Compressed Multi-Trace Pre-Stack Inversion with Elastic Half-Norm Regularization

Abstract

Multi-trace amplitude variation with angle inversion (MAVAI) is a vital tool for estimating the physical parameters of subsurface media, and it plays an important role in oil and gas exploration. However, the existing MAVAI method relies on the Kronecker product to construct an extremely large-scale inverse problem, and its computational inefficiency limits its widespread application. Furthermore, regarding regularization constraints, the existing MAVAI method only considers the smoothness of the inversion parameters, which leads to ambiguous formation boundaries and hinders accurate identification for complex reservoirs. To address these issues, a compressed MAVAI method with elastic half-norm regularization is proposed. Specifically, we first developed a compressed MAVAI (CMAVAI) framework that uses compressed measurements of seismic data and reference models in a sparse domain to construct the CMAVAI objective function, thereby reducing the scale of the inversion problem and improving inversion efficiency. Subsequently, the elastic half-norm is introduced into the CMAVAI framework as a regularization constraint for reservoir parameter estimation. Since the elastic half-norm can simultaneously characterize both the smoothness and blocky features of the subsurface medium, it effectively improves inversion accuracy compared to the MAVAI method. Finally, the performance of the proposed method is evaluated using a theoretical model and field data. The results demonstrate that, compared with the traditional MAVAI algorithm, the CMAVAI framework can effectively improve inversion efficiency while maintaining inversion accuracy. Moreover, the CMAVAI method regularized by the elastic half-norm can improve the accuracy of inversion parameters while retaining the high prediction efficiency of the CMAVAI framework.

1. Introduction

Pre-stack seismic inversion acts as a pivotal bridge connecting geophysics and geology, which can fully use various information contained in observed seismic data to obtain physical parameters reflecting the properties of subsurface media (such as velocity, impedance, modulus, Poisson’s ratio, density, etc.). These physical parameters are crucial data for characterizing the morphology and characteristics of oil and gas reservoirs, and they play an important role in identifying, locating, and developing oil and gas reservoirs [1,2,3,4,5].

Currently, pre-stack seismic inversion methods can be broadly categorized into two types: elastic impedance (EI) inversion [6,7] and amplitude variation with angle (AVA) inversion [8,9,10]. Among them, EI inversion mainly achieves physical parameter inversion through relatively mature post-stack impedance inversion algorithms, while AVA inversion requires geophysicists to design new algorithms. Buland et al. [11] designed a linearized AVA inversion method within a Bayesian framework. This method employs a Gaussian distribution as the posterior distribution for inversion parameters, enabling the simultaneous estimation of P-wave and S-wave velocities, as well as density parameters. However, because the Gaussian distribution imposes smoothness on the solution, this inversion method tends to produce overly smooth and low-resolution results. To improve inversion performance, geophysicists have subsequently proposed various optimization schemes. Karimi et al. [12] introduced a skewed Gaussian distribution as the posterior distribution for the natural logarithm of the three inversion parameters based on the linearized AVA inversion method, effectively improving the reliability of the inversion parameters. Downton [13] found that the correlations among physical parameters in the AVA approximation equations are a key factor affecting inversion accuracy. Therefore, a decorrelation matrix is introduced into the inversion function to decouple the parameters, thereby improving the accuracy of reservoir parameter inversion. Alemie et al. [14] applied a trivariate Cauchy distribution within the Bayesian inversion framework to model the prior distribution of the reflection coefficient of inverted parameters, obtaining high-resolution inversion results. Inspired by the work of Alemie et al., various sparse constraints are subsequently adopted in the AVA inversion. Typical sparsity constraints include the L0 norm [15], the L1 norm [16], the L1-2 norm [17], the L2,1 norm [18], group-sparse regularization [19], and total variation regularization [20]. These constraint methods invariably force pre-stack AVA inversion methods to obtain blocky solutions. Li et al. [21] noted that blocky characteristics are insufficient to accurately reflect the true structure of subsurface formations, as these formations contain both discontinuous boundaries and regions of smooth variation separating these boundaries (i.e., exhibiting both blocky and smooth structures). Thus, Li et al. [21] developed a hybrid regularization term combining total variation regularization and Tikhonov regularization to improve the accuracy of AVA inversion. However, all the above methods estimate physical parameters sequentially by inverting individual common-depth points (CDPs), which ignores the correlation between adjacent CDPs and can easily lead to lateral instability for inversion results. To effectively address the aforementioned issues, Hamid et al. [22] developed a multi-trace AVA inversion (MAVAI) method that constructs a regularization term using a second-order difference operator in the horizontal direction, thus enhancing the lateral continuity of the inversion results and effectively suppressing noise interference. Nevertheless, the MAVAI method involves an extremely large-scale inverse system, rendering the solution process very time-consuming. Furthermore, the MAVAI method imposes smoothing constraints only in the vertical direction of the inversion parameters, ignoring the actual variations in the reflection characteristics of strata. It tends to produce over-smoothed inversion results that blur stratigraphic boundaries and cause significant pseudo-layer artifacts parallel to the strata, hindering the accurate identification of complex reservoirs.

In order to effectively improve computational efficiency, experts have optimized the solution algorithms for pre-stack AVA inversion. Ahmed et al. [23] proposed a second-order optimization algorithm known as limited memory-BFGS (L-BFGS) to solve the objective function of AVA inversion, thereby improving the efficiency of elastic parameter estimation. In addition to the L-BFGS algorithm, the limited-memory quasi-Newton method [24], conjugate gradient approach [25], quantum annealing technique [26], improved differential evolution algorithm [27], and music-inspired hybrid optimization approach [28] were also applied to improve the efficiency of pre-stack inversion. Different from the above optimization strategies, this paper extends the multi-trace post-stack impedance inversion (MPII) method of Lan et al. [29] to include pre-stack seismic data, and it proposes an elastic half-norm regularized CMAVAI (RCMAVAI) method. In terms of algorithm architecture, the RCMAVAI method inherits the core concept of the MPII method, which involves constructing an objective function through compressed measurements to enhance computational efficiency and applying the elastic half-norm for vertical constraints to improve inversion accuracy. Specifically, this method first uses an orthogonal transformation to transform seismic data and the reference model into a sparse domain, and then, it designs an observation operator based on the restricted isometry property to obtain compressed measurements of seismic data and the reference model. Subsequently, these compressed measurements in the sparse domain are employed to construct an objective function for CMAVAI, thereby reducing the scale of the inverse problem and improving inversion efficiency. Moreover, to improve inversion accuracy, the elastic half-norm, defined as the weighted sum of the L1/2 and L2 norms, is introduced into the CMAVAI framework as a regularization constraint for predicting elastic parameters. As the L2 norm characterizes the smoothness of subsurface media and the L1/2 norm characterizes their blocky nature, the incorporation of the elastic half-norm can effectively improve the accuracy of parameter inversion. In terms of algorithm implementation, the RCMAVAI method upgrades both the forward modeling and solution approach compared to the MPII method. Specifically, the RCMAVAI method extends the convolution model to the pre-stack scale and constructs a coupled three-parameter (P-wave, S-wave, and density) pre-stack forward operator based on Fatti’s approximation equation [30], thereby establishing an effective link between the three types of elastic parameters and the multi-trace pre-stack records. Meanwhile, the RCMAVAI method abandons the general alternating direction method of multipliers and introduces the half-quadratic splitting algorithm to solve its optimization problem. This algorithm requires only two alternating minimization steps, and the closed-form solutions for the subproblems are simpler, enabling the simultaneous and efficient inversion of three elastic parameters. The performance of the proposed method is evaluated using both theoretical and field data, and the results confirm its feasibility and effectiveness.

2. Methods

2.1. MAVAI Method

Fatti’s approximation equation [30] defines seismic reflectivity as the weighted sum of the P-wave impedance, S-wave impedance, and density reflectivity:

$$R(\theta )={\sec }^2\theta ·R_p-8{\gamma }^2{\sin }^2\theta ·R_s+(4{\gamma }^2{\sin }^2\theta -{\tan }^2\theta )·R_{\rho }$$

(1)

where $R$ denotes seismic reflectivity; $\theta$ denotes the incident angle; $\gamma ={\nu }_s/{\nu }_p$ denotes the ratios of S-wave velocity to P-wave velocity; $R_p$, $R_s$, and $R_{\rho }$ represent the reflection coefficients for P-wave impedance, S-wave impedance, and density, respectively. For simplicity, $\gamma$ is usually set to a constant, i.e., $\gamma =0.5$. Under the small reflectivity hypothesis [31], the following relationship holds between the reflection coefficient and the impedance:

$$R_{p_i}\approx {\displaystyle \frac{\ln \left(Z_{p_{i+1}}/Z_{p_i}\right)}{2}}={\displaystyle \frac{1}{2}}\left(z_{p_{i+1}}-z_{p_i}\right)$$

(2)

where $Z_{p_i}$ denotes the P-wave impedance of the i-th subsurface layer, and $z_{p_i}=\ln \left(Z_{p_i}\right)$ denotes the natural logarithm of P-wave impedance. Rewriting Equation (2) in a more concise form yields the following:

$$R_p={\displaystyle \frac{1}{2}}D{z}_p$$

(3)

where $R_p={\left[R_{p_1},R_{p_2},\cdots ,R_{p_n}\right]}^T$, $D$ is the first-order derivative operator in the time direction, and $z_p={\left[z_{p_1},z_{p_2},\cdots ,z_{p_n}\right]}^T$. Similarly to the expressions of $R_p$, the reflection coefficient of S-wave impedance and density can also be expressed as

$$R_s={\displaystyle \frac{1}{2}}D{z}_s$$

(4)

$$R_{\rho }={\displaystyle \frac{1}{2}}D{z}_{\rho }$$

(5)

where $z_s$ and $z_{\rho }$ denote the natural logarithm of S-wave impedance and density. Based on the linear relationship given by Equations (3)–(5), Equation (1) can be rewritten as

$$R(\theta )={\displaystyle \frac{a_1}{2}}D{z}_p+{\displaystyle \frac{a_2}{2}}D{z}_s+{\displaystyle \frac{a_3}{2}}D{z}_{\rho }$$

(6)

where $a_1={\sec }^2\theta$, $a_2=-8{\gamma }^2{\sin }^2\theta$, and $a_3=4{\gamma }^2{\sin }^2\theta -{\tan }^2\theta$. Under the assumption of a linear time-invariant system, angle-dependent seismic records $y(\theta )$ can be expressed as the convolution of the seismic reflection coefficient $R(\theta )$ and a wavelet $w(\theta )$ as follows:

$$y(\theta )=w(\theta )\ast R(\theta )=W(\theta )R(\theta )={\displaystyle \frac{a_1}{2}}W(\theta )D{z}_p+{\displaystyle \frac{a_2}{2}}W(\theta )D{z}_s+{\displaystyle \frac{a_3}{2}}W(\theta )D{z}_{\rho }$$

(7)

$$W(\theta )=\left[\begin{array}{cccc}{w}_1& 0& \cdots & 0\\ ⋮& w_1& \ddots & ⋮\\ w_l& ⋮& \ddots & 0\\ 0& w_l& ⋮& w_1\\ ⋮& \ddots & \ddots & ⋮\\ 0& \cdots & 0& w_l\end{array}\right]$$

(8)

where $w(\theta )={\left[w_1,w_2,\cdots ,w_l\right]}^T$ denotes the seismic wavelet, $l$ is the length of the seismic wavelet, and $W(\theta )$ denotes the wavelet matrix that is non-singular. For multiple incidence angles, Equation (7) can be generalized to the following form:

$$\underset{{\mathbf{y}}_i}{\underbrace{\left[\begin{array}{c}\mathbf{y}({\theta }_1)\\ \mathbf{y}({\theta }_2)\\ ⋮\\ \mathbf{y}({\theta }_m)\end{array}\right]}}=\underset{P}{\underbrace{{\displaystyle \frac{1}{2}}\left[\begin{array}{ccc}{a}_{1}W({\theta }_1)\mathbf{D}& a_{2}W({\theta }_1)\mathbf{D}& a_{3}W({\theta }_1)\mathbf{D}\\ a_{1}W({\theta }_2)\mathbf{D}& a_{2}W({\theta }_2)\mathbf{D}& a_{3}W({\theta }_2)\mathbf{D}\\ ⋮& ⋮& ⋮\\ a_{1}W({\theta }_m)\mathbf{D}& a_{2}W({\theta }_m)\mathbf{D}& a_{3}W({\theta }_m)\mathbf{D}\end{array}\right]}}\underset{{\mathbf{v}}_i}{\underbrace{\left[\begin{array}{c}{\mathbf{z}}_p\\ {\mathbf{z}}_s\\ {\mathbf{z}}_{\rho }\end{array}\right]}}$$

(9)

where $y_i\in {ℝ}^{mn\times 1}$ denotes the set of m angle records at the i-th CDP, n denotes the number of sampling points, $\mathbf{P}\in {ℝ}^{mn\times 3n}$ is the forward operator, and ${\mathbf{v}}_i={\left[{\mathbf{z}}_p,{\mathbf{z}}_s,{\mathbf{z}}_{\rho }\right]}^T\in {ℝ}^{3n\times 1}$ is the inversion parameter composed of the natural logarithms of the P-wave impedance, S-wave impedance, and density. For multiple CDPs, Equation (9) can be rewritten as

$$\underset{{\mathbf{Y}}_{obs}}{\underbrace{\left[\begin{array}{c}{\mathbf{y}}_1\\ {\mathbf{y}}_2\\ ⋮\\ {\mathbf{y}}_N\end{array}\right]}}=\underset{Q}{\underbrace{\left[\begin{array}{cccc}P& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& P& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \ddots & \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& P\end{array}\right]}}\underset{\mathbf{v}}{\underbrace{\left[\begin{array}{c}{\mathbf{v}}_1\\ {\mathbf{v}}_2\\ ⋮\\ {\mathbf{v}}_N\end{array}\right]}}$$

(10)

where ${\mathbf{Y}}_{obs}\in {ℝ}^{mnN\times 1}$ denotes the set of all angle records, N denotes the number of CDP points, $Q=kron\left({\mathbf{I}}_{N\times N},P\right)\in {ℝ}^{mnN\times 3nN}$ is the forward operator in multi-trace scenarios, $kron\left(\mathbf{·},\mathbf{·}\right)$ denotes the Kronecker product, ${\mathbf{I}}_{N\times N}$ denotes the Nth-order identity matrix, $\mathbf{v}\in {ℝ}^{3nN\times 1}$ denotes the set of inversion parameters. In order to obtain the inversion parameter $\mathbf{v}$, the traditional pre-stack AVA inversion method needs to solve the following least-squares problem:

$${\mathbf{v}}^{*}=\arg \underset{\mathbf{v}}{\min }{‖W_d\left(Y_{obs}-\mathbf{Q}\mathbf{v}\right)‖}_2^2+\lambda {‖W_m\left({\mathbf{v}}_{ref}-\mathbf{v}\right)‖}_2^2$$

(11)

where ${\mathbf{v}}^{*}$ denotes the optimal solution of the inversion parameters, $\lambda$ is the regularization parameter, ${\mathbf{v}}_{ref}$ is the reference model that is used to compensate for missing low-frequency information in the observed data, and $W_d$ and $W_m$ represent the diagonal matrices of data error and model standard deviation, respectively. It should be noted that the reference model is typically generated by interpolating the low-frequency trends in well logs. The optimization problem shown in Equation (11) belongs to the typical CDP-by-CDP inversion method, which does not consider the lateral continuity of the parameters to be inverted. Therefore, its inversion results are prone to lateral instability, leading to severe “noodle-like” discontinuities in the solution.

Recently, Hamid et al. [22] proposed a MAVAI method with lateral constraints, which uses a second-order difference operator to introduce an additional constraint term in the horizontal direction, thereby effectively improving the lateral continuity and fidelity of the inversion results. Specifically, the objective function of the MAVAI method can be expressed as

$${\mathbf{v}}^{*}=\arg \underset{\mathbf{v}}{\min }{‖W_d\left(Y_{obs}-\mathbf{Q}\mathbf{v}\right)‖}_2^2+\lambda {‖W_m\left({\mathbf{v}}_{ref}-\mathbf{v}\right)‖}_2^2+\mu {‖L\mathbf{v}‖}_2^2$$

(12)

$$\mathbf{L}=\left[\begin{array}{ccccccccccc}{\mathbf{I}}_{n\times n}& \mathbf{O}& \mathbf{O}& -2{\mathbf{I}}_{n\times n}& \mathbf{O}& \mathbf{O}& {\mathbf{I}}_{n\times n}& \mathbf{O}& \mathbf{O}& \cdots & \mathbf{O}\\ \mathbf{O}& {\mathbf{I}}_{n\times n}& \mathbf{O}& \mathbf{O}& -2{\mathbf{I}}_{n\times n}& \mathbf{O}& \mathbf{O}& {\mathbf{I}}_{n\times n}& \mathbf{O}& \ddots & ⋮\\ \mathbf{O}& \mathbf{O}& {\mathbf{I}}_{n\times n}& \mathbf{O}& \mathbf{O}& -2{\mathbf{I}}_{n\times n}& \mathbf{O}& \mathbf{O}& {\mathbf{I}}_{n\times n}& \ddots & \mathbf{O}\\ ⋮& \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & ⋮\\ \mathbf{O}& \cdots & \mathbf{O}& \mathbf{O}& {\mathbf{I}}_{n\times n}& \mathbf{O}& \mathbf{O}& -2{\mathbf{I}}_{n\times n}& \mathbf{O}& \mathbf{O}& {\mathbf{I}}_{n\times n}\\ \mathbf{O}& \cdots & \mathbf{O}& \cdots & \mathbf{O}& \cdots & \mathbf{O}& \cdots & \mathbf{O}& \cdots & \mathbf{O}\\ \mathbf{O}& \cdots & \mathbf{O}& \cdots & \mathbf{O}& \cdots & \mathbf{O}& \cdots & \mathbf{O}& \cdots & \mathbf{O}\end{array}\right]$$

(13)

where $L\in {ℝ}^{3nN\times 3nN}$ denotes the second-order difference operator in the horizontal direction, and $\mu$ is the regularization parameter. Equation (12) is the general form of the Tikhonov regularization problem, and its optimal solution can be expressed as follows:

$${\mathbf{v}}^{*}={\mathbf{A}}^{-1}p$$

(14)

where $\mathbf{A}=\left[\begin{array}{c}{W}_d\mathbf{Q}\\ \sqrt{2\lambda }{W}_m\\ -\sqrt{2\mu }\mathbf{L}\end{array}\right]\in {ℝ}^{\left(m+6\right)nN\times 3nN}$ and $p=\left[\begin{array}{c}{W}_d{Y}_{obs}\\ \sqrt{2\lambda }{W}_m{\mathbf{v}}_{ref}\\ 0\end{array}\right]\in {ℝ}^{\left(m+6\right)nN\times 1}$. Equation (14) is computationally bottlenecked by the inversion of large matrices. To avoid inverting large matrices, the linear system shown in Equation (14) needs to be solved using the conjugate gradient method [32]. Nevertheless, the MAVAI method still suffers from low computational efficiency.

2.2. Compressed MAVAI Framework

To improve inversion efficiency, a compressed MAVAI (CMAVAI) framework is proposed based on the principles of compressed sensing. Specifically, the CMAVAI framework first uses a sparse transformation to map the MAVAI objective function to a sparse space, yielding the following optimization function:

$${\mathbf{v}}^{*}=\arg \underset{\mathbf{v}}{\min }{‖Y^{\prime }-{\Lambda }_1{W}_d\mathbf{Q}\mathbf{v}‖}_2^2+\lambda {‖{{\mathbf{v}}^{\prime }}_{ref}-{\Lambda }_2{W}_m\mathbf{v}‖}_2^2+\mu {‖L\mathbf{v}‖}_2^2$$

(15)

where ${\Lambda }_1=kron\left({\mathbf{I}}_{N\times N},kron\left({\mathbf{I}}_{m\times m},{\Phi }_1\right)\right)$; ${\Lambda }_2=kron\left({\mathbf{I}}_{N\times N},kron\left({\mathbf{I}}_{m\times m},{\Phi }_2\right)\right)$; ${\Phi }_1$ and ${\Phi }_2$ denote the sparse transformations; and $Y^{\prime }={\Lambda }_1{W}_d{Y}_{obs}$ and ${{\mathbf{v}}^{\prime }}_{ref}={\Lambda }_2{W}_m{\mathbf{v}}_{ref}$ denote the sparse-domain representations of the seismic data and the low-frequency reference model, respectively. Subsequently, the measurement matrix needs to be designed based on the restricted isometry property [33] to obtain compressed measurements of seismic data and the reference model. These compressed measurements in the sparse domain are then employed to construct the CMAVAI objective function, thereby reducing the scale of the inverse problem and improving inversion efficiency. Formally, the objective function of the CMAVAI framework can be described as

$${\mathbf{v}}^{*}=\arg \underset{\mathbf{v}}{\min }{‖\widehat{Y}-{\Omega }_1{W}_d\mathbf{Q}\mathbf{v}‖}_2^2+\lambda {‖{\widehat{\mathbf{v}}}_{ref}-{\Omega }_2{W}_m\mathbf{v}‖}_2^2+\mu {‖L\mathbf{v}‖}_2^2$$

(16)

where ${\Omega }_1=kron\left({\mathbf{I}}_{N\times N},kron\left({\mathbf{I}}_{m\times m},{\mathbf{\Psi }}_1{\Phi }_1\right)\right)$ and ${\Omega }_2=kron\left({\mathbf{I}}_{N\times N},kron\left({\mathbf{I}}_{m\times m},{\mathbf{\Psi }}_2{\Phi }_2\right)\right)$ denote the sensing matrices,${\mathbf{\Psi }}_1$ and ${\mathbf{\Psi }}_2$ denote the designed measurement matrices, and $\widehat{Y}={\Omega }_1{W}_d{Y}_{obs}$ and ${\widehat{\mathbf{v}}}_{ref}={\Omega }_2{W}_m{\mathbf{v}}_{ref}$ denote the compressed measurements of the observed data and the reference model in the sparse space, respectively. Since Equation (16) still belongs to the general form of the Tikhonov regularization problem, its optimal solution can still be expressed as follows:

$${\mathbf{v}}^{*}={\tilde{\mathbf{A}}}^{-1}\widehat{p}$$

(17)

where $\tilde{\mathbf{A}}=\left[\begin{array}{c}{\Omega }_1{W}_d\mathbf{Q}\\ \sqrt{2\lambda }{\Omega }_2{W}_m\\ -\sqrt{2\mu }L\end{array}\right]\in {ℝ}^{\left(m{\kappa }_1+3{\kappa }_2+3\right)nN\times 3nN}$; $\widehat{p}=\left[\begin{array}{c}\widehat{Y}\\ \sqrt{2\lambda }{\widehat{\mathbf{v}}}_{ref}\\ 0\end{array}\right]\in {ℝ}^{\left(m{\kappa }_1+3{\kappa }_2+3\right)nN\times 1}$; and ${\kappa }_1,{\kappa }_2\in \left(0,1\right]$ denotes the compression measurement ratios of the seismic data and the low-frequency reference model, respectively. As the compression measurement ratios are less than 1, the inversion problem of the proposed CMAVAI framework (i.e., Equation (17)) is significantly smaller in scale than that of the traditional MAVAI method (Equation (14)), which provides a reliable guarantee for rapid inversion. Similarly to the MAVAI method, the CMAVAI method also employs the conjugate gradient method to solve Equation (17) in order to avoid inverting large matrices, thereby efficiently obtaining multiple inversion parameters.

2.3. Compressed MAVAI with Elastic Half-Norm Regularization

Regarding vertical constraints on inversion parameters, the Tikhonov regularization problems shown in Equations (12) and (16) consider only the smoothness characteristics of the strata and do not account for their blocky features. Consequently, both the traditional MAVAI method and the CMAVAI method proposed in Section 2.2 tend to produce overly smooth inversion results, which not only blur stratigraphic boundary features but also introduce significant pseudo-layer artifacts parallel to the strata. To address the above issues, an elastic half-norm regularized CMAVAI method is developed. Specifically, the RCMAVAI method incorporates the elastic half-norm [34], defined as the weighted sum of the L1/2 and L2 norms, into the CMAVAI framework as a vertical constraint on the inversion parameters to achieve the simultaneous and accurate estimation of multiple formation parameters. The optimization function of the RCMAVAI method can be expressed in the following form:

$${\mathbf{v}}^{*}=\arg \underset{\mathbf{v}}{\min }{‖\widehat{Y}-{\Omega }_1{W}_d\mathbf{Q}\mathbf{v}‖}_2^2+\lambda {‖{\widehat{\mathbf{v}}}_{ref}-{\Omega }_2{W}_m\mathbf{v}‖}_2^2+\mu {‖L\mathbf{v}‖}_2^2+\eta {‖\mathbf{B}\mathbf{v}‖}_{1/2}^{1/2}+\beta {‖\mathbf{B}\mathbf{v}‖}_2^2$$

(18)

where $\mathbf{B}=kron\left({\mathbf{I}}_{N\times N},{\mathbf{D}}^{\prime }\right)$ and ${\mathbf{D}}^{\prime }=kron\left({\mathbf{I}}_{3\times 3},\mathbf{D}\right)$. As the L2 norm characterizes the smoothness of subsurface media and the L1/2 norm characterizes their blocky nature, the developed RCMAVAI method can yield inversion results that simultaneously preserve sharp interlayer boundaries and smooth intralayer structures. In other words, the proposed RCMAVAI method effectively eliminates interference from pseudo-layers while ensuring that the inversion results retain sharp boundary features.

To solve Equation (18), an auxiliary variable $\mathbf{B}\mathbf{v}=\mathbf{H}$ is introduced using the operator splitting technique [35] to reformulate it as the following optimization problem:

$$\begin{array}{c}\left({\mathbf{v}}^{*},H^{\ast }\right)=\arg \underset{\mathbf{v},H}{\min }{‖\widehat{Y}-{\Omega }_1{W}_d\mathbf{Q}\mathbf{v}‖}_2^2+\lambda {‖{\widehat{\mathbf{v}}}_{ref}-{\Omega }_2{W}_m\mathbf{v}‖}_2^2+\mu {‖F\mathbf{v}‖}_2^2+\eta {‖H‖}_{1/2}^{1/2}\\ \mathrm{s}.\mathrm{t}. H=B\mathbf{v}\end{array}$$

(19)

where $F=L+\sqrt{\beta /\mu }B$. Equation (19) can be optimized using the half-quadratic splitting method [36]. Specifically, the half-quadratic splitting method can decompose Equation (19) into the following two subproblems, which are updated iteratively:

$${\mathbf{v}}^{\left(j+1\right)}=\arg \underset{\mathbf{v}}{\min }{‖\widehat{\mathbf{Y}}-{\mathbf{\Omega }}_1{\mathbf{W}}_d\mathbf{Q}\mathbf{v}‖}_2^2+\lambda {‖{\widehat{\mathbf{v}}}_{ref}-{\mathbf{\Omega }}_2{\mathbf{W}}_m\mathbf{v}‖}_2^2+\mu {‖\mathbf{F}\mathbf{v}‖}_2^2+{\displaystyle \frac{{\varsigma }_j}{2}}{‖\mathbf{B}\mathbf{v}-{\mathbf{H}}^{\left(j\right)}‖}_2^2$$

(20)

$${\mathbf{H}}^{\left(j+1\right)}=\arg \underset{\mathbf{H}}{\min } \eta {‖\mathbf{H}‖}_{1/2}^{1/2}+{\displaystyle \frac{{\varsigma }_j}{2}}{‖\mathbf{B}{\mathbf{v}}^{\left(j+1\right)}-\mathbf{H}‖}_2^2$$

(21)

where j is the iteration index, ${\varsigma }_j=\xi {\varsigma }_{j-1}$, and $\xi >1$ is a scaling factor. The v-subproblem, i.e., Equation (20), is a typical quadratic optimization problem, and its optimal solution can be expressed as follows:

$${\mathbf{v}}^{\left(j+1\right)}={\mathbf{E}}^{-1}\mathbf{u}$$

(22)

$$\mathbf{E}=2{\mathbf{Q}}^T{\mathbf{W}}_d^T{\mathbf{\Omega }}_1^T{\mathbf{\Omega }}_1{\mathbf{W}}_d\mathbf{Q}+2\lambda {\mathbf{W}}_m^T{\mathbf{\Omega }}_2^T{\mathbf{\Omega }}_2{\mathbf{W}}_m+2\mu {\mathbf{F}}^T\mathbf{F}+{\varsigma }_j{\mathbf{B}}^T\mathbf{B}$$

(23)

$$\mathbf{u}=2{\mathbf{Q}}^T{\mathbf{W}}_d^T{\mathbf{\Omega }}_1^T\widehat{\mathbf{Y}}+2\lambda {\mathbf{W}}_m^T{\mathbf{\Omega }}_2^T{\widehat{\mathbf{v}}}_{ref}+{\varsigma }_j{\mathbf{B}}^T{\mathbf{H}}^{\left(j\right)}$$

(24)

To avoid matrix inversion, Equation (22) must be solved using the conjugate gradient method [32] to quickly obtain the optimal solution. The H-subproblem, i.e., Equation (21), is a typical non-convex L1/2 norm optimization problem [34,37,38], for which its solution can be obtained using the following function:

$${\mathbf{H}}^{\left(j+1\right)}={\mathcal{H}}_{\zeta }\left(\mathbf{B}{\mathbf{v}}^{\left(j+1\right)}\right)$$

(25)

$${\mathcal{H}}_{\zeta }\left(\mathbf{x}\right)=\left\{\begin{array}{ll}{f}_{\zeta }\left({\mathbf{x}}_i\right),& \left|{\mathbf{x}}_i\right|>{\displaystyle \frac{3\sqrt[3]{2}}{4}}{\zeta }^{{\displaystyle \frac{2}{3}}}\\ 0,& \mathrm{otherwise}\end{array}\right.$$

(26)

$$f_{\zeta }\left({\mathbf{x}}_i\right)={\displaystyle \frac{2{\mathbf{x}}_i}{3}}\left(1+\cos \left({\displaystyle \frac{2\pi }{3}}-{\displaystyle \frac{2{\phi }_{\zeta }\left({\mathbf{x}}_i\right)}{3}}\right)\right)$$

(27)

$${\phi }_{\zeta }\left({\mathbf{x}}_i\right)=\arccos \left({\displaystyle \frac{\zeta }{8}}{\left({\displaystyle \frac{g\left({\mathbf{x}}_i\right)}{3}}\right)}^{-3/2}\right)$$

(28)

$$g\left(x\right)=\left\{\begin{array}{ll}\left|x\right|,& \left|x\right|>a\\ -{\displaystyle \frac{1}{8a^3{x}^4}}+{\displaystyle \frac{3}{4a}}{x}^2+{\displaystyle \frac{3}{8}}a,& \left|x\right|\le a\end{array}\right.$$

(29)

where $\zeta =\eta /{\varsigma }_j$, and $a\in \left[{10}^{-4},{10}^{-6}\right]$ is a small positive constant. By alternately updating the original variable $\mathbf{v}$ and auxiliary variable $\mathbf{H}$ using Equations (22) and (25) until the stopping criteria are met, the inversion parameter ${\mathbf{v}}^{*}$ can be obtained. Finally, by performing the following exponential calculation, the P-wave impedance, S-wave impedance, and density of the subsurface medium can be acquired:

$$\left[Z_p Z_s \mathit{\rho }\right]=\exp \left({\mathbf{v}}^{*}\right)$$

(30)

where $Z_p$, $Z_s$, and $\mathit{\rho }$ denote the P-wave impedance, S-wave impedance, and density of the subsurface medium respectively.

3. Examples

3.1. Synthetic Example

In this section, the SEG/EAGE overthrust model is used to validate the feasibility of the proposed pre-stack AVA inversion method. Figure 1a–c show the P-wave impedance, S-wave impedance, and density profiles of this SEG/EAGE overthrust model, respectively. Using Figure 1 and Fatti’s approximation equation to calculate the reflection coefficients and convolving them with the 40 Hz Ricker wavelet, noise-free angle records were obtained (as shown in Figure 2, where the incident angles of the data shown in Figure 2a, Figure 2b and Figure 2c are 10°, 20°, and 30°, respectively). To obtain the elastic parameters of subsurface strata, we perform pre-stack inversions using the traditional MAVAI method and the proposed CMAVAI and RCMAVAI methods. Since all three of the above methods require introducing low-frequency reference models to reduce the ill-posed feature of multiparameter simultaneous inversion, we apply a low-pass filter to the data in Figure 1 to generate a low-frequency reference model (Figure 3). Figure 4 shows the P-wave impedance estimated by MAVAI, CMAVAI, and RCMAVAI, as well as the differences between these estimates and the true model (Figure 1a). Figure 5 shows the S-wave impedance estimated by MAVAI, CMAVAI, and RCMAVAI, as well as their differences from the true model (Figure 1b). Figure 6 shows the density estimated by MAVAI, CMAVAI, and RCMAVAI, as well as their differences from the true model (Figure 1c). In this case, both the proposed CMAVAI and RCMAVAI methods employ the discrete cosine transform and binary diagonal matrices as the domain transformation operator and measurement matrices, respectively. Moreover, the compression measurement ratios for the seismic data and reference model are set to 0.25 and 0.1, respectively. For the MAVAI and CMAVAI methods, regularization parameters $\lambda$ and $\mu$ are set to 0.7 and 1.2, respectively. In the RCMAVAI method, regularization parameters $\lambda$, $\mu$, $\eta$, and $\beta$ are set to 0.7, 1.2, 0.005, and 0.001, respectively. As shown in Figure 4, Figure 5 and Figure 6, the inversion results obtained using the traditional MAVAI method and the proposed CMAVAI are overly smoothed, with formation boundaries becoming blurred and false formation artifacts introduced (as indicated by the black arrows). Such issues arise because the Tikhonov regularized inversion model considers only the smooth characteristics of the formation while ignoring its blocky features. In contrast, since the RCMAVAI method simultaneously accounts for both the smooth and blocky characteristics of the strata, it yields more accurate inversion results. Specifically, these results effectively preserve clear stratigraphic boundaries while avoiding false pseudo-layer interference, which is highly beneficial for the accurate identification of subsurface reservoirs. Furthermore, as can be observed in Figure 4, Figure 5 and Figure 6, the MAVAI and CMAVAI methods exhibit significant residual leakage, whereas the RCMAVAI method produces relatively smaller residuals. This further demonstrates that, compared to the MAVAI and CMAVAI methods, the proposed RCMAVAI method can more accurately invert the elastic parameters of subsurface formations. For quantitative comparison,

Table 1. Quantitative evaluation and calculation time of the inversion results obtained by different inversion methods in the noise-free case.

Method		MSE		Run Time (s)
	P-Wave Impedance	S-Wave Impedance	Density
MAVAI	7.9321 × 104	4.4658 × 104	1.6662 × 10−4	518.99
CMAVAI	7.9501 × 104	4.4702 × 104	1.6668 × 10−4	167.98
RCMAVAI	3.7023 × 104	2.2719 × 104	0.8107 × 10−4	244.12

lists the mean squared error (MSE) and computational time for the inversion results obtained by the three methods. As shown in Table 1, the MSE values of the inversion results from the traditional MAVAI method and the proposed CMAVAI method are essentially consistent, while the computational time of the proposed CMAVAI method is significantly shorter than that of the traditional MAVAI method. This result effectively confirms that the proposed CMAVAI framework can significantly improve inversion efficiency while maintaining the same level of inversion accuracy. Additionally, Table 1 also shows that, compared to the traditional MAVAI method, the RCMAVAI method yields results with a smaller MSE and requires less computational time. These quantitative evaluation results further demonstrate the superiority of the proposed RCMAVAI method.

To further investigate the robustness of the proposed methods, we create noisy pre-stack data (as shown in Figure 7) by adding Gaussian noise with a signal-to-noise ratio of 2 to the data in Figure 2 for subsequent testing. Consistent with the noise-free case, the traditional MAVAI method and the proposed CMAVAI and RCMAVAI methods all use Figure 3 as the reference model. Furthermore, for the proposed CMAVAI and RCMAVAI methods, the compression measurement ratios of the seismic data and the reference model are still set to 0.25 and 0.1, respectively. For the regularization parameters, the values of $\lambda$ and $\mu$ in the MVAI and CMAVAI methods are set to 0.7 and 3, respectively. In the RCMAVAI method, regularization parameters $\lambda$, $\mu$, $\eta$, and $\beta$ are set to 0.7, 3, 0.01, and 0.004, respectively. Figure 8a, Figure 8c and Figure 8e show the P-wave impedance estimated by the MAVAI, CMAVAI, and RCMAVAI methods, respectively; Figure 8b,d,f show the differences between these inversion results and the true P-wave impedance (Figure 1a), respectively. Figure 9a,c,e show the shear wave impedance estimated by the MAVAI, CMAVAI, and RCMAVAI methods, respectively; Figure 9b,d,f show the differences between these inversion results and the true S-wave impedance (Figure 1b), respectively. Figure 10a, Figure 10c and Figure 10e show the density profiles estimated by the MAVAI, CMAVAI, and RCMAVAI methods, respectively; Figure 10b,d,f show the differences between these inversion results and the true density profile (Figure 1c), respectively. From Figure 8, Figure 9 and Figure 10, it can be seen that, compared with the MAVAI method and the CMAVAI method, the RCMAVAI method obtains better inversion results in the noisy case. These results are not only closer to the true model but also effectively preserve clear formation boundaries and significantly reduce interference from additional pseudo-layers (as shown by the black arrows of Figure 8e, Figure 9e and Figure 10e).

Table 2. Quantitative evaluation and calculation time of the inversion results obtained by different inversion methods in the noisy case.

Method		MSE		Run Time (s)
	P-Wave Impedance	S-Wave Impedance	Density
MAVAI	1.4569 × 105	7.7479 × 104	2.9764 × 10−4	535.58
CMAVAI	1.4590 × 105	7.7503 × 104	2.9771 × 10−4	174.42
RCMAVAI	1.1232 × 105	4.9417 × 104	2.5293 × 10−4	251.77

presents the quantitative evaluation and computational time of the inversion results obtained by the above three methods. As shown in Table 2, the inversion results obtained by the proposed CMAVAI method have basically the same MSE value as those obtained by the traditional MAVAI method, but its computational time is significantly lower than that of the traditional MAVAI method. This result further indicates that the proposed CMAVAI framework can effectively improve inversion efficiency while basically maintaining inversion accuracy. Additionally, from Table 2, it can also be found that, in the noisy case, the MSE values of the elastic parameters predicted by the RCMAVAI method are also smaller than those predicted by the traditional MAVAI method, and their computational time is shorter than that of the traditional MAVAI method. This result indicates that the RCMAVAI method offers greater robustness and faster inversion efficiency compared to the traditional MAVAI method. In other words, even in the presence of noise, the RCMAVAI method can rapidly produce more accurate inversion results.

3.2. Field Example

To further test the practicality of the proposed methods, this subsection applies these to field data (as shown in Figure 11). Specifically, Figure 11a shows the stacked profile with a small incidence angle (angle range of 1–9°), Figure 11b shows the stacked profile with a medium incidence angle (angle range of 10–18°), and Figure 11c shows the stacked profile with a large incidence angle (angle range of 19–25°). Note that there is a blind well at the purple dashed line in Figure 11, which is used to verify the reliability of the inversion results. Using the traditional MAVAI method, as well as the proposed CMAVAI and RCMAVAI methods, to simultaneously estimate P-wave impedance, S-wave impedance, and density parameters, the obtained results are displayed in Figure 12, Figure 13 and Figure 14, respectively. Within these, Figure 12a–c, respectively, show the P-wave impedance profiles predicted by the MAVAI, CMAVAI, and RCMAVAI methods. Figure 13a–c show the S-wave impedance profiles predicted by the MAVAI, CMAVAI, and RCMAVAI methods, respectively. Figure 14a–c show the density profiles predicted by the MAVAI, CMAVAI, and RCMAVAI methods, respectively. In this example, the low-frequency reference models used by these three methods are constructed by performing Kriging interpolation on well data (excluding the blind well) within the study area and subsequently applying a 10 Hz low-pass filter. Additionally, for the developed CMAVAI and RCMAVAI methods, the compression measurement ratios of the seismic data and the low-frequency reference model are set to 0.25 and 0.05, respectively. For the regularization parameters, the values of $\lambda$ and $\mu$ in the MVAI and CMAVAI methods are set to 1 and 3.7, respectively. In the RCMAVAI method, regularization parameters $\lambda$, $\mu$, $\eta$, and $\beta$ are set to 1, 3.7, 0.001, and 0.004, respectively. As shown in Figure 12, Figure 13 and Figure 14, both the MAVAI and CMAVAI methods are affected by pseudo-layer interference that blurs formation boundaries and causes aliasing between formations (as indicated by the ellipses). In contrast, the RCMAVAI method effectively avoids pseudo-layer interference, resulting in inversion results that not only exhibit clear boundary features but also have higher stratigraphic resolution. This result once again demonstrates that the proposed RCMAVAI method yields better inversion results compared to the traditional MAVAI method. To further verify the reliability of the inversion results, Figure 15, Figure 16 and Figure 17 show the comparison between the P-wave impedance, S-wave impedance, and density parameters obtained from these three methods and the actual logging data, respectively. In Figure 15, Figure 16 and Figure 17, the red curve represents the inversion result, the blue curve represents the logging data, and the green curve represents the low-frequency reference model. As observed in Figure 15, Figure 16 and Figure 17, the results obtained from the MAVAI and CMAVAI methods are largely consistent, but their consistency with logging data is poor due to the influence of pseudo-layer interference (as shown by the black arrow). Conversely, the RCMAVAI method avoids pseudo-layer interference, yielding inversion results that show a higher consistency with the logging data. For a quantitative comparison,

Table 3. Quantitative evaluation of the inversion results obtained by different inversion methods and the logging data in the field example.

Method	Correlation Coefficients			MSE
	P-Wave Impedance	S-Wave Impedance	Density	P-Wave Impedance	S-Wave Impedance	Density
MAVAI	0.7831	0.6307	0.7644	3.1601 × 105	1.0151 × 105	1.4455 × 10−3
CMAVAI	0.7825	0.6303	0.7639	3.1676 × 105	1.0160 × 105	1.4467 × 10−3
RCMAVAI	0.8649	0.7151	0.8476	2.0590 × 105	8.0234 × 104	8.1657 × 10−4

presents the correlation coefficients and MSE values between the inversion results and the true logging data. As shown in Table 3, the RCMAVAI method yields higher correlation coefficients and lower MSE values. This result further confirms that the RCMAVAI method exhibits higher reliability in estimating reservoir parameters. In this case, the computation times for the MAVAI, CMAVAI, and RCMAVAI methods are 186.23 s, 94.55 s, and 114.70 s, respectively. Clearly, compared to the traditional MAVAI method, both the CMAVAI and RCMAVAI methods have higher computational efficiency. The above results also demonstrate once again that the proposed CMAVAI framework is feasible and effective in reducing the scale of the inverse problem and improving pre-stack inversion efficiency by utilizing compressed measurements in the sparse domain.

4. Discussion

The proposed CMAVAI and RCMAVAI algorithms require compressive measurements of seismic data and low-frequency reference models using a measurement matrix to reduce the scale of the inverse problem and improve inversion efficiency. As a key parameter of the measurement matrix, the optimal selection of the compression measurement ratios is crucial for the proposed algorithms. Herein, we use Figure 2 as an example to investigate the impact of the compression measurement ratios on the proposed algorithms. Figure 18a–e illustrate the effects of the compression measurement ratio ${\kappa }_1$ of the seismic data on the information loss rate ${\chi }_{k_1}=1-{‖{\Omega }_1{Y}_{obs}‖}_2^2/{‖Y_{obs}‖}_2^2$, inversion accuracy, and inversion runtime, respectively, while keeping the compression measurement ratio of the low-frequency reference model constant (i.e., ${\kappa }_2=0.1$). As shown in Figure 18a, when ${\kappa }_1$ is less than 0.25, the information loss rate ${\chi }_{k_1}$ is significantly greater than 10⁻⁴, indicating that the compressed measurements cannot maximally preserve the information contained in the original signal. Therefore, the proposed CMAVAI and RCMAVAI methods do not obtain optimal inversion results. When ${\kappa }_1$ is greater than 0.25, the information loss rate ${\chi }_{k_1}$ gradually approaches 0. The result indicates that compression measurements greatly minimize the loss of information contained in the original signal. Under this condition, the proposed CMAVAI and RCMAVAI methods can effectively maintain the inversion accuracy of the elastic parameters. The quantitative evaluation metric of the inversion results (Figure 18b–d) also effectively confirms the above conclusion. Furthermore, Figure 18e reveals that an increase in the compression measurement ratio leads to an increase in computation time. The specific reason is that the scale of the inverse problem is positively correlated with the compression measurement ratio ${\kappa }_1$. Figure 19a–e illustrate the effects of the compression measurement ratio ${\kappa }_2$ of the low-frequency reference model on the information loss rate ${\chi }_{k_2}=1-{‖{\Omega }_2{\mathbf{v}}_{ref}‖}_2^2/{‖{\mathbf{v}}_{ref}‖}_2^2$, inversion accuracy, and inversion runtime, respectively, while keeping the compression measurement ratio of the seismic data constant (i.e., ${\kappa }_1=0.25$). Similarly to Figure 18, when the information loss rate ${\chi }_{k_2}$ is closer to 0, the proposed CMAVAI and RCMAVAI algorithms improve inversion accuracy but increase inversion time. To balance inversion accuracy and computational efficiency, the compression measurement ratio is generally set in practical applications to the value corresponding to the inflection point of the isometric constant.

The proposed RCMAVAI method incorporates three constraint terms: the reference model constraint, the lateral constraint, and the vertical constraint. A reasonable balance among the contributions of these three constraint terms is key to obtaining optimal inversion results. In practice, the magnitude of each constraint term’s contribution is controlled by the regularization parameter. Specifically, the regularization parameter $\lambda$ controls the contribution of the reference model. An excessively large value of $\lambda$ forces the inverted solutions to converge to the reference model, whereas an overly small value fails to constrain the solutions and thereby induces solution instability. The regularization parameter $\mu$ controls the weighting of horizontal constraints: an oversize $\mu$ leads to lateral smearing in inversion results, while an undersized $\mu$ cannot eliminate the lateral instability of the inverted solutions. Parameters $\eta$ and $\beta$ govern the weighting of vertical constraint terms. Among them, parameter $\eta$ controls the blocky characteristics of inversion results, while parameter $\beta$ determines their smoothness properties. In practical implementations, the regularization parameters $\lambda$ and $\mu$ need to be selected via the generalized cross-validation method [39]. After determining $\lambda$ and $\mu$, parameters $\eta$ and $\beta$ are optimized through the trial-and-error approach. Empirically, the ratio of $\eta$ to $\beta$ within the range of 1 to 5 often yields the optimal solution. In the future, how to achieve adaptive optimization of the regularization parameters $\eta$ and $\beta$ will become an important research direction.

5. Conclusions

To address the issues of low computational efficiency and insufficient prediction accuracy in traditional MAVAI algorithms, this paper proposes a compressed MAVAI (CMAVAI) method via elastic half-norm regularization. To effectively improve inversion efficiency, the CMAVAI framework in the sparse domain is developed. Specifically, this framework first employs an orthogonal transformation to map seismic data and the reference model into the sparse domain. Next, an observation operator is designed based on the restricted isometry property to perform compressed measurements. Finally, these compressed measurements are employed to construct an objective function for CMAVAI, which has a smaller scale and faster solving efficiency. To improve inversion accuracy, the elastic half-norm regularization is introduced as a vertical constraint on the inversion parameters within the CMAVAI framework, thereby constructing a regularized CMAVAI model. Since the regularized CMAVAI model simultaneously accounts for both the smooth and blocky characteristics of subsurface formations, it effectively improves the accuracy of reservoir parameter inversion compared to the traditional MAVAI model that considers only smoothness. The performance of the proposed method is tested using both theoretically generated data and field data, and the results confirm its feasibility and reliability.