Srinivas Bettadpur

A Reconciled Estimate of Ice-Sheet Mass Balance

Warming and Melting Mass loss from the ice sheets of Greenland and Antarctica account for a large fraction of global sea-level rise. Part of this loss is because of the effects of warmer air temperatures, and another because of the rising ocean temperatures to which they are being exposed. Joughin et al. (p. 1172) review how ocean-ice interactions are impacting ice sheets and discuss the possible ways that exposure of floating ice shelves and grounded ice margins are subject to the influences of warming ocean currents. Estimates of the mass balance of the ice sheets of Greenland and Antarctica have differed greatly—in some cases, not even agreeing about whether there is a net loss or a net gain—making it more difficult to project accurately future sea-level change. Shepherd et al. (p. 1183) combined data sets produced by satellite altimetry, interferometry, and gravimetry to construct a more robust ice-sheet mass balance for the period between 1992 and 2011. All major regions of the two ice sheets appear to be losing mass, except for East Antarctica. All told, mass loss from the polar ice sheets is contributing about 0.6 millimeters per year (roughly 20% of the total) to the current rate of global sea-level rise. The mass balance of the polar ice sheets is estimated by combining the results of existing independent techniques. We combined an ensemble of satellite altimetry, interferometry, and gravimetry data sets using common geographical regions, time intervals, and models of surface mass balance and glacial isostatic adjustment to estimate the mass balance of Earth’s polar ice sheets. We find that there is good agreement between different satellite methods—especially in Greenland and West Antarctica—and that combining satellite data sets leads to greater certainty. Between 1992 and 2011, the ice sheets of Greenland, East Antarctica, West Antarctica, and the Antarctic Peninsula changed in mass by –142 ± 49, +14 ± 43, –65 ± 26, and –20 ± 14 gigatonnes year−1, respectively. Since 1992, the polar ice sheets have contributed, on average, 0.59 ± 0.20 millimeter year−1 to the rate of global sea-level rise.

Ensemble prediction and intercomparison analysis of GRACE time‐variable gravity field models

Precise measurements of the Earths time-varying gravitational field from the NASA/Deutsches Zentrum fur Luft- und Raumfahrt Gravity Recovery and Climate Experiment (GRACE) mission allow unprecedented tracking of the transport of mass across and underneath the surface of the Earth and give insight into secular, seasonal, and subseasonal variations in the global water supply. Several groups produce these estimates, and while the various gravity fields are similar, differences in processing strategies and tuning parameters result in solutions with regionally specific variations and error patterns. This study examined the spatial, temporal, and spectral variations between the different gravity field products and developed an ensemble gravity field solution from the products of four such analysis centers. The solutions were found to lie within a certain analysis scatter regardless of the local relative water height variation, and the ensemble model is clearly seen to reduce the noise in the gravity field solutions within the available scatter of the solutions.

High‐resolution CSR GRACE RL05 mascons

The determination of the gravity model for the Gravity Recovery and Climate Experiment (GRACE) is susceptible to modeling errors, measurement noise, and observability issues. The ill-posed GRACE estimation problem causes the unconstrained GRACE RL05 solutions to have north-south stripes. We discuss the development of global equal area mascon solutions to improve the GRACE gravity information for the study of Earth surface processes. These regularized mascon solutions are developed with a 1° resolution using Tikhonov regularization in a geodesic grid domain. These solutions are derived from GRACE information only, and no external model or data is used to inform the constraints. The regularization matrix is time variable and will not bias or attenuate future regional signals to some past statistics from GRACE or other models. The resulting Center for Space Research (CSR) mascon solutions have no stripe errors and capture all the signals observed by GRACE within the measurement noise level. The solutions are not tailored for specific applications and are global in nature. This study discusses the solution approach and compares the resulting solutions with postprocessed results from the RL05 spherical harmonic solutions and other global mascon solutions for studies of Arctic ice sheet processes, ocean bottom pressure variation, and land surface total water storage change. This suite of comparisons leads to the conclusion that the mascon solutions presented here are an enhanced representation of the RL05 GRACE solutions and provide accurate surface-based gridded information that can be used without further processing.

Reducing errors in the GRACE gravity solutions using regularization

The nature of the gravity field inverse problem amplifies the noise in the GRACE data, which creeps into the mid and high degree and order harmonic coefficients of the Earth’s monthly gravity fields provided by GRACE. Due to the use of imperfect background models and data noise, these errors are manifested as north-south striping in the monthly global maps of equivalent water heights. In order to reduce these errors, this study investigates the use of the L-curve method with Tikhonov regularization. L-curve is a popular aid for determining a suitable value of the regularization parameter when solving linear discrete ill-posed problems using Tikhonov regularization. However, the computational effort required to determine the L-curve is prohibitively high for a large-scale problem like GRACE. This study implements a parameter-choice method, using Lanczos bidiagonalization which is a computationally inexpensive approximation to L-curve. Lanczos bidiagonalization is implemented with orthogonal transformation in a parallel computing environment and projects a large estimation problem on a problem of the size of about 2 orders of magnitude smaller for computing the regularization parameter. Errors in the GRACE solution time series have certain characteristics that vary depending on the ground track coverage of the solutions. These errors increase with increasing degree and order. In addition, certain resonant and near-resonant harmonic coefficients have higher errors as compared with the other coefficients. Using the knowledge of these characteristics, this study designs a regularization matrix that provides a constraint on the geopotential coefficients as a function of its degree and order. This regularization matrix is then used to compute the appropriate regularization parameter for each monthly solution. A 7-year time-series of the candidate regularized solutions (Mar 2003–Feb 2010) show markedly reduced error stripes compared with the unconstrained GRACE release 4 solutions (RL04) from the Center for Space Research (CSR). Post-fit residual analysis shows that the regularized solutions fit the data to within the noise level of GRACE. A time series of filtered hydrological model is used to confirm that signal attenuation for basins in the Total Runoff Integrating Pathways (TRIP) database over 320 km radii is less than 1 cm equivalent water height RMS, which is within the noise level of GRACE.

Geographical representation of radial orbit perturbations due to ocean tides: Implications for satellite altimetry

In analogy to the geographical representation of the zeroth-order radial orbit perturbations due to the static geopotential, similar relationships have been derived for radial orbit perturbations due to the ocean tides. At each location these perturbations are seen to be coherent with the tide height variations. The study of this singularity is of obvious importance to the estimation of ocean tides from satellite altimeter data. We derive analytical expressions for the sensitivity of altimeter derived ocean tide models to the ocean tide force model induced errors in the orbits of the altimeter satellite. In particular, we focus on characterizing and quantifying the nonresonant tidal orbit perturbations, which cannot be adjusted into the empirical accelerations or radial perturbation adjustments commonly used during orbit determination and in altimeter data processing. As an illustration of the utility of this technique, we study the differences between a TOPEX/POSEIDON-derived ocean tide model and the Cartwright and Ray 1991 Geosat model. This analysis shows that nearly 60 percent of the variance of this difference for M2 can be explained by the Geosat radial orbit error due to the omission of coefficients from the GEM-T2 background ocean tide model. For O1, K1, S2, and K2 the orbital effects account for approximately 10 to 40 percent of the variances of these differences. The utility of this technique to an assessment of the ocean tide induced errors in the TOPEX/POSEIDON-derived tide models is also discussed.

The Pole Tide and its Effect on GRACE Time‐Variable Gravity Measurements: Implications for Estimates of Surface Mass Variations

Gravity Recovery and Climate Experiment (GRACE) provides monthly solutions for the Earths gravity field in the form of spherical harmonic coefficients. These can be used to infer changes in mass at the Earths surface. The pole tide (the response of the Earth and oceans to polar motion) causes gravity signals dominated by harmonics of degree 2, order 1. If the pole tide is not removed from GRACE data, it affects the coefficients of those harmonics (C21, S21) and introduces errors when using those coefficients to determine surface mass variations. The pole tide is partially removed by GRACE processing centers before solving for the gravity field. But long-period pole tide signals are not usually included in the GRACE pole tide correction, and so those signals are still present in the GRACE coefficients. We discuss this issue from the standpoint of somebody who uses the GRACE gravity fields to infer changes in surface mass. We arrive at a recommendation for an optimal GRACE pole tide correction. We describe how to modify the C21, S21 coefficients provided by the processing centers, so that they conform with our recommendation. We discuss the size of the pole tide contributions to C21, S21, compared to those of the direct load-induced contributions. As an example, we show how an incompletely removed pole tide can impact GRACE results for the trend in ocean mass. We consider the impact of mantle anelasticity on long-period pole tide corrections and conclude that it is unlikely to affect those corrections by more than 20%.

Neutral Density Measurements from the Gravity Recovery and Climate Experiment Accelerometers

Predicting the orbits of space objects in low-altitude orbits requires an accuratemodel for the atmospheric neutral density. The current accuracy of semi-empirical models limits the prediction accuracy and impacts a number of operational decisions. The currentmodels are based on sparsemeasurements of the neutral density, collected over an extended period. One of the problems is observing the thermosphere density changes in response to the solar and geomagnetic variability on short temporal scales, such as those characterized by geomagnetic storms. The stochastic behavior of the solar forcing represents one of the major challenges in predicting satellite orbits. In situ measurements of thedensity canplay a significant role in improving the structure of the neutral densitymodels and in providing a timelymeasurement for enhancing the accuracy of the satellite predictions.Measurements fromorbiting accelerometers carried by the twin Gravity Recovery and Climate Experiment satellites have the potential for providing accurate and timelymeasurements to improve the satellite prediction accuracy. The objective of this paper is to describe the procedure for using the Gravity Recovery and Climate Experiment accelerometer measurements for determining accurate density measurements.

Temporal Variability of Earth’s Gravitational Field from Satellite Laser Ranging

Satellite laser ranging (SLR) observations of geodetic satellites (Degnan, 1993) have long been used to study the temporal variations of the Earth’s external potential through the long-period orbital perturbations they produce (Yoder et al., 1983; Rubincam, 1984; Cheng et al., 1989; Tapley et al., 1993; Gegout and Cazenave, 1993; Nerem et al., 1993; Chao and Eanes, 1995; Nerem and Klosko, 1995.) Secular and long period variations in the zonal Stokes coefficients and nearly diurnal and semidiurnal tidal variations in the order 1 and 2 coefficients can be monitored with this approach. In this paper we analyze SLR observations of Lageos-1 and Starlette and present new results for constraints on the secular and 18.6-year variations in the zonal harmonics. These results provide important information for improving geophysical models of glacial rebound, ice-sheet changes, and mantle anelasticity.

The TEG-3 Geopotential Model

A new solution for the static geopotential, TEG-3, complete to 70×70 in spherical harmonics, has been obtained. The solution represents one of the latest efforts to improve the Earth’s gravity model. The solution was obtained by combining inhomogeneous satellite and in situ data sets, and by simultaneously estimating the relative weights for individual satellite data sets. Data from over 20 satellites and terrestrial surface gravity data were used in the latest solution. The satellite data include groundbased satellite laser and radiometric (Doris and Tranet) tracking data, spaceborne GPS, and radar altimeter measurements. Analysis indicates that TEG-3 provides an incremental improvement in overall satellite orbit determination when compared with recent models, including JGM-3, GRIM4C4, and EGM96. In particular, notable improvement has been achieved for TEG-3 in reducing geographically-correlated gravity errors for orbit determination of altimetric satellites (Geosat and ERS-1). Error analysis indicates that there is no notable improvement in marine geoid accuracy in TEG-3 as compared to JGM-3, while the EGM-96 model represents an improvement in the marine geoid accuracy as indicated by comparing with ground-truth measurements (Levitus 94 hydrography) and mean topography from numerical ocean circulation model simulations (POCM_4B).

Spherical harmonic synthesis and least squares computations in satellite gravity gradiometry

The computational requirements in the simulations of geopotential estimation from satellite gravity gradiometry are discussed. Fast algorithms for spherical harmonic synthesis and least squares accumulation on a vectorizing supercomputers are presented. Using these methods, in a test case estimation of 2595 coefficients of a degree and order 50 gravity field, sustained program execution speeds of 275 Mflops (87 % peak machine speed) on a single processor of a CRAY Y-MP were achieved, with spherical harmonics computation accounting for less than 1 % of total cost. From the results, it appears that brute-force estimation of a degree and order 180 field would require 537 Million Words of memory and 85 hours of CPU time, assuming mission duration of 1 month, and execution speed of 1 Gflops. Both memory size and execution speed requirements are within the capabilities of modern multi-processor supercomputers.