*2.2. High-Resolution Sparse Parabolic Radon Transform*

The regularization method of using L-norm as a penalty term can make the result sparse and obtain a high-resolution parabolic Radon transform with sparse constraints. Common regularization methods include *L*<sup>0</sup> regularization, *L*<sup>1</sup> regularization, *L*<sup>2</sup> regularization, *Lq* regularization and the unit sphere geometry diagram of L-norm, shown in Figure 2. Since there are no corners in the constraint region of *L*<sup>0</sup> regularization, and it is difficult to have zero solutions, this method has sparsity. The constraint region of *L*<sup>2</sup> regularization also has no corners and it can avoid the overfitting of the model, but its solutions do not have the sparsity property. The constraint region of *L*<sup>1</sup> regularization is a square. The *L*<sup>1</sup> regularization is convex optimization problem, and it is sparse. It can be solved using the iterative soft threshold algorithm (ISTA) [38] and the fast iterative shrinkagethresholding algorithm (FISTA) [39]. However, some numerical experiments [21] showed that *Lq* (0 < *q* < 1) regularization has much better signal recovery capability than *L*<sup>1</sup> regularization minimization. As shown in Figure 2, the solution of *Lq* (0 < *q* < 1) regularization is more likely to be at a corner, which proves its sparsity.

**Figure 2.** Unit ball pictures for (**a**) *L*0, (**b**) *L*2, (**c**) *L*<sup>1</sup> and (**d**) *L*1/2 regularization.

The multiple suppression based on the parabolic Radon transform problem can be formulated as following minimization problem of *Lq* regularization:

$$\min\_{m} \{ \frac{1}{2} ||d - Am||\_2^2 + \alpha ||m||\_q^q \}\tag{11}$$

where *mq*=(∑*<sup>N</sup> <sup>i</sup>*=1|*mi*| *q* ) 1/*<sup>q</sup>* and *<sup>α</sup>* > 0 is the regularization parameter.

*Lq* (0 < *q* < 1) norm constrains whole seismic wavefield data to improve the inversion accuracy. However, in order not to cause signal damage and noise residue, sparse constraints of primary and multiple reflections are considered, respectively, according to the differences between them to achieve a better multiple suppression.
