Modeling Uncertainty in the Earth Sciences

Jef Caers

Stanford University

Modeling spatial uncertainty

A reminder: models of uncertainty

1 2 3

Models of uncertainty

( )

( | )

( )

Samples: , , ,...,

X

n

P A

P A B b

f X Y y

x x x x

A set of samples drawn by Monte Carlo simulation are a valid model of uncertainty

Motivation

Physical model

SpatialStochastic

model

SpatialInput

parameters

Forecast and

decision model

Physicalinput

parameters

Rawobservations

Datasets

response

uncertain

uncertain

uncertain certain or uncertain

uncertain/error

uncertain

uncertain

Spatial stochastic simulation

Modeling spatial uncertainty

Spatial stochastic simulation

Spatial Stochastic simulator

input

Seed 1

Seed 2

Earth models

(spatial uncertainty)

(input uncertainty)

Data (samples, well, geophysics…)

Conditional simulation = constrained to data Unconditional simulation = not constrained to any data

“Hard” data

Definition: Hard data = direct, exact information at the scale of modeling

All the rest is soft data

If you dig out a volume of this size at that location and measure the variable of interest than you have hard data

Example

Size of the core = 6 inch x 6 inch x 12 inch 1 grid cell = 200ft x 200 ft x 3 ft

5 miles

If we assume the core to be hard data, i.e. assign it to a grid cell then we basically assume that there is NO spatial variation in that grid cell

Grid cells

Example spatial stochastic simulation

Input uncertainty: Range horizontal= [15,25] Range vertical = 5 Isotropy or horizontal anisotropy max range = [20,25] min range = [10,15] Azm=45

?

Example spatial stochastic simulation

If I take a fixed input Range horizontal= 25, Range vertical = 5 Horizontal Isotropy

Result

The Earth model reflects our knowledge about The sampled values at their location The interpreted spatial continuity model

vertical horizontal

Spatial uncertainty

and infinitely more if you want

Input and spatial uncertainty

and infinitely more if you want

Object-based stochastic simulation

Modeling spatial uncertainty

Unconditional simulation

Principle mostly used : rejection sampler

Place an object drawn from the pdfs of the various parameters defining the object

Place the object, either randomly or according to trend

If object violates interaction or other rules Then reject

Else accept

This will make sure that the 3D Earth model created reflects exactly the Boolean model specified

Conditional simulation

Use rejection sampler: slow

Use “Metropolis sampler” Create an initial 3D Earth model: it probably violates data

constraints

Propose a perturbation Move an object

Remove an object

Add an object

Accept the change with a probability a, where a is dependent on how much improvement was made

These methods are “iterative” methods, hence slow and may not converge

Example: conditional simulation

After 198 iterations After 200 iterations

After 202 iterations After 204 iterations

Channel added Same channel removed

Channel position changes

Some well data still

not honored

Final result: constrained to all wells

Courtesy: Norwegian Computing Center

Conditional Boolean simulation

Too slow for practical Earth modeling involving uncertainty

Any conflict between model and data makes it even slower

Can only be constrained to

Limited amount of well data

Not to geophysical data or more complex data

Training-image-based simulation

Modeling spatial uncertainty

Idea

Generate a single unconditional Boolean Earth model

Anchor patterns to data

Conditional Earth model

Extract 3D Patterns

ActualData

Principle of sequential simulation

A reservoir

with a 2x2 grid

A reservoir

with a 2x2 grid

A training imageA training image

Step 1.

Pick a cell

Step 1.

Pick a cell

Step 2.

Assign probability

50%

Step 2.

Assign probability

50%50%

Step 3.

Assign color

Step 3.

Assign color

Step 4.

Pick a cell100%

Step 4.

Pick a cell100%

Step 5.

Assign color

Step 5.

Assign color

Step 6.

Final result

Step 6.

Final result

Another realizationAnother realization

In general

= data

data event

P ( A | B ) ?

Algorithm

Generic sequential simulation algorithm (1) Assign any hard data to grid cells if required

(2) Define a random path

(3) Loop over all grid cells

(1) Determine P(A|B) B=any data and previously simulated values

(2) Draw from P(A|B) a value (3) Add that value to the data set

How to use the training image ?

u1

u?u2

u4u3

Simulation grid with some data points

Using a training image

Training image

P ( A | B ) = 1 / 4 A = blue

Scanning is CPU-demanding

u1

u?

u2

u4 u3

Template

Simulation grid with some data points

Create a data-base

2

14 11

5 7

3 1 2 5 3 0

5 3

1 1

1 0 0 0 0 0

0 0 0 0 0

1 1 1 1 1 1

1 1 1 1 1

3 2

2

Training image

Search tree

Construction requires scanning training image one single time

Minimizes memory demand

Allows retrieving all training probabilities for the template adopted

Spatial continuity at large scale

Training image

Coarse simulation

grid

Freeze coarse grid nodes and use them as conditioning data to simulate finer grid nodes

Finer simulation grid

Coarse

template

fine

template

Example

Training image

Background

shales

Estuarine

sands

Tidal sand

flats Transgressive

lags Tidal bars

Top of

estuarines

Top of reservoir

Anywhere,

eroded by

sand bars

Anywhere

Stratigraphy

820004000Sheets (rectangles)Estuarine

sands

Sheets (rectangles)

Sheets (rectangles)

Elongated ellipses

w/ upper sigmoidal

cross-section

Conceptual

description

410003000Transgr.

Lags

610002000Tidal

sand

flats

3 to 75002000 to

4000

Tidal

bars

Thickness

(ft)

Width

(m)

Length

(m)

Facies

type

Top of

estuarines

Top of reservoir

Anywhere,

eroded by

sand bars

Anywhere

Stratigraphy

820004000Sheets (rectangles)Estuarine

sands

Sheets (rectangles)

Sheets (rectangles)

Elongated ellipses

w/ upper sigmoidal

cross-section

Conceptual

description

410003000Transgr.

Lags

610002000Tidal

sand

flats

3 to 75002000 to

4000

Tidal

bars

Thickness

(ft)

Width

(m)

Length

(m)

Facies

type

Example Plan view of stratigraphic

grid with location of

the 140 wells

N

1

0

Background shale

Sand bars

Estuarine sands

Aerial proportion maps

Facies model

Vertical proportion

curves

Back

gro

un

d s

hale

San

d b

ars

Est

ua

rin

e sa

nd

s

N

Variogram-based simulation

Modeling spatial uncertainty

Introduction to spatial estimation

ju ?

Data point

Spatial estimation = What is the best guess for the value at the location where no data was taken ? There is only one single guess that is the best Depends on what you determine as “best”

What is best?

Best = as close as possible to the unknown truth

Consider a situation where you want to estimate the total amount of pollution of Pb at a specific location. You have two methodologies do so. Consider that you apply these methodologies to 10 sites

Estimation: principles

(1) (2) (1) (2)1 1 1 1 1

(1) (2) (1) (2)2 2 2 2 2

(1) (2) (1) (2)3 3 3 3 3

(1) (2) (1) (2)4 4 4 4 4

(1) (2)5 5 5 5

estimation estimation unknown error errorsite

method 1 method 2 real# Pb method 1 method 2

ˆ ˆ1 m m m

ˆ ˆ2 m m m

ˆ ˆ3 m m m

ˆ ˆ4 m m m

ˆ ˆ5 m m m

e e

e e

e e

e e

e(1) (2)5

(1) (2) (1) (2)6 6 6 6 6

(1) (2) (1) (2)7 7 7 7 7

(1) (2) (1) (2)8 8 8 8 8

(1) (2) (1) (2)9 9 9 9 9

(1) (2) (1) (2)10 10 10 10 10

ˆ ˆ6 m m m

ˆ ˆ7 m m m

ˆ ˆ8 m m m

ˆ ˆ9 m m m

ˆ ˆ10 m m m

e

e e

e e

e e

e e

e e

Estimation: principles

Unbiased: the average error is zero (it is a property measured over many “trials”

Best ?

Average square error is zero ?

Absolute value of error is zero ?

loss

-error +error

Introduction to Kriging What is kriging ? Is an estimation method Finds the “best” (Least Square) linear estimate of the unknown Accounts for the variogram * Spatial correlation between unknown and data * Redundancy between data But is mostly used in sequential simulation

Linear estimation

z1=0.5

z2=0.9

z3=1.5

d

1 d

2 d

3

3

i i

i 1

z z

223

3 22 2

2

1 2 3

11

. .1

1 1 1

ii

i

dde g

d d d d

3

1

2 h13

h12

Layered system

Problem Inverse distance solution

Inverse distance mapping

1

1

i

i

i

d

d

1

3

2

1

3

2

Situation 1 Situation 2

Inverse distance

kriging

1=1/3

2=1/3

3=1/3

1=1/4

2=1/4

3=1/2

Linear estimation

Direction of major continuity

To be estimated

Data

a a a

a

n

*SK j

1

Z ( ) Z( ), what is ?u u

*SK jZ ( ) u

ju

a a Z( ), 1, ,n u

Principle 1: a datum close in “geological distance” to the unknown should get a large weight Principle 2: Data close together are redundant and should “share” their weight

?

How does kriging do it?

z(u1)=0.5

z(u2)=0.9

z(u3)=1.5

h13 h23

h12

3

i i

i 1

z(u) z

Record all the distance between Data locations Data location vs location of unknown Calculate the covariance function for those distances

How does kriging do it?

)(

)(

)(

)()()(

)()()(

)()()(

03

02

01

3

2

1

2313

2312

1312

hC

hC

hC

zVarhChC

hCzVarhC

hChCzVar

Solve this linear system of equations

a a

a

1

ˆ z( ) ( ) is the kriging estimaten

j zu u

Note: mean = assumed zero

What is the average error we make?

)(

)(

)(

)()()(

)()()(

)()()(

03

02

01

3

2

1

2313

2312

1312

hC

hC

hC

zVarhChC

hCzVarhC

hChCzVar

Solve this linear system of equations

2 20

1

ˆ ( ) ( ) is the kriging variancen

j C ha a

a

u

Note: mean = assumed zero

Example

Note: mean = assumed zero

? d d

1 1

2 2

3 3

( ) ( ) (2 ) ( ) 1 / 4

( ) ( ) (2 ) ( ) 1 / 4

(2 ) (2 ) ( ) ( ) 1 / 2

Var z Var z C d C d

Var z Var z C d C d

C d C d Var z C d

e

2 1 1 1ˆ ( ) var( ) ( ) ( ) ( ) var( ) ( )

4 4 2z C d C d C d z C d u

Various “flavors” of kriging

Simple kriging

You assume there is no trend

You assume you know the mean of the variable over the domain

Ordinary kriging

You don’t want to assume anything explicitly

Kriging with locally varying mean

You assume there is a trend and you know that trend exactly

Kriging with trend

You assume there is trend but you only know the type of trend (don’t know it exactly its magnitude)

Example

Back to sequential simulation

(1) Assign any hard data to grid cells if required

(2) Define a random path

(3) Loop over all grid cells

(1) Determine P(A|B) B=any data and previously simulated values

(2) Draw from P(A|B) a value (3) Add that value to the data set

Kriging is not simulation !

ju ?

ju ?

Kriging: what is the best guess?

Simulation: what is the uncertainty as expressed through a probability or probability distribution or a set of Possible outcomes

Gaussian simulation

ju ?

Mean: m

Variance: 2

a a

a

a a

a

*SK

1

2SK

1

mean: Z ( ) ( ) is the simple kriging mean

variance: ( ) 1 ( ) is the kriging variance

n

j

n

j j

Z

Cov

u u

u u u

Transformation of the data

Gaussian simulation assumes the data is Gaussian

Problem ? Prior to simulation, perform a normal score transform of the hard data After sequential visit of all cells, perform a back-transform which is the exact reverse of the normal score transform

Normal score transform

Unit free

Complete SGS algorithm

(1) Transform the data into normal score domain

(2) Assign any hard data to grid cells if required

(3) Define a random path

(4) Loop over all grid cells

(1) Determine, using kriging, the distribution P(A|B) B=any data and previously simulated values

(2) Draw from P(A|B) a value (3) Add that value to the data set

(5) Back-transform all simulated values (requires extra/interpolation)

Example