Pre-stack basis pursuit seismic inversion for brittleness of shale
Pre-stack basis pursuit seismic inversion for brittleness of shale
Xing-Yao Yin 0
Xiao-Jing Liu 0
Zhao-Yun Zong 0
0 School of Geosciences, China University of Petroleum (Huadong) , Qingdao 266580, Shandong , China
Brittleness of rock plays a significant role in exploration and development of shale gas reservoirs. Young's modulus and Poisson's ratio are the key parameters for evaluating the rock brittleness in shale gas exploration because their combination relationship can quantitatively characterize the rock brittleness. The highvalue anomaly of Young's modulus and the low-value anomaly of Poisson's ratio represent high brittleness of shale. The technique of pre-stack amplitude variation with angle inversion allows geoscientists to estimate Young's modulus and Poisson's ratio from seismic data. A model constrained basis pursuit inversion method is proposed for stably estimating Young's modulus and Poisson's ratio. Test results of synthetic gather data show that Young's modulus and Poisson's ratio can be estimated reasonably. With the novel method, the inverted Young's modulus and Poisson's ratio of real field data focus the layer boundaries better, which is helpful for us to evaluate the brittleness of shale gas reservoirs. The results of brittleness evaluation show a good agreement with the results of well interpretation.
with angle; Brittleness; Shale gas; Amplitude variation; Basis pursuit; Bayesian framework
Shale gas is a very important type of unconventional
resource. The term refers to the unconventional gas stored
in shale reservoirs. With the development of seismic
exploration, a large amount of practice in unconventional
shale reservoirs indicated that rock brittleness is one of the
critical parameters to be taken into consideration in the
evaluation of hydraulic fracturing. The study of shale
brittleness is very important for shale gas exploration and
development. An empirical brittleness cut-off is defined
based on Young’s modulus and Poisson’s ratio (Grieser
and Bray 2007; Rickman et al. 2008) as they control the
relationship between stress and strain given by Hooke’s
Law (Sena et al. 2011). A high-value anomaly of Young’s
modulus and a low-value anomaly of Poisson’s ratio can be
used to evaluate the rock brittleness and to infer ‘‘sweet
spots’’ of shale gas reservoirs (Harris et al. 2011; Zong
et al. 2013). Seismic inversion is the fundamental scientific
tool used to obtain parameters concerning lithology and
physical properties (Yin et al. 2015). Therefore, the
estimation of Young’s modulus and Poisson’s ratio from
prestack seismic data is a helpful guide for evaluating the
brittleness of shale.
Amplitude variation with angle (AVA) inversion can be
used to estimate the subsurface elastic properties from
prestack seismic reflection data. However, the geophysical
inversion problem in nature is an ill-conditioned problem
because slight noise contained in the observed data will
lead to enormous changes in the estimated parameters.
Another problem of AVA inversion is that there are many
models adequately fitting the data because the seismic data
are band limited. It is common to add additional constraints
to stabilize the inversion process and to reduce the number
of solutions. This is generally referred to as regularization.
The regularization method was proposed by Tikhonov
(1963), and the L-curve (Hansen 1992) was presented for
selecting the regularization parameters which balance the
data fitting term and trade-off function. A Bayesian
approach is another method for stabilizing the inversion
performance that treats the model parameter as a random
variable with a probability distribution (Duijndam 1987;
Buland and Omre 2003; Tarantola 2005; Yuan and Wang
2013; Yin and Zhang 2014). We seek the parameter
estimates with a maximum posterior distribution combined
with prior information of model parameters and the
likelihood function. The prior information can be the
probability distribution of the model parameters and the
geological information. In special cases, the regularization
function is equivalent to the prior information in the
The sparse solutions are full band; therefore, the sparse
estimations are often viewed as high-resolution estimations
(Levy and Fullagar 1981; Sacchi 1997; Alemie and Sacchi
2011). The inversion results via l2 norm regularization
(Tikhonov 1963) or assumption of Gaussian probability
distribution (Downton 2005) do not lead to high resolution
because the estimates lack sparsity. The sparse reflection
coefficients generate the blocky layer elastic parameters
which suppress the side lobes. The development of the
theory of sparse representation promotes the sparse
inversion method. Theune et al. (2010) investigated the Cauchy
and Laplace statistical distributions for their potential to
recover sharp boundaries between adjacent layers. Based
on the reflection dipole decomposition described by
Chopra et al. (2006), Zhang et al. (2009, 2011) studied the basis
pursuit inversion (BPI) of post and pre-stack seismic data,
respectively, and got the sparse reflection coefficients and
blocky layer elastic parameters, which is a high-resolution
inversion method. Pe´rez et al. (2013) proposed a hybrid
Fast Iterative Shrinkage-Thresholding Algorithm (FISTA)
least-squares strategy that inverts the location of reflection
by the FISTA algorithm (Beck and Teboulle 2009) first and
then reevaluates the sparse (high resolution) reflection
coefficients. However, this lacks the low-frequency
information if we only use the seismic data and sparse
regularization. The low-frequency information should be
incorporated into the objective function to enhance the
meaning of the inversion results and meanwhile promote
the stability of the inversion implementation (Yin et al.
2008, 2014; Zong et al. 2012a; Yuan et al. 2015).
The ultimate goal of pre-stack inversion is to obtain the
elastic information that can be used for evaluating
hydrocarbon potential and the brittleness of the reservoir and to
infer ideal drilling locations of ‘‘sweet spots’’ (Sena et al.
2011). Different linear approximations (Aki and Richards
1980; Gray et al. 1999; Russell et al. 2011) of the Zoeppritz
equation introduced by Zoeppritz (1919) help us to directly
estimate the elastic parameters (e.g., P-wave velocity,
S-wave velocity, Lame´ parameters, bulk modulus, density)
in which we are interested. The Young’s modulus and
Poisson’s ratio can be calculated from the P-wave velocity,
S-wave velocity, and density which can be inverted directly
via Aki-Richards approximation. The density is difficult to
invert, which will have a deleterious influence on the
estimation of Young’s modulus. Parameters estimated
indirectly will bring in more uncertainty in the inversion
results (Zhang et al. 2009). In order to estimate the
Young’s modulus (Y), Poisson’s ratio (P) and density
(D) directly, Zong et al. (2012b) derived the linear
approximation equation based on Young’s modulus,
Poisson’s ratio, and density and inverted the elastic parameters
by Bayesian framework via Cauchy distribution as prior
information. The approximation can be named as the YPD
approximation. Zong et al. (2013) reformulated the elastic
impedance equation in terms of Young’s modulus,
Poisson’s ratio and density based on the YPD approximation,
and introduced a stable inversion method named elastic
impedance varying with incident angle inversion with
damping singular value decomposition (EVA-DSVD)
inversion. In this study, we propose a model constrained
basis pursuit inversion method to estimate the Young’s
modulus, Poisson’s ratio, and density with the YPD
approximation. The model constraint term is added into the
objective function through a Bayesian framework. We also
take a decorrelation of model parameters before inversion.
The introduced model constraint promotes the stability of
the inversion. Basis pursuit ensures the sparsity of
reflection coefficients and the blocky structure of layer
parameters. Model synthetic gather data with different
signal-tonoise ratios are studied to test the proposed inversion
method. The application on real data from shale reservoirs
shows that Young’s modulus and Poisson’s ratio inverted
by the proposed inversion method are reasonable for
brittleness evaluation. The result of brittleness evaluation fits
well with well interpretation.
2 Theory and method
2.1 YPD approximation
Young’s modulus and Poisson’s ratio are the central
parameters in predicting the brittleness of the subsurface
layers. Young’s modulus can characterize the rigidity or
brittleness of rocks, and Poisson’s ratio can be regarded as
a kind of fluid factor which can be used for pore fluid
identification. The reflection coefficients approximate
equation was derived in terms of Young’s modulus,
Poisson’s ratio, and density (YPD approximation) with the
hypothesis of planar incident wave by Zong et al. (2012c):
where h is the incident angle; R(h) is the reflection
coefficients; k stands for the square of the average S-to-P
velocity ratio; and DE/E, Dr/r, and Dq/q represent the
reflection coefficients of Young’s modulus, Poisson’s ratio,
and density, respectively.
In order to perform the inversion for Young’s modulus,
Poisson’s ratio and density, we should firstly build the
forward model. Combining the convolution model and
YPD approximation, we can get the pre-stack data in angle
domain shown as Eq. (2).
2 Wðh1ÞCEðh1Þ Wðh1ÞCrðh1Þ Wðh1ÞCqðh1Þ 3
where WðhiÞ represents the wavelet matrix and CEðhiÞ,
CrðhiÞ , and CqðhiÞ represent the weighting coefficients of
Young’s modulus reflectivity vector rE, Poisson’s ratio
reflectivity vector rr, and density reflectivity vector rq,
respectively. The product of the wavelet matrix and
weighting coefficients makes up the kernel matrix G.
Setting the reflectivities as model vector r, the forward model
equation can be written in a linear equation as d = Gr.
2.2 Model parameters decorrelation
Decorrelation of model space parameters can enhance the
stability of the three-parameter AVA inversion (Downton
2005; Zong et al. 2012b). We took the singular value
decomposition (SVD) for covariance matrix Cr of model
where r2E is the variance of Young’s modulus; rEr is the
covariance of Young’s modulus and Poisson’s ratio, and so
on; v is a matrix made up of three eigenvectors; and S is the
diagonal matrix made up of the positive decreasing
For N samples, the inverse of the decorrelation matrix
V can be expressed as the Kronecker product of v-1 and an
N-order identity matrix. Therefore, the inverse of
decorrelation matrix V is expressed as Eq. (4):
In this case, the kernel matrix G becomes G~ ¼ GV 1,
and the model vector becomes r~ ¼ Vr. Therefore, the
forward modeling equation should be written as
2.3 Dipole decomposition and forward model
Then we used the dipole reflection coefficient
decomposition method to update the forward model of pre-stack BPI
inversion for Young’s modulus, Poisson’s ratio, and
density. The reflection coefficient decomposition method is
shown in Fig. 1. In this case, the vector of reflection
coefficients containing Young’s modulus, Poisson’s ratio,
and density can be written as Eq. (6).
where D stands for the dipole reflectivity decomposition
operator. mE, mr and mq are the sparse coefficients of
Young’s modulus, Poisson’s ratio, and density,
respectively, under the reflectivity decomposition,
mE ¼ ½ aET bTE T, mr ¼ ½ aTr bTr T , and
mq ¼ ½ aqT bqT T. D~ is a large scale matrix consisting of
three reflectivity decomposition operators. m is the sparse
coefficients vector consisting of mE, mr, and mq.
We can obtain the forward model shown as Eq. (7) by
substituting Eq. (6) into Eq. (5):
d ¼ G~D~m:
2.4 Bayesian inference and model constrained BPI
In this study, we constructed the objective function of the
AVA inversion under the Bayesian framework (Ulrych
et al. 2001). The Bayesian theorem is given by
where pðmjdÞ is the posterior probability distribution
function (PDF), pðdjmÞ is the likelihood function that is the PDF
of noise, pðmÞ is the prior information of the parameter m,
Fig. 1 The reflectivity decomposition (Zhang et al. 2013)
and pðdÞ is the normalization factor which can be ignored as
it is a constant value. Therefore, the Bayesian theorem is
often expressed as Eq. (9) without the scaling factor:
pðmjdÞ / pðdjmÞpðmÞ:
We suppose that the noise obeys a Gaussian distribution;
hence, the likelihood function should be
pðdjmÞ / exp
where Xd is the noise covariance matrix. For simplicity, we
suppose that the noise is uncorrelated, so the covariance
matrix should be Xd ¼ rd2Id, where Id is the identity matrix
and r2d is the variance of the Gaussian distributed noise.
In this paper, we assume that the prior distribution is
constructed by two terms:
pðmÞ ¼ ptðmÞplðmÞ;
where the first term, ptðmÞ, is the probability distribution of
m which represents the sparsity of the coefficients and the
second term, plðmÞ, is the low-frequency model
information that can enhance the lateral continuity. The prior
information stabilized the inversion process and provided a
principle to choose the ‘‘best’’ solution that can adequately
fit the observed data.
In order to recover the discontinuity of the layer
properties, the minimum l1 norm that works well in
selecting the sparse solution practically should be taken
into consideration to constrain the inversion. This l1 norm
regularization can be incorporated into the Bayesian
approach as the Laplacian distribution with a mean of
ptðmÞ / exp
In the lateral term, we suppose that the error between the
inversion and low-frequency model obeys a normal
where Xðm;nÞ is the covariance matrix associated with the
three elastic properties: Young’s modulus, Poisson’s ratio,
and density. Here we assume that the parameters at each
sample are independent as we took decorrelation of the
model parameters, and then Xðm;nÞ ¼ rðm;nÞIðm;nÞ, where
rðm;nÞ is the variance of the coefficients for estimation, and
Iðm;nÞ is the identity matrix. n is the vector made up of the
relevant Young’s modulus, Poisson’s ratio, and density; C~
is made up of diagonal integrated matrix C ¼ Rtt0 ds. The
expression of n and C~ can be written as Eq. (14):
Under Bayes’ framework, we can estimate the solution
as the maximum a posteriori (MAP) solution. Substituting
Eq. (12) and Eq. (13) into Eq. (11), the prior information
can be written as
pðmÞ / exp
Substituting the likelihood function Eq. (10) and prior
distribution Eq. (15) into the Bayesian theorem Eq. (9), we
get the objective function shown as Eq. (16) under the
d 22þkkmk1þl C~D~m
hwhere, k and l are the trade-off factors which balance the
overall impact of the regularization, and
k ¼ rd2=rm; and l ¼ r2d=rð2m;nÞ:
The objective function Eq. (16) can be viewed as the
normal expression for a basis pursuit problem via an
Accordingly, utilizing the Gradient Projection for Sparse
Reconstruction (GPSR) method (Figueiredo et al. 2007),
we minimized the objective function J(m) to obtain the
sparse estimates. After that, the reflection coefficients with
isolated spikes can be obtained by Eq. (6). The inverted
results contain low frequencies as the low-frequency trend
model data were added into the objective function as a
penalty function. Therefore, the output Young’s modulus,
Poisson’s ratio, and density with blocky boundaries can be
obtained by Eq. (18):
3 Model test
We tested the validity of our proposed inversion method
with well log data. The angle gather data with free noise
(Fig. 2a) were synthesized with Zoeppritz equations in the
time domain by utilizing the real well logs of P-wave
velocity, S-wave velocity, and density and a 35 Hz Ricker
wavelet for incident angles ranging from 0 to 35 .
Figure 2b–d displays the log curves of Young’s modulus,
Poisson’s ratio, and density. The blue curves shown in
Fig. 2b–d are the real models and the red curves are the
inversion results. From Fig. 2b–d, we can clearly see that
the Young’s modulus, Poisson’s ratio, and density can be
inverted reasonably with free noise. The error of the
Young’s modulus, 1010N/m2
Fig. 6 The partial stack seismic profile and initial model of Young’s
modulus, Poisson’s ratio, and density. a Partial stack seismic data
with a small incident angle range, the mean angle is 8 ; b Partial stack
seismic data with a medium incident angle range, the mean angle is
inverted density is a little bigger than that of the other two
parameters. In order to verify the stability of the inversion
method, we added random Gaussian noise to the synthetic
gather data with different signal-to-noise ratios (SNRs).
The gather traces are displayed in Figs. 3a, 4a and 5a, and
the SNRs are 4:1, 2:1, and 1:1, respectively. The inversion
results in different gather traces are shown in Figs. 3b–d,
4b–d and 5b–d, respectively. It is very clear that the
inversion results estimated by the proposed inversion
method match well with the real models as the
low-frequency model enables the inversion results to approximate
the real models. Although the inversion results are
16 ; c Partial stack seismic data with a large incident angle range, the
mean angle is 24 ; d–f The profiles of initial model of Young’s
modulus, Poisson’s ratio, and density, respectively
influenced by the noise, especially the density, the Young’s
modulus and Poisson’s ratio can match well with the real
models to some degree so that the inversion results are
helpful for us to evaluate the brittleness of the layer.
4 Real data example
The inversion method was applied to real partial angle
stack seismic data, and the sampling interval of the seismic
data is 2 ms. Figure 6 displays the used three partial angle
stack seismic sections, and the mean angles of the seismic
-0.03 0 0.03
Reflection of Young’s modulus
Reflection of Poisson's ratio
Fig. 7 The inverted reflection properties of Young’s modulus,
Poisson’s ratio, and density and the properties generated by the
reflections using Eq. (18). a Reflection of Young’s modulus,
data in Fig. 6a–c are 8 , 16 , and 24 , respectively. In
Fig. 6a–c, the green ellipse circles the target reservoir, and
a well is drilled through the target at CDP 156. Figure 6d–f
displays the initial models of Young’s modulus, Poisson’s
ratio, and density, respectively. The initial models are
established by spatial interpolation and extrapolation and
low-pass filtering. The inverted isolated reflectivity spikes
of rE, rr, and rq are shown in Fig. 7a–c, respectively. The
structure of the reflectivity estimates is similar to the partial
angle stack seismic profile and appears to have a better
resolving power for the layer boundaries. The blocky
Young’s modulus, Poisson’s ratio, and density displayed in
b Reflection of Poisson’s ratio, c Reflection of density, d Young’s
modulus, e Poisson’s ratio, f Density
Fig. 7d–f are obtained from the estimated sparse
reflectivity by using Eq. (18). The blocky results from model
constrained BPI focus on the layer boundaries well, which
is useful for us to interpret the inversion results. To some
extent, the lateral continuity is improved because the
lowfrequency trend model is continuous laterally. The
calculated logs of Young’s modulus, Poisson’s ratio and density
are inserted into the sections. Figure 8a–c plot the inverted
Young’s modulus, Poisson’s ratio, and density (red lines) at
the well location, aligning with the well logs (dark lines).
From Fig. 8, we can draw a conclusion that the inverted
results of Young’s modulus and Poisson’s ratio have a
Fig. 8 The comparison between inversion results and well logs. a Young’s modulus, b Poisson’s ratio, c density
good fit with the logs, while the density inversion does not
match as well as the other two elastic parameters as the
maximum angle is not large enough for us to invert the
density information. From the inverted results shown in
Fig. 7d–f, we can clearly see that the Young’s modulus
exhibits high anomalous values and Poisson’s ratio shows
low anomalous values in the target circled with dark
ellipses, which means that the brittleness of the circled
target is high. The drilling shows the high brittleness at
2.3 s. Therefore, the brittleness evaluated by the inverted
Young’s modulus and Poisson’s ratio using the proposed
inversion method is consistent with the drilling.
In this paper, we presented a novel stable inversion method
to estimate the Young’s modulus and Poisson’s ratio for
brittleness evaluation from pre-stack seismic data with the
YPD approximation. We introduced the low-frequency trend
model into the basis pursuit inversion implementation.
Therefore, we can call this method model constrained BPI.
We derived the objective function of model constrained BPI
by the Bayesian theorem. In the improved method, the
lowfrequency trend model as prior knowledge stabilized the
inversion and improved the lateral continuity because the
low-frequency trend model is continuous in the space axis.
The l1 norm of the model parameters and the GPSR
algorithm kept the sparsity of inversion results so that we can
obtain the isolated reflectivities and the elastic parameters
with discontinuous jumps in the time axis. The model test
results showed that we can obtain the high-precision
Young’s modulus, Poisson’s ratio, and density. The real data
application is performed to confirm the validity of the
proposed method, and it showed that the high anomalous value
of Young’s modulus and low anomalous value of Poisson’s
ratio which mean high brittleness matched well with the
brittleness interpretation of drilling.
Acknowledgments We would like to acknowledge the sponsorship
of the National ‘‘973 Program’’ of China (2013CB228604) and the
National Grand Project for Science and Technology
(2011ZX05030004-002), China Postdoctoral Science Foundation (2014M550379),
Natural Science Foundation of Shandong (2014BSE28009), Science
Foundation for Post-doctoral Scientists of Shandong (201401018),
and Science Foundation for Post-doctoral Scientists of Qingdao and
Science Foundation from SINOPEC Key Laboratory of Geophysics
(33550006-14-FW2099-0038). We also acknowledge the support of
the Australian and Western Australian governments and the North
West Shelf Joint Venture partners, as well as the Western Australian
Energy Research Alliance (WA:ERA).
Open Access This article is distributed under the terms of the
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