1. Uncertainty in Spatial Patterns: Generating Realistic Replicates for Ensemble Data Assimilation Problems D. 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 ?

2. Problem Formulation 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. 4.Bayes rule incorporates meas into conditional probability: These concepts form basis for most environmental data assimilation algorithms States & meas related by likelihood: Conditional probability describes everything we know about states, given meas 2. Unconditional (prior) “model” of state uncertainty: Conveys pattern information but is unwieldy for large problems – which aspects are most important ? pdf for one pixel ModelMeasMerged

3. 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 Likelihood Meas eq Meas State Eq Meas Eq t(k-1)t(k)t(k) Forecast State eq Update Bayes thm

4. Specifiying State & Input Statistics Intermittent or discontinuous processes are not necessarily described by simple pdfs & low-order moments … 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 PETPrecip. 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 PrecipitationGeological facies Generate pdf from primitive eq. atmospheric model ? Generate pdf from depositional model over geological time scales ? Convenient – but is it realistic ?

5. Ensemble Implementation Implicit input pdfs: Explicit input pdfs: Sample input replicates from specified pdf’s: 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 Specified input pdf 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.

6. Example: Ensemble Characterization of Petroleum Reservoirs Objective: Characterize petroleum reservoir properties for enhanced oil recovery States ( ): Saturation, pressure over 3D spatial grid Reservoir simulation model (ECLIPSE) Measurements: Injection well pressures, production well rates Well meas Augmented state vector also includes permeability, porosity Known inputs: injection well rates, production well pressures, initial saturation Enhanced recovery with water flooding Oil Water 0 Months6 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

7. 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 Field Powder River Basin

8. 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)

9. 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 meas Prior perm, porosity, IC replicates Time loop Updated perm, porosity, sat, pressure replicates Ensemble Kalman filter UpdateForecast

10. How Important is the Prior Ensemble – Poor Training Image ? True Log-perm Portion of training image Training image channels are too wide Poor prior  Initial channel estimate degrades over time Time True Saturation EnKF mean Log-perm EnKF mean sat

11. True Log-perm 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? Time True Saturation EnKF mean sat EnKF mean Log-perm

12. 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 100 ft 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

13. 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 repls Precip 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

14. 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

15. Typical Summer Storm 1 – Great Plains, US Intensity CDF Intensity Covariance NOWRAD Use ground radar to identify rainfall clusters within GOES features Rainfall intensity within cluster

16. Typical Summer Storm 2 – Great Plains, US Intensity CDF Intensity Covariance NOWRAD Rainfall intensity within cluster

17. 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 Rainfall intensity (mm/hr)

18. 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 ?

19. 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. Forecast Update 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

20. 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