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Full waveform inversion (FWI)

Time domain FWI was proposed by Tarantola (1984), and developed in Pica et al. (1990); Tarantola (1986). Later, frequency domain FWI was proposed by Pratt et al. (1998). Actually, many authors call it full waveform tomography. (tomography=fwi, imaging=migration) Here, we mainly follow two well-documented paper Pratt et al. (1998) and Virieux and Operto (2009). We define the misfit vector $\Delta \textbf{p}=\textbf{p}_{cal}-\textbf{p}_{obs}$ by the differences at the receiver positions between the recorded seismic data $\textbf{p}_{obs}$ and the modelled seismic data $\textbf{p}_{cal}=\textbf{f}(\textbf{m})$ for each source-receiver pair of the seismic survey. Here, in the simplest acoustic velocity inversion, $\textbf{m}$ corresponds to the velocity model to be determined. The objective function taking the least-squares norm of the misfit vector $\Delta \textbf{p}$ is given by

$\displaystyle E(\textbf{m})=\frac{1}{2}\Delta \textbf{p}^{\dagger}\Delta \textb... ...cal}(\textbf{x}_r, t;\textbf{x}_s)-p_{obs}(\textbf{x}_r, t;\textbf{x}_s)\vert^2$

(64)

where

and

are the number of sources and geophones, $\dagger$ denotes the adjoint and

the complex conjugate, while $\textbf{f}(\cdot)$ indicates the forward modeling of the wave propagation. The recorded seismic data is only a small subset of the whole wavefield.

The minimum of the misfit function $E(\textbf{m})$ is sought in the vicinity of the starting model $\textbf{m}_0$ . FWI is essentially a local optimization. In the framework of the Born approximation, we assume that the updated model $\textbf{m}$ of dimension can be written as the sum of the starting model $\textbf{m}_0$ plus a perturbation model $\Delta \textbf{m}$ : $\textbf{m}=\textbf{m}_0+\Delta \textbf{m}$ . In the following, we assume that $\textbf{m}$ is real valued.

A second-order Taylor-Lagrange development of the misfit function in the vicinity of $\textbf{m}_0$ gives the expression

$\displaystyle E(\textbf{m}_0+\Delta \textbf{m})=E(\textbf{m}_0) +\sum_{i=1}^M\f... ..._i \partial m_j}\Delta m_i \Delta m_j+O(\vert\vert\Delta\textbf{m}\vert\vert^3)$

(65)

Taking the derivative with respect to the model parameter

results in

$\displaystyle \frac{\partial E(\textbf{m})}{\partial m_i}=\frac{\partial E(\tex... ...artial^2 E(\textbf{m}_0)}{\partial m_j \partial m_i}\Delta m_j, i=1,2,\ldots,M.$

(66)

Briefly speaking, it is

$\displaystyle \frac{\partial E(\textbf{m})}{\partial \textbf{m}}=\frac{\partial... ...bf{m}}+\frac{\partial^2 E(\textbf{m}_0)}{\partial\textbf{m}^2}\Delta \textbf{m}$

(67)

Thus,

$\displaystyle \Delta \textbf{m}=-\left(\frac{\partial^2 E(\textbf{m}_0)}{\parti... ...ial E(\textbf{m}_0)}{\partial \textbf{m}}=-\textbf{H}^{-1}\nabla E_{\textbf{m}}$

(68)

where

$\displaystyle \nabla E_{\textbf{m}}=\frac{\partial E(\textbf{m}_0)}{\partial \t... ...}{\partial m_2}, \ldots, \frac{\partial E(\textbf{m}_0)}{\partial m_M}\right]^T$

(69)

and

$\displaystyle \textbf{H}=\frac{\partial^2 E(\textbf{m}_0)}{\partial\textbf{m}^2... ... m_2}&\ldots&\frac{\partial^2 E(\textbf{m}_0)}{\partial m_M^2}\\ \end{bmatrix}.$

(70)

$\nabla E_{\textbf{m}}$ and $\textbf{H}$ are the gradient vector and the Hessian matrix, respectively.

Subsections

Next: The Newton, Gauss-Newton, and Up: Pengliang Yang: Primer for Previous: Numerical examples

2021-08-31