Performing exactly the same calculation on the different GRACE

spherical harmonics leads to slightly different results.

it is possible to express the ODF f in a (generally infinite) sum of '

spherical harmonics', where the

spherical harmonics are the two-dimensional analogues to the sines and cosines in the Fourier series (see Section 2).

We compared the produced TWSA time series with three standard monthly gridded GRACE TWSA products from CSR, GFZ, and JPL, all based on RL05

spherical harmonics (Figure 2).

To alleviate this problem, a

spherical harmonics expansion-based MLFMA (SE-MLFMA) for the dyadic form of surface integral equation (SIE) was proposed [6].

(15) cannot be resolved by traditional expansion in

spherical harmonics with coefficients given by the monomials [r.sup.l].

In this work [S.sub.lm] ([theta], [phi]) is the complex ([S.sub.lm] = [Y.sub.lm]) or real

spherical harmonics. The definition of phases for the complex

spherical harmonics ([Y.sup.*.sub.lm] = [Y.sub.l-m]) differs from the Condon- Shortley phases [27] by the sign factor [(-1).sup.m] and can be defined as

On the one hand, there is a conceptually very appealing view on

spherical harmonics, due to Lebeau and Guillemin, based on the Szegoo kernel of the Fermat quadric.

It is easy to show that the [sup.1][Y.sup.ls]-RNSSTH and [sup.2][Y.sup.ls]- MRNSSTH harmonics for s = 0 are transformed into the scalar

spherical harmonics:

Based on the LO data, the

spherical harmonics of the Moon (degree 4 in the tesserals and degree 8 in the zonals) were first published in 1968 [16]; later that year, the existence of large mascons under the lunar ringed maria was also discovered [2].

Garboczi (2002) was probably the first employing the

spherical harmonics (SH) technique to mathematically describe in three-dimensions particle shapes applying it to aggregates used in concrete.

The ellipsoidal harmonics in these spheroconical coordinates replace the

spherical harmonics of spherical polar coordinates.

Among their topics are summaries of finite difference method and pseudospectral methods, global Radial basis functions for solving partial differential equations, applications to geo-modeling in spherical domains,

spherical harmonics, and some node distribution strategies.