The accuracy of Global Navigation Satellite System (GNSS), aimed to support precise positioning for aircraft navigation globally by coordinating different regional augmentation systems, is limited by the extent to which the atmospheric propagation delay of microwave signals can be modeled. An algorithm is developed for modeling the tropospheric delay based on mean meteorological parameters. A Region-specific Tropospheric Delay (RTD) model is developed exclusively for the Indian region using meteorological data from the Indian subcontinent, as a part of GPS Aided Geo Augmented Navigation (GAGAN) program. The applicability of this model is examined in the context of the global model used in Wide Area Augmentation System (WAAS), developed employing meteorological data mostly from North American continent, by comparing the estimated zenith tropospheric delay (ZTD) with those obtained from regional models employing measured atmospheric parameters at the surface. The rms deviation of ZTD estimated using RTD model from that of the surface model is found to be ∼5 cm. A further validation by comparing with GPS measurements from two IGS stations at Bangalore and Hyderabad showed that predictions made using the RTD model are within an rms deviation of ±5 cm while those using WAAS model is ±7 cm. Maximum value of the residual error for RTD model is ∼15 cm, which corresponds to a ∼0.5 m error in the vertical coordinates for the lowest satellite elevation angles usually encountered.
 The use of satellite for critical application like navigation is becoming a reality through the development of various augmentation systems like US-based Wide Area Augmentation System (WAAS), Russian System of Differential Corrections and Monitoring (SDCM) and European Geostationary Navigation Overlay Services (EGNOS), etc. [ICG, 2006; Kaplan and Hegarty, 2006]. While WAAS and SDCM are the independent stand-alone systems, EGNOS is designed to augment the position information available from the US-based Global Positioning System (GPS) and the Russian Global Navigation Satellite augmentation System (GLONASS). In total, the primary objective of these systems is to provide the essential integrity information needed for navigation service. Apart from these, many future space-based regional augmentation systems are being developed over different parts of the globe. Prominent such systems are GPS Aided Geo Augmented Navigation (GAGAN) and Indian Regional Navigation Satellite System (IRNSS) of India [Kibe, 2003], Multifunctional Transport Satellite (MTSAT) Satellite Augmentation System (MSAS) and Quasi-Zenith Satellite System (QZSS) of Japan, Compass/BeiDou Navigation Satellite System (CNSS) of China, etc. [ICG, 2006; Kaplan and Hegarty, 2006]. In the future these regional systems will operate simultaneously and emerge as a Global Navigation Satellite System (GNSS) to support a broad range of activities in the global navigation sector. Thus the regional augmentation systems will play a major role in GNSS. The potential of GNSS will be further enhanced through a focused attention on the coordination among these regional augmentation systems through maintaining its accuracy, integrity, reliability and inter-operability everywhere on the globe. Under the proposed framework for real-time position determination with high accuracy (∼0.2 – 0.5 m) for EUPOS (European Position Determination System) is aimed using DGNSS (Differential GNSS) by maintaining huge network (∼800 DGNSS reference stations with distance between them less than 100 km) of ground-based stations [Rosenthal, 2007]. Establishing and maintaining such network systems all-through the globe, especially over remote locations, is rather difficult, in which case proper models developed to account for the associated delays acts as a reprieve.
 The most challenging part in GPS-based aircraft navigation is in attaining the positioning accuracy to meet the criteria set for optimum positioning [RTCA, 1999]. Among different types of error sources in the GPS-based positioning, the propagation error due to the atmospheric refraction is very significant and large spatial and temporal variations in the earth's atmosphere refractivity (for microwave region) limit the accuracy that can be achieved. The atmospheric refraction introduces a time delay due to retardation and bending of the wave, leading to a pseudo range in GPS ranging. Major part of this delay comes from the ionized upper atmosphere region. However dual frequency measurements enable the user to minimize the ionospheric delay errors. This approach is not applicable for mitigating the delay in the neutral atmosphere owing to its non-dispersive nature at microwave frequencies. As majority of the neutral constituents resides in the troposphere, this pseudo range is termed as “tropospheric delay”. The tropospheric delay although only about 2 m in the zenith direction, increases with decreasing GPS-satellite elevation [Ramjee and Ruggieri, 2005] to over ∼20 m at lower elevation angle [Dodson et al., 1999], and exceeds the criteria set for optimum positioning for civil navigation [Kaplan and Hegarty, 2006].
 Wide Area Differential GPS (WADGPS), aimed at providing the GPS correction over a large area, splits the total error into different components, namely the orbit error, clock error, ionospheric error, tropospheric error, etc. Information on these errors is broadcasted via a geo-stationary satellite over a large region to enable the users to apply proper corrections appropriately. However large temporal and spatial variations in the troposphere often occur over WADGPS scale regions; thus a troposphere delay estimated on a local basis is needed to obtain the necessary accuracy. This led to the development of different models for estimating the tropospheric delay either using a profile-based method (ray-tracing) or employing conventional method based on surface atmospheric parameters [Bock and Doerflinger, 2000]. Both these require measured atmospheric parameters on a real-time basis as input which makes the task rather complex.
 Accurate estimation of tropospheric delay is achieved by ray-tracing the neutral atmospheric refractivity profile [Smith and Weintraub, 1953; Schueler, 2001]. The difficulties involved in getting these refractivity profiles on a real-time basis led to the development of other conventional methods based on the empirical relation connecting Zenith Tropospheric Delay (ZTD) and the atmospheric parameters at surface. [Hopfield, 1971; Saastamoinen, 1972; Bevis et al., 1992; Schueler, 2001; Saha et al., 2007]. These models are more suitable for the static GPS stations, where collocated atmospheric measurements are available. However in case of aircraft navigation where no direct access to measured atmospheric parameters is available a method to surrogate the measured atmospheric parameters based on the geographical location and day of the year is used [Collins and Langely, 1997]. Validity of this method is assessed in various studies over North America and Europe as a part of WAAS and EGNOS [Collins and Langley, 1998; Dodson et al., 1999; Penna et al., 2001; Farah et al., 2005]. In the context that India is implementing its satellite-based navigation using GPS under the program “GAGAN” (GPS Aided Geo Augmented Navigation) as well as “IRNSS” (Indian Regional Navigation Satellite System) as a part of GNSS [Kibe, 2003] development of a model for the Indian region becomes highly essential. While GAGAN uses the GPS Satellite constellation, IRNSS is an independent Navigation Satellite System intended for providing services over a limited region in and around the Indian subcontinent (40°E to 140°E and 40°S to 40°N). The space segment of IRNSS consists of seven satellites that include 3 in geo-stationary orbit and 4 in geo-synchronous orbit.
 The geographical features and weather conditions that prevail over the Indian sub-continent make it different from other regions where most of the tropospheric delay models are developed. While the temperature over northern region of India that lies in the temperate zone, undergoes large seasonal variation as well as large range of diurnal heating and cooling process, the southern peninsular region which lies south of the tropic of cancer, experiences a tropical climate characterized by warm and highly humid conditions with less seasonal and diurnal variations. Under this scenario, the tropospheric delay over different climatic zones (under selected Range-Integrity Monitoring stations) was examined and a model based on surface atmospheric parameters was proposed for the Indian region [Saha et al., 2007]. In the present study, a conventional model is developed analogous to the WAAS model for the Indian zone based on the regular radiosonde data collected by the India Meteorological Department (IMD). The applicability of this model was tested using the data from two Indian IGS (International GPS service) stations Bangalore and Hyderabad.
2. Tropospheric Delay for Microwave Frequencies
 Empirical relation connecting the atmospheric parameters with tropospheric refractivity (N), which in terms of refractive index (n) is expressed as (n − 1) × 106, developed based on Debye's expression for polarization of polar liquid under the influence of the external field is given by [Smith and Weintraub, 1953; Thayer, 1974]:
The first term in the right hand side of this equation is the refractivity due to the non-polar gaseous molecules in the atmosphere, which usually is referred to as the dry or hydrostatic component of refractivity (Ndry), and the second term is due to the polar component of the water vapor in the atmosphere called wet or non-hydrostatic component of refractivity (Nwet). In equation (1), P and e are expressed in hPa and T in Kelvin. The values of the constants K1, K2 and K3 as reported by Bevis et al.  are, respectively, 77.60 ± 0.05 K hPa−1, 70.4 ± 2.2 K hPa−1 and (3.739 ± 0.012) × 105 K2 hPa−1. In the zenith direction as there is no refractive bending for the ray, in the absence of strong horizontal gradients, the tropospheric delay is solely due to retardation along the propagation path which can be written as
where the integral is taken from the altitude of the GPS receiver antenna (hs) to the top of the Atmosphere (TOA). Using equation (1) the ZTD can be written as the sum of two components, one representing the Zenith Hydrostatic (dry) Delay (ZHD) and the other Zenith non-hydrostatic (Wet) Delay (ZWD) which can be written respectively as
The numerical computation of tropospheric delay based on these equations requires the altitude profile of N which in turn could be estimated only from the respective profiles of P, T, and e. Because of the difficulty in getting the altitude profile of N at every location, conventional models are developed for ZTD in terms of more easily obtainable surface atmospheric parameters [Davis et al., 1985]. Such models are extensively used in case of static stations/users where collocated meteorological measurements with GPS receivers are available. However implementation of such correction models in GPS-based aircraft navigation is limited by the availability of atmospheric parameters at fine spatial and temporal resolutions. This problem was addressed in WAAS and a methodology was evolved to model the spatial and temporal variations of different atmospheric parameters based on a look-up table and a periodic function.
3. The WAAS Model
 The WAAS model [Collins and Langely, 1997] intended to surrogate the measured atmospheric parameters, uses a look-up table (Table 1) primarily developed using standard meteorological data (US Standard Atmospheric Model 1966 COESA, ), containing the annual mean value of the five atmospheric parameters and the amplitudes of the respective annual variations at particular latitudes pertaining to the mean sea level. The latitude-dependent mean meteorological elements for 15° to 75° latitude are taken from the table, which subsequently is denoted as ξo. The seasonal variability of this parameter is taken into account by employing the corresponding amplitude values (Δξ) along with a periodic (cosine) function (of the day number). The phase is defined so that the minimum occurs at day number d0. Using this look-up table the value of the desired atmospheric parameter for any particular day of the year (d) is estimated as
where ϕ is the receiver's latitude. The atmospheric parameters for a location that lies in between the latitudes given in Table 1, is obtained through a linear interpolation as
where i and i + 1 refer to tabular latitudes below and above the receiver latitude, respectively. These values are further scaled up to the user altitude using a scaling factor derived based on the hydrostatic equation which employs temperature lapse rate (β) and water vapor lapse rate (λ). [Collins and Langely, 1997]. On the basis of these relationships the values of ZHD and ZWD are estimated [Saastamoinen, 1972; Davis et al., 1985; Askne and Nordius, 1987] as
where g is 9.80665 m/s2, H is the height (in meters) of the receiver above mean-sea level, Rd = 287.054 J/kg/K. P0, T0 and e0 are the values of P, T and e at MSL; PS, TS and eS are the values of P, T and e at surface; gm = 9.784 · (1 − 0.00266·cos2ϕ − 0.00028 · h) in m/s2, ϕ is the ellipsoidal latitude and h is the height of the antenna site above the ellipsoid. The value of C1 is same as that of K1 in equation (1) and the value of C2 is 3.82 × 105 K2 hPa−1.
Table 1. Look-Up Table for WAAS Model
 In line with the above, this model is evaluated for the Indian region using atmospheric data at the surface from 18 different locations (Figure 1) spread over the Indian subcontinent spanning the latitude 8°N to 32°N. The mean atmospheric model for WAAS is mainly governed by the values of the coefficients in Table 1 which does not show any seasonal variation for the five model parameters in the latitude region below 15°N. The mean values for Po, To and eo (at ≤15°N) are 1013.25 mb, 301.7°K and 26.31 mb, respectively. As a first step to examine the validity of this for the Indian stations the annual variation of these atmospheric parameters for all the 18 stations are examined using daily data for five years from 1995 to 1999 which showed that the observed annual mean and seasonal variation deviate significantly from the model values derived from Table 1. As a typical example the annual variations of PS, TS and eS (surface level) at Trivandrum and Chennai, the two stations located below 15°N, and Ahmedabad and Delhi located at 23°N and 28°N, respectively, are presented in Figure 2. As can be seen the annual variation is less at lower latitudes and increases with increase in latitude. For a comparison the values derived from the WAAS model are also presented in Figure 2 by a dashed curve. The most striking feature is that the values of PS, TS and eS deviate significantly from the respective values derived from WAAS model. The maximum deviation for PS is seen at Delhi (∼28 hPa). This large deviation in pressure can lead to an error of ∼6.5 cm in the estimated ZTD employing Saastamoinen model [Saastamoinen, 1972]. In case of eS the mean absolute deviation ranges from 4 to 12 hPa and does not show any systematic latitudinal trend. The maximum deviation in eS is observed for Delhi (∼33.95 hPa) which can introduce an error of ∼44 cm in the estimated value of ZWD. These analyses show that the mean atmospheric model employed in WAAS is not adequate for the Indian region and hence is not suitable for GAGAN. This prompted the development of a new model exclusively for this purpose, which we named as Region-specific Tropospheric Delay for GAGAN (RTD-GAGAN).
4. Region-specific Tropospheric Delay Model for GAGAN (RTD-GAGAN)
 A Region-specific Tropospheric Delay Model for GAGAN (RTD-GAGAN) applicable for the Indian Sub-continent between 5°N to 32°N latitude is developed using five years (1995–1999) of the radiosonde data from 18 selected IMD stations covering different climatic zones (Figure 1). All these IMD stations are in proximity of the airports. The altitude profiles of P, T and e obtained from the radiosonde data are used to develop monthly mean models. These are used to estimate the monthly mean values for β and λ. The values of β and λ in each month are used to scale the PS, TS, eS to the respective mean sea level (MSL) values (Pmsl, Tmsl, emsl). The time series of these values are Fourier analyzed to determine the mean (Po, To, eo, β and λ) as well as the annual amplitudes (P′, T′, e′, β′ and λ′, respectively) of these parameters for each of these stations. Once this is accomplished, these stations are arranged in different latitude sectors, by grouping stations located almost at the same latitude but in different longitudes and assigned a mean value for these parameters by averaging. These mean values in different discrete latitude sectors are further interpolated for a constant latitudinal interval of 5° and presented in the form of a look-up table as shown in Table 2. The phase offset epoch (do) is also estimated for each of these stations and optimized for a single value of 6 days for the whole sub-continent. Similar to WAAS, the values of these parameters at any location in between the 5° latitude interval can be obtained through linear interpolation (equations 5 and 6). These values could further be scaled to the user altitude employing appropriate scaling factors and using the corresponding values of β and λ [Collins and Langely, 1997]. The desired atmospheric parameter for any particular day of the year (d) then is estimated incorporating the periodic function (equation 4).
Table 2. Look-Up Table for RTD-GAGAN Model
 The atmospheric parameters thus estimated for a particular location is then incorporated into the unified surface model [Saha et al., 2007] developed exclusively for the Indian region. This model was further refined by Suresh Raju et al.  incorporating data from eighteen stations spread over the entire Indian subcontinent and refined the values of the relevant coefficients. The linear relationship thus established between ZHD and PS (in hPa) can be written as
Similarly a second-order polynomial established between ZWD with eS (in hPa) can be written as
 The values of ZHD and ZWD estimated using RTD-GAGAN and WAAS model are compared with those estimated using unified surface models using measured atmospheric parameters (for PS and eS) collected on a daily basis from different IMD stations listed in Figure 1 as input. As seen from Figure 1 while some of these stations are located near to sea coast, others are island or inland stations. They are distributed in different geographical zones varying from tropics to mid-latitudes. While most of the stations are near to MSL, Bangalore and Srinagar are located at higher altitude. The values of ZHD estimated using the RTD-GAGAN model reproduces the seasonal variations at all these stations. The WAAS model does not show any seasonal variation for the near equatorial stations. Though the WAAS model could reproduce the seasonal variations for the mid-latitude stations the amplitude of this was much smaller than the real observed annual variations. For near equatorial stations the rms deviation of RTD-GAGAN model is 0.3 ± 0.2 cm while that for mid-latitude region it is 0.5 ± 0.4 cm. In case of WAAS model, the rms deviation is more than twice that of the RTD-GAGAN model. For the near equatorial stations the amplitude seasonal variation in ZWD is very small (<20 cm) and increases with increasing latitude. Largest annual variation (≥50 cm) in ZWD is observed over Delhi. Though the WAAS model shows significant improvement in predicting the seasonal variation with increase in latitude it is inferior to the RTD-GAGAN model. Even though the RTD-GAGAN model could reproduce the annual variation to a reasonable extent it fails to capture the extremely high values encountered especially during the monsoon (July–September) period. The large deviation of this model (∼30 cm) at higher latitudes, especially over Delhi during the July–September period, which could be attributed to the advection of abundant water vapor associated with the onset of southwest monsoon. Collins and Langley  also reported similar extreme residuals (∼42 cm) at La Paz, Mexico, associated with the passage of tropical cyclones (Hurricane Flossie). In case of ZWD, the rms deviation of RTD-GAGAN model is 4 ± 3.1 cm for near-equatorial stations and >5 cm for mid-latitude region while the WAAS model deviates by 7.8 ± 4.1 cm for near-equatorial stations and >7.0 cm for mid-latitude region.
5. Comparison of RTD-GAGAN and WAAS Models With ZTD Derived From GPS Data
 Two years GPS data from the two IGS stations Bangalore (12.95°N, 77.68°E) and Hyderabad (17.45°N, 78.46°E) collected in RINEX format are used to derive the ZTD values. Quality checked GPS data from these stations are used for the present analysis. The GPS processing software GAMIT 10.2 [King and Bock, 2000] is employed for this purpose. The GAMIT incorporates a weighted least squares algorithm to estimate the relative positions of a set of stations, orbital and earth rotation parameters, zenith delays, and phase ambiguities by fitting to doubly differenced phase observations. Once the GPS data from all the stations are acquired (http://garner.ucsd.edu/pub/rinex/), ZTD is estimated in daily batch processing as piecewise constant parameters averaged for every two hours. An elevation angle of 10° is used as a cut-off angle. The new mapping function [Niell, 1996] is used to map the slant delay to corresponding zenith values. The accuracy of GPS data processing is further confirmed by comparing these values with those reported by the IGS on their sites at SOPAC/CSRC archive available at http://garner.ucsd.edu/pub/troposphere/ and found that the absolute difference is <8 mm.
 Time series plots of ZTD values derived from GPS data analysis (ZTDGPS) and that obtained from RTD-GAGAN (ZTDRTD) and WAAS model (ZTDWAAS) over Bangalore and Hyderabad for a period of two years are presented in Figure 3. Depending on the data quality and availability, the values of ZTDGPS for Bangalore are presented from January 2002 while that for Hyderabad from January 2003. As the GPS data are processed at 2 hr interval, all the 12 values of ZTDGPS derived from this data on each day are averaged and the mean value is plotted in this figure, with vertical bars representing the standard deviations indicating the amplitude of diurnal variability, while the model gives only one mean value for each day. As can be seen from the figure, for Bangalore the WAAS model does not show any seasonal variation for ZTD as depicted by the respective value derived from GPS data. The RTD-GAGAN model could reproduce the seasonal variation of ZTDRTD to a large extent notwithstanding the fact that the annual amplitude is marginally less (while the annual amplitude of ZTD derived from GPS data is >20 cm for ZTDRTD is around 7 cm). For Hyderabad, which is located ∼5° North of Bangalore and at a lower altitude (∼400 m below Bangalore), in general, the values of ZTD are larger than those for Bangalore. The amplitude of the annual variations also is large for Hyderabad. Being located at a latitude >15°N, the WAAS model could depict a small seasonal variation for ZTD with an amplitude of ∼2.5 cm. The RTD-GAGAN model shows significant improvement in representing the seasonal variations of ZTD (with amplitude of ∼5 cm) while the ZTDGPS shows an annual amplitude of ∼25 cm which is much smaller than the real (observed) amplitude. Quantitatively, the rms deviations of the ZTDRTD from ZTDGPS, for Bangalore and Hyderabad are ∼4.7 (±3.2) cm and ∼5.2 (±3.3) cm, respectively, while the rms deviations of ZTDWAAS are ∼6.0 (±3.6) cm and ∼7.0 (±3.6) cm.
6. Results and Discussions
 A detailed comparison of WAAS and RTD-GAGAN models on a day-to-day basis with ZTD estimated from GPS data shows that the latter model predicts the ZTD fairly well for both the IGS stations over the Indian sub-continent for different seasons. The rms deviation of ZTD is 4–5 cm for RTD-GAGAN and 6–7 cm for WAAS. However the phase of the annual variation predicted by WAAS leads approximately by a month. The RTD-GAGAN model is at par if not better than the WAAS model as far as the Indian sub-continent is considered.
 It would be worth in this context to note that since all these region specific models (WAAS, EGNOS, RTD-GAGAN, etc.) are derived based on the mean annual variation of surface atmospheric parameters and they could deviate more from real values under abnormal climatic conditions like prolonged drought, as well as during short-term weather changes like cyclones, cloud burst, strong thunder showers, etc. A typical example of this is observed at Hyderabad during May–June of 2003, when the region experienced an abnormal drought. As can be seen from Figure 3b, the models significantly overestimate the delay at Hyderabad during these months while no such abnormality is observed at Bangalore. In this context, we have examined the surface meteorological data at Hyderabad during 2003–04. Figure 4 shows the day-to-day variation of PS, TS and eS for Hyderabad for the years 2003 and 2004, which shows abnormally low values of eS and high values of TS during May–June 2003 (compared to those during the same period in a normal year as 2004). Low value of water vapor content in the atmosphere has significantly reduced the wet component of the zenith delay thereby resulting in an overestimate for the model. The maximum error in ZTD estimated from RTD-GAGAN model for Hyderabad during this 45 day period is ∼15 cm. A “rule of thumb” [MacMillan and Ma, 1994] states that an equivalent zenith atmospheric error will imply an error 3–4 times larger for the vertical coordinate. In this case 15 cm error in the ZTD means approximately 0.5 m error in the vertical positioning. For an aircraft flying above its intended height, given an unfavorable satellite constellation and unusual weather conditions, vertical position biases of up to 4 m are possible for wide-area differential users, solely due to mis-modeled tropospheric delays [Collins and Langley, 1998]. However, in such situation, the regional tropospheric delay models like Unified surface models (equations (9) and (10)) with near real time atmospheric parameters as input is a good option. However when such direct measurements at that location are not available another option available for this purpose would be the NCEP/NCAR reanalysis data. So as an alternate measure we have examined the applicability of the Unified surface model with NCEP/NCAR predicted surface values (PS and eS) as inputs for estimating ZTD. Figure 5 shows a time series plot for the ZTD estimated using six-hourly NCEP/NCAR reanalysis data for PS, TS and eS as input to the Unified surface model along with the ZTD estimated from GPS data for different days at Hyderabad for the year 2003. On an average the agreement is fairly good especially during the extreme weather condition prevailed during May–June 2003. However there is a general bias throughout the year, which amounts to 7 ± 4.5 cm. This bias could be attributed to the overestimation in the PS and eS values from NCEP/NCAR reanalysis, compared to real measurements near the surface at the GPS site. This shows that in case when measured surface meteorological data are not readily available the reanalysis data from NCEP/NCAR could be a good option. However further routines to provide the NCEP/NCAR prediction in real-time is to be developed for this purpose.
 The present study shows that on an average the region specific tropospheric delay model developed based on Indian atmospheric conditions could be incorporated in the GAGAN program within the acceptable error limits. This model performs much better than the currently available WAAS model over the Indian subcontinent and adjoining regions. A comparison of the model derived ZTD with GPS derived values from the two IGS stations, Bangalore and Hyderabad, shows a fairly good agreement, with a mean absolute difference of ∼5 cm and a maximum difference of ∼15 cm, which corresponds to a maximum vertical position bias of ∼0.5 m in aircraft navigation considering the lowest satellite elevation of 5°. This model is a better alternative for the GPS-based navigation applications at locations where directly measured surface atmospheric data are not available, even though like all other globally used periodic models (WAAS, EGNOS, etc.) this model also fails to respond to sudden atmospheric changes and abnormal weather conditions. The potential of using NCEP/NCAR reanalyzed surface atmospheric data to estimate the ZTD appears to be a better substitute in this context.
 The authors acknowledge the India Meteorology Department, Pune, for providing the radiosonde data for different stations, which were used for developing the region specific tropospheric delay model and other analyses. We also thank Prof. Robert W. King, MIT for providing GAMIT software support and the NCEP/NCAR data of reanalyzed atmospheric surface parameters.