Partial reconstruction of the trajectories of a discretely observed branching diffusion with immigration and an application to inference

In this treatise we consider finite systems of branching particles where the particles move independently of each other according to d-dimensional diffusions. Particles are killed at a position dependent rate, leaving at their death position a random number of descendants according to a position dependent reproduction law. In addition particles immigrate at constant rate (one immigrant per immigration time). A process with above properties is called a branching diffusion withimmigration (BDI). In the first part we present the model in detail and discuss the properties of the BDI under our basic assumptions. In the second part we consider the problem of reconstruction of the trajectory of a BDI from discrete observations. We observe positions of the particles at discrete times; in particular we assume that we have no information about the pedigree of the particles. A natural question arises if we want to apply statistical procedures on the discrete observations: How can we find couples of particle positions which belong to the same particle? We give an easy to implement 'reconstruction scheme' which allows us to redraw or 'reconstruct' parts of the trajectory of the BDI with high accuracy. Moreover asymptotically the whole path can be reconstructed. Further we present simulations which show that our partial reconstruction rule is tractable in practice. In the third part we study how the partial reconstruction rule fits into statistical applications. As an extensive example we present a nonparametric estimator for the diffusion coefficient of a BDI where the particles move according to one-dimensional diffusions. This estimator is based on the Nadaraya-Watson estimator for the diffusion coefficient of one-dimensional diffusions and it uses the partial reconstruction rule developed in the second part above. We are able to prove a rate of convergence of this estimator and finally we present simulations which show that the estimator works well even if we leave our set of assumptions.

Partial melting within the lower crust – Constraints from bulk rock geochemical data and trace element mineral analyses of granulites and migmatites from Central Finland

This PhD thesis concerns geochemical constraints on recycling and partial melting of Archean continental crust. A natural example of such processes was found in the Iisalmi area of Central Finland. The rocks from this area are Middle to Late Archean in age and experienced metamorphism and partial melting between 2.7-2.63 Ga. The work is based on extensive field work. It is furthermore founded on bulk rock geochemical data as well as in-situ analyses of minerals. All geochemical data were obtained at the Institute of Geosciences, University of Mainz using X-ray fluorescence, solution ICP-MS and laser ablation-ICP-MS for bulk rock geochemical analyses. Mineral analyses were accomplished by electron microprobe and laser ablation ICP-MS. Fluid inclusions were studied by microscope on a heating-freezing-stage at the Geoscience Center, University Göttingen. Part I focuses on the development of a new analytical method for bulk rock trace element determination by laser ablation-ICP-MS using homogeneous glasses fused from rock powder on an Iridium strip heater. This method is applicable for mafic rock samples whose melts have low viscosities and homogenize quickly at temperatures of ~1200°C. Highly viscous melts of felsic samples prevent melting and homogenization at comparable temperatures. Fusion of felsic samples can be enabled by addition of MgO to the rock powder and adjustment of melting temperature and melting duration to the rock composition. Advantages of the fusion method are low detection limits compared to XRF analyses and avoidance of wet-chemical processing and use of strong acids as in solution ICP-MS as well as smaller sample volumes compared to the other methods. Part II of the thesis uses bulk rock geochemical data and results from fluid inclusion studies for discrimination of melting processes observed in different rock types. Fluid inclusion studies demonstrate a major change in fluid composition from CO2-dominated fluids in granulites to aqueous fluids in TTG gneisses and amphibolites. Partial melts were generated in the dry, CO2-rich environment by dehydration melting reactions of amphibole which in addition to tonalitic melts produced the anhydrous mineral assemblages of granulites (grt + cpx + pl ± amph or opx + cpx + pl + amph). Trace element modeling showed that mafic granulites are residues of 10-30 % melt extraction from amphibolitic precursor rocks. The maximum degree of melting in intermediate granulites was ~10 % as inferred from modal abundances of amphibole, clinopyroxene and orthopyroxene. Carbonic inclusions are absent in upper-amphibolite facies migmatites whereas aqueous inclusion with up to 20 wt% NaCl are abundant. This suggests that melting within TTG gneisses and amphibolites took place in the presence of an aqueous fluid phase that enabled melting at the wet solidus at temperatures of 700-750°C. The strong disruption of pre-metamorphic structures in some outcrops suggests that the maximum amount of melt in TTG gneisses was ~25 vol%. The presence of leucosomes in all rock types is taken as the principle evidence for melt formation. However, mineralogical appearance as well as major and trace element composition of many leucosomes imply that leucosomes seldom represent frozen in-situ melts. They are better considered as remnants of the melt channel network, e.g. ways on which melts escaped from the system. Part III of the thesis describes how analyses of minerals from a specific rock type (granulite) can be used to determine partition coefficients between different minerals and between minerals and melt suitable for lower crustal conditions. The trace element analyses by laser ablation-ICP-MS show coherent distribution among the principal mineral phases independent of rock composition. REE contents in amphibole are about 3 times higher than REE contents in clinopyroxene from the same sample. This consistency has to be taken into consideration in models of lower crustal melting where amphibole is replaced by clinopyroxene in the course of melting. A lack of equilibrium is observed between matrix clinopyroxene / amphibole and garnet porphyroblasts which suggests a late stage growth of garnet and slow diffusion and equilibration of the REE during metamorphism. The data provide a first set of distribution coefficients of the transition metals (Sc, V, Cr, Ni) in the lower crust. In addition, analyses of ilmenite and apatite demonstrate the strong influence of accessory phases on trace element distribution. Apatite contains high amounts of REE and Sr while ilmenite incorporates about 20-30 times higher amounts of Nb and Ta than amphibole. Furthermore, trace element mineral analyses provide evidence for magmatic processes such as melt depletion, melt segregation, accumulation and fractionation as well as metasomatism having operated in this high-grade anatectic area.

An investigation into complex inorganic materials with Mössbauer spectroscopy

Mössbauer Spektroskopie ist ein unverzichtbares Instrument für die Bestimmung von Oxidationszuständen und für die Analyse von lokalen Ordnungsphänomenen von Mössbauer aktiven Atomen. Weil es sich um eine lokale Methode handelt können sowohl kristalline als auch amorphe Materialien untersucht werden. Die Kombination von lokaler Prüfung mit Mössbauer Spektroskopie und globaler Untersuchung z.B. mit Röntgendiffraktometrie ermöglicht die Studie von Ordnungseffekten von statistisch besetzten Positionen in einer geordneten Matrix. Das wurde hier eingesetzt um die lokale Umgebung in zwei Serien von Heuslerverbindungen, Co2-xFe1+xSi and Co2Mn1-xFexAl zu untersuchen. Für die Co2Mn1-xFexAl Serie wurde eine L21 geordnete Phase in einer insgesamt B2 geordneten Probe detektiert. Ein Wechsel von der AlCu2Mn zu der CuHg2Ti Struktur wurde für die Co2-xFe1+xSi Proben gefunden. Die Transformation von einem Glas zu einem keramischen Material wurde mit 119Sn Mössbauer Spektroskopie untersucht. Die höhere Ordnung in der Keramik wurde von einer kleiner werdenden Mössbauerlinienbreite begleitet. Demzufolge geben die Modifikationen der Sn Umgebungen klar die Transformation des gesamten Materials wieder. Ist die lokale Umgebung von unregelmäßig auftretenden Atomen in einer amorphen Matrix von Interesse, sind lokal prüfende Methoden die zuverlässigsten Methoden die zur Verfügung stehen. In dieser Arbeit wurde 119Sn Mössbauer Spektroskopie eingesetzt um die Oxidationszustände, die lokalen Umgebungen und relativen Intensitäten von Zinn Atomen in einer Silikatmatrix zu bestimmen. Modifikationen dieser Parameter als Funktion von Prozess bestimmenden Parametern wie der Sauerstoffpartialdruck, die Temperatur, die Behandlungsdauer und der Abkühlprozess genauso wie der SnO2 Gehalt sind von Interesse, weil durch Reduktions- und Diffusionsprozesse Änderungen des Koordinations- und des Oxidationszustands der Zinnatome auftreten. Da diese Änderungen in der Glasmatrix verursachen, die das fertige Produkt im industriellen Fertigungsprozess ruinieren können sind diese feinen Veränderungen sehr wichtig. Wenigstens zwei Mössbauerlinien korrespondierend mit zwei verschiedenen Umgebungen für Sn2+ und Sn4+ sind für eine Analyse mit ausreichender Qualität notwendig. Durch Vergleich von den bestimmten Hyperfein Parametern mit den Parametern von Modelsubstanzen werden lokale Umgebungen der Zinnatome entworfen. Für Sn2+ werden zwei auf einer trigonalen Pyramide basierende Umgebungen mit variierender Anzahl von bindenden und nicht-bindenden Sauerstoffatomen formuliert. Für Sn4+ wurde eine tetraedrische und eine oktaedrische Umgebung postuliert. Die relativen Intensitäten der vier Mössbauerlinien wurden um ein Diffusions- und Reaktionsmodell zu entwickeln und um einen Satz von Diffusions- und Transferkoeffizienten zu bestimmen eingesetzt. Die bestimmten Diffusionskoeffizienten stimmen mit den Literaturdaten überein. Der Massentransferkoeffizient ist kleiner als der bestimmte Wert, aber immer noch in der gleichen Größenordnung. Im Gegensatz zu den Erwartungen ist der präsentierte Diffusionskoeffizient für Sn4+ bestimmt als der von Sn2+. Das wiederum kann durch Berücksichtigung von Elektronhoppingprozessen erklärt werden.

Granulite facies metamorphism and partial melting processes in the Ivrea Zone, Northern Italy

The Ivrea Zone in northern Italy has been the focus of numerous petrological, geochemical and structural studies. It is commonly inferred to represent an almost complete section through the mid to lower continental crust, in which metamorphism and partial melting of the abundant metapelites was the result of magmatic underplating by a large volume of mantle-derived magma. This study concerns amphibolite and granulite facies metamorphism in the Ivrea Zone with focus on metapelites and metapsammites/metagreywackes from Val Strona di Omegna and metapelites from Val Sesia and Val Strona di Postua, with the aim to better constrain their metamorphic evolution as well as their pressure and temperature conditions via phase equilibria modelling.rnrnIn Val Strona di Omegna, the metapelites show a structural and mineralogical change from mica-schists with the common assemblage bi-mu-sill-pl-q-ilm ± liq at the lowest grades, through metatexitic migmatites (g-sill-bi-ksp-pl-q-ilm-liq) at intermediate grades, to complex diatexitic migmatites (g-sill-ru-bi-ksp-pl-q-ilm-liq) at the highest grades. Within this section several mappable isograds occur, including the first appearance of K-feldspar in the metapelites, the first appearance of orthopyroxene in the metabasites and the disappearance of prograde biotite from the metapelites. The inferred onset of partial melting in the metapelites occurs around Massiola. The prograde suprasolidus evolution of the metapelites is consistent with melting via the breakdown of first muscovite then biotite. Maximum modelled melt fractions of 30–40 % are predicted at the highest grade. The regional metamorphic field gradient in Val Strona di Omegna is constrained to range from conditions of 3.5–6.5 kbar at T = 650–730 °C to P > 9 kbar at T > 900 °C. The peak P–T estimates, particularly for granulite facies conditions, are significantly higher (around 100 °C) than those of most previous studies. In Val Sesia and Val Strona di Postua to the south the exposure is more restricted. P–T estimates for the metapelites are 750–850 °C and 5–6.5 kbar in Val Sesia and approximately 800–900 °C and 5.5–7 kbar in Val Strona di Postua. These results show similar temperatures but lower pressure than metapelites in Val Strona di Omegna. Metapelites in Val Sesia in contact with the Mafic Complex exhibit a metatexitic structure, while in Val Strona di Postua diatexitic structures occur. Further, metapelites at the contact with the Mafic Complex contain cordierite (± spinel) that overprint the regional metamorphic assemblages and are interpreted to have formed during contact metamorphism related to intrusion of the Mafic Complex. The lower pressures in the high-grade rocks in Val Sesia and Val Strona di Postua are consistent with some decompression from the regional metamorphic peak prior to the intrusion of the Mafic Complex, suggesting the rocks followed a clockwise P–T path. In contrast, the metapelites in Val Strona di Omegna, especially in the granulite facies, do not contain any cordierite or any evidence for a contact metamorphic overprint. The extrapolated granulite facies mineral isograds are cut by the rocks of the Mafic Complex to the south. Therefore, the Mafic Complex cannot have caused the regional metamorphism and it is unlikely that the Mafic Complex occurs in Val Strona di Omegna.