GS 388 Lab 2
Travel Times and Earthquake Locations
Introduction........................................................................................1
P phase arrival times..............................................................................1
Earthquake location ...............................................................................1
Summary of products for the write-up:.........................................................3
Appendix 1. Diagrammatic summary of location procedure (link to eq_loc.ppt)
Appendix 1. DISTAZ function (link to DISTAZ_function.pdf)
Appendix 2: Excel Macros in LOCATE (macro sheet LOCMACROS in workbook.......12
Introduction
The purpose of this exercise is to understand how travel times are used to locate
earthquakes. You will read seismogram records from the digital Global Seismographic Network
(GSN) to determine P wave arrival times, and then to use these times to locate the earthquake with
an Excel spreadsheet set up to do the locations. Although you do not have to create the
spreadsheet, it is important that you understand what is going on in it. If you do, you will learn
how earthquakes are located and a lot about how to use Excel.
P phase arrival times
Digital records from a set of seismological stations recording an earthquake on November
15, 1997 are provided. The records show three components of motion from the B H* or broadband high-gain channels which have a higher frequency response than the L H* channels you used
in Lab 1. This higher frequency response displays the P waves with more fidelity and exhibits them
as the sharp, pulse-like signals that they really are. A more precise determination of arrival times
can be made with these components.
In contrast to Lab 1, however, this collection of seismograms has horizontal components
oriented along N-S and E-W axes instead of the convenient "R" and "T" components. R and T are
actually computed from the EW and NS components using the station-to-source azimuth.
Obviously, no station recording earthquakes from all over the world can specify a single set of R
and T directions - those directions are specific to a particular station-earthquake pair. So, the
horizontal seismographs are oriented to record N-S and E-W components of ground motion.
In this exercise you are going to locate the earthquake using determinations of the arrival
times of P waves recorded by the GSN stations. You will be supplied with printouts of the
seismograms. Tabulate the data on a sheet of paper as you read the records, and carefully list the
following:
the station identifier (three or four letter code)
P-wave arrival time: hour, minute and second (pick to a tenth of a second); use the scale on the
records to measure time with a good ruler to measure between the marks. The 0 on the time
scale corresponds to a particular GMT (Greenwich Mean Time) that is listed on the record
with the station ID code. Thus you need to add the time measured on the record to the listed
time to get the arrival time. Sometimes the alpha-numeric printout is obscured by the trace
wiggles, but one of the components will usually be clear enough get it right.
For about 6 or so stations with a good global distribution (i.e., not all in the same region),
tabulate the amplitudes of the first and/or second half cyles of the P wave in order to determine
a rough location. For a given station these amplitudes will give the directions from the station
to the source. The intersection of the great circles from at least three well distributed stations
can give an approximate location (using a globe). This can be taken as your starting location
that you will need below.
After reading the records, enter the arrival time data in the Excel workbook called
STATIONS. This workbook includes the main entry spreadsheet, 'station selection', a table of
GS 388 Lab 2
Travel Times and Earthquake Locations
station locations called (not suprisingly) 'station locations', and a macro sheet called 'stamacros'.
Macro sheets in Excel are programs that you call from the regular spreadsheets (or from other
macro sheets). The programs are written in the special Excel macro language. You will work in the
spreadsheet 'station selection'. The first column will be the three letter station code, the second the
station latitude, the third column the station longitude, and the next three columns the arrival time in
terms of hour, minute, and second. The latitude and longitude columns have user-defined
functions which will take the three letter code (in capital letters please) in the first column and
search the table in 'station locations' of worldwide seismograph station locations to find the latitude
and longitude of the particular station. This is a nice example of how Excel can work with "lookup
tables".
Earthquake location
Now copy the data and "Paste Special..." as "values" into the second Excel workbook
called LOCATE2. It is important to paste “as values” and not simply just “paste”; otherwise the
things you put into LOCATE2 continue to reference the STATIONS workbook, which is a
nuisance. You can close the workbook STATIONS after this is done succesfully. LOCATE2 is the
main entry and analysis workbook with several specially defined functions.
Although I am not asking you to construct the formulas in LOCATE2, I do want you to
understand what is going on, and this is an excellent chance to see how a reasonably involved
Excel calculation system really works before you have to make one yourself. So, it's worth
delving into this one to figure out what is going on. The macro sheet called 'locmacros' shows the
functions with some comments (see Appendix 1). The key one, DISTAZ, calculates distances and
azimuths, and is discussed in Appendix 2. Believe it or not, we use this one twice more in the
course, in paleomagnetic and in plate tectonic calculations, so it is quite useful and worth getting to
know.
The travel time table, 'JBtables', is taken directly from the Jeffreys-Bullen tables, the
venerable and still used standard developed before WW II by the English geophysicist Harold
Jeffreys and the New Zealand seismologist (and student of Jeffreys) Keith Bullen. The J-B table
lists travel times as a function of distance in degrees (vertical, rows) and depth (horizontal,
columns). A user-defined function looks up and interpolates the J-B tables to yield a travel time as
output, with depth and distance as inputs.
The next entry required for the procedure is a test location, with origin time, latitude,
longitude and depth specified. This location will be changed until a good solution is found. With
the first motion data you obtained above you can "triangulate" on one of the National Geographic
globes to give a rough location. Then, use the travel time tables to find the travel time for a P phase
at one or several stations, subtract that travel time from your arrival time to get an estimate for the
origin time. The alternative method for choosing a trial starting location is simply to use the
location and arrival time of the station closest to the earthquake.
Now that you have entered the first test location, the spreadsheet goes about (1) calculating
distances and azimuths (earthquake-to-station) for each station, and from the distances and test
depth looks up values of travel times in the tables, and calculates observed travel times by
subtracting your arrival times from the test origin time. With both observed and calculated travel
times the spreadsheet computes an "observed minus calculated" time for each station. This is the
travel time residual, which is the main quantity of interest for earthquake locations. A cell near the
trial locations gives the square root of the sum of the square of the residuals (RMS), a useful
measure of how well the travel times together with the trial location fit the travel time tables. The
best location will minimize the RMS. Also included is a plot of residual (observed minus
GS 388 Lab 2
Travel Times and Earthquake Locations
computed, or O-C) versus azimuth, called 'OC vrs az', as part of the workbook. This plot is quite
useful in seeing how to move the test location to get a better location.
The classical method of location, done by the major international services (International
Seismological Centre or ISC in Britain and the USGS Preliminary Determination of Epicenters in
the US) is to perform a least square solution. This solution minimizes the sum of the squares of the
residuals and is considered the best estimate of the location. Excel has a rather elegant method to do
this, called "Solver", which is based on linear programming techniques. You can try this if you
want to, after reading the manual to see how it works and what it is, but do the trial and error
method described below first.
I want you first to change the location parameters to see what happens to the residual
versus azimuth relationship, and then to use this information to help find a solution by trial and
error. Initially fix the depth at 33 km as is often done in practice. First, if all the points are
systematically above or below zero as a function of azimuth, you can remove this bias simply by
changing the origin time. Next, use the plot of residual versus azimuth to show you what direction
to move the epicenter horizontally (in map view). The adjustment of epicenter and origin time can
be done more or less independently. If the epicenter is mislocated in a certain direction, the
residuals will have a characteristic sinusoidal variation with azimuth. Can you figure out what
direction to move the epicenter? By thoughtfully using trial and error with the 'OC vrs az' plot you
should be able to get a location which produces an RMS near a second.
When you have a good epicenter and origin time, you can then experiment with changing
the depth. Can you compensate for this by changing the origin time? What does this say about how
well depth is determined? How deep can you make it and still compensate by simply changing the
origin time? The plot of residual versus distance ('OC vrs dist') is useful for nailing the depth and
assessing its error.
Summary of products for the write-up:
1. original P readings;
2. calculations for a preliminary location and origin time;
3. copy of 'location' spreadsheet in theLOCATE workbook with your best location;
4. copy of 'OC vs az' plot from LOCATE for that best location;
5. copy of 'OC vrs dist' plot from LOCATE for the best location; and
6. discussion of how you proceeded, what happened and why.
user defined functions in 'locmacros' macro sheet in workbook LOCATE
A
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B
JB tt lookup and interpolation routines:
JBTT
=ARGUMENT("dist")
=ARGUMENT("depth")
these extract two
=VLOOKUP(dist,[LOCATE4]JBtables!table,DEPINT(depth))
distance and two
=VLOOKUP((dist+1),[LOCATE4]JBtables!table,DEPINT(depth))
depth values needed
=VLOOKUP(dist,[LOCATE4]JBtables!table,(1+DEPINT(depth)))
for interpolation
=VLOOKUP((dist+1),[LOCATE4]JBtables!table,(1+DEPINT(depth)))
=A5+((A6-A5)*(dist-INT(dist)))
Table in unit degrees, and
=A7+((A8-A7)*(dist-INT(dist)))
depth interpolation calculated
=A9+((A10-A9)*(HDEP(depth)-DEPINT(depth)))
in fraction of column number
=RETURN(A11)
C
these get depth as column
integer (DEPINT) and column
fractional number (HDEP)
DEPINT
=ARGUMENT("dep")
=(((dep-33)/(6371-33))*100)+3
=INT(C7)
=RETURN(C8)
HDEP
=ARGUMENT("dept")
=(((dept-33)/(6371-33))*100)+3
=RETURN(C14)