Spherical Harmonic Model Research Articles

Regional spherical harmonic model for large telescope pointing error correction

Regional spherical harmonic model for large telescope pointing error correction

Development of Precision Controllable Magnetic Field-Assisted Platform for Micro Electrical Machining.

In order to introduce the magnetic field into micro electrical machining technology to explore the influence of magnetic field on micro electrical machining, the development of a precision controllable magnetic field-assisted platform is particularly important. This platform needs to precisely control the spatial magnetic field. This study first completes the hardware design and construction of the magnetic field generation device, using electromagnetic coils with soft iron cores as the sources of the magnetic field. Mathematical models of the magnetic field are established and calibrated. Since the magnetic dipole model cannot effectively describe the magnetic field generated by the electromagnetic coil, this study adopts a more precise description method: the spherical harmonic function expansion model and the magnetic multipole superposition model. The calibration of the magnetic field model is based on actual excitation magnetic field data, so a magnetic field sampling device is designed to obtain the excitation magnetic field of the workspace. The model is calibrated based on a combination of the theoretical model and magnetic field data, and the performance of the constructed setup is analyzed. Finally, a magnetic field-assisted platform has been developed which can generate magnetic fields in any direction within the workspace with intensities ranging from 0 to 0.2 T. Its magnetic field model arrives at an error percentage of 2.986%, a variance of 0.9977, and a root mean square error (RMSE) of 0.71 mT, achieving precise control of the magnetic field in the workspace.

Dynamic Model Fidelity Effects on Covariance Based Track Association

This paper examines the performance of covariance-based track association with varying levels of dynamic model fidelity. To minimize the dynamic nonlinearities, the mean modified equinoctial (MME) elements are used. Ground-based radar measurements are generated by propagating the osculating orbits of 100 space objects in low Earth orbit using a 70×70 spherical harmonic gravity model (JGM-3). Atmospheric drag, solar radiation, and third-body gravity are not considered. To associate two tracks, the two sets of states and covariances are propagated to a common time. The Mahalanobis distance between the two tracks is then used to determine whether they associate. A higher-fidelity dynamic model for state propagation improves association performance, while the fidelity of the dynamic model for covariance propagation does not have a significant impact. The MME-based track association demonstrates excellent performance when incorporating J2, J22, J4, and J6 in the MME state propagation as well as the MME covariance propagation with J2. It achieves an accuracy of 99.92%, a true positive sensitivity of 92.43%, a false negative sensitivity of 7.57%, a false positive sensitivity of 0%, and a true negative sensitivity of 100% when associating tracks separated by 10 days.

The exact implementation of a spherical harmonic model for gravimetric quantities

Abstract This note presents formulas to evaluate a spherical harmonic model of Earth’s gravitational potential for essential gravimetric quantities without spherical and linear approximation. Typically, 10–13 significant digits of numerical accuracy for such computations are obtained over the globe using EGM2008 with FORTRAN 77 code that is also provided.

SHARAD Mapping of Mars Dayside Ionosphere Patterns: Relationship to Regional Geology and the Magnetic Field

AbstractThe electron density of the Martian ionosphere is modulated by solar wind forcing and crustal magnetic fields. Sounding observations from the orbital Shallow Radar (SHARAD) map the ionospheric total electron content (TEC) at a spatial resolution of ∼35 km. Averaging over a 250‐km diameter window from data collected weeks to years apart yields the first map of long‐term stable dayside martian TEC features. An extensive region of suppressed TEC in the southern highlands correlates with strong radial magnetic fields, but in other areas no simple correlation is observed. The TEC maps do follow the outlines of exposed Noachian crust and patches of magnetization in Tharsis not reset by volcanic activity. SHARAD TEC mapping may capture magnetic field strength at an intermediate height between the surface and the altitudes of orbital measurements underlying spherical harmonic models. Existing and future data will allow SHARAD TEC mapping to ∼100 km spatial resolution.

An Adjoint Method for High-Resolution EPMA Based on the Spherical Harmonics (PN) Model of Electron Transport

Electron Probe Microanalysis (EPMA) is a nondestructive technique to determine the chemical composition of material samples in the micro- to nanometer range. Based on intensity measurements of x-radiation, the reconstruction of the material composition and structure poses an inverse problem. The reconstruction methods currently applied are based on models that assume a homogeneous or a layered structure of the studied material. To increase the spatial resolution of reconstruction in EPMA the combination of a more sophisticated reconstruction method, which is based on a model that allows complex material structure, together with multiple measurements with varying beam configurations is required. We present a deterministic k-ratio model that is based on the PN model, an approximation of the radiative transfer equation for electron transport. Our goal is to approximate a maximum likelihood solution of the inverse problem using gradient-based optimization. We detail the application of the model in the context of algorithmic differentiation, in particular by deriving its continuous adjoint formulation. Algorithmic differentiation provides the flexibility to adapt the reconstruction method to various material parametrizations and thus to regularize and take into account prior knowledge. Through examples, we verify our implementation and demonstrate the flexibility of the reconstruction/differentiation method.

Local spherical harmonic power spectra from local magnetic or gravity data

SUMMARY We present a method to calculate local spherical harmonic power spectra directly from spacecraft magnetic and gravity data with varying spacecraft altitude. Previously published applications of local spherical harmonic power spectra have been formulated for data that are available at a single collection altitude, such as data evaluated from a global spherical harmonic model. Our approach consists of first solving for local models from local data and then obtaining local multitaper spectra from the local models. We demonstrate with numerical tests that this approach can produce reliable results. Our method is particularly useful in situations where data coverage does not allow for calculating global magnetic or gravity field models, or where data quality or quantity varies regionally and where local models could yield superior resolution or quality over global models.

Effects of Topside Ionosphere Modeling Parameters on Differential Code Bias (DCB) Estimation Using LEO Satellite Observations

Given the potential of low-earth orbit (LEO) satellites in terms of navigation enhancement, accurately estimating the differential code bias (DCB) of GNSS satellites and LEO satellites is an important research topic. In this study, to obtain accurate DCB estimates, the effects of vertical total electron content (VTEC) modeling parameters of the topside ionosphere on DCB estimation were investigated using LEO observations for the first time. Different modeling parameters were set in the DCB estimations, encompassing modeling spacing in the dynamic temporal mode and degree and order (D&O) in spherical harmonic modeling. The DCB precisions were then evaluated, and the impacts were analyzed. Thus, a number of crucial and beneficial conclusions are drawn: (1) The maximum differences in the GPS DCB estimates after adopting different modeling spacings and different D&Os exhibit that the different modeling spacings or D&Os both affect the GPS DCB estimates and their root-mean square (RMS), and the effects of the two are at the same level. (2) The maximum differences in receiver DCBs using different modeling spacings indicate that the modeling spacing has a significant impact on the receiver DCBs, compared with GPS DCBs. Whereas, the maximum differences in receiver DCBs with different modeling D&Os are inferior to the differences in the GPS DCBs. That is, the modeling spacing has a greater impact on the LEO DCBs than those of the modeling D&O. (3) The experimental results indicate that the GPS DCB estimates using a modeling spacing of 12H have higher precisions than the others, whereas LEO receiver DCBs using a spacing of 4H or 6H obtain optimal STD. In terms of modeling D&O, adopting 8D&O in the LEO-based VTEC modeling can attain superior estimates and precisions for both GPS and LEO DCBs. The research conclusions can provide references for LEO-augmented DCB estimation.

Comparison of polynomial and rational function cornea models for effective dimensionality reduction

This study aims to examine geometric models of the corneal surface that can be used to reduce in reasonable time the dimensionality of datasets of normal anterior corneas. Polynomial models (P) like Zernike polynomials (ZP) and spherical harmonic polynomials (SHP) were obvious candidates along with their rational function (R) counterparts, namely Zernike rational functions (ZR) and spherical harmonic rational functions (SHR, new model). Knowing that both SHP and ZR were more accurate than ZP for the modeling of normal and keratoconus corneas, it was expected that both spherical harmonic (SH) models (SHP and SHR) would be more accurate than their Zernike (Z) counterparts (ZP and ZR, respectively), and both rational (R) models (SHR and ZR) more accurate than their polynomial counterparts (SHP and ZP, respectively) for a low dimensional space (coefficient number J < 30). This was the case. The SH factor contributed more to accuracy than the R factor. Considering the corneal processing time as a function of J, P models were processed in quasi-linear time with a quasi-null slope and rational models in polynomial time. Z models were faster than SH models, and increasingly so in their R version. In sum, for corneal dimensionality reduction, SHR is the most accurate model, but its processing time is increasingly prohibitive unless the best coefficient combination is identified beforehand. ZP is the fastest model and is reasonably accurate with normal corneas for exploratory tasks. SHP is the best compromise between accuracy and speed.

Autonomous Asteroid Characterization Through Nanosatellite Swarming

This paper first defines a class of estimation problem called simultaneous navigation and characterization (SNAC), which is a superset of simultaneous localization and mapping (SLAM). A SNAC framework is then developed for the Autonomous Nanosatellite Swarming (ANS) mission concept to autonomously navigate about and characterize an asteroid including the asteroid gravity field, rotational motion, and 3D shape. The ANS SNAC framework consists of three modules: 1) multi-agent optical landmark tracking and 3D point reconstruction using stereovision, 2) state estimation through a computationally efficient and robust unscented Kalman filter, and 3) reconstruction of an asteroid spherical harmonic shape model by leveraging a priori knowledge of the shape properties of celestial bodies. Despite significant interest in asteroids, there are several limitations to current asteroid rendezvous mission concepts. First, completed missions heavily rely on human oversight and Earth-based resources. Second, proposed solutions to increase autonomy make oversimplifying assumptions about state knowledge and information processing. Third, asteroid mission concepts often opt for high size, weight, power, and cost (SWaP-C) avionics for environmental measurements. Finally, such missions often utilize a single spacecraft, neglecting the benefits of distributed space systems. In contrast, ANS is composed of multiple autonomous nanosatellites equipped with low SWaP-C avionics. The ANS SNAC framework is validated through a numerical simulation of three spacecraft orbiting asteroid 433 Eros. The simulation results demonstrate that the proposed architecture provides autonomous and accurate SNAC in a safe manner without an a priori shape model and using only low SWaP-C avionics.

The Amplitude and Timescales of 0–15 ka Paleomagnetic Secular Variation in the Northern North Atlantic

AbstractWe investigate the amplitude and frequency of directional geomagnetic change since 15 ka in the Northern North Atlantic (∼67°N) using five “ultra‐high” resolution continental shelf sediment cores deposited at rates greater than 1 m/kyr. The ages of these cores are constrained by 103 radiocarbon dates with reservoir ages assessed through tephra correlation to terrestrial archives. Our study aims to address many of the uncertainties that are common in sedimentary paleomagnetic studies, including signal attenuation in low to moderate resolution archives and difficulty to demonstrate reproducibility in higher resolution archives. The “ultra‐high” accumulation rates of our cores reduce “lock‐in” and smoothing uncertainties associated with magnetic acquisition processes. Abundant radiocarbon dates along with an objective alignment algorithm provide a test of signal reproducibility at sub‐millennial timescales. The paleomagnetic secular variation (PSV) signal, evaluated as individual records and as a new stack (GREENICE15k), validates prior results, but provides stronger geochronological constraints, demonstrates a reproducible PSV signal and amplitude, and extends through the abrupt Bølling–Allerød and Younger Dryas climate transitions of the latest Pleistocene. While broadly consistent with time‐varying spherical harmonic models and varve dated records from Northern Europe, we demonstrate greater variance and higher amplitudes—particularly at sub‐millennial timescales. This robust variability on centennial timescales is rarely observed or discussed, but is likely important to our understanding of some of the most intriguing aspects of the geodynamo.

Evaluation of Terrestrial Water Storage and Flux in North China by Using GRACE Combined Gravity Field Solutions and Hydrometeorological Models

To enrich the understanding of the dynamic evolution of the water resources in North China, terrestrial water storage anomalies (TWSA) from January 2003 to June 2017 are derived using the new GRACE time-variable gravity field model Tongji-GraceCom. Additionally, the spatiotemporal characteristics of terrestrial water fluxes (TWF) at multiple time scales are analyzed based on the water budget theory in conjunction with hydrometeorological and statistical data. The results show that the quality of the Tongji-GraceCom model is superior to the state-of-art spherical harmonic models (CSR RL06 and JPL RL06), with the signal-to-noise ratio improving by 10–16%. After correcting the leakage errors with a reliable correction method, the inferred TWSA in North China presents a significant downward trend, amounting to −1.61 ± 0.05 cm/yr, with the most serious TWSA depletion mainly clustering in the south-central area. The TWFs derived from GRACE and from hydrometeorological elements are in good agreement and both exhibit significant seasonal fluctuations induced by tracking the periodic movements of meteorological factors. However, unlike precipitation which manifests in an increasing trend, both TWFs reflect the obvious decreasing trends, indicating that North China is suffering from severe water deficits, which are mainly attributed to the enhanced evaporation and extensive groundwater pumping for agricultural irrigation.

Refined Spatio-Temporal Model of Accelerations of the Main Geomagnetic Field on the Earth’s Surface and Geomagnetic Jerks

Refined spatial spherical harmonic models of the piecewise constant field of secular geomagneticaccelerations have been constructed based on the concept of geomagnetic jerks. The data of the 189 magneticobservatories operating during the extended period of investigations have been analyzed, assuming that jerksare manifestation of sharp changes in regimes of quasi-stationary magnetohydrodynamic process of generationof geomagnetic secular variations in the high-frequency spectrum range. Qualitative and quantitativeanalyses have been carried out, showing the persistency of models of the current generation, and their consistencywith the acceleration field values obtained from direct observatory observations data. The epochs ofjerks that occurred in the extended time period were refined and it was found that on the global scale theyhave quasi-decadal repeatability, through which it is possible to estimate the time characteristics of regimesof the stationary evolution of the magnetic field of the Earth’s core. An analysis of the results of spherical harmonicmodeling using the maps of the global field of accelerations showed that the origins of all identifiedjerks are of an intraterrestrial character, which also provide the specific features of their morphological structure,as well as the degree of their distribution on the Earth’s surface

First evidence of the link between internal and external structure of the human inner ear otolith system using 3D morphometric modeling

Our sense of balance is among the most central of our sensory systems, particularly in the evolution of human positional behavior. The peripheral vestibular system (PVS) comprises the organs responsible for this sense; the semicircular canals (detecting angular acceleration) and otolith organs (utricle and saccule; detecting linear acceleration, vibration, and head tilt). Reconstructing vestibular evolution in the human lineage, however, is problematic. In contrast to considerable study of the canals, relationships between external bone and internal membranous otolith organs (otolith system) remain largely unexplored. This limits our understanding of vestibular functional morphology. This study combines spherical harmonic modeling and landmark-based shape analyses to model the configuration of the human otolith system. Our approach serves two aims: (1) test the hypothesis that bony form covaries with internal membranous anatomy; and (2) create a 3D morphometric model visualizing bony and membranous structure. Results demonstrate significant associations between bony and membranous tissues of the otolith system. These data provide the first evidence that external structure of the human otolith system is directly related to internal anatomy, suggesting a basic biological relationship. Our results visualize this structural relationship, offering new avenues into vestibular biomechanical modeling and assessing the evolution of the human balance system.

Filtering GRACE temporal gravity field solutions using ensemble empirical mode decomposition approach

Due to the strong noise that exists in GRACE (Gravity Recovery and Climate Experiment) temporal gravity field solutions, geophysical signals are normally drowned which need many effective filtering approaches. Considering the advantage of the ensemble empirical mode decomposition (EEMD) approach, we used the EEMD to filter the noise in this study together with the empirical mode decomposition (EMD) for comparisons. EMD method is a spectrum analysis method, which is very effective for non-stationary signals. EMD process is essentially a means to process non-stationary signals. It has been applied in many fields in recent years. Considering the characteristics of the spherical harmonic coefficient model that the noise level higher with the increasing degree, we divided the gravity field solutions into two parts (degrees 2–28 and degrees 29–60) based on the ratios of the latitude-weighted root mean square (RMS) over the land and ocean signals when adopting different truncated degrees. For the real GRACE solution experiments, the results show that the fitting errors of EEMD approach are always smaller than those of EMD approach, and the mean RMS ratio of EEMD is 3.45, larger than 3.40 of EMD. The simulation results show that the latitude weighted root mean square errors for EEMD approach are smaller than those of EMD, indicating that EEMD can extract the geophysical signals more accurately. Therefore, it is reasonable to conclude that EEMD performs better than EMD for filtering GRACE solutions.

Martian dayside magnetic topology and structures of the cross-terminator magnetic loops – multispecies magnetohydrodynamic simulations

ABSTRACTThe interaction between impinging magnetized solar wind and Martian crustal fields produces complexly distributed magnetic topologies in the dayside magnetosphere. This study focused on obtaining the distribution of Martian dayside magnetic topology and the structures of the cross-terminator magnetic loops. A 3D multispecies magnetohydrodynamic model was employed to simulate the interactions between Mars and solar winds, and a 110° spherical harmonic model was used to calculate the crustal fields. We randomly extracted more than 1000 magnetic field lines from the near-Mars region of the model results. These results indicate the existence of large-scale closed fields and high-inclination-angle open fields in the Southern hemisphere, exerting their influence even above the height of the ionopause, resulting in a complex relationship between plasma motion and magnetic topology distribution. In contrast, the plasma motion patterns in the Northern hemisphere are similar to those observed in unmagnetized planets. Furthermore, the model results show two types of cross-terminator magnetic loop. Small-scale cross-terminator magnetic loops connect the local atmosphere on the dayside and nightside, whereas many large-scale magnetic loops cross the centre–tail region and extend more than 2RM downstream of Mars, especially in the Southern hemisphere. Finally, the clock angle distribution shows magnetic field distortion at 1000 km altitude. This study provides a clearer and more detailed description of the Martian dayside magnetic topology and the structures of the cross-terminator magnetic loops.

Transformation from a global spherical to an adjusted local rectangular harmonic model

This work presents a technique to transform a global spherical to an adjusted local rectangular harmonic model. First, the mathematical form of a global spherical harmonic model is presented. Second, the necessary conversion from global (geocentric) into local rectangular coordinates is given. Third, Laplace’s equation is solved by the method of separation of variables in local rectangular coordinates and its solutions in different functional forms are presented. Then, the estimation of the coefficients of these mathematical models by a least squares’ adjustment process is described, using as data the values of the disturbing potential of the Earth’s gravity field. The strategy for the selection of the best mathematical model for a successful transformation is described and validated in different case studies. These refer to areas in Greece, China and Germany and include comparisons with other models or methods. The results show the applicability of the presented transformation and confirm its advantages.

Prediction of Wilms' Tumor Susceptibility to Preoperative Chemotherapy Using a Novel Computer-Aided Prediction System.

Wilms' tumor, the most prevalent renal tumor in children, is known for its aggressive prognosis and recurrence. Treatment of Wilms' tumor is multimodal, including surgery, chemotherapy, and occasionally, radiation therapy. Preoperative chemotherapy is used routinely in European studies and in select indications in North American trials. The objective of this study was to build a novel computer-aided prediction system for preoperative chemotherapy response in Wilms' tumors. A total of 63 patients (age range: 6 months-14 years) were included in this study, after receiving their guardians' informed consent. We incorporated contrast-enhanced computed tomography imaging to extract the texture, shape, and functionality-based features from Wilms' tumors before chemotherapy. The proposed system consists of six steps: (i) delineate the tumors' images across the three contrast phases; (ii) characterize the texture of the tumors using first- and second-order textural features; (iii) extract the shape features by applying a parametric spherical harmonics model, sphericity, and elongation; (iv) capture the intensity changes across the contrast phases to describe the tumors' functionality; (v) apply features fusion based on the extracted features; and (vi) determine the final prediction as responsive or non-responsive via a tuned support vector machine classifier. The system achieved an overall accuracy of 95.24%, with 95.65% sensitivity and 94.12% specificity. Using the support vector machine along with the integrated features led to superior results compared with other classification models. This study integrates novel imaging markers with a machine learning classification model to make early predictions about how a Wilms' tumor will respond to preoperative chemotherapy. This can lead to personalized management plans for Wilms' tumors.

Gravity degree–depth relationship using point mass spherical harmonics

SUMMARYRelationships between the degree of a spherical harmonic model of the gravitational field of a body and the depth of a source expressed as a density contrast can be used to study the structure of features. Here, we show that the gravitational acceleration per spherical harmonic degree of a constant density source has an extremum that depends on the depth of the source. Using the spherical harmonics expansion for a point mass source, we use this to derive a degree–depth relationship. Our relationship resembles an earlier one derived by Bowin, with substantial differences at the lower degrees. We also find that a recent relationship derived by Deng et al. overestimates the source depth. The relationship that we derive relates spherical harmonic degree n to depth d for a planet of radius R according to $d = (1-e^{\frac{-1}{n+1}})R$, which simplifies to d = R/(n + 1) for high degrees. We support our new relationship with synthetic models of a density contrast in a planet. We also show how the differences between our relationship and that of Bowin affect band-filtered gravity, for example when inspecting the upper 100 km of the Moon. Using point masses in our modelling results in an approximate relationship where in reality sources can be deeper than estimated, since any source contributes to all spherical harmonic degrees. The use of the contribution per individual degree however provides an intuitive relationship between spherical harmonic degree and depth that can be used to place relative bounds on source depths or to determine the bounds on spherical harmonic expansions when band-filtering gravity field models.

Updated Spherical Harmonic Magnetic Field Moments of Ganymede From the Juno Flyby

AbstractIn this study, we use data from the Juno and Galileo spacecraft to analyze the internal magnetic dynamo of Ganymede. As the only known moon with a strong internal magnetic field, Ganymede is a uniquely interesting object in the context of understanding the formation and structure of planetary magnetospheres. Using a spherical harmonic model centered on the moon, we report a dipole approximation for Ganymede of nT, nT, and nT. We find that using a quadrupole fit rather than a dipole fit provides only a marginal increase in accuracy and instead favor the use of a dipole approximation until more data can be obtained. The magnetic moment estimates provided here can be used as a baseline for interpreting data from future spacecraft flybys of the moon, and can serve as inputs into numerical models studying Ganymede's magnetosphere.