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# Introduction

## Geometric principle

The geometric principle of Very Long Baseline Interferometry (VLBI) is simple and straightforward. The radiation from extragalactic radio sources arrives on Earth as plane wavefronts. This is different from nearby Earth satellites such as those of the Global Navigation Satellite Systems (GNSS) where the finite distance to the emitter produces parallactic angles. The basic triangle for the determination of the baseline vector reduces to a rectangular one providing a direct relation between the baseline vector $b$ and the direction to the radio source $s_0$ (Campbell, 2000). The scalar product $\tau$ represents the observed delay between the reception times $t_1$ and $t_2$ at stations 1 and 2 (see Fig. and Equ. below) with the sign convention $\tau=t_2-t_1$ and the velocity of light $c$.

$$\tau=-\frac{b \cdot s_0}{c}=t_2-t_1$$

The delay $\tau$ is time-dependent, and the largest contribution to its variation is due to the fact that the interferometer is fixed to the Earth's surface and thus follows its diurnal rotation with respect to the celestial reference system that is realized by positions of radio sources. The geodetic VLBI concept uses two or more radio telescopes to observe numerous extragalactic radio sources distributed across the skies, mostly quasars or radio galaxies. In geodetic VLBI since the end of the 1970's the observations are done within S-band (2.3 GHz) and X-band (8.4 GHz) (A change of the frequency setup, e.g. observing on a frequency band between 2 and 14 GHz, is envisaged for the next VLBI generation, VLBI2010 (Petrachenko, 2009)) and the data are recorded and time-tagged using very stable and precise time signals obtained from hydrogen masers. These data are then sent to particular correlation centers for cross-correlation to generate so-called fringes and to obtain the group delay observable $\tau$ which is relevant for geodetic and astrometric applications. From these delays, the baseline lengths $b$ and other geodetic parameters can be derived nowadays with sub-centimeter accuracy. The VLBI technique measures very accurately the angle between the Earth-fixed baseline vector $b$ and the space-fixed radio sources $s_0$ which have to be transformed into a common system for the evaluation of the Eq. from above by parameter estimation techniques. Thus, even the most subtle changes in the baseline lengths and in the angles between the reference systems can be detected, and the main geodynamic phenomena such as Earth orientation parameters can be monitored with unprecedented accuracy (Schuh, 2000). However, '… if we leave the Euclidean geometry in empty space and return to the real world with curved space, flickering quasars, billowing atmospheres, wobbling axes, and drifting continents, we have to delve into layers of complexity, fortunately not only as a chore but also as an opportunity to gain a wealth of new knowledge about our system Earth.' (Campbell, 2000) More details about the complexity of VLBI are provided in the next sections.

## History and technological developments

In this section we give a summary of the early history of geodetic VLBI and of the VLBI technique. The interested reader may find further details on the history in (Sovers et al., 1998), (Campbell, 2000), and (Kellermann & Moran, 2001), and the references therein. For details on technology, we refer to the textbooks by (Thompson et al., 1986) and (Takahashi et al., 2000), and the references therein. Very Long Baseline Interferometry is an outgrowth of radio interferometry with cable-connected elements designed to overcome the limited resolution of single dish radio telescopes (Cohen et al., 1968). However, to reveal the structure of extremely compact radio sources, the resolving power of Connected Element Radio Interferometers (CERI) was insufficient, even at higher frequencies (Campbell, 2000). The advent in the late 1960s of high-speed tape recorders and high-stability atomic frequency/time standards made possible the construction of phase-coherent, Michelson-type interferometers whose elements required no physical connection between them and hence could be spaced arbitrarily far apart. In 1967, several groups working independently in Canada and the United States developed and successfully operated two-station interferometers (Bare et al., 1967; Broten et al., 1967; Moran et al., 1967; Brown et al., 1968). Signals received at each station were down-converted in frequency, time-tagged, and recorded on tape for subsequent playback at a correlator center, where the common signal received from a radio source at two (or more) antennas was detected by cross-correlation and integration although the signal itself is by far weaker than the background noise. This technique eliminated the need for a real-time phase-stable connection between radio telescopes. Potential geophysical applications of geodetic VLBI were recognized early (Gold, 1967; Shapiro & Knight, 1970). The first experiments that were explicitly aimed at achieving geodetic accuracy on long baselines were conducted by the Haystack/MIT group on the 845 km baseline between the Haystack Observatory in Northern Massachusetts and the National Radio Astronomy Observatory of Green Bank, West Virginia, U.S.A. (Hinteregger et al., 1972). Since that time the station position precision improved dramatically from a few meters to the current level of one centimeter or even better. A major factor in the improved precision was made possible by equipment improvements such as wider spanned and recorded bandwidths, dual-frequency observations, lower system temperatures, and phase calibration. As an example the first geodetic observations used the MkI system (Whitney et al., 1976) which could record only 0.72 Mbits/second, whereas modern systems allow to record at 1024 Mbits/second or even more. Other factors included improvements in observing strategies, analysis methods, and modeling of physical processes. The key to the high group delay precision of 1 ns (30 cm) attained in these experiments was the invention of the so-called bandwidth synthesis technique (Rogers, 1970), which helped to overcome the limitations of tape recording equipment in terms of recordable bandwidth (Campbell, 2000). A milestone was reached when the first significant estimates of the length change on the transatlantic baseline Haystack Onsala (Sweden) were announced. A baseline rate of 17 mm/yr with a statistical standard deviation of $\pm$2 mm/yr derived from 31 experiments between September 1980 and August 1984 was published by (Herring et al., 1986). However, they reported that the systematic error could be as large as 10 mm/year. In comparison, the Figure below indicates session-wise baseline length estimates from 1984 to 2011 between the stations Wettzell (Germany) and Westford (Massachusetts, U.S.A.) determined by the VLBI group at TU Wien, Vienna, Austria. Clearly visible is the continuously improving accuracy, in particular during the first decade of the time series, and the seasonal variation of the length estimates, which is either due to modeling deficiencies (e.g., of troposphere delays), due to unmodeled loading effects (e.g., atmosphere or hydrology loading), or a combination of both of them.

## Data Aquisition

Geodetic VLBI is an active observing technique which needs to control the radio telescopes and steer them to various positions on the sky in a predefined observing schedule; thus, scheduling is a very important part of VLBI. The package SKED (Vandenberg, 1999) developed at NASA Goddard Space Flight Center is widely used within the geodetic community to generate the observing plans for the radio telescopes. At any instant, different subsets of antennas will be observing different sources. (All observations to one source at a time form a so-called 'scan'.) The integration time varies from antenna to antenna to reach the required signal-to-noise-ratio (SNR) (Petrov et al., 2009). The elevation mask is usually set to $5^{\circ}$ but any obstacles or mountains have to be considered if they prevent observations at low elevation angles. There are various optimization criteria in SKED, but 'sky-coverage' is mostly selected. This strategy aims at filling large holes in the sky over the stations, which is important for the estimation of troposphere delays. More information about scheduling strategies can be found in (Vandenberg, 1999) or (Petrov et al., 2009).

The incoming signal first arrives at the primary paraboloidal dish of the radio telescope, then at the hyperboloidal subreflector, and finally it enters the feed horn (see Fig. above for an example of a Cassegrain antenna). The signal goes directly to the feed from the paraboloidal reflector in the case of prime focus antennas. Then, the signals are amplified before they are heterodyned from radio frequency to intermediate frequencies of several hundred MHz, and finally down-converted to baseband frequencies (simultaneously in multiple frequency sub-bands or channels), where the signal is band-limited to a width of a few MHz, sampled and digitized (Sovers et al., 1998). The system temperatures typically range from 20 to 100 K for S- and X-band. Finally, the signals are formatted and recorded on magnetic disks (or on tapes in the early years). Nowadays, data from shorter sessions can also be transferred to the correlators via high-speed broadband communication links. However, the majority of electronic transfer of the raw VLBI data is still asynchronous, i.e., the transfer is started during or after the observation but then needs more time than the actual observation (termed e-transfer). Only in a few experimental sessions, as e.g. described by (Sekido et al., 2008), the transmission was carried out in real-time (termed e-VLBI).

VLBI radio telescopes need to have large collecting areas as well as high sampling and recording rates because the signal flux density is in the order of 1 Jansky (1 Jy = $10^{-26}\rm{Wm}^{-2}\rm{Hz}^{-1}$) or even lower. On the other hand the structure of the antennas has to be sufficiently stable to allow slewing between widely separated sources within a few minutes or faster (Sovers et al., 1998). (See Section 'antenna deformation' for more information.) Phase shifts caused by the instrumentation have to be calibrated to take full advantage of the precision of current frequency standards (e.g., hydrogen masers stable to $10^{-14}$ at 50 minutes or better). Otherwise, those phase shifts can corrupt the estimated phase and group delay of the incoming signal. The technique of phase calibration (Rogers, 1975) compensates for the instrumental phase errors by generating a signal of known phase, injecting this signal into the front end of the VLBI signal path, and examining the phase after the signal has traversed the instrumentation. This calibration signal is embedded in the broadband VLBI data stream as a set of low-level monochromatic tones along with the signal of the radio source (Sovers et al., 1998). Those tones are used at a later stage by the post-correlation software. Furthermore, the length variations in the cables from the clocks to the antennas (called cable delays) have to be corrected properly. In the next step, the signals recorded at the antennas are combined pair-wise producing an interference pattern. These installations are called correlators, and they are presently run world-wide, e.g. in the U.S.A. (Haystack Observatory, Westford; U.S. Naval Observatory, Washington D.C.), in Germany (Max Planck Institute for Radio Astronomy, Bonn), and Japan (National Institute of Information and Communications Technology, Kashima). They are made up of special hardware that is used to determine the difference in arrival times at the two stations by comparing the recorded bit streams. If $V_1(t)$ and $V_2(t)$ are the antenna voltages as functions of time $t$, $T$ is the averaging interval, and the asterisk denotes the complex conjugate, the group delay $\tau$ can be determined by maximizing the cross-correlation function $R$ (Sovers et al., 1998):

$$R(\tau)=\frac{1}{T}\int^{T}_{0}V_1(t)\cdot V^{*}_2(t-\tau)\cdot dt.$$

Due to Doppler shifts caused by the Earth rotation, VLBI observations at X-band (8.4 GHz) would be oscillating at several kHz if not 'counter-rotated' first (Sovers et al., 1998). In recent years, also software correlators have been developed (e.g. Kondo et al., 2004, Tingay et al., 2009) because correlation algorithms for geodetic VLBI can be effectively implemented on parallel computers or on distributed systems. Software correlation is already well beyond the development stage. For example, the Bonn correlator has processed all IVS (International VLBI Service for Geodesy and Astrometry) sessions with the DiFX software correlator from November 2010 onwards. During the correlation process, amplitudes and phases are determined every 1 to 2 seconds in parallel for typically 14 frequency channels $\omega_i$. The post-correlation software applies corrections for the phase calibration and fits the phase $\Phi_0$, the group delay $\tau_{gd}$, and the phase rate $\tau'_{pd}$ to the phase samples $\Phi(\omega_i,t_j)$ from the various frequency channels $\omega_i$ and times $t_j$. The phase-derived observables are determined (for phase $\Phi$ and circular frequency $\omega$) from a bilinear least-squares fit to the measured phases $\Phi(\omega,t)$ with (Sovers et al., 1998)

$$\Phi(\omega,t)=\Phi_0(\omega_0,t_0)+\frac{d\Phi}{d\omega}(\omega-\omega_0)+\frac{d\Phi}{dt}(t-t_0),$$

where the phase delay $\tau_{pd}$, group delay $\tau_{gd}$, and phase delay rate $\tau'_{pd}$ are defined, respectively, as

$$\tau_{pd}=\frac{\Phi_0}{\omega_0},\tau_{gd}=\frac{d\Phi}{d\omega},\tau'_{pd}=\frac{1}{\omega_0}\frac{d\Phi}{dt}.$$

The group delay rate $\tau'_{gd}$ is not accurate enough to be useful for geodetic or astrometric purposes, however, it is needed to resolve group delay ambiguities in a first solution step. The amplitudes are usually not used in geodetic/astrometric VLBI. The natural ultra-wide band continuum radiation provides the means to use the essentially unambiguous wide band group delay as the prime geodetic VLBI observable. The group delay resolution is proportional to the inverse of the signal-to-noise-ratio (SNR) and the root mean square (rms) of the frequency about the mean (sometime called rms spanned bandwidth) $B_{eff}$ (Rogers, 1970):

$$\sigma_{\tau}=\frac{1}{2\pi}\cdot\frac{1}{{\rm{SNR}}\cdot B_{eff}}.$$

If we increase the rms spanned bandwidth $B_{eff}$ at a given SNR by a factor of ten, the group delay uncertainty will be reduced by the same factor, a relation with tremendous consequences (Campbell, 2000). On the other hand, the SNR is dependent on the recorded bandwidth $B$ with

$${\rm{SNR}}=\eta\cdot\rho_0\cdot\sqrt{2\cdot B\cdot T},$$

where $T$ is the so-called coherent integration time, $\eta$ is the digital loss factor, and $\rho_0$ is the correlation amplitude which depends on the system noise temperatures and on the equivalent noise temperature of the source signal (Takahashi et al., 2000). There are virtually no limitations to further improve the statistical precision of the geodetic group delay, except technological constraints and costs. For the upcoming VLBI2010 system, a four-band system is recommended that uses a broadband feed to span the entire frequency range from 2 to 14 GHz (see Section 'concluding remarks'). This will also allow the use of phase delays, which are still an issue of research with the current system where they provide very high accuracy, but only on very short baselines (Herring, 1992, Petrov, 1999) as on longer baselines the phase ambiguities are still an unsolved problem.

## Data Analysis

The VLBI data analysis model is developed using the best presently available knowledge to mathematically recreate, as closely as possible, the situation at the time of observation (see Section 'theoretical delays'). Then, a least-squares parameter estimation algorithm or other estimation methods can be used to determine the best values of the quantities to be solved for (Section 'least-squares adjustment in VLBI'). Before this process starts, the raw observations have to be cleaned from several systematic effects, which in fact limit the final accuracy of the results (Schuh, 2000).

The flow diagram of a geodetic VLBI data analysis (according to Schuh, 1987) is shown in the Fig. above. The system can be seen to have two main streams, one containing the actual observations which undergo instrumental and environmental corrections to obtain the reduced delay observables, and the other to produce the theoretical delays, starting with the a priori parameter values, a set of initial values for the parameters of the VLBI model. Both streams converge at the entrance to the parameter estimation algorithm, e.g. the least-squares fit, where the 'observed minus computed' values are formed. The instrumental effects include systematic clock instabilities, electronic delays in cables and circuitry, and the group delay ambiguities. The latter are due to observation by the multichannel frequency setup described in the section 'history and technological developments' covering the total spanned bandwidth around each of the two observing frequencies ($f_S = 2.3$ GHz (S-band) and $f_X = 8.4$ GHz (X-band)). As the group delay ambiguity spacing is comparably large and well-known the analyst can select - from a first solution using the observed group delay rates $\tau'_{gd}$ only - one level on which all residuals and thus the corresponding group delay observables are shifted. Care has to be taken that the group delay closure within each triangle of a multi-station VLBI network is zero (Schuh, 2000). The ionosphere, which is a dispersive medium in the radio frequency band, can be dealt with to first order by using two different observing frequencies, i.e. the ionosphere group delay corrections for the X-band observations are computed from the differences of group delay measurements at X-band and S-band:

$$\Delta\tau^{ion}_X=(\tau_X-\tau_S)\cdot f^2_S/(f^2_X-f^2_S).$$

In contrast to the Global Positioning System (GPS), where a very close frequency pair has been chosen, the factor in VLBI to convert the difference into a correction for the higher band is very small: 0.081, so that the error contribution from the S-band observations is marginal (Schuh, 2000). Unlike GPS, ionospheric second order effects can be neglected in VLBI analysis as was demonstrated by (Hawarey et al., 2005).

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