## 1 Introduction  
The full-waveform inversion (FWI) method can build high-precision velocity models of subsurface media using observed seismic data [1, 2]. The FWI method was originally proposed based on acoustic media [3, 4]. By matching the synthetic data simulated by the scalar acoustic wave equation with the observed data, the FWI method finally obtains a high-resolution P-wave velocity model of an underground medium [5]. The key point of the FWI algorithm is the calculation of the derivative of the objective function with respect to the model parameter to be determined, which is known as the gradient of FWI [6]. The adjoint-state method reduces the computational complexity of the gradient calculation, making practical application of the FWI algorithm possible [7]. In recent years, acoustic FWI has successfully inverted industrial-scale real 3D marine data [8] and refracted waves in ocean bottom mode data [9]. However, modeling of elastic wave propagation through elastodynamic theory is required for reflection analysis [10, 11]. In real cases, when seismic waves propagate in underground media, if they encounter a reflecting interface, energy conversion will occur and converted waves will be generated. The elastic wave equation can better describe the actual wavefield propagation characteristics and laws than the acoustic wave equation [12, 13]. The elastic full-waveform inversion (EFWI) method was first implemented in 1986 [14] and practical data applications appeared in 1990 [15]. Due to the consideration of multiple parameters, the nonlinearity of EFWI is stronger than that of acoustic FWI and the computational complexity is higher [16; 17]. To reduce the nonlinearity of EFWI, a good strategy is to choose a more convex objective function, which can help to mitigate the cycle-skipping issue during inversion [18]. The objective function of FWI is initially the L2 norm of the residuals between observed and synthetic data. When the travel time difference between the synthetic and observed data is greater than half a cycle, the cycle-skipping phenomenon will occur, resulting in the inversion falling into a local minimum. In addition to the differences in velocity models, many factors may result in cycle skipping during inversion, such as amplitude errors caused by data acquisition or processing, source wavelet errors, etc. To mitigate the cycle-skipping issue caused by amplitude errors, the cross-correlation objective function was proposed [19]. Using normalized global cross-correlation to construct the objective function can overcome the impact of the entire trace amplitude errors on FWI. The local cross-correlation objective function can further overcome the impact of local amplitude errors on inversion [20]. In addition, Laplace domain waveform inversion [21] and envelope inversion [22; 23] can construct large-scale velocity models of subsurface media when seismic data lack low-frequency information. The multiscale FWI strategy based on these methods can obtain high-precision velocity models [24]. [PERSON] and [PERSON] [25] combined the cross-correlation operation and the envelope operation to construct a new objective function, which can simultaneously overcome the effects of amplitude errors and missing low frequencies on the inversion. [PERSON] et al. [26] extended this method to elastic media and added the convolution operation to construct a new objective function, simultaneously overcoming the effects of amplitude errors, missing low frequencies, and source wavelet inaccuracy on the inversion. The source wavelet is another key factor affecting the accuracy of EFWI. For frequency-domain FWI, source wavelet estimation can be performed relatively simply according to the L2 norm inversion method [27; 28]. During the inversion process, the source wavelet needs to be updated simultaneously with the velocity.