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Automatic cross‐well tomography by semblance and differential semblance optimization: theory and gradient computation
Author(s) -
Plessix R.E.,
Mulder W.A.,
Kroode A.P.E. ten
Publication year - 2000
Publication title -
geophysical prospecting
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.735
H-Index - 79
eISSN - 1365-2478
pISSN - 0016-8025
DOI - 10.1046/j.1365-2478.2000.00217.x
Subject(s) - tomography , algorithm , computation , differential (mechanical device) , optimization problem , geology , computer science , physics , optics , thermodynamics
A technique for automatic cross‐well tomography based on semblance and differential semblance optimization is presented. Given a background velocity, the recorded seismic data traces are back‐propagated towards the source, i.e. shifted towards time zero using the modelled traveltime between the source and the receiver and corrected for the geometrical spreading. Therefore each back‐propagated trace should be a pulse, close to time zero. The mismatches between the back‐propagated traces indicate an error in the velocity model. This error can be measured by stacking the back‐propagated traces (semblance optimization) or by computing the norm of the difference between adjacent traces (differential semblance optimization). It is known from surface seismic reflection tomography that both the semblance and differential semblance functional have good convexity properties, although the differential semblance functional is believed to have a larger basin of attraction (region of convergence) around the true velocity model. In the case of the cross‐well transmission tomography described in this paper, similar properties are found for these functionals. The implementation of this automatic method for cross‐well tomography is based on the high‐frequency approximation to wave propagation. The wavefronts are constructed using a ray‐tracing algorithm. The gradient of the cost function is computed by the adjoint‐state technique, which has the same complexity as the computation of the functional. This provides an efficient algorithm to invert cross‐well data. The method is applied to a synthetic data set to demonstrate its efficacy.