Michael M. Watkins

Precise point positioning for the efficient and robust analysis of GPS data from large networks

Networks of dozens to hundreds of permanently operating precision Global Positioning System (GPS) receivers are emerging at spatial scales that range from 100 to 103 km. To keep the computational burden associated with the analysis of such data economically feasible, one approach is to first determine precise GPS satellite positions and clock corrections from a globally distributed network of GPS receivers. Then, data from the local network are analyzed by estimating receiver-specific parameters with receiver-specific data; satellite parameters are held fixed at their values determined in the global solution. This “precise point positioning” allows analysis of data from hundreds to thousands of sites every day with 40-Mflop computers, with results comparable in quality to the simultaneous analysis of all data. The reference frames for the global and network solutions can be free of distortion imposed by erroneous fiducial constraints on any sites.

The Crust of the Moon as Seen by GRAIL

The Holy GRAIL? The gravity field of a planet provides a view of its interior and thermal history by revealing areas of different density. GRAIL, a pair of satellites that act as a highly sensitive gravimeter, began mapping the Moons gravity in early 2012. Three papers highlight some of the results from the primary mission. Zuber et al. (p. 668, published online 6 December) discuss the overall gravity field, which reveals several new tectonic and geologic features of the Moon. Impacts have worked to homogenize the density structure of the Moons upper crust while fracturing it extensively. Wieczorek et al. (p. 671, published online 6 December) show that the upper crust is 35 to 40 kilometers thick and less dense—and thus more porous—than previously thought. Finally, Andrews-Hanna et al. (p. 675, published online 6 December) show that the crust is cut by widespread magmatic dikes that may reflect a period of expansion early in the Moons history. The Moons gravity field shows that the lunar crust is less dense and more porous than was thought. High-resolution gravity data obtained from the dual Gravity Recovery and Interior Laboratory (GRAIL) spacecraft show that the bulk density of the Moons highlands crust is 2550 kilograms per cubic meter, substantially lower than generally assumed. When combined with remote sensing and sample data, this density implies an average crustal porosity of 12% to depths of at least a few kilometers. Lateral variations in crustal porosity correlate with the largest impact basins, whereas lateral variations in crustal density correlate with crustal composition. The low-bulk crustal density allows construction of a global crustal thickness model that satisfies the Apollo seismic constraints, and with an average crustal thickness between 34 and 43 kilometers, the bulk refractory element composition of the Moon is not required to be enriched with respect to that of Earth.

Gravity Field of the Moon from the Gravity Recovery and Interior Laboratory (GRAIL) Mission

The Holy GRAIL? The gravity field of a planet provides a view of its interior and thermal history by revealing areas of different density. GRAIL, a pair of satellites that act as a highly sensitive gravimeter, began mapping the Moons gravity in early 2012. Three papers highlight some of the results from the primary mission. Zuber et al. (p. 668, published online 6 December) discuss the overall gravity field, which reveals several new tectonic and geologic features of the Moon. Impacts have worked to homogenize the density structure of the Moons upper crust while fracturing it extensively. Wieczorek et al. (p. 671, published online 6 December) show that the upper crust is 35 to 40 kilometers thick and less dense—and thus more porous—than previously thought. Finally, Andrews-Hanna et al. (p. 675, published online 6 December) show that the crust is cut by widespread magmatic dikes that may reflect a period of expansion early in the Moons history. The Moons gravity field reveals that impacts have homogenized the density of the crust and fractured it extensively. Spacecraft-to-spacecraft tracking observations from the Gravity Recovery and Interior Laboratory (GRAIL) have been used to construct a gravitational field of the Moon to spherical harmonic degree and order 420. The GRAIL field reveals features not previously resolved, including tectonic structures, volcanic landforms, basin rings, crater central peaks, and numerous simple craters. From degrees 80 through 300, over 98% of the gravitational signature is associated with topography, a result that reflects the preservation of crater relief in highly fractured crust. The remaining 2% represents fine details of subsurface structure not previously resolved. GRAIL elucidates the role of impact bombardment in homogenizing the distribution of shallow density anomalies on terrestrial planetary bodies.

Improved methods for observing Earth's time variable mass distribution with GRACE using spherical cap mascons

We discuss several classes of improvements to gravity solutions from the Gravity Recovery and Climate Experiment (GRACE) mission. These include both improvements in background geophysical models and orbital parameterization leading to the unconstrained spherical harmonic solution JPL RL05, and an alternate JPL RL05M mass concentration (mascon) solution benefitting from those same improvements but derived in surface spherical cap mascons. The mascon basis functions allow for convenient application of a priori information derived from near-global geophysical models to prevent striping in the solutions. The resulting mass flux solutions are shown to suffer less from leakage errors than harmonic solutions, and do not necessitate empirical filters to remove north-south stripes, lowering the dependence on using scale factors (the global mean scale factor decreases by 0.17) to gain accurate mass estimates. Ocean bottom pressure (OBP) time series derived from the mascon solutions are shown to have greater correlation with in situ data than do spherical harmonic solutions (increase in correlation coefficient of 0.08 globally), particularly in low-latitude regions with small signal power (increase in correlation coefficient of 0.35 regionally), in addition to reducing the error RMS with respect to the in situ data (reduction of 0.37 cm globally, and as much as 1 cm regionally). Greenland and Antarctica mass balance estimates derived from the mascon solutions agree within formal uncertainties with previously published results. Computing basin averages for hydrology applications shows general agreement between harmonic and mascon solutions for large basins; however, mascon solutions typically have greater resolution for smaller spatial regions, in particular when studying secular signals.

Progress in the determination of the gravitational coefficient of the Earth

In most of the recent determinations of the geocentric gravitational coefficient (GM) of the Earth, the laser ranging data to the Lageos satellite have had the greatest influence on the solution. These data, however, have generally been processed with a small but significant error in one of the range corrections. In a new determination of GM using the corrected center-of-mass offset, a value of 398600.4415 km3/sec2 (including the mass of the atmosphere) has been obtained, with an estimated uncertainty (1 σ) of 0.0008 km3/sec2.

Glacial isostatic adjustment observed using very long baseline interferometry and satellite laser ranging geodesy

In global space geodetic solutions, radial site motions are usually estimated relative to the geocenter (the center of figure of the solid Earth). Most geodesists estimate the motion of the geocenter assuming both that sites do not move radially and that sites move laterally as predicted by plate motion model NUVEL-1A [DeMets et al., 1990, 1994]. Here we estimate the motion of the geocenter assuming that the plate interiors deform radially and laterally as predicted by the postglacial rebound model of Peltier [1994] or that of Peltier [1996] without assuming a priori knowledge about relative plate motion. Radial site motions estimated relative to this rebound-adjusted geocenter are in the same reference frame as the rebound model predictions, whereas site motions estimated without adjusting for rebound are not. We further constrain the motion of the rebound-adjusted geocenter using satellite laser rangings sensitivity to the center of mass (of the solid Earth, the oceans, and the atmosphere) by assuming that the mean velocity between the rebound-adjusted geocenter and the center of mass is negligible over the time period of geodetic measurement. Twenty years of observation with satellite laser ranging and very long baseline interferometry record the isostatic response of the solid Earth to the unloading of the late Pleistocene ice sheets. The misfits of the postglacial rebound model of Peltier [1994] and that of Peltier [1996] are 34% and 16% less, respectively, than the misfit of the rigid plate model. Sites at Onsala (Sweden) and Algonquin Park (Ontario) are observed to be rising at 3 mm/yr and 2 mm/yr, respectively, reflecting unloading of the Fennoscandian and Laurentide ice sheets. Sites along the east coast of the United States are subsiding at <2 mm/yr, indicating that the forebulge produced by the Laurentide ice sheet is currently collapsing very slowly. Sites beneath the margins of the ice sheets during the last glacial maximum are currently moving laterally away from the ice sheet centers at <1.5 mm/yr, in disagreement with the moderately fast outward motion predicted by the model of Peltier [1996].

Lunar interior properties from the GRAIL mission

The Gravity Recovery and Interior Laboratory (GRAIL) mission has sampled lunar gravity with unprecedented accuracy and resolution. The lunar GM, the product of the gravitational constant G and the mass M, is very well determined. However, uncertainties in the mass and mean density, 3345.56 ± 0.40 kg/m3, are limited by the accuracy of G. Values of the spherical harmonic degree-2 gravity coefficients J2 and C22, as well as the Love number k2 describing lunar degree-2 elastic response to tidal forces, come from two independent analyses of the 3 month GRAIL Primary Mission data at the Jet Propulsion Laboratory and the Goddard Space Flight Center. The two k2 determinations, with uncertainties of ~1%, differ by 1%; the average value is 0.02416 ± 0.00022 at a 1 month period with reference radius R = 1738 km. Lunar laser ranging (LLR) data analysis determines (C − A)/B and (B − A)/C, where A < B < C are the principal moments of inertia; the flattening of the fluid outer core; the dissipation at its solid boundaries; and the monthly tidal dissipation Q = 37.5 ± 4. The moment of inertia computation combines the GRAIL-determined J2 and C22 with LLR-derived (C − A)/B and (B − A)/C. The normalized mean moment of inertia of the solid Moon is Is/MR2 = 0.392728 ± 0.000012. Matching the density, moment, and Love number, calculated models have a fluid outer core with radius of 200–380 km, a solid inner core with radius of 0–280 km and mass fraction of 0–1%, and a deep mantle zone of low seismic shear velocity. The mass fraction of the combined inner and outer core is ≤1.5%.

Shortening and thickening of metropolitan Los Angeles measured and inferred by using geodesy

Geodetic observations using the Global Positioning System (GPS) and other techniques record a high rate of north-south shortening in an east-southeast–trending, 5–40-km-wide belt in northern metropolitan Los Angeles, California. Downtown Los Angeles is observed to be converging upon the southern San Gabriel Mountains at 6 mm/yr. Aside from the elastic strain that will be released during earthquakes rupturing the San Andreas and San Jacinto faults, east-west lengthening across northern metropolitan Los Angeles is minor, <2.5 mm/yr. Therefore north-south shortening is accommodated mainly by vertical crustal thickening.

High‐resolution lunar gravity fields from the GRAIL Primary and Extended Missions

The resolution and accuracy of the lunar spherical harmonic gravity field have been dramatically improved as a result of the Gravity Recovery and Interior Laboratory (GRAIL) mission. From the Primary Mission, previous harmonic gravity fields resulted in an average n = 420 surface resolution and a Bouguer spectrum to n = 330. The GRAIL Extended Mission improves the resolution due to a lower average 23 km altitude orbit. As a result, new harmonic degree 900 gravity fields (GL0900C and GL0900D) show nearly a factor of 2 improvement with an average surface resolution n = 870 and the Bouguer spectrum extended to n = 550. Since the minimum spacecraft altitude varies spatially between 3 km and 23 km, the surface resolution is variable from near n = 680 for the central farside to near n = 900 for the polar regions. These gravity fields with 0.8 million parameters are by far the highest-degree fields of any planet ever estimated with a fully dynamic least squares technique using spacecraft tracking data.

Quantifying and reducing leakage errors in the JPL RL05M GRACE mascon solution

Recent advances in processing data from the Gravity Recovery and Climate Experiment (GRACE) have led to a new generation of gravity solutions constrained within a Bayesian framework to remove correlated errors rather than relying on empirical filters. The JPL RL05M mascon solution is one such solution, solving for mass variations using spherical cap mass concentration elements (mascons), while relying on external information provided by near-global geophysical models to constrain the solution. This new gravity solution is fundamentally different than the traditional spherical harmonic gravity solution, and as such, requires different care when postprocessing. Here, we discuss two classes of postprocessing considerations for the JPL RL05M GRACE mascon solution: (1) reducing leakage errors across land/ocean boundaries, and (2) scaling the solutions to account for leakage errors introduced through parameterizing the gravity solution in terms of mascons. A Coastline Resolution Improvement (CRI) filter is developed to reduce leakage errors across coastlines. Synthetic simulations reveal a reduction in leakage errors of ∼50%, such that residual leakage errors are ∼1 cm equivalent water height (EWH) averaged globally. A set of gain factors is derived to reduce leakage errors for continental hydrology applications. The combined effect of the CRI filter coupled with application of the gain factors, is shown to reduce leakage errors when determining the mass balance of large (>160,000 km2) hydrological basins from 11% - 30% (0.6-1.5 mm EWH) averaged globally, with local improvements up to 8% - 81% (9-19 mm EWH). This article is protected by copyright. All rights reserved.