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J. F. Reis et al.
when the correlation function is relative to different random variables, it is usually
referred to as cross-correlation function.
A classical example is a surface that is not perfectly flat, for instance, the Earth
crust. We could look at this as a stochastic field where the height varies according to
the terrestrial coordinates. To characterise the crust we could decide to sample the
height of the ground, with respect to the sea level, at different locations.
We will refer to three exemplary cases: a plain flat region, a hill country and a
rocky mountain. Once we map each region, we are left with three large data sets,
and, from each of them, we are able to compute the mean ground altitude and its
standard deviation.
If we look at the dataset related to the flat plain territory, we will find out that
the altitude does not vary significantly in space, and the deviation of each sample
is almost negligible (perhaps in the orders meters). This example is representative
of a very large correlation length; if the altitude at one point is 100 m above the sea
level, then the altitude of each point in the plain must be similar (otherwise it would
not be a plain).
We leave the plain, and we travel to a different location, where the land is
characterised by hills, and we repeat the very same procedure. We found out that
the altitude of the ground can vary quite significantly (even a few hundred meters)
on a very short distance (kilometers). So any measurement will give us an idea of
what is the altitude of the portion of the ground around us. The knowledge about the
altitude of the ground at one point tells us a lot about the altitude of the terrain in
the close proximity of such point, but that will be a meaningless information if we
are interested in the height of a point a few kilometers away. Correlation length is
thus smaller than in the previous case, as the terrain is now more irregular, though
the variation is still smooth.
We now move to the rocky mountains, and, once again, we repeat the measurements. In this case we will face edges, ridges, walls, cracks or cliffs. The
ground altitude can therefore change abruptly, by hundreds of meters on a very short
distance. In this rocky region, knowing the height of the crust at one point doesn’t
really tell us anything about the altitude of the ground around us. For instance, this is
what happens when we are close to the edge of a cliff. The correlation length is then
very small, and, to some approximation, we could assume that our measurements
are uncorrelated.
Therefore, a very short correlation length implies that the information collected
at one point (the altitude of a specific point over the Earth surface) doesn’t give
any information about the surrounding landscape. On the contrary, a very large
correlation length allows us to have a wider perspective on the territory.
Note that the correlation length is not a property of the problem, but it is a
parameter related to our knowledge. In this particular example, it represents the
information that we have regarding the sample set, i.e., the type of territory from
which the samples were taken. To make this point clear, one altitude sample would
be sufficient to characterise an entire plain region, but, to be able to do that, we first
need to know that the sample was taken in a flat territory. Therefore, if we were just
J. F. Reis et al.
when the correlation function is relative to different random variables, it is usually
referred to as cross-correlation function.
A classical example is a surface that is not perfectly flat, for instance, the Earth
crust. We could look at this as a stochastic field where the height varies according to
the terrestrial coordinates. To characterise the crust we could decide to sample the
height of the ground, with respect to the sea level, at different locations.
We will refer to three exemplary cases: a plain flat region, a hill country and a
rocky mountain. Once we map each region, we are left with three large data sets,
and, from each of them, we are able to compute the mean ground altitude and its
standard deviation.
If we look at the dataset related to the flat plain territory, we will find out that
the altitude does not vary significantly in space, and the deviation of each sample
is almost negligible (perhaps in the orders meters). This example is representative
of a very large correlation length; if the altitude at one point is 100 m above the sea
level, then the altitude of each point in the plain must be similar (otherwise it would
not be a plain).
We leave the plain, and we travel to a different location, where the land is
characterised by hills, and we repeat the very same procedure. We found out that
the altitude of the ground can vary quite significantly (even a few hundred meters)
on a very short distance (kilometers). So any measurement will give us an idea of
what is the altitude of the portion of the ground around us. The knowledge about the
altitude of the ground at one point tells us a lot about the altitude of the terrain in
the close proximity of such point, but that will be a meaningless information if we
are interested in the height of a point a few kilometers away. Correlation length is
thus smaller than in the previous case, as the terrain is now more irregular, though
the variation is still smooth.
We now move to the rocky mountains, and, once again, we repeat the measurements. In this case we will face edges, ridges, walls, cracks or cliffs. The
ground altitude can therefore change abruptly, by hundreds of meters on a very short
distance. In this rocky region, knowing the height of the crust at one point doesn’t
really tell us anything about the altitude of the ground around us. For instance, this is
what happens when we are close to the edge of a cliff. The correlation length is then
very small, and, to some approximation, we could assume that our measurements
are uncorrelated.
Therefore, a very short correlation length implies that the information collected
at one point (the altitude of a specific point over the Earth surface) doesn’t give
any information about the surrounding landscape. On the contrary, a very large
correlation length allows us to have a wider perspective on the territory.
Note that the correlation length is not a property of the problem, but it is a
parameter related to our knowledge. In this particular example, it represents the
information that we have regarding the sample set, i.e., the type of territory from
which the samples were taken. To make this point clear, one altitude sample would
be sufficient to characterise an entire plain region, but, to be able to do that, we first
need to know that the sample was taken in a flat territory. Therefore, if we were just