# Pseudo-3D Receiver Deghosting of Seismic Streamer Data Based on \(l_{1}\) Norm Sparse Inversion
## 1 Introduction  
For signal-to-noise ratio considerations, the receivers are sunk to a certain depth below the air-water interface during marine seismic acquisition [1]. When the upgoing wavefield propagates to the air-water interface, it reflects and propagates downward. The downgoing wavefield is called ghost [2], and it interferes with the upgoing wavefield, resulting in frequency notch and reducing the vertical resolution of the seismic dataset [3]. In marine acquisition, the sampling interval in the crossline direction is usually much larger than that in the inline direction (usually between 4:1 and 16:1) [4], and the sparse sampling poses a large challenge for deghosting [5]. Therefore, the 2D deghosting method for a densely sampled inline dataset is still the mainstream method for marine data processing. The current deghosting methods are mainly implemented in the frequency domain, and the idea of plane wave decomposition is directly or indirectly applied [6]. The deghosting methods based on plane wave decomposition can be roughly divided into two categories: the deconvolution-based method and the inversion-based method. The basic idea of the deconvolution-based method is to transform the seismic dataset containing ghost into other data domains (time domain [7], frequency domain [8], \(f\)-\(k\) domain [9], etc. ), construct a ghost delay operator, use the deconvolution algorithm to achieve deghosting in this domain, and transform it into the space-time domain. However,the deconvolution-based method requires the accurate estimation of ghost delay time, which cannot overcome the error of actual acquisition [10]. The basic idea of the inversion-based method is to combine the parameter of the observing system with the plane wave decomposition and express it as a transformation matrix. Due to the combination of a plane wave coding operator and a deghost delay operator, the ghost-free wavefield can be obtained by a linear inversion algorithm. The advantage of this method is the applicability of various observation systems (e.g., variable-depth streamer [10]) and the convenient introduction of an advanced inversion algorithm [11; 12; 13]. However, accurately obtaining a ghost-free wavefield by conventional 2D deghosting method technology is still difficult. Compressed sensing [14] is widely used in signal analysis [15; 16], noise suppression [17; 18], seismic acquisition [19], seismic imaging [20], and optimized inversion [21; 22]. Such methods decompose seismic datasets into different units with different characteristics (frequency, direction, wavelet, curvature, etc. ), and they extract corresponding units according to the characteristics of the target signal to achieve sparse representation, which means highlighting the useful information and removing the redundant information. [PERSON] et al. [23], [PERSON] et al. [24], and [PERSON] et al. [25] adopted the compressed sensing-based sparse inversion method to find the informative local minimum in the inversion of seismic signals. The key to these methods is to choose the appropriate data domain and sparse transformation for the sparse representation of seismic signals. By amplifying the coherence difference between the signal and the noise, the seismic signal can be highlighted in the inversion, thereby improving the inversion accuracy. In this paper, we propose a pseudo-3D deghosting framework based on an \(l_{1}\) norm sparse inversion to improve the deghosting effect. In this framework, we include the data coherence along the shot direction into inversion and introduce the idea of compressed sensing to further improve the inversion accuracy. Firstly, based on the consistency of the receiver depth in conventional streamer data, we derive a pseudo-3D ghost delay operator for multi-shot gathers for simultaneous deghosting. Then, we introduce an \(l_{1}\) norm sparse inversion in the pseudo-3D radon domain to improve the inversion accuracy.