Full Waveform Inversion Using Nonlinearly Smoothed Wavefields

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Full Waveform Inversion Using Nonlinearly Smoothed
Wavefields
KAUST
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Item type Conference Paper
Authors Li, Y.; Choi, Yun Seok; Alkhalifah, Tariq Ali; Li, Z.
Citation Li Y, Choi Y, Alkhalifah T, Li Z (2017) Full Waveform
Inversion Using Nonlinearly Smoothed Wavefields. 79th
EAGE Conference and Exhibition 2017. Available:
http://dx.doi.org/10.3997/2214-4609.201701342.
Eprint version Publisher's Version/PDF
DOI 10.3997/2214-4609.201701342
Publisher EAGE Publications BV
Journal 79th EAGE Conference and Exhibition 2017
Rights Archived with thanks to 79th EAGE Conference and
Exhibition 2017
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Link to item http://hdl.handle.net/10754/624987

79th EAGE Conference & Exhibition 2017
Paris, France, 12-15 June 2017
We P1 15
Full Waveform Inversion Using Nonlinearly Smoothed
Wavefields
Y. Li* (China University of Petroleum (East China)), Y. Choi (King Abdullah University of
Science & Technology), T. Alkhalifah (King Abdullah University of Science & Technology), Z.
Li (China University of Petroleum (East China))
Summary
The lack of low frequency information in the acquired data makes full waveform inversion (FWI) conditionally
converge to the accurate solution. An initial velocity model that results in data with events within a half cycle of
their location in the observed data was required to converge. The multiplication of wavefields with slightly
different frequencies generates artificial low frequency components. This can be effectively utilized by
multiplying the wavefield with itself, which is nonlinear operation, followed by a smoothing operator to extract
the artificially produced low frequency information. We construct the objective function using the nonlinearly
smoothed wavefields with a global-correlation norm to properly handle the energy imbalance in the nonlinearly
smoothed wavefield. Similar to the multi-scale strategy, we progressively reduce the smoothing width applied
to the multiplied wavefield to welcome higher resolution. We calculate the gradient of the objective function
using the adjoint-state technique, which is similar to the conventional FWI except for the adjoint source.
Examples on the Marmousi 2 model demonstrate the feasibility of the proposed FWI method to mitigate the
cycle-skipping problem in the case of a lack of low frequency information.

79th EAGE Conference & Exhibition 2017
Paris, France, 12-15 June 2017
Introduction
Full waveform inversion (FWI) updates the subsurface parameters by minimizing the difference
between the predicted and observed data. However, FWI still suffers from a cycle-skipping problem,
and thus, may converge to a solution corresponding to a local minimum, specifically when the
traveltime shift between the predicted and observed data is larger than a half of cycle. Data with
enough low frequency components, which has a long length of cycle, helps FWI avoid the cycle-
skipping problem, but an available frequency band of real seismic data is usually not low enough. On
the other hand, an initial model close enough to the true model can mitigate the cycle-skipping
problem in FWI, but obtaining a good initial model is not trivial task and requires elaborate
techniques, such as traveltime tomography and migration velocity analysis.
A lot of researches in FWI have been devoted recently to solving the cycle-skipping problem without
low frequency. Ma and Hale (2013) estimated the traveltime shift between the predicted and observed
data using dynamic warping method, and tried to minimize the time shift to update the subsurface
parameters. Warner and Guasch (2014) calculated the deconvolution filter between the predicted and
observed data and penalized the energy away from the optimal Dirac delta function to update velocity
model. Wu and Alkhalifah (2016) constructed the objective function based on the data extension and
data selective approach. Wu et al. (2014) estimated the envelope of wavefield and inverted the
artificial low frequency components included in the envelope of wavefield to update long wavelength
components of the model.
On the other hand, Hu (2014) proposed the beat tone inversion, where subtraction between slightly
different frequency wavefields carries apparent low-frequency information. However, since the
apparent frequency information is not real, it required an elaborate technique, such as the phase-
frequency differential and Hilbert transformation, to extract the apparent low frequency information.
The motivation of this abstract is that multiplication, instead of subtraction, between slightly different
frequency wavefields generates artificial low frequency components, which tends to represent the
kinematic components of the wavefield more accurately, thus we can easily extract it by applying a
low-pass filter. In order to generalize its process for the whole frequency-band of data, we use a
nonlinear operation of multiplying the wavefield with itself. We construct the objective function
based on the global-correlation norm (Choi and Alkhalifah, 2012) to deal with the energy (amplitude)
imbalance as a function of offset in the nonlinearly smoothed wavefield. We derive the gradient
expression of the objective function using the adjoint-state technique, which is similar to the
conventional FWI except for the adjoint source. Numerical examples show that the proposed FWI
method can mitigate the cycle skipping problem and generate a good convergent result in the case of
lack low frequency information.
Theory
A source of low frequency by nonlinearly smoothing wavefield
Hu (2014) proposed the beat tone inversion, where subtraction between slightly different frequency
wavefields results in apparent low-frequency information. However, this apparent frequency
information is not physical frequency, thus he needed an elaborate technique, such as the phase-
frequency differential, to extract low frequency information. The motivation of this abstract is to
multiply slightly different frequency wavefields instead of subtraction:
       
 
1
2 2 ( ) 2 (2 ) 2
2
cos f t cos f f t cos f f t cos f t
      
    (1)
where
f
and
t
are the frequency and time variables, respectively. We note that the real artificial low
frequency ( 2
f

) is generated through the multiplication. We can easily extract the artificial low
frequency component [


2
cos f t

] by applying a low-pass filter to the multiplied wavefield. In
order to obtain the artificial low frequency components from full frequency-band of data, we multiply

79th EAGE Conference & Exhibition 2017
Paris, France, 12-15 June 2017
a wavefield with itself (nonlinear operation). We also generalize the case by adjusting the power of
the multiplied wavefield:


2
( , ) ( , )
p
s x t S u x t
 

 
 
, (2)
where
( , )
s x t
is referred to as the nonlinearly smoothed wavefield,
S
indicates a triangle low-pass
filter and superscript
p
stands for the power of wavefield. In this abstract,
p
is 1 or 2.
Global-correlation-based objective function and its gradient
We construct the objective function using the nonlinearly smoothed wavefield in equation 2 following
the global-correlation norm approach (Choi and Alkhalifah, 2012):
2 2
( , ) ( , )
( )
( , ) ( , )
syn obs
t
xsyn obs
t t
s x t s x t dt
J
s x t dt s x t dt

  

 
m (3)
where
( , )
syn
s x t
and
( , )
obs
s x t
are the nonlinearly smoothed wavefields of the simulated and observed
wavefields, respectively. The gradient of the objective function can be derived by taking the partial
derivative of equation 3 with respect to the subsurface model parameter (
m
) as follows:
32 2
2 2
( , ) ( , )
( , ) ( , ) ( , ) ( , )
( )
( , ) ( , )
( , ) ( , )
syn syn
syn obs syn obs
t t t
xsyn obs
syn obs t t
t t
s x t s x t
s x t s x t dt s x t dt s x t dt
J
s x t dt s x t dt
s x t dt s x t dt
 
 
 
 


  
 
 
mm m
m , (4)
where
2
2
( , ) ( , )
( , ) ( , )
p
syn
s x t u x t
p u x t u x t




 
m m . (5)
The adjoint source
( , )
R x t
is derived from the above equations:
2 2
2 2
32 2
2 2
( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )
( , ) ( , )
( , ) ( , )
p p
syn obs syn obs
t t t
xsyn obs
syn obs t t
t t
s x t s x t dt s x t u x t u x t dt s x t u x t u x t dt
p
s x t dt s x t dt
s x t dt s x t dt
 
 


  
 
 
. (6)
Finally, the gradient is expressed as
( ) ( , )
( , )
t
x
J u x t
R x t dt
 

 

m
m m . (7)
We observe that the gradient has a similar form to the conventional FWI except for the computation
of adjoint source. Therefore, the proposed method has almost the same cost as the conventional FWI.
The envelope-based FWI makes the optimization problem less prone to cycle skipping. But when a
cycle-skip between the simulated and observed wavefield is larger than the dominant period, the
envelope-based FWI still may encounter a local minimum. However, the radius of attraction of our
objective function can be enlarged by increasing the triangle smoothing width. The computed gradient
is smoothed according to the smoothing radius of the filter as follows:
1
2
s
r = v f

 

 
 
, (8)
where

refers to the smoothing width,
v
is the average velocity, and
f
is the dominant frequency.
The larger the smoothing width (

) is, the larger the basin of attraction is. Based on the multiscale
strategy, we progressively reduce the smoothing width as the inversion proceeds.
Examples
We test the proposed method on the Marmousi 2 model. The true and initial velocity models are
shown in figure 1(a) and (b), respectively. A ricker wavelet with peak frequency of 7 Hz is used as the

79th EAGE Conference & Exhibition 2017
Paris, France, 12-15 June 2017
source. We filtered out the data below 3.5 Hz. The nonlinearly smoothed wavefields are computed
based on equation 2 with p=1. We start with a large smoothing window and progressively reduce it as
0.16, 0.12 and 0.08s. We also apply a Gaussian smoothing operator on the gradient to remove
potential artifacts. The inversion results with a smoothing width σ = 0.16, 0.12 and 0.08s are shown in
Figures 2a ~ 2c, respectively. As the smoothing width becomes smaller, the resolution is getting
higher. Figure 2d shows the subsequent FWI result starting from Figure 2c, which is compatible with
the true model. For comparison, we display the inverted models for the envelope-based FWI and
conventional FWI in Figures 2e ~ 2g, which show worse results than that of the proposed FWI
algorithm especially in this case where we lack the low frequencies. The velocity depth profiles also
show a good convergence of the proposed FWI results (Figure 3).
The examples demonstrate that the proposed FWI algorithm successfully generates a long wavelength
structure model without low frequency information, which can be used as a good starting model for a
subsequent conventional FWI.
Figure 1 The (a) true velocity model and (b) initial model.
(a) (b)
(a) (b)
(c) (d)
(e) (f)
(g)

79th EAGE Conference & Exhibition 2017
Paris, France, 12-15 June 2017
Figure 2 The inverted velocity models of the proposed FWI with the triangle smoothing width of (a)
0.16, (b) 0.12, and (c) 0.08 s, (d) subsequent FWI starting from Figure 2c, (e) envelope-based FWI, (f)
subsequent FWI starting from Figure 2e, and (g) conventional FWI starting from the initial model in
Figure 1b.
Conclusions
Multiplication between slightly different frequency wavefields generates artificial low frequency
components, which is easily extracted by applying a low-pass filter. For generalization in the whole
frequency-band of the data, we multiply a wavefield with itself to generate artificial low frequency
components. Also, we construct the objective function based on the global-correlation norm to
ameliorate the energy imbalance in the nonlinear smoothed wavefield. We calculate the gradient of
the objective function using the adjoint-state technique, which is similar to the conventional FWI
except for the adjoint source. Numerical examples demonstrate that the proposed FWI method
generates a convergent result for subsequent FWI even in the case of lack of low frequency.
Acknowledgements
We thank KAUST for its support and the SWAG for collaborative environment. Author Yuanyuan Li
wishes to thank the China Scholarship Council for support to study abroad.
References
Choi, Y. and Alkhalifah, T. [2012] Application of multi-source waveform inversion to marine
streamer data using the global correlation norm. Geophysical Prospecting, 60(4), 748–758.
Hu, W. [2014] FWI without low frequency data - beat tone inversion. SEG Technical Program
Expanded Abstracts 2014. 1116-1120.
Ma, Y. and Hale, D. [2013] Wave-equation reflection traveltime inversion with dynamic warping and
full waveform inversion. Geophysics, 78(6), R223–R233.
Warner, M. and Guasch, L. [2014] Adaptive waveform inversion: Theory. SEG Technical Program
Expanded Abstracts 2014. 1089–1093.
Wu, R. S., Luo, J. and Wu, B. [2014] Seismic envelope inversion and modulation signal model.
Geophysics, 79(3), WA13–WA24.
Wu, Z. and Alkhalifah, T. [2016] A selective extension of the data for full waveform inversion - an
efficient solution for cycle skipping. 78th EAGE Conference and Exhibition 2016.
Figure 3 The comparison of velocity
depth profiles at distances of 3 km
(a) and 5 km (b); the true model
(red), initial model (cyan), final
inverted model starting from the
proposed FWI result (blue) , and
final inverted model starting from the
envelope-based FWI result (green).
(a)
(b)