Spherical Harmonic Model Research Articles

Negative VTEC with spherical harmonic expansion model under different solar activity conditions

The Earth’s ionosphere can be described by a spherical harmonic (SH) expansion up to a specific degree. However, there exist negative vertical total electron content (VTEC) values in the global ionosphere map (GIM) with the SH expansion model. In this contribution, we specifically investigated the negative VTEC values that are induced by the SH expansion model and validated the performance of the inequality-constrained least squares (ICLS) method in eliminating the negative VTEC values. The GPS data from 2004 to 2017 was selected to cover one solar cycle and the experiments under different solar activity conditions were analyzed. The results in our work show that the occurrence of the negative VTEC values is attributed to the deficiency of the SH expansion model when the VTEC itself is small instead of the unevenly distribution of the GNSS stations. The negative VTEC values appear periodically in the temporal domain, showing apparently one year and half year periods. During one year, two peaks in June and December can be observed in the time series of the negative VTEC values. The number of negative VTEC values in June is obvious larger than that in December. During one solar cycle, the number of negative VTEC values under quiet solar activity condition is obvious larger than that under strong solar activity condition. In the spatial domain, the appearance of the negative VTEC values is strongly related with the movement of the subsolar point. In the latitude of the subsolar point has the largest magnitude, the negative values will appear on the opposite hemisphere and the further from the subsolar point the more negative values. The maximum number of the negative VTEC values in the southern hemisphere appears in June, while the peak value in the northern hemisphere appears in December. The maximum number of negative VTEC values in the southern hemisphere is generally larger than that in the northern hemisphere. In addition, the negative VTEC values are distributed both at middle latitude and high latitude in the southern hemisphere, while they are mainly distributed at high latitude in the northern hemisphere. When the ICLS method is used, the negative VTEC values can be eliminated efficiently and it has nearly no influence on the positive VTEC values. The ICLS method can also improve the receiver’s differential code bias (DCB) and significantly decrease the unreasonable negative slant TEC (STEC) values along the lines of sight. Using the final GIM product of the Jet Propulsion Laboratory (JPLG) as a reference, the root mean square (RMS) of the ICLS solution shows maximum 25%, 20% and 45% improvement relative to the least squares (LS) solution at northern high latitude, southern middle latitude and southern high latitude, respectively.

Neural network-based ionospheric modeling and predicting—To enhance high accuracy GNSS positioning and navigation

As a highly variable space plasma, the sophisticated monitoring and expression of the ionosphere are critically important for the performance of Global Navigation Satellite System (GNSS). To satisfy the requirement of more precise positioning and navigation, it is necessary to model and predict the ionosphere in a more accurate way. In this paper, the potential advantages of neural networks in ionosphere modeling and predicting are explored. Radial Basis Function Neural Network (RBF-NN) is used for modeling and the Long- and Short-Term Memory Neural Network (LSTM-NN) is used for predicting. Results validate that performance of the RBF-NN is superior to the traditional spherical harmonic and polynomial model. The average root mean square error of one-day modeling residuals by RBF-NN is 1.2951 Total Electron Content Unit (TECU, 1 TECU = 1016 electron/m2), which is reduced by 0.4172 and 0.2710 TECU respectively, compared with the traditional spherical harmonic and polynomial model. The predicting results show that the LSTM-NN has better short-time performance, compared with the finite Fourier series model and the autoregressive integrated moving average model. Finally, one-day predicted products are used as priori constraints for single-frequency static and dynamic precise point positioning (PPP). Positioning results show that constraint effect of the predicted products is equivalent to that of the International GNSS Services (IGS) final products. Compared with the results without ionospheric constraint, the PPP convergence time is obviously reduced, and the positioning performance in the horizontal direction is significantly improved.

Spherical Harmonic Model of Jupiter's magnetic field and secular variation

Spherical Harmonic Model of Jupiter's magnetic field and secular variation

Holocene Temperature Variations in Semi-Arid Central Mongolia—A Chronological and Sedimentological Perspective From a 7400-year Lake Sediment Record From the Khangai Mountains

Semi-arid Mongolia is a highly sensitive region to climate changes, but the region’s Holocene paleoclimatic evolution and its underlying forcing mechanisms have been the subject of much recent debate. Here we present a continuous 7.4 ka sediment record from the high-altitude Shireet Naiman Nuur (Nuur = lake) in the central Mongolian Khangai Mountains. We extensively dated the sediments and analyzed elemental composition and bulk isotopes for lake sediment characterization. Our results show that 14C-dating of bulk organic carbon and terrestrial macrofossils provide a robust and precise chronology for the past 7.4 ± 0.3 cal ka BP at Shireet Naiman Nuur and 14C-ages are mostly in stratigraphic order. The 14C-based chronology is confirmed by paleomagnetic secular variations, which resemble the predictions of spherical harmonic geomagnetic field models. The very good chronological control makes paleomagnetic secular variation stratigraphy a powerful tool for evaluating and refining regional 14C-chronologies when compared to the record presented here. The lake sediment proxies TOC, N, log (Ca/Ti) and log (Si/Ti) reveal increased lake primary productivity and high growing season temperatures from 7.4 ± 0.3 to 4.3 ± 0.2 cal ka BP, which is likely the result of stronger summer insolation and pronounced warming. Reduced summer insolation thereafter results in decreased productivity and low growing season temperatures at Shireet Naiman Nuur from 4.3 ± 0.3 cal ka BP until present day. The globally acknowledged 4.2 ka event also appears as a pronounced cooling event at Shireet Naiman Nuur, and additional abrupt cooling events occurred during minima in total solar irradiance at ∼3.4, 2.8 and 2.4 ka BP. Low lake primary productivity and growing season temperatures are likely the result of longer ice cover periods at the high-altitude (2,429 m a.s.l.) Shireet Naiman Nuur. This leads to shorter mixing periods of the lake water which is supported by more positive δ13CTOC because of increased incorporation of dissolved HCO3− by aquatic producers during periods of longer ice cover.

Analytical Radial Adaptive Method for Spherical Harmonic Gravity Models

Analytical Radial Adaptive Method for Spherical Harmonic Gravity Models

The accuracy of geostatistics for regional geomagnetic modeling in an archipelago setting.

Indonesia as an archipelago country relies on a limited number and clustered distributed repeat station networks. This paper explores the use of geostatistical modeling to overcome this data limitation. The model data set consisted of repeat station data from 1985 to 2015 epoch. The geostatistical methods utilized included ordinary kriging (OK), collocated cokriging (CC), and kriging with external drift (KED). The model generated using these geostatistical methods was then compared to spherical cap harmonic analyses (SCHA) and polynomial models. The geostatistical model was shown to perform better, with greater accuracy in declination, inclination, and total intensity, as indicated by the root mean square error (RMSE). We have demonstrated that the geostatistical method is a promising approach in the modeling of regional geomagnetic field, especially in areas with limited and clustered distributed data.

Late Holocene Paleomagnetic Secular Variation in the Chukchi Sea, Arctic Ocean

AbstractThe geomagnetic field behavior in polar regions remains poorly understood and documented. Although a number of Late Holocene paleomagnetic secular variation (PSV) records exist from marginal settings of the Amerasian Basin in the Arctic Ocean, their age control often relies on a handful of radiocarbon dates to constrain ages over the past 4,200 years. Here we present well‐dated Late Holocene PSV records from two sediment cores recovered from the Chukchi Sea, Arctic Ocean. The records are dated using 26 14C measurements, with local marine reservoir corrections calibrated using tephra layers from the 3.6 cal ka BP Aniakchak eruption in Northern Alaska. These 14C‐based chronologies are extended into the post‐bomb era using caesium‐137 dating, and mercury isochrons. Paleomagnetic measurements and rock magnetic analyses reveal stable characteristic remanent magnetization directions, and a magnetic mineralogy dominated by low‐coercivity minerals. The PSV records conform well to global spherical harmonic field model outputs. Centennial to millennial scale directional features are synchronous between the cores and other Western Arctic records from the area. Due to the robust chronology, these new high‐resolution PSV records provide a valuable contribution to the characterization of geomagnetic field behavior in the Arctic over the past few thousand years, and can aid in developing age models for suitable sediments found in this region.

Parallel Evaluation of Chebyshev Approximations: Applications in Astrodynamics

Approximations based on Chebyshev polynomials have several astrodynamic applications. The performance of these approximations can be improved by parallel implementations exploiting parallel architectures, such as OpenMP and CUDA. In this paper, we introduce the parallel implementation to two astrodynamic applications. The first is the gravitational finite element model (FEM): a piecewise Chebyshev approximation that replaces high degree and order gravitational spherical harmonic models (SHMs). Thus, much lower degree, locally valid functions can efficiently model and compute local gravity perturbations in parallel structure for efficient performance. For this model, the total gravity acceleration is split into a reference and disturbance term. The reference includes two-body plus J_2, which are relatively cheap to compute. The FEM approximates the higher-order gravity terms. It is developed from a 2D mesh grid covering a sphere of a specified radius, and a family of spherical shells is sampled using a cosine distribution in the radial direction. To reduce the required memory when seeking a specific accuracy, an adaptive version of the gravitational FEM is introduced. In addition, a parallel implementation of the FEM using OpenMP is preseneted. We show the runtime comparison for the 200 degree × 200 order EGM2008 SHM and the serial and parallel equivalent FEM algorithms. The other application is the Chebyshev-Picard method (CPM): a numerical integrator that solves an ordinary differential equation by approximating the integrand using a Chebyshev approximant and iterates over the trajectory via Picard iteration. A parallel CUDA implementation of the CPM method in conjunction with the EGM2008 SHM and the FEM is introduced. We present numerical examples for propagating four Earth-orbiting satellites considering both the 200times 200 EGM2008 SHM and the equivalent FEM representation to test the algorithm’s performance via parallel and serial computation (i.e., a single CPU thread).

An Approach for Predicting Global Ionospheric TEC Using Machine Learning

Accurate corrections for ionospheric total electron content (TEC) and early warning information are crucial for global navigation satellite system (GNSS) applications under the influence of space weather. In this study, we propose to use a new machine learning model—the Prophet model, to predict the global ionospheric TEC by establishing a short-term ionospheric prediction model. We use 15th-order spherical harmonic coefficients provided by the Center for Orbit Determination in Europe (CODE) as the training data set. Historical spherical harmonic coefficient data from 7 days, 15 days, and 30 days are used as the training set to model and predict 256 spherical harmonic coefficients. We use the predicted coefficients to generate a global ionospheric TEC forecast map based on the spherical harmonic function model and select a year with low solar activity (63.4 < F10.7 < 81.8) and a year with the high solar activity (79.5 < F10.7 < 255.0) to carry out a sliding 2-day forecast experiment. Meanwhile, we verify the model performance by comparing the forecasting results with the CODE forecast product (COPG) and final product (CODG). The results show that we obtain the best predictions by using 15 days of historical data as the training set. Compared with the results of CODE’S 1-Day (C1PG) and CODE’S 2-Day (C2PG). The number of days with RMSE better than COPG on the first and second day of the low-solar-activity year is 151 and 158 days, respectively. This statistic for high-solar-activity year is 183 days and 135 days.

A New Model of Jupiter's Magnetic Field at the Completion of Juno's Prime Mission

AbstractA spherical harmonic model of the magnetic field of Jupiter is obtained from vector magnetic field observations acquired by the Juno spacecraft during 32 of its first 33 polar orbits. These Prime Mission orbits sample Jupiter's magnetic field nearly uniformly in longitude (∼11° separation) as measured at equator crossing. The planetary magnetic field is represented with a degree 30 spherical harmonic and the external field is approximated near the origin with a simple external spherical harmonic of degree 1. Partial solution of the underdetermined inverse problem using generalized inverse techniques yields a model (“JRM33”) of the planetary magnetic field with spherical harmonic coefficients reasonably well determined through degree and order 13. Useful information regarding the field extends through degree 18, well fit by a Lowes' spectrum with a dynamo core radius of 0.81 Rj, presumably the outer radius of the convective metallic hydrogen region. This new model provides a most detailed view of a planetary dynamo and evidence of advection of the magnetic field by deep zonal winds in the vicinity of the Great Blue Spot (GBS), an isolated and intense patch of flux near Jupiter's equator. Comparison of the JRM33 and JRM09 models suggests secular variation of the field in the vicinity of the GBS during Juno's nearly 5 years of operation in orbit about Jupiter. The observed secular variation is consistent with the penetration of zonal winds to a depth of ∼3,500 km where a flow velocity of ∼0.04 ms−1 is required to match the observations.

Secular variation signals in magnetic field gradient tensor elements derived from satellite-based geomagnetic virtual observatories

SUMMARYWe present local time-series of the magnetic field gradient tensor elements at satellite altitude derived using a Geomagnetic Virtual Observatory (GVO) approach. Gradient element time-series are computed in 4-monthly bins on an approximately equal-area distributed worldwide network. This enables global investigations of spatio-temporal variations in the gradient tensor elements. Series are derived from data collected by the Swarm and CHAMP satellite missions, using vector field measurements and their along-track and east–west differences, when available. We find evidence for a regional secular variation impulse (jerk) event in 2017 in the first time derivative of the gradient tensor elements. This event is located at low latitudes in the Pacific region. It has a similar profile and amplitude regardless of the adopted data selection criteria and is well fit by an internal potential field. Spherical harmonic models of the internal magnetic field built from the GVO gradient series show lower scatter in near-zonal harmonics compared with models built using standard GVO vector field series. The GVO gradient element series are an effective means of compressing the spatio-temporal information gathered by low-Earth orbit satellites on geomagnetic field variations, which may prove useful for core flow inversions and in geodynamo data assimilation studies.

Application of spectral differential strategy in diffusion approximation model and simplified spherical harmonic approximation model

Application of spectral differential strategy in diffusion approximation model and simplified spherical harmonic approximation model

Statistical shape analysis of the tricuspid valve in hypoplastic left heart sydrome.

Hypoplastic left heart syndrome (HLHS) is a congenital heart disease characterized by incomplete development of the left heart. Children with HLHS undergo a series of operations which result in the tricuspid valve (TV) becoming the only functional atrioventricular valve. Some of those patients develop tricuspid regurgitation which is associated with heart failure and death and necessitates further surgical intervention. Repair of the regurgitant TV, and understanding the connections between structure and function of this valve remains extremely challenging. Adult cardiac populations have used 3D echocardiography (3DE) combined with computational modeling to better understand cardiac conditions affecting the TV. However, these structure-function analyses rely on simplistic point-based techniques that do not capture the leaflet surface in detail, nor do they allow robust comparison of shapes across groups. We propose using statistical shape modeling and analysis of the TV using Spherical Harmonic Representation Point Distribution Models (SPHARM-PDM) in order to generate a reproducible representation, which in turn enables high dimensional low sample size statistical analysis techniques such as principal component analysis and distance weighted discrimination. Our initial results suggest that visualization of the differences in regurgitant vs. non-regurgitant valves can precisely locate populational structural differences as well as how an individual regurgitant valve differs from the mean shape of functional valves. We believe that these results will support the creation of modern image-based modeling tools, and ultimately increase the understanding of the relationship between valve structure and function needed to inform and improve surgical planning in HLHS.

A new inclination-based method to evaluate the global geomagnetic configuration and axial dipole moment

The strength and configuration of the geomagnetic field control the average shape of the magnetosphere. The pure dipole assumption and the virtual dipole moment (VDM), determined by individual records, have been widely adopted to evaluate the strength of the geomagnetic field in geological time. However, such an assumption might not be valid during geomagnetic transitions, such as reversals and excursions. The traditional spherical harmonic modeling of the geomagnetic field could be difficult to implement because accurate global records are lacking. Here, we report that an empirical relationship exists between the ratio of the VDM to the true axial dipole moment (VDM/ADM) and the ratio of the power of the axial dipole to that of the non-axial dipole (AD/NAD) based on a new method utilizing globally distributed inclination records. The root mean square global deviation of inclination (RMSΔI) to the standard inclination distribution of the AD was fitted to the AD/NAD with a cubic polynomial by utilizing a large number of geodynamo simulations. Tests with geomagnetic field models showed that the AD/NAD derived from the RMSΔI agreed well with that calculated by using the Gauss coefficients, and the estimated ADM was consistent with the true value. Finally, the application of volcanic records during the Laschamp excursion showed the VDM might overestimate the ADM by a factor of 3. Our new method will be useful in future studies that characterize the configuration of the geomagnetic field and the strength of the axial dipole.

The Internal Structure of Mercury's Core Inferred From Magnetic Observations

AbstractPrevious models of Mercury's core magnetic field based on high altitude data from first MESSENGER flybys revealed an axisymmetric structure of the field. Here, we use low altitude MESSENGER data covering the entire mission period to construct spherical harmonic models based on various spatial norms. Although we find a dominantly axisymmetric field, our models nevertheless include detectable deviations from axisymmetry. These non‐axisymmetric features appear at high latitudes, resembling intense geomagnetic flux patches at Earth's core‐mantle boundary. Based on this core field morphology, we then attempt to infer Mercury's internal structure. More specifically, assuming that Mercury's high‐latitude non‐axisymmetric features are concentrated by downwellings at the edge of the planet's inner core tangent cylinder, and accounting for the presence of a stably stratified layer at the top of Mercury's core, we establish a relation between the inner core size and the thickness of the stratified layer. Considering plausible ranges, we propose that Mercury's inner core size is about 500–660 km, which corresponds to a stratified layer thickness of 880–500 km, respectively.

Establishment of European Regional Ionosphere Model Based on Spherical Harmonic Functions

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.

A Spherical Harmonic Model of Earth's Lithospheric Magnetic Field up to Degree 1050

AbstractDetailed mapping of the Earth's magnetic field brings key constraints on the composition, dynamics, and history of the crust. Satellite and near‐surface measurements detect different length scales and are complementary. Here, we build a model, selecting and processing magnetic field measurements from the German CHAMP and ESA Swarm satellites, which we merge with near‐surface scalar anomaly data. We follow a regional approach for modeling the magnetic field measurements and next transform the series of regional models into a unique set of spherical harmonic (SH) Gauss coefficients. This produces the first global model to SH degree 1050 derived by inversion of all available measurements. The new model agrees with previous satellite‐based models at large wavelengths and fits the CHAMP and Swarm satellite data down to expected noise levels. Further assessment in the geographical and spectral domains shows the model to be stable when downward continued to the Earth's surface.

Inside Front Cover

AbstractThis study adopted a hybrid diffusion equation with third order simplified spherical harmonics model and proposed a novel Least Square QR factorization based on the Lasso algorithm to overcome the limitation of the single proton propagation model and enable fast iteration and accurate sparse reconstruction for X‐ray Luminescence Computed Tomography (XLCT). It achieves superior reconstruction performance in terms of location accuracy, spatial resolution and materializes a balance between accuracy and efficiency.Further details can be found in the article by Hongbo Guo, Hengna Zhao, Jingjing Yu, Xiaowei He, Xuelei He, and Xiaolei Song (e202100089). image

A Spherical Harmonic Martian Crustal Magnetic Field Model Combining Data Sets of MAVEN and MGS

AbstractThis study presents a new spherical harmonic (SH) model of the crustal magnetic field of Mars, based on the magnetic field data set measured by the Mars Global Surveyor (MGS) and the Mars Atmosphere and Volatile EvolutioN (MAVEN) spacecraft. To minimize the influence of external fields due to solar wind interaction with Mars, we rejected data that were observed dayside and above an altitude of 500 km. The data points of MAVEN were reduced by using a proxy of solar wind activity that identified and rejected any data measured during magnetically disturbed intervals. We used a conventional least squares technique to estimate the Gauss coefficients fitted to the reduced data set and made a compromise between model misfit and model roughness by truncating the SH model at degree 110. This model is capable of representing crustal fields with a spatial resolution approaching ∼200 km at 120 km altitude and ∼260 km at the Martian surface. Since our model fits MAVEN's observational data better than previous models, especially the data obtained during MAVEN's low altitude periapsis passes, we conclude that it may more accurately approximate the low‐altitude crustal field. We calculate the crustal field power spectrum of various models and find that small‐scale fields at low altitudes were underestimated by most previous models. This new model could benefit future studies associated with the Martian crustal field and its interaction with the solar wind.

Dipole and Nondipole Evolution of the Historical Geomagnetic Field From Instrumental, Archeomagnetic, and Volcanic Data

AbstractAlthough a large number of magnetic declination and inclination measurements are available for the historical period from 1400 to 1900 CE, even the evolution of the axial dipole moment, the most prominent quantity of the geomagnetic field, is highly disputed for this time interval. Here, a new iterative Bayesian spherical harmonic model is constructed that combines historical (direct) and archeomagnetic or volcanic (indirect) records to better constrain the geomagnetic field evolution over this historical epoch. The resulting geomagnetic field reconstruction (BIGMUDIh.1) benefits from recently published and revised data for critical periods and regions. The highly variable data uncertainties and qualities are tackled with a weighting and selection scheme. Model uncertainties are mainly controlled by selection criteria applied to indirect intensity records and are estimated using a bootstrapping approach. A major improvement with respect to the established historical model gufm1 consists in the incorporation of archeointensities, which enables a direct estimation of the axial dipole evolution prior to 1840 CE. BIGMUDIh.1 yields a 2 µT (6%) decrease of the axial dipole component between 1500 and 1600 CE, followed by a relatively stable period until 1900 CE. Modeled declinations agree well with historical records and the gufm1 model derived from such records. Furthermore, the evolution of the South Atlantic Anomaly, driven by reverse flux patches (RFPs) at the core‐mantle boundary, is outlined. Periods of accelerated movement of the northern magnetic dip pole appear to be connected to the formation and changes of RFPs in the Northern Hemisphere.