Suppose that the water in a shallow lake flows only horizontally (i.e., in the (x, y) plane) and…

Suppose that the water in a shallow lake flows only
horizontally (i.e., in the (x, y) plane) and that the two components of fluid
velocity, vx and vy are measured at a set of N observation points, (xi, yi).
Water is approximately incompressible, so a reasonable type of prior
information is that the divergence of the fluid velocity is zero; that is
qvx/qx þ qvy/qy ¼ 0. Furthermore, if the grid covers the whole lake, then the
condition that no water flows across the edges is a reasonable one, implying
that the perpendicular component of velocity is zero at the edges. (A) Sketch
out how
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Suppose that the water in a shallow lake flows only
horizontally (i.e., in the (x, y) plane) and that the two components of fluid
velocity, vx and vy are measured at a set of N observation points, (xi, yi).
Water is approximately incompressible, so a reasonable type of prior
information is that the divergence of the fluid velocity is zero; that is
qvx/qx þ qvy/qy ¼ 0. Furthermore, if the grid covers the whole lake, then the
condition that no water flows across the edges is a reasonable one, implying
that the perpendicular component of velocity is zero at the edges. (A) Sketch
out how scripts eda05_07 and eda05_08 might be modified to fill in the gaps of
fluid velocity data.

Q.24:

In the example shown in Fig. 5.8, a two-dimensional pressure
field p(x, y) is reconstructed using sparse data and the prior information that
the field satisfies Laplace’s equation. Consider an alternative scenario in
which the pressure is believed to vary smoothly but the equation that it
satisfies is unknown. In that case, we might opt to use a combination of
flatness:

Q.25:

Write a MatLab script that uses the fft() function to
differentiate the Neuse River Hydrograph dataset. Plot the results.

Q.26:

What is the Fourier transform of sin(o0t)? Compare it to the
transform of cos(o0t).

Q.27:

 The MatLab script, eda06_15, creates a file, noise.txt,
containing normally distributed random time series, d(t), with zero mean and
unit variance. (A) Compute and plot the power spectral density of this time
series. (B) Create a second time series, a(t), that is a moving window average
of d(t); that is, each point in a(t) is the average of, say L, neighboring
points in d(t). (C) Compute and plot the power spectral density of a(t) for a
suite of values of L. Comment on the results.

Q.28:

Suppose that you needed to compute the DFT of the function

using MatLab’s fft() function. This function is centered on
t ¼ 0, and therefore has nonnegligible values for points to the left of the
origin. Unfortunately, we have defined the time and data column-vectors, t and
d, to start at time zero, so there seems to be no place to put these data
values. One solution to this problem is to shift the function to the center of
the time window, say by an amount, t0, compute its Fourier transform, and then
multiply the transform by a phase factor, exp(iot0), that shifts it back.
Another solution relies on the fact that, in discrete transforms, both time and
frequency suffer from aliasing. Just as the last frequencies in the transform
were large positive frequencies and small negative frequencies, the last points
in the time series are simultaneously

Therefore, one simply puts the negative part of d(t) at the
right-hand end of d. Write a MatLab script to try both methods and check that
they agree.

Q.29:

Compute and plot the amplitude spectral density of a cleaned
version of the Black Rock Forest temperature dataset. (A) What are its units?
(B) Interpret the periods of any spectral peaks that you find.

CH-7

Q.30:

Calculate by hand the convolution of a = [1, 1, 1, 1]T
and b = [1, 1, 1, 1]T . Comment on the shape of the function c = a *
b.

Q.31:

Plot the prediction error, E, of the prediction of the Neuse
River hydrograph as a function of the length, N, of the prediction error
filter. Is a point of diminishing returns reached?

Q.32:

What is the z-transform of a filter that delays a time
series by one sample?

Q.33:

 Note that any filter, g, with g1 =0 is a nonstationary
phase, as its z-transform is exactly divisible by z and so has a root at z = 0
that is not outside the unit circle. A simple way to change a stationary phase
filter into one that is nonstationary phase filter is to decrease the size of
its first element towards zero. Modify script eda07_07 to examine what happens
to the inverse filter when you decrease the size of g1 towards zero
in a series of steps. Increasing Ni might make the behavior clearer.

Q.34:

Generalize the recursive filter developed at the end of
Section 7.8 for the case g(t) / exp(t/t), that is, a smoothing filter of unit
area and arbitrary width, t. Start by writing gj / [1, c, c2 , …] T
with c ¼ exp(_t/t).

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