Earth’s
Interior Structure  Seismic Travel Times in a Constant Velocity Sphere
(L. BraileÓ, December, 2000)
http://web.ics.purdue.edu/~braile


Educational Objective: Provide experience with graphing, calculations and graphical analysis. Infer the structure of the Earth’s interior and gain experience with the methods used to study Earth structure. Optionally, provide an opportunity to make calculations using a calculator from an equation, or to write a computer program to perform seismic travel time calculations.
Possible Preparatory Lessons/Activities:
Seismic wave propagation
Elasticity
Materials:
2 11” x 17” sheets of photocopy paper
metric ruler
calculator
pencil or pen
40 cm long piece of string
protractor
scotch tape
copies of figures
Procedure:
1. Draw a cross section of half of the spherical Earth using a scale of 1:25,000,000 on the two sheets of paper taped together along the long side (Figure 1). Use the string with a loop at one end to draw the semicircle arc with a radius of 25.5 cm to represent the Earth’s surface in the cross section diagram. Alternatively, a template (Figure 2) is provided to make the halfsphere model. Copy the template (one half of figure at a time onto an 11x17 piece of paper using 200% enlargement. The two halves can then be taped together.
2. Draw raypaths associated with geocentric angles (a measure of distance from a source [earthquake] to a receiver [seismograph station]) of 40°, 80°, 120°, 150° and 180°. Use the protractor to measure the angles. One angle and raypath is shown in Figure 1 as an example.
3. Measure the distance (in cm, using the metric ruler) from source to receiver along the raypaths for each of the angles and write the result in Table 1. Convert these measurements to units of km for the real Earth by multiplying by 250 (accounting for our scale factor of 1 cm = 250 km or 1:25,000,000 in the scale model diagram) and write these numbers in the third column of the table. Determine the travel times for each raypath assuming a constant velocity of 11 km/s (divide length of raypath in km by velocity in km/s). Convert these times to minutes by dividing by 60. (These calculations could have been made more precisely by deriving a formula for the length of a chord of a circle using the geometry illustrated in Figure 3. However, the graphical solution is adequate to illustrate the concept of travel times through the Earth and is more concrete. We will use the formula which follows from the diagram in Figure 3 in the calculation of travel times using a computer program. You could also try other constant velocities, such as 10 km/s or 12 km/s, to see if the calculated travel times better fit the observed data.)
Table 1.
Calculation of travel times for raypaths in a spherical, constant
velocity Earth at various distances (geocentric angle).

Angle D (degrees) 
Length of Raypath measured on diagram (cm) 
raypath length (km) 
Time (s) 
Time (min) 


40 






80 






120 






150 






180 





4. Plot the calculated travel times at the five distances (place a dot at the distance and location given by the data in columns 1 and 5 of Table 1 on the graph shown in Figure 4. Draw a smooth curved line through the calculated travel times (from the graphical calculations, Table 1) beginning at zero distance (Delta) and zero time. Observed travel times for the compressional wave are plotted as solid lines. These lines correspond to three different phases (different paths through the Earth), but for now are simply used for a comparison with the calculated times. Do the observed and calculated times differ significantly? What does this imply about our assumption that the Earth’s interior might consist of a constant velocity of 11 km/s?
5. We can calculate the travel times in a homogeneous Earth more precisely (and thus compare observed and calculated travel times at all distances) using an equation corresponding to the geometry shown in Figure 3.
From the shaded triangle in the diagram in Figure 3, we have
_{} (1)
or,
_{} (2)
(R and S are in kilometers and D is in degrees)
from which the length of the raypath can be calculated for any distance (geocentric angle), D. The travel times for a constant velocity Earth can then be calculated by dividing by the assumed velocity. In order to compare the observed travel times with other possible raypaths, we also calculate the travel times for the path along the surface of the homogeneous Earth. For this path, the distance is just D in degrees, or D multiplied by 111.19 km/degree to obtain the distance in km, which can be used to determine travel time by dividing by velocity. A computer program (Table 2) to calculate and plot the travel times was prepared using MATLAB* . The calculations could be made by writing a program in any computer language. The results of the calculations from the program are illustrated in a travel time graph (Figure 5) and in the tabular output of the program (Table 3).
Questions:
Examine the observed and calculated travel times shown on the graph (Figure 5) and answer the following questions.
1. Do the observed travel times differ significantly (discuss what differ significantly means in this analysis; compare differences with the expected experimental error in the travel times) from the calculated times for the surface and direct paths? (The time of arrival of seismic waves at a seismograph station can usually be determined with an accuracy of better than 1 second.)
2. If the Earth is homogeneous but we selected a different velocity than 11 km/s, could we obtain an adequate fit between the observed and calculated travel times? Why or why not? (Note that we could easily try many different velocities using the graphical solution, calculations using a calculator and the formula given above for determining the raypath length, S, or by using the computer program.)
3. Now that we know that the Earth’s interior cannot consist of a constant velocity, we use the travel time information to make additional inferences about the velocity structure of the interior. Examine the raypaths that you constructed using the graphical analysis. By comparing the distance ranges where the observed and calculated travel times (for the P and P diffracted phases) differ significantly in Figure 5 with the “depth penetration” of the corresponding raypaths in the handplotted graph, can you infer the relative velocity (compared to our 11 km/s assumption) of the actual Earth’s interior for the shallow (about 0 to 1500 km depth) and deeper (greater than 1500 km depth) regions of the Earth? In other words, is the assumed constant velocity of 11 km/s too high or too low for the shallow or deep Earth?
4. What causes the travel time curves to “flatten” so strongly in the distance range of about 160 to 180 degrees?
5. From the simple analysis that we have performed, we can infer that Earth’s interior has a velocity structure in which the velocity varies with depth. How could we determine if the velocity also varies laterally (with location)?
Additional Information:
1. Examine Chapter 6 of Bolt (1993) to learn more about the actual velocity structure of the Earth’s interior and how this information has been obtained, both historically and utilizing modern and much more extensive observations and techniques.
2. The raypaths for compressional waves in a more accurate Earth model are shown in Figure 6.
Possible FollowUp Lessons/Activities:
Earth’s Interior Structure (Scale model “slice” through the Earth)
Reflection and Refraction
SEISMIC WAVES Program
Earthquake location techniques
References:
Bolt,
B.A., Earthquakes and Geological
Discovery, Freeman,
Richter,
C.F., Elementary Seismology, Freeman,
Figure 1. Cross section through half of a homogeneous,
constant seismic velocity, spherical Earth.
Sample seismic raypath from a surface source to a receiver at a distance
of D degrees geocentric angle (one degree corresponds to 111.19 km,
measured along the surface, for a 6371 km sphere) is shown by the heavy
line. For a graphical construction of
this model and measuring the distance along raypaths corresponding to various
geocentric angles (distances), the half circle can be drawn at a scale of
1:25,000,000 on two sheets of 11x17 inch paper taped together.

Figure 2. Template for making a 1:25 million scale
model of a cross section through the Earth.
Photocopy at 200% enlargement (or set the enlargement to the percentage
that will make the radius of the final halfcircle 25.5 cm) onto 11 x 17 inch
paper; then tape together to make a half circle with radius of 25.5 cm. Photocopy the top one quarter of the circle
on one sheet of paper (widthwise) and the lower one quarter of the circle on
another. Be sure that the tic mark
identifying the center of the half circle is visible on both sheets of
paper. Trim to make two quarter circles
and tape together.
Figure 3. Diagram illustrating the calculation of the
length of a chord, S, corresponding to a geocentric angle of D degrees.
Figure 4. Graph for plotting calculated travel times.
Figure 5.
Travel times of observed and calculated P waves.
Figure 6. Raypaths and wavefronts for selected primary
(compressional) wave phases which travel through the Earth. The travel times (in minutes) along the
raypaths and the corresponding wavefronts (short dashed lines; lines or surface
of equal travel time) are given by the small numbers adjacent to the
wavefronts. The raypaths are
perpendicular to the wavefronts and represent the direction that a specific
point on the wavefront is propagating.
The raypaths in this real Earth model are curved because the seismic
wave velocity varies with depth. Note
the strong refraction (bending) of the raypaths and wavefronts caused by the
velocity change across the coremantle boundary. The primary wave types (phases) illustrated
in this diagram are:
P Raypaths
for waves which travel through the mantle with a reltively direct path; 0°103° distance
range.
Pdiffracted Raypaths for
waves which travel through the mantle and are diffracted at the coremantle
boundary by the reduced outer core velocity; 103°150° distance range.
PKP Raypaths for
waves which travel through the mantle, are strongly refracted at the
coremantle boundary and travel through the outer core; 110°187° distance
range.
PKIKP Raypaths for
waves which travel through the mantle, the outer core and the inner core; 110°180° distance
range.
PKiKP Raypaths for
waves that are reflected from the inner core.
In more recent models of the Earth's interior, the PKiKP arrivals are
observed for distances less than about 120°.
Table 2. Program (written in MATLAB code) to calculate and plot seismic travel times in a spherical, constant velocity Earth. Observed Pwave travel time data for the real Earth are also included in the plot. Comments begin with a percent sign (%). The plot produced by the program is given in Figure 5. Calculated data from the program are shown in Table 3.
% Program to calculate and
plot seismic travel times
% in a spherical,
homogeneous velocity Earth.
%
% Program written in Matlab
code.
% (L. Braile,
%
%
R
= 6371; %
radius of Earth (km)
V
= 11; %
assumed constant seismic wave velocity (km/s)
%
%
% Calculate travel times
along Earth's surface
%
delta
= 0:5:180; %
distance in degrees, geocentric angle,
% %
from 0 to 180 in steps of 5 degrees;
% % on the surface, one degree = 111.19 km
delrad
= delta/57.2958; %
convert degrees to radians
S
= R*delrad; % calculate distance in km along surface
Ts
= S/V; %
travel time in seconds
Tm
= Ts/60; %
travel time in minutes
%
%
% Compute travel times through the Earth's interior (body waves)
%
S
= 2*R*sin(0.5*delrad);% calculate length of chords
(raypaths) in km
Ti
= S/V; % travel time in seconds
Ti
= Ti/60; % travel time in minutes
%
%
% Input observed travel times (min., first arrivals) for
comparison
% P and P(diffracted) arrivals.
(0 deg. to 105 deg for P, 105 deg
% to 150 deg. for P(diffracted).
% maintain equal size of all arrays and not be plotted on figure.
%
To
= [0 1.302 2.467 3.583 4.617 5.447 6.208 6.935 7.635 8.315];
To
= [To 8.967 9.590 10.178 10.733 11.257 11.750 12.212 12.642];
To
= [To 13.045 13.428 13.807 14.177 14.570 14.964 15.357 15.750];
To
= [To 16.144 16.537 16.930 17.324 17.717 NaN NaN NaN NaN NaN
%
% PKIKP observed arrivals
(110 deg. to 180 deg.)
%
To2
= [
To2
= [To2 19.508 19.653 19.790 19.908 20.013 20.097 20.153];
To2
= [To2 20.192 20.203];
%
%
% Print matrix of values
% Table 2. (continued)
format
compact
'Table 3. Travel times for
a homogeneous Earth, Velocity (km/s) ='
V
Heading=[' Delta T surface
T direct T observed T(PKIKP) observed',
' (deg)
(min) (min) (min) (min) ']
Data=[delta'
Tm' Ti' To' To2']
%
%
% Plot travel time curves
for Observed and Calculated arrivals
%
plot(delta,To,'r','linewidth',
2) % Observed P and P(diffracted)
hold
on % Keep all lines on
plot
plot(delta,To2,'r','linewidth',
2)
%
Observed PKIKP
plot(delta,Tm,'g','linewidth',
2) % Calculated, surface path
plot(delta,Ti,'b','linewidth',
2)
%
Calculated, direct path
%
%
% Label plot
%
set(gca,'fontsize',18,'fontweight',
'bold','linewidth',1)
text(75,1.9,'Delta (degrees)','fontsize',18,'fontweight','bold')
text(11,9,'Travel Time (minutes)','rotation',90,'fontsize',18,...
'fontweight','bold')
text(70,6,'Seismic travel times for a constant velocity,',...
'fontsize',16)
text(70,4.5,'spherical Earth:
Blue line = Direct path,',...
'fontsize',16)
text(70,3,'Green line = Surface path. Red line = Observed.',...
'fontsize',16)
vel=num2str(V);
text(70,1.5,'Assumed velocity (km/s) =','fontsize',16)
text(131,1.5,vel,'fontsize',16)
text(35,7.9,'P','fontsize',18,'fontweight','bold')
text(118,14.5,'P(diffracted)','fontsize',18,'fontweight','bold')
text(140,20.5,'PKIKP','fontsize',18,'fontweight','bold')
text(64,21,'Surface path, calculated','fontsize',16,'fontweight','bold')
text(60,9,'Direct path, calculated','fontsize',16,'fontweight','bold')
text(5,24,'Seismic Travel times in a constant velocity Earth model.',...
'fontsize',16)
text(5,23,' compared with
observed travel times.',...
'fontsize',16)
%
% Set axis size
axis([0
180 0 25]);
grid
on % Plot grid lines on graph
hold
off
% End of program
%
Table 3. Travel times for a constant velocity Earth (assumed velocity = 11 km/s). Output from the computer program given in Table 2.
Delta T surface
T direct T observed T(PKIKP) observed
(deg) (min)
(min) (min) (min)
0 0 0 0
5.0000 0.8424
0.8421 1.3020
10.0000 1.6848
1.6826 2.4670
15.0000 2.5272
2.5199 3.5830
20.0000 3.3695
3.3525 4.6170
25.0000 4.2119
4.1786 5.4470
30.0000 5.0543
4.9968 6.2080
35.0000 5.8967
5.8054 6.9350
40.0000 6.7391
6.6031 7.6350
45.0000 7.5815
7.3881 8.3150
50.0000 8.4239
8.1591 8.9670
55.0000 9.2662
8.9145 9.5900
60.0000 10.1086
9.6530 10.1780
65.0000 10.9510
10.3731 10.7330
70.0000 11.7934
11.0735 11.2570
75.0000 12.6358
11.7528 11.7500
80.0000 13.4782
12.4097 12.2120
85.0000 14.3206
13.0430 12.6420
90.0000 15.1629
13.6514 13.0450
95.0000 16.0053
14.2339 13.4280
100.0000 16.8477
14.7893 13.8070
105.0000 17.6901
15.3165 14.1770
110.0000 18.5325
15.8146 14.5700 18.5530
115.0000 19.3749
16.2826 14.9640 18.7170
120.0000 20.2173
16.7195 15.3570 18.8780
125.0000 21.0596
17.1247 15.7500 19.0400
130.0000 21.9020
17.4972 16.1440 19.2000
135.0000 22.7444
17.8365 16.5370 19.3570
140.0000 23.5868
18.1418 16.9300 19.5080
145.0000 24.4292
18.4125 17.3240 19.6530
150.0000 25.2716
18.6482 17.7170 19.7900
155.0000
26.1140 18.8484
160.0000 26.9563
19.0128
165.0000 27.7987
19.1409
170.0000 28.6411
19.2326
175.0000 29.4835
19.2877
180.0000 30.3259
19.3061
ÓCopyright 2000. L. Braile. Permission granted for reproduction for noncommercial uses.
*MATLAB
is a technical computing environment for numeric computation and
visualization. It operates on Macintosh,
Windows, UNIX and other computing platforms.
MATLAB is distributed by The MathWorks, Inc.,