Recently,the simplified spherical harmonics equations(SP)model has at tracted much att entionin modeling the light propagation in small tissue ggeometriesat visible and near-infrared wave-leng ths.In this paper,we rep...Recently,the simplified spherical harmonics equations(SP)model has at tracted much att entionin modeling the light propagation in small tissue ggeometriesat visible and near-infrared wave-leng ths.In this paper,we report an eficient numerical method for fluorescence moleeular tom-ography(FMT)that combines the advantage of SP model and adaptive hp finite elementmethod(hp-FEM).For purposes of comparison,hp-FEM and h-FEM are,respectively applied tothe reconstruction pro cess with diffusion approximation and SPs model.Simulation experiments on a 3D digital mouse atlas and physical experiments on a phantom are designed to evaluate thereconstruction methods in terms of the location and the reconstructed fluorescent yield.Theexperimental results demonstrate that hp-FEM with SPy model,yield more accurate results thanh-FEM with difusion approximation model does.The phantom experiments show the potentialand feasibility of the proposed approach in FMT applications.展开更多
As an emerging molecular imaging modality,cone-beam X-ray luminescence computed tomog-raphy(CB-XLCT)uses X-ray-excitable probes to produce near-infrared(NIR)luminescence and then reconst ructs three-dimensional(3D)dis...As an emerging molecular imaging modality,cone-beam X-ray luminescence computed tomog-raphy(CB-XLCT)uses X-ray-excitable probes to produce near-infrared(NIR)luminescence and then reconst ructs three-dimensional(3D)distribution of the probes from surface measurements.A proper photon-transportation model is critical to accuracy of XLCT.Here,we presented a systematic comparison between the common-used Monte Carlo model and simplified spherical harmonics(SPN).The performance of the two methods was evaluated over several main spec-trums using a known XLCT material.We designed both a global measurement based on the cosine similarity and a locally-averaged relative error,to quantitatively assess these methods.The results show that the SP_(3) could reach a good balance between the modeling accuracy and computational efficiency for all of the tested emission spectrums.Besides,the SP_(1)(which is equivalent to the difusion equation(DE))can be a reasonable alternative model for emission wavelength over 692nm.In vivo experiment further demonstrates the reconstruction perfor-mance of the SP:and DE.This study would provide a valuable guidance for modeling the photon-transportation in CB-XLCT.展开更多
To determine the Avogadro constant with a target relative uncertainty of 2 x 10-s, the uncertainty component of the silicon sphere＇s volume introduced by the spherical harmonics method, which is usually used in deter...To determine the Avogadro constant with a target relative uncertainty of 2 x 10-s, the uncertainty component of the silicon sphere＇s volume introduced by the spherical harmonics method, which is usually used in determining the sphere＇s volume, is reevaluated. By means of representing the shape of the silicon sphere by an ellipsoid with Gaussian white noise in its diameters, the uncertainty of the current mapping methods based on the spherical harmonics theory can be estimated theoretically. It is evidenced that the uncertainty component attributed to the current mapping method is underestimated. To eliminate this effect as much as possible, the number of mapping points should be increased to more than before. Moreover, a new mapping method is proposed to accomplish the equal-area mapping with large number points on the silicon sphere.展开更多
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
Let S^1-1,q≥2,be the surface of the unit sphere in the Euclidean space R^1,f（x）∈L^p（S^q-1）,f（x）≥0,f absohutely unegual to 0,1≤p≤＋∞,Then,it is proved in the present paper that there is a spherical harmonic...Let S^1-1,q≥2,be the surface of the unit sphere in the Euclidean space R^1,f（x）∈L^p（S^q-1）,f（x）≥0,f absohutely unegual to 0,1≤p≤＋∞,Then,it is proved in the present paper that there is a spherical harmonics PN（x） of order≤N and a constant C〉0 such that where ω（f,δ）L^p=sup 0〈t≤δ‖St（f）-f‖L^p is a kind of moduli of continuity and ^‖f-1/PN‖L^p≤Cω（f,N^-1）L^p,St（f,μ）=1/｜S^q-2｜Sin^2λt ∫-μμ’=t f（μ＇）dμ＇ is a translation operator.展开更多
Computer aided drug design is to develop a chemical that binds to a target macromolecule known to play a key role in a disease state. In recognition of ligands by their protein receptors, molecular surfaces are often...Computer aided drug design is to develop a chemical that binds to a target macromolecule known to play a key role in a disease state. In recognition of ligands by their protein receptors, molecular surfaces are often used because they represent the interacting part of molecules and they should reflex the complementarity between ligand and receptor. However, assessing the surface complementarity by searching all relative position of two surfaces is often computationally expensive. The complementarity of lobe hole is very important in protein ligand interactions. Spherical harmonic models based on expansions of spherical harmonic functions were used as a fingerprint to approximate the binding cavity and the ligand, respectively. This defines a new way to identify the complementarity between lobes and holes. The advantage of this method is that two spherical harmonic surfaces to be compared can be defined separately. This method can be used as a filter to eliminate candidates among a large number of conformations, and it will speed up the docking procedure. Therefore, it is possible to select complementary ligands or complementary conformations of a ligand and the macromolecules, by comparing their fingerprints previously stored in a database.展开更多
An algorithm is presented for estimating the direction and strength of point light with the strength of ambient illumination. Existing approaches evaluate these illumination parameters directly in the high dimensional...An algorithm is presented for estimating the direction and strength of point light with the strength of ambient illumination. Existing approaches evaluate these illumination parameters directly in the high dimensional image space, while we estimate the parameters in two steps： first by projecting the image to an orthogonal linear subspace based on spherical harmonic basis functions and then by calculating the parameters in the low dimensional subspace. The test results using the CMU PIE database and Yale Database B show the stability and effectiveness of the method. The resulting illumination information can be used to synthesize more realistic relighting images and to recognize objects under variable illumination.展开更多
We propose a new method to generate terahertz perfect vortex beam with integer-order and fractional-order. A new optical diffractive element composed of the phase combination of a spherical harmonic axicon and a spira...We propose a new method to generate terahertz perfect vortex beam with integer-order and fractional-order. A new optical diffractive element composed of the phase combination of a spherical harmonic axicon and a spiral phase plate is designed and called spiral spherical harmonic axicon. A terahertz Gaussian beam passes through the spiral spherical harmonic axicon to generate a terahertz vortex beam. When only the topological charge number carried by spiral spherical harmonic axicon increases, the ring radius of terahertz vortex beam increases slightly, so the beam is shaped into a terahertz quasi-perfect vortex beam. Importantly, the terahertz quasi-perfect vortex beam can carry not only integer-order topological charge number but also fractional-order topological charge number. This is the first time that vortex beam and quasi-perfect vortex beam with fractional-order have been successfully realized in terahertz domain and experiment.展开更多
Most GPS positioning errors can be eliminated or removed by the differential technique or the modeling method,but the multipath effect is a special kind of system or gross error,so it is difficult to be simulated or e...Most GPS positioning errors can be eliminated or removed by the differential technique or the modeling method,but the multipath effect is a special kind of system or gross error,so it is difficult to be simulated or eliminated.In order to improve the accuracy of GPS positioning,the single-epoch pseudorange multipath effects at GPS station were calculated,and firstly modeled based on the spherical cap harmonic（SCH）,which is the function of satellite longitude and latitude with the robust method.The accuracy of the kinematic point positioning technique was improved by correcting pseudorange observations with the multipath effect calculated by the SCH model,especially in the elevation direction.The spherical cap harmonic can be used to model the pseudorange multipath effect.展开更多
Nonlinear variations in the coordinate time series of global navigation satellite system(GNSS) reference stations are strongly correlated with surface displacements caused by environmental loading effects,including at...Nonlinear variations in the coordinate time series of global navigation satellite system(GNSS) reference stations are strongly correlated with surface displacements caused by environmental loading effects,including atmospheric, hydrological, and nontidal ocean loading. Continuous improvements in the accuracy of surface mass loading products, performance of Earth models, and precise data-processing technologies have significantly advanced research on the effects of environmental loading on nonlinear variations in GNSS coordinate time series. However, owing to theoretical limitations, the lack of high spatiotemporal resolution surface mass observations, and the coupling of GNSS technology-related systematic errors, environmental loading and nonlinear GNSS reference station displacements remain inconsistent. The applicability and capability of these loading products across different regions also require further evaluation. This paper outlines methods for modeling environmental loading, surface mass loading products, and service organizations. In addition, it summarizes recent advances in applying environmental loading to address nonlinear variations in global and regional GNSS coordinate time series. Moreover, the scientific questions of existing studies are summarized, and insights into future research directions are provided. The complex nonlinear motion of reference stations is a major factor limiting the accuracy of the current terrestrial reference frame. Further refining the environmental load modeling method, establishing a surface mass distribution model with high spatiotemporal resolution and reliability, exploring other environmental load factors such as ice sheet and artificial mass-change effects, and developing an optimal data-processing model and strategy for reprocessing global reference station data consistently could contribute to the development of a millimeter-level nonlinear motion model for GNSS reference stations with actual physical significance and provide theoretical support for establishing a terrestrial reference frame with 1 mm accuracy by 2050.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
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
As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the researc...As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.展开更多
We build a computer program to reconstruct convex bodies using even L_(p)surface area measures for p≥1.Firstly,we transform the minimization problem Pi,which is equivalent to solving the even L_(p)Minkowski problem,i...We build a computer program to reconstruct convex bodies using even L_(p)surface area measures for p≥1.Firstly,we transform the minimization problem Pi,which is equivalent to solving the even L_(p)Minkowski problem,into a convex optimization problem P4 with a finite number of constraints.This transformation makes it suitable for computational resolution.Then,we prove that the approximate solutions obtained by solving the problem P4 converge to the theoretical solution when N and k are sufficiently large.Finally,based on the convex optimization problem P_(4),we provide an algorithm for reconstructing convex bodies from even L_(p)surface area measures,and present several examples implemented using MATLAB.展开更多
Mie theory is a rigorous solution to scattering problems in spherical coordinate system. The first step in applying Mie theory is expansion of some arbitrary incident field in terms of spherical harmonics fields in te...Mie theory is a rigorous solution to scattering problems in spherical coordinate system. The first step in applying Mie theory is expansion of some arbitrary incident field in terms of spherical harmonics fields in terms of spherical which in turn requires evaluation of certain definite integrals whose integrands are products of Bessel functions, associated Legendre functions and periodic functions. Here we present analytical results for two specific integrals that are encountered in expansion of arbitrary fields in terms of summation of spherical waves. The analytical results are in terms of finite summations which include Lommel functions. A concise analytical expression is also derived for the special case of Lommel functions that arise, rendering expensive numerical integration or other iterative techniques unnecessary.展开更多
We used CHAMP satellite vector data and the latest IGRF12 model to investigate the regional magnetic anomalies over China's Mainland. We assumed satellite points on the same surface （307.69 km） and constructed a...We used CHAMP satellite vector data and the latest IGRF12 model to investigate the regional magnetic anomalies over China's Mainland. We assumed satellite points on the same surface （307.69 km） and constructed a spherical cap harmonic model of the satellite magnetic anomalies for elements X, Y, Z, and F over Chinese mainland for 2010.0 （SCH2010） based on selected 498 points. We removed the external field by using the CM4 model. The pole of the spherical cap is 36N° and 104°E, and its half-angle is 30°. After checking and comparing the root mean square （RMS） error of AX, AY, and AZ and X, Y, and Z, we established the truncation level at Kmax = 9. The results suggest that the created China Geomagnetic Referenced Field at the satellite level （CGRF2010） is consistent with the CM4 model. We compared the SCH2010 with other models and found that the intensities and distributions are consistent. In view of the variation off at different altitudes, the SCH2010 model results obey the basics of the geomagnetic field. Moreover, the change rate of X, Y, and Z for SCH2010 and CM4 are consistent. The proposed model can successfully reproduce the geomagnetic data, as other data-fitting models, but the inherent sources of error have to be considered as well.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
In order to study the temporal and spatial variation characteristics of the regional ionosphere and the modeling accuracy,the experiment is based on the spherical harmonic function model,using the GPS,Glonass,and Gali...In order to study the temporal and spatial variation characteristics of the regional ionosphere and the modeling accuracy,the experiment is based on the spherical harmonic function model,using the GPS,Glonass,and Galileo dual-frequency observation data from the 305th-334th day of the European CORS network in 2019 to establish a global ionospheric model.By analyzing and evaluating the accuracy of the global ionospheric puncture points,VTEC,and comparing code products,the test results showed that the GPS system has the most dense puncture electricity distribution,the Glonass system is the second,and the Galileo system is the weakest.The values of ionospheric VTEC calculated by GPS,Glonass and Galileo are slightly different,but in terms of trends,they are the same as those of ESA,JPL and UPC.GPS data has the highest accuracy in global ionospheric modeling.GPS,Glonass and Galileo have the same trend,but Glonass data is unstable and fluctuates greatly.展开更多
Given a continuous boundary value on the boundary of a ＂simply closed surface＂ as that encloses the whole Earth, a regular harmonic fictitious field V＊（P） in the domain outside an inner sphere Ki that lies inside...Given a continuous boundary value on the boundary of a ＂simply closed surface＂ as that encloses the whole Earth, a regular harmonic fictitious field V＊（P） in the domain outside an inner sphere Ki that lies inside the Earth could be determined, and it is proved that V＊（P） coincides with the Earth＇s real field V（P） in the whole domain outside the Earth. Since in the domain outside the inner sphere Ki and the fictitious regular harmonic function V＊（P） could be expressed as a uniformly convergent spherical harmonic series, it is concluded that the Earth＇s potential field could be expressed as a uniformly convergent spherical harmonic expansion series in the whole domain outside the Earth.展开更多
基金supported by the National Natural Science Foundation of China(Grant No.61372046)the Research Fund for the Doctoral Program of Higher Education of China(New Teachers)(Grant No.20116101120018)+6 种基金the China Postdoctoral Science Foundation Funded Project(Grant Nos.2011M501467 and 2012T50814)the Natural Science Basic Research Plan in Shaanxi Province of China(Grant No.2011JQ1006)the Fundamental Research Funds for the Central Universities(Grant No.GK201302007)Science and Technology Plan Program,in Shaanxi Province of China(Grant Nos.2012 KJXX-29 and 2013K12-20-12)the Science and Technology Plan Program in Xian of China(Grant No.CXY1348(2))the.GraduateInovation Project of Northwest University(Grant No.YZZ12093)the Seience and Technology Program of Educational Committee,of Shaanxi Province of China(Grant No.12JK0729).
文摘Recently,the simplified spherical harmonics equations(SP)model has at tracted much att entionin modeling the light propagation in small tissue ggeometriesat visible and near-infrared wave-leng ths.In this paper,we report an eficient numerical method for fluorescence moleeular tom-ography(FMT)that combines the advantage of SP model and adaptive hp finite elementmethod(hp-FEM).For purposes of comparison,hp-FEM and h-FEM are,respectively applied tothe reconstruction pro cess with diffusion approximation and SPs model.Simulation experiments on a 3D digital mouse atlas and physical experiments on a phantom are designed to evaluate thereconstruction methods in terms of the location and the reconstructed fluorescent yield.Theexperimental results demonstrate that hp-FEM with SPy model,yield more accurate results thanh-FEM with difusion approximation model does.The phantom experiments show the potentialand feasibility of the proposed approach in FMT applications.
基金the School of Life Science and Technology of Xidian University for providing experimental data acquisition system.This work was supported by the National Natural Science Foundation of China under Grant(Nos.61372046,61401264,11571012,61601363,61640418,61572400)the Science and Technology Plan Program in Shaanxi Province of China under Grant(Nos.2013K12-20-12,2015KW-002)+2 种基金the Natural Science Research Plan Program in Shaanxi Province of China under Grant(No.2015JM6322)the Scienti¯c Research Founded by Shaanxi Provincial Education Department under Grant No.16JK1772the Scienti¯c Research Foundation of Northwest University under Grant Nos.338050018 and 338020012.
文摘As an emerging molecular imaging modality,cone-beam X-ray luminescence computed tomog-raphy(CB-XLCT)uses X-ray-excitable probes to produce near-infrared(NIR)luminescence and then reconst ructs three-dimensional(3D)distribution of the probes from surface measurements.A proper photon-transportation model is critical to accuracy of XLCT.Here,we presented a systematic comparison between the common-used Monte Carlo model and simplified spherical harmonics(SPN).The performance of the two methods was evaluated over several main spec-trums using a known XLCT material.We designed both a global measurement based on the cosine similarity and a locally-averaged relative error,to quantitatively assess these methods.The results show that the SP_(3) could reach a good balance between the modeling accuracy and computational efficiency for all of the tested emission spectrums.Besides,the SP_(1)(which is equivalent to the difusion equation(DE))can be a reasonable alternative model for emission wavelength over 692nm.In vivo experiment further demonstrates the reconstruction perfor-mance of the SP:and DE.This study would provide a valuable guidance for modeling the photon-transportation in CB-XLCT.
基金Project supported by the National Key Technology Research and Development Program of the Ministry of Science and Technology of China (Grant No.2006BAF06B06)Tsinghua University Initiative Scientific Research Program,China (Grant No.2009THZ06057)
文摘To determine the Avogadro constant with a target relative uncertainty of 2 x 10-s, the uncertainty component of the silicon sphere＇s volume introduced by the spherical harmonics method, which is usually used in determining the sphere＇s volume, is reevaluated. By means of representing the shape of the silicon sphere by an ellipsoid with Gaussian white noise in its diameters, the uncertainty of the current mapping methods based on the spherical harmonics theory can be estimated theoretically. It is evidenced that the uncertainty component attributed to the current mapping method is underestimated. To eliminate this effect as much as possible, the number of mapping points should be increased to more than before. Moreover, a new mapping method is proposed to accomplish the equal-area mapping with large number points on the silicon sphere.
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 （No.10371024）, the Natural Science of Zhejiang Province （No. Y604003） and the Doctor Foundation of Ningbo City （No.2004A620017）.
文摘Let S^1-1,q≥2,be the surface of the unit sphere in the Euclidean space R^1,f（x）∈L^p（S^q-1）,f（x）≥0,f absohutely unegual to 0,1≤p≤＋∞,Then,it is proved in the present paper that there is a spherical harmonics PN（x） of order≤N and a constant C〉0 such that where ω（f,δ）L^p=sup 0〈t≤δ‖St（f）-f‖L^p is a kind of moduli of continuity and ^‖f-1/PN‖L^p≤Cω（f,N^-1）L^p,St（f,μ）=1/｜S^q-2｜Sin^2λt ∫-μμ’=t f（μ＇）dμ＇ is a translation operator.
文摘Computer aided drug design is to develop a chemical that binds to a target macromolecule known to play a key role in a disease state. In recognition of ligands by their protein receptors, molecular surfaces are often used because they represent the interacting part of molecules and they should reflex the complementarity between ligand and receptor. However, assessing the surface complementarity by searching all relative position of two surfaces is often computationally expensive. The complementarity of lobe hole is very important in protein ligand interactions. Spherical harmonic models based on expansions of spherical harmonic functions were used as a fingerprint to approximate the binding cavity and the ligand, respectively. This defines a new way to identify the complementarity between lobes and holes. The advantage of this method is that two spherical harmonic surfaces to be compared can be defined separately. This method can be used as a filter to eliminate candidates among a large number of conformations, and it will speed up the docking procedure. Therefore, it is possible to select complementary ligands or complementary conformations of a ligand and the macromolecules, by comparing their fingerprints previously stored in a database.
基金the National Natural Science Foundation of China (No. 60273005)
文摘An algorithm is presented for estimating the direction and strength of point light with the strength of ambient illumination. Existing approaches evaluate these illumination parameters directly in the high dimensional image space, while we estimate the parameters in two steps： first by projecting the image to an orthogonal linear subspace based on spherical harmonic basis functions and then by calculating the parameters in the low dimensional subspace. The test results using the CMU PIE database and Yale Database B show the stability and effectiveness of the method. The resulting illumination information can be used to synthesize more realistic relighting images and to recognize objects under variable illumination.
基金Project supported by the Fundamental Research Funds for the Central Universities,China (Grant No.2017KFYXJJ029)。
文摘We propose a new method to generate terahertz perfect vortex beam with integer-order and fractional-order. A new optical diffractive element composed of the phase combination of a spherical harmonic axicon and a spiral phase plate is designed and called spiral spherical harmonic axicon. A terahertz Gaussian beam passes through the spiral spherical harmonic axicon to generate a terahertz vortex beam. When only the topological charge number carried by spiral spherical harmonic axicon increases, the ring radius of terahertz vortex beam increases slightly, so the beam is shaped into a terahertz quasi-perfect vortex beam. Importantly, the terahertz quasi-perfect vortex beam can carry not only integer-order topological charge number but also fractional-order topological charge number. This is the first time that vortex beam and quasi-perfect vortex beam with fractional-order have been successfully realized in terahertz domain and experiment.
基金Project (41374009) supported by the National Natural Science Foundation of ChinaProjects (TJES1101,TJES1203) supported by the Key Laboratory of Advanced Engineering Surveying of NASMG,China+1 种基金Project (ZR2013DM009) supported by the Shandong Natural Science Foundation of ChinaProject (201412001) supported by the Public Benefit Scientific Research Project of China
文摘Most GPS positioning errors can be eliminated or removed by the differential technique or the modeling method,but the multipath effect is a special kind of system or gross error,so it is difficult to be simulated or eliminated.In order to improve the accuracy of GPS positioning,the single-epoch pseudorange multipath effects at GPS station were calculated,and firstly modeled based on the spherical cap harmonic（SCH）,which is the function of satellite longitude and latitude with the robust method.The accuracy of the kinematic point positioning technique was improved by correcting pseudorange observations with the multipath effect calculated by the SCH model,especially in the elevation direction.The spherical cap harmonic can be used to model the pseudorange multipath effect.
基金supported by the Basic Science Center Project of the National Natural Science Foundation of China(42388102)the National Natural Science Foundation of China(42174030)+2 种基金the Special Fund of Hubei Luojia Laboratory(220100020)the Major Science and Technology Program for Hubei Province(2022AAA002)the Fundamental Research Funds for the Central Universities of China(2042022dx0001 and 2042023kfyq01)。
文摘Nonlinear variations in the coordinate time series of global navigation satellite system(GNSS) reference stations are strongly correlated with surface displacements caused by environmental loading effects,including atmospheric, hydrological, and nontidal ocean loading. Continuous improvements in the accuracy of surface mass loading products, performance of Earth models, and precise data-processing technologies have significantly advanced research on the effects of environmental loading on nonlinear variations in GNSS coordinate time series. However, owing to theoretical limitations, the lack of high spatiotemporal resolution surface mass observations, and the coupling of GNSS technology-related systematic errors, environmental loading and nonlinear GNSS reference station displacements remain inconsistent. The applicability and capability of these loading products across different regions also require further evaluation. This paper outlines methods for modeling environmental loading, surface mass loading products, and service organizations. In addition, it summarizes recent advances in applying environmental loading to address nonlinear variations in global and regional GNSS coordinate time series. Moreover, the scientific questions of existing studies are summarized, and insights into future research directions are provided. The complex nonlinear motion of reference stations is a major factor limiting the accuracy of the current terrestrial reference frame. Further refining the environmental load modeling method, establishing a surface mass distribution model with high spatiotemporal resolution and reliability, exploring other environmental load factors such as ice sheet and artificial mass-change effects, and developing an optimal data-processing model and strategy for reprocessing global reference station data consistently could contribute to the development of a millimeter-level nonlinear motion model for GNSS reference stations with actual physical significance and provide theoretical support for establishing a terrestrial reference frame with 1 mm accuracy by 2050.
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 NSF of China under the Grant 10471010partially by the NSERC Canada under Grant G121211001
文摘As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.
文摘We build a computer program to reconstruct convex bodies using even L_(p)surface area measures for p≥1.Firstly,we transform the minimization problem Pi,which is equivalent to solving the even L_(p)Minkowski problem,into a convex optimization problem P4 with a finite number of constraints.This transformation makes it suitable for computational resolution.Then,we prove that the approximate solutions obtained by solving the problem P4 converge to the theoretical solution when N and k are sufficiently large.Finally,based on the convex optimization problem P_(4),we provide an algorithm for reconstructing convex bodies from even L_(p)surface area measures,and present several examples implemented using MATLAB.
文摘Mie theory is a rigorous solution to scattering problems in spherical coordinate system. The first step in applying Mie theory is expansion of some arbitrary incident field in terms of spherical harmonics fields in terms of spherical which in turn requires evaluation of certain definite integrals whose integrands are products of Bessel functions, associated Legendre functions and periodic functions. Here we present analytical results for two specific integrals that are encountered in expansion of arbitrary fields in terms of summation of spherical waves. The analytical results are in terms of finite summations which include Lommel functions. A concise analytical expression is also derived for the special case of Lommel functions that arise, rendering expensive numerical integration or other iterative techniques unnecessary.
基金supported by the National Natural Science Foundation of China(No.41404053)Special Project for MeteoScientifi c Research in the Public Interest(No.GYHY201306073)+2 种基金the Natural Science Foundation of Jiangsu Province(No.BK20140994)the Natural Science Foundation of Higher Education Institutions of Jiangsu Province(No.14KJB170012)the Training Program of Innovation and Entrepreneurship for Undergraduates of NUIST(No.201510300178)
文摘We used CHAMP satellite vector data and the latest IGRF12 model to investigate the regional magnetic anomalies over China's Mainland. We assumed satellite points on the same surface （307.69 km） and constructed a spherical cap harmonic model of the satellite magnetic anomalies for elements X, Y, Z, and F over Chinese mainland for 2010.0 （SCH2010） based on selected 498 points. We removed the external field by using the CM4 model. The pole of the spherical cap is 36N° and 104°E, and its half-angle is 30°. After checking and comparing the root mean square （RMS） error of AX, AY, and AZ and X, Y, and Z, we established the truncation level at Kmax = 9. The results suggest that the created China Geomagnetic Referenced Field at the satellite level （CGRF2010） is consistent with the CM4 model. We compared the SCH2010 with other models and found that the intensities and distributions are consistent. In view of the variation off at different altitudes, the SCH2010 model results obey the basics of the geomagnetic field. Moreover, the change rate of X, Y, and Z for SCH2010 and CM4 are consistent. The proposed model can successfully reproduce the geomagnetic data, as other data-fitting models, but the inherent sources of error have to be considered as well.
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Key Research and Development Program of Liaoning Province(2020JH2/10100044)National Natural Science Foundation of China(41904037)National Key Basic Research and Development Program(973 Program)(2016YFC0803102)。
文摘In order to study the temporal and spatial variation characteristics of the regional ionosphere and the modeling accuracy,the experiment is based on the spherical harmonic function model,using the GPS,Glonass,and Galileo dual-frequency observation data from the 305th-334th day of the European CORS network in 2019 to establish a global ionospheric model.By analyzing and evaluating the accuracy of the global ionospheric puncture points,VTEC,and comparing code products,the test results showed that the GPS system has the most dense puncture electricity distribution,the Glonass system is the second,and the Galileo system is the weakest.The values of ionospheric VTEC calculated by GPS,Glonass and Galileo are slightly different,but in terms of trends,they are the same as those of ESA,JPL and UPC.GPS data has the highest accuracy in global ionospheric modeling.GPS,Glonass and Galileo have the same trend,but Glonass data is unstable and fluctuates greatly.
基金Supported by the National Natural Science Foundation of China （No.40637034, No. 40574004）, the National 863 Program of China （No. 2006AA12Z211）. The author thanks Prof. Dr. Sjoberg for his valuable comments on the original manuscript.
文摘Given a continuous boundary value on the boundary of a ＂simply closed surface＂ as that encloses the whole Earth, a regular harmonic fictitious field V＊（P） in the domain outside an inner sphere Ki that lies inside the Earth could be determined, and it is proved that V＊（P） coincides with the Earth＇s real field V（P） in the whole domain outside the Earth. Since in the domain outside the inner sphere Ki and the fictitious regular harmonic function V＊（P） could be expressed as a uniformly convergent spherical harmonic series, it is concluded that the Earth＇s potential field could be expressed as a uniformly convergent spherical harmonic expansion series in the whole domain outside the Earth.
