Semiempirical ionospheric models are widely used in many applications. In this work we analyze the global representation of the vertical total electron content (VTEC) derived from the La Plata single-layer GPS ionospheric model (LPIM), the international reference ionosphere (IRI) model, and the Bent model. To perform this analysis, we compare the output from these models with VTEC determinations from the TOPEX/Poseidon (T/P) satellite. The analysis of the differences is performed over a large number of selected quiet geomagnetic days during the year 1997. From these comparisons we can conclude that LPIM gives the best global VTEC representation, even without modeling the ionospheric equatorial anomaly. The mean global difference between the VTEC determined by T/P and computed by LPIM is in the range of 2 and 3 TECU (1 TECU = 1016 el/m2). IRI and Bent models tend to show larger discrepancies with T/P during the summer solstices in both Northern and Southern Hemispheres.
 In this work we compare the vertical total electron content (VTEC), using the La Plata single-layer GPS ionospheric model [Brunini, 1998] (hereinafter LPIM), the international reference ionosphere model (IRI) [Rawer and Bilitza, 1990], and the ionospheric Bent model [Bent and Llewellyn, 1973], with direct VTEC measurements from the TOPEX/Poseidon (hereinafter T/P) altimeter [Imel, 1994]. The above-mentioned models describe the global behavior of the ionosphere at any geographic or geomagnetic coordinates and at any time.
 Among the great variety of ionospheric models it is possible to classify them as either theoretical or empirical. Although the former ones are generally more complicated and can describe qualitatively the main characteristics of the ionosphere, they lack quantitative precision. The empirical models are fitted using average values obtained from large databases that gather information collected from Earth, rockets, and artificial satellites at different times during the day, at different epochs during the year, and during varying levels of solar and geomagnetic activity [e.g., Bent and Llewellyn, 1973; Rawer et al., 1982].
 The first research that made use of GPS observations to determine empirical ionospheric models can be traced back to the late eighties [e.g., Georgiadou and Kleusberg, 1988]. These models can be considered as a particular type of empirical model as they describe current ionospheric conditions, while other ones provide only average values.
 The Global Positioning System (GPS) has become a powerful and mature geodetic tool widely used for a broad range of technological and scientific applications. The observations of permanent GPS tracking stations, under the management of the International GPS Service (IGS), seem suitable for ionospheric research, providing continuous and quite good worldwide coverage at low cost for the users.
 LPIM is based on the measurements of dual-frequency P-code GPS receivers. This model uses about 70 GPS receivers belonging to the global IGS tracking network. LPIM represents the global ionosphere with a spherical layer of infinitesimal thickness at about 450 km above the Earth's surface. When the GPS signals cross this layer, they suffer a delay that is proportional to the total electron content (TEC) found in their path. In the second section of this paper we describe this model, which basically maps the VTEC in two spherical coordinates: the geographic latitude ϕ and longitude λ. These maps represent the average behavior of the ionosphere for the time interval during which the GPS observations were performed (2 hours in our case).
Bent and Llewellyn  developed one of the more popular climatological models, known as the Bent model, which describes the ionosphere on a worldwide basis for any past or future date. Development of the Bent model involved fitting a theoretical electron density profile to a database of ionospheric measurements. For a given location, time, and date the Bent model describes the electron density. The resultant profile is composed of five sections: a biparabola, to model the lower ionosphere; a parabola, which joins together the topside and bottomside ionosphere; and three exponential functions, which are combined to model the topside ionosphere.
 The international reference ionosphere (IRI) is an international project sponsored by the Committee on Space Research (COSPAR) and the International Union of Radio Science (URSI). These organizations formed a working group in the late sixties to produce an empirical standard model of the ionosphere, based on all available data sources [Rawer et al., 1982]. One of the vital parts of the model is a “map” of critical frequencies. Using the international network of ionosondes, a spherical harmonic analysis is performed on each set of monthly mean critical frequencies over 24 hours of the day and for a range of sunspot numbers. Several steadily improved editions of the model have been released. For a given location, time, and date, the IRI95 version describes electron density in the altitude range from about 50 to 1000 km, electron temperature, ion temperature, and ion composition. The electron density profile consists of six height ranges.
 A comparison between the Bent model and the IRI model and their application for satellite orbit determination was discussed by Bilitza et al. . IRI showed better results, and they attribute it to the more detailed representation of the bottomside density structure.
 The IRI and Bent models provide averages values in the nonauroral ionosphere for magnetically quiet conditions. These models are ones of the classical global ionospheric models used as referent in much ionospheric research. LPIM provides a numerical ionospheric model at any time using GPS measurements. In disturbed magnetic activity it provides also a mean representation of ionosphere that is very different from that in quiet conditions [Brunini et al., 2000]. On some days the geomagnetic field varies smoothly and regularly solely because of currents in the ionosphere (quiet days), while on other days their changes are more or less irregular (disturbed days). Such irregular variations are a consequence of fluctuations in currents external to the Earth caused by quick and dense solar ejections moving into the regular solar wind, which can access the interior of the magnetosphere, even varying the ionospheric electron content. In order to compare LPIM with Bent and IRI models we have to work in the same conditions, where all the models have a good ionospheric representation. So in this study we chose about three T/P cycles with quiet geomagnetic conditions.
 T/P surveys sea level heights by measuring the time required for pulses generated by the onboard radar altimeters to bounce vertically back to the satellite from the sea surface [Fu et al., 1994]. To correct the Earth ionospheric delay on the radar pulse, the satellite's altimeter makes measurements in two channels. The difference between both measurements yields the estimated values of the integrated total electron content from the lower part of the ionosphere up to the height of the satellite (1330 km).
 In section 2 we describe LPIM and its general considerations. In section 3 we describe the VTEC T/P determinations. In section 4 we describe the methodology used to analyze the differences between the ionospheric models and T/P determinations. Sections 5 and 6 are the results obtained using statistical parameters and global map differences and the conclusions, respectively.
2. Brief Description of La Plata Ionospheric Model (LPIM)
 The Global Positioning System (GPS) consists of a constellation of 24 radio navigation satellites and a ground control subsystem, which continuously checks the satellite operations. The satellites have nearly circular orbits with altitudes of about 20,200 km, which is about 3 Earth's radii. The space vehicles are distributed in six orbital planes with inclinations of about 55° and with four satellites orbiting in each plane. This configuration provides global coverage with four to eight satellites over the horizon everywhere on the Earth, at any time. The satellites broadcast two carrier signals at 1.5754 GHz and at 1.2276 GHz named L1 and L2, respectively, both modulated in phase by two pseudorandom codes: the P code and the C/A code (for details, see Leick ). Using a dual-frequency receiver, it is possible to measure the pseudorange from it to a satellite, eliminating the major source of error, i.e., the ionospheric refraction.
 When simultaneous carrier phase observations in both frequencies are subtracted, the satellite-receiver geometrical range and all frequency independent biases are removed, and the so-called geometry-free linear combination, Φ4, is obtained [Kleusberg and Teunissen, 1996]:
where Φ1 and Φ2 are the raw carrier phase observations in both frequencies (in units of length); ΔTionos is the ionospheric range correction between the signals L1 and L2; τR and τS are electronic delays produced in the hardware of the receiver and the satellite, respectively; CRS is the combination of both carrier phase ambiguities; and ε is the error.
 Before being used, a preprocessing of the raw observations is done to correct cycle slips and determine the ambiguities for every satellite-receiver pair (for which P-code phase observations are used), and normal points every 8 min are created (the sampling interval of the raw observations is 30 s). The hardware delays remain as unknowns, and their values, assumed as constants, are estimated together with the sought ionospheric information.
 The speed of propagation of the signal emitted by the GPS satellite is affected by the ionosphere. In the frequency range of the GPS signals the ionosphere is a dispersive medium. Assuming the propagation theory of Appleton-Hartree (for details, see Bradley ), the phase refractive index for frequencies in the L band can be approximated as
where f is the frequency of the signal, expressed in hertz, and Ne is the electronic density in units of el/m3.
 The integration of the expression for nph along the radio signal path yields the electromagnetic path lengths ρph.
where ρG is the geometric range and ΔTionos,f is the ionospheric delay for a radio signal at frequency f. So, the delay of an electromagnetic wave through the ionosphere due to the change of speed in that medium depends on the frequency, as well as the total electron content, that is, the integrated electron density along the path from a source, satellite on board, to a ground receiver.
and STEC must be expressed in total electron content units (TECU, 1 TECU = 1016 el/m2).
 The ratio between the slant and the vertical total electron content is the so-called mapping function, which we assume, in a first approximation, to be elevation-dependent:
where z is the satellite zenith distance. A broad discussion about different kinds of ionospheric mapping functions is given by Schaer . In our model we assume a single-layer mapping function; this means that all free electrons are assumed to be concentrated in a shell of infinitesimal thickness at an altitude H. In this context the mapping function may be approximated by M(z) = sec [z′ (z)], where z represents the zenith distance of the satellite relative to the receiver and z′ is the zenith distance of the satellite referred to the ionospheric point (defined as the intersection of the signal path with the shell at height H). The dependence of M(z) on H is given by
where R is the mean Earth radius. The height H usually adopted lies between 350 km and 500 km, approximately the altitude of the F2 layer. Coco et al.  discussed this mapping function by comparing it with the Bent model's ionospheric profiles. This analysis showed that the mapping function given by expression (6) has an error of about 10% at 80° zenithal distance.
 As the state of the ionosphere varies with the geographic longitude λ and latitude ϕ, as well as time, a parameterization of the VTEC is needed. A spherical harmonics expansion is fitted for a global description of the space variation of VTEC:
where λSF represents the longitude referred to a Sun-fixed reference system and Plm (sinϕ) are the fully normalized Legendre-associated functions. The coordinate longitude relative to the Sun corotating system is computed as λSF = λ + UT − 12 hours, where UT is the universal time of the observation.
Equation (9) contains (L + 1)2 − (L − M) × (L − M + 1) expansion coefficients plus nR hardware receiver delays and nS hardware satellite delays, which are unknown. Using a large amount of observations providing global coverage, the parameters are solved by the least squares method. In this work we computed a set of expansion coefficients every 2 hours and a set of hardware delay every 24 hours.
 The VTEC calculated using LPIM, named VTEC(LPIM), was compared to that of other GPS-derived ionospheric models, like that computed by the Centre for Orbit Determination in Europe (CODE), named VTEC(CODE) [Schaer, 1999], and by the Jet Propulsion Laboratory (JPL), named VTEC(JPL) [Mannucci et al., 1998]. We analyzed 9 days with different geomagnetic conditions, and we detected systematic differences of about −1 TECU in VTEC(LPIM)-VTEC(CODE), −5 TECU in VTEC(LPIM)-VTEC(JPL), and −4 TECU in VTEC(CODE)-VTEC(JPL) [Brunini et al., 2000].
 We can then conclude that our VTEC model is as good as any current GPS VTEC model. Schaer  analyzed the agreement between the different GPS VTEC models, and the discrepancies are similar to ours. Diurnal variations of emission occurrence are examined using data observed at two stations.
3. Brief Description of VTEC TOPEX/Poseidon Determinations
 The mission of Ocean Topography Experiment (T/P) is to provide global sea level measurements with an accuracy of a few centimeters. This satellite is equipped with the National Aeronautics and Space Administration Altimeter (NASA ALT). NASA ALT surveys sea level heights, measuring the time it takes for a radar pulse emitted on board to bounce vertically back to the satellite from the sea surface [Fu et al., 1994].
 The satellite operates in a 66° inclination circular orbit at about 1336 km. The period is 1.87 hours, and the ground track velocity is 5.8 km/s. It is an exact repeat orbit with a 9.91 day/cycle. Each cycle samples two bands separated by approximately 12 hours, and the width of these bands is about 2 hours [Fu et al., 1994]. In one cycle the satellite records the sea level at any longitude and latitude between −66° and +66°. NASA ALT operates with Ku-band prime frequency 13.65 GHz and auxiliary C-band frequency 5.3 GHz. The difference in the signal delay of both frequencies provides an estimate of the ionospheric correction to the range measurements.
 The observation equation of NASA ALT can be written as [Imel, 1994]
where i = 1 or 2 denotes the Ku or C band, Ri is the range measurement, R0 is the true range, ΔRion(fi) is the ionosphere delay, fi is the frequency, bi is a frequency-dependent term, and c is a frequency-independent term.
 In the same way, GPS ionosphere delay ΔRion (fi) can be written proportional to the TEC that the signal encountered while passing through the ionosphere and inversely proportional to the square of the carrier frequency. From equation (4),
where fi must be expressed in hertz and VTEC must be expressed in electrons per square meter. So equation (10) can be rewritten as
and the range difference between the two frequencies is
The errors in VTEC determination from T/P data are due to the noise in the altimeter measurement and to the frequency-dependent term errors. Without losing accuracy in the ionosphere correction the noise in the altimeter data is reduced substantially at 0.2 cm (about 1.1 TECU) by averaging over 20 s [Imel, 1994]. The uncertainty in VTEC is dominated by uncertainty in the so-called electromagnetic bias (EMB), due to the difference in reflectivity of wave troughs versus wave peaks and to certain assumptions made in the onboard algorithm. This has been partially compensated for in the generation B data by adding a bias of 10 cm to the C-band range delay before differencing the C-band and Ku-band range delays to obtain VTEC. Nevertheless, it is believed that there remains a bias of about 0.5 cm (2.3 TECU) in the T/P-derived ionospheric range delay.
 The dual-frequency T/P VTEC determinations are sensitive to the ionosphere up to the altitude of the satellite, about 1330 km, so the higher protonosphere is not sensed by T/P. The Bent and IRI models compute the VTEC without taking into account the protonospheric contribution. The dual-frequency GPS VTEC determinations are sensitive to the ionosphere up to about 20,200 km, so the protonospheric contribution is included in the VTEC determined by GPS. This contribution is, however, only a few TECU and will be disregarded in our analysis when comparing LPIM, Bent, IRI, and T/P VTEC determinations.
4. Statistical Parameters
 To carry on the comparisons between the outcomes from LPIM, IRI model, and Bent model with T/P VTEC determinations, let us employ the following definitions: the VTEC difference, ΔVi = VTECiTP − VTECiM where VTECiTP is the VTEC measured by T/P and VTECiM is the VTEC computed by the ionospheric model (LPIM, IRI, and Bent) at the ith observation point associated to the geographic coordinates λi and ϕi. The average difference between VTEC values of T/P and VTEC computed from models, 〈ΔV〉, is the following weighted average:
and the standard deviation of ΔVi is
where cos ϕi is the weighting function for the geographic latitude ϕi.
 On account of the distinctive ionospheric behavior at different latitudes we analyze the 〈ΔV〉 value for low latitudes (|ϕ| ≤ 30°), for midlatitudes (Southern Hemisphere, ϕ < −30°, and Northern Hemisphere, ϕ > 30°), and for all latitudes (|ϕ| ≤ 66°).
 In our analysis we use three T/P cycles (Table 1). They have been chosen because of the quiet geomagnetic conditions and the epoch in the year: during the two solstices and one equinox.
Table 1. Geomagnetic Features of the Three TOPEX/Poseidon Cycles of Observation Processing
Interval Day of Year 1997
 During quiet geomagnetic conditions, changes in the magnetosphere are slow, and their parameters vary very slowly, so classical models are suitable for an ionospheric representation. As a measure of the ionospheric perturbation we have used the geomagnetic activity index Km [Menvielle and Berthelier, 1991], which is related to the middle energy density imbedded in the irregular magnetic variations at midlatitudes. Choosing intervals with low values of Km, we can assume that no energy source other than solar radiation is responsible for the regular ionization arriving in the high atmosphere.
 The differences of VTEC, ΔVi, for the three periods chosen, are mapped in Figures 1, 2, and 3 under sinusoidal projection. Some characteristics of these maps are as follows: (1) LPIM reaches the best agreement with T/P measurement. (2) The three models predominantly underestimate the VTEC value with respect to T/P measurements, except in the equatorial region, where there is an overestimation of the VTEC, especially by IRI and Bent models. (3) In T/P cycle 175 (positive Sun declination) the VTEC of the three models reach the best agreement between computed and T/P determinations. (4) In T/P cycle 187 (Sun's declination close to zero) all models reach the best agreement with T/P determinations in the Northern Hemisphere. (5) In T/P cycle 194 (negative Sun declination) all models reach the best agreement with T/P determinations in the Northern Hemisphere.
 The average differences between VTEC values of T/P and those computed from models, 〈ΔV〉, are plotted in Figure 4. LPIM shows a uniform bias (between 2 and 3 TECU) across time and practically independent of the latitude. The IRI and Bent models show a strong dependence on latitude, especially in the October and December periods.
 The standard deviations of ΔVi, σ, are plotted in Figure 5. The more relevant aspects are as follows: (1) LPIM reaches the smallest standard deviation at different analyzed periods and different latitude ranges. (2) The three models reach the largest standard deviation at low latitude (equatorial latitude) and at midlatitude, and the standard deviation in the Southern Hemisphere is larger than that in the Northern Hemisphere. (3) In T/P cycle 187 all models show the largest standard deviation.
 We performed a statistical study of LPIM behavior compared with Bent and IRI models. The last two models come from average values based on past measurements over several years. They provide a good long-term average but have less sensitivity to small spatial and temporal variations. LPIM uses GPS observations for the analyzed interval and generates daily maps every 2 hours giving actual VTEC maps. LPIM gives a better representation of the ionosphere than the IRI and Bent models do in quiet geomagnetic periods.
 IRI model, Bent model, and LPIM characterize the VTEC at northern midlatitudes better than they do at southern midlatitudes. The reason could be that data used to perform the Bent and IRI models come from a network of ionosonde and incoherent scatter measurements placed mostly in the Northern Hemisphere. Similarly, the measurements used to perform LPIM come from GPS receivers better distributed in the Northern Hemisphere.
 The standard deviations of VTEC differences between T/P and all models are larger at the equator and smaller at northern midlatitudes. During cycle 175 (June solstice) the standard deviations at northern midlatitudes are similar to the standard deviations at southern midlatitudes. During cycle 194 (December solstice) the standard deviations are clearly larger at southern midlatitudes than at northern midlatitudes. During cycle 187 (October equinox) the standard deviation shows the larger values, and they are comparatively larger at the equator. So we conclude that added to the problem of data coverage, the representation of the models is worse in the region where the Sun is placed.
 The agreement between the VTEC computed by LPIM and T/P determinations has a uniform behavior (between 2 and 3 TECU) during the three quite geomagnetic periods analyzed, and it does not show a strong dependence on the latitude. In the future we will compare our model with T/P data for geomagnetically disturbed periods.
 We wish to thank Wolfgang Bosch from the Deutsches Geodätisches Forschungsinstitut (DGFI), who provided us with the T/P data, and the Institute of Geophysics and Planetary Physics (IGPP) and the Scripps Institution of Oceanography of the University of California, San Diego, where we were provided with the GPS observations by anonymous ftp (ftp://lox.ucsd.edu). We also want to thank to the National Space Science Data Center (NSSDC), where the IRI software is available (ftp://nssdc.gsfc.nasa.gov/pub/models/ionospheric/iri/iri95/fortran_code). This work was partially funded by the Consejo Nacional de Investigaciones Cientificas y Tecnicas (CONICET, Argentina) and the Departamento de Ciencias Geologicas, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires (Argentina).