# An Application of Elastic-Net Regularized Linear Inverse Problem in Seismic Data Inversion
## 1 Introduction
Hence, Tikhonov regularization is an additional term after misfit in the objective function of inverse problems. In the early days, Tikhonov regularization was used to constrain seismic inverse problems [13; 14]. For example, pre-whitening least-square deconvolution is frequently used in seismic data processing [15]. The pre-whitening is, in fact, zero-order Tikhonov regularization. In addition, due to the band-limited feature of seismic data, i.e., lack of effective low-frequency and high-frequency information, the important low-frequency geological trend and high-frequency information characterizing thin layers will be missing in seismic inversion results. To solve this issue, a priori model constraint regularization was developed based on conventional Tikhonov regularization [16]. However, Tikhonov regularization yields a non-sparse solution. A dense reflectivity series will be obtained through Tikhonov regularization seismic inversion. The estimated impedance model from the dense reflectivity series is not blocky, i.e., the formation boundary is not sharp and precise. To invert a sparse reflectivity series with high resolution and estimate the blocky impedance model, sparse regularization is introduced in the seismic reflectivity inversion. The basic assumption behind sparse regularization in seismic inversion is that the reflectivity series of the underground formation is sparsely distributed. Hence, the final impedance from the sparse reflectivity series is blocky through the exponential integration transformation [2]. As early as the 1970s, sparse regularization has been used in seismic data inverse problems [17]. However, the impact is confined to the geophysical community. The extensive research of sparse regularization originates from the signal sparse representation and recovery and compressed sensing theory [18; 19; 20; 21]. There are many kinds of sparse regularization in the existing literature, such as Huber regularization [22], Cauchy regularization [23; 24; 25],modified Cauchy regularization [24], minimum entropy regularization [9; 26], t regularization [27], L1-norm regularization [4; 28], weighted L1-norm regularization [29], Lp-norm (\(0<p<1\)) regularization [30], L0-norm regularization [31; 32; 33], and so forth. Similarly, sparse regularization can also be interpreted from Bayesianism. In fact, the sparse regularization corresponds to a priori long-tail distribution of model parameters [12]. For example, L1-norm regularization corresponds to the bilateral exponential distribution. Other sparse regularizations include Lp-norm (\(0<p<1\)), Cauchy norm, and Huber norm, which, respectively, correspond to the generalized Gaussian distribution, Cauchy distribution, Huber distribution, and so forth. However, due to the inherent problems of sparse regularization, i.e., its tendency to protect large coefficients and suppress small coefficients [29; 34], the sparse regularization seismic inversion method cannot estimate some weak reflectors [1]. The high resolution of sparse-spike inversion is obtained at the cost of sacrificing small reflectivity. In fact, the reflectivity series from the actual impedance well log is not strictly sparse. It contains not only the sparse major large reflectivity, which corresponds to the main stratigraphic sequence interface of the underground formation, but also small reflectivity, which represents weak reflectors of a thin interbed between major reflectivity. To combat this issue, we use elastic-net regularization as an alternative to sparse regularization in seismic inversion. The elastic net is a hybrid regularization that combines sparse regularization and dense regularization. Next, the proposed inversion method is performed on a synthetic seismic trace, which is created from an actual well log. Then, a real seismic data profile is used to test the practice application.