# Presentation: 501883

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

**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 ?

## 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

- 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

- Model

- Meas

- Merged

## 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

**Forecast **State eq

**Update **Bayes thm

## 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**

- 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:**

- 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

## Example: Ensemble Characterization of Petroleum Reservoirs

- Objective: Characterize petroleum reservoir properties for** ****enhanced oil recovery**

- Enhanced recovery with **water flooding**

- Ensemble estimation:

## 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**

## 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 meas

- Prior perm, porosity, IC replicates

- Time loop

- Updated perm, porosity, sat, pressure replicates

- Ensemble Kalman filter

**Update**

**Forecast**

## 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**

## How Important is the Prior Ensemble – Good Training Image ?

- True Log-perm

- Portion of training image

- Training image channel widths **comparable to true**

- Poor good  Initial channel **estimate improves over time – robustness?**

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

- 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

- 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**)

## 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

## Typical Summer Storm 1 – Great Plains, US

- NOWRAD

- Use ground radar to identify rainfall clusters within GOES features

- Rainfall intensity within cluster

## Typical Summer Storm 2 – Great Plains, US

**Typical Summer Storm 2 – Great Plains, US**

- Intensity CDF

- Intensity Covariance

- NOWRAD

- 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

- Rainfall intensity (mm/hr)

## 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** ?

## 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.

- 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

## 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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