Estimating ionospheric delay using kriging: 1. Methodology

Authors


  • This is a commentary on DOI:

Abstract

[1] The Wide Area Augmentation System (WAAS) is an augmentation of the Global Positioning System (GPS) that provides safe and reliable use of GPS signals for airline navigation over much of North America. Ever since WAAS was first commissioned in July of 2003, the vertical delay estimate at each node in the WAAS ionospheric grid has 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, the vertical delay will be estimated by an established, geo-statistical technique known as kriging. When compared to the planar fit model, the kriging model is generally found to match better the observed random structure of the vertical delay. This paper presents the kriging methodology to be used to estimate the vertical delay and its uncertainty at each ionospheric grid point. In addition, it provides examples of the improvement in delay accuracy achieved. Under disturbed conditions, the implementation of kriging reduces the magnitude of the root mean square fit residual by up to 15%.

1. Introduction

[2] For single-frequency users of global navigation satellite systems (GNSS), ionospheric delay continues to be the largest source of positioning error. To guarantee the safety of airline navigation based upon GNSS signals, satellite-based augmentation systems have been developed to ensure the accuracy, integrity, availability, and continuity of user position estimates derived from GNSS measurements. In the United States, the Wide Area Augmentation System (WAAS) is an augmentation of the Global Positioning System (GPS) that 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 (see Figure 1). To allow the user to correct for the error due to ionospheric delay, WAAS derives from these measurements a vertical delay estimate at each ionospheric grid point (IGP) in a mask specified by the WAAS Minimum Operational Performance Standards [MOPS WAAS, 2001] (see Figure 2). The vertical delay at an IGP is designated the Ionospheric Grid Delay (IGD) at that IGP. In addition, WAAS computes at each IGP a safety-critical integrity bound called the Grid Ionospheric Vertical Error (GIVE). Integrity refers to the reliability and trustworthiness of the information provided by the navigation system and to the system's ability to deliver timely warnings to users when the system should not be used for navigation because of signal corruption or some other error or failure in the system. GIVEs are derived from inflated and augmented values of the formal estimation error. They protect the user from the effects of delay estimation error due to ionospheric irregularity, both sampled and undersampled.

Figure 1.

WAAS receiver sites in North America.

Figure 2.

WAAS mask of ionospheric grid points (blue squares identify IGPs in the IGP working set, i.e., IGPs at which IGDs and GIVEs are available to the user).

[3] From the Initial Operating Capability of WAAS in July of 2003 through Release 8/9 of the initial system to the current system, the vertical delay estimate and its integrity bound at each IGP have been calculated from a planar fit of slant delay measurements projected to vertical. The slant-to-vertical conversion is achieved by modeling the ionosphere as an infinitesimally thin shell at a representative ionospheric altitude (see Figure 3). In WAAS Follow-On Release 3, estimation of vertical delays will be performed by a geo-statistical technique known as kriging [Cressie, 1993; Webster and Oliver, 2001; Blanch, 2002; Wackernagel, 2003], a type of minimum mean square estimator, adapted to spatial data, that originated in the mining industry in the 1950's. Kriging provides a smoothed depiction of a spatially distributed variable that has been sampled by irregularly spaced measurements. Compared to the planar fit model, the kriging model generally achieves a better match to the observed random structure of the vertical delays (or it can be tuned to match these data better).

Figure 3.

Thin shell model of the ionosphere, used to define IPPs and the obliquity factor for converting slant delay to vertical delay. The shell height hi is 350 km.

[4] This paper represents the initial installment of a two-part analysis of ionospheric delay estimation using kriging that emphasizes, in particular, its implementation and its impact on system availability. The focus of this paper is on the kriging methodology, adopted in WAAS Follow-On Release 3, for estimating vertical delay and its uncertainty at each IGP. In a subsequent paper [Sparks et al., 2011], we present the second installment of the analysis, examining the influence of kriging on the WAAS strategy for protecting the user from estimation error due to ionospheric irregularities. An overview of the work discussed in this two-part analysis has been the subject of two prior presentations [Sparks et al., 2010a, 2010b].

[5] In a series of papers of which this is the first, it is our intent to present a comprehensive discussion of all aspects of the WAAS GIVE Monitor. The GIVE Monitor calculates the IGDs and GIVEs to be broadcast and ensures GIVE integrity. It is designed to handle a single threat, namely, the threat that local ionospheric behavior does not conform well to the assumed model. Associated with each satellite-to-receiver raypath is a vertical delay evaluated at the ionospheric pierce point (IPP) of the raypath, defined to be the location where the raypath penetrates the ionospheric shell. For each GPS signal detected by a user's receiver, the GIVE Monitor guarantees that the user's computed integrity bound on the vertical delay error at the signal IPP bounds, with a sufficiently high probability, the actual slant delay error converted to vertical.

[6] Our paper is comprised of five sections. Following the introduction, section 2 defines the ionospheric delay model. Section 3 derives the kriging estimation equations (part of this derivation is relegated to Appendix A). Section 4 discusses quantitatively the improvement in delay accuracy achieved by kriging. Section 5 concludes with a summary of the material presented.

2. Ionospheric Model

[7] As a GNSS signal passes through the ionosphere, it suffers a delay proportional to the number of free electrons along the signal's raypath. The slant total electron content (STEC) along the raypath may be written as a path integral of electron density along the line-of-sight from the receiver to the satellite:

display math

where the subscripts r and s identify, respectively, the receiver and satellite associated with the signal, ne is the ionospheric electron density, and xrs () identifies points along the raypath as parameterized by the path length . In a local Cartesian coordinate system, this integral may be redefined as an integral over altitude h:

display math

where

display math

Re is the earth radius, α is the elevation angle of the satellite as viewed from the receiver, and xrs is now treated as a function of the altitude h.

[8] To define IPPs and to convert slant delay to vertical delay, WAAS uses the standard thin shell model of the ionosphere, i.e., the electron density is taken to be non-negligible only within a thin shell at a given ionospheric reference height hi (see Figure 3). In WAAS, hi is specified to be 350 km, which is a typical value for the daytime peak of the F-layer electron density. Under this approximation, equation (2) simplifies to

display math

where

display math

is the vertical total electron content at the IPP. F(α, hi), the function used to convert slant delay to vertical delay, is designated the obliquity factor of the raypath.

3. Kriging Equations

[9] In this section we present a derivation of the equations used to calculate the kriging estimate of vertical delay. A more thorough discussion of this derivation and the assumptions on which it relies can be found in the work of Blanch [2003].

3.1. Coordinate System

[10] The kriging estimate of vertical delay at a given IGP is determined from observations whose IPPs lie in the vicinity of the IGP. In an earth-centered, earth-fixed (ECEF) Cartesian coordinate system, the coordinates of an IPP (or IGP) are given by

display math

where Re is the earth radius, hi is the height of the ionospheric shell, and ϕ and λ are, respectively, the latitude and longitude of the pierce point. Estimation is performed in a local up-east-north coordinate system whose origin coincides with the position of the IGP. In the ECEF coordinate system, the up, east, and north unit vectors of the local coordinate system are specified as follows:

display math

where ϕIGP and λIGP are, respectively, the latitude and longitude of the IGP.

3.2. Vertical Delay Model

[11] Our model for vertical ionospheric delay Itrue at an IPP near an IGP assumes

display math

where ΔxxxIGP is the Euclidean vector describing the distance separating the IPP from the IGP in ECEF Cartesian coordinates, the coefficients a0, aeast, and anorth specify the planar trend of the ionospheric delay, and rx) is a scalar field describing small irregularities that are superimposed on the planar trend. The scalar field characterizes the correlation between neighboring measurements. The vertical ionospheric delay measured at x can then be modeled as

display math

where ɛ represents the measurement error. The measurement noise is specific to each receiver-satellite pair and does not depend upon the ionosphere.

[12] Our vertical delay model should remain valid over spatial scales where the variation of the delay is approximately planar. Over larger scales, the magnitude of the scalar field rx) will no longer be small. In addition to the error due to non-linearity, however, there will also be error associated with the slant-to-vertical conversion (equation (4)). Two types of error restrict the accuracy of delay estimates based upon the thin shell model: (1) error arising from the implicit assumption that the electron density is independent of the azimuthal angle at the IPP serving as the fit center, and (2) error due to an invalid obliquity factor (e.g., error due to a suboptimal choice of shell height). Under nominal conditions at mid-latitudes, the magnitude of the error incurred from these sources is small. However, at low latitudes or at mid-latitudes under disturbed conditions, the error increases due to the presence of enhanced ionization, complex ionospheric structure, and large electron density gradients.

[13] The accuracy of delay estimation based upon kriging will also be limited by the error associated with the processing that generates the input observations used to perform the estimation. For example, our model assumes that the interfrequency hardware biases associated with each receiver and satellite have been removed from the slant delay observations prior to performing delay estimation. However, the software used to remove these biases also relies on the thin shell model to convert slant delay to vertical [Mannucci et al., 1998]. The obliquity error discussed above can be a source of systematic error in the estimation of the interfrequency biases. WAAS employs additional monitors to ensure that the interfrequency bias error is small.

3.3. IPP Search Algorithm

[14] To estimate the vertical delay in a given epoch at a given IGP, WAAS first selects a set of measurements whose IPPs reside in the region surrounding the IGP. The set of eligible measurements is restricted to those whose elevation angles are greater than or equal to 5°. (Other restrictions on eligible measurements will be discussed in a subsequent publication.) The fit domain on the ionospheric shell is defined to be a region within a circle centered upon the IGP, whose radius is Rfit as measured by the Euclidean (straight-line) separation between the location of the IGP and the points on the circle (see Figure 4). All eligible measurements that have IPPs within a minimum distance Rmin are included in the delay estimation. If the number of such measurements is less than Ntarget, the fit radius is expanded until it defines a circle that surrounds Ntarget points. If the fit radius attains a maximum value of Rmax and still fails to encompass Ntarget points, delay estimation is performed with fewer points, provided that at least Nmin points lie within a distance Rmax. If fewer than Nmin IPPs are within Rmax, no kriging estimate is computed, and the GIVE to be broadcast is set to Not Monitored.

Figure 4.

Selection of IPPs for performing estimation. The circle identifies the fit domain of the IGP. The IPP distribution within the fit domain is characterized by the fit radius (Rfit) and the relative centroid metric (RCM).

[15] If all IPPs lie within a distance Rmin of the IGP, the fit radius Rfit is defined to be Rmin. If there are fewer than Ntarget points in the fit domain, the fit radius Rfit is defined to be Rmax. Otherwise, the fit radius Rfit is defined to be the distance in kilometers from the IGP to the most distant IPP. In the undersampled threat model discussed by Sparks et al. [2011], Rfit is used as a metric characterizing the spatial density of IPPs within the fit domain. A large value of Rfit is associated with a higher probability that an ionospheric irregularity is undersampled.

[16] A second scalar that characterizes the spatial distribution of IPPs within the fit domain is the centroid radius (see Figure 4). In the undersampled threat model, the relative centroid metric (RCM), i.e., the ratio of the centroid radius to the fit radius, is used to describe the uniformity of the distribution of IPPs within the fit domain. The relative centroid metric ranges between 0 and 1. As values approach 1, they are associated with IPP distributions that are more highly skewed to one side of the IGP, increasing the probability that an ionospheric irregularity within the fit domain is undersampled.

[17] For all calculations discussed in this paper, the IPP selection parameters have the following values: Rmin = 800 km, Rmax = 2100 km, Ntarget = 30, and Nmin = 10.

3.4. Vertical Delay Estimation

[18] Define N to be the number of measurements whose IPPs lie within the fit radius of a given IGP at the locations xequation image, for equation image = 1 to N. Let equation imageκImeasxκ) identify the corresponding vertical delay values, i.e., slant delay measurements converted to vertical using the obliquity factor defined by equation (4). Then the kriging estimate equation image of the ionospheric vertical delay near this IGP will be determined from a linear combination of these delay values:

display math

where

display math

and

display math

Each vertical delay is computed from dual frequency measurements with the satellite and receiver interfrequency biases removed as follows:

display math

where fL12 and fL22 are, respectively, the L1 (1575.42 MHz) and L2 (1227.60 MHz) frequencies used by the Global Positioning System, PRL1,κ and PRL2,κ are the smoothed, multipath-corrected pseudo-range measurements (in meters) associated with the κth IPP at L1 and L2, respectively, BL1−L2,r and BL1−L2,s are the L1-L2 interfrequency bias estimates (in meters) associated with the measurement receiver and satellite, respectively, and F(ακ, hi) is the obliquity factor associated with the κth IPP. The variance characterizing the error in equation imageκ is calculated as follows:

display math

where σL1,κ2 and σL2,κ2 are the variances of the code noise and multipath error on, respectively, the L1 and L2 signals of the κth IPP, and σL1−L2,r2 and σL1−L2,s2 are the variances of the L1-L2 interfrequency bias estimates associated with the measurement receiver and satellite, respectively.

3.5. Estimation Constraint

[19] To determine the wκ coefficients, we first require that the estimate be unbiased, that is,

display math

where brackets indicate the expectation value. Substituting from equations (8) and (10) produces

display math

When equations (8) and (9) are substituted into the left-hand side, this equation implies that

display math

where

display math
display math

and we have assumed that the expectation values of the scalar field and the measurement noise vanish, i.e., 〈rx)〉 = 〈rxκ)〉 = 〈equation imageκ〉 = 0.

3.6. Minimizing the Estimation Variance

[20] The wκ coefficients are calculated by minimizing the estimation variance subject to the constraint imposed by requiring that the estimator be unbiased (i.e., equation (15)). The estimation variance is defined as follows:

display math

Requiring the estimator to be unbiased allows us to simplify this equation as

display math

where

display math

and

display math

[21] Since the ionospheric scalar field and the measurement noise are uncorrelated, equation (21) reduces to

display math

We can rewrite this equation for the estimation variance as

display math

where

display math
display math
display math
display math

that is, M is the N × N matrix describing the covariance of the measurement noise between measurement locations, C is the N × N matrix describing the covariance of the detrended ionospheric delay between measurement locations, c is an N-vector whose elements specify the covariance between the scalar field at a position Δx near the IGP and the detrended delays at the measurement locations, and c0 is the variance of the scalar field, henceforth assumed to be a constant independent of Δx.

[22] In Appendix A, we solve this constrained optimization problem by the method of Lagrange multipliers. Defining the weighting matrix W as

display math

the solution can be expressed as

display math

3.7. Measurement Noise Covariance

[23] To calculate a kriging estimate of the vertical delay using w as defined by equation (31), we need now to specify the vector c and the matrices C and M. The measurement noise matrix is constructed from the measurement error variances and the variances of the receiver and satellite biases:

display math

where

display math

The diagonal elements of this matrix are defined by equation (14). The off-diagonal elements describe the correlation of bias errors between measurements made with common receivers or common satellites.

3.8. Ionospheric Delay Variogram

[24] To model C and c requires a knowledge of the underlying spatial structure of the random scalar field rx). A useful means of characterizing this structure is the variogram [Wackernagel, 2003], defined as follows:

display math

The variogram measures the rate at which the values of a random function evaluated at two points become decorrelated as a function of the distance separating the two points, assuming that there is no deterministic underlying trend and that the random function is stationary. Expanding the right-hand side of equation (34) shows the relationship between the variogram and the covariance function Cxκ, Δxζ):

display math

where the covariance function (used to calculate the elements of the C matrix) is defined:

display math

The previously defined scalar field variance c0 is here designated the sill of the variogram and specifies the variance of two measurements that are completely decorrelated. In our model, the covariance function, like the variogram, will be a function only of the distance separating the points in question.

[25] The specification of the variogram is constrained by the requirement that any linear combination of residuals calculated using the variogram must have a positive variance. This is equivalent to requiring that C be positive definite. By definition, the variogram vanishes when its two arguments are identical. However, the variogram may be discontinuous where the two points coalesce (this is designated the nugget effect, a legacy of the use of kriging in the mining industry, where, for example, soil grade can be discontinuous due to the presence of gold nuggets). Experimental variograms based upon WAAS observational data sets [Blanch, 2003] show just such a discontinuity at the origin. The physical basis for this discontinuity, at least in part, is obliquity error. When the thin shell model is used to determine the obliquity factor for converting slant delay to vertical delay, azimuthal symmetry of the ionosphere about the IPP is assumed implicitly. Since this assumption is at best, only approximate, two measurements recorded at distinct receivers but with the same IPP will not, in general, even under quiet ionospheric conditions, give rise to the same vertical delay estimate.

[26] To match the observed behavior of our ionospheric data sets, we need a model variogram with a linear dependence on separation near the origin and a finite limit at infinity. When the point separation exceeds zero, we use an exponential model for the variogram:

display math

where

display math

and (σdecorrnominal)2 is the delay covariance associated with nearly coincident IPPs, c0 = (σdecorrtotal)2 is the delay covariance associated with widely separated (uncorrelated) IPPs, and ddecorr is the characteristic decorrelation distance. In this model, note that the nugget or discontinuity for zero separation is (σdecorrnominal)2.

[27] Figure 5 illustrates qualitatively the difference between the theoretical variograms used to model ionospheric delay in the WAAS Initial Operating Capability and in WAAS Follow-On Release 3. (Note: the values of σdecorrnominal actually used by WAAS in these two releases differ slightly, unlike their depiction in Figure 5.) In the planar fit model, σdecorrtotal is set equal to σdecorrnominal, i.e., the variogram is a constant independent of the separation between measurement IPPs, except at the origin where it is discontinuous (see the line of blue dots in Figure 5). The nugget-effect here is equivalent to modeling the concept of white noise in signal processing. The kriging model also displays a nugget-effect at the origin, but the behavior of the variogram near the origin is linear rather than constant (see the curve of red dots in Figure 5). The fact that the kriging model variogram describes the observed behavior of experimental variograms better than the planar fit variogram is the reason that our kriging model provides improved delay estimation accuracy.

Figure 5.

Theoretical variograms for the planar fit model of WAAS Initial Operating Capability (blue dots) and the kriging model of WAAS Follow-On Release 3 (red dots). The labeled value of (σdecorrtotal)2 applies only to the kriging model; for the planar fit model, this quantity is equal to (σdecorrrnominal)2. Note that both variograms are discontinuous at the origin where each has a value of zero.

[28] Blanch [2003] presents an extensive set of empirical variograms under both quiet and disturbed conditions. While the behavior of each variogram depends upon the overall level of ionospheric disturbance, each is generally characterized by linear growth on the spatial scale of the fit domain (<2100 km) and discontinuity at the origin. Thus, the two most important quantities for determining the model variogram constants are the intercept of the experimental variogram at zero and the slope at zero. Since delay estimation will be restricted to fit radii that are less than or equal to 2100 km, delay estimates are less sensitive to the choice of sill, i.e., (σdecorrtotal)2.

[29] The variogram constants are held fixed in WAAS Follow-On Release 3. This is an operational requirement, as the state of the ionosphere is not known prior to performing delay estimation, and there are usually not enough data in each epoch to determine the variogram constants. Keeping the constants fixed does not necessarily imply that the detrended observations for disturbed conditions exhibit the same characteristics as the detrended observations for nominal conditions. Contrary to the assumptions used to specify a variogram, a deterministic trend, highly variable with time and location, can remain after removing the planar trend from the delay residual differences. It should be noted, however, that when such a residual trend is detectable, the empirical variogram should increase faster than linearly, which is not generally observed in the empirical variograms. The optimal set of variogram constants for WAAS operation has been determined by trade studies that vary the value of the sill under a broad range of ionospheric conditions.

[30] If the state of the ionosphere were known prior to estimation, it would, in principle, be possible to use different sets of kriging parameters optimized for the known level of disturbance. Since this is not feasible, we adopt an alternative approach to account for the fact that the ionospheric behavior modeled by the scalar field is not stationary: the model variogram is inflated by a multiplicative factor, evaluated from observed data, that describes the roughness of the vertical ionospheric delay distribution [Blanch, 2003]. Thus, while the model variogram used in our delay estimation is not optimal under all ionospheric conditions, we contend in this paper that the kriging model with fixed variogram constants has been found to agree better with the observed data than the planar fit model.

3.9. Ionospheric Noise Covariance

[31] Since the model variogram is bounded for large point separations, we can use it and equation (35) to define model expressions for the matrix elements of C, the covariance of the detrended ionospheric delay between measurement locations, and the vector elements of c, the covariance between the scalar field rx) near the IGP and the detrended delays at the measurement locations:

display math
display math

where

display math

These equations reduce to their planar fit counterparts used in the WAAS Initial Operating Capability when (σdecorrtotal)2 approaches (σdecorrnominal)2.

3.10. Goodness-of-Fit

[32] The accuracy of a delay estimate will generally diminish as the level of ionospheric disturbance increases. To determine whether a given estimate is likely to satisfy our accuracy requirements, we need a measure of the quality of the estimation. For this purpose, the classical χ2 goodness-of-fit statistic has proven useful.

[33] According to equations (8) and (9), our slant delay measurements, converted to vertical, exhibit a trend and are correlated due to the scalar field rx) and the measurement error equation image. To derive a χ2 goodness-of-fit statistic from these measurements, it is first necessary to filter the trend from the estimation residuals and then decorrelate the resulting residuals. The χ2 goodness-of-fit statistic takes the form

display math

where each element of the vector Γequation imageIPP represents a reduced fit residual, i.e., a residual difference between the vertical delay at a given fit IPP and its kriging estimate (equation (10)), detrended, decorrelated, and normalized by the square root of its variance. Since the trend for the vertical delay at each fit IPP is given by Ga, where a = [a0aeastanorth]T, the trend for each residual can be filtered by requiring that Γ satisfy

display math

for any arbitrary choice of a, or equivalently,

display math

If F is a matrix with null space G, there must exist a matrix H such that Γ = HF. A matrix satisfying the condition imposed by equation (44) is:

display math

An expression for H can now be deduced by requiring the covariance of Γequation imageIPP be the identity matrix, ensuring that the reduced residuals are orthogonal [Blanch, 2003]. The resulting χ2 is

display math

The magnitude of this χ2 statistic provides a means of establishing whether the ionospheric measurements are consistent with the assumed model.

3.11. Irregularity Metric

[34] In practice, the best indicator of the local level of ionospheric disturbance has been found to be the χ2 goodness-of-fit statistic associated with the vertical delay estimate at a given IGP [Walter et al., 2000]. The expected distribution of χ2 is parameterized by the number of degrees of freedom of this statistic, i.e., the number of measurements minus the degrees of freedom in the model. From equation (8), we find that the number of degrees of freedom in the model is three. Given equation imageN − 3 and an allowable false alarm rate, we can calculate a threshold value χthreshold2 [Walter et al., 2000] using the inverse χ2 cumulative probability distribution [Abramowitz and Stegun, 1964]. If χ2 exceeds this value, it is likely that either the input variances or the ionospheric model (or both) are incorrect. Note, however, that a value of χ2 below χthreshold2 does not guarantee that the model is valid.

[35] The χ2 statistic provides a basis for defining a local irregularity metric to indicate whether it is safe for a WAAS user to calculate his or her position using the vertical ionospheric delay estimate associated with a given IGP [Blanch, 2003]:

display math

where

display math
display math

and χthreshold2 is obtained by evaluating the inverse χ2 cumulative probability distribution function for degrees of freedom and a probability of 99.9%, i.e., 1 minus a false alarm probability of 10−3. Rnoise is an inflation factor that prevents the presence of measurement noise from concealing the magnitude of an ionospheric irregularity.

4. Improvement in Delay Estimate Accuracy

[36] In this section we examine the improvement in delay estimate accuracy that can be achieved by implementing kriging. First we briefly review the observational data set from which we have taken the input data for our computations. Next, we quantify this improvement in accuracy by comparing distributions of the irregularity metric generated by a planar fit model and by a kriging model. Subsequently, we compare fit residuals produced by the two models.

[37] The results presented in this section have been produced using a software package developed at the Jet Propulsion Laboratory entitled Ionospheric Slant TEC Analysis using GNSS-based Estimation (IonoSTAGE) [Sparks, 2011]. IonoSTAGE is a Matlab platform for performing analysis and visualization of ionospheric slant total electron content (TEC) using GNSS measurements. (Matlab is a product of The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098.) IonoSTAGE can also generate the WAAS undersampled ionospheric irregularity threat model discussed by Sparks et al. [2011]. Threat model results produced by IonoSTAGE have been validated in extensive studies that have compared them to the corresponding results generated by the Raytheon Threat Model (TM) tool.

4.1. Supertruth

[38] The observational data processed to generate the results in this section are drawn from supertruth [Komjathy et al., 2005], a data set generated for WAAS and designed to provide a truth standard for ionospheric delay measurements. These data are derived from measurements recorded at the twenty-five reference stations of the WAAS Initial Operating Capability and are processed in receiver independent exchange (RINEX) format using the Jet Propulsion Laboratory's GNSS-Inferred Positioning System and Orbit Analysis Simulation Software package (GIPSY-OASIS [Zumberge et al., 1997] and the Global Ionospheric Model (GIM) software package [Mannucci et al., 1998]. The objectives of the post-processing are (1) to eliminate interfrequency hardware biases (assumed to be constant over durations of one day), (2) to remove the effects of cycle slips in carrier phase measurements, (3) to level the carrier phase measurements to the corresponding range measurements, and (4) to detect and eliminate spurious measurements using the redundancy provided by having three receivers at each site. The supertruth algorithm employs a voting scheme that compares redundant observations from the three receivers; an observation is included in a supertruth data set only if two of the leveled ionospheric observables agree to within a specified tolerance. The minimal error that these data contain is due primarily to multipath and receiver noise.

[39] For the computations reported in this paper, it should be noted that we use the most recent version of each input data file. At the time of our computations, the version number of the most recently revised supertruth data is 3. The older, version 1 supertruth data (data processed prior to 2003) have been reprocessed in subsequent versions to restore a large fraction of the observations that were eliminated in version 1. The version 2 and version 3 supertruth files are up to 50% larger than the version 1 files.

4.2. Distributions of the Irregularity Metric

[40] One means of assessing the improvement in accuracy afforded by kriging is to compare distributions of the irregularity metric (proportional to the goodness-of-fit statistic). As examples, we consider data from three days exhibiting distinct ionospheric conditions: a quiet day (9 November 2005), a day of moderate disturbance (5 November 2001), and a day on which an extreme storm occurred (30 October 2003). The level of ionospheric disturbance is reflected in the level of geomagnetic disturbance as indicated by two geomagnetic indices, the disturbance storm time index (DST) and the planetary K-index (Kp). For 9 November 2005, 5 November 2001, and 30 October 2003, the Dst index dropped to −10, −41, and −383 nT, respectively, while the Kp index attained maxima of 2+, 5+, and 9, respectively. In the examples that follow, the planar fit model uses σdecorrtotal = σdecorrnominal = 0.35 m, while for kriging, σdecorrnominal = 0.3 m, σdecorrtotal = 1.0 m, and ddecorr = 8000 km.

[41] Figures 6 and 7 display the distribution of χirreg2 for planar fit estimation and kriging estimation, respectively, under quiet ionospheric conditions. (Note: these results include contributions from estimation where data deprivation, as described by Sparks et al. [2011], has been imposed.) Figures 8 and 9 show the same for moderately disturbed conditions, while Figures 10 and 11 plot the same for extremely disturbed conditions. For each plot, the distribution has been normalized to unity, so that it becomes a probability distribution. Table 1 presents the maximum value, mean value, and standard deviation of each χirreg2 distribution for the planar fit model. Table 2 shows the corresponding values for the kriging model. Notice that the values for kriging estimation tend to be smaller than the corresponding values for planar fit estimation, especially under moderately disturbed or extremely disturbed conditions. Such statistics serve as an example of how estimation accuracy can improve when kriging is implemented.

Figure 6.

The distribution of χirreg2 for planar fit estimation based upon measurements under quiet ionospheric conditions (9 November 2005).

Figure 7.

The distribution of χirreg2 for kriging estimation based upon measurements under quiet ionospheric conditions (9 November 2005).

Figure 8.

The distribution of χirreg2 for planar fit estimation based upon measurements under moderately disturbed ionospheric conditions (5 November 2001).

Figure 9.

The distribution of χirreg2 for kriging estimation based upon measurements under moderately disturbed ionospheric conditions (5 November 2001).

Figure 10.

The distribution of χirreg2 for planar fit estimation based upon measurements under extremely disturbed ionospheric conditions (30 October 2003).

Figure 11.

The distribution of χirreg2 for kriging estimation based upon measurements under extremely disturbed ionospheric conditions (30 October 2003).

Table 1. The Mean, the Maximum, and the Root Mean Square of the χirreg2 Distribution for the Planar Fit Model Under Various Ionospheric Conditions
Conditions for Planar Fit χirreg2MeanMaximumσ
Quiet (9 Nov 2005)0.08963.520.0935
Moderately disturbed (5 Nov 2001)0.6424.21.46
Extremely disturbed (30 Oct 2003)6.5431.723.0
Table 2. The Mean, the Maximum, and the Root Mean Square of the χirreg2 Distribution for the Kriging Model Under Various Ionospheric Conditions
Conditions for Kriging χirreg2MeanMaximumσ
Quiet (9 Nov 2005)0.07481.710.0756
Moderately disturbed (5 Nov 2001)0.4713.30.90
Extremely disturbed (30 Oct 2003)4.9344.618.0

4.3. Fit Residuals

[42] A second means of quantifying the improvement in delay estimation accuracy achieved by kriging is to compare fit residuals, i.e., residual differences between estimated and observed values of the vertical delay at the IPPs of observations used to perform delay estimation. In Figures 12–14, we show fit residuals, each derived from an estimate where the fit center is the IGP nearest to the test IPP. For each of the data sets studied in the previous section, the results for the kriging model are superimposed on the corresponding results for the planar fit model. The plot presented in Figures 12a, 13a, and 14a shows the fit residuals obtained when the vertical delay observation associated with the residual is included as one of the measurements from which the residual is derived. Figures 12b, 13b, and 14b show the same set of residuals when the residual observation is excluded from the estimation. The latter, of course, is a better means of gauging the improvement in accuracy provided by kriging. Plotting the former, however, shows how much more sensitive the kriging model is than the planar fit model to the presence of observations near the location at which an estimate is sought.

Figure 12.

Fit residuals under quiet ionospheric conditions (9 November 2005), using kriging (red) and planar fit estimation (blue): (a) keeping in the estimation and (b) removing from the estimation the measurement at the test IPP.

Figure 13.

Fit residuals under moderately disturbed ionospheric conditions (5 November 2001), using kriging (red) and planar fit estimation (blue): (a) keeping in the estimation and (b) removing from the estimation the measurement at the test IPP.

Figure 14.

Fit residuals under extremely disturbed ionospheric conditions (30 October 2003), using kriging (red) and planar fit estimation (blue): (a) keeping in the estimation and (b) removing from the estimation the measurement at the test IPP.

[43] The magnitude of the fit residuals clearly correlates with the level of ionospheric disturbance. Table 3 shows the mean, maximum, and the root mean square of the distribution of fit residual magnitudes for the planar fit model. Table 4 shows the corresponding values for the kriging model. We can assess the improvement in accuracy provided by kriging if we compare the planar fit and kriging values of the root mean square of the fit residuals (with the fit residual observation excluded from the estimation). For quiet conditions, the reduction in this root mean square fit residual is only 5%. This is not surprising since ionospheric delay is well represented by a planar model under quiet conditions. Under disturbed conditions, however, the reduction in the root mean square fit residual is approximately 15%.

Table 3. The Maximum (x) and the Root Mean Square (σ) of the Fit Residual Magnitudes for the Planar Fit Model Under Various Ionospheric Conditionsa
Conditions for Planar Fit Residualsxkeepxomitxratioσkeepσomitσratio
  • a

    The subscripts keep, omit, and ratio indicate the residual measurement kept in the fit, the residual measurement omitted from the fit, and the ratio of the two (omit/keep), respectively. Tabulated values for all columns except those reporting ratios are specified in units of meters.

Quiet (9 Nov 2005)0.8620.9441.010.1840.1981.08
Moderately disturbed (5 Nov 2001)5.776.411.110.4850.5351.10
Extremely disturbed (30 Oct 2003)23.125.91.121.731.861.08
Table 4. The Maximum (x) and the Root Mean Square (σ) of the Fit Residual Magnitudes for the Kriging Model Under Various Ionospheric Conditionsa
Conditions for Kriging Residualsxkeepxomitxratioσkeepσomitσratio
  • a

    The subscripts keep, omit, and ratio indicate the residual measurement kept in the fit, the residual measurement omitted from the fit, and the ratio of the two (omit/keep), respectively. Tabulated values for all columns except those reporting ratios are specified in units of meters.

Quiet (9 Nov 2005)0.6551.011.540.1260.1891.50
Moderately disturbed (5 Nov 2001)4.546.151.350.310.4611.49
Extremely disturbed (30 Oct 2003)14.823.81.611.211.621.34

[44] The sensitivity of each model to having fit IPPs near the test IPP can be quantified by calculating the ratio of the residual root mean square when the test IPP is excluded from the fit IPPs to the residual root mean square when the test IPP is included in the estimation. For the planar fit model, this ratio is 1.1 in all three cases (see Table 3). For the kriging threat model, it ranges from 1.3 to 1.6 (see Table 4). Kriging effectively weights the contributions of vertical delays at fit IPPs near the test IPP more heavily in the estimation than does the planar fit model, a reason for its superior accuracy. This is a consequence of the assumed exponential decay of the covariance of the detrended ionospheric delay between measurement locations and the assumed exponential decay of the covariance between the scalar field rx) near the IGP and the detrended delays at the measurement locations (see equations (39) and (40)).

5. Summary

[45] 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 presented the kriging methodology to be followed. In addition, we have provided examples showing the magnitude of the improvement in delay accuracy thereby achieved. Under disturbed conditions, the implementation of kriging reduces the magnitude of the root mean square fit residual by up to 15%. A subsequent paper will show how the increase in delay accuracy provided by kriging results in improved system availability.

Appendix A: Derivation of the Kriging Estimation Coefficients

[46] To obtain an expression for the wκ coefficients in equation (25), we solve the constrained optimization problem by the method of Lagrange multipliers, that is, we define the Lagrangian

display math

where μ is a vector of Lagrange multipliers, and then we set to zero the partial derivatives of L with respect to the elements of w and μ. After some algebra, we arrive at the following system of equations:

display math

[47] The solution to this system of equations may be written formally as

display math

[48] It can be shown that the inverse matrix on the right-hand side can be expressed as

display math

where the weighting matrix W is defined in equation (30). Hence, from equation (A3), we find that the wκ coefficients are specified by equation (31).

[49] This derivation differs from standard derivations of the kriging coefficients [Webster and Oliver, 2001; Cressie, 1993] in two significant respects: (1) we have explicitly included the effect of measurement noise as distinct from the scalar field, and (2) standard derivations usually conclude at equation (A3). An advantage of deriving equation (A4) and expressing w in the manner of equation (31) is that this separates each wκ into two distinct components: a stochastic component dependent upon c and a deterministic component dependent upon s.

Acknowledgments

[50] 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.

Ancillary