Terms Of Spherical Harmonic Coefficients Research Articles

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.

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.

COMPUTATION OF GRAVITY FIELD FUNCTIONALS WITH A LOCALIZED LEVEL ELLIPSOID

The description of the earth’s gravity field is usually expressed in terms of spherical harmonic coefficients, derived from global geopotential models. These coefficients may be used to evaluate such quantities as geoid undulations, gravity anomalies, gravity disturbances, deflection of the vertical, etc. To accomplish this, a global reference normal ellipsoid, such as WGS84 and GRS80, is required to provide the computing reference surface. These global ellipsoids, however, may not always provide the best fit of the local geoid and may provide results that are aliased. In this study, a regional or localized geocentric level ellipsoid is used alongside the EGM2008 to compute gravity field functionals in the state of Johor. Residual gravity field quantities are then computed using GNSS-levelled and raw gravity data, and the results are compared with both the WGS84 and the GRS80 equipotential surfaces. It is demonstrated that regional level ellipsoids may be used to compute gravity field functionals with a better fit, provided the zero-degree spherical harmonic is considered. The resulting residual quantities are smaller when compared with those obtained with global ellipsoids. It is expected that when the remove-compute-restore method is employed with such residuals, the numerical quadrature of the Stoke’s integral may be evaluated on reduced gravity anomalies that are smoother compared to when global equipotential surfaces are used

A comparison of GRACE temporal gravity field models recovered with different processing details

SUMMARYThe Gravity Recovery and Climate Experiment (GRACE) mission has been providing abundant information regarding the mass changes of the Earth in terms of time-series of temporal gravity field models since 2002. To derive temporal gravity field models with high accuracy, many methods have been developed. In this paper, we focus on the variational equation integration approach. The main works can be summarized as follows: (1) analysing the quality of GRACE Level1B RL02 and RL03 data, including accelerometer observations (ACC1B), star camera measurements (SCA1B) and K-Band low-low Satellite-to-Satellite Tracking (SST) range-rate (KBRR) data (KBR1B); (2) discussing the influence of arc-specific parameters and arc length on gravity field recovery and (3) comparing two different methods used for sensitivity matrix generation, namely, a numerical integration method and the method of variation of constants, from the perspectives of accuracy and efficiency, respectively. Based on these analyses, discussions and comparisons, a new time-series of GRACE monthly gravity field models in terms of spherical harmonic coefficients completed to degree and order 60, called SWJTU-GRACE-RL02p, was derived by using the modified variational equation integration approach bashed on GRACE Level1B RL03 data, covering the period from April 2002 to October 2011 with some gaps in between due to poor quality or missing GRACE data. Thus we are looking at the results some 10yrs in the past. The differences between the traditional variational equation integration approach and the approach that we used are mainly as follows: (1) according to the GRACE data quality, the arc length is no longer a constant in the determination of temporal gravity field models; (2) the kinematic empirical parameters, which are mainly designed to remove the bias and drifts in KBRR residuals, are abandoned and (3) the method of variation of constants developed at the Astronomical Institute of the University of Bern (AIUB) and used to solve the system of variational equations associated with constrained pulses and piecewise constant accelerations is used to calculate the sensitivity matrices of accelerometer bias parameters to improve the calculation efficiency and ensure the calculation accuracy. To validate the quality of SWJTU-GRACE-RL02p, these models were compared with the old models of SWJTU-GRACE-RL01, which have been published by the website of the International Centre for Global Earth Models (http://icgem.gfz-potsdam.de/series), and the official products [i.e. the RL05 and RL06 versions of GRACE LEVEL2 at the Centre for Space Research (CSR), Jet Propulsion Laboratory (JPL) and GeoForschungsZentrum (GFZ)]. Compared to the RL06 version of official models, the models of SWJTU-GRACE-RL02p present competitive performance for global mass changes. Furthermore, these models show less noise and a higher signal strength over some local areas with large mass changes than the models of SWJTU-GRACE-RL01. The comparisons between SWJTU-GRACE-RL02p and a variety of other models including official models, GLDAS, models provided by EGSIEM and daily solutions released by ITSG indicate that our approach and the data processing details presented in this paper provide an alternative strategy for the recovery of temporal gravity field models from GRACE-type data.

Joint analysis of selected GRACE monthly spherical harmonic solutions and monthly MASCON solutions

Study presented in this paper is focused on comparison and statistical assessment of differences between the selected Level 2 products of the satellite mission Gravity Recovery and Climate Experiment (GRACE). Global monthly gravity field models in terms of spherical harmonic coefficients produced by three institutes of GRACE Science Data System are compared with the partially independent MASCON global gravity field model. Detailed comparison and statistical analysis of differences is performed in 5 selected river basins: Amazon, Congo, Danube, Yenisei and Lena. For each spherical harmonic solution, 8 different filtrations available at International Center for Global Gravity Field Models (ICGEM) are tested over the time span from April 2002 to July 2016. Fischer test at two significance levels 10% and 5% has been performed in order to qualify the statistical significance between the particular solutions.

Propagation and estimation of the dynamical behaviour of gravitationally interacting rigid bodies

Next-generation planetary tracking methods, such as interplanetary laser ranging (ILR) and same-beam interferometry (SBI) promise an orders-of-magnitude increase in the accuracy of measurements of solar system dynamics. This requires a reconsideration of modelling strategies for the translational and rotational dynamics of natural bodies, to ensure that model errors are well below the measurement uncertainties.The influence of the gravitational interaction of the full mass distributions of celestial bodies, the so-called figure-figure effects, will need to be included for selected future missions. The mathematical formulation of this problem to arbitrary degree is often provided in an elegant and compact manner that is not trivially relatable to the formulation used in space geodesy and ephemeris generation. This complicates the robust implementation of such a model in operational software packages. We formulate the problem in a manner that is directly compatible with the implementation used in typical dynamical modelling codes: in terms of spherical harmonic coefficients and Legendre polynomials. An analytical formulation for the associated variational equations for both translational and rotational motion is derived.We apply our methodology to both Phobos and the KW4 binary asteroid system, to analyze the influence of figure-figure effects during estimation from next-generation tracking data. For the case of Phobos, omitting these effects during estimation results in relative errors of 0.42% and 0.065% for the bar{C}_{20} and bar{C}_{22} spherical harmonic gravity field coefficients, respectively. These values are below current uncertainties, but orders of magnitude larger than those obtained from past simulations for accurate tracking of a future Phobos lander, showing the need to apply the methodology outlined in this manuscript for selected future missions.

Spherical Harmonic Smoothing for Localizing Coherent Sound Sources

Correlated or coherent sources can cause the localization performance of subspace-based beamformers to deteriorate. To solve this problem, various smoothing techniques have been proposed for the localization of multiple coherent sound sources. A common principle of smoothing techniques is to increase the rank of a covariance matrix by constructing multiple subarrays in the space, time, or frequency domain. The construction of such subarrays, however, requires the satisfaction of strong assumptions regarding the microphone positions or the temporal/spectral structures of the signals. In this paper, we propose a spherical harmonic smoothing technique that can perform smoothing in terms of spherical harmonic coefficients only. Unlike other smoothing techniques, the proposed technique constructs subarrays of spherical harmonic coefficients and uses them to increase the number of linearly independent observations in a signal subspace. Subarray construction in the spherical harmonics domain enables the accurate localization of multiple coherent sources even at a single frequency. The proposed technique can be applied to an arbitrary microphone array as long as the spherical harmonic coefficients can be measured up to a finite order.

Inverse Rendering and Relighting From Multiple Color Plus Depth Images.

We propose a novel relighting approach that takes advantage of multiple color plus depth images acquired from a consumer camera. Assuming distant illumination and Lambertian reflectance, we model the reflected light field in terms of spherical harmonic coefficients of the bi-directional reflectance distribution function and lighting. We make use of the noisy depth information together with color images taken under different illumination conditions to refine surface normals inferred from depth. We first perform refinement on the surface normals using the first order spherical harmonics. We initialize this non-linear optimization with a linear approximation to greatly reduce computation time. With surface normals refined, we formulate the recovery of albedo and lighting in a matrix factorization setting, involving second order spherical harmonics. Albedo and lighting coefficients are recovered up to a global scaling ambiguity. We demonstrate our method on both simulated and real data, and show that it can successfully recover both illumination and albedo to produce realistic relighting results.

First GOCE gravity field models derived by three different approaches

Three gravity field models, parameterized in terms of spherical harmonic coefficients, have been computed from 71 days of GOCE (Gravity field and steady-state Ocean Circulation Explorer) orbit and gradiometer data by applying independent gravity field processing methods. These gravity models are one major output of the European Space Agency (ESA) project GOCE High-level Processing Facility (HPF). The processing philosophies and architectures of these three complementary methods are presented and discussed, emphasizing the specific features of the three approaches. The resulting GOCE gravity field models, representing the first models containing the novel measurement type of gravity gradiometry ever computed, are analysed and assessed in detail. Together with the coefficient estimates, full variance-covariance matrices provide error information about the coefficient solutions. A comparison with state-of-the-art GRACE and combined gravity field models reveals the additional contribution of GOCE based on only 71 days of data. Compared with combined gravity field models, large deviations appear in regions where the terrestrial gravity data are known to be of low accuracy. The GOCE performance, assessed against the GRACE-only model ITG-Grace2010s, becomes superior at degree 150, and beyond. GOCE provides significant additional information of the global Earth gravity field, with an accuracy of the 2-month GOCE gravity field models of 10 cm in terms of geoid heights, and 3 mGal in terms of gravity anomalies, globally at a resolution of 100 km (degree/order 200).

Growth and Approximation of Entire Harmonic Functions in 𝑅𝑛, 𝑛 > 3

Abstract We study the growth of functions which are harmonic in any number of variables. The results are expressed in terms of spherical harmonic coefficients as well as by the approximation error of the harmonic function with (𝑝, 𝑞)-growth.

Satellite gradiometry using a satellite pair

The GRACE mission has substantiated the low–low satellite-to-satellite tracking (LL-SST) concept. The LL-SST configuration can be combined with the previously realized high–low SST concept in the CHAMP mission to provide a much higher accuracy. The line of sight (LOS) acceleration difference between the GRACE satellite pair, the simplest form of the combined observable, is mostly used for mapping the global gravity field of the Earth in terms of spherical harmonic coefficients. As an alternative observable, a linear combination of the gravitational gradient tensor components is proposed. Being a one-point function and having a direct relation with the field geometry (curvature of the field at the point) are two noteworthy achievements of the alternative formulation. In addition, using an observation quantity that is related to the second-instead of the first-order derivatives of the gravitational potential amplifies the high-frequency part of the signal. Since the transition from the first- to the second-order derivatives includes the application of a finite-differences scheme, the high-frequency part of the noise is also amplified. Nevertheless, due to the different spectral behaviour of signal and noise, in the end the second-order approach leads to improved gravitational field resolution. Mathematical formulae for the gradiometry approach, for both linear and higher-degree approximations, are derived. The proposed approach is implemented for recovery of the global gravitational field and the results are compared with those of LOS acceleration differences. Moreover, LOS acceleration difference residuals are calculated, which are at the level of a few tenths of mGal. Error analysis shows that the residuals of the estimated degree variances are less than 10−3. Furthermore, the gravity anomaly residuals are less than 2 mGal for most points on the Earth.

On the trends in global circulation in 50 years of NCEP/NCAR reanalyses

To represent objective characteristics of circulation patterns, the spectral structure of both stratospheric and tropospheric fields is analysed in terms of spherical harmonic coefficients. The aim of this study is to compare the long-term behaviour of circulation patterns and their connections to some “extra-terrestrial” influence and circulation indices as well. A huge database of reanalyses from NCEP is used with appropriate spectral coefficients available for period 1948-now four times per day in 28 levels for relative vorticity, divergence, temperature, etc. Temporal analysis of significant spherical harmonics is introduced as well as the comparison of their changes with respect to the changes of solar activity and the phase of QBO. Generally, these wavenumbers mark no or a little positive trends, but for some distinctive wavenumbers representing individual patterns of polar vortex shape negative trend can be seen as well at the first part of time series. Changes with respect to solar activity cycle and QBO phase are presented for whole spectral domain not too well expressed, quite significant results are analysed for a few distinct wavenumbers, mainly in case of separation of time series for 1948–1972 and 1973–1997 periods. Moreover, long-term natural variability connected to the extra-terrestrial influence is studied as well as interannual variability with the emphasis to the QBO. Cross-correlation analysis shows significant difference for selected wavenumbers against solar flux F10.7 in separated periods with some correlation for the first one and rather no significant correlation for the second period. Long-term changes in the variability of the circulation patterns are analysed by means of wavelet analysis access to the series of selected wavenumbers displaying development of some periodicities in time. The connection with the tropospheric circulation is also discussed in terms of coefficients of spherical harmonics mostly for all cases in this study. The paper does not provide definite evidence for connections of atmospheric circulation with decadal solar variability, finally, this could be hardly achieved by statistical analysis only, but the aim of this study is to show possible connections between some helio-geophysical or extra-terrestrial parameters and circulation parameters and their long-term changes with emphasis to some distinctive circulation patterns or shapes of polar vortex.

On the relationship between radiance and irradiance: determining the illumination from images of a convex Lambertian object.

We present a theoretical analysis of the relationship between incoming radiance and irradiance. Specifically, we address the question of whether it is possible to compute the incident radiance from knowledge of the irradiance at all surface orientations. This is a fundamental question in computer vision and inverse radiative transfer. We show that the irradiance can be viewed as a simple convolution of the incident illumination, i.e., radiance and a clamped cosine transfer function. Estimating the radiance can then be seen as a deconvolution operation. We derive a simple closed-form formula for the irradiance in terms of spherical harmonic coefficients of the incident illumination and demonstrate that the odd-order modes of the lighting with order greater than 1 are completely annihilated. Therefore these components cannot be estimated from the irradiance, contradicting a theorem that is due to Preisendorfer. A practical realization of the radiance-from-irradiance problem is the estimation of the lighting from images of a homogeneous convex curved Lambertian surface of known geometry under distant illumination, since a Lambertian object reflects light equally in all directions proportional to the irradiance. We briefly discuss practical and physical considerations and describe a simple experimental test to verify our theoretical results.

Spherical harmonic analysis, aliasing, and filtering

The currently practiced methods of harmonic analysis on the sphere are studied with respect to aliasing and filtering. It is assumed that a function is sampled on a regular grid of latitudes and longitudes. Then, transformations to and from the Cartesian plane yield formulations of the aliasing error in terms of spherical harmonic coefficients. The following results are obtained: 1) The simple quadratures method and related methods are biased even with band-limited functions. 2) A new method that eliminates this bias is superior to Colombo's method of least squares in terms of reducing aliasing. 3) But, a simple modification of the least-squares model makes it identical to the new method as one is the dual of the other. 4) The essential elimination of aliasing can only be effected with spherical cap averages, not with the often used constant angular block averages.

Annual Cycle and Spatial Spectra of Earth Emitted Radiation at Large Scales

Monthly averaged, resolution enhanced global distributions of the earth's emitted radiation, as measured by the Nimbus-6 Earth Radiation Budget (ERB) wide field of view radiometers, have been analyzed for 1 year of data from July 1975 to June 1976. These distributions are expressed in terms of spherical harmonic coefficients, and time and space variability of the emitted radiation field is studied in terms of these coefficients. The average annual distribution accounts for 78 percent of the space-time power, and the annual cycle accounts for 17 percent of the power. Spatial variations over the globe are described in terms of degree variance, and longitudinal variations are described in terms of spectral power as a function of latitude. The longitudinal spectra were found to vary strongly with time.

Spherical harmonic analysis of the geomagnetic field: an example of a linear inverse problem

The geomagnetic field at the Earth's surface may be represented in terms of spherical harmonic coefficients. The determination of the coefficients from observed values of the field, and the evaluation of a field value from a set of coefficients, may be regarded as linear inverse problems of discrete and continuous types respectively. Well-known procedures of inverse theory are found to provide new insight into some of the hazards of conventional methods of analysis, as well as suggesting some novel methods of approach. The methods are applied to a dataset of measurements of all three components of the field from 80 magnetic observations. Determining coefficients from observations is usually effected by least-squares analysis after truncating the harmonic series at some degree and order. Inverse methods all seem to have their counterpart in least-squares theory, but they do not help decide the best truncation level. An alternative approach, Parker's method of linear inference, has been developed to apply to spherical harmonic series. The results are similar to least squares but the error bounds are more pessimistic. In evaluating the field components from a finite set of coefficients it is only possible to find a weighted average of the field value over a certain area of a spherical surface. The averaging function appropriate to the usual spherical harmonic sum formula has oscillations and is not a well-localized average. Alternative formulae are suggested which give a better, localized average. The methods are applied to fields downward continued to the core-mantle boundary. Most striking results are obtained when the covariance matrix for the spherical harmonic coefficients is used to damp the field determination. This procedure retains the necessary information about the original distribution of the data and it is found to eliminate some of the features of magnetic charts in the oceans where data coverage is poor, showing that these anomalies are not real.