IEEE Signal Processing - March 2018 - 135

information buried in high-frequency seismic data? To find
the answer, first we need to investigate the spatial resolution of FWI.

Spatial resolution of FWI
For a specific source m and a receiver n, the Fréchet derivative
of the function S (v) in the cost function (1), with respect to the
velocity vl at the subsurface point rl , is
2S m, n 2vl = cG (rl , rm) G (rl , rn),

(6)

where G (rl , r) is the Green's function denoting the response
at rl to an impulse excited at location r, c is a complex coefficient. Discarding the amplitude and phase information and
assuming the far-field approximation, the sensitivity kernel in
(6) can be represented by
G (rl , rm) G (rl , rn) = A s e

ik s $ r l

Ar e

ik r $ r l

= Ae

ik eff $ rl

,

Ns

Nr

/ / G^rl , rshu r ^rlhe -4i

~

g

(9)

v^rlh ,

s=1 r=1

where the normalized time lag g = v (rl ) x/4, x is a time lag
variable, and u r is the backpropagated residual. We also have
cos 2 ^i 2h = k

2

k 2g,

(10)

where k is the effective wavenumber vector and k g is the
wavenumber corresponding to the normalized time lag g. The
formulation (10) serves as a scattering angle filter to identify
and exclude the information buried in the extended gradient
associated with small scattering angles. The extended gradient
after the filtering has the form
g FE ^rl, gh = F -1 w ^k, v, ~h Fg E ^rl, gh,

(11)

(7)

where A s and A r are the amplitude of the incident wave and
the scattered wave, k eff = k s + k r is the effective wavenumber dictating the model reconstruction spatial resolution. In
other words,
k eff = ^2~ vlh cos ^i 2h,

g E ^ rl , g h = ~ 2

(8)

where i is the subsurface scattering angle [11], as shown in
Figure 3. According to (8), frequency is not the only factor that determines the spatial resolution of FWI. The subsurface scattering angle i plays an important role too. With
a wide range of scattering angle (e.g., sources and receivers located around the targets), seismic data at a single frequency should be sufficient to cover a wideband spectrum of
wavenumbers. This observation directly led to the technique
of scattering angle-based filtering for low wavenumber component retrieval [17].

Scattering angle-based filtering method
Based on formulation (8), for monochromatic seismic waves,
the effective wavenumber can be precisely controlled by selecting the subsurface scattering angle. In other words, by allowing only a specific portion of the seismic energy with large
scattering angles to contribute to the velocity update, a subsurface reconstruction free of cycle-skipping might be possible. After that, more data with a wider scattering angle range
can be fed into the inversion engine to resolve the finer features. To apply this idea to practical scenarios, the scattering
angle-based filtering method was developed to precondition
the gradient, with the purpose of retaining only low wavenumber information [17]. The key component of the scattering
angle-based filtering method is the extension of the standard
FWI gradient by adding a velocity normalized time lag g.
This extra dimension in the gradient allows one to extract
the scattering angle information at any subsurface location.
With this technique, the FWI gradient g (rl ) becomes the
extended version

where F and F -1 are the Fourier and inverse Fourier transform, respectively, and the window function w is
w (k, v, ~) = )

cos ^i 2 h 2 v k ~
.
cos ^i 2 h # v k ~

1,
0,

(12)

After the filtering, the summation of the remaining energy in
the extended gradient is expected to contain only the low wavenumber information. An attractive feature of this method is the
quantitative control over the reconstruction resolution.
Instead of using the extended gradient approach, the
scattering angle filtering technique has also been practiced
through a local plane wave decomposition method [18]. To
be specific, after both the extrapolated source-side and the
receiver-side wavefields are decomposed into local plane
waves characterized by their propagation direction angles
i s and i r, a predefined angle domain selective filter is
applied to construct a gradient with controlled wavenumber components:

g ^rlh =

Ns

Nr

2 # / / # T / / A ^i , i h
s
r
v rlh s = 1 r = 1 0 is ir
# 6Gt ^rl , rs, t, i sh ut r ^rl , rr, t, i r h@ dt,
3^

Source

(13)

Receivers
Seismic Wave
θ

r′
ks

Reflector
kr

Keff = ks + kr

Figure 3. FWI resolution analysis.

IEEE Signal Processing Magazine

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March 2018

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135
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