We compared atmospheric excitations of the varying Earth rotation calculated from three different meteorological reanalysis data sets (NCEP/NCAR, NCEP-DOE and ERA-40) over 23 years, from 1979 to 2001. Atmospheric excitations are evaluated by atmospheric angular momentum (AAM) functions. We found that differences in AAM functions are primarily caused by differences in the wind data, especially the northward wind data, between the three reanalysis data sets. Differences between the NCEP-DOE and ERA-40 AAM functions are larger than those between the NCEP/NCAR and NCEP-DOE ones. The northward wind differences that have a large effect on the equatorial AAM functions originate from the upper troposphere at the tropics and midlatitudes in the southern hemisphere, especially in three regions: east Pacific off Latin America, Africa and the Indian Ocean. The eastward wind differences are seen at midlatitudes in the southern hemisphere. These differences are not confined to a single frequency. The difference spectra show a reddish frequency dependence with some peaks in seasonal bands and tidal diurnal bands. Since the eastward and northward winds over the tropics can most efficiently change the Earth's spin rate and polar motion, respectively, the wind accuracy over the tropics directly determines the accuracy in evaluations of the atmospheric excitations.
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 The variation of the Earth's rotation with time is driven by geophysical fluids, such as the atmosphere, oceans, and land water. Geodetic observation based on space geodetic techniques enables the detection of subtle variations in the Earth's rotation. On the other hand, evaluations of geophysical excitation on the Earth's rotation become more important not only for understanding the dynamical interactions between the geophysical fluids and the solid Earth but also for maintaining the geodetic reference frame [e.g., Chao et al., 2000].
 Among geophysical fluids, the atmosphere plays the most important role in the variation of the Earth's rotation with time. Therefore an accurate evaluation of the atmospheric excitation is necessary for an accurate evaluation of the geophysical excitations. The atmospheric excitation is evaluated by atmospheric angular momentum (AAM) functions [e.g., Barnes et al., 1983]. The functions are composed of two terms, the pressure term and the wind term, corresponding to the two mechanisms of the excitation regimes: the excitation via the change of the moment of inertia (that is, atmospheric mass redistribution on the Earth) and that via the angular momentum exchange between the atmosphere and the solid Earth. These functions are calculated from meteorological data sets that were analyzed on a global scale. Nowadays, various meteorological data sets based on operational analysis and data assimilation are available from meteorological institutions all over the world.
 Recently, reanalysis meteorological data sets have been favorably used as ‘reference’ or ‘standard’ data in studies on the atmospheric sciences. Because the reanalysis data are obtained using a frozen assimilation scheme and a fixed physical model over a few decades, they provide us time-sequence meteorological data of much more uniform quality than operational analysis data. Therefore to study the Earth's rotation over decades, we benefit from the uniform quality of the reanalysis data. Today, three reanalysis data sets are available. Since these three data sets have both merits and demerits, the feasibility of using them in the Earth rotation study and their physical characteristics should be examined.
 The first topic of this paper is the evaluation of the differences between meteorological reanalysis data sets that are commonly used today and their effects on geophysical excitations of the Earth's rotation by examining AAM functions.
 AAM functions calculated from different meteorological data sets have different values [e.g., Salstein et al., 1993]. For the axial AAM, many authors have examined their differences. See Hide and Dickey  for a summary of the reports. Rosen and Salstein [1983, 1985] calculated the seasonal differences contributing to axial AAM functions from the zonal band regions. Dickey et al.  also compared axial AAM functions calculated using four operational analysis data sets available in those days and showed their differences at intraseasonal bands with reddish spectra.
 For the equatorial AAM functions, several authors pointed out the importance of the wind term to the AAM differences. For example, Aoyama and Naito  studied the differences in AAM functions calculated from the NCEP/NCAR reanalysis data set (compiled by the National Centers for Environmental Prediction (NCEP) and the National Center for Atmospheric Research (NCAR)) and Japan Meteorological Agency (JMA) operational data set at seasonal bands. They pointed out that discrepancies arise from the tropospheric regional winds. At the diurnal frequency bands, de Viron and Dehant  compared AAM functions between different meteorological data and pointed out that cross correlations between wind AAM functions are worse than those between pressure AAM functions.
 Although the wind term of AAM functions has less power to excite the polar motion than the pressure term and is sometimes undervalued in its importance on the Earth's rotation, the differences in wind terms have crucial effects on the excitation of some phenomena. The winds also act on the Earth's rotation in indirect ways: For example, the winds transport water vapor, redistribute hydrological mass and drive ocean currents.
 The Chandler wobble, the free oscillation mode of the polar motion, shows relatively constant oscillation in observed data, despite the fact that, in theory, the wobble damps its oscillation through dissipation. Excitation sources of the Chandler wobble are still unknown, however, some geophysical fluids can be candidates for them. The wind has an enough power to excite the Chandler wobble at the Chandler frequency bands and is a possible candidate cause of the excitation of the Chandler wobble [Furuya et al., 1996; Aoyama, 2005]. Recent reports showed that oceanic excitation along with atmospheric excitation plays an important role on excitation of the Chandler wobble [Gross, 2000; Brzeziński and Nastula, 2002].
 In this paper, we mainly focus on differences in meteorological reanalysis data and their effects on the polar motion. However, such differences also affect the nutation. The nutation, the change in the Earth's orientation with periods longer than 2 days with respect to the celestial reference frame, is mainly driven by the astronomical interaction forces. However, geophysical fluids which have powers around retrograde diurnal frequency bands (with respect to the terrestrial reference frame) also excite the nutation. Brzeziński et al.  summarized the atmospheric contribution to the nutation. They showed that the diurnal retrograde variation in the equatorial wind term contributes as much as 80 percent to its total power and is much larger than variation in the equatorial pressure term.
 Their study also show that transfer functions converting from the geophysical excitations into the polar motion or nutation depend on frequency. Their results imply that differences in meteorological wind data at some resonant frequencies might produce large differences in the polar motion or nutation. Therefore accurate evaluation of atmospheric excitation is very important both for the polar motion and the nutation.
 The second topic of this paper is on the clarification of the properties (geographical distribution or frequency dependence) of differences in the atmospheric wind excitations calculated from different meteorological data sets.
Rosen et al.  compared axial AAM functions from the zonal wind databased on two operational analysis data sets compiled by the National Meteorological Center (NMC) and the European Center for Medium-range Weather Forecasting (ECMWF) in those days, and found that the differences mainly came from the region 33S–16N. They concluded that the differences lead to the annual differences. Their results are very intriguing because the error source of the atmospheric excitation is not uniformly distributed over the Earth, but arises from the specific regions. Aoyama and Naito  depicted wind differences that affect the evaluation of the atmospheric annual wobble excitation. About 20 years have passed after Rosen et al.'s publication, and meteorological analysis has been greatly improved. On-board satellite observation instruments and computer processing improve the data quality. How then have the modeled atmospheric excitations changed?
 On contribution from regional atmospheric excitation to the global excitation, Nastula et al.  showed that regional wind signals are several times larger than regional atmospheric pressure signals. They also noted that since wind signals in one region are out-of-phase to ones in another region, the net wind effect at a global scale are dilutized. However, such large regional wind signals can be potential error sources in evaluating atmospheric excitation.
 In the past few years, the satellite mission GRACE has revealed the temporal and spatial change of gravity on the Earth with high quality [Tapley et al., 2004; Rowlands et al., 2005]. The regional mass redistribution has been caught by observation and is expected to improve the accuracy of Earth rotation excitations via geographical mass redistribution over the Earth. However, atmospheric excitations via angular momentum exchange are only evaluated from such meteorological data sets, at present.
 For this reason, we should understand the potential errors and properties of the angular momentum data obtained from the currently available data sets.
 Soon after global data sets on oceans and land water became available in these several years, oceanic and hydrological excitations of the Earth's rotation were evaluated. Importance of the oceanic and hydrological excitations to the total geophysical excitation are shown by several authors [Ponte and Stammer, 1999; Gross et al., 2003; Chen et al., 2000]. The better the quality of geophysical excitation is required, the better the quality of geophysical data sets should be prepared. Since atmospheric excitation is largest among geophysical fluid excitation, we should examine the quality of the atmospheric data in detail.
 Herein, we will describe the methods of analysis in section 2 and the reanalysis data sets used in this study in section 3. The results and comparison between different AAM functions will be shown in section 4 and the discussion presented in section 5.
2. Methods of Analysis
 The atmospheric excitation of the Earth's rotation is evaluated using AAM functions. We set the Cartesian coordinates as follows: axis 1 is directed toward the Greenwich meridian on the equatorial plane, axis 2 is directed toward the 90°E meridian on the equatorial plane and axis 3 is directed toward the North Pole. The equatorial AAM functions (χ1, χ2) excite the polar motion, while the axial AAM function χ3 varies the spin rate. The AAM functions are composed of two terms: the pressure term and the wind term. In this paper, these terms are denoted by the superscripts p and w, respectively.
where ps and (u, v) are the surface pressure and (eastward, northward) wind velocities, respectively. R, Ω and (A, C) are the radius, the average spin rate and the principal moments of inertia about the (equatorial, polar) axes of the Earth. g is the mean surface gravity (9.8 m/s2). Cm denotes the C of the mantle. The right-hand side of these equations are integrated over the longitude λ, latitude ϕ and pressure level p.
 As we see in equations (1) and (2), the equatorial AAM functions have a sinusoidal dependence in longitude. This longitudinal dependence surpasses regional excitation and veils such regional patterns. We therefore introduce χeq such that
in order to eliminate the undesirable longitudinal dependence from the global map.
 In this paper, we adopt an inverse barometric (IB) hypothesis for the monthly averaged data in order to depict seasonal changes in section 4.1 and a noninverse barometric (non-IB) hypothesis for the data collected every six hours in order to catch rapid changes in sections 4.2 to 4.4. These hypotheses are adopted because of the following reasons: Generally speaking, for periods less than a week, the ocean surface response against the surface pressure is regarded as a non-IB response, while for periods longer than a week, ocean responses are inverted barometrically (for example, see Dickman ). However, our main focus in this paper is the wind term of AAM functions. Fortunately, the IB hypothesis affects the wind term only through the bottom surface changes in the pressure integrals of the wind field. Therefore regardless of which extreme case of the hypothesis is adopted in the evaluation of AAM functions, values of the wind term are not significantly affected by the choice.
 In our calculation, no winds blow inside mountains (i.e., the SP method, introduced by Aoyama and Naito ). The surface elevation data we used is obtained by modifying ETOPO-5 (5-minute-grid topographical data) distributed by the National Geophysical Data Center, the National Oceanic and Atmospheric Administration (NOAA) in order to adjust the grid size (2.5 by 2.5°) of the reanalysis data. We assigned zero elevation over the oceans.
 The two NCEP reanalysis data sets contain the meteorological data at 17 isobar levels from 1000 to 10 hPa while the ERA-40 contains data at 23 isobar levels from 1000 to 1 hPa. (See Table 1. The specifications of the reanalyses are summarized in the next section.) For the wind term calculation (equations (2) and (4)), we sum up the wind field vertically throughout the atmospheric layer. However, in order to make the column mass of the atmosphere equal for three AAM sets, we evaluate the wind term by integrating from the surface to 10 hPa for all three sets.
Table 1. Brief Summary of Reanalysis Data Sets Used in This Studya
Supplied Pressure Level Data
In the column Period, ‘+’ indicates that the period is extended to the present. Resolutions are expressed by the total wave number for the triangular truncation of the spherical harmonics (T) in the horizontal direction and the level number (L) in the vertical direction.
17 levels (up tp 10 hPa)
17 levels (up tp 10 hPa)
July 1957–June 2002
23 levels (up tp 1 hPa)
 The analysis period is for 23 years, from 1979 to 2001, because these three reanalysis data sets overlap during this period. Choosing this period, we benefit from meteorological observation with better quality, thanks to meteorological satellites.
 In order to compare the atmospheric excitations with the observed Earth rotation, we calculated the geodetic excitations from the Earth orientation data, the C04 series, compiled by the International Earth Rotation and Reference Services (IERS), using Wilson's  discrete formulae with the Chandler parameters P = 431 days and Q = 179 [Wilson and Vicente, 1990]. For the details of the IERS C04 series, see Gambis .
3. Reanalysis Data Sets
 In this study, we use three reanalysis meteorological data sets: the NCEP/NCAR reanalysis (hereafter, referred to as NCEP-1), NCEP-DOE reanalysis (NCEP-2) and ERA-40 reanalysis (ERA-40) data sets. Hereafter, we briefly summarize the specifications of these reanalysis data sets. See also Table 1.
 NCEP-1 was produced by the project team of NCEP and NCAR in order to complete a 40-year record of global meteorological analysis with a frozen state-of-the-art data assimilation system and a database [Kalnay et al., 1996; Kistler et al., 2001]. The data initially covered 40 years, from 1957 to 1996, but the period has subsequently been extended from 1949 to the present. The temporal resolution of the data is six hours. They adopted a model with spatial resolutions of T62 (triangular truncation at the wave number of 62, corresponding approximately to 210 km) in the horizontal direction and L28 (28 pressure levels) in the vertical direction. A 3-D variational method is adopted for the data assimilation scheme.
 However, after the public release of the NCEP-1 data, several bugs were found in the input data of PAOBS (the Southern hemisphere surface pressure data), the snow coverage, the oceanic albedo, and other components [Kistler et al., 2001; Kanamitsu et al., 2002]. The project team believes that the worse data were rejected as outliers during the data assimilation process and did not degrade the final outputs.
 The NCEP-2 reanalysis is a follow-up reanalysis of NCEP-1, with some corrections to the known errors [Kanamitsu et al., 2002]. The same observation data, assimilation scheme and model resolution used in NCEP-1 are adopted for NCEP-2.
 On the other hand, ECMWF had conducted the meteorological reanalysis project ERA-15 since 1993. ERA-15 covers meteorological data for 15 years from 1979 to 1993 [Gibson et al., 1999]. Recently, ECMWF has finished the follow-on reanalysis, ERA-40, covering over 40 years from 1957 to 2001 [Simmons and Gibson, 2000; Uppala et al., 2005]. The project team adopted a finer resolution model (T159, L60) than that used in the former reanalysis, ERA-15, and those used in the two NCEP reanalysis data sets [Simmons, 2002]. The observation data is assimilated by a 3-D variational method, which replaces the statistical optimum interpolation method adopted in ERA-15.
 Hereafter, we report our results of differences in AAM functions and their properties. The main topics of the following four subsections are (1) seasonal differences, (2) contribution to differences from eastward and northward wind components, (3) geographical distribution of the differences, and (4) spectral analysis of the differences.
4.1. Seasonal Differences in Wind Excitations
 First, we check differences in time series between the three AAM functions using monthly averaged data. Figure 1 shows differences between two pairs of the AAM functions, namely, NCEP-1 minus NCEP-2 and NCEP-2 minus ERA-40.
 In contrast to the small differences between the two NCEP AAM functions, the differences between the NCEP-2 and ERA-40 AAM functions are large. These differences can be explained by differences in the wind term of AAM functions. This figure also shows that the AAM differences are larger in the earlier period (before 1985) than in the later period. As Masaki and Aoyama  already reported, we can see a step-shaped offset in the difference in χ3 time series between the two NCEP AAM functions around 1996–1997. This offset is also attributed to the wind term. A similar but smaller offset is also seen in χ1 at the same instant. The differences in the χ3 component show clear seasonal signals, especially in the time series of NCEP-2 minus ERA-40, whereas the differences in the equatorial components have seasonal-like signals but with large modulations.
 Large differences between the wind terms are also supported by the statistical assessment. The RMS differences for three pairs of the AAM functions (Table 2a) show that, for any pair of meteorological data sets, the differences between the wind terms are several times larger than those between the pressure terms. For example, according to the simple linear transfer functions given by the equations (1) to (4), the RMS differences between the equatorial AAM functions of NCEP-2 and ERA-40, based on monthly averaged data, are (0.2220 × 10−7, 0.2133 × 10−7) corresponding to a polar motion of (4.579, 4.408) mas, and the RMS difference between the axial AAM functions is 0.0065 × 10−7 corresponding to a UT1 value of 0.056 ms. By considering the RMS statistics of wind excitations (Table 2b), we realized that the wind term of ERA-40 is about 1.5 times larger than those of the two NCEPs for any of the three components. This result suggests that the time variation of the ERA-40 wind field is larger than those of the two NCEP wind fields.
Table 2. RMSs of (a) the AAM Differences and (b) AAM Excitations of Three AAM Pairsa
The RMS values show magnitudes of the intraannual variations for the monthly averaged data. The ‘total’ indicates the sum of pressure and wind terms. The pressure terms are calculated under the assumption of the IB hypothesis. The values are in units of 10−7.
(b) Raw Time Series
 Next, we check seasonal AAM differences because we can observe seasonal-like features in the differences between AAM functions in Figure 1. Since the largest periodic excitation on the polar motion brought about by the atmosphere is the annual excitation, errors in annual excitation result in errors in the annual wobble. UT1 also shows eminent annual and semiannual excitations.
 We determine seasonal (annual and semiannual) components by fitting sinusoids of
where A, ω, t and δ are the amplitude, frequency, time and phase, respectively. We try to fit sinusoids using the least squares approach. The determined seasonal signals are shown in Figure 2. For comparison, we also show the seasonal geodetic excitations in the figure.
 The seasonal wind terms of the two NCEP AAM functions deviate markedly from those of the ERA-40 AAM functions. In all cases, the seasonal wind terms of ERA-40 have larger amplitudes than those of the two NCEPs. The seasonal AAM functions (pressure plus wind terms) calculated from ERA-40 are much closer to the geodetic excitation than those from the two NCEPs. The magnitudes of the legend arrows in Figure 2 indicate the equatorial and axial AAM functions of 0.2 × 10−7, 0.02 × 10−7, which correspond to amplitudes of 4.1 mas for the polar motion and 0.17 ms for UT1, respectively.
 If we compare our results with those of Aoyama and Naito , our annual signal for ERA-40 is close to the annual signals for the operational JMA data.
4.2. Contributions of Eastward and Northward Winds to Excitation Differences
 In the previous section, we showed that wind term differences between different meteorological data sets markedly affect the total AAM differences. Generally speaking, because the vertical component of wind vectors is negligibly small compared with the horizontal one, it is useful to express the wind field by decomposing it into two horizontal components; zonal (eastward) and meridional (northward) components. (Hereafter, the ‘eastward winds’ refer to winds decomposed into the east-west directions, that is, u-velocity winds. Similarly, the ‘northward winds’ refer to winds decomposed into the north-south directions, v-velocity winds.) Their contributions to the Earth system, however, differ: The zonal winds circulate around the Earth with large velocities and change the daily weather, whereas the meridional winds, usually weaker then the zonal winds, transport heat and vapor, among others, across latitudes and mitigate the inhomogeneity between the equator and the poles.
 In this section, we focus on wind differences that are detected in the AAM functions by decomposing winds into eastward and northward components (that is, the (u, v)-velocity components).
 Note that if we use meteorological data analyzed every six hours and evaluate differences between AAM functions in time series as in Figure 1, rapidly changing wind differences completely hide seasonal signals. This is why we adopt monthly averaged data in section 4.1. However, in the following sections, because we focus on wind differences, adopting a finer time resolution is appropriate for the purpose.
 Therefore we calculate wind field differences using data analyzed every six hours, with the non-IB hypothesis, hereafter. That is, results also include differences changing at the diurnal frequencies.
 We check the statistics of eastward and northward wind differences, separating them into constant biases and time-varying differences. The results are shown in Table 3.
Table 3. Wind Term Differences Contributed by the Eastward and Northward Components (u, v) of the Wind Velocity, Using Meteorological Data Analyzed Every Six Hoursa
(a) NCEP-1 minus NCEP-2
The standard deviations and biases of the differences are shown for (a) NCEP-1 minus NCEP-2 and (b) NCEP-2 minus ERA-40. The values are in units of 10−7.
(b) NCEP-2 Minus ERA-40
 For both NCEP-1 minus NCEP-2 and NCEP-2 minus ERA-40, the SD differences due to the northward winds are larger than those due to the eastward winds. The differences in the χ2 component are slightly larger than those in the χ1 component, suggesting that the wind field differences exhibit longitudinal inhomogeneity. Bias differences also exist, which are larger in the differences between NCEP-2 and ERA-40.
 For the axial component, the differences between the eastward wind components of NCEP-2 and ERA-40 show large time variations.
4.3. 3-D Map of Wind Excitation Differences
 Winds blowing on the Earth are not random but flow in a systematic way on global to regional scales and make a wind system. Since χ1 and χ2 have sensitivity to winds at different regions (see equations (2) and (4)), we envisage that the wind field differences are inhomogeneously distributed over the Earth. Then, where do these differences originate from? To answer this question, we study the 3-D (geographical and vertical) distribution of the wind term differences.
 In order to check the vertical distribution of differences, we divide the atmospheric layer into six for the two NCEP reanalysis data sets, or into seven for the ERA-40 reanalysis data set (Table 4). Since two NCEP data sets cover the lower six layers, we use these six layers for comparison. Caution is taken in dividing the atmosphere at the isobar levels such that each layer has almost the same mass. The tropopause does not necessarily coincide with the 100 hPa isobar level, but for convenience, we use the terms ‘troposphere’ and ‘stratosphere’ to refer to the atmosphere below the 100 hPa level (i.e., Layers 1 to 5) and above the 100 hPa level (Layer 6), respectively.
Because two NCEP reanalyses only supply the meteorological data for pressure levels of up to 10 hPa, we use the lower six layers for comparing three reanalyses in this study.
 Hereafter, we will focus on the wind excitation and their differences to specify areas in which large differences are generated.
 Before investigating the 3-D map of wind term differences, we check the 3-D map of the wind excitations. For the sake of brevity, we take the ERA-40 excitation as an example. (The overall characteristics of the 3-D excitation map of ERA-40 hold for the other two NCEP excitations.) Results are shown in Figure 3 for the equatorial component χeqw and Figure 4 for the axial component χ3w. (In order to eliminate undesirable longitudinal patterns in χ1 and χ2, we draw χeqw = instead of χ1 or χ2, although this procedure loses information on the phase.) The signals are decomposed in terms of their periodicities: First, we separate the time-varying component (depicted in the first column) from a constant bias (the third column) by subtracting mean values over the entire time span. Next, we decompose the time-varying component into seasonal parts (annual and semiannual; displayed only for the annual signal on the second column of the figure) by fitting seasonal sinusoids.
 Although both eastward and northward winds can contribute to the equatorial AAM functions, eastward winds largely dominate northward winds in power. The strong constant signals come from jet streams, strong zonal winds at midlatitudes in the upper troposphere (Layers 4 and 5). The jet streams also show strong annual variations. The strong signals at high southern latitudes in the stratosphere (Layer 6) are due to the polar night jet.
 The northward winds very weakly contribute to the equatorial AAM functions. Excitations due to the northward winds do not show a strong geographical correlation. Moreover, seasonal signals are also weak.
 The axial excitations have two strong zonal bands at midlatitudes due to the jet streams. Regions with positive biases indicate regions in which westerlies dominate, whereas those with negative biases indicate regions in which easterlies dominate.
4.3.2. Differences Between Reanalysis Data Sets
 Next, we draw similar 3-D maps but for the differences in wind AAM function between two pairs of reanalysis data sets, NCEP-1 minus NCEP-2 (Figures 5 and 6) and NCEP-2 minus ERA-40 (Figures 7 and 8) . Each column of figures shows periodicity as in the previous section, but the color scales used in these plots are new ones.
 Marked changes are found in these difference maps with respect to the previous excitation maps: As expected from the statistics in Table 3, northward wind differences largely dominate eastward wind differences for both NCEP-1 minus NCEP-2 and NCEP-2 minus ERA-40.
 Moreover, large differences do not necessarily originate from regions where strong winds are blowing (like jet streams, colored from yellow to red in Figures 3 and 4); they can also originate from regions in which only weak winds are observed (colored in green in these figures). Generally speaking, differences in eastward winds are observed along the so-called ‘barking 40°’ in the southern hemisphere, whereas those in northward winds are observed at low latitudes where no systematic meridional flows occur.
 The wind differences for NCEP-1 minus NCEP-2 are observed over the tropical regions, largely due to the northward winds. Among these regions, the larger difference signals are seen in the east Pacific off Peru, and the Atlantic and Indian Oceans. The differences have stronger power in the upper troposphere, Layers 4 and 5. Annual differences due to the northward winds sparsely spread near the equator. Wind term differences for NCEP-1 minus NCEP-2 due to eastward winds are seen in the east Pacific off Latin America.
 For the axial excitations (only driven by the eastward winds), larger differences come from the tropics. We also see a negative DC bias over the tropics in the mid-troposphere.
 Next, wind differences for NCEP-2 minus ERA-40 due to northward winds are larger and have a wider distribution over the tropics and also at midlatitudes of the southern hemisphere than in the case of NCEP-1 minus NCEP-2. As we also observed in the previous case, larger differences exist again in the east Pacific, and the Atlantic and Indian Oceans. These differences come from almost all the layers; in particular, larger differences come from the upper troposphere (i.e., Layers 3 to 5). Nonseasonal differences dominate; however, annual differences are also seen in the east Pacific and over Africa. Wind term differences of NCEP-2 minus ERA-40 due to eastward winds come from the wide region around ‘barking 40°’.
 We can clearly observe annual differences in the east Pacific for the χ3 component. A positive DC bias is observed at the lower stratosphere (Layer 6), whereas a negative DC bias is seen throughout the troposphere depending on the height. They appear to be systematic biases.
 Interestingly, differences over Australia are smaller than those in the surrounding areas or other areas of the same latitude. Smaller differences due to the northward winds between NCEP-2 and ERA-40 are clearly evident in the left panels of Figure 7b.
 Note that the geographic distribution of differences in the pressure term of the AAM functions shows subtle signals only over mountainous areas, although we did not show the figures in this article. Their differences are smaller than differences in the wind term, as we have expected from Figure 1 and Table 2.
 We also mention that these 3-D maps lost information on the phase. Since wind excitation in one region may be canceled by wind excitation in another region, a sum of wind difference magnitudes does NOT give the global wind excitation difference. The global value of the wind excitation difference can be read from Figure 1, Table 2 (based on the monthly data) or Table 3 (based on the four-times-daily data).
4.4. Wind Excitation Differences in Frequency Domain
 In intraseasonal bands (we consider a wide range of frequencies from seasonal to daily in this section), many excitation events accompanied by the dynamic process of the atmosphere are detected in time series of the Earth's rotation. For example, the Madden-Julian oscillation in the tropical winds with periods of around 40–50 days is a well-known atmospheric phenomenon in the frequency bands [Madden, 1987; Dickey et al., 1991]. Seasonal winds and monsoons also have some components in intraseasonal bands. Since these events occur primarily at low latitudes and are driven by winds, large differences between wind excitations (as we saw in the previous sections) might also distort the correct evaluation of the effects on the Earth's rotation. Moreover, some tidal modes and their modulated ones are also seen in the frequency bands. Therefore it is important to check the magnitude of wind differences at the intraseasonal bands and the accuracy of excitations induced by winds obtained from meteorological reanalysis data.
 In this section, we examine the frequency dependence of wind differences, including those in the three regions where large wind differences are observed: the east Pacific, the Atlantic, and the Indian Ocean at the tropics (0° to 30°S latitude). Hereafter, we refer to these regions as A, B and C, respectively. (See the definition in Table 5 and the inset in Figure 9.)
Table 5. Definition of the Three Specific Regions Where Large Differences are Observed in the Study
A (East Pacific)
Layers 3–5 (700–100 hPa)
Layers 3–5 (700–100 hPa)
C (Indian Ocean)
Layers 3–5 (700–100 hPa)
Figure 9 shows intraseasonal spectra of the wind AAM differences between NCEP1 minus NCEP2 and NCEP2 minus ERA40 on the whole Earth and in the three regions. We apply smoothing using a triangular filter with a base length of 0.0002 cycles per year (cpy). We also show wide-range difference spectra, up to diurnal frequencies, for the whole Earth in the left panel.
 The overall structures of the different spectra are reddish in the intraseasonal bands with some peaks at seasonal frequencies. The power of the difference spectra up to diurnal bands (for the whole Earth) increases at a frequency of around 310 cpy (0.85 cycles per day (cpd), corresponding to a period of 28.3 h) and has sharp peaks at 365.25 cpy (24 h) and 340 cpy (25.8 h). These peaks correspond to the S1 and O1 tidal frequencies, respectively.
 There is no significant peak in the difference spectra around the intraseasonal frequency bands, but the differences exhibit continuum-like spectra. The three regions also have similar difference spectra in the intraseasonal bands, except for seasonal peaks.
 As expected from the results of Table 3 and Figures 5 to 8, the northward winds produce larger difference powers than the eastward winds over almost the entire frequency range.
 Next, we show prograde and retrograde spectra in order to check true nature of the difference spectra. Figure 10 shows difference spectra of NCEP-1 minus NCEP-2 and NCEP-2 minus ERA-40. The power around the retrograde diurnal frequency is larger by one order and two orders than the power around the prograde diurnal frequency, for u(eastward)-wind and v(northward)-wind, respectively.
 The different bending structures near −310 cpy between NCEP-1 minus NCEP-2 and NCEP-2 minus ERA-40 are very intriguing. The difference spectrum of NCEP-1 minus NCEP-2 increases by one order of magnitude around this frequency, whereas that of NCEP-2 minus ERA-40 is relatively flat around this frequency. Despite the fact that the latter dominates the former in power, the power of the former is comparable to that of the latter around this frequency.
5.1. Geographical Distribution of Wind Excitation Differences and Their Backgrounds
 We have shown that wind AAM functions calculated from different meteorological data sets have different values and the largest differences are seen in the upper troposphere at the tropics or in the southern hemisphere. Hereafter, we will discuss potential problems in the reanalysis wind data.
 Our results on the axial AAM differences are concentrated in the tropics. Interestingly, Rosen et al.  also reported that the zonal wind differences between NMC and ECMWF operational data in those days also have maximum values at the tropics. The meteorological data sets and the aspects of the wind field being studied are different between these two studies; however, both studies commonly imply that the meteorological data in the tropics may contain larger potential errors.
 Some other studies also address the difficulties in determining wind fields at low latitudes. Kanamitsu et al. , the NCEP-2 project team, recognized some differences between the two NCEP reanalysis data sets. They found ‘minor but notable differences’ in the equatorial divergent winds, due to differences between the locations of the Hadley circulation and the southern jet stream. Kistler et al.  compared the wind fields of NCEP-1 and ERA-15 (the former reanalysis data set of ERA-40, provided by ECMWF) and found that the zonally averaged u (eastward) winds show differences near the equator (see Figure 17 of their paper). They suggested that model differences between them largely affect the determination of the wind field at the tropics where convergence flows are seen. Not only in the reanalysis computation but also in many operational analyses, the vertical motion of the atmosphere depends more on the models used than the horizontal motion does. Newman et al.  commented on the differences between the convergence flows of the NCEP(-1) and ECMWF(ERA-15) reanalysis data sets.
 These reports remind us that the precise determination of the wind field at low latitudes has potential difficulties. This is an ironical fact in the study of the Earth's rotation: The excitation of the spin rate variation is most sensitive to the motion of the geophysical fluids at the equator because the torque exerted on the solid Earth is largest since the radius from the rotational axis is largest. The polar motion excitation by u winds becomes maximum at midlatitudes, but the polar motion excitation by v (northward) winds becomes maximum at the equator.
 At the equator, the Coriolis parameter almost vanishes, the geostrophic relation (the balance between the pressure gradient and the Coriolis force) does not hold. The thermal wind relation (one typical case of the geostrophic relation, taking the thermal gradient into consideration) enables a better estimation of the upper wind field at midlatitudes; however, this relation does not hold near the equator.
 Note that the equatorial and axial AAM functions have different sensitivities to the wind direction. At the equator, the equatorial AAM functions χw are only sensitive to the meridional (northward) winds, whereas the axial AAM functions χ3w are only sensitive to the zonal (eastward) winds.
 In fact, as we observed in the previous sections, differences between the equatorial AAM functions at the equator are almost due to differences between northward winds.
 We suspect that the differences between wind fields are mostly due to differences between the compilation and analysis processes of meteorological models.
 Another problem on the accurate determination of the wind field at low latitudes is the limited number of meteorological observatories. To compensate the observational gap, aircraft have been collecting meteorological data en route. With the advent of the satellite era, meteorological satellites track cloud motion and provide vast amounts of upper wind data (SATOB), thereby significantly improving the accuracy of the upper wind data.
 However, satellite observation density is not uniformly distributed. Uppala  reported the amount of observation data (for example, wind observation data collected by radiosonde, aircraft, cloud motion, etc.) used in the ERA-15 reanalysis. The amount of original wind observation data used in the reanalysis shows a nonuniform distribution in longitude. (For example, for the tropics, see Figures 10, 68 and 70 in Uppala's  paper, and for the southern hemisphere, see Figures 14 and 69.) Since the available observation data in the past are limited, we suppose that larger differences in the earlier data in Figure 1 are attributed to the smaller amount of observation data used in the reanalysis. We speculate that the smaller differences over Australia mentioned in section 4.3 are attributable to the large amount of original observation data used in reanalysis data sets.
 Then, which reanalysis data set provides the most accurate wind data? It is difficult to answer this question, but comparing the reanalysis wind data with the observed data provides a hint.
 Cross-comparison between new reanalysis data and the observed data is still in progress. Some results using the former released data sets have already been reported. Uppala  evaluated RMS misfits of the ERA-15 minus observed wind data in order to assess the ERA-15 performance. He reported that the RMS misfits in wind velocity between the analysis and the SATOB observation are 1 to 3 m/s for both the eastward and northward winds. Hastenrath and Polzin  checked the validity of the NCEP-1 and ERA-15 reanalysis data sets by comparing the radiosonde data at Galapagos and found that the winds in ERA-15 are slower than the observed winds, whereas the winds in NCEP-1 are faster than the observed winds. Schafer et al.  compared wind profiler data with the NCEP-1 wind data over the tropical Pacific. They found that discrepancies between winds of NCEP-1 and those of observation tend to become larger as the distance from the rawinsonde site becomes larger. Their results show large differences at a site on the coast of the east Pacific and small differences near Australia.
 Differences in the pressure data are also reported. Bromwich and Fogt  compared the mean sea level pressures of ERA-40, NCEP-1 and observation. They found that in the modern satellite era (after 1978), ERA-40 is superior to NCEP-1. However, before 1973, both reanalyses had shortcomings over the southern high latitudes.
5.2. Relation Between Wind Excitation Accuracy and Wind Data Accuracy
 Then, what is the extent to which wind differences affect the atmospheric excitations of the Earth's rotation? To answer this question, we move on to a quantitative discussion of the effect of the accuracy of the wind data on that of atmospheric excitations.
 First, we estimate wind difference effects on the excitation functions χw and χ3w using equations (2) and (4). Let us assume a wind velocity difference of 4 m/s magnitude at the equator in a box region of 10° × 10° × 200 hPa (the pressure difference between the top and bottom of the box region). From equations (2) and (4), the northward wind difference of this velocity is equivalent to an equatorial AAM function χw = 5.1 × 10−9 whereas the eastward wind difference of this velocity is equivalent to an axial AAM function χ3w = 1.2 × 10−11.
 In the above discussion, we adopt a wind difference of 4 m/s, but we consider this value to be the typical one for errors in wind velocity for reanalysis data sets, as we discussed in section 5.1. For example, Schafer et al.  reported standard deviations of wind velocity differences between the profiler observation and the NCEP-1 reanalysis. The standard deviations of wind differences, read from their Figures 6, 8, 10 and 12, are approximately 2 to 6 m/s.
Equation (2) implies that, even if the magnitudes of wind errors in both northward and eastward directions are the same in the meteorological data at the tropics, the meridional wind data brings about critical issues on the quality of the atmospheric excitation models. In other words, wind data quality more easily diverges the equatorial AAM functions than it does the axial AAM functions. Such differences in the efficiency at which winds excite the Earth's rotation are explained by the differences in moment of inertia employed in the excitation calculation.
 Next, we estimate effects of the differences in the excitation functions on the polar motion or nutation. The true effects of the wind differences on the polar motion or nutation might be larger than those expected from the differences in the excitation functions because wind effects on the polar motion or nutation depend on frequency due to resonance regime.
 A transfer function converting the geophysical excitation into the polar motion or nutation is given by equations (3)–(9) of Brzeziński . The transfer function is given in the frequency domain. Since geophysical excitation functions are given in the time domain, we apply convolution calculation. Hereafter, numerical values of the effects on the polar motion are evalutated by standard deviations of time series of the polar motion after an inverted transformation from the frequency domain into the time domain is applied.
 If we use a set of constants appeared in equations (3)–(13) of Brzeziński  for the transfer function, and use the results of this study for differences between meteorological data sets (i.e., frequency dependence of the wind differences is given by Figure 10), effects on the polar motion (a set of standard deviations for the two directional axes 1 and 2) of wind differences of NCEP-1 minus NCEP-2 are (12.3, 12.4) mas due to u-wind, and (27.3, 27.3) mas due to v-wind. Also, those of wind differences of NCEP-2 minus ERA-40 are (76.4, 76.5) mas due to u-wind, and (53.9, 54.6) mas due to v-wind, respectively. These effects are mainly magnified through interaction of the Chandler resonance. On the other hand, effects of χ3w on UT1 can be evaluated in the same manner. However, because a transfer function is proportional to the reciprocal of frequency, differences in geophysical excitations at low frequencies are converted into long periodic signals (or features similar to a secular trend) with large amplitudes in UT1.
 Since the uncertainties of the polar motion and UT1 observation in the IERS C04 series data after 1997 are 0.2 mas and 20 μs, respectively [Gambis, 2004], the effect of the wind difference may be above the detectable level for the polar motion.
 If we apply a band-pass filter which passes only the effect on the nutation (i.e., a band-pass filter passing through retrograde diurnal bands) in the frequency domain, effects on the nutation are 4 μas (micro-arc-second) and 5μas due to u- and v-wind differences of NCEP-1 minus NCEP-2, 6 μas and 11 μas due to u- and v-wind differences of NCEP-2 minus ERA-40, respectively. Although differences in wind data have peaks at the retrograde diurnal frequency (Figure 10), their effects on the nutation are below the current detectable level. However, the maximum differences at a certain time can be larger than these standard deviations.
5.3. Interpretation of Spectral Differences
 The spectra of wind excitation differences shown in Figure 10 have very similar shapes to the wind excitation itself. This fact implies that the differences in wind data may appear in any frequency band.
 The difference spectra of NCEP-2 minus ERA-40 for the equatorial components show relatively flat profiles over wide frequency bands. However, those of NCEP-1 minus NCEP-2 bend upward around the frequency of −310 cpy (−0.85 cpd). What causes such a bending structure in difference spectra?
 The difference spectra between two NCEPs for the equatorial component show broad peaks around −0.85 cpd for both eastward and northward winds, but those for χ3 show no peak (Figure 9a). One possible interpretation of these features is the following: This ψ11 atmospheric mode is antisymmetric with respect to the equator [Hamilton and Garcia, 1986].
 On the other hand, weight factors in the AAM functions depend on their geographical locations (see equations (2) and (4)). The weight factor cos2 ϕ in χ3w is symmetric with respect to the equator, whereas those in χw for the northward winds cos ϕ and eastward winds sin ϕ cos ϕ are symmetric and antisymmetric with respect to the equator, respectively. Therefore an antisymmetric atmospheric mode like ψ11 remains in χw due to the eastward winds, but is canceled for others. Next, the zonal number 1 indicates that the meridional winds caused by this mode are maximum at the equator and the wind directions 180° apart in longitude are opposite. Therefore this atmospheric mode can also be detected in northward winds in χw. If there are some differences in describing this mode between meteorological data sets, the difference affects only χw.
 Note that since such modes are traveling around the Earth at their proper periods, we think that the differences that appeared on the global maps, as in Figures 5 to 8, do not contain these modes. In other words, the differences that appeared on these maps are fixed in location.
 This argument is based only on descriptive characteristics; to confirm the relationship between this atmospheric mode and the difference spectra, more detailed investigations are necessary.
 Nowadays, because the time resolution at which we can observe the Earth's rotation has improved, the rapid variation of geophysical fluids that excite the Earth's rotation becomes more important. Geophysical excitations at diurnal frequency bands (with respect to the terrestrial reference frame) are also important for geophysical excitations of nutation [e.g., Bizouard et al., 1998].
 Our results also imply that, even if differences in meteorological data are small at some frequency bands, differences are not necessarily small at other frequency bands. Despite the small differences between two sets of NCEP data at seasonal bands, differences at diurnal bands are comparable to the differences between NCEP-2 and ERA-40. This is a good example.
 We compared three AAM functions calculated from different meteorological reanalysis data sets, NCEP-1, NCEP-2 and ERA-40, for a period of 23 years (1979–2001). This work is the first report on differences in atmospheric Earth's rotation excitations calculated from three reanalysis data sets which are believed to be most reliable meteorological data sets available today, because of no changes in their analysis models. We found that these AAM functions have different values. The root-mean square differences in the wind term between NCEP-1 and NCEP-2 (based on monthly averaged data) are 0.0662 × 10−7, 0.0854 × 10−7 and 0.0035 × 10−7 for the χ1, χ2 and χ3 components, and those between NCEP-2 and ERA-40 are 0.2220 × 10−7, 0.2133 × 10−7 and 0.0065 × 10−7, respectively. These differences are largely due to differences in the wind term. If we decompose winds into two orthogonal horizontal directions (that is, eastward and northward), the northward wind differences have a larger effect on the equatorial AAM functions.
 Such differences in wind excitation do not necessarily accompany strong wind excitations (like jet streams); they can also accompany weak wind ones. In the case of northward winds, the upper troposphere at the tropics and midlatitudes in the southern hemisphere show large differences. Some regions (east Pacific off Latin America, Africa and the Indian Ocean) tend to have larger northward wind differences. Differences in eastward winds along the ‘barking 40°’ also affect the equatorial AAM functions. Differences between the NCEP-2 and ERA-40 excitations are larger than ones between the two NCEP excitations. The geographical distribution of such differences is considered to reflect the number of meteorological observation points.
 The spectrum of these differences is reddish and its powers are nonzero in all frequency bands. Some seasonal peaks are observed, but there is no significant peak in the intraseasonal bands. In the diurnal bands, the differences at some tidal frequencies are large. Despite the fact that the differences between the two NCEP wind excitations are smaller than those between NCEP-2 and ERA-40 wind excitations, these differences are almost comparable in the frequency range of 0.5 to 1 cpd.
 The quality of wind data tends to have larger effects on the equatorial AAM function than on the axial AAM function. For example, meridional and zonal wind velocity differences of 4 m/s at the equator in a box region of 10° × 10° × 200 hPa cause differences of 5.1 × 10−9 in the equatorial AAM function and 1.2 × 10−11 in the axial AAM function, respectively. The wind difference effect on the polar motion can be magnified because resonance mechanism at the Chandler frequency band amplifies the effect. Nowadays, Earth rotation observation is vigorously carried out via international cooperation. The high accuracy and fine temporal resolution of Earth rotation observation enable us to depict the global atmospheric motion. As the evaluation of atmospheric excitations from meteorological data sets becomes more powerful and helpful to Earth rotation studies, such differences between meteorological data sets become more important. Since the winds at the equator can most efficiently change the Earth's rotation, the accuracy of the wind data over the tropics directly determine the accuracy of AAM functions.
 We thank I. Naito and Y. Aoyama for their encouragement in this study. We acknowledge anonymous reviewers for improving the manuscript. The NCEP/NCAR and NCEP-DOE reanalyses used in this study were downloaded from the NCEP website. The ERA-40 reanalysis used in this study is compiled by the ECMWF. The IERS C04 data were downloaded from the Website of the IERS EOP product center. This study is financially supported by Special Coordination Funds for Promoting Science and Technology (Supporting Young Researchers with Fixed-term Appointments) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.