IEEE Signal Processing - March 2018 - 138

FWI has been applied to the field data acquired in the Gulf
of Mexico to reconstruct a large-scale salt structure, yielding
encouraging inversion results nearly independent of the frequency content of the data [29]. A successful ocean bottom
seismic data inversion case study using the logarithmic phase
misfit in the Laplace domain was reported in [30].
Another nonlinear operator-based method is envelope
inversion [31]. The envelope of seismic data shows relatively
less fluctuation than the original time domain seismic data,
implying that it may carry some ultralow-frequency information induced by the ultralow wavenumber structures in
the subsurface. The cost function of the envelope FWI is
defined as
C (v) =

Ns

Nr

/ / #0 T

s=1 r=1

E 6S r, s ^v, t h@ p - E 6M r, s ^ t h@ p 2 dt, (24)

where p is a data preconditioning factor to apply the weight as
a function of time. The instant envelope extraction operator E
applied on signal f(t) is defined by
E ^ f (t)h =

f 2^ t h + f H2 ^ t h ,

(25)

where fH (t) is the Hilbert transform of the real function f (t).
Again, as other approaches for cycle-skipping suppression, an
envelope FWI is usually followed by a standard FWI to build
the high wavenumber structures on top of the envelope inversion results with ultralow spatial resolution.
Relying on the dc and low-frequency information augmented by the nonlinear operators applied on the seismic
data amplitudes, Laplace domain FWI, and envelope FWI
are intrinsically sensitive to the signal-to-noise ratio of seismic data, casting doubts on their robustness in general field
data applications.
The FWI algorithm based on extrapolated low-frequency
data, the so-called EFWI, also falls into this category [32].
The EFWI exploits a nonlinear signal processing technique
to decompose recorded seismic data into atomic events, each
of which is defined by a smooth phase function and a smooth
amplitude function. This nonlinear signal processing technique, the so-called phase tracking, can be posed as a nonlinear least-square optimization problem
J ^a n, b n h = 1 u ^~, x h - d ^~, x h 2 + m 1 / d 2~ b n ^~, x h
2
n

2

+ m 2 / d x b n ^~, x h 2 + m 3 / d ~, x a n ^~, x h
n

u ^~, x h =

N

/ w^~ha n ^~, xhe

n

ib n ^~, x h

,

2

(26)
(27)

n

where d is the measured data, u is the predicted data constructed by the atomic events, d denotes the gradient, w (~)
is the known wavelet, and m 1, m 2, and m 3 are the empirically
determined regularization parameters. The cost function (26)
is minimized using a gradient-based algorithm to obtain the
138

amplitude function a n and the phase function b n. After that,
with the nondispersive earth assumption, another optimization is performed to obtain the following approximations:
a n (~, x) . a n (x);

b n (~, x) . ~b n (x) + z n (x) .

(28)

With the approximation (28), one is able to synthesize any flatspectrum atomic events outside of the frequency band of the
recorded seismic data by
N

u e ^~ e, x h = / w ^~ eh a n ^ x h e i^~ b n (x) + zn (x)h,
e

(29)

n

where ~ e is the frequency outside of the observed data frequency band and u e is the extrapolated low-frequency data.
Although the Marmousi model numerical experiment shows
some successful inversion results, the multiple assumptions
made in this method need to be further justified through extensive synthetic data and field data testing.

FWI immune to phase ambiguity
In the context of frequency-domain data fitting, the root
cause of cycle-skipping phenomenon in FWI can be partially
removed if both measured and simulated data are preprocessed
by a phase unwrapping tool. This group of approaches [33],
[34] resolving the ambiguity in the data domain without touching the FWI algorithm can potentially be effective but the
development of the phase unwrapping algorithm is extremely
challenging, especially for 3-D.
Inspired by interference beat tone, an acoustic phenomenon
commonly used by musicians for instrument tuning, the beat
tone FWI was developed to suppress the cycle-skipping phenomenon [35]. The idea of the beat tone FWI is based on the
phase modulation/demodulation concept. This method utilizes
two seismic data sets extracted at two slightly different high
frequencies. The interferences between the two data sets are
exploited by the cost function
C (v) =

Ns

Nr

//

s=1 r=1

U 6S r, s ^v, ~ 2 h S r, s ^v, ~ 1 h@

- U 6M r, s ^~ 2 h M r, s ^~ 1 h@ ,
2

(30)

where U is the phase extraction operator and ~ 1 and ~ 2
are two frequencies extracted from the measured data with
~ 2 - ~ 1 % min ^ ~ 2, ~ 1 h . To understand the mechanism
of the beat tone FWI, assuming there are two seismic signals with the plane wave approximation, u 1 = A 1 e i # k 1 dr and
u 2 = A 2 e i # k 2 dr, we have u 1 /u 2 = (A 1 /A 2) e i # (k 1 - k 2) dr . Since we
abandon the amplitude information in the cost function (30),
the spatial resolution of the gradient of the cost function is
mainly determined by the backpropagated data residual and
the subsurface scattering angle, replacing the effective wavenumber formulation (8) by
BT
k eff
= 2 ^k 1 (rl ) - k 2 (rl )h cos ^i 2h = ^2 ~ 1 - ~ 2

IEEE Signal Processing Magazine

|

March 2018

|

vlh cos ^i 2h .
(31)



Table of Contents for the Digital Edition of IEEE Signal Processing - March 2018

Contents
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