Spherical Harmonic Coefficients Research Articles

Different data-driven prediction of global ionospheric TEC using deep learning methods

Ionospheric Total Electron Content (TEC) is a crucial parameter for monitoring the ionosphere and space weather disasters. Its accurate prediction is vital for precise applications of Interferometric Synthetic Aperture Radar (InSAR) and Global Navigation Satellite System (GNSS). This study proposes a novel method for ionospheric TEC prediction that considers multiple TEC-related factors. We present the random forest method and the Autoformer deep learning with multilayer perceptron (Autoformer-MLP) to predict the global TEC by incorporating the geomagnetic and solar activity parameters. Two schemes with different ionospheric data, i.e., spherical harmonic coefficients (SHC) and vertical TEC (VTEC), are performed by Autoformer-MLP. Ionospheric products from 2009 to 2019 (Cycle 24), obtained from the Center for Orbit Determination in Europe (CODE), are collected to test and evaluate the proposed method. Experimental results demonstrate that the root mean square errors (RMSEs) of predicted global ionospheric maps (GIMs) using the SHC scheme are 3.79 and 1.39 TECU in 2015 and 2019, respectively, and those are 3.55 and 1.28 TECU for the VTEC scheme. Moreover, the RMSEs of the prediction results are 0.5 and 0.2 TECU lower than that of CODE's 1-day predicted global ionospheric map (GIM) product (C1PG) during high and low solar activity years, respectively. These analyses indicate that the proposed random forest and Autoformer-MLP deep learning methods exhibit high accuracy and stability for data-driven prediction of global ionospheric TEC in various scenarios.

A novel dynamic scale factor designed for recovering global TWS changes

For time-variable satellite gravity solutions of GRACE and GRACE-FO in terms of spherical harmonics coefficients, the Scale Factor (SF) is often used to recover the close to “true” signal of Terrestrial Water Storage (TWS) anomalies. However, the conventional SF method has some limitations that may hinder its effectiveness, including: (1) their dependency on input hydrological models that may lead to divergent estimations of SFs; (2) the arbitrary choice of filter strength, which may not be representative in different regions; (3) limited consideration of SF in the temporal dynamics (the conventional SF was fixed value) for monthly varied TWS. Here, we propose a new Dynamic SF method to overcome these limitations and increase accuracy of the restored global TWS changes. This method involves: (1) the Bayesian Three-Cornered Hat (BTCH) method is applied to merge three sets of hydrological products into an optimal hydrological dataset to be used for estimating a unique SF, (2) the anisotropic DDK3 filter (found to be numerically optimal) is applied to suppress the correlated noise, and (3) an iterative Kalman filter process is formulated and implemented to estimate monthly Dynamic SF corresponding to monthly global TWS fields. The Dynamic SF outperformed an ordinary SF method in terms of the Root Mean Squared Error (RMSE), the Mean Absolute Error (MAE), and the Signal to Noise Ratio (SNR), which were found to be improved by 30.8%, 32.2%, and 41.3%, respectively. Moreover, the recovered TWS using the Dynamic SF method showed a good agreement with the GRACE/GRACE-FO RL06 mascon solutions of the CSR and that of JPL regarding the long-term trend, seasonal, and interannual variations.

A Direct Approach for Local Quasi-Geoid Modeling Based on Spherical Radial Basis Functions Using a Noisy Satellite-Only Global Gravity Field Model

The remove–compute–restore (RCR) approach is widely used in local quasi-geoid modeling. However, the classical RCR approach usually does not take into account the noise of the satellite-only global gravity field model (GGM), which may lead to a suboptimal result. This paper presents an approach for local quasi-geoid modeling based on spherical radial basis functions that combines local noisy datasets and a noisy satellite-only GGM. This approach includes an RCR procedure using a satellite-only GGM. This is a direct approach that takes the spherical harmonic coefficients of satellite-only GGM as a noisy dataset and includes the corresponding full-noise covariance matrix in the least-squares estimation, aiming to obtain a statistically optimal local quasi-geoid model. The direct approach goes beyond the indirect approach, which treats the height anomalies generated from the satellite-only GGM as a noisy dataset. However, the generated GGM height anomaly dataset is not an equivalent representation of the satellite-only GGM, which may result in the loss of information from the satellite-only GGM. Through mathematical deduction, we demonstrate the theoretical consistency between the direct approach and the indirect approach. The direct approach also has an advantage over the indirect approach in terms of computational complexity due to the simpler algorithm. We conducted a synthetic closed-loop test with a real data distribution in Colorado, and numerical results demonstrated the advantage of the direct approach in local quasi-geoid modeling. In terms of the root mean square of the differences between the predicted values and the true reference values, the direct approach provided an improvement of approximately 14% compared to the indirect approach.

Research on ambiguity problem in magnetic characterization of spacecraft inside a magnetic shielded room

In space magnetic exploration missions, the measurements obtained from the magnetic sensors mounted on a spacecraft are often influenced by the spacecraft’s own magnetic interference. Therefore, to ensure precise magnetic exploration, it is imperative to conduct a thorough magnetic characterization of the spacecraft. Conducting the magnetic characterization of spacecraft within a magnetic shielded room (MSR) can mitigate the influence of environmental noise on the characterization process, thereby minimizing uncertainty and enhancing modeling accuracy. However, the utilization of high-permeability materials (such as permalloy) in the MSR introduces a disparity in magnetic characteristics between the spacecraft inside and outside the MSR. According to the method of images, the magnetic field generated by the spacecraft inside the MSR is the superposition of the magnetic fields produced by the spacecraft and that produced by several images of itself. Therefore, this may lead to an ambiguity problem, where spacecrafts with different magnetic characteristics and their corresponding images can generate identical magnetic field distribution in the MSR. To address the aforementioned issues, this paper proposes a novel method for magnetic characterization that integrates both the magnetic field vector data within the MSR and the magnetic field gradient data outside the MSR generated by the spacecraft, allowing for the elimination of the ambiguity problem without introducing environmental magnetic noise interference. As a result, it obtains a unique model for precisely representing the spacecraft’s magnetic characteristics in terms of the spherical harmonic coefficients.

VARIABILITY OF EQUIVALENT WATER HEIGHT (EWH) IN INDONESIA DURING 14 YEARS OF GRACE GRAVITY SATELLITE MISSION

GRACE Satellite (Gravity Recovery and Climate Experiment) is a gravity monitoring satellite, that is very sensitive to mass changes, especially the signals that produce by redistribution of water masses. Data from this satellite can be used to observe water distribution in the form of spherical harmonic coefficients. Water mass variations are presented as Equivalent Water Height (EWH). The results of GRACE processing showed that the largest EWH was 27.298 cm in January 2015 and the smallest EWH was 29.816 cm in June 2004 in Sumatera Island. The positive trend occurred in Sumatra island and the negative trend occurred in eastern Indonesia. Generally, the trend of Indonesian rainfall throughout 2002 to 2016 was constant. However, there was a change in seasonal patterns in 2014. This research also use Mascon data from GRACE satellite to determine the radius of the gaussian filter and TRMM satellite’s data to observe rainfall data to be compared with the EWH variability from GRACE.

A feasibility study on phase characterisation of nickel-based superalloys using ultrasound

The use of nickel-based superalloys, such as Inconel 718, Udimet 720Li and Waspaloy, is widespread across the engineering industry due to their superior mechanical properties at elevated temperatures and excellent resistance to creep and fatigue under harsh working conditions. The microstructure of these materials, such as the precipitates and grain sizes, strongly influence the materials’ performance. Currently, existing materials characterisation methods are time-consuming, costly and destructive, as well as only capable of taking measurements of surface or near-surface properties.This paper presents a novel ultrasonic technique for bulk materials characterisation of nickel-based superalloys, based on a spherical harmonic convolution theory which has been successfully used to quantify texture in various metals; it is applied here to study a nickel-based superalloy, with a specific focus on the phase changes when subject to thermal loading. The key parameter used to interpret phase information of a material is the zeroth spherical harmonic coefficient of ultrasonic wave velocities, V00, which corresponds to the average velocities across all directions in 3D space and is independent of crystallographic texture. Experimental investigation using samples of a representative material, Inconel 718, with contrasting microstructures, has shown a clear and quantifiable sensitivity of this coefficient to microstructural changes occurring within the material, and has thus laid the foundations for the development of a rapid phase characterisation technique for nickel-based superalloys using ultrasound.

Monsoon-Based Linear Regression Analysis for Filling Data Gaps in Gravity Recovery and Climate Experiment Satellite Observations

Over the past two decades, the Gravity Recovery and Climate Experiment (GRACE) satellite mission and its successor, GRACE-follow on (GRACE-FO), have played a vital role in climate research. However, the absence of certain observations during and between these missions has presented a persistent challenge. Despite numerous studies attempting to address this issue with mathematical and statistical methods, no definitive optimal approach has been established. This study introduces a practical solution using Linear Regression Analysis (LRA) to overcome data gaps in both GRACE data types—mascon and spherical harmonic coefficients (SHCs). The proposed methodology is tailored to monsoon patterns and demonstrates efficacy in filling data gaps. To validate the approach, a global analysis was conducted across eight basins, monitoring changes in total water storage (TWS) using the technique. The results were compared with various geodetic products, including data from the Swarm mission, Institute of Geodesy and Geoinformation (IGG), Quantum Frontiers (QF), and Singular Spectrum Analysis (SSA) coefficients. Artificial data gaps were introduced within GRACE observations for further validation. This research highlights the effectiveness of the monsoon method in comparison to other gap-filling approaches, showing a strong similarity between gap-filling results and GRACE’s SHCs, with an absolute relative error approaching zero. In the mascon approach, the coefficient of determination (R2) exceeded 91% for all months. This study offers a readily usable gap-filling product—SHCs and smoothed gridded observations—with accurate error estimates. These resources are now accessible for a wide range of applications, providing a valuable tool for the scientific community.

NVAM-Net: deep learning networks for reconstructing high-quality fiber orientation distributions.

Diffusion magnetic resonance imaging (dMRI) is a widely used non-invasive method for investigating brain anatomical structures. Conventional techniques for estimating fiber orientation distribution (FOD) from dMRI data often neglect voxel-level spatial relationships, leading to ambiguous associations between target voxels and their neighbors, which, in turn, adversely impacts FOD accuracy. This study aims to address this issue by introducing a novel neural network, the neighboring voxel attention mechanism network (NVAM-Net), designed to reconstruct high-quality FOD images. The NVAM-Net leverages a Transformer architecture and incorporates two innovative attention mechanisms: voxel attention and surface attention. These mechanisms are specifically designed to capture overlooked features among neighboring voxels. The processed features are subsequently passed through two fully connected layers, further enhancing FOD estimation accuracy by separately estimating spherical harmonics (SH) coefficients of varying orders. The experimental findings, based on the Human Connectome Project (HCP) dataset, reveal that the reconstructed super-resolution FOD images achieve results comparable to those obtained through more advanced dMRI acquisition protocols. These results underscore the NVAM-Net's robust performance in reconstructing multi-shell multi-tissue constrained spherical deconvolution (MSMT-CSD). In summary, this research underscores the NVAM-Net's advantages and practical feasibility in reconstructing high-quality FOD images. It provides a reliable reference point for clinical applications in the field of diffusion magnetic resonance imaging.

Fast computation of the observer motion effects induced on monopole frequency spectra for tabulated functions

Context. Various methods have been studied to compute the boosting effects produced by the observer peculiar motion, which modifies and transfers the isotropic monopole frequency spectrum of the cosmic background radiation to higher multipoles. Explicit analytical solutions for the spherical harmonic expansion coefficients were already presented and applied to different types of background spectrum, strongly alleviating the computational effort needed for accurate theoretical predictions. The frequency spectra at higher multipoles are inherently led by higher-order derivatives of the monopole spectrum. Provided that it can be well described by analytic or semi-analytic functions, the computation of its transfer is not affected by numerical instabilities when evaluated at the required level of numerical accuracy. Instead, monopole frequency spectra described by tabulated functions are computed with a relatively poor frequency resolution in comparison with the Doppler shift, which necessitates interpolation of the tabular representation. The spectra are also affected by uncertainties related to intrinsic inaccuracies in the modelling or in the related observational data as well as to limited accuracy in their numerical computation. These uncertainties propagate and increase with the derivative order, possibly preventing the trustworthy computation of the transfer to higher multipoles and of the observed monopole. Aims. We study methods to filter the original function or its derivatives and the multipole spectra, to mitigate numerical instabilities, and to derive reliable predictions of the harmonic coefficients for different cosmic background models. Methods. From the analytical solutions, and assuming that the monopole spectrum can be expanded in Taylor’s series, we derive explicit expressions for the harmonic coefficients up to the multipole ℓmax = 6 in terms of monopole spectrum derivatives. We then consider different low-pass filters: prefiltering in Fourier space of the tabular representation; filtering in both real and Fourier space of the numerical derivatives; interpolation approaches; and a dedicated method based on amplification and deamplification of the boosted signal. We study the quality of these methods when applied to suitable analytical approximations of the tabulated functions, possibly polluted with simulated noise. These methods are then applied to the tabulations. Results. We consider two very different types of monopole spectra superimposed to the cosmic microwave background: the (smooth) extragalactic source microwave background signal from radio-loud active galactic nuclei and the (feature-rich) redshifted 21 cm line, and present our results in terms of spherical harmonic coefficients. The direct prediction of these coefficients can be noisy at ℓ > 1 or, depending on the uncertainty level, even at ℓ ≤ 1. Without assuming a functional form for the extragalactic background spectrum, the Gaussian prefiltering coupled to the sequential real-space filtering of derivatives allows us to derive accurate predictions up to ℓ ∼ 6, while a log–log polynomial representation, which is appropriate over several decades, gives accurate solutions at any ℓ. Instead, it is difficult to characterise the 21 cm line model variety, and so it is relevant to work without assumptions about the underlying function. Typically, the prefiltering provides accurate predictions up to ℓ ≃ 3 or 4, while the further sequential filtering of the derivatives or the boosting amplification and deamplification method improves the results up to ℓ = 4, while also allowing reasonable estimations of the spectrum at higher ℓ. Conclusions. The proposed methods can significantly extend the range of realistic cosmic background models manageable with a fast computation, beyond the cases characterised a priori by analytical or semi-analytical functions. These methods require only an affordable increase in computation time compared to the direct calculation via simple interpolation.

Time-domain sound field reconstruction using a rigid spherical microphone array.

A time-domain approach for interior spherical near-field acoustic holography is proposed to achieve the low-delay reconstruction of time-domain sound fields using a rigid spherical microphone array. This reconstruction encompasses the incident pressure field, the incident radial particle velocity field, and the total pressure field, which includes scattering. The proposed approach derives time-domain radial propagators through the inverse Fourier transform of their frequency-domain counterparts. These propagators are then applied to the array measurements to obtain the time-domain spherical harmonic coefficients of the interior sound field. Given the fact that the time-domain radial propagators possess finite-time support and exhibit significant high-frequency attenuation characteristics, they can be efficiently implemented using finite impulse response (FIR) filters. The proposed approach processes the signal sample-by-sample through these FIR filters, avoiding a series of issues associated with time-frequency transformations in frequency-domain methods. As a result, the approach offers higher accuracy and lower latency in reconstructing non-stationary sound fields compared to its frequency-domain counterpart and thus holds greater potential for real-time applications. Additionally, owing to the scattering effect of the rigid sphere, the approach avoids the impact of spherical Bessel function nulls and does not require the measurement of particle velocities, which renders the measurements cost effective.

Orbital and Physical Characterization of Asteroid Dimorphos Following the DART Impact

The Double Asteroid Redirection Test (DART) mission impacted Dimorphos, the satellite of binary near-Earth asteroid (65803) Didymos, on 2022 September 26 UTC. We estimate the changes in the orbital and physical properties of the system due to the impact using ground-based photometric and radar observations, as well as DART camera observations. Under the assumption that Didymos is an oblate spheroid, we estimate that its equatorial and polar radii are 394 ± 11 m and 290 ± 16 m, respectively. We estimate that the DART impact instantaneously changed the along-track velocity of Dimorphos by −2.63 ± 0.06 mm s−1. Initially, after the impact, Dimorphos’s orbital period had changed by −32.7 minutes ± 16 s to 11.377 ± 0.004 hr. We find that over the subsequent several weeks the orbital period changed by an additional 34 ± 15 s, eventually stabilizing at 11.3674 ± 0.0004 hr. The total change in the orbital period was −33.25 minutes ±1.5 s. The postimpact orbit exhibits an apsidal precession rate of 6.7 ± 0.°2 day−1. Under our model, this rate is driven by the oblateness parameter of Didymos, J 2, as well as the spherical harmonics coefficients, C 20 and C 22, of Dimorphos’s gravity. Under the assumption that Dimorphos is a triaxial ellipsoid with a uniform density, its C 20 and C 22 estimates imply axial ratios, a/b and a/c, of about 1.3 and 1.6, respectively. Preimpact images from DART indicate Dimorphos’s shape was close to that of an oblate spheroid, and thus our results indicate that the DART impact significantly altered the shape of Dimorphos.

Surface-Based Probabilistic Fiber Tracking in Superficial White Matter.

The short association fibers or U-fibers travel in the superficial white matter (SWM) beneath the cortical layer. While the U-fibers play a crucial role in various brain disorders, there is a lack of effective tools to reconstruct their highly curved trajectory from diffusion MRI (dMRI). In this work, we propose a novel surface-based framework for the probabilistic tracking of fibers on the triangular mesh representation of the SWM. By deriving a closed-form solution to transform the spherical harmonics (SPHARM) coefficients of 3D fiber orientation distributions (FODs) to local coordinate systems on each triangle, we develop a novel approach to project the FODs onto the tangent space of the SWM. After that, we utilize parallel transport to realize the intrinsic propagation of streamlines on SWM following probabilistically sampled fiber directions. Our intrinsic and surface-based method eliminates the need to perform the necessary but challenging sharp turns in 3D compared with conventional volume-based tractography methods. Using data from the Human Connectome Project (HCP), we performed quantitative comparisons to demonstrate the proposed algorithm can more effectively reconstruct the U-fibers connecting the precentral and postcentral gyrus than previous methods. Quantitative validations were then performed on post-mortem MRIs to show the reconstructed U-fibers from our method more faithfully follow the SWM than volume-based tractography. Finally, we applied our algorithm to study the parietal U-fiber connectivity changes in autosomal dominant Alzheimer's disease (ADAD) patients and successfully detected significant associations between U-fiber connectivity and disease severity.

Cosmic topology. Part IIa. Eigenmodes, correlation matrices, and detectability of orientable Euclidean manifolds

If the Universe has non-trivial spatial topology, observables depend on both the parameters of the spatial manifold and the position and orientation of the observer. In infinite Euclidean space, most cosmological observables arise from the amplitudes of Fourier modes of primordial scalar curvature perturbations. Topological boundary conditions replace the full set of Fourier modes with specific linear combinations of selected Fourier modes as the eigenmodes of the scalar Laplacian. We present formulas for eigenmodes in orientable Euclidean manifolds with the topologies E 1–E 6, E 11, E 12, E 16, and E 18 that encompass the full range of manifold parameters and observer positions, generalizing previous treatments. Under the assumption that the amplitudes of primordial scalar curvature eigenmodes are independent random variables, for each topology we obtain the correlation matrices of Fourier-mode amplitudes (of scalar fields linearly related to the scalar curvature) and the correlation matrices of spherical-harmonic coefficients of such fields sampled on a sphere, such as the temperature of the cosmic microwave background (CMB). We evaluate the detectability of these correlations given the cosmic variance of the observed CMB sky. We find that topologies where the distance to our nearest clone is less than about 1.2 times the diameter of the last scattering surface of the CMB give a correlation signal that is larger than cosmic variance noise in the CMB. This implies that if cosmic topology is the explanation of large-angle anomalies in the CMB, then the distance to our nearest clone is not much larger than the diameter of the last scattering surface. We argue that the topological information is likely to be better preserved in three-dimensional data, such as will eventually be available from large-scale structure surveys.

Generation of 3D realistic geological particles using conditional generative adversarial network aided spherical harmonic analysis

The reconstruction of 3D realistic geological particles remains a significant challenge in the field of granular mechanics. Specifically, numerous spherical harmonic (SH) based generation frameworks have been proposed to synthetic new particle shapes retaining majority particle morphology yet having a certain variety. However, given the fact of assuming one or more established distributions or ignoring secondary particle features, the regenerated particles inevitably lose certain diversities. To address this issue, the deep learning method, conditional generative adversarial network (CGAN) was introduced to the SH analysis for particle shape regeneration. Three kinds of sand particles were synthesized and compared with their real mother particle samples concerning the distribution features of SH coefficients and particle shape parameters for validation. Results prove the proposed method has a good reliable and diverse regeneration performance. This approach is promising to facilitate a more reality closer research on 3D particle-related issues in the future.

High-resolution temporal gravity field data products: Monthly mass grids and spherical harmonics from 1994 to 2021

Since April 2002, Gravity Recovery and Climate Experiment (GRACE) and GRACE-FO (FollowOn) satellite gravimetry missions have provided precious data for monitoring mass variations within the hydrosphere, cryosphere, and oceans with unprecedented accuracy and resolution. However, the long-term products of mass variations prior to GRACE-era may allow for a better understanding of spatio-temporal changes in climate-induced geophysical phenomena, e.g., terrestrial water cycle, ice sheet and glacier mass balance, sea level change and ocean bottom pressure (OBP). Here, climate-driven mass anomalies are simulated globally at 1.0° × 1.0° spatial and monthly temporal resolutions from January 1994 to January 2021 using an in-house developed hybrid Deep Learning architecture considering GRACE/-FO mascon and SLR-inferred gravimetry, ECMWF Reanalysis-5 data, and normalized time tag information as training datasets. Internally, we consider mathematical metrics such as RMSE, NSE and comparisons to previous studies, and externally, we compare our simulations to GRACE-independent datasets such as El-Nino and La-Nina indexes, Global Mean Sea Level, Earth Orientation Parameters-derived low-degree spherical harmonic coefficients, and in-situ OBP measurements for validation.

Simulating room transfer functions between transducers mounted on audio devices using a modified image source method.

The image source method (ISM) is often used to simulate room acoustics due to its ease of use and computational efficiency. The standard ISM is limited to simulations of room impulse responses between point sources and omnidirectional receivers. In this work, the ISM is extended using spherical harmonic directivity coefficients to include acoustic diffraction effects. These effects occur in practice when transducers are mounted on audio devices of finite spatial extent, e.g., modern smart speakers with loudspeakers and microphones. The proposed method is verified using finite element simulations of various loudspeaker and microphone configurations in a shoebox-shaped room. It is shown that the accuracy of the proposed method is related to the sizes, shapes, number, and positions of the devices inside a room. A simplified version of the proposed method, which can significantly reduce computational effort, is also presented. The proposed method and its simplified version can simulate room transfer functions more accurately than currently available image source methods and can aid the development and evaluation of speech and acoustic signal processing algorithms, including speech enhancement, acoustic scene analysis, and acoustic parameter estimation.

Joint B0 and Image Reconstruction in Low-Field MRI by Physics-Informed Deep-Learning.

We present a model-based image reconstruction approach based on unrolled neural networks which corrects for image distortion and noise in low-field ( B0 ∼ 50mT) MRI. Utilising knowledge about the underlying physics, a novel network architecture (SH-Net) is introduced which involves the estimation of spherical harmonic coefficients to guarantee a spatially smooth field map estimate. The SH-Net is integrated in an end-to-end trainable model which jointly estimates the B0-field map as well as the image. Experiments were conducted on retrospectively simulated low-field data of human knees. We compare our model to different model-based approaches at distinct noise levels and various B0-field distributions. Our results show that our physics-informed neural network approach outperforms the purely model-based methods by improving the PSNR up to 11.7% and the RMSE up to 86.3%. Our end-to-end trained model-based approach outperforms existing methods in reconstructing image and B0-field maps in the low-field regime. low-field MRI is becoming increasingly more popular as it enables access to MR in challenging situations such as intensive care units or resource poor areas. Our method allows for fast and accurate image reconstruction in such low-field imaging with B0-inhomogeneity compensation under a wide range of various environmental conditions.

Analyzing Earth's Position based on the Anisotropic Characteristics of Cosmic Microwave Background Radiation

This paper elaborates on the results of an exhaustive study regarding the Earth's position in the universe based on the Cosmic Microwave Background (CMB) radiation map, which is the latest discovery in modern astronomy. CMB radiation provides crucial insights into the early distribution of mass and energy in the universe. The aim of this research is to understand the theory and mechanisms behind the formation of polarity structures in the CMB and analyze their correlation with Earth's position in the overall structure of the universe. Intensity measurement data of CMB radiation published by COBE, WMAP, and Planck present temperature distribution data in coordinates in FITS (Flexible Image Transport System) format files. Subsequently, a spherical harmonic transformation is performed to obtain spherical harmonic coefficients a_lm as equations that represent dipole, quadrupole, octupole models, and various other multipole models. The analysis of the correlation in the temperature distribution of CMB radiation involves detailing various patterns found in the dipole, quadrupole, and octopole models, demonstrating quasi-symmetry characteristics with Earth at its center. An analysis of anisotropic CMB data yields an interesting hypothesis that the Earth's position plays a role in shaping the structure of the universe on a certain scale. Even more extremely, it can be said that Earth is at the center of the universe. This finding prompts profound contemplation about Earth's position in the structure of the cosmos, opening the door for further research in this field.

Accuracy Assessment of High-Degree Geopotential Models in Peninsular Malaysia

The selection of the accurate geoid model is essential for geoid determination in specific regions. Many improvements to the basic theory of more reliable data are available for numerical modelling studies. All of the innovations have led to the development of a sequence of global geopotential models of increasing spherical harmonic degree and order and the resolution of the geopotential models. There are hundreds of geopotential models that can be downloaded in ICGEM. All geopotential ICGEM models have accuracy and resolution based on the degree and order of spherical harmonic coefficients. This study has two high-degree geopotential models: the latest one, XGM2019e_2159, and another geopotential model, EGM2008. The accuracy evaluation of two geopotential models needs to be evaluated based on terrestrial gravity data and existing GPS points data to choose which geopotential model is the best-fit geopotential models with different degrees and order in terms of spherical harmonic coefficients using the Root Mean Square method. The statistical analysis of the geopotential model derived based on the existing GPS points shows that the degree and order 720 for XGM2019e_2019 are the best-fit geopotential models in Peninsular Malaysia. The result also indicates that the gravity anomaly derived from EGM2008 with the maximum degree and order of 2190 regarding the spherical harmonic coefficient is the most accurate geoid model that can be used as a reference over Peninsular Malaysia. Overall, it can be concluded that XGM2019e_2019 and EGM2008 are the best-fit geopotential models for Peninsular Malaysia.

Deriving the Terrestrial Water Storage Anomaly From GRACE Spherical Harmonic Coefficients Using a Convolutional Neural Network

AbstractTerrestrial water storage anomaly (TWSA), derived from Gravity Recovery and Climate Experiment (GRACE) satellites, has been widely used in hydrology studies. The inversion is commonly achieved by truncating and filtering spherical harmonic coefficients (SHC), whereby the result is characterized by leakage error and low resolution. It remains unclear whether machine learning methods can help resolve this challenging issue. In this study, we present a convolutional neural network (CNN) approach to correct TWSA from GRACE SHC by leveraging the knowledge of the leakage effect determined from global hydrological models (GHMs) and land surface models (LSMs). The CNN approach is implemented in three representative regions in China, that is, the human‐impacted Haihe River Basin, the nature‐impacted Yangtze River Basin and the model‐limited Tibetan Plateau. The results show the following: (a) The recovery performance of CNN at the basin scale is better than that at the grid scale, and the grid‐scale recovery is significantly influenced by the spatial heterogeneity of TWSA and the input GHM/LSMs; (b) The more accurate the GHM/LSMs used for training, the better the recovery performance of CNN; and (c) The trained model retains comparable performance in deriving the TWSA time series from GRACE SHC when compared to that derived from other methods (i.e., scaling factor and mass concentration solutions) with average r = 0.90 and RMSE = 21.50 mm. This study highlights the potential of machine learning to supplement conventional correction methods when deriving the TWSA from GRACE SHC by utilizing signal restoration knowledge learned from multiple accurate GHMs/LSMs.