Uncertainty in Spatial Patterns: Generating Realistic Replicates for Ensemble Data Assimilation Problems D. McLaughlin – MIT, Cambridge, MA, USA Hurricane.

Uncertainty in Spatial Patterns: Generating Realistic Replicates for Ensemble Data Assimilation ProblemsD. McLaughlin – MIT, Cambridge, MA, USA

Hurricane Isabel – Sept 2003

Environmental data assimilation & forecasting often involve characterization of spatial features with distinctive but uncertain characteristics

A pattern or feature-based perspective can change the way we think about estimation, inversion, data assimilation

• What are the essential aspects of a particular type of spatial feature ?

• How can we best represent uncertainty about spatial features?

• How should we incorporate measurements into real-time predictions of changing features ?

Problem Formulation

)()|( )|( :1:1:1:1:1 TTmmT xpxzpczxp

3. Measurements of states (diverse types, scales, coverage, accuracy, etc):

1. We can describe spatial features in terms of vectors of space/time discretized states (e.g. a vector of pixel values):

The states are selected to reflect application needs.

),...,1(:1 Ttx T

4. Bayes rule incorporates meas into conditional probability:

These concepts form basis for most environmental data assimilation algorithms

States & meas related by likelihood: )|( :1:1 Tm xzp

Conditional probability describes everything we know about states, given meas

2. Unconditional (prior) “model” of state uncertainty:

)( :1TxpConveys pattern information but is unwieldy for large problems – which aspects are most important ?

mkkz ,,1

10 20

5

10

15

20-20

0

20

pdf for one pixel

10 20

5

10

15

20-20

0

20

1

0 2

0

51

01

52

0

-2

0

0

2

0

ModelMeasMerged

Application of the Bayesian Approach

It is often appropriate to derive prior state pdf & likelihood (or some of their distributional properties) from physically-based models of the system and measurement process.

Markovian models may be used to obtain recursive expressions convenient for real-time applications :

Derived distribution problems

)( )(:)1( ktktup

Likelihood Meas eq

)|( 1)( kkt zxp )|( 1)1( kkt zxp

)|( )( kkt zxp

)|( )(ktk xzp )( kvp

)( 0xp

Meas

),( 1 tttt uxfx

State Eq

Meas Eq

kktkk vxhz )( )(

t(k-1) t(k)

kz1kz

Forecast State eq

Update Bayes thm

Specifiying State & Input Statistics

Intermittent or discontinuous processes are not necessarily described by simple pdfs & low-order moments …

5 10 15 20 25 30

5

10

15

20

25

30

-2

-1

0

1

2

Gauss-Markov random field (defined by first 2 moments)

… yet it is not always practical to generate realistic pdfs from “first principle” models

Precip.

Land

Atmosphere

PET Precip.

Land

Atmosphere

PET

Defining states & inputs – selecting system boundaries

Land surface modeling is easier if precipitation is an input. But ….. then all the space-time complexity of precip must be captured in the input pdfs

Precipitation Geological facies

Generate pdf from primitive eq. atmospheric model ?

Generate pdf from depositional model over geological time scales ?

Convenient – but is it realistic ?

Ensemble Implementation

Implicit input pdfs:

Explicit input pdfs:

• Sample input replicates from specified pdf’s:jt

jt

j vux ,,0

• Devise a stochastic model that generates realistic input replicates – these replicates implicitly define input pdfs:

Ensemble approach offers more flexibility than classical inverse methods – we should exploit this capability

Gauss- Markov field

10 20

5

10

15

20-20

0

20

10 20

10

20-20

-10

010

10 20

5

10

15

20 -30-20-10010

10 20

5

10

15

20-20

-10

0

10

20

Specified input pdf

-99 -98.5 -98 -97.5

37

37.5

38

38.5

0

20

40

60

80

-99 -98

37

37.5

38

0

20

40

60

80

-99.5-99-98.5-98-97.536.5

3737.5

3838.5

0

20

40

60

80

-99 -98 -9736

37

38

0

20

40

60

80

Clustered Markov field

Implicit input pdf

Stocastic model

In either case, derive forecast replicates and updated replicates of states from state eq. and Bayes rule.

jkktx 1|)(

jkktx |)(

Example: Ensemble Characterization of Petroleum Reservoirs

Objective: Characterize petroleum reservoir properties for enhanced oil recovery

• States ( ): Saturation, pressure over 3D spatial gridtx

Reservoir simulation model (ECLIPSE)),( 1 tttt uxfx

• Measurements: Injection well pressures, production well rates

)( )( kktkk vxHz Well meas

Augmented state vector also includes permeability, porosityKnown inputs: injection well rates, production well pressures, initial saturation

Enhanced recovery with water floodingOil Water

0 Months 6 Months 18 Months

36 Months

Producer (8) Injector (15)

• Ensemble estimation:

Initially: Generate permeability, porosity replicates Periodically: Update forecast saturation & pressure replicates with meas ,

using ensemble Kalman filter

jx0 jkktx 1|)( kz

What Do Real Petroleum Reservoirs Look Like?

Difficult to say … we must generally rely on interpretations of limited borehole data

Cutaway of porosity distribution

Porosity > 0. 11

Areas most likely to contain oil are disconnected & irregular

House Creek Oil FieldPowder River Basin

How Should We Generate Realistic Permeability/Porosity Ensemble ?

The features that control flow can often be represented as distinct facies or channels.

This approach can account for relationships among groups of pixels

Infer pattern probabilities from training image

Generate replicates from pattern probabilities

Permeability replicates that produce channelized flow may be generated with a multipoint geostatistical algorithm that quantifies probabilities of particular patterns:

Training Image 1 (250×250) Problem domain (45×45)

Problem domain (45×45)

Ensemble Estimation/Inversion

Adopt an ensemble approach ….

Approximate Bayes rule with Kalman update

This approach updates perm & porosity at each meas time (filtering)

Results depend strongly on realism of prior ensemble

Test with simple synthetic experiment ……

ECLIPSE model

Forecast sat, pressure replicates

Well measPrior perm, porosity, IC replicates

Time loop

Updated perm, porosity, sat, pressure replicates

Ensemble Kalman filter

UpdateForecast

jkktx 1|)(

jkktx |)(

kz

How Important is the Prior Ensemble – Poor Training Image ?

Tru

e L

og-p

erm

Portion of training image

Training image channels are too wide

Poor prior Initial channel estimate degrades over time

Tim

e

Tru

e

Satu

rati

on

EnK

F m

ean

Log-p

erm

EnK

F m

ean

sat

Tru

e L

og-p

erm

Portion of training image

Training image channel widths comparable to true

How Important is the Prior Ensemble – Good Training Image ?

Poor good Initial channel estimate improves over time – robustness?

Tim

e

Tru

e

Satu

rati

on

EnK

F m

ean

sat

EnK

F m

ean

Log-p

erm

3D water flooding problem based on upscaled version of communiy (SPE10) geological model:

30 X 110 X 10 = 33,000 pixels

Work in Progress - Generating Prior Replicates for Realistic 3D Problems

10

0 f

t

For inverse problem: Parameterize all states with 3D discrete cosine transform (DCT)

This reduces dimensionality by ~ factor of 10

Composite fields

Gauss-Markov shale infill

Gauss-Markov sandstone infill

Shale & sandstone facies from training image

Sample Porosity

Sample Log-perm X,Y

Sample Log-perm Z

Generating perm & porosity replicates ….

Layer permeability fields

Example: Estimation of Hydrologic Fluxes over the Great Plains

Objective: Determine how land surface fluxes vary over time and space, in response to meteorological forcing (global perspective)

Other inputs

Updated evap & soil moisture repls

Time loop

Precip generator

Forecast precip repls

Updated precip replsPrecip ensemble

Kalman filter

Polar satellite meas

Land surface model

Land surface ensemble Kalman filter

Forecast evap & soil moisture repls

GOES satellite meas

Polar satellite meas

Available ~ globally: Surface meteorological meas, geostationary & polar-orbiting satellite meas, soil & vegetation data

Characterize precipitation, soil moisture, evapotranspiration over Great Plains, Summer 2004

Use ensemble Kalman filter to merge prior info. and satellite meas

GOES – Geostationary, cloud top temperature

0.05 degree (~4 km), 1 hr

SSMI – Polar, passive microwave

SSMI: 0.25 degree (~20 km), 2/day for one location

TRMM – Polar, passive and active microwave

0.05 degree (~5 km), 2/ day for one location

AMSU – Polar, passive microwave

0.15 degree, (~ 15km), 2/day for one location

Satellite-based Precipitation Data Sources

Typical Summer Storm 1 – Great Plains, US

GOES (K): 06/12/2004 22:00

-110 -105 -100 -95

30

35

40

45

220

240

260

280

300

320

0 20 40 600

0.5

106/12/2004 22:00; Cumulative Frequency

Rain Rate (mm/hr)

Cum

ulat

ive

Fre

quen

cy

Intensity CDF

0 10 20 30-1

0

1

2Covariance vs. Lag: 06/12/2004 22:00

Lag (pixels)

Cov

aria

nce Intensity

Covariance

GOES (K): 06/12/2004 22:00

-96 -95 -9436.5

37

37.5

38

38.5

39

220

240

260

280

300

320

NOWRAD (mm/hr): 06/12/2004 22:00

-96 -95 -94

37

37.5

38

38.5

39

0

20

40

60

NOWRAD

Use ground radar to identify rainfall clusters within GOES features

Rainfall intensity within cluster

0 10 20 30-0.5

0

0.5

1

1.5Covariance vs. Lag: 08/23/2004 11:00

Lag (pixels)

Cov

aria

nce

0 50 1000

0.5

108/23/2004 11:00; Cumulative Frequency

Rain Rate (mm/hr)

Cum

ulat

ive

Fre

quen

cy

GOES (K): 08/23/2004 11:00

-110 -105 -100 -95

30

35

40

45

210

220

230

240

250

260

270

280

290

Typical Summer Storm 2 – Great Plains, US

Intensity CDFIntensity Covariance

NOWRAD

NOWRAD (mm/hr): 08/23/2004 11:00

-100 -99 -98 -97

36.5

37

37.5

38

38.5

39

39.5

0

20

40

60

80

GOES (K): 08/23/2004 11:00

-100 -99 -98 -97

36

37

38

39

220

240

260

280

Rainfall intensity within cluster

Work in Progress - Constructing Prior Precipitation Replicates

Are these replicates realistic ?

Precipitation replicates should account for intermittency, spatial structure, non-Gaussian behavior observed in real storms

Divide process into two steps:

2. Generate continuous spatially correlated random rainfall intensity fields within clusters

1. Identify rain clusters where preciptation is likely

Initially use GOES cloud top temps and ground radar, eventually use only GOES

-95.5 -95 -94.5 -9437

37.5

38

38.5

0

20

40

60

80

-95.5 -95 -94.5 -94

37

37.5

38

38.5

0

20

40

60

80

-95.5 -95 -94.5 -94

37

37.5

38

38.5

0

20

40

60

80

-95.5 -95 -94.5 -9437

37.5

38

38.5

0

20

40

60

80

-95 -94.5 -9437

37.5

38

38.5

0

20

40

60

80

Rainfall intensity (mm/hr)

A Typical Rainfall Ensemble

Compare replicates to observed NOWRAD images – which one is the observed storm?

-93 -92 -9131.5

32

32.5

33

33.5

34

0

5

10

15

20

25

-93 -92 -91

31.5

32

32.5

33

33.5

34

0

5

10

15

20

25

-93 -92 -9131.5

32

32.5

33

33.5

34

0

5

10

15

20

25

-93 -92 -9131.5

32

32.5

33

33.5

34

34.5

0

5

10

15

20

25

-93 -92 -91

31.5

32

32.5

33

33.5

34

0

5

10

15

20

25

-93 -92 -91

31.5

32

32.5

33

33.5

34

0

5

10

15

20

25

Multivariate/marginal pdfs of the rainfall intensity are implicitly defined by replicates generated from our two step procedure ?

How can we assess whether the observed image and ensemble could have been drawn from the same distribution ?

Incorporating Polar-orbiting Satellite Measurements

At each meas time update the forecast precipitation replciates with new polar-orbiting satellite meas :

Particle update:

• Maintains realistic ensemble by reweighting rather than changing forecast replicates

• Currently not practical for large problems

Ensemble Kalman update:

• Simple and efficient

• Tends to distort replicate shapes, especially in the presence of position error.

Meas.

1 2 3 4 5

1

2

3

4

5

1 2 3 4 5

1

2

3

4

5

1 2 3 4 5

1

2

3

4

5

Forecast

UpdateIf forecast replicate and meas. are offset – updated storm is wider & less intense than either prior or meas.

• Kalman update needs to be constrained to yield realistic precipitation updates

Summary

• Environmental data assimilation is largely concerned with characterization and prediction of spatial patterns

• Uncertainties in spatial patterns are often best described by ensembles of replicates that reproduce the space-time structure of observations

• Realistic replicates can often be generated with stochastic models that implicitly define pdfs of the system states (and/or related inputs).

• Robust quantitiative methods are needed to assess realism of synthetically generated ensembles

• Ensemble measurement updates should preserve key structural properties of uncertain features while reducing uncertainty.

• Updating options for large real-time problems are limited – approximations are required.

• The Kalman update approach may need to be modified/supplemented to insure that updated replicates are physically reasonable.

Thanks to ….. :

NSF (ITR, CMG, DDDAS programs)

Shell Oil

Schlumberger Doll Research