Physics of the Earth and Planetary Interiors, 25 (1981) 297— 356 297 Elsevier Scientific Publishing Company, Amsterdam — Printed in The Netherlands Preliminary reference Earth model * Adam M. Dziewonski’ and Don L. Anderson 2 ‘Department of GeologicalSciences, Harvard University, Cambridge, MA 02138 (U.S.A.) 2 Seismological Laboratory, California Institute of Technology, Pasadena, CA 91125 (U.S.A.) (Received December 3, 1980; accepted for publication December 5, 1980) Dziewonski, A.M. and Anderson, D.L., 1981. Preliminary reference Earth modeL Phys. Earth Planet. Inter., 25: 297—356. A large data set consisting of about 1000 normal mode periods, 500 summary travel time observations, 100 normal mode Q values, mass and moment of inertia have been inverted to obtain the radial distribution of elastic properties, Q values and density in the Earth’s interior. The data set was supplemented with a special study of 12 years of ISC phase data which yielded an additional 1.75 X 106 travel time observations for P and S waves. In order to obtain satisfactory agreement with the entire data set we were required to take into account anelastic dispersion. The introduction of transverse isotropy into the outer 220 km of the mantle was required in order to satisfy the shorter penod fundamental toroidal and spheroidal modet This anisotropy also improved the fit of the larger data set. The horizontal and vertical velocities in the upper mantle differ by 2-4%, both for P and S waves. The mantle below 220 km is not required to be anisotropic. Mantle Rayleigh waves are surprisingly sensitive to compressional velocity in the upper mantle. High S~ velocities, low P~ velocities and a pronounced low-velocity zone are features of most global inversion models that are stqipressed when anisotropy is allowed for in the inversion. The Preliminary Reference Earth Model, PREM, and auxiliary tables showing fits to the data are presented. Preamble and his models A and B were employed exten- sively. The study of precession and nutation in astron- Seismological studies of the structure of the omy and geodesy, and of Earth tides and free Earth have developed rapidly since 1950, much oscillations in geophysics, need knowledge of the aided by the fast improvement in computer tech- internal structure of the Earth. The importance of niques. free and forced nutations, for instance, polar mo- Expansion in the utilization of computers made tion, Chandler and annual components, and di- it possible to construct many types of Earth mod- urnal motion of the Earth, in different fields of els. The consequent proliferation of Earth models science, emphasizes the value of the contribution had two consequences: of seismology for these researches. (I) There was a difficulty of choice of an ade- It was very difficult to set up models of the quate Earth model for researchers that depend on Earth’s structure before the advent of computers; the structure of the Earth, such as those listed in the more important ones were set up by Bullen, the first paragraph. (2) Several researchers adopted some properties • With a preamble by the Standard Earth Committee of the from one model and other properties from a sec- I.U.G.G. Followed by “A note on the calculation of travel ond model, with the consequence that their models times in a transversely isotropic Earth model” by L~ were not self-compatible. Woodhouse (this issue, pp. 357-359). These difficulties were pointed out, at a Sym-
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Physics of the Earth and Planetary Interiors, 25 (1981) 297— 356 297Elsevier Scientific Publishing Company, Amsterdam — Printed in The Netherlands
Preliminary reference Earth model *Adam M. Dziewonski’ and Don L. Anderson2
‘Departmentof GeologicalSciences, Harvard University, Cambridge, MA 02138 (U.S.A.)2 SeismologicalLaboratory, California Institute of Technology, Pasadena, CA 91125 (U.S.A.)
(Received December 3, 1980; accepted for publicationDecember 5, 1980)
Dziewonski, A.M. and Anderson, D.L., 1981. Preliminary reference Earth modeL Phys. Earth Planet. Inter., 25:297—356.
A large data set consisting of about 1000 normal mode periods, 500 summary travel time observations, 100 normalmode Q values, mass and moment of inertia havebeen inverted to obtain the radial distribution of elastic properties, Qvaluesanddensity in the Earth’s interior. The dataset was supplementedwith aspecial study of 12 yearsof ISC phasedatawhich yielded an additional 1.75X 106 travel time observations for P and S waves. In order to obtain satisfactoryagreement with the entire data set we were required to take into account anelastic dispersion. The introduction oftransverse isotropy into theouter 220 kmof the mantle wasrequired in order to satisfy the shorter penodfundamentaltoroidal and spheroidalmodet This anisotropy also improved the fit of the largerdata set. Thehorizontal and verticalvelocities in the upper mantle differby 2-4%, both for P and S waves.The mantle below 220 km is not required to beanisotropic. Mantle Rayleigh waves are surprisingly sensitive to compressional velocity in the upper mantle. High S~velocities, low P~velocities and a pronounced low-velocity zone are features of most global inversion models that arestqipressed when anisotropy is allowed for in the inversion.The Preliminary Reference EarthModel, PREM, and auxiliary tables showing fits to the data are presented.
Preamble and his models A and B were employed exten-sively.
The study of precession and nutation in astron- Seismological studies of the structure of theomy and geodesy, and of Earth tides and free Earth have developed rapidly since 1950, muchoscillations in geophysics, need knowledge of the aided by the fast improvement in computer tech-internal structure of the Earth. The importance of niques.free and forced nutations, for instance, polar mo- Expansion in the utilization of computers madetion, Chandler and annual components, and di- it possible to construct many types of Earth mod-urnal motion of the Earth, in different fields of els. The consequent proliferation of Earth modelsscience, emphasizes the value of the contribution had two consequences:of seismology for these researches. (I) There was a difficulty of choice of an ade-It was very difficult to set up models of the quate Earth model for researchers that depend on
Earth’s structure before the advent of computers; the structure of the Earth, such as those listed inthe more important ones were set up by Bullen, the first paragraph.
(2) Several researchers adopted some properties• With a preamble by the Standard Earth Committee of the from one model and other properties from a sec-I.U.G.G. Followed by “A note on the calculation of travel ond model, with the consequence that their modelstimes in a transversely isotropic Earth model” by L~ were not self-compatible.Woodhouse (this issue, pp. 357-359). These difficulties were pointed out, at a Sym-
298
posium on Earth Tides, during the 1971 IUGG arately and produce a consistent reference model.GeneralAssembly in Moscow (Vicente, R.O., 1973, Several approaches to the problem of setting up aBull. Geodesique No. 107, p. 105), and informal reference model were discussed, including al-discussions on the subject led to the setting up of a lowance for attenuation; it was decided to inviteworking group, composed of members of lAG and colleagues to produce and present complete mod-ISPEI, called the “Standard Earth Model Commit- els worked out by themselves and satisfying thetee”; the chairman was the late Professor K.E. guidelines laid down by the Committee. TheBullen. guidelines were published during 1976 in severalThe objective of the working group was to set scientific journals (Bull. Seismol. Soc. Am., Geo-
up a standard model for the structure of the Earth, phys. J., EOS, etc.) with the announcement thatfrom the center to the surface, defining the main proposed models should be presented during theparameters and principal discontinuities in such a IASPEI meeting in 1977.way that they could be adopted by the interna- The meeting of the Committee during thetional scientific community in any studies that IASPEI assembly in Durham (1977) was con-depended on the Earth’s structure. cerned with the presentation and discussion ofThe initial approach was to appoint several three different proposals, corresponding to re-
sub-committees dealing with different regions of searches done by D,L. Anderson, B. Bolt andthe Earth, composed of scientists specialising in A.M. Dziewonski. It appeared to be possible tothose areas. The original sub-committees were on: construct models taking account of damping, that(I) the hydrostatic equilibrium problem; (2) the is, of Q values. The Committee members presentcrust; (3) the upper mantle; (4) region D”; (5) core considered that the effects of attenuation wereradius; (6) P-velocity distribution in the core; and important and should be considered; but, since Q(7) density and rigidity of the inner core. was not well determined, instead of having oneDuring the meeting of the Symposium on reference model, we should have two reference
Mathematical Geophysics (Banff, 1972), three re- models—one including Q values’ and the othersearch groups, headed by D.L. Anderson, F. Gil- without Q values.bert and F. Press, presented models giving the At this meeting it was evident that the originalmain parameters; in spite of the fact that these reference model envisaged was growing more andmodels employed different data sets and computa- more detailed, thanks to the rapid progress oftion techniques, the values obtained for the core seismology. Several features that could not haveradius agreed within 0.2%, which was a remarkable been considered in 1971 were now feasible. It wasresult. The papers were published in Geophys. J., decided to entrust to D.L. Anderson and A.M.vol. 35 (1973); there was by then a general feeling Dziewonski the task of presenting a suitable refer-that it was possible to set up an adequate standard ence model.Earth model. The first report on the reference model ofDuring the IASPEI meeting in Lima (1973) Anderson and Dziewonski was presented at the
there was agreement about the need for a para- meeting of the Committee on Mathematical Geo-metrisation of the model to be adopted. It was physics (Caracas, 1978), with the statement thatdecided to call the model for the Earth’s structure the gross Earth data employed for the constructiona “reference model”, following the example of of the model was being enlarged, using the ISCgeodesy where there is a reference ellipsoid. In tapes. -
spite of this change in the name, it was agreed to During the 1979 IUGG General Assemblykeep the same name for the Committee. The re- (Canberra) a preliminary report on the Interimports of the sub-committees were published in Reference Earth Model was presented by A.M.Phys. Earth Planet. Inter., vol. 9 (1974). Dziewonski and D.L. Anderson. It was agreed byBy the time of the 1975 IUGG General Assem- the Committee that the report should be submitted
bly (Grenoble) it was evident that it would be to Physics of the Earth andPlanetary Interiors, anddifficult for several sub-committees to work sep- that interested seismologists should be encouraged
299
to use the Preliminary Earth Model and send within which the properties are assumed to varycomments to the authors. In order that the authors smoothlypredetermines the general features of theshould be able to consider such comments, and to final model andmust be done with care. In generalmodify themodel in the light of them if they think terms, we follow the recommendations of the Sub-fit, such comments should be in the hands of committee on Parainetenzation, whose findingsDziewonski andAnderson as soon as possible. were published among reports of the other sub-Professors Dziewonski and Anderson have been committees in Phys. Earth Planet. Inter., vol.9
asked to make a further Report, which the Corn- (1974).mittee hopes to be able to regard as the conclusion Section 3 contains a description of the datasetsof its work, at the IASPEI Assembly in 1981. that we use in inversion for the final model. Sec-
tion 4 describes procedures adopted in formula-E.R. Lapwood, Chairman tion of the starting model. In Section 6 weprovideR.O. Vicente, Secretary a brief outline of the inversion method used, the
For theStandard EarthMode~:n~m~~ final (in terms of this report only, of course)modeland discussion of the fit to the data.There have been a number of important prob-
1. Infroduction lems and decisions that we have faced during thecourse of this work. Some of these decisions re-
A variety of geophysical, geochemical and as- quired choices between conflictingdatasets. Otherstronomical studies require an accurate description were resolved by accepting that the velocities inof the variation of elastic properties and density in the real Earth are dispersive. The most difficultthe interior of the Earth. This paper contains a decision to make was whether we should drop thebrief description of a new Earth model, PREM, assumption of isotropy. A large amount of im-that satisfies the guidelines established by the portant data could not be fit adequately with theStandard Earth Model (S.E.M.) Committee at its preliminary isotropic Earth models. So we viewmeeting in Grenoble in 1975. In order to satisfy anisotropy and anelastic dispersion as essentialthe large amount of precise normal mode, surface complexities. We have, however, tabulated resultswave and body wave data, we have found it neces- for the “equivalent” isotropic Earth. “Equivalent”saiy to introduce anelastic dispersion and anisot- means that the model has approximately the sameropy. The model is therefore frequency-dependent bulk modulus and shear modulus as the aniso-and, for the upper mantle, transversely isotropic. tropic model, not that it provides an equivalent, orWe present tables containing the velocities, elastic satisfactory, fit to the data. The isotropic moduliconstants and Q as a function of radius, and are calculated with a Voigt averaging scheme and,auxiliary parameters such as gravity, pressure, therefore, represent the least upper bound. WedK/dP and the Bullen parameter, ilB. encourage the geophysical community to test andThe model is parametric in nature, a concept evaluate the model and the tables and to contrib-
discussed, and, in general terms, found acceptable ute to refinement of what we call the Preliminaryduring the meeting of the S.E.M. Committee, Reference Earth Model, or PREM.chaired by the late Professor Keith Bullen, inLima, 1973. We have provided analytic formulaefor the~seismic velocities, density and quality fac- 2. The concept of the modeltor Q as a function of radius. For this purposeweuse low-order polynomials in radius, of order be- An average Earth model, the subject of thistween zero and three, to describe seismic velocities, work, is a mathematical abstraction. The lateraldensity and attenuation in the various regions of heterogeneity in the first few tens of kilometers isthe Earth. so large that an average model does not reflect theSection 2 deals with the basic concept of the actual Earth structure at any point. In construc-
model. The division of the Earth into regions tion of the structure within the first 100 km we
300
have adopted the concept of weighted average: of free oscillations or travel times of body waves,assuming that oceanic crust covers two-thirds of but there are no equivalent expressions for dif-the Earth’s surface and that the average depth to fused transitions.the Moho is II km under oceans and 35 km under The preceding discussion dealt only with thecontinents, we arrive at a figure of 19 lan for the elastic properties of the Earth. One fundamentaldepth to the Moho for the average Earth. This is assumption that we make an interpretation of theused as the trial starting value, data in the seismic frequency band is that theWe recognize the following principal regions seismic quality factor Q is independent of the
within the Earth: frequency. This hypothesis seems to be consistent(I) Ocean layer. with most of the currently available information(2) Upper and lower crust. except perhaps for periods shorter than 5—10 s.(3) Region above the low velocity zone (LID), We in no way imply that Q is exactly independent
considered to be the main part of the seismic of frequency at all depths and over the entirelithosphere. When we finally dropped the as- seismic frequency band but only that most datasumption of isotropy the distinction between can be satisfactorily fitted with this assumption.LID and LVZ became less pronounced.
(4) Low velocity zone (LVZ).(5) Region between low velocity zone and 400 km 3. The gross Earth data set
discontinuity.(6) Transition zone spanning the region between There are three principal subsets of data:
the 400 and 670 km discontinuities.(7) Lower mantle. In our work we found it neces- 1. Astronomic-geodetic data
sary to subdivide this region into three parts Radius of the Earth: 6371 km. Mass: 5.974Xconnected by second-order discontinuities. 1024 kg. I/MR2: 0.3308. These values, listed in the(8) Outer core. guideline of the SEM Committee, were used as
(9) Inner core. constraints.While the existence of most of the regions listedabove has been recognized for some time, the 2. Free oscillation and long-period surface wavesubdivision of the upper mantle is still subject to datasome differences of opinion. We feel that evidence There is an impressive set of measurements offor the world-wide existence of a zone of low eigenfrequencies for over 1000 normal modes. Thevelocity gradient in the upper mantle is very strong precision ofmeasurementsvaries appreciably: fromand that the same is true with respect to at least up to 4 X l0~for some of the toroidal overtonestwo major discontinuities, although the actual to the 4X 106 recently reported for the funda-velocity gradients are still unresolved, mental radial mode.From a practical viewpoint, such a modular Qur set of normal mode data consists of about
construction is very convenient. It is much easier 900 modes collected from early observations corn-to perturb a particular feature of a model when it piled by Derr (1969), Dziewonski and Gilbertis separated from the remaining ones by clearly (1972, 1973), Mendiguren (1973), Gilbert anddefined discontinuities than to alter a model that Dziewonski (1975), and Buland et al. (1979). l’hisby definition is continuous. Other practical rea- dataset has been supplemented by measurementssons have to do with numerical applications. Many of dispersion of surface waves by Kanamori (1970),methods of construction of synthetic seismograms Dziewonski (1971), Dziewonski et al. (1972), Millscan satisfactorily treat abrupt discontinuities, but (1977), and Robert North (personal communica-arenot suitable for regions with very steep velocity tion, 1980).gradients. Another important issue is the inverse The data on attenuation are important in theproblem: there exist formulae for the effect of context of this report only to a limited extent. Thechange in the radius of a discontinuity on periods anelastic component of our model is primarily
301
meant as a tool for taking into account the small, equality of the area. Average travel times werebut important, frequency-dependence of the elastic computed only if at least five readings for a givenparameters. The published data on attenuation of 10 cell were available. Figures 1 and 2 show thenormal modes is of variable quality. We have deviations of our global average travel-time curvechosen a relatively small set of data based on from Jeffreys—Bullen travel times. Also shown aremeasurements of Kanamon (1970), Dratler et a!. residuals for other travel-time studies; these two(1971), Sailor (1978), Sailor and Dziewonski (1978), figures will be discussed more extensively later onStein and Geller (1978), Buland et al. (1979), Geller in the text.and Stein (1979), and Stein and Dziewonski (1980). All events analyzed were shallow, between 0
and 100 km in depth. For reasons that, at least3. Body waves originally, were not particularly relevant to thisObservations of travel times of body waves are report, we have also derived average travel-time
most numerous; there have been hundreds of curves separately for regions with shallow andstudies dealingwith this subject. The advantage of deepseismicity (but for shallow events only). Therethe body wave data over the normal mode ob- are significant and systematic differences betweenservations is that, because of shorter periods, they these two types of regions. The P-wave interceptare capable of higher radial resolution. The disad- time for shallow seismicity regions is earlier byvantage lies in the somewhat relative nature of about 0.6 s; the deep seismicity areas are earlier bytheir absolute values. The problem of base-line about 0.4 s at 90°.This is not purely a problem ofdifferences is well known and need not be dis- a tilt in the travel-time curves as there are at leastcussed here. There is also the question of small two intersections of the curves at intermediatedifferences in the overall slope of the teleseismic distances. While the broad-scale features for bothtravel-time curve; some of the controversies on curves are virtually identical, the short wavelength -
this subject are now over a decade old and still structure is markedly different.remain unresolved. The body wave dataset is im- For the S-wavedata, differences between travelportant in defining the regions of the mantlewhich times for regions of deep and shallow seismicitywe disc~issedabove and in improving the resolving are much greater. The difference in the interceptpower of the dataset. time is 5—6 s and, what is even more surprising,In order to obtain arepresentative global body- there is a systematic difference of 3—4 s in the
wave dataset we have studied the P- and S-wave distance range from 90 to 1000. The wide scattertravel times using the arrival time data from Bul- of data in the interval 80—90° can probably beletins of the International Seismological Centre for explained by interference with the SKS phase,years 1964—1975. Rejecting events with fewer than which begins to arrive before S at approximately30 stations reporting, one retains approximately 82°.26000 events with reports of nearly 2000000 P- It would appear that the events for regions withwave arrival times and 250000 S-wave arrival deep seismicity (trenches) are systematically mis-times. Since neither the distribution of stations nor located. The near-source stations tend to beearthquakes is uniform, there exists the important grouped only on one side of the event, and anquestion of an appropriate averaging method. After epicentral mislocation is very plausible, particu-many experiments, a decision was made to divide larly if S-wave data are not used. This error in thethe Earth into a number of sections of equal area, position and the origin tune must be compensatedderive a travel-time curve for sources in each of by an equivalent error in depth. This is reflected inthe areas and then average all these travel-time the baseline and tilt of aderived travel-time curve.curves. This would tend to eliminate the bias intro- Because of the very large differences betweenduced by unequal seismicity in the presence of various datasets, a meaningful average is difficultlateral heterogeneity. In actual experiments we to determine. Various datasets are shown in Fig. 2.have used 72 regions, each 30°wide in longitude Substantial adjustments in the absolute values ofand of the appropriate latitudinal extent to assure the travel times appear to be necessary.
302
0 I-v I I
—10C1~
— ci(a ci
- 00 ci8 ci ci ciw ci --
-2 - ci 0 ~D /
Es ~‘ ciI I -. ci Dcl ,~
Es ~ \~/ ~%%%%%%% 0 ci~
“.. ,/
4 I — I I I I I I I0 20 40 60 80 100
Epicentral distance (degrees)
Fig. 1. Surface focus P-wave travel-time residuals with respect to JB times. Crosses are ISC data. Boxes arefrom Hales et al. (1968).Dashed line is from Herrin et al. (1968). Solid line is anisotropic PREM. ISC times have been corrected with —1.88 s baseline and— 0.0085 s deg—‘ slope. Residuals calculated at a period of 1 s.
Because of the baseline and tilt uncertainty of P (Engdahl and Johnson, 1974) and ScS— S (Jordantravel-time observations from natural sources we and Anderson, 1974), are important for the con-use body-wave data as a constraint on the fine trol of the outer core radius.structure of velocity variations rather than as a If the issues are somewhat unclear with respectstrong constraint on absolute velocities. Thus, we to the mantle travel times, the difficulties increaseare mainly concerned with fitting the shape of the by an order of magnitude in the outer core. Totravel-time curves. Differential travel times and obtain reasonably good control over velocities inthe normal mode dataset provide constraints on the outer core, one must combine the data fromabsolute velocities. Even these datasets contain a four travel-time branches: SKKS, SKS, PKP(AB)source and path bias but wehave been able to find and PKP(BC). Some attempts to combine thesea spherically symmetric Earth model which satis- data have resulted in a marked roughness of thefies these data to high precision. derived model at depth intervals corresponding toOther subsets of teleseismic travel times, used junctures between segments. In addition, SKKS
mostly for comparison, include deep source P-wave data are likely to suffer from a sr/2 phase shiftdata of Sengupta and Julian (1976) and S-wave with respect to the SKS phase (Choy and Richards,data of Sengupta (1975), Hales et al. (1968), and 1975), unless SKKS is Hilbert transformed beforeGogna et al. (1981). Differential travel times, PcP— cross-correlation with SKS (or vice versa).
303
I ~‘ I I I I I I
6—I $
- St xit
4 xH ‘ Xix.
0$, t x-~ :: ~o t~9o x ~I : ~ :/~~ :~~s~*P~ ~‘?
o “ / ~+ + ~ %~S 0 0 0- b~ ~:J~+~~‘- ~
Es -2 - -
.1- -
+A ++
- + ++ +
-6- + + -
I I I I I I I I20 40 60 80 100
Epicentral distance (degrees)
Fig. 2. Surface focus S-wave travel-time residualswith respect to JB. Boxes are ISC data. +: from Gognaet at. (1981). X: from Halesand Roberts (1970). Solid line is for the vertically polarized (SV) shear wave and dashed line is for the horizontally polarized (SH)shearwave in the anisotropic PREM model. Nobaseline or slope corrections. Residuals calculated at a period of I S.
We use the SKS data of Hales and Roberts the ISC travel time data for distances up to 25°,(1970), core phase data for the AB, BC and DF allowing for an arbitrary base-line correction. Abranches of Gee and Dziewonski (unpublished) decrease in S-velocity gradient below 600 km wasand PKiKP-PcP differential travel times of Eng- dictated by the need to obtain intersection withdahl et a!. (1974). The latter study gives the best the teleseismic branch at 24°;without that featureavailable control of the inner core radius. the intersection occurs at 21.5°,which is distinctly
inappropriate.Once the starting velocity models for the upper-
4. The staim~gmo~iet most 670 km were designed, it was possible tostrip the upper mantle andinvert the stripped data
Design of the velocity models for the upper to obtain the lower mantle structure. It was at thismantle represented the most involved part of this stage that the need for introduction of the featuresstage of our work. Our decision to locate discon- in the lower mantle became obvious. One, andtinuities at 220, 400 and 670 km was based on perhaps the most important, is the second-orderresults of many other studies. The bottom of the discontinuity some 150 km above the core-mantlelithosphere was initially placed at a depth of 80 boundary. The velocity gradient at this depthkm. Then, the velocities were adjusted to satisfy changes abruptly and could become negative. This
304
feature is clear on a dT/d ~ plot, where a sudden Moho to the core). Our choice of the free parame-change in the slope of dT/d~occurs at 90°.The ters was —0.5, 5.55, and 3.32 g cm3, respectively.other feature is a region of steep velocity gradient This yielded a central density of 12.97 g cm3 andjust below the 670 km discontinuity extending to a a density jump at 670 km of —0.35 g cm3. Thesedepth that is not particularly easy to define ex- assigned and derived parameters were free toactly, but 771 km (5600 km radius) appeared to be change in the inversion. Derivation of the startinga reasonable estimate. The model of the lower density distribution completes this stage of ourmantle was formed by representing the velocity work.between 3485 and 3630 km as well as between5600 and 5701 km by linear segments and theregion between by a cubic in radius requiring that 5. Anisotropythere should be continuity at the points of junc-tion. The starting model for P-velocities predicts Global inversions of seismic data, such as pre-travel times that match observations with an r.m.s. sented here, usually give very high shear velocities,error of 0.06 s, roughly the average s.e.m. of a 4.8 km s~, in the uppermost mantle. Suchsingle observation in our global averaging proce- models do not satisfy short period (<200 s) Lovedure. and Rayleigh wave data or shear wave travel timesThe scatter of the S-wave data is larger by more at short distances (<20°).Very pronounced low-
than an order of magnitude, and these data could velocity zones (LVZ) are a prominent feature ofnot be expected to reveal independently the fine most models. We have found it impossible tofeaturesdemanded by the P-wavedata. The S-wave simultaneously satisfy the data which are relevantdatawere inverted assuming that first- and second- to the upper 200 km or so of the mantle with anorder discontinuities exist at the same depths as in isotropic model. The discrepancy between Lovethe P-velocity model, wave and Rayleigh wave data suggests that theIn view of the fact that our knowledge of the upper mantle is anisotropic (Anderson, 1966). The
structure of the inner and outer core is still rather discrepancy is also pronounced for relatively ho-poor, we began with the hypothesis that both cores mogeneous oceanicpaths (Forsyth, I975a,b; Schlueare individually homogeneous. For this reason we and Knopoff, 1977; Yu and Mitchell, 1979). Thhave used the results of fourth-order finite strain suggests that lateral variations are not the primarytheory to construct the starting model of P-velocity cause of the discrepancies. Although azimuthalin the outer core, and P- and S-velocities in the anisotropy is important just below the Moho ininner core. The starting density distribution was oceanic environments (Hess, 1964; Backus, 1965;obtained by a variation of the method proposed Raitt et a!. 1969), it appears to be less importantby Birch (1964). We assumed that the Adams— at surface wave periods (Forsyth, l975a; Yu andWilliamson equation is satisfied in each subregion Mitchell, 1979). Transverse isotropy, or polariza-from the center of the Earth up to the 670 km tion anisotropy, has been invoked to explain thediscontinuity. Following Birch, we assume that the Love wave—Rayleigh wave discrepancy. Since ourdensity in the upper mantle is linearly related to data represent an average over many azimuths anyP-velocity: p = a + bv~.Given the mass and the residual azimuthal anisotropy will be effectivelymoment of inertia of the Earth, we can find the averaged out. We therefore deal only with thedensity at the center of the Earth and the jump of spherical equivalent of transverse isotropy. Thedensity at the 670 kin discontinuity if we specify symmetry axis is vertical (radial).the following parameters: density jump across the For this type of anisotropy there are five elasticinner—outer core boundary; density at the base of constants, A, C, F, L and N, following the nota-the mantle; and density below the Mohoroviëiá tion of Love (1927, p. 196). A and C can bediscontinuity (Birch had only one free parameter, determined from measurements of the velocity ofbut he did not treat the inner core separately and P waves propagating perpendicular and parallel tohis density distribution was continuous from the the axis of symmetry. Since in our case the axis of
305
symmetry is vertical (radial) I I
A=pV~H 8.0U,
C—pvI~vwhere p is the density.
UIn general, the shear-wave velocity depends onpolarization and direction of propagation. In the ~ 1.1 0.90
-ldirection perpendicular to the axis of symmetry: 00
N PVIH U,U,
a,L=pV~~ ~7.8
UIn the radial direction, parallel to the symmetryaxis, there is no splitting and both polarizations I I I I I I
are controlled by the elastic constantL. Therefore, ________________________________bothhorizontally andvertically travelling SVwaveshave the same velocity. The elastic constant Ncontrols the propagation of fundamental modeLove waves. All five elastic constants enter intothe dispersion equation for Rayleigh waves but L ?
is the more important shear-type modulus (Ander- ~ 44v-~.III
son, 1965). For this reason, vertically travelling Sor ScS waves are controlled by the same set of ~ 4.2elastic constants that control Rayleigh wave dis-persion.ties at intermediate incidence angles. It is conveni- 4.1The fifth constant, F, is a function of the veloci-
ent to introducea non-dimensional parameter ~ = I I I I I I
F/(A — 2L) (Anderson, 1961; Harkrider and U 120 180Anderson, 1962; Takeuchi and Saito, 1972). InFig. 3 we show the P and S velocities as a function Angle of incidence (degrees)of incidence angle for five values of ,~ ranging Fig. 3. P- and S-velocities as a function of angle of incidencefrom 0.9 to 1.1. For an isotropic solid, A = C, and the anisotropicparameter~,which is varied from0.9 to 1.1L = N, and j= I. It is clear that variations in ~ at intervalsof 0.05c The values of velocities used in thecalcula-can lead to substantial differences in velocities at tion are Vpv = 7.752 km s~1 VPH= 7.994 km s- and V~v=4.343 km s~.intermediate incidence angles and also in averagevalues of velocity. Anderson (1966) showed thatfundamental mode Rayleigh wave dispersion isalso very sensitive to this parameter. almost independent of SV and SH, respectively,It is often assumed that Rayleigh waves are but Rayleigh waves are sensitive to tj, PV and PH.
controlled by the horizontal SV velocity so that In this sense an isotropic solid is a degeneratecase.isotropic programs can be used to compute disper- We shall show later that it is possible to satisfysion curves. It is also often assumed that the the global dataset with anisotropy restricted to thecompressional velocity is unimportant in Rayleigh upper 200 km of the mantle. The anisotropy re-wave inversion. These assumptions are not strictly quired is about 2—4% for both P and S waves. Thevalid (Anderson, 1966) and we have inverted for resulting models do not have the pronounced de-the five independent elastic parameters. The crease in velocity from the LID to the LVZ thatfundamental mode Love and Rayleigh modes are characterizes most surface wave models, particu-
306
larly for global and oceanic paths. In fact, the 0.217 DE a s 80variations with depth of all the velocities is rathermild in this region of the mantle. It appears thatsome of the features of isotropic or pseudo-isotropic (SH, SV) models are due to the neglect ofparameters, such as i~,PV and PH, which areimportant in anisotropic Rayleigh wave disper-sion.The anisotropic upper mantle reconciles the
ity data. ntis is important for surface wave studies .273 0 100 200 300 400 600Rayleigh and Love wave data and also permits afit of the short period Rayleigh wave group veloc -_______________________________of seismic sources. —vs ---- VSV ——-VSHIn the course of this study we have, of course, Fig. 4. Partial derivatives for a relative change in period of
calculated partial derivatives for anisotropic struc- mode oSso (T—. 120 s) as a function of depth. The short dashedtures. The results can be summarized as follows: line corresponds to perturbation in SV velocityand long dashed
line SH velocity; S~and ~H of eq. A6 of the Appendix. TheAs expected the fundamental toroidal mode is continuous line corresponds to the isotropic case (eq. A9).primarily controlled by SH. The toroidal over-tones, however, are sensitive to both SV and SH.The spheroidal modes are only slightly sensitive to important near the top of the structure. At depthSH. However, PV, PH, SV, and ij all have a the PV and PH partials are nearly equal andsignificant effect on the spheroidal modes and opposite. Individually they are significant but inthere is no a priori relationship between these the isotropic case they nearly cancel. Changes ofparameters. It is necessary, therefore, to invert for opposite sign of the component velocities cause anfive elastic parameters. We cannot assume, for additive effect and the net partial is nearly asexample, that P-wave velocities are isotropic and significant as the SV partial. The same effect per-invert only for P, SV, and SH. This would be a sists for all the spheroidal modes so that there isreasonable procedure only if the compressional good control on the anisotropic P-velocities andwave partials were very much less than the shear better control on P-velocities in the upper mantlewave partials or if naturally occurring upper man- in general than is usually considered to be thetle minerals had a more pronounced shear waveanisotropy. Neither is the case. MODE 0 S 80Figures 4, 5, and 6 show the effects of per- 1 /
turbing the shear and compressional velocities in / ./an anisotropic Earth model. The formulae used to /
evaluate these partials are given in the Appendix.As expected, fundamental mode Rayleigh waves, ‘ /in this case the mode 0S~with a period of 120 s, /a~emainly controlled by SV (short dashes) and are (___1~~little influenced by SH (long dashes). The totalshear wave partial derivative is shown as the solid I
curve. The parameter plotted is the relative change ________________________________in period of ~ for achange in shear velocity as a _2.438~ (00 200 300 400 600function of depth. —VP VPV ~“vPMA more surprising effect is the nature of per- Fig. 5. Partial derivatives for a relative change in period of
turbations in the PV and PH velocities, shown in mode oS’o (T—. 120 s) as a functionof depth. The short dashedline corresponds to perturbation inPV velocity and long-dashedFig. 5. The isotropic partial derivative (solid line) line in PH velocity; ~v and ~H of eq. A6of theAppendix. Theshows that compressional wave velocities are only solid line corresponds to the isotropic case (eq. A9).
307
MODE 0 S 80 where ~rrepresents either the period of free oscil-lation (T = Tin that case) or the appropriate periodfor the body wave under consideration. The per-~ (2)
It is clear that given the observed values for the, ,./ travel times, periods of free oscillations and their
/ attenuation factors, the inverse problems for theI ~ I I elastic and anelastic parameters can be solved
-s .273 0 100 200 300 400 500 simultaneously. An additional advantage of pro-ETA VP ~~VS ceeding in this manner is that presence of the
Fig. 6. Partial derivatives giving the relative change in period. 8q~terms in eq. 1 above may provide additionalwith respect to the anisotropic parametero~(solid line) and the resolution for the anelastic structure, since In ‘r inisotropic velocities V~(short-dashed line) and V~(long-dashed the seismic frequency band varies from 0 to 8,hne). See eqs. A6 and A9 of the Appendix.
roughly. Generalization of eq. 1 for the case oftransverse anisotropy is considered in the Appen-dix.
case. Mantle Rayleigh wave data are often in- Another feature of our particular inversion pro-verted for shear velocity alone. Even in the iso- cedure was that we optionally could introduce atropic case this is not good practice since a wrong baseline correction or linear slope for a givenP-velocity at shallow depths can cause a large branch of the travel times as additional unknowns.perturbation in shear velocity at greater depth. Our starting model and perturbations thereto wereThe isotropic P and S partials are shown in assumed to have, for each of the regions, a form of
Fig. 6 along with the q partial; it is clear that a low-order polynomial in radius. For exampleperturbations in the anisotropic parameter i~can — 2 3
OVp — a0 -r a1r +a2r + a3r ior r1 a~r ~lead to substantial changes m the penods of freeoscillations. Substitution into the integral leads to a familiar
form of the system of equations of conditionwhichthen can be solved by standard procedures. The
6. Inversion and the final model order of the polynomials needed to satisfy the datawas determined by trial and error.
Our starting model at a reference period of, say, The method was extended to the problem of1 s is defined by a set of five functions of radius transverse isotropy by modifying equations given(Vp, V~,p, q~,‘i~c)’where q Q~and q~,and q,, by Takeuchi and Saito (1972), as described in therelate to isotropic dissipation of the shear and Appendix and utilizing the formulas derived bycompressional energy, respectively. For an iso- Woodhouse (1981), in the note accompanying thistropic region of the Earth, perturbation in aperiod report, for the travel times. We solve for fiveof free oscillation or travel time of a body wave elastic constants which we take as the horizontalcan be expressed by P-velocity, PH; vertical P-velocity, PV; horizontal
and vertical S-velocities, SH and SV; and an an-— (I ( . - - - isotropic parameter (Anderson, 1966; Takeuchi
T J~ ‘ and Saito, 1972). We found it necessary to intro-+8q ~-lnT +8qJ.Inr) (1) duce anisotropy into the outer part of the upper
mantle but not elsewhere.+ (terms related to changes in radii of discos- Parameters of the final model are listed in
tinuities) Table I. Graphical representation of the model is
TABLE ICoefficients of the polyno~mialsdescribing the Preliminary Reference Earth Model (PREM). The variable x is the normalized radius:xr/a where a6371 km. The parameters listed are valid at a reference period of I s
Region Radius Density Vp V~(km) (gcm3) (kms~) (kms’)
* The region between 24.4 and 220 kin depths is transversely isotropic with the symmetry axis vertical. The effective isotropicvelocities over this interval can be approximated byVp 4.l875+3.9382x= 2. 15 19+ 2.348Ix
309
shown in Figs. 7 and 8. It is important to remem- more compatible with observations. The problember that these parameters are valid at a reference is highly non-unique and its early resolution is notperiod of 1 s. For otherperiods the velocities must likely.)be modified according to equations given in The velocities, density and several other param-Kanamori and Anderson (1977) eters of geophysical interest are listed in Table II.
I hi T \ In the depth range from 24.4 to 220 km. in whichv5(T) = V~(l). ~l — —~,L) our structure is anisotropic, we also give the values
hi T for the “equivalent” isotropic solid. This corre-V~(T)= V~(l).{1— —[(1 — E) q~+Eq,jJ sponds to an appropriate averaging over all anglesIT of incidence; the general equations havebeen given
(3) by Woodhouse and Dahlen (1978). For the case ofwhere transverse isotropy, the Voigt bulk and shear mod-
E=~(V5/V~)2 uliare
K=~(4A+C+4F—4N)(The particular distribution of bulk dissipation =J-( — )and shear dissipation m the inner core given mTable I should be only understood as a way to these represent upper bounds on the effectivelower the Q of radial modes in order to make them elastic moduli.
06~~~10~
7200 400 600 800
Depth (km)Fig. 7. Upper mantle velocities, density and anisotropic parameter ~iin PREM. The dashed lines are the horizontal components ofvelocity. The solid curves are ij, p and the vertical, or radial, components of velocity.
310
4:
-
11
I I I I I I0 2000 4000 6000
Depth (kin)Fig. 8. The PREM model. Dashed lines are the horizontal components of velocity. Where i~is I the model is isotropic. The core isisotropic.
One of the entries in Table II is the parameter be determined from eq. 3. Table IV lists the amso-tiB of Bullen (to be distinguished from the aniso- tropic and anelastic parameters in the crust andtropic parameter‘i), which represents ameasure of upper mantle computed at a reference frequencydeviation of a model from the Adams—Williamson of 1 s (above) and 200 s (below). Notice that theequation effect of the velocity dispersion due to anelasticity
dK 1 d~ leads to the development, at long periods, of a low+ ~ -~j-,- (4) velocity zone in a depth range from 80 to 220 km.This is due to the low ~ in this region.
For the most part ~ in the core and lower mantle In Fig. 9 the relative changes in P- and S-waveis very close tounity. Small deviations are, in some velocities are shown as a function of incidencecases, an artifact of the polynomial representation, angle for three depths: 24.4, 100 and 200 km. TheThe parameter dK/dP is another measure of ho- angular dependence of SV and SH explains themogeneity. The values for the lower mantle (except fact that the “effective” shear velocity at a depthfor the region inunediately above the core—mantle of 200 kin is lower than either SV or SH.boundary) and outer core can be considered nor- The Q distribution is modelled with a smallmal. number of homogeneous regions. The radial modesIn Table III we give the model parameters at a are the main control on QK and these essentially
period of 200 s; the velocities at other periods can constrain only the average value in large regions.
3Il
4 ferences are substantial. Group velocities corn-I I I I I
puted for the anisotropic model are consistentwith the observations of Mifis (1977) andKanamori (1970). There is satisfactory agreement
100 24 4 between the observed and predicted values of Q ofthe normal modes.The theoretical periods for the “equivalent” iso-
tropic model are systematically too short for thefundamental spheroidal mode.The reverse is thecase for the fundamental toroidal mode. The sametrend is evident for the first overtones. The iso-tropic model is an adequate fit to the longer period.;;i spheroidal overtones but the fit degenerates at theshorter periods. All of this is suggestive of an
________________________________ anisotropic upper mantle, such as wehave adopted1 ~.-—--~~ I I here. We see no need to invoke deep anisotropy.
14The highly anisotropic minerals olivine and pyrox-
~ 4 ‘~24.4vs /1 ene, in fact, are restricted to the upper mantle.4.,* //
100 The travel-time data and theoretical fits are/ ~ given in Table VI. The original data are also given./ .7 \ In some caseswe correct the data for an offset and
a tilt. The baseline for most travel-time studies is24
arbitrary and, for our purposes, adjustable.There are several effects which contribute to an
offset and tilt among various travel-time datasetsand between these and global models. First, thereare the well known source and receiver effects.h—200
________________________________ Secondly, the origin time and location of the eventI I I I I I-2 are in error if the travel-time table used in their
0 60 120 180 location is in error. An error in assigned depth ofAngle of incidence (degrees)
an event also causes an error in both baseline andFig. 9. Velocity as afunction of directionof angle of incidence tilt. Published depths are sometimes based onfor three depths in the anisotropic region of the PR.EM model, minimization of the residual vs. distance relative toUpper curves are compressional velocities; lower panel gives a standard curve. Thirdly, the effect of attenuationV~(solid) and VSH (dashed). makes the frequency content of the arrivals vary
with distance. In addition, in calculating theoreti-cal travel times we must assume a period and
Table V lists the observed and computed pert- correct for Q. Uncertainties in Q and a frequencyods of the normal mode data used in this study. dependence of Q give rise to a change in. baselineFor comparison we list the theoretical results for and tilt. The effects of dispersion and the depththe “equivalent” isotropic model discussed above, variation of Q give an offset and a variation ofIt may be noted that at high phase velocities or travel time with distance that depends on period.very long periods, the equivalent isotropic model Differential travel times also contain these effectsfits the data nearly as well as the anisotropic and have different effective Q ‘s for the two phasesmodeL For example, the periods of radial modes iii question. For these reasons we calculate allpredicted by both models are nearly identical and theoretical travel times at a period of one-secondthe same is true with respect to modes oS2-oSd. and, in Tables VIa-v, compute the baseline and,But for short-period fundamental modes the dif- in some cases, a tilt that gives the best match
312
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TABLE VObserved and theoretical parameters of the normal modes used in this study including period and Q. Periods are givenfor anisotropicPREM and an isotropic model having the bulk modulus and rigidity evaluated according to a Voigt averaging scheme. The groupvelocity is calculated for theanisotropic model; seeNote added in proof
M 0 D E OBSERVATION AMISOTROPIC ISOTROPIC Q OBS Q COM GROUPPERIOD S.D. PERIOD DEV PERIOD DEV VALUE S.D. VEL.
between theory and observation. The uncorrected mula isdata are also given. . lnfThe parameter T* is equal to the ratio of the Tcorr =
7~ef— — .
travel time to Q of a given phase. In addition to itsapplication in calculations of a change in the If we assume, for example, that the appropriatespectrum of a phase, it can be used to correct the frequency for observations of the travel times oftheoretical travel times computed at a certain ref- the S.waves by Hales and Roberts (1970) is 0.2erence frequency (1 s in our case) to the frequency Hz, then at 30°distance the correction is 2 s andappropriate for the observed waveform. The for- at 90°,3 s. The same approach can be used to
335
TABLE VI aP travel times: global average from ISC data. Baseline correction — 1.88 s; slope —0.0085 s deg —‘
DELTA OBSERVATION ANISOTROPIC ISOTROPICT OBS 7 CORR ERROR 7 COMP (0—C) P T COMP (0—C) P
correct the differential travel times, such as ScS— S, these “static” values from one travel-time study toand in this case the parameter 7’* has a meaning another.only in terms of this specific application. This The observed travel times from Tables VIa—cillustrates the large effect on both baseline and tilt are compared with the results of calculations forthat can occur by this process alone. Each source model PREM in Fig. I. After the global average ofregion has its own upper mantle velocity and Q the ISC data is corrected for baseline and slope,and these variations contribute to variations in the fit of the predicted travel times is very good to
336
TABLE VIa (continued)
DELTA OBSERVATION ANISOTROPIC ISOTROPIC 7’T OBS 7 CORR ERROR T COMP (0—C) P T COMP (0—c) P
distances as short as 20° ;in particular, the 24° et al. (1981, Table VIk) and Hales and Robertsdiscontinuity is well matched. At shorter distances, (1970, Table VIm) are compared with the globalthe difference between the upper mantle velocity average of the ISC data and SV and SH travelof model PREM (8.2 km s’) and the apparent times of the PREM model in Fig. 2. The overallvelocity of the P,~phase from the ISC data (—.‘ 8.0 shape of the SV first arrival travel times matcheskm s ~‘) leads to a several second difference. The that of the ISC data well up to a distance ofdata of Hales et al. (1968) and Herrin et al. (1968) approximately 80°.In a distance range from 30 toshow differences in slope of opposite sign with 80°all three data sets are in reasonable agreementrespect to PREM, but reproduce well the details of if allowance for tilt and baseline corrections arethe travel—time curve. made. Beyond 80°,the shape of the SV and SHThe surface focus S-wave travel times of Gogna travel— time curves are most consistent with the
data of Hales and Roberts.
337
TABLEVIbP travel times: Hales et al. (1968). Baseline correction — 1.18 s; slope 0.0085 s deg —‘
DELTA OBSERVATION ANISOTROPIC ISOTROPIC T’T OBS 7 CORR ERROR T COMP (0—C) P T COMP (0—C) P
7. Discussion inversion were 61 and 126 s for fundamentalmodelRayleigh and Love waves, respectively. These
When the upper mantle was allowed to be modes have wavelengths of > 240 km so we cananisotropic, the inversion decreased the shear only determine average properties of the uppervelocity of the LID and increased the velocity of mantle. Short-period Rayleigh waves, <20s, sug-the LVZ. The SH and SV velocities, although gest that sv in the uppermost mantle is about 4.6different, were individually almost continuous from km s-I, 5% greater than the average upper mantlethe Moho to 220 km. It appeared that the LID, or SV determined in this study.seismic lithosphere, could not be resolved with the Since the thickness of the LID and velocities indata being used. The shortest periods used in the the LID and LVZ can be expected to varywith the
338
TABLE VI cP travel times: Hemn et al. (1968). Baseline correction —0.94 s; slope —0.0141 s deg —
DELTA OBSERVATION ANISOTROPIC ISOTROPIC 7’T OBS 7 CORR ERROR 7 COMP (0—c) P T COMP (0—C) P
age of the lithosphere we have chosen to treat the scatter which is typical of S-wave studies. Otherentire upper mantle to a depth of 220 km as a contributors to S-wave scatter are: (I) the diffi-smooth entity. culty of identifying and picking later arrivals; (2)The final model predicts significantly different the longer period nature of 5; (3) the fact that
travel times for SV and SH waves. The effect is locations are based on P-wave times; and (4) themost pronounced at short distances, <25°,but is finiteness of natural sources and rupture velocitiesmaintained at teleseismic distances. SH is faster by which are close to shear velocities.1.16 s at 30°and 0.34 s at 90°.This, plus varia- As can be seen from Fig. 1 and the tables,tions in Q and period, may contribute to the large PREM is an excellent fit to P-wave travel-time
339
DELTA OBSERVATION ANISOTROPIC ISOTROPICT OBS 7 CORE ERROR 7 COMP (0—C) P T COMP (0—C) P
data from about 22 to 90°.The simplified upper- out to about 94°.At larger distances, dz/di~formantle structure we adopted is inappropriate for PREM is up to 0.1 s deg—‘, low compared to thelocal and regional travel-time studies. In addition majority of recent data. This indicates that veloci-to a good fit to the travel times, PREM is also an ties in the lowermost mantle should be decreasedexcellent fit to dt/di~data. Gogna et al. (1981), slightly.hereafter GJS, tabulate results from a recent study. The dt/d~for S-waves for PREM falls in thePREM fits d t/d ~ for P.waves, from this study, midst of the rather widely-scattered published val-with an average error of only 0.004 s deg — ~, over ues in the distance range 30—40°.Compared tothe interval 40—77°.Maximum isolated errors are GJS the errors are 0.03 s deg—‘ (36—40°), 0.04only 0.008 s deg~.Beyond 77°PREM deviates (41—51° ),0.03 (51—60°),0.02 (61—70°),0.05 (71—from GJS but is within the scatter of other studies 80°),0.11 (81—90°)and 0.16 (91—95°).The correc-
340
TABLE VI dDeep focus (550 km) P travel times: Sengupta and Julian (1976). Baseline correction 0.67 s; slope —0.0 173 s deg—‘DELTA OBSERVATION ANISOTROPIC ISOTROPIC
tion to a period of S s increases dt/d~ of PREM in the GJS study, P-wave times are up to 3.2 sby about 0.02 s deg ‘. Beyond 95°,PREM has short and S-times up to 6.9 s short in the Hindudt/di~values which are 0.17 to 0.38 s deg—‘ higher Kush area compared to previous travel-timethan GJS. This suggests that the shear velocities at studies, and the tilts from 30—90°differ by aboutthe base of the mantle should be increased or that 0.01 s deg ‘. This is of the order of the tilt correc-the structure in this region is more complicated tion required to reconcile PREM travel times withthan that given by PREM. observed travel times.Considering all data sets, the discrepancy starts
to set in at about 93°with dt/d~of 8.85 s deg~. Region D”This means that the error is in the lower 195 km ofthe mantle. The lowermost mantle, region D” in Bullen’sTravel times out to distances of about 20°vary notation, clearly has a different velocity gradient
substantially from region to region. For example, than the rest of the lower mantle. For simplicity
Li Zeng
343
TABLE VI kS travel times (SH): Gogjia et al. (1981). Baseline correction —1.87s; slope 0.0329 deg —lDELTA OBSERVATION ANISOTROPIC ISOTROPIC T’
7 OBS T CORE ERROR 7 COMP (0—C) P T COMP (0—C) PDEG SEC SEC SEC SEC SEC SEC/DEG SEC SEC SEC/DEG SEC
we have assumed that the top of this region is a velocity by about 0.04 km s— or by decreasing thesecond-order discontinuity. The inversion results velocity gradient somewhere near 3630 km radius.in a nearly constant velocity in the lower 150 km The study of amplitudes and wave-forms shouldof the mantle. The P-wave travel-times beyond 90° resolve the possibilities.are at least 0.08 s fast relative to baseline- andtilt-corrected travel-time data. This is a small error Radius of the corebut is indicated by all datasets. Apparently, theaverage time spent in region D” by rays at near The radius of the outer core in PREM is 3480grazing incidence should be longer by about 0.27%. km. It may be noted that PcP—P times for theThis can be accomplished by reducing the P. model are systematically slow with respect to the
344
TABLEVI k (continued)DELTA OBSERVATION ANISOTROPIC ISOTROPIC T’
T OBS 7 CORE ERROR T COMP (0—C) P T COMP (0—C) PPEG SEC SEC SEC SEC SEC SEC/DEG SEC SEC SEC/DEG SEC
observations, indicating that the core should be Woodhouse participated in numerous discussionsslightly larger. The ScS—S times are also margi- related to this work and assisted us in solvingnally too long. Taking into account the slower many problems. In particular, he derived equa-velocities which may exist in D”, good agreement tions for the travel times in a transversely isotropicwith these two datasets can be obtained with a medium and his note on this subject accompaniescore radius 1.7 km larger, or 3481.7 km. this report. Robert North made available to us his
Love wave data prior to publication. This research
Acknowledgements was supported by National Science FoundationGrants No. EAR78-05353 (Harvard) and EAR77-Anton Hales was an interested observer at all 14675 (California Institute of Technology). Con-
stages of this study and we gratefully acknowledge tribution No. 3531 of the Division of Geologicalhis advice. We also acknowledge helpful corre- and Planetary Sciences, California Institute ofspondence with Sir Harold Jeffreys. John Technology, Pasadena, California 91125.
345
TABLE VIIS travel times (SV): (3ogna et al. (1981). Baseline correction —0.29s; slope 0.0188 s deg~
DELTA OBSERVATION ANISOTROPIC ISOTROPICT OBS 7 CORE ERROR T COMP (0—C) P T COMP (0—C) P
Appendix. Differential kernels for perturbation of ~= (‘ri dr(8A.A +6C~(~+8F~freigenfrequencies of normal modes in a fransversely ~,2 J0isotropicmediuni +6L.L+6N.1~’+8p.E) (Al)
Equations for differential kernels given by whereA, C, F, N, and L are the five independentBackus and Gilbert (1967) can be easily expanded elastic constants as defmed by Love (1927, p. 196).to accommodate transversely isotropic medium. A The problem of differential kernels for this caserelative change in the squared eigenfrequency is has been presented by Takeuchi and Saito (1972),
346
TABLE VII (continued)
DELTA OBSERVATION ANISOTROPIC ISOTROPIC T*T OBS T CORE ERROR T COMP (0—C) P T COMP (0—C) P
but their expressions are inconvenient to apply in J~= r 2( (1 + 2)( 1 + 1)1(1— I) V2 — [2U— 1(1+ 1) V]2our formulation of the parameters sought in inver- . . . . .
where the dot signifies differentiation with respectto the radius and the eigenfunctions are normal-The expressions for the differential kernels in d
terms of the eigenfunctions for spheroidal modes izeare fl
~ii~Jp~U2+ 1(1+ 1)V2jr2 dr 1c=02 °
A = r2[2U— 1(1+ l)v]2 For toroidal modesF2r’U[2U—l(l+ 1)V] (A2) L(W— W/r)2
L1(i+ i)[J~+(u—V)/r]2 !1=(l+2)(!— 1)(W/r)2 (A3)
347
TABLE VImS travel times (SH): Hales and Roberts, (1970). Baseline correction — 1.14 s; slope —0.0068 s deg—
DELTA OBSERVATION ANISOTROPIC ISOTROPIC T*7 OBS T CORR ERROR T COMP (0—c) p T COMP (0—C) P
with the nonnalization non-dimensional parameter ~
w2fIpW2r2 dr = I VPH = (A/p)”2
The differential kernel for the density, 1~,is the ~pv = (C/p )h1~2same as given by Backus and Gilbert (1967). 1/2As in our inversion we consider simultaneously VSH (1’T/p) (A4)
periods of free oscillations and travel times of ~ = (L/p)’12body waves, it is desirable to recast the problem in svterms of the perturbations in velocities and, a ~g F/(A — 2L)
348TABLE VIm (continued)
DELTA OBSERVATION ANISOTROPIC ISOTROPIC TaT OBS 7 CORE ERROR T COMP (0—C) P T COMP (0—C) P
We seek an expression for a relative perturba- ~H = —T2PVPH(A +~P) (A6)tion in a period of a normal mode in the form6T ~ / , - - Svr2pVsv(L2~P)
dr~&pR+&VPVPv+&VPHPH
+&Vsv~Sv+8VSH..~H +&~.E) (AS) SH = —r2pV~~NAfter simple algebraic transformations, the ap- E = — ~r2PP(VP2H— 2P~)
propriate expressions are - - -
— — I 2F - 2 - I - \ 2 For toroidal modes, A, C, andF are set to zero,— ~r jR + Vpv~C+~A +7JF)VpH of course.+ (L— 2nP)V~~ + &V~~] In calculation of the kernels for and QK we
- - use the concept of an equivalent isotropic medium= r2pVpvC (Woodhouse and Dahlen, 1978) with the bulk and
349
TABLE YInS travel times (SV): Hales and Roberts (1970). Baseline correction 0.44 s; slope —0.021 s deg —‘
DELTA OBSERVATION ANISOTROPIC ISOTROPIC7 OBS T CORE ERROR T COMP (0—C) P T COMP (0—C) P
= — ~r2[E + (~siQ+ K~A~)/p] covered. The error was in a term associated with— — 2 - the gravitational potential in the fluid core and,— r pV~K (A9) therefore, it only affects results for the gravest
S —r2pv~(Jt~t—4~) modes. For example, for OS,, and all following0S,
modes the results in Table V are correct to alldecimal places listed. Also, none of the theoretical
Note added in ~ calculations of periods of normal modes have beencorrected for the second order effects. For exam-
An error in code for evaluation of group veloc- pie, the appropriate correction of the period of 0S0ity of spheroidal modes has recently been dis- brings it much closer to the observed value.
TABLE VI oDeep focus (550 km) S travel times (SH): Sengupta (1975). Baseline correction 2.28 s; slope —0.0433 s deg I
DELTA OBSERVATION ANISOTROPIC ISOTROPICT OBS T CORR ERROR T COMP (0—C) P T COMP (0—C) P
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