Spherical Harmonic Analysis Research Articles

Modeling geomagnetic spikes: the Levantine Iron Age anomaly

The Levantine Iron Age anomaly (LIAA) is a regional short-decadal geomagnetic strength field variation located at the Levantine region characterized by high intensities with maximum virtual axial dipole moments around 190 ZAm2. It has been constrained by archeomagnetic data coming from Eastern Europe and Western Asia between 1050 and 700 BC. The LIAA can be related to a fast and spatially localized geomagnetic positive anomaly (spike) at the Earth’s surface. In this study, we model the LIAA by using a Fisher–von Mises function that fits the most recent archeomagnetic intensity database in the region. A spherical harmonic analysis is implemented for this spike function to perturb a base model in order to build a global reconstruction (perturbed-model) that reproduces the spatial and temporal characteristics of the LIAA. Our results show the importance of harmonic degrees from n = 3–4 to n = 20 to reconstruct the anomaly extension suggested by the database. Two maxima linked with the LIAA are reproduced by our global perturbed-model at the Levantine region at 950 BC and 750 BC. A third maxima in intensity around 500 BC is also observed, affecting the whole Europe.Graphical

A grand spectrum of the geomagnetic field

The spectrum of geomagnetic variations is broad and reflects multiple internal and external physical Earth processes. The geodynamo generates an internal magnetic field, dominated by the low frequency contribution of the axial dipole whose temporal variations range from geomagnetic reversals, excursions, and large-scale paleosecular variations, down to decadal and sub-annual changes. Variations in the external field arise from high frequency interactions of the solar wind with Earth’s magnetosphere and ionosphere, along with excitation within the atmospheric cavity by lightning, power systems, and radio transmissions. External variations induce a field in the conductive Earth that adds to the internal signal. A spherical harmonic analysis of dipole terms in over 100 years of observatory data allows us to show that the external field is stronger than the internal field at periods of the 11-year sunspot cycle and shorter. Using spectral estimates derived from this and other data sets by adaptive multi-spectral time series analysis, we can create a composite power spectrum that spans frequencies from 10-15 Hz to 20 kHz (periods ranging from 10 million years to 5 × 10-5 s), and powers ranging from 10-9 (nT)2/Hz to 1021 (nT)2/Hz. The different processes contributing to the spectrum are characterized by various inverse power laws across frequency bands. This Grand Spectrum quantifies the successive dominance of the many different geophysical processes with frequency, but importantly provides a compelling graphic to illustrate the complexity of the geomagnetic field.

Regional geomagnetic core field and secular variation model over the Iberian Peninsula from 2014 to 2020 based on the R-SCHA technique

The Earth’s magnetic field originated in the fluid core, the so-called core field, is the dominant contribution to the geomagnetic field. Since ancient times, the core geomagnetic field has been used primarily for geographical orientation and navigation by means of compasses. Nowadays, thanks to the large amount of geomagnetic data available, core field models can be developed on a global or regional scale. Global models resolve large-scale geomagnetic field features, while regional models can resolve greater detail over a particular region. The spherical harmonic cap analysis is a widely used technique for regional-scale modelling of the geomagnetic field. In this work we have developed a regional model of the core field and its secular variation between 2014.5 and 2020.5 over the Iberian Peninsula, based on data from Swarm satellites, geomagnetic observatories and repeat stations. Its performance has been validated by comparing the fit to the available geomagnetic data using the regional model and the global models IGRF and CHAOS over the whole spatio-temporal range studied. In order to optimise the model, a detailed study of its input parameters has been carried out, showing that not all parameters have an equal influence on the modelling. This new model reproduces the input data with a root mean square error of 2.9 nT, improving the outcome of global models on this region. The results of this work will allow the Spanish Instituto Geográfico Nacional to produce the magnetic cartography of Iberia and the Balearic Islands in 2020.0, which for the first time will be based on a regional core field model, replacing the polynomial variation method used in the past.Graphical abstract

Tensor spherical harmonics analysis of electro-elastostatic fields in a spherically isotropic multiphase functionally graded piezoelectric medium

This paper is devoted to the study of the electro-elastostatic fields of a multiphase concentric spherical piezoelectric ensemble under general electromechanical loading in the presence or absence of body forces and body electric charges. Each phase is assumed to be spherically isotropic and at each point the poling direction is radially oriented. Thanks to the robustness of spherical harmonics, in this work we will obtain the exact solutions for the three-dimensional displacement and electric potential when the elastic, piezoelectric and dielectric constants of each phase vary as power-law functions of the radial distance.

Morphological variation of the Pan talus relative to that of Gorilla.

Differences in talar articular morphology relative to locomotion have recently been found within Pan and Gorilla. Whole-bone talar morphology within, and shared variation among, Pan and Gorilla (sub)species, however, has yet to be investigated. Here we separately analyze talar external shape within Pan (P. t. troglodytes, P. t. schweinfurthii, P. t. verus, P. paniscus) and Gorilla (G. g. gorilla, G. b. beringei, G. b. graueri) relative to degree of arboreality and body size. Pan and Gorilla are additionally analyzed together to determine if consistent shape differences exist within the genera. Talar external shape was quantified using a weighted spherical harmonic analysis. Shape variation both within and among Pan and Gorilla was described using principal component analyses. Root mean square distances were calculated between taxon averages, and resampling statistics conducted to test for pairwise differences. P. t. verus (most arboreal Pan) talar shape significantly differs from other Pan taxa (p < 0.05 for pairwise comparisons) driven by more asymmetrical trochlear rims and a medially-set talar head. P. t. troglodytes, P. t. schweinfurthii, and P. paniscus do not significantly differ (p > 0.05 for pairwise comparisons). All gorilla taxa exhibit significantly different talar morphologies (p < 0.007 for pairwise comparisons). The more terrestrial subspecies of G. beringei and P. troglodytes exhibit a superoinferiorly taller talar head/neck complex. P. t. verus exhibits talar morphologies that have been previously related to more frequent arboreality. The adaptations in the more terrestrial G. beringei and P. troglodytes subspecies may serve to facilitate load transmission.

Antarctic Time-Variable Regional Gravity Field Model Derived from Satellite Line-of-Sight Gravity Differences and Spherical Cap Harmonic Analysis

This study focuses on the development of a time-variable regional geo-potential model for Antarctica using the spherical cap harmonic analysis (SCHA) basis functions. The model is derived from line-of-sight gravity difference (LGD) measurements obtained from the GRACE-Follow-On (GFO) mission. The solution of a Laplace equation for the boundary values over a spherical cap is used to expand the geo-potential coefficients in terms of Legendre functions with a real degree and integer order suitable for regional modelling, which is used to constrain the geo-potential coefficients using LGD measurements. To validate the performance of the SCHA, it is first utilized with LGD data derived from a L2 JPL (Level 2 product of the Jet Propulsion Laboratory). The obtained LGD data are used to compute the local geo-potential model up to Kmax = 20, corresponding to the SH degree and order up to 60. The comparison of the radial gravity on the Earth’s surface map across Antarctica with the corresponding radial gravity components of the L2 JPL is carried out using local geo-potential coefficients. The results of this comparison provide evidence that these basis functions for Kmax = 20 are valid across the entirety of Antarctica. Subsequently, the analysis proceeds using LGD data obtained from the Level 1B product of GFO by transforming these LGD data into the SCHA coordinate system and applying them to constrain the SCHA harmonic coefficients up to Kmax = 20. In this case, several independent LGD profiles along the trajectories of the satellites are devised to verify the accuracy of the local model. These LGD profiles are not employed in the inverse problem of determining harmonic coefficients. The results indicate that using regional harmonic basis functions, specifically spherical cap harmonic analysis (SCHA) functions, leads to a close estimation of LGD compared to the L2 JPL. The regional harmonic basis function exhibits a root mean square error (RMSE) of 3.71 × 10−4 mGal. This represents a substantial improvement over the RMSE of the L2 JPL, which is 6.36 × 10−4 mGal. Thus, it can be concluded that the use of local geo-potential coefficients obtained from SCHA is a reliable method for extracting nearly the full gravitational signal within a spherical cap region, after validation of this method. The SCHA model provides significant realistic information as it addresses the mass gain and loss across various regions in Antarctica.

A global lithospheric magnetic field model between ± 65° latitude derived from CSES satellite scalar data

The China Seismo-Electromagnetic Satellite (CSES) was launched successfully in February 2018. It is China's first satellite to measure geophysical fields with scientific goals in both space and solid earth physics. In this work, we used CSES scalar magnetic data to derive a global lithospheric magnetic field model between ±65° geographic latitudes. The nightside data from March 2018 to November 2022 under quiet space weather conditions were selected. Then, the core and external fields were removed with the CHAOS-7 model. After further data quality control, the data were used to build a lithospheric magnetic field model using a spherical harmonic analysis. The obtained CSES model was compared with the CHAOS-7, CM6, and MF7 models in terms of power spectra and anomaly details, which confirmed that the CSES scalar data had good quality and could provide a reliable lithospheric magnetic field model up to degree 42.

Reconstruction of irregular elongated/flattened particles and generation of particle aggregates with customizable form distributions

In this study, we introduced a new method to address the limitations of the traditional spherical harmonic (SH) analysis when reconstructing the morphology of elongated/flattened particles. The proposed method combines SH analysis with a stretching algorithm (SH-stretching) to enable the reconstruction of irregular particles with arbitrary values of elongation index (EI) and flatness index (FI). Based on reconstruction of real cobble particles, the SH-stretching algorithm demonstrates superior performance in reconstructing elongated/flattened particles, and can ensure convergence. The SH-stretching algorithm is further integrated with principal component analysis (PCA) and coefficient standardization to develop a particle generation technique. This technique can strictly control the EI and FI distributions of the generated particles and is verified to be efficient in generating virtual cobble particles with realistic shape characteristics. Furthermore, the proposed technique allows the generation of particle aggregates with specified wide range distributions of EI and FI, while preserving their polydisperse characteristics.

The Study of the Lithospheric Magnetic Field over Xinjiang and Tibet Areas Based on Ground, Airborne, and Satellite Data

Combined with the ground, airborne, and CHAMP satellite data, the lithospheric field over Xinjiang and Tibet is modeled through the three-dimensional Surface Spline (3DSS) model, Regional Spherical Harmonic Analysis (RSHA) model, and CHAOS-7.11 model. Then, we compare the results with the original measuring data, NGDC720, LCS-1, and the newest SHA model with the degree to 1000 (SHA1000). Moreover, the error estimation and the geological analysis are carried out, and we investigate the possible correspondence between the lithospheric field and the surface heat flow. The results show that the 3DSS model can better describe the detailed distribution of the lithospheric field after comparing it with other models. Some new features are reflected, particularly in the areas of Southern Xinjiang and Tibet, such as a positive anomaly stripe in the southwest, its neighboring Tashkurgan–Hotan–Cele–Minfeng–Qiemo–Ruoqiang belt, and the middle edge of the Kunlun Mountains. The stripe, in terms of rock composition, has a shallow magnetic field source and is related to magnetic intrusions; the lithospheric field in Tibet is weak. Additionally, when the heat flow distribution is compared to our results, there is a good consistency between a positive stripe of heat flow and a positive stripe of the lithospheric field in southern Tibet. The large heat flow values may be related to the shallow Curie surface, which shows that demagnetization is happening close to the surface. However, more of a ferromagnetic mineral, Titanium magnetite, is found there.

Ionospheric Currents in the Equatorial and Low Latitudes of Africa

The magnetometer data obtained for 2008 from geomagnetic stations installed across Africa by magnetic data acquisition set (MAGDAS) have been used to study the ionospheric Sq current system in the equatorial and lowlatitudes of Africa. The aim of this work is to separate the quiet-day feld variations obtained in the equatorial and low latitude regions of Africa into their external and internal feld contributions and then to use the paired external and internal coeffcients of the SHAto determine the source current and induced currents. The method used involved a spherical harmonic analysis (SHA). This was applied in the separation of the internal and external field/current contribution to the Sq variations. The result shows that the variation in the currents is seen to be a dawn-to-dusk phenomenon with the variation in the external currents different from that of the internal currents both in amplitude and in phase. Furthermore, the seasonal variation in the external current maximizes during the March equinox and minimizes during the December solstice. The maximum current observed in AAB and ILR is due to the Equatorial Electrojet Current present in the AAB and ILR stations. Seasonal variation was observed in the geomagnetic component variations as well as in the currents. This is attributed to the position of the sun with respect to the earth at different months of the year. The equinoctial maximum is observed in external current intensity which occurred mostly during the March Equinox.

A spherical harmonic-random field coupled method for efficient reconstruction of CT-image based 3D aggregates with controllable multiscale morphology

The physical and mechanical performance of concrete is inherently dependent on aggregate morphology including the general shape at coarse-scale, the local roundness at medium-scale and the surface texture at fine-scale. This study develops a computational method for highly efficient generation of realistic 3D aggregates using micro X-ray Computed Tomography (μXCT) images, the spherical harmonic (SH) analysis and a random-field reconstruction algorithm. In this method, the real aggregate surface segmented from CT images is first decomposed and mapped by the SH function to radius fields at different scales, whose statistical data are then extracted to reconstruct new aggregates through a spectral representation method and an innovative folding algorithm. The proposed method is verified by comparison of four morphology indices (sphericity, convexity, roundness and roughness) of the real aggregate and the new ones, and its flexibility in reconstructing new sets of random aggregates with specified morphology indices at selected scales is also demonstrated. As an application, eight sets of Fuller-grade random aggregates with target morphology indices are constructed and packed into a cube container to generate digital concrete specimens. The aggregate sphericity and convexity are found to be linearly correlated with the maximum aggregate volume fraction that can be packed into the cube. The developed method can be also applied to other granular materials, such as pharmaceutical particles, colloids, ceramics, soils and coal.

Extensibility of External Magnetic Potential at High Latitudes - Antarctica

We investigated the external magnetic potential due to solar forcing, with nine years of data during 2001-2009, coveringthe deep solar minimum (2006-2009), at two stations: one is in the polar cap -Vostok (78º27'S, 106º52’E; mag. lat 83oS) andanother is in the subauroral region - Maitri (70º45'S, 11º43'E: mag. lat 67oS) in Antarctica. The significance of the work isassociated with space weather prediction and its impact on planet Earth. We used Advance Composition Explorer (ACE)satellite data for the aforesaid period for a thorough understanding of influences due to solar wind origin and to compare theparameter observed in these regions. We used the spherical cap harmonic analysis (SCHA) function as a tool. The inferenceindicates that at Vostok the magnitude is enhanced throughout and depicts a broad ambient external magnetic potential. Itseems to be essentially the intensification of the region 1 currents whereas at Maitri intense electric fields are producedduring geomagnetic perturbations which drive a system of disturbed time Region 2 currents over the quiet time currents.During this scenario in Maitri there are noticeable peaks or enhancements in the magnetic potential that can be observedmainly during geomagnetic disturbances. Hence the regression relation developed for external magnetic potentialcalculation, in terms of solar wind parameters agrees well with polar cap region and the area is relatively less exploredearlier, the present investigation can be expected to add knowledge about that regime.

A Novel Model Integrating the Spherical Cap Harmonic Analysis With the XGBoost Algorithm to Improve the MODIS NIR PWV

A Novel Model Integrating the Spherical Cap Harmonic Analysis With the XGBoost Algorithm to Improve the MODIS NIR PWV

Investigating the feasibility of an impact-induced Martian Dichotomy

A giant impact is commonly thought to explain the dramatic contrast in elevation and crustal thickness between the two hemispheres of Mars known as the “Martian Dichotomy”. Initially, this scenario referred to an impact in the northern hemisphere that would lead to a huge impact basin (dubbed the “Borealis Basin”), while more recent work has instead suggested a hybrid origin that produces the Dichotomy through impact-induced crust-production. The majority of these studies have relied upon impact scaling-laws inaccurate at such large-scales, however, and those that have included realistic impact models have utilised over-simplified geophysical models and neglected any material strength. Here we use a large suite of strength-including smoothed-particle hydrodynamics (SPH) impact simulations coupled with a more sophisticated geophysical scheme of crust production and primordial crust to simultaneously investigate the feasibility of a giant impact on either hemisphere of Mars to have produced its dichotomous crust distribution, and utilise spherical harmonic analysis to identify the best-fitting cases. We find that the canonical Borealis-forming impact is not possible without both excessive crust production and strong antipodal effects not seen on Mars’ southern hemisphere today. Our results instead favour an impact and subsequent localised magma ocean in the southern hemisphere that results in a thicker crust than the north upon crystallisation. Specifically, our best-fitting cases suggest that the projectile responsible for the Dichotomy-forming event was of radius 500–750 km, and collided with Mars at an impact angle of 15–30° with a velocity of 1.2–1.4 times mutual escape speed (∼6–7 km/s).

The l = 2 spherical harmonic expansion coefficients of the sky brightness distribution between 0.5 and 7 MHz

Context. The opacity of the ionosphere prevents comprehensive Earth-based surveys of low frequency ν ≲ 10 MHz astrophysical radio emissions. The limited available data in this frequency regime show a downturn in the mean sky brightness at ν ≲ 3 MHz in a divergence from the synchrotron emission power-law that is observed at higher frequencies. The turning over of the spectrum coincides with a shift in the region of maximum brightness from the Galactic plane to the poles. This implicates free-free absorption by interstellar ionized gas, whose concentration in the plane causes radiation that propagates in this region to suffer stronger absorption than radiation from the poles. Aims. Using observations from Parker Solar Probe (PSP), we evaluate the l = 0 and l = 2 spherical harmonic expansion coefficients of the radio brightness distribution at 56 frequencies between 0.5 and 7 MHz. These data quantify free-free absorption’s global effects on the brightness distribution, which provides new constraints on the distribution of free electrons in the Galaxy. Methods. The auto and cross spectra of the voltages induced on crossed short dipole antennas by radiation from a nonpolarized extended brightness distribution are linear combinations of the distribution’s l = 0 and l = 2 expansion coefficients. We extracted the least squares solution to these coefficients from PSP’s measurements of the radio background. Also, we generated hypothetical low frequency brightness maps that incorporated free-free absorption and tested their compatibility with the data. The maps primarily depended on models of the Galactic emissivity and distribution of free electrons. A comparison of the maps’ expansion coefficients with the empirical coefficients provided an indication of these input models’ accuracies. Results. An average reduced x2 ≈ 1.04 of the spherical harmonic analysis between 0.5 and 7 MHz indicates that PSP’s antennas act approximately as ideal short dipoles in this frequency band. The best-fit expansion coefficients show that, with decreasing frequency, the mean sky brightness decreases at ν &lt; 3 MHz and the Galactic plane darkens relative to the poles. At ν &gt; 0.6 MHz, these observations can be reproduced in synthetic brightness maps in which the Galactic emissivity maintains a power-law form and free-free absorption is modeled using free electron distributions derived from pulsar measurements. At lower frequencies, the empirical mean brightness falls below the mean in this model, possibly signifying a cutoff in the synchrotron power-law.

Equivalent Current Systems of Quiet Ionosphere during the 24th Solar Cycle Derived from the Geomagnetic Records in China

External and internal equivalent current systems of solar quiet (Sq) were obtained using the spherical harmonic analysis (SHA) method with a “mirror” technique based on geomagnetic records from 46 stations in China during 2008–2019. It is the first attempt to investigate Sq currents using so many stations in the China region for a long period. On the basis of criterion Kp ≤ 2+, geomagnetic vector data were selected to represent monthly Sq variations. After calculating the equivalent currents for each month and each Lloyd season, Sq variation was analyzed in relation to the solar cycle and season. The intensities of both external and internal Sq currents were found consistent with solar activity for the same month or season, while the positions of the current foci were evidently unaffected by solar activity. The intensities of Sq currents also exhibited primary semiannual (annual) variation in the periods of high (low) solar activity. The latitude of the internal current vortex showed evident seasonal variation in Lloyd seasons with high (low) values in the D (J) season, while the external current vortex exhibited no obvious seasonal variation. The strongest correlation between external and internal foci was found in D season, and the internal current foci usually appeared 20–40 min earlier than the external ones. Owing to the complex mechanisms behind Sq variation, some findings will need further analysis in the future.

DEM modeling of mini-triaxial test on soil-rock mixture considering particle shape effect

Soil-rock mixture (SRM) that is seriously affected by multiple factors such as particle morphology, grading, and composition has been widely used in geotechnical engineering. In this study, a series of discrete element method (DEM) simulations of mini-triaxial tests on SRM were carried out considering the particle shape effect. First, micro-computed tomography, advanced image processing, and spherical harmonics analysis techniques were successively used to obtain the natural particle shapes of grains which were used to build the particle shape template by the clumping algorithm in DEM. Then, a flexible membrane with constant confining pressure loading was made allowing free deformation of the SRM specimens. After that, a set of micro-parameters involved in the DEM model were calibrated by the uniaxial compression tests, and 9 DEM models of mini-triaxial tests were established to investigate the effects of particle shape, size (gradation), and the percentage of irregularly shaped particles on the deformation and failure characteristics of SRM under shearing. Lastly, the mechanical behaviors of SRM, e.g., stress–strain, macroscopic deformation, and cracks generation and evolutions, were comprehensively analyzed. The simulation results provide new insights into the discrete element study of SRM and contribute to the development of soil stabilization materials for geotechnical engineering.

The employment of quasi-hexagonal grids in spherical harmonic analysis and synthesis for the earth's gravity field

The employment of quasi-hexagonal grids in spherical harmonic analysis and synthesis for the earth's gravity field

Bouguer and mantle gravity maps of Nigeria

We used global gravitational models together with topographic, bathymetric, and lithospheric density structure datasets to compile Bouguer and mantle gravity maps of Nigeria, and then used these gravity maps to interpret the crustal and upper mantle structure in this country. We applied methods for a spherical harmonic analysis and synthesis of gravitational and lithospheric density structure models to produce both gravity maps. Whereas existing Bouguer gravity datasets for different parts of Nigeria were computed by applying only a simple planar Bouguer gravity reduction and a planar terrain gravity correction, our approach took into consideration the Earth’s spherical approximation and the spherical terrain correction, the ocean density contrast as well as density variations within sediments and the underlying consolidated crust. This study also provides the first mantle gravity map of Nigeria. Our results show that the free-air gravity pattern reflects mainly a topographic relief with prevailing positive values over topographically elevated regions, while Cretaceous sedimentary basins are characterized by low negative values. The pattern in the Bouguer gravity map exhibits mainly a Moho signature, with low negative values associated with a regional Moho deepening beneath the Sokoto Basin and the North Shield Belt. The pattern in the mantle gravity map exhibits a thermal signature of the lithospheric mantle. The high values (≥ 1000 mGal) of the mantle gravity disturbances detected over the Benin-Nigerian Block and the Phanerozoic cover of the West African Mobile Zone are possibly associated with a thermal signature of the lithospheric mantle. Another thermal signature of the lithospheric mantle along the Cameroon Volcanic Line partially outlines margins between the Oubanguides Neoproterozoic Belt and the southern extension of the West African Mobile Zone.

Solar and lunar daily geomagnetic variations and their equivalent current systems observed by Swarm

This paper describes solar and lunar daily variations of the geomagnetic field over low- and mid-latitude regions, using vector magnetometer data from Swarm satellites at altitudes of sim500 km during the solar minimum years of 2017–2020. The average solar variation of the geomagnetic field is within the range of ±14 nT, while the lunar variation is within ±2 nT. The latter is comparable to the ocean tidal field. A spherical harmonic analysis is performed on the solar and lunar variations to evaluate their internal and external equivalent current systems. The results show that both the solar and lunar variations are mainly of internal origin, which can be attributed to combined effects of ionospheric dynamo currents and induced underground currents. Global patterns of the internal solar and lunar current systems are consistent with the corresponding external current systems previously reported based on ground observations. The Swarm external currents are mainly in the meridional direction, and are likely associated with interhemispheric field-aligned currents. Both the internal and external current systems depend on the season and longitude.Graphical