scale matching with factorial kriging for improved porosity estimation from seismic data

Mathematical Geology, Vol. 31, No. 1, 1999

0882-8121/ 99/ 0100-0023$16.00/ 1 1999 International Association for Mathematical Geology

9369 Mathematical Geology PLENUM Aug′98 − Sep′98 963903ap 5/4/1999 [619] (PR1)

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Scale Matching with Factorial Kriging forImproved Porosity Estimation from Seismic Data1

Tingting Yao,2 Tapan Mukerji,3Andre Journel,2 and Gary Mavko3

When seismic data and porosity well logs contain information at different spatial scales, it isimportant to do a scale-matching of the datasets. Combining different data types with scalemismatch can lead to suboptimal results. A good correlation between seismic velocity and rockproperties provides a basis for integrating seismic data in the estimation of petrophysical properties.Three-dimensional seismic data provides an unique exhaustive coverage of the interwell reservoirregion not available from well data. However, because of the limitations of measurement frequencybandwidth and view angles, the seismic image can not capture the true seismic velocity at all spatialscales present in the earth. The small-scale spatial structure of heterogeneities may be absent in themeasured seismic data. In order to take best advantage of the seismic data, factorial kriging isapplied to separate the small and large-scale structures of both porosity and seismic data. Then thespatial structures in seismic data which are poorly correlated with porosity are filtered out prior tointegrating seismic data into porosity estimation.

KEY WORDS: spatial filtering, spectral coverage, cokriging.

INTRODUCTION

In earth sciences, precise measurements of petrophysical properties, such asporosity are always very sparse. We must resort to more abundant and acces-sible seismic data for estimation or simulation. Unfortunately, good correlationbetween seismic and porosity as observed in laboratory may not be reproducedon field data because of limitations of the measurement devices or differencein the measurement scales of porosity and seismic data. Therefore, in order to

1Received 30 September 1997; accepted 8 April 1998.2Stanford Center for Reservoir Forecasting, Department of Geological and Environmental Sciences,Stanford University, Stanford, California 94305. e-mail: [email protected]

3Rock Physics Laboratory, Department of Geophysics, Stanford University, California 94305. e-mail:[email protected]

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retrieve the good correlation, we may have to filter out those uncorrelated com-ponents and retain only the better correlated components.

When seismic data and porosity well logs contain information at differentspatial scales, it is important to do a scale-matching of these datasets. Combiningdifferent data types with scale mismatch can lead to suboptimal results. Theoverall correlation between seismic and well data comes from contributions atdifferent spectral frequencies. Using data containing frequencies at which theydo not have any useful signal can unnecessarily reduce the correlation and givepoor results.

We describe an approach based on factorial kriging (Sandjivy, 1984; Sand-jivy and Galli, 1984; Ma and Royer, 1988; Jaquet, 1989) for filtering and match-ing data at disparate scales having different spatial structures. Wen and Sinding-Larsen (1997) have analyzed the sensitivity of factorial kriging filters to filtershape. They also show that factorial kriging filtering can be superior to data-independent filters which are sensitive to noise.

We generated exhaustive synthetic seismic velocities based on a referenceporosity image on a 84 × 114 gridded area. The reference porosity data weretaken from a vertical section of a 3-D sequential Gaussian simulation realizationconditioned to facies data from wells in a Texas reservoir. From that referenceporosity dataset, true seismic velocity were simulated based on velocity-porosityregressions derived from laboratory ultrasonic measurements on core (Han, 1986).The seismic images measured at different frequency bandwidths, 5–25, 5–50, and5–100 Hz, respectively, were simulated from the seismic velocity data using afrequency-domain Born filter approach (Devaney, 1984; Harris, 1987; Wu andToksoz, 1987; Mukerji, Mavko, and Rio, 1997). The reference maps are shownin Figure 1. Because these reference data are assumed exhaustive, they providean opportunity to test the approach proposed.

HISTOGRAM AND SCATTERGRAM ANALYSIS

Histograms (Fig. 2) provide a first appreciation of the distributions of thedata. The histograms of porosity and velocity show typical asymmetry. Withdecreasing frequency bandwidth, the shape of the histogram of velocity becomesmore symmetric with smaller variance. This is because the limited measurementbandwidth filters out the high frequency components and smooths out extremevalues (Fig. 3). The Fourier support areas are limited within the low frequencyregion and shrink as the measurement frequency bandwidth decreases. There-fore, a potentially good correlation between porosity and true velocity does notguarantee a similar correlation between porosity and measured seismic velocity.

From the scattergrams of Figure 4, a good correlation is observed betweenporosity and true velocity, as expected from geophysical principles: the corre-lation coefficient is 0.991. Unfortunately, this is not anymore the case for seis-

Scale Matching with Factorial Kriging


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Figure 1. Grayscale maps of reference porosity, true velocity and seimic measurements with dif-ferent frequency range.

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Figure 2. Histogram of true porosity, velocity, and seismic measurements.



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Figure 2. Continued.

mic measurements. As the frequency bandwidth decreases from 5–100 to 5–25Hz, the scattergram becomes more “cloudy” and the correlation decreases cor-respondingly to 0.788 and 0.579.

VARIOGRAM INFERENCE AND MODELING

Prior to filtering spatial structures we must model the structures of spatialvariability and isolate those components that are shared by the porosity and theseismic image.

From a variogram map (not shown), zonal anisotropy is observed: the ver-tical range is shorter and its sill is higher. The variograms of both porosity andseismic are fitted with the same basic structures changing only the variance con-

Figure 3. Fourier support of seismic measurements at different frequency bandwidth. The spatialfrequency is denoted by kx and ky.

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Figure 4. Scatterplots of porosity vs. velocity and seismic measurements.



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tributions of each structure. For the practice of modeling a linear model of co-regionalization, refer to Isaaks and Srivastava (1988), Chu (1993), and Goovaerts(1997). The variogram models are as follows (Fig. 5):

For porosity:

g(hz, hy) c 0.1 + 0.195Sph

i1 hz

2.7 22

+ 1 hy

∞ 22

+ 0.315Sph

i1 hz

2.7 22

+ 1 hy

5.4 22

+ 0.205Sph

i1 hz

10.3 22

+ 1 hy

24.1 22

For seismic with bandwidth 5–100 Hz:

g(hz, hy) c 0.1085Sph

i1 hz

2.7 22

+ 1 hy

∞ 22

+ 0.1375Sph

i1 hz

2.7 22

+ 1 hy

5.4 22

+ 0.121Sph

i1 hz

10.3 22

+ 1 hy

24.1 22



i1 hz

2.7 22

+ 1 hy

∞ 22

+ 0.045Sph

i1 hz

2.7 22

+ 1 hy

5.4 22

+ 0.112Sph

i1 hz

10.3 22

+ 1 hy

24.1 22

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Figure 5.} Variogram inference and model fit for porosity and seismic measurement. In each subplot, the variogram with thelarger sill corresponds to the vertical direction.



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Figure 5. Continued.



i1 hz

2.7 22

+ 1 hy

∞ 22

+ 0.0223Sph

i1 hz

2.7 22

+ 1 hy

5.4 22

+ 0.061Sph

i1 hz

10.3 22

+ 1 hy

24.1 22

For the cross-variogram between porosity and seismic data with bandwidth 5–50Hz:

−g(hz, hy) c 0.138Sph

i1 hz

2.7 22

+ 1 hy

∞ 22

+ 0.04Sph

i1 hz

2.7 22

+ 1 hy

5.4 22

+ 0.13Sph

i1 hz

10.3 22

+ 1 hy

24.1 22



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Table 1. Contribution of Long Range vs. Short Range Structure Along (z) and ( y)Directions

Seismic Seismic SeismicParameter Porosity (5–100) (5–50) (5–25)

C1(z) 0.510 0.246 0.150 0.0708C2(z) 0.205 0.121 0.112 0.0833R(z) 0.40 0.49 0.75 1.18C1(y) 0.315 0.1375 0.045 0.0C2(y) 0.205 0.121 0.112 0.061R(y) 0.65 0.88 2.49 ∞

where hz is the vertical direction and hy is the horizontal direction.Let C1(z) and C2(z) be the variance contributions of the vertical short-range

structure (z-range 2.7) and the long-range structure (z-range 10.3), and R(z) cC2(z)/ C1(z) be their ratio. C1( y), C2( y), and R( y) are the corresponding valuesfor the horizontal direction ( y-short-range 5.4 and y-long-range 24.1). Table 1shows that the relative contributions R of the long-range structure increase asfrequency bandwidth decreases and that ratio R increases faster along h( y) thanin h(z) direction. Thus, there appears to be more smearing in the horizontal direc-tion than in the vertical direction. This is consistent with the scattering vector(which bisects the angle between the source-to-scatterer and scatterer-to-receivervectors) being near vertical for surface seismic reflection measurements (Muk-erji, Mavko, and Rio, 1997).

Next, we check that the model of linear coregionalization, for the pair poros-ity and 5–50 Hz seismic, is permissible (Journel and Huijbregts, 1978):

For the nugget effect:

det [ 0.1 0.00.0 0.0 ] c 0

For the second structure, Sph (f

(hz/ 2.7)2 + (hy/ ∞)2):

det [ 0.195 −0.138−0.138 0.105 ] c 0.001431 > 0

For the third structure, Sph (f

(hz/ 2.7)2 + (hy/ 5.4)2):

det [ 0.315 −0.04−0.04 0.045 ] c 0.012575 > 0

For the fourth structure, Sph (f

(hz/ 10.3)2 + (hy/ 24.1)2):



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det [ 0.205 −0.13−0.13 0.112 ] c 0.00606 > 0

Because the determinants of all the matrices of coefficients are nonnega-tive and all the diagonal elements of autovariograms are nonnegative, the linearmodel of coregionalization adopted is permissible. We need to check this per-missibility because these variograms with their structures will be used later inthe factorial kriging. In addition, from this model, we can calculate the corre-lation coefficient by adding the nugget and the sills of each structure. Becausethe first spherical structure has zonal anisotropy, its sill contribution to the totalcorrelation is strictly valid only for the vertical direction. The model correlationis:

r(0) c C(0)jpor . j seis

c −0.308f0.815 .

f0.262

c −0.6696

This model value almost identifies the sample linear correlation coefficient of−0.668 between porosity and seismic (5–50 Hz) data.

FACTORIAL KRIGING

Factorial kriging is now applied based on the previous variogram models.Consider a random field model Z(u) constituted of (K +1) independent stochasticcomponents (also called “factors,” hence the name “factorial kriging”):

Z(u) c Z0(u) + Z1(u) + . . . + ZK(u)

The Z-covariance is then the sum of the (K + 1) component covariances:

CZ(h) c K

∑k c 0

Ck(h)

The sum of the first k0 components is estimated as:

[Z(k0)(u)]*OK c n

∑a c 1

d(k0)a (u)Z(ua)

For example, these first k0 structures may correspond to noise or short-scale structures with zero mean. The factorial kriging system for the weightscorresponding to these components is:



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n

∑b c 1

d(k0)b (u)CZ(ub − ua) + m(k0)(u) c k0 − 1

∑k c 0

Ck(u − ua), a c 1, . . . , n

n

∑b c 1

d(k0)b (u) c 0

where m(k0)(u) is the Lagrangian parameter. For the complement large-scalestructures (the first k0 structures being filtered out) with mean equal to that ofthe original z(u), the system is:

n

∑b c 1

d(K − k0)b (u)CZ(ub − ua) + m(K − k0)(u) c K

∑k c k0

Ck(u − ua), a c 1, . . . , n

n

∑b c 1

d(K − k0)b (u) c 1

and the estimate is given by:

[Z(K − k0)(u)]*OK c n

∑a c 1

d(K − k0)a (u)Z(ua)

Thus, the algorithm of factorial kriging amounts to changing the right-handside covariance matrix in the usual normal equations for the kriging weights,and changing the constraint on the kriging weights (Deutsch and Journel, 1996;Goovaerts, 1997). In the right-hand-side matrix, we only retain the part of covari-ance associated with the components to be estimated, and we set different con-straints on the kriging weights (summing up to either 0 or 1) depending on whichcomponent is retained, nugget or large-scale structure. In the above equations,da represent the kriging weights. The superscripts (k0) and (K − k0) on theseweights indicate that they correspond to the first k0 components, and to theremaining components (with the first k0 structures filtered out), respectively. Thea c 1, . . . , n represent the known n data at locations ua. For a more detailedpresentation of factorial kriging, refer to Bourgault (1994), Sandjivy (1984),Wackernagel (1988), Ma and Myers (1994), and Deutsch and Journel (1996).Goovaerts (1992), Goovaerts, Sonnet and Navarre (1993), and Goovaerts andWebster (1994) present pioneering applications of factorial kriging in soil sci-ence and hydrogeology.

In this study, the search radius for kriging is set to 20.0 (unit c cell size)and the minimum and maximum data for kriging are 4 and 16, respectively. Westart by filtering out the short range structures only from the 5–50 Hz seismic



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Table 2. Correlations After Filtering out Short Range Components(Seismic Is 5–50 Hz)

1st struct. 1st + 2ndVariable Seismic filtered structs. filtered

Original porosity −0.67 −0.67 −0.65Nugget filtered −0.77 −0.77 −0.74Nugget + 1st + 2nd

structs. filtered −0.76 −0.85 −0.88

data (Table 2—1st row, Fig. 6). It appears that filtering does not improve theoriginal correlation (−0.67). Consider now filtering also the nugget effect fromthe porosity data, and then filtering the short range structures from the seismicdata (Table 2—2nd row, Fig. 7). The original correlation is improved to −0.77.Last, filter all short-scale structures from the porosity data, then filter the short-range structures from the seismic data (Table 2—last row, Fig. 8). Filtering theseismic now does help, increasing the correlation from −0.76 to −0.88.

It thus appears that the short-scale porosity structure is not reflected in theseismic data. The highest correlation is achieved when the short-scale structureis filtered from both porosity and seismic data. Hence, for utilization in porosityestimation, seismic data is better filtered to reflect only the large-scale structureof porosity. This is shown by comparing (Figs. 9–12) the cokriging estimates ofporosity using the raw, unfiltered seismic (5–50 Hz) vs. the factorial cokrigingporosity estimates using the filtered long range structure of the porosity and seis-mic data which are better correlated. Figure 9 shows the ordinary kriging (usingporosity alone) and cokriging (porosity and seismic) estimates of the porositywith the unfiltered seismic. Sample porosity data at three well locations wereretained from the complete reference data for the kriging. Two wells were atthe left and right edges of the vertical section, and a third well was at y c 10(pixel units) from the left edge. As is to be expected, both estimates are muchsmoother than the reference true porosity map, with the ordinary kriging merelybeing a horizontal interpolation of the porosity data from the three wells. Thescatterplots (Fig. 10) between the estimated and reference porosity show thatthe correlation between cokriging estimates and reference values is slightly bet-ter than the ordinary kriging estimates (0.55 vs. 0.45). Figures 11 and 12 show asimilar comparison, but now with the filtered porosity and seismic maps, keep-ing only the large-scale structure. Again the porosity data at the three wells, andthe seismic data at all grid nodes are used in the ordinary kriging and factorialcokriging to estimate the porosity. The scatterplot (Fig. 12) between the esti-mated porosity and reference porosity shows an increase in the correlation to0.73 for the factorial cokriging. This shows how factorial filtering of the appro-



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Figure 6. Scatterplot of original porosity vs. seismic (original and filtered out).



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Figure 7. Scatterplot of porosity with nugget filtered vs. seismic (original and filtered out).



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Figure 8. Scatterplot of porosity with nugget, first and second structures filtered vs. seismic (originaland filtered out).



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Figure 9. Ordinary kriging and cokriging porosity estimates using the unfiltered seismic at 5–50 Hz.Top: Reference porosity and seismic image. Bottom: Ordinary kriging (left) and cokriging (right)estimates.

priate structures can improve the cokriging estimates of the large scale structuresof porosity (see also Yao and Mukerji, 1996).

Tables 3–6 expand this correlation study to seismic true velocity and seis-mic measurements at different frequency bandwidths. From the above tables, weobserve that as the seismic measurement frequency bandwidth decreases, the cor-relation coefficient between original porosity and original seismic data decreasesfrom −0.99 to −0.79, −0.67, and −0.58; a similar pattern is observed for thecorrelation between porosity and seismic short-scale structures, from −0.79 to−0.57, −0.25, and −0.08. However, the correlation between large-scale struc-tures of porosity and seismic remain the same, −0.88, for different measurement



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Figure 10. Scatterplot between reference and kriging estimates of porosity.

bandwidths. Furthermore, for the true seismic velocity, the correlation betweenlarge-scale porosity and large-scale seismic velocity (−0.95) is lower than thatbetween full porosity and full seismic velocity (−0.99), while the contrary isobserved for seismic measurements. One explanation is that the true seismicvelocity relates to both the large-scale and the short-scale structures of poros-ity: when the short-scale seismic structure is filtered, the correlation decreases.Because the short-scale structures of seismic measurements are poorly correlatedwith that of porosity, this filtering increases the correlation.

We point out another important observation. For the 5–100 Hz seismicimage, filtering out the small-scale porosity and retaining only the large-scaleporosity decreases the correlation from −0.79 (original porosity) to −0.57. Butfor the 5–25 Hz seismic image, retaining only the large-scale porosity increasesthe correlation from −0.58 to −0.85. This can be understood in terms of the larger



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Figure 11. Ordinary factorial kriging and factorial cokriging porosity estimates using the filtered,large-scale structure of porosity and seismic at 5–50 Hz. Top: Reference porosity and seismic image.Bottom: Ordinary kriging (left) and cokriging (right) estimates of the large-scale porosity.

Fourier support of the wider bandwidth 5–100 Hz data. The important impli-cation is that retaining only the smooth large-scale structures does not alwaysincrease the correlation. The selection of the structures depends on the spectralcomponents present in the original image and the spectral support of the partic-ular seismic measurement. Knowing the spectral support domain therefore helpsin making a rational choice of the structures to be filtered out and the structuresto be retained in the kriging.

In summary, the general decrease in correlation between porosity and seis-mic measurements is caused by the small-scale structure of porosity being poorly



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Figure 12. Scatterplot between reference and factorial kriging estimates of porosity.

represented by the seismic measurements, which have a limited frequency rangeand imperfect measurement geometry.

Next, grey scale maps are plotted to depict the spatial structures of porosityand seismic (5–50 Hz). Seismic maps appear smoother with increasing filtering

Table 3. Correlations Between Different Scale Structures of Porosity and Seismic(True Velocity)

Large-scale Small-scaleVariable True porosity porosity porosity

True velocity −0.99 −0.66 −0.79Large-scale velocity −0.77 −0.95 −0.26Small-scale velocity −0.78 −0.08 −0.97



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Table 4. Correlations Between Different Scale Structures of Porosity and Seismic(5–100 Hz)

Large-scale Small-scaleVariable Original porosity porosity porosity

Original seismic −0.79 −0.57 −0.57Large-scale seismic −0.58 −0.88 −0.04Small-scale seismic −0.58 −0.06 −0.72

Table 5. Correlations Between Scale Structures of Porosity and Seismic(5–50 Hz)



Table 6. Correlations Between Different Scale Structures of Porosity and Seismic(5–25 Hz)



of short range structures (Fig. 13). The short scale components of porosity andseismic data do show residual spatial correlation. Thus, when retaining theseimperfectly filtered short-scale structures, we are losing some potentially valu-able information (structures).

CONCLUSIONS

When seismic data and porosity well logs contain information at differ-ent spatial scales, it is important to do a scale-matching of these datasets. The



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Figure 13. Greyscale maps of the results of factorial filtering applied on porosity and seismic (5–50Hz) data.

overall correlation between seismic and well data comes from contributions atdifferent spectral frequencies. Using data containing frequencies at which theydo not have any useful signal can unnecessarily reduce the correlation and givepoor results. We have shown how factorial kriging can be used to better integrateselected spatial structures best imaged by the seismic measurement, into porosityestimation techniques. The results of any factorial kriging depend on the decom-position of the random field model into the various structures or factors. Thisdecomposition is subjective and can be arbitrary, and the factors may not haveany physical significance (Deutsch and Journel, 1996). However, a knowledgeof the spectral support domain of the seismic measurement can sometimes helpin making a more rational selection of structures to filter out in the kriging andestimation algorithm.

One of the main goals of porosity simulation is to provide numerical modelsfor flow simulation. The flow simulation block scales are typically much largerthan the well log or seismic scales. Instead of combining seismic and log dataat mismatched scales to simulate porosity values at the well log scale and thenupscale these values to the flow simulator scale, we suggest that it might be more



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fruitful to “upscale” the porosity and seismic data through some sort of factorialkriging to the flow simulator scale, then directly simulate porosity at the flowsimulation scale. This concept of simulating already upscaled values has beenintroduced by Gomez-Hernandez (1991) and further developed by Tran (1995).

If the goal is to estimate or simulate porosity including its highest spatialfrequencies, the missing high frequency components of the seismic data spec-trum could be filled in. Approaches for completing the seismic data spectrumhave been suggested, based on various entropy or energy assumptions (Devaney,1984; Mavko and Burg, 1987). A better approach might be to fill in the missingcomponents of the seismic data spectrum by “borrowing” from the spatial spec-trum of the well-log porosity calibration data. As an example, the low spectralspatial frequencies could be obtained from the seismic while the higher frequen-cies could be filled in by iterated optimization to match dynamic well produc-tion data (e.g., Huang and Kelkar, 1996). Such spectrally enhanced seismic datamight help to improve the estimation or simulation of porosity at high spatialfrequencies.

ACKNOWLEDGMENTS

This work was sponsored by GRI contract 5093-260-2703, and supportedby the Stanford Rock Physics Project and Stanford Center for Reservoir Fore-casting. We thank Wenlong Xu for helping with the simulation of the porositycube. We acknowledge constructive reviews from P. Goovaerts and A. K. Singhwhich helped to improve the manuscript.

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scale matching with factorial kriging for improved porosity estimation from seismic data

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