(COMPUTER) You are given P-wave arrival times for two earthquakes recorded
by a 13-station seismic array. The station locations and times are listed in Table 5.2
and also given in the supplemental web material.
(a) Write a computer program that performs a grid search to find the best location
for these events. Try every point in a 100 km by 100 km array (x = 0 to
100 km, y = 0 to 100 km). At each point, compute the range to each of the
13 stations. Convert these ranges to time by assuming the velocity is 6 km/s
(this is a 2-D problem, don’t worry about depth). Compute the average sum
of the squares of the residuals to each grid point (after finding the best-fitting
origin time at the grid point; see below).
(b) For each quake, list the best-fitting location and origin time.
(c) From your answers in (b), estimate the uncertainties of the individual station
residuals (e.g., σ2 in 5.30) for each quake.
(d) For each quake, use (c) to compute χ2 at each of the grid points. What is χ2
at the best-fitting point in each case?
(e) Identify those values of χ2 that are within the 95% confidence ellipse. For
each quake, make a plot showing the station locations, the best quake location,
and the points within the 95% confidence region.
(f) Note: Don’t do a grid search for the origin time! Instead assume an origin
time of zero to start; the best-fitting origin time at each grid point will be the
average of the residuals that you calculate for that point. Then just subtract
this time from all of the residuals to obtain the final residuals at each point.