GPS Solut (2012) 16:95–104 DOI 10.1007/s10291-011-0214-z
ORIGINAL ARTICLE
The corridor correction concept Torben Schu¨ler • Stefan Junker • Yaw Poku-Gyamfi
Received: 21 December 2009 / Accepted: 3 February 2011 / Published online: 23 February 2011 Ó Springer-Verlag 2011
Abstract Atmospheric delays are contributors to the GNSS error budget in precise GNSS positioning that can reduce positioning accuracy considerably if not compensated appropriately. Both ionospheric and tropospheric delay corrections can be determined with help of reference stations in active GNSS networks. One approach to interpolate these error terms to the user’s location that is employed in Germany’s SAPOS network is the determination of area correction parameters (ACP, German: ‘‘Fla¨chenkorrekturparameter—FKP’’). A 2D interpolation scheme using data from at least 3 reference stations surrounding the rover is employed. A modification of this method was developed which only makes use of as few as 2 reference stations and provides 1D linear correction parameters along a ‘‘corridor’’ in which the user’s rover is moving. We present the results of a feasibility study portraying results from use of corridor correction parameters for precise RTK-like positioning. The differences to the reference coordinates (3D) attained in average for 1 h of data employing selected network nodes in Germany are between 0.8 and 2.0 cm, which compares well with the traditional area correction method that yields an error of 0.7 up to 1.1 cm.
T. Schu¨ler (&) S. Junker Department LRT9.2 - Space Geodesy, University of the Federal Armed Forces Munich, 85577 Neubiberg, Germany e-mail:
[email protected] S. Junker e-mail:
[email protected] Y. Poku-Gyamfi Council for Scientific and Industrial Research (CSIR), Accra, Ghana
Keywords GPS precise positioning RTK positioning Area correction parameters Active GNSS network corrections
Introduction Since the concept introduced is a derivation of existing approaches to real-time (RTK) positioning, we will first give a brief presentation of the evolution of the idea of corridor corrections, which can be considered a compromise between the simple single reference station DGNSS and the more sophisticated area correction approach (ACP, ‘‘Fla¨chenkorrekturparameter’’). The outline is followed by some motivation considerations outlining situations where the use of corridor corrections could be beneficial. Corridor corrections versus established concepts Three main concepts of RTK positioning are portrayed in Fig. 1: Single-station DGNSS is the traditional approach, which can be realized easily. The concept consists of a single reference station placed at a known point and a roving receiver for precise GNSS positioning. Influences like satellite orbit errors as well as ionospheric and tropospheric signal refraction and propagation delays can be minimized by differential GNSS. As Hu et al. (2003) and Wielgosz et al. (2005) state, one important disadvantage of the single-station DGNSS is that the maximum distance between the reference station and the rover receiver should not exceed 10–15 km, because the influences mentioned above are distance-dependent error source that may compromise a reliable ambiguity resolution with increasing distance from the reference site. Hence, accuracies at centimeter level cannot be achieved.
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Single Station DGNSS
RS User (“Rover”) RS: Reference Station
Area Correction Approach (“Flaechenkorrekturparameter”)
RS
ACP (FKP)
User
RS
RS
Corridor Correction Approach
RS User
RS
CCP Fig. 1 Illustration of several concepts employed in precise relative positioning
The second approach enables service providers to extent this critical distance considerably: According to an introduction to network RTK published by IAG Working Group 4.5.1 (http://www.wasoft.de/e/iagwg451/intro/introduction. html), distances up to approximately 50 km can be bridged. Ionospheric and tropospheric corrections, and if needed also orbit corrections, are derived from a network of reference stations surrounding the rover and are interpolated to its position. Schrock (2007) gives a basic overview of reference networks, and Dai et al. (2003a) covers the determination of atmospheric biases for real-time ambiguity resolution. A classical correction approach depicted by Wu¨bbena (1997) is called FKP (Fla¨chenkorrekturparameter) or ACP (area correction parameters). It is routinely employed in the network services offered by SAPOS, a national German RTK network provider, consisting of more than 250 stations (http://www.sapos.de). Using data of the nearest 3 reference stations around the rover, double-difference satellite-specific atmospheric corrections are derived using planar interpolation functions. A basic introduction to this concept can also be
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found in the study by Wanninger (2000), and a number of methods and variations to interpolate such correction parameters are provided by Fotopoulos and Cannon (2001). Dai et al. (2003b) present the results of performance comparisons; also refer to Willgalis (2005) for further information. Another network approach uses these correction parameters to generate a so-called virtual reference station (VRS) close to the user’s position. The synthetic observations of the VRS are then used to determine the short baseline between the VRS and the roving receiver (Wanninger 1997). This enables the user to make use of short-baseline processing software. Finally, the Master-Auxiliary-Concept (MAC) offers a relatively high degree of flexibility from the data processing point of view as outlined by Brown et al. (2005): Complete observation corrections are referenced to one master reference station; the observations of the other reference stations (auxiliary stations) are expressed as correction differences to the master station to keep the data volume moderate. The main goal of the presented reference networks is an area-wide availability. Therefore, extensive networks over entire continents are envisaged. On the other hand, there are areas, where an area-wide network is uneconomical or simply yet to be established, like some parts of Africa and Australia, where only narrow coastlines are of interest. In this study, we investigate the feasibility of a simplified approach: We only make use of two rather than three reference stations. In this case, it is of course mandatory that the user site is located within a narrow corridor between both reference sites. Atmospheric corrections— so-called corridor correction parameters (CCPs)—are interpolated using linear functions. This approach reduces the minimum number of reference receivers necessary for service operations to two compared to three for the ACP approach, but it will still allow ambiguity fixing over relatively long distances in contrast to the single-station DGNSS method. This corridor correction approach has not been investigated yet to our knowledge but can be useful for certain special applications as outlined in the next paragraph. Motivation The corridor correction approach is certainly reserved to special applications, because the areas will always be limited to a narrow corridor between the two reference stations. As a rule of thumb, we could tolerate a maximum corridor width of about 10 km similarly to the single-station DGNSS approach discussed earlier; also see ‘‘Ghana, Africa: geodetic network’’—Fig. 8—for a discussion of the distance off corridor on the accuracy of the positioning results.
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Fig. 2 Potential locations at cross-over points of the ENVISAT radar altimeter that could be occupied when using corridor correction parameters for atmospheric delay compensation
Fig. 3 A corridor line through points ‘PUMP’ and ‘PPRM’ and several more stations of Ghana’s renewed first-order national network that would be able to serve major parts of its coast in terms of precise correction parameters for RTK-like positioning
However, the corridor approach makes sense in a number of cases as portrayed in Figs. 2 and 3: A number of scientific projects are carried out in regions with insufficient existing network infrastructure. This may require the manual installation and operation of project-specific reference stations. The fewer stations are needed, the less the project costs. Figure 2 shows the ground tracks of the ENVISAT radar altimeter. GPS-equipped buoys were deployed at a well-suited cross-over point close to Minorca Island (small dot); see also Schu¨ler and Hein (2007). A reference station was established very close to the coast with baselines to the buoys being as short as 8–9 km, and data processing was essentially carried out in the singlestation DGNSS mode. Alternatively, additional locations of research buoys could have been realized (large dots) at moderate water depths to make use of conventional mooring systems, but they would have been too far away for reliable single-station DGPS operation. The corridor correction approach could have been a solution with one reference station being located near the coast of Spain and an additional one on the island of Ibiza/Mallorca.
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Fig. 4 Three-dimensional scheme for the derivation of a ‘‘corridor correction’’
Figure 3 gives an example for coastal DGNSS correction services in Ghana, Africa. A geodetic network renewal was recently accomplished in economically important parts of the country including most of its coastal areas. However, the network points were only occupied by GPS receivers in a campaign style mode. Permanent GNSS services are still developing. During this transitional phase, the concept of corridor corrections can be successfully applied to cover most parts of the coastal region with just two reference stations operating on request, e.g., ACRA and PPRM (35 km) or PUMP and ELMI (62 km). This example shows that—although only a narrow corridor can be covered—a correction service can cover most parts of the service area of interest, but reference station costs can be reduced by 1/3 compared to the area correction approach. ‘‘Ghana, Africa: geodetic network’’ illustrates positioning results obtained in that region of Ghana.
Solution strategy Corridor correction requires the establishment of two reference stations A and B in the project area as shown in Fig. 4. The line AB is the center of the corridor. Knowing the ionospheric and tropospheric delays at both stations, a constrained linear path between the two stations is assumed. This is usually not the real path as shown in Fig. 5, but we assume that the linear approximation is close enough to reality in order to enable precise carrier phase positioning. The probability for the deviation from the ideal assumed path is naturally increasing the longer the distance between the two stations becomes. This acts as a constraint to the length of the baseline AB. Although it is not possible to give exact numbers for a maximum baseline length, because atmospheric behavior depends on the location, season, and other parameters, the goal of the corridor correction approach is to show spatial
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Fig. 5 Diagram showing the impact of modeling atmospheric delays using corridor correction; Rd is the corridor correction computed for the user site
compensation characteristics similar to those commonly observed using area correction parameters in the German SAPOS network featuring baselines of 20–70 km. Let us now assume that the rover is located at position D. The atmospheric delay interpolated via AB will then be Rd (the letter ‘R’ is used because this quantity is normally a residual delay added to the value of an a` priori model). The interpolation process is illustrated in Fig. 5. Basically, the corridor correction concept is only valid in case the roving receiver is always located on the line AB—the vector defined by both reference stations. However, this goal cannot strictly be fulfilled in all cases. A user located off the corridor, e.g., at position C, must anticipate an increased error budget due to extrapolation: To generate CCP for an off-axis site, the correction interpolated for the projection on the line AB, i.e., D, will be extrapolated to a user located at C by keeping Rd constant, though some individual height reduction can be applied for the troposphere. The length of CD represents the horizontal projection of the distance off corridor. This permissible maximum length can be specified depending on the actual accuracy requirements, which can be guided by the severity of the ionosphere and troposphere at the time of observation. Our experience gained from the experiments conducted in this study, and presented in the following paragraphs, indicates that an off-corridor distance of about 10 km provides sufficient accuracy. Figure 5 demonstrates that interpolation using the linear corridor correction concept will hardly be error-free in an area with highly variable atmospheric delays. Corrections derived from single reference station RTK services are also depicted in this figure. In that case, however, the value of the atmospheric delay at station A is extrapolated to station
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D. Without any additional constraint as employed in the corridor correction method, the atmospheric delay computed does not improve for a rover at location D, but rather worsens due to the extrapolation error, when the baseline length between user and reference station increases. The introduction of a second reference station B provides the ‘true’ atmospheric condition at both sites, thereby providing the constrained expected path of the atmospheric delay between the reference stations. This linear trend, however, is not the real path, but there is an interpolation error that displaces the real delay from the constrained point. There is a greater probability of having an increased interpolation error for longer distances between the reference stations because the linear approximation is a simplification which works reasonably well for shorter inter-station distances but can be inadequate for longer ones. Depending on the different spatial characteristics of the error sources, larger inter-station spacing may require higher-order polynomials according to Wu¨bbena and Willgalis (2001). Insertion of more reference stations could improve interpolation accuracy but will increase the costs of equipment. Error model The in-house software is using double differences as observations, so for the carrier phase ui (in units of cycles) we have the double-difference phase observation rDui ¼
rDq rDIon rDTrop þ rDNi ki ki ki
ð1Þ
where the index i represents two different frequencies, L1 and L2 in case of GPS, q the geometric range, Ion the
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ionospheric propagation delay, Trop the tropospheric propagation delay, N the ambiguities, and k the corresponding wavelength. D denotes an inter-station and r an inter-satellite difference (double-difference operator). A dual-frequency receiver allows determining the atmospheric delay quantities. We limit the description to the ionospheric delays; the tropospheric delay corrections can be computed analogously. The dispersive part, i.e., the ionosphere is referenced to frequency f1 (1,575.42 MHz) and is derived and explained in detail by Schu¨ler (2008): a2
rDIonL1 ½m ¼ a1
rDu1 rDu2 þ aa21 rDN1 rDN2 a2
f2
ka11 þ k12 f12
ð2Þ
2
with a1 and a2 being the linear combination factors (77 and 60 for GPS L1/L2). The variance (in metric units) of these delays is computed following the law of error propagation: r2 þ r2 r2rDIonL1 ¼ U1 2 U22 f 1 f12
ð3Þ
2
where r denotes the standard deviation. Note that rU1 ¼ ru1 k1 and rU2 ¼ ru2 k2 and the covariance rU1 U2 ¼ 0, i.e., it is assumed that no significant correlation between the data on the first and second frequency is present. Reducing the classical approach of a planar approximation as depicted by Wanninger (2000), which is employed to derive area correction parameters, to a corridor between stations A and B with the distance s, we have R ¼ a0 þ ass s
ð4Þ
where R is the atmospheric correction quantity. Using double differences, the parameter a0 is set to zero, so we only have to compute the corridor correction parameter ass (gradient): ass ¼
rDIONL1AB sAB ) Rd ¼ RAD ¼ rDIONL1AD ¼ ass sAD
ð5Þ
which is given here for the ionospheric delay on L1.
Test cases The corridor correction parameter (CCP) approach has been compared with existing approaches like area correction parameters (ACP) in several environments. Test cases in the German SAPOS network and in Ghana, Africa, are summarized. Poku-Gyamfi (2009) provides additional information and details on the test results.
Fig. 6 Network sketch showing parts of the German SAPOS network with a corridor through reference stations WORB-SONN as well as MUEH-ILME and network triangle ERFU-MUEH-MEIN for derivation of area correction parameters (for purposes of comparison with the corridor correction approach)
Germany, Europe: SAPOS network The German SAPOS network is a relatively dense network for RTK services and therefore well suited for testing purposes using both stations nearby and further away from the site of interest. The sub-network of Thuringia was selected as test case. Figure 6 shows the main station GOTH acting as user site (‘‘rover’’) in these experiments. The long-range corridor is composed of reference stations WORB and SONN, abbreviated as GWS in some of the diagrams below, with a separation distance of about 130 km. In addition, a short-range corridor using stations MUEH and ILME, abbreviated as GMI, with an inter-station distance of slightly more than 60 km was selected. The distance off corridor is rather small in both cases so that extrapolation errors should be small. Furthermore, a network triangle (dotted lines) consisting of the stations ERFU, MUEH, and MEIN is used to derive area correction parameters, so that positioning results with that type of corrections could be directly compared with those from corridor corrections. The connecting baselines are between 47 and 69 km, similar to the MUEH-ILME baseline. We chose similar baseline lengths for the corridor and area correction calculations to facilitate comparison of the methods.
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Processing notes Data of 3 days in 2005 were analyzed (DoY—Day of Year—254, 255, and 256, i.e., September 11–13, 2005) with two windows per day starting at 00:00 a.m. and 03:20 a.m. GPS time. Each data batch covers 3,600 s; the sampling frequency of the data is 1 Hz. Though these data were collected during the solar minimum, DoY 254 and 255 featured some ionospheric disturbances: The I95 indices— please consult Wanninger (2004) for a description—were equal or higher than 4, but always lower than 8 on September 11. An I95 index smaller than 2 characterizes times without ionospheric disturbances that could be harmful to RTK positioning in an active GNSS network like SAPOS, whereas values between 2 and 4 designate ‘‘small,’’ indices between 4 and 8 indicate ‘‘strong,’’ and I95 values higher than 8 characterize ‘‘very strong’’ ionospheric disturbances. The ionospheric situation became quieter on September 12, but I95 still exceeded 4 during 29% of that day. It was below a value of 2 during almost the entire DoY 255 (see http://www.sapos.thueringen.de). Further details can be found in Schu¨ler (2008, p. 94ff and 167ff). Note that summarized results are presented in this paper rather than individual results for each data window. All position estimates were compared to a rigorous network solution derived in post-processing using all SAPOS stations of the region of Thuringia, which serve as reference coordinates. All positioning experiments were performed using the following three GPS signal selections or data combinations as well as ambiguity resolution strategies: 1.
2.
3.
Data analysis was carried out using L1 single-frequency data including an attempt to fix ambiguities, which is denoted as L1 (FX) in the diagrams. Alternatively, no ambiguity fixing was attempted, i.e., float ambiguities—L1 (FL)—were used for final positioning. Additionally, the ionosphere-free combination (LIF) was used which is formed as a linear combination of the L1 and L2 carrier-phase measurements such that the first-order ionospheric delay error cancels out (Leick 2004).
The reason for focusing on the use of L1 data and the corridor corrections is related to the fact that this concept is proposed to be used in Africa where continuously operating station often do not exist. The corridor correction concept, which requires fewer temporarily operating reference stations, is economically attractive. In addition, many users in Africa prefer single-frequency receivers because of their low costs compared with dual-frequency devices. Therefore, L1-based positioning is of major interest in such a situation.
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In addition, we present dual-frequency solutions, because these solutions are free of the first-order ionospheric effect. Although the standard deviation of doubledifference observations is approximately 3 times higher than those for undifferenced measurements, systematic ionospheric errors can be effectively mitigated. However, these residual systematic ionospheric errors will always remain in the area or corridor corrections. When these remainders are a significant burden on the positioning accuracy, the LIF results should show a better performance compared with L1 despite of the higher random noise level of the LIF combination. If this is not the case, then the ionospheric delay compensation with help of correction parameters is considered to be sufficiently precise, i.e., systematic residual errors seem to play a minor role or, at least, do not affect the position solution at a level higher than that of the random noise of the ionosphere-free combination. Results Two main parameters will have a major impact on the positioning accuracy with help of corridor corrections: Undoubtedly, the baseline length between the two reference stations in use for generation of the corrections is a limiting factor in terms of accuracy. The longer the baseline becomes, the less precise the corridor corrections will be. Secondly, the distance of the user site off the corridor— the center line between the two reference stations—is an important factor as it requires an extrapolation of the corrections. Results portraying the impact of these two parameters will be highlighted in this section. In addition, a comparison of results obtained using area correction parameters versus corridor correction parameters will be presented. Baseline length The standard deviation of the position is expected to be a function of the length of the corridor, i.e., of the distance between the two reference stations. Interpolation errors will increase the longer the distance becomes, because both the troposphere and the ionosphere can exhibit non-linear features, which cannot be modeled with the linear approach we are following here. For the moment, let us only have a look at the bars referring to the solutions ‘‘CCP (GMI)’’and ‘‘CCP (GWS)’’ of Fig. 7, which depicts the results for user-site GOTH using the short 62-km corridor MUEH-ILME (GMI) as well as the long 130-km corridor WORB-SONN (GWS). The 3D positional bias, i.e., the average unsigned difference between reference and computed coordinates, is between 0.8 and 2.0 cm in case of the GMI network using the L1 ambiguity-fixed (FX) data and between 1.1 and 3.0 cm for the long-distance GWS network. So, there is a
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Fig. 7 Comparison of corridor correction parameter (CCP) performance against area correction parameters (ACP/FKP). The short distance corridor is MUEH-ILME (GMI, 62 km), and the long-range one is WORB-SONN (GWS, 131 km)
maximum loss of precision of 1 cm or less, although the distance between both stations approximately doubles. The deterioration is clearly and expectedly stronger for the results obtained without fixing the ambiguities, see diagram referring to L1 (FL). From experience, we know that the minimum data batch required for a float solution is at least 2 h to compete with an ambiguity-fixed solution because the ambiguity parameter convergence is slow due
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to the relatively slow apparent motion of the satellites in the sky. We normally recommend 4 h of data; otherwise, this convergence error can be the main reason why a float solution will be less precise than the ambiguity-fixed one. In these experiments, a data batch of just 1 h each was intentionally chosen to emphasize the advantages of ambiguity fixing. The L1 float solution for the GMI network shows a difference to the reference solution between 1.8 and 2.4 cm, which is partially higher than the L1 fixed solution for the long-distance GWS network, and it grows to 2.1–4.5 cm for the GWS configuration. A second motivation to present float solutions is related to the fact the floating ambiguities could absorb systematic errors to a certain extent, e.g., ionospheric delay errors that are said to be more pronounced during days 254 and 255 as explained in Processing notes. However, we cannot state any positive effect of this kind for these positioning experiments, i.e., the convergence error seems to be the dominant factor. The results obtained using the ionosphere-free linear combination (LIF) are completely free of the first-order ionospheric delay. The positional biases are between 1.7 and 3.0 cm for the GMI setup and increase to 2.0–4.6 cm for the GWS configuration. This is a similar level or slightly higher than the L1 float solution, but clearly less precise than the L1 ambiguity-fixed solution. Our first interpretation is that ionospheric delay errors could be sufficiently compensated via the corridor correction approach so that systematic remainders are of minor concern here. This is an interesting statement, because we had expected a poorer L1 (FX) performance for DoY 254 and 255 due to the higher ionospheric activity. As a matter of fact, we used a rather long smoothing time of 1 h, which is relatively long compared to typical RTK operations, and apparently increased the precision of the corridor corrections to a comfortable level. The ionosphere-free solution slightly outperforms the L1 float solution on DoY 255. It is almost 2 cm more precise for the GWS setup, which could be attributed to a better compensation of ionospheric effects, but more positioning experiments would be required to underline this claim. Distance off corridor GOTH lies only 1.3 km off the center line. This user site was replaced by ERFU in a second experiment in order to demonstrate the effect of extrapolation in case of corridor corrections. ERFU is about 22 km off the corridor. Figure 8 shows a drastic decrease in positional bias from approximately 2–8 cm in case of the ambiguity-fixed single-frequency L1 solution. The float solution is marginally better. Apparently, the floating ambiguity parameters absorbed smaller parts of the extrapolation errors in tropospheric and ionospheric delays. A look on the ionosphere-free LIF resolutions illustrates
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Fig. 9 Network plot of the corridor ACRA-ASFO and user-site AKRO employed for positioning experiments in Ghana, Africa Fig. 8 Impact of the ‘‘distance off corridor’’ on positioning accuracy
that tropospheric delays play only a marginal role here: The LIF-derived position results almost show no difference for both GOTH and ERFU, because the ionospheric delay does not play a significant role any longer. Note that these ‘‘off corridor experiments’’ were only carried out for the first data batch of each of the 3 days, so the values for user-site GOTH may differ compared to Fig. 7. Area correction versus corridor correction parameters Finally, the reader might ask whether positions derived using corridor correction parameters (CCP) perform similarly to those derived using traditional area correction parameters (ACP). Figure 7 gives some light on this question. Area corrections were derived from the network ERFU-MUEH-MEIN. The differences to the reference coordinates are clearly smallest compared to the CCP-based results for both the short (GMI) and the long baseline results (GWS). Positional bias of between 0.7 and 1.1 cm could be obtained in the L1 ambiguity-fixed setup, whereas it is between 0.8 and 2.0 cm in the CCP case (GMI). We may conclude from these results that area corrections perform slightly better than corridor corrections. Although we do not want to doubt that this conclusion is correct, we would like to stress that the network configuration is somewhat fortunate for the ACP approach compared with the CCP setup: The baseline MEIN-ERFU is crossing the corridor MUEH-ILME at a location much closer to user-site GOTH. This is a clear advantage over the CCP setup, so that it is not unexpected that the ACPbased results outperform the CCP-derived coordinates even in the short baseline case. However, from our experience, this is the typical situation we encounter in such active networks that have an approximately homogenous distribution of reference stations.
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Ghana, Africa: geodetic network In addition to the German SAPOS test case, some positioning experiments were carried out using data from the Ghana Geodetic Network Renewal campaign performed during May 18–24, 2007, in order to establish a modernized network in the ‘‘Golden Triangle’’ of Ghana, i.e., that region in the country which is of highest economical importance. The sites shown in Fig. 9 are not continuously operating reference stations but were only temporarily occupied for a couple of hours using several sets of GPS receivers. For this reason, the selection of a suitable corridor with a user site not too far off the corridor was more difficult than in case of the German SAPOS network, in particular considering the constraint that three surrounding reference station shall be available in order to compare corridor with area corrections. The selected corridor constitutes a baseline larger than 120 km with the user site being located almost 24 km off the corridor, see Fig. 9. The average inter-station distance to derive the area correction parameters for AKRO is 94 km. Once again, the network solution separately processed for the Golden Triangle serves as reference solution. Although 2007 was a year of calm ionospheric behavior, the typical ionospheric propagation delay in this region close to the geomagnetic equator is approximately 2.5 times higher than what is anticipated in the SAPOS region (5.5 vs. 2 m). As shown in Fig. 10, two more correction approaches were experimented with in the Ghana case. Corrections for both the dispersive and non-dispersive propagation delays were extracted from the correction parameters in the ‘‘ACP’’ and ‘‘CCP’’ runs, just as it was done in the SAPOS experiments. In addition, a processing run employing independent corrections was carried out and named ‘‘NWM ? Klob.’’ in Fig. 10: The tropospheric corrections were integrated using refractivity profiles from 3D global numerical weather fields with a horizontal resolution of
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Fig. 10 Positioning results obtained for user-site AKRO following several types of correction data
1° 9 1° and a temporal resolution of 3 h with a re-analysis being carried out each 6 h and a 3-h prediction applied in between. The weather model in use was NOAA NCEP GDAS FNL (T170L42) with zenith total delays showing an RMS of approximately 2 cm in the tropical regions according to Schu¨ler et al. (2000). The ionospheric delay was compensated with help of the well-known ‘‘Klobuchar model’’; however, note that the model coefficients were obtained by a least-squares fit to IONEX (IONosphere EXchange) maps produced by the CODE computing center at the University of Berne, Switzerland, i.e., the precision can be expected to be higher than in case of the real-time broadcast coefficients. Finally, processing run ‘‘NWM ? CCP’’ employs ionospheric delays from the corridor correction parameters, but the tropospheric corrections are replaced by the weather mode–derived delays. Once again, coordinate differences are smallest for the results obtained with help of area correction parameters: Horizontal biases are slightly higher than 1 cm, whereas the corresponding results for the CCP approach are 1 cm higher. The 3D positional biases, which are not illustrated here, are significantly higher since AKRO is already rather far off the corridor’s center line and approach 6 cm for the CCP method. The ‘‘NWM ? Klob’’ run is slightly less precise in this case which is expected, because the Klobuchar model has strong limitations even if tuned with IONEX data. The horizontal biases significantly improve to approximately 1.3 cm if we replace the Klobuchar-style ionosphere corrections by the dispersive corridor corrections while maintaining the NWM-derived tropospheric delays. This is an interesting result which indicates that tropospheric corrections derived from numerical weather fields, i.e., from a source independent from any GPS network data, are quite valuable to improve positioning
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Fig. 11 Increase in ionospheric delay RMS with increasing offcorridor distance for stations within the Golden Triangle of Ghana between May 18–24, 2007 (grey bar: linear trend function)
performance. In this particular case, tropospheric corrections from the CCP approach are apparently a limiting factor as the CCP-only results are biased at a level higher than 2 cm. However, it should be stressed that it is not possible to deduce a general rule or recommendation from these few positioning experiments. This would require more measurements. Figure 11 shows a plot of the RMS of the ionospheric delay against the off-corridor distance for a number of corridor and user-site configurations in the Golden Triangle area, observed at various locations between May 18–24, 2007. The corridor corrections were generated for each of the setups and, afterward, compared with the double-difference ionospheric delays directly obtained from the primary reference station to the user site. The final result is a RMS value for each of these setups. The trend line shows a positive gradient of 0.66 ppm. Though the scatter of the individual baseline checks is high, the linear trend that indicates the increase in the error with increasing off-corridor distance can be clearly seen in this plot. A residual delay of 23 mm was obtained on the corridor, which is the difference between the actual delay and the modeled one. Similarly, the RMS of the tropospheric delays against the off-corridor distance was analyzed but is not plotted here. The tropospheric delay error will become larger with increasing distance, but the gradient is only 0.13 ppm, with a residual error on the corridor’s center line of 19 mm.
Conclusions The corridor correction concept presented is a simplification of the traditional area correction approach (‘‘Fla¨chenkorrekturparameter’’), which is well established
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in active GNSS networks like the German SAPOS net. However, no area elements are needed, but only two reference stations with the user site located close to the corridor’s center line. This method was developed as a special solution in regions where only temporarily operating reference stations are available. The results presented indicate that positions derived using corridor correction parameters (CCP) compare to the reference coordinates similarly as those derived using area correction parameters (ACP), though the ACP results are on average about 0.5–1.0 cm more precise for the experiments presented here. The off-corridor distance seems to be a critical configuration parameter: If the user position deviates from the corridor’s center line, defined by the two reference stations, extrapolation errors in the corridor corrections will increase. In the test cases presented, the increase in ionospheric error was particularly noticeable, so that the standard deviation of the 3D position deteriorated by a factor of four with respect to the L1-only position solution, whereas the ionosphere-free solution was not substantially affected. Future work should include a more detailed investigation into the corridor correction concept in parts of Africa and other sites under atmospheric conditions different from those encountered in Germany. A more comprehensive statistical analysis of the main impact parameters that limit CCP accuracy, i.e., the distance between the two reference stations and the off-corridor distance, should also be carried out. Acknowledgments The authors would like to thank Ch. Trautvetter from the Office of Surveying and Geo-Information of the German Federal State of Thuringia for providing data of the SAPOS subnetwork for this study.
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