p177 feasibility of full waveform inversion applied to sub ... - Earthdoc

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$Fayek & Kyser 2000 uraninite fractionations.pdf - Queen's University$

2InversionFigure 2 shows the result of a successful inversion test, in which we initiated the inversion at3 Hz (an admittedly unrealistic low frequency); in this case we find ourselves in the basin ofattraction of the global minimum. Provided this condition is met, the frequency domainapproach turns out to be particularly appropriate since it allows one to recover a continuouswavenumber coverage of the model with only 3 frequencies (3,4.5,and 7.5 Hz) (Sirgue andPratt, 2001).However, when we initiate the inversion at 5 Hz, the inversion fails to converge towards thesolution. The inversion fails to locate the sediment-basalt discontinuity at 2.5 km depth;instead the reconstruction creates a strong oscillation in velocity with depth. This oscillationoccurs in spite of the fact that the starting model (the “macro model”) contains the correct lowwavenumber components. Thus, the initial model is not accurate enough to begin a fullwaveform inversion at 5 Hz.We suggest that the convergence difficulty may be mitigated adopting a layer strippingstrategy. This approach consists in reducing the kinematic error of events from deeper parts ofthe model by first solving for the shallow part of the model, thus reducing the risks of cycleskipping. A suitable strategy is to decompose the model into sediment, the basalt, and the subbasaltlayers. However, some convergence difficulties remain and we propose to study thetopography of the misfit function with various degrees of smoothness of the starting model.Layer stripping and data misfit functionWe constructed a suite of models (Figure 4) and tested the evolution of the least-squares dataFigure 2: Successful waveform inversion resultafter inverting sequentially 3, 4.5 and 7.5 Hzshowing the true model (grey), the starting model(dashed), the result after 3 Hz (dotted) and after3,4.5 and 7.5 Hz (solid).Figure 3: A failed waveform inversion resultafter inverting only 5 Hz data. See Figure 2 forcurve legend.
3misfit function. Preliminary tests (not shown) led us to conclud that it is an categoricalrequirement that the sediment-Basalt interface should be discontinuous and located at anaccurate depth: the reflected and refracted data from the basalts are highly non-linear withrespect to the location and nature of the top basalt interface.Figure 5 depicts the data misfit function as the three layers in the true model are decreasinglysmoothed (i.e., the high wavenumbers are progressively introduced), starting in each casefrom a quasi-homogeneous medium. An increase followed by a decrease of the misfitfunction is evidence of the presence of a local minimum. We conclude from Figure 5 that theuse of 3 Hz data would set few conditions on the starting model. In contrast, in order to obtainconvergence within the sediment layer when using frequencies 5 and 7 Hz, we evidentlyrequire a more accurate starting model, since the topography of the misfit function is likely toguide the waveform inversion into a local minimum. A starting model for the sediments, atthese frequencies would need to come from the region of Figure 5a) in which the misfit isdecreasing. In contrast, Figure 5b) suggests that a less accurate (i.e., more smooth) startingmodel would suffice.One should however be cautious about misleading conclusions for the sub-basalt layer whenexamining Figure 5c). Although the results for 5 and 7 Hz show a constant decrease in themisfit functions, these curves are nevertheless extremely flat. In other words, the misfitfunctions show very little sensitivity to the low and intermediate wavenumber components ofthe model. We suggest that this is because of the fact that only narrow incidence anglesilluminate the sub-basalt sediment.These results lead us to suggest a design for the profile of an adequate starting modela) b) c)Figure 4: Velocity models illustrating the “ideal” layer stripping strategy showing the true model (black)and the true model increasingly smoothed (gray) for a) the sediment, b) the basalt only and c) the subbasaltonly.a) b) c)Figure 5: Misfit functions of 3,5 and 7 Hz as the true model is decreasingly smoothed as shown Figure 4 fora) the sediment and the top of the basalt, b) the basalt only, c) the sub-basalt only.EAGE 64th Conference & Exhibition — Florence, Italy, 27 - 30 May 2002
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