In geodetic very long baseline interferometry (VLBI), the observations are performed at two distinct frequencies (2.3 and 8.4 GHz) in order to determine ionospheric delay corrections. This allows information to be obtained from the VLBI observables about the sum of electrons per area unit (total electron content) along the ray path through the ionosphere. Because of the fact that VLBI is a differential technique, the calculated ionospheric corrections depend on the differences of the propagation media over the stations. Additionally, an instrumental delay offset per station causes a bias of the ionospheric measurements. This paper presents a method to estimate ionospheric parameters, that is, values of vertical total electron content from VLBI data, and compares the outcomes to results from other space geodetic techniques. As VLBI observations cover more than two complete solar cycles, the relation to space weather indices on long-term timescales can be shown.
 Since its development in the late 1960s (described by, e.g., Clark et al. ) the geodetic very long baseline interferometry (VLBI) technique has achieved a great improvement in precision and accuracy. As the first observations were carried out at single frequencies, ionospheric corrections could only be made by using external measurements. In the 1980s, stations were equipped with dual-frequency receiving systems and the ionospheric influence on the observed group delays could be measured directly and applied in astronomical, astrometric, or geodetic analyses. The routine analysis of GPS observations starting in the mid 1990s allowed the determination of ionospheric parameters all over the world. Since 2003, the International GNSS Service, formerly the International GNNS Service (IGS), has presented an official ionospheric product in the form of global two-dimensional maps of vertical total electron content (VTEC) values. Additionally, many other local, regional, and global solutions, tomographic inversions, high-rate observations, and satellite occultations using GPS have been investigated in the last years to determine ionospheric parameters. So far, the ionospheric correction that can be obtained from routine VLBI observations has not been included in any model of the ionosphere or investigated further. Except for one paper [Kondo, 1991], no group has focused how to gain ionospheric information from dual-frequency VLBI measurements. VLBI is only sensitive to the differences of the ionospheric influences at each station pair of a network; it is not a continuous observing technique and does not have as dense a distribution of stations as GPS. Nevertheless, VLBI provides capabilities to probe the ionosphere in an absolute sense, as will be described in this paper.
2. Ionospheric Information From VLBI Observations
2.1. Ionospheric Impact on VLBI Measurements
 Geodetic VLBI sessions are carried out at 2.3 GHz (S band) and 8.4 GHz (X band) in order to compensate for dispersive delays caused by the ionosphere. Higher-order ionospheric terms can still be neglected to get millimeter accuracy as discussed by Sovers and Jacobs  and recently confirmed by Hawarey et al. . We will start our derivation from a station-based model equivalent to that one used in GPS and then form differences to get VLBI observables. The delay di,j of the received signal at one station i on frequency j consists of a nondispersive part di, a contribution qi from the ionosphere depending on the observation frequency fj, and a constant instrumental delay Γi,j caused by the receiving system. For stations 1 and 2, observing at X and S bands, we obtain the following system of equations:
As VLBI observes delay differences τX = d2,X − d1,X and τS = d2,S − d1,S, equations (1)–(4) yield the VLBI measurements
Geodetic analysts are interested in an ionosphere-free delay which can be obtained by solving for the q term in equations (5) and (6) and equating them with each other:
Equation (9) is a linear combination of the delays (τX and τS) obtained at X and S bands, whereas the last two terms represent constant values and shift the ionosphere-free observable by an unknown constant offset, which is in geodetic parameter estimation absorbed by the clock model [Ray and Corey, 1991]. Reconstruction of the ionospheric delay (τX − τ) yields after doing some algebra
Equation (10) represents the ionospheric delay as stored in VLBI databases [e.g., Noll, 2003]. The first term contains information about the ionosphere, whereas the last two terms represent the constant instrumental effects at station i caused by instrumental delays in X band. These instrumental delays are contained in X band ionospheric corrections
As the left side of equation (10) represents a measured quantity, the parameters Δq, that is, the difference of the ionospheric contributions at stations 1 and 2 (see equation (7)) and τi,inst, will become the parameters to be solved for. The value of q used in equations (1)–(4) and (7) is directly related to the sum of electrons along the ray path and is assigned as slant total electron content (STECi)
Using a thin shell approximation of the ionosphere [Schaer, 1999] and an appropriate mapping function MF(ɛi), equation (13) can be written in terms of vertical total electron content VTEC
whereas MF(ɛi) is defined by
Re represents the radius of the Earth, H is the height of the ionospheric layer, and ɛi is the elevation angle. H is usually located around the height of the F2 peak of electron density [Schaer, 1999]. For our investigations a height of 450 km was chosen to be consistent with the heights used in ionospheric models from GPS measurements (see section 3.1). On the basis of equations (12), (14), and (15), a method was developed to estimate absolute values of vertical total electrons above each VLBI site. In section 2.2 we will discuss how such values can be derived and which simplifications have to be made in order to obtain absolute values.
2.2. Model Assumptions
 First, equation (12) is rewritten using equation (14), which shows the relation between the observables and unknown target parameters, that is, the vertical total electron content and instrumental offsets
VTEC1* and VTEC2* denote the values of the vertical total electron content at the intersection points of the ray paths with the (infinitesimally thin) shell for stations 1 and 2, respectively. The distance between the station and the intersection point is directly related to the elevation angle and can reach up to more than 1600 km for an elevation angle of 5° and an assumed height of the ionospheric layer of 450 km. By setting up a model that explains the relation between vertical total content above the station and at the intersection point, we were able to solve for station dependent values by an adjustment process, taking into account that spatially separated antennas will observe under different elevation angles when pointing to the same radio source. For the following considerations, we use (λi, ϕi) as geographical coordinates of station i and (λ*i, ϕ*i) as the location of the intersection point (see Figure 1). As a first step, it is taken into account that the diurnal variation of the ionosphere is mainly correlated with the position of the Sun. This leads us to the first assumption, that the thin shell representing values of vertical total electron content is following the apparent motion of the Sun. Or, in other words, it is assumed that the value VTEC*1 observed at time t* on longitude λ* will be the same, when it passes the meridian of the station, which means VTEC*1 can be observed at (λi, ϕ*i) at time t* + (λ*i − λi)/15. This first step is illustrated in Figure 1. The main task is now to determine the vertical total electron content above the station from the VTEC value at the rotated point. By applying the rough assumption of a linear north-south gradient ci, which is constant within the whole session,
station-dependent values can be obtained from information taken at the intersection points. The coefficients ci can be estimated for each station together with the other parameters, as described in section 2.3.
2.3. Estimation of Values of Vertical Total Electron Content (VTEC)
 The left side of equation (16) represents the measured ionospheric influence on a baseline formed by stations 1 and 2, whereas the right side shows the unknown target parameters, that is, values of vertical total electron content above the stations (by using relationships (17) and (18)) and instrumental offsets assigned to each site. The time-dependent variable behavior of the ionosphere is taken into account by setting up a piecewise linear function model
for station i. The time intervals between the VLBI scans vary according to geometry, slewing times of the antennas, flux densities of radio sources, and other reasons. Thus the time steps denoted in equation (19) should be set such that a given number n of observations defines the length of each interval. In our analysis, a value of n = 8 observations, contributing to the estimation of one interval, was chosen which corresponds to an average temporal resolution of the outcomes of about 30 min. This value also sets the lower limit of the detectable ionospheric periods. Changing the value n to a lower number of observations increases time resolution but decreases the redundancy of the estimation process and weakens the stability of the matrices used in the adjustment.
 The objective function tries to minimize the differences between the data and modeled values. Equation (20) aims at minimizing the vector of residuals by adjusting the unknowns Δx, using a linearized model represented by matrix A and a vector ΔY containing the values by means of “observed minus calculated.” Matrix A can be obtained easily as all model assumptions are already linear and iteration, caused by the neglect of higher order derivatives, will not be necessary. Minimizing the squared sum of the weighted residuals v = AΔx − ΔY, using the stochastic (weight) matrix P of the observations yields
Finding then the minimum of equation (20) by setting the derivative to Δx to zero would be equal to the Gauss-Markov model [Koch, 1997] used in classical geodetic adjustments. But we will let equation (20) remain unchanged and will find another condition which has to be fulfilled, too. Because of the physical nature of the ionosphere negative values of vertical total electron content are not possible. This knowledge can be added to the estimation algorithm by applying an in equation (expressed by equation (21)) to the vector of improvement, and as linearity of the model parameters is already given, matrices B and C can be set up easily [Hobiger and Schuh :
Solving equation (20) under condition (21) is carried out by a reflective Newton method [Coleman and Li, 1996; Gill et al., 1981] and is implemented in such mathematical analysis software as, for example, Matlab®. Before equations (20) and (21) can be solved, singularity of the A matrix, caused by the constant terms of the instrumental offsets (last two terms in equation (16)), has to be eliminated. One method to overcome this problem is to set the instrumental offset at one station within the network to zero and obtain the other values relative to that one. Another approach, equivalent to the first one, sets the sum of all offsets within a network of N stations to zero [Sekido et al., 2003]:
One has to be careful not to overparameterize the station-dependent VTEC models. Let the number of model intervals, set up for station i, be inti; then the redundancy R, using nobs observations and N stations, is given by
The last term in equation (23) results from the effect, that one artificial observation (equation (22)) increases the number of observations by 1. As the number of observations carried out by VLBI within one session is rather small compared to global GPS measurements, redundancy should be high enough to be able to detect outliers within the data.
3. Results and Comparison With Other Space Geodetic Techniques
 Applying the algorithms described in section 2 to VLBI databases of dual-frequency observations, let us estimate values of vertical total electron content and instrumental offsets together with their formal errors at each station. As VLBI is not observing every day and network stations change between the sessions, the outcomes cannot be provided on regularly time-spaced intervals. This has to be taken into account when doing frequency analysis.
3.1. VLBI Results and Their Comparison With GPS Data
 Using the whole available database of the International VLBI Service for Geodesy and Astrometry (International VLBI Service for Geodesy and Astrometry Web page, http://ivscc.gsfc.nasa.gov/), time series for 143 network stations plotted in Figure 2 were created. These series are of unequal lengths, as stations were constructed and dismantled at different epochs in time. Furthermore, some antennas contributed less than others, as they are not mainly dedicated to astrometric/geodetic observations and/or are not equipped permanently with dual-frequency receivers. As an example, the results for station Wettzell, Germany (20 m dish), shown in Figure 3 (top), will be discussed, as this station has a long history of observations going back to 1984. This long time span will become important when investigating long-term trends of the ionosphere. One can see clearly that the solar cycle dominates the overall shape of the total electron content results for this station, but an annual signal of which the amplitude is modulated by the long periodic variation can be found, too. For a cross-technique comparison we used results from a worldwide GPS station network. The International GPS Service (IGS) provides Global Ionospheric Models (GIM) of VTEC values on a geographical grid (Δλ = 5°, Δϕ = 2.5°) with a time resolution of 2 hours (International GNSS Service (IGS) Web page, http://igscb.jpl.nasa.gov/). Using a proper interpolation method suggested by Schaer et al. , values of VTEC were obtained for each VLBI station. When this paper was written, the availability of the official IGS files ranged back to the year 1998, which enables us to compare data on a rather short time span of only 6 years. Therefore we have chosen to use GIMs from the Astronomical Institute of Berne, Switzerland, which are also stored in ionospheric exchange (IONEX) format. As these data are provided back to the year 1995, comparison could be done for all observations after this date. Figure 3 (bottom) shows VTEC values for station Wettzell gained from GIM data. VLBI results contain some outliers after the year 2002, but in general, the agreement between both techniques is rather good. Interpolating VTEC values from GIM data to the epoch of each VLBI data point enables us to do descriptive statistics. Figure 4 shows a histogram of the differences between VLBI and GPS at station Wettzell. GPS seems to provide slightly higher values with a bias of 1.95 TECU and a standard deviation of about ±6 TECU. Figure 5 shows the differences between VLBI and GPS for all IVS network stations again displayed by a histogram. The overall mean difference between both techniques is about −2.8 TECU, whereas standard deviation is in the range of ±10 TECU. According to information on the accuracy of IGS products (see http://igscb.jpl.nasa.gov/), the GIM errors are within the range of ±2 to ±8 TECU. As the average formal error of the VLBI results is at about ±3 TECU, it can be concluded that both techniques agree well.
 As VLBI is not observing every day and data gaps might vary according to scheduling, dedication, maintenance, or budgets, the gained time series are not equally sampled. This has to be taken into account when computing the frequency spectra of the VLBI results. Two suitable algorithms that are able to treat unequally sampled data are mentioned by Foster [1996a, 1996b]. We have applied the CLEAN algorithm according to Baisch and Bokelmann  to the data.
3.2. Spectral Analysis
 CLEANed spectra of the VLBI outcomes were derived, using the algorithm described in section 3.1, and compared to GPS results. Furthermore, values of solar radio flux at 10.7 cm (F10.7) back to 1 January 1984 were downloaded from the World Data Center for Solar-Terrestrial Physics, Chilton, and were analyzed, too. Again, results for station Wettzell will be presented here, as this site can provide the longest time series. As values of F10.7 contain some data gaps, the CLEAN algorithm had also to be applied to this time series. Finally, all outcomes were normalized by the biggest amplitudes in order to compare TECU numbers to F10.7. Figure 6 shows the gained spectra from VLBI, GPS, and F10.7. Only VLBI and F10.7 show a sharp peak which can be assigned to the main solar cycle. The maximum amplitude from VLBI results is found at 10.5 years, which is identical to the maximum from F10.7. As also expressed by Figure 6, we see that GPS and VLBI have the same amplitudes for the diurnal, semiannual, and annual periods. As expected, GPS cannot detect longer periods than about 5 years in the ionosphere as time series are not long enough. The same series from VLBI and GPS were taken and wavelet spectrograms were obtained, as described, for example, by Foster [1996c] or Schmidt . The results are plotted in Figure 7. For our comparison we focused mainly on annual signals. The influence of the last two solar cycles can be seen clearly in the VLBI scalograms as the amplitude of the annual signal corresponds to the long-term variations of the activity of the Sun. Scalograms from GPS results confirm this variation in the annual period domain; both techniques differ less than half a day from the expected 365 day period.
 As described in the previous sections, the ionosphere, expressed by values of vertical total electron content, can be monitored by VLBI in an absolute sense. For the first time, VTEC values can be gained from VLBI without any external information using the algorithms presented here. Therefore VLBI can be used as a tool to study the ionosphere, but as VLBI is not a continuously observing technique, its preferable use with respect to ionospheric research will be to contribute to long-term studies or to validate theoretical or measurement-based local, regional, and global ionospheric models rather than providing results for routine monitoring of the ionosphere. At the moment, the combined IGS global ionospheric model which is derived from the results of five different GPS analysis centers is cross validated against Jason and/or Envisat measurements on a routine basis [Hernandez-Pajares, 2004], which shows an absolute bias of less than 1 TECU in most of the cases and a standard deviation of about ±5 TECU. We think that VLBI can be taken as a third independent technique for validation of IGS GIMs. As VLBI is the only technique which is covering more than two solar cycles, it is able to provide important input for theoretical models of the ionosphere and for modeling long-term trends.
 The final goal of ionospheric modeling using space geodesy should be the development of a global four-dimensional model of the ionosphere, assimilating several independent space techniques, obtaining the most robust and reliable solution of electron densities values. How VLBI can contribute to such a model depends not only on data quality and the number of observations but also on the ability to process the observations in near real time and to provide the measurements and/or ionospheric parameters as soon as possible to combination centers.
 We have presented a method to gain ionospheric parameters in terms of vertical total electron content in an absolute sense without any external information from geodetic VLBI measurements. The necessary model assumptions for the estimation of these parameters from differential measurements were presented. The results from VLBI agree well with GPS in the time and frequency domain. For globally distributed VLBI stations, the biases to GIMs are less than 3 TECU; the standard deviation is less than 10 TECU. Long-term trends in the results were compared to values of solar flux at 10.7 cm; for the last two solar cycles, the periods and intensity variations agree very well.
 We are very grateful to the Austrian Science Fund (FWF), which funded the research project P16136-N06 “Investigation of the ionosphere by geodetic VLBI.” Furthermore, the first author wants to thank the Japanese Society for the Promotion of Science (JSPS) (project PE 04023) and Kashima Space Research Center (NICT) for supporting his research and enabling him to get the necessary knowledge of VLBI technology. The International VLBI Service for Geodesy and Astrometry (IVS) and the International GPS Service (IGS) are acknowledged for providing data.