# Find the Volcano

## Previous Eruption

This is just so that you can see what uncorrupted data might look like. Here was the ground temperature just before an eruption that occured at x=66, y=40:

The functional form of the measured temperature $\displaystyle T$ is known to be a 2-dimensional Gaussian, centered on the predicted point of eruption $\displaystyle (x_0,y_0)$ ,

$\displaystyle T = T_0 + (T_1-T_0)\exp\left(-\frac{(x-x_0)^2+(y-y_0)^2}{2\lambda^2}\right)$

with unknown parameters $\displaystyle T_0, T_1, x_0, y_0, \lambda$ that vary with each eruption.

But we only get to measure the temperature around the perimeter of the fenced area (top, bottom, left, right). Here is what its actual values were (blue=top, green=bottom, red=left, black=right):

Worse yet, there is a lot of measurement noise, in fact, $\displaystyle \sigma(T) = 30$ . So these were the actual measurements of the previous eruption (blue=top, green=bottom, red=left, black=right):

## The Coming New Eruption

Uh-oh, it's about to erupt again, and we need to save the poor residents (but only in the correct village). Where is it going to erupt? The data (blue=top, green=bottom, red=left, black=right) is:

Or, numerically,

topdata =

        0   48.6345
10.0000   93.0313
20.0000  133.1038
30.0000  147.1216
40.0000  147.6922
50.0000  163.8339
60.0000  168.1475
70.0000  163.4634
80.0000  150.7994
90.0000  141.3370
100.0000  116.9523


botdata =

        0   54.9420
10.0000   36.9229
20.0000   70.5617
30.0000   30.5953
40.0000   34.5474
50.0000   47.0488
60.0000   11.4761
70.0000   37.6386
80.0000   20.1398
90.0000   22.9232
100.0000   29.1138


leftdata =

        0   54.9420
10.0000   -2.9088
20.0000   39.9287
30.0000   31.6726
40.0000   24.5548
50.0000  105.5842
60.0000  130.5641
70.0000  116.5640
80.0000   24.0367
90.0000   50.8359
100.0000   48.6345


rightdata =

        0   29.1138
10.0000   38.6237
20.0000   66.5028
30.0000   45.9631
40.0000   89.7711
50.0000   89.2857
60.0000  106.7689
70.0000   53.4243
80.0000  126.1221
90.0000  104.2912
100.0000  116.9523


Or, here is a link to a file with all the data as (x,y,T) triplets:

Or, the data repeated here,

        0  100.0000   48.6345
10.0000  100.0000   93.0313
20.0000  100.0000  133.1038
30.0000  100.0000  147.1216
40.0000  100.0000  147.6922
50.0000  100.0000  163.8339
60.0000  100.0000  168.1475
70.0000  100.0000  163.4634
80.0000  100.0000  150.7994
90.0000  100.0000  141.3370
100.0000  100.0000  116.9523
0         0   54.9420
10.0000         0   36.9229
20.0000         0   70.5617
30.0000         0   30.5953
40.0000         0   34.5474
50.0000         0   47.0488
60.0000         0   11.4761
70.0000         0   37.6386
80.0000         0   20.1398
90.0000         0   22.9232
100.0000         0   29.1138
0         0   54.9420
0   10.0000   -2.9088
0   20.0000   39.9287
0   30.0000   31.6726
0   40.0000   24.5548
0   50.0000  105.5842
0   60.0000  130.5641
0   70.0000  116.5640
0   80.0000   24.0367
0   90.0000   50.8359
0  100.0000   48.6345
100.0000         0   29.1138
100.0000   10.0000   38.6237
100.0000   20.0000   66.5028
100.0000   30.0000   45.9631
100.0000   40.0000   89.7711
100.0000   50.0000   89.2857
100.0000   60.0000  106.7689
100.0000   70.0000   53.4243
100.0000   80.0000  126.1221
100.0000   90.0000  104.2912
100.0000  100.0000  116.9523