structure of the earth's crust from spectral behavior of ...€¦ · resemblance to the...

JOUaN^L OF GEOPHVSIC^L RESg^aCH VOL. 69, No. 14 JuLy 15, 1964

Structure of the Earth's Crust from Spectral Behavior of Long-Period Body Waves

ROBERT A. [PHINNEY

Department o• Geology, Princeton University Princeton, New Jersey

Abstract. Long-period P waves from distant earthquakes have been analyzed from seismo- grams recorded at Albuquerque and Bermuda in the light of I-Iaskell's theory of the spectral response of a layered crust. By using the ratio of the vertical spectrum to the horizontal component spectrum, we obtain a function which depends on structure beneath the station. Be- cause of the poorly understood nature of the signal which follows the first P wave motion, the methods of power spectrum analysis are applied and a lag window selected to discriminate against long time correlations within the signal. Corrections for the differing responses of the three components are made by using the power spectral matrix of calibration signals. A range of crustal models has been found which agrees with the data. We are restricted primarily in choosing models for the lower half of the crust; variations of structure in the upper half do not noticeably affect the theoretical curves in the period range considered (0.02 to 0.20 cps). At Albuquerque the crust is about 40 km thick and the lower crust has velocities in the range 6.6 to 7.0 km/sec. The Mohorovicic (M) discontinuity under Bermuda is 12 km below sea level, and the structure appears to be a normal oceanic crust depressed elastically by the weight of the volcanics which compose the island.

List o• symbols. 0

,4(,0) b(t)

c

E[]

R•(r)

uo(t)

U(co)

0(t)

Spectrum of the incident pulse train. Source wavelet.

Spectrum of the source wavelet. Phase velocity--apparent velocity along

surface.

Expectation value (ensemble average) of[ ].

Lag window used in estimation of the power spectrum.

Fourier tra_u sform of g(r). Source wavelet (b) as distorted by the

crustal filter for horizontal (u) motion. Power spectrum of u0 (equation 9). Autocorrelation function of u0 (equa-

tion 8). Crustal transfer ratio, defined as W•/U•. Horizontal component of displacement

at the free surface.

Crustal transfer function: horizontal surface motion due to an incident P wave.

Vertical component of displacement at the free surface.

Crustal transfer function: vertical sur-

face motion due to an incident P wave.

Compressional wave velocity. Shear wave velocity.

Angle of incidence of plane wave in substratum.

Density. Apparent angle of incidence, defined as

tan-•(Uo/Wo). Angular frequency.

Introduction. The effect of a free surface on

the apparent polarization of an obliquely incident P wave is well known. The relation

2sin 2 0 = (a2/•)(1 -- cos •) (1) connects the angle of incidence of the P wave with the apparent angle of incidence •. This model, in which any layering near the surface is ignored, gives results which are independent of frequency.

Nutill and Whirmore [1961, 1962] studied the polarization angle of P and S waves by working directly with the vector time functions. Their conclusion that the polarization could be explained by a surface P wave velocity of over 7 km/sec seems to demand a frequency-dependent polarization which would, presumably, be consistent with lower velocities at short enough periods. The purpose of this paper is to estab- lish that the observed frequency-dependent polarization properties of teIeseismic P waves are

2997

2998 ROBERT A. PItINNEY

consistent with a layered model for the crust under the recording station and that the obser- vations can in fact be used to determine the crustal layering. Haskell [1962] demonstrated the applicability of the Thomson-Itaskell matrix method to the more general n-layered surface geometry (Figure l) and obtained, for one model, values of Uo and Wo as functions of frequency and angle of incidence in the semi-in- finite substratum.

In this paper I take the following points of view:

1. The theory predicts a frequency depend- ence; the data should consequently be analyzed for their frequency behavior, and some form of Fourier analysis is preferable to direct analysis of the time functions.

2. Seismic refraction studies, surface wave studies, and teleseismic travel-time studies have established that to a good approximation we can represent the crust and upper mantle as a structured stack of layers and the underlying lower mantle as an unstructured substratum showing only a smooth increase in velocity with depth. It would be surprising if the Haskell theory were not in some degree confirmed by analysis of the data.

Assuming an incident compressional plane wave in the substratum, we can write the vertical and horizontal components of motion at the free surface as

where A (to) is the spectrum of the incident wave and U• and W• are spectral transfer functions. Haskell's method gives a prescription for com- puting the transfer functions if the model is composed of homogeneous parallel layers; in that case U• and W• depend only on the frequency and apparent velocity of the wave. Re- gardless of the complexity of our model, however, these transfer functions still exist. There is the possibility, for example, that they may show a simple dipolar behavior with variations in the azimuth of approach of the P wave.

From (2), division yields

Wo/o = =

This implies that by dividing the observed spectrums, we obtain a response function which

n-I

onts r

Fig. 1. Obliquely incident P wave incident at base of layered crust.

does not depend on the spectrum of the incident pulse. In practice such a spectral ratio is of use only in the frequency range where Uo is sufficiently large compared with any noise contribu- tion to the spectrum of Uo.

It is therefore proposed that the spectral ratio, T•, be employed as a test function for the determination of earth structure immediately beneath a recording station. Observed values of Tp can be obtained by Fourier analysis of long- period P waves from distant earthquakes. For any given shock the wave will be very nearly plane as viewed from a given station. This makes it unnecessary to perform a wave number or velocity analysis of the pulse. Were this condition not satisfied, the method would require an array and would lose much of its attractiveness. Theo- retical curves expressing T•(o) can be calculated for any proposed earth model for various values of c and compared with the experimental curves. Eventually, by some method of variation of parameters, a theoretical curve will be found which matches the experimental result, and it will be inferred that this theoretical model repre- sents the velocity-density structure beneath the station.

This method, which involves the spectral ratio of two coupled wave fields, bears a strong formal

SPECTRAL BEHAVIOR OF LONG-PERIOD BODY WAVES 2999

resemblance to the magnetotelluric method for determining earth layering. Further comparison is of some interest, but space does not permit me to discuss the matter here.

In order that the proposed method be of practical use, both the theory and the data must satisfy certain conditions. Theoretical curves must be neither too sensitive nor too insensitive

to changes in the model parameters. A degree of uniqueness must be established if we are to put any confidence in the correctness of the chosen model. Enough data must be collected so that they can be assessed for repeatability and variability. Equation 3 implies that Tp is the same for different earthquakes having the same epicentral distance. The degree to which this is actually true should provide a measure of the validity of the analysis at any given station.

Behavior o• the theory. For purposes of illustration, several models have been devised (Table 1) to show the behavior of the spectral transfer ratio. The Thomson-Haskell matrix

method was the theoretical basis for the program used. Many more examples have been computed than are shown here; the curves shown merely illustrate types of behavior which have been verified by extensive calculations.

1. Characteristically, the spectral response, Tp, shows a sequence of peaks due to resonance effects in the crust. The frequency of the first peak is very close to 2•8, where •, is the transit time in the layer for a shear wave of the given phase velocity. This is more than coincidence; it can be shown that this frequency satisfies a destructive interference condition for a shear

wave generated at the base of the crust. The curve for model A in Figure 2 illustrates the effect. The curve for model B shows that adding layers to the crustal model changes the positions and amplitudes of the peaks but does not affect the general appearance of the transfer ratio. This is to be expected because both models have essentially the same surficial wave trap, namely the set of layers lying above the strongly defined M discontinuity. A much thicker structure would have the same transfer ratio, provided that the frequency were scaled accordingly.

Figure 3 demonstrates the slight sensitivity of the resonance frequencies to variations in phase velocity. The change in amplitude is due to the changing angle of incidence; as c becomes large, u must become small, and the ratio will become

quite large. Since the annual number of earthquakes generating useful long-period P waves is limited, we are forced to consider data from a variety of epicentral distances together in order to assess t•he repeatability of the results. The insensitivity of peak position to phase velocity permits us to do this.

In performing spectral analysis of a long P wave interval, it must be borne in mind that the spectral properties of the signal are less apt to be stationary for the longer intervals. It is therefore appropriate to consider the behavior of the transfer ratio when the signal contains a mixture of plane waves having values of c distributed around the value obtained from ray theory. Figure 4 shows the effect for model B. The expected smearing of the spectral peaks is insignificant, although the mixture does tend to reduce the amplitude of the peaks.

The effect of a low-velocity surface layer is shown in Figure 5. The first spectral peak due to the low-velocity layer is predicted at frequencies of 0.325 and 0.195 cps for the 3- and 5-km models, respectively. Only in the latter case is the superposed peak at 0.195 cps evident.

Finally, to demonstrate the effect of varying crustal structure, Figure 6 shows the theoretical transfer ratio for a selection of models which

represent reasonable alternative representations of a continental crust. This function is in fact

sensitive to our choice of model; the three peaks have varying positions and amplitudes.

We can summarize the theoretical properties of this transfer ratio which make it useful in crustal studies:

1. The effects of intermediate and deep crustal structure can be isolated in the behavior

of the three 'crustal' peaks in the transfer ratio. The positions of these peaks are neither too sensitive nor too insensitive to reasonable varia-

tions in structure. The upper frequency limit of 0.2 eps is sufficient to include all frequencies which can be reasonably investigated using standard long-period photographic recordings.

2. The insensitivity of peak position to changes in phase velocity makes it possible for us to ignore this factor and consider data from a wide range of distances.

3. A thin, low-velocity surface layer has no singular effect on the crustal peaks. Such a layer must be thicker than 3 or 4 km to be noticeable. If the velocities in the surface zone are not as

3OOO ROBERT A. PI-IINNEY

TABLE 1. Illustrative Crustal Models

Model •, l•, p, h,* z, •'

km/sec k/sec g/cm • km km Figure

Reference

6.15

8.14 3.55 2.74 35 4.70 3.30

35

6.15 6.95 8.14

3.55 2.74 22 4.00 3.00 13 4.70 3.30

22

35 2, 3, 4, 5

MB1 4.20

6.15 6.95 8.14

1.95 2.30 3 3.55 2.74 19 4.00 3.00 13 4.70 3.30

3

22

35

MB2 4.20 6.15 6.95 8.14

1.95 2.30 5 3.55 2.74 17 4.00 3.00 13 4.70 3.30

5 22

35

35CM2 3.85 1.75 2.15 6.05 3.60 2.85 6.10 3.70 3.15 7.80 4.10 3.20 8.10 4.55 3.25 8.00 4.45 3.30 7.90 4.35 3.30 7.80 4.25 3.30

1.50 13.5

10.0 24.0 16 20 20

1.50 15

25

49 65

85 105

AGN 4.93 2.72 2.67 4.2 4.2 6.14 3.40 2.91 15.0 19.2 6.72 3.72 3.14 11.9 31.1 7.10 3.93 3.29 19.7 50.8 8.23 4.55 3.46

T5 6.14 3.40 2.91 14 14.0 6.72 3.72 3.14 17.1 31.1 7.10 3.90 3.29 19.7 50.8 8.10 4.45 3.35

T6 6.14 3.40 2.91 12.0 12.0 6.30 3.55 3.00 7.2 19.2 6.72 3.72 3.14 11.9 31.1 7.10 4.10 3.29 19.7 50.8 8.10 4.45 3.35

T7 6.14 3.40 2.91 14.0 14.0 6.72 3.72 3.14 17.1 31.1 7.10 3.90 3.29 8.9 40.0 8.10 4.45 3.35

T8 6.14 3.40 2.91 19.2 19.2 6.72 3.72 3.14 11.9 31.1 7.10 4.10 3.29 12.9 44.0 8.10 4.45 3.35

6

* Thickness of layer. t Depth to bottom of layer.


I I I I I ,I ! I 1 o. o.I 0.2

FREQUENCY CPS

Fig. 2. Theoretical spectral ratio, T•, for models A and B. c -- 20 km/sec.

low as those shown for illustration, the zone must be even thicker.

In going from a set of data to a best-fitting model we must face the problem of uniqueness. The examples and discussion given above hint at some of the results. Obviously, in the limited frequency band under discussion, the data are unaffected by changes of structure in thin layers. Since a practical analysis scheme must involve some smoothing over neighboring frequencies, it may be impractical to resolve spectral behavior due to mantle structure. On the other

4

o. o.I 0.2

FREQUENCY CPS

Fig. 4. Theoretical spectral ratio, T•, for model B, showing the effect of assuming a mixture of plane waves containing values of c between 14 and 26 km/sec. Solid curve: pure 20-km/sec plane wave. Dashed curve: mixed wave.

ture on a scale of 5 km or more to give a degree of uniqueness which can be useful in discrimi- nating against incorrect crustal models.

Propagation of body waves in the mantle. Work to date has established the following model of crust and mantle. The crust is a zone

10 to 50 km thick, in which P wave velocities are primarily in the range 5.2 to 7.2 km/sec. In many cases a surface layer is present in which

hand, the 'crustal' peaks in the transfer function

are sufficiently sensitive to differences in struc-

4 MODEL B /

4--

2

2--

0 FREQUENcyOI CPS 0.2 o. o.I 0.2 Fig. 5. Theoretical spectral ratio, T•, for model

FREQUENCY CPS B, showing the effect of a low-velocity surface Fig. 3. Theoretical spectral ratio, T•, for model layer. Model MB1 has a 3-km surface layer, MB2

B. c ranging between 12 and 24 km/sec. a 5-km layer.

3002 ROBERT A. PItINNEY

6[ ' ' [ ' [ , .'It' ' [ [ [ ' ' I [ .";' , [

4

2

0

6

5

4

•• c-24 • I

O.

6

5

4

2

I

0 0 ' O.I 0.2

FREQUENCY CPS

Fig. 6. Theoretical spectral ratios for selected crustal models (Table 1).

the velocities are much lower. The base of the

crust normally involves a velocity jump to 7.8 km/scc or more. The crust seems to be charac- terized by a small number of velocity 'jumps,' and apparently no significant internal velociW minimum is present. The mantle low-velocity channel extends from the M discontinuity to more than 300 km and consists of a region of nearly constant velocity without evidence of any sharp jumps. From roughly 300 to 1000 km the velocity increases rapidly to about 11.2 km/sec; recent work with mantle surface waves suggests that at some places the velocity increase may be large enough to look like a discontinuity. We must keep this possibility in mind.

The lower mantle, from 1000 to at least 2700 km, appears to be chemically homogeneous [Birch, 1952], and the velocity increases smoothly and slowly from 11.2 km/see to about 13.6 kin/sec. No evidence has ever been adduced

to show velocity or density discontinuities in this region.

The method proposed requires that we have a nearly plane wave traveling upward from a homogeneous substratum into a layered zone. The nearest thing to a homogeneous substratum is the lower mantle with its smooth velocity increase. Since in this region the ray paths of waves in the period range 60 to I sec are smoothly curved according to the laws of geo- metrical optics, it can be shown that the distortion of a P wave signal produced by continuous generation of S waves in a smooth velocity gradient is small. Primarily we must require that the observed P wave be a pure compressional wave and contain no significant shear component due to interaction of the pulse with an embedded interface in the substratum. It is assumed that the lower mantle fits this defini-

tion of a 'homogeneous' substratum. It is per- missible to hope that the more structured upper mantle appears 'homogeneous' by comparison with the strongly inhomogeneous crust.

In this paper I consider P waves which have traversed a substantial portion of the lower mantle before becoming subject to transmission effects in the structured upper mantle and crust. Figure 7 shows that the relevant range of distances is from 50 ø to about 100 ø, corresponding to a range of phase velocities from 16 to 25 kin/sec. For a substratum velocity of 8.1 kin/see the angle of incidence ranges from 32 ø to 19 ø PcP follows P by less than a minute, at a higher phase velocity. The spectral ratio for PcP alone must closely resemble that of P. From the result of the next section, it is concluded that the spectral ratio of the sum is an adequate repre- sentation of that which would be derived from

P or PcP individually under ideal analysis conditions. In actual power spectrum analysis a perturbation is produced if PcP is sufi%iently strong.

Excluding for the moment multiple reflections and conversions near the source (such as pP, sP, etc.), we see that the event PP is the next to arrive, with a time 2 to 4 minutes later than P and a substantially different phase velocity. The earliest event which has traveled in the

lower mantle as a shear wave is either PcS, $, or, in the event of conversions taking place in the upper 1000 km, a weak precursor to $.

A typical long-period P wave is shown in Fig-


2• 19' '20' - 2500 km.

DEPTH OF RAY 20/22- - 2000 . 24-

PHASE _/y• 24- VELOCITY •/v•26' ANGLE - O F km/sec v•/'2"' INCIDENCE -1500

••.•o' IN MANTLE -I000 •3

-500

30 ø 60 ø 90' , , I I, ' I ' ' I ' '

EPICENTRAL DISTANCE

Fig. 7. Depth of penetration, phase velocity, and angle of incidence (assuming a: 8.1) for P waves as a function of epicentral distance.

ure 8. For this discussion the focus is assumed

to be shallow, since teleseismic deep-focus P wave is seldom strong enough at long periods to have a good signal-to-noise ratio. This is compared with a theoretical seismogram obtained from Fourier synthesis of a typical theoretical response function W•. In the observed P wave the reverberations are seen to persist many times longer than the theoretical pulse. An ex- planarion is given in Figure 9, which shows a few of the rays emanating from the earthquake focus. There is an n-fold infinity of these, resulting from multiple reflections and transmis- sions in the layered region near the source. In order for one of these near-source multiples to contribute to the signal in the time interval between P and PP, it must have traveled through the lower mantle as a P wave, along substantially the same path as the first P ray. It is not important that P-S conversions may take place near the source; what is important is that the lower mantle acts as a delay line to ensure that waves traveling as S waves in the lower mantle do not arrive until substantially later than the P waves. Each ray of the type shown is dis- •orted by the structure near the receiver in the same way, as in equation 2.

Spectrum analysis. The incident wave (that is, the one transmitted by the lower mantle) can be represented by

u( t) = • q, b( t -- ti) (4) where u is the motion along the ray, t• is the arrival time at the base of the crust of the ith

ray (due to near-source reflections), q• is a real transmission factor, quite small except for the first few rays, and b(t) is the form of the source impulse. Then A (•) (equation 2) is the transform of u(t). It is easily shown that a Fourier analysis of uo and Wo will give the result predicted by (3), although the incident pulse is formed as in (4). From (4),

A(w) = B(w) • q,e -iø"• (5) The data at the free surface, then, are

Uo(W) -- U•(w)B(w) • q,e -'ø'"

Wo(W) = W•(w)B(w) • q,e -'ø'" which, upon division, yield

(6)

Wo Uo

which is identical with (3).

(7)

3OO4

!

! 1 MIN. ,

ROBERT A. PHINNEY

have a Poisson distribution and the q• have a Gaussian distribution.

The autocorrelation function of u, for a data interval of length L, is given by

•L

= Jo Uo(t)Uo(t + (s) An estimate of the power spectrum of u, then, is

L

= f_ R•,('r)g('r) cos coz dr (9) L

where g(z) is a lag window which we apply to the autocorrelation before transforming.

To understand the utility of a window in analyzing data, we must consider what happens when a thick structure, such as the mantle low- velocity zone, and a thin structure, such as the crust, are coupled. The spectral transfer ratio for the crust has widely spaced peaks, due to the small time lags involved when waves are

. multiply reflected in the crustal layers. For the , mantle structure, Tp must have closely spaced

peaks owing to the large time lags involved. The combined system shows a superposition of peaks, giving the transfer ratio a very ragged shape. The data must then show the effects both of this

ragged theoretical curve and of any individual variability due to effects we have neglected, which we can call 'noise.' It is appropriate to separate in some way the broad crustal peaks from the denser set of mantle peaks and the noise. This makes it possible to assess separately the variability of our results for the two effects.

The correct procedure is to smooth the Wo and u• power spectrums by a transversal filter G(•o) in the frequency domain and then to divide the spectrums. Equivalent to this operation in the frequency domain is the operation of multi- plication in the • domain. A window g(•) can be chosen to pass only those parts of the

Fig. 8. Upper traces: P waves from earthquakes 90 ø distant as recorded by a standard 30-90 long- period vertical seismometer. Lower traces: Re- sponse of a vertical 30-90 seismometer to a P wave step and a P wave impulse, respectively, incident on the base of a three-layer continental crust.

In practice, however, some problems are as- sociated with obtaining the spectrums in the form of (6) and dividing, since the sum over i can be quite ill-behaved as a function of •o (because of the rapidly varying phase of the sum). This is especially so when we are analyzing a long interval contributing many terms to the reverberation sum. Since there is no way of knowing all the t• and q•, there is no. way of assessing this factor independently. The method of power spectrums presents an alternative to the straight Fourier technique. It is assumed that we have a data interval composed of wave- lets arriving at random times and having random amplitudes, but with the same spectral behavior. We may assume specifically that the t,

Fig. 9. Schematic illustration of the generation of multiple P waves near the source.


autocorrelation which result from time lags in the range of interest. The cosine transform of this windowed autocorrelation gives a spectrum showing the response of just that part of the crust-mantle under investigation. Since G(•) should be without significant side lobes, the choice of g(•) is severely restricted. A Gaussian function has been chosen as the low-lag window for crustal studies (Figure 10). This gives us a G(o•) without side lobes; the slight sacrifice of frequency resolution is unimportant at this stage. Selection of a window which both rejects low lags and has no side lobes is more complicated; additional assumptions about the composition of the autocorrelation are required. Knowing that information in the high-lag region is statistically less reliable than the low-lag crustal data, we defer this type of analysis until the variability of the crustal results is better understood. If the

t• and q• are random, an ensemble average of P•(•) will equal the squared amplitude spectrum of hb• as seen through the window:

= ]L © e-i'•'hbu(t)g(t) dt

We now require that the frequency window G(o•) be effect.ively zero for Io•] • Am, where Ao• is small. By assuming that B changes little over a frequency range A,% we can write

G(f) 0.8

g('•) 0.6

0.4

O2

f $e½. "!

.Ol .o• I I

,,,,

_

I I I ! • IO •o •0 40

SeC,

Fig. 10. Gaussian lag window g(•) and its transform G(j•) used in the data analysis.

- (11)

Any individual estimate of P•(o•) will have a variability which depends on the shape of the window G.

Having similarly obtained P•(o•), we can write the ratio

TM f f -

(12)

Equation 12 will be the basis for determining the spectral transfer ratio from observed P waves. Theoretical values for comparison are obtained by using G((o) to smooth the theoretical curves of Up and Wp. If one is interested only in crustal (low-lag) effects, it becomes unnecessary to include mantle structure when com- puting Up and Wp.

The signal will fit the assumption of randomness if it consists of a single strong pulse fol- lowed by many reverberations which are individually much weaker than the first arrival. When a second equally strong pulse arrives in the middle of the analysis interval the spectrums and the spectral ratio show a perturbation ripple with a frequency spacing equal to the reciprocal of the time lag. Since pP, with an often poorly known lag time, and PcP are to be expected, this perturbation is going to be the main reason that spectral ratios from various events observed at a single station do not agree exactly.

The effect of pP can be removed by taking the ensemble average. That is, we must have as many events as possible and form a normal- ized average of the spectrums before dividing. Since the time lag between P and PcP is much better known, that perturbation may be treated by more direct methods. These refinements will not be treated in this paper, however. They await a slightly better empirical knowledge of the peculiarities of the data. Display of spectral ratios for a number of earthquakes permits a rough averaging by inspection, which sui•ces for the present. In many of the cases considered here, the relevant lags are great enough so that

3006 ROBERT A. PI-IINNEY

3008 ROBERT A. PHINNEY

the 24-sec low-lag window completely eliminates the problem.

Experimental results. Smoothed spectral ratios have been computed for the records from the U.S. Coast and Geodetic Survey stations in Albuquerque, N.M., and Bermuda-Columbia. Long-period signals most suitable for analysis come from earthquakes in the distance range 50 ø to 100 ø and the magnitude range 6.4 to 7.2. Tables 2 and 3 list the earthquakes used in this study. Some of the events are beyond 103ø; assuming that diffraction along the core boundary or transmission through the core does nothing unusual, we can still make tentative use of this information. Results for events at less than 50 ø

are also included, although they are of ques- tionable reliability.

It is necessary that the components be reasonably well matched. Working with uncom- pensated spectral values is especially hazardous when rotation of the horizontal axes is involved

and the two horizontal components have different phase responses. The following method is used in this study. Each record contains a calibration pulse, obtained from the application of a step force (or its equivalent) to the seismometer mass. If P•jC(•o) is the (3 X 3) spectral matrix of the calibration pulses, then corrected values P•/ of the data spectral matrix can be obtained by

Pii ! = piiøb"(paa•/pii •) Each matrix element is equalized to the response of the z component (P• -- P•). The corrected radial power is then

p• = p•t sin" qb q- p•t cos" qb

q- 2 Re P•/ cos 4 sin

and P/ - P,. q• is the azimuth from station to source, measured from the y axis (north) in a clockwise sense. The directions north, east, and up are assumed to be positive. The calibration spectrums are viewed through the same lag window as the data spectrums.

All records were hand digitized from photo- graphically enlarged copies of the original records. The sampling interval was 1.25 see for the Albuquerque data and 0.75 see for the Bermuda data.

Figures 11 and 12 and 15 and 16 show the smoothed spectral ratios. The data for each sta-

tion have been grouped according to their azimuth of approach to bring out any differences which may be due to this factor. The upper-frequency limit of 0.2 cps is taken to be about half the Nyquist frequency. It is also the frequency below which the signal power is sufficiently above the noise. For this purpose, we assume that the power level near the Nyquist frequency is all noise.

Bermuda. The north-south data group (Fig- ure 11) shows the best repeatability. More distant events show up generally with larger values of the transfer ratio, as should be expected because of their steeper angles of incidence. BEC001 and BEC003 at 71ø and 101 ø, respectively, give the most unequivocal expression of the peak around 0.16 cps. The east-west data group is represented by fewer events, more widely dispersed in azimuth; the consistency within the group is not as good. By throwing out BEC004 and BEC014A for lying outside the preferred range from 50 ø to 100 ø, and by setting BEC008 aside, we are left with two spectral ratios which agree quite well. Our ex- cuse for leaving out BEC008 is its intermediate focal depth; the matter will be described in con- nection with the Albuquerque results.

The smoothed 'best fits' to the north-south

and east-west data are shown in Figures 13 and 14. It appears that the difference is not due to simple dip in the structure, since data from azimuths nearly 180 ø apart show better internal consistency than data from azimuths roughly 90 ø apart. If dip were a determining factor, the results up and down dip would differ most noticeably, while the results along the strike would agree. Our results could indicate an effective horizontal anisotropy in the transmission characteristics. Whether this is due to velocity anisotropy or structural anisotropy cannot be determined here.

In the theoretical models selected for com-

parison velocities as reported by Officer ei al. [1952] from seismic refraction studies are used. We have arbitrarily assumed that Poisson's ratio is 0.25 (a/fi -- 1.73) to reduce the number of parameters which must be determined. This restriction is probably satisfactory (for the crust) in areas showing little or no seismicity or volcanism at the present time. I have computed many more cases than can be reported here, and, in general, the shear velocities and layer


005 •.. 001 .

oo• oo• :/ o,o_..__•_

O,Oo, • ........ ..;::.,::.:.... ...... .... ........... '•--•••'• ........... ............... O0?B .........

I I I I I i I I I / O. 0.1 0.2

FREQUENCY CPS

Fig. 11. Smoothed experimental spectral ratios at Bermuda for P waves arriving along azimuths near north or south. Table 2 lists earthquakes used.

I I

] I i I I I I I

/ • 014 '

: " :/•. \ ,.-!

,:'

o" ' ......... \• I I :"• 0.1 0.::'

FREQUENCY CPS

Fig. 12. Smoothed experimental spectral ratios at Bermuda for P waves arriving along azimuths near east or west. Table 2 lists earthquakes used.

I I I I I I i

NS

O. 0.1

FREQUENCY CPS

I i I i

0.2

Fig. 13. Best fits to the Bermuda data for NS and EW azimuths of approach compared with theoretical spectral ratios for models 1, 1', 2, and 3. c = 20 km/sec. Table 4 lists model parameters.

3010 ROBERT A. PI-IINNEY

• I i i I I' ' ! ' I !

3-

• • oQO • o

o. o.I o .2

FREQUENCY CPS

Fig. 14. Best fits to the Bermuda data for NS and EW azimuths of approach compared with theoretical spectral ratios for models 5 and 7. c ---- 20 km/sec. Table 4 lists model parameters.

thicknesses are primarily responsible for variations in the spectral transfer ratio. A phase velocity of 20.0 kin/see is used throughout. Table 4 summarizes the models considered.

Models 1 and i t represent a 10-kin crust and differ in regard to the surface 5-km layer of 'volcanics.' The resultant transfer ratios are, for all practical purposes, identical in the frequency range under discussion. Models 2 and 3 deal with a total crustal thickness of 15 and 20 kin, respectively. Model 4, where an additional layer

has been added to model 3, gives a spectral ratio (not shown) which differs from that of model 3 only in the amplitudes of the peaks. As might be expected, the positions of the peaks are determined primarily by the total crustal thickness.

In developing model 5 (Figure 14) a serious attempt is made to obtain a fit with a 12-km crust. The fit to the data is better than for the

10-kin crust and is as good as can be obtained without using more adjustable parameters. (The 5.43-layer is included only for agreement with

I I I I I i I I I

005 ..........706 A 4 I• oo• • / \ oo•

008 ......'"" '" 015 "" .... '" ...........

013

Fig. 15.

I ! I I I I ! I I O. 0.1 0.2

FREQUENCY CPS

Smoothed experimental spectral ratios at Albuquerque for P waves arriving along azimuths SW and NE. Table 3 lists earthquakes used.


Fig. 16.

i I

OlO

OOl 011

Ol4 .."" '\ o01 o •'..

o. o.I o.• I I I

FREQUENCY CPS

Smoothed experimental spectral ratios at Albuquerque for P waves arriving along azimuths NW and SE. Table 3 lists earthquakes used.

refraction results; it cannot be resolved in this study.) Replacing the single crustal layer with a simulated gradual velocity increase (models 6 and 7) is unsatisfactory (Figure 14).

Further attempts to refine this result, and possibly to probe the difference betw•n the experimental curves, are only partly satisfactory. Table 4 summarizes the results of this investigation, where I have assigned a rating of the quality of fit to the data for each model. Models 9, 11, and 14 fit the data about as well as model 5 and demonstrate that I cannot

say much about the top 5 km or the region between the M discontinuity and the mantle low-velocity zone. Lowering the velocity at the M discontinuity produces little effect on the theoretical curve; the two models considered (8 and 10) are marginally less satisfactory than model 5. Models 12 and 13 are ruled out, primarily because the 10-km crustal thickness forces the peak to a frequency of 0.19 cps. Perturbing the velocities slightly (models 15 to 18) enables us to rule out only the increased crustal velocity (16).

To summarize, I have determined that the M discontinuity lies about 12 km under Ber- muda. No distinction can be made between models with and without a 5.43-km/sec surface

layer. The lower crust is a distinct layer whose velocity is roughly determined to be between 6.6 and 6.85 km/sec. Tentative preference is given to a mantle velocity (immediately below M) over 8.0 km/sec. The total crustal thickness is our best result, and it compares favor- ably with Woollard's [1954] determination of 11.7 km, which is based on regional Bouguer anomalies and seismic control in the adjacent ocean basins. I-Iis suggestion that the crust at Bermuda is a normal oceanic structure which has been depressed elastically under the load of the overlying volcanics would seem to be further borne out by this study. The section can be regarded as being in isostatic balance with the normal oceanic column if the load is assumed

to be distributed over an area roughly 55 km in diameter.

Albuquerque. Large earthquakes in the distance range of interest naturally fall into two groups: (1) South American and Kurile Islands at azimuths of 145 ø and 315 ø at distances

around 68 øand (2) Fiji-Kermadec and Near East shocks lying at azimuths of about 240 ø and 40 ø at distances around 90 ø . Since the continental crust is about 3 times as thick at

Albuquerque as it is at Bermuda, the resultant scaling in frequency brings three peaks into the


TABLE 4. Theoretical Models for Bermuda

Model a, km/sec /?, km/sec p, g/cm • z,* km Fit to Data i Figure

I 5.43 3.15 2.70 5 C 13 6.81 3.90 2.95 10 8.10 4.55 3.33

10

11

12

13

14

6.81 3.90 2.95 10 C 13 8.10 4.55 4.33

2 5.43 3.15 2.70 5 C 6.81 3.90 2.95 15 8.10 4.55 3.33

3 5.43 3.15 2.70 5 C 6.81 3.90 2.95 20 C

4 5.43 3.15 2.70 5 C 6.81 3.90 2.95 10 7.45 4.10 3.10 20 8.10 4.55 3.33

5 5.43 3.15 2.70 5 A 6.81 3.90 2.95 12

8.10 4.55 3.33

6 5.43 3.15 2.70 5 C 6.81 3.15 2.95 8 7.10 4.10 3.10 12 8.10 4.55 3.33

5.43 6.81 6.90 7.10 7.27 8.10

5.43 6.81 7.53

6.81 8.10

6.81 7.53

6.81 8.10 8.00 7.90

6.81

8.10 8.00 7.90

6.81 8.10

8.00 7.90

6.81 8.10 8.00 7.90

3 15 3 90

4 00 4 10 4 20

4 55

3.15 3.90 4.35

3.90 4.55

3.90 4.35

3.90 4.55 4.45

4.35

3.90 4.55 4.45

4.35

3.90 4.55 4.45

4.35

3.90 4.55 4.45

4.35

2.70 5 2.95 6 3.00 8 3.10 10 3.10 14 3.33

2.70 5 2.95 12 3.20

2.95 12 3.33

2.95 12 3.20

2.95 12 3.33 25

3.30 45 3.3O

2.95 10 3.33 25 3.30 45 3.30

2.95 10 3.33 4O 3.30 60 3.30

2.95 12 3.33 40 3.30 60 3.30

13

14

14

SPECTRAL BEHAVIOR OF LONG-PERIOD BODY WAVES

TABLE 4. (Continued)

3013

Model a, km/sec /•, km/sec p, g/cm 8 z,* km Fit to Datat Figure

15 5.43 3.15 2.70 5 C 6.81 3.90 2.95 12 8.40 4.75 3.33

16 5.43 3.15 2.70 5 C 6.95 4.00 2.95 12 8.10 4.55 3.33

17 5.43 3.15 2.70 5 A 6.65 3.80 2.95 12 8.10 4.55 3.33

18 5.43 3.15 2.70 5 A 6.81 3.90 2.95 12 8.40 4.50 3.33

* Depth to bottom of layer. T Best fits, A; satisfactory, B; not satisfactory, C.

range 0 to 0.2 cps. The data (Figures 15 and 16) have the familiar property oœ being most. consistent at the longer periods. My interpreta- tion oœ the best fit is given in Figure 17, with some possible alternatives. For the southwest- northeast group one may argue the details of the second peak and the amplitude oœ the third. The northwest-southeast group contains three events which suggest the existence oœ a com- pound peak around 0.10 to 0.14 cps instead of two discrete peaks at 0.095 and 0.155. The geo- graphical distribution of the events which do or do not show this rather singular effect seems to rule out simple structural dip as the cause.

SW- NE ............... ...."' ';..., ':..

O. C.I

FREQUENCY CPS 0.2

Fig. 17. Best fits to the Albuquerque data for SW-NE and NW-SE data groupings. Preferred in- terpretations are the heavy lines; alternative in- terpretations are shown as lighter lines.

No explanation will be attempted at this time. The• event in Greece on August 28, 1962

(ALQ006), shows the effect of • conspicuous departure from our assumption of randomness. The focal depth oœ 120 km yields • strong pP 27 sec after P. Consequently, the unsmoothed spectrums of w and u contain a series. of holes spaced at intervals oœ 0.037 cps. If smoothing were unnecessary and everything else were ideal, division would suffice to eliminate the effect.

Division of smoothed spectrums, however (equa- tions 11 and 12), does not give exactly the same result as dividing and smoothing; the error involved is unimportant if the raw spectrum contains no sharp lines or holes. Thus in this one case we can see distinctly • cause oœ variability in the observed spectral transfer ratio. In other applications of data from this earthquake the effect is not as pronounced; the PP result (Figure 15) does not show it, and the result at Bermuda (Figure 12) appears to have been perturbed, although this is not as clear-cut as at Albuquerque.

The difference between the best fits for the

two data groups as shown in Figure 17 can be attributed entirely to the difference in phase velocity; the northeast-southwest group involves epicentral distances clustered around 90 ø , and the northwest-southeast distances are around

68 ø . This has been verified by plotting the theoretical spectral ratio against c for various frequencies, using one of the better fitting struc-


ture models, and for comparison adding points taken from Figure 17, assuming phase velocities of 24 and 18 km/sec for the two curves. The 'experimental' curve in Figures 18 and 19 is gotten by interpolation to 20 km/see in Fig- ure 17. All theoretical curves for comparison similarly result from taking c equal to 20.

Theoretical curves were computed for 51 models in an attempt to try systematically a variety of reasonable structures. Since the spectral peaks for a continental crust are nearly as narrow as the frequency window employed on the data (Figure 10), we must take account of the resultant distortion. This has been done by using the

TABLE 5. Theoretical Medels for Albuquerque

1. THL Series

Layer km/sec km/sec g/cm •

6.14 3.50 2.75 6.80 3.85 2.95 8.10 4.55 3.32

Model

Depth to Bottom of Layer

Figure z,, km z2, km Reference Fit

THL I 15 3O 18 C THL 2 15 35 18 C THL 3 15 4O 18 C THL 4 15 45 18 C THL 5 15 50 18 C THL 6 20 30 19 C THL 7 20 35 19 B THL 8 20 40 19 B THL 9 20 45 19 C THL 10 20 50 19 C THL 11 25 30 20 C THL 12 25 35 20 A THL 13 25 40 20 A THL 14 25 45 20 C THL 15 25 50 20 C THL 2A 16.3 38.0 B

2. PRM Series

Layer km/sec km/sec g/cm 3

TABLE 5. (Continued)

Model

Depth to Bottom of Layer

z,, km z2, km z3, km Fit

PRM 1 PRM 2 PRM 3 PRM 4 PRM 5 PRM 6 PRM 7 PRM 8 PRM 9 PRM 10 PRM 11 PRM 12 PRM 13 PRM 14 PRM 15 PRM 16 PRM 17 PRM 18 PRM 19 PRM 20 PRM 21 PRM 23 PRM 24 PRM 25 PRM 26 PRM 27

26

26

26

26

26 26 26

26

26 22

22

22

22

22

22

22

22

22

18 18

18

18 18

18

18 18

38 38 38 35

35 35 32

32 32 38 38 38 35

35

35

32 32

32 38 38 38 35

35 32

32 32

5 47.O C 5 43 C 5 39.5 A 5 47 C 5 43 C 5 39 A 5 47 C 5 43 C 5 39 C 5 47 C 5 43 C 5 39 B 5 47 C 3 43 B 5 39 A 5 47 C 5 43 C 5 39 B 5 47 C 5 43 C 5 39 A 5 43 C 5 39 B 5 47 C 5 43 C 5 39 C

3. VEL Series

o•, l•, p, h, z, Layer km/sec km/sec g/cm 3 km km

i 6.14 3.50 2.91 22.5 22.5 2 3.14 13. 35.5 3 3.20 7.5 43. 4 8.10 4.50 3.35

Model •, •, •, •,

km/sec km/sec km/sec km/sec Fit

VEL 1 VEL 2 VEL 3 VEL 4 VEL 5

6.58 7.45 3.80 4.30 B 6.58 7.45 3.95 4.40 C 6.84 7.62 3.67 4.17 B 6.84 7.62 3.80 4.30 C 6.84 7.62 3.95 4.40 C

I 6.14 3.50 2.91 2 6.70 3.80 3.14 3 7.60 4.30 3.20 4 8.10 4.50 3.35

VEL 6 VEL 7 VEL 8 VEL 9

6.58 7.45 3.67 4.17 A 6.35 7.23 3.95 4.40 C 6.35 7.23 3. $0 4.30 B 6.35 7.23 3.67 4.17 B


ALBUQUERQUE THL MODELS

2 I

O. 0.1 0.2.

FREQUENCY CPS

Fig. 18. Best fit for Albuquerque interpolated from Figure 17 for a value of c -- 20 km/sec, compared with theoretical spectral ratios for models THL I through THL 5 (Table 5).

i I I I I I I I I

ALBUQUERQUE TH L//•Mi•DE LS

I I I i [ I I I I O. 0.1 0.2

FREQUENCY CPS

Fig. 20. Best fit for Albuquerque interpolated from Figure 17 for a value of c -- 20 km/sec, compared with theoretical spectral ratios for models THL 11 through THL 15 (Table 5).

frequency window to smooth the theoretical spectrums before dividing. While this tends to remove detail and thus reduce the resolution

between different models, the alternative of using a much narrower frequency window (i.e. a broader lag window) reduces the consistency of the data.

In Table 5 I have listed the theoretical struc-

tures considered for comparison with the data. The models labeled THL involve a fairly con- ventional set of velocities for a two-layer crust. A comparison of the THL theoretical spectral ratios with the data is offered in Figures 18 to 20. The column labeled 'fit' in Table 5 is a

graded estimate of the degree to which the

I i ! { I i I I

ALBUQUERQUE THL MODELS

_

O. 0.1 0.2

FREQUENCY CPS

Fig. 19. Best fit for Albuquerque interpolated from Figure 17 for a value of c ---- 20 km/sec, compared with theoretical spectral ratios for models TIlL 6 through TIlL 10 (Table 5).

model fits the data. In assigning this I have placed more weight on agreement at longer periods because of the better data consistency there.

The series of models labeled PRM is an inde-

pendent attempt to find a fit. Since studies in the Great Basin have indicated a layer of anomalously high-velocity crustal material (or low-velocity mantle), I have tested this possibility here. For reasons of space the curves are not shown, but a graded degree of 'fit' has been assigned. A shorter series of models, labeled VEL, is included to test possible variations in velocity.

No single-layer structures have been used for comparison. When such structures are calculated without smoothing, the peaks in the spectral ratio are very sharp (Figure 2). Of course this cannot be seen if both data and theory are passed through our 24-sec lag window. However, if the data are analyzed with a 60-sec lag window, a corresponding increase in the amplitudes of the peaks is not observed. Instead, the data merely show more nonsystematic raggedness, which is due to the larger lags being admitted into the spectrum.

The best-fitting ('A') models are shown in Figure 21. With the help of this figure and the list of models that do not fit (Table 5) a few conclusions can be drawn about the structure

at Albuquerque. 1. The M discontinuity lies between 35 and

3016

6

IO

20

7

ROBERT A.

8 km/sec • I

I I

4.4 •

PRM 21

""'"""• ,,.---P R M 15 [,

PRM I:• "" " • ...... 40 - THL I• ':

50- ;

Fig. 21. Structure sections for models in Table 5 which agree most closely with the data ('A' rating).

2. The 6.14-km/sec layer appears to extend to around 25 km, although two exceptions can be cited from the PRM models.

3. If a 7.6-km/sec layer exists at the base of the crust, it cannot be much thicker than 5 or 6 km.

Since the resonance frequencies are determined by total travel times between the surface and the main structural discontinuities, it appears to be possible to adjust velocities and thicknesses jointly in such a way that the reso- nances remain unchanged. Therefore it is diffi- cult to assess the range of permitted structures when the velocities are permitted to vary. The one VEL model which fits well does, however, look very much like the other models shown in Figure 21.

Discussion. Determination of crustal struc-

ture has always rested on a combination of methods. Body wave spectral ratios appear to be most critical with respect to structure in the lower half of the crust (in the period range 5

PHINNEY

to 150 see). Seismic refraction studies are especially good for the upper half of the crust. Grav- ity data are useful for obtaining averages over the entire structure. It appears that the spectral ratio method can be used very effectively in combination with either of these familiar methods. The principal drawback is one of time; a station must be occupied for about a year and a half in order that enough earthquakes may be recorded. The technique can now be applied at any station having a routinely operating long-period seismometer.

One of the most attractive features of using spectral ratios is that ordinary photographic records can be used. This advantage is lost, however, if high standards of station mainte- nance and recording are not adhered to. In particular, daily calibration signals must be avail- able, and the paper speed must be fast enough to resolve all details of the signal at periods around 3 sec. The photographic contrast must be high and the light beam properly focused. Signal fidelity, a necessary goal for stations de- siring to use modem methods of analysis, can be attained on paper records if sufficient care is exercised.

The use of long-period surface waves for structure determination differs from the spectral ratio method in that it deals with structures

averaged over hundreds or thousands of kilometers. The spectral ratio method deals with horizontal distances of the same dimension as

the crust under the station. The effects of struc-

tural inhomogeneity are much less with the body waves because of the near-vertical angles of incidence of the constituent plane wave functions. We can see this, for example, at Bermuda, which is a steep truncated pile of volcanics about 50 km across at the base. Of approximately 30 examples studied, only one Rayleigh wave train recorded there fails to be badly distorted by 'beats.' It is reasonable to attribute this to the

effect of the island's conical shape. In contrast, P waves have spectral ratios which agree quite well over a broad frequency range.

Determination of crustal structure using P wave spectral ratios is only the first step in a sequence of investigations of the distortion of body waves. After we have obtained the structure and the transfer function for a number of

stations, the following further investigations look very promising.


1. Assume that the principal cause of non- agreement between spectral ratios is the non- stationary character of the signal due to superposition of pP on the record. Use this to determine the depth of focus. Several stations must be used as an independent check.

2. Introduce a correction to observed travel

times of short-period P waves, using the crustal structure determined with the long-period P waves. Any significant delays in such corrected travel times must then be due to differences in

mantle velocities. Analysis of P wave delays can be quite ambiguous if one does not know how to allot the time lags to crust and mantle.

3. Perform relative transmission studies of

ray pairs emanating from a given source. For example, P at 95 ø and PKP at 150 ø involve transmission paths which are practically identical in regard to the mantle. The PKP has, in addition, passed twice through the core-mantle boundary. Since we are able to remove the signal distortion due to the crust near the two

receiving stations, we can determine whether the spectrum of PKP has been distorted by transmission through the core-mantle boundary.

4. Determine crustal structures near an

earthquake. After removing the distorting effect of the receiving station, we would attribute any remaining structure in the spectrum of a P wave to the summed contributions from rays multiply reflected in the crust above the source.

All the experiments proposed above require a good knowledge of crustal structure at the receiver. To increase the confidence that may be placed in that result, we can perform the following check. Determine the structures at a pair of stations a few hundred kilometers apart.

The vertical components at each station can then be compensated for this structure and their spectrums then divided. If the structure determination is correct, the resulting ratio must have constant amplitude and a phase equal to ,•(Ax/c). If results of such a test of a group of stations in a particular region are favorable, we should be able to place a high degree of confidence in the transfer ratios and structures.

Acknowledgments. This research was supported by the Air Force Ofi%e of Scientific Research under contract AF 49(638)-1243 as a part of the Ad- vanced Research Projects Agency's Vela-Uniform Program.

Use was made of computer facilities supported in part by National Science Foundation grant NSF-GP579.

REFERENCES

Birch, Francis, Elasticity and constitution of the earth's interior, J. Geophys. Res., 57, 227-286, 1952.

Haskell, Norman A., Crustal reflection of plane P and SV waves, J. Geophys. Res., 67, 4751-4767, 1962.

Nuttli, Otto, and John D. Whitmore, An observa- tional determination of the variation of the

angle of incidence of P waves with epicentral distance, Bull. $eismol. $oc. Am., 51, 269-276, 1961.

Nuttli, Otto, and John D. Whitmore, On the determination of the polarization angle of the $ wave, Bull. $eismol. $oc. Am., 52, 95-108, 1962.

Ofi%er, Charles B., Maurice Ewing, and Paul C. Wuenschel, Seismic refraction measurements in the Atlantic Ocean, Bull. Geol. $oc. Am., 63, 777-808, 1952.

Woollard, G. P., Crustal structure beneath oceanic islands, Proc. Roy. $oc. London, A, 222, 361- 387, 1954.

(Manuscript received February 28, 1964.)

structure of the earth's crust from spectral behavior of ...€¦ · resemblance to the...

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