A note on the equilibrium points of the restricted problem of three bodies

Using Sylvester's theorem on matrices, an elegant expression is obtained for the solutions of the restricted problem of three bodies in the neighborhood of the equilibrium points.     

Selection criteria for linear regression models to estimate individual tree biomasses in the Atlantic Rain Forest,Brazil

Background

Biomass models are useful for several purposes, especially for quantifying carbon stocks and dynamics in forests. Selecting appropriate equations from a fitted model is a process which can involves several criteria, some widely used and others used to a lesser extent. This study analyzes six selection criteria for models fitted to six sets of individual biomass collected from woody indigenous species of the Tropical Atlantic Rain Forest in Brazil. Six models were examined and the respective fitted equations evaluated by the residual sum of squares, adjusted coefficient of determination, absolute and relative estimates of the standard error of estimate, and Akaike and Schwartz (Bayesian) information criteria. The aim of this study was to analyze the numeric behavior of these model selection criteria and discuss the ease of interpretation of them. The importance of residual analysis in model selection is stressed.

Results

The adjusted coefficient of determination (\( R^{2}_{adj.} \)) and the standard error of estimate in percentage (Syx%) are relative model selection criteria and are not affected by sample size and scale of the response variable. The sum of squared residuals (SSR), the absolute standard error of estimate (Syx), the Akaike information criterion and the Schwartz information criterion, in turn, depend on these quantities. The best fit model was always the same within a given data set regardless the model selection criteria considered (except for SSR in two cases), indicating they tend to converge to a common result. However, such criteria are not always closely related across different data sets. General model selection criteria are indicative of the average goodness of fit, but do not capture bias and outlier effects. Graphical residual analysis is a useful tool to this detection and must always be used in model selection.

Conclusions

It is concluded that the criteria for model selection tend to lead to a common result, regardless their mathematical formulation and statistical significance. Relative measures of goodness of fitting are easier to interpret than the absolute ones. Careful graphical residual analysis must always be used to confirm the performance of the models.

     

Intercomparison of Boron Isotope and Concentration Measurements. Part I: Selection, Preparation and Homogeneity Tests of the Intercomparison Materials

In 1999 the Istituto di Geoscienze e Georisorse (IGG), with the support of the International Atomic Energy Agency (IAEA), undertook the collection, preparation and distribution of eight geological materials intended for a blind interlaboratory comparison of measurements of boron isotopic composition and concentration. The materials came from Italian sources and consist of three natural waters (Mediterranean seawater and two groundwaters) and five rocks and minerals (tourmaline, basalt, obsidian, limestone and clay). The solid materials were crushed, milled and mixed, in preparation for distribution. Extensive assays performed at the IGG on these materials demonstrated that their boron isotopic and chemical compositions are homogeneous.
Additional homogeneity tests were carried out on solid material fragments at the GeoForschungsZentrum Potsdam, with the specific objective of investigating the suitability of some of them for the calibration in situ of micro-analytical techniques. Two materials, B4 (tourmaline) and B6 (obsidian), proved to be isotopically homogeneous and may become excellent references for in situ microanalyses of boron isotopes.
The materials described here were used as the basis of a major laboratory intercomparison study and are now available for further distribution from either the IAEA (solid materials) or the IGG (waters).     

The mafic-ultramafic complex near Finero (Ivrea-Verbano Zone), II. Geochronology and isotope geochemistry

Whole-rock Nd and Sr isotopic compositions of the mafic-ultramafic complex near Finero demonstrate that the magma was derived from a depleted, perhaps MORB-type mantle reservoir. The Sm-Nd data for the Amphibole Peridotite unit can be interpreted as an isochron with an apparent age of 533 ± 20 Ma, which is consistent with a ²⁰⁷Pb/²⁰⁶Pb evaporation age of 549 ± 12 Ma of a single zircon grain from the Internal Gabbro unit. However, the interpretation of these apparent ages remains open to question. We therefore retain the alternative hypotheses that the intrusion occurred either about 533 or 270 Ma ago, the latter being the most likely age of emplacement of the much larger magma body near Balmuccia (Val Sesia). The implication of the older emplacement age (if correct) would be that the igneous complex may be related to the numerous amphibolite units, which are intercalated with the metapelites of the overlying Kinzigite Formation, and together with them may constitute an accretionary complex. In this case, the mafic-ultramafic complex itself might also be part of such an accretionary complex (as has been proposed for the Balmuccia peridotite).

Internal Sm-Nd isochrons involving grt, cpx, plag and amph from the Internal Gabbro unit yield concordant ages of 231 ± 23, 226 ± 7, 223 ± 10, 214 ± 17, and 203 ± 13 Ma. These results confirm published evidence for a separate, regional heating event about 215 ± 15 Ma ago.

Initial _Nd(533) values average +6.3 ± 0.4 for six samples of the Amphibole Peridotite unit and +6.0 ± 1.2 for ten samples of the External Gabbro unit. ⁸⁷Sr/⁸⁶Sr ratios require little or no age correction and range from 0.7026 to 0.7047 (with two outliers at 0.7053 and 0.7071). Strong correlations between ⁸⁷Sr/⁸⁶Sr and K₂O and weaker correlations between initial _Nd and K₂O imply a comparatively minor (≤ 10%) contamination of the External Gabbro magma by crustal material and a later alteration by a crustal or seawater-derived fluid. These results contrast sharply with the isotopic composition (negative _Nd and high ⁸⁷Sr/⁸⁶Sr values) of the associated mantle rocks, the Phlogopite Peridotite unit, which has been pervasively metasomatized by crustal fluids. This type of metasomatism and its isotopic signature are never seen in the magmatic complex. This evidence rules out any direct genetic relationship between the igneous complex and the mantle peridotite. The crust-mantle interaction is the opposite of that seen at Balmuccia, where the mantle peridotite is essentially ‘pristine’ and the magmatic body has been extensively contaminated by assimilation of crustal rocks.     

Potential evidence for slab detachment from the flexural backstripping of a foredeep: Insight on the evolution of the Pescara basin (Italy)

The discrepancy between the size of the Apenninic chain and the depth of the Adriatic foredeep is investigated using 2D flexural backstripping on well‐constrained depth‐converted cross‐sections in the Pescara basin (Central Italy). The procedure consisted of removal, uplift, unfolding and unfaulting of the Pliocene–Pleistocene foreland deposits to produce a palaeogeographic map of the basin at the end of the Messinian and to constrain sedimentary rates since the Miocene. Results are found to support the contribution of an external load to the foreland evolution together with the Apenninic chain load. The interplay of the two types of loads resulted in spatial and temporal variations of the foredeep evolution that are quantified by palaeogeographic maps and sedimentation rates obtained through backstripping. Results are interpreted as representing the effects of a southward‐migrating wave linked to slab detachment beneath the Adriatic foredeep. This procedure can be useful to investigate similar problems on other chains worldwide.     

Reconstruction of hydrocarbons accumulation in sediments affected by the oil refinery industry: the case of Tehuantepec Gulf (Mexico)

The Isthmus of Tehuantepec corresponds to the shortest distance (~200?km) between the Gulf of Mexico and the Pacific Ocean in Southern Mexico, and the main economical activity of this region is oil extraction and refining. Polycyclic aromatic hydrocarbons (PAHs) and total petroleum hydrocarbons (TPHs) were determined in a ²¹⁰Pb dated sediment core collected from the continental shelf of Tehuantepec Gulf, in the vicinity of the oil refinery of Salina Cruz, Oaxaca, the main oil refining facility of the country. The sediments were mostly of coarse nature and hence PAHs and TPHs concentrations throughout the core (61?C404???g?g^?1 and 29?C154?mg?kg^?1, respectively) were below international quality benchmarks. Depth profiles of both PAHs and TPHs concentrations showed increasing trends since the early 1900s but the higher values were found from the 1950s to present. PAH congener ratios showed that these contaminants had both petrogenic and pyrolitic sources, although the former has been predominant since the 1970s. The Salina Cruz refinery started operations in 1978 but the oil industry activities in the Tehuantepec Isthmus go back to the beginning of the twentieth century with the operation of Minatitlan refinery in the Gulf of Mexico, and the Gulf of Tehuantepec being the main conduit for oil distribution in the Pacific coast. The observed changes in contaminant distributions described well the oil industry development in the area.     

Mantle rheology: radial and lateral viscosity variations inferred from microphysical creep laws

Inferences on the rheology of the mantle based on theoretical and experimental rate equations for steady state creep are discussed and compared with results from geophysical models. The radial increase of viscosity by one to three orders of magnitude across the mantle, required by inversion of postglacial rebound and geodynamic data, is confirmed by microphysical models based on the estimation of continuous and discontinuous changes of creep parameters with depth. The upper mantle (viscosity 10²⁰–10²¹ Pa s) is likely to show non-Newtonian rheology (power-law creep) for average grain sizes larger than 0.1 mm as an order of magnitude. Given the variability of both grain size and stress conditions, local regions of linear rheology can be present. The rheology of transition zone and lower mantle (viscosity 10²²–10²⁴ Pa s) cannot be definitely resolved at present. Estimation of creep parameters leads to possible nonlinear or mixed rheology, if grain sizes are not lower than 0.1 mm and flow conditions can be approximated by a constant strain rate of about 10⁻¹⁵ s⁻¹. This conclusion can be modified by different flow conditions (e.g. a decrease in strain rate or constant viscous dissipation). Furthermore, experiments on fine-grained garnetites and perovskite analogues have shown that diffusion creep is predominant at laboratory conditions. However, the pressure dependence of creep in these phases is unknown, and therefore direct extrapolation to lower mantle conditions is necessarily speculative. Lateral variations of viscosity, largest in the upper and lowermost mantle (up to 2–4 orders of magnitude) are predicted by models based on lateral temperature anomalies derived from seismic tomographic models.     

On the Vp/Vs–Mg# correlation in mantle peridotites: Implications for the identification of thermal and compositional anomalies in the upper mantle

We use thermodynamically self-consistent and hybrid methods to analyze the correlation of important physical parameters (e.g. bulk density, elastic moduli) with bulk Mg# and modal composition in mantle peridotites at upper mantle conditions. Temperature (anharmonic and anelastic), pressure and compositional derivatives for all these parameters are evaluated. The results show that the widely used correlations between Vp/Vs and Mg# in peridotites are strictly valid only for garnet-bearing assemblages at temperatures < 900 °C. The correlation breaks down when: i) spinel is the stable Al-rich phase in the assemblage and ii) when anelastic attenuation of seismic velocities becomes important (T ? 900 °C). This implies that the range of applicability of published Vp/Vs–Mg# correlations for the upper mantle is limited to a depth interval between the spinel–garnet phase transition and the 900 °C isotherm. We use numerical simulations to show that this depth interval is virtually nonexistent in lithospheres thinner than ～ 140 km and can comprise up to ～ 50% of the lithospheric mantle in thick (> 220 km) lithospheric domains. In addition, we show that for most of the upper mantle the expected Δ(Vp/Vs) values associated with compositional variations are smaller than the resolution limit of current seismological methods. All these considerations suggest that the Vp/Vs ratio is not a reliable measure of compositional variations and that for large parts of the upper mantle compositional anomalies cannot be separated from thermal anomalies on the basis of seismological studies only. We further confirm that the only reliable indicator of compositional anomalies in a peridotitic mantle is the ratio of density to shear wave velocities (ρ/Vs). Our results demonstrate that geophysical–petrological models (forward or inverse) that model these two fields (i.e. density and Vs) self-consistently within a robust thermodynamic framework are necessary for characterizing the small-scale thermal and compositional structure of the lithosphere and sublithospheric upper mantle.