Uncertainty in Spatial Patterns: Generating Realistic Replicates for Ensemble Data Assimilation Problems

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Uncertainty in Spatial Patterns Generating
Realistic Replicates for Ensemble Data
Assimilation Problems D. McLaughlin MIT,
Cambridge, MA, USA
Environmental data assimilation forecasting
often involve characterization of spatial
features with distinctive but uncertain
characteristics

• 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 ?

A pattern or feature-based perspective can change
the way we think about estimation, inversion,
data assimilation
Hurricane Isabel Sept 2003
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Problem Formulation
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.
2. Unconditional (prior) model of state
uncertainty
pdf for one pixel
Conveys pattern information but is unwieldy for
large problems which aspects are most important
?
3. Measurements of states (diverse types, scales,
coverage, accuracy, etc)
4. Bayes rule incorporates meas into conditional
probability
Conditional probability describes everything we
know about states, given meas
Model
Meas
Merged
These concepts form basis for most environmental
data assimilation algorithms
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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.
Derived distribution problems
Markovian models may be used to obtain recursive
expressions convenient for real-time applications

Forecast State eq
Update Bayes thm
Likelihood Meas eq
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Specifiying State Input Statistics
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
Intermittent or discontinuous processes are not
necessarily described by simple pdfs low-order
moments
Convenient but is it realistic ?
Gauss-Markov random field (defined by first 2
moments)
Precipitation
Geological facies
yet it is not always practical to generate
realistic pdfs from first principle models
Generate pdf from primitive eq. atmospheric model
?
Generate pdf from depositional model over
geological time scales ?
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Ensemble Implementation
Explicit input pdfs
Implicit input pdfs

• 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
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Example Ensemble Characterization of Petroleum
Reservoirs
Objective Characterize petroleum reservoir
properties for enhanced oil recovery
Enhanced recovery with water flooding

• Ensemble estimation

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What Do Real Petroleum Reservoirs Look Like?
Difficult to say we must generally rely on
interpretations of limited borehole data
House Creek Oil Field Powder River Basin
Cutaway of porosity distribution
Porosity gt 0. 11
Areas most likely to contain oil are disconnected
irregular
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How Should We Generate Realistic
Permeability/Porosity Ensemble ?
The features that control flow can often be
represented as distinct facies or channels.
Permeability replicates that produce channelized
flow may be generated with a multipoint
geostatistical algorithm that quantifies
probabilities of particular patterns
Infer pattern probabilities from training image
Generate replicates from pattern probabilities
Problem domain (4545)
Problem domain (4545)
Training Image 1 (250250)
This approach can account for relationships among
groups of pixels
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Ensemble Estimation/Inversion
Adopt an ensemble approach .
Approximate Bayes rule with Kalman update
Well meas
Prior perm, porosity, IC replicates
Forecast sat, pressure replicates
Updated perm, porosity, sat, pressure replicates
ECLIPSE model
Ensemble Kalman filter
Update
Forecast
Time loop
This approach updates perm porosity at each
meas time (filtering)
Results depend strongly on realism of prior
ensemble
Test with simple synthetic experiment
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How Important is the Prior Ensemble Poor
Training Image ?
Training image channels are too wide
True Log-perm
Portion of training image
Poor prior ? Initial channel estimate degrades
over time
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How Important is the Prior Ensemble Good
Training Image ?
Training image channel widths comparable to true
True Log-perm
Portion of training image
Poor good ? Initial channel estimate improves
over time robustness?
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Work in Progress - Generating Prior Replicates
for Realistic 3D Problems
Layer permeability fields
3D water flooding problem based on upscaled
version of communiy (SPE10) geological
model 30 X 110 X 10 33,000 pixels
100 ft
For inverse problem Parameterize all states with
3D discrete cosine transform (DCT) This reduces
dimensionality by factor of 10
Composite fields
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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)
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Satellite-based Precipitation Data Sources
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
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Typical Summer Storm 1 Great Plains, US
Use ground radar to identify rainfall clusters
within GOES features
NOWRAD
Rainfall intensity within cluster
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Typical Summer Storm 2 Great Plains, US
Intensity Covariance
Intensity CDF
NOWRAD
Rainfall intensity within cluster
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Work in Progress - Constructing Prior
Precipitation Replicates
Precipitation replicates should account for
intermittency, spatial structure, non-Gaussian
behavior observed in real storms
Rainfall intensity (mm/hr)
Are these replicates realistic ?
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A Typical Rainfall Ensemble
Compare replicates to observed NOWRAD images
which one is the observed storm?
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 ?
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Incorporating Polar-orbiting Satellite
Measurements
At each meas time update the forecast
precipitation replciates with new polar-orbiting
satellite meas

• Ensemble Kalman update
• Simple and efficient
• Tends to distort replicate shapes, especially in
  the presence of position error.

If 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

• Particle update
• Maintains realistic ensemble by reweighting
  rather than changing forecast replicates
• Currently not practical for large problems

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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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