Contributions of terrestrial and GRACE data to the study of the secular geoid changes in North America

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Abstract

This paper tests and discusses different statistical methods for modelling secular rates of change of the geoid in North America. In particular, we use the method of principal component/empirical orthogonal functions (PC/EOF) analysis to model the geoid rates from Gravity Recovery and Climate Experiment (GRACE) satellite data. As demonstrated, the PC/EOF analysis is useful for studying the contributions from different signals (mainly residual hydrology signals and leakage effects) to the GRACE-derived geoid rates. The PC/EOF analysis leads to smaller geoid rates compared to the conventional least-squares fitting of a trend and annual and semi-annual cycles to the time series of the spherical harmonic coefficients. This is because we filter out particular spatiotemporal modes of the regional geoid changes.

We apply the method of least-squares collocation with parameters to combine terrestrial data (GPS vertical velocities from the Canadian Base Network and terrestrial gravity rates from the Canadian Gravity Standardization Net) with the GRACE-derived vertical motion to obtain again the geoid rates. The combined model has a peak geoid rate of 1.4 mm/year in the southeastern area of Hudson Bay contrary to the GRACE-derived geoid rates that show a large peak of 1.6–1.7 mm/year west of Hudson Bay. We demonstrate that the terrestrial data, which have a longer time span than the GRACE data, are important for constraining the GRACE-derived secular signal in the areas that are well sampled by the data.

Introduction

The objective of this paper is to study and compare different statistical approaches (summarized in the flowchart in Fig. 1) used to model the empirical rates of change of the geoid in North America that are mainly due to changes in the Earth potential associated with postglacial rebound (Peltier and Wu, 1982). These methods include the principal component/empirical orthogonal functions (PC/EOF) analysis, least-squares fitting, and least-squares collocation with parameters. The PC/EOF analysis and least-squares fitting are applied to the time series of the geoid change derived from the Gravity Recovery and Climate Experiment (GRACE) satellite mission (Tapley et al., 2004a). The least-squares collocation with parameters is used to combine terrestrial data (rates of change of the terrestrial gravity and GPS vertical velocities) and GRACE-derived vertical motion data in Canada.

There are two cases of main interest when analyzing mass variations derived from the GRACE data in North America. On the one hand, water mass variations can be analyzed and compared with hydrology model outputs (e.g., Rangelova et al., 2007a) provided that the postglacial rebound (PGR) signal is removed. On the other hand, the GRACE-derived geoid variations can be analyzed in terms of PGR (e.g., Tamisiea et al., 2007, van der Wal et al., 2008). In this case, the hydrology signal should be removed from the GRACE time series. These corrections are necessary because GRACE observes gravity changes that are integrated in a vertical column (Tapley et al., 2004b). In addition, studies have been devoted to the GRACE-derived mass loss signal in southeastern Alaska due to glaciers’ melting (e.g., Tamisiea et al., 2005, Chen et al., 2006a). Another group of studies focuses on estimating the rate of the Greenland ice sheet melting (e.g., Velicogna and Wahr, 2005, Chen et al., 2006b, Luthcke et al., 2006). We have excluded Greenland from our study, but we take care of the leakage of the Greenland ice sheet melting signal, which spreads out over the northeastern parts of the studied region.

Although we focus on North America only, we should also refer to relevant studies in different regions to highlight the excellent capabilities of the GRACE mission to detect temporal gravity variations. These studies include (but are not limited to) the GRACE-determined PGR signal in Fennoscandia (e.g., Steffen et al., 2008), Antarctic mass changes (e.g., Velicogna and Wahr, 2006, Chen et al., 2006c), and Patagonian icefield melting (Chen et al., 2007).

The rates of change of the geoid are computed using the time derivatives (rates) of the fully normalized GRACE spherical harmonic (SH) coefficients, C˙lm and S˙lm, as follows (see, e.g., Velicogna and Wahr, 2002):N˙(φ,λ)=Rl=2lmaxm=0lPlm(sinφ)C˙lmcos(mλ)+S˙lmsin(mλ)where Plm (sin φ) is the fully normalized associated Legendre function of spherical harmonic degree l and order m; R is a mean radius of the Earth, φ and λ are the latitude and longitude of the computational point, and lmax is the maximum spherical harmonic degree of the GRACE gravity field model.

The rate of any coefficient Klm = {Clm, Slm} is computed by least-squares fitting to the time series of Klm using the following parameterization for the epoch t:Klm(t)=k1t+k2cos(ωt)+k3sin(ωt)+k4cos(ωt/2)+k5sin(ωt/2)+ν(t)with the mean of the series removed. The parameter k1 is the rate, k2 and k3 are the annual cosine and sine amplitudes, respectively, and k4 and k5 are the amplitudes of the semi-annual components; ν(t) is a random error and ω = 2π/T, where T = 1 year, is the frequency of one cycle per year. The unknown parameters ki, i = 1, …, 5, are determined by solving the over-determined system produced by Eq. (2) after introducing weights for the coefficients at every epoch by means of their calibrated standard deviations. Alternatively to the least-squares fitting in the spectral domain, the geoid rate can be computed by means of fitting the time series of the geoid changes at grid points using the parameterization in Eq. (2).

In contrast to least-squares fitting, which requires parameterization of the trend and the periodic variability of the geoid, the PC/EOF analysis is a non-parametric method whose base functions adapt to the particular spatiotemporal data. This allows for studying inter-annual and long-term signals, which is the main advantage of the method of the PC/EOF analysis compared to least-squares. As we show, the PC/EOF analysis is also useful for separating trend signals with different geographical footprints.

To the best of the authors’ knowledge, Wiehl et al. (2005) and Viron et al. (2006) were one of the first to apply the PC/EOF analysis in relation to the GRACE-derived mass variations. The first work studies the low frequency sea water mass variability in the Baltic Sea, which could possibly mask the PGR geoid signal. The second study deals with climate signals derived from GRACE data. Other recent examples for global applications of the PC/EOF analysis include modelling the main signals of the seasonal steric sea level variations by Chambers (2006) and optimizing the smoothing of errors in the GRACE-derived mass variations (Schrama et al., 2007). In contrast to these studies, we show that the PC/EOF analysis is an efficient method for modelling the temporal geoid variations on a continental scale. This method provides a means to study the effect of the residual hydrology signal and the spatial leakage of the climate Alaskan and Greenland signals on the secular rate of change of the GRACE-derived geoid in North America.

While the GRACE data are homogeneously distributed both in space and time, the terrestrial data are irregularly distributed (scattered). Thus, modelling approaches that account for the irregular density of the terrestrial data are necessary to describe a surface of vertical deformation. These approaches include functional modelling, stochastic signal modelling, hybrid modelling, and dynamic Kalman filtering (Liu and Chen, 1998).

In the functional approach, a deformation surface is parameterized by analytical functions, such as bi-variate polynomials (e.g., Koohzare et al., 2008) or radial base functions (Holdahl and Hardy, 1979). The advantage of the functional approach is the ease of its implementation. The disadvantage is that the modelled vertical deformation surface depends largely on the measured deformation at a single point. Therefore, an erroneous observation occurring in an area weakly constrained by measurements will distort the modelled surface.

In the stochastic signal approach, the vertical deformation is a zero-mean signal defined by its covariance matrix computed from the local empirical signal covariance function. The deformation at new locations is obtained by the method of least-squares prediction (Moritz, 1980). This approach has a limited implementation in crustal deformation modelling because of the assumption for an isotropic and homogeneous deformation field, which, in reality, may not be fulfilled. Nevertheless, the stochastic signal approach has been implemented in earthquake studies for modelling migrating vertical displacements near convergent plate boundaries (Fujii and Xia, 1993) and detecting temporal changes in vertical displacement rates (El-Fiky et al., 1997). Egli et al. (2007) have recently developed an adaptive least-squares collocation procedure that accounts for the inhomogeneity and anisotropy of the vertical crustal deformation.

The functional approach can be combined with the stochastic signal modelling in a hybrid approach which is identical with least-squares collocation with parameters. Using the hybrid approach, Vestøl (2006) has recently modelled the Fennoscandian uplift from levelling, tide gauge and GPS data. In addition to the local solutions, this approach provides a means for error propagation and stepwise computations. The most important characteristic, however, is the heterogeneous data input and output that allow different functionals of the gravity field and its temporal variations to be included in the empirical deformation modelling.

The basic equation of the least-squares collocation with parameters is (Moritz, 1980):l=Ax+s+vwhere l = [l1l2ln]T is a vector of geodetic observations, x = [x1x2xk]T is a vector of unknown parameters, A is a (n × k) coefficient matrix, and v = [v1v2vn]T is a vector of normally distributed zero-mean errors with a covariance matrix Cvv.

In addition to the parametric part Ax, the observations, by definition, contain zero-mean stochastic signal components collected in the vector s = [s1s2sn]T with a covariance matrix Css. In this study, we assume that the signals are linear functionals of the rate of change of the Earth potential (Heck, 1984). These functionals include rates of change of the terrestrial gravity and the geoid as well as rates of the absolute vertical (radial) displacement.

The observations li, i = 1, …, n are assumed to be normally distributed variables with a covariance matrix Cll = Css + Cvv.

Let the zero-mean signal vector s¯ contain the vector s and the predictions at new locations collected in the vector sp, i.e., s¯=[sTspT]T. The signal covariance matrix is given as followsCs¯s¯=CssCsspCspsCspspwhere Csps=CsspT is the cross-covariance matrix of the predicted and given signals and Cspsp is the covariance matrix of the predicted signals. The estimated vector of parameters x and the signal vector s¯ are obtained by minimizing the norms¯TCs¯s¯s¯+vTCvvvunder the condition in Eq. (3). The signal and noise are assumed uncorrelated, i.e., Cs¯v=0. The estimated vector of parameters isxˆ=(ATCll1A)1ATCll1lwith an error covariance matrixExˆxˆ=(ATCll1A)1where the diagonal elements are the error variances of the estimated parameters.

The vector of predicted signals issˆp=CspsCll1(lAxˆ)with an error covariance matrixEsˆpsˆp=CspspCspsCll1Cssp+CspsCll1AExˆxˆATCll1Cssp.

The empirically predicted observations at new locations are computed aslˆp=Apxˆ+sˆpwith an error covariance matrixElˆplˆp=ApExˆxˆApT+ApExˆsˆp+EsˆpxˆApT+EsˆpsˆpwhereExˆsˆp=EsˆpxˆT=ExˆxˆATCll1Cssp.

Two issues need to be addressed in more detail. In a statistical sense, we have defined the signal as a spatially zero-mean variable described by its covariance matrix. In a broad geophysical sense, signal could be anything different from noise possessing even a deterministic description. One example is the annual continental hydrology cycle defined by its amplitude and phase. Often, this difference should be clear from the context of our discussion. Next, we use empirical predictions to denote the geoid rates computed at new locations by means of an optimal combination of the terrestrial and GRACE data. In parallel, we use model predictions to denote the geoid rates computed by postglacial rebound modelling.

This paper is organized as follows. Section 2 describes the data used in this study. This includes the rates of change of the terrestrial gravity and the GPS vertical velocities available in Canada. Also, the GRACE data and the post-processing procedures applied in this study are described. The paper proceeds with a description of the PC/EOF analysis and its application for computing the GRACE-derived geoid rates in Section 3. This is followed by a description of the optimal combination of the GRACE and terrestrial data and the combined model for the geoid rates. The paper concludes with a comparison of our empirical models with PGR model geoid rates and a discussion of the results obtained by the different statistical methods.

Section snippets

Description of geodetic data

Refinements of PGR models rely largely on the geodetic data constraints in North America because this signal is the largest one in magnitude and spatial scale. Sites well distributed across Canada and the northern parts of the US are required to accurately sample the PGR signal. In principle, collocated continuously operating GPS stations and absolute gravity sites can provide the necessary information for empirical geoid rate modelling. A considerable step towards providing consistent in space

GRACE-derived rates of the geoid computed by means of PC/EOF analysis

In the PC/EOF analysis, zero-mean GRACE-derived geoid changes are organized in the data matrix D = [dij], i = 1, …, n, j = 1, …, p, where n is the number of the epochs and p is the number of points on a 1° × 1° grid. The data covariance matrix is defined asS=DTD.

The data matrix D can be decomposed by singular value decomposition (SVD) as (Jolliffe, 2002):D=ULVTwhere U and V are (n × n) and (p × p) orthonormal matrices. V contains the eigenvectors of S and provides the EOF patterns. L is a diagonal matrix

Combining GRACE and terrestrial data

In order to combine the terrestrial data (rates of change of terrestrial gravity and GPS vertical velocities) and the GRACE-derived vertical motion data, we start with the basic model given by Eqs. (3), (4), (5), (6a), (6b), (7a), (7b), (8a), (8b), (8c), (9a), (9b), (10), (11), (12), (13), (14), (15), (16), (17a), (17b), (17c), (18), (19), (A.1a), (A.1b), (A.2a), (A.2b), (A.2c), (A.2d), (A.2e), (A.2f), (A.2g), (A.3a), (A.3b), (A.3c), (A.3d), (A.3e), (A.3f), (A.3g). We define the observation

Discussion and conclusions

The empirical geoid rates derived in Sections 3 GRACE-derived rates of the geoid computed by means of PC/EOF analysis, 4 Combining GRACE and terrestrial data are compared with the smoothed ICE-5G (VM2) rates along the two profiles shown in Fig. 7a. The north-south profile has a longitude of 80 °W and passes through Foxe basin in the north and James Bay in the south. In Fig. 8a, the smoothed ICE-5G (VM2) rates show a small peak of 1.0 mm/year in Foxe basin and a larger peak of 1.4–1.5 mm/year in

Acknowledgements

We would like to thank the Center for Space Research at the University of Texas for the GRACE data, Dr. S. Pagiatakis for providing the CGSN terrestrial gravity rates, and Dr. J. Henton for the CBN GPS data. We also thank Dr. R. Peltier for the ICE-5G (VM2) model outputs. We are very thankful to the two anonymous reviewers and Dr. E. Ivins for the careful and useful comments and suggestions that improved the quality of this paper. The funding for this research is from GEOIDE NCE and NSERC

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