Terms Of Spherical Harmonic Expansion Research Articles

Effects of observer peculiar motion on the isotropic background frequency spectrum: From the monopole to higher multipoles

Context.The observer peculiar motion produces boosting effects in the anisotropy pattern of the considered background with frequency spectral behaviours related to its frequency spectrum.Aims.We study how the frequency spectrum of the background isotropic monopole emission is modified and transferred to the frequency spectra at higher multipoles,ℓ. We performed the analysis in terms of spherical harmonic expansion up to a certain value ofℓmax, for various models of background radiation, spanning the range between the radio and the far-infrared.Methods.We derived a system of linear equations to obtain the spherical harmonic coefficients and provide the explicit solutions up toℓmax = 6. These are written as linear combinations of the signals atN = ℓmax + 1 colatitudes. We take advantage of the symmetry property of the associated Legendre polynomials with respect toπ/2, which allows for the separation of the system into two subsystems: (1) forℓ = 0 and even multipoles and (2) for odd multipoles. This improves the accuracy of the solutions with respect to an arbitrary choice of the adopted colatitudes.Results.We applied the method to different types of monopole spectra represented in terms of analytical or semi-analytical functions, that is, four types of distortions of the photon distribution function of the cosmic microwave background and four types of extragalactic background signals superimposed onto the cosmic microwave background’s Planckian spectrum, along with several different combinations of these types. We present our results in terms of the spherical harmonic coefficients and of the relationships between the observed and the intrinsic monopole spectra, as well as in terms of the corresponding all-sky maps and angular power spectra. For certain representative cases, we compare the results of the proposed method with those obtained using more computationally demanding numerical integrations or map generation and inversion. The method is generalized to the case of an average map composed by accumulating data taken with sets of different observer velocities, as is necessary when including the effect of the observer motion relative to the Solar System barycentre.Conclusions.The simplicity and efficiency of the proposed method can significantly alleviate the computational effort required for accurate theoretical predictions and for the analysis of data derived by future projects across a variety of cases of interest. Finally, we discuss the superposition of the cosmic microwave background intrinsic anisotropies and of the effects induced by the observer peculiar motion, exploring the possibility of constraining the intrinsic dipole embedded in the kinematic dipole in the presence of background spectral distortions.

Geometrical Modeling of Concrete Microstructure for the Assessment of ITZ Percolation

Percolation is considered to be a critical factor affecting the transport properties of multiphase materials. In the case of concrete, the transport properties are strongly dependent on the interfacial transition zone (ITZ), which is a thin layer of cement paste next to aggregate particles. It is not computationally simple to assess ITZ percolation in concrete, as the geometry and topology of this phase is complex. While there are many advanced models that analyze the behavior of concrete, they are mostly based on the use of spherical or ellipsoidal shapes for the geometry of the aggregate inclusions. These simplified shapes may become unsatisfactory in many simulations, including the assessment of ITZ percolation. This paper deals with geometrical modeling of the concrete microstructure using realistic shapes of aggregate particles, the geometry of which is represented in terms of spherical harmonic expansion. The percolation is assessed using the hard core – soft shell model, in which each randomly-placed aggregate particle is surrounded by a shell of constant thickness representing ITZ.

On inversion for mass distribution from global (time-variable) gravity field

The well-known non-uniqueness of the gravitational inverse problem states that the external gravity field, even if completely and exactly known, cannot uniquely determine the density distribution of the body that produces the gravity field. In this paper, we provide conceptual insight by examining the problem in terms of spherical harmonic expansion of the global gravity field. By comparing the multipoles and the moments of the density function, we show that in 3-D the degree of knowledge deficiency in trying to inversely recover the density distribution from an external gravity field solution is (n + 1)(n + 2)/2 − (2n + 1) = n(n − 1)/2 for each harmonic degree n. On the other hand, on a 2-D spherical shell we show via a simple relationship that the inverse solution of the surface density distribution is unique. The latter applies quite readily in the inversion of time-variable gravity signals (such as those observed by the GRACE space mission) where the sources largely come from the Earth's surface over a wide range of timescales.

High resolution global phase velocity distributions

We present global phase velocity distributions for fundamental mode Love and Rayleigh waves in the period range 40 to 150 seconds. The models, expressed in terms of spherical harmonic expansions up to degree and order 40, have beed derived from 28,479 Love wave and 33,662 Rayleigh wave measurements. The measurements were made using an automatic procedure based on non‐linear waveform inversion. We show that the results are characterized by lateral resolving radii between 500 and 850 km, a significant reduction from previous work. The improved global resolution is largely due to the inclusion of many more major arc measurements. The power spectra of the models are much whiter than has previously been found. The results show great similarity to the phase velocity distributions derived from the recent a priori model 3‐SMAC, in which the primary relevant features are the crustal thickness and crustal velocity distributions, the cooling oceanic lithosphere and the thick (300km) lithosphere of the continental cratons. The results thus confirm these features. Differences, for example, in the strengths of the crust, ridge and craton signatures should lead to refinement in the thermal and constitutive parameters upon which the a priori model depends.

Topography of the core–mantle boundary and lateral homogeneity of the liquid core

Separate inversions of travel-time residuals of waves reflected from and transmitted through the core-mantle boundary yield similar results in terms of spherical harmonic expansion of its topography up to degree and order 4. This indicates absence of detectable lateral heterogeneity in the liquid core. The relief of the boundary is ±6km but its flattening, determined by the c02 term, does not depart significantly from the hydrostatic theory.

Anomalies in the time‐averaged paleomagnetic field and their implications for the lower mantle

The proposal that the time‐averaged paleomagnetic field is not strictly that of a geocentric axial dipole but that of an axial dipole displaced slightly northward from the geocenter is examined in terms of spherical harmonic expansions. Standard procedures for spherical harmonic expansions such as are applied to the present instantaneous geomagnetic field are not necessarily applicable to paleomagnetic data. A technique is proposed that enhances the time‐averaging process that is a necessary part of determining the paleomagnetic field. This involves analyzing the inclination anomaly ΔI around latitude strips to determine the zonal harmonics and the declination anomaly ΔD around longitude sectors to determine the first nonzonal harmonics. The technique is applied to a carefully selected data set covering the last 5 m.y. All data are divided into normal or reversed sets providing 266 land‐based points (171 normal, 95 reversed) and 100 deep‐sea core results with inclination data only (50 normal, 50 reversed). These data demonstrate clearly that the time‐averaged field is not simply that of a geocentric axial dipole and also that it is different for the normal and reversed fields. With respect to the present field the zonal harmonics of the time‐averaged field are reduced less than the nonzonal harmonics, the indication being that westward and/or eastward drift of the nondipole field is dominant over 5 m.y. The magnitude of the second zonal harmonic suggests that on the average, paleomagnetic poles could be in error by about 3° when calculated by the usual geocentric axial dipole assumption. The data show clear asymmetries between the northern and southern hemispheres and possibly between the oceanic (Pacific) and continental hemispheres. Possible explanations of these asymmetries include variations in topography and/or temperature at the core‐mantle interface. Large‐scale asymmetries in the boundary conditions at this interface could affect the magnetic field through alteration of the convection pattern in the core. The asymmetries correlate with other global asymmetries such as the distribution of continents and oceans and the locations of subduction zones. They also apparently correlate with seismic velocity anomalies observed in the lower mantle. All these may reflect large‐scale inhomogeneities related to dynamic processes that have occurred in the lower mantle. Although all the Gaussian coefficients determined change sign when the axial dipole field reverses, there are significant differences between the normal and reversed fields, suggesting that the distribution of sources for these fields is not identical. This is consistent with the cyclonic convection models for reversal of the geomagnetic field proposed by Parker and Levy.