2D phase unwrapping by DCT method
Kui Zhang, Marcilio Castro de Matos, and Kurt J. Marfurt
ConocoPhillips School of Geology & Geophysics, University of Oklahoma
Phase dispersion correction
B(1,1)
A(1,1)
Seismic waves propagating through the subsurface undergo strong energy dissipation and velocity
dispersion due to both anelasticity and heterogeneity within the earth. High frequency data
components suffer more loss than low frequency components that traveled along the same ray path,
resulting in a relatively narrow-band, low-frequency spectrum. In general, the high frequencies travel
at different velocities than the low frequencies resulting in a significant change in waveform shape.
Due to the above two effects, the seismic wavelet become noticeably stretched, and displays
“ringing” characteristic as the travel-time increases.
The traditional approach to compensate seismic wave attenuation and dispersion is inverse Q
filtering by assuming the earth Q model to be a multilayed structure. Inverse Q filtering is
implemented in a layer stripping manner(Wang, 2002) by downward continuation. However, very
few papers have mentioned about how to estimate Q values of the shallow earth effectively. Noted
exceptions are work by Singleton and Taner (2006) who use logs and Chopra and Alexeev (2003)
who use VSPs.
In our study, we provide a different approach to eliminate velocity dispersion. Without considering
attenuation, a plane wave at time t that has undergone dispersion can be formulated as:
U ( f , t )  U ( f ,0) exp[  ( f , t )]
(1)
Figure 1. (a) 2D Gaussian model
where ( f , t )  2ft  G ( f , t )  Q ( f , t ). (2)
2πft is the linear phase part caused by the time delay, φQ(f,t) is a nonlinear phase delay that
depends on both intrinsic and geometric Q dispersion, and φG(f,t) is directly related to the
underlying geology and impedance of the layers, including π phase changes associated with
negative reflection coefficients,  π/2 phase changes associated with thin bed tuning and upward
fining/coarsening, as well as more complicated phase changes due to stratigraphic layering.
Our objective is to retain the linear part and φG(f,t), and remove the effects of φQ(f,t) to compensate
for the dispersion to obtain φ(f,t):
 ( f , t )   ( f , t )  Q ( f , t ).
(3)
The phase dispersion corrected signal in time domain can be obtained by summing all plane waves.
Why phase unwrapping?
In order to estimate φQ(f,t), we average phase over a finite time window of thickness T over the
entire seismic survey so that equation 2 becomes:
 Q ( f , t )   ( f , t )  2ft    G ( f , t )  (4)
Where, the sign < > denotes the average operator. Assuming that the reflectivity character of the
earth is white,
 G ( f , t )   0.
(5)
Combining (4), (5), and assuming the invariablity of Q model horizontally, we get
Q ( f , t )   ( f , t )  2ft 
(6)
In order to calculate the average of equation (6), all the phases ψ(f,t) must be unwrapped .
Itoh’s 1D phase unwrapping
In 1982 Itoh proposed that the unwrapped phase can be obtained by integrating wrapped phase
differences (Itoh, 1982). The unwrapped phase will equal the true phase provided the true phase
differences are less than π radians in magnitude everywhere. If W is the wrapped operator, Φ is the
true phase, and φ is the wrapped phase (φ (n)=W{Φ(n)}, n=0,… N-1). The 1D phase unwrapping
can be realized by:
n 1
 ( n)   (0)   W { ( m)}
(7 )
m 1
Since the spectral phase of a seismic trace after spectral decomposition will be a 2D panel, if we
apply 1D phase unwrapping to every component, the output will have some vertical stripes because
spectral phase also change laterally. Considering this fact, 2D or multi-dimensional phase
unwrapping is essential.
(b)The wrapped phases of (a)
(c)The unwrapped result from (b)
(using B(1,1) as the reference)
(d)The unwrapped result from (b)
(using A(1,1) as the reference)
(e) 1D unwrapped result from (b)
(notice its big difference with (c))
Real data example
2D phase unwrapping by Discrete Cosine Transform
Ghiglia (1994; 1998) gave several discrete approaches to unwrap the phase by solving Poisson’s equation:
(i 1, j  2i , j  i 1, j )  (i , j 1  2i , j  i , j 1 )   i , j
(8)
where i , j  (x i , j  x i 1, j )  (x i , j  x i , j 1 )
(9)
and  i , j  W{i 1, j  i , j }, 
 W{i , j 1  i , j }.
(10)
x 1, j  0, x M 1, j  0, y i , 1  0, y i , N 1  0.
(11)
x
y
i, j
The least squares approach requires Neumann boundary condition on Poisson’s equation, that is:
Applying the two dimensional Fourier transform to the two sides of (8), we get
 m,n 
Pm , n
(12)
4  2 cos(m / M )  2 cos(n / N )
Where Ψ and P are the two-dimensional Fourier transform of ϕ and ρ respectively. The solution ϕ can
then be obtained by inverse Fourier transform.
To satisfy the Neumann boundary condition (11), we need either mirror the 2D Laplacian ρi j, and then
apply the fast Fourier transform, or apply the two dimensional forward DCT (Discrete Cosine Transform) to
,
ρi j thereby eliminate the need for mirroring.
Figure 2. (a) Wrapped phase
(b) Unwrapped result from 2(a)
,
In our study, we implemented the algorithm using the DCT method. To test the algorithm, we designed a
2D Gaussian model, wrapped it, and then unwrapped the phase and compared with the original model.
The least squares solution described above is relative such that one reference point needs to be set by the
user to calibrate it. Figure 1d shows that if we know only one true absolute phase of the model and use it
as the reference, the unwrapped result is exactly the same as the original model (Figure 1a).
Conclusions
The statistical phase compensation after 2D phase unwrapping will be investigated using real seismic
data.
 The
DCT method of 2D phase unwrapping provides a robust result, but it still needs to be tested by
more models and real data examples.
 Some attributes based on the unwrapped phase will be analyzed to map structural and stratigraphic
information.
References
Chopra, S., A., Alexeev, V. Sudhakar, 2003, High frequency restoration of surface seismic data: The
leading Edge, 22, 730-738.
Ghiglia, D. C., and M. D. Pritt, 1998, Two-dimensional phase unwrapping: Theory, algorithm, and
software: John Wiley & Sons, Inc.
Itoh, K., 1982, Analysis of the phase unwrapping problem: Applied Optics, 21, p2740.
Lomask, J., 2007, Seismic volumetric flattening and segmentation, Ph.D. thesis, Stanford University.
Singleton,. S., M. T., Taner, 2006, Q estimation using Gabor-Morlet joint time-frequency analysis
techniques: 76th Annual meeting SEG, Expanded abstracts, 1442-1446.
Wang, Y., 2002, A stable and efficient approach to inverse Q-filtering: Geophysics, 67, 657-663.
Acknowledgement
We thank all the sponsors of Attribute Assisted Seismic Processing and Interpretation
Consortium for their support. We also thank Tim Kwiatkowski, Jesse Lomask for their
helpful discussions.
11/18/2008