Radio Science

A correction model of dispersive troposphere delays for the ACES microwave link

Authors


Corresponding author: T. Hobiger, Applied Electromagnetic Research Institute, National Institute of Information and Communications Technology (NICT), Tokyo, Japan. (hobiger@nict.go.jp)

Abstract

[1] The Atomic Clock Ensemble in Space (ACES) will be a future ESA experiment which utilizes ultra-stable clocks on-board the International Space Station (ISS). This mission is expected to perform tests of fundamental physics (relativity, possible drift of fundamental constants with time) and at the same time allows to compare the ACES time reference with respect to ground stations by using a novel microwave link concept. However, uncorrected dispersive troposphere delays pose the risk of degrading the performance of this microwave link over longer integration periods. Thus, a semi-empirical correction model has been developed which is only based on input from meteorologic sensors at the ground stations. The proposed model has been tested with simulated ISS overflights at different potential ACES ground station sites, and it could be demonstrated that this model is capable to remove biases and elevation dependent features caused by the dispersive troposphere delay difference between the uplink and downlink. The model performs well at all sites by reducing the impact on all reasonable averaging time scales by at least 1 order of magnitude. Similar studies like this might be of importance for other time and frequency transfer instruments or future space geodetic instruments.

1 Introduction

[2] Two-way satellite time and frequency transfer (TWSTFT) is a technique commonly used to compare frequency standards which are separated over very long distances. Thereby, one takes advantage of that almost all of the propagation effects cancel after differencing the travel times in both directions. Due to the fact that uplink and downlink frequencies are slightly different, one usually accounts for dispersive delay effects related to the propagation of the electromagnetic waves through the ionosphere [Hargreaves, 1992]. It is assumed that troposphere delays are non-dispersive and thus completely cancel out. This assumption works well for the currently operational TWSTFT system performance but might need to be revised in the case of next-generation instruments or missions like the Atomic Clock Ensemble in Space (ACES) which targets at much higher measurement precision.

1.1 The ACES Microwave Link

[3] ACES is an ESA mission in which an ensemble of atomic clocks will be installed on the International Space Station (ISS) [Laurent et al. 2008]. Those clocks will be compared with clocks distributed on ground via a dedicated microwave link as the primary tool [Hess et al. 2011]. Such clock comparisons are an essential tool to perform fundamental tests, e.g., of Einstein's gravitational frequency shift, or the search for a variation of fundamental physical constants [Cacciapuoti and Salomon, 2009]. The ACES microwave link comprises a flight segment on the ISS and a number of distributed ground terminals around the globe at sites where high-performance clocks are available. Up to four ground terminals can be connected simultaneously to the flight segment for comparisons of the space clocks with ground clocks via a dedicated two-way link using frequencies in the Ku-band (f1 = 13.475 GHz for uplink, f2 = 14.703 GHz for downlink). The Ku-band carrier frequencies are phase modulated by a pseudorandom noise (PRN) code of 100 Megachips per second (Mcps) with a corresponding bandwidth of 200 MHz. A second downlink in the S-band (f0 = 2.248 GHz) is used to determine the total electron content (TEC) and hence the ionospheric delay. Its PRN code has a lower chip rate (2.5 Mcps, 5 MHz bandwidth). The microwave link is designed to enable clock phase difference measurements with an instability of only 0.3 ps in less than 100 s integration time. Thus, it is of special interest to check whether limiting factors exist which could deteriorate the envisaged performance of the ACES microwave link. The refractivity of the troposphere, even at Ku-band frequencies, has been identified as a possible cause, and the impact was estimated for operational two-way satellite time and frequency transfer via geostationary satellites [Jesperson, 1989; Piester et al., 2008]. The effect of the dispersive troposphere is rather small and negligible for operational TWSTFT but could be of importance for next-generation techniques such as TWSTFT carrier phase or the ACES microwave link.

1.2 Dispersive Troposphere Delays

[4] Microwave techniques work under the assumption that only dispersive, i.e., frequency-dependent delay contribution is caused by the ionosphere. In general, the refractivity [see Liebe 1989] for the definition), even for the troposphere, is a complex quantity which can be denoted as

display math(1)

where N0 is a frequency-independent term, and N ′ ( f ) and N ′ ′ ( f ) represent the complex frequency dependence. The imaginary part can be used to derive the loss of energy (absorption), and the real part can be assigned to the changes in the propagation velocity (refraction) and thus describes the delay of an electromagnetic wave which propagates through that medium. Liebe [1989] as well as Liebe et al. [1993] describes in detail how to derive the constituents listed in equation (1) based on laboratory measurements of the most important absorption lines. By the use of this model, it is possible to compute the complex refractivity based on atmospheric quantities like pressure, temperature, and relative humidity. Although the frequency-dependent terms appear to be of small size (e.g., N ′ ( f ) ∼ 0.2 ps/km for surface conditions), one has to consider that signals are propagating through several kilometers of troposphere and even longer at low elevations. The model described by Liebe et al. [1993] has been adopted in the ITU-R P.676-8 [ 2009] for which only the dry air module and the water vapor module were taken from Liebe et al. [1993] and the other, much smaller effects (water droplets and ice crystals), were not considered for the recommendations. Thus, for the following section, we refer to the coefficients from the ITU-R recommendation [ITU-R P.676-8, 2009] for the calculation of the frequency-dependent complex refractivity values.

2 Dispersive Troposphere Delays on the ACES Microwave Link

[5] In order to evaluate how dispersive troposphere delays can affect the performance of the ACES microwave link, simulations with numerical weather model data were carried out. Thereby, two effects are considered: first, the direct effect, i.e., the troposphere delay difference between the Ku-band uplink and downlink paths; and second, an indirect effect caused by the absorption of dispersive troposphere delays into the ionosphere correction based on the S-band and Ku-band downlink signals. In order to model both effects properly, we start with the uplink (f1) and downlink (f2) timing equations:

display math(2)
display math(3)

where TIC(S) and TIC(G) are the time interval counter (TIC) readings of the space-based (S) and ground clock (G), τ denotes all non-dispersive delays (in seconds), and τ(f) represents a dispersive troposphere delays at a given frequency f (Hz). Path asymmetry, relativistic, or other effects are thought to be of much smaller order and thus are not relevant for this study. First-order ionosphere corrections are considered in the last term of equations (2) and (3), respectively, where the quantity B ( [e −  / m2]) represents the slant total electron content (STEC) measured in total electron content units (TECU) [Schaer, 1999] and the constant

display math(4)

relates to units of time. The time difference between clocks S and G can be obtained from differencing equations (2) and (3), yielding

display math(5)

[6] If uplink and downlink signals are using the same frequency, the last two terms will cancel out and one can directly obtain the time comparison result from the TIC readings. Since uplink and downlink frequencies are not identical, one needs to compensate the direct and indirect effects caused by this feature. The dispersive troposphere correction Δτ( f1,f2) needs to be estimated and compensated by a semi-empirical model which is derived in the following sections. The ionosphere and thus the total electron content B is thought to be known by deriving its quantity via the S-band downlink signal at frequency f0. However, one needs to take care of an indirect effect of the dispersive troposphere delay which is absorbed into the quantity B. Using the last term of expression (5), one can derive a timing error of 0.6 ps/TECU between the Ku-band uplink/downlink frequencies (see section 2.1.2).

[7] Ignoring higher-order ionospheric effects, one can write the equations for the two downlink frequencies:

display math(6)
display math(7)

\where τi is the measured travel time at frequency fi and the other notations are identical to equations (2) and (3). At the time when the ionosphere is determined, no other dispersive effects are assumed, and thus, equations (6) and (7) change to

display math(8)
display math(9)

where B ′ is the slightly biased ionosphere contribution. Solving for B ′ yields

display math(10)

[8] If we substitute now the “correct” expressions (6) and (7) into the equations, we obtain

display math(11)

which describes how much the ionosphere estimation is biased by ignoring the dispersive troposphere delay between the S-band and Ku-band downlink. If we have an empirical model for the inter-frequency dispersive troposphere delay Δτ(f0,f2), we can compute the observed ionosphere by

display math(12)

and use this value in equation (5) to remove the ionosphere influence correctly. Thus, in the following, we need to derive a model for the inter-frequency delay between the Ku-band uplink and downlink frequencies as well as a model for the two downlink frequencies. In order to achieve this, numerical weather model data from the European Center for Medium-Range Weather Forecasts (ECMWF) have been utilized and ray-traced zenith delays at more than 3,000,000 cases at different spatial locations and epochs were computed for the ACES uplink and downlink frequencies. In total, 168 hourly epochs and vertical meteorological profiles within ± 7° around potential ACES ground stations were considered, whereas it was not distinguished if the grid point is located on land or the sea. Based on these ray tracings, models are derived which correct for the frequency dependence of the troposphere delay in the ACES microwave link. In order to apply corrections at arbitrary elevation angles, a basic mapping function is used to translate the model's zenith correction into the direction towards the ISS. The complete correction model is then tested with respect to simulated ISS overflights at five anticipated ACES ground stations for which uplink/downlink troposphere delays should be ray traced based on the 4-D information given in the numerical weather models.

2.1 Correction Model

[9] A correction model for dispersive troposphere delays on the ACES microwave link should satisfy three basic criteria:

  1. [10] It should predict the frequency-dependent troposphere delay for the nominal ACES uplink/downlink frequencies based on a minimal number of input parameters. Additionally, it is assumed that ACES ground stations are equipped with meteorologic sensors which provide surface temperature, pressure, and relative humidity (or an equivalent) at the site.

  2. [11] The model should be applicable to any ACES ground terminal independent of the location where it is hosted.

  3. [12] The model should compensate the dispersive troposphere delays so that the remaining uncorrected troposphere delays are well below the noise floor of the instrument at all averaging periods. However, given that one needs to model also the indirect effect introduced via the ionosphere correction, two separate models have to be setup. One for the direct effect, which corrects the dispersive delay for the Ku uplink/downlink path, and a second model which adds a correcting term to the derived ionosphere delay, expressed in units of total electron content (TEC).

2.1.1 Numerical Weather Model and Ray Tracing

[13] ECMWF analysis data between 1 and 7 August 2011 for five areas on the globe (around potential ACES ground terminal sites) were obtained. The weather model data with a temporal resolution of 6 h, a horizontal grid spacing of 0.2 × 0.2°, and 25 pressure levels between 1000 hPa and 1 hPa allow to derive the 4-D complex refractivity field which is necessary to compute the frequency-dependent propagation path as well as the troposphere delay along that path. Although the vertical resolution in the lower height domain might not resolve the water vapor distribution well, it is anticipated that the contained information is sufficient for the derivation of a dispersive troposphere delay correction model. Since height ranges between sea level and 3000 m as well as different environmental conditions are represented in the selected areas, one can expect to be able to derive a model which provides dispersive troposphere delay corrections that are valid in other regions as well. Figure 1 shows the locations of the five areas for which numerical weather model (NWM) data have been obtained. Since the refractivity fields are obtained solely from the pressure level data set, downward continuation is necessary in case that the location is below that model domain. The preparation 4-D complex refractivity field and all ray-tracing calculations have been carried out with the Kashima Ray-tracing Tools (KARAT) [Hobiger et al. 2008].

Figure 1.

Overview of the five areas around potential ACES ground station sites for which numerical weather model data from the ECMWF were obtained. The area covered with NWM data is highlighted by its topography. The maps are labeled by identifiers of IGS GPS receivers at the chosen sites.

2.1.2 Dispersive Troposphere Zenith Delays at Various Locations and Epochs

[14] In order to get a significant amount of data, frequency-dependent zenith troposphere delays were computed at all grid points (with a spacing of 0.2 × 0.2°) with time steps of 1 h between 1 August 2011 0 UT and 7 August 2011, 18 UT. Calculations were carried out for the ACES nominal uplink/downlink frequencies, i.e., 13.475 GHz and 14.703 GHz, as well as for the second downlink in the S-band (2.248 GHz). All ray tracings were done within the KARAT package [Hobiger et al. 2008] which has been modified to enable the consideration of a frequency-dependent refractivity as described in section 1.2. Zenith delays were computed between the Earth surface as deduced from shuttle radar topography mission (SRTM) data [Rabus et al. 2003] and an upper (ellipsoidal) height of 86 km which made it necessary to extend the NWM by the US Standard Atmosphere [U.S. Standard Atmosphere, 1976] as described by Hobiger et al. [2008]. In total, zenith troposphere delay differences between the uplink and downlink troposphere delays and the ionosphere impact on the two downlink frequencies have been computed at 3,801,812 profiles. In addition, meteorological parameters (temperature, dry air pressure, and water vapor pressure) at the ray-tracing initial point (i.e., the SRTM topography) were extracted from the ECMWF data set. Since almost all environmental conditions, including high mountain ranges, coastal sites, and highly humid areas were covered by these calculations, it is anticipated that a model which is derived from these data can be applied as correction at an ACES ground terminal at any given locations around the globe.

[15] Figures 2-4 show the ray-traced differential troposphere delays (in units of mm) between uplink and downlink Δτ(f1,f2) in zenith direction in dependence of dry air pressure, water vapor pressure, and temperature at the surface. Figures 5-7 depict how much the total electron content would be biased (i.e., the term B * ) by ignoring the dispersive troposphere delay influence on the S- and Ku-band downlink paths. The data points are color coded by their relative frequency of occurrence in order to identify atmospheric ground conditions which are more likely (red) than others (black). As for Figures 2 and 5, a somewhat linear relation between the direct/indirect dispersive troposphere delays and the dry pressure can be seen. However, a quite large scattering around that linear trend and the fact that scattering broadens in the domain between 950 and 1000 hPa, where most of the data comes from, do not permit to set up a correction model that relies on ground pressure input solely. Figures 3 and 6 reveal a stronger linear correspondence between surface water vapor pressure at the ground site and the magnitude of the ACES dispersive troposphere delays. Although there is some scattering in the order of about ± 10% of the magnitude, a linear model based only on water vapor pressure at the ground site should according to the ECMWF fields be capable to take out most of the dispersive troposphere delay that remains in the time difference between the uplink and downlink paths. As water vapor pressure cannot be directly measured at the site but needs to be calculated from relative humidity and temperature, a meteo sensor which logs these two quantities should be placed at an ACES ground terminal. The relation between surface temperature and the dispersive troposphere delays, as shown in Figures 4 and 7, does not reveal a clear correlation. Thus, temperature measurements alone appear to be not useful for correcting the ACES dispersive troposphere delays, and pressure and humidity information are required. In order to derive a simple correction model for such delays, it will be investigated which combination of the three input parameters gives a useful but simple formula for easy and precise correction of the measurements.

Figure 2.

Dispersive zenith troposphere delays Δτ(f1,f2) plotted with respect to the dry air pressure at the surface level (SRTM topography).

Figure 3.

Dispersive zenith troposphere delays Δτ(f1,f2) plotted with respect to the water vapor pressure at the surface level (SRTM topography).

Figure 4.

Dispersive zenith troposphere delays Δτ(f1,f2) plotted with respect to the temperature at the surface level (SRTM topography).

Figure 5.

Impact of dispersive zenith troposphere delays on the ionosphere correction plotted with respect to the dry air pressure at the surface level (SRTM topography).

Figure 6.

Impact of dispersive zenith troposphere delays on the ionosphere correction plotted with respect to the water vapor pressure at the surface level (SRTM topography).

Figure 7.

Impact of dispersive zenith troposphere delays on the ionosphere correction plotted with respect to the temperature at the surface level (SRTM topography).

2.1.3 Fitting a Correction Model

[16] In the following, we are going to fit different models as linear functions of dry pressure (Pd), water vapor pressure (Pw), and absolute temperature (T)

display math(13)

to the information derived in the prior section and judge their performance to correct the dispersive delays and the ionosphere correction, respectively. Thereby, all possible linear combinations for the three constituents are used, and it is tested how well each model can represent the dispersive troposphere delays obtained from the ray tracing. Both root mean square (RMS) and its equivalent in relative error in percent with respect to the mean delay are computed. This information can be used to find the most suitable model, which is capable to represent the ray-traced information with the highest accuracy. Table 1 lists the fitting results for the (direct) dispersive troposphere delays in units of length, and Table 2 shows the estimated parameters for each model in TEC units. Formal errors of the fitted parameters are given at the one-sigma level, and the corresponding relative errors are expressed as percentage.

Table 1. Various Model Fits Which Describe the Dispersive Troposphere Delays (in Units of Meter) as a Function of Atmosphere Surface Parameters
Modela0a1a2b0RMS[%]
M1 − 2.94 · 10 − 8 ± 2.8 · 10 − 116.13 · 10 − 6 ± 2.7 · 10 − 83.74 · 10 − 617.35
M2 − 6.41 · 10 − 7 ± 1.4 · 10 − 10 − 1.09 · 10 − 5 ± 2.5 · 10 − 91.59 · 10 − 67.40
M3 − 4.33 · 10 − 7 ± 3.7 · 10 − 101.06 · 10 − 4 ± 1.1 · 10 − 73.62 · 10 − 616.79
M4 − 4.34 · 10 − 9 ± 1.4 · 10 − 11 − 6.18 · 10 − 7 ± 1.6 · 10 − 10 − 7.15 · 10 − 6 ± 1.2 · 10 − 81.57 · 10 − 67.29
M5 − 2.98 · 10 − 8 ± 2.3 · 10 − 11 − 4.38 · 10 − 7 ± 3.0 · 10 − 101.35 · 10 − 4 ± 9.2 · 10 − 82.93 · 10 − 613.59
M6 − 6.08 · 10 − 7 ± 1.6 · 10 − 10 − 7.60 · 10 − 8 ± 1.9 · 10 − 101.10 · 10 − 5 ± 5.3 · 10 − 81.56 · 10 − 67.22
M7 − 6.82 · 10 − 9 ± 1.4 · 10 − 11 − 5.59 · 10 − 7 ± 1.8 · 10 − 10 − 1.06 · 10 − 7 ± 1.9 · 10 − 102.54 · 10 − 5 ± 5.9 · 10 − 81.50 · 10 − 66.97
Table 2. Same as Table 1 but for Models Which Describe the Indirect Troposphere Delay Effect on the Ionosphere Correction (in TEC Units)
Modelinline imageinline imageinline imageinline imageRMS[%]
inline image − 1.63 · 10 − 6 ± 1.8 · 10 − 91.95 · 10 − 4 ± 1.7 · 10 − 62.30 · 10 − 417.22
inline image − 3.85 · 10 − 5 ± 8.5 · 10 − 9 − 6.93 · 10 − 5 ± 1.5 · 10 − 79.57 · 10 − 57.16
inline image − 2.69 · 10 − 5 ± 2.2 · 10 − 86.58 · 10 − 3 ± 6.5 · 10 − 62.14 · 10 − 416.03
inline image − 8.03 · 10 − 8 ± 8.3 · 10 − 10 − 3.81 · 10 − 8 ± 9.6 · 10 − 9 − 6.24 · 10 − 4 ± 7.3 · 10 − 79.56 · 10 − 57.15
inline image − 1.65 · 10 − 6 ± 1.4 · 10 − 9 − 2.72 · 10 − 5 ± 1.9 · 10 − 88.21 · 10 − 3 ± 5.6 · 10 − 61.79 · 10 − 413.40
inline image − 3.61 · 10 − 5 ± 9.4 · 10 − 9 − 5.78 · 10 − 6 ± 1.1 · 10 − 89.65 · 10 − 4 ± 3.2 · 10 − 69.20 · 10 − 56.88
inline image − 2.40 · 10 − 7 ± 8.4 · 10 − 10 − 3.43 · 10 − 5 ± 1.1 · 10 − 8 − 6.84 · 10 − 6 ± 1.1 · 10 − 81.47 · 10 − 3 ± 3.6 · 10 − 69.08 · 10 − 56.80

2.1.4 Model Choice

[17] As discussed in the prior section, all models which have the water vapor pressure as one of the input parameters are expected to perform approximately at the same level. This can be explained by the existence of a single water vapor absorption line around 22 GHz which leads to a stronger slope of the dispersive delay that extends also to much lower frequencies [Liebe et al. 1993]. When all three constituents (Pd, Pw, and T) are given, the best performance by means of RMS with respect to the computed delays is achieved. Since water vapor pressure cannot be measured directly at the site, it is more convenient to derive this quantity from relative humidity measurements. Thus, the pressure of dry air should be obtained by applying the relationship Pd = P − Pw. Therefore, in an intermediate step, the saturation vapor pressure Pv, which depends only on the absolute air temperature T, is computed as follows:

display math(14)

which denotes the well-known equation from Goff [1957]), incorporating the corrections proposed by the World Meteorological Organization [WMO, 2000]. Finally, Pw can be obtained from

display math(15)

where RH is the relative humidity (expressed as a percentage). Since total pressure (P), temperature (T), and relative humidity (RH) can be easily obtained by a simple meteo sensor at the ACES ground station, it is recommended to use models M7 and inline image for correcting the dispersive troposphere delays and their effect on the ionospheric correction, respectively, after converting to Pd and Pw. This ensures that the approach works also well in dry regions where almost no water vapor can be seen around the station and most of the refractivity depends only on dry pressure and temperature.

2.1.5 Corrections at Arbitrary Elevation Angles

[18] In the prior section, only zenith delays have been considered for the correction model. However, as the ISS is seen at elevation angles ε between 10° and 90°, one needs to consider a mapping function that accounts for the growth of the path length with zenith distance. Although a variety of such mapping functions exists for space geodetic applications [e.g., Boehm et al., 2006a], we use a simple first-order approximation in the form of 1 / sinε. Moreover, we neglect azimuthal dependence of the delays as this simple model is applicable to all sites without external information and/or download of model coefficients. Although one can replace the simple mapping function by any modern mapping function used in space geodesy [Boehm et al. 2006b], the accuracy is expected to be almost identical for elevation angles above 10°. Thus, the proposed model for correcting the ACES specific dispersive troposphere delays reads as

display math(16)

where c represents the speed of light which is necessary for the conversion to time delay. The parameters a0, a1, a2, and b0 should be taken from model M7 in section 2.1.3 (Table 1) or re-determined at a later stage if necessary. In a similar way, the correcting term for the ionosphere delay can be denoted as

display math(17)

where the coefficients corresponding to model inline image (Table 2) are recommended.

2.2 Correction to ACES Ground Stations

[19] In order to test the usefulness of the correction model, five potential locations for ACES ground stations were selected and ISS overflights between 1 and 7 August 2011 were simulated. Each time the ISS became visible above an elevation angle of 10°, tracking parameters, i.e., time, local azimuth, and elevation angle were obtained until the ISS settles below the 10° elevation mask. As the ISS ground tracks are varying with each overflight, the geometry and the length of each passage change between two trackings. All ISS passages at the five ACES ground stations were then used to compute dispersive troposphere delays using a full 4-D ray tracer as described in Hobiger et al. [2008]. Thus, the proposed correction model can be validated for different elevation angles, and it can be checked whether an azimuth dependency of the dispersive delay can be neglected or not.

2.2.1 Ground Stations

[20] Five globally distributed sites were considered as potential ACES ground stations. Table 3 lists these stations together with their geodetic coordinates which were taken from the IGS GPS receiver placed at the timing laboratory at each site.

Table 3. Potential ACES Ground Station Sites Considered for This Study
IDCityCountryLongitude [deg]Latitude [deg]Ellipsoidal height [m]
nistBoulderUSA254.73739.9951648.5
ptbbBraunschweigGermany10.46052.296130.2
kgniKoganeiJapan139.48835.710123.5
opmtParisFrance2.33548.836124.2
pertPerthAustralia115.885-31.80212.9

2.2.2 ISS Orbit

[21] ISS orbit parameters were obtained in the form of two-line element (TLE) data for the period between 1 and 7 August 2011. The ISS orbit is permanently monitored, and TLE data sets are updated at least 3 times per day. Thus, we need to deal with as many different TLEs for the 1 week period. However, as the computed orbit only differs by a few hundred meters, it was decided to carry out the overflight simulations with a single TLE from the middle of the week that is assumed to be valid over the time span of the whole week. This approximation is sufficiently accurate to get the overflight geometry with an accuracy of better than 0.001° in azimuth and elevation. Thus, the following TLE parameters [CelesTrak, 2012] were used to simulate the location of the ISS at any given epoch between 1 and 7 August 2011.

display math

2.2.3 Overflight Simulations

[22] The geocentric position of the ISS has been computed every second based on the TLE data set described in the prior section. For each ground station, it was checked whether the space station rises above the local horizon or whether its elevation angle is larger than the one defined by the sky mask. The ISS is then tracked until it sets below the sky mask again, whereas every second azimuth and elevation are computed. Figures 8-12 depict the visibility of the ISS above 10° elevation as sky plots at each station for all days of the simulation period. The time of the passages is color coded within each 24 h batch.

Figure 8.

Sky plot of the ISS overflights between 1 (DOY 213) and 7 (DOY 219) August 2011 at Koganei.

Figure 9.

Sky plot of the ISS overflights between 1 (DOY 213) and 7 (DOY 219) August 2011 at Boulder.

Figure 10.

Sky plot of the ISS overflights between 1 (DOY 213) and 7 (DOY 219) August 2011 at Paris.

Figure 11.

Sky plot of the ISS overflights between 1 (DOY 213) and 7 (DOY 219) August 2011 at Perth.

Figure 12.

Sky plot of the ISS overflights between 1 (DOY 213) and 7 (DOY 219) August 2011 at Braunschweig.

2.2.4 Dispersive Troposphere Delays and Their Correction With the Proposed Models

[23] Ray tracing between the ground stations and the ISS at the frequencies f1, f2, and f3 allows to access the impact of the dispersive troposphere delay effect on differential delay Δτ(f1,f2) as well as on the TEC measurements. Dispersive troposphere delays are plotted for each site in dependency of the elevation angle (black dots in Figures 13 and 14). These delays are growing roughly with 1 / sinε and are in general differing between each site and epoch. For example, at Koganei (kgni), high humidity leads to generally larger delays than those observed at other stations. At Boulder (nist), which is a site at higher altitude, the atmosphere appears to be less humid which causes the dispersive delays to be of smaller magnitude. Thus, one needs to verify whether the proposed correction model works for all sites. Surface dry pressure (Pd), water vapor pressure (Pw), and absolute temperature (T) were taken from the numerical weather models and interpolated to the time of each ISS overflight. This allows us to simulate a meteo sensor that provides these quantities at the ACES ground terminal. Based on these three input parameters and the knowledge of the elevation angle at each data point, it is straightforward to apply the suggested dispersive troposphere correction. As shown in the plots, the usage of the semi-empirical model is feasible in order to model the dispersive troposphere delays well at all sites. The corrective approach is capable to remove the biases as well as the elevation dependency of the dispersive delays. At all stations, epochs, and elevation angles between 10 and 90°, the model guarantees that the dispersive troposphere delays are removed with a precision of 0.1 ps or better. Total electron content values can be corrected with an uncertainty of 0.001 TECU which is expected to be well below the formal error of this correction. Since the elevation dependency is reduced significantly, it can be assumed that the influence from the remaining (un-modeled) troposphere is well below the noise floor of the instruments for integration periods during an ISS overflight. This will be verified in the next section.

Figure 13.

Time difference between the uplink and downlink signals due to dispersive troposphere (black dots) and time difference after the correction with the suggested model M7 (red dots).

Figure 14.

Wrongly modeled total electron content caused by the dispersive troposphere of the ACES microwave link before (black dots) and after (red dots) applying the suggested correction.

2.2.5 Time Deviation of the Ground-Space Link Before and After the Corrections Being Applied

[24] In order to access how much the ACES microwave link would be impacted by neglecting any kind of dispersive troposphere delay, the results from the overflight simulations in the prior section were taken and time deviation (TDEV) [Riley, 2008] characteristics for each passage and station were computed. One has to consider that for such an evaluation, one needs to calculate the impact of the dispersive troposphere on the difference between the uplink and downlink measurements. Thus, both direct dispersive troposphere correction and the indirect effect via the ionosphere correction are scaled by 0.5 (equation (5)). Figures 15-19 show the obtained TDEV results before and after the suggested correction were applied. It can be seen that the correction models are capable to remove more than 90% of all dispersive troposphere delays of the ACES microwave link. The semi-empirical models perform at the same level at all sites tested in this study. In general, the impact of dispersive troposphere delays on the systems time transfer performance is reduced by at least 1 order of magnitude, which ensures that at all relevant averaging times, elevation angles, and locations, the anticipated time stability of the microwave link is not compromised.

Figure 15.

Time deviation of all dispersive troposphere delays before and after applying the suggested corrections at Koganei.

Figure 16.

Time deviation of all dispersive troposphere delays before and after applying the suggested corrections at Boulder.

Figure 17.

Time deviation of all dispersive troposphere delays before and after applying the suggested corrections at Paris.

Figure 18.

Time deviation of all dispersive troposphere delays before and after applying the suggested corrections at Perth.

Figure 19.

Time deviation all dispersive troposphere delays before and after applying the suggested corrections at Braunschweig.

3 Conclusions

[25] We propose the semi-empirical model as described in equation (16) with the coefficients given in 2.1.3. Tests with simulated ISS overflights at different ACES ground stations have shown that this model is capable to remove biases and elevation-dependent features caused by the dispersive troposphere delay difference between uplink and downlink. The model performs well at all sites and reduces the impact on all averaging time scales by at least 1 order of magnitude. Simulations revealed that the impact of the dispersive troposphere delays is below the stability (noise) of the ACES microwave link for averaging times up to 100 s but the corresponding instability may surpass the microwave link specification between 200 and 300 s averaging time, which is the longest period covered when the ground station sky mask is set to a cut-off elevation angle of 10°. In that case and moreover when lower elevation angles are considered or the performance of the microwave link is better than anticipated, dispersive troposphere correction can bias the obtained results. Thus, by applying the suggested correction model, it can be ensured that the remaining (un-modeled) dispersive troposphere delays are well below the noise floor of the microwave link and thus should not impact the anticipated performance of the instrument. The suggested correction scheme does not require any changes of the system, and corrections can be made easily within the post-processing chain. The only requirements are that each site is equipped with a small meteorological sensor that measures surface temperature, pressure and relative humidity, and the knowledge of elevation angles under which the ISS is seen from the ground during its paths. Moreover, the suggested model allows to correct for the implicit effect of ionosphere mismodeling, which is direct consequence of the frequency-dependent troposphere delay characteristics. Similar studies like this might be of importance for other time and frequency transfer instruments or future space geodetic instruments like VLBI2010 [Behrend et al., 2009] which operate over a broad frequency range.

Acknowledgments

[26] We thank the three anonymous reviewers for their valuable comments which help improve the paper. The present work has benefited from the input of L. Cacciapuoti, P. Wolf, O. Montenbruck, and other members of the ACES science team, who provided valuable comments to the undertaking of the research summarized here. We acknowledge the support extended by Research Group Advanced Geodesy at Vienna University of Technology in providing the weather model data.

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