We provide an enhanced model of the errors induced by deviations of ionospheric delays from those estimated by the planar model used by the GPS-based Wide Area Augmentation System (WAAS). To a first approximation the nominal ionospheric spatial decorrelation of vertical equivalent signal delays, σdecorrnom, is roughly constant over the whole of the WAAS service volume. However, significant gains may be achieved by including a more detailed description of σdecorrnom as a function of various metrics such as geomagnetic latitude, time of day, as well as the quality of the planar fit as characterized by the radius, relative centroid of the fit, and the density of delay data. We take the first step in the development of this more sophisticated model by determining which of these parameters is best suited for use as a metric for determining σdecorrnom. This allows us to construct a first-order model of the ionospheric decorrelation which depends on the local density of ionospheric pierce points. Our preliminary study indicates that this first-order model will result in a better than 20% reduction in the values of broadcast grid ionospheric vertical errors (GIVEs) within the coterminous United States. We also observe a better than 50% reduction in trips of the ionospheric irregularity detector in the Alaska region, which will lead to significant improvements to continuity, although this comes at the cost of a roughly 20% increase in the median GIVE in the Alaska region.
 The Wide Area Augmentation System (WAAS) [Federal Aviation Administration, 1999; RTCA, Inc., 2001] is an enhancement to the Global Positioning System (GPS) [Assistant Secretary of Defense, 2001] that provides sufficient accuracy, availability, continuity, and integrity of positioning data to meet the stringent requirements of the civil aviation community. WAAS operates by broadcasting a GPS-like signal from geostationary satellites which contains differential corrections [Kee et al., 1991] to GPS satellite orbits and clocks as well as ionospheric delay corrections. In addition, WAAS transmits integrity information in the form of bounds on the differential errors remaining after the WAAS corrections have been applied.
 The integrity data consists of the user differential range error (UDRE) and the grid ionospheric vertical error (GIVE). The UDRE is broadcast for each satellite and bounds the residual error in the satellite's orbit and clock. The GIVE is broadcast based upon a 5° ×5° grid in latitude and longitude; for satellite signals passing through the ionosphere in the neighborhood of a grid location, the GIVE provides a bound on the vertical equivalent delay error remaining after the WAAS ionospheric correction is applied. A user combines this information for each satellite to form an integrity bound which has at least 1–−10−7 probability of bounding that user's true position.
 The WAAS ionospheric algorithms are based on a local decorrelated planar model [Hansen et al., 2000a, 2000b]. Using a ground network of dual frequency receivers, accurate delays of the GPS satellite signals are determined throughout the coterminous United States (CONUS) and Alaska. These are converted to vertical equivalent delays under the assumption of a thin-shell ionospheric model. At each grid location a planar fit of these data is performed to provide a vertical delay estimate for that location which is broadcast to WAAS users. The GIVE is constructed from four terms. The first is the formal error term which arises from the geometry of the data used in the planar fit. Next is the ionospheric decorrelation which describes the deviation expected from the planar model. The final two terms account for inaccuracies owing to undersampling of ionospheric features and owing to changes in the ionosphere which may occur between message broadcasts.
 In addition to the nominal GIVE computation, WAAS includes an ionospheric irregularity detector [Walter et al., 2000] which is used to determine whether the true ionosphere is sufficiently planar that the planar WAAS ionospheric model can be relied upon with confidence. The detector is based upon a chi-squared test using the residual errors of the data off of the best fit plane. If the chi-squared statistic is greater than a certain threshold, then the detector is said to have tripped and the corresponding GIVE is set to a value large enough to meet the specified integrity levels.
 The present Wide Area Augmentation System design, when fully implemented, is expected to meet all accuracy, availability, continuity, and integrity requirements for the Lateral Navigation/Vertical Navigation (LNAV/VNAV) service volume [Federal Aviation Administration, 1999]. That is to say, no further algorithm changes are strictly necessary before WAAS certification, which is expected in July 2003. Nonetheless, there are two important aspects of WAAS performance under the present design which might be improved further. First, it may be possible to reduce the magnitude of GIVEs in the CONUS, where we have the most thorough coverage. Second, continuity and availability in Alaska might be improved by reducing the number of trips of the chi-squared irregularity detector and/or by lowering the estimates of the ionospheric errors.
 The goal of this study is to try to determine if and how these two factors might be addressed through a relatively simple change in the WAAS operational system parameters (OSPs). As such, we do not consider the more serious algorithm changes which may be introduced in later stages of WAAS, focusing our attention instead on simple improvements that can be made for the LNAV/VNAV service.
2. Constructing the Model
 To improve WAAS performance, it is clear that we should concentrate on the ionospheric delay estimation process carried out by the grid ionospheric vertical error algorithm, which outputs both delay estimates at the ionospheric grid points (IGPs) and the GIVEs. The reason we look at this particular algorithm is because the GIVEs contribute the greatest portion of the user integrity bounds and because the chi-squared irregularity detector acts as a test of the validity of the planar model of the ionosphere upon which the delay estimates and GIVEs are based.
 To determine the approach to take, we need to better understand why these issues arise. To this end we compute a set of estimates of ionospheric delay values using the WAAS ionospheric planar fitting algorithm [Walter et al., 2000], with which we can compare postprocessed truth data from measurements at the ionospheric pierce points (IPPs). We have processed a 30 day data set from the period of 15 December 2000 to 20 January 2001 (note that owing to outages this is not a continuous data set). This is a nominal data set from a period of solar maximum without significant geomagnetic activity and is representative of the ionosphere observed by WAAS a majority of the time. The data consists of dual frequency GPS measurements from the WAAS ground network. For each pierce point in the data set we have an associated truth delay value, which has been determined by removing associated receiver and satellite biases and by leveling the data to remove noise while accounting for cycle slips. The data is converted from slant delays to vertical equivalent delays by means of a thin-shell model at an ionospheric shell height of 350 km. To produce our delay estimates, we perform a planar fit of the truth data about each of the IPPs. Note that the ionospheric delay truth value at the IPP in question is excluded from the planar fit to provide an unbiased delay estimate. Next, a vertical equivalent delay residual, ΔI, at the IPP is constructed by subtracting the estimated delay from the truth value at that same point. This process produces a delay residual for each of the IPPs in the data set over the 30 day period.
 To understand the results, it is helpful to visualize how these delay residuals vary over the WAAS coverage region. We therefore bin the delay residuals at intervals of 5 degrees in both latitude and longitude and plot in Figure 1 the standard deviation of delay residuals, σΔI, in each of the geographic bins. There are several features to note from Figure 1.
 First, in central CONUS, the standard deviation of delay residuals dips down to a value close to 20 cm. This is particularly relevant since an important OSP in the WAAS system is the spatial decorrelation under nominal conditions, denoted as σdecorrnom. This OSP is set to a constant value of 35 cm, a value which was derived from the study of a few days' worth of ionospheric delay data using a somewhat different, albeit still planar, model for fitting the ionosphere [Hansen et al., 2000a, 2000b]. σΔI effectively combines contributions due to both the actual decorrelation from a plane, σiono, as well as a contribution denoted as σsystem, which is introduced into WAAS planar fit by system errors such as noise, residual biases and multipath, and effects due to sparsity of data. It should therefore be clear that σΔI ≥ σiono. This suggests that the current value, σdecorrnom = 35 cm, is larger than necessary in central CONUS.
 The next point to notice from Figure 1 is that σΔI is not constant. In particular, it becomes large in the far northwest around Alaska and in the extreme southeast, peaking at values exceeding a meter. Now, since these correspond to regions of poor WAAS coverage relative to CONUS, a portion of the contribution to σΔI is certainly attributable to nonzero σsystem owing to inaccuracies in the planar fit as a result of sparse and/or skewed pierce point distributions. However, it is also true that these two regions are expected to have a more active ionosphere than CONUS because of the more intense ionosphere in the southeast and because of auroral activity in the northwest. It is therefore very reasonable to expect that the contribution σiono grows significantly beyond its CONUS value in these regions.
 Given these results, our strategy is to develop a varying σdecorrnom such that a value appropriate to the prevailing conditions is used in both the planar fits and the computation of the GIVEs. One possibility would be to simply use Figure 1 to derive a value of σdecorrnom associated with each WAAS IGP. However, this approach is somewhat problematic due to the sparsity of data at the fringe IGPs. It also presents a number of issues from the perspective of integrity analysis, as such a model is too strongly tied to the current configuration of Wide-area Reference Stations (WRSs), and is not robust in the event of station outages or added stations. Rather than take this approach, it is more convenient to identify a simple metric which correlates well with the behavior of σΔI shown in Figure 1.
 To accomplish this task, we examine the behavior of σΔI as a function of a number of relevant parameters, including geomagnetic latitude, local time of day, the number of points in the planar fit, the radius of the fit, the relative centroid of the pierce points in the fit, and the overall pierce point density of the fit. For each of these parameters we bin the delay residuals to determine the behavior of σΔI. The results for each case are shown in Figures 2a–2f.
 Clearly σΔI depends upon each of these parameters to lesser or greater extent. While each could be used as a metric for determining appropriate values of σdecorrnom, the behavior of σΔI as a function of the IPP density of the planar fit, shown in Figure 2f, is of particular note. Importantly, the relationship is very smooth, with σΔI growing quickly as the density drops and flattening out in well sampled regions. In fact, the curve shown in Figure 2f is remarkably close to a hyperbola for which the least squares best fit form is
where the IPP density is measured in units of the number of IPPs per square megameter (or one million square kilometers). In these units, ρIPP has an effective range in WAAS of about 0.78 to 24.4. The fit is shown in Figure 3, which also indicates a variation of roughly ±5 cm from the best fit value.
 To determine whether ρIPP truly is an appropriate metric to use for characterizing σdecorrnom, we compare σΔI from Figure 1 with the corresponding map σΔI(ρIPP) derived from the mean IPP density at each location. This comparison is shown in Figure 4. The similarity between these two maps indicates that ρIPP will act as a reasonably accurate metric for determining σdecorrnom.
 Having determined an appropriate metric to base the model upon, we now construct our next-to-leading order model of σdecorrnom. We recall that σΔI is effectively made up of two components, σiono and σsystem, where σiono is the uncertainty inherent in using a planar model of the ionosphere while σsystem is the error introduced by the our system's planar fitting procedure. Ideally, we would eliminate the contribution of σsystem to σΔI in order that we may be able to set σdecorrnom = σiono. However, this separation is not possible because we have a limited set of truth data. The approach we therefore take is to use an overestimate of σiono by setting σdecorrnom = σΔI. In regions where σsystem is small, this will give us a best fit σdecorrnom, while in fringe regions where σsystem is expected to be significant, this gives a σdecorrnom, which is greater than the actual ionospheric contribution to the error. Our model is therefore defined to be
3. Impact to WAAS Performance
 To determine how this relation will affect the system compared to the current system for which σdecorrnom has a constant value of 35 cm, we note that σdecorrnom appears in two places in the GIVE algorithm. First, it appears as part of the formal error term in a sum of squares with the integrity monitor noise and multipath uncertainty σmon on the diagonals of the planar fitting weighting matrix. An important property of σmon is that it is defined as an overbound of the true monitor noise uncertainty. Therefore each of the diagonal elements in the weighting matrix represents an overestimate of the true uncertainty. For this reason, using an overestimate of the ionospheric uncertainty as in equation (2) does not in any way change the character of the weighting matrix. While it can be argued that using uncertainty overestimates in a least squares weighting matrix is not ideal, using a quantity other than the overbound σmon would involve system changes more extensive than we wish to consider here.
 The second place where σdecorrnom appears in the GIVE algorithm is as an explicit contribution to the GIVE equation. The consequence here of using the function (2) rather than a constant value of 35 cm is a reduction in the value of the GIVE in solid coverage regions such as central CONUS due to values σdecorrnom (ρIPP) < 35 cm. Of course, there is also an increase in GIVEs in the more fringe regions where σdecorrnom(ρIPP) > 35 cm. The algorithm therefore is expected to achieve at least one of the goals described at the onset of this study: a reduction of the GIVEs in CONUS. This reduction is found to be approximately at the 25% level.
 The question as to whether the algorithm change would improve availability in Alaska is somewhat more involved. Making σdecorrnom a function of IPP density in Alaska has two primary effects. First, it dramatically reduces the number of trips of the chi-squared irregularity detector owing to the increase in the allowed margin of uncertainty in the planar fits tested by the detector. This reduction can be upward of 50%, which will significantly improve continuity in Alaska, and everything else being constant, would also lead to improved availability. However, this modification in σdecorrnom also leads to larger GIVEs owing to sparse IPP density in the region. In particular, GIVEs grow by roughly 20%, and in many cases this additional contribution to the GIVE may be enough to push the user range error beyond the LNAV/VNAV service limit, which would negatively impact availability. To determine which of these factors is most significant will require a full study of availability in Alaska using the updated model of σdecorrnom.
 The results using the new model of the ionospheric decorrelation compared with the nominal constant model for the 30 day data set are displayed in Table 1. Although it is not ideal to use the same data set as was used to derive the decorrelation function, this very representative nominal data set was the only substantial data set available at the time, and the results were validated further by performing cross-validation using subsets of the full data set. Note that these data are derived by implementing the respective model in a mock up of the full WAAS GIVE algorithm where each IPP is now treated as a virtual user and delay values are interpolated from the values at the IGPs to these user positions. The appropriate error on the delay correction is therefore the user ionospheric vertical error (UIVE) which has been interpolated from the GIVE values from the 3 or 4 nearest IGPs which surround the user location. Note that the values listed are interpolated from unquantized GIVEs, while WAAS uses an indexing system to broadcast GIVE values over a limited bandwidth.
Table 1. Comparison of Results of the Nominal and Improved Modelsa
The negative value for the “improvement” of the median UIVE in Alaska indicates that this quantity gets worse with the model change.
Chi-squared trips (AK)
Median UIVE (AK)
Median UIVE (CONUS)
 Although it offers significant advantages over the model of a constant ionospheric decorrelation, the first-order model presented here could be extended further to more precisely reproduce the observed distribution of uncertainties. Ideally, we would like to construct a model which will be entirely robust to arbitrary changes in the system, including extensions of WAAS into Mexico and Canada, and which would also be applicable to other satellite-based augmentation systems in regions like Japan, India, and South America. The next step in achieving such a model is to incorporate additional metrics, such as those shown in Figure 2, into the functional form of the σdecorrnom.
 Even if the simplistic model presented here is not implemented in its present form, it acts as a demonstration of the potential gains that can be achieved in system performance by going beyond a model of constant ionospheric decorrelation. As such it will make an important component of future WAAS improvements.
the nominal decorrelation of ionospheric delays from a planar model.
the deviation of a vertical ionospheric delay from the planar fit model (a delay residual).
the standard deviation of delay residuals.
the standard deviation of decorrelation errors attributed to the ionosphere.
the standard deviation of system contributions to errors in the planar fits.
the number of ionospheric pierce points per square megameter.
the monitor noise uncertainty including multipath and noise.
 The authors thank Helena Go for her help in determining the dependencies of the delay residuals on the various metrics. The contents of this material reflect the views of the authors. Neither the Federal Aviation Administration nor the Department of Transportation make any warranty or guarantee, or promise, expressed or implied, concerning the content or accuracy of the views expressed herein. Export authority for this document is U.S. Department of Commerce exemption EAR 99/NLR for 7D994.