Spherical harmonic analysis of faceted spheroids identifies shaping strategies and standardisation at Qianshangying (North China)

Models of the geomagnetic field are mathematical expressions able to represent the Earth's magnetic field space and time variations. Time variations on the long term basis are represented by the so-called secular variation. This paper describes and reviews recent activities of the Italian group at the Istituto Nazionale di Geofisica in regional magnetic field modelling. The models are introduced starting from the classical technique of Spherical Harmonic Analysis (SHA) undertaken for the first time by Gauss, the polynomial analysis and the regional harmonic analysis, specifically introduced as a regional analogue of SHA. In this last group the recent techniques of Spherical Cap Harmonic Analysis (SCHA), Translated Origin Spherical Cap Analysis (TOSCA) and Adjusted Spherical Cap Harmonic Analysis (ASHA) are also described and discussed.

Spherical Harmonic Analysis (SHA) is normally used to model the three‐dimensional global geomagnetic field. To address the same problem in regional modelling, Haines (1985) proposed Spherical Cap Harmonic Analysis (SCHA). This regional technique involves the computation of more complex Legendre functions with real (generally non‐integer) harmonic degree. Here a new more practical technique is described; it is called Adjusted Spherical Harmonic Analysis (ASHA) because it is based on the expansion of conventional spherical harmonics after the colatitude interval is adjusted to that of a hemisphere. This kind of analysis can also be applied to modelling general two‐dimensional functions.

There is the need for robust alternatives to the widely used spherical harmonic analysis when measurements are restricted to a region, or when high spatial frequency fields with much less parameters are required. Spherical cap harmonic analysis (SCHA) is one of the preferred alternative regional modelling techniques over the last decades. This paper presents a comprehensive and systematic review of the SCHA literature, underlining the respective merits and weaknesses of the ways in which the technique has been used since it was proposed in the context of geomagnetic field modelling. It reflects the multidisciplinary use of this technique and examines the evidences presented mainly in Earth and planetary science journals. Some bibliometric parameters are provided to understand how the technique and the knowledge of its limitations have progressed and improved, and some avenues for future research are highlighted.

Spherical harmonic analysis of the earth's magnetic field is limited in the resolution that can be obtained. This limitation is caused by inadequacies of computers and of available data sets. The fundamental wavelength in spherical harmonic analysis is the circumference of the earth. To resolve wavelengths as short as 100 km would require a spherical harmonic analysis carried out to a degree and order 400 involving 160,800 coefficients. This is impractical even with modern computers. This limitation of spherical harmonic analysis can be overcome by using rectangular harmonic analysis in successively smaller areas so that the data are more fully utilized. Rectangular harmonic analysis is illustrated for data from Europe and then again for a subset of the data from a smaller area of Europe. The magnetic field at 15 observatories for the smaller area can be computed to an rms residual of only 7 nT for all three components using two sets of rectangular harmonic coefficients and the AWC/75 world chart model. Rectangular harmonic analysis and spherical harmonic analysis are complementary.

Sexual dimorphism, a widespread phenomenon, has been extensively researched in extant and fossil crustaceans. However, identifying sexual dimorphism in phyllocarid fossils preserved as isolated parts is often challenging, except in cases where the specimens are exceptionally well preserved, including those with soft tissues. This study proposes a novel approach by introducing the use of geometric morphometric techniques to identify sexual dimorphism in phyllocarid fossils based on carapace morphology. It presents a comprehensive re-analysis of Soomicaris ordosensis Liu et al., 2023a, carapaces from the Upper Ordovician in North China and Tarim Plates. Elliptic Fourier analysis was applied to quantify the size and shape variation in nearly 100 specimens. The results demonstrate the presence of significant sexual dimorphism in the length and shape of the S. ordosensis carapace. The carapace shape exhibited variation between the sexes: the posterodorsal margin of one group of carapaces gradually extends backward to form a posterodorsal spine; the carapaces of the other group have a convex posterior margin and lack a posterodorsal spine. Additionally, both types manifest an overall allometric growth pattern, albeit with distinct growth coefficients. Furthermore, the observed approximately 1:1 ratio between the two forms suggests that the population of S. ordosensis may have exhibited a dioecious mating system. Geometric morphometrics are a highly effective method for elucidating the subtle variations in the carapace morphology of S. ordosensis , thereby underscoring the cryptic dimorphism characteristics of fossil animals. This finding offers the first indirect evidence for egg-brooding behavior within the extinct order Archaeostraca.

Ionospheric mapping by regional spherical harmonic analysis: New developments

Ionospheric currents are driven by several different physical processes and exhibit complex spatial and temporal structure. Magnetic field measurements of ionospheric sources are often spatially sparse, causing significant challenges in visualizing current flow at a specific time. Standard methods of fitting equivalent current models to magnetic observations, such as line currents, spherical harmonic analysis, spherical cap harmonic analysis, and spherical elementary current systems (SECS), are often unable to capture the full spatial complexity of the currents or require a large number of parameters which cannot be fully determined by the available data coverage. These methods rely on a set of generic basis functions which contain limited information about the geometries of the various ionospheric sources. In this study, we develop new basis functions for fitting ground and satellite measurements, which are derived from physics‐based ionospheric modeling combined with principal component analysis (PCA). The physics‐based modeling provides realistic current flow patterns for all of the primary ionospheric sources, including their daily and seasonal variability. The PCA technique extracts the most relevant spatial geometries of the currents from the model run into a small set of equivalent current modes. We fit these modes to magnetic measurements of the Swarm satellite mission at low and middle latitudes and compare the resulting model with independent measurements and with the SECS approach. We find that our PCA method accurately reproduces features of the equatorial electrojet and Sq current systems with only 10 modes and can predict ionospheric fields far from the data region.

The microscopic characteristics of skeletal particles in rock and soil media have important effects on macroscopic mechanical properties. A mathematical procedure called spherical harmonic function analysis was here developed to characterize micromorphology of particles and determine the meso effects in a discrete manner. This method has strong mathematical properties with respect to orthogonality and rotating invariance. It was used here to characterize and reconstruct particle micromorphology in three-dimensional space. The applicability and accuracy of the method were assessed through comparison of basic geometric properties such as volume and surface area. The results show that the micromorphological characteristics of reproduced particles become more and more readily distinguishable as the reproduced order number of spherical harmonic function increases, and the error can be brought below 5% when the order number reaches 10. This level of precision is sharp enough to distinguish the characteristics of real particles. Reconstructed particles of the same size but different reconstructed orders were used to form cylindrical samples, and the stress-strain curves of these samples filled with different-order particles which have their mutual morphological features were compared using PFC3D. Results show that the higher the spherical harmonic order of reconstructed particles, the lower the initial compression modulus and the larger the strain at peak intensity. However, peak strength shows only a random relationship to spherical harmonic order. Microstructure reconstruction was here shown to be an efficient means of numerically simulating of multi-scale rock and soil media and studying the mechanical properties of soil samples.

Initial SCHA.DI.00 regional archaeomagnetic model for Europe for the last 2000 years

The results of a spherical harmonic analysis and a sector spherical harmonic analysis of the solar magnetic field on the photosphere, source surface, and in the Earth’s orbit on July 10–20, 2004, were compared. It was found that the field values according to a sector harmonic analysis are an order of magnitude as large as the same values according to a spherical harmonic analysis and differ in the configuration. A twocomponent magnetic field structure was revealed: short-range sources are better described by a sector spherical harmonic analysis; long-range sources are better described by a spherical harmonic analysis. This is caused by the different depths of the occurrence of sources below the photosphere.

Chimpanzees, bonobos (Pan), and gorillas (Gorilla) are the only extant knuckle-walking primates. Two hypotheses have been proposed to explain the evolution of knuckle-walking in Pan and Gorilla. The knuckle-walking hypothesis holds that the last common ancestor of Pan, Gorilla, and Homo knuckle-walked. The alternative hypothesis holds that Pan and Gorilla evolved knuckle-walking independently. One unresolved issue is that, despite sharing knuckle-walking as a common locomotor mode, Pan and Gorilla exhibit different carpal morphologies. This morphological variation has been used to support the theory that Pan and Gorilla evolved knuckle-walking independently. To date, however, little work has focused on the effect of body mass on this functional complex. Here, we compare shape changes in the carpals of Pan and Gorilla to those of other quadrupedal mammals with similar body size differences. If morphological trends in the carpus of Pan and Gorilla mirror those of other mammals with similar body mass variation, then carpal variation in African apes may be attributed to differences in body mass, rather than the independent evolution of knuckle-walking. More specifically, we hypothesize that the carpals of high body mass taxa will be relatively proximo-distally shorter and medio-laterally/antero-posteriorly wider than those of low body mass taxa. The capitate, hamate, and triquetrum of high and low body mass taxa from the mammalian families Hominidae, Cercopithecidae, Bovidae, Canidae, and Myrmecophagidae were surface scanned. A weighted spherical harmonic (SPHARM) analysis was used to quantify the shape of each carpal. During the SPHARM analysis, surface models were mapped onto the surface of a sphere, and then decomposed into a weighted sum of spherical harmonic functions. Shape variation of each carpal (within each mammalian family) was summarized using a principal component analysis (PCA) of coefficients from spherical harmonic functions. The log-centroid size of each surface model was added to each PCA to determine the effect of size on shape variation. Shape changes were visualized by comparing surface models of the average specimen from species with the highest and lowest body mass within each family. Species separate by body mass in all PC plots, and comparison of average specimen surface models indicates that across all mammalian families included in the analysis, the capitate, hamate, and triquetrum exhibit similar shape changes between high and low body mass taxa (i.e., the carpals of high body mass taxa are proximo-distally shorter and medio-laterally wider than those of low body mass taxa). These distinctions are likely caused by the need to accommodate the higher forelimb loading associated with an increase in body mass. Furthermore, because these patterns of shape change are seen in the carpals of multiple mammalian families, morphological variation in the carpals of chimpanzees and gorillas is more likely due to differences in body size than the independent evolution of knuckle-walking.

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Winch D. E. 1981Spherical harmonic analysis of geomagnetic tides, 1964-1965Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences3031–104http://doi.org/10.1098/rsta.1981.0193SectionRestricted accessArticleSpherical harmonic analysis of geomagnetic tides, 1964-1965 D. E. Winch Google Scholar Find this author on PubMed Search for more papers by this author D. E. Winch Google Scholar Find this author on PubMed Search for more papers by this author Published:22 October 1981https://doi.org/10.1098/rsta.1981.0193AbstractThe mathematical forms chosen for analyses of Sq and L by seventeen different authors are reviewed. On the basis of the review and on consideration of the ionospheric dynamo theory and the Hough function structure of wind velocities in the upper atmosphere, a mathematical model is chosen that includes two more terms than the recent analysis by Malin (1973). Hourly mean values from 130 magnetic observatories for the I.Q.S.Ys (International Quiet Solar Years) 1964 and 1965 were prepared in machine-readable form and analysed. The solar and lunar transient magnetic variations were evaluated, together with the lunar elliptic magnetic tide and the seasonal change at one and two cycles per year for all magnetic tides. Spherical harmonic analyses were made of the phase-law tides and the smaller partial tides. Altogether 94 different spherical harmonic analyses were completed for a total of ten different magnetic tides. Two of the ten tides are the solar and lunar magnetic tides usually denoted S and L respectively. The results are tabulated in a form that distinguishes between eastward- and westward-moving terms, which is suitable for the evaluation of phase angle differences and amplitude ratios between internal and external parts. Malin’s ocean dynamo calculation has been applied to the lunar and lunar elliptic magnetic tides. The spherical harmonic coefficients are compared in some detail with those of Malin (1973) for years of high sunspot activity. The magnetic tidal potential associated with a given atmospheric tidal mode is derived theoretically and used to obtain an estimate of the Hough function components of the solar and lunar semi-diurnal atmospheric tides at ionospheric levels from the corresponding magnetic tidal potential.FootnotesThis text was harvested from a scanned image of the original document using optical character recognition (OCR) software. As such, it may contain errors. Please contact the Royal Society if you find an error you would like to see corrected. Mathematical notations produced through Infty OCR. VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetails This Issue22 October 1981Volume 303Issue 1473 Article InformationDOI:https://doi.org/10.1098/rsta.1981.0193Published by:Royal SocietyPrint ISSN:0080-4614Online ISSN:2054-0272History: Manuscript received07/03/1980Published online01/01/1997Published in print22/10/1981 License:Scanned images copyright © 2017, Royal Society Citations and impact

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.

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.

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.