next up previous [pdf]

Next: Numerical results Up: Full waveform inversion (FWI) Previous: Fréchet derivative

Gradient computation

According to the previous section, it follows that

$\displaystyle \frac{\partial p_{cal}}{\partial v_i(\textbf{x})} =\int_V \mathrm...
...partial^2 p(\textbf{x},t;\textbf{x}_s)}{\partial t^2}\frac{2}{v^3(\textbf{x})}.$ (99)

The convolution guarantees

$\displaystyle \int \mathrm{d}t [g(t)*f(t)]h(t)=\int \mathrm{d}t f(t)[g(-t)*h(t)].$ (100)

Then, Eq. (71) becomes

\begin{displaymath}\begin{split}\frac{\partial E(\textbf{m})}{\partial m_i} &=\s...
...ght)^*p_{res}(\textbf{x}_r,t;\textbf{x}_s)\right]\\ \end{split}\end{displaymath} (101)

where $ p_{res}(\textbf{x},t;\textbf{x}_s)$ is a time-reversal wavefield produced using the residual $ \Delta p(\textbf{x}_r,t;\textbf{x}_s)$ as the source. It follows from reciprocity theorem

$\displaystyle p_{res}(\textbf{x},t;\textbf{x}_s) =\int_V \mathrm{d}\textbf{x}\G...
...bf{x}\Gamma(\textbf{x},0;\textbf{x}_r,t)*\Delta p(\textbf{x}_r,t;\textbf{x}_s).$ (102)

satisfying

$\displaystyle \frac{1}{v^2(\textbf{x})}\frac{\partial^2 p_{res}(\textbf{x},t;\t...
...bla^2 p_{res}(\textbf{x},t;\textbf{x}_s)=\Delta p(\textbf{x}_r,t;\textbf{x}_s).$ (103)

It is noteworthy that an input $ f$ and the system impulse response function $ g$ are exchangeable in convolution. That is to say, we can use the system impulse response function $ g$ as the input, the input $ f$ as the impulse response function, leading to the same output. In the seismic modeling and acquisition process, the same seismogram can be obtained when we shot at the receiver position $ \textbf{x}_r$ when recording the seismic data at the position $ \textbf{x}$ .


next up previous [pdf]

Next: Numerical results Up: Full waveform inversion (FWI) Previous: Fréchet derivative

2021-08-31