A tidal gauge (see figure below) is a device used by coastal engineers to measure sea-surface elevation over the course of half-to-one day. Numerical differentiation may then be used to compute the up-down velocity of the surface and its associated acceleration. A typical signal recorded over the course of one day is always a near-perfect sinusoid.

Now, lets assume we have a sensor that measures sea surface elevation in meters at a sampling rate of 1 Hz. A tide has a period of T=12 hours and an amplitude of A=1 m. If our instrument were noise-free it would have measured an elevation described by the exact expression f=A cos(2t/T). The sensor has a noise level of 0.01 cm.

A file with actual data will be posted on the Canvas website. This file will contain a timeseries of noisy data on which you will apply numerical differentiation.

a) Plot the 12-hr experimental data timeseries and zoom in to better show the noise.

b) Use an O(h2)-accurate centered finite difference approximation to numerically compute the velocity df/dt at every point in your sampled timeseries, with points spaced apart by h=1 sec (neglect the end-points for which you might not be able to form centered differences). Plot your computed result against the exact one (the analytical derivative of f=A cos(2t/T)). Compute the maximum absolute error within your numerically differentiated dataset.

c) Comment on your result in Part (b). Why does the approximated df/dt look noisier than your measured signal of f ? Are those errors associated with truncation error, round-off error or something else? Does the textbook paradigm of ``total error = truncation error + round-off error'' completely describe the picture ?

HINT: The values that enter your finite difference formula are contaminated by inaccuracies of your measurement system.

d) In the spirit of minimizing the error, repeat the above exercise using different values of h each time. Specifically, sample 50 equally spaced values of h on logscale (Matlabs built-in function logspace may help you here) in the range of [1,104] s. For each value of h sample the original dataset accordingly (see below). Feel free to round a particular value of h, as computed above, such that the code built for part (b) may be flexibly used when working with the down-sampled signal.

CAREFUL:

• Dont plot your numerically computed timeseries for each value of h. You simply want

  to hold on to the maximum absolute error for each value of h.

• This part of the problem involves your downsampling your existing dataset. Directions

  towards this will be given in recitation.

  What is the value of h value at which your total error is minimized ? For this value of h (about 360 s), generate the same plot as in part (b). Draw the log-log plot of error as a function of h. Does this plot remind you of a figure in both the textbook and your lecture notes ? What is the dominant component of the error curve to the left of its observed minimum? To the right ?

  e) Using the analysis similar to what is done in section 4.4.1 of the textbook (and in your lecture notes), compute the theoretical prediction of the optimal h value that minimizes your total error.