# Sensitivity Analysis of P-Wave Polarization Direction and Velocity Gradient Inversion
###### Abstract  
Velocity gradient is an important parameter for interpreting tomographic velocity field and identifying geological boundaries. It is usually translated from the results of seismic travel-time tomography. Recent researches show seismic polarization direction appears to be a promising data source for obtaining velocity gradient field directly. However, what remains unclear is the sensitivity of polarization direction to velocity gradient, which causes difficulty for correctly inverting polarization direction data. To clarify this problem, the sensitivity of velocity gradient parameters on polarization direction is discussed in this paper. It was found that the sensitivity of polarization direction is related to the spatial position of the model parameter. The further the parameter position is from the sensor, the lower the sensitivity is. Such nonuniform distribution of sensitivity may cause distortion of inversion results with incomplete projection data. Based on this analysis, adjustment factors are introducing to the polarization direction inversion algorithm for correctly inverting polarization direction data. Numerical tests are conducted to verify our theoretical analysis and inversion algorithm. Test results show that our theoretical analysis is accurate in both homogeneous velocity field and near velocity interfaces. The inversion method with the adjustment factor can more accurately recover the velocity gradient, offering a promising approach for geological boundary imaging. v 2020  
In the field of geology, various methods have been employed to generate gradient maps. However, the majority of these methods rely on travel-time data to invert velocity images, thereby obtaining gradient maps. For instance, [PERSON] converted the velocity gradient map from the travel time inversion results using finite difference approximation [8]. [PERSON] et al. obtained a vertical velocity gradient map by using the P-wave velocity structure determined by first-arrival P-wave travel-time tomography [10]. Travel-time data are invariant to perturbations along the ray paths, which makes them insensitive to the precise locations of velocity heterogeneities [11; 12]. In some cases, the rays used in the inversion process may deviate from their true paths, leading to biases in the data kernel. These biases can cause velocity anomalies to be misallocated in the imaged velocity structure [11]. Consequently, the gradient maps generated from the resulting velocity model may exhibit deviations. This indicates that the reliability of these gradient maps hinges on the accuracy of the travel-time inversion results. The existence of errors in the inversion results introduces a level of uncertainty. Polarization (the slowness vector of a ray) is another fundamental characteristic of seismic waves [13]. Polarization inversion is significantly more sensitive to local velocity changes than travel-time inversion [12]. This sensitivity allows for better depiction of the edges of velocity anomalies, compensating for the limitations of traditional travel-time methods. Polarization data have been widely used to explore the velocity structure of subsurface media [11; 14; 15; 16]. However, accurate velocity structures at the receiver locations are often difficult to obtain [17]. In contrast, polarization direction data involve only directional information, thereby simplifying the complexity of the problem. [PERSON] and [PERSON] first established a relationship between P-wave polarization direction and relative velocity gradients, and attempted to directly obtain relative velocity gradients from polarization direction data [18]. This approach surpasses the limitations of gradient maps constructed from travel time data. The utilization of polarization direction data offers new insights for identification of geological structures. This research breakthrough opens new avenues for in-depth analysis of seismic wavefields. Unfortunately, inversion results exhibit poor performance [19].