SKP AND RELATED PHASES* By R. D. FORESTER ABSTRACT For shallow shocks, multiple SKP phases are observed after the initial SKP motion as long as 54 seconds on short-period instruments and as long as 87 seconds on long-period instruments. Ampli- tude data indicate that each multiple phase has a focal point similar to that of the initial SKP phase. The focal point for waves having periods of 1 to 5 seconds occurs at 131~ °, and that for waves having periods of 5 to 10 seconds is broadly defined between 130 ° and 131 °. Short-period SKP waves extend from 129 ° to at least 140°; long-period SKP waves, from 125 ° to 145°. The long- period waves are believed to be diffracted from the caustic in accordance with Airy's hypothesis. For all types of SKP phases the energy content of the short-period waves is several times less than that of the long-period waves. For the vertical component the agreement between theoretical and observed values of energy of long-period waves is good. For the horizontal component the observation of too little energy is not satisfactorily explained. SKP", the SKP phase associated with the inner core, is observed between 114° and 125 °. It records with very short periods. The observations of SKP" present additional support for the hypothesis of large but continuous increase of velocity at the transitional boundary of the inner core. K INNER CORE OUTEN CORE MANTLE Fig. 1. Ray paths of PKS. INTRODUCTION THE RAY components of the core phase PKS are shown in figure 1. The path rela- tionship of SKP to PKS is reversed. For a shock of zero focal depth PKS and SKP should have identical travel times. For increasing focal depths the time interval by which SKP leads PKS increases. * Manuscript received for publication March 8, 1955. [ 185 ]
Transcript
SKP AND RELATED PHASES*
By R. D. FORESTER
ABSTRACT
For shallow shocks, multiple SKP phases are observed after the initial SKP motion as long as 54 seconds on short-period instruments and as long as 87 seconds on long-period instruments. Ampli- tude data indicate that each multiple phase has a focal point similar to that of the initial SKP phase. The focal point for waves having periods of 1 to 5 seconds occurs at 131~ °, and that for waves having periods of 5 to 10 seconds is broadly defined between 130 ° and 131 °. Short-period SKP waves extend from 129 ° to at least 140°; long-period SKP waves, from 125 ° to 145 °. The long- period waves are believed to be diffracted from the caustic in accordance with Airy's hypothesis.
For all types of SKP phases the energy content of the short-period waves is several times less than that of the long-period waves. For the vertical component the agreement between theoretical and observed values of energy of long-period waves is good. For the horizontal component the observation of too little energy is not satisfactorily explained.
SKP", the SKP phase associated with the inner core, is observed between 114 ° and 125 °. I t records with very short periods. The observations of S K P " present additional support for the hypothesis of large but continuous increase of velocity at the transitional boundary of the inner core.
K
INNER CORE
OUTEN CORE
MANTLE
Fig. 1. Ray paths of PKS.
INTRODUCTION
THE RAY components of the core phase PKS are shown in figure 1. The path rela- tionship of SKP to PKS is reversed. For a shock of zero focal depth PKS and SKP should have identical travel times. For increasing focal depths the time interval by which SKP leads PKS increases.
* Manuscript received for publication March 8, 1955.
[ 185 ]
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SKP AND RELATED PHASES 187
In figure 2, travel times for PKS computed by Jeffreys (1939b) can be compared with those synthesized by the method of Wadati and Masuda (1934) from travel times given by Jeffreys (1939a) for PcS and from travel times given by Gutenberg (1951a) for K. In the two figures, "A," "B," "C," and "D" designate corresponding ray paths and travel times.
The difference between the two travel-time curves in figure 2 results primarily from the difference in core velocities computed by Jeffreys (1939b) and Gutenberg (1951a). (See fig. 3.)
D E P T H IN K M
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R A D I U S IN KM Fig. 3. Calculated velocity of longitudinal waves in the core.
I 5OO 0
The slope of the upper focal branch, "A-B," is similar for the two curves. The focus occurs at 130 ° for Jeffreys' curve and at about 131 ° for the New Curve. The distance at which the "P" component of PKS grazes the outer core is 148 ° for Jeffreys' curve and 143 ° for the New Curve. Which of these two values is more nearly correct is difficult to verify because of the low amplitudes associated with rays which have a travel-time curve the slope of which approaches that of a straight line.
The apparent surface velocity (reciprocal of the slope of the travel-time curve) of the main focal branch, "B-C," is larger for Jeffreys' curve than for the New Curve because Jeffreys computes higher velocities for the outer core than does Gutenberg.
The main focal branch of PKS extends from 130 ° to 140 ° for Jeffreys' curve and from 131 ° to 149 ° for the New Curve. "C" represents the ray which grazes the inner core. The distance for "C" is shorter for the curve of Jeffreys than for the New Curve, partly because Jeffreys (1939b) calculates 130 km. less depth to the inner core than does Gutenberg (1951a), and pkrtly because Jeffreys calculates a larger velocity gradient for the outer core than does Gutenberg.
188 BULLETIN OF TItE SEISMOLOGICAL SO,CIETY OF AMERICA
The purpose of the present investigation is to study the travel times and energy of SKP and related phases, and to determine if the features observed for SKP are consonant with those observed by Denson (1952) for PKP. Associated with SKP, there should be an inner core phase, SKP", which is analogous to P", the inner core phase associated with PKP.
MATERIALS
Seismograms recorded at Pasadena, at some of its auxiliary stations (Tinemaha, Haiwee, Riverside, Mount Wilson, and China Lake), and at Huancayo comprised the main source of data for this study. Some data were obtained from seismograms recorded at Tucson, Weston, and Harvard.
Readings were taken from seismograms recorded as early as 1928 at Pasadena and as early as 1932 at Huancayo. No shocks recorded after 1950 were investigated. The shocks examined ranged in magnitude from 6 to 8~ . Shocks of large magni- tudes were frequently rejected on account of their complexity.
An epicentral range from 105 ° to 155 ° was investigated. The distribution of data with epicentral distance was uneven. A serious lack of data existed at 135 °. Most of the examined shocks which were recorded at Pasadena, a t its auxiliary stations, and at Tucson originated in the Sunda Arc. At Pasadena, Sumatra shocks yielded a large amount of data for the study of the SKP focal point. The examined shocks which were recorded at Huancayo originated in the western half of the Pacific Island Belt.
TRAVEL TIMES AND AMPLITUDES
Shallow shocks.--Unsmoothed travel-time data indicate that six rather definite SKP phases appear on both long- and short-period instruments for both vertical and horizontal components. These six phases agree closely with the times given for them by Gutenberg and Richter (1934). Two later phases are sometimes well re- corded on long-period instruments. Multiple SKP phases are classified below accord- ing to their arrival time after the initial SKP phase, 1SKP:
Phase . . . . . . . . . . . . . . . . . 2SKP 3 S K P 4 S K P 5 S K P 6 S K P 7 S K P 8SKP Arrival time after 1SKP
It is rare that all the phases classified above can be clearly discerned on one seis- mogram, or even on a group of seismograms of one earthquake.
The initial SKP motion observed on horizontal instruments appears to lag behind that observed on vertical instruments by 2 or 3 seconds. Actually, the initial motion generally observed on horizontal instruments is not 1SKP, but 1PKS, which has a slightly longer travel time and a much greater horizontal component of amplitude than 1SKP.
3SKP is strongly related to prominent multiple phases associated with P" and PP. The time interval between such multiple phases and the initial motion for P" and for PP is smaller than that between ~SKP and ~SKP. ~SKP probably represents a phase reflected from the earth's surface. 3SKP often appears to be 180 ° out of phase with ~SKP, and is therefore suspected to be sSKP.
On all instruments the wave periods are noticeably shorter for ~SKP than for the later phases. There is only a slight increase in wave period from 2SKP to sSKP.
S K P A N D R E L A T E D P H A S E S 189
Short-period S K P waves could not be followed with certainty beyond 140°; long- period S K P waves, beyond 145 °. Short-period S K P waves could not be followed to distances less than 129°; long-period S K P waves could be followed as far back as 125 ° .
Although the t ravel- t ime curves of all S K P phases appear to parallel one another, there is some tendency for the later S K P phases to have relatively larger ampli tudes
a
Fig. 4. a. Shock of September 27, 1937 at 08:55:10, G.C.T., normal depth, epicenter 9~/~ ° S, 111 ° E,
magnitude 7.2. Recorded at Tucson on Benioff short-period vertical, epicentral distance 135 °. .b. Shoc k of September 21, 1934, at 12:38:54, G.C.T., depth 100 kin., epicenter 2 ° N, 99 ° E, mag-
nl~uoe 6~. Recorded at Pasadena on Benioff short-period vertical, epicentral distance 130 °. c. Shock of March 22, 1944, at 00:43:I8. G.C.T., depth 220 km., epicenter 8~/~ ° S, 123~/~ ° E,
magnitude 7.5. Recorded at Pasadena on Benioff short-period vertical, epicentral distance 118 °.
as the distance from the focus increases. The general diminution of ampli tudes for successively later S K P phases is more pronounced for short-period than for long- period waves.
Evidence for the upper focal branch, SKP~, exists in the fact tha t the multiple phases, which appear to be uni tary near the focus, often appear as couplets be- tween 135 ° and 138 °. The two par ts of the couplets are separated by intervals of from 2 to 5 seconds. (See fig. 4, a.) The later par t of each couplet has the same period, but lesser ampli tude than the first part . Because of their small t ime separa- t ion the two focal branches can be discerned only on short-period instruments.
190 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA
Shocks of 60 km. depth.--Owing to the interference from reflected PP phases, the initial SI4P motion is not easily discerned at distances greater than 140 ° for shocks of 60 kin. depth. Short-period SKP waves are not observed at distances less than 129 °. Long-period SKP waves appear at distances as short as 127 °. PKS appears at distances as short as 125 ° on the very long-period Huancayo records.
For SKP the amplitude is larger for the vertical than for the horizontal compo- nent. For PKS the reverse is true.
pPKS has high amplitudes on the horizontal instruments. Its time twin, pSKP, is usually prominent on the vertical instruments. On the vertical instruments sSKP often has amplitudes exceeding those of SKP. sPKS, which is the time twin of sSKP, usually has high amplitudes on the horizontal instruments.
As a rule, strong PKS motion is followed by strong pPKS motion and strong SKP motion. The reversal in phase between sSKP and SKP is usually evident when the pulses are distinct. (See fig. 4, b.)
OBSERVED ENERGY
Shallow shocks.--The equation used to investigate the observed energies is based upon theory presented by Gutenberg (1945a) :
A0 = M - log ~TT - K ( M - 7) - G (1)
A 0 = energy parameter. The larger the value of A 0, the less is the energy. M = earthquake magnitude W = trace amplitude. T = wave period. K = a residual correction factor given by Gutenberg (1945a) for the magni-
rude determination of earthquakes from seismic body waves. G --station ground factor as given in table I of Gutenberg (1945b).
Figures 5 and 6 give values for A0 as a function of the epicentral distance for various phases and recording instruments. A hollow circle denotes a short-period seismometer; a half-filled circle, a long-period seismometer; and a solid circle, a very long-period seismometer. (A seismometer is classified as short-period if the product of the pendulum and galvanometer periods in seconds is less than 1; as long-period, if between I and 10; and as very long-period, if greater than 10.)
Both the long-period and the short-period data indicate that the focal point occurs at 131~/~ °. Very long-period data from horizontal instruments indicate a broad focus between 130 ° and 131 ° . The increase of energy at the focus is more marked for the short-period than for the long-period data. Short-period focal energy could not be detected at distances less than 129 ° . Very long-period focal energy persists as far back as 125% The appearance of a definite focus of energy for each of the later phases corroborates the assumption that they are SKP phases.
A gradual decrease in energy with increasing distance from the focal point is evident on all plots up to about 145 °. On the vertical instruments the energy in- creases a~ distances beyond 145 °. This may result because some late PP phases may have been mistaken for SKP phases. Theoretically, PP energy should rise to a focus
SKP AND I%ELATED PHASES 191
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E P I C E N T R A L D ISTANCE
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BULLETIN O~ T I l E SEISMOLOGICAL SOCIETY OF AMERICA
Fig. 6. Values for A0 as a function of the epicentral distance for various phases and recording horizontal instruments. H = N.
IS0 °
S K P A N D R E L A T E D P H A S E S 193
as the antipode of the shock epicenter is approached, whereas SKP energy should drop to zero.
Over the entire range of epicentral distance investigated the short-period energy is significantly less than the long-period energy. This is true for all types of SKP phases.
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E P I C E N T R A L D I S T A N C E
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Fig. 7. Values for A0 for shocks about 60 kin. depth.
SKP V E R T I C A L
1 4 0 ° 1 4 2 °
1SKP contains the most energy; however, ~SKP is nearly as strong. The decrease in energy for the successively later phases is more marked for the short- than for the long-period records.
Shocks of 60 kin. depth.--For shocks of about 60 km. depth, values of A0 were determined for the vertical component of SKP and sSKP and for the horizontal component of PKS. (See fig. 7.)
All three plots in figure 7 indicate that the focal point lies between 131 ° and 132 °. For SKP and PKS the focal point is more prominent for short- than for long-period w a v e s .
194 B U L L E T I N OF T I I E SEISMOLOGICAL SOCIETY OF AMERICA
For SKP and PKS the energy is less for the short- than for the long-period waves. For sSKP the reverse is true. The energy is nearly as large for sSKP as for SKP.
SKP"
SKP '1 was observed sporadically between distances of 114 ° and 125 ° for shocks ranging in depth from normal to 2 2 0 km.
The fact that SKP" records with short periods on both long- and short-period instruments indicate that it has a narrow spectrum of energy.
A striking feature of SKP" is that for distances less than 120 ° it frequently ap- pears as a set of twin pulses separated by an interval of from 1.6 to 3.9 seconds. (See fig. 4, c.) Such twins may signify the two branches of a focus.
The slope of the New Curve for SKP" ranges from 1.9 sec/deg, at a distance of 115 ° to 1.6 sec/deg, at a distance of 125 °. The slope actually observed for the SKP" travel-time curve ranges from 1.5 to 2.4 sec/deg. The data for the observed slopes were too sparse to permit noting whether or not they decrease with epicentral dis- tance. The observed slopes are definitely too low to associate SKP" with principal SKP motion, which has a slope of 3.9 sec/deg, near the focal point.
It was desired to check the observed SKP" travel times against those of the New Curve, which was constructed for shocks of zero depth. Use was made of the deep- focus travel times given by Gutenberg and Richter (1936) for SKP at a distance of 145 °. The difference between the SKP travel time for zero depth and that for the depth of interest was subtracted from the SKP" travel times for the New Curve. The travel times of SKP" adjusted in this way for depth checked the observed travel times to within a few seconds. No consistent deviation between the observed and the computed times was noted.
The energy observed for SKP" is low relative to that for SKP. The inability to identify SKP" between 125 ° and the SKP focal point may result from the fact that SKP" motion in this distance range is too weak to be distinguished from micro- seisms. Beyond the SKP focal point, SKP" is most probably much too weak to be distinguished from the strong SKP motion.
THEORETICAL ENERGY
The equation used to determine the theoretical energy content of SKP and related phases was:
] F 1 F 2 F 8 • • • e -~KeD d cos i h / d A At C log Q si~ X co~ i~ (2)
It is derived from equations presented by Gutenberg (1944) and is based on a theory originated by Zoeppritz (1912). The basic assumption in its use is that energy is governed by the geometry of the ray paths, an assumption which, as pointed out by Fu (1947), is not inherent in Fermat's principle.
C is a constant related to the amount of energy imparted to a particular wave type at its source. Gutenberg (1945a) has found that C is approximately the same for both longitudinal and transverse waves. His value of 6.3 was used for all calcu- lations.
S K P A N D R E L A T E D P H A S E S 195
Q is the ratio of ground displacement to incident amplitude. It was determined from equations given by Gutenberg (1944).
/'1 F2 F~ • • • are factors which indicate the energy ratios of refracted and reflected waves to incident waves. The energy ratios for refraction at the core boundary were taken from Dana (1944). For the reflected phases (pSKP, pPKS, sSKP, and sPKS) surface-energy ratios were computed from equations given by Gutenberg (1944).
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I H O R I Z O N T A L
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1300 1320 134 ° 1310 |~l ° 140 ° 14~0 144e 1410
EPIGENTRAL DISTANCE
Fig. 8. Theoretical energy parameters for various SKP phases.
e -st:dD accounts for absorption along the ray path, D. The absorption coefficient, K, was set equal to O.O0012/km., the value determined by Gutenberg (1945a) for core waves. The attenuation factor obtained for SKP amplitudes averaged about 1/~.
A is the epicentral distance. i~ is the angle of emission of a ray from the hypocenter. i~ is the angle of surface incidence at the recording station.
The prediction of energy by eq. (2) requires that travel times be known to a high degree of precision. Observed SKP data are too few to yield a travel-time curve of the required precision. To attain such precision the travel times of SKP were calcu- lated by integration along its component travel paths in the mantle and the core. For this integration the velocity distribution within the earth was divided into seg- merits represented by continuous functions. Use was made of the velocities for the mantle and the core listed in tables 69 and 73, respectively, of Gutenberg (1951b).
The slopes of the calculated SKP travel-time curve were used to determine d cos ih/dA, cos is, and the angles of incidence necessary to evaluate Q. Little error
196 BULLETIN OF TItE SEISMOLOGICAL SOCIETY OF AMER,ICA
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E P I C E N T R A L D I S T A N G E
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Fig. 10. Theoretical energy parameters for various sSKP phases.
t 4 4 0 146 o
S K P AND I~ELATED P H A S E S 197
should result from the fact that the slopes of the calculated SKP travel-time curve were used not only for SKP and PKS, but also for pSKP, pPKS, sSKP, and sPKS.
At the hypocenter, a velocity of 7.5 km/sec, was used for compressional waves and a velocity of 4.0 kin/see, for shear waves. At the earth's surface, a velocity of 5.5 km/sec, was used for compressional waves and a velocity of 3.2 kin/see, for shear waves.
In figures 8, 9, and 10 the theoretical energy parameters are plotted against epi- central distances for the various SKP phases. Dashed lines denote the upper focal branch; solid lines, the main focal branch.
The graphs indicate that the horizontal components of PKS and pPKS, and the vertical components of SKP and sSKP, have the largest amplitudes. Although the amplitudes of the two focal branches for each phase are the same at the focal point, amplitudes for the upper focal branch fall off more rapidly with increasing distance from the focal point. This may well explain why the upper focal branch was not detected at distances greater than 138 °. However, the energy parameters calculated for the upper focal branch lose physical significance for distances greater than 135 °, because eq. (2) is not valid for rays which graze the core.
The initial velocities of 7.5 kin/see, used for PKS and 4.0 kin/see, for SKP are probably too high for shocks originating in the upper granitic layers and too low for shocks originating below the Mohorovi~i6 discontinuity. Higher velocities at the hypocenter mean higher values for d cos ih/dA, and hence more energy for the core phases. Accordingly, the energy calculated for very shallow shocks may be too large and that for shocks of intermediate depths too small.
Table 1 gives the residuals for the observed minus the theoretical "A" values for some of the more prominent SKP phases. The residuals are given for shocks of normal and 60 km. depths. Data are presented for the long- and short-period instru- ments in separate rows. Question marks signify doubtful data; blanks, that no data are available.
The energies of pPKS and pSKP, and of sSKP and sPKS, are inseparable by virtue of identical travel times. Since the horizontal amplitudes of pPKS should be twenty times larger than those of pSKP, little error should result in attributing the combined energies wholly to pPKS. Since the vertical amplitudes of sSKP should be twenty-five times larger than those of sPKS, little error should result in attributing the combined energies wholly to sSKP. For shallow shocks, the values of A0 for sSKP and pPKS were assumed to be those computed for 3SKP in figures 5 and 6. The energy comparisons for A = 132 ° give a crude measure of the "focal energy," and have significance because the actual focus is rather broad. The comparisons for A = 135 ° to 140 ° have, of course, more physical meaning.
The vertical residuals are relatively small; hence the observed and theoretical energies of SKP and sSKP are in good agreement. The large positive values for the horizontal residuals indicate that the observed energies of PKS and pPKS are several times smaller than the theoretical energies. The problem of the recording of too little energy by horizontal instruments is not satisfactorily explained:
1. The hypothesis that too little energy was transmitted across the core boun- daries would contradict the fact that the residuals for the SKP phases are small.
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S K P A N D RELATED P H A S E S J99
Because of the reciprocity of the transmission coefficients associated with the core, they should affect the energy content of PKS and SKP phases equally.
2. The hypothesis that the energy imparted to the PKS phase at the hypocenter is smaller than that assumed would contradict the observation of too much energy on both horizontal and vertical instruments for PeP by Martner (1950) and for PKP by Denson (1952).
3. The hypothesis might be advanced that the ratio of the horizontal to the verti- cal component of amplitudes of shear waves incident to the surface is smaller than that implied. However, this would tend to contradict the observation of too much energy for the horizontal relative to the vertical component found by Martner for PeP and by Denson for PKP.
SKP, PKP, AND THE INNER CORE
Observed energy parameters for SKP indicate that the focal point for short-period waves occurs at 1311/~ ° and that the focal point for long-period waves is broadly defined between 130 ° and 131 °. For PKP, Denson (1952) notes that the focal point for short-period waves occurs at 147 ° and that for long-period waves at 143 ° . Thus, the dependence of the position of the focal point upon the wave periods is less marked for SKP than for PKP.
Denson suspects that the PKP1 caustic may be affected by diffraction in accord- ance with Airy's theory. For SKP, the backward extension from the focus of short- period waves to only 129 ° and that of long-period waves to 125 ° strongly suggests a diffraction phenomenon.
Denson observed that long-period PKP waves persist with large amplitudes to distances as far as 157 °. The New Curve for SKP, which is based indirectly upon some of Denson's results for PKP, indicates that SKP1 should extend to 149 °. Actually, long-period SKP phases could not be observed with certainty beyond 145 °. This may be due partly to the interference from late PP phases.
Denson notes that the amplitudes of short-period PKP waves fall off rapidly between 146 ° and 157 °. Amplitudes of short-period SKP1 waves fall off rapidly be- tween 131~ ° and 140 °. Gutenberg and Richter (1938) suggest that PKP1 may ex- tend to epicentral distances greater than that which is subtended by the ray which grazes the inner core. The observed data for both PKP and SKP indicate that as a result of diffraction the longer-period waves may extend into the shadow zone produced by the inner core.
Denson notes that P" records with highest amplitudes between 120 ° and 125 °. Correspondingly, SKP" would be expected to record with highest amplitudes be- tween 114 ° and 119 °. Highest amplitudes were observed for SKP" between 114 ° and 122 ° .
At distances less than 120 ° or 125 °, p1, records with only short periods. At dis- tances between 114 ° and 122 °, SKP 1' records with only short periods.
The caustics occur at the same distance for the later SKP phases as for the initial SKP phase. Some of the multiple phases have travel times nearly independent of the wave periods. The en ~chelon nature of the travel-time curves for the multiple
200 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA
phases are less apparent for SKP than for PKP, although a shift of SKP energy to the later phases for increasing distance from the focal point is observed.
For PKP, Denson suspects that short-period waves arriving between 120 ° and 147 ° and long-period waves arriving between 145 ° and 157 ° have separated from each other during their passage through the earth. For SKP, a smaller magnitude of separation is indicated. Possibly, short-period SKP" waves arriving between 114 ° and 122 ° are separated from long-period waves arriving between 140 ° and 145 °.
The twin pulses observed for SKP" may signify the two branches of a focal point. However, the two branches would not be expected to be distinguished from each other unless the focal point occurs at a distance several degrees shorter than the 1 1 5 ~ ° indicated by the New Curve.
The observations of SKP" are essentially in harmony with those for P" and the data for the New Curve; and hence, corroborate to a limited degree Gutenberg's hypothesis of a large, but continuous increase in velocity in the transition zone at the boundary of the inner core.
ACKNOWLEDGMENTS
The writer is grateful to Dr. B. Gutenberg and Dr. C. F. Richter for inspiration and guidance at all stages of the work. Appreciation is extended to the Rev. F. J. Donohoe, Mr. R. W. Knox, and Dr. L. Don Leet for supplying additional seismo- grams.
CALIFORNIA INSTITUTE OF TECHNOLOGY, PASADENA~ CALIFORNIA. (Division of the Geological Sciences, contribution no. 715.)
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SKP AND RELATED PHASES 201
JEFFREYS, H. 1939a. "The Times of PcP and ScS," Mort. Not. Roy. Astron. Soc., Geophys. Suppl., 4: (7) 537-547. 1939b. "The Times of Core Waves (Second Paper)," ibid., 4: (8) 594-615.
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