Download - Lecture 2C Cdf-PDF
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Geostatistics for Reservoir
Characterization
Lecture 2C - What is a Random Variable and
How Do We Describe It?
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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)
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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
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A Continuous RV CDF
F(x)
x
1
0
F(x) = Prob( X < x)
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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
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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
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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
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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
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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
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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
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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)
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Uses of CDFs and PDFs . . .
Modelling . . . Kriging and Monte Carlo
Estimation . . . Averages and variabilities
Analysis . . . Diagnosis of important features
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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
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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
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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
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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)
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Box and Whisker Plot ExampleFracture Spacing vs Fold Angle vs Bed Thickness
Bui et al, 2003
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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
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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
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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