# An Application of Elastic-Net Regularized Linear Inverse Problem in Seismic Data Inversion
## 1 Introduction  
Seismic inversion is a way to estimate the impedance of underground formations from post-stack seismic data. The inverted impedance can be used in rock characterization, stratigraphic correlation, lithology identification and further reservoir prediction, and many other areas in the oil and gas industry [1; 2; 3]. Usually, the post-stack seismic data are first inverted to obtain the reflectivity series of the underground formation. Then, under a simple exponential integration transformation, the inverted reflectivity series can give the final impedance [4; 5]. Hence, the quality of the inverted reflectivity series directly affects the quality of impedance. In seismology, we think the observed seismic data are the convolution result between reflectivity series and a seismic source wavelet. A seismic pulse incident at an interface is partly reflected and partly transmitted into the following layer. In a multi-layer medium, this is repeated at each interface present in the way of the advancing wave. A detector placed on the surface receives the reflected wave sequentially, at time intervals depending on the depths of the reflectors. The source pulse reaches the detector after being affected by various types of modifications, including the process of reflection, in which amplitude and polarity are governed by the reflectivity of the reflector. From each reflector, a waveletrepresenting the seismic source wavelet, scaled by the reflectivity, will arrive at the detection point. Thus, a series of such wavelets, each of which is shifted in time with respect to the preceding wavelet, are superimposed on each other to form a record that is a seismic data trace [6]. This process, which involves multiplication of the source wavelet by the reflectivity, time shifting, and superposition, is similar to the mathematical convolution process taking place between the source wavelet, in this case, and the reflectivity series. This is the construction mechanism model which is accepted by geophysicists for the formation of the seismic trace. The concept that the seismic data can be represented as the convolution of the seismic source wavelet with the earth reflectivity series was developed by [PERSON] [7]. Hence, we can say that a seismic wavelet is a waveform with the bulk of its energy confined to a finite interval on the time scale. It can be defined as a source energy signal that is concentrated in a certain time interval [8]. A seismic wavelet, or simply a wavelet, is one of the basic building blocks that is used for the construction of seismic signal models. These models are used as a basis for various seismic data inversions or processing algorithms. Hence, the reflectivity series inversion is, in fact, seismic data deconvolution, which is a geophysical inverse problem. [PERSON] gave the definition and conditions of the well-posedness for the geophysical inverse problem [9]. The well-posed inverse problem possesses all the properties of the \"good\" solution: the solution exists, is unique, and is stable [9]. However, the majority of geophysical inverse problems are ill posed, because at least one of the well-posedness conditions fails. Like many other geophysical inverse problems, the seismic data deconvolution is also ill posed [10]. To solve the ill-posed inverse problem, the most effective way is regularization constraint [10]. The first used regularization method in geophysical inverse problems is Tikhonov regularization, which is the L2-norm of the model parameter or its attributes [9; 11; 12]. The regularized inverse problem theory can also be interpreted from Bayesianism. The misfit between observed data and synthetic data corresponds to the likelihood function, and the regularization corresponds to a priori distribution. The solution of the regularized inverse problem corresponds to maximum posterior estimation [9; 12]. From this point of view, the Tikhonov regularization corresponds to a priori Gaussian distribution of model parameters.