Backpropagated Residuals $\otimes$ Direct Waves

Now we are ready for physically interpreting equation 3.7, which can be rearranged as:

 

$$\Delta s(\bf {r}) A \sum_{\omega} \omega^2 \sum_g \sum_s [\Delta \tilde{P} (\bf {r}... ...G(\bf {r}\vert \bf {r}_g)^* ]^* [\tilde{W}(\omega) G(\bf {r}\vert \bf {r}_s) ],$$ =

where $A=-2 \kappa s(\bf {r})$. Recalling Parseval's theorem which says $b(t) \otimes f(t)\vert _{t=0} = \int_{-\infty}^{\infty} \tilde B(\omega) \tilde f(\omega)^* d\omega$, equation 3.17 can be compactly represented by the following formula:

 

$$\Delta s(\bf {r}) \sum_g \sum_s b(\bf {r}_g, \bf {r}_s, \bf {r}, t) \otimes f(\bf {r},\bf {r}_s,t)\vert _{t=0}.$$ =

Here the forward modeled wavefield and backward modeled residuals are given by:

 

$$f(\bf {r},\bf {r}_s,t) A \int g_c(\bf {r},t\vert\bf {r}_s,t_s) \dot w(t_s) dt_s ,$$ =

$$b(\bf {r}_g,\bf {r}_s,\bf {r},t) -\int g_a(\bf {r},t\vert\bf {r}_s,t_s) \Delta \dot p(\bf {r}_g,\bf {r}_s,t_s) dt_s ,$$ = (3.15)

Equation 3.18 says the backpropagated residual wavefield $b(\bf {r}_g,\bf {r}_s,\bf {r}, t)$is crosscorrelated with the forward propagated wavefield $f(\bf {r},\bf {r}_s, t)$to give the slowness update.

To illustrate the physical meaning of equation 3.19, the top picture in Figure 3.8 depicts the wavefronts emanating from a source in a homogeneous medium with an embedded point scatterer. The combined direct and scattered wavefields can be represented as $b(\bf {r}_g, \bf {r}_s, \bf {r}, t)+f(\bf {r},\bf {r}_s,t)$. If the homogeneous medium is used as the model velocity then the middle figure depicts the forward propagated field, where no scattering from the point scatterer is extant. This field is represented by $f(\bf {r},\bf {r}_s, t)$. The bottom figure represents the backpropagated scattered field (i.e., the residual field $\Delta \dot p(\bf {r}_g,\bf {r}_s,t_s)$) denoted by $b(\bf {r}_g,\bf {r}_s,\bf {r}, t)$. The operation of crosscorrelation at zero lag is equivalent to multiplying each snapshot in the middle figure by the corresponding snapshot in the bottom figure to give a product snapshot, and these product snapshots are then summed over the time to give the image panel. This image panel is zero everywhere except at the location of the scatterer because that is the only location where the downgoing direct wavefront is coincident in both space and time.

  

Figure 3.8: Snapshots of wavefronts for a point source located at the origin, and a buried scatterer denoted by *. Top figure depicts the total wavefield snapshots, middle figure depicts the forward propagated wavefield in a homogeneous medium, and bottom figure depicts the backpropagated scattered field.

The update formula for many different inversion algorithms reduces to "migrating" data residuals. The data residuals might be associated with the traveltimes, autocorrelograms, phases, or amplitudes.