III. CHARACTER MODELING FROM SYNTHETIC SEISMOGRAMS 8 …

SYNTHETIC SEISMOGRAMS AND CHARACTER MODELING

AN AID TO THE DETERMINATION OF THE EARTH'S

STRUCTURE FROM RAYLEIGH WAVES

by

RICHARD L. CRIDER, B.S.

A THESIS

IN

GEOSGIENCES

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

Approved

Accepted

Dean of the Graduad^e" School

May, 1980

7 w

ACKNOWLEDGMENTS

I am very grateful to Professor D. H. Shurbet for

his direction of this thesis.

My appreciation is expressed to Sunmark Exploration

Company, Irving, Texas, for permitting me to use their

computer facilities, without which this study would not

have been possible.

11

CONTENTS

ACKNOWLEDGEMNTS ii

LIST OF FIGURES iv

I. INTRODUCTION 1

II. THEORY 3

III. CHARACTER MODELING FROM SYNTHETIC

SEISMOGRAMS 8

IV. CORRELATION OF SYNTHETIC SEISMOGRAMS

WITH EARTHQUAKE SEISMOGRAMS 41

V. CONCLUSIONS 76

VI. RECOMMENDATIONS 77

LIST OF REFERENCES 79

111

LIST OF FIGURES

FIGURE

1. Configuration of the basic Earth model

2. Synthetic seismograms generated for the basic model Ml and the model M2 . . .

5.

6.

3. Synthetic seismograms generated for the models M3 and M4


Synthetic seismograms generated for the models M7 and M8

Synthetic seismograms generated for the models M9 and MlO

7. Synthetic seismograms generated for the models Mil and M12


9. Theoretical dispersion data for the basic model Ml

10. Theoretical dispersion data for the model M2








PAGE

9

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

IV

FIGURE

1 8 .

1 9 .

2 0 .

2 1 .

2 2 .

2 3 .

2 4 .

2 5 .

2 6 .

2 7 .

2 8 .

PAGE

Theoretical dispersion data for the model MlO. 27

Theoretical dispersion data for the model Mil. 28

Theoretical dispersion data for the model M12. 29


Theoretical dispersion data for the model M14. 31 (•

Synthetic seismograms generated for models M15 and M16 35

Synthetic seismogram generated for model M17 . 36




Long period vertical components of seismograms for Event 1 and Event 2 43

29. Locations of the epicenters for Event 1 and Event 2 44

30. Cross correlation function for Event 1 correlated with the synthetic seismogram for basic model Ml 46

31. Cross correlation function for Event 1 correlated with the synthetic seismogram for the model M2 47




35. Cross correlation function for Event 1 correlated with the synthetic seismogram for the model Mb 51

V

FIGURE PAGE




39. Cross correlation function for Event 1 correlated with the synthetic seismogram for the model MlO 55

40. Cross correlation function for Event 1 correlated with the synthetic seismogram for the model Mil 56




44. Cross correlation function for Event 2 correlated with the synthetic seismogram for the basic model Ml 61


46. Cross correlation function for Event 2 correlated with the synthetic seismogram for the model M3 6 3

47. Cross correlation function for Event 2 correlated with the synthetic seismogram for the model M4 6 4

vi

FIGURE PAGE






53. Cross correlation function for Event 2 correlated with the synthetic seismogram for the model MlO 70

54. Cross correlation function for Event 2 correlated with the synthetic seismogram for the model Mil 71




Vll

CHAPTER I

INTRODUCTION

Direct and indirect modeling techniques are commonly

used in the determination of the earth's interior from

seismic surface wave dispersion. The most widely used

modeling technique is an indirect trial-and-error procedure

in which dispersion is computed for models in an effort to

duplicate observed dispersion. Indirect modeling has been

applied to Rayleigh and Love wave phase and group velocity

dispersion in the fundamental and higher modes (Dorman and

Ewing, 1962; Dorman et al., 1960). Theoretically exact

methods for the computation of dispersion have been pre

sented by Haskell (1953) and by Alterman and others (1959).

Direct modeling is used to determine the correspond

ing earth model from observed surface wave dispersion

(Dorman and Ewing, 1962; McEvilly, 1964; Braile and Keller,

1975; Bloch et al., 1969). Current direct modeling tech

niques place stringent requirements on data quality (Braile

and Keller, 1975) and these requirements limit the appli

cation of direct modeling.

Character modeling is a type of direct modeling which

utilizes the theoretical computations aspect of indirect

modeling to make a direct determination of earth structure

from surface wave dispersion. Character modeling is accom

plished by comparing synthetic seismograms for earth models

with earthquake seismograms. The identification of similar

character between the synthetic and true seismograms pro

vides a direct determination of the earth model.

Character modeling is an important tool in exploration

seismology. The synthesis of dispersed wave trains is also

well known (Sato, 1960; Brune et al., 1960; Aki, 1960), but

there has been no direct extension of character modeling

into earthquake seismology.

The purpose of this paper is to present the results

obtained from the application of character modeling to the

determination of a model for the earth's crust beneath

Mexico. A rapid method for the synthesis of dispersed waves

is presented. Variations in seismogram character are il

lustrated by the computation of synthetic seismograms for

a basic model and for permutations of the basic model. The

suite of synthetic seismograms produced are compared to

Rayleigh waves from two earthquakes in Mexico to determine

a crustal model.

The technique for the analysis of surface wave disper

sion presented in this paper is not intended to replace

other direct or indirect modeling tactics. This procedure

is presented as a means of enhancing the other techniques by

more effectively utilizing current data processing methods.

CHAPTER II

THEORY

The behavior of sound waves in the earth is described

by the one-dimensional time dependent wave equation

9^ y(x,t) _ 1 8^ y(x,t) ,,, 2 2~ 2 ^•^'

dx v^ 9t

where v is the velocity of propagation of the disturbance

y(x,t), x is distance, and t is time. A solution to the

wave equation is expressed as

y(x,t) = A 6 '' -' = ' (2)

where A is the amplitude, w is the angular frequency, and

k is the propagation constant or wave number.

Equation (2) represents a propagating wave with a well-

defined frequency w. However, if the frequency is to re

main well-defined, the wave train y(x,t) must be infinite

in length (Marion, 1967). For y(x,t) to be finite in

length, the wave train must be represented as the super

position of an infinite number of wave functions with

unique frequencies and wave numbers differing from each

other by some small amount Aw and Ak. The amplitude of the

function y(x,t) will then be a function of the wave number

and y(x,t) is expressed as

(x,t) = r A(k) ei''"*=-'^^'dx. (3)

The superposition of these wave functions is demon

strated by assuming that two of the functions have the form

,, / x.\ TV i(a),t-kTx) y, (x,t) = A e ' 1 1

y2(x,t) = A e^'"2*-'^2=^'

where w- = o), + Aw and k^ = k + Ak and, for simplicity,

the amplitude A is the same for each function. Summing

these two functions and retaining only the real parts of

the complex result, the total wave function is obtained.

,./ i.> 17V _„ (Aw) t-(Ak) x „ L ^ Awv. ,, ,Ak, I ,., y(x,t) = 2A cos - —j^ — cos (w+ -2-) t-(k+-2-) x (4)

Equation (4) for the resultant wave function is comprised

of two factors: the first represents an envelope, while

the second represents an average component wave.

The velocities of propagation of the envelope and com

ponent wave are determined by requiring that the arguments

of the two cosine terms, the phase, remain constant.

This requirement is expressed as

r(Aw)t - (Ak)x]_ . . ,, |_ 2 J •'^ ^^ envelope.

and dl(w+ "J") t - (k + "J") x = 0 for the component wave

Taking the derivatives we find, if we assume that w+ Aw

and w are not much different from each other, the results

(Aw)dt - (Ak)dx = 0 for the envelope,

and wdt - kdx = 0 for the component wave.

The velocities then are determined to be

V-dx dt

Aw Ak for the envelope.

= «^ TT - dx _ w. "• " 2 - dt - k for the component wave

The velocity v is known as the group velocity and the com

ponent wave velocity v^ as the phase velocity. The group

velocity is the velocity with which the energy in the wave

is transmitted.

The wave function described by equation (3) then is a

packet of waves whose velocities are a function of frequen

cy. This dependency of velocity upon frequency causes a

distortion of the wave form, a phenomenon known as

dispersion.

The displacement of the wave function at a particular

distance x, arbitrarily chosen as x = 0, is described by

the equation

y(t) = r A(w) e^'^^dw. (5)

This function can be computed by an inverse Fourier

Transform when the spectral distribution A(w) is known

(Marion, 1967).

Synthetic seismograms may be generated by approxi

mating the spectral distribution of the function y(t) by

a simple rectangle of the form

•TV / \ -i'l' (W) J

A(w) = e and A(w) = 1 (6)

for all frequencies in a band between w, and w^ (Aki, 1960).

Here (J) (w) = ^ ^ + *,- (w) (7) ^ Vw J in

where A is distance, C(w) is phase velocity, and <t> - (w) is

instrumental phase delay. The spectral distribution func

tion is then defined by the phase velocities of the frequen

cy band of interest.

Phase velocities for the frequency band used were

computed using Haskell's (1953) method. The computations

were accomplished using a FORTRAN IV computer program made

available to Texas Tech University by the University of

Texas at Dallas. The original program was modified to run

on the Sunmark Exploration Company's RDS 500 computer. The

spectral distribution was computed in equations (6) and (7)

for arbitrary distance. Since most seismograms reflect

particle velocity, the synthetic seismograms were computed

for velocity rather than displacement by taking the time

derivative of both sides of equation (6).

CHAPTER III

CHARACTER MODELING FROM SYNTHETIC SEISMOGRAMS

The basic earth model used in this study is shown in

Figure 1. The crustal section of this model is 41.3 km.

thick and is comprised of single-layer sedimentary and

granitic portions and a multi-layer basaltic portion. The

mantle section of the model is a generalized model common

to many areas of the world (Dorman et al., 1960; Brune and

Dorman, 1963) ; it approxim.ates a Gutenberg model. Vari

ations in this model for this study are restricted to the

crustal section. Additionally, synthetic seismogram

generation is limited to the Rayleigh mode since this

is the m.ost commonly recorded surface wave mode.

Synthetic seismograms for the basic model and thirteen

variations of the model are shown in Figures 2 to 8. The

theoretical phase and group velocity data for these syn

thetics are given in Figures 9 to 22. These seismograms

and data will be referred to as Ml through M14. A summary

of the model variations is given in Table I.

The synthetic seismograms were computed using a FOR

TRAN IV program written for the Raytheon RDS 500 computer.

This program accepts frequency, phase velocity, and arbi-

8

h

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5 .0

9 .0

12.0

30 .0

20 .0

35 .0

50 .0

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0 .0

V

p

^ . 9 3

6 . 1 ^

6.52

6.65

6.72

6.77

8.27

7.98

7.82

8.02

8.11

8.50

V s

2 .85

3.^9

3 .73

3.79

3.83

3.86

h.6k

hr.kS

^ .39

4 .50

^'55

^.77

p

2.62

2 .65

2 .87

2 .95

3 .0^

3 . 1 ^

3.57

3.69

3 .7^

3.79

3.80

3.69

Figure 1. Configuration of the basic Earth model. Layer thickness H is given in km, the compressional and shear velocities Vp and Vs are given in km/sec, and the density p is given in gm/cc. The dashed line indicates the boundary between the 41.3 km-thick crust and the mantle.

10

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11

fV^'

90 sec.

Figure 2. Synthetic seismograms generated for the basic model Ml and the model M2 having a thinned sedimentary layer and a thickened granitic layer. The theoretical dispersion data for these models are given in Figures 9 and 10, respectively.

12

90 sec.

Figure 3. Synthetic, seismograms generated for models M3 and M4. M3 has a thickened sedimentary layer and a thinned granitic layer. The sedimentary layer velocity was reduced in M4. The theoretical dispersion data for these models are given in Figures 11 and 12, respectively.

13

90 sec.

Figure 4. Synthetic seismograms generated for models M5 and M6. Each model has a thinned granitic layer and a thickened basaltic layer. The theoretical dispersion data for these models are given in Figures 13 and 14, respectively.

14

90 sec.

Figure 5. Synthetic seismograms generated for models M7 and M8. These models have reduced velocities in either the second basaltic layer (M7) or in the first and second basaltic layers (M8). The theoretical dispersion data for these models are given in Figures 15 and 16, respectively.

15

90 sec.

Figure 6. Synthetic seismograms generated for models M9 and MlO. M9 has a thinned crust (33.3 km). MlO has a thickened crust (45.3 km). The theoretical dispersion data for these models are given in Figures 17 and 18, respectively.

16

90 sec

Figure 7. Synthetic seismograms generated for models Mil and M12. Mil has the same velocity structure as the basic model Ml but a much thicker granitic layer. M12 has a different velocity structure, including a very low velocity in the second basaltic layer. The theoretical dispersion data for these models are given in Figures 19 and 20, respectively.

17

90 sec.

Figures. Synthetic seismograms generated for models M13 and M14. These models are identical to M12 with the exception of a thickened granitic layer in M13 and a thickened sedimentary layer in M14. The theoretical dispersion data for these models are given in Figures 21 and 22, respectively.

18

MODEL I

4.5-

4.0-

o a>

3.5

u o - J >

3.0-

H

4.3

5.0

6.0

5.0

9.0

12.0

Vp

4.93

6.14

6.52

6.65

6.72

6.77

Vs

2.85

3.49

3.73

3.79

3.83

3.86

P

2.62

2.65

2.87

2.95

3.04

3.14

PHASE VELOCITY

GROUP VELOCITY

10 20 30 40 50

PERIOD (sec)

Figure 9. Theoretical dispersion data for the basic model Ml.

19

4.5-

MOOEL 2

H

2.3 7.0

6.0

5.0

9.0

12.0

Vp

4.93 6.14

6.52

6.65

6.72

6.77

Vs

2.85 3.49

3.73

3.79

3.83

3.86

P 1 2.62 2.65

2.87

2.95

3.04

3.14

4.0-u

£

>•

h: o o - J UJ >

3.5-

PHASE VELOCITY

3.0-

GROUP VELOCITY

- r — 10 20^ 'zo 40* 50

PERIOD (sec)

Figure 10. Theoretical dispersion data for the model M2. The sedimentary layer in this model is 2 km. thinner than that of the basic model. The crustal thickness is unchanged.

20

MODEL 3

4.5

4.0-

o CO PHASE

VELOCITY

3.5-

o o -J UJ >

3.0-

GROUP VELOCITY

PERIOD (sec)

'h^^^f^/^\ Theoretical dispe: M3. The sedimentary layer in thir^^i"? ^^^^ -°^ ^^^ " odel than that of the baLc m odel ? 5.\^ " ^ ^ ^ ' ^ l ^ ^ ^ ^ ^ ^ ^ ^ r unchanged.

21

4.5

4.0

o

E 2C

O O

UJ >

3.5-

3.0

MODEL 4

H 4.3

5.0

6.0

5.0

9.0

12.0

Vp 4.13

6.14

6.52

6.65

6.72

6.77

Vs 2.38

3.49

3.73

3.79

3.83

3.86

P 2.62

2.65

2.87

2.95

3.04

3.14

PHASE VELOCITY

GROUP VELOCITY

10 20 30 40 50

PERIOD (sec)

Figure 12. Theoretical dispersion data for the model M4. The velocity in the sedimentary layer is reduced from that of the basic model.

22

4.5

4.0-

o

>•

O O _l UJ >

3.5

3.0-

MODEL 5

H

4.3

3.5

7.5

5.0

9.0

12.0

Vp 4,93

6.14

6.52

6.65

6.72

6.77

Vs 2.85

3.49

3.73

3.79

3.83

3.86

P 2.62

2.65

2.87

2.95

3.04

3.14

PHASE VELOCITY

GROUP VELOCITY

10 20 30 40 50

PERIOD (sec)

Figure 13. Theoretical dispersion data for the model M5. The first basaltic layer is thicker in this model than in the basic.model. The crustal thickness remains unchanaed.

23

4.5"

4.0-

o (A

E

>-

o o _l UJ >

3.5-

3.0-

MODEL 6

H 4.3

3.5

6.0

6.5

9.0

12.0

Vp 4.93

6.14

6.52

6.65

6.72

6.77

Vs 2.85

3.49

3.73

3.79

3.83

3.86

P 2.62

2.65

2.87

2.95

3.04

3.14

PHASE VELOCITY

GROUP VELOCITY

10 20 30 40 50

PERIOD (sec)

Figure 14. Theoretical dispersion data for the model M6. The second basaltic layer is thicker in this model than in the basic model. The crustal thickness remains unchanged.

24

4.5-

4.0-o a>

£

>-

o - I UJ >

3.5-

3.0-

MODEL 7

H

4.3

5.0

6.0

5.0

9.0

12.0

Vp

4.93

6.14

6.52

6.25

6.72

6.77

Vs

2.85

3.49

3.73

3.61

3.83

3.86

P

2.62

2.65

2.87

2.85

3.04

3.14

PHASE VELOCITY

GROUP VELOCITY

10 20 30 40 50

PERIOD (sec)

Figure 15. Theoretical dispersion data for the model M7. The second basaltic layer is replaced by a low velocity layer of the same thickness.

25

4.5-

4.0-o

MODEL 8

H

4.3

5.0

6.0

5.0

9.0

12.0

Vp

4.93

6.14

6.40

6.25

6.72

6.77

Vs

2.85

3.49

3.66

3.61

3.83

3.86

P 2.62

2.65

2.87

2.85

3.04

3.14

PHASE VELOCITY

3.5-

3 UJ >

3.0

GROUP VELOCITY

10 20 30 40 50

PERIOD (sec)

Figure 16. Theoretical dispersion data for the model M8. The velocities in the first and second basaltic layers are reduced.

26

4.5-

4.0-o (A

o o - i UJ >

3.5

3.0-

MODEL 9

H

3.3

4.0

6.0

4.0

7.0

9,0

Vp

4.93

6.14

6.40

6.25

6.72

6.95

Vs

2.85

3.49

3.66

3.61

3.83

3.96

P 2.62

2.65

2.87

2.85

3.04

3.14

PHASE VELOCITY

GROUP VELOCITY

10 20" 30 40 50

PERIOD (sec)

Figure 17. Theoret ica l d ispers ion data for the model M9. The c r u s t a l th ickness of t h i s model i s 33.3 km.

27

MODEL 10

4.5-

4.0

E

3.5"

u o UJ >

3.0-

H

3.3

4.0

6.0

4 .0

7.0

9.0

Vp

4.93

6.14

6.52

6.65

6.72

6.77

Vs

2.85

3.49

3.73

3.79

3.83

3.86

P

2.62

2.65

2.87

2.95

3.04

3.14

PHASE VELOCITY

GROUP VELOCITY

10 20 30 40 50

PERIOD (sec)

Figure 18. Theoretical dispersion data for the model MlO. The crustal thickness of this model is 45.3 km.

28

MODEL 11

4.5-

4.0-

o

O o -J UJ >

3.5-

3.0-

H

4.3

10.0

6.0

5.0

9.0

12.0

Vp

4.93

6.14

6.52

6.65

6.72

6.77

Vs

2.85

3.49

3.73

3.79

3.83

3.86

P 2.62

2.65

2.87

2.95

3.04

3.14

PHASE VELOCITY

GROUP VELOCITY

10 20 30 40 50

PERIOD (sec)

Figure 19. Theoretical dispersion data for the model Mil. The granitic layer in this model is 5 km. thicker than in the basic model. The crustal thickness is increased to 46.3 km.

29

4.5-

4.0-o

E JiC

>

O o - J UJ >

3.5-

MODEL 12

H 4.3

5.0

6.0

5.0

9.0

12.0

Vp

5.3

6.35

6.75

6.05

7.4

7.57

Vs

3.06

3.62

3.84

3.49

4.21

4.31

P 2.62

2.65

2.87

2.95

3.04

3.14

PHASE VELOCITY

3.0-

GROUP VELOCITY

lo" 20 30 40 50

PERIOD (sec)

Figure 20. Theoretical dispersion data for the model M12. Velocities in all crustal layers are greater than in the basic model.

30

4.5-

4.0-

o (A

E 2C

3.5-

o 3 UJ >

3.0-

MODEL 13

H

4.3

10.0

6.0

5.0

9.0

12.0

Vp 5.3

6.35

6.75

6.05

7.40

7.67

Vs 3.06

3.62

3.84

3.49

4.21

4.31

P 2.62

2.65

2.87

2.95

3.04

3.14

PHASE VELOCITY

GROUP VELOCITY

"l^ 20 30 40 50

PERIOD (sec)

Figure 21. Theoretical dispersion data for the model M13. This model has a thicker granitic layer and higher velocities in all crustal layers than in the basic model.

31

MODEL 14

4.5-

4.0-u

E

o o -J UJ >

H

8.3

5.0

6 .0

5.0

9.0

12.0

Vp

5.3

6.35

6.75

6.05

7.40

7.57

Vs

3.06

3.62

3.84

3.49

4.2J

4.31

P 2.62

2.65

2.87

2.95

3.04

3.14

PHASE VELOCITY

3.5-

3.0-

GROUP VELOCITY

10 20 30 40 50

PERIOD (sec)

Ml 4 Th?c I ^ Jh^^^^tical dispersion data for the model M14. This model has a thicker sedimentary layer and higher velocities in all crustal layers than in the basic model

ear

va

32

trary distance and time references for phase computations.

Particle velocity is computed as a function of frequency by

Fourier inversion of phase information computed from the

phase velocities.

The source function for the synthetic seismograms is

assumed to approximate a step function. This function is

typical of source functions obtained for South American

thquakes (Aki, 1972), and is characterized by a slowly

rying frequency spectrum as observed for earthquake seis

mograms (Goforth, 1976).

Examination of the synthetic seismograms shows that

the character of each seismogram is different. The

theoretical dispersion data reveals that little change in

the phase velocities is caused by variations in the models.

However, marked differences are observed in the group vel

ocity data, particularly for periods less than 20 seconds.

Character change may be interpreted in terms of group vel

ocity variation and, hence, variations in earth structure.

The synthetic seismograms for M3, M4, MlO, and M14

(Figures 3, 6, and 8) exhibit a ringing appearance not so

pronounced in the synthetic for the basic model Ml. The

models have a thicker sedimentary layer (M3, MlO, and M14),

a thicker crust (MlO and M14), or a lower velocity in the

sedimentary layer (M4), than included in the basic model Ml.

The thick sedimentary layer and the low velocity in the

33

sedimentary layer causes the slope of the group velocity

curve to increase for periods less than about 12 seconds.

The increased slope for the group velocity data indicates

strong dispersion for waves with periods less than about

12 seconds. This causes the ringing appearance in the

synthetic seismograms.

The remaining synthetic seismograms exhibit large

increases in the amplitude of wave groups with periods of

about 12 seconds. Relative to the basic model Ml, the

models for which these synthetics are generated have the

following variations: thick basaltic layers (M5 and M6,

illustrated in Figure 4); a thick granitic layer (M2, Mil,

and M13, illustrated in Figures 2, 7, and 8, respectively);

a low velocity layer in the crust (M7, M8, M12, and M13,

illustrated in Figures 5, 7, and 8, respectively); a thin

crust (M9, illustrated in Figure 6); increased velocities

in the crust (M12 and M13, illustrated in Figures 7 and 8).

The amplitude increase is attributed to constructive

interference between wave groups having nearly the same

period and velocity. For M2 and M5 through M9, this inter

ference arises from the increase in the length of the

inverse dispersion branch of the group velocity curve for

periods between about 18 and 10 seconds. For Mil through

M13, the interference is due to the nearly zero slope in

the group velocity curve for periods between about 18 and

10 seconds.

34

The synthetics for M2, Mil, M12, and M13 show the

greatest amplitude increase, indicating that the primary

factors contributing to this change in character are a

thickened granitic layer, increased crustal velocity, and

a low velocity layer. The synthetic for M13 shows that

the amplitude increase is most significant when faster

crustal velocities accentuate the effects of a low velocity

zone.

The amplitude increase in the synthetics at about 12

seconds wave period is similar to the earthquake phase R

(Press and Ewing, 1952). Evidence from the synthetic seis

mograms indicates that the phase R is a phenomenon of ar-

rival of wave groups with periods between 18 and 10 seconds

and nearly the same group velocity. This phenomenon of

near equal group velocity for waves of a range of periods

is due primarily to the effect of the granitic layer in the

crust.

The effect of each portion of the crust on the R^

phase is illustrated by the synthetic seismograms in

Figures 23 and 24 for models containing a thick basaltic

crust only (M15, illustrated in Figure 25), a thin sedi

mentary layer overlaying thick basalt (M16, illustrated in

Figure 26), and a thin granitic layer overlaying thick

basalt (M17, illustrated in Figure 27). Figures 23 and 24

show that the R phase is produced only in the synthetic y

for model M17. The synthetics for models M15 and M16 show

35

90 sec

Figure 23. Synthetic seismograms generated for models M15 and M16. M15 has a basaltic crust. A thin layer of sedimentary material overlays the basaltic material in M16.

36

lyVX/—

90 sec,

Figure 24. Synthetic seismogram generated for model M17. A thin layer of granitic material overlays this basaltic material in the crustal portion of this model.

37

MODEL 15

H

40.0

Vp

7.57

Vs

4.31

P 3.14

4.5-

4.0-

o a> (A

3.5-

3 UJ >

3.0-

PHASE VELOCITY

GROUP VELOCITY

lo" 20 30 40 7o

PERIOD (sec)

Figure 25. Theoretical dispersion data for the model M15. The crustal portion of this model is comprised of a single layer of basaltic material.

38

MODEL 16

H

5.0 35.0

Vp

5.30 7.57

Vs

3.06 4.31

P 2.62

3.14

4.5-

4.0 o (A

3.5-

3 UJ >

3.0-

PHASE VELOCITY

GROUP VELOCITY

10 20 30 40 50

PERIOD (sec)

Figure 26. Theoret ical d ispers ion data for the model M16. The c r u s t a l por t ion of t h i s model i s comprised of a th in layer of sedimentary mater ia l (overlaying a thick b a s a l t i c l aye r . )

39

MODEL 17

H

5.0

35.0

Vp 6.35

7.57

Vs 3.62

4.31

P 2.65

3.14

4.5-

4.0-o 

3.5-

3.0-

ROUP VELOCITY

10 20 30 40 50

PERIOD (sec)

Figure 27. Theoretical dispersion data for the model M17. The crustal portion of this model is comprised of a thin layer of granitic material (overlaying a thick basaltic layer.)

40

only weak dispersion for model M15 and the pronounced ring

ing due to the sedimentary layer in M16.

The synthetic seismograms presented in Figures 2 to 8

and in Figures 23 and 24 clearly illustrate that small

changes in crustal structure cause significant changes in

the characteristics of Rayleigh waves. Increased thickness

of, or decreased velocity in, the sedimentary layer causes

the seismogram to assume a ringing appearance. Thickening

of the granitic layer, the introduction of a low velocity

layer into the crustal model, and increased crustal velo

cities result in amplitude increases in the seismogram re

sembling the earthquake phase R . Identification of these y

characteristics in earthquake seismograms can provide im

portant information about the crustal structure of the

earth.

CHAPTER IV

CORRELATION OF SYNTHETIC SEISMOGRAMS

WITH EARTHQUAKE SEISMOGRAMS

The application of character modeling to the identi

fication of crustal structure requires cross correlation of

synthetic seismograms with earthquake seismograms. When

the earthquake (true) seismograms are reasonably noise free

the correlation may be made visually. However, true seis

mograms are rarely noise free, and numerical cross correla

tion is usually required.

Numerical cross correlation is a procedure commonly

used in digital signal processing to measure the similarity

between time series (Lee, 1960, Kanasewich, 1975: Gold and

Rader, 1969). The function produced by cross correlation

is represented as

n g d ) = Z f (t)f^(t+T) 0 ^T<m+n

t=l ^ ^

where f, (t) and £9^^^ ^"^^ time series of length m and n,

respectively, and x is a time displacement. This equation

is equivalent to a series of dot products between the

members f, and t,. of the correlation.

41

42

The cross correlation function produced from two time

series is a measure of the similarity in frequency content

and the phase difference at each frequency. Time series

which have similar frequency and phase components produce

cross correlation functions which have large maxima and

good symmetry about the maxima.

The Rayleigh waves of two earthquakes with epicenters

in Mexico and off the western coast of Mexico are shown in

Figure 28. Locations of the epicenters are shown in

Figure 29. These Rayleigh waves were recorded at the WWSSN

station in Lubbock, Texas. The travel path between Lubbock

and Event 2 has previously been examined by the author

using Love waves (Crider, 1975), and the crustal model de

rived in that study is the basic model used in the synthe

tic seismogram generation described in Chapter III.

Examination of the earthquake seismograms reveal that

the Rayleigh waves recorded for Event 2 are considerably

different from those recorded for Event 1. The primary

cause for this difference is interference between Rayleigh

wave groups arriving at Lubbock along different azimuths

(e.g. Evernden, 1953). This type of interference is also

expected in Rayleigh waves for Event 1. Visual correlation

between these true seismograms and the synthetic seismo

grams is not possible and numerical cross correlation must

be performed to find any similarity between the true and

synthetic seismograms.

43

—^A

90 sec.

Figure 28. Long period vertical components of seismograms recorded at Lubbock, Texas from two earthquakes along the western coast of Mexico. The locations and times for the earthquakes are: Event 1 (upper) - July 2, 1972, Ooh 34m 42.8^; 6 = 16.4° N; X= 98.5° W. Event 2 (lower) -May 28, 1972, 23^ 11 ^ 36.7S; 9 = 19.3° N; X = 108.3° W.

44

Figure 29. Location of the epicenters for Event 1 and Event 2 and the receiving station. The dashed lines indicate the assumed travel paths between the epicenters and the receiving station. Contours are in fathoms.

45

Numerical cross correlation between the true and syn

thetic seismograms was accomplished on the Sunmark Explor

ation Company's Wang 2200 VP computer. All seismograms

were digitized using a Summagraphics digitizing board inter

faced with the Wang computer. A one-second sample interval

was used in all digitizing.

The cross correlation functions obtained between Event

1 and the suite of synthetic seismograms shown in Chapter

III are illustrated in Figures 30 to 43. The correlation

functions obtained for the models Ml, M9, and MlO, re

presenting different crustal thicknesses, show the crust

between Lubbock and Event 1 to be at least as thick as that

for the basic model Ml. The maximum in the correlation

functions for Ml and the thickened crust model MlO are

much greater than that for the thinned crust model M9.

The maximum of the function for MlO is the greatest,

suggesting that the crustal thickness is somewhat greater

than that for the basic model.

The greatest maximum among all the cross correlation

functions is found in the correlation function for M4 which

corresponds to the same crustal thickness as in Ml but with

a reduced velocity in the sedimentary layer. However, the

poor symmetry observed in the function is indicative of

poor correlation between the seismograms for Event 1 and

M4.

The cross correlation function obtained by comparina

u

46

<D CO

u 0)

cn

o

w u

A4

0) 0 s:

4-1

c 0

•H 4-» U c p

iw

C 0

•H +J to

rH 

^ 0

MH

g nJ }H

0 e Ui

•H o; (0

u •H +J <\)

u i : cn cn 0 M u

4J c >. cn

the

o s: m

0) SH

•H I?

3 'C cn (U

•H fa

•P •

rH (0 S

rH (1) iH U d) U TJ 0 0

0 E

47

SI cn j o I

i H CN

s +> iH

c 0) Q) 73 > »

U

0

e 0)

0 x: IW

c 0

•H •P

u c 3

« 4 - l

c 0 •H +J (TJ

i H (U ^ u 0

+J

^ 0

* 4 - l

e fC ^ cn 0 e tn.

•H (U cn u

•H 4J (U

u ^ cn CO 0 u u

•

•P

c > 1

cn (1) x: 4J

.H J : ro

0) }-(

-p •H ^

D 73 D (U

•H fa

V (0

i H 0) ^ U 0 u

48

cn'

o

4J

c • S W

i H U (U 0 73

IM

c 0

0 e Q)

•H .C 4J U c p M-(

c 0

•H 4J to

i H

Q) ^ U 0 0

cn cn 0

+J

>H 0

M-l

B (d M CP 0 B cn •H (U cn U

•H JJ (U

u s: u

t

CN ro

(U JH

4J

c > 1

cn (U .c 4-»

:3 73 D Q)

•H fa

4-> (0

i H <U u u 0 u

49

u

cn

o

^ • H S •P rH

c (U 

w u

0

e (U

0 x: ^

c 0

• H 4-» U c o

««

c 0

• H +J (0

i H (U J-l

u 0

4J

U 0

MH

e nj

u CP 0

e cn • H 0) cn u

• H +J (U

U JS

cn cn 0 u u

ro ro

H

4J

c > 1

cn Q)

x: 4J

x: •p • H

15 3 73 O (U

• 1 -

fa 4J fd

i H (I) }-l >H 0 o

50

(U

cn o

in

rH S

JJ rH

0) ^3

> o w e >H QJ

ox: C U o o

• H M-t 4J

3 e td }H MH CT

o c o

•H

e cn

•H «d cn

(U u }H -H U -P O Q)

u x: cn c cn > i 0 cn 5H U d)

x:

ro 4J •H

<D 5 5H

D 0) •H 4J fa Id

0)

}H O U

51

cn I

o

yo .H s 4-> r-i C w M

(U T3 0

e (U

0 x: »+-(

c 0

•H 4-> U c 3

^-^

>•* 0

•H +J (d

rH (U }H M 0

•p

u 0

4-4

e (d M tj^ 0

e cn •H (U cn u

•H +J (U

u x: cn cn 0 u u

4J

c > 1

cn 0) x: •p

in ^ ro

(U }H

+j •H >

3 73 tP <U

•r-

fa 4J fd r-\ Q) U U 0 U

52

C 0) 0) 73

> o w e }H 0) o x :

C SH O O

•H MH

U c :3

e fd u o

o -H

e cn

• H 0) (d in

rH (U u J-i -H U 4-> o i O m

x: 4->

ro -P •H

Q) ^ U O 73 cn (U

•H 4-) fa fd

rH (U }H >H O U

53

o cn o

00 -H S •P r-^ C 0) 0 W E

U <D

ox: 4-( + j C JH

o o •H U-t -P u c: p

B fd

4 H C7I

o o

•H 4J

cn •H

fd cn r-\ <D u ^ -H o i o cn u

x: 4-»

ro x: -p •H

U D TJ cn Q)

•H 4J fa fd

rH <D U u o u

54

t • I

U ; Q)! cn i

(T\

<y» i H s 4J i-i C 0) 0) 73 > W

}-j

0 B 0)

0 x: 4H

c 0

•H -p u c :3

MH

C 0

•H 4J Id

rH H CP 0 B cn

•H (U cn u

•H +J 0)

u x: cn cn 0 u u

00 ro

dJ }-

4-)

c > l

cn (U x: 4J

x: +j •H ^

3 'd C7 0)

• r

CL 1 -P ( fd

i H (1) }H }H 0 O

55

cn

o

^ S

4J c

rH fa $H

0

e 0)

0 x: 4H

c 0

•H 4-> U c p

<4H

C 0

•H 4-» fd r-*. Q) U u 0

+J

u 0

MH

e fd u cn 0 e cn

•H 0) cn u

•H +J (U

0 x: cn cn 0 u u

+J c > i

cn 0) £ j j

cr> jC ro

(U 5H

+J •H 5

P 73 cr 0)

•H fa

JJ fd

rH 0) M M 0 u

56

o cn o

+J c Q) > W

u

i H (U Ti 0 E Q)

0 J: yu

G 0

•H 4-> U C P

4H

C 0

•H JJ fd

rH (U U u 0

4J

}H 0

t+H

E Id k CP 0 E cn

•H

 1

cn (U x: 4-!

ox: ^

0) ^

4J •H >

P 'C c: o

•H fa

4J Id

f-i <D U U 0 u

57

CN

U

cn

o CJ^

4J r-\ C QJ > H

»H

QJ 73 0 E QJ

0 x: MH

c 0

•H +J u c p

4H

c 0

•H 4-J fd r-\ i

cn QJ

£ 4J

rH ^ ^

QJ U

• P • H

15 P TS cn QJ

•H fa

4J fd

rH QJ U U 0 ^

58

ro

u QJ cn

o ,

•P >H C QJ QJ T l > fa U

0 E QJ

0 x: 4H

<-• 0

•H •P U C P

4-1

C 0

•H 4J fd

PH QJ »H ^ 0

+J

^ 0

4H

E .•0 u cn 0 E cn

• H QJ cn u

• H +J QJ

u x: cn cn 0 u u

•p c >t CO

QJ x: 4-»

<N x : "^

Q) 1

4-1 • H

^

P T5 CP QJ

•H fa

4-> fd

l-i QJ 4

U 0 u

59

u QJ

cn o

I-H

• • ^

t-i

S •p >-{

c QJ QJ TJ > fa

U

0 E QJ

0 x: 4H

C 0

•H 4J U C P

4H

c 0 •H 4J fd

r-i QJ }H ^ 0

-P

U 0

m

E (d M cn 0 E cn

•H QJ cn

U •H 4J QJ

0 x:

cn cn 0 u u

ro ^

QJ U

4J

c > 1

cn

QJ x: 4J

£ 4-) •H ^

P 73 Cn QJ

•H fa

•p Id

rH QJ V u 0 u

60

Event 1 and the models M2 and M7 exhibit large amplitudes

and good symmetry indicating good correlation. These cor

relation functions'suggest that the crust between Lubbock

and the Event 1 epicenter contains either a low velocity

layer or a thicker granitic layer than included in the basic

model Ml. The larger amplitude of the M2 cross correlation

suggests that a thicker granitic layer is the more likely

of the two alternatives.

Models Mil and M13 contain granitic layers thicker

than that in the basic model Ml. However, the crustal vel

ocity structure in these two models is significantly dif

ferent from that of the basic model Ml. The nearly zero

correlations for these models demonstrate the failure of

these models to describe the crustal structure between

Lubbock and the Event 1 epicenter.

The cross correlation functions obtained between Event

2 and the suite of synthetic seismograms are shown in

Figures 44 to 57. Large maxima are observed in the cross

correlation functions for models M2, M3, M4, M7, and MlO

and the Event 2 Rayleigh waves (Figures 45, 46, 47, 50, and

53, respectively). The greatest maximum among these cor

relation functions is observed in the function for the

thickened crust model, MlO, suggesting that the thickness of

the crust between Lubbock and Event 2 is greater than that

included in Ml. However, this function lacks the symmetry

expected of good correlation.

61

o '• QJ

cn

o

CM

4J U

QJ cn > Id W Xi

u QJ

o x: 4H 4J

C }H

o o •H 4H 4J U E c Id p u

4H CP o o

•H 4J

E cn

• H QJ

fd cn rH QJ U ». -H J-1 4J O QJ

JJ

cn

cn cn O u U QJ

X 4J

^' X 'd' -P

•H QJ ^ >-i P 73 cn QJ

•H 4-1 fa fd

QJ r-{ U Q) U 73 O O U E

62

u QJ

cn

o

CN CN S

4J r-< C QJ QJ 73 > O W E

}H QJ ox 4H 4->

C }H O O

• H 4H 4->

p

c (d }H

4H cn o

c o

• H 4J

E cn

• H QJ

fd cn r-\ QJ u }H -H U V O QJ O X

4-» cn c cn >i O cn >H U QJ

X 4J

• in X •^ 4->

•H QJ ^ JH P T5 CP QJ

•H 4J fa fd

QJ U U O U

63

u QJ

cn

o

CN ro S

4-> ^ C QJ QJ T i > fa U

0 E QJ

0 X 4H

c 0

• H •p

u c p

4H

C 0

• H P fd

t-\

QJ }H ^ 0

4J

^ 0

4H

E fd u cn o E cn

•H QJ cn u

- H 4J QJ

U X

cn cn 0 u u

VO '3 '

QJ U

4-> c > 1

cn QJ X 4-1

X 4J •H ^

P TJ cn QJ

•H

fa 4J Id

rH QJ U U 0 U

64

ul Q J i cn o! o i

' CN

«:r S

4J r-\

c QJ QJ T3 > fa U

0 £ QJ

0 X 4H

c 0

•H 4-1 U c p

4H

c 0

•H 4-J fd

i H QJ U U 0

+J

SH 0

4-1

E fd u CP 0 E cn

•H QJ cn u

•H 4J QJ

0 X

cn cn 0 u u

r ^

QJ u

4J c > l

cn QJ X •p

X 4-1 •H ?

P 73 cn QJ

•H fa

4-1 fd

rH QJ JH JH 0 U

65

in CN X

4J r-i C QJ Q) 73 > O W E

J H QJ O X

4 H 4->

c u o o

•H 4H 4J U

p

E fd u

4H CP

o

O I QJ

cn

o

o • H 4J

E cn

• H QJ

fd tn H QJ U JH - H JH 4J O QJ O X

4J cn C cn >i O tn JH U GJ

X 4->

• 00 X -a * 4->

• H QJ ^ JH p T3 cn d)

•H +J fa Id

r-{ QJ J H J H O U

66

o QJ

cn

o

VO CN 2

4J r^ C QJ QJ TJ > O W E

JH O

4H

QJ X

C JH

o o •H 4H 4-> U

p

E fd JH

4H CP O

o •H 4J

E cn H QJ

fd tn i H QJ o JH - H JH 4J O QJ U X

4J cn C cn > i O tn JH U QJ

X 4J

• <j\ X r r 4->

•H QJ > JH p 73 CP QJ

•H 4J fa Id

r-i QJ JH JH O U

67

cn!

O !

CN

4-)

c QJ QJ 7 3 > O fa E JH O

4H

QJ X 4J

J : JH o o •H 4H 4J u c p

E Id JH

4H CP O

c o

•H 4J

E cn

• H QJ

fd tn rH QJ u JH - H V4 4J O QJ U X

4J tn c cn > i O cn JH CJ QJ

X 4->

O X in 4J

•H QJ 5 JH p 73 cn QJ

•H 4-1 fa Id

rH QJ JH U o o

68

u QJ cn o as

CO CN S

4J r^

c QJ QJ 73 > fa JH

0 E QJ

0 X 4H

c 0

•H 4-> U G P

4H

C 0

•H 4-> Id

r-^ QJ JH JH 0

4J

JH 0

4H

E Id JH CP 0 E cn

•H QJ cn U

•H 4-> d)

U X

cn cn 0 JH

u

p c >^ cn QJ

X 4->

r^ X i n

QJ U

4J •H 5

P 73 cn QJ

• r

fa 4J Id

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69

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70

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71

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CN

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72

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CN S

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75

Symmetry and large amplitude are observed in the

cross correlation function for the basic model Ml (Figure

39) and the Event 2 Rayleigh waves. It should be noted

that the maximum of the correlation function for the model

Ml is nearly as large as that in the MlO correlation. This

indicates that the correct model for the crustal structure

between Lubbock and the Event 2 epicenter is probably only

slightly different from the model Ml. The large amplitude

observed in the M2, M3, M4, and M7 correlation functions

suggest that the model should include a thicker granitic or

sedimentary layer than included in the model Ml, in addi

tion to a crustal low velocity layer.

Cross correlation of the suite of synthetic seismo

grams with Rayleigh waves from two earthquakes in Mexico

shows that the crustal structures between the epicenters

for the two events and Lubbock are similar. These struc

tures are slightly different from that determined using

Love waves from Event 2 in that they probably have thicker

granitic and sedimentary layers than included in the basic

model Ml in addition to a low velocity layer in the crust.

CHAPTER V

CONCLUSIONS

1. Synthetic seismograms that reveal the character

of seismic waves in a given layered earth model may be gen

erated from theoretical phase velocities for the model.

2. The earthquake phase R may be explained as the g

result of waves with periods between 18 and 10 seconds

having nearly the same group velocity due to the presence

of the granitic layer in the crust.

3. Cross correlation of synthetic seismograms for

several earth models with true earthquake seismograms

verify that the crust beneath Northern Mexico is 41.3 km.

thick and is comprised of a slow velocity, 4.3 km. thick

sedimentary layer, a 5.0 km. thick granitic layer, a

crustal low velocity layer, and 32 km. of basaltic material

4. Visual or numerical correlation of synthetic

seismograms for varying earth models with earthquake

seismograms may give a direct determination or a verifi

cation of an earth model.

76

CHAPTER VI

RECOMMENDATIONS

The results of this study have been limited to the

explanation of the characteristics of seismograms of earth

quake Rayleigh waves having periods greater than about 9

seconds. However, true seismograms record Rayleigh waves

and other surface waves with shorter periods as well.

These shorter periods may have pronounced effects on the

characteristics of the seismograms. For this reason the

following recommendations for further study are made.

1) Extend the range of wave periods included in

the synthetic seismogram to include periods

at least as low as 5 seconds. The inclusion

of these short wave periods may provide useful

data on the propagation of microseisms in the

period range of 5-6 seconds.

2) Include higher mode Rayleigh waves in the con

struction of the synthetic seismograms. In

clusion of these high velocity, short period

waves may provide clarification for the

77

^fWB^W

78

characteristics of the early portions of

earthquake Rayleigh wave seismograms which have

not been investigated in this study.

LIST OF REFERENCES

Aki, K., 1960. Study of Earthquake Mechanism by a Method of Phase Equalization Applied to Rayleigh and Love Waves, Journal of Geophysical Research, 65, p. 729-7T0: ~ ~

1972. "Earthquake Mechanism." In: A. R. Ritsema (Editor), The Upper Mantle. Tectonophysics, 13 p. 423-446:

Alterman, Z., Jarosch, H., and Pekeris, C. L., 1959. Oscillations of the Earth, Proceedings of the Royal Astronomical Society of London, A, 252, p. 80-95.

Bloch, L., Hales, A. L., and Landisman, M., 1969. Velocities in the Crust and Upper Mantle of Southern Africa from Multi-Mode Surface-Wave Dispersion, Bulletin of the Seismological Society of America, 59, p. 1599-1630.

Braile, L. W. , and Keller, G. R. , 1975. Fine Structure of the Crust Inferred from Linear Inversion of Rayleigh-Wave Dispersion, Bulletin of the Seismological Society of America, 65, p. 71-83.

Brune, J. N., Nafe, J. E., and Oliver, J. E., 1960. A Simplified Method for the Analysis and Synthesis of Dispersed Wave Trains, Journal of Geophysical Research, 65, p. 287-303.

Brune, James, and Dorman, J., 1963. Seismic Waves and Earth Structure in the Canadian Shield, Bulletin of the Seismological Society of America, 43, p. 167-210.

Bullen, K. E., 1965. An IntrocSuction to the Theory of Seismology, Cambridge University Press.

Crider, R. L., 1975. Structure of the Crust and Upper Mantle Beneath Mexico from Love Wave Dispersion, Unpublished Manuscript, Texas Tech University.

79

80

Dorman, J., Ewing, M., and Oliver, J., 1960. Study of Shear-Velocity Distribution in the Upper Mantle by Mantle Rayleigh Waves, Bulletin of the Seismological Society of America, 50, p. 87-115.

Dorman, J. , and Ewing, M., 1962. Numerical Inversion of Seismic Surface Wave Dispersion Data and Crust-Mantle Structure in the New York-Pennsylvania Area, Journal of Geophysical Research, 67, p. 5227-5241.

Evernden, J. F., 1953. Direction of Approach of Rayleigh Waves and Related Problems, Bulletin of the Seismological Society of America, 43, p. 335-374.

Goforth, Tom T. , 1976. A Model Study of the Effect on the Rayleigh Spectrum of Lateral Heterogeneity in Earthquake Source Regions, Journal of Geophysical Research, 81, p. 3599-3606.

Gold, B., and Rader, C. M., 1969. Digital Processing of Signals, McGraw-Hill Book Company.

Haskell, N. A., 1953. Dispersion of Surface Waves on Multi-layered Media, Bulletin of the Seismological Society of America, 43, p. 17-43.

Kanasewich, E. R. , 1975. Time Sequence Analysis in Geophysics, The University of Alberta Press, Edmonton, Alberta, Canada.

Lee, Y. W., 1960. Statistical Theory of Communication, John Wiley and Sons, Inc., New York, New York.

Marion, J. B., 1967. Classical Dynamics, Academic Press, New York, New York.

McEvilly, T. v., 1964. Central U. S. Crust-Upper Mantle Structure from Love and Rayleigh Wave Phase Velocity Inversion, Bulletin of the Seismological Society of America, 54, p. 1997-2015.

Press, Frank, and Ewing, Maurice, 1952. Two Slow Surface Waves Across North America, Bulletin of the Seismological Society of America, 42, p. 219-228.

Richter, C. F., 1958. Elementary Seismology, W. H. Freeman and Company, Inc., San Francisco, California.

Sato, Y., 1960. Synthesis of Dispersed Surface Waves by Means of Fourier Transform, Bulletin of the Seismological Society of America, 50, p. 417-426.

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