Elasticity of ilmenites

Ultrasonic data for the velocities of the ilmenites MgTiO₃ and CoTiO₃ have been determined as a function of pressure to 7.5 kbar at room temperature for polycrystalline specimens hot-pressed in a piston-cylinder apparatus at pressures up to 30 kbar. Titanate and germanate ilmenites define divergent isostructural trends on a Birch diagram of bulk sound velocity (υ_φ) vs. density (ρ). On a υ_φ vs. mean atomic weight ($\text{M}$) diagram, however, all of the ilmenite consistent with a single $\text{υ}{}_{\mathrm{\phi }}\text{M}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{=}\text{constant trend}$. Elasticity systematics for isostructural sequences are used to e the bulk modulus (2.09 Mbar) and bulk sound velocity (7.4 km/sec) of MgSiO₃-elmenite.     

Physical constraints for the analysis of the geomagnetic secular variation

Slow changes in the magnetic field are believed to originate in the core of the Earth. Interpretation of these changes requires knowledge both of the vertical component of the field and of its rate of change at the core-mantle boundary (CMB). While various spherical harmonic models show some agreement for the field at the CMB, those for secular variation (SV) do not. SV models depend heavily on annual means at relatively few and poorly distributed magnetic observatories. In this paper, the SV at the CMB is modelled by fitting 15-year differences in the annual means of the X, Y and Z components (from 1959 to 1974). The model is made unique by imposing the constraint that $\text{?}{}_{\mathrm{CMB}}\text{B}\text{?}{}_{\mathrm{r}}{}^2\text{d}\text{S}$ be a minimum, using the method of Shure et al. (1982). If SV is attributed to motions of core fluid, then this model will yield, in some sense, the slowest core motions. The null space is determined by the distribution of observations, and therefore, to be consistent, only those observatories have been retained which recorded almost continuously throughout the interval 1959–1974.The method allows misfit between the model and the observations. The best value for the misfit can be derived from estimates of errors in the data, or alternatively, because larger misfit leads to smoother models (i.e., smaller $\text{?}\text{B}\text{?}{}_{\mathrm{r}}{}^2\text{d}\text{S}$), the best value can be estimated subjectively from the final appearance of the model. Both procedures have their counterparts in the conventional spherical harmonic expansion approach, when smoothing is achieved by lowering the truncation level. The new proposal made in this paper is to use objective criteria for determining the misfit, based on the assumption that diffusion is negligible, in which event all integrals $\text{∫}\text{B}\text{?}{}_{\mathrm{r}}{}^2\text{d}\text{S}$ will vanish when S_i is a region on the CMB bounded by a contour of zero vertical component of field. For the 1965 definitive model which is adopted here, and for most other contemporary models, there are six such areas, giving five independent integrals (the integrals over the six regions must sum to zero if ? · B = 0). Tabulating these integrals for various choices of the misfit gives minimum values near 2 nT y^?1. It is impossible to achieve this good a fit to the data using a reasonable model derived by truncating the spherical harmonic expansion. The value 2 nT y^?1 corresponds to errors of ～ 20 nT in individual annual means, which is rather larger than expected from the scatter in the data.     

General relationships among sound speeds: II. Theory and discussion

The dependence of bulk sound speed V_φ upon mean atomic weight $\text{m}$ and density ρ can be expressed in a single equation:

$\text{V}{}_{\mathrm{\phi }}\text{=Bρ}{}^{\mathrm{\lambda }}\text{(}\text{m}{}_0\text{m}{}^{\mathrm{<mtext>[</mtext><mtext>1</mtext><mtext>2</mtext><mtext>+\lambda (1?c)]</mtext>}}\text{(}\text{km/sec}\text{)}$

Here B is an empirically determined “universal” parameter equal to 1.42, $\text{m}{}_0\text{= 20.2}$, a reference mean atomic weight for which well-determined elastic properties exist, and λ = 1.25 is a semi empirical parameter equal to $\text{γ ?}\text{1}\text{3}$ where γ is a Grüneisen parameter. The constant c = (? ln V_M/? ln $\text{m}\text{)}{}_{\mathrm{X}}$, where V_M is molar volume, is in general different for different crystal structure series and different cation substitutions. However, it is possible to use c_Fe = 0.14 for Fe²⁺Mg²⁺ and GeSi substitutions and c_Ca ? 1.3 for CaMg substitutional series. With these values it is pos to deduce from the above equation Birch's law, its modifications introduced by Simmons to account for Ca-bearing minerals, variations in the seismic equation of state observed by D.L. Anderson, and the apparent proportionality of bulk modulus K to V_M^?4.     

The caustic and other properties of SKP

The caustic of SKP is found at an epicentral distance Δ_C = 129.5° for surface foci and at Δ_C = 128.9° for foci at 400 km depth, by means of amplitude-distance graphs based upon short-period time-domain measurements. These results are essentially confirmed by long-period time-domain measurements of SKP as well as by frequency-domain studies, even though the spectra are less accurate for such determinations. The average period of SKP is $\text{T}\text{= 1.45 ± 0.45}\text{sec}$ from short-period records, significantly different from the corresponding PKP-period of 1.00 ± 0.31 sec. Likewise, the long-period averages of SKP = 10.8 ± 4.5 sec and of PKP = 7.7 ± 3.0 sec are significantly different from each other. A travel-time table of SKP1 is deduced, covering the epicentral distance range of 130–143° and the focal depth range of 100–700 km. All results are based on measurements on seismograms of the Swedish network of stations, deriving almost exclusively from earthquakes in the southwest Pacific area.     

Q-structure beneath the Tibetan Plateau from the inversion of Love- and Rayleigh-wave attenuation data

The fundamental mode Love and Rayleigh waves generated by ten earthquakes and recorded across the Tibet Plateau, at QUE, LAH, NDI, NIL, KBL, SHL, CHG, SNG and HKG are analysed. Love- and Rayleigh-wave attenuation coefficients are obtained at time periods of 5–120 s using the spectral amplitudes of these waves for 23 different paths. Love wave attenuation coefficient varies from 0.0021 km^?1, at a period of 10 s, to 0.0002 km^?1 at a period of 90 s, attaining two maxima at time periods of 10 and 115 s, and two minima at time periods of 25 and 90 s. The Rayleigh-wave attenuation coefficient also shows a similar trend. The very low value for the dissipation factor, Q_β, obtained in this study suggests high dissipation across the Tibetan paths. Backus-Gilbert inversion theory is applied to these surface wave attenuation data to obtain average Q_β^?1 models for the crust and uppermost mantle beneath the Tibetan Plateau. Independent inversion of Love- and Rayleigh-wave attenuation data shows very high attenuation at a depth of ～50–120 km ($\text{Q}{}_{\mathrm{\beta }}\text{? 10}$). The simultaneous inversion of the Love and Rayleigh wave data yields a model which includes alternating regions of high and low Q_β^?1 values. This model also shows a zone of high attenuating material at a depth of ～40–120 km. The very high inferred attenuation at a depth of ～40–120 km supports the hypothesis that the Tibetan Plateau was formed by horizontal compression, and that thickening occurred after the collision of the Indian and Eurasian plates.     

Applications of liquid state physics to the Earth's core

By use of the modern theory of liquids and some guidance from the hard-sphere model of liquid structure, the following new results have been derived for application to the Earth's outer core. (1) dK/$\text{d}\text{P ? 5 ? 5. 6P}$/K, where K is the incompressibility and P the pressure. This is valid for a high-pressure liquid near its melting point, provided that the pressure is derived primarily from a strongly repulsive pair potential φ. This result is consistent with seismic data, except possibly in the lowermost region of the outer core, and demonstrates the approximate universality of dK/dP proposed by Birch (1939) and Bullen (1949). (2) dlnT_M/dlnρ = (γC_V ? 1)/($\text{C}{}_{\mathrm{<mtext>V</mtext>}}\text{?}\text{3}\text{2}\text{),}\text{where}\text{T}{}_{\mathrm{<mtext>M</mtext>}}$ is the melting point, ρ the density, γ the atomic thermodynamic Grüneisen parameter and C_V the atomic contribution to the specific heat in units of Boltzmann's constant per atom. This reduces to Lindemann's law for C_V = 3 and provides further support for the approximate validity of this law. (3) It follows that the “core paradox” of Higgins and Kennedy can only occur if $\text{γ <}\text{2}\text{3}$. However, it is shown that $\text{γ <}\text{2}\text{3}\text{? ∫}{}^{\mathrm{\infty }}{}_0\text{(?g/?T)}{}_{\mathrm{\rho }}\text{r(}\text{d}\text{/}\text{d}\text{r)(r}{}^2\text{φ)}\text{d}\text{r > 0}$, which cannot be achieved for any strongly repulsive pair potential φ and the corresponding pair distribution function g. It is concluded that $\text{γ >}\text{2}\text{3}$ and that the core paradox is almost certainly impossible for any conceivable core composition. Approximate calculations suggest that γ ～ 1.3–1.5 in the core. Further work on the thermodynamics of the liquid core must await development of a physically realistic pair potential, since existing pair potentials may be unsatisfactory.     

Transient and steady-state creep of pure forsterite at low stress

Thirty-one single crystals of synthetic forsterite, Fo₁₀₀, were deformed in 69 compressional creep tests in a 0.1-MPa confining atmosphere of H₂ + Ar. Temperature ranged from 1753 to 2023 K and stress σ (= σ₁ - σ₃) from 1.5 to 37.8 MPa. Steady-state creep under these conditions follows an empirical law of the form: strain rate $\text{?}\text{?}\text{= A}{}_{\mathrm{zigma;}}{}^{\mathrm{n}}\text{exp}\text{(}\text{?Q}\text{RT}\text{)}$ where A, n, and Q are constants. General characteristics of Fo₁₀₀ creep — uniformity of strain, shape change as a function of orientation of σ, relative deformation resistance of different orientations — match those of natural olivine single crystals of composition Fo₉₂. Specific constants in the flow law, however, are distinctly new: for σ oriented along [111]_c (equidistant from the three principal crystallographic axes), values for Fo₁₀₀ are n = 2.9 ± 0.2 and Q = 0.67 ± 0.03 MJ/mol (160 ± 7 kcal/mol). A single law covers the range 3 < σ < 30 MPa and 1753 < T < 1953 K. Steady-state deformation is preceded by a transient period of strain softening. High strain rates at σ ? 10 MPa render the transient barely resolvable; it apparently displaces the steady-state flow law by approximately ?0.5% in strain. At σ ? 7.8 MPa, the amount of strain imparted to a sample of the [101]_c orientation is typically <0.1% after one hour.     

Creep response of the lunar crust in mare regions from an analysis of crater deformation

The settling trends of 318 lunar mare craters are compared with predictions of numerical finite-element models in order to determine the creep response of the upper lunar mare crust. No settling is evident in craters smaller than 5 km in diameter. Settling rates of larger craters increase as function of crater size in a manner suggesting a non-linear lunar creep response corresponding to the power law $\text{ε}\text{?}\text{= 8.3 · 10}{}^{\mathrm{?34}}\text{σ}{}^2$ where ε&#x0301; is the strain rate and σ is the differential stress. However, the observed nonlinearity is probably an apparent nonlinearity resulting from the temperature induced viscosity decrease with depth due to a lunar crustal temperature gradient of 3° C/km and a creep activation energy of 20 kcal/mole. It is concluded that creep in the lunar medium is essentially Newtonian, and that the effective viscosity of the upper lunar mare crust is (1.6 ± 0.3) · 10²⁵ poise.     

Seismic array evidence for a two-layer core transition zone

Measurements of traveltimes and traveltime gradients for PKP phases recorded at the Warramunga seismic array from distances between 113° and 176° provide evidence for a two-layer transition zone in the earth's core. The basic data consist of paper recordings played out at 40 mm/sec from analog magnetic tape with amplitude gain control and narrow bandpass filters. Measured values of $\text{d}\text{T}\text{dΔ}$ are perturbed by structure beneath the array and it is necessary to correct for this effect by an empirical approach. On the basis of the $\text{d}\text{T}\text{dΔ}$ values and traveltimes, two precursor branches to PKP_DF for Δ < 143° are identified consistently. These continue as branches intermediate to PKP_DF and PKP_AB for Δ > 145°. The corrected $\text{d}\text{T}\text{dΔ}$ values of all phases are smoothed and used to derive a velocity model for the core. The preferred model includes two velocity discontinuities of less than 0.1 km/sec each in the core transition zone to generate the two additional PKP branches. Together with the velocity jump at the outer-inner core boundary, these discontinuities define two layers each a few hundred kilometers thick. It is argued that recent proposals concerning inhomogeneities at the base of the mantle or within the transition zone as the source of PKP precursors do not agree with our observations nor those of other researchers. Inhomogeneities are not excluded but are considered a secondary effect for PKP phases, the primary effect being due to two discontinuous velocity increases in the transition zone.     

The Vrancea,Romania, earthquake of March 4, 1977 — a quite successful prediction

The available data on the destructive intermediate earthquakes ($\text{M ? 6}\text{3}\text{4}$) in the Vrancea, Romania, region have been examined with the aim of revealing some time-magnitude regularities. The basic idea is that the total sequence (? 1100–1973 yr.), which appears as random, could be decomposed in some regular source-components which, by extrapolation, are superimposed to predict the future total sequence.The common nature of faulting (reverse dip-slip) and inferred regularities in the time-magnitude pattern of destructive Romanian earthquakes — (a) three active (seismic) time-bands alternating with quiet periods, the existence of (b) “quasicycles” and of (c) “supercycles” — led to the following predictions: (1) the occurrence of a shock with $\text{M ≈ 6}\text{3}\text{4}\text{? 7}$ in 1980 ± 13 years; and (2) later earthquakes are predicted in 2005, in 2030–2040 ($\text{M ≈ 6}\text{3}\text{4}\text{? 7}$), and one with nearly maximum magnitude ($\text{M = 7}\text{1}\text{2}\text{?7}\text{3}\text{4}$) in 2070–2090.In every century, about 40 years represent a time interval of high seismic danger for Romania and, according to the proposed long-term time-magnitude model, three destructive earthquakes arc to occur in (and/or near) the evidenced seismic periods P₁, P₂ and P₃.It is shown that, taking into account the actual difficulties involved in the earthquake prediction, the Vrancea destructive earthquake of March 4, 1977 (M = 7.1) was quite successfully predicted.     

Orthogonality of spherical harmonic coefficients

An essentially arbitrary function V(θ, λ) defined on the surface of a sphere can be expressed in terms of spherical harmonics $\text{V(θ, Λ) = a}\text{∑}\text{n=1}\text{∞}\text{∑}\text{m=0}\text{n}\text{p}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{(cos θ) (g}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{cos mΛ + h}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{sin mΛ)}$ where the P_n^m are the seminormalized associated Legendre polynomials used in geomagnetism, normalized so that $\text{〈[P}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{(cos θ) cos mΛ]〉}{}^2\text{=1/}\text{(2n+1)}$ The angular brackets denote an average over the sphere. The class of functions V(θ, λ) under consideration is that normally of interest in physics and engineering. If we consider an ensemble of all possible orientations of our coordinate system relative to the sphere, then the coefficients g_n^m and h_n^m will be functions of the particular coordinate system orientation, but $\text{〈:(g}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{〉) = 〈(h}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{=}\text{S}{}_{\mathrm{n}}\text{/}\text{(2n=1)}$ where $\text{S}{}_{\mathrm{n}}\text{=}\text{∑}\text{m=0}\text{n}\text{[(g}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{+ (h}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{]}$ for any orientation of the coordinate system (S_n is invariant under rotation of the coordinate system). The averages are over all orientations of the system relative to the sphere. It is also shown that 〈g^m_ng^l_p〉 = 〈h^m_nh^l_p〉 = 0 for l ≠ m or p ≠ n and 〈g^m_nh^l_p〉 = 0 fro all n, m, p, l.     

Intrinsic Q and its frequency dependence

Various techniques for estimating $\text{t1}$ (travel time/quality factor Q) from short-period seismic-array records of body waves have been investigated. Spectral analysis in the frequency domain seems to be more appropriate for this purpose than time domain methods, because of the relative ease with which source and instrument effects can be removed. Of the techniques available, those based on maximum likelihood and homomorphic deconvolution give estimates of relative power versus frequency which best represent the power contained in a time-domain wavelet of short duration. The latter technique seemed to have better noise-eliminating properties than the former. Therefore, homomorphic deconvolution was used to obtain estimates of $\text{t1}$ values from P, PcP, ScP and S phases recorded at the Warramunga array in the Northern Territory of Australia. The source regions for the event studied were the Sunda, Mariana, New Hebrides, Kermadec and Tonga trench zones.The short-period $\text{t1}$ estimates obtained using the above method were much smaller than estimates from published free-oscillation Q models, indicating that the values of Q for compressional and shear waves are frequently-dependent. It was found that short-period $\text{t1}$ values and free-oscillation Q models could be made consistent with one another by assuming Q = Q₀(1+τω) where Q₀ and τ are constants. The results of this investigation suggest another approach to how the Q structure of the mantle can be investigated.     

A preliminary geoelectrical model of the Karelian megablock of the Baltic shield

Three groups of electromagnetic data have been considered in order to construct a preliminary geoelectrical depth model of the old basement of the Baltic shield: (i) audiomagnetotelluric data in the period range $\text{1}\text{3700}\text{?}\text{1}\text{8}\text{s;}$ (ii) magnetotelluric data in the period range 25–1000 s; and (iii) global magnetovariation-sounding data in the period range >6 h.These data sets fit together on a common apparent-resistivity curve, indicating the existence of a unified geoelectrical model. This model does not support the existence of conducting crustal or asthenospheric layers.     

Effect of oxygen partial pressure on the dislocation recovery in olivine: a new constraint on creep mechanisms

The dislocation annihilation rate in experimentally deformed olivine single crystals was measured as a function of oxygen partial pressure (P_O2). It was shown that the dislocation annihilation rate decreased with increasing P_O2. This result is inconsistent with the reported P_O2 dependence of creep rate () in single olivine crystals, thus indicating that the creep in single olivine crystals is not rate-controlled by recovery, under the experimentally investigated conditions.     

Source mechanism of the 1906 San Francisco earthquake

Surface-wave amplitudes in the period range 50–100 s at eight European and North American stations, horizontal slip profiles along the rupture zone and the timing of certain events along the fault during rupture time are all engaged in unison to reconstruct the motion at the source. A modified source model is used to accommodate a moving rupture with variable dislocation in the direction of propagation.It is inferred that the rupture started at about 13 h 11 m 55 s GMT near San Juan Bautista and propagated unilaterally northwestward along N35°W over 400 km with an average rupture velocity of 3.5 km/s. At 13 h 12 m 12 s, the dynamic shear front, moving with the rupture speed, hit the Lick Observatory. Then, at 13 h 12 m 18 s, the rupture arrived to the vicinity of the epicenter in the Santa Cruz Mountains given by B. Bolt. There the slip changed sharply from an average of 0.5 m to a high value of 3 m causing extensive landslides and avalanches. At 13 h 12 m 32.5 s two railroad clocks at San Rafael were stopped. Finally, at 13 h 12 m 36 s the offset front hit the Naval Observatory at Mare Island and stopped the astronomical clocks there. Conspicuous surface waves, visible on Wiechert seismograms in Europe in the period range 55–65 s, reflect the true rupture time.The seismic data inversion yields an effective radiation source some 240 km long with an average vertical extent of some 34 km over a total fault length of 400 km ($\text{U}\text{d}\text{S ? 29,000}\text{m km}{}^2\text{or}\text{μ}\text{U}\text{d}\text{S ? 9 · 10}{}^{27}\text{dyn cm}$). It began at the Santa Cruz Mountains and ended some 20 km off coast Point Arena. Thus, due to the nonuniform slip profile, only $\text{3}\text{5}$ of the total fracture length contributed to the far radiation field.Although the product of the average source displacement (over the entire fault) and the vertical extent appears to be fairly well determined from the surface-wave spectrums, the separate values of these entities cannot be uniquely determined. If the average surface displacements (～ 3.2 m) are diagnostic of the entire fault, a vertical extent of H = 34 km is required.Finally, a new analysis of surface waves from the Alaska earthquake of July 10, 1958, the Queen Charlotte Islands earthquake of August 22, 1949 and the Kern County shock of July 21, 1952, enables us to draw parallels between the three biggest major events which occurred along the NE Pacific coast during 1906–1958. A common feature of all of these earthquakes is that vertical failure extents of 30–40 km are implied.     

The thermal boundary-layer interpretation of D″ and its role as a plume source

The “anomalous” layer in the lowermost mantle, identified as D″ in the notation of K.E. Bullen, appears in the PREM Earth model as a 150 km-thick zone in which the gradient of incompressibility with pressure, $\text{d}\text{K}\text{d}\text{P}$, is almost 1.6, instead of 3.2 as in the overlying mantle. Since PREM shows no accompanying change in density or density gradient, we identify D″ as a thermal boundary layer and not as a chemically distinct zone. The anomaly in $\text{d}\text{K}\text{d}\text{P}$ is related to the temperature gradient by the temperature dependence of K_s, for which we present a thermodynamic identity in terms of accessible quantities. This gives the numerical result (?K_s/?T)_P=?1.6×10⁷ Pa K^?1 for D″ material. The corresponding temperature increment over the D″ range is 840 K. Such a layer cannot be a static feature, but must be maintained by a downward motion of the lower mantle toward the core-mantle boundary with a strong horizontal flow near the base of D″. Assuming a core heat flux of 1.6 × 10¹² W, the downward speed is 0.07 mm y^?1 and the temperature profile in D″, scaled to match PREM data, is approximately exponential with a scale height of 73 km. The inferred thermal conductivity is 1.2 W m^?1 K^?1. Using these values we develop a new analytical model of D″ which is dynamically and thermally consistent. In this model, the lower-mantle material is heated and softened as it moves down into D″ where the strong temperature dependence of viscosity concentrates the horizontal flow in a layer ～ 12 km thick and similarly ensures its removal via narrow plumes.     

Great-circle Rayleigh wave attenuation and group velocity,Part III: Inversion of global average group velocities and attenuation coefficients

Average shear-velocity models for the upper mantle have been derived by controlled Monte Carlo inversion of global average Rayleigh wave group velocity (GAGV) data for periods between 50 and 300 seconds. GAGV data have been corrected for attenuative dispersion using a method based on the theory of Liu, Anderson and Kanamori. Two types of model bounds have been used with one- or two-layer low-velocity zones beginning at depths of 70 and 100 km. All models fitting GAGV data within one standard deviation have low-velocity zones in the 100–200 km depth range. Models with low-velocity zones beginning at 70 km, as well as 100 km, fit GAGV data within one standard deviation, so the average thickness of the lithosphere (taken as the depth to the top of the low-velocity zone) cannot be determined with precision.Global average models for shear-wave attenuation (Q^?1_β) have been derived from global average Rayleigh wave attenuation coefficients for periods between 50 and 300 s and average shear-velocity models. Zones of high Q^?1_β coincide with the low-velocity zones of all shear-velocity models, however, models with low-velocity zones beginning at a depth of 70 km have the highest-attenuation layer in the lower half of the low-velocity zone. Resolution kernels for these attenuation models show that parameters for layers shallower than the lower part of the low-velocity-high-attenuation zone are strongly coupled but are distinct from the lower part of this zone. This suggests that the deeper part of the low-velocity-high-attenuation zone is the most mobile part of the zone or that on the average, the top of the zone is deeper than 70 km.The average Q_β of the lithosphere, low-velocity zone, and sub-low-velocity layer (asthenosphere) are approximately 200, 85–110 and 170–200, respectively.