Benchmark forward gravity schemes: the gravity field of a realistic lithosphere model WINTERC-G

Several alternative gravity forward modelling methodologies and associated numerical codes with their own advantages and limitations are available for the Solid Earth community. With the upcoming state-of-the-art lithosphere density models and accurate global gravity field data sets it is vital to understand the opportunities and limitations of the various approaches. In this paper, we discuss the four widely used techniques: global spherical harmonics (GSH), tesseroid integration (TESS), triangle integration (TRI), and hexahedral integration (HEX). A constant density shell benchmark shows that all four codes can produce similar precise gravitational potential fields. Two additional shell tests were conducted with more complicated density structures: lateral varying density structures and a Moho density interface between crust and mantle. The differences between the four codes were all below 1.5 percent of the modeled gravity signal suitable for reproducing satellite-acquired gravity data. TESS and GSH produced the most similar potential fields (< 0.3 percent). To examine the usability of the forward modelling codes for realistic geological structures, we use the global lithosphere model WINTERC-G, that was constrained, among other data, by satellite gravity field data computed using a spectral forward modeling approach. This spectral code was benchmarked against the GSH and it was confirmed that both approaches produce similar gravity solution with negligible differences between them. In the comparison of the different WINTERC-G-based gravity solutions, again GSH and TESS performed best. Only short-wavelength noise is present between the spectral and tesseroid forward modelling approaches, likely related to the different way in which the spherical harmonic analysis of the varying boundaries of the mass layer is performed. The Spherical harmonic basis functions produces small differences compared to the tesseroid elements especially at sharp interfaces, which introduces mostly short-wavelength differences. Nevertheless, both approaches (GSH and TESS) result in accurate solutions of the potential field with reasonable computational resources. Differences below 0.5 percent are obtained, resulting in residuals of 0.076 mGal standard deviation at 250 km height. The biggest issue for TRI is the characteristic pattern in the residuals that is related to the grid layout. Increasing the resolution and filtering allows for the removal of most of this erroneous pattern, but at the expense of higher computational loads with respect to the other codes. The other spatial forward modelling scheme HEX has more difficulty in reproducing similar gravity field solutions compared to GSH and TESS. These particular approaches need to go to higher resolutions, resulting in enormous computation efforts. The hexahedron-based code performs less than optimal in the forward modelling of the gravity signature, especially of a lateral varying density interface. Care must be taken with any forward modelling software as the approximation of the geometry of the WINTERC-G model may deteriorate the gravity field solution.

Regional Crustal Vertical Deformation Driven by Terrestrial Water Load Depending on CORS Network and Environmental Loading Data: A Case Study of Southeast Zhejiang

Monitoring regional terrestrial water load deformation is of great significance to the dynamic maintenance and hydrodynamic study of the regional benchmark framework. In view of the lack of a spatial interpolation method based on the GNSS (Global Navigation Satellite System) elevation time series for obtaining terrestrial water load deformation information, this paper proposes to employ a CORS (Continuously Operating Reference Stations) network combined with environmental loading data, such as ECMWF (European Centre for Medium-Range Weather Forecasts) atmospheric data, the GLDAS (Global Land Data Assimilation System) hydrological model, and MSLA (Mean Sea Level Anomaly) data. Based on the load deformation theory and spherical harmonic analysis method, we took 38 CORS stations in southeast Zhejiang province as an example and comprehensively determined the vertical deformation of the crust as caused by regional terrestrial water load changes from January 2015 to December 2017, and then compared these data with the GRACE (Gravity Recovery and Climate Experiment) satellite. The results show that the vertical deformation value of the terrestrial water load in southeast Zhejiang, as monitored by the CORS network, can reach a centimeter, and the amplitude changes from −1.8 cm to 2.4 cm. The seasonal change is obvious, and the spatial distribution takes a ladder form from inland to coastal regions. The surface vertical deformation caused by groundwater load changes in the east–west–south–north–central sub-regions show obvious fluctuations from 2015 to 2017, and the trends of the five sub-regions are consistent. The amplitude of surface vertical deformation caused by groundwater load change in the west is higher than that in the east. We tested the use of GRACE for the verification of CORS network monitoring results and found a relatively consistent temporal distribution between both data sets after phase delay correction on GRACE, except for in three months—November in 2015, and January and February in 2016. The results show that the comprehensive solution based on the CORS network can effectively improve the monitoring of crustal vertical deformation during regional terrestrial water load change.

Multiscale Three-Dimensional Morphological Characterization of Calcareous Sand Particles Using Spherical Harmonic Analysis

Particle morphology is a fundamental inherent property that substantially affects the macroscopic behavior of granular materials. The division and separation of particle morphology at different scale levels contributes to the further multiscale morphology related orthogonal researches. In this context, the high-resolution X-ray micro-computed tomographic (X-CT) scanning and spherical harmonic (SH) analysis were combined to complete the precise and digitized reconstruction of sand particles. The 3D sphericity, roundness and roughness were introduced to define the particle morphology at three scale levels (the general shape, local angularity and surface textures). Two typical sand particles, Calcareous sand (CS) and Fujian sand (FS), were tested in this study. The results showed that the irregularity, angularity and roughness of CS is higher than that of FS, and the multiscale morphological features of the two types of natural sand were given and compared digitally.

Geomagnetic Virtual Observatories: monitoring geomagnetic secular variation with the Swarm satellites

We present geomagnetic main field and secular variation time series, at 300 equal-area distributed locations and at 490 km altitude, derived from magnetic field measurements collected by the three Swarm satellites. These Geomagnetic Virtual Observatory (GVO) series provide a convenient means to globally monitor and analyze long-term variations of the geomagnetic field from low-Earth orbit. The series are obtained by robust fits of local Cartesian potential field models to along-track and East–West sums and differences of Swarm satellite data collected within a radius of 700 km of the GVO locations during either 1-monthly or 4-monthly time windows. We describe two GVO data products: (1) ‘Observed Field’ GVO time series, where all observed sources contribute to the estimated values, without any data selection or correction, and (2) ‘Core Field’ GVO time series, where additional data selection is carried out, then de-noising schemes and epoch-by-epoch spherical harmonic analysis are applied to reduce contamination by magnetospheric and ionospheric signals. Secular variation series are provided as annual differences of the Core Field GVOs. We present examples of the resulting Swarm GVO series, assessing their quality through comparisons with ground observatories and geomagnetic field models. In benchmark comparisons with six high-quality mid-to-low latitude ground observatories we find the secular variation of the Core Field GVO field intensities, calculated using annual differences, agrees to an rms of 1.8 nT/yr and 1.2 nT/yr for the 1-monthly and 4-monthly versions, respectively. Regular sampling in space and time, and the availability of data error estimates, makes the GVO series well suited for users wishing to perform data assimilation studies of core dynamics, or to study long-period magnetospheric and ionospheric signals and their induced counterparts. The Swarm GVO time series will be regularly updated, approximately every four months, allowing ready access to the latest secular variation data from the Swarm satellites.

An upward continuation method based on spherical harmonic analysis and its application in the calibration of satellite gravity gradiometry data

The ground gravity anomalies can be used to calibrate and validate the satellite gravity gradiometry data. In this study, an upward continuation method of ground gravity data based on spherical harmonic analysis is proposed, which can be applied to the calibration of satellite observations from the European Space Agency's Gravity Field and Steady-State Ocean Circulation Explorer (GOCE). Here, the following process was conducted to apply this method. The accuracy of the upward continuation method based on spherical harmonic analysis was verified using simulated ground gravity anomalies. The DTU13 global gravity anomaly data were used to determine the calibration parameters of the GOCE gravitational gradients based on the spherical harmonic analysis method. The trace and the tensor invariants I2, I3 of the gravitational gradients were used to verify the calibration results. The results revealed that the upward continuation errors based on spherical harmonic analysis were much smaller than the noise level in the measurement bandwidth of the GOCE gravity gradiometer. The scale factors of the Vxx, Vyy, Vzz, and Vyz components were determined at an order of magnitude of approximately 10−2, the Vxz component was approximately 10−3, and the Vxy component was approximately 10−1. The traces of gravitational gradients after calibration were improved when compared with the traces before calibration and were slightly better than the EGG_TRF_2 data released by the European Space Agency (ESA). In addition, the relative errors of the tensor invariants I2, I3 of the gravitational gradients after calibration were significantly better than those before calibration. In conclusion, the upward continuation method based on spherical harmonic analysis could meet the external calibration accuracy requirements of the gradiometer.

APPLYING THE METHOD OF MAXIMUM CONTRIBUTIONS TO THE MAGNETOGRAM INVERSION TECHNIQUE

Fundamentals of the spherical harmonic analysis (SHA) of the geomagnetic field were created by Gauss. They acquired the classical Chapman — Schmidt form in the first half of the XXth century. The SHA method was actively developed for domestic geomagnetology by IZMIRAN, and then, since the start of the space age, by ISTP SB RAS, where SHA became the basis for a comprehensive method of MIT (magnetogram inversion technique). SHA solves the inverse problem of potential theory and calculates sources of geomagnetic field variations (GFV) - internal and external electric currents. The SHA algorithm forms a system of linear equations (SLE), which consists of 3K equations (three components of the geomagnetic field, K is the number of ground magnetic stations). Small changes in the left and (or) right side of such SLE can lead to a significant change in unknown variables. As a result, two consecutive instants of time with almost identical GFV are approximated by significantly different SHA coefficients. This contradicts both logic and real observations of the geomagnetic field. The inherent error of magnetometers, as well as the method for determining GFV, also entails the instability of SLE solution. To solve such SLEs optimally, the method of maximum contribution (MMC) was developed at ISTP SB RAS half a century ago. This paper presents basics of the original method and proposes a number of its modifications that increase the accuracy and (or) speed of solving the SLEs. The advantage of MMC over other popular methods is shown, especially for the Southern Hemisphere of Earth.