The understanding how the power is distributed in the wavenumber domain is essential for the proper parameterization of the inverse problem for three dimensional structure. Also, the distribution of the spectral power as a function of depth in the mantle may be an extremely valuable diagnostic in assessing the applicability of various mantle convection models, for example.

In studies such as those of Dziewonski [1984], Woodhouse and Dziewonski [1984] or Tanimoto [1990] it was assumed a priori that the lateral variations in the structure could be represented by a spherical harmonic expansion truncated at some low degree such as 6 or 8. But, if this assumption were not true, then these truncated solutions could be aliased by the presence of the signal with a considerable power at the higher wavenumbers.

Tanimoto [1991] was among the first to point out the significance of the predominance of the large-scale heterogeneity in the earth. Su and Dziewonski [1991, 1992] collected a large set of the absolute SS travel times with the globally distributed mid-path reflection points. The spherical harmonic expansion of these residuals shows that on the log-log scale, the power is approximately constant up to degree 6, but then it decreases rapidly---roughly as ---for the higher orders. A very similar spectrum (up do degree 8) is obtained by predicting the SS data from 3-D models such as an upper mantle model M84C [ Woodhouse and Dziewonski, 1984].

Therefore, the upper mantle structure retrieved in the low order expansion need not be significantly biased by the higher wavenumbers structure. The low wavenumber corner ( 5--8) has also been identified in the P-velocity model of Inoue et al. [1990], particularly in the lowermost mantle; in the middle mantle the corner is shifted towards a higher degree (10--12). Other global geophysical observables also have the same character: Love wave phase velocities at 100 seconds [ Zhang and Tanimoto, 1991], free air gravity and the continent-ocean function.

In geometrical terms, this indicates that the heterogeneity is organized in a form of `patches' characterized by the same sign of the anomaly, having a rather regular shape and predominant dimensions of several thousand kilometers. The smaller wavelength heterogeneity is superimposed on these patches, but does not obscure the planetary scale pattern.