# Sensitivity Analysis of P-Wave Polarization Direction and Velocity Gradient Inversion
## 4 Inversion Method Based on Polarization Direction Sensitivity  
It is commonly accepted that the Tuy-Smith data completeness condition must be satisfied by the projection data in order to achieve high-quality reconstructed images, regardless of the reconstruction algorithm employed [27]. However, in practical applications, the acquired projection data are frequently incomplete due to a range of subjective and objective factors [28; 29]. Subjective factors may include operator errors or limitations in measurement techniques, while objective factors might encompass equipment malfunctions, environmental conditions, or inherent limitations in data acquisition systems. When the number of projections is limited and the sensitivity of P-wave polarization data to the relative velocity gradient is unevenly distributed, it can lead to distortions in the inversion results. Specifically, the update of the relative velocity gradients farther from the receiver may become weaker, resulting in insufficient updates that fail to effectively capture the actual relative velocity gradient changes. Consequently, the reconstructed velocity gradient model may be inaccurate, which ultimately affects the accuracy and stability of the reconstruction results. In this paper, our approach is based on a commonly used method: the Algebraic Reconstruction Technique (ART) for inverting relative velocity gradients. The ART method allows us to achieve high-quality images with relatively fewer projections. Specifically, this method obtains the correction values for the image by comparing the estimated projection data with the actual projection data obtained [30]. Its traditional iterative formula is  
\[\mathbf{G}^{k+1}(j)=\mathbf{G}^{k}(j)+\lambda_{k}\frac{\mathbf{n}_{e}(i)-\sum_{j=1}^{N} \mathbf{W}_{i,j}\mathbf{G}^{k}}{\sum_{j=1}^{N}\mathbf{W}_{i,j}^{2}}\mathbf{W}_{i,j},\lambda_{ k}\in(0,2),i=1,2,\cdots,M,j=1,2,\cdots,N, \tag{21}\]  
where \(\mathbf{G}\) is the inversion parameter, which is the relative velocity gradient. \(k\) is the number of iterations, \(\lambda_{k}\) is the relaxation factor in the \(k\)th iteration process. \(M\) and \(N\) are the number of the data and the size of the model, respectively. \(\mathbf{W}_{i,j}\) is the weight coefficient, which is numerically equal to the partial derivative of the polarization direction of the \(i\)th ray with respect to the relative velocity gradient at the \(j\)th grid node. The weight coefficient in Equation (21) is an important factor influencing the degree of correction applied to the relative velocity gradient values. Although ART enables us to achieve high-quality images with relatively fewer projections, there remains the observation that fewer projections lead to poorer image quality in ART reconstructions [31]. To address the problem where the nonuniform distribution of sensitivity may cause distortion of inversion results with incomplete projection data, it is crucial to incrementally enhance the correction level of the relative velocity gradient from receiver to source positions. By reducing the update magnitude at positions with large derivatives and increasing the update magnitude at positions with small derivatives, it helps to balance the update magnitudes of the inversion values [32]. This process involves adding a factor prior to the weight coefficient \(W_{i,j}\) in Equation (21). By adjusting the update magnitude to increase for positions with smaller derivatives while decreasing for positions with larger derivatives, we aim to balance the update amplitude of the relative velocity gradient values at each position. The following details the approach to determine this factor. Initially, we extract the partial derivative values for each source-receiver pair (starting from the first source to the first receiver, the second source to the second receiver, and so on, up to the \(I\)th source and \(J\)th receiver, totaling \(I\) or \(J\) source-receiver pairs). Each source-receiver pair contributes \(JJ\) partial derivative values, resulting in a total of \(II\times J\) values.