Spherical Harmonic Model Research Articles

Utility of Slepian basis functions for modeling near-surface and satellite magnetic anomalies of the Australian lithosphere

The utility of frequency- and space-limited spherical harmonic Slepian basis functions for magnetic anomaly modeling over restricted spherical patches of the Earth was investigated using combined near-surface scalar and CHAMP satellite vector observations from Australia and adjacent marine areas. In particular, Slepian spherical harmonic models up to degree 360 were studied for modeling anomaly features of 1° (~111 km) and longer over a 25°-radius cap centered on Australia. Relative to the roughly 130,000 coefficients required for global spherical harmonic modeling, less than 5% of this number of coefficients is sufficient for effective localized Slepian modeling. Slepian coefficients have maximum power over the spherical cap and may be exploited for estimating the magnetic anomaly vectors and gradients to all orders within the working precision of the observations. The Earth cap modeled by Slepian coefficients is also more efficient in accommodating local crustal constraints from drilling and other geological and geophysical studies for interpreting the associated magnetic anomaly data registered in spherical coordinates. In general, Slepian spherical harmonic modeling is well suited for combining spectrally diverse compilations of near-surface and satellite magnetic observations over any spatially restricted spherical cap of the Earth or other planetary body.Graphical The utility of frequency- and space-limited Slepian spherical harmonic basis functions up to degree 360 was studied for modeling near-surface scalar and CHAMP satellite magnetic anomalies of 1° (~111 km) and longer over a 25°-radius cap centered on Australia. Slepian spherical harmonic modeling is well suited for combining spectrally divorce compilations of near-surface and satellite magnetic observations. It is also very efficient for updating global spherical harmonic models for new regional data and providing perspectives on how magnetic lithospheric anomalies vary up to satellite altitudes that are not available from standard upward and downward anomaly continuations. For example, Map A shows the Australians magnetic anomaly estimates at 10 km altitude from the Slepian model jointly constrained by near-surface and CHAMP satellite magnetic observations that minimize the differences between Maps B and C of upward continued near-surface and downward continued CHAMP data, respectively. Map D, on the other hand, shows the Slepian model estimates at 275 km altitude that minimize differences between Maps E and F of downward continued CHAMP and upward continued near-surface data, respectively. Any continuous, however, is not unique and subject to measurement and modeling errors so that its interpretation at location lacking observations requires considerable care.

Modeling real shim fields for very high degree (and order) B0 shimming of the human brain at 9.4 T.

To describe the process of calibrating a B0 shim system using high-degree (or high order) spherical harmonic models of the measured shim fields, to provide a method that considers amplitude dependency of these models, and to show the advantage of very high-degree B0 shimming for whole-brain and single-slice applications at 9.4 Tesla (T). An insert shim with up to fourth and partial fifth/sixth degree (order) spherical harmonics was used with a Siemens 9.4T scanner. Each shim field was measured and modeled as input for the shimming algorithm. Optimal shim currents can therefore be calculated in a single iteration. A range of shim currents was used in the modeling to account for possible amplitude nonlinearities. The modeled shim fields were used to compare different degrees of whole-brain B0 shimming on healthy subjects. The ideal shim fields did not correctly shim the subject brains. However, using the modeled shim fields improved the B0 homogeneity from 55.1 (second degree) to 44.68 Hz (partial fifth/sixth degree) on the whole brains of 9 healthy volunteers, with a total applied current of 0.77 and 6.8 A, respectively. The necessity of calibrating the shim system was shown. Better B0 homogeneity drastically reduces signal dropout and distortions for echo-planar imaging, and significantly improves the linewidths of MR spectroscopy imaging. Magn Reson Med 79:529-540, 2018. © 2017 International Society for Magnetic Resonance in Medicine.

Frequency-domain optical tomographic image reconstruction algorithm with the simplified spherical harmonics (SP3) light propagation model

We introduce here the finite volume formulation of the frequency-domain simplified spherical harmonics model with n-th order absorption coefficients (FD-SPN) that approximates the frequency-domain equation of radiative transfer (FD-ERT). We then present the FD-SPN based reconstruction algorithm that recovers absorption and scattering coefficients in biological tissue. The FD-SPN model with 3rd order absorption coefficient (i.e., FD-SP3) is used as a forward model to solve the inverse problem. The FD-SP3 is discretized with a node-centered finite volume scheme and solved with a restarted generalized minimum residual (GMRES) algorithm. The absorption and scattering coefficients are retrieved using a limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm. Finally, the forward and inverse algorithms are evaluated using numerical phantoms with optical properties and size that mimic small-volume tissue such as finger joints and small animals.The forward results show that the FD-SP3 model approximates the FD-ERT (S12) solution within high accuracy; the average errors in the phase (<3.7%) and the amplitude (<7.1%) of the partial current at the boundary are reported. From the inverse results we find that the absorption and scattering coefficient maps are more accurately reconstructed with the SP3 model than those with the SP1 model. Therefore, this work shows that the FD-SP3 is an efficient model for optical tomographic imaging of small-volume media with non-diffuse properties both in terms of computational time and accuracy as it requires significantly lower CPU time than the FD-ERT (S12) and also it is more accurate than the FD-SP1.

Airborne geoid mapping of land and sea areas of East Malaysia

Abstract This paper describes the development of a new geoid-based vertical datum from airborne gravity data, by the Department of Survey and Mapping Malaysia, on land and in the South China Sea out of the coast of East Malaysia region, covering an area of about 610,000 square kilometres. More than 107,000 km flight line of airborne gravity data over land and marine areas of East Malaysia has been combined to provide a seamless land-to-sea gravity field coverage; with an estimated accuracy of better than 2.0 mGal. The iMAR-IMU processed gravity anomaly data has been used during a 2014-2016 airborne survey to extend a composite gravity solution across a number of minor gaps on selected lines, using a draping technique. The geoid computations were all done with the GRAVSOFT suite of programs from DTU-Space. EGM2008 augmented with GOCE spherical harmonic model has been used to spherical harmonic degree N = 720. The gravimetric geoid first was tied at one tide-gauge (in Kota Kinabalu, KK2019) to produce a fitted geoid, my_geoid2017_fit_kk. The fitted geoid was offset from the gravimetric geoid by +0.852 m, based on the comparison at the tide-gauge benchmark KK2019. Consequently, orthometric height at the six other tide gauge stations was computed from HGPS Lev = hGPS - Nmy_geoid2017_.t_kk. Comparison of the conventional (HLev) and GPS-levelling heights (HGPS Lev) at the six tide gauge locations indicate RMS height difference of 2.6 cm. The final gravimetric geoidwas fitted to the seven tide gauge stations and is known as my_geoid2017_fit_east. The accuracy of the gravimetric geoid is estimated to be better than 5 cm across most of East Malaysia land and marine areas

Gradient nonlinearity calibration and correction for a compact, asymmetric magnetic resonance imaging gradient system

Due to engineering limitations, the spatial encoding gradient fields in conventional magnetic resonance imaging cannot be perfectly linear and always contain higher-order, nonlinear components. If ignored during image reconstruction, gradient nonlinearity (GNL) manifests as image geometric distortion. Given an estimate of the GNL field, this distortion can be corrected to a degree proportional to the accuracy of the field estimate. The GNL of a gradient system is typically characterized using a spherical harmonic polynomial model with model coefficients obtained from electromagnetic simulation. Conventional whole-body gradient systems are symmetric in design; typically, only odd-order terms up to the 5th-order are required for GNL modeling. Recently, a high-performance, asymmetric gradient system was developed, which exhibits more complex GNL that requires higher-order terms including both odd- and even-orders for accurate modeling. This work characterizes the GNL of this system using an iterative calibration method and a fiducial phantom used in ADNI (Alzheimer’s Disease Neuroimaging Initiative). The phantom was scanned at different locations inside the 26 cm diameter-spherical-volume of this gradient, and the positions of fiducials in the phantom were estimated. An iterative calibration procedure was utilized to identify the model coefficients that minimize the mean-squared-error between the true fiducial positions and the positions estimated from images corrected using these coefficients. To examine the effect of higher-order and even-order terms, this calibration was performed using spherical harmonic polynomial of different orders up to the 10th-order including even- and odd-order terms, or odd-order only. The results showed that the model coefficients of this gradient can be successfully estimated. The residual root-mean-squared-error after correction using up to the 10th-order coefficients was reduced to 0.36 mm, yielding spatial accuracy comparable to conventional whole-body gradients. The even-order terms were necessary for accurate GNL modeling. In addition, the calibrated coefficients improved image geometric accuracy compared with the simulation-based coefficients.

Gravity Compensation Using EGM2008 for High-Precision Long-Term Inertial Navigation Systems.

The gravity disturbance vector is one of the major error sources in high-precision and long-term inertial navigation applications. Specific to the inertial navigation systems (INSs) with high-order horizontal damping networks, analyses of the error propagation show that the gravity-induced errors exist almost exclusively in the horizontal channels and are mostly caused by deflections of the vertical (DOV). Low-frequency components of the DOV propagate into the latitude and longitude errors at a ratio of 1:1 and time-varying fluctuations in the DOV excite Schuler oscillation. This paper presents two gravity compensation methods using the Earth Gravitational Model 2008 (EGM2008), namely, interpolation from the off-line database and computing gravity vectors directly using the spherical harmonic model. Particular attention is given to the error contribution of the gravity update interval and computing time delay. It is recommended for the marine navigation that a gravity vector should be calculated within 1 s and updated every 100 s at most. To meet this demand, the time duration of calculating the current gravity vector using EGM2008 has been reduced to less than 1 s by optimizing the calculation procedure. A few off-line experiments were conducted using the data of a shipborne INS collected during an actual sea test. With the aid of EGM2008, most of the low-frequency components of the position errors caused by the gravity disturbance vector have been removed and the Schuler oscillation has been attenuated effectively. In the rugged terrain, the horizontal position error could be reduced at best 48.85% of its regional maximum. The experimental results match with the theoretical analysis and indicate that EGM2008 is suitable for gravity compensation of the high-precision and long-term INSs.

Hybrid coupling of a one-dimensional energy-transport Schrödinger system

We consider one dimensional coupled classical-quantum models for quantum semiconductor device simulations. The coupling occurs in the space variable: the domain of the device is divided into a region with strong quantum effects (quantum zone) and a region where quantum effects are negligible (classical zone). In the classical zone, transport in diffusive approximation is modeled through diffusive limits of the Boltzmann transport equation. It can lead to an energy-transport model, obtained using a Spherical Harmonic Expansion model as intermediate step. The quantum transport is described by the Schrodinger equation. The aim of this work is to focus on the derivation of boundary conditions at the interface between the classical and quantum regions. Numerical simulations are provided for a resonant tunneling diode with the energy-transport model.

A comparison of navigators, snap-shot field monitoring, and probe-based field model training for correcting B0 -induced artifacts in T2*-weighted images at 7 T.

To compare methods for estimating B0 maps used in retrospective correction of high-resolution anatomical images at ultra-high field strength. The B0 maps were obtained using three methods: (1) 1D navigators and coil sensitivities, (2) field probe (FP) data and a low-order spherical harmonics model, and (3) FP data and a training-based model. Data from nine subjects were acquired while they performed activities inducing B0 field fluctuations. Estimated B0 fields were compared with reference data, and the reductions of artifacts were compared in corrected T2* images. Reduction of sum-of-squares difference relative to a reference image was evaluated, and Method 1 yielded the largest artifact reduction: 27 ± 15%, 20 ± 18% (mean ± 1 standard deviation) for deep breathing and combined deep breathing and hand motion activities. Method 3 performed almost as well (24 ± 18%, 15 ± 17%), provided that adequate training data were used, and Method 2 gave a similar result (21 ± 16%, 19 ± 17%). This study confirms that all of the investigated methods can be used in retrospective image correction. In terms of image quality, Method 1 had a small advantage, whereas the FP-based methods measured the B0 field slightly more accurately. The specific strengths and weaknesses of FPs and navigators should therefore be considered when determining which B0 -estimation method to use. Magn Reson Med 78:1373-1382, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

Upward continuation of Dome-C airborne gravity and comparison with GOCE gradients at orbit altitude in east Antarctica

An airborne gravity campaign was carried out at the Dome-C survey area in East Antarctica between the 17th and 22nd of January 2013, in order to provide data for an experiment to validate GOCE satellite gravity gradients. After typical filtering for airborne gravity data, the cross-over error statistics for the few crossing points are 11.3 mGal root mean square (rms) error, corresponding to an rms line error of 8.0 mGal. This number is relatively large due to the rough flight conditions, short lines and field handling procedures used. Comparison of the airborne gravity data with GOCE RL4 spherical harmonic models confirmed the quality of the airborne data and that they contain more high-frequency signal than the global models. First, the airborne gravity data were upward continued to GOCE altitude to predict gravity gradients in the local North-East-Up reference frame. In this step, the least squares collocation using the ITGGRACE2010S field to degree and order 90 as reference field, which is subtracted from both the airborne gravity and GOCE gravity gradients, was applied. Then, the predicted gradients were rotated to the gradiometer reference frame using level 1 attitude quaternion data. The validation with the airborne gravity data was limited to the accurate gradient anomalies (TXX, TYY, TZZ and TXZ) where the long-wavelength information of the GOCE gradients has been replaced with GOCO03s signal to avoid contamination with GOCE gradient errors at these wavelengths. The comparison shows standard deviations between the predicted and GOCE gradient anomalies TXX, TYY, TZZ and TXZ of 9.9, 11.5, 11.6 and 10.4 mE, respectively. A more precise airborne gravity survey of the southern polar gap which is not observed by GOCE would thus provide gradient predictions at a better accuracy, complementing the GOCE coverage in this region.

Исследование вариаций гравитационного поля Земли по данным космической геодезии

Abstract: The article considers the results of the analysis of temporal variations of the geopotential coefficients obtained under Gravity Recovery and Climate Experiment (GRACE) mission operations. For this purpose were used spherical harmonic models for the time span from 2002 to 2015 years. Methods of the spectral analysis were used for determination of type of the parametric model which was used for computations of temporal variations of the geopotential coefficients. The estimates of temporal variations were investigated by Fisher test. The article shows the geoid height variations caused by changes in the harmonic coefficients. For verification of the results were performed comparisons with outcomes in the other researches.

An Online Gravity Modeling Method Applied for High Precision Free-INS.

For real-time solution of inertial navigation system (INS), the high-degree spherical harmonic gravity model (SHM) is not applicable because of its time and space complexity, in which traditional normal gravity model (NGM) has been the dominant technique for gravity compensation. In this paper, a two-dimensional second-order polynomial model is derived from SHM according to the approximate linear characteristic of regional disturbing potential. Firstly, deflections of vertical (DOVs) on dense grids are calculated with SHM in an external computer. And then, the polynomial coefficients are obtained using these DOVs. To achieve global navigation, the coefficients and applicable region of polynomial model are both updated synchronously in above computer. Compared with high-degree SHM, the polynomial model takes less storage and computational time at the expense of minor precision. Meanwhile, the model is more accurate than NGM. Finally, numerical test and INS experiment show that the proposed method outperforms traditional gravity models applied for high precision free-INS.

Spherical cap harmonic analysis of regional magnetic anomalies based on CHAMP satellite data

We used CHAMP satellite vector data and the latest IGRF12 model to investigate the regional magnetic anomalies over mainland China. We assumed satellite points on the same surface (307.69 km) and constructed a spherical cap harmonic model of the satellite magnetic anomalies for elements X, Y, Z, and F over Chinese mainland for 2010.0 (SCH2010) based on selected 498 points. We removed the external field by using the CM4 model. The pole of the spherical cap is 36N° and 104°E, and its half-angle is 30°. After checking and comparing the root mean square (RMS) error of ΔX, ΔY, and ΔZ and X, Y, and Z, we established the truncation level at K max = 9. The results suggest that the created China Geomagnetic Referenced Field at the satellite level (CGRF2010) is consistent with the CM4 model. We compared the SCH2010 with other models and found that the intensities and distributions are consistent. In view of the variation of F at different altitudes, the SCH2010 model results obey the basics of the geomagnetic field. Moreover, the change rate of X, Y, and Z for SCH2010 and CM4 are consistent. The proposed model can successfully reproduce the geomagnetic data, as other data-fitting models, but the inherent sources of error have to be considered as well.

Persistent high paleosecular variation activity in southern hemisphere for at least 10 000 years

Direct observations of the geomagnetic field show that secular variation is strong in the Atlantic hemisphere, and comparatively reduced in the Pacific region. The dipole has been decaying since at least 1840 AD, driven by growth and migration of reverse flux patches in the southern hemisphere. We investigate whether anything like this modern pattern of geomagnetic secular variation persists and can be detected in global paleomagnetic field models. Synthesis of results from two new time-varying spherical harmonic models shows that geographically distinct geomagnetic secular variation extends to at least 10 000 BP. The models use the same database but differ in methodology, leading to some regional differences in results. Consistent large-scale surface features include strong average fields in the northern hemisphere and weaker fields with greater overall variability in the south. Longitudinal structure is present, with weaker average fields in the western Pacific than in the east, and prominent negative inclination anomalies extending beneath Indonesia, across Africa and to Brazil, but weaker anomalies in the central Pacific. Marginally positive inclination anomalies occur west of the Americas. Paleosecular variation activity peaks at high southern latitudes, and there is a pattern of reduced activity at equatorial and mid-latitudes beneath the Pacific. Although the dipole has exhibited both growth and decay over the interval 0–10 000 BP, our results show that geomagnetic paleosecular variation is preferentially focused in similar geographic regions to secular variation seen in the modern field.

Application of the spherical harmonic gravity model in high precision inertial navigation systems

The spherical harmonic gravity model (SHM) may, in general, be considered as a suitable alternative to the normal gravity model (NGM), because it represents the Earth’s gravitational field more accurately. However, the high-resolution SHM has never been used in current inertial navigation systems (INSs) due to its extremely complex expression. In this paper, the feasibility and accuracy of a truncated SHM are discussed for application in a real-time free-INS with a precision demand better than 0.8 nm h−1. In particular, the time and space complexity are analyzed mathematically to verify the feasibility of the SHM. Also, a test on a typical navigation computer shows a storable range of cut-off degrees. To further evaluate the appropriate degree and accuracy of the truncated SHM, analyses of covariance and truncation error are proposed. Finally, a SHM of degree 12 is demonstrated to be the appropriate model for routine INSs in the precision range of 0.4–0.75 nm h−1. Flight simulations and road tests show its outstanding performance over the traditional NGM.

A Swarm lithospheric magnetic field model to SH degree 80

The Swarm constellation of satellites was launched in November 2013 and since then has delivered high-quality scalar and vector magnetic field measurements. A consortium of several research institutions was selected by the European Space Agency to provide a number of scientific products to be made available to the scientific community on a regular basis. In this study, we present the dedicated lithospheric field inversion model. It uses carefully selected magnetic field scalar and vector measurements from the three Swarm satellites between March 2014 and December 2015 and directly benefits from the explicit expression of the magnetic field gradients by the lower pair of Swarm satellites. The modeling scheme is a two-step one and relies first on a regional modeling approach that is very sensitive to small spatial scales and weak signals which we seek to describe. The final model is built from adjacent regional solutions and consists in a global spherical harmonics model expressed between degrees 16 and 80. The quality of the derived model is assessed through a comparison with independent models based on Swarm and the CHAMP satellites. This comparison emphasizes the high level of accuracy of the current model after only 2 years of measurements but also highlights the possible improvements which will be possible once the lowest two satellites reach lower altitudes.

Stability of the Divergent Barotropic Rossby-Haurwitz Wave

Stability of the barotropic Rossby-Haurwitz wave is investigated using the numerical models on the global domain. The Rossby-Haurwitz wave under investigation is composed of the basic zonal flow of super-rotation and a finite amplitude spherical harmonic wave. The Rossby-Haurwitz wave is given as either steady or unsteady wave by adjusting the strength of the super-rotating zonal flow. Stability as well as the growth rate of the wave in the numerical simulation is determined by comparing the perturbation amplitude at two different time stages. Unstable modes of the Rossby-Haurwitz wave exhibited a horizontal structure composing of various zonal-wavenumber components. The vorticity perturbation for some modes showed a discontinuity around the area of weak flow, which was found robust regardless of the horizontal resolution of the model. Fourier finite element model was shown to generate the unstable mode in earlier stage of the time integration due to less accuracy compared to the spherical harmonic spectral model. Taking the overall accuracy of the models into consideration, the time by which the unstable mode begin to dominate over the spherical harmonic wave was estimated.

Modeling the Jovian magnetic field and its secular variation using all available magnetic field observations

AbstractWe present new models of Jupiter's internal magnetic field and secular variation from all available direct measurements from three decades of spacecraft observation. A regularized minimum norm approach allows the creation of smooth, numerically stable models displaying a high degree of structure. External field from the magnetodisk is modeled iteratively for each orbit. Jupiter's inner magnetosphere is highly stable with time, with no evidence for variation with solar activity. We compare two spherical harmonic models, one assuming a field constant in time and a second allowing for linear time variation. Including secular variation improves data fit with fewer additional parameters than increasing field complexity. Our favored solution indicates a ∼0.012% yr−1 increase in Jupiter's dipole magnetic moment from 1973 to 2003; this value is roughly one quarter of that for Earth. Inaccuracies in determination of the planetary reference frame cannot explain all the observed secular variation. Should more structure be allowed in the solutions, we find the northern hemispherical configuration resembles recent models based on satellite auroral footprint locations, and there is also evidence of a possible patch of reversed polar flux seen at the expected depth of the dynamo region, resembling that found at Earth and with implications for the Jovian interior. Finally, using our preferred model, we infer flow dynamics at the top of Jupiter's dynamo source. Though highly speculative, the results produce several gyres with some symmetry about the equator, similar to those seen at Earth's core‐mantle boundary, suggesting motion on cylinders aligned with the rotation axis.

Application of Slepian theory for improving the accuracy of SH‐based global ionosphere models in the Arctic region

AbstractDue to significant energy resources in polar regions, they have emerged as strategic parts of the world. Consequently, various researches have been funded in order to study these areas in further details. This research intends to improve the accuracy of spherical harmonic (SH)‐based Global Ionospheric Models (GIMs) by reconstructing a new map of ionosphere in the Arctic region. For this purpose, the spatiospectral concentration is applied to optimize the base functions. It is carried out using the Slepian theory which was developed by Simons. Here the new base functions and the corresponding coefficients are derived from the SH models for the polar regions. Then, VTEC (vertical total electron content) is reconstructed using Slepian functions and the new coefficients. Reconstructed VTECs and the VTECs derived from SH models are compared to the estimates of this parameter, which are directly derived from dual‐frequency GPS measurements. Three International Global Navigation Satellite Systems Service stations located in the northern polar region have been used for this purpose. The starting and ending day of year of adopted GPS data are 69 and 83, respectively, (totally 15 successive days) of the year 2013. According to the obtained results, on average, application of Slepian theory can improve accuracy of the GIM by 1 to 2 total electron content unit (TECU) (1 TECU = 1016 el m−2) in the Arctic region.

Local Ionospheric Modeling Using the Localized Global Ionospheric Map and Terrestrial GPS

global ionosphere maps are generated on a daily basis at CODE using data from about 200 GPS/GLONASS sites of the IGS and other institutions. The vertical total electron content is modeled in a solargeomagnetic reference frame using a spherical harmonics expansion up to degree and order 15. The spherical Slepian basis is a set of bandlimited functions which have the majority of their energy concentrated by optimization inside an arbitrarily defined region, yet remain orthogonal within the spatial region of interest. Hence, they are suitable for decomposing the spherical harmonic models into the portions that have significant strength only in the selected areas. In this study, the converted spherical harmonics to the Slepian bases were updated by the terrestrial GPS observations by use of the least-squares estimation with weighted parameters for local ionospheric modeling. Validations show that the approach adopted in this study is highly capable of yielding reliable results.

简化球谐近似模型的图形处理器加速求解

简化球谐近似模型的图形处理器加速求解