Introduction P phase arrival times

Transcription

1 Introduction...1 P phase arrival times...1 Earthquake location...1 Summary of products for the write-up:...3 Diagrammatic summary of location procedure...4 Appendix 1. DISTAZ function...8 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

2 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

3 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.

4 Appendix 1. DISTAZ function The DISTAZ function takes the latitudes and longitudes of two points on a spherical earth and computes (1) the angular distance along the great circle connecting the two points and (2) the azimuth of the great circle as measured from one of the points towards the other. These quantities are shown in Figure 1 below, where the azimuth is measured from the earthquake towards the station. STA azimuth distance EQ lines of longitude Figure 1. view of earth from space showing two locations, EQ and Station, the great circle connecting them, and longitudinal lines (dashed) going through the two locations. The azimuth of the station measured at the EQ is the angle between the longitudinal great circle through EQ and the great circle connecting the EQ and the station.

5 cos(90-lat) to north pole z 90 - LAT unit vector at the center of the earth pointing to a location on the surface at latitiude LAT and longitude LON center of earth cos(lat) cos(lon) cos (LAT) sin (LON) y to equator at 90 E LON equatorial plane x to equator at 0 E Figure 2. The cartesian coordinate system placed at the earth's center to transform a location on the surface at latitude LAT and longitude LON into coordinates x, y, and z as shown.

6 For the calculation we will locate a point on the sphere by means of a unit vector placed at the center of the earth and which aims at the point in question. Latitude and longitude are the angular components of the vector in a spherical coordinate system. We will transform these components to a Cartesian coordinate system which is defined as follows. The z axis points towards the north pole, the x axis points towards the equator at 0 longitude, and the y axis points towards the equator at 90 East longitude, as shown in Figure 2 on the preceding page. The components of a unit vector pointing to the location at latitude, LAT, and longitude, LON, are x = cos(lat) cos(lon) y = cos(lat) sin(lon) z = sin(lat) = cos (90-LAT) Now we have two such vectors, one pointing to the earthquake and one pointing to the station. They are both of unit length. In vector notation let's call these EQ and STA. Let us label the latitude and longitude of the earthquake epicenter as LATE and LONE, respectively, and the latitude and longitude of the station as LATS and LONS, respectively. Each will also have x, y and z coordinates obtained by substituting the appropriate latitude and longitudes into the equations above. In the Excel macro sheet for LOCATE2, the function called XYZ calculates the x, y and z coordinates given latitude and longitude. The angle between the two vectors is the angular distance,, between the two points on the sphere (the variable "DISTANCE" is used in the macro instead of ). This distance is obtained by taking the dot product between the two unit vectors: EQ STA = (1) (1) cos( ) or = cos -1 ( EQ STA) The azimuth is more involved. Azimuth is defined as the angle between two great circles: one is the line of longitude through the point of interest (the earthquake location in this case) and the other great circle is the one joining the earthquake location to the station. Now a great circle on a sphere defines a plane through that sphere. The angle of intersection between these two great circles is given by the angle between the normals to the corresponding planes of the great circles. Thus to calculate this angle we need to construct the vectors perpendicular to the two great circle planes. The angle between the two vectors is again simply obtained by using the dot product between the two vectors. The way to get the normal to a great circle plane is to take the vector cross product of two vectors in the plane itself. Since both the z axis and the EQ location vector are in the plane of the earthquake longitude great circle, we can take the cross product of these two to get a (non-unit) vector that is perpendicular to one of the two planes. For the other plane we can take the EQ location vector and the STA location vector which are both in the plane of the great circle connecting the earthquake and the station. The unit vector pointing in the z direction is denoted by z, and the azimuth that we are trying to get is called AZ. Thus we find EQ xz EQ xsta =EQ xz EQ xsta COS(AZ) where the x denotes vector cross product and denotes vector dot product, or

7 AZ =COS 1 EQ xz EQ xsta EQ xz EQ xsta DISTAZ takes advantage of the array functions offered by Excel to do these calculations. We then need to figure out whether the angle AZ as calculated is positive or negative (the COS -1 function doesn't do this!) which depends upon where the two points are relative to each other. We have to figure out which of the two possible ways to go along the great circle connecting the earthquake and station, the short way or the long way. We should use the shortest route ( less than 180 ) to compute the azimuth, since that is what we use for the distance. When = 180 exactly, then it doesn't make any difference. The "IF" statements in the DISTAZ function take care of all this. See if you can figure it out (or figure out more elegant code)! This calculation, a central one for earthquake location, also turns out to be a central one both for geomagnetism and plate tectonics, (the same spherical earth!) so it's worth learning how it works and how to use it.

8 user defined functions in 'locmacros' macro sheet in workbook LOCATE JB tt lookup and interpolation routines: JBTT =ARGUMENT("dist") =ARGUMENT("depth") =VLOOKUP(dist,[LOCATE4]JBtables!table,DEPINT(depth)) =VLOOKUP((dist+1),[LOCATE4]JBtables!table,DEPINT(depth)) =VLOOKUP(dist,[LOCATE4]JBtables!table,(1+DEPINT(depth))) =VLOOKUP((dist+1),[LOCATE4]JBtables!table,(1+DEPINT(depth))) =A5+((A6-A5)*(dist-INT(dist))) =A7+((A8-A7)*(dist-INT(dist))) =A9+((A10-A9)*(HDEP(depth)-DEPINT(depth))) =RETURN(A11) A B C Table in unit degrees, and depth interpolation calculated in fraction of column number these extract two distance and two depth values needed for interpolation these get depth as column integer (DEPINT) and column fractional number (HDEP) DEPINT =ARGUMENT("dep") =(((dep-33)/( ))*100)+3 =INT(C7) =RETURN(C8) HDEP =ARGUMENT("dept") =(((dept-33)/( ))*100)+3 =RETURN(C14)

9 user defined functions in 'locmacros' macro sheet in workbook LOCATE E F G H rpd DISTAZ =PI()/180 =RESULT(64) z is towards north pole =ARGUMENT("ELAT",1) XYZ =ARGUMENT("ELON",1) =RESULT(64) =ARGUMENT("SLAT",1) =ARGUMENT("lat") =ARGUMENT("SLON",1) =ARGUMENT("lon") =XYZ(ELAT,ELON) =COS(lat*rpd)*COS(lon*rpd) =XYZ(ELAT,ELON) EQ =COS(lat*rpd)*SIN(lon*rpd) =XYZ(ELAT,ELON) =COS((PI()/2)-(lat*rpd)) =XYZ(SLAT,SLON) =RETURN(E8:E10) =XYZ(SLAT,SLON) STA =XYZ(SLAT,SLON) =SUMPRODUCT(EQ,STA) =ACOS(G13)*(1/rpd) DISTANCE in degrees =IF((SLON-ELON)=0) this part tests whether =IF(SLAT>ELAT) station is due north or AZ=0 south of epicenter, =ELSE.IF(SLAT<ELAT) and avoids having to AZ=180 go thru rest of calculations =ELSE.IF(ABS(SLON-ELON)=180) =IF(SLAT>(-ELAT)) AZ=0 =ELSE.IF(SLAT<(-ELAT)) AZ=180 =CROSPROD(EQ,{0;0;1}) =CROSPROD(EQ,{0;0;1}) =CROSPROD(EQ,{0;0;1}) =CROSPROD(EQ,STA) =CROSPROD(EQ,STA) =CROSPROD(EQ,STA) =DOTPROD(G28:G30,G31:G33) =ACOS(G34)/rpd az eq to sta =IF(ELON>=0) =IF(SLON>=0) =IF(SLON>ELON) AZ=G35 =ELSE.IF(SLON<ELON) AZ=-G35 =IF(SLON<0) =IF(SLON<(ELON-180)) AZ=G35 =ELSE.IF(SLON>(ELON-180)) AZ=-G35

10 user defined functions in 'locmacros' macro sheet in workbook LOCATE E F G H =IF(ELON<0) =IF(SLON<0) =IF(SLON>ELON) AZ=G35 =ELSE.IF(SLON<ELON) AZ=-G35 =IF(SLON>=0) =IF(SLON<(ELON+180)) AZ=G35 =ELSE.IF(SLON>(ELON+180)) AZ=-G35 =DISTANCE =AZ =RETURN(TRANSPOSE(G68:G69))

11 user defined functions in 'locmacros' macro sheet in workbook LOCATE I CROSPROD =RESULT(64) =ARGUMENT("A",64) =ARGUMENT("B",64) =(INDEX(A,2)*INDEX(B,3))-(INDEX(A,3)*INDEX(B,2)) =(INDEX(A,3)*INDEX(B,1))-(INDEX(A,1)*INDEX(B,3)) =(INDEX(A,1)*INDEX(B,2))-(INDEX(A,2)*INDEX(B,1)) =RETURN(I5:I7) DOTPROD =RESULT(1) =ARGUMENT("AA",64) =ARGUMENT("BB",64) =SQRT(SUMPRODUCT(AA,AA)) =SQRT(SUMPRODUCT(BB,BB)) =SUMPRODUCT(AA,BB)/(I16*I17) =RETURN(I18) all this to figure whether the azimuth is positive or negative!