## 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 (*f*_{1} = 13.475 GHz for uplink, *f*_{2} = 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 (*f*_{0} = 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

where *N*_{0} 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.