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
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pdf for one pixel
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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 …
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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
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Specified input pdf
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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
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Portion of training image
Training image channels are too wide
Poor prior Initial channel estimate degrades over time
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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
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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
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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
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Rain Rate (mm/hr)
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Intensity CDF
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Lag (pixels)
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Covariance
GOES (K): 06/12/2004 22:00
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NOWRAD (mm/hr): 06/12/2004 22:00
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NOWRAD
Use ground radar to identify rainfall clusters within GOES features
Rainfall intensity within cluster
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GOES (K): 08/23/2004 11:00
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Typical Summer Storm 2 – Great Plains, US
Intensity CDFIntensity Covariance
NOWRAD
NOWRAD (mm/hr): 08/23/2004 11:00
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GOES (K): 08/23/2004 11:00
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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
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Rainfall intensity (mm/hr)
A Typical Rainfall Ensemble
Compare replicates to observed NOWRAD images – which one is the observed storm?
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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.
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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
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