# Amplitude-Versus-Angle (AVA) Inversion for Pre-Stack Seismic Data with L0-Norm-Gradient Regularization

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. AVA Inversion for Pre-Stack Seismic Data

_{pp}(θ) is the P-wave reflection coefficient at incident angle θ; $a(\theta )={\mathrm{sec}}^{2}\theta $; $b(\theta ,\gamma )=-8{\gamma}^{2}{\mathrm{sin}}^{2}\theta $; $c(\theta ,\gamma )=(1-4{\gamma}^{2}{\mathrm{sin}}^{2}\theta )$; γ is the ratio of background S-wave velocity to P-wave velocity; and R

_{vp}, R

_{vs}and R

_{ρ}are the relative variation rates of P-wave velocity, S-wave velocity and density, respectively.

**d**are well described by the synthetic data given by

**Wr**and an additive noise

**n**in this paper.

**Φ**=

**WA**$\in {R}^{MN\times 3N}$, which is the combined matrix of

**W**and

**A**and serves as the forward operator of conventional AVA inversion.

#### 2.2. L0-Norm-Gradient Regularization

**m**, its L0-norm can be expressed as [12]

**m**.

**D**is the first-order difference operator matrix.

#### 2.3. AVA Inversion with L0-Norm-Gradient Regularization

**R**

_{vp}$\in {R}^{N}$ represents the vector of relative variation rates of P-wave velocity at N samples, i.e.,

**u**$\in {R}^{N}$ is the vector with the following elements:

**u**is the vector of logarithmic P-wave velocity.

**R**

_{v}

_{s}$\in {R}^{N}$ and

**R**

_{ρ}$\in {R}^{N}$ represent the vector of relative variation rates of S-wave velocity and density at N samples, and

**v**$\in {R}^{N}$ and

**w**$\in {R}^{N}$ are the vectors of logarithmic S-wave velocity and density, respectively.

**R**

_{v}

_{p},

**R**

_{v}

_{s}and

**R**

_{ρ}, one can obtain

**r**in Equation (2) and give the following equation:

**m**$\in {R}^{3N}$ is the vector combined

**u**,

**v**and

**w**, and

**D**$\in {R}^{3N\times 3N}$ represents a combined difference matrix for

**u**,

**v**and

**w**.

**G**=

**ΦD**$\in {R}^{MN\times 3N}$, which is the combined matrix of

**Φ**and

**D**.

**Gm**is the forward equation for the merged AVA inversion, and

**m**is the model parameters. Based on Equation (24), we can directly estimate P-wave velocity, S-wave velocity and density.

**m**contains logarithmic P-wave velocity, logarithmic S-wave velocity and logarithmic density. Generally, there are correlations between different parameters. In addition, the contributions from the relative change rates of different parameters to the P-wave reflection coefficient are different. Compared to the relative change rates of P-wave velocity and S-wave velocity, the contribution from the relative change rate of density is very small [5]. Hence, the estimation of density is extremely unstable. To deal with this issue in AVA inversion, we can introduce a weighting matrix

**W**

_{m}$\in {R}^{3N\times 3N}$ for model parameters.

**W**

_{d}$\in {R}^{MN\times MN}$ for seismic data.

**W**

_{m}and

**W**

_{d}, the objective function (25) can be updated as

**W**

_{m}and

**W**

_{d}usually take the following forms [9]:

**C**

_{m}$\in {R}^{3N\times 3N}$ and

**C**

_{d}$\in {R}^{MN\times MN}$ are the covariance matrix of model parameters and seismic data, respectively.

**C**

_{d}can be obtained from the observed pre-stack seismic data through statistical estimation. The covariance matrix

**C**

_{m}can be obtained from well log data or by means of empirical petrophysical relationships between model parameters [5,6]. For example, the Gardner formula represents the statistical relation between P-wave velocity and density [29], the Castagna formula represents the statistical relation between P-wave velocity and S-wave velocity [30], etc.

**W**

_{m}at each iteration using inverted model parameters from the last iteration. This is the so-called strategy of iterative re-weighting, and AVA inversion becomes a nonlinear problem. However, the computational cost has increased accordingly.

#### 2.4. Split-Bregman-like Algorithm

**a**$\in {R}^{3N}$; hence, the objective function (26) can be re-written as

**β**$\in {R}^{3N}$ can be interpreted as a Lagrangian multiplier vector.

**a**and

**DW**

_{m}

**m**.

**a**and

**m**. In each sub-problem, one variable is fixed with values obtained from the previous iteration.

**a**fixed and estimate

**m**. The objective function for sub-problem 1 takes the following form:

**m**fixed and update

**a**. The objective function for sub-problem 2 takes the following form:

**DW**

_{m}

**m**)(i) are the ith element of

**a**and

**DW**

_{m}

**m**, respectively.

## 3. Synthetic Data Tests

**C**

_{m}and data covariance matrix

**C**

_{d}. The inverse of these two covariance matrices is served as the weighting matrices.

**m**is true model parameters. The calculated REs of inverted model parameters shown in Figure 2 are listed in Table 1. We can see that the RE for L0-AVA is much smaller than for C-AVA.

## 4. Field Data Applications

**C**

_{m}and data covariance matrix

**C**

_{d}, and the inverse of these two covariance matrices was served as the weighting matrices.

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**The true and inverted model parameters: (

**a**) P-wave velocity (m/s); (

**b**) S-wave velocity (m/s); (

**c**) density (g/cm

^{3}). The red curves are true model parameters, the green curves are inverted model parameters by L0-AVA inversion and the black curves are inverted model parameters by C-AVA inversion.

**Figure 3.**The real seismic data profiles with different partial angle stack: (

**a**) 2–11°; (

**b**) 12–21°; (

**c**) 22–31°.

**Figure 4.**The inverted model parameters profiles: (

**a**) P-wave velocity (m/s); (

**b**) S-wave velocity (m/s); (

**c**) density (g/cm

^{3}). The locations of white arrows and ellipses shown in the inverted S-wave velocity profiles are oil-bearing sandstone from the interpretation of the well log.

**Figure 5.**The single trace comparison between well log data and inverted model parameters: (

**a**) P-wave velocity (m/s); (

**b**) S-wave velocity (m/s); (

**c**) density (g/cm

^{3}). The blue curves are well log, the red curves are inverted model parameters and the green curves are initial model in the process of inversion.

Inversion Method | RE |

C-AVA | 0.2223 |

L0-AVA | 0.1305 |

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**MDPI and ACS Style**

Dai, R.; Yang, J.
Amplitude-Versus-Angle (AVA) Inversion for Pre-Stack Seismic Data with L0-Norm-Gradient Regularization. *Mathematics* **2023**, *11*, 880.
https://doi.org/10.3390/math11040880

**AMA Style**

Dai R, Yang J.
Amplitude-Versus-Angle (AVA) Inversion for Pre-Stack Seismic Data with L0-Norm-Gradient Regularization. *Mathematics*. 2023; 11(4):880.
https://doi.org/10.3390/math11040880

**Chicago/Turabian Style**

Dai, Ronghuo, and Jun Yang.
2023. "Amplitude-Versus-Angle (AVA) Inversion for Pre-Stack Seismic Data with L0-Norm-Gradient Regularization" *Mathematics* 11, no. 4: 880.
https://doi.org/10.3390/math11040880