Nonstationary seismic inversion: joint estimation for acoustic impedance, attenuation factor and source wavelet

Abstract

Seismic signal can be expressed by nonstationary convolution model (NCM) which integrates acoustic impedance (AI), attenuation factor (AF) and source wavelet (SW) into a single formula. Although it provides attractive potential to invert AI, AF and SW, simultaneously, effective joint inversion algorithm has not been developed because of the extreme instability of this nonlinear inverse problem. In this paper, we propose an alternating optimization scheme to achieve this nonlinear joint inversion. Our algorithm repeatedly alternates among three subproblems corresponding to AI, AF and SW recovery until changes in inverted models become smaller than the user-defined tolerances. Also, when we optimize one parameter, other two parameters are fixed. NCM is an explicit linear formula for AI; therefore, AI recovery is accomplished by linear inversion which is regularized by low-frequency model and isotropy total variation domain sparse constraints. However, NCM is a complicated nonlinear formula for AF. To facilitate the AF inversion, we propose a centroid frequency-based attenuation tomography method whose forward operator and observations are acquired from the time-varying wavelet amplitude spectra which is estimated according to Gabor domain factorization of NCM. SW is decoupled from NCM based on Toeplitz structure constraint, and we obtain an orthogonal wavelet transform domain sparse regularized SW inverse subproblem. Split Bregman technique is adopted to optimize AI and SW inverse subproblems. Numerical test and field data application confirm that the proposed nonstationary seismic inversion algorithm can stably generate accurate estimates of AI, AF and SW, simultaneously.

Appendices

Appendix 1

Reformulation of NCM Based on Toeplitz structure constraint

Since W is a Toeplitz matrix, it can be written as

$${\mathbf{W = }}\left[ {\begin{array}{*{20}c} {w_{0} } & {w_{ - 1} } & \cdots & {w_{ - J} } \\ {w_{1} } & {w_{0} } & \cdots & {w_{ - J + 1} } \\ \vdots & \vdots & \ddots & \vdots \\ {w_{J} } & {w_{J - 1} } & \cdots & {w_{0} } \\ \end{array} } \right]$$

(35)

Therefore, ${\mathbf{w}} = \left[ {w_{ - J} ,w_{ - J + 1} , \ldots ,w_{0} , \ldots w_{J} } \right]^{T}$ can represent the time domain SW. Then, we rewrite W as

$${\mathbf{W}} = \sum\limits_{n = - J}^{J} {w_{n} {\mathbf{I}}_{n} }$$

(36)

where ${\mathbf{I}}_{n}$ is a shift matrix whose elements are zero except that the $n^{th}$ sub-diagonal is one, for example ${\mathbf{I}}_{0} = \left[ {\begin{array}{*{20}c} 1 & {} & {} \\ {} & \ddots & {} \\ {} & {} & 1 \\ \end{array} } \right],\begin{array}{*{20}c} {} \\ \end{array} {\mathbf{I}}_{ - 1} = \left[ {\begin{array}{*{20}c} 0 & 1 & {} & {} \\ {} & \ddots & \ddots & {} \\ {} & {} & \ddots & 1 \\ {} & {} & {} & 0 \\ \end{array} } \right],\begin{array}{*{20}c} {} \\ \end{array} {\mathbf{I}}_{1} = \left[ {\begin{array}{*{20}c} 0 & {} & {} & {} \\ 1 & \ddots & {} & {} \\ {} & \ddots & \ddots & {} \\ {} & {} & 1 & 0 \\ \end{array} } \right]$. Substituting Eq. (36) into Eq. (27), we can obtain the following formula deduction process:

$$\begin{gathered} {\mathbf{b}}_{3} = vec({\mathbf{S}}){\mathbf = }vec\left( {\sum\limits_{n = - J}^{J} {w_{n} {\mathbf{I}}_{n} {\overline{\mathbf{R}}}} } \right){\mathbf{ + }}vec({\mathbf{N}}) = w_{n} \sum\limits_{n = - J}^{J} {(vec({\mathbf{B}}_{n} ))} + {\mathbf{n}} \hfill \\ = [vec({\mathbf{B}}_{ - J} )|vec({\mathbf{B}}_{ - J + 1} )| \cdots |vec({\mathbf{B}}_{0} )| \cdots |vec({\mathbf{B}}_{J} )]{\mathbf{w}} + {\mathbf{n}} \hfill \\ = {\mathbf{A}}_{3} {\mathbf{w}} + {\mathbf{n}} \hfill \\ \end{gathered}$$

(37)

where ${\mathbf{B}}_{n} = {\mathbf{I}}_{n} {\overline{\mathbf{R}}}$; ${\mathbf{A}}_{3}$ is a reformulated matrix which is expressed by

$${\mathbf{A}}_{3} = [vec({\mathbf{B}}_{ - J} )|vec({\mathbf{B}}_{ - J + 1} )| \cdots |vec({\mathbf{B}}_{0} )| \cdots |vec({\mathbf{B}}_{J} )]$$

(38)

Appendix 2

Split Bregman method for solving hybrid sparse constraint problem

Here, we rewrite Eq. (33):

$${\mathbf{z}} = \arg \mathop {min}\limits_{{\mathbf{z}}} \{ {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}\left\| {{\mathbf{y}} - {\mathbf{Az}}} \right\|_{2}^{2} + \mu |\varphi ({\mathbf{z}})|_{1} + \kappa |{\mathbf{Cz}}|_{1} \} \begin{array}{*{20}c} {} \\ \end{array} s.t.\begin{array}{*{20}c} {} \\ \end{array} {\mathbf{z}}_{{{\text{low}}}} \le {\mathbf{z}} \le {\mathbf{z}}_{{{\text{up}}}}$$

(39)

Then, we define $|({\mathbf{d}}_{x} ,{\mathbf{d}}_{y} )|_{1} = \left| {\sqrt {({\mathbf{D}}_{x} {\mathbf{m}})^{2} + ({\mathbf{D}}_{y} {\mathbf{m}})^{2} } } \right|_{1}$, ${\mathbf{d}}_{x} = {\mathbf{D}}_{x} {\mathbf{z}}$, ${\mathbf{d}}_{y} = {\mathbf{D}}_{y} {\mathbf{z}}$ and ${\mathbf{d}}_{c} = {\mathbf{Cz}}$. Therefore, Eq. (39) can be iteratively solved by (Goldstein and Osher 2009)

$$\begin{gathered} ({\mathbf{z}}^{ik + 1} ,{\mathbf{d}}_{x}^{ik + 1} {,}{\mathbf{d}}_{y}^{ik + 1} {,}{\mathbf{d}}_{c}^{ik + 1} ) = \arg \mathop {min}\limits_{{{\mathbf{z}},{\mathbf{d}}_{x} {,}{\mathbf{d}}_{y} {,}{\mathbf{c}}}} \{ \mu |({\mathbf{d}}_{x} {,}{\mathbf{d}}_{y} )|_{1} + \kappa |{\mathbf{d}}_{c} |_{1} + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}\left\| {{\mathbf{y}} - {\mathbf{Az}}} \right\|_{2}^{2} \hfill \\ + {\eta \mathord{\left/ {\vphantom {\eta 2}} \right. \kern-\nulldelimiterspace} 2}\left\| {{\mathbf{d}}_{c} - {\mathbf{Cz}} - {\mathbf{b}}_{c}^{ik} } \right\|_{2}^{2} + {\nu \mathord{\left/ {\vphantom {\nu 2}} \right. \kern-\nulldelimiterspace} 2}\left\| {{\mathbf{d}}_{x} - {\mathbf{D}}_{x} {\mathbf{z}} - {\mathbf{b}}_{x}^{ik} } \right\|_{2}^{2} + {\nu \mathord{\left/ {\vphantom {\nu 2}} \right. \kern-\nulldelimiterspace} 2}\left\| {{\mathbf{d}}_{y} - {\mathbf{D}}_{y} {\mathbf{z}} - {\mathbf{b}}_{y}^{ik} } \right\|_{2}^{2} \} \begin{array}{*{20}c} {} \\ \end{array} s.t.\begin{array}{*{20}c} {} \\ \end{array} {\mathbf{z}}_{{{\text{low}}}} \le {\mathbf{z}} \le {\mathbf{z}}_{{{\text{up}}}} \hfill \\ \end{gathered}$$

(40)

$${\mathbf{b}}_{c}^{ik + 1} = {\mathbf{b}}_{c}^{ik} + ({\mathbf{Cz}}^{ik + 1} - {\mathbf{d}}_{c}^{ik + 1} ),$$

(41)

$${\mathbf{b}}_{x}^{ik + 1} = {\mathbf{b}}_{x}^{ik} + ({\mathbf{D}}_{x} {\mathbf{z}}^{ik + 1} - {\mathbf{d}}_{x}^{ik + 1} ),$$

(42)

$${\mathbf{b}}_{y}^{ik + 1} = {\mathbf{b}}_{y}^{ik} + ({\mathbf{D}}_{y} {\mathbf{z}}^{ik + 1} - {\mathbf{d}}_{y}^{ik + 1} ).$$

(43)

where “ik” denotes the iteration number; $\eta$ and $\nu$ are regularization parameters. Also, decoupling the l₁ and l₂ norm in Eq. (40), we will yield

$$\begin{aligned} {\mathbf{z}}^{ik + 1} & = \arg \mathop {min}\limits_{{\mathbf{z}}} \{ {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}\left\| {{\mathbf{y}} - {\mathbf{Az}}} \right\|_{2}^{2} + \{ {\eta \mathord{\left/ {\vphantom {\eta 2}} \right. \kern-\nulldelimiterspace} 2}\left\| {{\mathbf{d}}_{c}^{ik} - {\mathbf{Cz}} - {\mathbf{b}}_{c}^{ik} } \right\|_{2}^{2} + {\nu \mathord{\left/ {\vphantom {\nu 2}} \right. \kern-\nulldelimiterspace} 2}\left\| {{\mathbf{d}}_{x}^{ik} - {\mathbf{D}}_{x} {\mathbf{z}} - {\mathbf{b}}_{x}^{ik} } \right\|_{2}^{2} \\ & \quad + {\nu \mathord{\left/ {\vphantom {\nu 2}} \right. \kern-\nulldelimiterspace} 2}\left\| {{\mathbf{d}}_{y}^{ik} - {\mathbf{D}}_{y} {\mathbf{z}} - {\mathbf{b}}_{y}^{ik} } \right\|_{2}^{2} \} \begin{array}{*{20}c} {} \\ \end{array} s.t.\begin{array}{*{20}c} {} \\ \end{array} {\mathbf{z}}_{{{\text{low}}}} \le {\mathbf{z}} \le {\mathbf{z}}_{{{\text{up}}}} \\ \end{aligned}$$

(44)

$${\mathbf{d}}_{c}^{ik + 1} = \arg \mathop {min}\limits_{{{\mathbf{d}}_{c} }} \{ \kappa |{\mathbf{d}}_{c} |_{1} + {\eta \mathord{\left/ {\vphantom {\eta 2}} \right. \kern-\nulldelimiterspace} 2}\left\| {{\mathbf{d}}_{c} - {\mathbf{Cz}}^{ik + 1} - {\mathbf{b}}_{c}^{ik} } \right\|_{2}^{2} \}$$

(45)

$$({\mathbf{d}}_{x}^{ik + 1} {,}{\mathbf{d}}_{y}^{ik + 1} ) = \arg \mathop {\min }\limits_{{{\mathbf{d}}_{x} {,}{\mathbf{d}}_{y} }} \{ \mu |({\mathbf{d}}_{x} {,}{\mathbf{d}}_{y} )|_{1} + {\nu \mathord{\left/ {\vphantom {\nu 2}} \right. \kern-\nulldelimiterspace} 2}|{\mathbf{d}}_{x} - {\mathbf{D}}_{x} {\mathbf{z}}^{ik + 1} - {\mathbf{b}}_{x}^{ik} |_{2}^{2} + {\nu \mathord{\left/ {\vphantom {\nu 2}} \right. \kern-\nulldelimiterspace} 2}|{\mathbf{d}}_{y} - {\mathbf{D}}_{y} {\mathbf{z}}^{ik + 1} - {\mathbf{b}}_{y}^{ik} |_{2}^{2} \}$$

(46)

Now, we have split the l₁ and l₂ penalty terms and yielded three simple subproblems. Equation (B3a) can be solved via the following linear equation with inequality constraint:

$$({\mathbf{A}}^{T} {\mathbf{A}} + \eta {\mathbf{C}}^{T} {\mathbf{C}} + \nu {\mathbf{D}}_{x}^{T} {\mathbf{D}}_{x} + \nu {\mathbf{D}}_{y}^{T} {\mathbf{D}}_{y} ){\mathbf{z}}^{ik + 1} = {\mathbf{U}}^{ik} \begin{array}{*{20}c} {} \\ \end{array} s.t.\begin{array}{*{20}c} {} \\ \end{array} {\mathbf{z}}_{low} \le {\mathbf{z}} \le {\mathbf{z}}_{up}$$

(47)

where ${\mathbf{C}}^{T} {\mathbf{C}} = {\mathbf{I}}$ is an identity matrix; ${\mathbf{U}}^{ik} = {\mathbf{A}}^{T} {\mathbf{y}} - \eta {\mathbf{C}}^{T} ({\mathbf{b}}_{c}^{ik} - {\mathbf{d}}_{c}^{ik} ) - \nu {\mathbf{D}}_{x}^{T} ({\mathbf{b}}_{x}^{ik} - {\mathbf{d}}_{x}^{ik} ) - \nu {\mathbf{D}}_{y}^{T} ({\mathbf{b}}_{y}^{ik} - {\mathbf{d}}_{y}^{ik} )$. The bound constrained problem in Eq. (42) is solved by bounded variable least-squares (BVLS) algorithm (Stark and Parker 1995). Both Eq. (41b) and (41c) are standard l₁ norm based denoising problems whose solutions can be explicitly expressed by shrinkage formulas (Goldstein and Osher 2009):

$${\mathbf{d}}_{c}^{ik + 1} = \max \{ {{\varvec{\uprho}}}_{1} - {\kappa \mathord{\left/ {\vphantom {\kappa \eta }} \right. \kern-\nulldelimiterspace} \eta },{\mathbf{0}}\} \circ {\text{sig}}n({{\varvec{\uprho}}}_{1} )$$

(48)

$${\mathbf{d}}_{x}^{ik + 1} = \max \{ 1 - {\mu \mathord{\left/ {\vphantom {\mu {(\nu {{\varvec{\uprho}}}_{2} )}}} \right. \kern-\nulldelimiterspace} {(\nu {{\varvec{\uprho}}}_{2} )}},{\mathbf{0}}\} \circ ({\mathbf{D}}_{x} {\mathbf{z}}^{ik + 1} + {\mathbf{b}}_{x}^{ik} )$$

(49)

$${\mathbf{d}}_{y}^{ik + 1} = \max \{ 1 - {\mu \mathord{\left/ {\vphantom {\mu {(\nu {{\varvec{\uprho}}}_{2} )}}} \right. \kern-\nulldelimiterspace} {(\nu {{\varvec{\uprho}}}_{2} )}},{\mathbf{0}}\} \circ ({\mathbf{D}}_{y} {\mathbf{z}}^{ik + 1} + {\mathbf{b}}_{y}^{ik} )$$

(50)

where symbol “$\circ$” denotes Hadamard product between two vectors, ${{\varvec{\uprho}}}_{1} = {\mathbf{Cz}}^{it + 1} + {\mathbf{b}}_{c}^{it}$ and ${{\varvec{\uprho}}}_{2} = \sqrt {({\mathbf{D}}_{x} {\mathbf{z}}^{it + 1} + {\mathbf{b}}_{x}^{it} )^{2} + ({\mathbf{D}}_{y} {\mathbf{z}}^{it + 1} + {\mathbf{b}}_{y}^{it} )^{2} }$.