## 1. Introduction

[2] It is well known that microseisms are excited at random by standing ocean surface waves. The typical frequency of microseisms at about 0.2 Hz approximately doubles the typical frequency of ocean surface waves through nonlinear interactions [*Hasselman*, 1963; *Longuet-Higgens*, 1950]. Microseisms are the main source of noise for seismic observation because they mask seismic signals from earthquakes.

[3] Using their random excitation properties by contraries, one-dimensional (1-D) *S* wave velocity structures at shallow depth (≤5 km) have been explored since an early work by *Aki* [1957, 1965]. In the method known as spatial autocorrelation method, the dispersion curves of surface waves are obtained from the cross spectra between many pairs of stations of an array. The measured dispersion curves are inverted for obtaining a 1-D *S* wave velocity structure under the array.

[4] Recently, *Shapiro et al.* [2005] performed a cross-correlation analysis of long sequences of the ambient seismic noise at around 0.1 Hz to obtain a group velocity anomaly of Rayleigh waves due to the lateral heterogeneity of the crust in Southern California. They inverted the measured anomalies for obtaining a group velocity map. This method is called ambient noise surface wave tomography. The ambient noise tomography is theoretically justified by the fact that a cross correlation function between two stations provides its Green's function between the stations [e.g., *Snieder*, 2004]. These studies resulted in group speed maps at short periods (7.5–15 s) that display a striking correlation with the principal geological units in California with low-speed anomalies corresponding to the major sedimentary basins and high-speed anomalies corresponding to the igneous core of the main mountain regions. Group velocity maps have also been obtained at larger scales and longer periods across much of Europe [*Yang et al.*, 2007], in South Korea at very short periods [*Cho et al.*, 2007], and in Tibet at long periods [*Yao et al.*, 2006]. However, three-dimensional (3-D) *S* wave velocity inversion has not been performed because of complex propagation of the observed waves.

[5] For performing 3-D inversion, the phase information of the observed surface waves is important. However, only group velocity maps were obtained by the ambient noise tomography in most cases because the propagation of short-period surface waves, which are most sensitive to the crust, is too complicated. The waves are preferentially attenuated and scattered; therefore, the propagation distance exceeding 100 km distorted their waveforms significantly. In order to use the phase information, both dense instrumentation and widely distributed stations are required. The densed Hi-net array data of tiltmeters in Japan [*Obara et al.*, 2005] enables us to use their phase information.

[6] For performing 3-D wave velocity inversion, we developed a new method to fully utilize the waveform information. For modeling the observed cross spectra, we formulated synthetic cross spectra based on a normal mode theory with an assumption of stochastic stationary excitation of surface waves [*Fukao et al.*, 2002; *Nishida and Fukao*, 2007]. The method we have used is similar to partitioned waveform inversion [*Nolet*, 1990; *van der Lee and Nolet*, 1997]. Our method has three steps: (1) measurement of dispersion curves using many pairs of the observed cross spectra as in the spatial autocorrelation method [*Aki*, 1957], and inversion of the dispersion curves for obtaining local 1-D *S* wave velocity models; (2) estimation of path-averaged 1-D *S* wave velocity structures by modeling observed cross spectra; and (3) inversion of path-averaged structures for obtaining 3-D *S* wave velocity structure (0.1° × 0.1° × 1 km grid from the surface to a depth of 50 km) using ray approximation.