To gain a predictive understanding of water in Earth’s changing environments, atmosphere-geology-hydrology and potential hazards to human communities, infrastructures and sustainable living, our ability to measure terrestrial water transient storage (a.k.a. changes of terrestrial water storage) must evolve. Earth environment monitoring by satellite began with the NASA Earth Resources Technology Satellite launched in 1972 and lead to the joint NASA-USGS Landsat Program  . Notions of terrestrial water transient storage, a component of the Earth’s energy and mass balance have been at the core of NASA’s Decadal Survey Global Cycles of Energy and Water and other governmental agencies both national and international starting with NASA’s Mission To Planet Earth (1989) for more than two decades  . These notions have driven the development of Electromagnetic-Optical sensors and satellites missions in attempts to measure and assess potential feedbacks, human and natural, in global and regional meteorology and climatology of Earth. In particular feedbacks, which may be caused by human activities and potential run-away changes (amplification or reduction) of natural cycles have become the interest of many national and international activist and governance organizations  - .
Our knowledge of climatology, the causes and workings of Earth’s past, present and possible future “climate” come directly from empirical observations of the Earth’s rocks, those exposed on the continents and islands and cores recovered from land and in the depths of ocean basins, i.e. Geology, spaning a history of 4.5 Ga. Examples include the Geological Sea Level curves, the Phanerozoic Eon atmosphere CO2 concentration, trangression and regressions on the continents and inferred surface continental temperature curves and the empirical linkage of the 405 Ky Malankovitch to radiometric-magnetostratigraphy of the last 215 million years of Earth history  - . In these renderings, CO2 concentration, sea level and land temperature are proxy-parameters that may be associated to “inferred climate” during the Eras of Earth’s history.
With the advent of satellite technologies and sensor-systems in near-Earth orbit, and the growth of electronic computational methods and techniques, attempt to measure meteorological parameters and possibly “climate parameters” has been a growing enterprise fraught with many failures and miss-interpretations. The emphasis from the ideas of the 1950s was to use numerical machine codes, i.e. computer models, to ingest meteorological measurements and over a sufficiently long period of time, “climate” would emerge from the model output, i.e. both the match of current “climate” and a prediction of future “climate”. This leads to the creation of the United Nations sponsored World Meteorological Office, International Panel on Climate Change and the Framework Convention on Climate Change. The term “Climate Change” and physical “climate” parameters are to this day still undefined (i.e. undiscovered), though meteorological parameters are assumed by necessity. Of note, the “satellite era” measurements currently cover less than 1 millionth of Earth’s history.
In the 1980s various ideas were moving among researchers in academia and government agencies about employing gravity sensors, i.e. gravimeters, and magnetometers to measure any changes of the Earth’s gravity and magnetic fields from near-Earth orbit, using knowledge from exploration geophysics methods and techniques. Germany and the United States developed several missions. The space agencies of both countries partnered in the mid-1990s to develop such a joint mission to measure the time-variable gravity field from near-Earth orbit. This became the Gravity Recovery and Climate Experiment (GRACE) program of Germany’s DLR and U.S.A.’s NASA. In both countries several universities and other national agencies partnered as well as those from France and Italy. GRACE was launched in March 2002 (end-of-mission March 2018) and GRACE Follow-On was launched in May 2018.
This report gives details of: 1) A critical flaw in the GRACE and GRACE Follow-On mission design; 2) The affect of the flaw on derived global datasets regarding Terrestrial Water Transient Storage interpretations; and 3) The Chandler Wobble measured by GRACE and GRACE Follow-On. The purpose of the research is to assess 1, 2 and 3 and recommend a solution to re-compute the GRACE and GRACE Follow-On Datasets.
2. Data, Methods and Techniques, and the GRACE Flaw
Data for water equivalent mass change (i.e. terrestrial water transient storage) comes from Release-05 (R5) Level-3 products provided by the GRACE Science Team centers and Release-06 from GRACE Follow-On Science Team center. Previous releases R2 through R4 and R5 have been investigated  - . Grids are produced at 1-arc-degree global coverage complete to degree and order 40. The GRACE solution to the gravity potential formulated as water equivalent mass change (length scale, Δh) can be expressed in the harmonic expansion given by
and coefficients Plm: Normalized Legendre polynomials, ΔClm(t), and ΔSlm(t): Normalized time-varying Stokes spherical harmonic geopotential coefficients, ae: Earth mean radius, r: spatial radius, kl: Love numbers, ρe: Earth mean density, ρw: fresh water density, t: time, and φ, λ are latitude and longitude . Beyond degree (order) 40 to 70, the inherent noise level in the mass change signal becomes significant . Processing includes downward propagation and adjustments to remove the time-variable mass change effects from ocean tides, atmosphere variance and mean variation (the GRACE geoid model). Low-order Stokes spherical harmonic geopotential coefficient derived by the International Lunar Ranging Service (NASA-Goddard and International Earth Rotation Service) and International Terrestrial Reference Frame are used, Figure 1  . A normalized Gaussian smoother filter mitigates striping artifacts produced by the orbit non-crossing and control-descent geometry  . Differences in processing (de-aliasing) and error sources and products are attributable to differences in assumed zero-degree and order Stokes harmonics, tide (ocean) models and the modeled atmosphere mass change removal, respectively in decreasing order of magnitude  .
GRACE global coverage presented here is from August 2002 through December 2020. Glacial Isostatic Adjustment (GIA) is a global phenomenon by way of mantle flow following the decay of the Pleistocene ice sheets in North America and Euro-Scandinavia . GIA is removed from the GRACE grids .
The GRACE satellites (A and B of GRACE and C and D of GRACE Follow-On) cover their twin-orbit spheres in about 28-days (i.e. close to the Sidereal month). This global “measure-month” represents a periodic sampling signal (i.e. alias) that repeats every 12 months   .
The flaw in GRACE, illustrated in Figure 2, is that the harmonic expansion is an Earth-centric, i.e. Geocentric, field reference with mean radius, mean density and Love numbers pertaining to only the Earth. GRACE assumes a One-Body gravity field, not the real Two-Body gravity field. The Stokes harmonic field potentials for the Moon had been estimated from early satellite tracking data and laser ranging up to the 1980s . However, quality of the harmonics of degree 4 and higher were considered low and many questions remained regarding the
Figure 1. ITRF Stations. The ITRF combines multi-satellite-sensors and ground stations laser measurements from the international services of the Global Navigation Satellite System, Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS), Very-Long Baseline Interferometry (VLBI)  and the Lunar Ranging Service (LR) that was established by Apollo Missions 11, 14 and 15 with Russian Space Missions Luna 17 and 21 .
Figure 2. Barycenter-Geocenter Comparison. The illustration compares the gravity fields and orbit geometries of the Barycenter versus the Geocenter Systems (not to scale), with the GRACE and GRACE Follow on (A and B and C and D, respectively) managed geocentric orbits. Notation as follows: N-rotation axis, G-Geocenter, β-Barycenter, re-radius of earth, reβ-radius of Barycenter in earth’s mantle, β-Barycenter rotation axis and rβ-radius of Moon to Barycenter in Earth’s mantle.
offset of the Moon’s gravity center to the center of figure, the topography and density variations of the farside and nearside . Recall that the Moon is in a locked-orbit with Earth with mass concentrated on the nearside. Therefore during the GRACE mission development the Stokes harmonic field potentials of the Moon were not used.
Consider the following thought experiment. Assume the Earth is a perfect and homogenous sphere (no internal mass variation, not visco-elastic) and there is no Moon. As the GRACE satellites orbit at about 400 km above the Earth, their path is a geodesic (Earth-centered) defined by their angular momentum. As the GRACE satellites range their intra-satellite distance (by KBR), there is no variation with the exception of electronic-system induced errors. Introduce a mass, the Moon, at a distance (ellipsoidal orbit) of roughly 240,000 km away from Earth with a mass of about 1/81 Earth mass. The gravity field in this case is no longer Geocentric it is Barycentric. The GRACE satellite’s orbits are maintained Geocentric. The KBR-measured intra-satellite range varies, because the GRACE satellite orbits cross Barydesics, equipotentials of the Barycentric Earth-Moon gravity field . In this case the perfect Earth and Moon co-orbit the Barycenter, 1650 km within the perfect Earth (about 4728 km outward along the Earth’s radius) on a Sidereal monthly rate of 27.32 days . If the GRACE satellites as they are ranging by KBR the intra-satellite distance could cover the perfect Earth in their orbits within about 1 hour, the principle harmonics they would “measure” would be 1 hour, 24 hours, one Sidereal month and one Sidereal year, repeated over the mission life time. Yet, as the GRACE satellite’s KBR’s measure the inter-satellite distance, the GRACE orbits are crossing Barydesic equipotentials, causing apparent accelerations and decelerations caused by the maintained-Geocentric orbits. Sampling errors, i.e. aliases (by way of the Shannon-Nyquist theorem), are not geophysical signals.
The Chandle Wobble (CW) is an excited resonance and the Annual Wobble (AW) a forced resonance of the Earth’s rotation axis . Both are recognized as periodic variations of Polar Motion . Polar motion, a.k.a. the “variation latitude” and the variation of star zeniths (local plumb line from observer to star), have been known since antiquity by astrometry. L. Euler mathematically solved the problem of polar motion in 1765. His analytical Free-unforced nutation of the rotation axis solution gave a period of 305 days and would dissipate in 68 years  . The empirical measurement and physical cause remained elusive until F. Küster in 1888 and S. Chandler in 1891 produced the first accurate measurements, 428 days (Chandler, 1891/92, with Free and Forced-annual components), and a hypothesis  .
During the 20th century refinements by way of advances in instrumentation and ground networks (including the Lunar Ranging sites on the Moon) have focused analysis on the components (prograde and retrograde) and variability (amplitude and period) of the CW and AW of Polar Motion  . Recent measurements have shown the instantaneous periods of the CW and AW (prograde) are 392 to 441 days and 359 to 370 days, respectively . Interestingly, the amplitude of CW has varied from 43 to 287 mas (milli-arc seconds) and is following a decreasing trend since 1995 . The amplitude of AW (prograde) has varied from 65 to 180 mas and is following a decreasing trend since 2010 (the AW retrograde component has been oscillating since 1960) .
Chandler (1893) followed his discovery of the CW and AW components of Polar Motion as observed by star zeneth variations, i.e. the “latitude variation”, with his hypothesis that the CW was caused by the “Free nutation motion” of the rotation axis within the Earth  . However, a “Free nutation” of the rotation axis would quickly dissipate as Euler (1765) analytically calculated  . Investigations in the first half of the 20th century proposed hypotheses for AW as “logically evident” but untestable such as mantle anelasticity, inner and outer core variable rotations, tectonic motions of convecting-mantle and outer core mantle-jets, i.e. hot-spot plumes, solar and jupiter-saturn planetary gravitational torques, electromagnetic torques generated by the liquid outer core producing interference with the lower mantle magnetic field, and perturbations by large magnitude earthquakes  . Late 20th and early 21st century investigations looked to causative interpretations of AW by variations of atmospheric mass, glacial-eustacy, ocean bottom pressure and terrestrial water transient storage  . However, as testable hypotheses and valid measurements remain elusive, such interpretations particularly with regard to terrestrial water transient storage are suspect
3.1. A Test of GRACE
A question now arises regarding the GRACE mission. Other than the KBR measured inter-satellite range and range rate, what physical property of the Earth can the GRACE satellites possibly measure? To evaluate this question three test regions of Earth have been selected for investigation: the Alaska North Slope, Amazon River Basin and Qinghai-Tibet Plateau. These regions have unique environments, geology-tectonics, hydrology and topography against which the GRACE R06 (described earlier) will be evaluated for regionalized spectral power and harmonic content, Figure 3.
Figure 3. Global Test Regions. Seismicity, Holocene volcanos and tectonic plate boundries coutersy of UNAVCO (EarthScope GEON, NSF, NASA) and USGS illustrated in Google Earth.
3.2. Alaska North Slope
The Alaska North Slope is the most extensive arctic environment region of the United States. The surface from the Brooks Range to the Chukchi and Beaufort Sea coasts is one of continuous permafrost up to 600 meters thick, tundra vegetation and extensive rivers that is home to indigenous peoples, Caribou and Muskox herds, summer wetlands supporting wild fowl and extensive winter thaw lakes     . Beneath the Holocene-Pleistocene permafrost (gravels, silts and ground ice) is a vast Permian-to-Miocene foreland fault/fold orogen structurally connected to the Paleozoic-Mesozoic formations of the Brooks Range  - . At depths of in excess of 600 meters, the geologic structures are sources of gas and oil that support operational productions as at Purdue Bay and exploration activates   . Petroleum reserves make this region one of America’s largest and of National strategic and world importance . Since the Quaternary the Alaska North Slope has been a passive continental margin of the North American Plate. The ocean portion of the North American plate meets the divergent boundary at the Gakkel Ridge (spreading rift) in the Arctic Ocean basin, Figure 4(A)  . North Slope earthquake activity is typically in the 1 to 3-magnitude range and at depths to 33 kilometers occurs daily distributed mostly in the Brooks Range and elsewhere  .
3.3. Qinghai-Tibet Plateau
The Hindu Kush-Karakoram-Himalayas, Tian Shan Mountains with the Qinghai-Tibet Plateau and Tarim Basin form a region of more than 3.4 million square kilometers  -  . Home to numerous large lakes and tarns (glacier lakes), and to more than 50,000 glaciers and high-elevation snowfields, continuous permafrost and the Taklimakan desert this region is the source of the Indus, Ganga, Brahmaputra, and Yamuna Rivers, the Indo-Gangetic River system   . The Himalayan Mountains and associated ranges create a boundary separating westerly continental air masses and southerly marine air masses of the summer South Asian monsoon  . The long-term seasonal stationarity of these air masses against the high mountains gives rise to anomalous atmospheric mass variations with abundant precipitation . The mean elevation of the plateau is about 4500 meters above mean sea level, which affects broadband solar and infrared-thermal fluxes of the atmosphere above the plateau . The broad plateau high elevation is due to the subduction of the India plate beneath the Eurasia plate from about 55 million years ago (Late Cretaceous) and is on going, Figure 4(B)  . With more than 30-years of GPS and recently GNSS (GPS and GLONASS) geodetic measurements it is know that the plateau is rising at about 8 mm per year, on average, and which in southern Tibet becomes about 17 to 22 mm/yr   . However, failure to account for the anomalous atmospheric mass variations and the mantle mass transport has given rise to many dubious conclusions raised in poorly researched scientific papers .
Figure 4. Geologic-Plate Tectonic cross-sections of the Alaska North Slope, Qinghai-Tibet Plateau and the Amazon Basin (South American Plate).
3.4. Amazon Basin
The Amazon basin, the Amazon River and tributaries, drains an area in excess of 35% of the continent of South America the land portion of the South American Plate  . Its area is estimated at 6,300,000 km2. From the high elevation Andes Mountains, the result of subduction of the Fallon (former) and Nazca Plates in the west, the modern Amazon spans to the Atlantic Ocean in the east where the ocean portion of the plate continues to the boundary with the Mid-Atlantic Ridge, Figure 4(C)   .
The Amazon rainforest and transcontinental river system are the largest on Earth . Many of the flora and fauna of modern Amazonia date from lineages as old as Late Cretaceous to Paleogene, 100 to 23 million years ago, coeval with the subduction of the India Plate with the Eurasia Plate and the rising of the Himalayas and Qinghai Plateau. The main geological events associated with stream capture that formed the modern transcontinental rives system are estimated to have occurred in the Neogene. While much of the present day topography of basins and mountains can be explained by Plate Tectonics there is mounting evidence that viscous flow within the mantle, a transient mass, plays a significant role in the topography of river systems such as the Amazon, the Brahmaputra and others .
3.5. Tests by Regional Power Sectral Density
Figure 5 shows the GRACE water equivalent mass change time-series, Periodograms and Power Spectral Density of the test regions from August 2002 through January 2017. The time-series are the regionalized (volumes) for the Alaska North Slope, Qinghai-Tibet Plateau and Amazon, respectively. Inspection of the time-series, periodograms and power spectra reveals remarkable similarities! The Periodogram and Powe Spectral Density are Red-Noise on each of the Test Regions. The reader will note the magnitude (Periodogram score and Power dB) of the changes rise as the area of regionalization (area) increases, from Alaska North Slope to Amazon. The Power Spectral Density of the time-series black line and the estimate of significance (90%) using Chi-square χ2 process red line with the time-series covariance shows very low significance except for the 12 sideralmonth signal. The 12-month signal may be an alias of the Chandler Wobble (14.23-month period)    .
4.1. Results: Alaska North Slope
The highest Periodogram scores occur near the 2-month (Sideral Months) start of the series: an artifact of GRACE orbit sampling. Two components at 11.636 and 12.190 months have low scores 75.703 and 25.476 (at 0.082 and 0.086 frequency), respectively. Power Spectral Density of these two components is 19.912 and 21.285 dB, and above the model-noise level (significance better than 90%). These components are not the AW. Components where CW should be in the Periodogram at 14.222 sideral months has a very low score of 0.478 and in the Power Density Spectrum at 0.070 frequency has 14.823 dB of low significance, Figure 5(A).
4.2. Results: Qinghai-Tibet Plateau
Components in the Periodogram at 2 and 2.032 Siderial months have score of 4.5E5 and 8.5E5, respectively. Power Spectral Density at 0 and 0.012 frequency is 37.285 and 34.801 dB, respectively. These components are aliases produced by the GRACE orbital sampling. Where the AW should be in the Periodogram at 11.326 and 12.190 Siderial months have scores of 2.0E7 and 1.6E7, respectively. Power Spectral Density of these components is 39.23 and 39.35 dB (at 0.082 and 0.086 frequency), respectively. In the Periodogram where the CW should be at 14.222 Siderial months the component has a score of 4.7E4. In Power Spectral Density at corresponding frequency 0.070 the power is 31.473 dB, Figure 5(B).
Figure 5. GRACE Regional Time Series, Periodogram and Power Spectral Density of the Test Regions Alaska North Slope, Qinghai-Tibet Plateau and Amazon Basin. The Red-series in the Power Spectral Density represents the Chi-Square (χ2) red-noise significance estimate (90%) based on the covariance of the regional time series. For processing, the GRACE time series are zero-padded to 256 (28) to satisfy the Discrete Fast Fourier Transform requirement.
4.3. Results: Amazon Basin
Components in the Periodogram at 2 and 2.016 Sidereal months have score of 4.9E4 and 2.3E5, respectively. Power Spectral Density at 0 and 0.004 frequency is 41.311 and 40.300 dB, respectively. As with the Qinghai-Tibet Plateau, these components are aliases produced by the GRACE orbital sampling. Where the AW should be in the Periodogram at 11.326 and 12.190 Siderial months have scores of 2.0E7 and 5.7E7, respectively. Power Spectral Density of these components is 49.317 and 47.049 dB (at 0.082 and 0.086 frequency), respectively. In the Periodogram where the CW should be at 14.222 Sidereal months the component has a score of 6.6E4. In Power Spectral Density at corresponding frequency 0.070 the power is 39.899 dB, Figure 5(C).
4.4. Summary of Results
The actual physical dissimilarities of the regions are stark: two regions, the Alaska North Slope and the Qinghai Plateau have minimal vegetation and substantial permafrost and snow cover whereas the Amazon region has no permafrost, almost no snow cover except for the high elevations and very substantial vegetation! Furthermore, the near-surface and crustal geology of each region is very different, and they reside on different tectonic plates with very different velocity fields!
The only physical property of Earth that is the same in the three test regions that GRACE can measure is the Chandler Wobble, the variation of Earth’s rotational axis, excited and forced, from torques primarily in the Earth’s mantle and gravity fields (primarily the Barycenter field). The AW is unfortunately not resolvable, i.e. separately from the CW, by GRACE due to aliasing (the GRACE Flaw) and not significant in comparison to inherent noise (Figure 5).
This report assesses the GRACE flaw in the GRACE and GRACE Follow-On mission design, the CW and the aliased AW (Polar Motion components) and their effects on the current global GRACE datasets. Furthermore, at present the only physical property of the Earth that GRACE can possibly measure the Chandler Wobble, though at this time it is not measured accurately. From the analysis presented here of the GRACE satellite orbits and spectral analysis of the datasets, our main conclusion is that the annual harmonic component, Figure 5, is not the AW of Polar Motion, but rather an alias artifact of orbital sampling from the KBR measurements (i.e. orbital mechanics and instrument ranging sampling rates). By deduction, we must conclude that interpretations of terrestrial water transient storage derived from GRACE are currently wrong.
Therefore, to correct the GRACE Flaw, a new two-body Barycentric field solution and two-body harmonic expansion are called for. It is proposed that such is achievable using the Moon Stokes coefficients and parameters derived from the Gravity Recovery Interior Laboratory (GRAIL) mission . Furthermore, this will resolve the GRACE Flaw in the original GRACE missions and allow for the refinement of the CW and resolution of the AW in GRACE datasets following the works of Lambeck  , Xie and Kopeikin , Adhikari and Ivins  and Lambert and Sottili .
Research was supported by grants NASA NNOG6M48G, NNX17AC57A and NSF ARC0632400, ARC-0612533, ARC0856864 and EPS-0701898. Thanks to the Geophysical Institute and the International Arctic Research Center colleagues and employees at the University of Alaska Fairbanks and colleagues at the State Key Laboratory of Frozen Soil Engineering, Chinese Academy of Sciences at Lanzhou, Gansu, China.
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