FORWARD MODELING IN THE FREQUENCY-SPACE DOMAIN*

Forward modeling of zero-offset data is performed in the frequency-space domain using a one-way extrapolation equation. The use of the frequency domain offers several advantages over conventional time domain methods. The greatest advantage of the frequency domain is that all time derivatives are evaluated exactly by a simple multiplication. Synthetic zero-offset sections are computed with a high degree of accuracy for arbitrary velocity and reflectivity structures. Examples are shown for realistic complicated models and compared with results from physical modeling.     

Fractional Crystallization and Liquid Immiscibility Processes in the Alkaline-Carbonatite Complex of Juqui (So Paulo, Brazil)

The Juqui circular intrusion, which is Cretaceous in age (130–135Ma), crops out in the Precambrian gneissic basement in Brazilover an area of 14 km2. It consists of olivine clinopyroxen-itecumulates (with minor olivine gabbros) in the northeastern sector(74 vol.%), whereas ijolites-melteigites-urtites (4%) and nephelinesyenites with minor essexites and syenodiorites (21%) outlinesubannular concentric patterns with an Mg-carbonatite core (1%), in the southwestern part of the complex. Petrographical, bulk rock, and mineral compositional trendsindicate that the origin of the complex can be largely accountedfor by shallow-level fractional crystallization of a carbonatedbasanitic parental magma. Such a magma was generated deep inthe subcontinental lithosphere by low-degree partial meltingof a garnet-phlogopite peridotite source. Mass-balance calculations in agreement with field volume estimatespermit definition of several fractionation stages of the magmaticevolution under nearly closed-system conditions, with inwarddevelopment of zonally arranged side-wall cumulates. These stagesinvolved: (1) fractionation from basanite to essexite magma(liquid fraction F = 33–5%) by crystallization of olivineclinopyroxenite plus minor olivine alkali gabbro cumulates;(2) derivation of the least differentiated mafic nepheline syenite(F = 5–5 %) from essexitic magma by subtraction of a syenodioriteassemblage; (3) exsolution of a carbonatite liquid (5%) froma CO₂-enriched mafic nepheline syenite magma, which also underwentcontinuous fractionation giving rise to ijolite-melteigite-urtitecumulates. The proportion of cumulus clinopyroxene and biotiteand intercumulus nepheline and alkali feldspar in these lastrocks, as well as the absence of alkalis in carbonatite, maybe attributed, at least in part, to loss of alkali-rich hydrousfluids released during and after the unmixing formation of thetwo conjugate liquids. The KD values determined for Mg-carbonatite/nepheline syeniteare lower (1–4–2–9) for light rare earth elements(LREE) than for REE from Eu to Yb (4–6–7–8),in contrast to recent experimental results (Hamilton et al.,1989). A possible explanation is that Juquia Mg-carbonatiterepresents an alreadydifferentiated magma, which underwent extensivefractionation of LREE-enriched calcite. In this way, the highvariability of K₀ REE patterns observed in several alkaline-carbonatitecomplexes can also be accounted for. The remarkably constant initial ⁸⁷Sr/⁸⁶Sr ratios (mostly between0–7052 and 0–7057) support the interpretation ofthe intrusion as having been generated by fractrional crystallizationand liquid immiscibility from a common parental magma. Iligherisotopic ratios (0–7060–0–7078), found mainlyin dykes and in the border facies of the intrusion, may be dueto contamination by the gecissic basement.     

Spatial Breakdown Point of Variogram Estimators

In the context of robust statistics, the breakdown point of an estimator is an important feature of reliability. It measures the highest fraction of contamination in the data that an estimator can support before being destroyed. In geostatistics, variogram estimators are based on measurements taken at various spatial locations. The classical notion of breakdown point needs to be extended to a spatial one, depending on the construction of most unfavorable configurations of perturbation. Explicit upper and lower bounds are available for the spatial breakdown point in the regular unidimensional case. The difficulties arising in the multidimensional case are presented on an easy example in IR² , as well as some simulations on irregular grids. In order to study the global effects of perturbations on variogram estimators, further simulations are carried out on data located on a regular or irregular bidimensional grid. Results show that if variogram estimation is performed with a 50% classical breakdown point scale estimator, the number of initial data likely to be contaminated before destruction of the estimator is roughly 30% on average. Theoretical results confirm the previous statement on data in IR^d , d 1.     

REE constraints on fractionation processes of massive-type anorthosites on the Lofoten Islands, Norway

Summary Rare Earth Element (REE) data of 34 samples of magmatic rocks from the Lofoten Islands in Norway lend support to the derivation of anorthosites, ferrodiorites and jotunites by fractionation and cumulus processes from typical basaltic magma. Both REE concentration and Eu anomalies (expressed as Eu/Eu^*) form continuous linear trends from anorthosite towards gabbro, ferrodiorite and jotunite in discrimination diagrams against molar CaO/Al₂O₃ ratios indicating the predominant accumulation of plagioclase. Eu/Eu^* decreases from about 4 in the cumulates (anorthosites) to around 1 in the fine-grained gabbroic dikes and to below 1 in some ferrodiorites and the jotunite. The various types of ferrodiorites and the jotunite are regarded as residual liquids, in some cases with variable amounts of cumulus plagioclase. The whole fractionation series from gabbro towards anorthosites and ferrodiorites can be observed in a single intrusion. With increasing fractionation, the REE patterns generally change from flat, slightly LREE-enriched or LREE-depleted to steep and strongly LREE-enriched. These changes and the REE abundances are mainly controlled by the abundance of apatite. Temporally and spatially related mangerites and charnockites form a trend from low-SiO₂ mangerites with Eu/Eu^* > 1 to intermediate-SiO₂ acidic mangerites with Eu/Eu^* ≈ 1 and charnockites with Eu/Eu^* < 1. Accordingly, the low-SiO₂ mangerites are interpreted as alkali feldspar-rich cumulates and the charnockites as residual liquids derived from the acidic mangerites. The mangerites with Eu/Eu^* around 1 have patterns similar to those of some highly evolved ferrodiorites possibly indicating a genetic link. Received December 12, 1999; revised version accepted November 15, 2000     

The almost constant-speed two-body problem with resistance

The almost constant-speed motion of a mass acted upon by a Newtonian attraction and a resisting force is treated. The equation of orbit is derived for a specific type of resistance which covers the familiar case of Danby's drag(=r ^–2) whilst the vector invariants are obtained by direct operation on the vector form of the equation of motion.     

Powder- and single crystal Mössbauer spectroscopy on synthetic fayalite

Mössbauer measurements on synthetic iron orthosilicate Fe₂SiO₄ (fayalite) were carried out in the antiferromagnetic spin state below T _N 65 K. The Mössbauer parameters isomer shift , inner magnetic field H(0), angle between H(0) and the z-component of the electric field gradient (efg), quadrupole splitting QS and asymmetry parameter were determined as a function of temperature. These parameters could be attributed to the two crystallographic sites M1 and M2.The smaller isomer shift on M1 with respect to M2 displays the more covalent character of the Fe-O bond on M1, which is supported by previous neutron diffraction experiments. H(0) shows a Brillouin-type behaviour with different fields on the two crystallographic sites (stronger on M1) and a small discontinuity at T = 23 K which corresponds with previous magnetic measurements. The quadrupole splitting is equal on both sites within error bars, in agreement with previous theoretical results and in contradiction to previous Mössbauer refinements published elsewhere.     

Carbon and Noble Gas Isotopes in the Tengchong Volcanic Geothermal Area, Yunnan, Southwestern China

Carbon and noble gas isotope analyses are reported for bubbling gas samples from the Tengchong volcanic geothermal area near the Indo-Eurasian suture zone. All samples contain a resolvable component of mantle-derived 3He. Occurrence of mantle-derived 3He coincides with surface volcanism. However, 3He occurs over a larger geographic areathan do surface volcanics. δ13C values for CO2 and CH4 vary from -33.4‰ to 1.6 ‰ and from -52.8‰ to -2.8‰, respectively. He and C isotope systematics indicate that CO2 and CH4 in the CO2-rich gases originated predominantly from magmatic component mixed with crustal CO2 produced from carbonate. However, breakdown of organic matter and near-surface processes accounts for the CH4 and CO2 in N2-rich gases. 3He/4He ratio distribution pattern suggests that mantle-derived He and heat sources of high-temperature system in central Tengchong originate from a hidden magma reservoir at subsurface. CO2-rich gases with the highest 3He/4He ratio (5.2 Ra) may be representative of the