# Seismic Multi-Parameter Full-Waveform Inversion Based on Rock Physical Constraints
## 2 Materials and Methods
### Construction of Rock Physical Prior Information
#### 2.1.2 Wave Velocity and Formation Density  
For the strata situated in an identical sedimentary environment, this study constructs the mathematical expression of prior information with the help of the petrophysical relationship between physical parameters. In practical engineering, petrophysical relationships are derived from well logging data, which are usually obtained in the ultrasonic frequency band (kHz level). Due to the petrophysical dispersion effect [20; 21; 22; 23], the logging data obtained in the ultrasonic frequency band (kHz level) are different from the data obtained in the seismic frequency band (0-100 Hz). Consequently, when transferring petrophysical relationships from well logging data to seismic FWI workflows, two strategies can be adopted: (1) implementing frequency-dependent dispersion corrections to the logging data [24] or (2) utilizing frequency-independent empirical relationships. [PERSON]'s equation [25] represents an empirical model for estimating bulk density from P-wave velocity. Derived through the statistical fitting of a large dataset of experimental measurements, this relationship assumes a stable power-law correlation between density and velocity under specific lithological conditions. The robustness of [PERSON]'s equation in seismic applications stems from three key attributes: (1) the statistical consistency of its empirical foundation; (2) the inherently low frequency sensitivity of bulk density; and (3) parameter insensitivity to viscoelastic dispersion mechanisms. These characteristics collectively ensure its applicability across the seismic frequency spectrum. [PERSON]'s equation is expressed as follows:  
\[\ln(\rho)=a\ln(v_{p})+b \tag{1}\]  
where \(v_{p}\) represents P-wave velocity, \(\rho\) is rock density, and \(a\) and \(b\) are constants. Parameters \(a\) and \(b\) in Equation (1) exhibit regional variations. With reference to the measured data (presented in Table 1) collected in [26] from a particular outcrop in New Zealand, the measured data were plotted within a logarithmic coordinate system (presented in Figure 1) to validate the empirical equation proposed in [25]. Red dots represent the measured data, and the blue solid line represents the fitted line of the velocity and density of the measured data. The following parameters were selected for a quantitative assessment of the goodness of fit:  
\[ \begin{gathered}\text{\emph{Sum~{}Squard~{}Error~{}(SSE):S=\sum \limits_{i=1}^{m}\left(y_{i}-\widehat{y}_{i}\right)^{2}}}\\ \text{\emph{R-square}:R^{2}=1-\frac{\sum\limits_{i=1}^{m}\left(y _{i}-\widehat{y}_{i}\right)^{2}}{\frac{1}{m}\sum\limits_{i=1}^{m}\left(y_{i}- \overline{y}_{i}\right)^{2}}}\\ \text{\emph{Adjusted~{}R-square}:R^{2}_{adjusted}=~{}1-\frac{ (1-R^{2})(m-1)}{m-p-1}}\\ \text{\emph{Root~{}Mean~{}Squard~{}Error~{}(RMSE):R=\sqrt{\frac{1}{m} \sum\limits_{i=1}^{m}\left(y_{i}-\widehat{y}_{i}\right)^{2}}}}\end{gathered}  \tag{2}\]  
where \(y\) represents the real value, \(\widehat{y}\) represents the predicted value, \(m\) is the number of samples, and \(p\) is the number of features. Through calculation, the evaluation results were as follows: the SSE value is 0.0007242, R-square is 0.9911, adjusted R-square is 0.9889, and RMSE is 0.01346. The results show that the above relationship between velocity and density is applicable to the outcrop data and has a high degree of fitting. This provides a theoretical basis for subsequent rock physical constraints based on the relationships among physical parameters. the equation for calculating the gradient in the multi-parameter FWI of seismic waves.