Prestack AVO Inversion of Exact Zoeppritz Equation Using Adaptive Edge Preserving Filter

Abstract

Amplitude variation with offset (AVO) pre-stack inversion is an effective method for reservoir prediction. The traditional AVO method is based on the Zoeppritz approximate formula, which will lead to inaccurate inversion results when the difference of incident angle and formation impedance are large. In addition, it will be affected by noise and the formation boundary may be blur. To overcome these shortcomings, a new method is proposed. First, the exact Zoeppritz equation prestack inversion (EZPI) is used. Second, the L2 norm of the second order difference matrix is added to the objective function to enhance the anti-noise performance (EZL2). Finally, the adaptive edge-preserving filter (Ad-EPS) is used in every iteration (EZL2AEPS) to make the formation boundary clearer and obtain block structure. Ad-EPS scans windows automatically to find the most suitable filter window for each sampling point. Levenberg-Marquart algorithm is used to implement the inversion process. The synthetic data test shows the superiority of EZL2AEPS, which not only has excellent anti-noise ability but also can recover the formation boundary accurately. The example of the real data from a coal mine is also proposed. The parameters of the inversion need to be continuously tested. EZL2AEPS is also able to characterize thin coal seams. The boundaries of upper and lower bedrock can be distinguished, which demonstrates the proposed approach is feasible.

Appendix

The most critical part of the LM algorithm is the calculation of the Jacobian matrix, and the calculation accuracy directly affects the final result of the inversion. The partial derivative relationship between each parameter is very important. The specific expression of the Jacobian matrix G(m) is as follows:

$${\mathbf{G(m)}} = \sum\limits_{k = 0}^{n - 1} {\frac{{\partial d_{i} }}{{\partial m_{k} }}} = {\mathbf{W}} * \sum\limits_{k = 0}^{n - 1} {\frac{{\partial R_{i} }}{{\partial m_{k} }}} ,$$

(17)

where d = W ⊗ R, W is the seismic wavelet, and n is the number of sampling points.

$${\mathbf{G(m)}} = \left[ {\begin{array}{*{20}c} {{\mathbf{G}}_{{{\mathbf{11}}}} } & {{\mathbf{G}}_{{{\mathbf{12}}}} } & \cdots & {{\mathbf{G}}_{{{\mathbf{1l}}}} } \\ {{\mathbf{G}}_{{{\mathbf{21}}}} } & {{\mathbf{G}}_{{{\mathbf{22}}}} } & \cdots & {{\mathbf{G}}_{{{\mathbf{2l}}}} } \\ \vdots & \vdots & \ddots & \vdots \\ {{\mathbf{G}}_{{{\mathbf{i1}}}} } & {{\mathbf{G}}_{{{\mathbf{i2}}}} } & \cdots & {{\mathbf{G}}_{{{\mathbf{il}}}} } \\ \end{array} } \right]$$

(18)

In the G(m) matrix, l is the type of inversion parameter. As this study pertains to a three-parameter inversion, l = 3. i is the number of incident angles or the number of angle gathers.

$${\mathbf{G}}_{i1} = {\mathbf{W}}*\left[ {\begin{array}{*{20}c} {\frac{{\partial R_{i} }}{{\partial vp_{1} }}} & {\frac{{\partial R_{i} }}{{\partial vp_{2} }}} & 0 & 0 & \cdots & 0 \\ 0 & {\frac{{\partial R_{i} }}{{\partial vp_{2} }}} & {\frac{{\partial R_{i} }}{{\partial vp_{3} }}} & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & {\frac{{\partial R_{i} }}{{\partial vp_{n - 1} }}} & {\frac{{\partial R_{i} }}{{\partial vp_{n} }}} \\ 0 & 0 & 0 & 0 & \cdots & {\frac{{\partial R_{i} }}{{\partial vp_{n} }}} \\ \end{array} } \right]$$

(19)

$${\mathbf{G}}_{i2} = {\mathbf{W}}*\left[ {\begin{array}{*{20}c} {\frac{{\partial R_{i} }}{{\partial vs_{1} }}} & {\frac{{\partial R_{i} }}{{\partial vs_{2} }}} & 0 & 0 & \cdots & 0 \\ 0 & {\frac{{\partial R_{i} }}{{\partial vs_{2} }}} & {\frac{{\partial R_{i} }}{{\partial vs_{3} }}} & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & {\frac{{\partial R_{i} }}{{\partial vs_{n - 1} }}} & {\frac{{\partial R_{i} }}{{\partial vs_{n} }}} \\ 0 & 0 & 0 & 0 & \cdots & {\frac{{\partial R_{i} }}{{\partial vs_{n} }}} \\ \end{array} } \right]$$

(20)

$${\mathbf{G}}_{i2} = {\mathbf{W}}*\left[ {\begin{array}{*{20}c} {\frac{{\partial R_{i} }}{{\partial \rho_{1} }}} & {\frac{{\partial R_{i} }}{{\partial \rho_{2} }}} & 0 & 0 & \cdots & 0 \\ 0 & {\frac{{\partial R_{i} }}{{\partial \rho_{2} }}} & {\frac{{\partial R_{i} }}{{\partial \rho_{3} }}} & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & {\frac{{\partial R_{i} }}{{\partial \rho_{n - 1} }}} & {\frac{{\partial R_{i} }}{{\partial \rho_{n} }}} \\ 0 & 0 & 0 & 0 & \cdots & {\frac{{\partial R_{i} }}{{\partial \rho_{n} }}} \\ \end{array} } \right]$$

(21)

Let m = [vp₁, vp₂, vs₁, vs₂, ρ₁, ρ₂]. Find the partial derivative with respect to m:

$${\mathbf{A}}\frac{{\partial {\mathbf{R}}}}{{\partial {\mathbf{u}}}} + \frac{{\partial {\mathbf{A}}}}{{\partial {\mathbf{u}}}}{\mathbf{R}} = \frac{{\partial {\mathbf{B}}}}{{\partial {\mathbf{u}}}}$$

(22)

Simplify Eq. (20) to obtain the exact partial derivative matrix (Eq. (21)) of the reflection coefficient with respect to the three parameters:

$$\frac{{\partial {\mathbf{R}}}}{{\partial {\mathbf{m}}}} = {\mathbf{A}}^{ - 1} \left( {\frac{{\partial {\mathbf{B}}}}{{\partial {\mathbf{m}}}} - \frac{{\partial {\mathbf{A}}}}{{\partial {\mathbf{m}}}}{\mathbf{R}}} \right)$$

(23)

Calculate the partial derivatives of the matrices A and B with respect to the parameters as follows:

• (1) Calculate the matrix of partial derivatives of matrices A and B with respect to vp₁:

  $$\frac{{\partial {\mathbf{A}}}}{{\partial vp_{1} }} = \frac{1}{{vp_{1} }}\left[ {\begin{array}{*{20}c} 0 & {\tan \beta_{1} \sin \beta_{1} } & {\sin \alpha_{2} } & {\sin \beta_{2} \tan \beta_{2} } \\ 0 & {\sin \beta_{1} } & {\sin \alpha_{2} \tan \alpha_{2} } & {\sin \beta_{2} } \\ 0 & {1 + 2\sin^{2} \beta_{1} } & {2\frac{{\rho_{2} }}{{\rho_{1} }}\frac{{vs_{2}^{2} }}{{vs_{1}^{2} }}\sin \alpha_{1} \sin \alpha_{2} \tan \alpha_{2} } & { - \frac{{\rho_{2} }}{{\rho_{1} }}\frac{{vp_{1} }}{{vs_{1} }}(1 + 2\sin^{2} \beta_{2} )} \\ { - 4\sin^{2} \beta_{1} } & {2\frac{{vs_{1} }}{{vp_{1} }}(\sin^{2} \beta_{1} \tan \beta_{1} - \sin 2\beta_{1} )} & {\frac{{\rho_{2} }}{{\rho_{1} }}\frac{{vp_{2} }}{{vp_{1} }}(6\sin^{2} \beta_{2} - 1)} & {2\frac{{vs_{2} }}{{vp_{1} }}\frac{{\rho_{2} }}{{\rho_{1} }}(\sin^{2} \beta_{2} \tan \beta_{2} - \sin 2\beta_{2} )} \\ \end{array} } \right]$$

  (24)
  $$\frac{{\partial {\mathbf{B}}}}{{\partial vp_{1} }} = \frac{1}{{vp_{1} }}\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & {4\sin^{2} \beta_{1} } \\ \end{array} } \right]^{T}$$

  (25)

• (2) Calculate the matrix of partial derivatives of matrices A and B with respect to vp₂:

  $$\frac{{\partial {\mathbf{A}}}}{{\partial vp_{2} }} = \frac{1}{{vp_{2} }}\left[ {\begin{array}{*{20}c} 0 & 0 & { - \sin \alpha_{2} } & 0 \\ 0 & 0 & { - \tan \alpha_{2} \sin \alpha_{2} } & 0 \\ 0 & 0 & { - 2\frac{{\rho_{2} }}{{\rho_{1} }}\frac{{vs_{2}^{2} }}{{vs_{1}^{2} }}\sin \alpha_{1} \sin \alpha_{2} \tan \alpha_{2} } & 0 \\ 0 & 0 & {\frac{{\rho_{2} }}{{\rho_{1} }}\frac{{vp_{2} }}{{vp_{1} }}\cos 2\beta_{2} } & 0 \\ \end{array} } \right]$$

  (26)
  $$\frac{{\partial {\mathbf{B}}}}{{\partial vp_{2} }} = \frac{1}{{vp_{2} }}\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ \end{array} } \right]^{T}$$

  (27)

• (3) Calculate the matrix of partial derivatives of matrices A and B with respect to vs₁:

  $$\frac{{\partial {\mathbf{A}}}}{{\partial vs_{1} }} = \frac{1}{{vs_{1} }}\left[ {\begin{array}{*{20}c} 0 & { - \sin \beta_{1} \tan \beta_{1} } & 0 & 0 \\ 0 & { - \sin \beta_{1} } & 0 & 0 \\ 0 & { - \frac{{vp_{1} }}{{vs_{1} }}(1 + 2\sin^{2} \beta_{1} )} & { - \frac{{2\rho_{2} }}{{\rho_{1} }}\frac{{vp_{1} }}{{vp_{2} }}\frac{{vs_{2}^{2} }}{{vs_{1}^{2} }}\sin 2\alpha_{2} } & {\frac{{\rho_{2} }}{{\rho_{1} }}\frac{{vp_{1} }}{{vs_{1} }}\cos 2\beta_{2} } \\ {4\sin^{2} \beta_{1} } & {\frac{{2vs_{1} }}{{vp_{1} }}(\sin 2\beta_{1} - \sin^{2} \beta_{1} \tan \beta_{1} )} & 0 & 0 \\ \end{array} } \right]$$

  (28)
  $$\frac{{\partial {\mathbf{B}}}}{{\partial vs_{1} }} = \frac{1}{{vs_{1} }}\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & { - 4\sin^{2} \beta_{1} } \\ \end{array} } \right]^{T}$$

  (29)

• (4) Calculate the matrix of partial derivatives of matrices A and B with respect to vs₂:

  $$\frac{{\partial {\mathbf{A}}}}{{\partial vs_{2} }} = \frac{1}{{vs_{2} }}\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & { - \sin \beta_{2} \tan \beta_{2} } \\ 0 & 0 & 0 & {\sin \beta_{2} } \\ 0 & 0 & {\frac{{2vp_{1} }}{{vp_{2} }}\frac{{vs_{1}^{2} }}{{vs_{2}^{2} }}\frac{{\rho_{2} }}{{\rho_{1} }}\sin 2\alpha_{2} } & {4\frac{{\rho_{2} }}{{\rho_{1} }}\frac{{vp_{1} }}{{vs_{1} }}\sin^{2} \beta_{2} } \\ 0 & 0 & { - \frac{{4\rho_{2} }}{{\rho_{1} }}\frac{{vp_{2} }}{{vp_{1} }}\sin^{2} \beta_{2} } & {\frac{{2\rho_{2} }}{{\rho_{1} }}\frac{{vs_{2} }}{{vp_{1} }}(\sin 2\beta_{2} - \sin^{2} \beta_{2} \tan \beta_{2} )} \\ \end{array} } \right]$$

  (30)
  $$\frac{{\partial {\mathbf{B}}}}{{\partial vs_{2} }} = \frac{1}{{vs_{2} }}\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ \end{array} } \right]^{T}$$

  (31)

• (5) Calculate the matrix of partial derivatives of matrices A and B with respect to ρ₁:

  $$\frac{{\partial {\mathbf{A}}}}{{\partial \rho_{1} }} = \frac{1}{{\rho_{1} }}\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & { - \frac{{\rho_{2} }}{{\rho_{1} }}\frac{{vp_{1} }}{{vp_{2} }}\frac{{vs_{2}^{2} }}{{vs_{1}^{2} }}\sin 2\alpha_{2} } & {\frac{{\rho_{2} }}{{\rho_{1} }}\frac{{vp_{1} }}{{vs_{1} }}\cos 2\beta_{2} } \\ 0 & 0 & { - \frac{{\rho_{2} }}{{\rho_{1} }}\frac{{vp_{2} }}{{vp_{1} }}\cos 2\beta_{2} } & { - \frac{{\rho_{2} }}{{\rho_{1} }}\frac{{vs_{2} }}{{vp_{1} }}\sin 2\beta_{2} } \\ \end{array} } \right]$$

  (32)
  $$\frac{{\partial {\mathbf{B}}}}{{\partial \rho_{1} }} = \frac{1}{{\rho_{1} }}\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ \end{array} } \right]^{T}$$

  (33)

• (6) Calculate the matrix of partial derivatives of matrices A and B with respect to ρ₂:

  $$\frac{{\partial {\mathbf{A}}}}{{\partial \rho_{2} }} = \frac{1}{{\rho_{1} }}\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & {\frac{{vp_{1} }}{{vp_{2} }}\frac{{vs_{2}^{2} }}{{vs_{1}^{2} }}\sin 2\alpha_{2} } & { - \frac{{vp_{1} }}{{vs_{1} }}\cos 2\beta_{2} } \\ 0 & 0 & {\frac{{vp_{2} }}{{vp_{1} }}\cos 2\beta_{2} } & {\frac{{vs_{2} }}{{vp_{1} }}\sin 2\beta_{2} } \\ \end{array} } \right]$$

  (34)
  $$\frac{{\partial {\mathbf{B}}}}{{\partial \rho_{2} }} = \frac{1}{{\rho_{1} }}\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ \end{array} } \right]^{T}$$

  (35)

Therefore, the reflection coefficient is the same as the three parameters, the P-wave velocity, S-wave velocity, and partial derivative of the density, expressed as follows:

$$\frac{{\partial {\mathbf{R}}}}{{\partial vp_{1} }} = {\mathbf{A}}^{ - 1} \left(\frac{{\partial {\mathbf{B}}}}{{\partial vp_{1} }} - \frac{{\partial {\mathbf{A}}}}{{\partial vp_{1} }}{\mathbf{R}}\right), \, \frac{{\partial {\mathbf{R}}}}{{\partial vp_{2} }} = {\mathbf{A}}^{ - 1} \left(\frac{{\partial {\mathbf{B}}}}{{\partial vp_{2} }} - \frac{{\partial {\mathbf{A}}}}{{\partial vp_{2} }}{\mathbf{R}}\right)$$

(36)

$$\frac{{\partial {\mathbf{R}}}}{{\partial vs_{1} }} = {\mathbf{A}}^{ - 1} \left(\frac{{\partial {\mathbf{B}}}}{{\partial vs_{1} }} - \frac{{\partial {\mathbf{A}}}}{{\partial vs_{1} }}{\mathbf{R}}\right), \, \frac{{\partial {\mathbf{R}}}}{{\partial vs_{2} }} = {\mathbf{A}}^{ - 1} \left(\frac{{\partial {\mathbf{B}}}}{{\partial vs_{2} }} - \frac{{\partial {\mathbf{A}}}}{{\partial vs_{2} }}{\mathbf{R}}\right)$$

(37)

$$\frac{{\partial {\mathbf{R}}}}{{\partial \rho_{1} }} = {\mathbf{A}}^{ - 1} \left(\frac{{\partial {\mathbf{B}}}}{{\partial \rho_{1} }} - \frac{{\partial {\mathbf{A}}}}{{\partial \rho_{1} }}{\mathbf{R}}\right), \, \frac{{\partial {\mathbf{R}}}}{{\partial \rho_{2} }} = {\mathbf{A}}^{ - 1} \left(\frac{{\partial {\mathbf{B}}}}{{\partial \rho_{2} }} - \frac{{\partial {\mathbf{A}}}}{{\partial \rho_{2} }}{\mathbf{R}}\right)$$

(38)