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Unformatted text preview: Florida International University Department of Civil and Environmental Engineering CWR6117 – Statistical Hydrology Assignment No. 3 – Due March 31, 2009 Problem 1: For the 2 rainfall time series discussed in class (Nylsvley savanna and Everglades National Park), do the following: (a) Compute the mean and variance of the temporal data. (b) Compute the covariance function R() and the autocorrelation function for both time series. What shape do they have? Is it reasonable to fit any of the autocorrelation models discussed in class (exponential, harmonic, white noise) to these data sets? Explain any differences found between the 2 data sets analyzed. (c) Compute the correlation time (integral scale) for both time series; use numerical integration. (d) Are these time series stationary? Explain your answer in specific terms (first and second‐order stationarity). (e) Compute the mean‐removed covariance function R’() and the mean‐removed autocorrelation function ’ for both time series. Use these for part (f). (f) Compute the spectrum of both time series. What shape do they have, and how does this shape compare to the models discussed in class? Explain your answers. (g) Verify that the area under the spectrum vs. frequency curves is equal to the variance of the mean‐removed time series X’(t). How close is it? Explain. 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 1
f Z ( z) 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: h
a ( h) e h c) Second‐order autoregressive: h (h) (1 )e a 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. ...
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This note was uploaded on 07/09/2011 for the course CWR 6117 taught by Professor Miralles during the Spring '09 term at FIU.
- Spring '09