Application of new Monte Carlo method for inversion of prestack seismic data. Yang Xue Advisor: Dr. Mrinal K. Sen

Transcription

1 Application of new Monte Carlo method for inversion of prestack seismic data Yang Xue Advisor: Dr. Mrinal K. Sen

2 Overview Motivation Introduction Bayes theorem Stochastic inference methods Methodology Greedy annealed importance sampling (GAIS) Application of GAIS on seismic inversion Example 1: HRS demo data Example 2: China data set Conclusions

3 Motivation Goal: estimate a log of rock properties as a function of two-way vertical travel time or depth Problem: limited wells Purpose of seismic inversion: derive a pseudolog to fill in the gaps between wells Geophysical inversion has non unique solutions Derive models with uncertainties

4 Bayes theorem: P( d) E x = P ( d x) P( x) P( d) x T 1 ( x) = ( d g( x) ) C ( d g( )) P Introduction Posterior pdf D Likelihood ( d x) exp( E( x) ) Prior pdf Draw samples from Prior distribution Stochastic inference methods Samples in Posterior distribution x ( x ) = x P d 4

5 Introduction Stochastic inference methods Independent Monte Carlo Dependent Markov Chain Monte Carlo Importance Sampling Greedy Importance Sampling (GIS) Metropolis Algorithm Very Fast Simulated Annealing (VFSA) (Southey and Shuurmans 2002) 5

6 Introduction Metropolis-Hastings Start at x 0 with Error E(x 0 ) x 1 = x 0 + x; E(x 1 ) Accept X 1 with Prob = exp(- E/ T) Weigh each model with probability and then evaluate the integrals to estimate the marginal PPD, posterior mean, covariance and correlation matrices

7 Introduction Very Fast Simulated Annealing Start at x 0 with Error E(x 0 ) x 1 = x 0 + x; E(x 1 ) Reduce temperature temperature dependent Accept X 1 with Prob = exp(- E/ T) Weigh each model with probability and then evaluate the integrals to estimate the marginal PPD, posterior mean, covariance and correlation matrices

8 Importance Sampling Draw samples from prior distribution Q, assign weights w i = P / Q, estimate expectation: Problem: Introduction Q misses high probability regions of P i n EP x) f ( x) = f ( xi ) w( xi ) / n ( = i= 1

9 Stochastic inference methods Methods Algorithm Acceptance Rule Pros and cons Metropolis Algorithm Updating sample using Markov Chain Change of misfit ΔE Accurate estimation, Computationally slow Very Fast Simulated Annealing Updating sample using Markov Chain, prior distribution temperature dependent Change of misfit ΔE Fast, biased estimation of uncertainties due to continuous change of proposal distribution with iterations Importance Sampling Randomly drawing samples from prior distribution All accepted, but weighted w = P Q ( x) ( x) Fails, if prior distribution Q miss high probability region of posterior distribution P 9

10 Methodology Greedy Importance Sampling - GIS VFSA(T1) VFSA(Ti) ( x 1,.. x i., x n ) Q B i = ( xi 1,.. xi, j., xi,, m local maximum of or m-1 steps E ) f ( x) P( x) VFSA(Tn) w ( x ) = i j x i X α ij P( x ) j αij Q ( x ) i I = 1 i = n m = P( x) f ( x) 1 f ( xi, k ) wi ( xk ) n i= 1 k= 1 (Southey and Shuurmans 2002) ij Methodology: Greedy search for important region of X Each independent block contains one or two important points from X Advantages: Minimized variance while maintaining unbiasedness 10

11 Methodology Greedy Annealed Importance Sampling - GAIS Generate fractal initial model (Srivastava and Sen, 2010) Employ multiple VFSA with different starting temperatures and a small number of iterations Starting from the best fit model of each VFSA, greedy search important region ( ± ΔZp, ±ΔZs),( ΔZp, ΔZs) Summation of all weighted samples 11

12 Zs:[m/s*g/cc] Zs:[m/s*g/cc] Zs:[m/s*g/cc] Methodology Visualization of Greedy Search Well log Samples Accepted samples End of searching Zp:[m/s*g/cc] Zp:[m/s*g/cc] Zp:[m/s*g/cc] Zp:[m/s*g/cc] Marginal probability map of one layer at the well location with multiple runs of VFSA (left box) and zoom in (right). GIS searches within the high probability area step by step. At each step, we compare the importance of the square s ends, whose middle point is our starting point and choose the most important end as a new starting point. Calculation of gradient matrix can also provide the direction of search Zs:[m/s*g/cc] Well log Samples Accepted samples

13 Application of GAIS Example 1: HRS demo data (prestack seismic) Well logs plot from Hampson Russell Strata Guide

14 Application of GAIS Example 1: HRS demo data (prestack seismic) (Angle gather from Hampson Russell Strata Guide) 14

15 Application of GAIS Example 1: HRS demo data (prestack seismic) Quality control at the well location TWT:[ms] Well580 GAIS VFSA600 Initial model TWT:[ms] GAIS 560 VFSA Zp:[m/s*g/cc] Zs:[m/s*g/cc] σ(zp)/zp σ(zs)/zs 15

16 Application of GAIS Example 1: HRS demo data (prestack seismic) 550 inverted Zp GAIS 550 VFSA inverted Zs GAIS 550 VFSA TWT:[ms] CDP Number CDP Number CDP Number CDP Number

17 Application of GAIS Example 1: HRS demo data (prestack seismic) TWT:[ms] inverted ZpZs ratio GAIS VFSA GAIS shows superior Performance than VFSA alone CDP Number CDP Number 17

18 Application of GAIS Example 2: Ordos Basin, China (prestack seismic) Study area Zoomed study area

19 Application of GAIS Example 2: Ordos Basin, China (prestack seismic)

20 Application of GAIS Example 2: Ordos Basin, China (prestack seismic)

21 Application of GAIS Example 2: Ordos Basin, China (prestack seismic) 1200 Quality control at the well location GAIS VFSA 1300 TWT:[ms] GAIS Well 1400 Initial model VFSA 1450 TWT:[ms] TWT:[ms] TWT:[ms] Zp:[m/s*g/cc] x Zs:[m/s*g/cc] σ(zp)/zp σ(zs)/zs

22 Application of GAIS Example 2: Ordos Basin, China (prestack seismic) MAP of Zp MAP of Zs 22

23 Application of GAIS Example 2: Ordos Basin, China (prestack seismic) 1400 MAP of inverted ZpZs ratio TWT: [ms] Gas sand

24 Conclusion GAIS attempts to explore important regions starting with models that are close to the important regions already located by VFSA and estimates the expectation value very accurately. The example of pre-stack inversion demonstrates superior performance of GAIS compared to VFSA alone

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