Tropospheric Delay: Prediction for the WAAS User Paul Collins and Richard B
INNOVATION
Tropospheric Delay: Prediction for the WAAS User Paul Collins and Richard B. Langley University of New Brunswick
The weather — it affects us all. Sometimes The ideal GPS measurement is the true range of the delay occurs within the troposphere, disastrously with vicious storms; sometimes or distance between the receiver’s antenna the whole delay is often referred to solely as pleasantly with sunshine and warm breezes. and the GPS satellite antenna. But as with the tropospheric delay. It also affects GPS. But, whereas bad most things in our imperfect world, we can- By assuming that the neutral atmosphere weather might disrupt our lives, causing not achieve the ideal. A number of error is both horizontally stratified and azimuthally us to curtail or postpone an activity, GPS sources bias GPS measurements, and pro- symmetric, we can model the tropospheric continues to perform — it’s an all-weather cessing software must account for these delay in two parts: the delay experienced in system. Rain, snow, fog, and clouds all errors to obtain accurate position, velocity, or the zenith direction and the scaling of that have a negligible effect on GPS. However, time information. One of these biases is the delay to the one experienced at the raypath’s unseen weather — temperature, pressure, tropospheric propagation delay, for which all zenith angle (referred to as either the map- and humidity variations throughout GPS receivers and postprocessing software ping function or obliquity factor). The com- the atmosphere — does affect GPS employ a delay algorithm of some kind that mon formulation of zenith delays and observations. These parameters determine attempts to model or predict, and thus mini- mapping functions seen in space geodetic the propagation speed of radio waves, an mize, its impact. Depending on the model’s and navigation literature derives from model- important factor that must be accounted sophistication, however, some error or resid- ing tropospheric delay in this way. It can be for when processing GPS or other ual delay will remain. described mathematically as radiometric observations. Because we Users often treat the residual tropospheric z z cannot predict their exact values ahead of delay in GPS position estimation in a very off- dtrop = dhyd á mhyd + dwet á mwet time, these invisible weather variables are hand manner — they assume that the effect is a source of error in GPS positioning and small or that a simple estimation technique in which total delay (dtrop) is a function of the navigation. In this month’s column, we will resolve the problem. But what is the true delays in the zenith direction caused by the examine the atmosphere’s effect on GPS impact? Just how much does the lower, electri- atmospheric gases in hydrostatic equilibrium z z and discuss how we’ve attempted to model cally neutral, atmosphere vary in terms of and those that are not (dhyd and dwet, respec- it for the users of the forthcoming Wide refractivity and its effect on GPS applications? tively) as well as their corresponding map-
Area Augmentation System. The aim of this article is to provide answers to ping functions (mhyd and mwet), which project I am joined this month by Paul Collins. these questions and offer a quantification of the zenith delay into the line-of-sight delay. Paul graduated from the University of the neutral atmospheric effect for users of Gases not in hydrostatic equilibrium are pri- East London in 1993 with a B.Sc. (Honors) wide-area differential GPS systems, such as marily water vapor, and the mapping func- degree in surveying and mapping sciences. the Federal Aviation Administration’s (FAA’s) tions are usually described as functions of the He is currently enrolled in the M.Sc.E. Wide Area Augmentation System (WAAS), satellite elevation angle — the complement degree program in the Department of who seek meter-level accuracy or better and of the zenith angle. Geodesy and Geomatics Engineering at operate within a position critical environment. Because this customary formulation of the University of New Brunswick, where tropospheric delay assumes horizontal stratifi- he is investigating the tropospheric THE TROPOSPHERIC DELAY cation and azimuthal symmetry, it precludes effects on kinematic GPS positioning. Constituent gases affect an electromagnetic the existence of gradients in the atmosphere. “Innovation” is a regular column signal propagating through the neutral atmo- Researchers have derived more sophisticated featuring discussions about recent advances sphere. Their combined refractive index, models to try to account for atmospheric gradi- in GPS technology and its applications as slightly greater than unity (nominally 1.0003 ents caused by pressure slopes and passing well as the fundamentals of GPS positioning. at sea level), causes the signal’s velocity to be weather fronts, but the effects are typically at The column is coordinated by Richard lower than it would be in a vacuum and the centimeter level or less and are accordingly Langley of the Department of Geodesy and increases the time it takes a signal to reach a insignificant for most navigation applications. Geomatics Engineering at the University of GPS receiver’s antenna, thus extending the New Brunswick, who appreciates receiving equivalent path length. Both these effects are DELAY MODELS your comments as well as topic suggestions often referred to as delay. Refraction also Researchers have developed many different for future columns. To contact him, see the bends, and thereby lengthens, the raypath, algorithms over the years in an effort to “Columnists” section on page 4 of this issue. further increasing the delay. Because the bulk empirically model the tropospheric delay
52 GPS WORLD July 1999 www.gpsworld.com with varying degrees of accuracy. When pro- from the GPS data itself. This procedure, cessing GPS observations, a receiver or post- however, is typically only carried out for processing software program predicts a value such high-precision GPS applications as for the tropospheric delay from such a model deformation monitoring. using assumed or real-time values for the ambient temperature as well as total baromet- DEVELOPING A NEW MODEL ric and partial water-vapor pressures. Unfor- Although the geodetic community has been tunately, even with accurate real-time acutely aware of the propagation delay prob- measurements, models can rarely predict the lem and has striven over the years to improve true total delay with a degree of accuracy the accuracy of tropospheric delay models, much better than a few percent. the navigation community has overlooked, Although, we can determine the delay’s for the most part, these developments and hydrostatic component in the zenith direction continued to use simpler, less-accurate at the millimeter level, the highly variable models. The need for higher positioning nature of atmospheric water vapor degrades accuracies, however, concurrent with the the accuracy of the wet delay prediction to development of augmented GPS has necessi- the centimeter or even decimeter level. This tated a fresh look at tropospheric delay occurs because the hydrostatic delay in the models for navigation applications. WAAS in zenith direction is a function of the total sur- particular requires a model, as aircraft face pressure only, which, under conditions employing WAAS-augmented receivers need NovAtel of hydrostatic equilibrium, represents the the resulting increase in accuracy. total weight of the column of air above the At the University of New Brunswick 1/3 Page Vert user. Analogously, the zenith wet delay is a (UNB), we have developed a series of com- function of the total precipitable water — the posite or hybrid models for GPS navigation, amount of vapor present in the column of air culminating in UNB3, which the FAA and Ad Goes above the user. Nav Canada have adopted for the WAAS user Tropospheric delay models express these receiver. We based our original definition of Here two parameters in various ways. The most the model on the zenith delay algorithms of common method of describing the wet delay is Jouko Saastamoinen (whose work at the Keyline does through a combination of surface parameters National Research Council of Canada in the (water-vapor pressure, or relative humidity, early 1970s has withstood the test of time), not print and temperature) and some kind of water- the recent, high-accuracy mapping functions vapor lapse rate (commonly known as the of Haystack Observatory’s Arthur Niell, and page 53 lambda parameter). Not all models explicitly a table of atmospheric parameter values parameterize the wet delay as such, but they derived from the U.S. 1966 Standard Atmos- often do so implicitly. phere Supplements. The problem with modeling the wet delay In the interest of computational simplicity, in this way is that, unlike the hydrostatic a subsequent proposal was made to replace delay, no simple physical law governs the the Niell mapping functions with the com- distribution of water vapor in the lower bined function of Harold Black and Arie Eis- atmosphere; hence, a precise definition and ner, who both worked at the Johns Hopkins evaluation of the lambda parameter is not University Applied Physics Laboratory. This possible. As a consequence, the only way to change has a negligible impact on most of the accurately measure the lambda parameter is results presented in this article, because we to employ some technique that attempts to deal mainly with the zenith delay residual sample the whole atmospheric column, such error. (We did calculate our position error as a radiosonde or a radiometer. As these are simulations described later in this article impractical for real-time GPS users (and using the Black and Eisner mapping func- indeed for most other GPS users), the lambda tion; however, we do not expect the results to parameter can only be represented empiri- be significantly different when using the cally and will consequently always be associ- Niell functions.) ated with some error in the determination of UNB3. We designed the UNB3 model to the wet zenith delay. improve on first-generation navigational-use We should point out that it is often pos- tropospheric delay algorithms, such as the sible to improve tropospheric delay model- Altshuler model (developed by Edward Alt- ing by estimating a zenith-delay correction shuler at the Air Force Cambridge Research Circle 25 www.gpsworld.com GPS WORLD July 1999 53 INNOVATION
90ûN 90ûN
75ûN 75ûN
60ûN 60ûN
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0û 0û 180û 165ûW 150ûW 135ûW 120ûW 105ûW 90ûW 75ûW 60ûW 45ûW 180û 165ûW 150ûW 135ûW 120ûW 105ûW 90ûW 75ûW 60ûW 45ûW
175 180 185 190 195 200 205 210 215 220 225 230 235 240 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
Figure 1. Mean hydrostatic (left) and wet (right) zenith delays in centimeters computed from radiosonde soundings made in 1992. Triangles indicate station locations.
Laboratories in the early 1970s), as well as METHODOLOGY temperature profile model, namely the Com- the simple constant-value models attributed Because UNB3’s look-up table is based on mittee on Space Research (COSPAR) Inter- to NATO and Stanford Telecommunications, average climatological behavior, the model national Reference Atmosphere (CIRA) Inc. The latter two essentially use standard, will not perform as well during anomalous model, which provides monthly mean tem- predetermined parameter values at mean sea weather conditions. We carried out a series of peratures at 5-kilometer intervals for alti- level and a constant atmospheric profile tests to gauge the model’s accuracy. For com- tudes as high as 120 kilometers for every 10 across the whole latitude range. They only parison purposes, we also tested the Altshuler degrees of latitude. By computing a weighted account for the vertical variation to represent and UNB1 models. average of the four CIRA profiles surround- the change in a user’s height (in an aircraft, To provide the benchmark data for our ing a radiosonde launch site and epoch (two for example). investigation of propagation delay errors, we in latitude and two in time) and then offset- In this article, our prototype model, UNB1, ray-traced a portion of a four CD-ROM set of ting it to match the radiosonde temperature represents delay algorithms of this type. It radiosonde data for North American sites that value at the truncation height, we obtained uses U.S. Standard Atmosphere parameter covers the years 1946Ð1996. The National the required time and location profiles. values of 1013.25 millibars (total pressure), Oceanic and Atmospheric Administration We subtracted the zenith delay values 288.15 kelvins (temperature), 11.7 millibars (NOAA) compiled the data set, which con- computed by the various models from the (water-vapor pressure), 6.5 kelvins per kilo- sists of radiosonde soundings at mandatory zenith ray-trace values to obtain the residual meter (temperature lapse rate), and 3 (lambda and significant pressure levels as high as 100 tropospheric delay and hence the zenith parameter). (We developed two other models millibars (equivalent to a height of about 16 model errors. The results are presented in — UNB2 in which we first tested latitudinal kilometers) from almost all the sites operat- three parts: the average statistical perfor- variation of the surface parameters and UNB4 ing in the United States, Canada, Mexico, the mance of the tropospheric delay models; the which has slightly improved performance at Caribbean, and Central America. For our extreme delay errors; and the impact of those high altitudes at the expense of reduced per- analyses, we used the last 10 full years of extremes on position computations simulated formance near the ground. For details on available data (1987Ð1996). This represents with real GPS ephemerides. these other models, consult the report listed in an average of 173 stations per year and the Further Reading sidebar.) approximately one million total soundings. AVERAGE MODEL PERFORMANCE The kernel of the UNB3 model expands Figure 1 shows a sample of mean hydrostatic To test the distributions of the residuals, we the representation of these five UNB1 atmo- and wet zenith delays across North America can use Gaussian plots to compare them to a spheric parameters into a look-up table that from this 10-year data set. standardized or normal distribution (see Fig- provides values that vary with respect to lati- Almost all atmospheric water vapor is ures 2 and 3). Figure 2 indicates the degree to tude and day of year. Linear interpolation is found well below the 100-millibar level and which some tropospheric delay models can applied between latitudes, and a day-of-year thus the CD-ROM data sets were sufficient be biased and skewed toward large residuals. sinusoidal function attempts to model the for our purposes in this respect. To ray-trace The mean of the Altshuler model residuals is seasonal variation. Users employ the lapse hydrostatic delay, however, the temperature 15.9 centimeters, with a standard deviation rates to scale the sea-level pressures and tem- profile must be extended above this height. (σ) of 8.1 centimeters. Carefully choosing perature to their altitude. We accomplished this by using a suitable parameter values, as was done for UNB1, can
54 GPS WORLD July 1999 www.gpsworld.com These plots indicate that characterizing 70 tropospheric delay errors using a normal dis- 60 Altshuler σ UNB1 tribution beyond the 4 level cannot be rec- 50 ommended, especially with simpler models 40 because the true distribution will be underes- 30 timated. The probability level equivalent 20 to 4σ in a normal distribution is 99.994 per- 10 cent. Safety-critical systems, however, may 0 demand even higher levels. -10 The mean and standard deviation statistics Residual error (centimeters) -20 for UNB3 based on the entire 10-year data -30 set are Ð1.9 and ±4.9 centimeters respec- -5 -4 -3 -2 -1 0 1 2 3 4 5 Reduced variate (normal distribution) tively. These statistics are very consistent from year to year, indicating that it is suffi- Figure 2. Gaussian plot of zenith delay cient to use one year’s worth of data (with residuals for the Altshuler model and wide geographical coverage) to quantify the UNB1. Thin lines represent best-fit average, or typical, tropospheric delay model normal distributions. performance. EXTREME DELAY ERRORS NovAtel 50 We will no longer consider the Altshuler and 40 UNB3 N(0,5) UNB1 models in this article because their 1/3 Page Vert 30 performance with regard to large, or extreme, 20 residuals is poor. Comparing Figures 2 and 3 10 reveals that the UNB3 model provides a Ad Goes 0 56-percent improvement over the Altshuler -10 model at the upper 4σ level. Here -20 It is convenient to use a nonextreme cut- -30 off range of ±20 centimeters for the UNB3 Keyline does Residual error (centimeters) -40 model, based on the results shown in Figure -50 3. A zenith delay error of this magnitude, not print -5 -4 -3 -2 -1 0 1 2 3 4 5 Reduced variate (normal distribution) however, could lead to a potential 2-meter bias in height (see later discussion). In total, page 55 Figure 3. Gaussian plot of zenith delay 76 residuals from 1,011,651 profiles in the residuals for UNB3. The thin line shows 10-year data set exceed the ±20-centimeter the zero-mean, 5-centimeter standard range for the UNB3 model. This is equivalent deviation normal distribution. to approximately 0.00751 percent. Corre- spondingly, 99.99249 percent of the residuals provide a near-zero-mean model, but the dis- are within the nonextreme range. tribution is highly skewed and has a large stan- We checked all the initially detected dard deviation. UNB1 has a 1.9-centimeter extreme profiles, rejecting several as unlikely, mean error, with 8.5-centimeter standard supposing radiosonde or other errors. Unfortu- deviation. nately, without detailed knowledge of each Figure 3 shows the distributions of UNB3 station’s climatic conditions, we cannot be model residuals and a theoretical, zero-mean completely sure about the remaining extremes. distribution with 5-centimeter standard devi- We believe, however, that our results are a ation. This normal distribution characterizes good representation of the true errors. the residuals of the UNB3 model quite well Extreme Locations. Figure 4 displays the loca- up to approximately ±4σ where the value of tions of stations having extreme residuals. the residuals is almost exactly ±20 centime- Those with at least one positive extreme (a ters. Beyond the 4σ level, the lower bounds residual of more than +20 centimeters zenith for UNB3 become progressively conserva- delay error) are marked with a triangle and tive because the magnitude of the negative labeled with a station identification number residuals appears to level off. A normal distri- to the right. Stations marked with an inverted bution also quite drastically underestimates triangle and labeled to the left have at least the residuals beyond the upper bound. one negative extreme (a residual magnitude Circle 26 www.gpsworld.com GPS WORLD July 1999 55 INNOVATION
Return period (years) 135ûW 120ûW 105ûW 90ûW 75ûW 60ûW 45ûW 50 10 5 3 2 1.5 1.1 60ûN 60ûN -16 -17 -18 -19 14918 94983 (2) 04734 (2) 14685 -20 45ûN 45ûN 14898 14733 (2) -21 13840 14684 (2) 23154 93755 -22 23230 03879 (2) 93734 93214 (2) 03948 -23 93116 (3) 03131 03940 13601 (5) Zenith delay error (centimeters) -24 03125 (13) 30ûN 03937 30ûN 0.02 0.10 0.20 0.33 0.50 0.67 0.91 22104 (5) 22012 Frequency (year-1) 92803 22103 (16) 22103 12919 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 21001 (3) Reduced variate (Weibull distribution) 21101 21001 11641 15ûN 15ûN Figure 5. Extreme value cumulative 135ûW 120ûW 105ûW 90ûW 75ûW 60ûW 45ûW probability plot for negative extremes Figure 4. Location of radiosonde stations providing data between 1987 and 1996. In addition to their identification numbers, stations with positive extremes from model (with a 63-percent probability). Both the UNB3 are labeled ᮡ and stations with negative extremes are labeled ᮢ. Parentheti- Frechet and Weibull distributions have cal digits indicate the number of extremes for sites having more than one. threshold values beyond which the expected return period is infinite. The Frechet thresh- greater than Ð20 centimeters zenith delay we can examine the extreme residuals sepa- old for positive extremes is approximately 11 error). Parenthetical numbers indicate more rately using Gumbel, Frechet, and Weibull centimeters, indicating that an error at least than one extreme during the 10-year period. probability distributions. These functions this large will occur every year. The negative extremes are confined to the represent the limiting forms that most com- If we assume no error in the hydrostatic Baja California, Sonora, and Sinaloa regions mon distributions take when only consider- delay, the maximum wet zenith delay value of Mexico as well as the southern tips of Cal- ing the largest or smallest values in a sample of the UNB3 model (approximately 27 cen- ifornia and Arizona. The number and location set. It is not necessary to know the underlying timeters) limits the magnitude of the negative of the positive extremes are geographically distribution to consider the distribution of the extremes. This error would occur with a dry, more scattered than the negative extremes, yet extreme values. or nearly dry, atmosphere in the tropics. concentrations do occur. Bermuda (station An extreme value probability plot has the Thus, we could specify this value as the number 13601), for example, seems particu- potential to provide an abundance of useful threshold in the Weibull distribution; how- larly prone to extreme conditions, a possible information. As an example, Figure 5 shows ever, we tried to use the data itself to identify consequence of its mid-Atlantic location. the extreme value probability plot for the the value. This approach appears to work Forecasting Extremes. We examined the largest yearly negative extremes. Of the three because the Weibull distribution best fit spec- extremes as a function of day of year to distributions, the positive extremes fit a ifies a cut-off value of Ð23.5 centimeters. understand, in particular, the pattern of nega- Frechet distribution best, and the negative The slight difference between this value tive extremes. We found that most negative extremes agree most closely with a Weibull and the theoretical one suggests either insuf- extremes occur during late spring. Examin- distribution. ficient data or indicates that the maximum ing the residual time series for stations in the The power in these plots lies in the ability zenith wet delay error will never be reached west of Mexico indicates that the climate is to extrapolate and compute the return periods because some water vapor (and, therefore, a relatively constant through this time of year. of future extremes. The return period merely few centimeters’ worth of wet zenith delay) is Unfortunately, UNB3 has a mean bias that, denotes the inverse of the occurrence proba- always present in the atmosphere. The fore- when combined with the increasing sinu- bility. By using the largest value in each year, cast return period for an extreme zenith delay soidal seasonal variation, can result in an we can ascertain the return periods in years. error less (that is, greater in absolute value) error for these stations that exceeds the It should be pointed out that the use of than Ð23 centimeters is 50 years. ±20-centimeter error limit. extreme value statistics usually requires a set Look-up Table. The UNB3 model uses a look- The positive extremes are more likely in of at least 20 samples. With only 10 years of up table to obtain values of the meteorologi- the summer — although the second and third data in our sample, the confidence in these cal parameters because most WAAS aircraft largest extremes occurred in the winter. results is not high. We can use our data, how- receivers will not have access to real-time Unlike the majority of negative extremes, the ever, as a good example of the application of meteorological data. We did investigate positive extremes are outliers in the overall extreme value statistics. replacing the look-up table with real data time series, suggesting the influence of short- If the positive extremes do follow the extracted from the radiosonde data set. We period, transient weather systems. Frechet distribution, then an average return discovered that, although the mean error of Given that Figure 3 indicates different tail period of 25 years is forecast for an extreme the model (at sea level) is reduced to 0.2 ± 3.4 distributions from that of the Gaussian curve, zenith delay value of at least 55 centimeters centimeters, there is no apparent benefit, with
56 GPS WORLD July 1999 www.gpsworld.com regard to extreme errors, in replacing a good and unweighted position solutions converged meteorological look-up table with real-time toward the same value (∼1.5 meters). Figure values. The small improvement in overall 6 illustrates this phenomenon and shows the bias is negligible in comparison to other correlation between the unweighted vertical potential error sources, such as multipath and position bias and the maximum tropospheric the ionosphere. delay error. Figure 6 also clearly demon- strates that any function of simple VDOP will POSITION DETERMINATION IMPACT not correctly model this kind of error. The Elevation-angle dependence complicates the residual zenith error for this station was impact of an unmodeled tropospheric range approximately 21 centimeters, with the delay on GPS positions. The elevation angle unweighted vertical bias approaching 2 will not be the same for all the satellites in meters during one time period. view, and thus users cannot rely on simple This analysis indicates that the amount of vertical dilution of precision (VDOP) calcu- vertical position bias resulting from unmod- lations to model these errors. The only reli- eled tropospheric range delay is approxi- able way to study the impact of troposphere mately equal to the maximum residual model errors is to undertake position simula- tropospheric delay present in the solution. tions, replacing the GPS range with the Because of the tropospheric delay’s eleva- unmodeled delay. In this way, we are able to tion-angle dependence, this position bias will predict how the error manifests itself in posi- essentially be the delay error from the lowest- NovAtel tion coordinates. elevation-angle satellite. Given the expected We computed position solution biases for zenith delay error, an approximate mapping 1/3 Page Vert all the stations with extreme residuals, using function value will give the correct result. At broadcast ephemerides from 1997 to repre- a 5-degree elevation angle, the mapping func- sent satellite constellations for six hours tion value is roughly 10. With this figure, the Ad Goes around the time of each radiosonde launch. maximum possible vertical height error value We assumed that the extreme residuals could can be easily calculated. Here occur at any time of year because performing The size of the maximum bias in the com- position simulations for each day of the year puted position will be limited in one direction Keyline does would be extremely time consuming. We also because of the lower limit of the UNB3 presumed that the tropospheric error remains model. A negative error indicates that the not print constant for several hours. In any case, we tropospheric delay model prediction was too were able to derive a general relationship large. By effectively shortening the range, the page 57 between receiver-satellite geometry, tropo- computed position will be higher than the true spheric delay error, and the resulting position position. An aircraft flying below its intended bias. height can therefore only ever be approxi- We performed two kinds of position solu- mately 3 meters too low because of UNB3 tion simulations: a regular, unweighted least- tropospheric delay mismodeling. For an air- squares solution and a weighted least-squares craft using wide-area differential GPS, flying solution using the squared inverse of the above its intended height, and having an unfa- mapping function to down-weight any low- vorable satellite constellation in unusual elevation-angle errors. The position biases weather conditions, 4-meter vertical position were computed every two minutes. For biases are possible solely from mismodeled almost all of the position solutions, the tropospheric delays. The extreme value the- weighted vertical biases were one-third to ory predicts errors as high as 5 meters. two-thirds less than the unweighted biases. In general, the weighted solution reduced the CONCLUSIONS extreme vertical bias to the meter level or Our results indicate that the maximum size of less. The horizontal biases for both solutions the residual tropospheric delay is not too were always much smaller — at or below the large, as long as a good model is used. For the decimeter level. UNB3 model, only seven or eight in 100,000 Maximum Bias. We discovered that for one predictions resulted in residual zenith delay particular time period at one station, the errors outside the range of ±20 centimeters. satellite constellation was dominated by low- A zero-mean normal distribution, with 5- elevation-angle satellites. During this interval centimeter standard deviation up to the ±4σ (approximately 10 minutes), the weighted point can adequately represent the distribu- Circle 27 www.gpsworld.com GPS WORLD July 1999 57 INNOVATION
tion of the majority of the errors. Beyond that meters). There is more potential for greater resulting from receiver-satellite geometry, point, the normal distribution is too conserv- positive extremes, but our small sample size assuming a constant average range error for ative for the positive extreme residuals. (10 values) limits us from providing a reli- all satellites in view. A better indication is Of the extreme values beyond ±20 cen- able forecast. More data would be required to provided by the maximum residual error — timeters, there are slightly more negative improve the confidence of future predictions. that is, the unmodeled range on the lowest extremes than positive. The negative Mismodeled tropospheric range delays elevation satellite. This error maps almost extremes appear more consistent with pre- primarily impact a computed position’s directly into the vertical position component. vailing weather conditions and are limited in height component. VDOP values are not a WAAS will augment GPS to provide the magnitude by the maximum wet zenith delay good indicator of vertical bias in this case required accuracy improvement for precision value of UNB3 (approximately 27 centi- because they only model the position error aircraft approaches to airports, as well as navigation integrity, continuity, and availabil- ity for all phases of flight. Although tropos- 90 pheric delay is only one component of the WAAS error model, it is an important one 70 that demands an accurate, reliable user model to ensure that WAAS achieves its operational 50 objectives. As we’ve described in this article, the UNB3 composite model fulfills these 30 requirements. Other satellite-based wide- Elevation angle (degrees)
10 area augmentation systems such as the Euro- 9 10 11 12 13 14 15 pean Geostationary Navigation Overlay 2.5 System or Japan’s Multifunctional Trans- portation Satellite system could also make 2.0 use of UNB3, although further testing with 1.5 radiosonde data covering the intended ser- vice regions would be needed. 1.0 ACKNOWLEDGMENTS 0.5 Our work was undertaken on behalf of Nav 0.0 Canada in conjunction with the U.S. Federal VDOP, vertical position bias (meters), maximum tropospheric error (meters) 9 10 11 12 13 14 15 Coordinated Universal Time (hours) Aviation Administration, who are coopera- tively developing WAAS for North America. Figure 6. Satellite constellation at station 12919, Brownsville, Texas, on July 18, This article is based on the paper “The Resid- 1997, and simulated vertical position biases from a 21-centimeter zenith delay ual Tropospheric Propagation Delay: How error. The solid and dashed lines show the unweighted and weighted solution Bad Can It Get?” presented by the authors biases respectively. The dashed and dotted line denotes maximum tropospheric at ION GPS ’98, in Nashville, Tennessee, error (the error at lowest elevation satellite), and the dotted line displays the vertical September 1998. ■ dilution of precision (VDOP).
Ⅲ Minimum Operational Performance Standards for Global FURTHER READING Positioning System/Wide Area Augmentation System Airborne For a thorough discussion of tropospheric propagation delay and Equipment, RTCA/DO-229A, prepared by Special Committee 159, the models used to account for it in GPS and other space-based RTCA, Inc., Washington, D.C., June 1998. radiometric data, see Ⅲ A Tropospheric Delay Model for the User of the Wide Area Ⅲ Modeling the Neutral-atmosphere Propagation Delay in Augmentation System, by J.P. Collins and R.B. Langley, Final Radiometric Space Techniques, by V. de Brito Mendes, a Ph.D. contract report for Nav Canada Satellite Navigation Program dissertation published as Department of Geodesy and Geo- Office and published as Department of Geodesy and Geomatics matics Engineering Technical Report No. 199, University of New Engineering Technical Report No. 187, University of New Brunswick, Brunswick, Fredericton, New Brunswick, Canada, April 1999. Fredericton, New Brunswick, Canada, September 1997. Electronic supplement:
58 GPS WORLD July 1999 www.gpsworld.com