Value of Information Analysis and Bayesian Inversion for Closed Skew-Normal Distributions: Applications to Seismic Amplitude Versus Offset Data

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1 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 assumptions. This paper presents results for Bayesian inversion and value of information calculations when the input distributions are skewed and non-gaussian. Reservoir parameters and seismic amplitudes are often skewed and by using models which capture the skewness of distributions, the input assumptions are less restrictive and the value of information analysis is more reliable. We use a Closed Skew Normal distribution for the saturation and porosity variates and the seismic amplitude data. Sensitivity of the value of information analysis to skewness, mean values, accuracy and correlation parameters 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 structures, lithologies and hydrocarbon indicators. However, these interpretations are associated with uncertainty. In evaluating the value of data under uncertainty, along with the cost of the data, it is the impact of the data on the underlying development decision that has to be critically considered. The goal is to maximize expected revenue, not to reduce uncertainty per se. A decision-theoretic value of information (VOI) framework provides a very useful tool for evaluating seismic data in this context. VOI analysis relates to making better decisions under uncertainty (Raiffa, 968; Howard, 996). It is an old concept in the petroleum industry (Grayson et al., 962), and it seems to havegainedmoreinterestinrecentyears(branco et al.,25;bickel et al.,28;bratvold et al., 29; Bhattacharjya et al., 2). The VOI is useful in several petroleum applications where one 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. If the price of data is larger than the VOI, the data are not worth purchasing (Bratvold et al., 29). Consider the Expected revenues WITH additional information minus the Expected revenues WITHOUT additional information. The former is sometimes called posterior value, while the latter is the prior value. Then, VOI = Posterior Value Prior Value. () Our main example in this paper concerns the decision of drilling / not drilling new wells

2 Rezaie, Eidsvik & Mukerji 2 Geophysics at a number of selected reservoir units (Eidsvik et al., 28). We assume the following scenario: Initial reservoir analysis has been done with post-stack data, and now the decision is whether or not to purchase better processed pre-stack seismic data for amplitude versus offset (AVO) based reservoir characterization. Good AVO data, calibrated and interpreted with appropriate rock physics models might help with better drilling decisions by indicating presence or absence of hydrocarbons and reservoir quality. The VOI analysis is useful for helping with this auxiliary decision about obtaining AVO datathatislinkedtothemaindecisionaboutdrilling. IftheVOIissmall, thenewamplitude dataisunlikely tohelpusmuchinmakingthedrillingdecisions. Inthis situation theseismic re-processing may be too expensive compared with its actual added value. In practice a well can of course be dry, even though the data VOI looks promising. This happens as a result of variability in the data which arises from several factors including subsurface heterogeneity and noise. The VOI performs an average calculation over all possible datasets we might get in the seismic re-processing, given our prior knowledge about the reservoir rock and fluid properties, as well as the seismic model. As we will show, the reservoir parameters and seismic amplitude measurements are skewed 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 VOI analysis are then more reliable. We introduce a Closed Skew Normal (CSN) family of distributions for the saturation and porosity variates, and for the seismic amplitude data. This extension provides flexibility in the working assumptions. In fact, if the Gaussian approximation is the best, the fitted CSN distribution converges to the Gaussian, otherwise it works better by capturing higher order moments of the data. Our approach extends that of Eidsvik et al. (28). Notation NOTATION AND BACKGROUND MATERIAL Throughout this paper, we use x R nx as a n x dimensional distinction of interest. In the numeric example this will contain variables related to the saturation and porosity in reservoir units. Using appropriate transformations of variables, the vector x has real entries. The notation x π( ) is used to show that x is distributed according to probability density function (pdf) π(x). For a Gaussian model we have pdf π(x) = φ nx (x;µ,σ) and the associated cumulative distribution function (cdf) is denoted Φ nx (x;µ,σ). Here, the mean is µ R nx and the positive definite covariance matrix is Σ R nx nx. Besides, the notation π(x) = CSN nx,q x (µ,σ,γ,v, ) means a CSN pdf with parameters µ R nx, Σ R nx nx, Γ R qx nx, v R qx and R qx qx, where Σ and are positive definite matrices. A random vector x R nx is CSN distribution with q x,µ,σ,γ,v, parameters if its pdf is as follows (See Appendix A for details): x CSN nx,q x (x;µ,σ,γ,v, ) = [ Φ qx ( ;v, +ΓΣΓ T )] Φqx (Γ(x µ);v, )φ nx (x;µ,σ). (2)

3 Rezaie, Eidsvik & Mukerji 3 Geophysics We use d R n d to denote the seismic amplitude data. The data consist of indirect observations of the reservoir parameters; the zero offset and seismic AVO gradients at all reservoir units. We assume a weakly nonlinear relationship d = h(x)+e. Here, e is additive measurement noise with pdf π(e). We will linearize the measurement equation using first order Taylor series expansion to get h(x) h +Hx, where H R n d n x. The likelihood model for the data is denoted π(d x), with parameters derived from d = h +Hx+e. The marginal distribution for the data is π(d), obtained by integrating out all reservoir variables from the joint model for x and d. Seismic data relations The seismic amplitude versus offset (AVO) data is related to the reservoir saturation, s, and porosity, ϕ, through rock physics relations. Here, the data consist of zero-offset reflectivity and AVO gradient at top reservoir. We assume the elastic properties of the cap rock are fixed. The elastic properties of the reservoir layer are uncertain because they depend on saturation and porosity. The likelihood model combines the seismic measurement equations with associated uncertainties. Saturation and porosity are two key reservoir parameters which directly relate to the drilling decision. The prior information about these parameters provides rough knowledge with large uncertainty. These parameters are limited to change between some lower/upper limits, say s min s s max and ϕ min ϕ ϕ max. Logistic transformations define x s = log ( s smin s max s ) and x ϕ = log ( ϕ ϕmin ϕ max ϕ ), and we have x s and x ϕ on the real line. The inverse logistic function transforms back to their actual range: s = exp(x s ) +exp(x s ) s max and ϕ = +exp(x ϕ ) ϕ min + exp(x ϕ) +exp(x ϕ ) ϕ max. +exp(x s ) s min+ The rock physics models consist of relations for bulk modulus and shear modulus as a function of porosity for % brine saturation, which are then converted to other saturations using Gassmann s equations. There are various functional forms for the modulus-porosity relations, depending on sorting, cementation, and diagenesis. For example, Bachrach (26) 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 reasonable 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(s max,ϕ). (3) The effective fluid bulk modulus for mixed saturations is given by the Reuss average of the individual fluid moduli: ( s K f = + s ), (4) K b K o where K b and K o are the fixed bulk modulus of oil and brine, respectively. The rock bulk

4 Rezaie, Eidsvik & Mukerji 4 Geophysics 2 8 Moduli porosity relations for % brine saturation K G 6 4 K, G (GPa) Porosity Figure : Brine-saturated bulk modulus, K, and shear modulus, G, versus porosity. modulus for saturation and porosity is obtained by Gassmann s formula: K = BK q +B, B = K(s max,ϕ) K q K(s max,ϕ) K b ϕ(k q K b ) + K f ϕ(k q K f ), (5) where K q is the fixed quartz bulk modulus and K(s max,ϕ) is the bulk modulus fitted from well data in brine sand (see Figure for the values we use here). The density is given by: ρ = ϕsρ b +ϕ( s)ρ o +( ϕ)ρ q, (6) where ρ b, ρ o and ρ q 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 density through: K + 4 V p = 3 G G, V s = ρ ρ. (7) Finally, the zero-offset reflectivity d and AVO gradient d 2 can be evaluated from the elastic constraints in the caprock and the reservoir. We use the Aki and Richard s formula to

5 Rezaie, Eidsvik & Mukerji 5 Geophysics Zero offset Reflectivity..5 AVO Gradient Porosity Saturation Porosity Saturation.8 Figure 2: Zero-offset reflectivity (left plot) and AVO gradient (right plot) versus saturation and porosity. evaluate the expected values of d and d 2 (Mavko et al., 23): E[d x] = ( δvp + δρ ρ ), 2 V p E[d 2 x] = δv p 2 V p 2 Vs 2 V p 2 ( ) δρ ρ +2δV s. (8) V s where conditioning on x means using all the formulas in (2)-(6). In addition, for general property ǫ, ǫ = ǫ+ǫ cr and δǫ = ǫ ǫ cr are the average and the difference between reservoir 2 and caprock property, respectively. Based on the presented relations, we see that seismic attributes relate to the saturation and the porosity through nonlinear relationships. Figure 2 shows the modeled zero-offset reflectivity and AVO gradient versus the saturation and porosity where. s.9 and. ϕ.5. The Society of Petroleum Engineers (SPE) organized a series of projects, known as SPE comparative projects, in order to provide benchmark data sets which can be used to compare the performance of different algorithms and methods. The th SPE comparative project is the latest one in this series and known as the SPE data set (Christie and Blunt, 2). The SPE data set consist of porosity for Cartesian grid cells. By using this data set as the input to a reservoir flow solver (simulator), the saturation and porosity of that reservoir can be used for further evaluations. For our simulations, we use the porosity and permeability from the SPE data set as the reservoir parameters. Then we use MATLAB Reservoir Simulator Toolbox (MRST), see Lie et al. (22), as flow

6 Rezaie, Eidsvik & Mukerji 6 Geophysics.5 Empirical Skew Normal Normal.8.6 Empirical Skew Normal Normal π(x s ) π(x φ ) Logistic Brine Saturation (x ) s Logistic Porosity x φ 25 2 Empirical Skew Normal Normal 4 3 Empirical Skew Normal Normal π(r ) 5 π(g) Zero offset reflectivity AVO gradient Figure 3: A graphical description of distribution fitting based on the SPE data, for saturation (upper left plot), porosity (upper right plot), zero-offset reflectivity (lower left plot) 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 at Figure 3 shows the empirical marginal distribution of the logistic transformed saturation and porosity (top) and the zero-offset reflectivity and AVO gradients (bottom). Then the reservoir porosity and permeability are selected from the SPE data set and we run the simulations on MRST for days and record the saturation. By using the saturation and porosity 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 (blocks with poor permeability and porosity). We first approximate these empirical distributions with the Gaussian distributions, and then with an skewed distribution (Figure 3). According to these plots, the distribution of the logistic porosity seems to be Gaussian, since the fitted results for Gaussian and skewed normal models are similar. For the rest, their marginal distributions are not symmetric and consequently neither are their joint distributions. Thus, modeling them with the Gaussian distribution may result in inaccurate calculations. This motivates the use of the CSN distribution as a more sophisticated and flexible class. DISTRIBUTION ASSUMPTIONS AND CLOSED SKEW NORMAL Statistical modeling is an important step in dealing with uncertainties. The statistical distributions can be divided in three general categories: i) Continuous, (Karimi et al., 2),

7 Rezaie, Eidsvik & Mukerji 7 Geophysics.4 π(x ).35 π(x 2 ).3.25 π (x,x 2 ) x 2 x Figure 4: The bivariate CSN distribution (3D plot) and its marginals (2D plots on the sides) which are also skewed. ii) Discrete, (Ulvmoen and Hammer, 2) and iii) Mixed, (Grana et al., 22), consisting of both continuous and discrete components. Continuous distributions are the most useful in our applications. Closed skew normal distribution The Skew Normal (SN) random variable is an extension of the Gaussian. The SN distribution has one more parameter for capturing the skewness (Azzalini and Dalla-Valle, 996; Arellano-Valle et al., 22; Liseo and Loperdo, 23; Gupta et al., 24). The SN distribution has some properties similar to the Gaussian distribution; unimodal, support is the real line and the distribution of the square of a SN random variable is χ 2 with one degree of freedom (Azzalini and Capitanio, 999). Thedistributionof asn randomvector x R nx is: x SN nx (x;µ x,σ x,γ x ) = 2φ nx (x;µ x,σ x )Φ (Γ x (x µ x );,), (9) where Γ x R nx. For Γ x =, it is identical to the Gaussian pdf with mean µ x and covariance Σ x. The SN distribution extends to the more general family of CSN. This is done by adding two more parameters, and via these additions, we get nice analytical properties (Domínguez-Molina et al., 23; Genton, 24; González-Farías et al., 24). For instance, they are closed under addition, and the general full rank linear transformation of a CSN random vector is also CSN. Consequently, the marginal distributions of a multivariate CSN are also CSN distributed (González-Farías et al., 24). Figure 4 shows a bivariate CSN distribution and its marginals. We see that both joint and marginals are skewed.

8 Rezaie, Eidsvik & Mukerji 8 Geophysics The CSN distribution is more general than the SN and the Gaussian. For instance, SN distribution is an special case of the CSN when v = and =. Also, the Gaussian distribution is a special case of the CSN if q x = or Γ = (See Appendix A for details). Besides, similar to the Gaussian distributions if the prior and likelihood are both CSN, the posterior is also CSN (See Appendix B for details). This is one important property of the CSN distribution which is very useful for Bayesian inversion problems (Karimi et al., 2). BAYESIAN INVERSION Assume we have some prior knowledge about the distribution of interest. In addition, we can acquire measurements which directly or indirectly relate to these variables. In our setting π(x) forms the prior model for reservoir variables, while π(d x) is the likelihood of the seismic AVO data, given the reservoir variables. Bayes theorem combines prior information and current observations, in the probability domain, in order to get the posterior distribution of the variables of interest: π(x d) π(d x)π(x). () Figure 5 shows a schematic diagram of a Bayesian inversion problem from a distributional point of view. The prior knowledge (upper left plot) is here a bivariate distribution. Bayes rule adds the information content of the likelihood distribution (lower left plot) to it. The posterior distribution is shown in the right plot, and we see that the uncertainties are reduced. If the prior distribution is a Gaussian distribution, ( x φ ) nx (x;µ x,σ x ), and the likelihood is a Gauss linear distribution, d x φ nd d;hx,σd x, the posterior distribution ( is also Gaussian, x d φ nx x;µ x d,σ x d ), and there are analytical formulations for its parameters (Buland and Omre, 23): µ x d = µ x +Σ x H T [ HΣ x H T +Σ d x ] (d Hµx ), Σ x d = Σ x Σ x H T [ HΣ x H T +Σ d x ] HΣx. () Another example of closed-form solutions for the posterior is when the distributions are modeled as a Gaussian mixture distribution (Grana et al., 22). In most other situations, there are no closed-form solutions for the posterior distribution π(x d). In Figure 5, we have used a CSN distribution for the prior and likelihood model. The posterior model is then also CSN in this situation, and this makes the CSN very applicable. Precisely speaking, if theprioris x CSN nx,q x (µ x,σ x,γ x,v x, x ) andthelikelihood is d x CSN nd,q d (Hx,Σ d x,γ d x,v d x, d x ), thenx d CSN nx,q x+q d (µ x d,σ x d,γ x d,v x d, x d ),

9 Rezaie, Eidsvik & Mukerji 9 Geophysics Figure 5: Illustration of Bayesian inversion. The upper left plot is the prior distribution which is CSN, the lower left plot is the likelihood which is CSN and the right plot is the resulting posterior which is CSN too. where (See Appendix B for details): µ x d = µ x +Σ x H T [ HΣ x H T +Σ d x ] (d Hµx ), Σ x d = Σ x Σ x H T [ HΣ x H T ] HΣx +Σ d x, [[ ] [ Γx Σ Γ x d = x Γx Σ x H T ] ] [HΣx H T HΣx +Σ Γ d x Σ d x] Σ x d, d x [ ] [ vx Γx Σ v x d = + x H T ] [HΣx H T ] (d Hµx +Σ v d x Γ d x Σ d x ), d x [ x +Γ x ΣΓ T ] x x d = d x +Γ d x Σ d x Γ T, d x [ Γx Σ x H T ] [ [HΣx H T Γx Σ +Σ x H T ] T Γ d x Σ d x] Γ d x Γ d x Σ x dσ x dγ T x d. (2) d x These relations are also valid when the prior and/or likelihood are Gaussian. For instance, if thepriordistributionisgaussian,x φ nx (x;µ x,σ x ),theposterioriscsnwithparameters by Γ x =, v x = and x = I. Thus, we can get the Gaussian inversion results, equation (), as a special case of CSN inversion by setting Γ x =, v x =, x = I, Γ d x =, v d x = and d x = I. Although CSN random variables are more general than the Gaussian one, and they extend the Gaussian distribution for handling asymmetry and skewness, the added flexibility comes at the cost of more computational tasks: We must use the maximum likelihood for parameter estimation, but faster methods exist for some predefined structures (Flecher et al., 29). The sampling problem also becomes harder

10 Rezaie, Eidsvik & Mukerji Geophysics (See Appendix A for details). Simulations is done by rejection sampling which means we generate 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 of these calculations are readily available for the CSN distribution. We will next outline methods for computing the VOI related to drilling / not drilling at prospects depending on the oil saturation and porosity. The VOI is computed for seismic AVO attributes, which are informative of the saturation and porosity values. We next introduce the detailed requirement assumptions for this setting, see also Eidsvik et al. (28). In addition to the saturation and porosity variables at all prospects, we use a scaling term toconvert tomonetary units. We have a cost C of drillinganewwell to hiddenpockets of oil, and we are interested in maximizing expected profit. Assume we have M prospects, the value of a prospect i =,...,M, is Rφ i ( s i ), where ( s i ) is the oil saturation, and R is a factor defined as the product of the following: The area of the prospect A i, the thickness of the reservoir h i, and a factor R which includes the oil price, recovery rates, and net-to-gross. Other values are possible here. Note that only saturation and porosity are taken as key uncertain reservoir parameters, while the others are kept fixed. We define the VOI as: VOI = Posterior value Prior value, M Prior value = max{re[φ i ( s i )] C,}, Posterior value = i= M i= max{re[φ i ( s i ) d] C,}π(d)dd. (3) Note that the prior value for prospect i is the expected value over units E[Rφ i ( s i )] = RE[φ i ( s i )], minus the cost of drilling. The decision maker will only drill if the expected profits are positive. We assume a risk-neutral decision maker and hence can define value in terms of expectations. In general this may not be true and the decision maker s utility function should then be used in defining values. Considering one saturation and one porosity variable for each prospect like in equation (3), the prior is CSN with n x = 2M and q x = 2M. AVO data is considered as potentially measured data, d. By considering zero-offset reflectivity and AVO gradient as measurements for each prospect, the likelihood is CSN with n d = 2M and q d = 2M. The expectation in the posterior value is a conditional expectation over the seismic observations. According to the previous section, the posterior distribution of logistic saturation x s and porosity x ϕ is CSN with n x = 2M and q x = q x + q d, and the posterior parameters are analytically available by equation (2). Note that the posterior value is an

11 Rezaie, Eidsvik & Mukerji Geophysics Computing time Monte Carlo estimates Mean Upper 95 Lower Monte Carlo sample size.5.5 With GPU Without GPU Monte Carlo sample size Figure 6: Illustration of, MC approximation of the VOI. Top: MC estimates and MC uncertainty as a function of sample size. Bottom: A comparison between CPU computation time and GPU computation time. integral expression over all possible seismic AVO data d with distribution π(d) (See Appendix C for details). This integral is estimated using approximations that will be explained in the next section. Since we condition on various data, the posterior value is larger than the 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 have an expectation over the product of oil saturation and porosity E[ϕ( s)]. For evaluating these expectations we need the related distributions, then these expectations can be calculated numerically, by Monte Carlo (MC) simulation or analytically. There is an analytical formulation for calculating the mean value of a CSN random variable. Thus, we know that the posterior distribution of the logistic saturation x s and porosity x ϕ is CSN, but the posterior of the original saturation s and porosity ϕ is not CSN. One straightforward way to handle this problem is generating many samples from transformed variables (x s and x ϕ ), then using the inverse transformation to get samples of s and ϕ and finally calculate the empirical mean. This method is time consuming. Another way is using a reasonable

12 Rezaie, Eidsvik & Mukerji 2 Geophysics approximation for the expected value. For any function f (x), if the firstorder Taylor series expansion is a good approximation, the calculation of E[f (x)] is easy. Because, if f (x) = a +a x + a 2 x 2 + a + a x, then E[f (x)] f (E[x]). For our problem, f(x) = f(x,x min,x max ) = +exp(x) x min + exp(x) +exp(x) x max. We analytically calculate the expectation of transformed saturation and porosity, then put this expectation into the inverse logistic transformation: E[ϕ( s)] = E[f(x ϕ,ϕ min,ϕ max )( f(x s,s min,s max ))] f(e[x ϕ ],ϕ min,ϕ max )( f(e[x s ],s min,s max )). (4) This trick reduces the computation cost dramatically and for our application the approximation seems to work very well. The integral is over seismic AVO data d approximated by MC integration by generating many i.i.d. samples from the observation distribution and then take the average of conditional values. More precisely, if we have W i.i.d. samples, d,d 2, d W from pdf π(d) then π(d) W W w= 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) W W δ(d d w )dd = W w= W g(d w ), (5) w= Here, g(d) = max{re[φ( s) d] C,}. Note that we calculate the posterior parameters for each d w. But, by looking at the posterior parameters formulations in equation (2), we see that all parameters are constant for all d w s, except µ x d and v x d. Thus, we just need to calculate the posterior parameters one time and then use them, updating only the d w in µ x d and v x d. The organized version of this VOI calculation is presented in Algorithm. Algorithm VOI Calculation : 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 = to W do 6: Generate a sample from observation distribution, d w π(d). 7: Construct the posterior distribution for the logistic parameters given d w (equation (2)). 8: Construct the value according to equation (4). 9: end for : Approximate the integral part in the posterior value as an average over all W runs (equation (5)). : 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 rejection sampling 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) for generating many samples from the associated Gaussian distribution and then apply rejection for choosing correct samples. Figure 6 (bottom) shows the computation time trends when we generate samples by CPU and GPU processors and compute the associated VOI (top) for a CSN model of dimension n x = n d =, with MC uncertainty bounds. According to this plot, we see that by increasing the number of samples, the computation time for the

13 Rezaie, Eidsvik & Mukerji 3 Geophysics VOI.35 VOI Γ x Σ x.8.5 No Γ x Very Low Σ x.6 Low Γ x Medium Γ x Low Σ x Medium Σ x π (x).4 High Γ x π (x) High Σ x x 5 5 x Figure 7: VOI calculation with perfect information: effect of Γ x and Σ x on the VOI (upper plots) and on the pdf (lower plots). CPU increases with a linear rate, but it is almost constant when we use the GPU. Here, we use the gfor routine in Jacket MATLAB software, which runs a for loop in parallel on the GPU. The comparison was done on an 2 x 6-core Intel Xeon X GHz CPU with 96 GB 333 MHz DDR3 memory. The GPU is a nvidia Tesla C25. Perfect information NUMERIC EXAMPLES Inthissimpleexample, weanalyze thesensitivity of VOIas afunctionofscale parameter Σ x andskewnessparameterγ x. Consideraunivariaterandomvariablex CSN, (µ x,σ x,γ x,v x, x ). Then VOI = max(x,)π(x)dx max[e(x),]. Assume that E[x] = for all Σ x and Γ x. Figure 7 shows the sensitivity of VOI for a range of. Σ x 2 and. Γ x 2. The left plots in Figure 7 show the effect of Γ x parameter on the VOI (upper left plot) and on the pdf π(x) (lower left plot) for Σ x =. As we see in the upper left plot, by increasing the value of Γ x the VOI decreases. As you see in lower left plot, by increasing Γ x 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 Σ x for Γ x =. By increasing Σ x the prior distribution becomes wider and the VOI increases.

14 Rezaie, Eidsvik & Mukerji 4 Geophysics 6 x 5 5 CSN Normal 4 VOI x Data Error Level CSN without approximation CSN with approximation in mean CSN with median VOI Data Error Level Figure 8: VOI for the Gaussian and CSN modeling assumptions: The top plot is the comparison between VOI with Gaussian and CSN input assumption. The bottom plot is the VOI for different approximations based on equation (4) and the exact solution based on MC sampling, and also using the median instead of the mean. One prospect case Consider M =, let (s,ϕ) T be the reservoir parameters of interest and d = (d,d 2 ) T the AVO measurements. The saturation is generated by MRST based on the SPE data, which also input porosities. In addition, the seismic data d is generated for the given saturation and porosity, and in accordance with the relations presented above. Now, we consider two cases for the prior and likelihood fitting, i) prior modeled with CSN distribution, ii) prior modeled with Gaussian distribution. For both cases, the posterior distributions are analytically calculated and we approximate the VOI. We use a Gaussian distribution to model the likelihood for both cases. The reason for choosing this likelihood model 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 the measurement error variance. If the data processing price is over and under both plots, the resulting decision for both is similar, although the predicted final revenue is different. But if this price is in between two curves then the resulting decision about purchasing data is different. Although the VOI calculation in the CSN case is higher, it does not mean that the CSN case is better. The CSN case will however give more reliable decisions, because it is a more accurate description of the data. This is because of structure of CSN distribution. Recall that the CSN automatically sets Γ = if the data are Gaussian. From now on, we focus only on the CSN case. The upper right plot represent the results of using different methods for calculating the expectations in the prior and posterior

15 Rezaie, Eidsvik & Mukerji 5 Geophysics 6 x 5 VOI CSN Data Price Scenario 7 x 5 VOI CSN Data Price Scenario VOI 3 VOI Z 2 Z Z Z 4 Z Data Error Level Z Data Error Level Figure 9: The effect of the data price on the decision about purchasing various data processing scenarios. values. According to this plot we see that using the fast first order Taylor series expansion of the inverse logistic transformations works very well,since the VOI based on this analytical approach is almost equal to the MC approximation. In addition, we see that the result of using the median statistic instead of the mean also gives similar VOI. The reason for choosing the median statistic is that the median of a monotone function of a random variable is the same 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. As you see, increasing the data accuracy, increases the VOI. On the other hand, more accurate data needs more data processing and it increases the cost of data. Thus, we can consider a one to one relation between accuracy and costs, and having the relative position of VOI and data costs as a function of data processing level. If the price of processing data is less than VOI, it is worth buying it. For understanding the idea behind required data processing level, consider scenario (left), where the cost of data processing is defined to be linearly dependent on the processing level. When the level of seismic AVO data processing is in zone Z, it is not worthwhile to buysuch data, becausethe VOI is less than thedata cost. By increasing thedata processing level and entering zone Z 2 the VOI increases with higher rate than the processing cost and it is worthwhile purchasing data in this zone. In reality there are often more limitations. It is obvious that more data processing needs more expert people, more time, more resources etc. These limitations may change the scenario of data processing price. Consider that the time is one more limitation, and after a maximum number of days we are penalized step by step. This penalty represents itself in the data processing cost by increasing the rate of change of costs (right plot of

16 Rezaie, Eidsvik & Mukerji 6 Geophysics Figure 9). For this case, when we are in zone Z, it is not worth buying data. By increasing the seismic AVO data processing level (zone Z 2 ), it is worthwhile. The most revenue is achieved at the end of this zone. If we need more time for data processing, we are penalized with new data processing cost rates. In zone Z 3, data are valueable, but the final revenue reduces. In zone Z 4 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 we consider the likelihood to be CSN. Table shows the sensitivity of VOI with respect to the Σ d x and Γ d x parameters of data likelihood distribution. As is shown, by increasing Σ d x and Γ d x the VOI decreases but with different rate. The rate of the VOI decreases as a function of Σ d x is higher than Γ d x. This fact means the sensitivity of VOI to Σ d x is higher than Γ d x. In addition, when we increase Σ d x and decrease Γ d x with the same scale, the VOI decreases which is another indicator for the superior effect of Σ d x on the VOI. Γ d x Σ d x No Low Medium High Low Medium High Table : The effect of data processing level on the VOI for one prospect case (in units of thousand dollar). In the above analysis we have assumed drilling costs C = 2 million dollar. We will now study the influence of drilling cost on the VOI. Simulation results show that for the drilling cost from zero up to.9c, the VOI is zero and the decision maker decides to drill without purchasing any additional data. The VOI is also zero when the drilling cost is higher than.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 VOI is 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 VOI using CSN distribution. We consider two different cases i) dependent prior and ii) independent prior. By dependent prior, we assume an exponentially decreasing spatial correlation between grid cells as follows: c(,)σ x c(,m)σ x c(2,)σ x c(2,m)σ x Σ =....., (6) c(m,)σ x c(m,m)σ x where Σ x is the 2 2 covariance between logistic porosity and saturation in one reservoir unit, c(i, j) = exp(.3distance(i, j)) and distance(i, j) is the Euclidean distance between prospect i and j. Other distances, for example, following stratigraphic horizons, can also be used. Assume cell i and j has (xx i,yy i ) and (xx j,yy j ) as their positions in a two

17 Rezaie, Eidsvik & Mukerji 7 Geophysics dimensional Cartesian plane, then distance(i,j) = (xx i xx j ) 2 +(yy i yy j ) 2. This correlation assumption means over the top-reservoir that closer prospects are more correlated. The SPE data are used for the prior mean and the seismic AVO data are generated according to equation (7). Γ d x Prior Σ d x No Low Medium High Correlated Low Medium High Independent Low Medium High Table 2: VOI results. The effect of prior correlation and skewness between prospects (in million dollars). Table2presentsthesimilarresultsasTablefortheVOIforvariousscales, skewnessand for low versus high spatial correlation. We see that for correlated and uncorrelated prospects by increasing Σ d x and/or Γ d x, the VOI decreases. In addition, the VOI decreases more slowerbyincreasingγ d x incomparisontoσ d x. Inaddition, correlationassumptionbetween prospects in the prior results in higher VOI than independent prior. When we have some information about prospect i (through measurement data for prospect i), and it is highly correlated to prospect j, we have information about prospect j indirectly. Consequently, the seismic AVO data is valueable and results in higher VOI. Finally, simulation results show that by considering correlation between observations, the resulting VOI is lower than for independent likelihood. This means that the observations are more valueable when they are independent, providing more information about the prospects. Note that for all cases in this simulation, when we change the Γ d x and Σ d x, we adapt the mean of the likelihood distribution to be of the same order to remove the effect of the mean on the VOI. CLOSING REMARKS We introduce skewed distributions for the modeling of reservoir parameters and geophysical measurements. The CSN distributions are closed under linear combinations and conditioning. 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 problems. CSN distribution is more flexible than the classical Gaussian distribution for data with skewed distributions. We fit CSN models to the SPE data set and check the sensitivity of the VOI to the model parameters. The Gaussian assumption directs the decision maker to wrong decisions when the data processing cost is in between the evaluated Value Of Information of the Gaussian and CSN. Simulation results show that by increasing the Σ d x and/or Γ d x parameters of data distribution, the VOI decreases, and it is more sensitive to Σ d x than Γ d x.

18 Rezaie, Eidsvik & Mukerji 8 Geophysics One of the computational challenges in the VOI calculation is evaluation of the integrals. 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, variance and skewness on the VOI in real systems. Seismic data sets are massive in size, checking the effect of data reduction algorithms on the efficiency of the proposed algorithm would be useful. Another interesting topic will be using the concept of VOI in the spatio-temporal Bayesian 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 Norwegian University of Science and Technology (NTNU). We further thank Stanford Center for Reservoir Forecasting (SCRF), Department of Energy Resources Engineering (ERE) at Stanford University for providing the facilities for working together. We thank SINTEF for the MATLAB reservoir simulation toolbox. APPENDIX A PROPERTIES OF THE CLOSED SKEWED NORMAL DISTRIBUTION [ Assume ] x and y ([ are ] respectively [ n x ]) and n y random vectors and are jointly normal, x µx Σx Σ N y nx+n y, x,y Then, we have: µ y Σ y,x Σ y π(y x)π(x) π(z) = π(x y ) = π(y ) = [ ( )] [ ( Φ ny ;µy,σ y Φ ny ;µ y x,σ y x )]φ nx (x;µ x,σ x ). (A-) where µ y x = µ y +Σ y,x Σ x (x µ x ), Σ y x = Σ y Σ y,x Σ x Σ x,y. By re-writing the above equation to get a standard form, we have: z π(z) = CSN nz,q z (µ,σ,γ,v, ) = [ Φ qz ( ;v, +ΓΣΓ T )] Φqz (Γ(z µ);v, )φ nz (z;µ,σ). (A-2) Where n z = n x, q z = n y, µ = µ x, Σ = Σ x, Γ = Σ y,x Σ x, v = µ y and = Σ y Σ y,x Σ x Σ x,y. Similarly, if we have a CSN distribution we can construct the original unconditional jointly normal distribution: [ ] ([ ] x µz Σz Σ φ y nz+q z,[ z Γ T ]) z v z Γ z Σ z z +Γ z Σ z Γ T. (A-3) z CSN distributions have some properties that are similar to Gaussian.

19 Rezaie, Eidsvik & Mukerji 9 Geophysics Lemma : if x CSN nx,q x (µ,σ,γ,v, ) and x 2 CSN nx,q x2 (µ 2,Σ 2,Γ 2,v 2, 2 ) areindependentcsnrandomvariables, thenx = x +x 2 isalsocsn,x CSN nx,q x +q x2 (µ x,σ x,γ x,v x, x where: ( Γ Σ µ x = µ +µ 2, Σ x = Σ +Σ 2, Γ x = (Σ +Σ 2 ) ) ( ) v Γ 2 Σ 2 (Σ +Σ 2 ), v =, v 2 ( ) x = 2, 2 = +Γ Σ Γ T Γ Σ (Σ +Σ 2 ) Σ Γ T, = 2 +Γ 2 Σ 2 Γ T 2 Γ 2Σ 2 (Σ +Σ 2 ) Σ 2 Γ T 2, 2 = Γ Σ (Σ +Σ 2 ) Σ 2 Γ T 2. (A-4) Lemma 2: if x CSN nx,q x (µ x,σ x,γ x,v x, x ) and A is n y n x matrix (n y n x ) then y = Ax is CSN, y CSN ny,q y (µ y,σ y,γ y,v y, y ), where q y = q x, µ y = Aµ x, Σ y = AΣ x A T, Γ y = Γ x Σ x A T Σ y, v y = v x and y = x +Γ x Σ x Γ T x Γ x Σ x A T Σ y AΣ x Γ T x. Lemma 3: if x CSN nx,q x (µ x,σ x,γ x,v x, x ) then E[x] = µ x + Σ x Γ T xψ. Where Ψ = Φ ( q x ;V, x +Γ x Σ x Γ T ) x ( Φ qx ;V, x +Γ x Σ x Γ T ) and Φ q x (x;µ,ω) = [ x Φ qx (x;µ,ω)] T. x Sampling from a CSN distribution is relatively easy by rejection sampling. For example, let E φ nx (E ;,Σ x ) and E 2 φ qx (E 2 ;, x ) be independent random vectors, then x = µ x +E CSN nx,q x (µ x,σ x,γ x,v x, x ) if y = v x +Γ x E +E 2. APPENDIX B BAYESIAN INVERSION IN CSN SETTING Assume x CSN nx,q x (µ x,σ x,γ x,v x, x ) and also linear measurement ( with CSN ) distributed errors as d = Hx + e d x, where e d x CSN nd,q d,σd x,γ d x,v d x, d x, let x = [t u ] and e d x = [s v ]. By using the previous formula for finding the joint distribution of all random vector and this assumption that t, s and v are mutually independent, and also u and v are mutually independent, we have: t r = Ht+s u φ n x+n d +q x+q d ( v µ x Hµ x v x v d x Σ x Σ x H T Σ x Γ T x, HΣ x HΣ x H T +Σ d x HΣ x Γ T x Σ d x Γ T e Γ x Σ x Γ x Σ x H T x +Γ x Σ x Γ T x Γ d x Σ d x d x +Γ d x Σ d x Γ T d x ). (B-) Besides, from analysis of multivariate normal distribution: [ ] ([ ] [ ]) x µ Σ Σ φ x nx +n x2, 2. (B-2) 2 µ 2 Σ 2 Σ 2

20 Rezaie, Eidsvik & Mukerji 2 Geophysics [x x 2 ] φ nx ( x ;µ +Σ 2 Σ 2 (x 2 µ 2 ),Σ Σ 2 Σ 2 Σ 2). (B-3) By re-arranging equation (B-) in accordance with equation (B-2) and using equation (B-3) and matrix multiplication we have: t r µ t r Σ t r Σ tu r Σ tv r u r φ nx+q x+q d µ u r, Σ ut r Σ u r Σ uv r,. v r µ v r Σ vt r Σ vu r Σ v r (B-4) Let w = µ t r = µ x +Σ x H T [ HΣ x H T +Σ d x ] (r Hµx ), µ u r = v x +Γ x Σ x H T [ HΣ x H T +Σ d x ] (r Hµx ), µ v r = v d x +Γ d x Σ d x [ HΣx H T +Σ d x ] (r Hµx ), Σ t r = Σ x Σ x H T [ HΣ x H T +Σ d x ] HΣx, Σ u r = [ x +Γ x Σ x Γ T x] Γx Σ x H T [ HΣ x H T ] HΣx +Σ d x Γ T x [ ], Σ v r = d x +Γ d x Σ d x Γ T [ d x Γ d x Σ d x HΣx H T ] Σd x +Σ d x Γ T d x, Σ ut r = Γ x Σ x Γ x Σ x [ HΣx H T +Σ d x ] HΣx, Σ vt r = Γ d x Σ d x [ HΣx H T +Σ d x ] HΣx, Σ vt r = Γ x Σ x H T [ HΣ x H T +Σ d x ] Σd x Γ T d x. [ ] u, then: v [ ] ([ ] t r µt r φ w r nx+q x+q d, µ w r From the definition of the CSN model: [ Σt r Σ tw r Σ wt r Σ w r (B-5) ]). (B-6) π(x d) = π(t r,w ) = CSN nx,q x+q d (µ x d,σ x d,γ x d,v x d, x d ). (B-7) where parameters are as defined in equation (2). APPENDIX C OBSERVATION DISTRIBUTION Weassumedatad = Hx+e,wherex CSN nx,q x (µ x,σ x,γ x,v x, x )ande CSN nd,q d (,Σ d x,γ d x,v d x, ByusingLemma2, Hx CSN nd,q x (µ p,σ p,γ p,v p, p ), whereµ p = Hµ x, Σ p = HΣ x H T, Γ p = Γ x Σ x H T Σ p, v p = v x and p = x +Γ x Σ x Γ T x Γ xσ x H T Σ p HΣ xγ T x.

21 Rezaie, Eidsvik & Mukerji 2 Geophysics From Lemma and previous result, we have d CSN nd,q x+q d (µ d,σ d,γ d,v d, d ): ( Γp Σ µ d = µ p +µ d x, Σ d = Σ p +Σ d x, Γ d = p Σ ) ( ) d vp Γ d x Σ d x Σ, v d =, d v d x ( ) A A d = 2, A A 2 A = p +Γ p Σ p Γ T p Γ p Σ p Σ d Σ pγ T p, 22 A 22 = d x +Γ d x Σ d x Γ T d x Γ d xσ d x Σ d Σ d xγ T d x, A 2 = A T 2 = Γ p Σ p Σ d Σ d xγ T d x. (C-)

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