The complete Algorithm: Brute-force Hypocenter Estimation

As state earlier, the algorithm is a 4 step process done in real time on each and every time step.

Gridify

3.2.3.1 Step 1: Gridify

Divide the region under observation into finite number of points, at which we will test the hypothesis. The project mainly observed the area of south California, dividing the entire region in 20 x 20 Km grid, and each grid point being an interest point. A total of 28 x  28 points were taken into consideration. Note that this is a pre-evaluation procedure and needs to be done only once.

3.2.3.1 Step 2 and 3: Pick Patterns

For each time step, we collect picks from each and every sensor in past T seconds within D km of a grid point. This is done for each and every grid point. The project experimented with T = 10 seconds, since 10 seconds from occurrence of event is a good time frame for detecting quake. By fixing T, D can be found out by:

D=T VP wave=10 ×5=50 Kms

Since earthquake can only travel as fast as fastest P waves, and area outside 50 KM range should not have seen any shaking if earthquake occurred 10 seconds ago.

Let A(x,y) be grid point under consideration B [sensor, Observed Pick time] be a set of all picks within 50 km of A(x,y). Using TauP model we calculate arrival times for waves on all of these sensors within 50Km that picked, and constructed a set C [sensor, Theoretical Pick Time]. This procedure is done for each and every grid point A(x,y).

For a point where earthquake actually occurred 10 seconds ago, B should be very close to C. At other points, there will be large discrepancy between B and C. There B can be termed as Observed Pick Pattern (OPP) whereas C can be termed as Actual Pick Pattern (APP).

Note that we don’t allow one sensor to pick more than once in 20 seconds. Moreover, this calculation only needs to done, if and only if more than certain percentage of sensors in 50 Km radius. Otherwise, we declare that no earthquake occurred, saving memory and cost of processing.

3.2.3.1 Step 2 and 3: Error Calculation

When we plot B vs C scatter plot for a grid point near hypocenter, it is expected to look like the one below:

Note how nicely all points arrange themselves around m = 1 slope line for T =20. If this were not actual hypocenter, the cluster would be away from m = 1 slope line. Thus error can be calculated by taking mean of all deviation from the y = x line.

Let Bi and Ci be the observed and actual pick times for ith sensor that picked in 50 km radius.

Let h be the percentage of sensor that picked. Then error can be given by:

Error (e)= f(x)={( Bi- Ci)2,  h>30% inf ,  otherwise

We calculate Error (e) only if more than h =30% of total sensors pick in past 10 seconds; otherwise we assign very high value to the Error, saving computation cost.

For further understand, this error can be plotted versus time.

Note that when grid point is farther from actual hypocenter, error increases (left figure) on even onset. On the other hand, when grid point is close to point under consideration, error decreases, since OPP and APP begin to match. This fact can be used to locate points close actual hypocenter in real time, where we find the grid points with least error in time, and observe how pattern radiates henceforth. The absolute error levels vary from event to event; it’s the inverse peak that matters!

We ran simulation for more than 30 events, and tried to estimate epicenter, while assuming the depth as given variable. However, finding depth is not alien to this model, all that is required is to run several models parallel for different depths and find the closest match. The algorithm remains the same. The screen below sums up whatever has been said in above paragraph.

In case we get multiple hits, i.e. multiple points show up at different time instants as event progresses, we simply do a circular fitting assuming each of them hypocenter one by one, and fit with least error most is likely to be the hypocenter.

We only consider points that have been caught after the point under consideration for circular fitting. Such a case arises when the propagating wave causes sensors to be affected in such a manner that it raises a wrong solution. However, since we know distance and the speed of wave, we can rule out few possibilities since earthquake can’t travel more distance than what fastest wave can.

Using the above described algorithm, we can solve for hypocenter in linear time in best case, and squared time in worst case.