# Radon Transform Based on Waveform for AVO-Preserving Data Construction
## 1 Introduction  
Over the past two decades, Radon transform has been widely used in seismic data processing, especially in multiple attenuation and seismic data interpolation [1, 2, 3, 4, 5, 6]. Radon transform can be divided into three different types based on its shape: line, parabolic, or hyperbolic Radon transform. [PERSON] et al. [8] proposed a time-domain hyperbolic least-squares method, which led to the high-resolution transform at the expense of a very large matrix computation. [PERSON] [9] (1986) suggested a frequency domain least-squares parabolic transform, which is more efficient. Therefore, the frequency domain quickly became the industry standard. [PERSON] and [PERSON] [10] (1995) implemented the sparse Radon transform in the frequency domain, which obtained a high-resolution result. Their frequency domain technique is now generally used in seismic processing. Sparse Radon transform improves the resolution but introduces new problems that make it difficult to preserve the AVO. When the resolution is high enough, there is only one sample in the Radon model associated with every reflection event, and the inverse transform amplitude is constant. Thus, the resolution and AVO characteristics are conflicting. [PERSON] [11] (1999) pointed out that Radon transform is not an orthogonal transform, and the inverse transform cannot completely reconstruct the original seismic data, especially the amplitude distortion at the near offset, resulting in the AVO analysis error. [PERSON] [12] (1995) used the orthogonal polynomials to fit the AVO characteristics of the CDP gathers, which showed that AVO can be characterized by fewer coefficients of low-order orthogonal polynomials. [PERSON] [13] (2004) proposed a high-resolution Radon transform in the \(t-x\) domain using the \"intelligent\" prioritization of the Gauss-Seidel estimation sequence. [PERSON] [14] (2009) proposed a de-multiple method based on Radon and orthogonal transform which showed a good performance in retaining the AVO properties. [PERSON] [15] (2014) split the seismic gather into two Radon gathers, which reduced the amplitude distortion and preserved the AVO phenomena compared with the classical sparse Radon transform. Another problem of high-resolution Radon transform is the large computation time. Compared with the frequency-domain Radon transform, the time-domain Radon transform achieves a higher resolution because the Radon model has a higher sparseness in the time domain. The downside of the time-domain Radon transform is its significantly increased computation burden. [PERSON] et al. [16] (2003) proposed a method to compute the time-invariant Radon transform in the time domain applying frequency domain operators. This approach combines the high resolution of the time-domain Radon transform and the computational efficiency of the frequency-domain Radon transform. However, this technique cannot implement the hyperbolic Radon Transform, which is time-variant. In general, a true hyperbolic Radon transform produces better results for data with large moveout than the parabolic version. In addition, the hyperbolic Radon transform can be directly applied to CMP data without NMO-corrected data. The large computation cost is the main problem of the hyperbolic Radon transform. Due to its time-variant kernel, it is not possible to use fast solutions instead of iterative methods such as the conjugate gradient, which increase the computation time [17; 18; 19; 20; 21; 22; 23]. This paper describes a technique for implementing a high-resolution hyperbolic Radon transform-preserving AVO. To obtain the high-resolution hyperbolic Radon transform, we fully consider the characteristics of the seismic data. Previous research on the Radon transform has mainly focused on the kinematic characteristics of the seismic data signal, rarely considering the waveform properties containing energy and cycle. This paper uses a new method to improve the resolution based on the waveform energy and cycle.