Markov random field modeling for mapping geofluid distributions from seismic velocity structures
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We applied the Markov random field model, which is a kind of a Bayesian probabilistic method, to the spatial inversion of the porosity and pore shape in rocks from an observed seismic structure. Gaussian Markov chains were used to incorporate the spatial continuity of the porosity and the aspect ratio of the pore shape. Synthetic inversion tests were able to show the effectiveness and validity of the proposed model by appropriately reducing the statistical noise from the observations. The proposed model was also applied to natural data sets of the seismic velocity structures in the mantle wedge beneath northeastern Japan, under the assumptions that the fluid was melted and the temperature and petrologic structures were uniformly distributed. The result shows a significant difference between the volcanic front and the forearc regions, at a depth of 40 km. Although the parameters and material properties will need to be determined more precisely, the Markov random field model presented here can serve as a basic inversion framework for mapping geofluids.
KeywordsBayesian estimation Markov random field Geofluid Mantle wedge Data-driven science
In order to understand the various dynamic processes in the earth, it is important to understand the distribution of geofluids. Recent developments in the technology for geophysical observations, such as seismic tomography and geomagnetic methods, provide detailed images of the earth’s interior (Nakajima et al. 2001; Ogawa et al. 2001; Takahashi et al. 2009). Additionally, there has been increased understanding of the constitutive relationships between the physical variables, such as lithology, the porosity of rocks, and the observational data, such as seismic velocity and resistivity (Glover et al. 2000; Takei 2002). Against this background, a pioneering study by Nakajima et al. (2005) used the constitutive function proposed by Takei (2002) to evaluate the effective aspect ratio and the volume fraction (porosity) of the fluid-filled pores in the observed low-velocity anomalies in the mantle wedge beneath northeastern Japan.
Recently, several studies have attempted to make a quantitative and detailed map of the spatial distribution of geofluids (Hoshide and Nakamura 2013; Iwamori et al. 2011). However, this remains difficult, because there is still much uncertainty in the available data and assumptions. In order to overcome the difficulties arising from this noise and uncertainty, a statistical and probabilistic analysis of the geophysical data is essential.
The main purpose of the present study is to construct an inversion framework that can be used to estimate precisely the distributions of various physical properties from observed spatial data sets; we do this by developing the Markov random field (MRF) model, which is a kind of a Bayesian statistical model. The Bayesian approach enables us to incorporate a forward model and prior information into a data-driven inversion analysis.
The MRF model uses Markov chains to describe the properties of an image, and it is often used in the field of information science for image restoration and pattern recognition (Geman and Geman 1984; Li 2009; Tanaka 2002). In the MRF model, the spatial variations in physical properties are assumed to be generally smaller than the noise in the data and the analytical uncertainty. If this assumption is valid, then by using the Bayesian approach, the MRF model appropriately filters out the high-frequency noise, and we can obtain the accurate spatial distributions of the physical properties. Recent papers in the natural sciences have applied the MRF model to inversion problems for various observational data sets (Kuwatani et al. 2012; Watanabe et al. 2009).
Here, we develop a Gaussian MRF model to reconstruct the spatial distribution of geofluids from the seismic velocity structure. On the basis of the Bayesian framework, the process for generating the velocity structure and the spatial continuity of the distribution of geofluids are introduced into the stochastic inversion analysis in accordance with the law of causality. In order to deal with the nonlinear relationship between the target physical variables and the observed data, a Markov chain Monte Carlo (MCMC) algorithm was incorporated into the MRF model (Metropolis et al. 1953). An application of the method to synthetic data showed that the spatial distributions of porosity and the aspect ratio could be reliably estimated, and this supports the effectiveness of the MRF model. We also applied the model to the velocity structure of the mantle wedge beneath northeastern Japan, which was obtained by 3-D tomography (Matsubara et al. 2008), under the simple assumption that variables other than the porosity and aspect ratio were known and spatially uniform. Finally, we will discuss the validity of our assumptions, the effectiveness and applicability of the MRF model, and the geophysical implications. Although many parameters and material properties remain to be determined more precisely, the proposed framework will be very effective for determining the distribution of geofluids.
where N is the total number of grid cells measured, and V P , V S , ϕ, and α indicate the respective set of variables VP, VS, ϕ, and α for the observed grid cells i=1,…N.
where is the summation of all pairs of neighboring grid cells, is the variance of the change in ϕ between two adjacent grid cells, and is a normalization coefficient. The prior probability for α can be also written as Equation 6. The denominator of the right-hand side of the equation p(V P ,V S ) is invariant for changes in ϕ and α, so this is negligible for our analysis.
where θ indicates the set of parameters , and C is a constant that is independent from ϕ, α, and θ. Due to the monotonicity of the logarithm function, the minimization of the evaluation function E(ϕ,α;θ) is equivalent to the maximization of the posterior probability p(ϕ,α|V P ,V S ).
In the evaluation function, the first and second terms indicate the reproducibility of the observation, respectively, whereas the third and forth terms indicate the spatial continuity of the porosity and the aspect ratio, respectively. Minimization of E(ϕ,α;θ) satisfies the requirements of both the reproducibility of the observed data and the spatial continuity of the physical variables.
where we assume that the prior probability p(θ) is uniformly distributed, and C is a constant that is independent from θ. In this study, the free energy F(θ) was minimized by the steepest descent method, using the MCMC method [see Additional file 1]. A maximum a posteriori (MAP) solution set of ϕ and α can also be obtained from numerous candidates which are generated by the MCMC calculations.
Synthetic inversion test
In addition to the synthetic distributions of ϕ and α, which have Gaussian inhomogeneities, we also checked the effectiveness of the proposed method using a power-law inhomogeneity, since this is considered to be the typical distribution of actual inhomogeneities in the earth (Sato et al. 2012). The details are in Additional file 1. Although there is difference between the prior probabilities, which were assumed to be Gaussian, and the actual power-law inhomogeneities, the estimated variances of continuity, and , were approximately the same as the true values obtained by estimating them from the hyperparameters. We can also estimate the spatial distributions of ϕ and α, which supports the effectiveness of the proposed method for actual non-Gaussian distributions in natural systems. Although further investigation is needed, due to the versatility of the Gaussian distributions, the proposed method is approximately valid for natural continuous distributions.
Application to the mantle wedge beneath northeastern Japan
The ϕ values estimated by the MRF model range from 0 to 0.01. By comparison with the original mappings of VP and VS (Figure 6), the regions of large ϕ values correspond to small VP and VS, reflecting that both VP and VS decrease monotonically with increasing ϕ. In particular, the ϕ values are relatively large (>0.002) beneath the Quaternary volcanoes and Hokkaido. On the forearc side, the value of ϕ is generally low, ranging from 0 to 0.002. However, several anomalous, large values of ϕ are found in the east, off Fukushima and between the Honshu and Hokkaido islands.
The α values are 0.01 to 0.03 on the back-arc side and 0.001 to 0.01 on the forearc side. The regions of small α are roughly consistent with the regions that have a high VP/VS ratio. Beneath the Quaternary volcanoes, the α value is generally high. In particular, near Chubu and west Hokkaido, the value may be as large as ∼ 0.1. On the forearc side, the α value is generally low, and small values of α, about 0.001, are detected in the Hidaka region and westward.
In terms of reducing high-frequency noise, the role and efficiency of the MRF model appear to be similar to those of a smoothing filter applied to the observational data. In a smoothing method such as a moving average, too large of a filter will cause excessive smoothing and blur the details of the image. In actual analyses for natural systems, however, the true distribution and magnitude of the noise are unknown, so we cannot make a prior determination of the appropriate filter size. Thus, it is difficult to use an ordinary filtering (averaging) method to determine the precise physical values from noisy observations. However, even without a priori information about the magnitude of the noise, the MRF model can determine the variance of the noise from the data. This is the most significant advantage of the MRF model, that it enables us to analyze the data objectively and quantitatively.
When applied to the actual data, the deterministic method, which uses the data and the inverse function to obtain an analytic solution, results in a very jagged and incomplete estimate. When observational data is converted to physical parameters, the results are sometimes beyond the scope of the model, and thus no solution can be derived. This is caused by a combination of observational noise and uncertainty, which highlights the importance of using a statistical or probabilistic analysis. The proposed method was able to image the continuous distribution of fluid because it did not take a deterministic approach but a probabilistic approach, and it was thus able to avoid perturbations due to noise.
This study used the seismic velocities estimated by tomography, V P and V S , and realistic observations were simulated by adding uncorrelated Gaussian noise with zero mean to each of the grid cells. At present, the proposed model cannot deal with the cross-correlated errors of seismic velocities that are derived from a tomographic inversion. For a more accurate estimation of the distribution of geofluids, a ray-path matrix, which relates the travel time to the inverse of the seismic velocity, can be incorporated into the probabilities of the generated observations, Equation 3. In our current research, due to the flexibility of the MRF model, we were able to successfully apply it to a seismic tomographic inversion by using the ray path matrix to generate the probabilities and continuity of the rate to obtain the prior probability. The inversion of physical properties directly from the observed travel time is an important issue that needs to be addressed in future work.
The calculated spatial variations in the porosity ϕ and the aspect ratio α of the geofluids show a significant difference between the forearc regions and the volcanic front, at a depth of 40 km. On the forearc side, the values are generally low, with ϕ ranging from 0 to 0.2 vol.%, and α ranging from 0.001 to 0.01; this indicates that the fluid is not intergranular but is between thin cracks. There is little melting in this region, and even if melt exists, it is not textually equilibrated to the surrounding rocks. The small amount of melt is consistent with other geophysical observations which indicate weak inhomogeneities and weak attenuation on the forearc side (Takahashi et al. 2009; Umino and Hasegawa 1984; Yoshimoto et al. 2006).
Beneath the Quaternary volcanoes, on the other hand, the large amount of geofluid (> 0.2 vol.%) indicates partial melting of the rock. The α values are generally high (> 0.02), so the melt is considered to fill oblate spheroid cracks or dikes. In particular, in the region of Chubu and west Hokkaido, the value is up to ∼ 0.1, which indicates that the pore geometry is near equidimensional, and the fluid is distributed in the spaces between the grains (Waff and Bulau 1979; Takei 2002). The estimated amount and shape of geofluid are considered to be closely related to the magmatic process of the Quaternary volcanoes (Tamura et al. 2002; Nakajima et al. 2005).
At the same depth beneath the volcanic front, Nakajima et al. (2005) estimated the porosity at 1 to 2 vol.% with an aspect ratio of 0.02 to 0.04. Although the values of the aspect ratios are consistent with each other, the porosities differ. It is not possible to make a simple comparison between this study and that of (Nakajima et al. 2005), because different seismic velocity structures were used in their analyses (Nakajima et al. 2005; Matsubara et al. 2008). There are also several sources of uncertainty in natural systems, which affect the values of the parameters used in the analysis; this is discussed below. Further studies are necessary to evaluate the validity of the obtained distributions of geofluids.
In order to match the number of unknown parameters to the number of observable parameters, we have assumed that the parameters other than porosity and aspect ratio of the geofluids are known and uniformly distributed. In the actual mantle wedge zone, however, the spatial variations of temperature and composition overlap with geofluid distributions. For more realistic imaging of the distribution of geofluids, it is necessary to introduce a priori information and to introduce other models, such as those for thermal or petrological structure, into the analysis (Iwamori et al. 2011). However, many of these models are still poorly constrained, and thus, in order to obtain reliable distributions of geofluids, some of the parameters should be probabilistic variables. The MRF model may have the potential to overcome these difficulties due to its Bayesian approach and flexible formalism. By allowing us to add terms to the evaluation function, it allows us to incorporate other geophysical observational data sets and various types of prior information as probabilistic constraints. Although additional theoretical improvements are needed for individual problems, the MRF model presented here can serve as a basic inversion framework for the mapping of geofluids.
We would like to thank the editor and two anonymous reviewers for their valuable comments. Discussions with H. Iwamori and A. Okamoto were thoughtful and constructive. We used the seismic velocity structure of (Matsubara et al. 2008), provided by the National Research Institute for Earth Science and Disaster Prevention (NIED). This study was financially supported by the research project ‘Evaluation and disaster prevention research for the coming Tokai, Tonankai, and Nankai earthquakes’ from the MEXT of Japan. This work was also supported by JSPS KAKENHI Grant Numbers 25120005, 25120009, and 25280090.
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