Enhancing the resolution of seismic data using improved ...

Enhancing the resolution of seismic data using improved time-frequency spectral modeling Huailai Zhou*, Changcheng Wang, State Key Laboratory of Oil and Gas Reservoir Geology and Exploration,

Chengdu University of Technology; Kurt J. Marfurt, University of Oklahoma; Yiwei Jiang, and Jianxia Bi,

Zhongyuan Oilfield of SINOPEC

Summary

Maximizing vertical resolution is a key objective in seismic

data processing. In this study, we propose an improved

time-frequency spectral modeling deconvolution method,

which firstly uses a generalized S-transform (GST) to

decompose each seismic trace, then models smoothly

varying spectra of the seismic wavelet using nonstationary

polynomial fitting (NPF) from the seismic trace, and finally

performs deconvolution and the inverse GST to obtain the

seismic data with enhanced resolution. Application to a

synthetic and a large 3D survey from onshore China shows

the proposed method is superior to conventional spectral

modeling method in improving vertical resolution, and

confirms that the proposed method is stable in practice and

robust in the presence of noise.

Introduction

The need to properly “land” horizontal wells in resource

plays has reinvigorated the need to improve the vertical

resolution of 3D surface seismic surveys. Conventional

enhancing resolution methods are based on a stationary

convolutional model (Robinson and Treitel, 1967) that

assumes the propagating wavelet does not evolve with

travel time. In reality, seismic energy spreads geometrically

through a heterogeneous, anelastic subsurface which gives

rise to both attenuation and dispersion effects, deforming

the seismic wavelet with travel time, and giving rise to a

nonstationary signal (Ziolkowski, 1991; Margrave and

Lamoureux, 2001). The problem of wavefront divergence

is commonly addressed by applying a simple geometric

spreading correction (Yilmaz, 1987, 2001). The more

challenging frequency-dependent attenuation is usually

referred to as the "Q-effect". While many authors,

including Futterman (1962), Kjartansson (1979), Hale

(1982), Hargreaves and Calvert (1991), Hargreaves (1992),

Zhang et al. (2007) and Wang (2010), have proposed the

design of inverse Q-filters to deal with the attenuation

component of nonstationarity, however, the estimation of

an accurate Q-model is difficult in practice. Margrave et al.

(2011) presented a time-varying deconvolutional method

that estimates the (attenuating) wavelet spectrum and

reflectivity spectrum directly from the Gabor transform

spectrum of the measured seismic data. Wang et al. (2013)

developed an adaptive spectral-broadening method to

improve the resolution of seismic data, based on the

assumption that the seismic data can be split into

approximately stationary segments.

Seismic interpreters routinely use the spectral response of

seismic reflections to detect subtle changes in stratigraphy.

Such nonstationary changes in the spectral response are

quantitatively estimated using a series of nonstationary

time-frequency analysis techniques including Gabor

transform (Gabor, 1946), the short-time Fourier transform

(STFT) (Partyka et al., 1999), wavelet transform (Rioul and

Vetterli, 1991), empirical mode decomposition (Huang et

al., 1998), and seislet transform (Fomel and Liu, 2010).

One of the more popular techniques is the S-transform

(Stockwell et al., 1996) which avoids many of the

shortcomings of the short-time Fourier transform. While

the original S-transform used a fixed mother wavelet,

McFadden et al. (1999), Pinnegar and Mansinha (2003) and

Gao et al. (2003) chose the decomposition parameters

based on the demands of the seismic data, giving rise to the

generalized S-transform (GST). Liu et al. (2011) proposed

a local attribute analysis method for nonstationary signals

using an iterative inversion framework, which calculates

time-varying Fourier coefficients by solving a least-squares

problem based on regularized nonstationary regression

(Fomel, 2007, 2009, 2013).

We begin this study with a review of the expanded

generalized S-transform used to compute the time-

frequency components of the seismic trace. Next, we use

the improved spectral modeling method using NPF (Fomel,

2009) to estimate time-varying wavelet spectra from

seismic trace and perform the deconvolution for seismic

data. Finally, we apply this workflow to a synthetic and a

3D land survey acquired in China to illustrate the

adaptability and effectiveness in improving vertical

resolution.

Methodology

The generalized S-transform

Time-frequency analysis is powerful to characterize the

nonstationarity of seismic signal using the generalized S-

transform (GST), which is defined as 2

''2

( )

22

''ˆ( , ) ( ) e e

2

rr t f

i ft

GST

fx f x t dt

, (1)

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Enhancing the resolution of seismic data using improved time-frequency spectral modeling

where x(t) is the original seismic signal, is the time shift

parameter, ˆ( , )x f is the expression of a two-dimensional

time-frequency variable with regard to f and , '' and r are

adjustable parameters. Examining equation (1), we see that

the Gabor transform (Gabor, 1946) is a special case of the

GST when r=0. Similarly one obtains the S-transform

(Stockwell et al., 1996) by setting ''=1, and r=1.

Deconvolution using the improved spectral modeling

Based on the assumption of statistically white reflectivity,

Rosa and Ulrych (1991) represented to model the amplitude

spectra of the wavelet from seismic trace using a smooth

polynomial, in time-frequency domain (or generalize S-

domain), which can be defined as

0

( , ) exp ( ) , (2)N

k n

a m m n m

n

W f f a f

where Wa(,fm) denotes a smooth amplitude spectrum of a

time-varying wavelet, an() is the nth polynomial coefficient

independent on frequency at time , N is the polynomial

order, and fm is the mth frequency. The conventional method

is to use equation (2) to model the amplitude spectrum of a

wavelet. In our study, we think that the polynomial

coefficients varying with time and frequency represent the

nonstationarity of seismic data more precisely than

conventional fitting method. Hence, we can redefine the

equation (2) as

0

( , ) exp ( , ) , (3)N

k n

a m m n m m

n

W f f a f f

As illustrated in the context, we can estimate the

polynomial coefficients by solving the following least-

squares problem:

2

2

( , )1 1 0 2

min ln ( , ) ln ( , ) ( , ) , (4)n m

M M Nn

m m m n m m n ma f

m m n

A f k f a f f R a f

where 2

2denotes the squared L-2 norm of a function,

A(,fm) denotes the time-frequency amplitude spectra of

seismic trace, an(,fm) expresses the polynomial coefficients

varying with time and frequency, R denotes the

regularization operator, and εm is the misfit error to be

minimized. Here we use Fomel’s (2007) “shaping

regularization” which uses an adjustable radius Gaussian

smoothing operator to control the smoothness of the

coefficients an(, fm). We note that time-frequency

amplitude spectra Wa(,fm) of a wavelet, can be estimated

by iterating the procedure of equation (3) and (4) over all

the frequencies at every time-sample . As discussed above,

once getting the nonstationary polynomial coefficients an(, fm), we can model wavelet spectra between equation (3) and

the amplitude of seismic trace, and then design the

deconvolution operator to complete the processing of

resolution enhancement.

Application

In this section, we demonstrate the effectiveness and

advantage of the improved spectral modeling

deconvolution method proposed in this study on synthetic

data, a 2D seismic line, and a large 3D data volume

acquired from onshore China.

Synthetic data example

In order to verify the effects of the improved spectra

modeling method based on the nonstationary polynomial

fitting (NPF) in resolution enhancement for seismic data,

we use the nonstationary convolution model (Margrave and

Lamoureux, 2001) to produce a synthetic for testing

purpose. Figure 1 shows the comparison results using

respectively stationary polynomial fitting (SPF) and

nonstationary polynomial fitting (NPF) deconvolution

(improved spectra modeling method) to process the

stationary (Figure 1b) and nonstationary synthetic (Figure

1b). Comparing Figure 1d, 1e, 1f and 1g in the red dashed-

line boxes, we find that the adjacent reflection events

centered about 200ms and 250ms cannot be distinguished

in Figure 1d and 1e processed by SPF deconvolution

method, but resolved in Figure 1f and 1g applied by NPF

deconvolution method. In comparison of these results with

the reflectivity (Figure 1a) we again confirm that the

improved spectra modeling deconvolution is superior to

conventional method based on the SPF.

Figure 1: A synthetic seismic trace example. (a) The

reflectivity spike series. (b) The stationary seismic trace

obtained by convolving (a) with a single wavelet. (c) The

nonstationary seismic trace obtained by convolving (a) with

a time-varying wavelet defined by a value of Q=50. Results

of SPF deconvolution method applied to (d) the stationary

and (e) nonstationary synthetic traces. Results of our

improved NPF deconvolution applied to (f) the stationary

and (g) nonstationary synthetic traces. Notice the

enhancement in resolution using our proposed method.

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Field data examples

The previous synthetic example is noise free. In order to

compare the effectiveness and stability of the SPF (Figure

2b) vs the NPF (Figure 2c) deconvolution algorithm we

apply them to a vertical slice (Figure 2a) from a large 3D

poststack seismic volume acquired from an area in Western

China. Comparing the events within yellow ovals of

Figures 2a and 2b, we see an improvement in vertical

resolution that brings out more detailed reflections. The

thin beds indicated by the red arrow in Figure 2c are

unresolved in Figure 2b. While both SPF and NPF improve

the spectral content of the signal, NPF method which more

accurately accounts for the nonstationarity of the seismic

wavelet is significantly better. In addition, the NPF

deconvolution algorithm avoids increasing the spectral

content of the noise.

Figure 2: A vertical slice through a 3D seismic amplitude

volume (a) before and after deconvolution using the (b)

SPF and (c) NPF algorithms. Location of the vertical slice

is shown in Figure 4. Yellow ovals and red arrows indicate

areas of improved vertical resolution using the NPF

algorithm.

In Figure 3 we illustrate the comparison of the average

amplitude spectra for the three images shown in Figure 2.

As expected from our previous inspection of Figure 2, the

spectrum of the NPF deconvolution data (red dot-dashed

curve) is broader than that of the SPF deconvolution data

(green dot-dashed curve), and broader still than the original

data (black curve).

Seismic coherence is a measure of lateral changes in the

seismic response caused by variation in structure,

stratigraphy, lithology, porosity and the presence of

hydrocarbons (Marfurt et al., 1998). To see if we have

improved the lateral delineation of any thin stratigraphic

features, we calculate coherence for the three seismic

volumes shown in Figure 2. Figure 4 shows representative

time slices at t=2920 ms through the amplitude and

coherence volumes before and after SPF as well as NPF

deconvolution. Figures 4b and 4c show amplitude time

slices applying the conventional method and improved

method for input data (Figure 4a). We find that the

improved method (NPF deconvolution) provides higher

temporal and spatial resolution. Time slices from the

coherence attribute volumes shown respectively in Figures

4d, 4e and 4f corresponding to Figures 4a, 4b and 4c

manifest that the high resolution volume generated from the

improved NPF deconvolution serves to show the increased

details and more subtle structure characters.

Figure 3: Comparison of average amplitude spectra of three

images shown in Figure 2 for the original data (black dot-

dashed curve), data after SPF deconvolution (green dot-

dashed curve) and data after NPF deconvolution (red dot-

dashed curve). Note that the dominant frequency is higher

and bandwidth is broader for the seismic section processed

by NPF algorithm compared to the original and SPF

deconvolution data.


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Figure 4: Time slices at t=2920ms through the (a) original

data volume and data volumes after (b) application of SPF

deconvolution, and (c) NPF deconvolution. (d)-(f) Time

slices through the corresponding coherence volumes. Black

line indicates location of the vertical slice shown in Figure

2. Notice the enhancement in reflection detail on the time

slice of amplitude and coherence attribute data after

applying the improved method.

Figure 5: Stratal slices through the (a) original data volume,

and after (b) application of SPF deconvolution and, (c) NPF

deconvolution. (d)-(e) Stratal slices through the

corresponding coherence volumes. Note that the stratal

slices from the amplitude and coherence volumes show

much more stratigraphic features (green arrows and red

ovals) and subtle faults (red arrows) with improved method

than the other two.

In Figure 5, we show stratal slices of amplitude volumes

from the original seismic data (Figure 5a) as well as

resolution-enhanced data generated using the SPF

algorithm (Figure 5b) and NPF deconvolution method

(Figure 5c). The stratal slices through the corresponding

coherence volumes are shown in Figures 5d-f. In

comparison areas (indicated by red ovals and green arrows),

we observe that the two methods can greatly improve the

seismic resolution and efficiently identify the faults. Note

that the result using the improved NPF deconvolution is

superior in delineating stratigraphic features and subtle

faults (show in red arrow) than SPF deconvolution method.

This example manifests that the enhanced resolution data

using the improved method is helpful to extract the better

seismic attribute and facilitate interpreters to accurately

depict the structural and stratigraphic features. Notice that

the amplitude and coherence slices show improved

resolution in the order stated above, with the highest lateral

resolution observed from slices computed on improved

spectral modeling algorithm.

Conclusions

Time-frequency analysis using the generalize S-transform

provides a basis for more precise and adaptive

deconvolution (or spectral modeling) algorithms. Previous

algorithms based on stationary polynomial fitting of the

spectrum assumed fitting coefficients to be invariable over

all the frequencies. By using nonstationary polynomial

fitting concepts described by Fomel (2009) we develop a

deconvolution algorithm that adjusts adaptively according

to nonstationary seismic wavelets. Unlike earlier

algorithms, our nonstationary polynomial fitting (NPF)

deconvolution algorithm circumvents the need to explicitly

choose spectral modeling parameters k and N and avoids an

explicit parameterization of Q. Application to synthetic and

field data show that the improved method not only

enhances the seismic resolution without boosting the noise

more effectively and also performs better compensation for

attenuation than conventional spectra modeling.

Furthermore, application to a large 3D survey from onshore

China shows an improvement in vertical resolution that

significantly facilitates stratigraphic interpretation, and also

confirms that the proposed method in practice is stable and

robust in the presence of noise.

Acknowledgement

This work is financially supported by the National Natural

Science Foundation of China (Grant No. 41204091), and

Science and Technology Support Program (Grant No.

2011GZ0244) from Science and Technology Department of

Sichuan Province of China. The first author appreciates the

Chinese Scholarship Council for financial support. We also

thank Puguang Branch of Zhongyuan Oilfield of SINOPEC

for permission to present these results and want to honestly

thank Ph.D. students Shiguang Guo, Fangyu Li and Tao

Zhao in the University of Oklahoma for their inspiring

discussions.


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