# Multiple Elimination Based on Mode Decomposition in the Elastic Half Norm Constrained Radon Domain
## 1 Introduction  
The ocean holds substantial reserves of petroleum and gas resources. Seismic exploration serves as the initial step in offshore petroleum and gas exploration and development, and the quality of seismic data directly impacts the subsequent petroleum and gas hydrate exploration [1; 2; 3; 4]. When conducting offshore seismic acquisition, due to the strong wave impedance difference between seawater and air, seismic acquisition data contain multiples in addition to primaries. Multiples can cover the effective signals and hinder the imaging and identification of subsurface geological structures. To enhance data quality and achieve a more precise identification of subsurface targets, it becomes imperative to suppress multiples. One commonly employed technique for multiple suppression is the Radon transform-based filtering method. The Radon transform (RT) is a classical transformation [5] and has been extensively used in seismic exploration for various purposes such as multiple elimination, denoising, deblending, and interpolation [6; 7; 8; 9; 10; 11]. Primaries and multiples exhibit different move-out characteristics in a Common Midpoint (CMP) gather. After applying the normal move-out (NMO) correction, primaries are corrected from hyperbolic to linear, whereas multiples are corrected from hyperbolic to parabolic,exhibiting a residual move-out. Using the parabolic RT, multiples and primaries are located in variant regions in the Radon domain due to their variation in curvature. Subsequently, a filter is utilized to mute the multiple coefficients, and the final demultiple result is obtained by performing the inverse RT to the filtered Radon model [12; 13]. This is the fundamental theory of why and how the Radon filter method can be used for multiple suppression. However, when using the filter method for multiple elimination, it is crucial to set appropriate filtering parameters. These parameters are heavily dependent on the experience of geophysicists. If the filtering parameters are set improperly, the filtered result will either contain multiple residuals or hurt primaries [6]. Complicating matters further, the optimal filtering parameters for one area may not be suitable for another due to the inherent differences in seismic data. It requires processors to pay sufficient attention to ensure whether the processing parameters are appropriate for the input data. Therefore, developing a demultiple method that does not rely on manual parameters setting is currently an urgent need in the petroleum industry. Empirical mode decomposition (EMD) [14; 15; 16; 17] and variational mode decomposition (VMD) [18; 19; 20] are two signal-processing methods that can automatically decompose signals into several intrinsic mode functions based on a predefined number of modes. Geometric mode decomposition (GMD) [21], inspired by VMD, is an innovative approach in signal analysis that enables the decomposition of a 2D seismic data into several modes with various geometric characteristics. These geometric features include principal slopes and curvatures. GMD designs a new objective function and adaptively obtains geometric parameters by minimizing the variational energy along different directions. Using the alternating direction method of multipliers (ADMM) algorithm [22; 23], the optimization problem of GMD can be solved, and the mode decomposition can be achieved. GMD contains two components: GMD-F, which performs decomposition in the frequency domain, and GMD-R, which performs decomposition in the Radon domain. The GMD-F algorithm can be utilized to separate diffractions based on the kinematic and dynamic disparities between reflections and diffractions in the f-k domain [24]. GMD-R leverages the principle of the Wiener filter to iteratively decompose the input data into modes with varying curvatures and intercept times. After NMO correction, there are curvature differences between multiples and primaries.