Zenith neutral delay (ZND) estimates derived from ground-based GPS receivers exhibit variations at harmonics of the solar day. The aim of this work is to characterize the semidiurnal (S2) variation and determine its probable origin. Data from 100 GPS sites are compared with surface pressure measurements to reveal close agreement between the estimated ZND S2 variation and the S2 surface pressure tide. Error analysis suggests that the S2 variation in ZND estimates is not due primarily to orbit, solid earth, or Earth orientation modeling errors. Atmospheric loading and mapping function errors are each expected to contribute less than 11% to the estimated ZND S2 amplitude. Local incongruities reflect the influence of water vapor or site-dependent errors.
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 Since 1998 the International GPS Service (IGS) has included an estimated zenith neutral delay (ZND) (sometimes called tropospheric delay) among its official products. The ZND measures the amount by which the neutral atmosphere delays a GPS signal in excess of the free-space delay. ZND is conventionally measured in meters, with a nominal value of 2.4 m at sea level. Current ZND accuracy is about 4 mm [Byun et al., 2005].
 Spectral analysis of IGS ZND data reveals mm-level variations at harmonics of the solar day and at the lunar tidal frequency. An obvious explanation is that these variations reflect the regular tides induced on the atmosphere by solar and lunar thermal and gravitational excitation. Indeed, Dai et al.  showed that a solar diurnal (S1) variation in the wet component of GPS-derived ZND agrees roughly with that in microwave radiometer (MWR) and radiosonde measurements. However, Humphreys et al.  showed that the amplitude of the lunar tide in IGS ZND data prior to 2000 is drastically reduced in post-2000 data, and that high-frequency solar harmonics extend well beyond S6, contrary to the behavior of the surface pressure tides. Such results encourage careful scrutiny of the periodic variations in IGS ZND data if these are to be accepted as manifestations of actual atmospheric tides. Moreover, a close examination of the variations in GPS-derived ZND will be useful for detecting persistent periodic errors in the ZND record.
 The S2 variation in IGS ZND data will be the focus of this study. The approach taken will be to compare its global phase and amplitude distribution with that of the semidiurnal surface pressure tide, S2(p). With long wavelengths excited by ozone and water vapor absorption of solar radiation, the 12-hour S2(p) tide is characterized by a strong, zonally homogeneous surface pressure variation with tidal maximum occurring about 2 hours before local noon and midnight [Chapman and Lindzen, 1970; Dai and Wang, 1999; Ray, 2001]. A strong correlation between the global distribution of the two signals will provide evidence for the authenticity of the S2 variation in IGS ZND estimates. Precision and likely errors will also be considered.
2. Models, Data, and Analysis Technique
 ZND is the integrated refractivity along a vertical path through the neutral atmosphere:
where τz is ZND measured in units of distance, c is the speed of light in a vacuum, and tz is the delay measured in units of time. Neutral atmosphere refractivity N is approximately related to the total mass density of moist air ρ (kg m−3), temperature T (K), and partial pressure of water vapor e (mb) by the relation [Davis et al., 1985]
Here, Zw is a factor near unity that accounts for the small departure of moist air from an ideal gas. The integral of the first term in equation (2) is designated the hydrostatic component, τhz; the integral of the remaining two terms is the wet component, τwz. Thus τz = τhz + τwz.
 The hydrostatic component dominates τz, accounting for roughly 90% of the total delay. It follows that τz is strongly correlated with surface pressure p0. Disregarding the change in the acceleration of gravity g with height, the hydrostatic approximation dp = −gρdz relates τhz to p0 by τhz = (2.28 mm/mb)p0. Davis et al.  note that this scale factor varies less than 1% under extreme weather conditions. Assuming a typical 1-mb semidiurnal surface pressure tide S2(p), this scale factor predicts a semidiurnal ZND variation S2(τz) with an amplitude of 2.28 mm. Thus, it is unsurprising to note variations in IGS ZND data at frequencies corresponding to the atmospheric tides (Figure 1, left). Peaks at S1 to S6 are visible as well as a peak at the lunar tidal frequency, L2 (nearly coincident with S2). This paper will focus only on S2, which, as observed in the IGS ZND data, will be denoted S2(z) to distinguish it from the error-free S2(τz) variation. This is done to concede the possibility that the observed variations in IGS ZND data are due to causes other than actual tides in τz.
 Though small on average, the wet component is highly variable, contributing to τz significant dynamics that are not present in surface pressure. This explains some of the irregularity of the IGS ZND data relative to surface pressure (Figure 1, right). The effect of water vapor dynamics on S2(τz) has not been well established. Dai et al.  detected a ∼0.1-mm semidiurnal variation in GPS-derived precipitable water vapor (PWV) estimates over North America, amounting to a ∼0.7-mm semidiurnal variation in τwz—comparable in amplitude to the semidiurnal variation in τhz. On the basis of this estimate, one would expect τwz to significantly influence the response of τz to S2 forcing. But whereas Dai et al.  conclusively demonstrated a large diurnal variation in PWV, the reported semidiurnal variation is much smaller than the rms errors for GPS-derived PWV and was not visible in independent radiosonde and MWR measurements. As will be shown subsequently, the irregularities in S2(z) at some IGS sites are consistent with the hypothesis that there exist significant local semidiurnal signals in τwz. However, in general, the S2(z) variation is well predicted by its hydrostatic component alone.
 This study analyzes the new IGS ZND product (available at ftp://cddis.gsfc.nasa.gov/gps/products/trop_new) for the interval October 2000 to June 2005 and for 100 sites distributed globally as indicated by the dots in Figures 2 and 3. The new IGS ZND product is based on the precise point positioning technique. It has a higher sampling rate and lower formal errors than the legacy IGS ZND product [Byun et al., 2005]. Most (79) of the sites are Reference Frame sites, which are subject to strict standards of data quality and continuity. Gaps are common in the data, but at least 2 years of ZND estimates are available for each site. The IGS data are downsampled from 5- to 15-min intervals and the harmonic coefficients a2 and b2 of the oscillation at the S2 frequency are estimated by least squares from the ZND time series for each site. These are used to calculate the mean amplitude A2 and phase σ2 of S2(z) over the data interval:
Here, t′ is local mean solar time (LST) in degrees. Zonal harmonic analysis of the harmonic coefficients follows the procedure outlined by Haurwitz and Cowley . First, a2 and b2 are interpolated onto a regularly-spaced 5° lat by 10° long grid. Next, the gridded coefficients around each line of latitude θi are expanded using a trigonometric series of the longitude λ. This yields a decomposition of S2 into wave components S2(t, θi) = S2s(t, θi), where s is wave number and t is Universal Time. Predominant among the wave components is the migrating tide S22, which moves westward at the speed of the mean Sun. All zonal wave components with wave numbers s ≠ 2 are termed nonmigrating wave components; s = 0 is a standing wave; wave numbers s < 0 move eastward.
 The large-scale features of the S2(z) amplitude distribution, which forgive deficiencies in the spatial resolution of the 100 sites used, correlate well with large-scale features of S2(p); namely, S2(p)'s characteristic zonal homogeneity and its amplitude increase toward the equator. A comparison of the latitudinal distribution of S2(z)'s and S2(p)'s migrating components is presented in the AM, leading to an estimated S22(z)/S22(p) scale factor of 1.9 mm mb−1—near the scale factor predicted by the hydrostatic approximation. This suggests that, on average, S2(z) is dominated by its hydrostatic component.
 The S2(z) phase distribution exhibits a predominant phase near 150°, corresponding to tidal maximum at 1000 LST. This is consistent with the global average phase for S2(p), which is 158° (0940 LST). The phase of S2(z) is more irregular at middle and low latitudes than that of S2(p) (irregularity at high latitudes is expected—a result of the increasing dominance of the S20 standing wave toward the poles). This is due in part to systematic errors in ZND and to the short ZND data record, but may also indicate wet component influences on S2(z).
 The global mean wave components of S2(z) (inset of Figure 3) agree closely with surface pressure data. The migrating wave component, S22, predominates as expected for the S2 tide. The next largest contribution for both data sources is the standing wave, S20. As a consequence of amplitude and phase irregularities, the non-migrating wave components are more pronounced in the ZND data than in the surface pressure field [e.g., S22(z)/S20(z) ≃ 4 vs. S22(p)/S20(p) ≃ 10].
4. Error Analysis and Discussion
 The results of the previous section are encouraging insofar as they suggest that S2(z) behaves generally as expected: it is dominated by its hydrostatic component. The ‘fingerprint’ of the hydrostatic tide in the amplitude and phase data (most striking in the wave component analysis) makes it unlikely that S2(z) is caused primarily by variations in EOP, ocean tide loading, or solid earth tides. Each of these has its own unique global phase and amplitude fingerprint, different from that of the S2(z) and S2(p) tides. (For example, the anelastic response of the solid Earth to the solar tide potential causes the maximum solid-earth deformation to occur after local noon, in contrast to the phase shown in Figure 3.) Similarly, it is unlikely that errors in the final IGS orbit estimates couple into the ZND estimates in such a way as to closely replicate the hydrostatic tide (except in the case of atmospheric pressure loading, which will be discussed subsequently). Nonetheless, incongruities between the global distributions of S2(z) and S2(p) warrant closer examination. It is the purpose of this section to further assess the validity of S2(z) by examining its precision and considering likely sources of error.
 A tidal determination's precision is calculated by dividing the data record into smaller subrecords that are assumed to be stochastically independent. The mean amplitude and phase of these samples are considered significant if the mean tidal amplitude is at least 3 times greater than the semi-major axis of its 1σ error ellipse [Chapman and Lindzen, 1970]. To assess S2(z) precision, 13 sites with nearly continuous data records from 2000 to 2005 were chosen for statistical analysis. Of these sites, 9 were equipped with a meteorological (MET) package providing surface pressure measurements, enabling a site-by-site comparison of S2(z) and S2(p) (MET data for site MKEA were taken from the MET package of a nearby telescope). The ZND and surface pressure records are divided into 4 one-year subrecords, each beginning on the same day of the year. From these, a four-sample amplitude and phase standard deviation is computed. The entire 5-year interval is used to evaluate the mean amplitude and phase. Results are reported in Table 1. All the determinations of S2(z) are significant except for those corresponding to sites YELL and ALGO, whose high latitudes explain the difficulty in the determination. All the determinations of S2(p) from MET data are significant. In all comparable cases, the precision of the MET determination is superior to that of ZND. Possible sources of random errors in S2(z) are phase range measurement errors and random fluctuations in τwz.
Table 1. ZND and Surface Pressure (MET) Determinations of S2a
Longitude in deg. E, latitude in deg. N, height in m, ZND amplitude in 10−2 mm, pressure amplitude in 10−2 mb, phase in deg. Error bounds are 1σ values.
27 ± 5
157 ± 29
27 ± 6
148 ± 8
32 ± 1
142 ± 2
60 ± 9
134 ± 16
36 ± 1
143 ± 1
48 ± 14
200 ± 20
149 ± 12
148 ± 3
131 ± 14
84 ± 9
69 ± 1
150 ± 1
111 ± 18
91 ± 8
70 ± 1
154 ± 1
249 ± 19
156 ± 15
106 ± 2
157 ± 1
190 ± 12
162 ± 1
87 ± 1
160 ± 1
224 ± 6
92 ± 2
86 ± 1
164 ± 1
96 ± 10
118 ± 3
83 ± 1
161 ± 1
303 ± 18
154 ± 1
125 ± 2
163 ± 1
107 ± 16
135 ± 7
 Two features of Table 1 invite further attention. First, phase estimates for sites JPLM, MDO1, KOKB, and MKEA are low compared with MET data. Their phases correspond to tidal maxima at about local solar noon. In each case the good phase precision makes it unlikely that random instrument or meteorological noise is to blame. A glance at Figure 3 shows that, at mid to low latitudes, such departures from S2(p)'s global mean phase (158° or 0940 LST) are exceptional, indicating site-dependent systematic errors or a significant semidiurnal variation in τwz at these sites. A second anomaly in Table 1 is the sharp decrease in S2(z) amplitude from KOKB to MKEA, i.e., from the northwest to the southeast extremes of the Hawaiian archipelago—a short span on the scale of the S2 tide. No other pair of similarly proximate sites with statistically significant determinations of S2(z) manifests such a disparity in amplitude. The disparity is unlikely to be caused by errors in the IGS orbits, the solid earth tide models, the transmitter clock biases, or the EOP models since these errors would be approximately common to both KOKB and MKEA. Furthermore, the MET data indicate that the hydrostatic component of the semidiurnal variation is nearly equivalent at the two sites, despite a large difference in height.
 The S2(z) amplitude at KOKB is within 14% of the value predicted by the hydrostatic scale factor, but the S2(z) amplitude at MKEA is 50% lower than predicted. A close examination of MKEA's ZND frequency spectrum in the neighborhood of S2 reveals that a considerable fraction of S2(z) energy is contained in sidebands resulting from a seasonal modulation. The same sidebands are a smaller fraction of S2(z) amplitude at KOKB and a much smaller fraction at LHAS, an inland site on the Tibetan Plateau whose height is nearly that of MKEA. One explanation might be that at MKEA's height the marine-influenced atmosphere produces a seasonally-varying semidiurnal oscillation in τwz.
 Atmospheric pressure loading (APL), the slight (mm-scale) deformation of the flexible Earth caused by redistribution of atmospheric mass [Petrov and Boy, 2004], is a possible error source for S2(z). Because APL is site-dependent it contributes to irregularities in S2(z); and because APL is caused by variations in surface pressure it is capable of masquerading as an atmospheric tide. None of the analysis centers that contribute to the IGS final orbits currently includes APL in its measurement model, nor is it included in the models used to generate the new IGS ZND product.
 ZND is sensitive to APL through direct and indirect effects. The direct effect occurs simply because more atmosphere separates a depressed station from a GPS satellite at zenith. This effect is small: a 1-cm (worst-case) depression corresponds to a 0.003-mm increase in τz. The indirect effect arises because the site position errors caused by APL couple into satellite orbit and ZND estimates. This effect is larger and more difficult to evaluate because it depends on many factors, including: (1) the spatial extent and amplitude of the pressure anomaly, (2) the spatial distribution of the ground sites, (3) the APL response at each site, (4) the elevation cutoff angle, and (5) the estimation strategy used to determine ZND.
 A study of the effects of APL on ZND was carried out by simulation. Realistic GPS satellite orbits and phase range measurements were generated for a globally-distributed set of ground stations. The effects of atmospheric loading were simulated by varying station heights in response to a global model of S2(p). APL sensitivity at each site was specified by regression coefficients from the International Earth Rotation Service Special Bureau for Loading (available at http://www.sbl.statkart.no). The coefficients were doubled to account for local departures from the inverted barometer assumption used in their calculation. Simulation revealed that APL contamination of ZND estimates amounts to less than 11% of the amplitude of the hydrostatic oscillation in ZND. Simulation details and plots are found in the AM.
 A final error source considered here is the ZND mapping function (MF). The IGS currently uses the Niell MF to convert neutral slant delays to zenith delays. Niell  showed that radiosonde-derived (“truth”) MFs exhibit a diurnal variation at low elevation angles—a consequence of temperature-driven changes in the atmospheric scale height. This variation gives rise to a ∼0.1% diurnal error at 5° in the hydrostatic Niell MF because the latter does not account for MF variations on time scales less than one year. If only 5° elevation slant delays were used to estimate ZND, a spurious 1.7-mm diurnal variation in estimated ZND would result. Accounting for the average distribution of elevation angles at each site and the 7° IGS elevation cutoff angle, the effect is reduced to 0.5 mm. A semidiurnal error in ZND caused by the same process can be estimated to be ≤0.17 mm (or ≤11% of the global average S22(z) amplitude) by recognizing that the semidiurnal atmospheric temperature variation is at least factor of 3 smaller than the diurnal variation. Improved MFs based on numerical weather models [see Byun et al., 2005] can be expected to reduce these periodic errors in ZND estimates.
 Conclusion: A strong global correlation with S2(p) suggests that the semidiurnal variation in IGS ZND data, S2(z), is due primarily to the actual semidiurnal variation in ZND, S2(τz), and not to other geophysical signals or to orbit errors. Atmospheric loading and mapping function errors each contribute less than an estimated 11% of S2(z) amplitude. Local incongruities between S2(z) and S2(p) may indicate lingering site-dependent errors or result from a semidiurnal variation in water vapor. These incongruities invite further study.
 The authors are grateful to Jim Ray and Paul Tregoning for their careful comments. Research at Cornell was supported in part by ONR and NSF.