In Global Navigation Satellite Systems (GNSS) using L band frequencies, the ionosphere causes signal delays that correspond to link-related range errors of up to 100 m. Whereas this error can be corrected in dual-frequency measurements by a linear combination of L1 and L2 phases, in single-frequency measurements, additional information is needed to mitigate the ionospheric error which is proportional to the total electron content (TEC) of the ionosphere. This information can be provided by TEC maps deduced from corresponding GNSS measurements or from model values. Besides direct range error correction in navigation and remote sensing applications, TEC or electron density models play a key role in ionospheric monitoring and forecasting. In this paper we discuss the development and use of TEC models for calibrating TEC, reconstructing reliable TEC maps, and forecasting TEC behavior based on GNSS measurements. European and global TEC maps and corresponding 1 h ahead forecasts are distributed via the operational space weather and ionosphere data service (http://swaciweb.dlr.de) to the international community. The Neustrelitz TEC Model is a basic approach for a family of regional and global TEC models used in different types of applications. The model approximates typical TEC variations depending on the location, time, and level of solar activity with only a few coefficients.
 When traveling through the ionosphere, L band signals of Global Navigation Satellite Systems (GNSS) are delayed. The delay corresponds to range errors of up to 100 m. In a first-order approximation the range error di is proportional to the integral of the electron density along the raypath (total electron content (TEC)) and may be approximated by:
where K = 40.3 m3s−2 and the integral of the electron density ne along the raypath s defines the slant total electron content (TECslnt).
 Whereas this error can be corrected in dual-frequency measurements by a linear combination of L1 and L2 phases, in single-frequency measurements, additional information is needed to mitigate the ionospheric error. This information can be provided by TEC maps deduced from corresponding GNSS measurements or by model values. Hence, single-frequency systems used for example in positioning and in remote sensing radars rely on ionospheric models which provide a climatological estimation of the ionospheric impact. For estimating the transionospheric time delay or range error, several ionospheric models are currently available.
 Whereas single-frequency GPS users may correct ionospheric range errors by the broadcasted GPS correction or Klobuchar model [ARINC Research Corporation, 1993; Klobuchar, 1987], Galileo users will apply the NeQuick model [Hochegger et al., 2000; Radicella and Leitinger, 2001; Leitinger et al., 2005; Nava et al., 2008]. The Klobuchar model describes the diurnal variation of vertical ionospheric delay by a cosine function with varying amplitude and period, depending on the geomagnetic latitude. For nighttime hours the vertical ionospheric delay is approximated to a constant value fixed at 5 ns. The related third-order polynomial is determined by eight coefficients which are broadcasted via the GPS navigation message. Thus, any GPS user, knowing the satellite geometry can reduce the link-related ionospheric range error easily by about 50% [Klobuchar, 1987]. Single-frequency users of the European global navigation satellite system ‘Galileo’ are offered to use the electron density model NeQuick for range error corrections.
 NeQuick, developed at the International Centre for Theoretical Physics in Trieste and at the University of Graz, is a quick run 3-D electron density model from which TEC can be determined along any ground-to-satellite or satellite-satellite raypath by means of numerical integration. Electron density models such as International Reference Ionosphere [Bilitza, 2001] or the Bent model [Bent et al., 1972] may represent the peak density very well but may fail in estimating TEC [e.g., Brown et al., 1991; Batista et al., 1994; Mazzella et al., 2002; Coisson et al., 2006].
 The other option to correct ionospheric propagation errors relies on current monitoring data as demonstrated by Satellite Based Augmentation Systems (SBAS). The ionospheric correction information provided by SBAS systems such as the European Geostationary Navigation Overlay Service [Ventura-Traveset and Flament, 2006] and the Wide Area Augmentation System [El-Arini et al., 2001] consists of vertical ionospheric delays and associated residual error bounds called grid ionospheric vertical errors at the nodes, or ionospheric grid points, of a predefined ionospheric grid with a grid spacing of 5° by 5° in latitude and longitude. As we will discuss in the subsequent chapters, also this type of TEC data provision to users may benefit from TEC modeling. Thus, TEC models may assist the calibration of TEC measurements, may help to generate operational TEC maps and finally may be used to forecast the ionospheric behavior. All these aspects are taken into account in the operational ionosphere data service provided by the Space Weather Application Center–Ionosphere (SWACI, http://swaciweb.dlr.de) and will be described in sections 2–4.
2. Modeling Activities
 GNSS-based TEC monitoring is carried out in German Aerospace Center (DLR) Neustrelitz routinely since 1995 over the European area with a temporal resolution of 10 min [e.g., Jakowski, 1996]. The database is mainly provided by the European GPS ground station network of the International GNSS Service [Dow et al., 2009]. To ensure a high quality of the TEC maps also in case of only a few measurements or at greater distances from the measuring points, the measured data are combined with an empirical TEC model which has been developed specifically for the region in view [e.g., Jakowski, 1996; Jakowski et al., 1998]. Besides the European area also Northern Polar cap area (ϕ > 50°N) has started to be monitored in 2002 and continued since that time (see http://www.kn.nz.dlr.de/daily/tec-np/). Although the data coverage is extremely poor, also monitoring of the Southern Polar cap area (ϕ > 50°S) is carried out for DLR internal use.
 For all these regions a family of empirical regional ionospheric models has been developed. The different regional TEC models (Neustrelitz TEC Model (NTCM)) are developed for using them as background models in regional TEC mapping procedures, for Europe NTCM-EU, for North and South Pole areas NTCM-NP and NTCM-SP, respectively.
 Following the same principles of mathematical approach, recently also a global TEC model (NTCM-GL) has been developed [Jakowski et al., 2011].
here Hi (h) denotes the diurnal and semidiurnal variation, Yj (d) the annual and semiannual variation, Lk (ϕ,λ, h, d) the dependence on the latitude and the solar zenith angle, Sl (F10) the dependence on the solar activity and cijkl the coefficients.
 The linear model coefficients are determined by a least squares fitting procedure. The solar activity level dependence is controlled by the solar radio flux index F10.7. This parameter varies from about 70 at low solar activity (LSA) up to more than 200 at high solar activity (HSA) conditions.
 To convert the vertical TEC to slant TEC along the raypath, a common single layer mapping function M(ɛ) is applied according to:
height of ionosphere single-layer approximation [km];
slant TEC along the raypath;
vertical TEC at subionospheric or pierce point;
Earth radius [km];
elevation angle [rad].
For reasons of simplicity, in the following, if not specified, TEC means vertical TEC. The RMS error of the vertical TEC data referred to monthly medians is generally better than 3 TECU (1 TECU = 1016m−2, equivalent to a range error of 16.2 cm at L1 frequency) under HSA conditions over the European area. At LSA conditions the standard deviation from monthly medians is below 1 TECU (<16 cm). In the operational service SWACI we use a simplified TEC model for bias calibration and TEC mapping as will be described in section 3. Neglecting annual and solar cycle dependencies, such an operational model may sufficiently work with only 12 coefficients. To further optimize the instantaneous modeling approach, we have reduced Hi to 3 terms and increased Lk to 4 terms.
Figure 1 indicates also that the Klobuchar model approach deviates significantly stronger from TEC map reconstructions of DLR and CODE and from the NTCM-EU60 model. In the climatological sense the latter is comparable with the data-based reconstructions at CODE and DLR. This is certainly not the case under perturbed conditions.
 At CODE the vertical TEC is modeled with a spherical harmonic expansion up to degree 15 and order 15 referring to a solar geomagnetic reference frame. The two-hourly map sets are derived from GPS data of the International GNSS Service (IGS) network [e.g., Schaer et al., 1998]. The CODE has been routinely generating global ionosphere maps (GIM) on daily basis since 1 January 1996 by using more than 130 IGS station's data. The GIM/CODE is regarded as one of the most precise TEC maps generated from GPS observations. CODE contributes to the IGS model [Hugentobler et al., 2000]. The global IGS TEC grid, provided with a latency of 11 days, has a time resolution of 2 h and an estimated accuracy of 2–8 TECU, grid spacing 5° by 2.5° in longitude and latitude.
 Considering the fact that CODE data are available since 1996 on global scale, we started developing a global TEC model following the same mathematical approach as described above [Jakowski et al., 2011]. This global TEC model provides a simple and easily accessible representation of temporal and spatial variations of global TEC under permanently varying conditions of solar activity. The model requires only 12 coefficients and may be used for a full solar cycle using the F10.7 cm radio flux as input parameter. Due to the simplicity of the model some application advantages exist in operational systems, in particular no integration of electron density profiles is needed.
 To check the quality of the model, TOPEX and JASON altimeter data has been used as independent reference data source (ftp://cddis.gsfc.nasa.gov/gps/products/ionex/). For May 2002 and December 2006 NTCM-GL and NeQuick vertical TEC estimates are calculated at coincident location and time windows. Subsequently, their differences from the altimeter TEC, i.e., TECNTCM-GL - TECaltimeter and TECNeQuick - TECaltimeter, are computed. The histogram plots of differences are shown in Figure 2 [cf. Jakowski et al., 2011].
 The corresponding root-mean-squares (RMS), mean deviation and standard deviation (std) of differences are given in the upper left corner in Figure 2. Principally, we expect mean positive deviations of a few TEC units characterizing the plasmaspheric electron content between 1330 and 20.000 km height. The observed negative values indicate erroneous TEC calibration of TOPEX/Poseidon altimeter data in the order of 5–10 TECU.
 Considering these HSA and LSA data sets, we see that both models perform very similar.
3. Application of TEC Models
3.1. TEC Calibration of GNSS Measurements
 An operational ionosphere data service like SWACI requires the operational determination and control of the interfrequency biases in order to compute absolute TEC values from the input GNSS data.
 To estimate TEC from GNSS observations, instrumental biases at the receiver and satellites must be determined. Dual-frequency phase measurements at L1 and L2 carrier frequencies of GPS provide only accurate relative TEC values: The observation equations for carrier phase measurements are given by:
where ρ denotes the geometric distance between satellite and receiver, including the remaining nondispersive error contributions, TECslnt denotes the (true) slant TEC, and N1,2 denotes the integer ambiguities on the L1 and L2 carrier frequencies; λ1,2 are the corresponding wavelengths; by ɛL1,2 we denote collectively the remaining error sources as, e.g., the phase noise. The analogous description for the code measurements reads
 In order to obtain calibrated TEC data from carrier phase measurements alone, the phase ambiguities and biases have to be determined. Since it is hard to accurately determine phase ambiguities and phase biases, we determine absolute TEC data from both, differential code and carrier phase measurements as illustrated in Figure 3. We use the low-noise carrier phase derived relative TEC to smooth the code-derived relative TEC by
with the offset TECoffslnt between differential code and carrier phases given by
This smoothing is performed in near real time.
 By 〈TECoffslnt〉 we denote the mean value of TECoffslnt. It is updated in each epoch i; that is, for epoch number i + 1 it is given in terms of the old mean 〈TECoffslnt〉i and the new measurement TECoff,i+1slant by
An analogous formula holds for the RMS of TECoffslnt which is used to estimate the measurement error. Cycle slips are detected as jumps in TECoffslnt exceeding a threshold proportional to the RMS of this difference. When a cycle slip is detected, the calculation of the mean is reset and started again. The computation of the RMS is started with an initial value of 100 TECU.
 The measured differential time delay is a measure of the absolute TEC after instrumental signal delays (biases) at satellite and receiver level have been removed.
 Principally, the calibration requires an ionospheric model for which usually simple approaches are used. Instead of using a simple polynomial approach for modeling the ionosphere [e.g., Sardón et al., 1994], here we propose using a well qualified ionospheric model which can better adapt to the real ionospheric behavior. Thus, we follow the approach described by Jakowski and Jungstand  and use the NTCM for the calibration procedure. Note, however, that unlike Jakowski and Jungstand  where the NTCM is used to provide fixed reference TEC values, here the NTCM is continuously updated in the same process in which the biases are determined.
 The calibration approach uses the NTCM TEC model at epoch i:
where bRX and bSAT represent the interfrequency or differential code biases of GNSS receiver and corresponding GNSS satellite. The term ɛN explicitly keeps track of the measurement errors which are determined from the code carrier phase smoothing as described above. TECNTCMslnt stands for slant TEC derived from a regional or global NTCM TEC model depending on time, latitude, and longitude as indicated in equation (2). In operational applications long-term dependencies such as seasonal and solar activity impact can be ignored, i.e., the number of basic functions discussed above can be reduced, thus lowering the computational complexity of the model.
 The required slant TEC values are obtained from the model by applying the elevation-dependent mapping function M(ɛ) given in equation (3). The spherical ionospheric shell height is fixed at 400 km. Finally we determine the dependence of the operational TEC model NTCMop − XX on elevation (or more general on the geometry) which allows us to separate the slowly varying biases from the more rapidly varying TEC model. XX stands here for the selected region, e.g., EU for European or GL for global area.
 The interfrequency biases (IFB) bRX and bSAT are obtained by least squares fit of the model coefficients and biases to the observation data. Once every 24 h we perform weighted least squares estimation of model coefficients and biases using the linear approach indicated in equation (2) which can be rewritten as
with the slant TEC measurements y, the model coefficients x, the measurement errors ɛ, and the matrix A which contains the TEC model basic functions and entries ±1 for the biases. It is instructive to derive the weighted least squares (WLS) solution from a cost function of the form
The WLS solution minimizing this cost function is given by
 This completes our discussion of the WLS solution which is performed every 24 h using the complete data set of the preceding day.
 To get solutions every 5 min, a linear least squares–based recursive filter is used for estimating current model coefficients and biases. The equations for this filter can be derived from a cost function of the form
The difference between this cost function and the cost function for the WLS solution is the second summand: it says that the solution of the last epoch, xm, projected to the current epoch with the projector Pm+1,m, should agree to the current solution, xm+1, weighted by Cm+1 = Cm + R, with the matrix R parametrizing the mismatch of the model to the data. A sample of the derived operational TEC model for 4 December 2010 at 15:05 UT is shown in Figure 4.
 Since all measurements are carried out along radio links between one of the GPS satellites and one of the globally distributed receivers, the absolute values of all satellite and all receiver biases remain uncertain. To get comparable results for the biases, we follow the common practice determining the biases in such a way that the condition ∑ bSAT = 0 is fulfilled [e.g., Hernández-Pajares et al., 2009].
 Thanks to the model-assisted calibration technique, the calibration of IFBs and the subsequent generation of TEC maps can be carried out in near real time. Samples for GPS satellite biases monitored over some days in October 2010 are shown in Figure 5.
3.2. TEC Mapping
 To reconstruct TEC maps over a selected area, the observation data are assimilated into a specific TEC background model NTCM-XX [cf. Jakowski, 1996; Jakowski et al., 1998]. This model assisted approach has the advantage that in case of only a few measurements (e.g., over Antarctica) or even in case of total loss of input data, the operational data service is maintained by providing model values. Since ground-based GNSS data are often unevenly distributed, the inclusion of a background model in the TEC reconstruction helps to overcome such data gaps which predominately occur over the oceans.
 The TEC data assimilation procedure starts with the adjustment of the background model with respect to all observations in a least squares sense.
 Assuming that the adjustment value at measurement epoch i is TECadji, we compute the deviations ΔTECji at all N ionospheric piercing points j according to:
Here TECNTCM–XXi corresponds with the current model approach using coefficients from the same day. Due to the model adjustment, the deviations ΔTECji fulfill the condition:
To describe the influence of the TEC value measured at piercing point j at a distance Djkli in grid point (k,l) at the map area, we apply a Gaussian-type weighting function WF which is defined as:
This is done for each of the N observations. The free parameter sw controls the horizontal extension of the influence of the deviations upon the final result which can be tuned according to data density; that is, bigger sw is chosen if data coverage is poor.
 The resulting deviation of all measurements from the adjusted TEC model at the grid point (k,l) of the TEC map are then computed by:
where γ is a small value to prevent singularity of equation (18) in operational mode.
 Taking into account the model adjustment value ΔTECadji, the final TEC value at grid point (k,l) is then given by:
Applying this procedure for all grid points, the complete TEC map is generated. The final results represent measured TEC values in the vicinity of the measurement points whereas at greater distances from measurements somewhat modified model values are provided. Depending on the number of available stations and the correlation length, the width sw of the weighting function can be modified in a proper way.
 Comparing the NTCMop − EU model and the corresponding assimilated TEC map for 5 December 2010 in Figure 6, we see that the model already provides a good mapping quality what is expected under geomagnetically quiet conditions.
 The advantage of the model-assisted technique is seen in Figure 7. The right panel indicates the RMS deviations of the TEC measurements from the background model as useful indicator of the TEC error. The influence of the weighting functions defined in equation (17) can clearly be seen around the piercing points of GNSS satellite – receiver links. Outside this area the impact of measurements is small, i.e., model values dominate. Nevertheless, as the map at the left panel shows, there is a smooth transition of TEC between well covered areas and those areas where no data are available.
 To demonstrate the effect of the assimilation from another point of view, Figure 8 shows four simultaneous time series of TEC at 50°N; 10° E for some days in February 2002 obtained from operational background models NTCMop − EU and NTCMop − GL and related TEC assimilation products, i.e., European and global TEC maps. Although the background models may deviate over the selected point, the assimilated data products are rather similar. Obviously the data coverage over Europe is so good that generated TEC becomes independent from the model. Validation of TEC maps will be continued.
3.3. TEC Forecast and Data Service
 Although nowcast information of ionospheric propagation errors helps already in GNSS application practice, forecast of errors and their development are of particular interest for GNSS customers. Since there is always a time delay between time of measurements and service provision, the derived propagation errors do not exactly correspond to actual propagation conditions. Thus, to give an example, the Space Weather Application Center provides TEC maps with an update rate of 5 min. Considering the delay due to computation time and data management, the provided information is delayed by about 8 min at user level. To overcome this operational gap, corresponding forecast tools can be applied. Doing so, the user can work with an estimated real-time value. If applications require some warning or planning time to be well prepared when ionospheric perturbations are approaching, the forecast must exceed the aforementioned time interval. Since currently no reliable forecasts of ionospheric behavior (TEC) during perturbations are available, much work must still be done.
 In this section, we present a simple model-assisted forecast algorithm which may help to provide some preliminary results and surely helps to learn about ionospheric forecast problems. To give an example, Figure 9 provides a 1 h sample forecast.
 Our TEC map–related forecast algorithm takes benefit from actual trends of the TEC behavior at each grid point. During perturbations, characterized by large TEC fluctuations or ionisation fronts, this approach may seriously fail. So we merge the trend information with the current background model which provides a stable climatologically TEC behavior. In average such a forecast will be better than applying the trend or model approach alone. The data reconstructed at grid point (k,l) at epoch i, are merged in the following way:
where TECfci (k,l) is the forecasted TEC at grid point (k,l) for Tfc hours ahead, the parameter η (0 ≤ η ≤ 1) is a weight factor controlling weight of model and actual trend and ti is the time at measurement epoch i.
 The presented solution is a first step to regularly provide forecasted TEC maps. Much work is still needed to develop a reliable ionospheric storm models for TEC which may be used to forecast TEC also under perturbed conditions. Due to the complex interaction of the ionosphere, thermosphere and magnetopsphere, composition changes and highly dynamic perturbation forces such as neutral winds and electric fields are extremely difficult to model.
 The described model assisted procedures for TEC calibration, mapping and forecast are all used in the ionospheric data service SWACI in DLR Neustrelitz.
 DLR is establishing an operational ionosphere data service via the project Space Weather Application Center – Ionosphere (http://swaciweb.dlr.de) since 2006. SWACI offers regional and global TEC maps, corresponding model information and hourly forecasts using ground- and space-based GNSS measurement techniques [e.g., Jakowski et al., 2005].
4. Summary and Conclusions
 Ionospheric TEC and electron density models are not only needed for correcting range errors and travel time delays in navigation and remote sensing applications, but can also be used for calibration and model improvement tasks. Based on currently applied procedures in the ionosphere data service SWACI, it has been shown that models can essentially help to calibrate TEC retrievals from GNSS data, to improve TEC mapping and to estimate hourly TEC forecasts. Consequently, a family of regional empirical TEC models has been developed in DLR Neustrelitz which provides climatological information on TEC behavior.
 With the exception of the Klobuchar model used as the single-frequency ionospheric correction model by GPS, a simple global TEC model is not yet available. The global TEC model, recently developed by Jakowski et al.  might help to overcome this gap. Since this model using only 12 coefficients is easy to handle, it is well suited to be used in operational services and near-real-time applications.
 It has been shown that the model-assisted mapping technique provides consistent results. If data coverage is good, the assimilated results are independent from the background model. The big advantage of the model-assisted technique is evident over areas with low or small data density. Ionospheric forecasts rely on ionospheric modeling. Short-term forecasts up to about 1 h in advance may be assisted by a climatologically background model. Long-term forecasts require the application of an ionospheric storm model for TEC which is not yet available. Best solutions will certainly be achieved in the future by data driven physical models.
 The authors thank IGS and CODE for making available high-quality GPS observation data and modeling results, respectively. The authors thank the reviewers for their valuable comments and recommendations. This research was primarily supported by the Ministry of Education and Science of Mecklenburg-Vorpommern under grant AU 07 008.