The non-uniformity and incompleteness of seismic data in space have long been one of the significant challenges affecting seismic imaging. The factors that cause the non-uniform spatial distribution of seismic data are primarily twofold: Firstly, the non-uniform distribution of earthquakes, which are the sources of seismic waves; Secondly, the non-uniform distribution of seismic stations used to record these waves. In areas where stations are difficult to built, such as mountainous regions and offshore locations, uniform array deployment becomes impractical. These factors lead to the acquisition of non-uniform and incomplete data, thereby reducing the resolution and accuracy of the imaging results.

The core principle of compressed sensing theory is to exploit the sparse nature of the signal and then employ a non-linear reconstruction algorithm to recover the original signal. Since seismic wavefields exhibit continuity, then missing or irregularly sampled seismic data can potentially be recovered through compressed sensing techniques. In the time domain, seismic observation data usually contain rich frequency information. However, due to the filtering effects of subsurface layers, the bandwidth of the actual recorded seismic data is limited, and the data exhibit sparsity in the frequency domain. Based on this characteristic, under limited acquisition conditions, the reconstruction of missing seismic data utilizing compressed sensing theory can, to some extent, mitigate the problem of insufficient seismic data coverage.

In oil and gas seismic exploration, data reconstruction methods based on compressed sensing theory have been widely applied to address the issue of insufficient sampling. However, relevant discussions are not yet common in the acquisition and processing of natural earthquake data. In fact, due to environmental and site constraints, the acquisition of natural earthquake data also suffers missing and irregular sampling issues. Moreover, owing to the irregular distribution of acquisition stations, the reconstruction of natural earthquake data encounters greater challenges. Natural earthquake waveforms are more complex, and proposing a reliable scheme for reconstructing natural earthquake data under the condition of irregularly distributed acquisition stations is a relatively difficult problem.

For artificial seismic exploration data reconstruction, the trace spacing is small, and the waveforms of adjacent traces exhibit good consistency. In Chinese continental natural earthquake observation stations, the minimum station spacing is less than 30 km, and the teleseismic waveforms recorded at different stations also exhibit certain similarities. This study commences with natural teleseismic observation data, draws upon the experience of artificial seismic data reconstruction, and attempts to conduct research on the reconstruction of teleseismic data. Relevant experimental verifications have been carried out.

To verify the reconstruction effect of the algorithm, we first use the reflectivity method to calculate the theoretical seismogram, simulating the teleseismic P-wave data recorded by the Inner Mongolia movable array. Thirty-six seismic stations receive the data, with a sampling interval of 2 ms and 5501 sampling points. The simulated seismic data are obtained through calculation of seismic travel time, and the missing teleseismic P-wave data are recovered, verifying the effectiveness of the method. Compared with the original record, the reconstructed seismic record signals can restore the waveform relatively well; the reconstructed result is generally consistent with the original record. In terms of amplitude, the reconstructed result is significantly smaller, and the frequency is also higher, which shows that the reconstruction method has the effect of suppressing random noise. Simultaneously, there are certain differences in the waveforms across different stations, and the analysis suggests that the number of iterations for certain station data is insufficient, or there are discrepancies in parameter adjustments.

We applied the seismic data reconstruction method that based on compressed sensing to process P-wave arrival times of teleseismic events. Firstly, the curvelet transform is utilized as a sparse transform, and a regularized inversion model based on the L₁ norm is established. The iterative shrinkage-thresholding algorithm （ISTA） is employed to solve this model. Subsequently, the compressed sensing theory is applied to the reconstruction of teleseismic data observed by the portable seismic array in the Inner Mongolia region. On this basis, data reconstruction is performed on the acquired real earthquake data, and the reconstructed P-wave arrival time data are picked to verify the effectiveness of the method through teleseismic tomography.

The practice of natural earthquake data reconstruction demonstrates that natural earthquake data exhibit a sparse representation in the curvelet transform domain. Based on the seismic data itself, when seismic data are missing or the natural earthquake observation data are incomplete, the compressed sensing theory can be employed to reconstruct the seismic data. Concurrently, this method can be utilized to reconstruct data with poor signal-to-noise ratio and obtain high signal-to-noise ratio seismic data. In this study, the fast marching teleseismic tomography （FMTT） method is used to invert the velocity structure from the original data, sampled data, and reconstructed seismic data, respectively. The results indicate that when data are missing, the calculated tomographic velocity structure deviates significantly from that calculated with complete data. The relative error between the tomographic result obtained from the reconstructed data and that from the complete data is within 0.02%. The vertical cross-section shows that the tomographic results obtained from the original data and reconstructed data exhibit highly similar travel time residuals, and the spatial positions and shapes of the main high and low velocity anomaly zones are highly consistent. This signifies that the compressed sensing based theory reconstruction method can recover the imaging results from the sampled data. The practice of three-dimensional P-wave imaging proves that the data reconstruction technique based on compressed sensing theory can improve the resolution of seismic tomography. The results also demonstrate the potential application value of compressed sensing acquisition technology in natural earthquake data, which can provide and broaden the research ideas for exploring new array observation methods and data acquisition strategies for natural earthquakes.