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The Fukushima nuclear fallout.

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On March 11, 2011, after an earthquake and

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the tsunamis had devastated
the northeast coast of Japan,

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a multiple core meltdown happened at
the Fukushima Daiichi Nuclear Power plant,

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leading to a substantial release of
radioactive material into the environment.

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Due to a general lack of detailed
radiation measurement, SafeCast,

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a citizen science group based in Japan,
started building instruments for

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geo-localized measurements
of radioactivity.

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The ideas of project was
to reconstruct a map of

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radioactivity from these measurements.

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Let us first consider the following
simple, one-dimensional problem.

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Suppose that you're interested
in a continuous time function,

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f of t, that is only known through
some discreet set of measurements.

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The known samples here on the right side.

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In addition, these measurements
might not be uniformly sampled.

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That is,
they do not lie on the regular grid.

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We would like nonetheless, to be able
to say something about the value of

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the function in areas where
no data is available.

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If it possible to make a few
assumptions about the function itself,

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it is impossible to make an educated
guess about the value of

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the function at locations
where the signal is unknown.

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This process is referred
to as interpolation.

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In our example, we will consider the
particular case of band limited functions.

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We can see here a band limited
interpolation from the known samples.

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As you can see, the dotted line,
or the continuous time function,

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goes through all the red samples so it
does a perfect match at the known values.

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And it interpolates in between.

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Let us consider a continuous time,
t-periodic, and

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band limited function f of t.

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Since it is band limited,
this function can be

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parameterized by its Fourier expansion
coefficients up to a finite order m.

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This means there are two N plus 1 free
coefficients that are conveniently

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represented in vector form by vector c.

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So here we have the expression
of the periodic function,

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which is the expression of the Fourier's
series that we study in this class.

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And the vector is given here
by 2 N plus 1 coefficients.

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We assume now that we
know capital end pairs,

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ti, zi, where zi is a value
of the function at time ti.

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We do not make any assumptions
about the locations of the ti's.

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Given N large enough with respect to
the maximum free coefficient order M,

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we can make a least
squares fit of the data to

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find the free expansion coefficient
that best approximates data collected.

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We need at least M equal
2M plus 1 measurements.

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But in general, we would like
a much larger number because of

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possible measurement noise.

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The least square fit can be rewritten
matrix form and solved efficiently.

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But let's just skip the details here.

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For a full demonstration,
see the corresponding IPython notebook.

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Once we have recovered the best guess
of the Fourier expansion coefficients,

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it is possible to resample the function
f of t at any location we like.

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Often, the function is
resampled on a regular grid.

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Let us give a first example.

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The function we chose here satisfies
perfectly the assumptions of

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periodicity and band limitedness.

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It just the sum of two signs of frequency,
one hertz and four hertz.

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The true function is represented
by the thick black line.

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We pick at random 20 samples,
represented here by red dots.

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The model order is N is equal to 6.

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The interpolation given by our least
squares fit is a dotted red line.

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Since all the assumptions are satisfied,
and because there is no noise,

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the interpolation recovers
perfectly the original signal.

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You might be wondering what happens if the
original signal does not satisfy some of

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the assumptions, as we might very
well be the case in practice.

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For example,
assume the single is not band-limited.

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We take here the example of signal,

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continuous everywhere, but
not differentiable at zero.

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Such a signal is not band-limited.

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However, it's Fourier coefficients
decay proportionally to the square of

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the frequency.

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That is quite fast.

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The non-band limitedness of
the function can be somewhat

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compensated by picking a larger order for
our model and having more samples.

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We choose here the ordinal
to be m is equal to 10,

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and take n is equal to 20 samples,
picked, again, at random.

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We can observe that our interpolation is
still very close to the true function.

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The approximation is,

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however, not so tight around
the sharp angle at t is equal to 0.

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Let us now move on to the real
world example of Fukushima.

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We are interested in measuring
radiation levels all over Japan, and

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beyond after the tsunami of 2011 and
the nuclear accident.

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Instruments were loaned to
volunteers in the field and

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measurements were carried out
by fixing the sensors on cars.

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The sensors would then take a measurement
of the radiation every five seconds.

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Therefore, sampling is generally
limited to areas accessible by car.

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A subset of the measurements is shown
here on the map of the Fukushima

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prefecture area.

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Each colored dot is a measurement
with yellow being low intensity and

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red high intensity.

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You can observe some fairly high
measurement levels on the path of

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the plume, starting around the plant,

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situated at the far right of
the mountain and going northeast.

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This is a typical example where
the location of measurements is

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constrained by available infrastructure.

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Since the measurements are done with
a car, we can only measure on roads,

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as plainly visible on the map.

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It is thus necessary to apply
interpolation to get an estimate of

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the intensity of radiation in areas where
no samples are available, such as forests,

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fields, and so on.

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The interpolation is done using
a model with m is equal to 441 Fourier

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coefficients and a number of
samples that is close to a million.

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The function so obtained is then
resampled on the regular grid and

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displayed here on the same map as before.

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A decent estimate of the total radiation
field is obtained in this way.

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A few caveats are to mention.

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First, radiation is notoriously
non-bandlimited, and

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we would probably need the larger
order model for better approximation.

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Second, large measurement
noise is present and

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also degrades the quality
of the estimation.

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A few tricks were used in this
reconstruction to improve the quality of

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the estimation.

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The code and

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detailed explanations are available
in the companion IPython Notebook.

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Check it out.