This document evaluates the performance of the Huber function in 1D frequency-domain full waveform inversion (FWI). It introduces FWI and discusses the l1 and l2 norms commonly used, as well as the Huber function which combines the two. The methodology section describes implementing the Huber function for 1D FWI, including gradient formulae. Results show FWI using the Huber function produces acceptable models for two synthetic velocity models, with the threshold parameter affecting the models differently based on their smoothness. The Huber function performs well in the presence of noise and balances the benefits of the l1 and l2 norms.

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
2

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
4

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