IEEE Signal Processing - March 2018 - 139

In other words, the mapping from the low-frequency data to the
low wavenumber model established by the wave equation is now
replaced by the mapping from the frequency difference in the
data to the low wavenumber model. The effect of phase ambiguity resolving of this method can be observed in Figure 5, where
S 1 is the data recorded at 8.5 Hz and S 2 is recorded at 7 Hz. The
number of phase-wrapping occurrences is significantly reduced
with the beat tone FWI strategy, and low wavenumber structures
are expected to be reconstructed reliably. Accordingly, the number of local minima may reduce as well, as shown in Figure 6.
In Figure 7, the velocity models reconstructed by a conventional
FWI method and the beat tone FWI are compared, where the
strong artifacts induced by the cycle-skipping are observed in the
conventional FWI result. On the other hand, with the same seismic data sets, the beat tone FWI successfully mitigated the cycleskipping issue. One main disadvantage of the beat tone FWI is
the amplified scattering effect observed as a noise-like pattern
existing in the objective function when the two selected frequencies are too close.

Discussion and conclusions
In this article, we reviewed an advanced seismic data processing technology, the FWI, with a focus on one of its main challenges, the cycle-skipping phenomenon, which prevents the FWI
technology from being widely applied for subsurface exploration.
After analyzing the relationship between cycle-skipping, local
minimum, and nonlinearity, and investigating the spatial resolution formulation of the conventional FWI algorithm, we identify that one of the solutions to the cycle-skipping problem is to
retrieve the low wavenumber information buried in the band limited recorded seismic data without resorting to extra low-frequency data. Although discussing all of the methods of cycle-skipping
mitigation is not possible, this article gives a reasonable review of
five main categories of these approaches.
The first category is the FWI incorporated with a subsurface
scattering angle-based filter. This method was inspired by the
observation that the effective wavenumber of a subsurface geophysical model (i.e., the wavenumber of model reconstruction),
is mainly determined by the frequency of the seismic data, the

Cost Function
Receivers

S1
S2
S1/S2

Conventional
FWI

Source

Local Minimum
True Solution

Beat Tone FWI
Velocity Model

Standard FWI
4
3
2
1
0
-1
-2
-3
-4

Phase of S2
Phase of S1

Figure 6. A diagram showing the mechanism of local minimum mitigation using beat tone FWI.

Beat Tone FWI
4
3
2
1
0
-1
-2
-3
-4

Phase Wrapping

(a)

100

Receiver Index
Phase of S1/S2

(b)

Figure 5. A true velocity model: phases of data S 1 measured at 8.5 Hz and

S 2 measured at 7 Hz; and phase difference between S 1 and S 2, serving as
beat tone FWI data.

Figure 7. (a) The beat tone FWI and (b) conventional FWI.

IEEE Signal Processing Magazine

March 2018

139

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