Value of Information Analysis and Bayesian Inversion for ... · for the saturation and porosity...

Value of Information Analysis and Bayesian Inversion for

Closed Skew-Normal Distributions: Applications to Seismic

Amplitude Versus Offset Data

Javad Rezaie∗, Jo Eidsvik† and Tapan Mukerji‡

ABSTRACT

The evaluation of geophysical information sources depends on input modeling assump-tions. This paper presents results for Bayesian inversion and value of informationcalculations when the input distributions are skewed and non-Gaussian. Reservoir pa-rameters and seismic amplitudes are often skewed and by using models which capturethe skewness of distributions, the input assumptions are less restrictive and the valueof information analysis is more reliable. We use a Closed Skew Normal distributionfor the saturation and porosity variates and the seismic amplitude data. Sensitivity ofthe value of information analysis to skewness, mean values, accuracy and correlationparameters is performed.

INTRODUCTION

Geophysical data are directly or indirectly informative of important subsurface parameters.Seismic measurements, for example, may provide a rich source of information about struc-tures, lithologies and hydrocarbon indicators. However, these interpretations are associatedwith uncertainty. In evaluating the value of data under uncertainty, along with the cost ofthe data, it is the impact of the data on the underlying development decision that has to becritically considered. The goal is to maximize expected revenue, not to reduce uncertaintyper se. A decision-theoretic value of information (VOI) framework provides a very usefultool for evaluating seismic data in this context.

VOI analysis relates to making better decisions under uncertainty (Raiffa, 1968; Howard,1996). It is an old concept in the petroleum industry (Grayson et al., 1962), and it seems tohave gained more interest in recent years (Branco et al., 2005; Bickel et al., 2008; Bratvold et al.,2009; Bhattacharjya et al., 2010). The VOI is useful in several petroleum applications whereone considers purchasing more data before making a decision. The data comes with a price,and one might ask if it is really worth it, or which data to acquire at the current stage?

The VOI is defined as the maximum cost that we should pay for new information. Ifthe price of data is larger than the VOI, the data are not worth purchasing (Bratvold et al.,2009). Consider the Expected revenues WITH additional information minus the Expectedrevenues WITHOUT additional information. The former is sometimes called posteriorvalue, while the latter is the prior value. Then,

VOI = Posterior Value− Prior Value. (1)

Our main example in this paper concerns the decision of drilling / not drilling new wells

Value of Information analysis for Closed Skew Normal Seismic Data

Rezaie, Eidsvik & Mukerji 2 Geophysics

at a number of selected reservoir units (Eidsvik et al., 2008). We assume the followingscenario: Initial reservoir analysis has been done with post-stack data, and now the decisionis whether or not to purchase better processed pre-stack seismic data for amplitude versusoffset (AVO) based reservoir characterization. Good AVO data, calibrated and interpretedwith appropriate rock physics models might help with better drilling decisions by indicatingpresence or absence of hydrocarbons and reservoir quality.

The VOI analysis is useful for helping with this auxiliary decision about obtaining AVOdata that is linked to the main decision about drilling. If the VOI is small, the new amplitudedata is unlikely to help us much in making the drilling decisions. In this situation the seismicre-processing may be too expensive compared with its actual added value. In practice a wellcan of course be dry, even though the data VOI looks promising. This happens as a result ofvariability in the data which arises from several factors including subsurface heterogeneityand noise. The VOI performs an average calculation over all possible datasets we might getin the seismic re-processing, given our prior knowledge about the reservoir rock and fluidproperties, as well as the seismic model.

As we will show, the reservoir parameters and seismic amplitude measurements areskewed and non-Gaussian. By using models which capture the skewness of distributions,the input assumptions are better-fitting. Bayesian inversion of seismic data and the VOIanalysis are then more reliable. We introduce a Closed Skew Normal (CSN) family ofdistributions for the saturation and porosity variates, and for the seismic amplitude data.This extension provides flexibility in the working assumptions. In fact, if the Gaussianapproximation is the best, the fitted CSN distribution converges to the Gaussian, otherwiseit works better by capturing higher order moments of the data. Our approach extends thatof Eidsvik et al. (2008).

NOTATION AND BACKGROUND MATERIAL

Notation

Throughout this paper, we use x ∈ ℜnx×1 as a nx dimensional distinction of interest. In thenumeric example this will contain variables related to the saturation and porosity in reservoirunits. Using appropriate transformations of variables, the vector x has real entries. Thenotation x ∼ π (·) is used to show that x is distributed according to probability densityfunction (pdf) π(x). For a Gaussian model we have pdf π(x) = φnx(x;µ,Σ) and theassociated cumulative distribution function (cdf) is denoted Φnx(x;µ,Σ). Here, the meanis µ ∈ ℜnx×1 and the positive definite covariance matrix is Σ ∈ ℜnx×nx . Besides, thenotation π(x) = CSNnx,qx (µ,Σ,Γ,v,∆) means a CSN pdf with parameters µ ∈ ℜnx×1,Σ ∈ ℜnx×nx , Γ ∈ ℜqx×nx , v ∈ ℜqx×1 and ∆ ∈ ℜqx×qx , where Σ and ∆ are positive definitematrices.

A random vector x ∈ ℜnx×1 is CSN distribution with qx,µ,Σ,Γ,v,∆ parameters if itspdf is as follows (See Appendix A for details):

x ∼ CSNnx,qx (x;µ,Σ,Γ,v,∆)

=[

Φqx

(

0;v,∆+ ΓΣΓT)]−1

Φqx (Γ (x− µ) ;v,∆)φnx (x;µ,Σ) . (2)



We use d ∈ ℜnd×1 to denote the seismic amplitude data. The data consist of indirectobservations of the reservoir parameters; the zero offset and seismic AVO gradients at allreservoir units. We assume a weakly nonlinear relationship d = h (x)+e. Here, e is additivemeasurement noise with pdf π(e). We will linearize the measurement equation using firstorder Taylor series expansion to get h (x) ≈ h0 +Hx, where H ∈ ℜnd×nx . The likelihoodmodel for the data is denoted π(d|x), with parameters derived from d = h0+Hx+e. Themarginal distribution for the data is π(d), obtained by integrating out all reservoir variablesfrom the joint model for x and d.

Seismic data relations

The seismic amplitude versus offset (AVO) data is related to the reservoir saturation, s, andporosity, ϕ, through rock physics relations. Here, the data consist of zero-offset reflectivityand AVO gradient at top reservoir. We assume the elastic properties of the cap rock arefixed. The elastic properties of the reservoir layer are uncertain because they depend onsaturation and porosity. The likelihood model combines the seismic measurement equationswith associated uncertainties.

Saturation and porosity are two key reservoir parameters which directly relate to thedrilling decision. The prior information about these parameters provides rough knowledgewith large uncertainty. These parameters are limited to change between some lower/upperlimits, say smin ≤ s ≤ smax and ϕmin ≤ ϕ ≤ ϕmax. Logistic transformations define

xs = log

(

s− smin

smax − s

)

and xϕ = log

(

ϕ− ϕmin

ϕmax − ϕ

)

, and we have xs and xϕ on the real line.

The inverse logistic function transforms back to their actual range: s =1

1 + exp(xs)smin +

exp(xs)

1 + exp(xs)smax and ϕ =

1

1 + exp(xϕ)ϕmin +

exp(xϕ)

1 + exp(xϕ)ϕmax.

The rock physics models consist of relations for bulk modulus and shear modulus as afunction of porosity for 100% brine saturation, which are then converted to other saturationsusing Gassmann’s equations. There are various functional forms for the modulus-porosityrelations, depending on sorting, cementation, and diagenesis. For example, Bachrach (2006)fits a nonlinear function for relating brine-saturated bulk modulus, K, and shear modulus,G, to the porosity. We simplify these relations and use a linear fitting function with rea-sonable accuracy. The parameters of the linear fit are determined from well-log data.

According to Gassmann’s formula, the shear modulus does not change with saturation;it is a function of porosity alone:

G = G (s, ϕ) = G (smax, ϕ) . (3)

The effective fluid bulk modulus for mixed saturations is given by the Reuss average of theindividual fluid moduli:

Kf =

(

s

Kb

+1− s

Ko

)−1

, (4)

where Kb and Ko are the fixed bulk modulus of oil and brine, respectively. The rock bulk



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Porosity

K ,

G (

GP

a)Moduli−porosity relations for 100% brine saturation

KG

Figure 1: Brine-saturated bulk modulus, K, and shear modulus, G, versus porosity.

modulus for saturation and porosity is obtained by Gassmann’s formula:

K =BKq

1 +B, B =

K (smax, ϕ)

Kq −K (smax, ϕ)−

Kb

ϕ (Kq −Kb)+

Kf

ϕ (Kq −Kf ), (5)

where Kq is the fixed quartz bulk modulus and K (smax, ϕ) is the bulk modulus fitted fromwell data in brine sand (see Figure 1 for the values we use here).

The density is given by:

ρ = ϕsρb + ϕ (1− s) ρo + (1− ϕ) ρq, (6)

where ρb, ρo and ρq are defined as fixed density of brine, oil and quartz, respectively.

P-wave and S-wave velocities relate to the bulk modulus, shear modulus and densitythrough:

Vp =

√

√

√

√

K +4

3G

ρ, Vs =

√

G

ρ. (7)

Finally, the zero-offset reflectivity d1 and AVO gradient d2 can be evaluated from the elasticconstraints in the caprock and the reservoir. We use the Aki and Richard’s formula to



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Figure 2: Zero-offset reflectivity (left plot) and AVO gradient (right plot) versus saturationand porosity.

evaluate the expected values of d1 and d2 (Mavko et al., 2003):

E[d1|x] =1

2

(

δVp

Vp

+δρ

ρ

)

,

E[d2|x] =1

2

δVp

Vp

− 2V 2s

V 2p

×

(

δρ

ρ+ 2

δVs

Vs

)

. (8)

where conditioning on x means using all the formulas in (2)-(6). In addition, for general

property ǫ, ǫ =ǫ+ ǫcr

2and δǫ = ǫ− ǫcr are the average and the difference between reservoir

and caprock property, respectively.

Based on the presented relations, we see that seismic attributes relate to the saturationand the porosity through nonlinear relationships. Figure 2 shows the modeled zero-offsetreflectivity and AVO gradient versus the saturation and porosity where 0.1 ≤ s ≤ 0.9 and0.1 ≤ ϕ ≤ 0.5.

The Society of Petroleum Engineers (SPE) organized a series of projects, known asSPE comparative projects, in order to provide benchmark data sets which can be used tocompare the performance of different algorithms and methods. The 10th SPE comparativeproject is the latest one in this series and known as the SPE10 data set (Christie and Blunt,2001). The SPE10 data set consist of porosity for 60 × 220 × 85 Cartesian grid cells. Byusing this data set as the input to a reservoir flow solver (simulator), the saturation andporosity of that reservoir can be used for further evaluations. For our simulations, weuse the porosity and permeability from the SPE10 data set as the reservoir parameters.Then we use MATLAB Reservoir Simulator Toolbox (MRST), see Lie et al. (2012), as flow



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π(G

)


Figure 3: A graphical description of distribution fitting based on the SPE10 data, forsaturation (upper left plot), porosity (upper right plot), zero-offset reflectivity (lower leftplot) and AVO gradient (lower right plot), where the solid curve is the empirical distribution,dot curve is the fitted Gaussian and dash-dot is the fitted skewed.

solver for simulating the saturations (More details about the simulator are available atwww.sintef.no/Projectweb/MRST/).

Figure 3 shows the empirical marginal distribution of the logistic transformed saturationand porosity (top) and the zero-offset reflectivity and AVO gradients (bottom). Then thereservoir porosity and permeability are selected from the SPE10 data set and we run thesimulations on MRST for 100 days and record the saturation. By using the saturation andporosity as inputs to equation (7), zero-offset reflectivity and AVO gradients are calculated.From the total number of grid cells, we choose a subset by discarding inactive blocks (blockswith poor permeability and porosity).

We first approximate these empirical distributions with the Gaussian distributions, andthen with an skewed distribution (Figure 3). According to these plots, the distribution ofthe logistic porosity seems to be Gaussian, since the fitted results for Gaussian and skewednormal models are similar. For the rest, their marginal distributions are not symmetric andconsequently neither are their joint distributions. Thus, modeling them with the Gaussiandistribution may result in inaccurate calculations. This motivates the use of the CSNdistribution as a more sophisticated and flexible class.

DISTRIBUTION ASSUMPTIONS AND CLOSED SKEW NORMAL

Statistical modeling is an important step in dealing with uncertainties. The statisticaldistributions can be divided in three general categories: i) Continuous, (Karimi et al., 2010),


http://www.sintef.no/Projectweb/MRST/


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π(x2)

x1

π(x1)

x2

π (x

1,x2)

Figure 4: The bivariate CSN distribution (3D plot) and its marginals (2D plots on thesides) which are also skewed.

ii) Discrete, (Ulvmoen and Hammer, 2010) and iii) Mixed, (Grana et al., 2012), consistingof both continuous and discrete components. Continuous distributions are the most usefulin our applications.

Closed skew normal distribution

The Skew Normal (SN) random variable is an extension of the Gaussian. The SN distri-bution has one more parameter for capturing the skewness (Azzalini and Dalla-Valle, 1996;Arellano-Valle et al., 2002; Liseo and Loperdo, 2003; Gupta et al., 2004). The SN distribu-tion has some properties similar to the Gaussian distribution; unimodal, support is the realline and the distribution of the square of a SN random variable is χ2 with one degree offreedom (Azzalini and Capitanio, 1999). The distribution of a SN random vector x ∈ ℜnx×1

is:x ∼ SNnx (x;µx,Σx,Γx) = 2φnx (x;µx,Σx)Φ1 (Γx (x− µx) ; 0, 1) , (9)

where Γx ∈ ℜ1×nx. For Γx = 0, it is identical to the Gaussian pdf with mean µx andcovariance Σx.

The SN distribution extends to the more general family of CSN. This is done byadding two more parameters, and via these additions, we get nice analytical properties(Domınguez-Molina et al., 2003; Genton, 2004; Gonzalez-Farıas et al., 2004). For instance,they are closed under addition, and the general full rank linear transformation of a CSNrandom vector is also CSN. Consequently, the marginal distributions of a multivariate CSNare also CSN distributed (Gonzalez-Farıas et al., 2004). Figure 4 shows a bivariate CSNdistribution and its marginals. We see that both joint and marginals are skewed.



The CSN distribution is more general than the SN and the Gaussian. For instance, SNdistribution is an special case of the CSN when v = 0 and ∆ = 1. Also, the Gaussiandistribution is a special case of the CSN if qx = 0 or Γ = 0 (See Appendix A for details).Besides, similar to the Gaussian distributions if the prior and likelihood are both CSN, theposterior is also CSN (See Appendix B for details). This is one important property of theCSN distribution which is very useful for Bayesian inversion problems (Karimi et al., 2010).

BAYESIAN INVERSION

Assume we have some prior knowledge about the distribution of interest. In addition, wecan acquire measurements which directly or indirectly relate to these variables. In oursetting π(x) forms the prior model for reservoir variables, while π(d|x) is the likelihood ofthe seismic AVO data, given the reservoir variables.

Bayes theorem combines prior information and current observations, in the probabilitydomain, in order to get the posterior distribution of the variables of interest:

π(x|d) ∝ π(d|x)π(x). (10)

Figure 5 shows a schematic diagram of a Bayesian inversion problem from a distributionalpoint of view. The prior knowledge (upper left plot) is here a bivariate distribution. Bayesrule adds the information content of the likelihood distribution (lower left plot) to it. Theposterior distribution is shown in the right plot, and we see that the uncertainties arereduced.

If the prior distribution is a Gaussian distribution, x ∼ φnx (x;µx,Σx), and the like-lihood is a Gauss linear distribution, d|x ∼ φnd

(

d;Hx,Σd|x

)

, the posterior distribution

is also Gaussian, x|d ∼ φnx

(

x;µx|d,Σx|d

)

, and there are analytical formulations for its

parameters (Buland and Omre, 2003):

µx|d = µx +ΣxHT[

HΣxHT +Σd|x

]−1(d−Hµx) ,

Σx|d = Σx −ΣxHT[

HΣxHT +Σd|x

]−1HΣx. (11)

Another example of closed-form solutions for the posterior is when the distributions aremodeled as a Gaussian mixture distribution (Grana et al., 2012). In most other situations,there are no closed-form solutions for the posterior distribution π(x|d). In Figure 5, wehave used a CSN distribution for the prior and likelihood model. The posterior model isthen also CSN in this situation, and this makes the CSN very applicable.

Precisely speaking, if the prior is x ∼ CSNnx,qx(µx,Σx,Γx,vx,∆x) and the likelihood isd|x ∼ CSNnd,qd(Hx,Σd|x,Γd|x,vd|x,∆d|x), then x|d ∼ CSNnx,qx+qd(µx|d,Σx|d,Γx|d,vx|d,∆x|d),



Figure 5: Illustration of Bayesian inversion. The upper left plot is the prior distributionwhich is CSN, the lower left plot is the likelihood which is CSN and the right plot is theresulting posterior which is CSN too.

where (See Appendix B for details):

µx|d = µx +ΣxHT[

HΣxHT +Σd|x

]−1(d−Hµx) ,

Σx|d = Σx −ΣxHT[

HΣxHT +Σd|x

]−1HΣx,

Γx|d =

[[

ΓxΣx

0

]

−

[

ΓxΣxHT

Γd|xΣd|x

]

[

HΣxHT +Σd|x

]−1HΣx

]

Σ−1x|d,

vx|d =

[

−vx

−vd|x

]

+

[

ΓxΣxHT

Γd|xΣd|x

]

[

HΣxHT +Σd|x

]−1(d−Hµx) ,

∆x|d =

[

∆x + ΓxΣΓTx 0

0 ∆d|x + Γd|xΣd|xΓTd|x

]

,

−

[

ΓxΣxHT

Γd|xΣd|x

]

[

HΣxHT +Σd|x

]−1[

ΓxΣxHT

Γd|xΣd|x

]T

− Γx|dΣx|dΓTx|d. (12)

These relations are also valid when the prior and/or likelihood are Gaussian. For instance, ifthe prior distribution is Gaussian, x ∼ φnx (x;µx,Σx), the posterior is CSN with parametersby Γx = 0, vx = 0 and ∆x = I. Thus, we can get the Gaussian inversion results, equation(11), as a special case of CSN inversion by setting Γx = 0, vx = 0, ∆x = I, Γd|x = 0,vd|x = 0 and ∆d|x = I. Although CSN random variables are more general than theGaussian one, and they extend the Gaussian distribution for handling asymmetry andskewness, the added flexibility comes at the cost of more computational tasks: We mustuse the maximum likelihood for parameter estimation, but faster methods exist for somepredefined structures (Flecher et al., 2009). The sampling problem also becomes harder



(See Appendix A for details). Simulations is done by rejection sampling which means wegenerate samples from a Gaussian distribution and then discard some of them.

VALUE OF INFORMATION IN A CSN SETTING

Problem specification: the VOI of AVO data

To compute the VOI we have to assess the prior and posterior value. Most elements ofthese calculations are readily available for the CSN distribution. We will next outlinemethods for computing the VOI related to drilling / not drilling at prospects depending onthe oil saturation and porosity. The VOI is computed for seismic AVO attributes, whichare informative of the saturation and porosity values. We next introduce the detailedrequirement assumptions for this setting, see also Eidsvik et al. (2008).

In addition to the saturation and porosity variables at all prospects, we use a scalingterm to convert to monetary units. We have a cost C of drilling a new well to hidden pocketsof oil, and we are interested in maximizing expected profit. Assume we have M prospects,the value of a prospect i = 1, . . . ,M , is Rφi (1− si), where (1− si) is the oil saturation,and R is a factor defined as the product of the following: The area of the prospect Ai, thethickness of the reservoir hi, and a factor R0 which includes the oil price, recovery rates,and net-to-gross. Other values are possible here. Note that only saturation and porosityare taken as key uncertain reservoir parameters, while the others are kept fixed.

We define the VOI as:

VOI = Posterior value− Prior value,

Prior value =

M∑

i=1

maxRE[φi(1− si)]− C, 0,

Posterior value =

M∑

i=1

∫

maxRE[φi(1− si)|d]−C, 0π(d)dd. (13)

Note that the prior value for prospect i is the expected value over units E[Rφi (1− si)] =RE[φi (1− si)], minus the cost of drilling. The decision maker will only drill if the expectedprofits are positive.

We assume a risk-neutral decision maker and hence can define value in terms of expec-tations. In general this may not be true and the decision maker’s utility function shouldthen be used in defining values.

Considering one saturation and one porosity variable for each prospect like in equation(13), the prior is CSN with nx = 2M and qx = 2M . AVO data is considered as potentiallymeasured data, d. By considering zero-offset reflectivity and AVO gradient as measurementsfor each prospect, the likelihood is CSN with nd = 2M and qd = 2M .

The expectation in the posterior value is a conditional expectation over the seismicobservations. According to the previous section, the posterior distribution of logistic sat-uration xs and porosity xϕ is CSN with nx = 2M and qx = qx + qd, and the posteriorparameters are analytically available by equation (12). Note that the posterior value is an



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integral expression over all possible seismic AVO data d with distribution π(d) (See Ap-pendix C for details). This integral is estimated using approximations that will be explainedin the next section. Since we condition on various data, the posterior value is larger thanthe prior value, and the VOI is positive.

Computational aspects

VOI consists of two parts: i) prior value ii) posterior value, and in both of them we havean expectation over the product of oil saturation and porosity E [ϕ (1− s)]. For evaluatingthese expectations we need the related distributions, then these expectations can be calcu-lated numerically, by Monte Carlo (MC) simulation or analytically. There is an analyticalformulation for calculating the mean value of a CSN random variable. Thus, we knowthat the posterior distribution of the logistic saturation xs and porosity xϕ is CSN, butthe posterior of the original saturation s and porosity ϕ is not CSN. One straightforwardway to handle this problem is generating many samples from transformed variables (xs andxϕ), then using the inverse transformation to get samples of s and ϕ and finally calculatethe empirical mean. This method is time consuming. Another way is using a reasonable



approximation for the expected value.

For any function f (x), if the first order Taylor series expansion is a good approximation,the calculation of E [f (x)] is easy. Because, if f (x) = a0 + a1x + a2x

2 + · · · ≈ a0 + a1x,

then E [f (x)] ≈ f (E [x]). For our problem, f(x) = f(x, xmin, xmax) =1

1 + exp(x)xmin +

exp(x)

1 + exp(x)xmax. We analytically calculate the expectation of transformed saturation and

porosity, then put this expectation into the inverse logistic transformation:

E [ϕ (1− s)] = E [f(xϕ, ϕmin, ϕmax) (1− f(xs, smin, smax))]

≈ f(E [xϕ] , ϕmin, ϕmax) (1− f(E [xs] , smin, smax)) . (14)

This trick reduces the computation cost dramatically and for our application the approxi-mation seems to work very well.

The integral is over seismic AVO data d approximated by MC integration by generatingmany i.i.d. samples from the observation distribution and then take the average of condi-tional values. More precisely, if we have W i.i.d. samples, d1,d2, · · · dW from pdf π (d) thenπ (d) ≈ 1

W

∑Ww=1 δ (d− dw). These two distributions are equivalent when W → ∞. Now,

for calculating∫

g (d) π (d) dd we have:

∫

g (d)π (d) dd ≈

∫

g (d)1

W

W∑

w=1

δ (d− dw) dd =1

W

W∑

w=1

g (dw) , (15)

Here, g(d) = maxRE[φ(1− s)|d]−C, 0. Note that we calculate the posterior parametersfor each dw. But, by looking at the posterior parameters formulations in equation (12), wesee that all parameters are constant for all dw’s, except µx|d and vx|d. Thus, we just needto calculate the posterior parameters one time and then use them, updating only the dw inµx|d and vx|d. The organized version of this VOI calculation is presented in Algorithm 1.

Algorithm 1 VOI Calculation1: Fitting a CSN prior distribution.2: Estimate the prior value.3: Model the likelihood as a CSN distribution.4: Approximate the posterior value by Monte Carlo integration:5: for w = 1 to W do

6: Generate a sample from observation distribution, dw ∼ π (d).7: Construct the posterior distribution for the logistic parameters given dw (equation (12)).8: Construct the value according to equation (14).9: end for

10: Approximate the integral part in the posterior value as an average over all W runs (equation (15)).11: The VOI is the difference between the expected posterior and prior values.

We should mention that the most time consuming part of this algorithm is the rejectionsampling method for simulating samples from the data distribution, π (d), which is CSN.We can use the power of parallel computing and Graphical Processing Units (GPUs) forgenerating many samples from the associated Gaussian distribution and then apply rejectionfor choosing correct samples. Figure 6 (bottom) shows the computation time trends whenwe generate samples by CPU and GPU processors and compute the associated VOI (top)for a CSN model of dimension nx = nd = 100, with MC uncertainty bounds. According tothis plot, we see that by increasing the number of samples, the computation time for the



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)

No Γx

Low Γx

Medium Γx

High Γx

−5 0 5 100

0.5

1

1.5

x

π (x

)

Very Low Σx

Low Σx

Medium Σx

High Σx

Figure 7: VOI calculation with perfect information: effect of Γx and Σx on the VOI (upperplots) and on the pdf (lower plots).

CPU increases with a linear rate, but it is almost constant when we use the GPU. Here, weuse the gfor routine in Jacket MATLAB software, which runs a for loop in parallel on theGPU. The comparison was done on an 2 x 6-core Intel Xeon X5680 3.33 GHz CPU with 96GB 1333 MHz DDR3 memory. The GPU is a nVidia Tesla C2050.

NUMERIC EXAMPLES

Perfect information

In this simple example, we analyze the sensitivity of VOI as a function of scale parameter Σx

and skewness parameter Γx. Consider a univariate random variable x ∼ CSN1,1(µx,Σx,Γx, vx,∆x).Then VOI =

∫

max(x, 0)π(x)dx −max[E(x), 0]. Assume that E[x] = 0 for all Σx and Γx.Figure 7 shows the sensitivity of VOI for a range of 0.1 ≤ Σx ≤ 2 and 0.1 ≤ Γx ≤ 2.

The left plots in Figure 7 show the effect of Γx parameter on the VOI (upper left plot)and on the pdf π(x) (lower left plot) for Σx = 1. As we see in the upper left plot, byincreasing the value of Γx the VOI decreases. As you see in lower left plot, by increasingΓx the variance of x decreases which results in more focused prior, which is a little skewed.

The right plots in Figure 7 show the sensitivity of VOI (upper right plot) to the Σx forΓx = 1. By increasing Σx the prior distribution becomes wider and the VOI increases.



0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40

1

2

3

4

5

6x 10

5

Data Error Level

VO

I

CSNNormal

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40

1

2

3

4

5

6x 10

5

Data Error Level

VO

I

CSN without approximationCSN with approximation in meanCSN with median

Figure 8: VOI for the Gaussian and CSN modeling assumptions: The top plot is thecomparison between VOI with Gaussian and CSN input assumption. The bottom plot isthe VOI for different approximations based on equation (14) and the exact solution basedon MC sampling, and also using the median instead of the mean.

One prospect case

Consider M = 1, let (s, ϕ)T be the reservoir parameters of interest and d = (d1, d2)T the

AVO measurements. The saturation is generated by MRST based on the SPE10 data, whichalso input porosities. In addition, the seismic data d is generated for the given saturationand porosity, and in accordance with the relations presented above.

Now, we consider two cases for the prior and likelihood fitting, i) prior modeled withCSN distribution, ii) prior modeled with Gaussian distribution. For both cases, the posteriordistributions are analytically calculated and we approximate the VOI. We use a Gaussiandistribution to model the likelihood for both cases. The reason for choosing this likelihoodmodel is comparing the results by changing the variance of the measurement error.

The upper plot of Figure 8 shows the results of both assumption for a range in themeasurement error variance. If the data processing price is over and under both plots, theresulting decision for both is similar, although the predicted final revenue is different. Butif this price is in between two curves then the resulting decision about purchasing data isdifferent. Although the VOI calculation in the CSN case is higher, it does not mean thatthe CSN case is better. The CSN case will however give more reliable decisions, because itis a more accurate description of the data. This is because of structure of CSN distribution.Recall that the CSN automatically sets Γ = 0 if the data are Gaussian.

From now on, we focus only on the CSN case. The upper right plot represent theresults of using different methods for calculating the expectations in the prior and posterior



0.5 1 1.5 2 2.50

1

2

3

4

5

6x 10

5

Data Error Level

VO

I

VOI − CSNData Price Scenario 1

0.5 1 1.5 2 2.50

1

2

3

4

5

6

7x 10

5

Data Error Level

VO

I

VOI − CSNData Price Scenario 2

Z1Z

1Z

2Z

3Z

4Z2

Figure 9: The effect of the data price on the decision about purchasing various data pro-cessing scenarios.

values. According to this plot we see that using the fast first order Taylor series expansionof the inverse logistic transformations works very well,since the VOI based on this analyticalapproach is almost equal to the MC approximation. In addition, we see that the result ofusing the median statistic instead of the mean also gives similar VOI. The reason for choosingthe median statistic is that the median of a monotone function of a random variable is thesame as the function of the median, i.e. median [f (x)] = f (median [x]).

The two plots of Figure 9 represent the VOI and different data processing scenarios. Asyou see, increasing the data accuracy, increases the VOI. On the other hand, more accuratedata needs more data processing and it increases the cost of data. Thus, we can considera one to one relation between accuracy and costs, and having the relative position of VOIand data costs as a function of data processing level. If the price of processing data is lessthan VOI, it is worth buying it.

For understanding the idea behind required data processing level, consider scenario 1(left), where the cost of data processing is defined to be linearly dependent on the processinglevel. When the level of seismic AVO data processing is in zone Z1, it is not worthwhile tobuy such data, because the VOI is less than the data cost. By increasing the data processinglevel and entering zone Z2 the VOI increases with higher rate than the processing cost andit is worthwhile purchasing data in this zone.

In reality there are often more limitations. It is obvious that more data processingneeds more expert people, more time, more resources etc. These limitations may changethe scenario of data processing price. Consider that the time is one more limitation, andafter a maximum number of days we are penalized step by step. This penalty representsitself in the data processing cost by increasing the rate of change of costs (right plot of



Figure 9). For this case, when we are in zone Z1, it is not worth buying data. By increasingthe seismic AVO data processing level (zone Z2), it is worthwhile. The most revenue isachieved at the end of this zone. If we need more time for data processing, we are penalizedwith new data processing cost rates. In zone Z3, data are valueable, but the final revenuereduces. In zone Z4 it is not worthwhile, because the price of data is higher that VOI.

In previous discussions, we assumed that the likelihoods are Gaussian but for Table 1we consider the likelihood to be CSN. Table 1 shows the sensitivity of VOI with respect tothe Σd|x and Γd|x parameters of data likelihood distribution. As is shown, by increasingΣd|x and Γd|x the VOI decreases but with different rate. The rate of the VOI decreases asa function of Σd|x is higher than Γd|x. This fact means the sensitivity of VOI to Σd|x ishigher than Γd|x. In addition, when we increase Σd|x and decrease Γd|x with the same scale,the VOI decreases which is another indicator for the superior effect of Σd|x on the VOI.

Γd|x

Σd|x No Low Medium High

Low 74 74 69 67Medium 39 37 32 29High 25 23 20 15

Table 1: The effect of data processing level on the VOI for one prospect case (in units ofthousand dollar).

In the above analysis we have assumed drilling costs C = 2 million dollar. We willnow study the influence of drilling cost on the VOI. Simulation results show that for thedrilling cost from zero up to 0.9C, the VOI is zero and the decision maker decides to drillwithout purchasing any additional data. The VOI is also zero when the drilling cost ishigher than 1.3C and the decision maker decides not to drill without any data processing.For in-between situations, the data is valuable for making the decision if the positive VOIis larger than the processing cost. The VOI is the largest for C = 2.24 million dollar.

Spatial dependency

In this section we explore the effect of spatial dependency between prospects on the VOIusing CSN distribution. We consider two different cases i) dependent prior and ii) indepen-dent prior. By dependent prior, we assume an exponentially decreasing spatial correlationbetween grid cells as follows:

Σ =

c (1, 1)Σx · · · c (1,M)Σx

c (2, 1)Σx · · · c (2,M)Σx

.... . .

...c (M, 1)Σx · · · c (M,M)Σx

, (16)

where Σx is the 2 × 2 covariance between logistic porosity and saturation in one reservoirunit, c (i, j) = exp(−0.3distance(i, j)) and distance(i, j) is the Euclidean distance betweenprospect i and j. Other distances, for example, following stratigraphic horizons, can alsobe used. Assume cell i and j has (xxi, yyi) and (xxj , yyj) as their positions in a two



dimensional Cartesian plane, then distance(i, j) =√

(xxi − xxj)2 + (yyi − yyj)2. This cor-relation assumption means over the top-reservoir that closer prospects are more correlated.The SPE10 data are used for the prior mean and the seismic AVO data are generatedaccording to equation (7).

Γd|x

Prior Σd|x No Low Medium High

Correlated Low 23.6 21.1 18.3 16.1Medium 19.6 15.8 12.8 10.8High 16.4 12.4 9.5 7.3

Independent Low 2.9 2.7 2.6 2.4Medium 1.6 1.4 1.3 1.2High 1.1 0.98 0.86 0.80

Table 2: VOI results. The effect of prior correlation and skewness between prospects (inmillion dollars).

Table 2 presents the similar results as Table 1 for the VOI for various scales, skewness andfor low versus high spatial correlation. We see that for correlated and uncorrelated prospectsby increasing Σd|x and/or Γd|x, the VOI decreases. In addition, the VOI decreases moreslower by increasing Γd|x in comparison toΣd|x. In addition, correlation assumption betweenprospects in the prior results in higher VOI than independent prior. When we have someinformation about prospect i (through measurement data for prospect i), and it is highlycorrelated to prospect j, we have information about prospect j indirectly. Consequently,the seismic AVO data is valueable and results in higher VOI. Finally, simulation resultsshow that by considering correlation between observations, the resulting VOI is lower thanfor independent likelihood. This means that the observations are more valueable when theyare independent, providing more information about the prospects.

Note that for all cases in this simulation, when we change the Γd|x and Σd|x, we adaptthe mean of the likelihood distribution to be of the same order to remove the effect of themean on the VOI.

CLOSING REMARKS

We introduce skewed distributions for the modeling of reservoir parameters and geophysicalmeasurements. The CSN distributions are closed under linear combinations and condition-ing. This means that if the prior and likelihood are both CSN, the posterior is also CSN.This important property of the CSN distribution is very useful for Bayesian inversion prob-lems. CSN distribution is more flexible than the classical Gaussian distribution for datawith skewed distributions.

We fit CSN models to the SPE10 data set and check the sensitivity of the VOI tothe model parameters. The Gaussian assumption directs the decision maker to wrongdecisions when the data processing cost is in between the evaluated Value Of Informationof the Gaussian and CSN. Simulation results show that by increasing the Σd|x and/or Γd|x

parameters of data distribution, the VOI decreases, and it is more sensitive to Σd|x thanΓd|x.



One of the computational challenges in the VOI calculation is evaluation of the inte-grals. We use analytical approximations, and Monte-Carlo approximations for calculation.Besides, we use the power of parallel computing and GPUs for computation speed up.

Main future challenges are finding the reason behind the effect of correlation, varianceand skewness on the VOI in real systems. Seismic data sets are massive in size, checkingthe effect of data reduction algorithms on the efficiency of the proposed algorithm would beuseful. Another interesting topic will be using the concept of VOI in the spatio-temporalBayesian inversion for finding the optimal time for conditioning on geophysical data.

ACKNOWLEDGMENTS

We thank the sponsors of the Uncertainty in Reservoir Evaluation (URE) project at Nor-wegian University of Science and Technology (NTNU). We further thank Stanford Centerfor Reservoir Forecasting (SCRF), Department of Energy Resources Engineering (ERE) atStanford University for providing the facilities for working together. We thank SINTEF forthe MATLAB reservoir simulation toolbox.

APPENDIX A

PROPERTIES OF THE CLOSED SKEWED NORMAL

DISTRIBUTION

Assume x and y are respectively nx × 1 and ny × 1 random vectors and are jointly normal,[

x

y

]

∼ Nnx+ny

([

µx

µy

]

,

[

Σx Σx,y

Σy,x Σy

])

Then, we have:

π (z) = π (x|y ≥ 0) =π (y ≥ 0|x)π (x)

π (y ≥ 0)

=[

1− Φny

(

0;µy,Σy

)]−1[

1− Φny

(

0;µy|x,Σy|x

)]

φnx (x;µx,Σx) . (A-1)

where µy|x = µy +Σy,xΣ−1x (x− µx) , Σy|x = Σy −Σy,xΣ

−1x Σx,y.

By re-writing the above equation to get a standard form, we have:

z ∼ π (z) = CSNnz,qz (µ,Σ,Γ,v,∆)

=[

Φqz

(

0;v,∆+ ΓΣΓT)]−1

Φqz (Γ (z − µ) ;v,∆)φnz (z;µ,Σ) . (A-2)

Where nz = nx, qz = ny, µ = µx, Σ = Σx, Γ = Σy,xΣ−1x , v = −µy and ∆ = Σy −

Σy,xΣ−1x Σx,y.

Similarly, if we have a CSN distribution we can construct the original unconditionaljointly normal distribution:

[

x

y

]

∼ φnz+qz

([

µz

−vz

]

,

[

Σz ΣzΓTz

ΓzΣz ∆z + ΓzΣzΓTz

])

. (A-3)

CSN distributions have some properties that are similar to Gaussian.



Lemma 1: if x1 ∼ CSNnx,qx1(µ1,Σ1,Γ1,v1,∆1) and x2 ∼ CSNnx,qx2

(µ2,Σ2,Γ2,v2,∆2)are independent CSN random variables, then x = x1+x2 is also CSN, x ∼ CSNnx,qx1+qx2

(µx,Σx,Γx,vx,∆x

where:

µx = µ1 + µ2, Σx = Σ1 +Σ2, Γx =

(

Γ1Σ1 (Σ1 +Σ2)−1

Γ2Σ2 (Σ1 +Σ2)−1

)

, v =

(

v1

v2

)

,

∆x =

(

∆11 ∆12

∆21 ∆22

)

, ∆11 = ∆1 + Γ1Σ1ΓT1 − Γ1Σ1 (Σ1 +Σ2)

−1Σ1Γ

T1 ,

∆22 = ∆2 + Γ2Σ2ΓT2 − Γ2Σ2 (Σ1 +Σ2)

−1Σ2Γ

T2 ,

∆12 = −Γ1Σ1 (Σ1 +Σ2)−1

Σ2ΓT2 . (A-4)

Lemma 2: if x ∼ CSNnx,qx(µx,Σx,Γx,vx,∆x) and A is ny×nx matrix (ny ≤ nx) then y =Ax is CSN, y ∼ CSNny,qy(µy,Σy,Γy,vy,∆y), where qy = qx, µy = Aµx, Σy = AΣxA

T ,

Γy = ΓxΣxATΣ−1

y , vy = vx and ∆y = ∆x + ΓxΣxΓTx − ΓxΣxA

TΣ−1y AΣxΓ

Tx .

Lemma 3: if x ∼ CSNnx,qx(µx,Σx,Γx,vx,∆x) then E [x] = µx + ΣxΓTxΨ. Where Ψ =

Φ∗qx

(

0;V ,∆x + ΓxΣxΓTx

)

Φqx

(

0;V ,∆x + ΓxΣxΓTx

) and Φ∗qx (x;µ,Ω) = [∇xΦqx (x;µ,Ω)]T .

Sampling from a CSN distribution is relatively easy by rejection sampling. For example,let E1 ∼ φnx (E1; 0,Σx) and E2 ∼ φqx (E2; 0,∆x) be independent random vectors, thenx = µx +E1 ∼ CSNnx,qx(µx,Σx,Γx,vx,∆x) if y = −vx + ΓxE1 +E2 ≥ 0.

APPENDIX B

BAYESIAN INVERSION IN CSN SETTING

Assume x ∼ CSNnx,qx(µx,Σx,Γx,vx,∆x) and also linear measurement with CSN dis-tributed errors as d = Hx + ed|x, where ed|x ∼ CSNnd,qd

(

0,Σd|x,Γd|x,vd|x,∆d|x

)

, letx = [t|u ≥ 0] and ed|x = [s|v ≥ 0]. By using the previous formula for finding the jointdistribution of all random vector and this assumption that t, s and v are mutually inde-pendent, and also u and v are mutually independent, we have:

t

r = Ht+ s

u

v

∼ φnx+nd+qx+qd(

µx

Hµx

−vx

−vd|x

,

Σx ΣxHT ΣxΓ

Tx 0

HΣx HΣxHT +Σd|x HΣxΓ

Tx Σd|xΓ

Te

ΓxΣx ΓxΣxHT ∆x + ΓxΣxΓ

Tx 0

0 Γd|xΣd|x 0 ∆d|x + Γd|xΣd|xΓTd|x

). (B-1)

Besides, from analysis of multivariate normal distribution:

[

x1

x2

]

∼ φnx1+nx2

([

µ1

µ2

]

,

[

Σ1 Σ12

Σ21 Σ2

])

. (B-2)



[x1|x2] ∼ φnx1

(

x1;µ1 +Σ12Σ−12 (x2 − µ2) ,Σ1 −Σ12Σ

−12 Σ21

)

. (B-3)

By re-arranging equation (B-1) in accordance with equation (B-2) and using equation (B-3)and matrix multiplication we have:

t|ru|rv|r

∼ φnx+qx+qd

µt|r

µu|r

µv|r

,

Σt|r Σtu|r Σtv|r

Σut|r Σu|r Σuv|r

Σvt|r Σvu|r Σv|r

, . (B-4)

µt|r = µx +ΣxHT[

HΣxHT +Σd|x

]−1(r −Hµx) ,

µu|r = −vx + ΓxΣxHT[

HΣxHT +Σd|x

]−1(r −Hµx) ,

µv|r = −vd|x + Γd|xΣd|x

[

HΣxHT +Σd|x

]−1(r −Hµx) ,

Σt|r = Σx −ΣxHT[

HΣxHT +Σd|x

]−1HΣx,

Σu|r =[

∆x + ΓxΣxΓTx

]

− ΓxΣxHT[

HΣxHT +Σd|x

]−1HΣxΓ

Tx ,

Σv|r =[

∆d|x + Γd|xΣd|xΓTd|x

]

− Γd|xΣd|x

[

HΣxHT +Σd|x

]−1Σd|xΓ

Td|x,

Σut|r = ΓxΣx − ΓxΣx

[

HΣxHT +Σd|x

]−1HΣx,

Σvt|r = 0− Γd|xΣd|x

[

HΣxHT +Σd|x

]−1HΣx,

Σvt|r = 0− ΓxΣxHT[

HΣxHT +Σd|x

]−1Σd|xΓ

Td|x. (B-5)

Let w =

[

u

v

]

, then:

[

t|rw|r

]

∼ φnx+qx+qd

([

µt|r

µw|r

]

,

[

Σt|r Σtw|r

Σwt|r Σw|r

])

. (B-6)

From the definition of the CSN model:

π(x|d) = π(t|r,w ≥ 0) = CSNnx,qx+qd(µx|d,Σx|d,Γx|d,vx|d,∆x|d). (B-7)

where parameters are as defined in equation (12).

APPENDIX C

OBSERVATION DISTRIBUTION

We assume data d = Hx+e, where x ∼ CSNnx,qx(µx,Σx,Γx,vx,∆x) and e ∼ CSNnd,qd(0,Σd|x,Γd|x,vd|x,

By using Lemma 2, Hx ∼ CSNnd,qx(µp,Σp,Γp,vp,∆p), where µp = Hµx, Σp = HΣxHT ,

Γp = ΓxΣxHTΣ−1

p , vp = vx and ∆p = ∆x + ΓxΣxΓTx − ΓxΣxH

TΣ−1p HΣxΓ

Tx .



From Lemma 1 and previous result, we have d ∼ CSNnd,qx+qd(µd,Σd,Γd,vd,∆d):

µd = µp + µd|x, Σd = Σp +Σd|x, Γd =

(

ΓpΣpΣ−1d

Γd|xΣd|xΣ−1d

)

, vd =

(

vp

vd|x

)

,

∆d =

(

A11 A12

A21 A22

)

, A11 = ∆p + ΓpΣpΓTp − ΓpΣpΣ

−1d ΣpΓ

Tp ,

A22 = ∆d|x + Γd|xΣd|xΓTd|x − Γd|xΣd|xΣ

−1d Σd|xΓ

Td|x,

A12 = AT21 = −ΓpΣpΣ

−1d Σd|xΓ

Td|x. (C-1)



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