## 1. Introduction

[2] In geophysical tomography, model parameterization is important for its influences on results of forward modeling and inversion. For forward modeling, it could be a critical factor for accuracy and efficiency in the computations of data prediction and sensitivity matrices. For inversion, an ideal model parameterization is expected to match the desired data-adaptive resolution with minimum effective elements in the sensitivity matrix. These arguments become more evident for large-scale seismic tomography, in which massive data are usually involved, and the spatial sampling of data is often highly nonuniform as inherited from the nature of earthquake distribution and mostly on-land seismic observatories.

[3] Conventional parameterization schemes tend to fall into two extreme categories that emphasize either spatial resolution, such as boxcar functions, or spectral resolution such as spherical harmonics (SpH). Among them, SpH are natural bases for the Earth; accordingly, they are often used as lateral model bases in three-dimensional (3-D) global mantle tomography, particularly for those models developed using normal-mode asymptotic theories and long-period waveform data. Here we discuss models of this kind. Comparisons between them and models developed with primarily traveltime data can be found in a review article by *Romanowicz* [2003].

[4] The first 3-D global mantle model of shear wave velocity (Vs) was parameterized in terms of SpH up to degree 8 [*Woodhouse and Dziewonski*, 1984]. Since then, with quickly accumulated seismic data and improved computation capability, the degree of SpH of global tomographic models has been increasing steadily [e.g., *Li and Romanowicz*, 1996; *Masters et al.*, 1996; *Ekström and Dziewonski*, 1998; *Mégnin and Romanowicz*, 2000].

[5] Another important reason for the popularity of SpH in global waveform tomography is likely due to its computational advantage in the forward stage, as they can be converted to Fourier series along the source-receiver great circle path, thus providing efficient and accurate analytical solutions for path integrals [e.g., *Woodhouse and Dziewonski*, 1984].

[6] At the current stage, major long-wavelength features among various global mantle models are relatively consistent and interest has now focused on the finer details of Earth structure. Nevertheless, limitations of using SpH as model bases are apparent when high-degree models are desired. For global function bases like SpH, no component can be ignored when constructing a local sensitivity kernel. In other words, the resulting sensitivity matrix will be fully loaded with the size proportional to (ℓ_{max} + 1)^{2}, where *l*_{max} is the maximum degree used in model expansion, and the follow-up inversion will be greatly hampered with increasing degree and/or amount of data. Furthermore, it has been pointed out that some features in the SpH-based models may be biased from spectral leakage [*Trampert and Snieder*, 1996; *Chiao and Kuo*, 2001], which is similar to the aliasing effect when truncated Fourier series is adopted to expand a function with high-degree signals.

[7] An alternative approach to obtain finer details of the mantle structure is high-resolution regional tomography in areas with dense data coverage. Instead of SpH, local function bases are commonly invoked for such studies.

[8] Clearly, a proper parameterization scheme for forward modeling is not necessarily suitable for a follow-up inversion, and vice versa. To maximize parameterization merits in both stages, and add flexibilities to inversion schemes, we propose a two-step model parameterization approach in which different model bases of sensitivity matrix are used in each stage through a simple matrix transformation.

[9] We demonstrate this approach through an experimental study. We first introduce theories to be used in forward computation, inversion and basis transformation of sensitivity matrices, and discuss benefits provided by the two-step model parameterization. We then detail its application to a regional tomography, waveform tomography for Vs structure of the Pacific upper mantle using long-period Rayleigh waves. Finally, results derived from a simple damping scheme and wavelet-based inversion are compared and discussed.