introducing the Huber Function and it's performance in 1D FWI in frequency-domain

1. Evaluation of the Huber Function
Performance in 1D Frequency-Domain Full
Waveform Inversion
Kamal Aghazade1, Navid Amini2
1Msc student, Institute of Geophysics, University of Tehran,
aghazade.kamal@ut.ac.ir
2Assistant professor, Institute of Geophysics, University of Tehran,
navidamini@ut.ac.ir

2. Presentation outline
Introduction Methodology Results Conclusion
Introduction Methodology Results Conclusions
•FWI problem
•Literature Review
• Huber Function
• FWI problems Based
on Huber Function
• implementation of The Huber
Function in 1-D Frequency-
Domain
• numerical tests
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3. • Full Waveform Inversion (FWI) is a data fitting
procedure which is used to obtain high resolution
quantitative models of the subsurface by
exploiting the full information content of the data
recorded by seismograms.
• FWI is an ill-posed and high nonlinear problem.
• The presence of errors (e.g. noise) in the data can
lead to undesirable results.
• The question is which data residual norm has a
good performance in FWI problem?
Introduction Methodology Results Conclusion
𝑙2 − 𝑛𝑜𝑟𝑚 , 𝑙1 − 𝑛𝑜𝑟𝑚 , 𝐻𝑦𝑏𝑟𝑖𝑑
𝑙1
𝑙2
𝑜𝑟 𝐻𝑢𝑏𝑒𝑟?
• Huber, 1973 : introduced a new Criteria
to incorporate l1 − norm for large
residuals and l2 − norm for small
residuals.
• (Guitton and Symes , 2003), used
Huber Function for linear inverse
problem for velocity analysis with both
synthetic and field
data.
• Bube and Nemeth, 2007, studied fast
line search method for Huber and
hybrid l1
l2
norms.
• Ha et al, 2009, studied Huber function
in Frequency Domain FWI.
• (Brossier et al, 2010), studied the
robustness of different data residual
norms in elastic-domain FWI.
Literature Review
3

4. • in the presence of noise in data, 𝑙1_𝑛𝑜𝑟𝑚 is more
robust than 𝑙2_𝑛𝑜𝑟𝑚.
• The main difficulty with 𝑙1_𝑛𝑜𝑟𝑚 is it’s gradient
singularity when data residuals tends toward zero.
• Huber Function :
Combining the features of 𝑙1 𝑎𝑛𝑑 𝑙2 norms in theory.
𝑙1_𝑛𝑜𝑟𝑚 for large residuals and 𝑙2_𝑛𝑜𝑟𝑚 for small
residuals.
• the transition between two types of norms
controls by some threshold.
Introduction Methodology Results Conclusion
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5. FWI problem in frequency-domain by using
Huber Function ( synthetic data) :
• Employing 1D frequency domain Forward
modeling to generate synthetic data .
• extracting the gradient formulae for Huber
function in frequency- domain.
• optimization procedure.
Because 𝒍 𝟏_𝒏𝒐𝒓𝒎 is always greater than 𝒍 𝟐_𝒏𝒐𝒓𝒎 we
eq.1 can be replaced by :
The Gradient formulae :
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Introduction Methodology Results Conclusion

6. Frequency-Domain FWI procedure
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Introduction Methodology Results Conclusion

7. Numerical results for 2 velocity models:
Source : Ricker wavelet with dominant
frequency 25 Hz.
Receiver spacing : 5 m
Frequency range : 0 – 60 Hz
Employing ABC in frequency-domain for
eliminating undesirable reflections from
boundaries.
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Introduction Methodology Results Conclusion

8. Inversion results for velocity model 1 :
group 1 2 3 4 5 6 7 8 9
frequency 6.8Hz 11.4 Hz 16 Hz 20.6 Hz 25.2 Hz 34.4 Hz 39 Hz 48.2 Hz 52.8 Hz
0.3 ( )thr std r
Introduction Methodology Results Conclusion
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9. Introduction Methodology Results Conclusion
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0.4 ( )thr std r
Inversion results for velocity model 1 :
group 1 2 3 4 5 6 7 8 9
frequency 6.7Hz 12.6 Hz 24.4 Hz 30.3 Hz 42.1 Hz 53.9 Hz - - -

10. • in the presence of noise in data Huber function
based FWI in frequency-domain leads to
acceptable results.
• the main challenge with Huber function is it’s
threshold, need consideration for kinds of velocity
models.
• For the smooth velocity model (model 1) we can
not see the importance of the threshold, but for
smooth-blocky model(model 2) we can see the
balancing effect of Huber function.
Introduction Methodology Results Conclusion
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11. Good Luck