Early Assessment of NTG Uncertainty:

A workflow and Application to Stanford VI Reservoir

Amisha Maharaja and Andre G. Journel.

Stanford Center for Reservoir Forecasting

May 2005

Abstract

Scarce and preferentially located well data at early appraisal stage of a reservoirtend to give an upwardly biased estimate of Net-to-Gross (NTG). Seismic data providethe most extensive coverage of the reservoir which helps to correct the biased estimate,but they need to be calibrated with well data. A Bayesian approach is used to calibrateacoustic impedance with core facies data in the synthetic Stanford VI reservoir. Sucha calibration calls for a prior probability of NTG, which itself is highly uncertain andtherefore treated as a random variable. Multiple-point sequential simulation honoringall available data is used to obtain multiple equi-probable realizations of the reservoir.Spatial bootstrap done on the realizations is used to obtain uncertainty about the NTGestimate.

1 Introduction

1.1 Problem Statement

When a new hydrocarbon reservoir is discovered, many factors influence the decision ofwhether to develop or abandon that reservoir. These factors include the Hydrocarbons inPlace (HIP), the location of the reservoir (e.g. deep water or shallow water), the type ofhydrocarbon (e.g. oil or gas), the demand for that hydrocarbon, the price of the hydrocar-bon, etc. Many of these factors are determined by the existing economic conditions, whileothers are a direct result of the geological setting and processes that occured in the past.HIP is one such parameter. In expensive, deepwater reservoirs it is crucial to know thedistribution of HIP in the reservoir because a reservoir with likely low HIP might not beprofitable to develop. The goal of this study is to present a rigorous and practical workflowbased on firm theoretical foundations to assess uncertainty about HIP.

HIP or more specifically Oil in Place (OIP) is given by (1).

OIP = V ∗ NTG ∗ φ ∗ So (1)

where V is the reservoir volume, NTG is the Net-to-Gross ratio, φ is porosity of the reservoirquality rock, and So is the oil saturation. To obtain the uncertainty about OIP, one mustassess the uncertainty about each of these quantities and combine them accounting for any

1

dependencies between them. The reservoir volume and the NTG have the largest impact onthe volumetrics, hence the corresponding uncertainties should be carefully assessed beforeproceeding into elaborate porosity and saturation models. Volume uncertainty is directlyrelated to seismic processing and the subsequent interpretation of reservoir boundaries, theresponsibility of which is entrusted to the geophysicist. In this paper, we address the secondmost consequential uncertainty, that related to the NTG.

1.2 Net-to-Gross Uncertainty

Net-to-Gross (NTG) is defined as the ratio of volume of reservoir quality rock to total rockvolume in the reservoir. The former is related to the geological setting, for e.g., turbiditychannel reservoirs typically have lower NTG than reservoirs containing massive sheet-likesands. In the early stage of reservoir development the uncertainty about the geologicalsetting is high as the well data are sparse. Hence, it is essential to use different geologicalscenarios that fit all available data until some could be eliminated by subsequently acquireddata. We denote geological scenario by S, and sk, k = 1, . . . ,K are its realizations.

True NTG of the reservoir is unknown. One can only estimate it through the availabledata such as cores, well logs, and seismic attributes. Denote the true value by A and theestimate by A?. By definition NTG is a global parameter, hence it has no replicate, whichmakes it difficult to assess the uncertainty about it. But, had the wells been drilled else-where or had a different velocity model been used for seismic inversion, one would get adifferent estimate of NTG. Hence, the data are treated as random variables and represent asource of uncertainty in the estimation of NTG. These data are denoted in bold as a vectorD since we have different types of data. The available data, d0, is one realization of D.

Bitanov and Journel (2004) proposed a workflow (Figure 1) to assess uncertainty inNTG by randomizing the well locations. In their workflow the geological scenario, seismicdata, and initial drilling strategy are frozen. Facies are simulated conditional to the avail-able well and seismic data, then several different sets of synthetic wells are sampled fromthe corresponding realizations. Each new set yields a new NTG estimate, which providesa measure of uncertainty about the NTG. This idea of re-sampling from stochastic realiza-tions is known as spatial bootstrap (Journel, 1993; Norris et al., 1993) and unlike classicalbootstrap (Efron, 1979), it does not make the assumption of independence between dataalong the wells or between wells.

1.3 Estimation of Net-to-Gross

Applying some function on the data d gives an estimate of the NTG a?. This functioncan be as simple as averaging the sand/non-sand values at the wells. However, in the earlystages of reservoir development, this estimate will not be representative of the reservoirbecause the wells may be preferentially located in high pay zones. Seismic data, which sam-ples the reservoir more exhaustively, should be used to correct for the potentially biasedestimate obtained from well data. Since seismic data is an indirect indicator of facies, it

2

must be calibrated with the well data first.

Many calibration techniques are found in the literature. Examples include indicatorco-kriging of well data and seismic impedance (Goovaerts, 1997, p. 297); co-located orBayesian calibration of impedance data based on well calibration (Bitanov and Journel,2004; Caumon et al., 2004); neural-net fitting (Caers and Ma, 2002); PCA-based calibra-tion (Strebelle et al., 2002); and canonical correlation between facies and various seismicattributes (Fournier and Derain, 1995). Some of these techniques are not robust in situationsof sparse data. For e.g., PCA-based calibration technique is shown to be unreliable becausethe NTG estimate fluctuates depending on how many PCs are retained for calibration (Xuet al., 2004). More sophisticated techniques such as canonical correlation and neural-net fit-ting require too many parameters, which might not be correctly estimated from sparse data.

Bayesian calibration of acoustic impedance data with core data was used by Bitanovand Journel (2004). In this simple technique, the observed impedance-facies pair along theavailable wells are used to obtain the likelihood of observing an impedance value (I) given aparticular rock type. For simplicity consider that all the reservoir units can be grouped intosand (R) and non-sand (R). Then the likelihood functions are P (I | R) and P (I | R). Bayesinversion (2) is then applied at all locations sampled by seismic to obtain the probabilityof observing a particular rock type given the observed impedance value, P (R | I). Thisprobability, when averaged over all locations, gives a NTG estimate, hopefully correctedfrom the bias induced by preferential location of the wells.

P (R | I) =P (I | R) ∗ P (R)

P (I)(2)

P (I) is obtained by applying the total probability rule (3).

P (I) = P (I | R) ∗ P (R) + (1 − P (I | R)) ∗ (1 − P (R)) (3)

Bayes inversion calls for a prior probability of the given rock-type, P (R), which could be ob-tained from any prior information we might have about the reservoir NTG, e.g. as obtainedfrom analog reservoirs or outcrops. However simple this method might be, its effectivenessin calibrating well data with seismic data still depends on the quality of data. For e.g., if, ina certain reservoir, acoustic impedance is not a good discriminator of sand/non-sand, thenthe calibration will be poor. In such a scenario, the ratio P (I | R) to P (I) will be close to1 and the prior P (R) will dominate the results of spatial bootstrap (see Table 1) and givea false sense of certainty. Caumon et al. (2004) therefore proposed to use a distribution ofpossible prior NTG values instead of using a single prior NTG as in Bitanov and Journel(2004).

3

Prior NTG Corrected NTG

0.20 0.23

0.35 0.36

0.50 0.47

Table 1: Sensitivity to prior NTG. Bayes inversion done using Stanford VI data: 4 wellswith NTG of 0.54 and acoustic impedance. True Stanford VI NTG is 0.34

2 Net-to-Gross Workflow

The workflow proposed by Caumon et al. (2004) (Figure 2) is a significant update of theone proposed by Bitanov and Journel (2004) (Figure 1). A more detailed verion of theworkflow is shown in Figure 3. They also advocate the use of several different geologicalscenarios to obtain a more realistic range of uncertainty about NTG. For a given geologicalscenario a distribution of NTG P (A | S = sk) is selected, which is necessarily a subjective,expert task. The median value of that distribution is used as the prior sand probability inthe Bayes inversion (2) to obtain initial best NTG estimate a?

0from the original data d0.

The goal is to update this prior distribution given this NTG estimate a?0.

For this purpose, the NTG distribution is discretized into M classes resulting in M

realizations of the NTG, a1, . . . , aM . Corresponding to each class of NTG L realizationsof the reservoir facies are generated using some simulation algorithm, preferably one thatutilizes multiple-point statistics because it is better at honoring the data patterns foundin the geological scenario. Freezing the seismic data and initial drilling strategy, N sets ofsynthetic wells are sampled from the L realizations by randomizing well locations, whichprovides N ∗ L realizations of the NTG estimate, a?

1, . . . , a?

NL. From this distribution the

likelihood of observing the initial best NTG estimate for the given class of true NTG andgiven geological scenario, P (A? = a?

0| A = am, S = sk), is obtained. Bayes inversion is

then used to obtain the probability of the true NTG being in that class given the observedNTG estimate and geological scenario, P (A = am | A? = a?

0, S = sk) (4).

P (A = am | A? = a?

0, S = sk) =

P (A? = a?0| A = am, S = sk) ∗ P (A = Am | S = sk)

P (A? = a?0| S = sk)

(4)

P (A? = a?0| S = sk) is obtained by applying the total probability rule (5).

P (A? = a?

0| S = sk) =

M∑

m=1

P (A? = a?

0| A = am, S = sk) ∗ P (A = am | S = sk) (5)

This procedure, when repeated for each class of true NTG, gives the posterior distributionof true NTG given the initial NTG estimate for a given geological scenario,P (A = am | A? = a?

0, S = sk). Finally, the true NTG given initial best estimate

P (A = am | A? = a?0) is obtained by total probability rule (6).

P (A = am | A? = a?

0) =

K∑

k=1

P (A = am | A? = a?

0, S = sk) ∗ P (S = sk) (6)

4

Assigning prior probability to a geological scenario is a difficult task. A convenient solutionconsists of assuming each scenario to be equally likely, which might be reasonable in sparsedata situation. To summarize, Caumon et al. (2004) considered three jointly related sourcesof uncertainty: the geological scenario S, the true NTG value A, and the global NTG es-timate A? obtained from the available data. Seismic data, the well-to-seismic calibrationalgorithm, and the initial drilling strategy were frozen throughout, however these can berandomized. Although Bayesian calibration was used for estimation, the workflow does notcall for any specific well-to-seismic calibration algorithm.

3 Application to Stanford VI Reservoir

The workflow of Caumon et al. (2004) (Figure 2) is demonstrated using the Stanford VIreservoir. Stanford VI is a prograding fluvial reservoir which is later deformed into anasymmetric anticline (Castro et al., 2005). It is 3.75km × 5.0km × 0.2km and contains150 × 200 × 200 gridblocks of dimensions 25m × 25m × 1m. It consists of three layers withdistinct geological setting and NTG (Table 2). A prograding delta in the bottom layertransitions into meandering channels in the middle layer, which themselves transition intolow-sinuosity channels in the top layer (Figure 4). The top two layers consist of four facies,namely channel sand, point bar, channel mud drapes, and floodbasin clays. The bottomlayers consists of channel sands and floodbasin clays. In this study, we group the channelsand and point bar into a single “sand” facies and group the channel mud drapes and flood-basin clays into “non-sand”. For details about the petrophysical properties of the variousfacies and seismic data generation refer to Castro et al. (2005).

Layer True NTG

bottom 0.44

middle 0.25

top 0.27

overall 0.34

Table 2: True NTG of Stanford VI reservoir

Filtered acoustic impedance in the depth domain mimicking realistic seismic quality isused as seismic data. The time-to-depth conversion is an important source of uncertaintyand should be accounted for during the geophysical inversion. For this study we assumethat it has been accounted for. To mimic an early development stage four wells are drilled inthe 20 percent lowest, vertically-averaged impedance area, which represents high pay zones(Figure 5). The NTG obtained by averaging sand/non-sand values at the four wells is 0.53,which is much higher than the true Stanford VI NTG of 0.34. The likelihoods of impedancegiven sand and non-sand are computed from the well data (Figure 6). Since there is a finiteamount of well data, we need to apply a smoothing function to these likelihoods.

5

The geological interpretation is done mainly from the seismic data. Channels are seenin the horizon slices of seismic data indicating that this is a channelized reservoir (Figure7). Consequently, we use a channel training image (Ti) for the entire reservoir (Figure 8).This Ti was generated using the object-based simulation algorithm fluvsim (Deutsch andWang, 1996). It is seen from the seismic data (Figure 7) that the channels in the bottomregion fan out and are thinner than the ones in the upper region. Hence, we use an affinitytransform and angle constraint during simulation in addition to conditioning to the welland seismic data.

Since the well NTG is very high, we suspect a bias, hence choose a prior NTG distri-bution with mean lower than the well NTG. First we retain a triangular distribution withmode 0.40, minimum 0.15, and maximum 0.60 (Figure 9). This prior distribution is quiteconservative and includes the well NTG of 0.53, however its mean 0.38 is much lower. Ide-ally, the prior NTG distribution should be obtained from a database of analog reservoirs,which would vary from one company to another and encapsulates each company’s explo-ration strategy. Using the median value of the prior distribution as prior sand probability inexpression (2) and using the likelihood distributions shown in Figure 6, we obtain the prob-ability of observing sand at each location sampled by seismic data. Taking the average ofthese sand probabilities gives the initial best NTG estimate a?

0= 0.41. The goal is to update

the prior distribution of NTG given that value a?0

and given the geological scenario retained.

The prior NTG distribution (Panel B, Figure 3) is discretized into 10 classes. Themedian NTG value from each class is retained to simulate a facies realization using themultiple-point simulation algorithm snesim (Strebelle, 2002). Within each such realization,using the initial drilling strategy, 200 synthetic sets of 4 wells are sampled and the aforemen-tioned Bayesian calibration procedure is applied to each such set to obtain a distributionof NTG estimates (see Panel C, Figure 3). From this distribution, the probability of theNTG estimate a?

0is obtained. Then using Bayes inversion (4) the probability of the NTG

value being in a given class is computed (see Panel D, Figure 3).

The entire previous workflow is repeated using a uniform prior distribution for NTGinstead of the triangular. The minimum and maximum values of this uniform distributionare 0.15 and 0.60 respectively, the same as for the triangular distribution. The mean of thisuniform distribution is 0.37, again close to the mean of the triangular prior. However, thisis a very different prior distribution because with a uniform distribution we are assumingthat each NTG value between its bounds is equally likely.

4 Results

The final posterior NTG distributions are obtained by applying the workflow of Figure 3,using successively the triangular and uniform prior distributions. The results are shownin Figures 11 and 12, respectively. The summary statistics of the posterior distributionresulting from the triangular prior are given in Table 3. Those resulting from the uniformprior are given in Table 4.

6

Statistics Prior Posterior

Mean 0.38 0.33

Std. Dev. 0.09 0.05

P10 0.26 0.26

Median 0.39 0.35

P90 0.51 0.40

Table 3: Summary statistics of the triangular prior distribution and the resulting posteriordistribution obtained by running the workflow on Stanford VI using an initial best NTGestimate of 0.41. The true NTG of Stanford VI is 0.34.

Statistics Prior Posterior

Mean 0.38 0.31

Std. Dev. 0.13 0.06

P10 0.20 0.22

Median 0.38 0.31

P90 0.56 0.40

Table 4: Summary statistics of the uniform prior distribution and the resulting posteriordistribution obtained by running the workflow on Stanford VI using an initial best NTGestimate of 0.40. The true NTG of Stanford VI is 0.34.

The standard deviation of the triangular prior distribution is reduced by 42 percentfrom 0.09 to 0.05, which is significant. The true NTG value 0.34 of Stanford VI is includedin the [P10, P90] interval and both the posterior mean and median are close to that true value.

When considering a uniform prior distribution, its standard deviation is reduced byover 50 percent from 0.13 to 0.06. This is a significant reduction of the extreme uncertaintycarried by such a uniform prior. The posterior mean and median are 0.31, lower than thetrue NTG value 0.34, but fairly close. The posterior mean NTG (0.33) obtained by usingthe triangular prior was much closer to that true value (0.34).

The mean and median of both prior NTG distributions were higher than, yet relativelyclose, to the true Stanford VI NTG. Using these median values, the corresponding bestinitial NTG estimates were 0.40 and 0.41, values also higher than the true NTG. What ifour initial best NTG estimate was highly under or over-estimated? Figure 13 shows theposterior NTG distributions obtained if the initial best NTG estimate is 0.19, a severeunder-estimation case. Figure 14 shows the result if the initial best NTG estimate is 0.57, asevere over-estimation case. In none of the resulting posterior distributions, the true NTGof Stanford VI is included in the [P10, P90] interval. This demonstrates that the choice of aprior NTG distribution is a critical issue. If the mean of the prior distribution is reasonablyclose to the true mean the proposed workflow reduces significantly the uncertainty intervalin the right direction.

7

5 Conclusions

We have demonstrated the NTG workflow of Caumon et al. (2004) using the Stanford VIreservoir. This workflow is based on the Bayesian concept of updating prior knowledge withnew information. In case of NTG estimation, the prior information is a prior distribution forthat NTG, which ideally comes from a database of analog reservoirs. The new informationis the actual well data and seismic data available from the reservoir. The resulting posteriorNTG distribution is an update of the prior distribution given all available actual reservoirdata.

The best estimated NTG value is obtained by calibrating the well core data with theseismic data using a Bayesian technique. We chose not to use the well NTG value becausethe wells were drilled in the high pay zones and we suspected a bias in that naive estimate.The seismic data helped in correcting that bias. Note that any other relevant well-to-seismiccalibration procedure could have been used.

A triangular and a uniform prior distribution for the NTG were used. Both distrib-utions, although very different in nature, had conservative bounds and roughly the samemean NTG. In both cases, the resulting posterior distribution showed an excellent reduc-tion of uncertainty and the mean of these posteriors were close to the true NTG of StanfordVI. However, were the initial best NTG estimate greatly under or over-estimated, the biaswould not be corrected.

Although we did not demonstrate the impact of geological uncertainty in this paper, weemphasize that in real cases this is likely the largest source of uncertainty and should beaccounted for using expression (6) as discussed in Section 2, and developed in Caumon etal. (2004).

6 Acknowledgements

This work is supported by ChevronTexaco Energy Technology Company (ETC). We wantto thank Sebastien Strebelle from ETC for his useful feedback.

References

Bitanov, A. and Journel, A.G., Uncertainty in N/G ratio in early reservoir development.Journal of Pet. Sci.& Engr., vol.44(1-2), pp.115-131 , 2004.

Caers, J. and Ma, X., Modeling conditional distributions of facies from seismic using neuralnets. Mathematical Geology, vol.34(2), pp.139-163, 2002.

Castro, S., Caers, J., Mukerji, T., The Stanford VI reservoir. 18th Annual Report, StanfordCenter for Reservoir Forecasting, Stanford University., 2005.

Caumon, G., Strebelle, S., Caers, J., and Journel, A.G., Assessment of Global Uncertainty

8

for Early Appraisal of Hydrocarbon Fields. In SPE Annual Technical Conference andExhibition, SPE paper 89943, 2004.

Deutsch, C.V. and Wang, L., Hierarchical object-based stochatic modeling of fluvial reser-voirs. Mathematical Geology, vol.28(7), pp.857-880, 1996.

Efron, B., Bootstrap methods: Another look at the jackknife. Ann. Stat., vol.7, pp.1-26,1979.

Fournier, F. and Derain, J.F., A statistical methodology for deriving reservoir propertiesfrom seismic data. Geophysics, vol.60(5), pp.1437-1450, 1995.

Goovaerts, P., Geostatistics for natural resources evaluation. Oxford University Press, NewYork, NY, p.483, 1997.

Journel, A. G., Resampling from stochastic simulations. Environ. Ecol. Stat, vol.1, pp.63-83,1993.

Norris, R., Massonat, G., Alabert, F., Early quantification of uncertainty in the estima-tion of oil-in-place in a turbidite reservoir. In SPE Annual Technical Conference andExhibition, SPE paper 26490, 1993.

Strebelle, S., Conditional simulation of complex geological structures using multiple-pointstatistics. Mathematical Geology, vol.34(1), pp.1-21, 2002.

Strebelle, S. and Payrazyan, K., and Caers, J., Modeling of a Deepwater Turbidite ReservoirConditional to Seismic Data Using Multiple-Point Geostatistics. In SPE Annual TechnicalConference and Exhibition, SPE paper 77425, 2002.

Xu, Y., Caers, J and Arroyo-Garcia, C., Automatic geobody detection from seismic usingminimum message length clustering. Computers and Geosciences, 30(7), 741-751, 2004.

9

Figures

Figure 1: Simplified workflow of Bitanov and Journel (2004). The red line in the NTGdistribution indicates the prior NTG value used for simulating the realizations

Figure 2: Simplified workflow of Caumon et al. (2004).

10

Figure 3: Workflow of Caumon et al. (2004) shown in greater detail.

11

(a) Bottom Layer NTG=0.44 (b) Middle Layer NTG=0.25 (c) Top Layer NTG=0.27

Figure 4: Representive slices showing the channel styles in the three layers of Stanford VIreservoir. Overall NTG of Stanford VI is 0.34.

Figure 5: Figure showing location of 4 initial wells and vertically averaged acoustic im-pedance.

Freq

uenc

y

Value

Sand Impedance likelihood

4.595 5.646 6.698 7.750 8.802

mean: 6.101std. dev.: 0.608

minimum: 4.595P10: 5.296

median: 6.137P90: 6.979

maximum: 8.802

(a) Sand

Freq

uenc

y

Value

Non-Sand Impedance likelihood

4.595 5.646 6.698 7.750 8.802

mean: 6.805std. dev.: 0.714

minimum: 4.595P10: 5.857

median: 6.698P90: 7.820

maximum: 8.802

(b) Non-sand

Figure 6: Sand and non-sand impedance likelihoods obtained from the four well data.

12

(a) Bottom Layer (b) Middle Layer (c) Top Layer

Figure 7: Representive horizon slices of acoustic impedance in the three layers of StanfordVI reservoir.

Figure 8: Training Image used for Stanford VI. Ti has 150 x 200 x 25 grid blocks and itsNTG is 0.29.

13

Freq

uenc

y

Value

Prior NTG distribution

0.150 0.263 0.375 0.488 0.600

mean: 0.383std. dev.: 0.092

minimum: 0.150P10: 0.256

median: 0.387P90: 0.505

maximum: 0.600

Figure 9: The triangular prior true NTG distribution used for Stanford VI. It is discretizedinto 10 classes

Freq

uenc

y

Value

Prior NTG distribution

0.150 0.263 0.375 0.488 0.600

mean: 0.375std. dev.: 0.130

minimum: 0.150P10: 0.195

median: 0.375P90: 0.555

maximum: 0.600

Figure 10: The uniform prior true NTG distribution used for Stanford VI. It is discretizedinto 10 classes

14

Freq

uenc

y

Value

Posterior NTG Probability given Estimate = 0.407097

0.150 0.263 0.375 0.488 0.600

mean: 0.333std. dev.: 0.053

minimum: 0.150P10: 0.263

median: 0.353P90: 0.398

maximum: 0.600

Figure 11: The posterior true NTG distribution obtained starting from a triangular priorof [0.15, 0.40, 0.60] and initial best estimate a?

0= 0.41.

Freq

uenc

y

Value

Posterior NTG Probability given Estimate = 0.39653

0.150 0.263 0.375 0.488 0.600

mean: 0.305std. dev.: 0.064

minimum: 0.150P10: 0.218

median: 0.308P90: 0.398

maximum: 0.600

Figure 12: The posterior true NTG distribution obtained starting from a uniform prior of[0.15, 0.60] and initial best estimate a?

0= 0.40.

15

Freq

uenc

y

Value

Posterior NTG Probability given Estimate = 0.190476

0.150 0.263 0.375 0.488 0.600

mean: 0.237std. dev.: 0.024

minimum: 0.150P10: 0.218

median: 0.218P90: 0.263

maximum: 0.600

(a) With Triangular Prior

Freq

uenc

y

Value

Posterior NTG Probability given Estimate = 0.190476

0.150 0.263 0.375 0.488 0.600

mean: 0.222std. dev.: 0.029

minimum: 0.150P10: 0.173

median: 0.218P90: 0.263

maximum: 0.600

(b) With Uniform Prior

Figure 13: The posterior distributions of true NTG obtained conditional to an initial bestNTG estimate of 0.19.

16

Freq

uenc

y

Value

Posterior NTG Probability given Estimate = 0.571429

0.150 0.263 0.375 0.488 0.600

mean: 0.479std. dev.: 0.050

minimum: 0.150P10: 0.398

median: 0.488P90: 0.533

maximum: 0.600

(a) With Triangular Prior

Freq

uenc

y

Value

Posterior NTG Probability given Estimate = 0.571429

0.150 0.263 0.375 0.488 0.600

mean: 0.490std. dev.: 0.058

minimum: 0.150P10: 0.398

median: 0.488P90: 0.578

maximum: 0.600

(b) With Uniform Prior

Figure 14: The posterior distributions of true NTG obtained conditional to an initial bestNTG estimate of 0.57.

17