This question is about the earth's axial precession. Recently it has been changing as measured by aquifer depletion using NASA's GRACE mission.
"A study by Rodell et al. [2009] in northwest India used terrestrial water storage-change observations from GRACE and simulated soil-water variations from a data-integrating hydrological modeling system to show that groundwater is being depleted at a mean rate of 4.0 +/- 1.0 cm yr-1 equivalent height of water (17.7 +/- 4.5 km3 yr-1) over the Indian states of Rajasthan, Punjab and Haryana (including Delhi). During the study period of August 2002 to October 2008, groundwater depletion was equivalent to a net loss of 109 km3 of water, which is double the capacity of India's largest surface-water reservoir" as reported by the NASA Jet Propulsion GRACE and Tellus Gravity Recovery and Climate Experiment on their website
https://grace.jpl.nasa.gov/applications/groundwater/
The scientific term for Earth's combined gravitational center with the moon is called the baycenter. It is what creates our four seasons.
https://spaceplace.nasa.gov/seasons/en/
The question being presented is how much the baycenter has deviated due to human activity which redistributes mass globally across the earth thus changing this gravitational center which determines earth's rotational axis that generate the four seasons. This deviation is termed precession. Fortunately this science has already been done. In a research paper titled Measuring the De Sitter precession with a new Earth’s satellite to the ≃10−5 level: a proposal by Lorenzo Iorio his abstract states:
"The inclination I of an Earth’s satellite in polar orbit undergoes a secular De Sitter precession of −7.6 milliarcseconds per year for a suitable choice of the initial value of its non-circulating node Ω. The competing long-periodic harmonic rates of change of I due to the even and odd zonal harmonics of the geopotential vanish for either a circular or polar orbit, while no secular rates occur at all. This may open up, in principle, the possibility of measuring the geodesic precession in the weak-field limit with an accurately tracked satellite by improving the current bound of 9×10−4 from Lunar Laser Ranging, which, on the other hand, may be even rather optimistic, by one order of magnitude, or, perhaps, even better. The most insidious competing effects are due to the solid and ocean components of the K1 tide since their perturbations have nominal huge amplitudes and the same temporal pattern of the De Sitter signature. They vanish for polar orbits. Departures of ≃10−5∘to10−3∘ from the ideal polar geometry allow to keep the K1 tidal perturbations to a sufficiently small level. Most of the other gravitational and non-gravitational perturbations vanish for the proposed orbital configuration, while the non-vanishing ones either have different temporal signatures with respect to the De Sitter effect or can be modeled with sufficient accuracy. In order to meet the proposed goal, the measurement accuracy of I should be better than ≃35 microarcseconds=0.034 milliarcseconds over, say, 5 year."
This research helped contribute to new investigations using GEM-TI and its predecessors GEM-T3 and GEM-T3S by NASA to calculate a new gravitational model for the earth.
https://doi.org/10.1029/JB093iB06p06169
Abstract
A major new computation of a terrestrial gravitational field model has been performed by the Geodynamics Branch of Goddard Space Flight Center (GSFC). In the development of this new model, designated Goddard Earth Model GEM‐T1, the design decisions of the past have been reassessed in light of the present state of the art in satellite geodesy. With GEM‐T1 a level of internal consistency has been achieved which is superior to any earlier Goddard Earth Model. For the first time a simultaneous solution has been made for spherical harmonic parameters of both invariant and tidal parts of the gravitational field. The solution of this satellite model to degree 36 is a major factor accounting for its improved accuracy. The addition of more precise and previously unused laser data and the introduction of consistent models were also accomplished with GEM‐T1. Another major factor allowing the creation of this model was the redesign and vectorization of our main software tools (GEODYN II and SOLVE) for the GSFC Cyber 205 computer. In particular, the high‐speed advantage (50:1), gained with the new SOLVE program, made possible an optimization of the weighting and parameter estimation scheme used in previous GEM models resulting in significant improvement in GEM‐T1. The solution for the GEM‐T1 model made use of the latest International Association of Geodesy reference constants, including the J2000 Reference System. It provided a simultaneous solution for (1) a gravity model in spherical harmonics complete to degree and order 36; (2) a subset of 66 ocean tidal coefficients for the long‐wavelength components of 12 major tides. This adjustment was made in the presence of 550 other fixed ocean tidal terms representing 32 major and minor tides and the Wahr frequency dependent solid earth tidal model; and (3) 5‐day averaged Earth rotation and polar motion parameters for the 1980 period onward. GEM‐T1 was derived exclusively from satellite tracking data acquired on 17 different satellites whose inclinations ranged from 15° to polar. In all, almost 800,000 observations were used, half of which were from third generation (<5 cm) laser systems. A calibration of the model accuracies has been performed showing GEM‐T1 to be a significant improvement over earlier GSFC “satellite‐only” models based purely on tracking data for both orbital and geoidal modeling applications. For the longest wavelength portion of the geoid (to 8×8), GEM‐T1 is a major advancement over all GEM models, even those containing altimetry and surface gravimetry. The radial accuracy for the anticipated TOPEX/POSEIDON orbit was estimated using the covariances of the GEM‐T1 model. The radial errors were found to be at the 25‐cm rms level as compared to 65 cm found using GEM‐L2. This simulation evaluated only errors arising from geopotential sources. GEM‐L2 was the best available model for TOPEX prior to the work described herein. A major step toward reaching the accuracy of gravity modeling necessary for the TOPEX/POSEIDON mission has been achieved.
Using this information one can calculate the earth's gravitational center using gravity wave mechanics and over time derive a histogram of these measured changes.
Unfortunately at this time no published models citing the change in earth's precession or displaying how much it has deviated due to human activity exist for public use as this information would be vital in the development of ICBM targeting technology. While NASA has researched and modeled these gravitational changes extensively due to issues involving national security this information is not currently available to the general public. Perhaps in the future under the freedom of information act it may be possible to obtain a general estimate. Until then we are obliged to do our own calculations and measurements.
