} in W^{−1} with (σ_{decorr}^{RMS})^{2} and moving the (constant) offdiagonal elements (σ_{bias}^{satellite})^{2} + (σ_{bias}^{receiver})^{2} to the diagonal. From equation (8) we then obtain
Solving for σ_{1}^{2} produces equation (12).
[23] We use a convergence criterion of
which is generally found to be satisfied within three to eight iterations. Occasionally, we find that the NewtonRaphson iteration produces a negative estimate for σ^{2} (usually when χ^{2} is anomalously small). When this occurs, we replace the NewtonRaphson iteration with a false position search [see Press et al., 1988]. The false position method converges more slowly to the root of f(σ^{2}), but the search interval can be constrained so that σ^{2} ≥ 0. This iteration is stopped when
is satisfied. Thereafter the NewtonRaphson iteration is resumed.
[24] Note that when satellite and receiver biases are neglected and ,
where
with = Gx. For the data sets processed in this study, we find that this approximate equation for σ_{decorr}^{2} generally holds.
7. Data Processing
 Top of page
 Abstract
 1. Introduction
 2. Bounding the Probability of Transmitting HMI
 3. Review of WAAS Ionospheric Algorithms
 4. The Probability of Sudden Ionospheric Decorrelation
 5. Method of Determining an Upper Bound on P_{D}(t,T_{ns})
 6. Algorithm for Calculating σ_{decorr}
 7. Data Processing
 8. Results
 9. Conclusions
 Acknowledgments
 References
 Supporting Information
[25] To determine the _{D}(t,T_{ns}∣K_{i}), we have tabulated values of the decorrelation ratio as a function of the Kp index for delay data from days on which moderate to severe ionospheric storms occurred. These data consist of postprocessed slant delay measurements collected by the existing 25 WAAS reference stations. The intent of the postprocessing is to eliminate interfrequency biases, to remove the effects of cycle slips in carrier phase measurements, to level the carrier phase measurements to the corresponding range measurements, and to filter spurious measurements using the redundancy provided by the presence of three receivers at each station. Such data contain minimal error, and it is due primarily to multipath and data noise.
[26] To identify days on which significant storms have occurred, we have examined the variation of the Kp index over the time period extending from 1 January 2000 to 1 April 2001. The following criteria were used to select data sets: (1) all days on which the Kp index equaled or exceeded 8; (2) days on which the index equaled 7 for an extended duration (typically half the day or more); and (3) storms with Kp values greater than or equal to 5 that have been found to stress WAAS algorithms (e.g., data sets where planar fits have resulted in unusually high values of χ^{2}). The intent of the selection process has been to find storm data representative of the worst conditions that WAAS is likely to encounter. Data from the following 17 storm days have been processed: 11 January 2000, 12 February 2000, 6 and 7 April 2000, 25 May 2000, 8 June 2000, 15 and 16 July 2000, 11 and 12 August 2000, 19, 20, 28, 29, 30, and 31 March 2001, and 1 April 2001.
[27] The parameters used in the analysis are presented in Table 2. Fit residuals are tabulated for each epoch of data that follows a nonstorm duration of at least 900 s.
Table 2. Parameters Used in the AnalysisParameter  Value 

Ionospheric reference height (h_{i})  350 km 
Minimum permitted fit radius (R_{min})  800 km 
Maximum permitted fit radius (R_{max})  2100 km 
Minimum number of points in fit (N_{min})  30 
Maximum number of points in fit when R_{fit} > R_{min}  30 
Standard deviation of nominal ionosphere (σ_{decorr}^{nom})  35 cm 
Data epoch interval (t′)  100 s 
Nonstorm duration (T_{ns})  900 s 