1. More on Monte Carlo simulation in SR GEOL/CE/EEB 8601 Intro to Stream Restoration

2. Peter Wilcock Geography and Environmental Engineering National Center for Earth-surface Dynamics Johns Hopkins University SEDIMENT TRANSPORT IN STREAM RESTORATION 19 September 2012

3. Transport model for a threshold channel is based on a definition of incipient sediment motion Uncertainty Exercise For a simple, wide, prismatic channel, find critical discharge Q c for incipient motion hydraulic geometry momentum Manning’s eqn. continuity 2/19/20163

4. What if you are not too sure about some of the values needed to determine Q c ? Like n, D, and  * c – what do you do? 2/19/20164

5. Suppose your best estimate of Manning’s n is 0.035 and that you are pretty sure that the real value falls between 0.03 and 0.04. We could approximate your assessment of the value 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, so we 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 = 0.0379 for 0.23, n = 0.0332 2/19/20165

6. 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 gives estimate of the effect of input uncertainty on calculated critical discharge and transport rate. 1.1. 2.2. 4.4.  Monte Carlo 2/19/20166

7. Threshold Channel Find critical discharge Q c at which grain motion begins Mobile Channel Find transport capacity for different water discharge Q Estimating uncertainty in sediment transport It’s the input, not the formula !!! These terms have lots of uncertainty !!

8. 2x 2x – 10x

9. No point being normal… Log-normal: log(x) is normally distributed Exponential: D(x) = 1 – e -kx

10. No point being normal… Pareto: “long tail”

11. 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)