Satellite laser ranging (SLR) data were used to determine the variations in the Earth's principal figure axis represented by the degree 2 and order 1 geopotential coefficients: C21 and S21. Significant variations at the annual and Chandler wobble frequencies appear in the SLR time series when the rotational deformation or “pole tides” (i.e., the solid Earth and ocean pole tides) were not modeled. The contribution of the ocean pole tide is estimated to be only ∼8% of the total annual variations in the normalized coefficients: / based on the analysis of SLR data. The amplitude of the nontidal annual variation of is only ∼ 30% of from the SLR time series. The estimates of the annual variation in from SLR, the Gravity Recovery and Climate Experiment (GRACE) and polar motion excitation function, are in a good agreement. The nature of the linear trend for the Earth's figure axis determined by these techniques during the last several years is in general agreement but does not agree as well with results predicted from current glacial isostatic adjustment (GIA) models. The “fluid Love number” for the Earth is estimated to be ∼0.9 based on the position of the mean figure axis from the GRACE gravity model GGM03S and the mean pole defined by the IERS 2003 conventions. The estimate of / from GRACE and SLR provides an improved constraint on the relative rotation of the core. The results presented here indicate a possible tilt of the inner core figure axis of ∼2° and ∼3 arc sec displacement for the figure axis of the entire core.
 In the Euler's dynamical equations of the rotating Earth in the body-fixed reference frame, the angular momentum of the nonrigid Earth is expressed as H = I(t)ω(t) + h(t), where I(t) is the time-varying second-degree symmetric inertia tensor, ω(t) is the angular velocity vector of the body-fixed axes, h(t) is the relative angular momentum. The time-independent parts of the diagonal components of I(t) correspond to the principal axes of inertia for an undeformed Earth. The figure axis of the Earth is the axis of maximum inertia for the deformed (oblate) Earth. Neglecting the second-order terms, the unit vector of the figure axis can be defined by the products of inertia, (Ixz, Iyz, C − A)/(C − A), where C and A are the Earth's polar and equatorial moments of inertial, respectively [Moritz and Mueller, 1987]. The Earth's external gravitational potential is conveniently expressed in body fixed coordinates through a spherical harmonic expansion as
where r is the geocentric distance, ϕ is the latitude, and λ is the longitude of the point at which U is determined. G is the gravitational constant, M is the mass of the Earth, and R is the equatorial radius of the Earth. Pnm(sin ϕ) are the Legendre polynomials, and Cnm and Snm are the Stokes geopotential coefficients of degree n and order m as function of the mass distribution within the Earth. and are hereafter referred to as the normalized Stokes coefficients. The unnormalized Stokes coefficients, Cnm and Snm, are related to the normalized coefficients by the relation (, ) = (Cnm, Snm)/Nnm, where Nnm is the normalizing factor expressed as Nnm = [(2n + 1)(2 − δ0m)(n − m)!/(n + m)!]1/2, where δ0m = 1 for m > 0, and = 0 for m = 0. The second-degree Stokes coefficients have physical meaning and relate to Earth's inertia tensor. In particular, the degree 2 and order 1 geopotential coefficients can be expressed as C21 = −Ixz/MR2 and S21 = −Iyz/MR2. Thus, the Earth's principal figure axis can be determined by the degree 2 and order 1 geopotential coefficient, and , where and .
 In the absence of external torques, the principal figure axis of the Earth would be very close to the rotational axis [Wahr, 1987]. However, as a consequence of excitations due to tidal and nontidal mass redistribution within the Earth system (dominated by seasonal variations), the two axes are not completely aligned, and the rotational axis precesses around the Earth's principal figure axis (causing the ∼435 day Chandler “wobble”). Although the frame defined by the principal axes of the Earth is convenient for theoretical discussion of the Earth's rotation, measuring the figure axis and its variations can improve the understanding of the relation of the figure axis oscillation to changes in the Earth's mass redistribution and provide an important global constraint on the properties of the core and core-mantle boundary as shown by Wahr [1987, 1990].
 Recent improvement in the accuracy of space geodetic measurements and related force modeling suggest that it is possible to directly measure the position of the figure axis and its variations rather than relying on calculations based on current solid Earth and ocean tide models. At present, the gravity field to degree and order, 60, along with its seasonal variations, is well determined by the Gravity Recovery and Climate Experiment (GRACE) mission [Tapley et al., 2004], and a low-degree model can also be accurately determined from analysis of satellite laser ranging (SLR) data. The long-term SLR analyses from multiple geodetic satellites and results from GRACE measurements spanning more than 7 years are examined here to study the mean Earth's figure axis and its variations through the determination of spherical harmonic coefficients, and . Theoretically, the excitation of polar motion is caused by changes in the products of the inertia due to the relative motion and the surface mass redistribution, which is proportional to the changes in / [Munk and MacDonald, 1960; Lambeck, 1980]. More accurate Earth orientation parameters (EOP) determined from space geodetic techniques, such as GPS, SLR, LLR and VLBI, and the surface mass loading and motion (i.e., the atmospheric winds and ocean currents) data are also available for exploring the variations of /.
 As a means of validating the GRACE results, Chen and Wilson  compared the seasonal variation in / determined by GRACE with results from SLR data and polar motion excitation function. The monthly SLR estimates available at that time for / were part of the routine estimation of the second-degree zonal harmonic coefficient () for GRACE science applications. A 5 year time series, spanning April 2002 to May 2007, was based on the GRACE RL01 standards where (1) the IERS1996 solid Earth pole tide model [McCarthy, 1996] was used and (2) the ocean pole tide model was not a part of the GRACE model standards at that time. The most important influence of the ocean pole tide is on /, while the effect on is small. Chen and Wilson  found a poor agreement between the two results, with the SLR-derived annual variations differing from the GRACE results in amplitude (∼51% smaller) for and in phase (∼100°) for . Since there are significant rotational variations at the Chandler and annual frequencies, it is not unexpected that there would be poorer agreement with no ocean pole tide included and a less accurate solid Earth pole tide model used for the SLR data analysis. In addition, the time series of 5 years was relatively short and, over this period, the Chandler variation is the dominant signal. Separation of the Chandler and annual frequencies requires a time span longer than ∼6.4 years, so estimating the annual variation in / from a 5 year data set would be difficult. Understanding the contribution of the pole tides to the variations in / is necessary to correctly interpret the estimates of the seasonal variations in these harmonics.
 This paper presents an updated analysis of estimates of / determined from SLR tracking data of multiple satellites using models consistent with the current GRACE data processing (RL04) [Bettadpur, 2007], including the ocean pole tide and Atmosphere-Ocean De-aliasing (AOD) model [Flechtner, 2007]. Section 2 briefly reviews the model for the pole tides (rotational deformation), and the variations induced in the Earth's figure axis. Constraints on the “fluid Love number” and core rotation will be inferred based on the mean pole and the mean figure axis determined from SLR and GRACE data. Section 3 introduces the GRACE estimate of / and the AOD model used in the SLR and GRACE data processing. Analysis of SLR data is given in section 4. Section 5 analyzes the spectrum of variations in / based on the time series of weekly estimates of / from SLR data spanning approximately 17 years. In addition, the annual variation for a period of 8 years is determined based on the monthly estimates from SLR, GRACE, and Polar motion excitation function. The effects on / from the AOD background model are also compared. Section 6 discusses the secular change in / inferred from the motion of the mean pole, SLR, and GRACE, and section 7 summarizes the results.
 The time variable geopotential coefficient (normalized) of degree 2 and order 1 can be expressed as
where j = 1 for , j = 2 for , the superscript st and ot refer to the lunisolar-induced solid Earth and ocean tides (noted as attraction tides), spt and opt refer to the rotational deformation induced variations in the solid Earth and ocean, called the solid Earth and ocean pole tides. Modeling of the attraction tides and rotational pole tides in the solid Earth and ocean is addressed in sections 2.1 and 2.2. The superscript L refers to the loading effects due to the mass redistribution within the Earth system, including the atmosphere, ocean and surface water storage. The and coefficients describe the instantaneous position of the Earth's figure axis along the x and y axis of the conventional reference frame (the Terrestrial Reference Frame, or TRF). and represent the “mean” position of the Earth's figure axis at an epoch t0 defined in the reference gravity models, such as the mean field of the GRACE solution determined from the data over several years.
 In historical studies of Earth's rotational dynamics, polar motion was defined as the departure of the rotation axis from the figure axis [Munk and MacDonald, 1960; Lambeck, 1980; Moritz and Mueller, 1987; Chao, 1989]. Such a definition can simplify the discussion of the geophysical influence on the rotation, but it does not permit a ready comparison with observation [Lambeck, 1980] and can lead to some confusion [Eubanks, 1993]. Currently, polar motion is defined as the motion of the rotation axis of the Earth crossing its surface and measured with respect to a conventional pole, adopted as part of the definition of the terrestrial reference frame (TRF). The adopted z axis is oriented along the mean rotation axis for the period of 1900–1905 (i.e., the Conventional International Origin (CIO) pole of 1903 [see, e.g., Torge, 2001]). The coordinates of the pole of rotation axis, xp(t) and yp(t) can be approximated by
In equation (3), j = 1 accounts for the annual period of 365.25 day j = 2 accounts for the Chandler period of 434.2 sidereal days, i = 1 for xp(t) and i = 2 for yp(t). The power at these two frequencies accounts for more than 70% of the variations in the polar motion. However, in addition to the Chandler and annual periods, there are a few other long-period signals appearing in the spectrum of both xp and yp that depend on the time interval used to fit the data. For example, there is a fluctuation at a period of ∼484 days with an amplitude of ∼50% of the annual signal during the time interval from 1993 to 2010. Periodic terms with amplitudes larger than 20% of the annual signal must be included to obtain a better fit over the selected time interval.
 In most discussions of nonrigid Earth rotation, the position of the instantaneous rotation axis relative to the z axis is defined by m1 and m2, which represent the polar motion or “wobble.” In this notation, m1 and m2, are π/2 out of phase with the definition of xp and yp [Lambeck, 1988]. In the modeling of the solid Earth and ocean pole tide, m1 and m2 are the wobble variables representing the deviation of the instantaneous pole (xp, yp) from the time varying mean pole () and are expressed as m1 = xp − and m2 = −(yp − ). The time varying mean pole has been conventionally defined by its position at a fixed epoch and the linear drift rate (,) as and , which are the first two terms in the right side of equation (3) [McCarthy and Petit, 2004]. However, this linear model may be valid only over a relatively short interval of time; the motion of the mean pole over decades has significant departures from a purely secular model [Petit and Luzum, 2010].
2.1. Modeling the Solid Earth and Ocean Tides
 The contribution of the solid Earth and ocean tides to the changes in / behave primarily as a diurnal oscillation. The solid Earth tides are modeled with frequency-independent and frequency-dependent parts in terms of the Love number as described in the IERS 2003 conventions [McCarthy and Petit, 2004]. The frequency-independent part for / is proportional to the degree 2 and order 1 terms in the tide generation potential with the nominal Love number; 48 diurnal tide terms were used for frequency-dependent part based on the IERS 2003 conventions (section 6.1).
 In earlier analyses of SLR data [Cheng et al., 1990; Tapley et al., 1993] prior to the TOPEX era, only diurnal ocean tides from the model given by Schwiderski  were accounted for in the variations in /. Later, studies of the perturbations on geodetic satellites due to ocean tides indicated that the orbits are also sensitive to the degree 2 and order 1 harmonics for some long-period and semidiurnal tides. Those terms must be included in the ocean tide model for the analysis of SLR data. In the current GRACE RL04 model, the ocean tide model contains terms up to degree and order 100, while ocean tides are modeled to degree and order 50 for the SLR data analysis for those satellites in the lower altitude of ∼800 to 1500 km and to degree and order 20 for the LAGEOS-1 and -2 satellites at the altitudes of ∼6000 km (satellites at altitudes higher than GRACE are less sensitive to the higher-degree tidal variations). The long-period tides are assumed to be equilibrium tides (self-consistent) [Egbert and Ray, 2003], while the diurnal and semidiurnal bands are based on the FES2004 ocean tide model [Lyard et al., 2006]. The atmospheric S1 and S2 tide are also modeled [Ray and Ponte, 2003].
2.2. Modeling of the Solid and Ocean Pole Tides
 Rotation of the Earth causes the pole tides in the solid Earth and ocean. With the differential centrifugal potential [Wahr, 1985], the changes in the / coefficients due to rotation deformation in the solid Earth and ocean are expressed
where R[…] represents the real part of the complex variation in the bracket, j = 1 for , j = 2 for , the are the respective ocean pole tide coefficients for the degree 2 and order 1, ae is the equatorial radius of the Earth, Ω is the nominal mean angular velocity of the Earth, and GM is the geocentric gravitational constant (the product of the gravitational coefficient G and the mass of the Earth M). The values for ge (the mean equatorial gravity) and ρw (the density of seawater) are given in Table 1.1 of the IERS 2003 conventions [McCarthy and Petit, 2004]. The value of 0.3077 +i0.0036 is the anelastic Love number used for the solid Earth pole tide and γ2R + iγ2i = 0.705+i0.0036 for the ocean pole tide model [Desai, 2002] in the IERS 2003 conventions (the value of γ2R is updated to be 0.687 in the IERS 2010 convention [Petit and Luzum, 2010]). The IERS 2003 conventions provide numerical expression in terms of the wobble variables (m1 and m2) for the solid and ocean pole tides based on equations (4) and (5). Based on the IERS 2003 conventions, the contribution of the ocean pole tide, at both the annual and Chandler frequencies, is ∼17% of the solid Earth pole tide for and ∼13% for .
 The motion of the mean pole is not linear, varying with timescales of seasonal to decadal and secular. The estimates of the polar drift rate depend on the running average time interval. It was estimated to be 0.83 and 3.95 milliarc seconds (mas)/yr based on a linear fit to the time series of the polar motion data over the period spanning from 1976 to 1999, but the drift rate is 1.08 and 2.91 mas/yr when the polar motion data period is extended to 2009. The rate changes will result in a bias in calculating the mean pole and relative polar motion (m1,m2) [see McCarthy and Petit, 2004, equations 22 and 23a, section 7.1.4] and in turn, in / through the modeling of the pole tide.
2.3. Nontidal Variations in C21/S21 and Excitation Function of Polar Motion
 The mass redistribution induced changes of the inertia tensor (Ixz,Iyz) result in the excitation function of the polar motion. Denote ψxmass and ψymass as the excitation function from mass redistribution only, this mechanism can be described as [Gross, 2007, equation 41–42]
where A′ = (A+B)/2, A, B, C refer to the least, intermediate and greatest principal moment of inertia of the Earth, k′2 and Δk′an are the degree 2 load Love number of the Earth and the modification due to the mantle anelasticity. Ac and ɛc are the least principal moment of inertia and the ellipticity of the surface of the core. Ω is the mean angular velocity of Earth, and ωcw is the frequency of Chandler wobble (with period of 434.2 sidereal days). The factor Fσ = 1.608 [Gross, 2007] accounts for the effects of the surface loading and rotational deformation of the Earth, the passive response of equilibrium to changes in rotation (or “rotational feedback”) and the relative angular momentum of the core caused by the changes in rotation.
 Surface mass redistributions change the inertia tensor (Ixz,Iyz) of the Earth. Earth's elastic response to the loading also results in additional changes in the (Ixz,Iyz) as characterized by the loading Love number, k′2, which is also modified by the Earth's mantle anelasticity, represented by Δk′an [Gross, 2007]. Thus, the changes in the Stoke coefficients, C21 and S21, are related to the excitation function of the polar motion due to the nontidal mass redistribution and can be expressed as follows [Munk and MacDonald, 1960; Lambeck, 1980]:
where C′20 = (C − A′)/MR2. The load factor (1 + k′2 + Δk′an) is canceled in equation (7), after substituting equation (6), when converting the polar motion excitation function to ΔC21 and ΔS21. The loading effects are reflected in the observations from both polar motion and geodetic ΔC21 and ΔS21. Thus, equation (1) of Chen and Wilson  would overestimate the ΔC21 and ΔS21 from the excitation function of the polar motion by Δk′an/(1 + k′2 + Δk′an) = 1.6%.
 The daily polar motion (geodetic) excitation function from the IERS C04 polar motion time series and the modeled motion (physical) excitation term from atmosphere winds and ocean currents (assuming Inverted Barameter (IB) ocean response) can be obtained using the IERS online interactive tools (http://hpiers.obspm.fr/eop-pc/analysis) with a Chandler period of 433 days and Q = 175. As can be seen there, after 1984 the EOP C04 series contains large variations at the shorter timescales, due to the introduction of VLBI observations in the combination. Consequently, the equatorial geodetic excitation appears more “noisy,” but it is better correlated with the Atmosphere Angular Momentum (AAM) (C. Bizouard, IERS Earth Orientation Parameter Center, personal communication, 2009). The mass redistribution induced polar motion excitation is the difference between the geodetic and physical (motion) excitation, or can be directly obtained using IERS online tool by selecting the option: difference. To compare with the observation from SLR and GRACE, the polar motion excitation derived variations of and (denoted as EOP-EF) are inferred using equation (7) for the monthly or weekly mean values averaged from the daily polar motion excitation function (mass part). The average is carried out over the period from 1993 to September 2009 (when the ECCO (http://www.ecco-group.org/) ocean excitations are available). In averaging, the effects of the atmosphere and ocean motion terms on and are estimated to be less than ∼6 × 10−12 over the period from 2002 to March 2008. A significant drift appears in the atmosphere wind-induced motion term from 31 March 2008 when the operational NCEP model starts to be used to compute the effective AAM functions.
2.4. Mean Figure Axis and Geophysical Significance
2.4.1. Fluid Love Number Relates the Mean Figure Axis to the Rotation Axis
Equation (4) implicitly relates the rotational deformation induced changes in the figure axis to the rotation axis for the nonrigid rotating Earth. After averaging data over several years, the periodic variations will be eliminated. The mean figure axis ( and ) (as a net change) corresponding to the mean rotation axis (or mean pole: and ) can be expressed as
In equation (8), and C20 =(C − A)/MR2, ko = 3C20GM/Ω2R3 ≈ 0.942 [Lambeck, 1980, equation 3.4.4], and k is the “Love number” characterizing the response of the Earth to the perturbing force (which is frequency independent or depends on the stress state of the planet). Equation (8) indicates that the orientation of the Earth's figure axis is proportional to the rotation axis with a dynamic factor of k/ko as the consequence of the bulge adjusting itself to the continuously changing position of the motion of the rotation axis (see discussion by Munk and MacDonald , Lambeck , and Moritz and Mueller ). For long time averages, the shape of the solid Earth with its fluid surface is approximately an ellipsoid of revolution near hydrostatic equilibrium. Thus, the Love number k is also called as the “fluid Love number,” k = 0.96 for a rotating Earth in hydrostatic equilibrium [Munk and MacDonald, 1960] and a value of 0.934 is recommended for computation of the variation of the polar moment of inertia due to secular change in the rotation [Lambeck, 1980]. Those estimates for k depend on estimates of the hydrostatic flattening.
 The geopotential coefficients and are determined with respect to the Earth fixed ITRF system. The “pseudo-body-fixed” system (with the z axis as the instantaneous rotation axis) is related to the ITRF system through the coordinate transformation R1(yp)R2(xp) (where Rj(x) denotes the rotation abort the j axis with angle x). The products of inertia with respect to the pseudo-body-fixed system (denoted as Ix′z′r and Iy′z′r) can be expressed in terms of the ITRF based inertia tensor of order 2 (neglecting the higher-order terms depending on xp2, yp2 and xpyp). The relation between the ITRF based second-degree geopotential coefficients ( and , m = 0, 1, 2) and polar motion angles can be approximated as
In equation (9), and . Reigber  considered the z axis of the pseudo-body-fixed system to be the Earth's figure axis corresponding to an averaged pole. For this case, Ix′z′r and Iy′z′r (as well as and ) become zero by definition, which leads to a simple geometric relation between the second-degree geopotential coefficients and the mean pole. Neglecting the contribution from the and terms, the geometric relation is equivalent to kf = k/k0 = 1 in equation (8). The geometric relation developed by Reigber  is referred to the specific pseudo-body-fixed system, but the dynamic relation in equation (8) is referred to the ITRF system. The IERS 2003 conventions used this geometric relation to predict the mean figure axis from the corresponding mean pole position, assuming the figure axis “closely” coincides with the rotation pole averaged over many years. It is not clear that this assumption is valid for decadal time spans and if there is a difference between the two mean axes, which could be caused by the motion in the Earth's core [Wahr, 1987]. Currently, only the measurements of the mean pole are available for determining the equatorial polar offset, d, of the mean figure axis from the mean rotation axis using the expression
With the dramatic improvement in the determination of the geoid from the GRACE mission and the accuracy of current SLR data analysis, satellite observations with a time span of several years might provide a constraint on the fluid Love number k and the equatorial polar offset of the mean figure axis.
 In the GRACE-derived gravity solutions for the mean field, the geopotential coefficients, / were estimated with an assigned uncertainty ranging of 7 × 10−11 to 8 × 10−12. Those models include several of the CSR and GFZ satellite-only mean gravity fields [Tapley et al., 2004, 2005, 2007; Reigber et al., 2005; Förste et al., 2008]. Table 1 compares the estimates of , and from different gravity mean fields and the estimates from the ∼17 years SLR data used in this study. The row for the mean of gravity models refers to the mean of four GRACE solutions in Table 1. The later GRACE models, such as GGM03S and EIGEN-GL05S, provide the best estimates of the geoid at the longest wavelengths. Consequently, they can be used to estimate the k relating the Earth's figure axis to the continuously changing position of the rotation axis. Using equation (8) and the mean pole (, ) at the epoch of 2000.0 from the IERS 2003 conventions, the value of k can be inferred from and , separately, and denoted as kc and ks. Table 1 summarizes results of the inferred kc and ks from GRACE models and SLR solutions.
Table 1. Comparison of Various Estimates for , , and (at Epoch 2000)
Mean of gravity models
−2.47 ± 0.32
14.13 ± 0.48
−484.16948 ± 0.00028
1.06 ± 0.14
0.92 ± 0.03
SLR (this study)
−2.39 ± 0.1
14.24 ± 0.1
−484.16948 ± 0.00002
 Comparison shows a good agreement between the mean values of GRACE-derived gravity models, and the estimate from approximately 17 years of SLR data from five geodetic satellites. The mean value of kc is generally larger than ks and is scattered, with the RMS about the mean of 0.14 and 0.03, respectively. The factor ks may mainly reflect the Earth's response to the east-west latitudinal variations related to the Earth's rotation, while the factor kc reflects the Earth's response to the meridianal mass transformation. The mean pole is estimated to be ∼0.054 and ∼0.350 arc sec at the epoch 2004.0 (denoted as the GGM03S mean pole) corresponding to the middle point of the period from 2003 to 2006, over which the GRACE data were used to obtain the GGM03S solution. The values for kc and ks are estimated to be 0.89 and 0.92, respectively. The equatorial polar offset of the mean figure axis is to be ∼697 mas based on the GGM03S mean pole over the 4 year period. A consistent estimate of k might be approximately 0.9 ± 0.02 from the later GRACE gravity solution. This estimate implies that the kf is ∼0.96, and that the approximation of kf = 1 in equation (8) may be an adequate approximation for relating the position of the mean figure axis to the mean pole of the rotation axis, as a validation for gravity models.
 It is noted here that the internal consistency of the individual estimates for was better than 3 × 10−11 from the SLR tracking data of LAGEOS-1, LAGEOS-2 and Starlette [Ries, 2009]. Consequently, an estimate of the uncertainty from the combination of SLR data from multiple satellites of ∼2 × 10−11 for is a reasonable, if not conservative, error estimate. This is at least an order of magnitude better than the consistency between the GRACE estimates, and it should be considerably more reliable due to significant long-period tidal aliases in the GRACE estimates [Ries et al., 2008]. In addition, the SLR data spans a period that is close to being centered on the epoch 2000.0, which effectively decorrelates the estimate for from errors in the linear rate model for that term. The GRACE estimates must be mapped from the mean epoch of each solution back to epoch 2000.0, and thus may be biased due to errors in the rate model.
2.4.2. Global Constraints on the Core-Mantle Dynamics
 By definition, the and would be zero if the figure axis coincides with the z axis of the ITRF frame used to process SLR tracking data. It has been of considerable interest to understand what forces maintain the offset between the mean figure axis and the Earth fixed z axis of ITRF system other than temporal variations. The Earth-body-fixed z axis (defined by the mean rotation axis of the Earth toward the CIO) is measured on the mantle, while the estimates for and determine the mean figure axis of the entire Earth. Using the GEM-T1 model [Marsh et al., 1988], Wahr [1987, 1990] shows that the measurements of and could provide important information on the core's dynamics.
 Here we give only a brief summary for the theory presented by Wahr . Denote H = I · Ω + hc as the angular momentum for the Earth and Hc = Ic · Ω + hc for the core. I is the Earth's inertia tensor, Ω is the time-averaged mean angular velocity of the mantle. hc and Ic are the relative angular momentum and inertia tensor of the entire core, respectively. For time-independent displacement (∂tH = 0) while the atmosphere and ocean contribution can be negligible (the effects are within the range of the uncertainty of ∼8 × 10−12 for and solution), conservation of angular momentum (Ω × H = 0) leads to that the observed and are proportional to the equatorial component of the mean rotation of the core with respect to the mantle [Wahr, 1987, equation (10)], assuming there are no time-independent external torques on the Earth, the Earth's relative angular momentum vector results only from hc of the core with no contributions from the atmosphere and ocean, and the core is approximately spherically symmetric. Conservation of core angular momentum in the mantle fixed frame (Ω × Hc = Nc) leads to that the observed mean figure axis is caused by a dynamic equatorial torque on the core (Nxc and Nyc) from the mantle and the misalignment of the core figure axis (represented by Ixyc and Ixzc), as described as follows [from Wahr, 1987, equations (10) and (17)]:
In equation (11), Fe = MR2. The equatorial torque is assumed to be due to the dynamic pressure (acting against the topography at the CMB) resulting from dynamic processes inside the core and the hydrostatic response of the core to its diurnal rotation. Thus, it is possible to determine the tilt of the core's figure axis with knowledge of the pressure field at the CMB. Using the and with an accuracy of level of 8 × 10−12 from GGM03S solution [Tapley et al., 2007] as listed in Table 1, an equatorial component of the relative rotation of the core is estimated to be ∼5×10−8 times the Earth's diurnal spin rate based on equation (10) of Wahr ; the equivalent equatorial component of dynamic torque is estimated to be Nx = −Fe Ω2 = −2.3 × 1021 N m and Ny = Fe Ω2 = −0.4 × 1021 N m assuming the products of the core inertia are zero in equation (11).
 The geostrophic pressure field at the CMB was derived from the “frozen flux” core surface flow estimates based on the observations of the magnetic field and its secular variation near the Earth's surface with certain approximations for the core dynamics [Fang et al., 1996; Greff-Lefftz et al., 2004]. The mean value of the degree 2 and order 1 spherical harmonic term (normalized p21c and p21s) for the pressure field at the CMB are estimated to be −112 and −77 Pa from Fang's model with unknown uncertainty. The pressure induced equatorial torque at the CMB can be Nxp = −5.4 × 1019 N m and Nyp = −7.8 × 1019 N m based on equation (22) of Wahr  for the topography of the CMB with a hydrostatic elliptical bulge with the radius of 3480 km and the hydrostatic ellipticity of 1/393. The modeled pressure torque is only ∼4% of the equivalent torque inferred from the observed and . This fact suggests that the geodetic values for and contain other significant geophysical contributions, such as the misalignment of the core's figure axis with the mantle rotation axis in the case for the non strict hydrostatic equilibrium core. In fact, the equatorial component of the tensor of the core, Ixzc and Iyzc, can be estimated based on equation (11) using the values of / from GGM03S model and the pressure torque from Fang's model. The tilt of figure axis of the core (represented by Ixzc/(Cc − Ac) and Iyzc/(Cc − Ac) [Wahr, 1987]) is estimated to be ∼ 0.36 arc sec in the x direction and ∼3.07 arc sec in the y direction, where Cc = 9.14 × 1036 kg/m3 and Ac = 9.1168 × 1036 kg/m3 are used for the principal moments of inertia of the core. The direction of the mean figure axis of the entire core (arctan(Ixzc/Iyzc) is estimated to be ∼277° eastward, opposite to the direction of the mean figure axis of the entire Earth. The tilt of the core would be −0.47 and 3.15 arc sec in the x and y direction, respectively, if the pressure torque was not applied. To determine whether this estimate is reasonable for the tipped figure axis of the core with respect to the mantle rotation axis, a different theory is used in the following discussion to estimate the tilt of the core.
Mathews et al.  developed a theory of the forced nutations of the Earth in accounting for the influence of inner core by modeling of the internal gravitational and pressure coupling between Earth's interior regions in a body fixed reference frame with its z axis along the “instantaneous figure axis.” The dynamic system is described by the dynamical equations for the angular momentum for the whole Earth, fluid outer core, and solid inner core with an additional kinematic equation accounting for the tilt of the inner core relative to the mantle, where the tilt vector of the inner core is defined as the difference between the figure axis of the core and mantle. Thus, the tilt of the inner core can be solved directly from four linear algebraic equations [Mathews et al., 1991, equation (26)]. Based on those equations, Dumberry and Bloxham  show that in the static case (frequency independent) and conservation of angular momentum of the whole Earth, the rotation axis of the inner core is aligned with that of the whole Earth, a polar offset is balanced by a titled figure of the inner core and a misaligned rotation of the fluid core based on equation (16) of Dumberry and Bloxham . The amplitude of the tilt of the inner core () is relative to the polar offset () by the ratio: = −104 (mas) [Dumberry and Bloxham, 2002, equation (17)] (hereinafter denoted as DB ratio).
 Averaging over a time interval of years, the observed mean figure axis can be considered to be the time averaged “instantaneous figure axis” of the reference frame of Mathews' theory [Mathews et al., 1991]. The mean polar offset (defined as the difference between the observed mean pole and the mean figure axis) can be used to estimate the mean tilt of the inner core based on the DB ratio of −104. The mean polar offset is estimated to be ∼697 mas based the observed values of the and from the GGM03S and mean pole over the GRACE period. This polar offset implies that a mean tilt of the figure axis could be ∼1.9° for the inner core and ∼2.9 arc sec for the entire core based on the equation (28) of Wahr . This estimate is in good agreement with the result of ∼3.09 arc sec based on the Wahr's theory as described in equation (11) while the pressure toque was applied. Because the pressure torque is not involved in the DB ratio, the difference of the estimates of the mean tilt of the entire core from two theories suggests that the contribution is expected to be no more than ∼9% from the pressure torque at the CMB in the Whar's theory if the GRACE or SLR estimates of and are adopted, and an error could be as large as by 70% in the model of pressure torque from Fang's geostrophic pressure field at the CMB.
 The agreement from the two theories suggests that the contribution of the tilt of the core is dominant to the observed nonzero and defining the mean figure axis, and results in the difference between the mean figure axis and mean pole. However, the result presented here is twice as large as that from the analysis of polar motion data by Greiner-mai and Barthelmes . The cause for this discrepancy is unknown. The tilt of the inner core may be underestimated by Greiner-mai and Barthelmes . The value of the factor used in equation (7) of Greiner-mai and Barthelmes  needs to be clarified.
3. Atmosphere-Ocean Dealiasing Model and GRACE Estimates
 In the GRACE and SLR data processing, the Atmosphere and Ocean Dealiasing Level-1B (AOD) data were used to remove a priori the short-term nontidal atmospheric and oceanic induced variations in the gravity field. These data reflect the spatiotemporal mass in the atmosphere and ocean derived from the meteorological field with 0.5° spatial resolution at 6-hourly intervals of the integrated forecast system of the European Center for Medium-range Weather Forecast (ECMWF) data and corresponding ocean dynamics simulated as response to wind stress, atmospheric pressure as well as heat and freshwater fluxes provided by the baroclinic ocean model (OMCT) [Flechtner et al., 2007].
 In this comparison, the GRACE time series of and from 91 monthly RL04 solutions from the Center for Space Research (CSR) over the period of April 2002 to January 2010 were used. In addition, the values of −0.337 × 10−11/yr and 1.606× 10−11/yr for and were adopted as modeled rates in the GRACE data analysis [McCarthy and Petit, 2004]. The monthly average of the AOD and above modeled rates were restored to the estimates of and from GRACE data for an internally consistent comparison.
4. SLR Data Analysis
 Laser ranging is the only means of obtaining precise and unambiguous range measurements for the various passive geodetic satellites. This accurate range information allows the determination of even very small gravitational forces acting on the geodetic satellite. The satellites used in this study (Starlette, Ajisai, Stella, LAGEOS-1 and LAGEOS-2) all have spherical shapes, which simplifies the modeling of the nongravitational forces. In addition, except for Ajisai, they have dense metal cores and very low area-to-mass ratios, which further reduces the impact of the nongravitational force modeling errors. The orbit inclination ranges from 50° to 109°, and the altitudes range from 700 km to 6000 km. To accommodate residual dynamic modeling errors, 12 h drag coefficients (CD) or empirical along-track acceleration parameters (CT) were estimated as part of the precision orbit determination process. While these parameters have the potential to affect the determination of the gravitational forces, the orbit fits and resulting gravity estimates are much worse without adjusting them.
 Two SLR time series were obtained in this study. The first SLR time series (denoted as SLR-1, hereinafter) consists of the weekly solutions for the geopotential coefficients, up to degree and order 5 and geocenter parameters spanning ∼17 years from 1 November 1992. The EGM08 gravity model [Pavlis et al., 2008] and LPOD2005 station coordinates [Ries, 2008] are used without the AOD model. The rates of and were not modeled in SLR data analysis for both time series. The time series (SLR-1) of the weekly estimates of the variations in and from ∼17 years of SLR data were obtained in the three cases: (1) without modeling of the pole tides for both solid Earth and ocean, (2) with modeling of the solid Earth pole tide only based on the IERS 2003 conventions, and (3) with modeling of the pole tides in the solid Earth and ocean. For comparison in Figures 1 and 2, the time series in case 3 were smoothed using a wavelet analysis [Cheng and Tapley, 2004]. The second SLR time series (hereinafter denoted as SLR-2) consists of 97 monthly solutions from five geodetic satellites over the period from January 2002 to January 2010 (M. K. Cheng and J. Ries, Monthly estimates of C20 from 5 SLR satellites, GRACE Technical Note 5, 2009, available at ftp://podaac.jpl.nasa.gov/pub/grace/doc/TN-05_C20_SLR.txt). All geopotential coefficients up to degree 3, including the three components of the geocenter offset, were estimated. The C20 estimates from this series are used to replace the GRACE derived C20 for GRACE applications in extracting the mass variation signal in the hydrological, ocean and polar ice sheets. The background gravity models, and modes for the solid Earth and ocean pole tides as well as the atmosphere and ocean dealiasing (AOD) employed in this analysis are generally consistent with those for the GRACE RL04 products.
5. Spectrum of the Variations in the /
Figures 1 and 2 compare the weekly solution from case 1 and the long-period variations from the case 3 (smoothed) with a linear fit. Quite prominent variations appear in / in the case 1 when the pole tides are not modeled for SLR data analysis. Significant linear trend show up for both and . Figures 3 and 4 compare the spectrum of the SLR derived variations in and from the three cases. Figures 3 and 4 show the / undergo significant variations at the annual, Chandler frequency and a period of ∼490 days in the case 1 without modeling of the pole tide. The dominant power of the variations in the / are at ∼ Chandler period induced by the solid Earth pole tide. The variation with a period of ∼490 days is significantly reduced by ∼30% for , and is not apparent for when the solid Earth pole tides were modeled. The source of the signal at ∼490 days is not known yet; it may be that the time series is not yet long enough to fully resolve and separate the various long-period signals. Table 2 compares (1) the amplitude and phase of annual and Chandler variation from a least square fit to the estimates from 3 cases; (2) the amplitude and phase from the weekly averaged polar motion excitation function (mass) of IERS (denoted as EOP-EF) based on equation (7); and (3) the amplitude and phase from the AOD data over the same period for SLR data. The phase shown Table 2 is referred to the epoch of 2002.0.
Table 2. Amplitude and Phase of the Variations at Annual and Chandler Period in / From ∼17 Years SLR Data (SLR-1)a
Amplitudes (Amp) are in units of 10−11 and phases are in degrees. Phase is referred to 1 January 2002. The standard deviation (1 sigma) for the three cases is estimated to be ∼0.5 × 10−11 for and for the amplitude, and ∼3° for the phase from SLR-1.
 The amplitude and phase of case 1 shown in Table 2 represents the integrated effects in the SLR residual without modeling of the pole tides. The difference between case 1 and 2 represents the contribution from the solid Earth's pole tide. The difference between case 2 and 3 represents the contribution from the ocean pole tide. The amplitude and phase of case 3 represents the nontidal variations in / that remain in the SLR measurement residual after modeling the pole tides in analysis of SLR data.
 When the solid Earth pole tide model was applied in the orbits fitting SLR data, the power of the Chandler variation was reduced by more than 80% in and , the amplitude of annual variation was reduced by 86% in , and 21% in . When the ocean pole tide model is applied, the signal of the variations at the Chandler period become negligible in the spectrum for both of and . This fact reflects the accuracy of the pole tide models given the dominance of the Chandler wobble variations in the observed polar motion. The magnitude of the annual variation in decreased when applying the solid Earth pole tide but increased when applying the ocean pole tide, and become comparable with that without modeling pole tides in , with a phase shift of ∼210°. The including ocean pole tide improved the agreement in phase with EOP-EF. The agreement between SLR (case 3) and EOP-EF is within ∼8% for as shown in Table 2. This fact suggests that the pole tides has small contribution to change the power, but does shift the phase of the annual variation in . The contribution of the ocean pole tide is less than 8% of total variations in from analysis of SLR data. When both of the solid Earth and ocean pole tides were modeled as in case 3, the amplitude of the nontidal annual variation is considerably smaller than that of the variations at other longer period in as shown in Figure 3. This fact implies that either the signal is too small to be detected or the SLR data is relatively insensitive to the variations along the meridian due to the poor distribution of the SLR tracking stations between the southern and northern hemispheres.
Figures 5 and 6 compare the monthly solutions of the and from GRACE and SLR (SLR-2) data during the GRACE mission. The two solutions are in a good agreement, and the significant drift is evident for . Figures 7 and 8 compare the spectrum of variations in and from monthly solutions of GRACE, SLR (SLR-2), monthly averaged AOD and EOP Excitation of the IERS polar motion (based on equation (7)) over the GRACE period. Figure 7 shows that the dominant power of the variations in the is at the annual band. The variations at the period > ∼ 400 days are evident in the spectrum of from GRACE, SLR and EOP-EF. The annual power from SLR is almost the same as from AOD. Much of the annual variation in and from SLR can be explained by the AOD model. The annual variation is the dominant signature in the spectrum of the , the signal for the other long-period variation is weaker than the annual variation as shown in Figure 8.
Table 3 compares the amplitudes and phase for the annual variation from SLR (SLR-2), GRACE, polar motion excitation function (mass) (EOP-EF) and AOD. The results for the amplitude and phase in Tables 2 and 3 (case 3) are the estimates, in average sense, for the climate mass redistribution induced temporal variations in / at annual frequency. The amplitudes and phases are expected to be time dependent and to depend on the period and the time interval to be considered, as can be seen from the difference in the results of annual variation from SLR-1 and SLR-2 in the Tables 2 and 3 (case 3). The magnitude of the estimated annual variation is 1 order above the noise level (∼2.0 × 10−12 from GRACE). It requires a great deal of efforts to obtain a reliable estimate of the uncertainty of the time-variable gravity field, for which the GRACE teem have taken year's efforts to obtain and improve the time series of the GRACE monthly solutions, from which the / coefficients were used in this study. For estimating the error bar from individual technique, the error bar remarked in the Tables 2 and 3 is based on the standard deviation determined by the weighting RMS obtained from a least square fit to the time series based on the uncalibrated uncertainties of the coefficients from SLR and GRACE solution (for which a factor of 14 was applied to the uncertainties of GRACE coefficients of the regular RL04 solutions (S. Bettadpur, personal communication, 2009)), while the errors are unknown for the polar motion excitation (mass part) and AOD model. This standard deviation reflects not only the noise level, but also the signals of high frequencies contained in the time series and the variability of annual variations. Thus, the quoted “standard deviation” in Tables 1–3 requires a calibration to represent a realistic error estimate. As an assessment, the inter comparison is considered to provide an external validation for the estimates of the averaged seasonal annual variations from the independent observations, such as GRACE, SLR and the polar motion excitation. Comparison indicates that the annual variation in from SLR, GRACE, and EOP-EF data is in good agreement within ∼1.5% for amplitude and phase. The difference in the amplitudes from three techniques is smaller than the standard deviation. The amplitude for annual variation in the from AOD model is larger than the observed variations from SLR and GRACE. This suggests the AOD model may have overestimated the variations in ; both SLR (SLR-2) and GRACE data corrected the error in the AOD model and restored the annual signal of the variation in . The power of the annual signal in is only ∼35% of that in . Estimates of the annual variation in from three techniques appear scattering for the phase; in particular, there is a phase shift of ∼60° for SLR (SLR-2) from EOP-EF and GRACE. The amplitude deviates by ∼5% for SLR-2 and ∼14% for EOP-EF from that of GRACE. Those differences in the annual amplitudes for are within the range of the standard deviation (1 sigma) cited for SLR and GRACE as shown in Table 3.
Table 3. Comparison of the Annual Variation for /a
The amplitude (a) in units of 10−11 and the phase (ϕ) in unit of degrees with the convention a cos (ωt − ϕ), where t is time past 1 January 2002. The standard deviation (1 sigma) is estimated to be ∼0.8 × 10−11 from SLR-2, ∼0.4 × 10−11 from GRACE for amplitude, and ∼5° from SLR-2, ∼3° from GRACE for phase.
The annual variation from SLR (SLR-2) and GRACE without restoring the monthly averaged AOD.
 The seasonal variations in and are well known to be due to the climate mass redistributions in the atmosphere, ocean and surface water storage. A reliable hydrological model is not available at present. However, the results in the last two rows in Table 3 are expected to be mainly due to the hydrological contributions to the annual variation. It is estimated to be no more than ∼50% for and ∼25% for of the total annual variation if AOD model is correct. The amplitude of the semiannual variation is estimated to be no more than approximately 60% and 30% of the annual variation in and , respectively.
6. Rates for and and the Mean Pole
 A least squares fit was performed for the bias, trend and periodic terms at the annual and semiannual frequencies to the / time series from SLR, GRACE and EOP (IERS polar motion series) with additional Chandler period and other terms based on equation (3). Table 4 compares the rates of / from SLR, GRACE, IERS polar motion data (denoted as EOP, hereafter) and the prediction from a GIA model. The rates from the polar motion variables (xp and yp) are computed from the derivative of equation (8) with kf = 1. Four time spans were used to compute the rates of the mean pole from IERS polar motion series: (1) from 1 January 1976 to 1 January 2000 consistent with the IERS 2003 conventions, (2) the same time period extended to June 2009, (3) for the period from 1 November 1992 to 1 June 2009, and (4) the GRACE mission period from the 1 January 2002 to January 2010. Two SLR rates were obtained over periods 3 and 4 based on the SLR-1 and SLR-2 time series, respectively. The rate from GRACE was based on the monthly solutions from CSR (RL04) with the modeled rate restored. For comparison, a prediction for GIA is also given by J. M. Wahr (personal communication, 2009).
In unit of 10−12/yr. The standard deviation (1 sigma) is estimated to be ∼1 and 2 × 10−12/yr for SLR-1 and SLR-2, respectively, and ∼2 × 10−12/yr for GRACE. The uncertainty for the EOP is not available.
 The behavior of the linear trend in / appear to be in general agreement between the rates determined from EOP, GRACE and SLR data, but the agreement with the prediction of the GIA model is poor. The GIA model is based on mean polar motion rates considerably smaller than the recent rates and would not be expected to agree particularly well. The motion of mean pole is not linear and is subject to significant decadal and other long-period fluctuations. Consequently, the rate from EOP will depend on the period over which the polar motion (EOP) data were fit as shown in Table 4. The SLR derived rates are smaller for than rates from the EOP and the CSR GRACE solution over the same period. There is good agreement between the EOP and GRACE over the GRACE period, but, due to the effects of the interannual variations, the time period is too short to place much confidence in the secular term.
 On timescales of thousands of years, the most important mechanism for the linear trend in / is glacial isostatic adjustment (GIA). The contributions to the secular changes in the / have been considered including the relative precession of the inner core [Greiner-mai and Barthelmes, 2001] and the continental drift [Dickman, 1977]. The rates of / from geodetic measurements and GRACE contain those excitation as well as the significant contribution from the present-day mass changes in glaciers, ice sheets and the accompanying change in the nonsteric sea level. The discrepancy between geodetic results and GIA model suggest there is significant contribution from the present-day mass changes. For example, significant ice mass loss over Greenland and Antarctica is observed from GRACE [Velicogna and Wahr, 2005, 2006], and recent results indicate a likely acceleration of that mass loss [Velicogna, 2009; Chen et al., 2009].
 The Earth's figure axis is subjected to significant variations due to the pole tides and other mass redistribution in the atmosphere, ocean and continents. The dominant contributions are due to the rotational deformation in the solid Earth at the annual and Chandler periods for ; the variations at the Chandler period are dominant for . The annual variation in is mainly due to the nontidal mass redistribution based on the analysis of SLR data spanning 17 years. Analysis of recent estimates of / from SLR, GRACE and polar motion excitation function data spanning more than 7 years shows a good agreement within ∼1.5% for the annual variations in . The amplitude of the annual variations in is only ∼30% of the variation. The deviations in the annual amplitude of is ∼5% for SLR-2 and ∼14% for EOP-EF from GRACE; the seasonal variations are smaller in amplitude, making it more difficult to separate it from the long-period aliases and other errors. The dominant contribution to the observed annual variation appears to be due to the mass variations in the atmosphere and oceans as described by the AOD model. The AOD model could overestimate the seasonal variations and leaves a margin of less than ∼ 25% for and ∼50% for accounting for the contribution from the hydrological excitation. Analysis of ∼17 years of SLR data shows the variation of the figure axis in the latitudinal direction (the y direction, 90° east) is larger than in the meridional direction (x direction, north to south) for the annual period.
 The fluid Love number characterizes the response of the nonrigid Earth to the slow changes in the perturbing forces with a timescale of several years, and relates the mean figure axis and the mean pole of rotation axis. The mean figure axis determined from the GRACE and SLR data provides a constraint on the fluid Love number, which is estimated to be 0.9 based on the GGM03S gravity solution. The longitude of the mean figure axis is estimated to be 100°E.
 The estimates of / from GRACE and SLR data provide an improved constraint on the equatorial components of the relative rotation of the core of 5 × 10−8 times the Earth's diurnal spin rate. An equatorial polar offset of ∼ 697 mas for the mean figure axis implies a tilt of the inner core figure axis of ∼1.9°, and ∼3.1 arc sec displacement for the figure axis of the entire core. The results from this study suggest that the dominant contribution to the observed mean figure axis is due to the tipped figure axis of the core. The contribution of the pressure torque at the CMB is expected to be no more than ∼9%. The direction of the mean figure axis of the entire core is estimated to be 277° east, which is opposite to the direction of the mean figure axis of the entire Earth.
 This research was supported in part by NASA grants NNX08AE99E and JPL1368074. We thank M. Fang for helpful discussion and providing the estimate of pressure field at the CMB. We thank two anonymous reviewers for helpful review and suggestions.