More on Monte Carlo simulation in SR

GEOL/CE/EEB 8601 Intro to Stream Restoration

Peter Wilcock Geography and Environmental EngineeringNational Center for Earth-surface Dynamics

Johns Hopkins University

SEDIMENT TRANSPORT IN STREAM RESTORATION

19 September 2012

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*

15 / 3 1*

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/

so

or

( 1) so

( 1)

b

b

b

c c

bc c

Q BhU

B aQh gS

SU hn

SQ aQgS n

a SQn gS

s gD

aQ s DnS

Transport model for a threshold channel is based on a definition of incipient sediment motion

Uncertainty ExerciseFor a simple, wide, prismatic channel, find critical discharge Qc for incipient motion

*

Your transport model:

0.045( 1)

cc s gD

hydraulic geometry

momentum

Manning’s eqn.

continuity

05/04/23 3

What if you are not too sure about some of the values needed to determine Qc?

Like n, D, and *c –what do you do?

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so

or

( 1) so

( 1)

b

b

b

c c

bc c

Q BhU

B aQh gS

SU hn

SQ aQgS n

a SQn gS

s gD

aQ s DnS

05/04/23 4

Suppose your best estimate of Manning’s n is 0.035and that you are pretty sure that the real value falls between 0.03 and 0.04.

We could approximate your assessment of thevalue of n with a normal distribution with mean = 0.035 & standard deviation = 0.0025.

95% of this distribution falls between 0.03 and 0.04,as can be seen in the cumulative frequency plot, sowe are saying that the real value of n is 95% likely to fall between 0.03 and 0.04 and that it is more likely to be around the center of the distribution (0.035) than in the tails. We use this distribution to pick values of n in our Monte Carlo simulation.

How does that work? We use a random number generator to pick a number between 0 and 1 and then use this number to find a value of n for the cumulative frequency distribution. For example,for 0.88, n = 0.0379for 0.23, n = 0.0332

n n 0.02 0.03 0.04 0.05

Freq

uenc

y

Manning's n

2 n

0.0350.0025n

n

0

0.2

0.4

0.6

0.8

1

0.02 0.03 0.04 0.05

Cum

ulat

ive

Freq

uenc

y

2 n

Manning's n

0

0.2

0.4

0.6

0.8

1

0.02 0.03 0.04 0.05

Cum

ulat

ive

Freq

uenc

y

Manning's n05/04/23 5

The Monte Carlo simulation

1. Pick values of n, , and D from specified frequency distributions.

2. Calculate critical discharge and transport rate.

3. Repeat 1000 times.

4. Distribution of calculated values givesestimate of the effect of input uncertaintyon calculated critical discharge and transport rate.

*c

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or

( 1) so

( 1)

b

b

b

c c

bc c

Q BhU

B aQh gS

SU hn

SQ aQgS n

a SQn gS

s gD

aQ s DnS

0

100

200

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400

0.02

0

0.02

4

0.02

8

0.03

2

0.03

6

0.04

0

0.04

4

0.04

8

Manning's n(a)

0

50

100

150

200

0.03

0

0.03

4

0.03

8

0.04

2

0.04

6

0.05

0

0.05

4

0.05

8

*c(b)

0

50

100

150

200

250

14 20 26 32 38 44 50 56

Grain Size D(mm)(c)

050

100150

200250

300

0 3 6 9 12 15 18 21

Critical Discharge Qc (m^3/s)(d)

Manning's n

n

Manning's n

D

Manning's n

*c

1.

2.

4.

Monte Carlo05/04/23 6

3/23/51 0.73 *( 1)

( 1)

bs o c

nQ SQ cB s gDa s D

1

5/3 1*7/6 ( 1)

bc c

aQ s DnS

Threshold ChannelFind critical discharge Qc at which grain motion begins

Mobile ChannelFind transport capacity for different water discharge Q

Estimating uncertainty in sediment transportIt’s the input, not the formula !!!

These terms have lots of uncertainty !!

0

50

100

150

200

250

300

0.026 0.028 0.030 0.032 0.034 0.036 0.038 0.040

Manning's n

(a)

0

50

100

150

200

250

300

0.023 0.027 0.031 0.035 0.039 0.043 0.047 0.051

*c

(b)

0

50

100

150

200

250

44 49 55 60 66 71 76 82

Grain Size D (mm)

(c)

0

50

100

150

200

250

13.0 17.0 21.0 25.0 29.0 33.0 37.0 41.0

Critical Discharge (m^3/s)

(d)

5.011.017.023.029.035.0

0 187 374 561 748 935 1122

Discharge (cms)

Time (hrs)

Dis

char

ge (m

^3/s

)

(e)

050

100150200250300

0 100 200 300 400 500 600 700

Cumulative Transport (metric tons)

(f)

2x

2x – 10x

No point being normal…

• Log-normal: log(x) is normally distributed

• Exponential: D(x) = 1 – e-kx

No point being normal…

• Pareto:“long tail”

( ) 1 mxD x

x

http://en.wikipedia.org/wiki/Weibull_distribution

http://en.wikipedia.org/wiki/Exponential_distribution

All types of distributions in Wikipedia

http://en.wikipedia.org/wiki/Gamma_distribution

http://en.wikipedia.org/wiki/L%C3%A9vy_distribution

http://en.wikipedia.org/wiki/Poisson_distribution

http://en.wikipedia.org/wiki/Pareto_distribution

http://en.wikipedia.org/wiki/Normal_distribution

http://en.wikipedia.org/wiki/Log-normal_distribution

http://en.wikipedia.org/wiki/Uniform_distribution_(continuous)

http://en.wikipedia.org/wiki/Uniform_distribution_(discrete)