# Two-Dimensional Full-Waveform Joint Inversion of Surface Waves Using Phases and Z/H Ratios
## 1 Introduction  
Surface waves are essential for resolving crust and upper mantle structures at various length scales. Surface waves have dispersive characteristics; that is, a surface wave travels with different velocities at different frequencies in the wave-propagation process. This feature has been well studied and widely used to image Earth structures, especially for the S-wave velocity model. Teleseismic surface-wave tomography was used in earlier studies, and it offered a good resolution for the upper mantle [1, 2, 3]. However, it is difficult to constrain structures in the shallow to middle crust due to the low frequency of such waves. In recent decades, ambient-noise tomography was developed, which can extract the Green's function of a surface wave from ambient-noise data, and it significantly improved the surface-wave-dispersion data coverage (mostly less than a 40 s period). Short- to intermediate-period surface waves have been widely used for imaging regional- or local-scale crustal structures [4, 5, 6, 7]. However, the surface-wave-dispersion curve is only sensitive to the integral feature of the velocity structure of a depth range, resulting in ambiguities during inversion. Studies have shown that the amplitude information of a surface wave can also help improve the velocity structure. The Z/H ratio (or ellipticity) of a Rayleigh wave is always defined as the amplitude ratio of the vertical and horizontal components of the fundamental mode of the Rayleigh wave. The Z/H ratio of a Rayleigh wave is only controlled by the structurebeenath the recording seismic station [8; 9], and it is more sensitive to shallow structures than dispersion in the same period range. Therefore, the Rayleigh wave's \(Z/\mathrm{H}\) ratio can provide extra constraints to reduce the ambiguity of surface wave tomography, and in particular, it can improve the imaging of shallow sedimentary structures. Because of the resolution gap caused by the different data sensitivities, independent inversions using the surface-wave-dispersion curve or the \(Z/\mathrm{H}\) ratio cannot provide a unified S-wave velocity model. The joint inversion of the surface-wave-dispersion curve and the \(Z/\mathrm{H}\) ratio has been proposed for imaging the complementary velocity structure. In particular, joint inversion based on ray theory has been widely used in imaging crust and upper mantle structures of various length scales [10]. However, this approach does not account for the complexities of wave propagation in complex media, and may lead to inaccurate tomographic results. Recent advances in the numerical methods involving the wave equation combined with developments in high-performance computation (HPC) have enabled the routine simulation of seismic wave propagation in realistic 2D and 3D Earth models based on the spectral-element method (SEM) ([PERSON] and [PERSON], 2002a, 2002b). Utilizing these seismic-wave simulations, adjoint-state methods can efficiently incorporate the full nonlinearity of the wave propagation into iterative seismic inversions [11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22]. Based on these advances, [PERSON] et al. [23] proposed ambient-noise adjoint tomography for 3D crustal structures by matching the empirical Green's function (EGF) and the synthetic Green's function (SGF) calculated based on the SEM. In addition, full-wave 3D ambient-noise tomography has been developed, and has been used to image the structure of the Cascades, the Ontong Java Plateau, and the eastern North American margin [24; 25; 26]. [PERSON] et al. [27] derived the equations for 3D full-wave sensitivity kernels of the \(Z/\mathrm{H}\) ratio of a Rayleigh wave using the scattering-integral method [28; 29; 30; 31]. However, the computational cost and storage requirement of 3D adjoint tomography are large. In contrast, 2D adjoint tomography is much more computationally efficient.