In this work we present a simple technique to estimate the medium-scale traveling ionospheric disturbances (MSTIDs) characteristics (such as occurrence, velocity, vertical propagation) with periods lower than 20 min and its application to a set of GPS data both temporally and spatially representative (near one solar cycle and four local networks in the Northern and Southern Hemispheres, respectively). Some of the main results presented in this paper are the MSTIDs which occur at daytime in local winter and nighttime in local summer, related to the solar terminator and modulated by the solar cycle. They present equatorward (from ∼100 to 400 m/s) and westward (∼50 to 200 m/s) horizontal propagation velocities, respectively. The corresponding periods are compatible (higher) with the theoretical prediction, which is given by the neutral atmosphere buoyancy period associated with the Brunt-Väisälä frequency (about 600 s). Moreover, higher TIDs productivity is mainly associated with the downward vertical propagation. Finally, the results obtained in this study suggest the possibility of developing future MSTID models to mitigate its impact in applications like precise satellite navigation.
 Some authors [e.g., Hunsucker, 1982; Jacobson et al., 1995] distinguish between large-scale TIDs (LSTIDs) with a period greater than 1 hour and moving faster than 0.3 km/s, and medium-scale TID (MSTIDs) with shorter periods (from 10 min to 1 hour) and moving slower (0.05–0.3 km/s). The LSTIDs seem to be related with geomagnetic disturbances (i.e., aurora, ionospheric storms, etc.) that in high latitudes heat the thermosphere by Joule effect. This heating of neutral atmosphere produces an energy transfer toward lower latitudes in the form of thermospheric waves that in high altitudes, interact with the ions. Some authors have pointed out that the origin of MSTIDs is more related with meteorological phenomena like neutral winds, eclipses, or solar terminator (ST) that produce atmospheric gravity waves (AGW) manifesting them as TIDs at ionospheric heights [Somsikov, 1995], due to the collision between the neutral and ionized molecules. Under these circumstances, AGW can be produced with frequencies lower than the Brunt-Väisälä frequency (νB = 1.7 mHz) moving with phase velocities lower than the ST velocity.
 Despite the small amplitude of the MSTIDs, typically of tenths of a TECU (1 TECU = 1016 electrons/m2), several authors [Chen et al., 2003; Wanninger, 2004; Orus et al., 2003] have shown that the presence of such ionospheric disturbances causes a decrease (sometimes dramatic) on the performance of precise navigation strategies. This is because the slant differential ionospheric delays should be predicted (interpolated) with a high precision (better than 0.25 TECU) in order to navigate with few centimeters of error by fixing the carrier phase ambiguities in real-time [see Hernandez-Pajares et al., 2000]. Also, in this ionospheric interpolation process, the TIDs can introduce a significant error within the networks with baselines from tens to hundreds of kilometers, significant distances compared with the MSTID wavelengths. In Figure 1, an example of this relationship between the presence of MSTIDs and the error in the ionospheric correction is depicted (in a similar way as the I95 index [Wanninger, 2004]), for a rover navigating on a midlatitude RTK network during the year 2002. As it is shown, large errors on the ionospheric correction correspond to large amplitudes of the TIDs for both local time and seasonal dependences.
 A statistical study of the presence of these ionospheric perturbations and its main propagation characteristics is made. Section 2 describes the methods that we have developed for such purpose, and section 3 is devoted to the application of these methods along several years and several geographical locations.
2. Detecting and Measuring MSTIDs
 The basic observable that we use to detect TIDs is the geometry-free (or ionospheric) combination of the two GPS carrier phases, that, as it is well known, is proportional to the total electron content (TEC) plus an unknown bias that includes the carrier phase ambiguity and the instrumental delays, in such a way that for every satellite-receiver pair we have
where Li is the geometry-free combination, L1 and L2 are the two GPS carrier phases (in meters), α is a proportionality factor (α ≃ 10.5 m/TECU), and b is an unknown bias constant for each different satellite-receiver continuous arch of data.
 We will assume that TIDs are produced by an interaction (ion drag) between the neutral particles under an AGW with the ionized particles that are constrained to move along the magnetic field lines. The importance of such interaction will depend on both the neutral and ionized particle densities. Assuming an exponential density variation for the neutral particles and a typical ion/electron vertical distribution with a pronounced maximum (hmF2), the maximum of this interaction (i.e., the TID generation) will occur at an altitude below hmF2, depending on the wideness of the ion/electron maximum and the scale height of the neutral particles density. Taking this into account, we have assumed that this interaction occurs on a thin layer with an altitude of 200 km, but we have also considered the effect of assuming other heights.
 In order to detect such TIDs, the first step is to detrend the data from well-known dependencies, such as diurnal variation and elevation angle dependences. This detrending can be done in several ways, for example, Tsugawa et al.  makes this detrending by looking for vertical TEC perturbations from a dense GPS network, whereas other authors make the detrending using a band filtering or a polynomial adjust of the ionospheric data over a period of 1 hour of data. However, the measurement noise is low enough (typically few hundreds of TECU, equivalent to few millimeters of Li) when it is compared with the natural variation of Li (i.e., Li is smooth, also when TIDs occurs), that is possible to detect such fluctuations in a shorter period of time. Considering this, the detrending is done simply by subtracting from each value an average value of the previous and a posterior measurements (i.e., the curvature of the Li temporal dependency).
where τ has been chosen in order to have a significant variation on Li (300 s in our case). It is interesting to note that this detrending procedure can be used in real time with a single receiver, so it is adequate to identify these ionospheric perturbations in navigation applications. This method can play a similar role to the I95 defined by Wanninger , but in our case a single receiver is only needed to define it. This is the reason to call this new index as Single Receiver TID Index (SRTI).
 With this detrending, the amplitude (A′) of δL will differ with the amplitude (A) of the TID by a factor that depends on the TID period (T):
In this sense, the method will have the major sensitivity for TID periods of 600 s (A′ = 2A), and will be quite efficient between 400 s and 1200 s (at these periods A′ = A).
 This problem is common in all detrending procedures, for example, in the Figure 2, it is shown how this method works on two perturbations of different periods and it is compared with the detrending using polynomial fit with different degrees. It can be seen that the sensitivity of the SRTI is comparable with the sensitivities of the polynomial adjusts and also that the sensitivity (in all detrending) depends on the perturbation periods. In our case, the SRTI with a sampling rate of 300 s will be more efficient to detect perturbations between 400 s and 1200 s. Taking into account that the buoyancy period of the neutral atmosphere is typically around 600 s, one should not find many perturbations with periods under 400 s if these perturbations are related to AGWs. Longer period TIDs (such as 1 hour) will be quite undetectable with this sampling rate of 300 s, beside this, such long-period perturbations would be mixed with the variation of the slant TEC (STEC) due to the obliqueness factor or mapping function (i.e., the relationship between the vertical and slant TEC) and to the movement of the GPS satellite that in this long periods of time, will cause a large variation in the position of the pierce point (i.e., the intersection of the receiver-GPS line of sight with the ionosphere, assumed as a thin layer at a fixed height). In any case, the determination of the TIDs characteristics (such as period, velocity, azimuth) are not affected, once detected, by this sensitivity factor.
 Once the data are detrended, the following step is to detect TID-like perturbations. This is done through a FFT over the detrended data. As the International GNSS Service (IGS) data has a typical temporal resolution of 30 s, we have chosen arc windows of 3840 s that corresponds to 128 observations, with steps of 900 s. We decide the presence of a TID if there exists any mode between 5 and 30 min with an amplitude greater than 0.1 TECU (≃1 cm Li). To avoid any significant effect from the mapping function uncertainty, an elevation mask of 50 degrees is applied (i.e., data below 50 degrees are excluded), in such a way that the STECs can be used directly before the FFT filtering, in order to obtain the corresponding TID parameters.
 An estimation of the propagation parameters (i.e., velocity and direction) can be done using small networks of receivers with distances of up to several tens of kilometers, lower than the expected wavelength (horizontally projected): choosing a reference receiver, one can determine these propagation parameters through cross-correlating the TEC perturbations with the perturbations suffered by the other network receivers.
 For a correct estimation of the propagation parameters, it has to be taken into account that the relative movement of the GPS satellite and the receiver makes that observations, for the same satellite-receiver pair, will occur in different ionospheric pierce points. This introduces a Doppler effect that must be corrected in the observational equations. The Doppler effect depends on the altitude where the TID occurs. In Figure 3 the ionospheric pierce point velocities observed from a reference GPS receiver, and considering two different heights (100 and 300 km), are depicted. However, we will show below (Figure 4) that this impact is in general small in the final results. We have assumed an altitude of 200 km because it is a compromise between the need of high enough neutral particle density where the AGW propagates and also a high enough ionized particle density interacting with the neutral atmosphere.
 Assuming the TID propagates as a planar wave, for an instant t and a pierce point rpp, we can express the ionospheric disturbance (δLi) as:
where F is an arbitrary function, ω is the frequency, and k is the propagation vector. Thus defining, as Jacobson et al. , the slowness vector s = , δLi can be expressed as:
Then, looking for correlation between this ionospheric perturbation for different receivers, it is easy to see that the temporal delay for maximum correlation (Δtmax) between two receivers, with a relative vector between its pierce points Δrpp moving with a relative velocity vpp, must verify that
Applying this equation in a network of receivers, one can obtain the components of s and, as a consequence, the horizontal velocity.
 Alternatively to this method, the relationship between the derivatives of equation (5) can be used:
that also allows to compute the components of s. This method, which is based on the δLi derivatives as Afraimovich et al.  does, however predicts no Doppler effect when vpp is perpendicular to the propagation vector, as it is expected.
 The result of such cross-correlation will not be only the horizontal propagation velocity but also the frequency of the TID (ω). This can be done by comparing the maximum and minimum correlations (Δtmin), from equation (6) taking into account the Doppler effect:
From equation (6), it can be seen that the importance of the Doppler effect depends on the ratio between the velocity of the pierce point and the velocity of the TID. As it is shown in Figure 3, this pierce point velocity will depend on the assumed altitude for the ionospheric layer: it can be of 50 m/s for an altitude of 300 km, or 20 m/s for an altitude of 100 km. This means that if the Doppler effect is not considered, a mean error of about a 20% can be introduced in the parameters estimation (velocities and frequencies), but in some cases, this error can be greater than the 100%. Therefore the model for the ionosphere can have some influence in the velocity estimation, mainly if the propagation velocity is comparable to the pierce point velocity. However, in general, we will see that its influence is not significant in the final results (Figure 4).
 Once ω has been estimated, the vertical component of the propagation vector using the AGWs dispersion relationship derived by Hines  can be estimated.
where c is the speed of sound, ωa the acoustic-cutoff frequency, and ωB the Brunt-Väisälä frequency (all these three parameters must be computed at the assumed AGW height) and x and z denote the horizontal and vertical directions, respectively.
 From equation (9), the propagation elevation angle can be obtained:
where vH is the measured phase velocity horizontal component (i.e., vH = ).
Equation (10) does not allow to determine the sign of θ (i.e., one can not distinguish between upward and downward propagations). This propagation direction can be obtained through an indirect way by looking at the angle β (see Figure 5) between the local magnetic field and the direction of the displacements of the neutral particles (un, as transversal wave) and taking into account that the ion motion (uion) is constrained to moving along the geomagnetic field lines. In this way, one should not expect TIDs at values of β close to 90 deg. In Figure 5, it can be seen that in this particular example, assuming a polarward TID detection, an AGW propagating upward would be more effective in the TID production than an AGW propagating downward (not represented in the figure for clarity reasons). This is because the displacement of the neutral particles associated with an upward AGW will show a higher projection on the magnetic field direction, compared to an AGW assumed propagating downward. However, we have to emphasize that this is just an example, and we will show that below most of the detected TIDs are equatorward and westward and compatible to downward propagation.
 To analyze the occurrence and parameters of MSTIDs, the global network of IGS receivers is used. They are constituted by survey-grade dual frequency GPS receivers on permanent sites, providing, in this way, direct biased STEC observation of several satellites simultaneously, typically between 4 and 8, each 30 s.
3.1. Occurrence of MSTIDs
 Following the procedure described in the last section, we are able to easily analyze a great number of data in order to characterize the presence of the MSTIDs. In Figure 6, we represent the measured amplitudes for an IGS receiver EBRE (40.6N,0.5E) as a function of the year from 1997 to 2004, and the local time in the vertical axis. From this image, it is clear that MSTIDs are more evidently close to the solar cycle maximum. This solar cycle dependency can be expected if one assume that TIDs are produced by interactions (collisions) between neutral particles and ionized particles; owing to that, these collisions depend on the ionization degree, the AGW effectiveness on producing TIDs will be greater when it is closer to the solar cycle maximum.
 A seasonal dependency noticed by some authors is also clear: more frequent during winter at daytime ([Jacobson et al., 1995] and [Warnant and Pottiaux, 2000]), but they are also present during summer at nighttime ([Shiokawa et al., 2003]). This local time dependency appears to be confined by the Solar Terminator, which is also represented.
 These relationships (seasonal and with the LT) can be also seen in Figure 7 for the networks of Table 1, only for 1 year of data but covering different coordinates. In this figure, the TIDs with amplitude greater than 0.1 TECU, and observed with an elevation angle greater than 50 deg, are only plotted (see details below).
Table 1. Location of the Four Networks Studied in This Papera
First column: network label; second and third column: starting and ending date of the analysis in years and days of year; fourth column: receiver ids; fifth to eighth columns, receiver spherical and geomagnetic longitude and latitude in degrees; last column, and for each network, distance regarding to the reference station in kilometers.
 From these two figures, we can conclude that in general, MSTIDs occur at daytime (approximately from 0700 to 1700 of LT) during local winter and at nighttime during local summer. This occurrence has been observed in the four studied networks and appears related to ST. In order to quantify the occurrence we have defined the corresponding probability as the percentage of events detected as TIDs (amplitude greater than 0.1 TECU) over the total number of considered observations (with elevations greater than 50 degrees). The corresponding plots for day and night time can be seen in Figures 8 and 9, respectively.
3.2. Propagation Parameters
 In order to characterize the propagation parameters of the MSTIDs, we have done this study for four different networks at different coordinates (see Table 1). A European network (EU), another located on North America (NA, at similar geomagnetic latitude as EU), a Middle East (ME) network, and a Southern Hemisphere network placed in New Zealand (NZ).
 Since we are only interested in MSTIDs for this propagation parameter study, we have only considered data gathered during the days with relatively low geomagnetic activity (Kp < 4) and without the presence of solar flares: in both cases, one can expect the presence of LSTIDs and/or other space weather phenomena that are detected with our method as ionosphere perturbations, similar to MSTIDs, but with higher velocities (apparently greater than several kilometers per second in the case of solar flares, actually detected simultaneously in all the networks receivers). We have considered only those TIDs with an amplitude greater than 0.1 TECU for the Li combination. Also, in order to diminish the effect of the obliqueness factor on the ionospheric observations, we have excluded the data with an elevation below 50 deg. Finally, the original data has a time interval of 30 s. With this time interval and applying equation (6) to a network with baselines of, for instance, 10 km and with a TID velocity of 100 m/s, a sampling error of about a 30% would be obtained. To avoid this, we have interpolated, in time, the data series to an interval of 3.75 s. In this way, the delays between signals (Δtmax in equation (6)) will be obtained in steps of 3.75 s, instead of 30 s. As can be seen in Figure 10, the interpolated data (small points in the lower plot) has the same relative delays as the non-interpolated data (big points in the lower plot), in such a way that is not expected an associated significant error in the Δtmax estimation.
 Another important issue in the parameter estimation is the distribution of the array receivers: on one hand, in order to solve equation (6), the horizontal distribution should allow to cover as many different directions as possible with respect to a station used as reference for the correlation procedure, explained above. On the other hand, the typical distances should be smaller than the expected wavelengths (few tens of kilometers, see Table 1). In addition to this, the correlation between the two STEC signals, is taken into account in order to weight the corresponding equation (6).
Figures 11 and 12 show the horizontal velocities and azimuths (measured clockwise from the north), respectively, for each one of these networks as a function of the local time. From these figures, one can see some common features for all the four networks: (1) daytime and nighttime MSTIDs appear in all the networks, which presence seems to be related to the relative orientation of the magnetic field and hence to the magnetic latitude (see below), and (2) the daytime MSTIDs (i.e., winter MSTIDs) propagate faster and with a wider range of velocities than the nighttime ones, (3) in all the networks, daytime MSTIDs seem to propagate mainly equatorward while nighttime ones propagate mainly westward. In some of these plots, one can see tracks of the Coriolis effect on the azimuth, i.e., the azimuth increases at a rate of twice the Earth rotation. This effect is not considered in the model, so it appears in the results, consistently to the neutral atmosphere origin of the MSTIDs.
Figures 13 and 14 show the corresponding TID observed periods and wavelengths, respectively, calculated from the maximum and minimum correlation between network receivers as was explained before. In all the four networks, one can find a cutoff of the observed periods, these cutoffs (about 700 s for the daytime TIDs and about 600 s for the nighttime ones) provide an estimation of the mean buoyancy periods (corresponding to the Brunt-Väisälä frequencies) at the TID altitudes. These results agree at altitudes above 200 km and with mid-solar cycle conditions with those values deduced from the atmospheric models such as the “Neutral Atmosphere Empirical Model from the Surface to Lower Exosfere” model (hereinafter MSISE90, see Hedin  and Figure 15).
 The low value of the period for night time TIDs in the NZ network could be related with a TID production at an altitude different to the one assumed initially in this work (200 km). For example, taking into account Figure 3 and equation (8), and assuming that (1) these TIDs are mostly westward (as it has been typical for such season/local time), (2) the pierce point velocity is mostly eastward from a fixed ground site, and (3) that this velocity increase with the assumed ionospheric height (see Figure 3), it is clear that a greater value of the ionospheric height should be assumed in order to increase the estimation of the TID periods for these westward TIDs, being this estimation more compatible with the predicted buoyancy period.
 The estimated horizontally projected wavelengths are observed from about 80 km at noon and shorter in the night (see again Figure 14). Similar results are obtained for both TID periods and velocities when different ionospheric effective heights are considered, especially in daytime (see Figure 4). In nighttime there is an slight increase of estimated periods, this is because the Doppler effect in equation (8) increase its importance with height, and the typical TIDs and pierce point velocities are westward and eastward, respectively (Figure 3).
 As it was commented before, with equation (10), one cannot distinguish between upward and downward propagation, but some authors [see e.g., Davies, 1990, p. 245, and references therein] have indicated the downward TID propagation.
 Using standard parameters from the MSISE90 model at an altitude of 200 km and assuming a buoyancy period of 700 s, we can obtain an estimation of the AGW elevation angle by substituting ω and vH in (10). In the first row of Figure 16, this elevation angle for every network is depicted as a function of the local time, one can see that AGWs occur with an open range of elevation angles in both daytime and nighttime.
 We can check the downward propagation by looking the second row of Figure 16; in this figure, the number of the detected TIDs is plotted, in logarithmic scale, as a function of the angle between the neutral particle displacements (assuming that AGWs are transversal waves) and the local magnetic field (angle β in Figure 5), this local magnetic field has been computed using the International Geomagnetic Reference Model (IGRM) [Tsyganenko, 2003]. The angle β has been computed assuming upward and downward propagation for nighttime and daytime TIDs (see caption of Figure 16). From this figure, one can see that day time AGWs (assuming downward propagation) would be more effective in TIDs production than assuming upward propagation. This is because the angle β is closer to 180 deg under the first assumption.
 From the three networks in the Northern Hemisphere, we cannot distinguish a preferable direction for nighttime perturbations because there is no a clear maximum close to 0 or 180 deg. But looking at the network in the Southern Hemisphere, one can see that the maximum of the night time perturbations is close to 90 degrees assuming an upward propagation and close to 130 degrees assuming downward propagation. This would explain, on one hand, the abundance of such kind of TIDs in this network, in spite of they occur at nighttime (i.e., under low TEC), and, on the other hand, that the direction of these perturbations is also downward. The minor presence of daytime TIDs in this network, compared to the Northern Hemisphere ones, can be explained if we see that, for these perturbations, the peak is not so close to 180 deg like in the other three networks.
 This can be confirmed as well if we look at Figure 17 where the β angle that would have an AGW is depicted, propagating downward with an elevation angle of 45 deg, as a function of the propagation azimuth for the four networks. With this figure, one can justify the absence of TIDs propagating polarward (zero azimuth) in all four networks (see Figure 12) with values of β close or relatively close to 90 deg.
 In conclusion, if we assume that these kind of TIDs are produced by AGWs, i.e., TIDs reproduce the AGW signatures, from the horizontal parameters (velocity and azimuth), we can estimate the vertical ones by using the AGW dispersion relationship. In this sense, we find that TIDs propagates mainly downward in nightime and daytime when a high enough productivity exists, coinciding with a higher AGW displacement alignment to the magnetic field (which depends on the geomagnetic latitude). These last results, together with the above mentioned compatibility between TIDs periods and neutral atmospheric periods (and also the Coriolis effect traces), are compatible with the assumption of AGW origin of this MSTIDs.
 Medium-scale TIDs (MSTIDs) can affect different Global Navigation Satellite System (GNSS) applications such as precise navigation. To detect and characterize them (including TIDs velocity and period), simple procedures have been developed in this work, allowing the study of MSTIDs for long periods of time and from several networks of ground GPS receivers. In the approach presented here, all the relevant dependences, such as the Doppler effect, have been taken into account.
 In particular, MSTIDs present common properties in four different studied regions (California, Venice, Middle East, and New Zealand) for a whole year, and for single midlatitude receivers for more than half solar cycle. Occurrences happen mostly in winter during the daytime and in summer during the nighttime, in both cases appearing to be confined by the Solar Terminator (see Figure 11). The corresponding amplitude appears as well modulated by the solar cycle for both north and south midlatitudes. TID velocities are mostly equatorward during winter daytime (∼100–400 m/s) and toward the west during summer nighttime (∼50–200 m/s). Assuming that TID occurs at 200 km, TID detected periods are greater than 700 s in winter/daytime and than 600 s in summer/nighttime. This lower limit is compatible with an AGW origin with a cutoff period of 600–700 s (buoyancy period).
 It has been proven, taking into account the relative geometry of the magnetic field, the predominant downward propagation in all the scenarios, as far as the higher night time productivity of NZ due to the more favorable magnetic field geometry.
 As the main conclusion, the results shown in this work (downward propagation, productivity correlated to magnetic field, compatibility to neutral atmospheric periods, Coriolis effect traces) are compatible with the initial assumption of AGW as the main origin of the studied MSTIDs.
 In the opinion of the authors, these results make a potential real-time TIDs modeling from permanent networks more feasible, to be applied, for instance, in accurate GNSS navigation.
 This work has been partially supported by the Galileo Joint Undertaking/6FP project WARTK-EGAL and by the Spanish Project ESP2004-05682-C02-01.
 Shadia Rifai Habbal thanks both referees for their assistance in evaluating this paper.