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Potential risks when using seismic attributesas predictors of reservoir properties

CYNTHIA T. KALKOMEY, Mobil E&P Technical Center, Dallas, Texas

The advance of our ability to generate seismic attrib-utes and the growing emphasis on production geo-physics has led to the widespread use of seismic attrib-utes as predictors of reservoir properties. In manycases, we can show using seismic modeling or rockphysics a physically justifiable relationship betweena seismic attribute and the reservoir property of inter-est. When this is true, we are able to greatly reducethe uncertainty of interwell predictions of reservoirproperties.

The simple flow diagram in Figure 1 illustrates the

basic process. The first critically important step is accu-rately tying the well and seismic both vertically andareally. We then must choose which seismic attributeswe believe to be related to the reservoir property. Usingthat attribute or set of attributes, we use the dense seis-mic data to guide the prediction of reservoir properties

between sparse well data. A number of methods can beused for the prediction step linear or nonlinearregression, geostatistics, or neural networks. The pur-pose is estimating in-place hydrocarbon volumes ormaking reservoir management decisions such aslocation and number of wells, depletion strategy, or gasand water injection operations. The problem with theprocess is: How do we identify which seismic attribut-

es to use?

The problem. All of the prediction methods (regres-sion, geostatistics, and neural networks) require mak-ing inference from seismic data at a small number ofwells to a larger population which we assume is rep-

resented by that sample. Enthusiastic practitioners cannow generate 10s even 100s of seismic attributes.An all too common practice is identifying which seis-mic attributes to use solely on the strength of theirobserved correlations with reservoir properties mea-sured at the wells. Often these correlations are esti-mated by a very small number of observations. As withany parameter, when the sample size is small, theuncertainty about the value of the true correlation can

be quite large.To get an idea of how large this uncertainty can be,

consider a sample correlation (r) of 0.80 calculated from10 data points. The 95% confidence interval estimate ofthe true correlation is [0.34, 0.95]. The only statementwe can make with (95%) confidence is that the true cor-relation between the reservoir property and seismicattribute is between 0.34 and 0.95. If we have only fivedata points instead of 10, the 95% confidence intervalwidens to [-0.28, 0.99]. Since this interval contains zero,we cannot say with confidence that there is any corre-lation between the reservoir property and the seismicattribute.

Most practitioners are already aware that the small-er the sample, the greater the uncertainty about thetrue value of the correlation. A lesser known phenom-

ena is something statisticians call experiment-wiseerror rates. This just means that as we generate an everincreasing number of attributes, the greater the chanceof observing at least one spuriously large sample cor-relation. You can get the idea of this if you consider anexperiment where you draw a small number of obser-vations from two completely unrelated variables say the annual birthrate of elephants and pounds con-sumed of a food substance. If we randomly select fiveyears and observe the birthrate of elephants andpounds of sugar consumed, we would probably cor-rectly infer that there is no relationship between thetwo variables. But if we continue to test this for all pos-sible food types, by chance alone we will likely find at

least one food type that appears, on the basis of thesample correlation, to be related to the birthrate of ele-phants. When this happens, we say weve encoun-tered a spurious correlation.

Definition: A spurious correlation is a samplecorrelation that is large in absolute valuepurely bychance.

For seismic attributes and reservoir properties thismeans we have happened to sample at locations thatyield a large sample correlation when in truth they are

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Figure 1. Simple flow diagram showing use of seismicattributes to predict reservoir properties.

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actually uncorrelated. How likely is it that this couldoccur?

Probability of observing a spurious correlation. Con-sider a single seismic attribute as a possible predictor.The probability of observing the absolute value of thesample correlation (r) greater than some constant (R)given the true correlation (r) is zero can be found from

where n is the sample size or number of locations with

measurements of both the reservoir property and theseismic attribute, and t is distributed as a Students tvariate with n-2 degrees of freedom.

Note that the probability of a spurious sample cor-relation depends only on R (the magnitude of the spu-rious sample correlation) and n (the number of wellmeasurements). An assumption of this relationship isthat the n observations are drawn randomly. The rea-son for a random sample is to insure that the sampleis representative of the population under study and

that the amount of independent information availableto estimate the true correlation is maximized. If in factthe data are spatially correlated, the relationship givesa conservative estimate of the probability of a spuriouscorrelation. This is because the effective sample size will

be smaller than the actual sample size; and as ndecreases, the probability of a spurious correlationincreases. Table 1 shows the probability of observinga spurious correlation for different magnitudes of thesample correlation (R) and different sample sizes (n).

Now that weve quantified the probability of a spu-rious correlation when considering a single seismicattribute, we can calculate the probability of at least onespurious correlation when considering a set of k inde-

pendent attributes. The probability of observing at leastone spurious correlation when considering a set of kindependent attributes is simply 1 minus the probabili-ty that none of the sample correlations are spurious. Thiscan be expanded to the summation

248 T HE LEADING EDGE MARCH 1997

Table 1. Probability of spurious correlation between areservoir property and a single seismic attribute

Sample size

R 5 10 15 20 25 35 50 75 100

0.1 0.87 0.78 0.72 0.67 0.63 0.57 0.49 0.39 0.32

0.2 0.75 0.58 0.47 0.40 0.34 0.25 0.16 0.09 0.05

0.3 0.62 0.40 0.28 0.20 0.15 0.08 0.03 0.01 0.00

0.4 0.50 0.25 0.14 0.08 0.05 0.02 0.00 0.00 0.00

0.5 0.39 0.14 0.06 0.02 0.01 0.00 0.00 0.00 0.00

0.6 0.28 0.07 0.02 0.01 0.00 0.00 0.00 0.00 0.00

0.7 0.19 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.8 0.10 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.9 0.04 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Table 2. Five independent seismic attributes

Sample size

R 5 10 15 20 25 35 50 75 100

0.1 1.00 1.00 1.00 1.00 0.99 0.98 0.97 0.92 0.86

0.2 1.00 0.99 0.96 0.92 0.87 0.76 0.59 0.36 0.21

0.3 0.99 0.92 0.80 0.67 0.54 0.34 0.16 0.04 0.01

0.4 0.97 0.77 0.53 0.34 0.22 0.08 0.02 0.00 0.00

0.5 0.92 0.53 0.26 0.12 0.05 0.01 0.00 0.00 0.00

0.6 0.81 0.29 0.09 0.03 0.01 0.00 0.00 0.00 0.00

0.7 0.65 0.12 0.02 0.00 0.00 0.00 0.00 0.00 0.00

0.8 0.42 0.03 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.9 0.17 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Table 3. Ten independent seismic attributes

Sample size

R 5 10 15 20 25 35 50 75 100

0.1 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.98

0.2 1.00 1.00 1.00 0.99 0.98 0.94 0.83 0.59 0.38

0.3 1.00 0.99 0.96 0.89 0.79 0.57 0.29 0.09 0.02

0.4 1.00 0.95 0.78 0.57 0.39 0.16 0.04 0.00 0.00

0.5 0.99 0.78 0.45 0.22 0.10 0.02 0.00 0.00 0.00

0.6 0.96 0.50 0.17 0.05 0.02 0.00 0.00 0.00 0.00

0.7 0.88 0.22 0.04 0.01 0.00 0.00 0.00 0.00 0.00

0.8 0.67 0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.9 0.32 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Table 4. Twenty independent seismic attributes

Sample sizeR 5 10 15 20 25 35 50 75 100

0.1 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.2 1.00 1.00 1.00 1.00 1.00 1.00 0.97 0.83 0.61

0.3 1.00 1.00 1.00 0.99 0.96 0.81 0.50 0.16 0.05

0.4 1.00 1.00 0.95 0.81 0.62 0.29 0.08 0.01 0.00

0.5 1.00 0.95 0.70 0.39 0.20 0.04 0.00 0.00 0.00

0.6 1.00 0.75 0.31 0.10 0.03 0.00 0.00 0.00 0.00

0.7 0.98 0.39 0.07 0.01 0.00 0.00 0.00 0.00 0.00

0.8 0.89 0.10 0.01 0.00 0.00 0.00 0.00 0.00 0.00

0.9 0.53 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Table 5. Forty independent seismic attributes

Sample sizeR 5 10 15 20 25 35 50 75 100

0.1 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.2 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.97 0.85

0.3 1.00 1.00 1.00 1.00 1.00 0.95 0.75 0.30 0.09

0.4 1.00 1.00 1.00 0.97 0.86 0.50 0.15 0.01 0.00

0.5 1.00 1.00 0.91 0.63 0.36 0.09 0.01 0.00 0.00

0.6 1.00 0.94 0.52 0.19 0.06 0.01 0.00 0.00 0.00

0.7 1.00 0.62 0.14 0.02 0.00 0.00 0.00 0.00 0.00

0.8 0.99 0.20 0.01 0.00 0.00 0.00 0.00 0.00 0.00

0.9 0.78 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00

p r tn

sc = ( ) = -

-

Pr R Pr

R

R2

2

1

Tables 2-5. Probability of observing at least one spurious sample correlation from agiven number of independent seismic attributes.

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wherepsc is the probability of a spurious correlation fora single attribute and k is the number of independentseismic attributes considered.

From the summation we can see that the penalty forincreasing the number of attributes considered fromk-1to k is

Tables 2-5 give the probability of observing a spuri-ous sample correlation for one or more seismic attrib-utes from a set of k = 5, 10, 20, and 40 independentattributes. Each table gives the probability of observinga sample correlation greater than or equal to 0.1 to 0.9when the reservoir property is actually uncorrelatedwith the attributes, given a sample size of 5, 10, 15, 20,25, 35, 50, 75, or 100. Note, we are assuming that theseismic attributes are independent. If in fact the attrib-utes considered are correlated with each other, then theprobabilities of observing at least one spurious correla-tion shown in the tables will be too large. This is becausethe effective number of independent seismic attributes will

be smaller than the actual number of attributes consid-ered and as k decreases, the probability of at least one

spurious correlation decreases. To avoid this, oneshould use the table where k equals the number of inde-pendent linear combinations of the set of seismic attrib-utes considered.

These tables can be used to assess the risk of select-ing a seismic attribute that is actually uncorrelated withthe reservoir property being predicted. For example,

look at the column headed 5 and row labeled 0.60 ofTable 3. We see that, given we have only five well mea-surements, there is a 96% chance of observing a samplecorrelation coefficient of 0.60 for at least one of the 10seismic attributes considered, when no correlation actu-ally exists between the attributes and the property. If wedouble the number of wells to 10, the probability of atleast one spurious correlation of magnitude 0.6 or moredrops to 0.50.

What happens if we increase the number of inde-pendent seismic attributes considered from 10 to 20?Table 4 shows that the chance of observing at least onesample correlation of 0.60 or more in absolute valuefrom a sample of size five, when no correlation actually

exists between the attributes and the reservoir property,is almost 100% (its actually 0.999). Even if we double thenumber of wells to 10, in this case there is still a 75%chance of at least one spurious correlation.

Factors influencing the probability of a spurious cor-relation. Figure 2 reveals the factors that influence theprobability of observing spurious correlations. Figure 2ashows the probability of observing at least one samplecorrelation greater than or equal to 0.6 when the actualcorrelation is zero. Figure 2b presents the same resultsfor an observed sample correlation of 0.4.

From Figure 2, we see that the probability of observ-ing one or more spurious correlation increases as

the number of independent well measurementsdecreases, or

the number of independent seismic attributes con-sidered as potential predictors increases, or

the absolute value of the observed sample correla-tion decreases.

Associated risks. Statistical decision theory defines riskas the expected loss due to our uncertainty about the truestate of nature under the possible decision alternativesor actions. Figure 3 illustrates the possible outcomeswhich must be considered in order to assess the risk

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Figure 2. Probability of observing at least one spurious sample correlation of magnitude (a) 0.6, and (b) 0.4when actually the well data and seismic attributes are uncorrelated.

Figure 3. Possible outcomes when selecting a seismicattribute as a predictor of a reservoir property.

a) b)

p psc sc

k-( )

-( )1

1.

p p psck

i

k

sc sc

i- -( ) = -( )

=

-( )1 1 1

1

1

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associated with choosing seismic attributes to predict a

reservoir properties.A Type I Error occurs if no relationship truly existsbetween the seismic attribute and the reservoir proper-ty of interest, yet we select the seismic attribute as a pre-dictor. A Type II Error occurs if a physical relationshipdoes exist between the seismic attribute and the reser-voir property of interest, but we fail to use the seismicattribute as a predictor.

The previous section quantified the probability of aType I Error occurring when the selection criteria is

based solely on the magnitude of the sample correlationbetween a seismic attribute and a reservoir property.(Calculating the probability of a Type II Error occurring

requires additional assumptions about the true magni-

tude of the correlation between the well property andthe seismic attribute.) To quantitatively assess risk, wewould have to assign the cost or economic consequenceof each of the four possible outcomes. The costs willobviously be situation dependent; however, we can gen-eralize the cost in a qualitative sense.

The cost of a Type I Error (using a seismic attributeto predict a reservoir property when actually uncorre-lated) is: Inaccurate prediction biased by the attribute. Inflated confidence in the inaccurate prediction

apparent prediction errors are small.

The cost of a Type II Error (rejecting a seismic attribute

for use in predicting a reservoir property when in factthey are truly correlated) is: Less accurate prediction than if wed used the seismic

attribute. Larger prediction errors than if wed used the attribute.

I believe that for most cases the economic conse-quences of making highly confident but inaccurate pre-dictions are more severe than the consequences of a TypeII Error.

A simple example. To illustrate the consequences of aType I Error, consider a simple example. Suppose youhave measurements of porosity and a seismic attribute

250 T HE LEADING EDGE MARCH 1997

Figure 4. Scatter plot of porosity versus seismic ampli-tude at the 10 well locations. Figure 5. True spatial distributions. (a) True distribu-

tion of porosity ranging from 0 (dark blue) to 20%(red). Well locations are indicated are indicated byblack dots. (b) Seismic attribute over the same area.

Figure 6. (above)Scatter plot ofporosity versusseismic amplitudeat alllocations.

Figure 8. Estimation errors. (a) Areas overestimated bymore than five porosity units. (b) Areas underestimat-ed by more than five porosity units.

Figure 7. (right)Estimate of poros-ity generatedusing collocatedcokriging with theseismic attributeas the covariate.

p

o

r

o

s

it

y

seismic attribute

a) b)

a) b)

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extracted at 10 wells. Figure 4 shows a scatter plot ofporosity versus the seismic attribute with a sample cor-relation of 0.82. It could be very tempting to proceedwith the prediction using this relationship.

The true spatial distribution of porosity over thearea is shown in Figure 5a; Figure 5b shows a seismicattribute over the same area. If we look at the completescatter plot over the entire area shown in Figure 6 wesee a shotgun pattern indicating the true correlation ofzero. By happenstance, the field has been drilled in

locations which makes it appear that there is a strongrelationship between this attribute and porosity whenin fact there is none. We have observed a spurious sam-ple correlation.

If we use this attribute for prediction, our resultingestimate of porosity will bear a strong resemblance tothe seismic attribute regardless of whether we usedregression, geostatistics or neural networks. Figure 7shows such an estimate, generated using collocatedcokriging.

This prediction results in large actual estimationerrors. Figure 8a highlights areas where the porosity isoverestimated by 5-12 porosity units. Figure 8b showsareas where the porosity is underestimated by 5 -12

porosity units. As you can imagine, drilling locationschosen on the basis of this prediction will be less thanoptimal. This could also be disastrous for other reser-voir management decisions. For example, if an injec-tion strategy was designed on the basis of this predic-

tion, it would likely be chosen about 90 from the opti-mal orientation.

Conclusions. There are two main points to rememberfrom this work. First, the probability of observing spu-rious sample correlations between a seismic attributeand well data can be quite large if the number of inde-pendent well measurements is small or the number ofindependent attributes considered is large.

Secondly, when the probability of a spurious cor-

relation is large, then selection of seismic attributesbased solely on empirical evidence is risky it canlead to highly confident, but highly inaccurate predic-tions and thus poor business decisions. Therefore, it isstrongly recommended, especially when the numberof wells is small, that only those seismic attributesthat have a physically justifiable relationship withthe reservoir property be considered as candidates forpredictors.

Cynthia Kalkomey received a doctorate in statistics from SouthernMethodist University (1991). She joined Mobil in 1979 and is cur-rently manager of Reservoir Characterization, Mobil Exploration andProducing Technical Center.

Corresponding author: Cynthia Kalkomey, Mgr., Reservoir Char-acterization, Mobil Exploration and Producing Technical Center,13777 Midway Rd., Dallas, TX 75244; phone 972-851-8598; fax972-851-8703.

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