A numerical and phenomenological model of global ionospheric electron density (Ne) is investigated. The three-dimensional Ne model has been named the Taiwan Ionospheric Model (TWIM) and constructed from monthly weighted and hourly vertical Ne profiles retrieved from FormoSat3/COSMIC GPS radio occultation measurements. The TWIM exhibits vertically fitted Chapman layers, with distinct F2, F1, E, and D layers, and surface spherical harmonics approaches for the fitted Chapman layer parameters including peak density, peak density height, and scale height. These results are useful in investigation of near-Earth space and large-scale Ne distribution with diurnal and seasonal variations, along with geographic features such as the equatorial anomaly (EA). This paper also investigates the diurnal and seasonal variations of EA within different ionospheric layers and specifically attempts to account for the latitudinal and longitudinal structures caused by atmospheric tides.
 For the past 40 years, the International Reference Ionosphere (IRI) model [Bilitza, 2001] is the most used empirical model for the specification of ionospheric Ne and ion and electron temperatures. Here we have attempted to construct another numerical and phenomenological model of global ionospheric Ne using monthly and hourly ionospheric sounding data from the Taiwan FormoSat3/Constellation Observing System for Meteorology, Ionosphere and Climate (FS3/COSMIC) data. The primary propose of this construction is to provide a simple and easily accessible representation of temporal and synoptic variations in the ionospheric Ne, which is frequently required as a subsidiary input in a number of dynamical calculations of upper atmospheric interests and applications in space communication and wave propagation. The numerical model reproduces the three-dimensional (latitude, longitude, and altitude) Ne encountered by FS3/COSMIC. This match should be maintained in the first and second derivatives of practical schemes for providing reliable radio propagation predictions and electrostatic field determinations. We shall also consider the fundamental parameters of the ionospheric layers and their spatial and temporal distribution. The concentrations of Ne at certain altitudes have characteristic maximum peaks, forming the ionospheric F2, F1, E, and D layers. As a result, the stratified structure of the ionosphere may therefore be characterized by a set of parameters: the concentrations at layer maximum, the peak density altitudes, and the layer scale heights. In addition to variation with the altitude, ionospheric parameters also vary with the geographical coordinates (latitude and longitude) and describe the special features of Ne distribution. Herein we report the validation and performance of the TWIM, especially the statistics of occurrence of the layer parameter mapping results, including the F2, F1, and E layers.
2. FS3/COSMIC Ne Profiling and Layer Parameter Mapping
 In the FS3/COSMIC mission, all six spacecraft (FM1 to FM6) were integrated and launched together into a parking orbit at 515 km altitude; whereupon each spacecraft was separated and then transferred from the parking orbit to their final orbits at ∼800 km. Since 28 July 2006, raw excess phase data and the retrieved Ne profiles are available from the Taiwan Analysis Center for COSMIC (TACC, http://tacc.cwb.gov.tw/en/) and the COSMIC Data Analysis and Archive Center (CDACC, http://www.cosmic.ucar.edu/cdacc/). For further details about the FS3/COSMIC program, see Rocken et al.  and Hajj et al. . Generally FS3/COSMIC can perform over 2500 radio occultation (RO) measurements per day, and on average, 75% of the RO measurements, roughly 1900 measurements, could be successfully retrieved into Ne profiles prior to the middle of 2007, but the number of profiles retrieved has declined to about 1500 Ne profiles since that point. Failed measurements are usually caused by locking onto the GPS carrier signals too late or ending too early, within an RO observation.
 Each GPS RO observation consists of a set of limb-viewing links with tangent points ranging from the LEO satellite orbit altitude to the Earth's surface. Under the assumptions of straight-line radio propagation and spherical symmetry for the local ionosphere of each RO observation, the “calibrated” TEC values (TEC’) along radio paths and below the LEO orbital altitude can be derived from differential GPS phase measurements, due to the dispersive propagation properties of L1 and L2 radio signals. The Ne values may then be retrieved using the Abel integral transform given by Tricomi  defined as:
The Ne value at a tangent point's radial distance rt can be computed recursively starting from the outer rays. The Ne profile may then be obtained [Schreiner et al., 1999]. In general, the assumption of spherical symmetry of the local ionosphere used in the Abel integral transform is not realistic. Tsai and Tsai  and Tsai et al.  proposed an improvement on the classical approach of the Abel inversion that considers the effects of large-scale horizontal gradients and/or inhomogeneous ionospheric electron densities (Ne) by developing an iterative scheme to determine “compensated” total electron content (TEC) values through nearby RO observations. Validation checks with independent ionospheric F2 layer peak Ne and height data from ionosonde systems show successful improvement and fundamental suitability of the iteration scheme for the inversion of compensated TEC values into Ne information.
 The terrestrial ionosphere at all latitudes has a tendency to separate into layers, despite the fact that different physical processes dominate in different latitudinal domains. Specifically, the Ne profiles exhibit layered structures, with distinct F2, F1, E, and D layers. Each layer may be generally characterized by a Chapman-type function described by the parameters of peak Ne, peak density height, and scale height. A Chapman-type layer [Chapman, 1931] is predicted by a simplified aeronomic theory, assuming photoionization in a one-species neutral gas, and neglecting transport processes. Thus, NemaxF2 (NemaxF1, NemaxE, and NemaxD), hmF2 (hmF1, hmE, and hmD), and HF2 (HF1, HE, and HD) represent the F2 layer (F1, E, and D layer) and can be obtained with least squares error fitting of the observed profile to the Chapman functions.
where each i means a physical layer of F2, F1, E, or D layer. Other methodologies involve using the Chapman-like functions to constrain the vertical Ne structure including: Rishbeth and Garriott  suggested a Chapman function with continuously varying scale height for the description of F region profiles; Semeter and Mendillo  successfully inverted ground-based brightness measurements to Chapman-like parameterized tomography; Reinisch and Huang  used an α–Chapman function to represent the bottomside profiles inverted from ionograms, and then used the scale height value at the F2 peak to extrapolate the profile to the topside. More recently, Reinisch et al.  used an α–Chapman function with a continuously varying scale height, dubbed a vary-Chap function, to represent the entire topside profile from the F2 peak to the plasmasphere; Meza et al.  also applied the vary-Chap approach to reconstruct the topside ionosphere using GPS and ionosonde measurements. We note that all of the ionospheric layers are usually present during the daytime. The F1 and D layers decay at night and could be hidden within the other layers, but the F1 and D layer parameters are still derivable across all times by least squares error fitting. Figure 1 shows an example of daytime Ne profile fitting and its results.
 A two-dimensional (latitude and longitude) numerical map to fit derived layer parameter values can be constructed by spherical harmonic analysis of the Laplace's partial differential equation. The resulting real functions of θ (latitude) and λ (longitude or the local time angle) are combinations of the surface spherical harmonics [Davis, 1989] defined by
where Pnm() is the familiar associated Legendre polynomial of the first kind of degree n and order m. We note that the fitted ionospheric parameter values can be affected by noise produced from a number of sources, including limitations of equipment, retrieval techniques such as the Abel inversion method, errors from the Ne profile fitting, and intrinsic random fluctuations of the physical phenomena being measured. We assume that these noise values are small compared to the main physical parameter variation, distributed independently of time of day, and normally have a mean of zero and a standard deviation σi. Thus the maximum likelihood estimate of surface spherical function coefficients is obtained by minimizing the chi-square error average:
where yi is a Nemax, hm, or H measurement, N is the number of RO measurements, K is the total term number, and (N-K-1) is the number of degrees of freedom remaining after subtraction of one degree for each term in the series Unm(θ, λ) and Vnm(θ, λ). At specified order and degree for the surface spherical functions the coefficients Anm and Bnm can be determined recursively starting from the lower order and degree. Davis  showed that if the numerical mapping results have continuous derivatives of the αth order, the root-mean-square (RMS) amplitudes of Anm and Bnm can approach zero at least as fast as n−α. The criterion adopted for determining the optimum order cutoff is therefore to locate a tendency to level off the RMS amplitudes. Figure 2 shows average RMS amplitudes of NemaxF2 mapping analyses against the degree number n in a log-log scale at each of the first four orders. It shows that the smoothed slopes of the RMS amplitudes for the first three order analyses (not including the third-order analysis) appear to be less than −1 but greater than −2. This implies that the geographic variation of NemaxF2 has at least a piecewise continuous first derivative. After choosing a cutoff order of 3, the optimum degree number can be determined by truncating the high spectrum harmonics based on a least squares error fitting of equation (4). As shown in Figure 2, in the zero-order analysis the graph of RMS amplitude against degree number shows a tendency to level off around degree 24, which is thus optimum cutoff q0. In a similar manner we determine the optimum cutoffs to be degrees 20 and 15 for the first- and second-order terms q1 and q2 respectively. We note the spatial resolution (or horizontal scale) is about 4 degrees in the latitude and longitude scales solved by a degree number of 24, revealing large-scale representation in Ne. The archived data and the complete structure of the model coefficients may be downloaded from “http://isl.csrsr.ncu.edu.tw/” hosted by the Center for Space and Remote Sensing Research, National Central University, Taiwan.
3. Validation of the TWIM Using the Ionosonde Data
 The evaluation of the TWIM has been done with the ionogram parameters scaled from 58 ionosonde stations and provided by the Space Physics Interactive Data Resource (SPIDR), National Geophysical Data Center (NGDC), NOAA (http://spidr.ngdc.noaa.gov/spidr/). Coincidences of the TWIM parameters to the scaled ionogram parameters include foF2, hmF2, foF1, foE, and hmE. Here we verify the TWIM only on the foF2 and foE results from July 2006 to October 2008 but not the other hmF2, hmE, and foF1 because it is difficult to directly and accurately scale the last three parameters from ionograms. Figure 3 shows a scatterplot of the TWIM foF2 and foE at 9°N and 167°E versus the scaled foF2 and foE from the Kwajalein island (9°N, 167°E) ionosonde data, their RMS foF2 and foE difference curves, and their fractional mean foF2 and foE difference curves as a function of the ionosonde foF2 or foE. The foF2 and/or foE values are monthly averages within every local or universal hour. It is noticed that for foF2 being larger than 3 MHz the TWIM foF2 results are accurate and within a 10% fractional mean difference comparable to the Kwajalein island ionosonde foF2 observations. At the low foF2 side, the TWIM foF2 results are biased high, i.e., overestimated, with respect to the ionosonde foF2 observations with a maximum ∼30% mean difference at 2 MHz. The RMS foF2 difference curve in red color shows results of< 1 MHz at most of foF2 values. For foE determinations at Kwajalein island, it is noticed that the RMS foE differences are all less than 1 MHz too but the fractional mean differences are much worse than the TWIM foF2 results. It is possible to happen when a retrieving error is accumulated and then larger from the top to the bottom of each occultation observation or when a scaling error is produced because of spread and/or weak E layer echoes or strong Es layer echoes in ionograms.
Figure 3 depicts an example evaluation by one ionosonde station at Kwajalein island. Figure 4 shows a worldwide evaluation of the TWIM foF2 and foE using all 58 ionosonde station data. Each RMS or mean foF2 (foE) difference has been obtained by averaging all RMS or mean foF2 (foE) differences from one ionosonde station and has shown as a function of the dip latitudes of ionosonde sites. The mean difference results show that except for the high dip-latitude regions (>70° dip-latitude) the TWIM foF2 values are accurate to the ionosonde observations with less than 0.5 MHz averaged difference, but the TWIM foE values are underestimated, i.e., biased low, by ∼0.5 MHz and typically worse at the southern hemisphere. The RMS foF2 differences are less at the middle dip-latitude regions (30°∼60° dip latitudes) than at the low and high dip-latitude regions, i.e., the TWIM foF2 values are more reliable at the middle dip-latitude regions. The RMS foE difference results show that the TWIM foE values are more reliable at the northern hemisphere than at the southern hemisphere.
4. Discussion of the TWIM Results
 We have applied spherical harmonics analyses to the monthly layer parameter values (peak Ne, peak density height, and scale height) at the geographic coordinates and within each local time (LT) or universal time (UT) hour. The corresponding peak Ne is converted to critical frequency for communication and wave propagation purposes in accordance with the conversion formula fc2 (MHz) = 80.6 Nemax (cm−3). Figures 5, 6, and 7show the noontime foF2, hmF2, and HF2 maps, respectively, in both March (left) and June (right) 2008. Figure 8 shows the noontime foF1 maps in March and June 2008. Figure 9 shows the noontime foE and hmE maps in June 2008. In these map images, the colored solid points denote the line-of-viewing tangent point locations at the F2 peak of RO observations, and the red curves present positions with modified magnetic dip latitudes of +18°, 0°, and −18°.
4.1. Diurnal and Seasonal Variations of F2 Layer Parameter Mapping Results
 In this paper, we focus on the seasonal variations and geographic features of the EA and atmospheric tide coupling discussed in the sections below. The F2 layer is the most heavily ionized layer, with a maximum of Ne in the range of 200 to 400 km. Features of the diurnal F2 layer variations (not shown here) can be easily obtained and directly described as follows. Except for the low-latitude and dip equatorial regions, the foF2s are greater during the daytime than at night; foF2 increases promptly after sunrise, continues to increase for a few hours after noon, and thereafter usually decreases. Except for the dip equatorial region, there is a general tendency for hmF2 to fall after sunrise and then to rise during the afternoon or evening, while the F2 layer is thicker during the day than at night, and in particular, much thicker near the dip equator.
 The June noontime foF2 map in Figure 5 shows that the north pole area has higher values (∼5 MHz) than the south pole area (∼2 MHz), which is in continuous night during the winter. Similarly, the midlatitude area of the northern hemisphere has higher noontime foF2 values (by ∼1 MHz) than those of the midlatitude area of the southern hemisphere, in general. The June noontime hmF2 map of the midlatitude and high-latitude hmF2s in the northern hemisphere generally have relatively higher values (by 30–40 km) than the southern hemisphere (Figure 6). Similarly, the June noontime HF2 map in Figure 7 shows that the midlatitude and high-latitude HF2s in the northern hemisphere generally have relatively thicker values (by 10–20 km) than the southern hemisphere. Overall, the noontime HF2 values display peaks along the dip equator and are largest in March and lowest in June.
4.2. F2 and F1 Layer Equatorial Anomaly
 It is well known that the EA densities are the result of the combined effects of ExB drift and ambipolar diffusion. During the daytime, there are eastward dynamo electric fields generated in the equatorial E region by thermospheric winds and transmitted along the dipole magnetic field lines to F region altitudes because of the high parallel conductivity. The dynamo action causes F region plasma to drift upward with a velocity E × B/B2. At the same time, gravitational and pressure gradient forces move ionization downward along the magnetic field lines, thus transporting plasma away from the dip equator toward higher latitudes and producing two crests in Ne. This is called the fountain effect [Hanson and Moffett, 1966]. Theoretically, in the absence of neutral winds, the upward E × B drift produces almost identical effects at conjugate-hemisphere points with the same magnetic dip. As shown in the low-latitude regions of Figure 5, both the noontime foF2 maps in March and June show one equatorial Ne trough along the dip equator and two Ne crest traces at ±∼18° magnetic dip latitudes. However, neutral winds can cause conjugate asymmetry by modulating the fountain and moving ionospheric electrons at conjugate hemispheres to different altitudes. Abur-Robb and Windle  and Sterling et al.  demonstrated the asymmetry of crest Ne and peak density height resulting from meridional neutral winds blowing from the summer hemisphere to the winter hemisphere. These north-south asymmetry effects are reproduced by our model as shown in the June mapping results of Figures 5, 6, and 7. At noon of the June solsticial conditions, the effect of the meridional neutral wind blowing from the north to south across the dip equator can decrease plasma transport from the dip equator to the northern hemisphere, thus reducing Ne at the north crest. Such summer-to-winter wind effects can also transport plasma up the field line and produce higher hmF2s and thicker HF2s in the northern hemisphere and low dip-latitude regions, as shown in the June hmF2 and HF2 maps in Figures 6 and 7; there are lower hmF2s and HF2s in the southern hemisphere and low dip-latitude regions vice versa.
 For seasonal variations of EA at noon, the foF2 values along the two crests and the HF2 values along the dip equator are highest at the March equinox and lowest at the June solstice. Although less pronounced, the variation in the noontime crest foF2 and trough HF2 at the September equinox and the December solstice is intermediate between the June solstice and the March equinox, and the December solstice noontime foF2 is basically larger than that of the September equinox. The higher daytime Ne in winter than in summer is a manifestation of the well known winter anomaly.
 The EA can produce one equatorial trough and two crests at ±∼18° magnetic dip latitudes in F2 region Ne. The gravitational and pressure gradient forces of EA can move ionization toward middle latitudes and downward further along the magnetic field lines, increasing the F1 region Ne. As shown in Figure 8, the noontime foF1 maps of March and June 2008 show a wider depletion region between the ±18° dip latitudes and two peak traces as the foF2 EA features, but locating at higher dip latitudes of ±∼32°. Even though the September and December foF1 maps are not shown, Figure 8 displays the seasonal variation of this latitudinal distribution of EA. Overall, the crest foF1 values are highest at the March equinox and lowest at the June solstice, similar to the foF2 EA seasonal variations. Figure 8 specifically attempts to account for the longitudinal differences observed in the equatorial and low-latitude regions. From the March noontime foF1 map the Asia and the Pacific Ocean sectors, where the foF1 values are less than 2 MHz in the equatorial regions, have a wide and complete foF1 depletion feature, but the other regions at the Atlantic Ocean and the Africa sectors, where the equatorial foF1 is enhanced to ∼3 MHz, could produce two separated F1 layer Ne caves locating at the two sides of the dip equator. From the June noontime foF1 map the two-cave feature was observed at different location of the East Pacific Ocean and the American sectors.
4.3. The foE and hmE Mapping Results
 In the E layer, Ne has a maximum within the range 90 to 140 km. Figure 9 shows the June noontime foE and hmE maps of 2008. In high-latitude regions (45° ∼ 75° latitudes), the noontime foE mapping results present lower values (mostly < 1.5 MHz) in the southern hemisphere than in the northern hemisphere (mostly > 2 MHz), similar to the high-latitude June foF2 distributions. In the low-latitude and midlatitude regions, previous studies [Talaat and Lieberman, 1999; Forbes et al., 2003] have shown that DW1 (diurnal and westward tide with a zonal wave number of 1) and DE3 (diurnal and eastward tide with a zonal wave number of 3) tides are the predominant atmospheric solar tides observed in the D and E regions. The latitude structure of DW1 Hough functions reveals distinctive signatures with maxima at the equator, and the DE3 latitude structure reveals signatures with maxima at the equator and between ∼20 and 30° colatitudes as well [Kato, 1980]. Figure 9 shows that the latitude structures of the June noontime foE map have a center peak along the ∼8° dip latitude trace and two side peak traces at +8° ± 32° dip latitudes, i.e., +40° and −24° dip latitudes, and represent a good approximation of the total coupling response from the DW1 and DE3 atmospheric tides. The latitude difference of the TWIM foE peak traces and the computed Hough structures may be due to the summer season in the northern hemisphere and the impact from the wind interactions. Although less pronounced, similar peak latitude structures are obtained from the December noontime foE map but now moved to about −8°, 24°, and −40° dip latitudes for the one main and two side peak traces. The June noontime hmE and HE values have peaks along the ∼8° dip latitude trace as well, but the HE map is not shown in this paper.
 The solar tidal fields can be represented in the form
where tUT = universal time (in days) = tLT (local solar time in days) − λ/Ω, Ω = rotation rate of the Earth = 2π day−1, n denotes a subharmonic of a solar day and is equal to 1 for diurnal waves, 2 for semidiurnal waves, and so forth, s is the zonal wave number and is <0 for eastward propagating waves, or >0 for westward propagating waves, and the amplitude Ans and phase ϕns are functions of height and latitude. These solar tides may be migrating (Sun synchronous) when n = s or nonmigrating when n ≠ s, and at any height and latitude the total tidal response is obtained as a sum over n and s. From equation (5), DE3 reveals a wave-4 longitude structure in constant local time observations because ∣s − n∣ = 4. The June noontime foE and hmE maps in Figure 9 show that the +8° dip-latitude foE and hmE traces present an alternative depiction of the four-peaked longitudinal structure in the regions: east of the Pacific Ocean (∼−100°E), west of the Africa (∼−10°E), India (∼80E), and the Pacific Ocean Center (∼180°E). The −24° dip-latitude foE trace shows that there also exists a light and four-peaked longitudinal structure but the peak longitudes move eastward, about 10° in longitude. It is noted here that the June foF2 maps in Figure 5 and the June HF2 maps in Figure 7 also present similar four-peaked longitudinal structures at the two EA crest traces and the dip equator, respectively, and match the four-peaked longitudinal structures as shown in Figure 9.
5. Final Remarks
 The TWIM model is described in this paper. It is intended to model large-scale and statistical Ne variations, and exhibits three-dimensional Ne distributions with vertically fitted Chapman layers that have distinct F2, F1, E, and D layers, using surface spherical harmonics approaches for the fitted Chapman layer parameters. This match should be maintained in the first and second derivatives of practical schemes for providing reliable radio propagation prediction and electrostatic field determination. The layer-parameter mapping results can be used to investigate the diurnal and seasonal variations of the latitudinal distribution of EA in different ionospheric layers. Our study specifically attempts to account for the longitudinal differences of mapped EA. The mapping results also investigate the latitudinal and longitudinal structures caused by atmospheric tides. We conclude that the global and phenomenological representations of the ionospheric layer parameter median may make use of GPS RO data, with the advantage that data compilation includes areas over the oceans, the southern hemisphere, and the polar regions, where few ground-based stations are positioned.
 Concerning the future of the TWIM, three main tasks are currently being done: (1) improving the accuracy of the topside F2 layer of TWIM by using the vary-Chap functions; (2) increasing the temporal and spatial resolutions of the TWIM; and (3) doing the TWIM parameter predictions.
 This work has been supported by projects 97-NSPO(B)-SP- FA07–02(J), NSC97-2623-7-008-001-D, and NSC97-2111-M008-008-024-MY2.