A numerical tour of wave propagation

Fréchet derivative

Recall that the basic acoustic wave equation can be specified as

$$\frac{1}{v^2(\textbf{x})}\frac{\partial^2 p(\textbf{x},t;\textbf{... ...tial t^2}-\nabla^2 p(\textbf{x},t;\textbf{x}_s)=f_s(\textbf{x},t;\textbf{x}_s).$$ (91)

where $f_s(\textbf{x},t;\textbf{x}_s)=f(t')\delta(\textbf{x}-\textbf{x}_s)\delta(t-t')$ . The Green's function $\Gamma(\textbf{x},t;\textbf{x}_s,t')$ is defined by

$$\frac{1}{v^2(\textbf{x})}\frac{\partial^2 \Gamma(\textbf{x},t;\te... ...ma(\textbf{x},t; \textbf{x}_s,t') =\delta(\textbf{x}-\textbf{x}_s)\delta(t-t').$$ (92)

Thus the integral representation of the solution can be given by

$$\begin{split}p(\textbf{x}_r,t; \textbf{x}_s)&=\int_V \mathrm{... ...f{x}_r,t;\textbf{x},0)*f(\textbf{x},t;\textbf{x}_s) \end{split}$$ (93)

where $*$ denotes the convolution operator.

A perturbation $v(\textbf{x})\rightarrow v(\textbf{x})+\Delta v(\textbf{x})$ will produce a field $p(\textbf{x},t;\textbf{x}_s)+\Delta p(\textbf{x},t;\textbf{x}_s)$ defined by

$$\frac{1}{(v(\textbf{x})+\Delta v(\textbf{x}))^2}\frac{\partial^2 ... ...xtbf{x}_s)+\Delta p(\textbf{x},t;\textbf{x}_s)] =f_s(\textbf{x},t;\textbf{x}_s)$$ (94)

Note that

$$\frac{1}{(v(\textbf{x})+\Delta v(\textbf{x}))^2} =\frac{1}{v^2(\textbf{x})}-\frac{2\Delta v(\textbf{x})}{v^3(\textbf{x})}+O(\Delta^2 v(\textbf{x}))$$ (95)

Eq. (94) subtracts Eq. (91), yielding

$$\frac{1}{v^2(\textbf{x})}\frac{\partial^2 \Delta p(\textbf{x},t;\... ...x},t;\textbf{x}_s)]}{\partial t^2}\frac{2\Delta v(\textbf{x})}{v^3(\textbf{x})}$$ (96)

Using the Born approximation, Eq. (96) becomes

$$\frac{1}{v^2(\textbf{x})}\frac{\partial^2 \Delta p(\textbf{x},t;\... ...{x},t;\textbf{x}_s)}{\partial t^2}\frac{2\Delta v(\textbf{x})}{v^3(\textbf{x})}$$ (97)

Again, based on integral representation, we obtain

$$\Delta p(\textbf{x}_r,t; \textbf{x}_s)=\int_V \mathrm{d}\textbf{x... ...x},t;\textbf{x}_s)}{\partial t^2}\frac{2\Delta v(\textbf{x})}{v^3(\textbf{x})}.$$ (98)

A numerical tour of wave propagation