Spherical harmonic analysis(SHA)and synthesis(SHS)are widely used by researchers in various fields.Both numerical integration and least-squares methods can be employed for analysis and synthesis.However,these approach...Spherical harmonic analysis(SHA)and synthesis(SHS)are widely used by researchers in various fields.Both numerical integration and least-squares methods can be employed for analysis and synthesis.However,these approaches,when calculated via summation,are computationally intensive.Although the Fast Fourier Transform(FFT)algorithm is efficient,it is traditionally limited to processing global grid points starting from zero longitude.In this paper,we derive an improved FFT algorithm for spherical harmonic analysis and synthesis.The proposed algorithm eliminates the need for grid points to start at zero longitude,thereby expanding the applicability of FFT-based methods.Numerical experiments demonstrate that the new algorithm retains the computational efficiency of conventional FFT while achieving accuracy comparable to the summation method.Consequently,it enables direct harmonic coefficient calculation from global grid data without requiring interpolation to align with zero longitude.Additionally,the algrithm can generate grid points with equi-angular spacing using the improved FFT algorithm,starting from non-zero longitudes.To address the loss of orthogonality in latitude due to discrete spherical grids,a quadrature weight factor-dependent on grid type(e.g.,regular or Gauss grid)-is incorporated,as summarized in this study.展开更多
As a means of quantitative interpretation,forward calculations of the global lithospheric magnetic field in the Spherical Harmonic(SH)domain have been widely used to reveal geophysical,lithological,and geothermal vari...As a means of quantitative interpretation,forward calculations of the global lithospheric magnetic field in the Spherical Harmonic(SH)domain have been widely used to reveal geophysical,lithological,and geothermal variations in the lithosphere.Traditional approaches either do not consider the non-axial dipolar terms of the inducing field and its radial variation or do so by means of complicated formulae.Moreover,existing methods treat the magnetic lithosphere either as an infinitesimally thin layer or as a radially uniform spherical shell of constant thickness.Here,we present alternative forward formulae that account for an arbitrarily high maximum degree of the inducing field and for a magnetic lithosphere of variable thickness.Our simulations based on these formulae suggest that the satellite magnetic anomaly field is sensitive to the non-axial dipolar terms of the inducing field but not to its radial variation.Therefore,in forward and inverse calculations of satellite magnetic anomaly data,the non-axial dipolar terms of the inducing field should not be ignored.Furthermore,our results show that the satellite magnetic anomaly field is sensitive to variability in the lateral thickness of the magnetized shell.In particular,we show that for a given vertically integrated susceptibility distribution,underestimating the thickness of the magnetic layer overestimates the induced magnetic field.This discovery bridges the greatest part of the alleged gap between the susceptibility values measured from rock samples and the susceptibility values required to match the observed magnetic field signal.We expect the formulae and conclusions of this study to be a valuable tool for the quantitative interpretation of the Earth's global lithospheric magnetic field,through an inverse or forward modelling approach.展开更多
Based on the CHAMP Magsat data set, spherical cap harmonic analysis was used to model the magnetic fields over China continent. The data set used in the analysis includes the 15′×15′ gridded values of the CHAMP...Based on the CHAMP Magsat data set, spherical cap harmonic analysis was used to model the magnetic fields over China continent. The data set used in the analysis includes the 15′×15′ gridded values of the CHAMP anomaly fields (latitude φ=25°N to 50°N and longitude λ=78°E to 135°E). The pole of the cap is located at φ=35°N and λ=110°E with half-angle of 30°. The maximum index (K max) of the model is 30 and the total number of model coefficients is 961, which corresponds to the minimum wavelength at the earth's surface about 400 km. The root mean square (RMS) deviations between the calculated and observed values are ～ 4 nT for ΔX, ～ 3 nT for ΔY and ～ 3.5 nT for ΔZ, respectively. Results show that positive anomalies are found mainly at the Tarim basin with ～6- 8 nT, the Yangtze platform and North China platform with ～4 nT, and the Songliao basin with ～4-6 nT. In contrast, negative anomaly is mainly located in the Tibet orogenic belt with the amplitude ～ (-6)-(-8) nT. Upward continuation of magnetic anomalies was used to semi-quantitatively separate the magnetic anomalies in different depths of crust. The magnetic anomalies at the earth's surface are from -6 to 10 nT for upper crust, middle crust -27 to 42 nT and lower crust -12 to 18 nT, respectively. The strikes of the magnetic anomalies for the upper crust are consistent with those for the middle crust, but not for the lower crust. The high positive magnetic anomalies mainly result from the old continental nucleus and diastrophic block (e.g. middle Sichuan continental nucleus, middle Tarim basin continental nucleus, Junggar diastrophic block and Qaidam diastrophic block). The amplitudes of the magnetic anomalies of the old continental nucleus and diastrophic block are related to evolution of deep crust. These results improve our understanding of the crustal structure over China continent.展开更多
The CUG_CLMFM3D series comprises high-resolution three-dimensional lithospheric magnetic field models for China and its surroundings.The first version,CUG_CLMFM3Dv1,is a spherical cap harmonic model integrating the WD...The CUG_CLMFM3D series comprises high-resolution three-dimensional lithospheric magnetic field models for China and its surroundings.The first version,CUG_CLMFM3Dv1,is a spherical cap harmonic model integrating the WDMAMv2(World Digital Magnetic Anomaly Map version 2)global magnetic anomaly grid and nearly a decade of CHAMP(Challenging Minisatellite Payload for Geophysical Research and Application)satellite vector data.It achieves a~5.7 km resolution but has limitations:the WDMAMv2 grid lacks high-resolution data in the southern Xinjiang and Tibet regions,which leads to missing small-to medium-scale anomalies,and unfiltered CHAMP data introduce low-frequency conflicts with global spherical harmonic models.Above the altitude of 150 km,correlations with global models drop below 0.9.The second version,CUG_CLMFM3Dv2,addresses these issues by incorporating 5-km-resolution aeromagnetic data and rigorously processed satellite data from CHAMP,Swarm,CSES-1(China Seismo-Electromagnetic Satellite 1),and MSS-1(Macao Science Satellite 1).The comparison analysis shows that the CUG_CLMFM3Dv2 captures finer high-frequency details and more stable long-wavelength signals,offering improved magnetic anomaly maps for further geological and geophysical studies.展开更多
In this paper,the X-ray micro-computed tomography(X-rayμCT),spherical harmonical-based principal component analysis(SH-PCA),and discrete element method(DEM)were incorporated to generate virtual samples with morpholog...In this paper,the X-ray micro-computed tomography(X-rayμCT),spherical harmonical-based principal component analysis(SH-PCA),and discrete element method(DEM)were incorporated to generate virtual samples with morphological gene mutation at different length scales.All samples were subjected to axial compression and constant confining stress.The effects of multiscale particle morphology on the stress-strain and energy storage/dissipation responses of granular soils were investigated.It is found that:(a)the effects of particle morphology on the initial stiffness,stress-strain,volumetric strain,and frictional energy dissipation behaviours are more pronounced for looser samples than for denser ones;(b)among different length scales,the particle morphology at the local roundness-level outperforms the one at the general form-level in dictating the macro-scale responses of granular soils;(c)the energy dissipation of a granular assemblage is a result of competition between particle morphology and initial void ratio.展开更多
In this study,discrete element method(DEM)simulations of a biaxial test were used to examine the effect of particle roundness on the mechanical behavior of sands at both the macro and micro scales.First,a stack of mic...In this study,discrete element method(DEM)simulations of a biaxial test were used to examine the effect of particle roundness on the mechanical behavior of sands at both the macro and micro scales.First,a stack of microcomputed tomography images were binarized,segmented,and labeled using advanced image processing and analysis techniques.Second,a spherical harmonic(SH)analysis,which involves a complete set of orthogonal functions,was implemented to rebuild the natural particle shape.Then,five templates of virtual particles were built in a DEM simulation,four of which were obtained from SH degrees of 3,8,12,and 15,and one template was an elementary sphere.A flexible membrane was numerically generated to allow the material to deform freely under a prescribed confining stress.Finally,the effect of particle roundness on the mechanical properties of granular materials was investigated and discussed.Two shear bands were found to intersect,forming an X shape in both the rotation and displacement fields.Moreover,a lower particle roundness results in higher deviatoric stress and stronger dilation in the volumetric change.A decrease in particle roundness leads to less rotation of particles despite a higher displacement value.In addition,a larger SH degree leads to smaller normalized contact forces of the particles.This implies that decreasing the roundness results in higher anisotropy of the contact forces.展开更多
CHAMP satellite data and ground-based magnetic observations are used and combined to map the lithospheric magnetic field over China by means of the revised spherical cap harmonic analysis(R-SHCA)modeling technique.The...CHAMP satellite data and ground-based magnetic observations are used and combined to map the lithospheric magnetic field over China by means of the revised spherical cap harmonic analysis(R-SHCA)modeling technique.The magnetic field is described to a spatial resolution of 150 km at the mean Earth’s radius,which represents a good compromise between the resolutions afforded by surface and satellite data.We compare the magnetic anomalies modeled at the regional scale with composite regions containing large-scale of tectonic structures.These regions,including the Tarim Basin and the Tibetan Plateau,are correlated with regional magnetic anomalies at satellite altitude but contain a significant number of small-scale and complex magnetic structures at the mean Earth’s radius.These magnetic anomalies are globally consistent with the known geological features in China but also offer a way to delineate the contours of the geological blocks and to discuss the connection between magnetic anomalies and the heat flow distribution in this region.展开更多
基金supported by The National Natural Science Foundation of China(42374004).
文摘Spherical harmonic analysis(SHA)and synthesis(SHS)are widely used by researchers in various fields.Both numerical integration and least-squares methods can be employed for analysis and synthesis.However,these approaches,when calculated via summation,are computationally intensive.Although the Fast Fourier Transform(FFT)algorithm is efficient,it is traditionally limited to processing global grid points starting from zero longitude.In this paper,we derive an improved FFT algorithm for spherical harmonic analysis and synthesis.The proposed algorithm eliminates the need for grid points to start at zero longitude,thereby expanding the applicability of FFT-based methods.Numerical experiments demonstrate that the new algorithm retains the computational efficiency of conventional FFT while achieving accuracy comparable to the summation method.Consequently,it enables direct harmonic coefficient calculation from global grid data without requiring interpolation to align with zero longitude.Additionally,the algrithm can generate grid points with equi-angular spacing using the improved FFT algorithm,starting from non-zero longitudes.To address the loss of orthogonality in latitude due to discrete spherical grids,a quadrature weight factor-dependent on grid type(e.g.,regular or Gauss grid)-is incorporated,as summarized in this study.
基金supported by the National Natural Science Foundation of China(Grant Nos.42250103 and 42174090)the Opening Fund of Key Laboratory of Geological Survey and Evaluation of Ministry of Education(Grant No.GLAB2023ZR02)the Ministry of Science and Technology(MOST)Special Fund from the State Key Laboratory of Geological Processes and Mineral Resources(Grant No.MSFGPMR2022-4)。
文摘As a means of quantitative interpretation,forward calculations of the global lithospheric magnetic field in the Spherical Harmonic(SH)domain have been widely used to reveal geophysical,lithological,and geothermal variations in the lithosphere.Traditional approaches either do not consider the non-axial dipolar terms of the inducing field and its radial variation or do so by means of complicated formulae.Moreover,existing methods treat the magnetic lithosphere either as an infinitesimally thin layer or as a radially uniform spherical shell of constant thickness.Here,we present alternative forward formulae that account for an arbitrarily high maximum degree of the inducing field and for a magnetic lithosphere of variable thickness.Our simulations based on these formulae suggest that the satellite magnetic anomaly field is sensitive to the non-axial dipolar terms of the inducing field but not to its radial variation.Therefore,in forward and inverse calculations of satellite magnetic anomaly data,the non-axial dipolar terms of the inducing field should not be ignored.Furthermore,our results show that the satellite magnetic anomaly field is sensitive to variability in the lateral thickness of the magnetized shell.In particular,we show that for a given vertically integrated susceptibility distribution,underestimating the thickness of the magnetic layer overestimates the induced magnetic field.This discovery bridges the greatest part of the alleged gap between the susceptibility values measured from rock samples and the susceptibility values required to match the observed magnetic field signal.We expect the formulae and conclusions of this study to be a valuable tool for the quantitative interpretation of the Earth's global lithospheric magnetic field,through an inverse or forward modelling approach.
基金ThispaperissupportedbytheChinese" 973"Program (No .2 0 0 3CD71 650 6)
文摘Based on the CHAMP Magsat data set, spherical cap harmonic analysis was used to model the magnetic fields over China continent. The data set used in the analysis includes the 15′×15′ gridded values of the CHAMP anomaly fields (latitude φ=25°N to 50°N and longitude λ=78°E to 135°E). The pole of the cap is located at φ=35°N and λ=110°E with half-angle of 30°. The maximum index (K max) of the model is 30 and the total number of model coefficients is 961, which corresponds to the minimum wavelength at the earth's surface about 400 km. The root mean square (RMS) deviations between the calculated and observed values are ～ 4 nT for ΔX, ～ 3 nT for ΔY and ～ 3.5 nT for ΔZ, respectively. Results show that positive anomalies are found mainly at the Tarim basin with ～6- 8 nT, the Yangtze platform and North China platform with ～4 nT, and the Songliao basin with ～4-6 nT. In contrast, negative anomaly is mainly located in the Tibet orogenic belt with the amplitude ～ (-6)-(-8) nT. Upward continuation of magnetic anomalies was used to semi-quantitatively separate the magnetic anomalies in different depths of crust. The magnetic anomalies at the earth's surface are from -6 to 10 nT for upper crust, middle crust -27 to 42 nT and lower crust -12 to 18 nT, respectively. The strikes of the magnetic anomalies for the upper crust are consistent with those for the middle crust, but not for the lower crust. The high positive magnetic anomalies mainly result from the old continental nucleus and diastrophic block (e.g. middle Sichuan continental nucleus, middle Tarim basin continental nucleus, Junggar diastrophic block and Qaidam diastrophic block). The amplitudes of the magnetic anomalies of the old continental nucleus and diastrophic block are related to evolution of deep crust. These results improve our understanding of the crustal structure over China continent.
基金supported by the National Natural Science Foundation of China(Grant Nos.42250103,42174090,42250101,42250102,and 41774091)the Macao Foundation+1 种基金the Opening Fund of Key Laboratory of Geological Survey and Evaluation of Ministry of Education(Grant No.GLAB2023ZR02)the MOST Special Fund from the State Key Laboratory of Geological Processes and Mineral Resources(Grant No.MSFGPMR2022-4)。
文摘The CUG_CLMFM3D series comprises high-resolution three-dimensional lithospheric magnetic field models for China and its surroundings.The first version,CUG_CLMFM3Dv1,is a spherical cap harmonic model integrating the WDMAMv2(World Digital Magnetic Anomaly Map version 2)global magnetic anomaly grid and nearly a decade of CHAMP(Challenging Minisatellite Payload for Geophysical Research and Application)satellite vector data.It achieves a~5.7 km resolution but has limitations:the WDMAMv2 grid lacks high-resolution data in the southern Xinjiang and Tibet regions,which leads to missing small-to medium-scale anomalies,and unfiltered CHAMP data introduce low-frequency conflicts with global spherical harmonic models.Above the altitude of 150 km,correlations with global models drop below 0.9.The second version,CUG_CLMFM3Dv2,addresses these issues by incorporating 5-km-resolution aeromagnetic data and rigorously processed satellite data from CHAMP,Swarm,CSES-1(China Seismo-Electromagnetic Satellite 1),and MSS-1(Macao Science Satellite 1).The comparison analysis shows that the CUG_CLMFM3Dv2 captures finer high-frequency details and more stable long-wavelength signals,offering improved magnetic anomaly maps for further geological and geophysical studies.
基金supported by the General Research Fund(Nos.CityU 11201020 and CityU 11207321)the Research Grant Council of the Hong Kong Special Administrative Region(SAR),China+2 种基金the Contract Research Project(No.9211295)the Geotechnical Engineering Office of the Civil Engineering Development Department of the Government of the Hong Kong SARthe financial support from the Hong Kong PhD Fellowship Scheme(HKPFS)。
文摘In this paper,the X-ray micro-computed tomography(X-rayμCT),spherical harmonical-based principal component analysis(SH-PCA),and discrete element method(DEM)were incorporated to generate virtual samples with morphological gene mutation at different length scales.All samples were subjected to axial compression and constant confining stress.The effects of multiscale particle morphology on the stress-strain and energy storage/dissipation responses of granular soils were investigated.It is found that:(a)the effects of particle morphology on the initial stiffness,stress-strain,volumetric strain,and frictional energy dissipation behaviours are more pronounced for looser samples than for denser ones;(b)among different length scales,the particle morphology at the local roundness-level outperforms the one at the general form-level in dictating the macro-scale responses of granular soils;(c)the energy dissipation of a granular assemblage is a result of competition between particle morphology and initial void ratio.
基金supported by General Research Fund Grant(Nos.CityU 11201020 and CityU 11213517)from the Research Grants Council of the Hong Kong SARResearch Grant(No.51779213)from the National Science Foundation of China.
文摘In this study,discrete element method(DEM)simulations of a biaxial test were used to examine the effect of particle roundness on the mechanical behavior of sands at both the macro and micro scales.First,a stack of microcomputed tomography images were binarized,segmented,and labeled using advanced image processing and analysis techniques.Second,a spherical harmonic(SH)analysis,which involves a complete set of orthogonal functions,was implemented to rebuild the natural particle shape.Then,five templates of virtual particles were built in a DEM simulation,four of which were obtained from SH degrees of 3,8,12,and 15,and one template was an elementary sphere.A flexible membrane was numerically generated to allow the material to deform freely under a prescribed confining stress.Finally,the effect of particle roundness on the mechanical properties of granular materials was investigated and discussed.Two shear bands were found to intersect,forming an X shape in both the rotation and displacement fields.Moreover,a lower particle roundness results in higher deviatoric stress and stronger dilation in the volumetric change.A decrease in particle roundness leads to less rotation of particles despite a higher displacement value.In addition,a larger SH degree leads to smaller normalized contact forces of the particles.This implies that decreasing the roundness results in higher anisotropy of the contact forces.
基金supported by National Natural Science Foundation of China(Grant Nos.4117412240890163&41031066)OPWSRP(Grant No.201005017)
文摘CHAMP satellite data and ground-based magnetic observations are used and combined to map the lithospheric magnetic field over China by means of the revised spherical cap harmonic analysis(R-SHCA)modeling technique.The magnetic field is described to a spatial resolution of 150 km at the mean Earth’s radius,which represents a good compromise between the resolutions afforded by surface and satellite data.We compare the magnetic anomalies modeled at the regional scale with composite regions containing large-scale of tectonic structures.These regions,including the Tarim Basin and the Tibetan Plateau,are correlated with regional magnetic anomalies at satellite altitude but contain a significant number of small-scale and complex magnetic structures at the mean Earth’s radius.These magnetic anomalies are globally consistent with the known geological features in China but also offer a way to delineate the contours of the geological blocks and to discuss the connection between magnetic anomalies and the heat flow distribution in this region.
