where . Recalling Parseval's theorem which says , equation 3.17 can be compactly represented by the following formula:

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

Equation 3.18 says the backpropagated residual wavefield is crosscorrelated with the forward propagated wavefield 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
.
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
.
The bottom figure represents the backpropagated
scattered field (i.e., the residual field
)
denoted
by
.
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.

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.