# Multi-Component Temporal-Correlation Seismic Data Compression Algorithm Based on the PCA and DWT
## 2 Related Work  
Encoding through transformation represents a highly effective approach to data compression that is widely utilized in various compression tools like JPEG, JPEG-2000, and HEVC [8; 9; 10]. The encoding strategies differ based on the type of transform utilized. Commonly employed transforms in data compression include the discrete cosine transform (DCT) [11; 12], the discrete wavelet transform (DWT) [6], and the family of lapped transforms [13; 14]. Moreover, in seismic data compression, domain-specific transforms such as the seislet transform [15], the dreamlet transform [16], and the unvelet transform [17] still play significant roles; however, they are also very costly operations.  
Dimensionality reduction-based compression techniques involve converting data from an \(N\)-dimensional space to an \(M\)-dimensional space, where \(M<N\). This transformation exploits redundancy within the original \(N\)-dimensional representation. By reducing dimensionality, the method lowers the number of bits needed to encode the initial data while retaining a substantial portion of its information. Consequently, it falls under the category of lossy compression schemes.  
Although principal component analysis (PCA) is limited to linear transformations, when applied to compression, it achieves reduced storage demands with a lower computational burden compared to intensive training procedures, such as machine learning-based algorithms. In seismic data compression, PCA has been effectively utilized alongside specific domain-specific enhancements. These include incorporating a pre-processing stage where signals are aligned to optimize cross-correlation [7] and implementing distributed approaches for real-time compression during the acquisition phase [18].  
[PERSON] et al. [7] explore PCA as a dimensionality reduction solution for seismic data. Their results point to a compression ratio of 12:1 while preserving 99% of the signal energy with 0.8% reconstruction error. Most of the compression error prevails from aligning traces with reference data, which allows for reducing the number of principal components. Compression to PCA comprehends finding a pair of eigenvectors and eigenvalues and selecting the highest eigenvalues to promote a projection of the values into the vector. Datadecompression can be obtained by projecting the projection coefficients back to the original dimension and adding the average vector. Aligning seismic traces, for instance, shifting traces over time such that patterns are in the same time index (e.g., reflections), promotes a higher uniformity that can improve the compression ratio. In this sense, the authors apply a circular time shift to move patterns inside data (i.e., the trace length is unchanged). Pattern alignment is promoted by maximum amplitude and cross-correlation techniques. Pattern alignment is also an interesting technique that can be applied to other approaches, such as autoencoders, and when searching for compression over time or space. Nevertheless, data alignment based on linear events may lead to the loss of alignment of hyperbolic events, possibly affecting further analysis.  
Principal component analysis (PCA) can be enhanced through various combinations with transform encoding techniques. In the literature, PCA has been integrated with the discrete wavelet transform (DWT) as a preliminary filter before transform encoding [19]. Additionally, the DWT has been utilized as a preprocessing stage for PCA, leveraging its ability to decompose the original signal into smaller-dimensional signals to enhance the computational efficiency of PCA. Similarly, PCA has served as a preprocessing step in an image compression workflow, functioning as a pre-filter before employing DCT-based transform encoding, along with quantization and thresholding. Conversely, the DCT has been employed as a preprocessing step for PCA, resulting in loadings that may offer more meaningful insights for operations on compressed data, such as classification purposes [20].