# Seismic Multi-Parameter Full-Waveform Inversion Based on Rock Physical Constraints
## 1 Introduction  
Full-waveform inversion (FWI) for multi-parameter seismic wavefield reconstruction presents a strongly nonlinear and ill-posed optimization challenge, wherein conventional gradient-based algorithms frequently converge to local minima [1, 2, 3, 4]. Furthermore, multi-parameter FWI is significantly influenced by crosstalk between parameters, meaning it is easy to map the error of one parameter to the update of other parameters [5, 6, 7, 8], thus undermining its reliability as a basis for lithological interpretation. The regularization methodology enhances the stability of ill-posed inverse problems by integrating prior information about the model during the inversion process and takes the estimation of model parameters as constraint conditions [9, 10].  
The Tikhonov regularization method constitutes one of the most frequently utilized approaches in addressing ill-posed inversion problems. By integrating a regularization term with prior model constraints, it enhances stability to the inversion procedure. However, its isotropic smoothing nature inevitably introduces edge smear artifacts across geological interfaces. The Total Variation (TV) model constraint methodology can effectively mitigate the over-smoothing of boundary values that transpires during Tikhonov regularizationinversion [11; 12]. However, the improvement effect is highly contingent upon the selection of the regularization factor. When the regularization factor is properly chosen, more accurate inversion results can be obtained. In contrast, an improper selection of the regularization factor undermines the equilibrium between model parameters and residuals, consequently impairing the inversion accuracy. In practical engineering applications, seismic data volumes are customarily of considerable magnitude. The selection experiment of regularization factors not only demands copious amounts of time but also depends on the experience of professionals, which restricts the practical deployment of FWI in practical engineering to a certain extent. Domestic and foreign scholars have developed a variety of methods for the selection of regularization parameters. Among them, the Generalized Cross-Validation (GCV) method [13] and the L-curve method [14] are the most commonly adopted. Nevertheless, the application of adaptive regularization parameter selection in practical applications still confronts several predicaments. For example, the minimization of the GCV function is typically extremely smooth, rendering it difficult to locate mathematically [15]. Additionally, when errors are highly correlated, it becomes exceedingly difficult to obtain accurate regularization parameters [11]. With regard to the L-curve method, it requires a large number of experiments to determine the appropriate selection of regularization parameters [16]. Hence, it is necessary to study the optimization theory and method for the multi-parameter FWI of seismic waves.  
Based on geological information, this paper considers regularized FWI based on geological prior information. In practical engineering, the prior information of underground strata, such as lithological information and petrophysical relationship information, can be obtained in various ways, such as drilling, laboratory analysis, and acoustic logging in research areas [17]. Such information, sourced directly from underground media, holds substantial referential significance and serves as dependable prior knowledge for inversion. The approach of introducing the above prior information as constraints into the inversion can be termed \"rock physical constraint inversion\". The multi-parameter FWI expounded in this paper is intended to realize simultaneous high-precision imaging of wave velocity and formation density. To this end, the focus of this paper is primarily focused on prior information relevant to wave velocity and formation density and the conversion of such information into constrained information to achieve constrained inversion.