A local area differential GPS (DGPS) method applies corrections from a reference GPS receiver to improve positioning accuracy for a roaming GPS receiver. Increasing separation between reference and roaming receivers dilutes this improvement, largely because ionospheric and tropospheric effects differ between their two locations. We correct differential corrections for this difference and determine the improvement with this “atmospheric” DGPS method at roaming receiver positions that are separated from a Coast Guard reference receiver at Annapolis, Maryland, by 44, 67, and 228 km. For ionospheric corrections we use our Raytrace–Ionospheric conductivity and electron density–Bent–Gallagher ionospheric propagation model with driving parameters obtained from two-frequency data of surveyed reference GPS receivers. For tropospheric corrections we use the Hopfield model and weather station data for surface temperature, pressure, and relative humidity. Internet delivery of atmospheric differential corrections is used to avoid blockage or range cutoff of radio transmissions. Some comparisons are made with Wide Area Augmentation System GPS receiver performance.
 A standard Global Positioning System (GPS) receiver processes L band radio signal pseudoranges from four or more GPS satellites to determine the receiver's position with a 1σ precision of about 5–10 m in real time. The receiver may also be equipped for radio reception of differential corrections to satellite pseudoranges from GPS reference receivers at known positions. Differential corrections supplied by a local area differential GPS (LDGPS) service, such as the U.S. Coast Guard (USCG) Maritime DGPS service, are supplied by the nearest reference receiver. A satellite differential correction is calculated by subtracting the associated pseudorange at the reference receiver from the known free-space range. This correction is then transmitted to the roaming receiver, where it is added to its corresponding measured pseudorange. The differential corrections tend to cancel out error biases (departures from actual free-space ranges and perfect timekeeping) that the reference and roaming receivers have in common. Pseudorange error biases are associated with the satellite clock residual error, satellite position error, ionosphere mismodeling, troposphere, and multipath and have approximate 1σ values of 2.0, 2.1, 4.0, 0.5, and 1.0 m, respectively, according to a standard error model [Parkinson, 1996]. The preceding list of error biases is usually in order of increasing spatial and temporal decorrelation tendency for increasing space-time separation between the roaming and reference receivers. Increasing this baseline separation decreases the effectiveness of differential corrections. We seek to restore this effectiveness by accounting for the difference in atmospheric delay contributions at the reference and roaming receiver positions. Although ionosphere error bias dominates the tropospheric error bias in magnitude, the tropospheric differences can be more significant than ionospheric differences because of their smaller decorrelation length scales and timescales [Misra and Enge, 2001].
 One can also purchase GPS receivers with access to a wide-area DGPS service (WADGPS), such as the U.S. Wide Area Augmentation System (WAAS). The WAAS system consists of 25 GPS reference receivers that are spread over the United States and Puerto Rico. These receivers correct their pseudoranges for tropospheric effects and transmit the constituent error biases for satellite clocks, satellite ephemeris, and ionospheric contributions to a central processing station. The station processes this data for use by WAAS-capable GPS receivers and uploads the information to geostationary satellites. WAAS GPS receivers download this information at the GPS L1 frequency and compute corrections to their pseudorange data, adding tropospheric corrections from a climatological model [Collins et al., 1996]. DGPS is further discussed by Misra and Enge .
 There are some potentially important issues with existing DGPS systems. One is coverage. In particular, ionospheric spatial and decorrelation lengths and times sometimes decrease beneath the resolution limits of the reference receiver distribution in the case of a WADGPS system. In the case of LADGPS, ionospheric conditions can differ substantially between reference and roaming receivers for large separations. Such instances that compromise the performance of a WADGPS system are shown to occur, especially at low and high latitudes, particularly during periods of high solar and magnetic activity [Skone and Shrestha, 2002; Skone et al., 2003, 2004; Yousuf and Skone, 2005]. Unsettled tropospheric weather can also be a problem. In the case of LADGPS, knowledge of the ionospheric and tropospheric conditions at or near the reference and roaming receiver sites can be used to help correct these DGPS problems, especially if these data are used to update effective correction models. We test this approach on Virginia and Maryland data sets. If we find it to be effective at midlatitudes, we expect it would also be effective at low and high latitudes, except for the effect of irregularities and the sharp gradients that often are not included in models. In this case, there is probably no substitute for increasing the number of reference stations and observations.
 Another issue, particularly for WADGPS systems, is the approximation used to represent the ionospheric delay component of pseudorange, which is proportional to slant total electron content (TEC). Slant TEC, determined from two-frequency reference receiver data, is represented in a thin-shell approximation as the product of a geometrical obliquity factor and a vertical TEC value where the straight ray intersects the height of the electron density thin shell, called the ionospheric pierce point. This type of approximation and the associated interpolation of vertical TEC data at the pierce points have been found to be a problem at low latitudes [Paul et al., 2005]. We attempt to avoid these approximations by processing the two-frequency data of GPS reference receivers to determine driving parameters [Reilly and Singh, 2004] of our Raytrace–Ionospheric conductivity and electron density–Bent–Gallagher (RIBG) ionospheric model [Reilly, 1993], which is then used with fast, straight ray-tracing computation through the RIBG height versus electron density profiles to determine ionospheric contributions. This type of approach, with interpolation of model-driving parameters at reference stations, should often improve the resolution capability and accuracy of a WADGPS GPS reference receiver network. For the troposphere we use weather station data for surface temperature, pressure, and relative humidity inputs to the Hopfield model; see equation (4.41) and the separate mapping functions for dry and wet delays given in the book by Misra and Enge . The Hopfield model is relatively accurate for the dry component, which accounts for about 90% of the tropospheric effect. The wet component, which is adequate by the Hopfield model, contributes the remaining amount.
 A third issue with DGPS systems is communication of the differential correction information to the roaming receiver. The roaming receiver's radio reception of differential corrections may sometimes be range limited or blocked by buildings or other obstructions. We avoid this difficulty by using the Internet to transmit differential corrections to the GPS roaming receiver.
 We investigate the effect of different ionospheric and tropospheric conditions at reference and roaming receivers and accordingly modify the DGPS corrections from the USCG LADGPS system, which then become what we call atmospheric DGPS (ADGPS) corrections. We process roaming GPS receiver data in section 2 for Alexandria and Fairfax in northern Virginia, as well as Driver in southeastern Virginia. The latter is also a reference station. These sites are separated by about 44, 67, and 228 km from the USCG GPS reference station in Annapolis, Maryland, respectively. Two-frequency data from the reference stations are processed [Reilly and Singh, 2004] to yield effective sunspot number drivers for RIBG, which are then used to compute the difference between ionospheric contributions to pseudoranges at roaming and reference receiver locations. Weather station data are used to compute differences in tropospheric contributions. We evaluate performance enhancement from using the ADGPS corrections. We also compare our results with WAAS GPS receiver results in Alexandria and Fairfax. Section 3 discusses results and proposes some ADGPS modifications for WADGPS and LADGPS systems.
2. Data Analysis and Results
 L1 (1575.42 MHz) code pseudorange, which is normally used to compute GPS receiver positions, is the group path length between satellite and receiver plus the errors due to satellite and receiver clocks. The observed pseudorange between satellite s and receiver r is given by
where the terms on the right represent, in order, the free-space range; receiver clock error; satellite clock error; and ionosphere, troposphere, and multipath bias errors. The last term represents random measurement errors. Frequency dependence could be represented by another subscript that would appear on all terms except the free-space range. The GPS satellites transmit ephemeris data, which are used to calculate their positions. A typical GPS position solution first corrects pseudorange data for the estimated satellite clock errors. It also corrects for the ionosphere from the Klobuchar model [e.g., see Misra and Enge, 2001], except when the DGPS method is used, and possibly also for tropospheric delays from a climatological tropospheric model. The solution algorithm then neglects all other terms on the right, except for the free-space range and the receiver clock error. Hence corrected pseudorange data from at least four satellites are required for a single-epoch solution for the three receiver coordinates and clock error. As an example, we process L1 data from the two-frequency GPS reference receiver at Annapolis, Maryland, operated by the U.S. Coast Guard. This receiver is part of the Continuously Operating Reference Stations (CORS) network, which includes about 200 receivers worldwide that transmit data to world centers for public use. Data from several stations are uploaded every hour for near-real-time applications. Figure 1 shows the position solution errors we obtained from the L1 data from seven to nine satellites in view, relative to the known receiver position. This scatterplot is in the XY or equatorial plane, where we refer to standard Earth-centered Earth-fixed geocentric equatorial coordinates. The X and Y axes are in the equatorial plane, with the X axis in the Greenwich meridian plane, and the Z axis is the Earth's rotation axis. The data were collected at 30 s intervals for 2 hours on 28 February 2005 from local noon to 2:00 p.m. (1700–1900 UT). No atmospheric or DGPS corrections were applied to the pseudorange data for this plot. We refer to this as the free-space approximation. The pronounced error bias is principally due to ionospheric effects, which also contribute to the scatter.
 DGPS corrections are transmitted by U.S. Coast Guard reference stations near waterways, including the Annapolis station, in a specified format by low-frequency transmitters. GPS receivers that receive these corrections are commercially available. They lock onto the DGPS station with the strongest signal. We collect DGPS corrections with an MBX-3 receiver purchased from Communication Systems International, Inc. The message collected supplies both pseudorange corrections and their change rates, which we upload to our Web page, www.adgps.com, every half minute.
 The ionospheric delay (m) of a GPS signal is given by the familiar formula
where f is the frequency (Hz) of radio waves and TEC (electrons m−2) is the total electron content along the slant path. We fit two-frequency GPS data from reference receivers to the RIBG model in order to determine its effective sunspot number driving parameter [Reilly and Singh, 2004]. We use the effective sunspot number in RIBG and Internet values of the planetary magnetic activity index Kp (see, e.g., www.sec.noaa.gov/ftpmenu/indices/old_indices.html) to compute ionospheric delays at the reference and roaming receivers. To correct DGPS corrections, we subtract ionospheric contributions to pseudoranges at the reference receiver and add corresponding contributions at the roaming receiver.
 The tropospheric contributions are accounted for by the Hopfield model, which is driven by data for surface temperature, pressure, and relative humidity. These atmospheric parameters are available from the Internet for all the airports and normally are updated every hour. On a smaller scale of distances, most local TV broadcast stations have set up weather stations at public schools that can be accessed by the Internet. Corresponding tropospheric corrections to pseudoranges are obtained for reference and roaming receiver positions.
 We first processed GPS measurements at Fairfax, Virginia, on 21 September 2004, using differential corrections from Annapolis, Maryland. The observations were made with a Marconi Allstar single-frequency GPS receiver and development kit. The Fairfax control point, located in an open area behind a Fairfax County government building, has been surveyed by National Geodetic Survey. For the ionospheric specification, we used the two-frequency data from Annapolis. The single-frequency Fairfax GPS data were taken at 1 s intervals for about 20 min, starting at 1630 UT. The effective sunspot number from Annapolis data for the period from 1400 to 1600 UT was found to be 58.3. The surface temperature 63.2°F (17.3°C), pressure 1100 mb, and relative humidity 62% were taken from Reagan National Airport in Alexandria, Virginia. During the experiment, DGPS data were received in real time via the Internet from our Web page with a wireless modem. The Web page is automatically updated every half minute. The free-space approximation for six to eight satellites in view produces results for Fairfax similar to Figure 1 with large bias errors. When we include the Annapolis DGPS corrections, we compute the Fairfax equatorial plane position errors shown in Figure 2 as light triangles. With ADGPS corrections, the results are shown in Figure 2 as small, dark points. Statistical parameters for the scatterplots are shown in Table 1, including mean values and associated standard deviations for variables X and Y in the equatorial plane. Inclusion of atmospheric corrections to the DGPS corrections effectively removes an average bias error of about 1.3 m. The distance between the Annapolis reference receiver and the Fairfax receiver is about 67 km. Although we assumed the same driving parameters for the ionospheric and tropospheric models applied to Annapolis and Fairfax, the models produced useful atmospheric corrections to the Annapolis DGPS corrections. This would be less likely to occur for larger receiver separations, or smaller ionospheric and tropospheric correlation lengths, as one would find for increased weather activity, both ionospheric and tropospheric. In this case, it becomes important to measure and process atmospheric data at both the roaming and reference receiver sites in obtaining the ADGPS corrections.
Table 1. Statistical Parameters of the XY Scatterplots
Fairfax, 21 September 2004
X standard deviation
Y standard deviation
Driver, 28 February 2005
X standard deviation
Y standard deviation
 A reference station in Driver in southeastern Virginia serves as our next roaming receiver, separated from Annapolis by about 228 km. A position solution in the free-space approximation for seven to nine satellites in view produces results like Figure 1, but this time with a substantial bias in the +Y direction. Figure 3 shows the Driver position errors as light triangles with Annapolis DGPS corrections. These data were taken at 30 s intervals on 28 February 2005 from 1900 to 2100 UT. Figure 3 also shows the corresponding results with ADGPS corrections as small, dark points. Statistical parameters for the Driver scatterplots are shown in Table 1. The effect of atmospheric corrections to DGPS is to reduce an average DGPS bias error of about 1.3 m to an average ADGPS bias error of about 0.3 m. In this case, there is also a reduction in the breadth of scatter. We collected data and processed atmospheric effects at both the Driver and Annapolis reference receivers in computing atmospheric corrections to DGPS. Sunspot numbers at Driver and Annapolis were 61.1 and 59.7, respectively. Surface temperature, pressure, and relative humidity at Driver were 38.3°F (3.5°C), 1080 mb, and 45%, and Annapolis values were 41.2°F (5.1°C), 1080 mb, and 40%.
 The next data set of interest is in Alexandria, Virginia, 44 km west of Annapolis. Data were taken in a car parked next to the curb, pointing north, in a residential/commercial area near the Potomac River. The receiver antenna was placed on the dashboard behind the steering wheel. The WAAS GPS receiver we used was Novatel's Superstar II Development Kit, which not only provides raw code pseudorange and carrier phase data at L1 for our processing with Annapolis differential corrections but also provides the WAAS position solutions. The time period was 13 January 2006 between 1843 and 1901 UT at 1 s intervals. Annapolis two-frequency data provided the effective sunspot number 56.3 for RIBG. The surface pressure, temperature, and relative humidity were 43.5°F (6.4°C), 1080 mb, and 100% in Alexandria and 48.3°F (16.3°C), 1075 mb, and 58% in Annapolis. The ground truth position was not available at this location, so we selected a reference position for the scatterplots that approximately coincided with the mean position of the ADGPS scatterplots, of which there are three: east-north (EN), east-up, and north-up, with respect to the local horizontal plane. We show the EN scatterplot in Figure 4, where we see the DGPS solution as the light triangles, the ADGPS solution as the darker points, and the WAAS solution as the small black splotch in the southeast quadrant. The statistical parameters of the scatterplots are given in Table 2. The first thing we notice is that the DGPS and ADGPS distributions are much more spread out than before. We have a different receiver and an apparently pronounced multipath effect at this location. On the other hand, the WAAS distribution is much smaller and undoubtedly uses carrier phase smoothing of code pseudoranges to drastically reduce multipath and measurement noise [Misra and Enge, 2001]. We have not put this smoothing into our algorithms yet. Its only drawback is that it can cause an error that arises from the fact that the ionosphere term enters with different signs in the code pseudorange and carrier phase measurements. However, in the epoch differences of phase data used for smoothing, this effect builds up only very slowly. Table 2 indicates that the mean of the WAAS scatter lies east of the ADGPS mean by about 2.5 m. We know from plotting our results on calibrated aerial imagery maps that the mean point of the WAAS distribution lies on the sidewalk to the east of the curb. Hence the ADGPS mean point is correct with regard to its westward displacement from the WAAS mean point, which is the only ground truth information we have for this data set. The main effect of ADGPS relative to DGPS is a 1 m downward displacement in the height or up direction. It is found for this case that ionospheric contributions are much less than tropospheric contributions to the atmospheric corrections to DGPS. Tropospheric delay differences between Alexandria and Annapolis range up to about 1.4 m, whereas ionospheric delays range up to about 0.1 m. Tropospheric effects were large because of differing weather conditions in Alexandria and Annapolis. There were other parked car positions in this area, where the WAAS receiver was unable to receive WAAS corrections, because of building obstructions in the southward direction.
Table 2. Statistical Parameters of the East-North-Up (ENU) Scatterplots
Alexandria, 13 January 2006
E standard deviation
N standard deviation
U standard deviation
Fairfax, 20 January 2006
E standard deviation
N standard deviation
U standard deviation
 In the final data set we return to the Fairfax ground truth reference point with our Superstar II WAAS receiver. Data were collected at this point on 20 January 2006 between 1833 and 1901 UT at 1 s intervals. Annapolis two-frequency data provided the effective sunspot number 49.1 for RIBG. The surface pressure, temperature, and relative humidity were 60°F (15.6°C), 1070 mb, and 40% in Fairfax and 45°F (7.2°C), 1070 mb, and 76% in Annapolis. We show the EN scatterplot in Figure 5, where we see the DGPS solution as the light triangles, the ADGPS solution as the darker points, and the WAAS solution as the small black splotch in the northwest quadrant. The statistical parameters of the scatterplots are given in Table 2. We see that the WAAS mean value results are in error by about 0.4, 3.5, and 1.7 m in the east, north, and up directions, respectively, whereas corresponding figures for ADGPS are 0.4, 1.7, and 0.6 m. In this case, the DGPS figures are slightly better than the ADGPS figures, unlike the noticeable improvement found with ADGPS in Table 1. The tropospheric part of the atmospheric corrections ranged up to about 0.15 m for seven of the satellites and up to about 0.8 m for one of the satellites. The ionospheric part ranged up to about 0.05 m for the preceding seven satellites and up to 0.15 m for the eighth satellite. Small atmospheric corrections to DGPS explain why DGPS and ADGPS were not much different in this case. The DGPS and ADGPS scatterplots in Figure 5 are somewhat broader than in Figure 2, which suggests that multipath is more of an effect for our Superstar II receiver than for our older Allstar receiver. We plan to implement carrier phase smoothing, which should largely remove multipath effects.
3. Discussion and Conclusions
 We considered position errors in various approximations. Ignoring ionospheric effects altogether results in large bias errors of order 10 m. The Klobuchar ionospheric compensation model, which we did not consider, is included in conventional GPS receiver solutions. The usual estimate is that it removes roughly 50% of the ionospheric error, which is the largest part. A climatological tropospheric model can similarly remove a substantial portion of the tropospheric error. DGPS corrections produce smaller bias errors, but our results indicate that these can amount to several meters for large separations between roaming and reference receivers, relative to ionospheric and tropospheric decorrelation length scales. Timeliness of the differential corrections is also an issue for short-decorrelation timescales. GPS reference station and weather station data, if they are timely and in the vicinity, can be processed to obtain atmospheric corrections to DGPS. We have found for midlatitude cases that these corrections to DGPS can remove a large portion of residual DGPS bias errors. At high and low latitudes, where irregularities can cause small decorrelation lengths and times, it is especially important to correct DGPS corrections. Our ADGPS approach was applied to a USCG LADGPS system and was found to yield results in the cases we have considered that are superior to the WAAS DGPS results from the Superstar II receiver, at least with regard to mean position errors relative to a ground truth position. Part of the problem for WAAS may be in the ionospheric algorithms, to which we alluded in section 1, and to the use of a climatological tropospheric model for the roaming receiver. Otherwise, we would have expected comparable performance, since there is a WAAS GPS reference station in Leesburg, Virginia, which is about as close to Fairfax and Alexandria as the USCG station in Annapolis.
 Years from now, we will be able to buy inexpensive WADGPS-capable GPS receivers that process more than just the single L1 frequency and are capable of receiving and processing real-time tropospheric data from the nearest weather station. Then the ionospheric components can be directly measured and processed by the receiver, and the driving parameters of a tropospheric model, like the Hopfield model, can be updated in the receiver for compensation of tropospheric effects. The WADGPS corrections for satellite clock error and ephemeris would still be useful. For the near term, however, WADGPS GPS receivers and LADGPS differential corrections provide the best accuracy for low-cost applications. We may consider what some ADGPS improvements might be. A possible upgrade in the WADGPS (e.g., WAAS) receiver would provide Internet access. This would facilitate access to weather data from the nearest station for the purpose of updating the driving parameters of an internal tropospheric compensation model, like the Hopfield model. It would also make reception of WAAS data more reliable. A WAAS reference station could compute driving parameters of a model, as we did the effective sunspot number for RIBG, and a central processing station could interpolate model-driving parameters at reference stations to a user latitude-longitude grid. After downloading this information from the Internet, the receiver would use the appropriate model-driving parameters to internally select the appropriate tabulation of height versus electron density on a latitude-longitude grid and ray trace through it to remove ionospheric contributions to its pseudoranges. This is a fast calculation for straight rays.
 A possible upgrade in the LADGP (e.g., USCG) system is similar. The reference station would transmit modified differential corrections from which its ionospheric and tropospheric contributions had been removed. Two-frequency data would be used to remove the ionospheric effect, and an updated Hopfield-like model could be used to remove the tropospheric effect. The station could also transmit ionospheric model-driving parameters, or possibly the associated tabulation of height versus electron density profile on a latitude-longitude grid. The receiver would access this information over the Internet, along with data from the nearest weather station, and then remove tropospheric and ionospheric contributions from its pseudoranges, similarly to the preceding paragraph.