Date post: | 18-Jan-2018 |
Category: |
Documents |
Upload: | aubrey-harvey |
View: | 221 times |
Download: | 0 times |
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
2 / 3
5 / 3
15 / 3 1
*
15 / 3 1*
7 / 6
/
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?
2 / 3
5 / 3
15 / 3 1
*
15 / 3 1*
7 / 6
/
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
2 / 3
5 / 3
15 / 3 1
*
15 / 3 1*
7 / 6
/
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
0
100
200
300
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)