Lecture 2C Cdf-PDF

8/13/2019 Lecture 2C Cdf-PDF

1/20

Geostatistics for Reservoir

Characterization

Lecture 2C - What is a Random Variable and

How Do We Describe It?


2/20

Cumulative Distribution Function (CDF)

Another way to present prob behaviour of RV

Simpler to express than PDF

Uses same info as PDF

For an RV X with CDF F(x0):

F(x0) = Prob(X < x0)


3/20

Discrete RV CDF's

Given three facies 0 = A; Prob(X = 0) = 0.1

1 = B; Prob(X = 1) = 0.6

2 = C; Prob(X = 2) = 0.3

So

Prob(X < 0) = 0.1

Prob(X < 1) = 0.1+0.6 = 0.7

Prob(X < 2) = 0.1+0.6+0.3= 1

00.10.20.30.40.50.60.7

0 1 2

Probability

ofFacies

Facies

Facies PDF

00.10.20.30.40.50.60.70.8

0.91

-1 0 1 2

CumulativeP

rob

Facies

Facies CDF


4/20

A Continuous RV CDF

F(x)

x

1

0

F(x) = Prob( X < x)


5/20

CDF Properties . . .

1. 0 < F(x) < 1

2. F(- ) = 0

3. F(+ ) = 1

4. F(x+h) > F(x) for h>0

5. F is a continuous function from the right

Note similarity to Prob properties because F is a prob


6/20

Relation of CDF to PDF

For an RV X with CDF F(x0) and PDF f(x0):

For discrete RV's, the integral becomes a sum

ox

o dttfxF )()(

)max( where)()(1

oim

mi

iio xxxxpxF


7/20

Example PDF to CDF

11

10

00

11

101

00

)()(

then

10

101

00

)(Let

0

o

oo

o

x

o

o

xo

o

o

o

o

o

x

xx

x

x

xdt

x

dttfxF

x

x

x

xf

o o

0 1

f(x)

x

0 1

F(x)

x

1


8/20

Comparing the PDF and CDF

0

0.2

0.4

0.6

0.8

1

-3 -2 -1 0 1 2 3

x

F(x)

f(x)

F(x) or f(x)

Where fis large, F is steep

Where fis small, F is flat


9/20

Some CDF Features: Quantiles

1.0

0.0

F(x)

0.75

0.50

0.25

XMedian

x

Median = F-1(0.5) = X0.50

Lower quartile = F-1(0.25) = X0.25

Interquartile range = F-1

(0.75) - F-1

(0.25)

X0.50 X0.75X0.25


10/20

Creating a Sample CDF

Order data X1< X2< < XN

Assign probability to each datum

Several possible formulas

I like pi= (i - 0.5)/N

Plot up Xs versus ps Caution estimating quantiles when p is near 0 or 1

Example using Excel


11/20

CDF vs PDF

CDF

Doesn't require binning

Easy identification of quantiles

PDF

More sensitive to subtle changes in prob

Easier detection of mode(s)


12/20

Uses of CDFs and PDFs . . .

Modelling . . . Kriging and Monte Carlo

Estimation . . . Averages and variabilities

Analysis . . . Diagnosis of important features


13/20

Analysis and CDFs: Shale Length CDF

0

20

40

60

80

100

0 500 1000 1500 2000

Shale Intercalation Length, f t.

coarsept. b ars

dist. channel

de lta fringe & delta p la in

de ltaic, barr ier

m a rine

*

*

PercentageShorterThan

)ob(LPr)(

F

Weber, 1982


14/20

The Complementary CDF . . .

0

20

40

60

80

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

B

D

FH

J

Frequency,

%

Pore Throat Size, microns

s)Prob(S-1)ob(SPr)( ssFc


15/20

Complementary CDF of Transformed RV . . .

0

20

40

60

80

100

0.001 0.01 0.1 1

B

D

F

H

JFrequency,

%

Pore Throat Size, microns


16/20

Box and Whisker Plots -- Mini CDF's

Graphical display

Ordered data

3 quartiles shown

Upper and lower fences

Beware of differing

versions!

Show Assymetry

Extremes

X0.25

X0.50

X0.75

X0.75+1.5(X0.75-X0.25)

X0.25-1.5(X0.75-X0.25)


17/20

Box and Whisker Plot ExampleFracture Spacing vs Fold Angle vs Bed Thickness

Bui et al, 2003


18/20

Monte Carlo Modelling - Overview

Principles

Uses computer random number generator

Each number generated is a realisation

Numbers can have any specified CDF/PDF

Applications

Reserves estimates

Facies distributions

Fractures or shale positions

Petrophysical parameter assignments

Any use where uncertainty effects are evaluated


19/20

Monte Carlo Modelling - Stochastic Shales

L

Inter-well RegionShale location CDF

x, y, z

Shale size CDF

w, d, t

w

d

t

along-strike

along-dip


20/20

Summary Points . . .

Random variable

discrete and continuous

sample and population

CDFs and PDFs are probabilities CDFs/PDFs do not measure natural order

Uses for CDFs/PDFs: modeling, estimation,

analysis

Documents

Lecture 2C Cdf-PDF