A numerical tour of wave propagation

Gradient computation

According to the previous section, it follows that

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

The convolution guarantees

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

Then, Eq. (71) becomes

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

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

satisfying

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

A numerical tour of wave propagation