Seismic fluid identification using a nonlinear elastic impedance inversion method based on a fast Markov chain Monte Carlo method
Seismic fluid identification using a nonlinear elastic impedance inversion method based on a fast Markov chain Monte Carlo method
Guang-Zhi Zhang 0
Xin-Peng Pan 0
Zhen-Zhen Li 0
Chang-Lu Sun 0
Xing-Yao Yin 0
0 School of Geosciences, China University of Petroleum (East China) , Qingdao 266580, Shandong , China
Elastic impedance inversion with high efficiency and high stability has become one of the main directions of seismic pre-stack inversion. The nonlinear elastic impedance inversion method based on a fast Markov chain Monte Carlo (MCMC) method is proposed in this paper, combining conventional MCMC method based on global optimization with a preconditioned conjugate gradient (PCG) algorithm based on local optimization, so this method does not depend strongly on the initial model. It converges to the global optimum quickly and efficiently on the condition that efficiency and stability of inversion are both taken into consideration at the same time. The test data verify the feasibility and robustness of the method, and based on this method, we extract the effective pore-fluid bulk modulus, which is applied to reservoir fluid identification and detection, and consequently, a better result has been achieved.
Elastic impedance; Nonlinear inversion; Fast; Markov chain Monte Carlo method; Preconditioned conjugate gradient algorithm; Effective pore-fluid bulk modulus
Compared to amplitude versus offset (AVO) inversion
based on common mid-point (CMP) gathers, elastic
impedance inversion based on partial angle-stack gathers
& Xin-Peng Pan
has the advantages of high computational efficiency, high
stability, high noise immunity, and low dependence on the
quality of seismic data. This has been widely used in
reservoir fluid identification and detection, and has become
one of the main directions of pre-stack inversion (Downton
2005; Yin et al. 2014).
Connolly (1999) proposed the concept of elastic
impedance on the basis of acoustic impedance for the first time.
Cambois (2000) considered that the high noise immunity of
elastic impedance could avoid ‘‘leakage’’ between the
various AVO attributes generated by noise. This is more
advantageous in the extraction of pre-stack parameters.
Whitcombe (2002) first applied a new normalized form of
elastic impedance to improve the stability of parameter
extraction. Additionally, Martins (2006) and Cui et al.
(2010) introduced P- and P-SV wave elastic impedance in
weakly anisotropic media, respectively. In recent years,
fluid indicators estimated from seismic data play important
roles in reservoir characterization and prospect
identification, so many methods, such as elastic impedance
inversion, have been introduced to extract a variety of fluid
factors directly to avoid the cumulative error generated by
indirect combination of parameters in the process of
reservoir fluid identification (Ma 2003; Peng et al. 2008;
Zong et al. 2011, 2012; Yin et al. 2013b; Zhang et al. 2013;
Chen et al. 2014a, b; Li et al. 2014). Goodway et al. (1997)
proposed kq and lq as fluid indicators. Russell et al. (2011)
used f as a fluid indicator based on the poroelastic theory.
However, the sensitivity of these fluid indicators is
dependent on the mixed effect of pore fluid and rock
matrix. To improve the sensitivity of reservoir fluid
identification, we use the effective pore-fluid bulk modulus as
the fluid indicator, which is related only to pore fluid and
may diminish the rock-matrix effect (Han and Batzle 2004;
Yin and Zhang 2014).
The present elastic impedance inversion techniques are
mostly based on linear or quasi-linear inversion methods,
not only losing the precision of inversion in the process of
linearization but also strongly relying on the accuracy of
initial model, while convergence to the global optimum is
difficult in the usage of these techniques (Su et al. 2014).
However, the nature of most inverse problems is nonlinear
and multi-extremum, so in terms of the complex pre-stack
reservoir elastic parameter inversion, the development of
elastic impedance with high efficiency and high stability
based on the nonlinear inversion method has become more
significant. We propose a nonlinear elastic impedance
inversion method based on a fast Markov chain Monte
Carlo (MCMC) method, and validate the feasibility and
robustness of the method by testing the noise immunity of
well data. Meanwhile, on the basis of two-phase medium
theory for elastic impedance equation, we extract the
effective pore-fluid bulk modulus from seismic data to
apply to reservoir fluid identification and detection (Russell
et al. 2003; Yin et al. 2013a).
2 Fast MCMC method
Hastings (1970) proposed an extended form of the
Metropolis algorithm (Metropolis et al. 1953), the
Metropolis–Hastings algorithm, laying the foundation for
the development of the MCMC method. The MCMC
method was first applied to fully nonlinear inverse
problems by Malinverno (2002). Zhang et al. (2011a, b)
studied post-stack and pre-stack seismic inversion
methods based on the MCMC method. In recent years, the
MCMC method has been applied to sample the posterior
distribution of reservoir parameters for identifying
reservoir lithology and fluid on the basis of a Bayesian
framework (Bachrach et al. 2009; Grana and Rossa 2010;
Rimstad and Omre 2010; Ulvmoen and Omre 2010;
Ulvmoen et al. 2010; Rimstad et al. 2012). However, the
conventional MCMC method has the defects of low
computational efficiency and a low convergence rate
toward the global optimum for multi-extremum inverse
problems, so we propose a faster MCMC method than the
conventional MCMC method in this paper.
2.1 Metropolis–Hastings algorithm MCMC can sample posterior probability distribution converging to inverted parameters in Bayesian inference,
and then make some statistical analysis of random
samples to obtain useful properties of parameter posterior
The construction methods of the transition kernel play
important roles in the MCMC method, including the
Metropolis–Hastings (M–H) algorithm and Gibbs
algorithm. We choose the M–H algorithm, and after a
sufficiently long iteration, stable Markov chains form, which
can be used in making the statistical analysis for random
samples of parameter posterior distribution, and which
satisfy the detailed balance condition.
2.2 Principle of fast MCMC method
Conventional MCMC can be applied efficiently to
nonlinear and single-extremum inverse problems, but it is
difficult to converge to global optimums of
multi-extremum inverse problems. Thus, we propose a fast
MCMC method, which integrates an efficient
optimization algorithm into the MCMC method to improve the
convergence rate to the globally optimal solution, and
greatly raises the computational efficiency of the MCMC
The PCG algorithm is considered an efficient and
stable optimization algorithm (Stefano et al. 2013), so we
combine the MCMC method based on the global optimal
M–H algorithm with the PCG algorithm equipped with
local search ability. Overall, the central idea of the fast
MCMC method proposed in this paper is the integration
of the more efficient PCG algorithm in the process of
random search in whole space based on the global
optimal MCMC method. The method can search the solution
of inverted parameters for fast objective function
convergence, which saves the huge computational cost of
large numbers of iterations compared to the conventional
MCMC method. We call the more efficient algorithm the
‘‘fast MCMC method’’, and its specific process is shown
in Fig. 1.
To test the feasibility of the fast MCMC method, we
designed a simple bimodal probability density function
(PDF) pðxÞ ¼ 0:3 eax2 þ 0:7 eaðx 10Þ2 , and the
estimated parameter is a, whose true value is -0.2. Figure 2
shows two inverted results of 250 iterations from the
conventional MCMC method (blue solid line) and the fast
MCMC method (red solid line), respectively. From Fig. 2,
we find that the fast MCMC method converges to a global
optimal solution quickly, but conventional MCMC
method has not reached convergence condition after 250
Output inversion parameter r
(Maximum likelihood solution)
Fig. 1 Process of the fast MCMC method
Fig. 2 Comparison of the fast MCMC result (red) and the
conventional MCMC result (blue) within 250 iterations
iterations, verifying that the fast MCMC method is more
efficient than conventional MCMC method to converge to
a global optimum.
The value of x
Fig. 3 Statistical characteristics of bimodal PDF sampled by the fast
The maximum likelihood solution of unknown
parameter a is regarded as the estimated value, and then, we
sample the bimodal PDF p(x). As shown in Fig. 3, the
statistical characteristics of random samples sampled by
the fast MCMC method are quite consistent with the
characteristics of bimodal PDF, further verifying the
feasibility and reliability of the fast MCMC method and laying
the foundation for the fast algorithm of the nonlinear elastic
impedance inversion method.
3 Nonlinear elastic impedance inversion method based on the fast MCMC method
The elastic inversion method used mostly now is the
constrained sparse spike linear inversion method developed in
the 1980s, so we propose a nonlinear elastic impedance
inversion method based on the fast MCMC method to
improve the accuracy of elastic impedance inversion and
the reliability of reservoir prediction and fluid
3.1 Elastic impedance equation based on two-phase
To improve the reliability of reservoir pore-fluid
discrimination, we choose the normalized elastic impedance
equation on the basis of two-phase medium theory, which
highlights the effective pore-fluid bulk modulus Kf (Yin
et al. 2013a; Yin and Zhang 2014):
EIðhÞ ¼ EI0
EI0¼Kf 1=2fm/ 3=2q1=2csat csat
where Kf is the effective pore-fluid bulk modulus term,
fm = /l is the dry rock matrix term, q is density, and /
is porosity value; Kf0, fm0, q0, and /0 are the average
effective pore-fluid bulk modulus, dry rock matrix term,
density, and porosity value of well data; EI0 is the
normalization coefficient; h is the average of the incident and
refracted angles; cdry and csat are the square of the P- to
S-wave velocities of the dry rock and saturated rock,
3.2 Nonlinear elastic impedance inversion method
based on the fast MCMC method
Based on Bayes’ theorem, the posterior probability density
of inverted parameter r is expressed as
pðrjdÞ / pðrÞ pðdjrÞ;
where d is the seismic observation data, r is the reflection
coefficient sequence, p(r|d) is the posterior probability of
the reflection coefficient, p(r) is the prior information, and
p(d|r) is a likelihood function.
Assuming seismic background noise obeys an
independent Gaussian distribution with a zero mean and rn2
variance, the seismic observation likelihood function can be
Meanwhile, based on the assumption that the prior
information obeys the Cauchy distribution to stand out
weak reflection of the underground media, the prior
information is expressed as
To obtain the posterior probability distribution p(r|d),
we apply the Metropolis–Hastings algorithm to generate
stable Markov chains converging to inverted parameter
However, the conventional MCMC method results in
huge computational cost and there is likely to be
instability in the inverted results. Therefore, having taken the
computational efficiency and inversion stability into
account, we apply the fast MCMC method (shown as
Fig. 1) to generate stable Markov chains converging to
the posterior probability density distribution of inverted
parameter and invert the elastic impedance using a
nonlinear method based on the fast MCMC method to
In the Metropolis–Hastings algorithm, we choose a
symmetrical distribution satisfying a symmetric random
walk as the proposal distribution of the algorithm, so the
proposal distribution and acceptance rate are expressed
delta; rt þ deltaÞ
p r Þqðr ; rtÞ
aðrt; r Þ ¼ min 1; ð
pðrtÞqðrt; r Þ
In the PCG algorithm, on the Bayesian framework, we
construct the objective function expressed as
where l ¼ 2 rn2 is the sparse constraint factor, and the larger
its value is, the more sparse the reflection coefficient is; the
last term in the equation is the elastic impedance constraint
term, C and g are integral operator matrix and relative
impedance value, respectively, and a is the elastic
impedance constraint factor, and the larger the value is, the
more stable and accurate the inverted results are.
Optimizing the objective function, we get the ultimate equation
3.4 Process of fluid identification using
the nonlinear elastic impedance inversion
method based on the fast MCMC method
GTG þ lQ þ aCTC r ¼
where Q = diag ; 1 , and it adds the
denominator squared term to reduce the effect of
strength contrast and to highlight weak reflection of the
underground media, and it can be termed the modified
Cauchy constraint (Alemie and Sacchi 2011). Eventually,
we apply the fast MCMC method to invert the reflection
coefficient of different angles based on the convergence
judgment of Eq. (7), and then, we obtain the elastic
impedance of different angles by using the path integral
method or the recursion method.
3.3 Direct extraction of the effective pore-fluid bulk
Expecting to apply elastic impedance volume of at least
four angles to extract the effective pore-fluid bulk modulus
directly following Eq. (1), we must turn the equation into a
log-domain equation at first, and then obtain 16 fitted
regression coefficients (a(hi), b(hi), c(hi), d(hi), i = 1, 2,
3, 4) via the least square method or the singular value
decomposition method by using the elastic impedance data
and well log data from nearby wells, and finally the
inverted elastic impedance data volume of four angles is
put into the equation expressed as
Synthesizing the research and analysis above, we
conclude that the whole process of fluid identification using
the nonlinear elastic impedance inversion method based
on the fast MCMC method proposed in this paper is as
1. Pretreatment of pre-stack seismic data and well log
2. Accurate extraction of four angle wavelets from
prestack seismic data;
3. Inversion of elastic impedance of four angles based on
the fast MCMC method by using four abstracted
anglestack seismic gathers and four angles extracted from
seismic data from nearby wells;
4. Extraction of effective pore-fluid bulk modulus from
inverted elastic impedance of four angles on the basis
of two-phase medium theory for the elastic impedance
equation as the representation of reservoir pore-fluid
5. Application of extracted effective pore-fluid bulk
modulus to identify fluid and predict reservoir parameters.
4 Model test
To test the feasibility of fluid identification using elastic
impedance based on the fast MCMC method, we carry
>>>>>>>>>>>8>< llnn EEIIððEEttII;;00hh21ÞÞ ¼¼ aaððhh21ÞÞ llnn KKKKff ððff00ttÞÞ þþ bbððhh21ÞÞ llnn ffmmffmmðð00ttÞÞ þþ ccððhh21ÞÞ llnn qqqqðð00ttÞÞ þþ ddððhh21ÞÞ llnn ///ðð0ttÞÞ
>>>>>:>>>>>>> llnn EEIIððEEttII;;00hh43ÞÞ ¼¼ aaððhh43ÞÞ llnn KKKKff ððff00ttÞÞ þþ bbððhh43ÞÞ llnn ffmmffmmðð00ttÞÞ þþ ccððhh43ÞÞ llnn qqqqðð00ttÞÞ þþ ddððhh43ÞÞ llnn ////ðð00ttÞÞ
So we can obtain the effective pore-fluid bulk modulus
at any sampling point.
out the feasibility and noise immunity test of a well in
one work area in the eastern China. The target reservoir
Fig. 4 Comparison of synthetic angle gathers using well log data and gathers by using inverted results in a noise-free situation. a synthetic angle
gathers by using well data, b synthetic angle gathers by using inverted results in noise-free situation, c residual error between two synthetic angle
2.9 2 4 6 8 2.9 2 4 6 8 2.92350
x 1010 x 108
Kf , N/m2 fm , N/m2
Fig. 5 Inverted parameter results from using the nonlinear elastic
impedance inversion method based on the efficient MCMC in a
noisefree situation (black and red solid lines indicate true values and
inverted results, respectively; black dashed ellipse shows the
oilbearing reservoir from logging interpretation)
is a clastic reservoir at about 2.85 s in the seismic
As shown in Figs. 4a and 6a, we apply well log data to
make a synthetic seismogram in noise and noise-free
situations, and then generate high-accuracy and
high-resolution elastic impedance of four angles based on the fast
MCMC method for the direct extraction of effective
porefluid bulk modulus parameters, testing the feasibility and
noise immunity of the method.
From Fig. 5, we find that inverted effective pore-fluid
bulk modulus, dry rock matrix term, density, and porosity
values in the noise-free situation are consistent with the
true values, and synthetic angle gathers using these
inverted results produce fewer errors compared with real angle
gathers, so the inverted parameters can obviously reflect
the major characteristics of the oil-bearing reservoir,
agreeing with the results of oil-bearing reservoir from
From Fig. 7, we find that when adding random noise in
SNR = 3 to the synthetic seismograms, inverted effective
pore-fluid modulus, dry rock matrix term, density, and
porosity values are also consistent with the true values,
reflecting the major characteristics of the oil-bearing
reservoir well. Therefore, it validates the robustness and
noise immunity of the nonlinear elastic impedance
inversion method based on the fast MCMC in fluid
identification, and the inverted parameters are relatively accurate,
which can reflect the major characteristics of the
oilbearing reservoir well and be applied to identify fluid and
5 Application of real seismic data
The real work area is selected from an exploration area
in eastern China, and the target is a clastic reservoir. As
shown in Fig. 8, the logging interpretation results
indicate that the clastic reservoir at 2.85 s shows an oil layer
with thickness up to 13 m. To verify the application
effect of the method of fluid identification using the
nonlinear elastic impedance inversion based on the fast
MCMC method, we apply this method to real seismic
First of all, we invert elastic impedance based on the fast
MCMC method using angle-stack seismic profiles with
four different angles, and the inverted elastic impedance
profiles of four angles are shown in Fig. 9, in which the
well logs are elastic impedance curves of four different
angles. Then, we extract the effective pore-fluid bulk
modulus based on inverted elastic impedance data volume
of four angles directly, and apply them to fluid
identification and reservoir prediction. With the two effective
porefluid bulk modulus logging curves in Fig. 10, the figure
shows the extracted effective pore-fluid bulk modulus
profile and enlarged partial profile.
From Fig. 9, we find that the inverted elastic impedance
profiles based on the fast MCMC method with four
different angles are highly precise, which agree with the
results from the oil-bearing reservoir logging
Figure 10 shows that the extracted effective pore-fluid
bulk modulus is also consistent with the logging
interpretation results, presenting low values and reflecting the
characteristics of the oil-bearing reservoir, so it further
Fig. 6 Comparison of synthetic angle gathers using well data and gathers by using inverted results in a noise situation (SNR = 3). a synthetic
angle gathers using well data, b synthetic angle gathers by using inverted results in the noise situation, c residual error between two synthetic
2.9 2 4 6 8 2.9 2 4 6 8 2.9 2350
Fig. 7 Inverted parameter results from the nonlinear elastic
impedance inversion method based on the efficient MCMC in the noise
situation (SNR = 3) (black and red solid lines indicate true values
and inverted results respectively; the black dashed ellipse shows the
oil-bearing reservoir from the logging interpretation)
validates the reservoir fluid detection and identification
from seismic data by using the nonlinear elastic impedance
inversion method based on the fast MCMC method, which
is promising in practical applications.
We introduce a more reliable fluid identification method
using nonlinear elastic impedance inversion based on the
fast MCMC in this paper, and the fast MCMC method is
the key of foundation for the nonlinear elastic impedance
inversion algorithm. Based on the Bayesian framework,
we can identify the reservoir hydrocarbon more accurately
owing to the more sensitive reservoir fluid indicator, that
is the effective pore-fluid bulk modulus, and this method
may improve the reliability and stability of fluid
identification. Application research based on actual logging and
Acknowledgments We would like to express our gratitude to the
sponsorship of the National Basic Research Program of China (973
Program, 2013CB228604, 2014CB239201) and the National Oil and
Gas Major Projects of China (2011ZX05014-001-010HZ,
2011ZX05014-001-006-XY570) for their funding of this research.
We also thank the anonymous reviewers for their constructive
Open Access This article is distributed under the terms of the
Creative Commons Attribution 4.0 International License (http://crea
tivecommons.org/licenses/by/4.0/), which permits unrestricted use,
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appropriate credit to the original author(s) and the source, provide a
link to the Creative Commons license, and indicate if changes were
Fig. 10 Extracted effective pore-fluid bulk modulus profile and
enlarged partial profile
seismic data shows that the nonlinear elastic impedance
fluid identification method based on the fast MCMC is a
practical method for reservoir fluid identification.
However, the method has a limitation of obtaining only one
inverted result of the effective pore-fluid bulk modulus,
and may lack the uncertainty evaluation about reservoir
fluid identification. So further in-depth research and
analysis is needed, and we are intending to continue our
research in this field.
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