# Sensitivity Analysis of P-Wave Polarization Direction and Velocity Gradient Inversion
## 2 Theoretical Foundation  
Seismic inversion is a geophysical technique that infers important information such as velocity structure and other physical parameters of underground media. Its foundation lies in establishing mathematical relationship between observed data and subsurface media physical parameters [20; 21]. To obtain the relative velocity gradient from the P-wave polarization direction data, understanding the relationship between the polarization di rection and the relative velocity gradient is essential. In this section, we will introduce the mathematical relationship between P-wave polarization direction data and relative velocity gradient. The propagation of seismic rays in continuous media can be described by the Hamiltonian Equation [22]:  
\[\frac{\partial\mathbf{r}}{\partial t}=v^{2}(\mathbf{r})\mathbf{p}, \tag{1}\]  
\[\frac{\partial\mathbf{p}}{\partial t}=-\mathbf{p}^{2}v(\mathbf{r})\
abla v(\mathbf{r}), \tag{2}\]  
where \(\mathbf{r}\) denotes the position, \(v\) the velocity, and \(\mathbf{p}\) the slowness vector. \(t\) describes the travel time along the trajectory of seismic ray \(\mathrm{y}(t)=[\mathbf{r}(t),\mathbf{p}(t)]\). Utilizing ray-tracing techniques, we can obtain model data and Frechet derivatives needed in tomographic inversion according to Equations (1) and (2). For models with given source and receiver positions, there are [18]  
\[\mathbf{r}_{e}=\mathbf{r}_{b}+\int_{L}v^{2}(\mathbf{r})\mathbf{p}dt, \tag{3}\]  
\[\mathbf{p}_{e}=\mathbf{p}_{b}-\int_{L}\mathbf{p}^{2}v(\mathbf{r})\
abla v(\mathbf{r})dt, \tag{4}\]  
where \(L\) represents the ray path; the subscripts \(b\) and \(e\) represent that the vector is located at the source and receiver positions, respectively. Equations (3) and (4) provide an explicit integral relationship between the slowness vector and the velocity distribution in the time domain. By utilizing this relationship and P-wave polarization (slowness vector) data, velocity fields have been obtained through inversion algorithms [11; 14; 15; 16]. However, the slowness vector contains both the magnitude and direction information of the wave speed, requiring higher measurement precision, being more sensitive to noise, and having a stronger dependence on the assumptions of the subsurface medium model [15]. Moreover, the computational effort and processing time for the inversion process are also greater. In contrast, the polarization direction data involves only directional information, which simplifies the problem. By introducing the polarization direction vector \(\mathbf{n}\), which is the unit vector of the slowness vector, Equations (3) and (4) are rewritten in the following form [19]:  
\[\mathbf{r}_{e}=\mathbf{r}_{b}+\int_{L}\mathbf{m}dl, \tag{5}\]  
\[\mathbf{n}_{e}=e^{\int_{\mathbf{r}_{b}}^{\mathbf{r}_{e}}[\
abla v(\mathbf{r})/v(\mathbf{r})]\cdot d \mathbf{r}}\mathbf{n}_{b}-\int_{L}\frac{\
abla v(\mathbf{r})}{v(\mathbf{r})}e^{\int_{\mathbf{r}} ^{\mathbf{r}_{e}}[\
abla v(\mathbf{r})/v(\mathbf{r})]\cdot d\mathbf{r}}dl, \tag{6}\]  
where \(\
abla v(\mathbf{r})/v(\mathbf{r})\) is the relative velocity gradient, and \(dl\) is defined as \(v(\mathbf{r})dt\). \(\mathbf{n}_{e}\) and \(\mathbf{n}_{b}\) are the polarization direction vectors of seismic ray at \(\mathbf{r}_{e}\) and \(\mathbf{r}_{b}\), respectively. The endpoint position of the ray can be derived using Equation (5). Equation (6) describes the relationship between the polarization direction and the relative velocity gradient, indicating that the polarization direction is a function of the relative velocity gradient and the direction vector of the seismic ray at the source position. Based on this relationship, relative velocity gradients can be inverted using P-wave polarization direction data.