A Geostatistical Approach to the Characteristic Values. Horatiu Corbeanu.pdf · the dataset hole...

A Geostatistical Approach to the Characteristic Values

Horatiu V. Corbeanu Halcrow/CH2M HILL

07/11/12 – 12:15

Pro Statistics

• “Statistical methods . . . constitute the science of collecting, analysing and interpreting data in the best possible way” C. Chatfield (Statistics of Technology, 3rd ed. 1983)

• “The product of an arithmetical computation is the answer to an equation; it is not the solution to a problem” - G.O. Ashley

• “Years ago a statistician might have claimed that statistics deals with the processing of data. . . today’s statistician will be more likely to say that statistics is concerned with decision making in the face of uncertainty” – H.Chernonff & L.E. Moses (Elementary Decision Theory, 1959)

Abuse of Statistics

• “Attempts by statisticians to tackle geotechnical design have often ended in ridicule, and it is very difficult for one person to have sufficient grasp of both disciplines that he can use them sensibly”, Decoding Eurocode 7, A.Bond, A.Harris, 2008

• “There are lies, damned lies, and statistics”, Benjamin Disraeli

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1

1

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022

22

Twisted Example

Objective

• The Problem

• The Dataset

• General presentation of geotechnical data

• Distribution of geotechnical data

– Descriptive statistical techniques

– Transformations to a normal distribution

• Conclusions

1) Provide a reliable characteristic value

2) Identify the mathematical tool that can provide confidence to the engineering judgement.

The Problem

The Dataset HOLE EASTING NORTHING GEOL_GEOL ELEVATION ROCK_UCS ROCK_E ROCK_MU ROCK_MC ROCK_BDEN

01.BH-353 228,131 393,303 SL -6.09 13.851 13100 0.265 1.7 21

01.BH-353 228,131 393,303 RUS -19.59 5.678 8300 1.4 17.6

01.BH-536 229,228 393,111 RUS -15.52 7.18 1.8 18.9

01.BH-536 229,228 393,111 RUS -21.52 41.884 26900 0.172 1.2 26.2

01.BH-570 225,071 394,022 MSH -3.80 25.599 22800 0.6 18

01.BH-570 225,071 394,022 RUS -12.80 18.3 17600 1.1 17.4

01.BH-570 225,071 394,022 RUS -23.82 12.85 11700 0.206 1.3 20.2

01.BH-572 226,451 393,594 RUS -10.49 7.957 1.6 20.4

01.BH-572 226,451 393,594 RUS -13.49 8.156 7700 0.212 1.9 20.5

01.BH-572 226,451 393,594 RUS -22.49 12.174 11800 0.238 2.1 21.6

01.BH-755 224,397 394,360 SL -6.49 11.736 2 22.9

01.BH-755 224,397 394,360 MSH -14.54 40.022 1.4 24

01.BH-755 224,397 394,360 RUS -17.54 15.158 2.8 21.9

01.BH-756 220,978 395,899 RUS 0.81 15.197 19500 2.6 17

01.BH-756 220,978 395,899 RUS -5.19 12.452 16400 2.1 17.1

01.BH-756 220,978 395,899 RUS -12.69 10.523 14100 0.215 2.5 25.1

01.BH-757 221,265 395,957 MSH 1.38 16.331 17000 0.200 3.9 22.6

Count :

Minimum :

Maximum :

Mean :

5% Fractile :

226

2.83

12,567

2,531

240

MPa

MPa

MPa

MPa

-42

-40

-38

-36

-34

-32

-30

-28

-26

-24

-22

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

16

18

20

0 5000 10000 15000 20000 25000

Intact Elastic Modulus (MPa)

Ele

va

tio

n (

m Q

NH

D)

WSL Lab SL & BSL Lab M SH Lab RUS Calcareous Lab RUS Clayey Lab RUS Gypsum Lab

WSL Derived SL & BSL Derived M SH Derived RUS Calcareous Derived RUS Clayey Derived RUS Gypsum Derived

-42

-40

-38

-36

-34

-32

-30

-28

-26

-24

-22

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

16

18

20

1 10 100 1000 10000 100000


Ele

va

tio

n (

m Q

NH

D)



General Presentation of Data

Shapes of Data Distribution

Triangular

Data

Fre

qu

en

cy (

%)

Uniform

Data

Fre

qu

en

cy (

%)

Exponential

Data

Fre

qu

en

cy (

%)

Normal

Data

Fre

qu

en

cy (

%)

Lognormal

Data

Fre

qu

en

cy (

%)

Fact for Log Normal Distribution

c = a + b Normal

C

Fre

qu

en

cy (

%)

Two geological processes: a & b (i.e. rate of deposition, duration of deposition)

Log (c) = Log (a*b)

Log (c) = Log (a) + Log (b)

c = a * b Lognormal

C

Fre

qu

en

cy (

%)

How does it look like?

-42

-40

-38

-36

-34

-32

-30

-28

-26

-24

-22

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

16

18

20

1 10 100 1,000 10,000 100,000

Intact Elastic Modulus (MPa)E

lev

ati

on

(m

QN

HD

)



The distribution of Data

- +

68.26%

99.95%

- +

Fre

qu

en

cy (

%)

Parameter

Count :

Minimum :

Maximum :

Mean :

Standard Deviation :

226

2.83

12,567

2,531

2,445

MPa

MPa

MPa

MPa

- = 86 MPa + = 4976 MPa

- 2 = -2359 MPa + 2 = 7421 MPa

- 3 = -4804 MPa + 3 = 9866 MPa

95.44%

- +

-15

-13

-11

-9

-7

-5

-3

-1

1

3

5

7

9

11

13

15

17

19

1 10 100 1,000 10,000 100,000


Ele

vati

on

(m

QN

HD

)

SL & BSL Lab SL & BSL Derived

-15

-13

-11

-9

-7

-5

-3

-1

1

3

5

7

9

11

13

15

17

19

-5 -4 -3 -2 -1 0 1 2 3 4 5

Z-Score

Ele

va

tio

n (

m Q

NH

D)


Z = (x - ) /

- +

68.26%

+

Fre

qu

en

cy (

%)

Parameter

+

Z-Score

Log-normal Distribution

0

10

20

30

40

50

60

70

3

90

0

1,7

98

2,6

95

3,5

93

4,4

90

5,3

88

6,2

85

7,1

82

8,0

80

8,9

77

9,8

75

10

,77

2

11

,67

0

12

,56

7

Intact Elastic Modulus

Nu

mb

er

of

Sa

mp

les

Ways Forward

• Ignoring the mathematical tools, introducing further descriptive statistical parameters (median, skewness, kurtosis, coefficient of variation, etc.)

• Applying transformations (rotations, translations, multiplications, etc.) to the existing data in such a way to be represented as a normal distribution

Descriptive Statistics Approach • Median

The midpoint of the observed values if they are arranged in increasing order (ex. No of Samples = 226 – median = sample no 113)

0

10

20

30

40

50

60

70

3

90

0

1,7

98

2,6

95

3,5

93

4,4

90

5,3

88

6,2

85

7,1

82

8,0

80

8,9

77

9,8

75

10

,77

2

11

,67

0

12

,56

7


Nu

mb

er

of

Sa

mp

les

Count :

Minimum :

Maximum :

Mean :

Median :

226

2.83

12,567

2,531

1,710

MPa

MPa

MPa

MPa

Descriptive Statistics Approach • Median

• Deciles, Percentile, Quartiles

• Interquartile Range (IQR)

Splitting the data into tenth (deciles), hundredths (percentile) or any other fraction (quartiles)

IQR = difference between q0.75 and q0.25

Attention: Excel works a bit different – percentile for any fraction, quartile for min, 25%, median, 75%, max

Box Plot

1.5 * IQR 1.5 * IQR

IQR = q0.75 – q0.25= 2400

q0.25 = 880 q0.75 = 3280

q0.50 = 1710

880-1.5*2400 = -2720

q0.00 = 3 Q1.00 = 12567

3280+1.5*2400 = 6880

6,876.88

3.00

1,711.50

878.75

3,278.00

9,758

9,170

10,188

8,366

10,256

12,153

12,567

11,724

8,489

9,932

8,579

8,885

7,0017,091

9,211

9,049

7,281

-15

-13

-11

-9

-7

-5

-3

-1

1

3

5

7

9

11

13

15

17

19

1 10 100 1,000 10,000 100,000


Ele

vati

on

(m

QN

HD

)


Distribution of Data

Descriptive Statistics in Excel

Descriptive Statistics Approach

0

10

20

30

40

50

60

70

3

90

0

1,7

98

2,6

95

3,5

93

4,4

90

5,3

88

6,2

85

7,1

82

8,0

80

8,9

77

9,8

75

10

,77

2

11

,67

0

12

,56

7


Nu

mb

er

of

Sa

mp

les

Transformations to a Normal Distribution

• Applying any mathematical calculation over the existing dataset (square, log, exponential, etc.)

• Applying rotation and translation matrices

Transformations

0

5

10

15

20

25

30

35

40

1.37 2.48 3.60 4.71 5.83 6.94 8.06 9.17 10.28 11.40 12.51 13.63 14.74 15.86 16.97

Nu

mb

er

of

Sam

ple

s

0

10

20

30

40

50

60

1.04 1.64 2.24 2.84 3.44 4.04 4.64 5.24 5.84 6.44 7.04 7.64 8.24 8.84 9.44

Nu

mb

er

of

Sam

ple

s

Descriptive Statistics

Count 226.000

Minimum 1.041

Maximum 9.439

Range 8.398

Mean 7.392

Median 7.445

Mode 7.706

Standard Deviation 1.060

Sample Variance 1.123

Kurtosis 5.192

Skewness -1.184

Confidence Level (95%) 0.004

x1 = ln(x) x2= x0.3

Descriptive Statistics

Count 226.000

Minimum 1.366

Maximum 16.973

Range 15.607

Mean 9.613

Median 9.333

Mode 10.092

Standard Deviation 2.777

Sample Variance 7.709

Kurtosis 0.098

Skewness 0.336

Confidence Level (95%) 0.012

Box Plots 9.44

4.80

7.45

6.78

8.10

4.58

1.04

3.81

x1 = ln(x) 16.89

2.09

9.33

7.64

11.34

1.37

17x2= x0.3

0

5

10

15

20

25

30

35

40

1.37 2.48 3.60 4.71 5.83 6.94 8.06 9.17 10.28 11.40 12.51 13.63 14.74 15.86 16.97

Nu

mb

er

of

Sam

ple

s

0

10

20

30

40

50

60

1.04 1.64 2.24 2.84 3.44 4.04 4.64 5.24 5.84 6.44 7.04 7.64 8.24 8.84 9.44

Nu

mb

er

of

Sam

ple

s

x1 = ln(x)

x2= x0.3

-15

-13

-11

-9

-7

-5

-3

-1

1

3

5

7

9

11

13

15

17

19

1 10 100 1,000 10,000 100,000


Ele

vati

on

(m

QN

HD

)


Histograms

6,876.88

3.00

1,711.50

878.75

3,278.00

9,758

9,170

10,188

8,366

10,256

12,153

12,567

11,724

8,489

9,932

8,579

8,885

7,0017,091

9,211

9,049

7,281

9.44

4.80

7.45

6.78

8.10

4.58

1.04

3.81

Exp(1.04) = 2.82 MPa

Exp(4.58) = 97 MPa

Exp(4.80) = 120 MPa

Exp(3.81) = 45 MPa

Exp(7.45) = 1,711.5 MPa

Exp(6.78) = 878.75 MPa

Exp(8.10) = 3,278 MPa

Exp(9.44) = 12,580 MPa

Back calculation

Median Ln : 7.45

Back calculated median : Exp(7.45) = 1,711.5 MPa

Conclusion

Statistical assessment:

Minimum = 120 MPa

Maximum = 12,567 MPa

Characteristic value = 1,600 – 1700 MPa

5% Fractile = 380 MPa

95% Fractile = 8,570 MPa

-15

-13

-11

-9

-7

-5

-3

-1

1

3

5

7

9

11

13

15

17

19

1 10 100 1,000 10,000 100,000


Ele

vati

on

(m

QN

HD

)


Outliers

Original assessment:

Minimum = 2.83 MPa


Characteristic value = 2,531 MPa



Simsima Limestone

• Earth data are generally characterized by log normal distributions

• The basic statistical tools available are used to quantify our engineering judgment and to provide confidence in our assessment

• Applying blind mathematical tools without the understanding of our dataset can result in misleading answers.

Conclusion

0

5

10

15

20

25

30

200

2,1

07

4,0

14

5,9

21

7,8

29

9,7

36

11,6

43

13,5

50

15,4

57

17,3

64

19,2

71

21,1

79

23,0

86

24,9

93

26,9

00

Midra Shale

0

10

20

30

40

50

60

70

100

2,8

00

5,5

00

8,2

00

10,9

00

13,6

00

16,3

00

19,0

00

21,7

00

24,4

00

27,1

00

29,8

00

32,5

00

35,2

00

37,9

00

Midra Shale and Rus Formation

Rus Formation

-40

-35

-30

-25

-20

-15

-10

-5

0

5

10

15

1 10 100 1000 10000 100000


lev

ati

on

(m

QN

HD

)

MSH Lab RUS Lab

0

1

2

3

4

5

6

7

8

9

2.30

1

2.45

3

2.60

5

2.75

7

2.90

9

3.06

1

3.21

3

3.36

5

3.51

7

3.66

9

3.82

2

3.97

4

4.12

6

4.27

8

4.43

0

Log (E)

Nu

mb

er o

f sa

mp

les

Midra Shale

200 MPa 26,910 MPa

Cluster 1

0

1

2

3

4

5

6

7

2.3

0

2.3

9

2.4

7

2.5

5

2.6

4

2.7

2

2.8

1

2.8

9

2.9

7

3.0

6

3.1

4

3.2

3

3.3

1

3.3

9

3.4

8

Log (E)

Nu

mb

er

of

Sam

ple

s

Cluster 2 - 3

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

3.7

7

3.8

2

3.8

6

3.9

1

3.9

6

4.0

1

4.0

5

4.1

0

4.1

5

4.1

9

4.2

4

4.2

9

4.3

4

4.3

8

4.4

3

Log (E)

Nu

mb

er

of

Sam

ple

s

0

0.5

1

1.5

2

2.5

3

3.5

3.4

91

3.5

69

3.6

47

3.7

24

3.8

02

3.8

80

3.9

57

4.0

35

4.1

13

4.1

90

4.2

68

4.3

46

4.4

23

4.5

01

4.5

79

0

1

2

3

4

5

6

2.0

00

2.1

00

2.2

00

2.3

00

2.3

99

2.4

99

2.5

99

2.6

99

2.7

99

2.8

99

2.9

99

3.0

98

3.1

98

3.2

98

3.3

98

5,900 MPa 10,100 MPa

200 MPa

3,000 MPa

Midra Shale

17,400 MPa 26,910 MPa

MEAN : 905 MPa

MEDIAN: 850 MPa

MEAN : 21,380 MPa

MEDIAN: 22,800 MPa MEAN : 7,465 MPa

MEDIAN: 7,160 MPa

0

2

4

6

8

10

12

14

16

2.00

0

2.18

4

2.36

8

2.55

3

2.73

7

2.92

1

3.10

5

3.28

9

3.47

4

3.65

8

3.84

2

4.02

6

4.21

0

4.39

4

4.57

9

Log (E)

Num

ber o

f Sam

ples

Rus Formation

100 MPa 37,930 MPa

Conclusion Original assessment:

Minimum = 200 MPa


Characteristic value = 5.813 MPa



Confidence Level (95) = 2,370 MPa

Midra Shale


Lithology 1 (Possible Shale)

Minimum = 200 MPa

Maximum = 3,000 MPa

Characteristic value = 850 - 1170 MPa



Confidence Level (95) = 295 MPa

Rus Formation

Lithology 2 (Possible Dolomitic Limestone)

Minimum = 16,400 MPa


Characteristic value = 21,655 – 22,800 MPa




Original assessment:

Minimum = 100 MPa


Characteristic value = 4,750 MPa





Lithology 1 (Possible Chalk/Calcisiltite)

Minimum = 100 MPa

Maximum = 2,500 MPa

Characteristic value =1,000 – 1,050 MPa



Confidence Level (95) = 160 MPa

Lithology 2 (Possible Limestone)

Minimum = 3,100 MPa


Characteristic value = 9,600 – 11,900 MPa




-20

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-10

-5

0

5

10

1 10 100 1000 10000 100000


lev

ati

on

(m

QN

HD

)

-40

-35

-30

-25

-20

-15

-10

-5

0

5

10

1 10 100 1000 10000 100000


Ele

va

tio

n (

m Q

NH

D)

Conclusion

Midra Shale Rus Formation

Date post:	18-Oct-2020
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A Geostatistical Approach to the Characteristic Values. Horatiu Corbeanu.pdf · the dataset hole...

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