the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 21 Oct 2024
Bayesian inference based on algorithms: MH, HMC, Mala and Lip-Mala for Prestack Seismic Inversion
Abstract. Seismic data inversion for estimating elastic properties is a crucial technique for characterizing reservoir properties post-drilling. The choice of inversion method significantly impacts results. Markov chain Monte Carlo (MCMC) algorithms enable Bayesian inference, incorporating seismic data uncertainty and expert information via prior distribution. This study compares the performance of four inversion methods—Metropolis-Hastings (MH), Hamiltonian Monte Carlo (HMC), and two Lagrangian Diffusion variants (MALA and Lip-MALA)—in prestack seismic inversion, using synthetic and real-world data from an eastern Venezuelan hydrocarbon reservoir. All four methods show acceptable performance but differ in specific strengths and weaknesses. Gradient-based methods (HMC, MALA, and Lip-MALA) outperform MH in velocity estimation. Density estimation is more challenging; MH and HMC yield unsatisfactory results, whereas MALA and Lip-MALA show promise. Execution time varies significantly: MH and MALA are substantially faster than HMC and Lip-MALA. Therefore, both accuracy and computational efficiency should be considered when choosing a method. The study evaluates the mean values and standard deviations of the subsequent parameters: P-wave (Vp), S-wave velocity (VS) and density (ρ). The quality of the MCMC sample is checked using correlations, objective function plots, seismic trace and Root Mean Square Error (RMSE) estimation. Acceptance rate and execution time assessments reveal HMC has the lowest acceptance rate, and MH the shortest execution time. Future research aims to extract additional elastic parameters and reservoir properties, enhancing subsurface understanding. Integrating well log conditioning into the model could improve vertical resolution near wells and align the model with well data at drilling locations.
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RC1: 'Comment on egusphere-2024-2694', Anonymous Referee #1, 04 Nov 2024
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This is mostly a survey paper on MCMC methods that does not contain much novelty, especially when considering the target (4) is a generalised Normal model. The style is poor at times (too many times to point all difficulties) The authors are mistakenly using the symmetric version of MH, which invalidates their comparisons.
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(p2) the HMC was not popularised by Betancourt (2018), there were already books on the topic by that year
(p4) one does not "estimate several samples in the parameter space"
(p4) it is unclear why the forward function moves from F(m) to g(m)
(p5) the first sentence of 3.1 misses a principal verb
(p5) one does not have to "assume that we have a chain that converges to the source distribution"
(p5) the "transition rule of the convergent chain to the source probability145 density" is not defined and given (5)
it should further be symmetric
(p5) the wording "If [the new] m̃ is better than the [old] m" is unclear and unnecessary
(p7) it is not only "in this work, we use the leapfrog method for numerical integration" since this is the default sc
heme as e.g. in Stan. Furthermore, the leapfrog steps are not provided
(p7) as described, the HMC algorithm changes the momentum m at each iteration (in Step 1), which is not the case in g
eneral
(p8) the Langevin algorithm with the Metropolis correction is incorrect since the acceptance ratio does not involve t
he assymmetric proposals. Langevin diffusion is spelled Langivin diffussion
(p9) ULA was created to avoid rejection and has been deeply investigated in the past years, which makes one wonder at
the appeal of MALA-MCMC (missing again the proposals in the acceptance ratio (20)
(p11) the initial stage is not "called the burn stage" but the burn-in or warm-up stage
(p17) comparing raw acceptance rates in Table 3 is not appropriate since the different algorithms have different optimal acceptance ratesCitation: https://doi.org/10.5194/egusphere-2024-2694-RC1 -
RC2: 'Comment on egusphere-2024-2694', Anonymous Referee #2, 15 Nov 2024
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Review comments on ‘Bayesian inference based on algorithms: MH, HMC, Mala and Lip-Mala for Prestack Seismic Inversion’
Amplitude versus offset (AVO) inversion is an efficient method for fluid identification in geophysical prospecting. Under the Bayesian framework, using Markov chain Monte Carlo (MCMC) methods, multiple parameters (P- and S-wave velocities, density, porosity, etc.) can be estimated simultaneously, and the method has an extensive applications. In this paper, the authors compared four MCMC algorithms (Metropolis-Hastings, Hamiltonian Monte Carlo, and two Langevin dynamics based algorithms). Also the paper compared the accuracy, efficiency, and uncertainty of the four algorithms from both synthetic seismic data and real data tests. However, there are many issues in this article, and need necessary major corrections.
Please carefully review and revise the article, to ensure consistency in the expression of the same parameter all over the paper, and make sentences more concise, so that readers can better understand the meaning of the sentences.
Major comments:
- Line 15: the full name of ‘MALA’ and ‘Lip-MALA’ should be provided as they firstly appear in the article.
- Line 24: ‘…extract additional elastic parameters and reservoir properties…’ What additional elastic parameters and reservoir properties? Please clarify this.
- Line 25: ‘Integrating well log conditioning into the model could improve vertical resolution near wells and align the model with well data at drilling locations’ Integrating well log data, not only can improve the vertical resolution near wells, but also can improve the vertical resolution far away from wells. Cited here: e.g., Shi et al., 2024.
- Line 44: ‘…the parameters of the medium 𝑚 with the observed data…’ Is parameters m or medium m? And please clarify whether m is a scalar or a vector? There are many similar issues in other parts of the manuscript. Please carefully review the manuscript and unify the expression format.
- Line 54: ‘The approach of this work is based on computer statistics, which allows to include uncertainty in seismic data…’ what is its meaning?
- Line 66: ‘Bosch et al., (2007) solve an inverse problem…’should be modified to ‘Bosch et al., (2007) solved an inverse problem…’ There are many similar issues in the article, please revise.
- Line 83: Please provide additional literature related to MALA and Lip-MALA, and briefly describe them.
- Line 104: Why do you emphasize the application in machine learning? Bayesian seismic inversion has always been a major topic.
- Please standardize the formulas in the article. E.g., g:m → dobs indicates a forward model?
- Line 184: ‘particle’ is it a model parameter? And what does ‘artificial time variable’ mean?
- Equation (14): Does ‘ϵ’ have the same meaning as ‘ϵ’ in Formula 1? And please explain n.
- Please confirm if formula 18 is written correctly.
- Line 263: What do ‘β0’ and ‘𝐿c’ represent?
- In synthetic data test, the synthetic seismic trace is noise free or noisy?
- Please explain any figure to make it easier for readers to understand, e.g., what do Figures 1a-d represent respectively? And add legends.
- The table name should be above the table.
- For Figure 1, how does authors distinguish between the sampling phase and the burn-in phase? During the sampling phase, the decrease of the objective function does not tend to a steady state. Will this sampling influence the accuracy of the inversion results? Please explain.
- Is Figure 2 the initial samples obtained from a prior distribution? Are the average values of these samples the same as the initial model?
- In Figure 3, does the gray line represent the synthetic seismic records corresponding to the initial samples (Figure 2)?
- In Figure 4, does the red line represent a random inversion result or the mean of multiple inversion results?
- What does ‘corr’ mean in Table 2? Does it mean correlation coefficient? If it is the correlation coefficient, it can be seen that the correlation coefficients of the density inversion results obtained by the four algorithms are very low, especially based on the MH algorithm. I think the correlation coefficient based on the MH algorithm is relatively very low, this may unreasonable. Moreover, based on the MH algorithm, the correlation coefficient of the obtained three parameter inversion results may not be so low.
- Please show the seismic gathers before and after well seismic tie?
- Line 82: How are ‘uncertainty bounds’ determined?
- Line 82: The realizations of velocities and density better illustrates their characteristics and variability.
- Can authors provide a two-dimensional data test?
Some minor corrections:
- Line 36: this -> those
- Line 55: ‘…model parameters and, through…’ -> model parameters, and through
- Line 55: solution of the inverse problem -> solutions of the inverse problem
- Line 56: combined -> combing
- Line 62: ‘A more general algorithm is the Hamiltonian Monte Carlo (HMC) is another approach’ -> A more general algorithm is the Hamiltonian Monte Carlo (HMC) which is another approach
- Line 68: ‘In Wu et al., (2019) propose…’ -> Wu et al. (2019) proposed…
- Line 70: ‘(Gaussian MH sampling with data driving (GMHDD) approach)’ -> [Gaussian MH sampling with data driving (GMHDD) approach]
- Line 72: ‘use’ compute’ -> using computing
- Line 89: ‘understand about’ -> understand
- Line 89: ‘in our subsurface model parameter estimate’ -> in subsurface model parameter estimate
- Line 117: ‘and ρ(m) prior probability density’ -> and ρ(m) is prior probability density
- Lines 118-120: ‘In other words, the prior probability density tells us what we think we know about the subsurface before we look at the data. The likelihood function tells us how much the data changes our mind about the subsurface. And the posterior probability density tells us what we think we know about the subsurface after we have looked at the data’. It is suggested to delete.
- Line 140: ‘a source density’ -> a prior model
References:
Y Shi, B Yu, H Zhou et al., FMG_INV, a Fast Multi-Gaussian Inversion Method Integrating Well-Log and Seismic Data. IEEE Transactions on Geoscience and Remote Sensing 2024;62:1-12.https://doi.org/doi: 10.1109/TGRS.2024.3351207.
Citation: https://doi.org/10.5194/egusphere-2024-2694-RC2
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