Comput Geosci (2009) 13:13–27DOI 10.1007/s10596-008-9101-2

ORIGINAL PAPER

Improved initial sampling for the ensemble Kalman filter

Dean S. Oliver · Yan Chen

Received: 2 April 2008 / Accepted: 29 July 2008 / Published online: 29 August 2008© Springer Science + Business Media B.V. 2008

Abstract In this paper, we discuss several possible ap-proaches to improving the performance of the ensem-ble Kalman filter (EnKF) through improved samplingof the initial ensemble. Each of the approaches ad-dresses a different limitation of the standard method.All methods, however, attempt to make the resultsfrom a small ensemble as reliable as possible. Thevalidity and usefulness of each method for creatingthe initial ensemble is based on three criteria: (1) doesthe sampling result in unbiased Monte Carlo estimatesfor nonlinear flow problems, (2) does the samplingreduce the variability of estimates compared to ensem-bles of realizations from the prior, and (3) does thesampling improve the performance of the EnKF? Ingeneral, we conclude that the use of dominant eigen-vectors ensures the orthogonality of the generated real-izations, but results in biased forecasts of the fractionalflow of water. We show that the addition of high fre-quencies from remaining eigenvectors can be used toremove the bias without affecting the orthogonality ofthe realizations, but the method did not perform sig-nificantly better than standard Monte Carlo sampling.It was possible to identify an appropriate importanceweighting to reduce the variance in estimates of thefractional flow of water, but it does not appear to bepossible to use the importance weighted realizations instandard EnKF when the data relationship is nonlinear.The biggest improvement came from use of the pseudo-

D. S. Oliver (B) · Y. ChenMewbourne School of Petroleum Engineering,The University of Oklahoma, 100 East Boyd Street,Room SEC T301 Norman, Oklahoma 73019, USAe-mail: dsoliver@ou.edu

data with corrections to the variance of the actualobservations.

Keywords Ensemble Kalman filter · Monte Carlo ·Variance reduction · Importance sampling

1 Introduction

Almost all history matching algorithms, including theensemble Kalman filter, rely on Monte Carlo meth-ods to quantify the uncertainty in predictions. BecauseMonte Carlo methods are inherently random, the qual-ity of the results are variable and one collection of real-izations may be better than another. The goal is alwaysto get good assessments of uncertainty from as fewrealizations as possible. Standard methods for reducingthe variance of estimates using Monte Carlo methodsare described in Kalos and Whitlock [11] and Caflisch[4]. Berliner and Wikle [2] describes approximate im-portance sampling methods that can be used to improvethe generation of the initial ensemble for particle filtersin high dimensioned systems and van Leeuwen [23]describes an importance resampling algorithm for usewith particle filters when observations are available.Importance sampling has also been applied to reduc-tion in the variance of estimates of state variables foruncertainty quantification in porous media flow [13].

The problem of improving the performance of EnKFthrough improvement of the initial ensemble of realiza-tions has been previously examined by Evensen [7] whosuggested a method for generating ensemble membersthat best approximate the model covariance. Evensen’smethod has been used by Skjervheim et al. [20] fortime-lapse seismic data assimilation and is closely

14 Comput Geosci (2009) 13:13–27

related to the SEIK filter of Pham [17] that has beenused successfully in oceanography [10, 14]. Becausethis method has become somewhat standard, we willexamine it in some detail. Turner et al. [22] describe amethod that differs from Evensen’s [7] method primar-ily in the generation of the large initial ensemble fromwhich dominant eigenvectors are extracted. Wen andChen [24] propose what appears to be a form of strati-fied sampling for improvement of the initial ensemble.Their method requires that all realizations be simulatedfor a short period, after which an initial ensemble isselected based on production forecasts. The method ofSchubert et al. [19] selects initial perturbations basedon a stratification of the error distribution in a weatherprediction application. The stratification thus obtainedis used to choose a small subsample of initial stateswith which to perform the dynamical Monte Carloforecasts. Stratified importance sampling and controlvariates were shown by Oh and Berger [15] to be par-ticularly effective variance reduction techniques whenthe density can be modeled as a mixture of t densityfunctions.

In this paper, we will examine three fundamentallydifferent approaches to improvement of EnKF throughimprovement of the initial ensemble.

1. Reduce redundancy. It seems intuitively obviousthat the repetition of a reservoir realization in anensemble will not add to the predictive power of theensemble, or its ability to assimilate data. Evensen[7] suggested that the initial ensemble should besampled such that the members are orthogonal inthe subspace spanned by the dominant eigenvec-tors of the model covariance. By making each ofthe initial realizations orthogonal to all others, themembers are as dissimilar as possible with respectto this particular measure.

2. Reduce sampling error (variance of estimates).Since most of the cost in data assimilation is asso-ciated with numerical simulation of flow and trans-port, the efficiency of the ensemble Kalman filtercan be greatly increased if small ensembles can beused to represent the PDF of the state and modelvariables. For small ensembles, however, the vari-ability between ensembles can become quite large.Variance reduction methods, such as importancesampling, can be used to reduce variability amongthe estimates.

3. Reduce variability far from the observed behavior.A reduction in variability improves the linearity ofthe problem and the applicability of the ensembleKalman filter. In other words, the assumption oflinearity inherent to the analysis step in EnKF is

better satisfied when the magnitude of the datamismatch is small.

We generally require that the Monte Carlo estimatesof the mean and variance resulting from an “improvedsampling method” should be unbiased when the prob-lem is linear and that the bias be small when theproblem is nonlinear. We also seek sampling methodsfor which the variance of the Monte Carlo estimatesare small and for which EnKF can be used for dataassimilation.

Although the focus of this paper is on initial sam-pling for EnKF, it is useful to consider the particlefilter, which is a very different ensemble-based MonteCarlo method for data assimilation. In both the en-semble Kalman filter and the ensemble particle filter,the ensemble of states or particles provides a MonteCarlo approximation to the PDF, and in both methodsthe forecast step is identical. The two methods differgreatly, however, in the analysis step where new dataare assimilated. In a particle filter [6], when data are as-similated, the probability density for the model parame-ters is estimated based on an assumption that the priorPDF is adequately represented by a discrete PDF. Theconditional probability of each of the forecast states iscomputed based on the previous probability and thelikelihood of the variables. In the basic application, onlythe probability of each of the states is adjusted, not thestates themselves. There is no assumption of linearitymade when computing the weights of the particles, butas data are assimilated and the region of significantconditional probability in model space becomes small,it is not uncommon for all particles to be located inregions of low probability density.

In the analysis step for the ensemble Kalman fil-ter, each of the state vectors (with model variablesincluded) is adjusted for consistency with the data.The states themselves are assumed to be Monte Carlosamples from the PDF so there is no need for weighting.The adjustment is, however, based on the assump-tions of linearity and normality so the adjustments arenot optimal for nonlinear problems. A combination ofaspects of both methods (weighting and adjustment)provides the basis for a hybrid method that uses impor-tance weighting with EnKF.

2 Effect of initial sampling on forecast

2.1 Orthogonal ensemble members

Evensen [7] suggested that the initial ensemble begenerated such that, for a given ensemble size, it best

Comput Geosci (2009) 13:13–27 15

represents the model covariance. To do this, he pro-posed that the realizations in the initial ensembleshould be linear combinations of the dominant eigen-vectors of the covariance matrix. If the realizations arethen generated in such a way that they are random andorthogonal in the subspace spanned by the dominanteigenvectors, then we conclude that they are as dissim-ilar as possible. This method of construction eliminatesthe possibility of generating redundant ensemble mem-bers, at least with respect to the covariance measure inthe subspace.

Assume the dimension of a realization is Ns. If thereare only p realizations, we require that the projectionsof the realizations onto a subspace Sp spanned by thethe p dominant eigenvectors of Cy should be orthogo-nal with respect to the inner product

(u, v)p = uTC−1yp

v. (1)

Here, C−1yp

= U p�−1p UT

p is the restriction of C−1y to the

subspace spanned by the first p eigenvectors of Cy

(columns of U p) associated with the p largest eigen-values (diagonal elements of �p). Note that the pro-jection of a vector onto Sp (in the original coordinatesystem) is

wp =(

U pUTp

)w

and the restriction of the covariance matrix Cy to Sp is

Cyp =(

U pUTp

)Cy

(U pUT

p

)T

=(

U pUTp

) [U p U0

] [�p 00 �0

] [UT

p

UT0

](U pUT

p

)

= U p�pUTp , (2)

where �0 is a diagonal matrix with the remaining Ns− peigenvalues on its diagonal and U0 contains columns ofeigenvectors associated with �0.

2.1.1 Method 1 — smooth orthonormal

Evensen’s [7] suggestion is to generate an improvedinitial ensemble, Y(⊥), from the dominant singular val-ues of a reduced rank approximation to the modelcovariance matrix. In this investigation, however, toreduce variability in results we will use the dominanteigenvectors of the true covariance matrix instead ofthe singular vectors of the approximate covariance.

Y(⊥) =(

U p�1/2p

)Z (⊥) (3)

where Z (⊥) is a vector of zero-mean univariateGaussian random variables and the columns of Z (⊥) areconstructed to be orthogonal. Note that this is simplya square root method for generating Gaussian randomvariables with an additional orthogonality constraint onthe columns of Z (⊥).

Figure 1 shows a set of realizations from the priorPDF and a set sampled from the dominant eigenvec-tors using Eq. 3. It is clear that realizations generatedfrom the first few eigenvectors of the covariance aremuch smoother than realizations generated from thecomplete set of eigenvectors.

It is straightforward to establish that all membersof the initial ensemble are orthogonal with respect tothe inner product of Eq. 1. Let Z (⊥)

i and Z (⊥)

k be two

20 40 60 80 100

-2

-1

1

2

20 40 60 80 100

-2

-1

1

2

20 40 60 80 100

-2

-1

1

2

20 40 60 80 100

-2

-1

1

2

20 40 60 80 100

-2

-1

1

2

20 40 60 80 100

-2

-1

1

2

Fig. 1 Three realizations from the true PDF (top), three realizations from the first 10 eigenvectors (bottom)

16 Comput Geosci (2009) 13:13–27

columns of Z (⊥) and let Y(⊥)

i and Y(⊥)

k be the corre-sponding initial state vectors. The inner product is

(Y(⊥)

i

)TC−1

ypY(⊥)

k =[(

U p�1/2p

)Z (⊥)

i

]T

×[U p�

−1p UT

p

] [(U p�

1/2p

)Z (⊥)

k

]

=[

Z (⊥)

i

]TZ (⊥)

k

= 0 if i �= k (4)

because of the orthogonality of the columns of Z (⊥).

2.1.2 Method 2 — rough orthonormal

The realizations from Method 1 are too smooth to berandom draws from the prior PDF, so the samplingappears to be at least qualitatively incorrect [9]. It is,however, straightforward to modify the method so thatthe orthogonal realizations in the initial ensemble areindistinguishable from realizations sampled from theprior PDF. Let

Y(⊥) = [U p U0

] [�

1/2p 00 �

1/20

][Z (⊥)

Z

](5)

where Z (⊥) is the same p × p matrix whose columns areorthogonal random vectors and Z is an (Ns − p) × pmatrix of independent standard Gaussian random vari-ables. Figure 2 shows three realizations from Method 2corresponding to the three realizations from Method 1.Note that the new realizations are virtually indistin-guishable from realizations drawn from the prior PDF(top row of Fig. 1) while containing all of the same largescale features as the corresponding realizations fromMethod 1 (bottom row of Fig. 1).

Although the realizations generated from Eq. 5 havegreater roughness, it is straightforward to establish that

these realizations are orthogonal with respect to thesame measure that was used in Method 1 (Eq. 1).(

Y(⊥)

i

)TC−1

ypY(⊥)

k

=([

U p U0] [

�1/2p 00 �

1/20

][Z (⊥)

iZi

])T

×[U p�

−1p UT

p

] [U p U0

] [�

1/2p 00 �

1/20

] [Z (⊥)

kZk

]

=[

Z (⊥)

iZi

]T[�

1/2p 00 �

1/20

]

×[

UTp

UT0

] [U p�

−1p UT

p

] [U p U0

] [�

1/2p 00 �

1/20

] [Z (⊥)

kZk

]

=[

Z (⊥)

iZi

]T [I 00 0

] [Z (⊥)

kZk

]

= 0 if i �= k (6)

2.1.3 Method 3 — rough orthogonal with lengthsampling

In the original presentation of the improved samplingmethod using dominant eigenvectors [7], the methodthat was used to generate an ensemble of randomorthogonal realizations, Z (⊥), generated columns thatare not only orthogonal, but also orthonormal. Theelements of the normalized vector were multiplied bythe square root of the dimension of the vector so thatthe elements of Z (⊥) would have variance equal to one.If, however, the samples are assumed to be realizationsfrom a multivariate Gaussian PDF,

p(Y) ∝ exp

[−1

2(Y − μy)

TC−1y (Y − μy)

],

which is equivalent to saying that the realizations canbe generated as

Y = μy + LZ with LLT = Cy

20 40 60 80 100

-2

-1

1

2

20 40 60 80 100

-2

-1

1

2

20 40 60 80 100

-2

-1

1

2

Fig. 2 Three realizations from Method 2, corresponding to realizations from Method 1 (bottom row of Fig. 1)

Comput Geosci (2009) 13:13–27 17

where Z is a vector of iid standard normal deviates,then the squared length of the vector Z is a randomvariable W = ‖Z‖2 whose distribution is chi-square or

‖Z‖2 ∼ χ2Ns

.

The realizations from Method 2 can be modified tohave the correct distribution of lengths in the subspaceSp simply by multiplying each orthonormal realizationby the square root of a sample from the chi-squaredistribution with p degrees of freedom to generateZ (⊥). In our implementation of Method 3, the samplingis stratified to ensure a good distribution of lengths.

2.2 Importance sampling for reduction in variabilitybetween ensembles

The ultimate goal of any Monte Carlo method, of whichEnKF is an example, is to estimate properties of the un-certain probability densities for variables and forecastsby computing integrals of the form

V = Ep[g] =∫

g(x)p(x) dx

where p(x) is the probability density for x. In petroleumengineering, x might be the permeability and porosityfields, and g(x) might be the fractional flow of water at aproducing well, in which case the value of the integral isthe expectation of the fractional flow of water. Becausethe dimension of x is typically very large, the integral isapproximated by drawing Np samples of xi from p(x)

and then computing

V ≈ VNp = 1

Np

Np∑i=1

g(xi).

One weakness of this approach is that when Np is small,the variability of the estimates can be large, especially ifg(x) is highly variable in the regions of significant prob-ability. In some cases, however, it is possible to reducethe variability of the estimates through an improvedsampling of the evaluation locations.

Using an importance sampling form of Monte Carlo,we draw samples from a probability distribution p(x),and then compute

V =∫

g(x)p(x)

p(x)p(x) dx

≈ 1

Np

Np∑i=1

g(xi)p(xi)

p(xi)

= 1

Np

Np∑i=1

aig(xi)

where xi is drawn from p(x) and

ai = p(xi)

p(xi).

The choice of the probability distribution from whichto sample is the key to good performance of thisalgorithm. Clearly, if p(x) = c g(x)p(x) there would beno variability in the estimate of the expectation of g(although the variability in the estimate of some otherquantities might actually increase). Although methodsfor estimating the optimal PDF for importance sam-pling have been suggested (e.g. Capriotti [5]), mostappear to be impractical for problems with very largenumbers of variables. Instead, we will focus on sam-pling from a PDF that has approximately the correctstructure, and that might be easy to sample from. Onestraightforward approach to sample generation that canbe easily applied even when the dimension of the modelspace is very large is sampling-resampling [21] in whicha large number of realizations are initially sampledfrom one distribution, then resampled to representsamples from another.

2.2.1 Importance sampling for the initial ensembleto estimate the fractional flow of water

Here we illustrate the application of importance sam-pling for reducing variability in Monte Carlo estimatesof production forecasts. We first generate a large num-ber, Nbe, of realizations of the model variables (e.g.the permeability field). This large ensemble serves as adiscrete approximation to the prior probability densityp(m). The objective is to sample from some otherprobability density, p(m), that is more heavily weightedtowards the “important” regions of the model space.Assume that p(m) is related to p(m) by the importanceweighting w(m),

p(m) = cw(m)p(m).

Since p(m) must be a density, we require∫

p(m) dm =∫

cw(m)p(m) dm = 1.

A Monte Carlo approximation to the integral canbe used to estimate the value of the normalizationconstant:

c = Nbe∑Nbei=1 w(mi)

,

where mi is sampled from p(m). Because computationof the weighting constant, c, requires the evaluation of

18 Comput Geosci (2009) 13:13–27

the importance factor for all realizations in the largeensemble, a practical importance factor must be in-expensive to compute (i.e. it should not require largesimulation runs).

Although restrictions on possible variables to use asthe importance weight are fairly mild, the ability toreduce the variability of the estimate depends stronglyon the quality of the choice. If, for example, the goalis to predict the fractional flow of water, then a goodimportance variable should behave like the fractionalflow of water, in that realizations with small fractionalflow of water (at a particular time) should have acorrespondingly small value of the importance variableand vice versa. For a 1D waterflood with fixed pressuresat the ends, the velocity of the water front is propor-tional to the harmonic average of permeability, so theharmonic average of permeability would be a goodimportance variable if the objective was to estimatefront velocity. The water-oil ratio (ratio of water pro-duction rate to oil production rate) is a more complexfunction, but for the particular relative permeabilitycurves and viscosities that we use here, the ratio ofproducing water rate to oil rate (Fwo) at day 180 isapproximately proportional to the cube of the har-monic average of permeability (see Fig. 3). The variablethat we wish to estimate, however, is the fractionalflow of water at day 180, so instead of using k3

H as animportance variable, we use

wi = 5.5 k3H,i

10−6 + 5.5 k3H,i

, (7)

where kH,i is the harmonic mean permeability of the ithrealization in units of μm2.

After computing the importance weights, wi, foreach realization in the large ensemble, we resample Ne

0.0000 0.0002 0.0004 0.0006 0.0008 0.00100

100

200

300

400

500

kH cubed

wat

eroi

lrat

io–

Fig. 3 Water-oil-ratio at 180 days versus the cube of the har-monic average of permeability for 10,000 realizations

realizations from the large ensemble, using wi/∑Nbe

i=1 wi

as the probability density for resampling. For the ithresampled realization, we run the simulator to computethe fractional flow of water, fw,i. The estimate of meanand variance of the fractional flow of water at day 180from ensemble k are

fw,k = 1

Ne

Ne∑i=1

fw,i

cwi(8)

and

σ 2fw,k = 1

Ne − 1

Ne∑i=1

(fw,i − fw,k

)2

cwi, (9)

where for simplicity of notation, the index for fw refersonly to realizations in the kth ensemble.

2.3 Evaluation of initial sampling methodsfor forecast estimation

We compare the four methods described above tostandard Monte Carlo sampling, for a nonlinear testproblem, to determine (1) if the ensemble means offorecasts are biased and (2) if any of the methodsprovides a reduction in variance of the means.

The problem tests the ability to estimate the frac-tional flow of water at day 180 in a 1-D two-phasesynthetic reservoir model. The model is uniformly dis-cretized into 100 gridblocks, each with size 5 × 5 × 5 ft3.The prior PDF for the logarithm of permeability ismultivariate Gaussian with mean equal to 4 and stan-dard deviation equal to 2. The covariance is exponentialwith practical range equal to 25 gridblocks. Porosityis known and is equal to 0.15 in all gridblocks. Thereis an injector in the first gridblock and a producer inthe last gridblock. The injector is controlled by thepressure at the perforations pbh of 5000 psi and theproducer is controlled by pbh of 2000 psi. The end pointrelative permeabilities are 0.55 and 0.8 for water and oilrespectively. Residual oil and irreducible water satura-tion are both 0.2. Capillarity is neglected. Viscosity andformation volume factor for oil and water are both veryclose to 1. Initial reservoir pressure is 4500 psi.

A commercial simulator was used to forecast thefractional flow of water at day 180 for each realization.We generate 4000 ensembles of size 4, 10, 40, and 100and report the ensemble mean prediction and varianceof the ensemble means for each of the four methodsdescribed above and for a traditional Monte Carloapproach in which the realizations for the ensembleare generated independently from the prior PDF. Forvalidation of importance sampling, we ran the reservoir

Comput Geosci (2009) 13:13–27 19

0.2 0.4 0.6 0.8 1

50

100

150

200

0.2 0.4 0.6 0.8 1

50

100

150

200

250

0.2 0.4 0.6 0.8 1

100

200

300

400

500

0.2 0.4 0.6 0.8 1

100

200

300

400

500

600

700

(a) (b) (c) (d)

Fig. 4 Histogram of ensemble mean values of fractional flow of water at 180 days for standard Monte Carlo sampling. a Four members.b Ten members. c Forty members. d One hundred members

simulator to compute the water-oil ratio at day 180for all the 400,000 realizations of the permeability field(the entire large ensemble) to choose the importancevariable (Eq. 7). In practice, one might only run thesimulator for a small number of realizations to assist inidentifying an importance variable.

Figure 4 shows histograms of 4000 ensemble meansof the fractional flow of water at day 180 when differentsizes of ensemble were used. Because most realizationsof the fractional flow of water at day 180 are eitherclose to zero or close to 0.9, the histogram of ensem-ble averages is clearly multimodal when Ne = 4 (seeFig. 4(a)). The first peak (near fractional flow of waterof zero) corresponds to ensembles in which all realiza-tions have very small fractional flow of water, while thepeak fractional flow of water near 0.9 corresponds toensembles in which all realizations have high fractionalflow of water. The intermediate peaks correspond tovarious combinations of realizations with high and lowfractional flow of water. When the ensemble size islarger, the distributions are more nearly Gaussian.

Table 1 compares estimates of the fractional flow ofwater at day 180 from the different sampling methodsas a function of the fraction of the cumulative eigen-values used in the expansion. The true expectation offractional flow of water is approximately 0.531. Theerror ranges in Table 1 are the standard deviations ofthe estimates from different ensembles. If a methodis unbiased, it should give estimates that are closeto 0.531. The fractional flow of water estimate fromMethod 1, however, is far too high for ensembles ofsize 4. The estimate from Method 1 is also high when

10 realizations are used in each ensemble. Only whenthe number of ensemble members approaches the sizeof the model space does Method 1 give approximatelycorrect estimates of fractional flow of water. Resultsare not shown for Ne = 100 for Methods 2 and 3 inTable 1 because the size of the model space is the sameas the size of the ensemble so there are no additionaleigenvectors to add.

Only the importance-weighted estimates of themean fractional flow of water and the estimates fromMethod 3 (orthogonal dominant eigenvectors withrandom lengths) are consistent with estimates fromstandard Monte Carlo sampling for the smallest num-ber of included eigenvectors. The differences betweenMethods 2 and 3 is insignificant when the ensemblesize is 10 or greater and both are close to the resultsfrom importance sampling. It seems unlikely that thisdifference will be important in practical applications.Note, however, that the variability of the estimates isgreatly reduced when importance sampling is used withthe function in Eq. 7 as the importance variable. In thiscase, only one tenth as many realizations are requiredusing importance sampling to achieve the same level ofprecision as standard Monte Carlo.

The comparison of the estimates of the mean frac-tional flow of water from the various methods aresummarized graphically in Fig. 5. In this figure, thebox spans from the 0.25 quantile to the 0.75 quantile,the whiskers span the entire range of the set and thehorizontal line in the box indicates the median. It iseasy to see from Fig. 5 the bias in Method 1 and thereduction in variance in importance sampling.

Table 1 Ensemble estimates of the mean forecast fractional flow of water at 180 days. Cum EV is the fraction of the sum of theeigenvalues contained in the Ne largest eigenvalues

Ne Cum EV Trad MC Method 1 Method 2 Method 3 Importance

4 0.50 0.532 ± 0.202 0.903 ± 0.037 0.516 ± 0.190 0.527 ± 0.190 0.531 ± 0.07110 0.76 0.533 ± 0.129 0.716 ± 0.093 0.522 ± 0.121 0.529 ± 0.114 0.531 ± 0.04440 0.95 0.532 ± 0.063 0.573 ± 0.056 0.534 ± 0.059 0.533 ± 0.057 0.531 ± 0.023

100 1.00 0.531 ± 0.040 0.531 ± 0.036 0.531 ± 0.014

20 Comput Geosci (2009) 13:13–27

Fig. 5 Comparison ofensemble means of forecastfractional flow of water at day180 from different samplingmethods

4 10 40 100 4 10 40 100 4 10 40 4 10 40 4 10 40 100

0

0.2

0.4

0.6

0.8

1

MC Method 1 Method 2 Method 3 Importance

The ensemble of forecasts is also used to estimatethe uncertainty in the forecast (shown in Table 2). Thetrue expectation of standard deviation of fractional flowof water at day 180 is approximately 0.402. Again, theresults from Method 1 with only 4 ensemble membersare far from the truth (0.077 vs. 0.402). Importancesampling, using the same weighting that was used forthe estimate of the mean, seems to be somewhat biased,presumable because the constant used for weightingis not quite correct. Note, also, that the importance-weighted estimate of the standard deviation of frac-tional flow of water (Table 2) is less precise (largervariability of the estimate) than the estimate from stan-dard Monte Carlo. This is simply a reflection of thefact that importance sampling may improve estimatesof some variables while the estimates of others mayworsen [1]. The best estimates of standard deviation offractional flow of water in this example are obtainedusing Method 3.

3 Effect of initial sampling on analysis

If the estimate from the ensemble of realizations at theend of the forecast step is severely biased, it is unlikelythat the analysis step will be useful. It is possible, how-ever, that even methods that are unbiased in the fore-cast and that reduced the variance substantially maynot work well with the EnKF analysis step. Conversely,methods that gave only modest improvement in the es-timates from the forecast may give large improvementsin the EnKF analysis step because of improvements in

the approximation of the Kalman gain, or because ofa decrease in the nonlinearity of the problem due to areduced variance within the ensemble.

3.1 Orthogonal dominant eigenvectors with EnKF

Initial ensembles were generated using Method 1(smooth realizations) and Method 2 (rough realizationswithout length sampling). The realizations in each en-semble were updated using standard EnKF analysiswith perturbed data [3]. Because the ensemble meanfrom the forecast was significantly biased, we testedthe methods with two different fractional flow of waterobservations: fw = 0.687 and fw = 0.368. In both cases,we created 4000 ensembles each of which contained 10realizations of the model variables.

Results from the investigation of the use of the re-alizations generated using orthogonal dominant eigen-vectors with EnKF are shown in Table 3. In one testthe fractional flow of water at day 180 was observed tobe 0.687 ± 0.020 and in the other test it was observedto be 0.368 ± 0.020. After assimilation of the observa-tions, the ensemble estimate of fractional flow of waterobtained from rerunning the models should be close tothe observed values. In the case of the high fractionalflow of water measurement, we see that standard EnKFis marginally better than either Method 1 or Method 2.When the low fractional flow of water observation isassimilated, Method 1 appears to give the best result. Infact, however, none of the methods gave good results.Figure 6 shows the distribution of ensemble meansfrom the various methods. Note that approximately onetenth of the ensembles that used Method 1 resulted

Table 2 Ensemble estimates of the standard deviation of forecast fractional flow of water at 180 days. Cum EV is the fraction of thesum of the eigenvalues contained in the Ne largest eigenvalues

Ne Cum EV Trad MC Method 1 Method 2 Method 3 Importance

4 0.50 0.377 ± 0.137 0.077 ± 0.022 0.379 ± 0.114 0.385 ± 0.121 0.362 ± 0.18410 0.76 0.397 ± 0.055 0.316 ± 0.082 0.395 ± 0.047 0.401 ± 0.046 0.363 ± 0.17140 0.95 0.401 ± 0.021 0.393 ± 0.022 0.399 ± 0.019 0.403 ± 0.018 0.383 ± 0.140

100 1.00 0.402 ± 0.012 0.401 ± 0.011 0.388 ± 0.087

Comput Geosci (2009) 13:13–27 21

Table 3 Ensemble estimates of fractional flow of water at day 180 for two different measurements

fw,obs = 0.687 fw,obs = 0.368Ne EnKF M1 + EnKF M2 + EnKF EnKF M1 + EnKF M2 + EnKF

10 mean .752 ± .099 .789 ± .138 .778 ± .080 .440 ± .126 .378 ± .189 .437 ± .11910 std .202 ± .103 .116 ± .058 .153 ± .083 .339 ± .056 .286 ± .107 .378 ± .189

in mean fractional flow of water estimates of 0 afterassimilation of data (Fig. 6(b)). Because the differencebetween Method 2 and Method 3 is largely a functionof ensemble size and appears to become insignificantfor ensemble size of about 10, we did not evaluate ithere or in the final test problem.

3.2 Data assimilation for weighted realizations

In the previous section, we addressed the use of impor-tance sampling to improve the estimate of random vari-ables from a small ensemble of realizations. In order tobe useful for data assimilation, it is necessary to eithercontinue to adjust the weights (as in particle filters) orto adjust the variables themselves (and possibly theirweights) as data are assimilated. In this section, weinvestigate a hybrid method that has some similaritieswith particle filters in that the model realizations areweighted, and yet uses the Kalman gain to update themodel variables.

We consider a problem for which sampling uncon-ditional realizations from a prior PDF, p(m) is rela-tively easy, but for which sampling from the posterior,p(m|dobs), is more difficult. We would also like toapply importance sampling to allow the use of smallensembles. In bootstrap resampling [21] or sampling/importance resampling [18] a large ensemble is sampledfrom the prior PDF for the model variables, p(m).After identifying an appropriate importance variable,smaller ensembles are resampled using the importance

factor w(m) such that the resulting realizations aredrawn from a PDF

p(m) ∝ w(m) p(m).

If the ensemble Kalman filter is then used to producerealizations from the posterior PDF, conditional to datad, then the realizations should be samples from a PDF

p(m|d) = c w(m) p(m|d).

When we compute expectation of some functiong(m) with respect to the posterior PDF p(m|d), we mustevaluate

Ep(m|d)[g] =∫

g(m)p(m|d) dm

=∫

g(m)

c w(m)c w(m) p(m|d) dm

=∫

g(m)

c w(m)p(m|d) dm

≈ 1

Ne

∑ g(mi)

c w(mi)for mi from p(m|d).

(10)

The constant appearing in the formula for computationof the expectation can be approximated by the require-ment that both p(m|d) and p(m|d) must be densities. Itmeans that∫

p(m|d) dm =∫

a p(d|m) p(m) dm = 1

0.2 0.4 0.6 0.8

50

100

150

0.2 0.4 0.6 0.8

50

100

150

200

250

300

350

0.2 0.4 0.6 0.8

50

100

150

(a) (b) (c)

Fig. 6 Histograms of ensemble means for 4000 ensembles of10 realizations. The means are estimated after the analysis stepwhen the observed fractional flow of water equals 0.368 (indi-

cated by the vertical line) with standard deviation equal to 0.020.a EnKF. b Method 1 with EnKF. c Method 2 with EnKF

22 Comput Geosci (2009) 13:13–27

and∫p(m|d) dm =

∫c w(m) p(m|d) dm = 1.

We use Monte Carlo integration over the large ensem-ble to approximate the constants, so

a = Nbe∑p(d|mi)

(11)

and

c = Nbe

a∑

w(mi) p(d|mi), (12)

where mi are samples from p(m). Combining Eqs. 11and 12, we obtain an expression for c that can be usedin Eq. 10 to compute expectations from EnKF withimportance sampling of the prior ensemble:

c =∑

p(d|mi)∑w(mi) p(d|mi)

. (13)

In this method of estimating c, it is necessary to com-pute p(d|m) for all realizations in the large ensemble.

3.2.1 Importance sampling EnKF appliedto a linear problem

Consider a linear problem in which the permeability ona 1D grid of 100 blocks is a sample from a Gaussianrandom field with mean 10 and exponential covariancewith practical range of 25 gridblocks and variance equalto 1.0. We make one measurement of the sum of allgridblock permeabilities (dobs = 1020.) with standarderror of the measurement equal to 10. We wish to es-timate the permeability in the first gridblock and its as-sociated uncertainty. Because the relationship betweenthe observations and the model variables is linear, andthe prior PDFs for model variables and the likelihoodare Gaussian, the posterior PDF is also Gaussian and itis straightforward to compute the true posterior meanand variance (Table 4).

After assimilation of the measurement of the sumof all permeabilities, the true value of the posteriormean permeability in the first gridblock is 10.109 withstandard deviation 0.976. Because we wish to estimate

the value of the permeability of the first gridblock, weshould choose the permeability of the first gridblockas the importance variable. Importance resampling forthe initial ensemble and EnKF for data assimilationgives an estimate of 10.1093 ± 0 for the mean and0.993 ± 0.224 for the standard deviation. These are themean results from 1000 ensembles each of which has 40members. There is no variability in the estimate of themean between ensembles because the permeability ofthe first gridblock was used as the importance variable.

Simple EnKF, in which the initial ensemble is drawnfrom the prior PDF without importance weighting,gives an estimate of 10.099 ± 0.175 for the mean and0.974 ± 0.220 for the standard deviation from 1000 en-sembles when the ensemble size is 40. The uncertaintiesrefer to the variability among ensembles. Clearly, wecan say that importance sampling can be used withEnKF for linear Gaussian problems to reduce the vari-ability in the estimates of particular variables.

3.2.2 Importance sampling EnKF appliedto a nonlinear problem

The same nonlinear test problem as in the previoussection is used to test the use of EnKF with weightedrealizations. The data to be assimilated in this caseis the producing fractional flow of water at 180 days.Because a high fractional flow of water is correlatedwith a high value of harmonic mean permeability, wedecided to use a function of the harmonic mean ofpermeability (Eq. 7) for importance weighting. Thesame 400,000 realizations of the permeability field fromthe prior PDF were used. For each realization, we com-puted the harmonic mean of permeability, which wethen used to compute importance weights (Eq. 7) forresampling 4000 ensembles of 40 members. StandardEnKF updating (see e.g. Evensen, [7]) was used to as-similate the observation of fractional flow of waterat day 180. Following the update, the importanceweighted ensemble approximation to the mean frac-tional flow of water was computed using Eq. 10. In thiscase, we need to recalculate the importance weightswith the updated model variables and rerun the sim-ulator with updated model variables to compute the

Table 4 Ensemble estimatesof the permeability ofgridblock 1 after dataassimilation

True population EnKF estimate EnKF withimportance

Mean 10.109 10.099 ± 0.175 10.109 ± 0.000Standard dev. 0.976 0.974 ± 0.220 0.993 ± 0.224

Comput Geosci (2009) 13:13–27 23

updated realizations the fractional flow of water. Thedetailed results will be shown later.

3.3 Assimilation of pseudo-observation

One of the difficulties with the use of the EnKF fordata assimilation in reservoir production is that whilethe analysis (or updating) step is useful for linearand slightly nonlinear relationships, the relationshipsbetween production data and reservoir model vari-ables are inherently nonlinear for multiphase flow.Although the precision of the measurements does notaffect the nonlinearity of the relationship, it does limitthe applicability of the EnKF analysis step for up-dating, i.e. highly precise measurements in nonlinearsystems are more difficult to assimilate using EnKF.One possible solution is to assimilate multiple less pre-cise measurements instead of a single highly precisemeasurement.

For linear problems, it is difficult to imagine a sit-uation for which it might be advantageous to replacea single measurement with two less precise measure-ments. For nonlinear problems, however, there may beadvantages, especially when a linear analysis step, suchas in EnKF, is used for data assimilation.

Consider a system in which the relationship betweenan observation dobs and the model variables m is linearand the errors are additive, i.e. dobs = Gm + ε. The dataerrors are normally distributed with mean zero andvariance σ 2

d . The prior PDF for the model variables isalso Gaussian with mean mpr and covariance CM. Afterassimilation of the single observation, the posteriorPDF is Gaussian with mean

mpost = mpr +(

GTG

σ 2d

+ C−1M

)−1GT

σ 2d

(dobs − Gmpr).

(14)

The posteriori model covariance is

CM′ =(

GTG

σ 2d

+ C−1M

)−1

. (15)

Let us consider the same system, but with twoobservations,

dobs,1 = Gm + ε1 and dobs,2 = αGm + ε2.

The relationship of the first observation to model vari-ables is the same as in the system with only one mea-surement, but the precision of the measurement may

be different, i.e. ε1 ∼ N[0, σ 2

d1

]. The sensitivity of the

second observation to model variables is proportionalto the sensitivity of the first observation and the vari-ance of the measurement error is σ 2

d2. The errors inthe two measurements are assumed to be uncorrelated.An example of two observations that are related in thisway is breakthrough time of tracer at a producer, andaverage porosity between the injector and producer.

If we would like to have the mean and the covarianceestimates after two pseudo-observations be equivalentto the estimate after a single actual observation, thenwe must select dobs,1, dobs,2, σd1, and σd2 to give consis-tent results with the original problem. After assimila-tion of the two data, the posteriori mean is

mpost = mpr +(

GTG

σ 2d1

+ α2 GTG

σ 2d2

+ C−1M

)−1

×[

GαG

]T ([dobs,1

dobs,2

]−

[GαG

]mpr

), (16)

and the posteriori covariance is

CM′ =(

GTG

σ 2d1

+ α2 GTG

σ 2d2

+ C−1M

)−1

. (17)

The posterior covariance will be the same as the pos-terior covariance for the one-observation system if werequire that

σ−2d1 + α2σ−2

d2 = σ−2d (18)

and the means of the two distributions will be the sameif we require

dobs,1 = dobs and dobs,2 = αdobs

Although the preceding analysis was based on theassumptions of linearity and normality, these are infact the same assumptions that we implicitly use inthe EnKF analysis step. We consider the same testproblem with measurement of fractional flow of waterto investigate the potential advantage of assimilating apseudo-observation with nearly the same informationcontent as fractional flow of water to improve the initialensemble. For this simple 1-D linear flood, fractionalflow of water is related to breakthrough time, andbreakthrough time is proportional to the inverse of theharmonic mean permeability. Examination of a few re-alizations of fractional flow of water versus time deter-mines that dfw/dt ≈ 0.0031 day−1 in the neighborhoodof the observation (fractional flow of water equal to0.687). Also, the sensitivity of arrival time to harmonic

24 Comput Geosci (2009) 13:13–27

Table 5 Ensemble estimates of fractional flow of water at day 180 for an observed fractional flow of water of 0.687 ± 0.020

Ne EnKF EnKF+Imp kH+EnKF EnRML RML

40 mean 0.735 ± 0.043 0.568 ± 0.025 0.712 ± 0.016 0.711 ± 0.014 0.688 ± 0.00340 std 0.232 ± 0.049 0.220 ± 0.042 0.097 ± 0.035 0.069 ± 0.017 0.020 ± 0.003

average of permeability is dt/dkH ≈ 14, 600 day/μm2.The factor α relating the sensitivities of fractional flowof water and harmonic mean permeability to modelvariables is

α = (0.0031 × 14, 600)−1 = 0.022 μm2

Thus, instead of sampling the initial realizations fromthe prior PDF and conditioning them to an observationof fractional flow of water at day 180 with a standarderror of 0.020, we sample the initial realizations from aPDF conditional to a pseudo-observation of harmonicmean of permeability equal to 0.0123 ± 0.0018 μm2. Tomake the total variance correct, we must adjust thevariance of the actual measurement of fractional flowof water to be

σd1 = (σ−2

d − α2σ−2d2

)−1/2 = 0.0207

instead of the actual value of 0.020 to ensure that we donot overweight the value of the data. Because we useda relatively large variance for the pseudo-observation,the correction to the variance of the actual data is quitesmall.

In certain applications, the assimilation of pseudo-data would not necessarily be considered an improvedsampling method. It might, for example be used dur-ing the assimilation process in place of actual data.Examples might include the use of seismic time-lapse‘saturation changes’ (which are not actually measured)or ‘arrival times’ for water. In our application, however,we have chosen to assimilate pseudo-data that do notrequire running the reservoir simulator and we do thisassimilation before we ever begin using EnKF to assim-ilate the production data. The purpose is to improve the

initial ensemble so that the assumptions of linearity arenot violated so badly in the analysis step.

3.4 Evaluation of initial sampling methodsfor analysis estimation

The same 1-D two-phase problem is used to evalu-ate the improvement to EnKF with different initialsampling methods. The relationships between modelvariables (log-permeability) and the observation (frac-tional flow of water) are highly nonlinear, so the truemean and variance are not easily computed. We canbe sure, however, that the fractional flow of waterfrom the model should be close to the observation(0.687) and that the standard deviation in the pre-dicted fractional flow of water should probably be closeto the standard error in the measurement of frac-tional flow of water (0.020). We therefore computedensemble means using several different methods, in-cluding the standard EnKF; EnKF with weighted real-izations (EnKF+Imp); EnKF with pseudo-observation(kH+EnKF); randomized maximum likelihood (RML),an approximate Monte Carlo method based on mini-mization of a stochastic objective function [12, 16]; andEnRML, an iterative form of the ensemble Kalman fil-ter [8] based on RML. For EnKF, EnKF with weightedrealizations and EnKF with pseudo-observation, weused 4000 ensembles of 40 members. For both RMLand EnRML, we used 1000 ensembles of 40 members.Both iterative methods are computationally expensiveas they require the reservoir simulator to be run multi-ple times to achieve a single conditional realization.

Table 5 and Fig. 7 summarize the results of these fivemethods. Although EnKF with importance sampling

EnKF EnKF+ Imp kH+EnKF EnRML RML0.2

0.4

0.6

0.8

1.0

EnKF EnKF+Imp kH+EnKF EnRML RML

0.0

0.1

0.2

0.3

0.4

(a) (b)

Fig. 7 Box and whisker plots of ensemble mean and ensemble standard deviation of updated fractional flow of water at 180 days.a Ensemble mean. b Ensemble standard deviation

Comput Geosci (2009) 13:13–27 25

(EnKF+Imp) worked well for the linear problem, itseems to have done a poor job of updating modelvariables in this nonlinear problem. The importanceweighted estimate of the mean fractional flow of waterafter assimilation of the fractional flow of water obser-vation is quite far from the correct value. The reason forthe failure of importance weighted EnKF in this non-linear problem is almost certainly the assumption thatthe realizations in the updated ensemble are samplesfrom the posterior PDF, p(m|dobs). Note that standardEnKF also does a poor job and for a very similarreason; the assumption that the updated realizationsare samples from p(m|dobs) is not correct. EnKF withinitial conditioning to pseudo-data (kH+EnKF) gavethe best results among the non-iterative methods. Themean ensemble estimate of fractional flow of water isapparently biased, but much less so than the estimatesfrom EnKF or EnKF with importance sampling. In fact,the estimate from EnKF with a pseudo-observation ofthe harmonic mean of permeability is very close to theestimate from the iterative ensemble filter, EnRML.

4 Discussion

In this paper, we investigated one method that has beenproposed for generating improved initial ensemblesfor EnKF. Based on our investigation, we developedimprovements to the method, and have investigatedseveral other possible methods that might sometimesbe appropriate. The evaluation of the utility of themethods is based primarily on magnitude of the bias inthe estimation of the ensemble mean and the ensemblespread of predictions. Secondly, we seek methods forwhich the variability of predictions between ensemblesis small.

It seems intuitively obvious that there should be anefficiency advantage in generating the initial ensemblesuch that all of the members of the ensemble are asdissimilar as possible. One way to enforce dissimilarityis by requiring all realizations to be orthogonal withrespect to a meaningful metric. Method 1 [7] does pro-duce ensembles of realizations that are dissimilar butthe realizations are too smooth, and produce biasedforecasts in multiphase flow and transport. This methodcannot be recommended.

Method 1 can be easily modified so that the initialrealizations contain contributions from all eigenvectorsof the model covariance, while retaining orthogonalitywith respect to the same metric used for Method 1. Thisstep alone (Method 2) removes most of the bias in theforecasts, and the remainder can largely be removed

by ensuring that the lengths of the random vectorsthat multiply the dominant eigenvectors are sampledfrom the chi-square distribution. For the particular flowproblems that we investigated, however, the differencebetween results from either Method 2 or Method 3 andstandard Monte Carlo sampling from the prior are quitesmall. It is possible that other problems might benefitfrom this type of sampling but our test case did not.

If prediction or estimation of one particular vari-able is of primary interest, then it might be feasible toidentify an importance variable for importance sam-pling that allows ensemble estimates to have greatlyreduced variability. We showed that by using a functionof the harmonic mean of permeability as an importancevariable, the ensemble estimates of fractional flow ofwater were much more precise than the estimates fromstandard Monte Carlo with the same ensemble size.Hence importance sampling without data assimilationcan be useful for reducing the number of simula-tion runs required if an importance variable can beidentified.

When production data are available, it is natural thatthey should be used to improve the ensemble estimateby conditioning all realizations in the ensemble to theavailable data. In this paper, we showed how to useEnKF to update an importance-weighted ensemble.The example in which the data relationship was linearresulted in an importance-weighted posterior PDF forwhich unbiased estimates of the mean with negligiblevariability were obtained. When importance samplingwas used with EnKF for multiphase flow data assim-ilation, the resulting estimates of the mean were sig-nificantly biased, even more so than occurred whenonly EnKF was used without importance sampling. Itdoes not appear that importance sampling can be usedwith EnKF for data assimilation in multiphase flowproblems.

Finally, we investigated the possibility of generatingan initial ensemble that is conditional to pseudo-datasuch as a nonlinear function of the property field. If thepseudo-data are appropriately chosen, this initial con-ditioning makes the magnitude of the corrections in theEnKF analysis step smaller than they would be withoutthe improved initial ensemble. The result is that theassumptions that justify the use of EnKF (linear datarelationships and Gaussian priors) are not so badlyviolated. Of course, when pseudo-data are introduced,it is necessary to make an adjustment to the procedurefor assimilating the actual data, so that the data are notoverweighted. Results from this method were almostidentical to results from EnRML, an iterative form ofthe ensemble Kalman filter for which the cost wouldgenerally be higher.

26 Comput Geosci (2009) 13:13–27

5 Conclusions

Although the improved sampling method that gener-ates orthogonal realizations from the dominant eigen-vectors of the model covariance matrix is useful forreducing the variance in linear and nearly linear sys-tems, it produces severely biased estimates for porousmedia multiphase flow transport problems. The biascan be removed by adding heterogeneity at smallerscales, but the variance of the ensemble estimateswas only slightly smaller than estimates from standardMonte Carlo sampling.

Importance sampling can be very useful for reducingthe variance in Monte Carlo estimates, but when usedwith data assimilation it requires an accurate analysisscheme. The results can be very good when used withEnKF for linear data relationships, but not when therelationships are nonlinear.

Our general advice for generating the initial ensem-ble is to use Methods 2 or 3 (variants of Evensen’smethod). For a fixed ensemble size, these realizationswill be as diverse as possible (orthogonal with respect tothe subspace spanned by the dominant eigenvectors).Although these methods were not particularly usefulin our nonlinear examples, it is clear from previousreports that the orthogonality can be useful in someinstances. The addition of the heterogeneity from theremaining eigenvectors reduces the bias that wouldotherwise result from the smoothness of the propertyfields.

If the observations do not lie within the range ofthe forecasts from the initial ensemble, then we rec-ommend the use of pseudo-data and an increase inthe variance of the actual data for the analysis step.Using this procedure reduced both the bias and thevariability between ensembles compared to EnKF withstandard Monte Carlo sampling of the initial ensemble.We cannot recommend the use of importance samplingwith EnKF for nonlinear problems because weightscomputed from the posterior distribution (after analy-sis) are correct only when the relationship between dataand model variables is linear.

Acknowledgements The support from member companies ofthe OU Consortium on Ensemble Methods is gratefully acknowl-edged. We are also grateful to Schlumberger for the donationof multiple Eclipse licenses that were used for simulation. Com-putation on the Linux cluster was provided by the OklahomaSupercomputer Center for Education and Research.

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