Parameterization of tropospheric delay correction for mobile GNSS positioning: a case study of a cold front passage

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1 METEOROLOGICAL APPLICATIONS Meteorol. Appl. 15: (2008) Published online 11 August 2008 in Wiley InterScience ( Parameterization of tropospheric delay correction for mobile GNSS positioning: a case study of a cold front passage R. Eresmaa, a * M. Nordman, b M. Poutanen, b J. Syrjärinne, c J.-P. Luntama a and H. Järvinen a a Finnish Meteorological Institute, P.O. Box 503, FI-00101, Helsinki, Finland b Finnish Geodetic Institute, P.O. Box 15, FI-02431, Masala, Finland c Nokia Technology Platforms, P.O. Box 1000, FI-33721, Tampere, Finland ABSTRACT: Numerical weather prediction (NWP) model output can be used for derivation of tropospheric delay correction in order to decrease positioning errors of Global Navigation Satellite System (GNSS) applications. In precise geodetic positioning it is possible to make use of corrections that are derived separately for all GNSS signal paths. Due to the large amounts of data, this approach is considered impractical from real-time mobile positioning point of view. This article introduces several parameterization schemes that provide a zenith- and azimuth-angle-dependent description of tropospheric delay correction at a single observation site. These parameterization schemes are compared in a case study of a cold front passage. Inclusion of zenith-angle dependency in the parameterization, instead of applying a prescribed mapping function, is found to be crucially important for accuracy. Parameterization can be further improved by applying a horizontal delay gradient and by separating the total tropospheric delay into hydrostatic and wet components. It is concluded that any increase of sophistication in the parameterization results in an increased parameterization accuracy. Copyright 2008 Royal Meteorological Society KEY WORDS data compression; Global Positioning System; numerical weather prediction Received 28 January 2008; Revised 10 June 2008; Accepted 11 June Introduction Global Navigation Satellite Systems (GNSS), such as the Global Positioning System (GPS), constitute the basis for modern positioning applications. A dense network of ground-based GNSS receivers, used in a measurement campaign of sufficient length, allows relative positioning with millimetre accuracy, which is acknowledged by geophysical research. Atmospheric effects, in particular the tropospheric delay, are nowadays among the largest sources of error in precise geodetic positioning (Bar- Sever et al. 1998). Recent results show that tropospheric delay corrections derived from a numerical weather prediction (NWP) model output for actual GNSS signal paths can effectively mitigate tropospheric impacts in geodetic observation time series (Nordman et al. 2007). The precise geodetic GNSS measurements can, on the other hand, be made use of in short-range weather forecasting and climate studies (Elgered et al. 2005). This is possible through advanced data processing software, that allow estimation of the tropospheric delay on GNSS signal paths as by-products of the geodetic solution. With certain assumptions and using additional data, it is possible to convert the estimated tropospheric delay to an integral measure of atmospheric humidity (Bevis et al. 1992; Pramualsakdikul et al. 2007). Alternatively, sophisticated * Correspondence to: R. Eresmaa, Finnish Meteorological Institute, P.O. Box 503, FI-00101, Helsinki, Finland. reima.eresmaa@fmi.fi observation modelling algorithms allow direct use of the tropospheric delay estimates in variational data assimilation systems (Järvinen et al. 2007). Tropospheric delay modelling provides an example of how NWP model output can be applied to simulate propagation of electromagnetic signals. In a similar manner, ongoing activities exist that aim at simulating propagation of weather radar pulses (Atkinson and Zhu, 2006; Bech et al. 2007). The number of mobile GNSS receivers is expected to increase in future together with demands for high mobile positioning accuracy. The mobile positioning applications have to manage with limitations that are due to signalling capacity in the wireless communication channel, modem power consumption and memory capacity. These aspects make it impractical to make use of large NWP model output in order to derive the tropospheric delay correction in the mobile handsets. Application of the NWP model output for practical mobile positioning therefore necessitates some kind of a parameterization for tropospheric delay correction. Such a parameterization compresses the tropospheric delay information into only a few parameters and avoids transmission of the full NWP model output with only a minimal loss of accuracy. In this article, performances of several parameterization schemes for tropospheric delay correction are studied in a case of a cold front passage. The study is conducted under the assumption of a perfect NWP model. Moreover, all error sources of GNSS positioning, except the tropospheric delay, are neglected. These assumptions are Copyright 2008 Royal Meteorological Society

2 448 R. ERESMAA ET AL. justified in the context of evaluating the characteristics of different parameterization schemes. The reference atmosphere for the study is provided by the High Resolution Limited Area Model (HIRLAM) (Undén et al. 2002). The article is structured as follows. Principles of deriving the tropospheric delay correction from NWP model output are outlined in Section 2. This is followed by a description of the studied parameterization schemes in Section 3. The parameterization schemes are intercompared in Section 4, with special attention on parameterization accuracy and number of parameters required by each scheme. Significance of the results is discussed in Section 5. Section 6 summarizes the conclusions. 2. Determination of the tropospheric delay correction GNSS data processing algorithms make use of pseudorange measurements. A pseudorange measurement can be considered as a first-order approximation of geometric distance between a GNSS satellite and a ground-based receiver. Pseudorange is contributed by a number of error sources that need to be taken into account in the data processing. These include, e.g. receiver and satellite clock biases, ionospheric and tropospheric delays and multipath propagation. Tropospheric delay T is often approximated as an integral: T = (n r 1)ds = 10 6 Nds (1) s where s is the signal path through the atmosphere, n r is the real part of atmospheric refractive index, and N = 10 6 (n r 1) is the refractivity. This approximation neglects the effect of geometric delay that arises from bending of the GNSS signal path at very large satellite zenith angles; at zenith angles smaller than 80, for instance, the effect of geometric delay contributes less than 0.3% to the total value of tropospheric delay. Contribution of T in GNSS pseudorange measurement is typically about 2.5 m if the GNSS satellite is at the local zenith and the receiver is at the mean sea level. At the microwave frequency domain, N is obtained through: p d e N = k 1 T Z 1 d + k 2 T Z 1 w + k 3 s e T 2 Z 1 w (2) where p d and e are the partial pressures of dry air and water vapour, T is the temperature, k 1, k 2 and k 3 are the empirical refractivity coefficients, and Z d and Z w are the compressibility factors of dry air and water vapour, respectively (Bevis et al. 1992), making the ideal gas assumption such that Z d = Z w = 1. Substituting p d = p e (3) where p is the barometric pressure of compound air, allows Equation (2) to be written as p N = k 1 T + (k 2 k 1 ) e T + k e 3 T 2 (4) Since the HIRLAM model operates with specific humidity q, rather than with e, it is necessary to formulate Equation (4) accordingly. This is achieved through applying the relation q = 0.622e p 0.378e (5) where the numeric values follow from the molar masses of water vapour and dry air. Equation (5) is equivalent to: pq e = (6) q Introducing now allows substitution of: ɛ = q (7) e = pq ɛ (8) in Equation (4) to yield: p N = k 1 T + (k 2 k 1 ) qp ɛ T + k qp 3 ɛ T 2 (9) Equation (9) provides the core to calculation of the refractivity from the HIRLAM model variables (p, T, q). This study applies the values suggested by Bevis et al. (1994) for the empirical refractivity coefficients. These are k 1 = K hpa 1, k 2 = 70.4 KhPa 1 and k 3 = K 2 hpa 1. The signal path s is defined as a set of coordinates indicating the intersections of s with each of the horizontal levels of the NWP model grid. The intersections follow from the observation geometry given by azimuth and zenith angles of the satellite as viewed from the ground-based receiver. The effect of the refractive bending on the signal path is taken into account by an explicit refractivity-dependent correction. Once the signal path is known, the tropospheric delay correction is obtained by numerical integration of Equation (1). Tropospheric delay modelling in the framework of HIRLAM data assimilation system (Gustafsson et al. 2001; Lindskog et al. 2001) is discussed in more detail in Eresmaa and Järvinen (2006). 3. Parameterization schemes 3.1. Zenith angle dependency The tropospheric delay correction is parameterized by least-squares fitting the parameters of the chosen scheme to tropospheric delay corrections determined from the NWP model output at different azimuth and zenith

3 PARAMETERIZATION OF TROPOSPHERIC DELAY CORRECTION 449 angles. Since the tropospheric delay is known to depend primarily on satellite zenith angle ζ, and only secondarily on azimuth angle, all tested schemes involve parameters for zenith angle dependency. This study applies the threeparameter continued fraction: a b f(ζ)= 1 + c a cos ζ + b cos ζ + cos ζ + c (10) for the zenith-angle dependency. The parameterization schemes differ from each other in terms of the way how the coefficients a, b and c are determined. Equation (10) has earlier been used by a number of authors including Niell (1996) and Boehm et al. (2006) Use of prescribed mapping function coefficients One alternative that one might consider is to apply prescribed mapping function coefficients. Most of the mapping functions discussed in the literature consist of separate sets of coefficients for the hydrostatic and wet components of the tropospheric delay. Consequently, parameterization with prescribed mapping functions would imply two tuning parameters, that need to be passed to the mobile receiver. These parameters are the Zenith Hydrostatic Delay (ZHD) and the Zenith Wet Delay (ZWD). In this study the widely-used and referenced mapping function coefficients proposed by Niell (1996) and Boehm et al. (2006) are chosen to be tested. This choice is supported by the fact that these coefficients do not depend on surface weather. Since there are no surface measurements involved in this study, choosing these mapping function coefficients is considered to be appropriate in order to obtain conclusive results Fitted mapping function coefficients Another alternative for parameterization of the zenithangle dependency is to fit the mapping function coefficients to modelled tropospheric delay corrections at different satellite zenith angles. It is likely that this approach provides a more accurate description of tropospheric delay than the use of prescribed coefficients, because fitting of these coefficients allows much more flexibility in taking the spatial and temporal atmospheric variations into account. In this study, two specific alternatives are explored; the coefficients are fitted either to the total delay corrections or separately to the hydrostatic and wet components of the total correction Azimuth angle dependency Some of the studied parameterization schemes involve dependency on azimuth angle in addition to the zenith angle dependency. The azimuth angle dependency is achieved by adopting the tropospheric delay gradient model of the form T,a (α, ζ ) = m(ζ ) tan ζ (n cos α + esinα) (11) where T,a is the azimuthally asymmetric delay component and α is the azimuth angle. In this study, the mapping m(ζ ) is chosen to equal the hydrostatic Vienna mapping function (Boehm et al. 2006). The delay component T,a is obtained as a residual term consisting of that part of the tropospheric delay, which is not resolved by those schemes that involve no dependency on azimuth angle. The functional model (11) is similar to the one used by MacMillan (1995) and Bar-Sever et al. (1998) for linear horizontal tropospheric delay gradients. The gradient model parameters n and e are interpreted as the meridional (north) and zonal (east) components of the horizontal gradient vector. Inclusion of the gradient model increases the number of needed parameters by two Summary of the parameterization schemes Based on the considerations above, seven parameterization schemes are selected for comparison. The schemes differ from each other in terms of the treatment of the tropospheric delay components (either total or separated into hydrostatic and wet components), the nature of the mapping function coefficients (either prescribed or fitted), and the possibility to model azimuthal variations (whether horizontal gradient vector is applied or not). The properties of the tested schemes are listed in Table I, and they are also briefly discussed below. The simplest alternative is to use prescribed mapping function coefficients and tune only the hydrostatic and wet zenith delay components. The two-parameter schemes corresponding to this alternative are referred Table I. Summary of the parameterization schemes that are used in this study. Scheme ID Delay components Coefficients of Equation (10) Gradient Parameter number SNN Separated Niell (1996) 2 SVN Separated Boehm et al. (2006) 2 TFN Total Fitted 4 SFN Separated Fitted 8 SVG Separated Boehm et al. (2006) Included 4 TFG Total Fitted Included 6 SFG Separated Fitted Included 10

4 450 R. ERESMAA ET AL. to as SNN (Separated delay components, Niell mapping functions and No gradients) and SVN (Separated delay components, Vienna mapping functions and No gradients), which apply mapping function coefficients suggested by Niell (1996) and Boehm et al. (2006), respectively. Most of the remaining schemes involve fitting of mapping function coefficients. The schemes TFN (Total delay, Fitted mapping function coefficients and No gradient) and SFN (Separated delay components, Fitted mapping function coefficients and No gradient) require four and eight parameters, respectively. The schemes SVG (Separated delay components, Vienna mapping functions and Gradient included), TFG (Total delay, Fitted mapping function coefficients and Gradient included) and SFG (Separated delay components, Fitted mapping function coefficients and Gradient included) are modified on top of the schemes SVN, TFN and SFN, respectively, by including the horizontal gradient vector in the parameterization. The most sophisticated scheme is SFG, which requires 10 tuning parameters. Figure 1. Six hour HIRLAM forecast of mean sea level pressure valid at 0600 UTC on 27 May Units are in hpa. Metsähovi GNSS receiver station is denoted by a dot. 4. Evaluation of the parameterization schemes The parameterization schemes are evaluated by assessing that part of the tropospheric delay correction that is not resolved by each scheme. On the one hand this is done by plotting the unresolved component of the tropospheric delay correction as a function of azimuth and elevation angles. On the other hand, the influence of the unresolved component to the estimated receiver position is approximated in a realistic GPS observation geometry. The case of interest in this study took place on 27 May 2005 in southern Finland. During the course of the day, a depression moved over northern Sweden towards the northeast, associated with a cold front that passed the central parts of Finland. Figures 1 and 2 show the 6 h forecasts of mean sea level pressure and 700 hpa specific humidity, respectively, valid at 0600 UTC, as forecast by the hydrostatic HIRLAM NWP model with a 9 km horizontal resolution on 40 levels in vertical. The cold front is revealed by a strong humidity gradient that extends from eastern Finland through the Baltic Sea to southern Sweden in Figure Experiment design A hypothetical mobile GNSS receiver is assumed to be located at Metsähovi, Finland (60.22 N, E). This location is shown by a dot in Figures 1 and 2. The tuning parameters of each parameterization scheme are determined as a least-squares fit to tropospheric delay correction data that corresponds to a 6 h HIRLAM NWP forecast valid at 0600 UTC on 27 May The tropospheric delay corrections are determined to Metsähovi for 67 hypothetical satellites, that are shown in Figure 3 (circles). This type of skyplot is oriented such that the Figure 2. Six hour HIRLAM forecast of specific humidity at 700 hpa valid at 0600 UTC on 27 May Units are in g kg 1.Metsähovi GNSS receiver station is denoted by a dot. northern (eastern) horizon is on top (right) and the zenith is in the middle of the panel. In addition to the 67 hypothetical satellite positions, Figure 3 shows a realistic GPS satellite constellation (dots) that is later used for simulating the positioning errors due to parameterization (Section 4.3). It can be argued that the hypothetical GNSS satellite constellation of Figure 3 is unrealistically evenly distributed in azimuth and zenith angles. In reality, there is a relatively large segment where GPS satellites are never seen due to the properties of the satellite orbits. The exact position of this segment depends on the latitude of the receiver. Therefore one might prefer to use a more realistic satellite geometry for determination of the parameters. The motivation for the use of the unrealistic constellation is that it is considered to provide a homogeneous scan of the tropospheric delay in different

5 PARAMETERIZATION OF TROPOSPHERIC DELAY CORRECTION 451 W N S Figure 3. Hypothetical GNSS satellite positions (circles) used in parameter fitting and real GPS satellite positions (dots) used for simulation of positioning errors. azimuth and zenith angles. It is believed that the parameters derived with the homogeneous scan are more robust than those parameters that would be obtained from using a realistic satellite constellation. Whether the approach of using the homogeneous scan results in significantly different parameters as compared with the approach of using a realistic satellite constellation, is beyond the scope of this article Unresolved component of tropospheric delay correction Once the tuning parameters are determined, it is straightforward to produce a parameterized tropospheric delay correction to an arbitrary satellite at any azimuth and zenith angles. In order to study the distribution of the unresolved component of tropospheric delay correction, it is useful to provide these corrections in a regular (α, ζ ) grid that allows contour plotting, e.g. in the form of a E skyplot. Figure 4 shows such contoured skyplots for the parameterization schemes TFN and TFG. The delay that is not resolved by the TFN scheme (Figure 4(a)) is positive to the south and negative to the north of the receiver. More precisely, the unresolved delay maximum (215.1 mm) and minimum ( mm) are aligned approximately in azimuth angles 165 and 345, respectively. This agrees roughly with the gradients of mean sea level pressure and 700 hpa specific humidity shown in Figures 1 and 2, which indicate that both surface pressure and atmospheric humidity increase to the southeast from the receiver. In this case it seems to be important to supply the parameterization with the horizontal delay gradient. Figure 4(b) shows that the delay that is unresolved by the scheme TFG is very close to symmetric with azimuth. Moreover, the maximum (28.9 mm) and minimum ( 43.3 mm) valuesof the unresolved delay are reduced by a factor of 5 7 as compared with the TFN parameterization scheme. It is concluded that applying a horizontal gradient in the parameterization efficiently allows to model the azimuthal delay variations in this case that is characterized by a cold front passage. A closer look to the unresolved component of tropospheric delay correction is provided in Figure 5 in the form of a cross-section along azimuth angles 0 (north) and 180 (south). In principle, the closer the curves are to zero, the better is the accuracy of the parameterization. Panel 5(a) shows that those schemes that apply prescribed mapping function coefficients (SNN, dash-dotted line, and SVN, solid line) lead to positive unresolved delay components of the order of mm for nearly the whole range of zenith angles in both north and south. Such a behaviour of unresolved delay correction implies that the tropospheric delay corrections that are passed to the receiver are too small. This results in too large estimates of the satellite-receiver distances. Qualitatively speaking, the effect of this is that the receiver position is estimated to be lower than the real position. Those schemes that involve the fitting of the mapping function coefficients (TFN, dashed line, and SFN, dotted line) outperform the simpler two-parameter schemes. Since the SFN scheme is mostly closer to zero than the TFN Figure 4. Skyplots of the unresolved delay component in a single case of a cold front passage. (a) Scheme TFN, (b) scheme TFG. Contours are ±1, ±10 and ±100 mm, and negative contours are dashed. Maximum and minimum values of each plot are denoted by circles.

6 452 R. ERESMAA ET AL. (a) UNRESOLVED DELAY [mm] (b) NORTH SOUTH NORTH SOUTH ZENITH ANGLE [ ] UNRESOLVED DELAY [mm] ZENITH ANGLE [ ] Figure 5. Cross-sections of the unresolved delay component in a single case of a cold front passage. (a) Schemes SNN (dash-dotted line), SVN (solid line), TFN (dashed line) and SFN (dotted line), (b) schemes SVG (solid line), TFG (dashed line) and SFG (dotted line). Note that the scale of the y-axis is logarithmic. scheme, it is concluded to be very beneficial to apply separate parameters for the hydrostatic and wet components of the tropospheric delay correction, instead of providing the corresponding parameters for total delay correction only. Finally, it is noted that the unresolved component of all of the schemes plotted in Figure 5(a) is asymmetric with respect to the zenith. Therefore, all of these parameterization schemes are expected to contribute to the horizontal (meridional) component of positioning error in this specific case. Figure 5(b) plots the cross-section of the unresolved delay component for those schemes that involve a horizontal delay gradient. In general it is noted that the unresolved delay component becomes more symmetric around the zenith as the horizontal delay gradient is taken into account. This means that the inclusion of the horizontal delay gradient improves the parameterization in particular through reducing the horizontal components of positioning error. However, as the unresolved delay component is greatly reduced near the southern horizon, that in the northern end of the cross-section turns out to increase as the horizontal delay gradient is taken into account. It is concluded that the inclusion of the horizontal delay gradient has very little effect on the vertical positioning error that is attributed to the parameterization Simulation of positioning error In order to quantify the qualitative conclusions that are drawn above, the propagation of the inaccuracies of the parameterization into estimated receiver position is simulated next in a realistic observation geometry. The real GPS satellite constellation as viewed at Metsähovi at 0000 UTC on 1 October 2005 is used for this purpose. Applying a zenith angle cutoff of 85 results in 11 satellites being visible at the receiver location. The satellite constellation is shown by dots in Figure 3. The positioning error that is attributed to the parameterization of tropospheric delay correction is simulated by calculating the zonal (east), meridional (north) and vertical (up) error components using the following procedure: 1. Calculate the unparameterized tropospheric delay correction T (using the algorithm described in Section 2), the parameterized tropospheric delay correction P, and the unresolved part of the tropospheric delay correction U = T P at the azimuth α and zenith angles ζ of each satellite. 2. Interpret the unresolved part of the tropospheric delay correction U as a projection of the positioning error along the line that is specified by α and ζ. This allows to project the positioning error in zonal, meridional and vertical components (ɛ e, ɛ n and ɛ u ) through ɛ e = U sin ζ sin α (12) ɛ n = U sin ζ cos α (13) ɛ u = U cos ζ (14) separately for each satellite. 3. Obtain the simulated components of the positioning error by summing the contributions of all satellites. Table II shows the components of the simulated positioning error for different parameterization schemes at Metsähovi on 27 May 2005 at 0600 UTC. It should be noted that these numbers are interesting only in a relative sense, i.e. as compared with each other. The total positioning error (last column) generally decreases as more parameters are included in the parameterization. Those schemes that involve no horizontal delay gradient are responsible for positioning error of mm, and this error is dominated by the meridional component. Inclusion of the horizontal gradient in parameterization reduces the zonal and meridional error components considerably, but increases the vertical error component. It is worth noting that in the case of the most sophisticated parameterization scheme, SFG, the three components of the positioning error are at the same order of magnitude with each other. This is actually not the case with any of the other parameterization schemes.

7 PARAMETERIZATION OF TROPOSPHERIC DELAY CORRECTION 453 Table II. Zonal, meridional, vertical and total positioning error that is attributed to each parameterization scheme on 27 May 2005 at 0600 UTC at Metsähovi. Units are in millimetres. Scheme ID Zonal Meridional Vertical Total SNN SVN TFN SFN SVG TFG SFG Discussion In previous sections it has been shown that the tropospheric delay correction can be compressed efficiently into only a few parameters that need to be passed to mobile GNSS receivers. Practical questions remain on spatial and temporal representativity of the parameters. For instance, how large is the area that can be thought to be represented by a single value of each parameter, and what should be the temporal update interval of each one of the parameters? These questions need to be investigated thoroughly prior to implementation of a tropospheric modelling and delay correction in mobile GNSS receivers. It is not obvious that the accurate tropospheric modelling results in a significant improvement in mobile positioning accuracy. While tropospheric delay is among the largest error sources in precise geodetic positioning applications (Bar-Sever et al. 1998), other considerable error sources, such as multipath propagation and ionospheric refraction, also affect mobile positioning. Moreover, atmospheric and oceanic loading and tidal displacements of the solid Earth are phenomena that need to be taken into account in the precise applications of GNSS positioning. In order to benefit from the tropospheric delay corrections that are derived from NWP model output, it is essential to take control over the other error sources as well. At geodetic GNSS receiver stations these error sources are controlled by dual frequency receivers, measurement sessions of extended length and sophisticated geophysical modelling as a part of the geodetic data processing. One could perhaps consider implementing a parameterization for ionospheric delay correction in a similar manner to the one proposed for tropospheric delay correction in this article. However, this would require existence of a relatively accurate numerical ionospheric forecasting model. 6. Conclusions This article introduces several parameterization schemes that can be applied for tropospheric delay correction for mobile GNSS positioning. The parameters for these parameterization schemes are obtained as a result of a least-squares fit to tropospheric delay corrections that are derived from a NWP model output. Determination of the parameters is demonstrated with the HIRLAM limited area NWP model, and the accuracies of the parameterization schemes are intercompared in a case study of a cold front passage. The following conclusions are drawn: 1. The determination of the mapping function coefficients (coefficients a, b and c of Equation (10)) from NWP model output provides a remarkable improvement in the parameterization accuracy, as compared with the use of prescribed mapping function coefficients. 2. Use of a gradient model in order to account for the azimuthal asymmetry of tropospheric delay correction clearly improves the parameterization accuracy. 3. Separation of mapping function coefficients into hydrostatic and wet components provides a further improvement in the parameterization accuracy as compared with the use of mapping function coefficients for total tropospheric delay correction only. It is summarized that any increase in the number of parameters reflects in an inevitable increase in the parameterization accuracy. The parameters used by the most complete parameterization scheme discussed in this article (SFG) include the zenith hydrostatic and wet delays, mapping function coefficients a, b and c separatelyfor the hydrostatic and wet delay components, and the zonal and meridional components of the tropospheric delay gradient vector. In the case study of a cold front passage of 27 May 2005 at Metsähovi, this parameterization scheme is found to be responsible of 18.7 mm for positioning error. Acknowledgements The funding from the EU FP5 project Targeting Optimal Use of GPS Humidity Measurements in Meteorology (TOUGH) in and from the TEKES project Geophysically Assisted Satellite Positioning in is thankfully acknowledged. TOUGH is a shared-cost project (contract EVG1-CT ) cofunded by the Research DG of the European Commission within the RTD activities of the Environment and Sustainable Development sub-programme (5th Framework Programme). TEKES is the Finnish Funding Agency for Technology and Innovation. References Atkinson BW, Zhu M Coastal effects on radar propagation in atmospheric ducting conditions. Meteorological Applications 13: Bar-Sever YE, Kroger PM, Borjesson JA Estimating horizontal gradients of tropospheric path delay with a single GPS receiver. Journal of Geophysical Research 103: Bech J, Gjertsen U, Haase G Modelling weather radar beam propagation and topographical blockage at northern high latitudes. Quarterly Journal of the Royal Meteorological Society 133: Bevis M, Businger S, Chiswell S, Herring TA, Anthes RA, Rocken C, Ware RH GPS meteorology: Mapping zenith wet delays onto precipitable water. Journal of Applied Meteorology 33:

8 454 R. ERESMAA ET AL. Bevis M, Businger S, Herring T, Rocken C, Anthes R, Ware R GPS meteorology: Remote sensing of atmospheric water vapor using the global positioning system. Journal of Geophysical Research 97: Boehm J, Werl B, Schuh H Troposphere mapping functions for GPS and very long baseline interferometry from European Centre for Medium-Range Weather Forecasts operational analysis data. Journal of Geophysical Research 111: B02406, DOI: /2005JB Elgered G, Plag H-P, van der Marel H, Barlag S, Nash J (eds.) COST Action Exploitation of Ground-Based GPS for Operational NumericalWeather Prediction and Climate Applications, Report No. EUR European Union: 234. Eresmaa R, Järvinen H An observation operator for groundbased GPS slant delays. Tellus A 58: Gustafsson N, Berre L, Hörnquist S, Huang X-Y, Lindskog M, Navascués B, Mogensen KS, Thorsteinsson S Threedimensional variational data assimilation for a limited area model. Part I: General formulation and the background error constraint. Tellus A 53: Järvinen H, Eresmaa R, Vedel H, Salonen K, Niemelä S, de Vries J A variational data assimilation system for ground-based GPS slant delays. Quarterly Journal of the Royal Meteorological Society 133: Lindskog M, Gustafsson N, Navascués B, Mogensen KS, Huang X-Y, Yang X, Andræ U, Berre L, Thorsteinsson S, Rantakokko J Three-dimensional variational data assimilation for a limited area model. Part II: Observation handling and assimilation experiments. Tellus A 53: MacMillan DS Atmospheric gradients from very long baseline interferometry observations. Geophysical Research Letters 22: Niell A Global mapping functions for the atmosphere delay at radio wavelengths. Journal of Geophysical Research 101: Nordman M, Eresmaa R, Poutanen M, Järvinen H, Koivula H, Luntama J-P Using numerical weather prediction model derived tropospheric slant delays in GPS processing: a case study. Geophysica 43: Pramualsakdikul S, Haas R, Elgered G, Scherneck HG Sensing of diurnal and semi-diurnal variability in the water vapour content in the tropics using GPS measurements. Meteorological Applications 14: Undén P, Rontu L, Järvinen H, Lynch P, Calvo J, Cats G, Cuxart J, Gerola K, Fortelius C, Garcia-Moya JA, Jones C, Lenderlink G, McDonald A, McGrath R, Navascues B, Woetman Nielsen N, Ødegaard V, Rodriguez E, Rummukainen M, Rõõm R, Sattler K, Hansen Sass B, Savijärvi M, Wichers Schreur B, Sigg R, The H, Tijm A HIRLAM-5 Scientific Documentation. Hirlam-5 Project, c/o Per Undén, SMHI, SE-60176, Norrköping,. Available online at publications/scidoc Dec2002.pdf.