


 Ignoring density in waveform inversion  

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Application of the Born approximation as expressed in (14)
requires knowledge of the residual wavefield everywhere in the image space.
Unfortunately, the full wavefield, and thus the residual wavefield, is only
known at the receivers. For simplicity we assume that receivers are located at
all xlocations on the surface, or that the receiver wavefield is unaliased and
can be perfectly recovered. We define
as the background
wavefield for shot at the iteration. This wavefield is computed by
forward modeling the shot field through the slowness model. The data
residual
is computed by selecting the background
wavefield at and subtracting from the recorded data , or

(15) 
The objective of the inversion is to minimize the norm of .
Our implementation uses the linear forward operator to compute a step length at
each iteration. Substituting for the background wavefield and selecting
only the scattered field at the receivers, the frequency domain expression for
the linear forward operator becomes

(16) 
Since we use timedomain finitedifference modeling, it is useful to express
the operator in the time domain. The factor is applied as a
second time derivative to the wavefield and the multiplication of ,
and becomes a convolution along the time dimension, yielding

(17) 
The forward operator is implemented in two steps. First, the background
wavefield is computed by propagating the source field forward in time.
Next, the background wavefield is scaled by
and used
as a new source field that is also propagated forward in time.
The gradient direction for each step of the inversion is
computed using the adjoint of the forward operator. The independent variables
used in the forward operator are , , , and . The
forward operator integrates over and selects data at , so the
adjoint is expressed by integrating over the remaining variables and injecting
data, expressed here as multiplying with a delta function, at :

(18) 
This integral represents reversetime migration of the data residual. We show a
simplified expression by defining a new wavefield that represents the
propagation of the data residual. The time axis of the Green's function is
reversed due to the complex conjugate in the frequency domain:

(19) 
In practice, the integral is computed by forward propagating the timereversed
data residual. Due to Green's function reciprocity, integration over
is equivalent to the integration over in (17).
The wavefield is then substituted into the timedomain expression for
the adjoint operator where integration over frequencies is exchanged for
integration over time, and the time axis of the background wavefield is
reversed:

(20) 
With both the forward and adjoint linear seismic modeling expressions
defined, we have all of the building blocks needed to invert for . We
use a nonlinear variation of conjugate gradients following
Claerbout (2004). The method differs from linear conjugate gradients in
that for each iteration the operators, which depend on , change and the data
residual is recomputed.



 Ignoring density in waveform inversion  

Next: Application to models with
Up: Review of waveform inversion
Previous: The Born approximation
20070918