Ignoring density in waveform inversion

The Born approximation

Given a solution $\Psi(\vec{r},t)$ to equation (2), is it possible to recover $\sigma(\vec{r})$? The Born approximation, named after physicist Max Born, was first developed for scattering theory in quantum mechanics. Applied to seismology, the first-order approximation provides a linear, and thus invertible, relationship between a small change in the slowness model and a resulting small change in the wavefield. We split the model into a background slowness $\sigma_0(\vec{r})$ and a small slowness perturbation $\Delta\sigma(\vec{r})$, where

$$\sigma(\vec{r}) = \sigma_0(\vec{r}) + \Delta\sigma(\vec{r}).$$ (3)

The wavefield depends on slowness squared, so here we bring in the first approximation, which is not yet the Born approximation:

$$\sigma(\vec{r})^2 \approx \sigma_0(\vec{r})^2 + 2\sigma_0(\vec{r})\Delta\sigma(\vec{r}).$$ (4)

To achieve an approximate relation linear with $\Delta\sigma$, we first substitute (4) into the wave equation:

$$\left(\nabla^2 + \omega^2\sigma(\vec{r})^2\right)\Psi(\vec{r... ...2\sigma_0(\vec{r})\Delta\sigma(\vec{r})\right)\Psi(\vec{r},t).$$ (5)

This approximation is then divided into halves, with only one side depending on $\Delta\sigma$:

$$(\nabla^2 + \omega^2\sigma_0(\vec{r})^2)\Psi(\vec{r},t) \app... ...\omega^2\sigma_0(\vec{r})\Delta\sigma(\vec{r})\Psi(\vec{r},t).$$ (6)

Much as the slowness field was split into two parts, the wavefield now is divided into a background wavefield $\Psi_0$ and a scattered wavefield $\Delta\Psi$ such that

$$\Psi(\vec{r},\omega)=\Psi_0(\vec{r},\omega)+\Delta\Psi(\vec{r},\omega),$$ (7)

where, by definition, $\Psi_0$ is the solution for the background wavefield, or

$$\left(\nabla^2 + \omega^2\sigma_0(\vec{r})^2\right)\Psi_0(\vec{r},\omega)=0.$$ (8)

Substituting the divided wavefield (7) into the approximate wave equation (6), and using the fact that the background wavefield is an exact solution for the background velocity, we can write

$$\left(\nabla^2 + \omega^2\sigma_0(\vec{r})^2\right)\Delta\P... ...\omega^2\sigma_0(\vec{r})\Delta\sigma(\vec{r})\Psi(\vec{r},t).$$ (9)

At this point, we have an implicit relation between a small change $\Delta\sigma$ in the model and the resulting scattered wavefield $\Delta\Psi$. Ideally, we would like to have an explicit expression for $\Delta\sigma$ as a function of the background and scattered wavefields. Such an expression cannot be written directly; instead, we can find an expression for $\Delta\Psi$ as a function of $\Delta\sigma$. This expression is an integral over potential scatterers convolved with the Green's function $G_0(\vec{r},\omega;\vec{r}')$, the response at point $\vec{r}'$ and frequency $\omega$ for a point source at point $\vec{r}$. The subscript indicates that the Green's function is defined for the background wavefield. We build up the integral expression by starting with the formal definition of the Green's function, which is the solution of the wave equation with a delta-function source:

$$\left(\nabla^2 + \omega^2\sigma_0(\vec{r})^2\right)G_0(\vec{r},\omega;\vec{r}') = \delta(\vec{r}-\vec{r}').$$ (10)

Both sides of this definition are multiplied by $-2\omega^2\sigma_0\Delta\sigma\Psi$ and integrated with respect to $\vec{r}'$:

$$-\int d\vec{r}'2\omega^2\sigma_0(\vec{r}')\Delta\sigma(\vec... ...0(\vec{r},\omega;\vec{r}') = \delta(\vec{r}-\vec{r}')\right].$$ (11)

The Laplacian operator is taken with respect to $\vec{r}$, not $\vec{r}'$, so the left side of the expression can be simplified by moving the integral inside the operator; on the right side, the delta function sifts the original function out of the integral, leaving

$$\left(\nabla^2 + \omega^2\sigma_0(\vec{r})^2\right)-\int d\... ...a^2\sigma_0(\vec{r})\Delta\sigma(\vec{r})\Psi(\vec{r},\omega).$$ (12)

Comparing with (9), we can see that the integral represents a solution for $\Delta\Psi$, allowing us to write

$$\Delta\Psi(\vec{r},\omega)\approx -\int d\vec{r}'2\omega^2\... ...(\vec{r}')G_0(\vec{r},\omega;\vec{r}') \Psi(\vec{r}',\omega).$$ (13)

Unfortunately, the scattered wavefield is still a function of the entire--and unknown--wavefield $\Psi$. The first-order Born approximation asserts that when the scattered wavefield is small compared to the background wavefield, the interaction between scattering points can be ignored. This is equivalent to replacing $\Psi$ with $\Psi_0$ on the right-hand side, leaving

$$\Delta\Psi(\vec{r},\omega)\approx -\int d\vec{r}'2\omega^2\... ...vec{r}')G_0(\vec{r},\omega;\vec{r}') \Psi_0(\vec{r}',\omega).$$ (14)

This approximation now provides a linear relationship between a small change in the model and the resulting small wavefield perturbation.

Ignoring density in waveform inversion