Resume 1: Introduction To Bayesian Statistic

Introduction to My Future-Thesis

Wuah, for those of you who are expert on this stuff, please guide me and correct this resume. I’m on my way to write a thesis (is it really a thesis?) about *direct* petrophysical properties estimation from *pre-stack* seismic data. What I was thinking is put Seismic Inversion, Bayesian statistic, maybe some smart optimization algorithm to solve the Inversion & Bayesian problem using something like Genetic Algorithm or Markov Chain Monte Carlo.

Attempts on extracting petrophysical properties (Water Saturation, porosity, etc.) from seismic data has become a new paradigm and breakthrough in rock physics, seismic inversion research. I noticed a few papers that good in explaining this problem, such as Eidsvik, et al (2002)[1], which then explained further by Eidsvik in 2004[2]. Fugro-Jason already include commercial version of what I want in their latest software, Jason Geosciences Workbench 2008 [3]. A thesis (or disertation, I forget), about probabilistic seismic inversion based on Rock-Physics Models also has been published by Kyle Spikes[4] at Stanford. Coleu of CGGVeritas also published an insight for Petrophysical Seismic Inversion[5].

So this is a next step of seismic inversion, at first, we only invert seismic data for it’s acoustic properties through post stack seismic inversion. Then, the research continues to separately invert partial stack seismic data, in order to get shear wave information, this is a lead to extract fluid properties from seismic. The limitation of separate inversion of partial stack, lead to simultaneous partial stack inversion, to solve Zoepprits equation, therefore we can extract compressional, shear, and density information. Next, enhancement of simultaneous seismic inversion is, we invert not only partial stack data, but now we invert *real* pre-stack seismic data. Along with continous research in rock physics, maybe I’ll write about it next time, sinc I need to read alot about it.

So, let’s start by understanding piece by piece of this petrophysical properties estimation from seismic data, by looking of wonderful statistical theorem proposed by Thomas Bayes, from a publication titled, “An Essay towards solving a Problem in the Doctrine of Chances“.

Intro to Bayesian Statistic

Integration of many surface/downhole measurement to characterize reservoir with statistic method is a notable method. The fields of geostatistic will tell you a lot about it. I don’t really understand at first about how to integrate this Bayesian Theorem for reservoir characterization. If you look at Wikipedia entry for Bayes’s Theorem, there are two possibilities that you can get. A headache because you see uncommon mathematical notation throughout the page, or you really understand about it. I understand this by reading Data Mining book by Budi Santosa [6].

My understanding about this theorem is simply like this, given a set of observation of two (or n-number of possibilities) possibilities, you will have initial guess or you may say initial probability or in Bayesian Statistic Term, Prior Distribution. For example, given two sets of data of weather in Jakarta for the past 100 days, you have 40 days of rain, 60 days of not raining. Then you will have prior distribution of 40% of raining and 60% not-raining. Then to guess, whether today will be raining or not, you will 40% maybe it’s raining, or 60% of chances today will not-raining.

Now, besides observation of two probabilites, you will have knowledge about each probability. Let’s say, the cloud type. Maybe, if the cloud type is nimbus, it will not raining, and if it’s cummolunimbus, it will definitely raining. (Look, I didn’t pay attention alot to meteorology, so, I don’t really know nor I don’t googling a lot about what cloud makes what?, so let’s just continue). This often called likelihood function.

Next, what we are after is, the possiblity of what is next event? This possibility is often called posterior distribution. Bayes theorem stated, the adding of new knowledge from likelihood function, will change your initial probability to become posterior probability. Thus will change your decision on what is next event. In mathematical this can be written as (from Wikipedia),

explanation of each terms (also from Wikipedia),

Each term in Bayes’ theorem has a conventional name:

• P(A) is the prior probability or marginal probability of A. It is “prior” in the sense that it does not take into account any information about B.
• P(A|B) is the conditional probability of A, given B. It is also called the posterior probability because it is derived from or depends upon the specified value of B.
• P(B|A) is the conditional probability of B given A.
• P(B) is the prior or marginal probability of B, and acts as a normalizing constant.

Intuitively, Bayes’ theorem in this form describes the way in which one’s beliefs about observing ‘A’ are updated by having observed ‘B’.

Hehehe, I don’t think, I explained it well :D. Some of good explanation about this theorem was published by Prof. Tarantola, Popper, Bayes and the inverse problem, you can download it from his page, which also contain alot of useful information about inverse theory, probability theory, etc. [7].

Let’s summarize then, Bayesian Theorem is a statistical theorem, that say, given a initial probability of observations (prior distribution), having a new knowledge on observed data (likelihood function), will change our point of view of initial probability, we will get a new probability distribution (posterior distribution).

Application of Bayesian Theorem for Seismic Inversion

Eidsvik (2002, 2004)[1,2] explain this application quite well in the papers. I will try to summarize few things about his flow chart from his papers,

1. Classes Definition
   This is pretty straightforward, like what we done for any kind of seismic inversion. Looking at dowhole measurements (well log, core, etc.) we can define certain facies, such as sand-shale, oil-brine, etc.
2. Build Bayesian Network
   We were talking about likelihood function, before we can build this certain function, we can define which variable is affecting which. For example, if we want to estimate posterior probability of acoustic impedance, then variable that will affect AI will be seismic data, compressional sonic, and density.
   We define this network for all properties that we want to estimate.
3. Build Prior Probabilistic Distribution Function
   Having defined and zone the facies, and properties, we can define prior probability model. This can be done using some deterministic method between each variables.
   It also necessary to define spatial continuity for facies.
4. Defined Likelihood Function
   Define functon that relate rock properties (density, compressional, and shear velocity) to observed seismic attribute (zero offset reflectvity, and AVO Gradient).

Eidsvik [1,2] noted 6 points in his paper about stochastic reservoir characterization, the first 4 that we discussed above was about Bayesian theorem implementation for seismic inversion. The next two steps is how to sample the posterior distribution, or in my understanding, how to defined the posterior distribution using Monte Carlo Markov Chain. We will discuss these 2 steps, later on :D.

Conclusion

I hope you are interested in Bayesian Theorem, as much as I do. Sounds simple, but the ground truth philosophy for that is so deep. And also, the application in estimating petrophysical properties from seismic data seems so brilliant. And also after implementing smart algorithm such as Markov Chain Monte Carlo in sampling posterior distribution.

Petrophysical properties is high order dimension, and of course, inverse problem always have il-pose problem. But exploring it with Markov Chain Monte Carlo (MCMC), seems fair enough to exploit anything we can do about the data. But, MCMC itself need guidance, or ground science about estimating it’s solution.

Umm, last paragprah was written by limited understanding about MCMC and Bayesian, so please don’t pay attention a lot on that thing 😀

Well, folks, thats all for today.

For DCS-ING’ers who read this, hahahaha :), I know, you will get more fascinated about me

References

[1]

Seismic reservoir prediction using Bayesian integration of rock physics and Markov random fields: A North Sea example

The Leading Edge 21, 290 (2002)

http://link.aip.org/link/?LEEDFF/21/290/1

[2]

Stochastic reservoir characterization using prestack seismic data

Geophysics 69, 978 (2004)

http://link.aip.org/link/?GPYSA7/69/978/1

[3]Release note on JGW 8.0
[4] Thomas, Probabilistic Seismic Inversion Based on Rock-Physics Models for Reservoir Characterization (PDF)
[5]Petrophysical Seismic Inversion (PDF)
[6]Buku Data Mining: Data Mining,Teknik Pemanfaatan Data untuk Keperluan Bisnis
[7]Poper, Bayes, Inverse Problem (PDF)