Fully normalized spherical cap harmonics: application to the analysis of sea-level data from TOPEX/POSEIDON and ERS-1

We introduce two sets of fully normalized harmonics for the spectral analysis of functions defined on a spherical cap. The harmonics are the products of Fourier functions and the fully normalized associated Legendre functions of non-integer degree. Using Sturm-Liouville theory for boundary-value problems, we present two convenient and stable formulae for computing the zeros of the associated Legendre functions that form two sets of orthogonal functions. Formulae for the stable numerical evaluation of the fully normalized associated Legendre functions of non-integer degree that avoid the gamma function are also derived. The result from the expansions of sea-level anomaly from altimetry into Set 2 fully normalized cap harmonics shows fast convergence of the series, and the degree variances decay rapidly without aliasing effects. The zero-degree coefficients (Set 2) of sea-level anomaly from TOPEX/ POSEIDON (T/P) and ERS-1 indicate an El Niño event during 1993 January-1993 July, and a La Niño event during 1993 November-1994 July, although the ERS-1 result is less obvious. Ocean circulations over the South China Sea and the Kuroshio area are clearly identified with the low-degree expansions of sea-surface topography (SST) from T/P and ERS-1. A cold-core eddy of 4° in diameter centred at 17.5°N, 118 E was detected with the expansion of SST from T/P cycle 47, and a property of the cap harmonics is used to compute this eddy's kinetic energy. The kinetic energy is at a low in winter and high in summer, and its variation seems to be periodic with an amplitude of 0.4 m² S^-2.     

A fast converging and anti-aliasing algorithm for Green's functions in terms of spherical or cylindrical harmonics

Geophysical observables are generally related to earth structure and source parameters in a complicated non-linear way. Consequently, a large number of forward modelling processes are commonly necessary to obtain a satisfactory estimate of such parameters from observed data. The most time-consuming part of the forward modelling is the computation of the Green's functions of the different earth models to be tested. In this study, we present a fast converging algorithm: the differential transform method for the computation of Green's functions in terms of spherical or cylindrical harmonics. In this method, a deconvolutable high-pass filter is used to enhance the numerical significance of the far-field spectrum of Green's functions. Compared with existing fast converging algorithms such as the Kummer's transformation and the disc factor method, the differential transform method is more efficient except for the extremely near-source region. The new method can be used to suppress numerical phases (non-physical seismic signals) associated with the aliasing effect that may arise in synthetic seismograms when the latter are computed from a windowed wavenumber (or slowness) spectrum. The numerical efficiency of the new method is demonstrated via two representative tests.     

Orthogonality and mean squares of the vector fields given by spherical cap harmonic potentials

It is well known that the vector fields derived from spherical harmonics are orthogonal over the sphere. It is now shown that the vector fields derived from spherical cap harmonics are orthogonal over the cap, to the same extent as the cap potentials are, and expressions are given for their mean squares.     

Modelling of mantle postglacial relaxation in axisymmetric geometry with a composite rheology and a glacial load interpolated by adjusted spherical harmonics analysis

Although studies on glacial isostatic adjustment usually assume a purely linear rheology, we have previously shown that mantle relaxation after the melting of Laurentide ice sheet is better described by a composite rheology including a non-linear term. This modelling is, however, based on axially symmetric geometry and glacial forcing derived from ICE-3G and suffers from a certain amount of arbitrariness in the definition of the ice load. In this work we apply adjusted spherical harmonics analysis to interpolate the ice thicknesses of ICE-3G and ICE-1 glaciological models. This filters out the non-axisymmetric components of the ice load by considering only the zonal terms in the spherical harmonics expansion. The resulting load function is used in finite-element simulation of postglacial rebound to compare composite versus purely linear rheology. Our results confirm that composite rheology can explain relative sea level (RSL) data in North America significantly better than a purely linear rheology. The performance of composite rheology suggests that in future investigations, it may be better to use this more physically realistic creep law for modelling mantle deformation induced by glacial forcing.     

Spectral-finite-element approach to two-dimensional electromagnetic induction in a spherical earth

We present a spectral-finite-element approach to the 2-D forward problem for electromagnetic induction in a spherical earth. It represents an alternative to a variety of numerical methods for 2-D global electromagnetic modelling introduced recently (e.g. the perturbation expansion approach, the finite difference scheme). It may be used to estimate the effect of a possible axisymmetric structure of electrical conductivity of the mantle on surface observations, or it may serve as a tool for testing methods and codes for 3-D global electromagnetic modelling. The ultimate goal of these electromagnetic studies is to learn about the Earth's 3-D electrical structure.
Since the spectral-finite-element approach comes from the variational formulation, we formulate the 2-D electromagnetic induction problem in a variational sense. The boundary data used in this formulation consist of the horizontal components of the total magnetic intensity measured on the Earth's surface. In this the variational approach differs from other methods, which usually use spherical harmonic coefficients of external magnetic sources as input data. We verify the assumptions of the Lax-Milgram theorem and show that the variational solution exists and is unique. The spectral-finite-element approach then means that the problem is parametrized by spherical harmonics in the angular direction, whereas finite elements span the radial direction. The solution is searched for by the Galerkin method, which leads to the solving of a system of linear algebraic equations. The method and code have been tested for Everett & Schultz's (1995) model of two eccentrically nested spheres, and good agreement has been obtained.