HW2 - Florida International University Department of Civil...

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Unformatted text preview: Florida International University Department of Civil and Environmental Engineering CWR6117 – Statistical Hydrology Assignment No. 2 – Due February 17, 2009 Problem 1: Suppose that X is a zero‐mean normally distributed random variable and that Y=X2. Show that X and Y are uncorrelated. How do you reconcile this with the fact that they are functionally related? Problem 2: Suppose that the hydraulic conductivity in a soil can be characterized as a spatially dependent random variable log K=f(z). You have a number of hydraulic conductivities along a vertical transect and you wish to determine if there is a large scale trend. In order to investigate this issue, you can construct a “moving window” average of the measurements in the following way: z f Z ( z) 1 H H 2 f ( z ' )dz ' z H 2 where z is a specified location along the transect, and H defines the size of the averaging window. Show that the variance of the averaged log hydraulic conductivity is given by: h 2 H fZ 2 2 1 (h)dh f H 0 H where (h) is the log conductivity spatial correlation function expressed as a function of the relative displacement h. Evaluate this expression for the following hypothesized log conductivity correlation functions: h a) Triangular: ( h) 1 for h a , 0 otherwise. a b) Exponential: c) Second‐order autoregressive: h a ( h) e h a h (h) (1 )e a Plot the value of (fZ2/f2) obtained for each case for different values of a, and also provide a plot of all three alternatives as a function of H for a given a. What do your results imply about the behavior of the moving window average as H increases? Explain and discuss clearly your results. Problem 3: Suppose that a small particle moves along a one‐dimensional trajectory in accordance with the following equation of motion: x k 1 x k vt s ( k 0.5) where xk is the position of the particle at time k, v is the flow velocity, t is a small time step, s is a small displacement and k is a random number uniformly distributed between 0 and 1. a) What is the pdf of the particle position xk? What are its mean, variance and covariance functions? b) Suppose that the displacement and time step are related as s2=t. How does the pdf behave as t0? This limit defines a “random walk” process. c) Use a synthetic simulation to confirm the results found in the two questions above. Use a random number generator of your choice (search the web, or use the random number utilities found in MATLAB or in EXCEL); be sure to generate enough random numbers (1000 or more). Plot your results in an informative way. ...
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