


 Targetoriented wavefield tomography using demigrated Born data  

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The objective function of imagedomain wavefield tomography can be defined as the
norm of either
an imagedomain residual field (Shen, 2004; Sava, 2004) or
the negative imagestack power (or image coherence) (Toldi, 1985; Soubaras and Gratacos, 2007) or both (Shen and Symes, 2008).
A widely used residual operator is the subsurfaceoffsetdomain differential semblance optimization (DSO) operator,
where the velocity model is optimized by penalizing energy at nonzero subsurface offset, utilizing the fact that
the SODCIGs should be focused at zero subsurface offset if migrated using an accurate velocity model.
The DSO objective function reads (Shen, 2004)



(9) 
where
is the subsurfaceoffsetdomain image migrated using velocity
(equation 5)
and the Bornmodeled data described in the previous section.
In contrast, minimizing the negative imagestack power (ISP) utilizes the fact
that the stacked image should achieve maximum energy (or focus) when migrated with an accurate velocity (Toldi, 1985; Soubaras and Gratacos, 2007).
Since the zerosubsurfaceoffset image is the stacked image, the ISP objective function reads



(10) 
The above two objective functions can also be combined into one single objective function as follows (Shen and Symes, 2008):



(11) 
where
is a constant to trade off these two objective functions.
The gradients can be calculated using the adjointstate method without explicitly building the sensitivity matrix,
see, e.g. Sava and Vlad (2008); Tang et al. (2008), for implementation details.
The gradient is then used to update the velocity model with a suitable step length chosen by a linesearch step.
We iterate this process until an acceptable velocity model is obtained.
Figures 5 and 6
compare the normalized DSO and ISP gradients
obtained using the original data set with those obtained using the new data set. These gradients are computed using the
starting velocity model (Figure 1(b)).
Note that both DSO and ISP gradients correctly identify those two anomalies
and they also give correct directions for velocity updates. Also note that the gradients obtained using the new data
set (Figures 5(b) and 6(b))
are very similar to those obtained using the original data set
(Figures 5(a) and 6(a)), except that they are obtained
with only
planewavesource gathers recorded at depth
ft instead of
pointsource gathers recorded at the surface (
ft).
These examples demonstrate that the new data set can be used for wavefield tomography with much lower computational cost.



 Targetoriented wavefield tomography using demigrated Born data  

Next: tomographic inversion results
Up: Tang and Biondi: Imagedomain
Previous: Targetoriented Born wavefield modeling
20100519