Closed-form solution of P4P or the three-dimensional resection problem in terms of Möbius barycentric coordinates

 The perspective 4 point (P4P) problem - also called the three-dimensional resection problem - is solved by means of a new algorithm: At first the unknown Cartesian coordinates of the perspective center are computed by means of M?bius barycentric coordinates. Secondly these coordinates are represented in terms of observables, namely space angles in the five-dimensional simplex generated by the unknown point and the four known points. Substitution of M?bius barycentric coordinates leads to the unknown Cartesian coordinates (2.8)–(2.10) of Box 2.2. The unknown distances within the five-dimensional simplex are determined by solving the Grunert equations, namely by forward reduction to one algebraic equation (3.8) of order four and backward linear substitution. Tables 1.–4. contain a numerical example. Finally we give a reference to the solution of the 3 point (P3P) problem, the two-dimensional resection problem, namely to the Ansermet barycentric coordinates initiated by C.F. Gau? (1842), A. Schreiber (1908) and A.␣Ansermet (1910). Received: 05 March 1996; Accepted: 15 October 1996     

Stable carbon isotope variations in northwest Europe during the last glacial–interglacial transition

Stable carbon isotope data that span part of the last glacial–interglacial transition (ca. 14-9 ka ¹⁴C BP; ca. 15–11 ka cal. BP), and which derive from organ-specific plant macrofossils recovered from two lake sediment profiles in the UK and one in Norway, are compared. The recorded temporal variations show similar trends, which, over a millennial time-scale appear to parallel the main drift in δ¹⁸O as determined for the GRIP ice-core. It is postulated that some smaller scale variations in the δ¹³C profiles may reflect the shorter term oscillations in δ¹⁸O values evident in the GRIP record, although this is less certain. Overall, however, the results suggest that stable carbon isotope measurements based on organ-specific terrestrial plant macrofossils may provide (i) a means for establishing correlations between terrestrial successions and (ii) additional paleoenvironmental information, as the apparent ‘shadowing’ of the GRIP record indicates a common forcing mechanism for both Greenland δ¹⁸O and northwest European δ¹³C variations. From the evidence available we suggest that the recorded δ¹³C variations reflect fluctuations in air temperature and/or changes in water vapour pressure in the atmosphere. © 1997 John Wiley & Sons, Ltd.     

Experiments on steady buoyancy transfer through turbulent fluid layers separated by density interfaces

A laboratory experiment was performed to investigate mixing across a density interface which separates two turbulent fluid layers and coexists with a stabilizing buoyancy flux. It was found that the buoyancy flux (q₀) across the interface and through the turbulent layers (of depth D) becomes steady and constant in magnitude in the vertical direction, only when , where u is the horizontal r.m.s. velocity at the base of the mixed layers. The results suggest that mixing across the density interface is controlled by a dynamically important buoyancy gradient induced in the turbulent layers and that parameters such as the bulk Richardson number, , where Δb is the interfacial buoyancy jump, are of secondary importance. Measurements are used to infer the mixing mechanism at the interface, the mixing efficiency of stratified fluids and the entrainment law. Some geophysical applications of the results are also discussed.     

Silica depleted melting of pelites. Petrogenetic grid and application to the Susqueda Aureole, Spain

Metapelitic rocks in the low pressure contact metamorphic aureole around the Susqueda igneous complex, Spain show a number of features that make them an ideal testing ground for the modelling of silica‐undersaturated melting. Rocks in the aureole experienced localized depletion in silica by the segregation of quartz veins during a pre‐anatectic, regional cordierite‐andalusite grade metamorphic event. These rocks were then intruded by gabbroic to dioritic rocks of the Susqueda igneous complex that formed a migmatitic contact metamorphic aureole in the country rocks. This migmatisation event caused quartz‐saturated hornfels and restite formation in rocks that had experienced no quartz vein segregation in the previous regional metamorphic event, but silica‐undersaturated melting in those rocks that were previously depleted in silica. Silica‐undersaturated melting is investigated using a new petrogenetic P–T projection and equilibrium pseudosections calculated in the KFMASH and NCKFMASH systems, respectively. The grid considers quartz absent equilibria and a range of phases that form typically in silica‐undersaturated bulk compositions, for example corundum. It is shown that the quartz‐rich precursors in the Susqueda contact aureole produced about 10% melt during contact metamorphism. However, most of this melt was extracted leaving behind rocks with restitic bulk compositions and minor leucosome segregation. It is suggested that the melt mixed with the host igneous rocks causing an apparent magmatic zoning from diorite in the centre of the complex to tonalite at the margins. In contrast, the quartz‐poor precursors (from which the quartz veins segregated) melted in the silica‐undersaturated field producing a range of assemblages including peritectic corundum and spinel. Melting of the silica‐undersaturated rocks produced only negligible melt and no subsequent melt loss.     

Pillow talk

Three distinct types of pillows and pillow lava sequences with different modes of origin have been recognized in the extrusive sequences comprising the upper parts of ophiolite complexes that represent the mafic portion of the floor of an Early Cretaceous back-arc basin in southern Chile. One type of pillow formed by non-explosive submarine effusion. A second type formed by magmatic intrusion into pre-existing aquagene tuff formed by explosive eruption. The third type of pillow occurs within dikes, forming pillowed dikes, possibly as a result of vapor streaming within a cooling dike. Where studied in southern Chile, aquagene tuffs and intrusive pillows decrease and water-lain pillows increase in relative abundance from north to south. This variation corresponds with a north-to-south decrease in both the relative volume of extrusives to extensional dikes and the range and volume of differentiated rocks, suggesting a southward increase in rate of extension relative to rate of magma supply within the spreading ridges at which the ophiolites formed. In the northern part of the original basin where the rate of extension was small relative to the rate of magma supply, magma remained in magma chambers longer, resulting in a greater range and volume of differentiated rocks. The larger volume of more differentiated, cooler and more viscous magmas, in conjunction with the likelihood of more violent eruption of volatile-rich differentiates, may have been responsible for the large volume of aquagene tuff in the northern part of the original basin. These observations in southern Chile suggest that ophiolites which contain a great abundance of aquagene tuffs and intrusive pillow lavas formed in tectonic environments in which the rate of extension was small relative to the rate of magma supply (island arcs, embryonic marginal basins). Ophiolites with predominantly water-lain pillowed and massive lavas formed in tectonic environments in which the rate of extension was large relative to the rate of magma supply (mid-ocean ridges, mature back-arc basins). Thus geologic field data may supplement geochemical data as a tool in distinguishing the original igneous-tectonic environments in which ophiolites originate.     

Using amphibole phenocrysts to track vapor transfer during magma crystallization and transport: An example from Mount St. Helens,Washington

In order to evaluate and further constrain models for volatile movement and vapor enrichment of magma stored at shallow levels, amphibole phenocrysts from 2004–2005 Mount St. Helens dacite were analyzed for major and selected trace elements (Li, Cu, Zn, Mn, and REE) and Li isotopes. Several recent studies have examined fluid-mobile trace element abundances in phencryst phases and melt inclusions as a means of tracking volatile movement within subvolcanic magmatic systems, and high Li contents in plagioclase phenocrysts from 1980 and 2004 Mount St. Helens dacites have been interpreted as evidence that shallow magma was fluxed by a Li-bearing vapor phase prior to eruption.     

Ocean tide loading (OTL) displacements from global and local grids: comparisons to GPS estimates over the shelf of Brittany, France

In this paper we examine OTL displacements detected by GPS stations of a dedicated campaign and validate ocean tide models. Our area of study is the continental shelf of Brittany and Cotentin in France. Brittany is one of the few places in the world where tides provoke loading displacements of ∼10–12 cm vertically and a few cm horizontally. Ocean tide models suffer from important discrepancies in this region. Seven global and regional ocean tide models were tested: FES2004 corrected for K2, TPXO.7.0, TPXO.6.2, GOT00.2, CSR4.0, NAO.99b and the most recent regional grids of the North East Atlantic (NEA2004). These gridded amplitudes and phases of ocean tides were convolved in order to get the predicted OTL displacements using two different algorithms. Data over a period of 3.5 months of 8 GPS campaign stations located on the north coast of Brittany are used, in order to evaluate the geographical distribution of the OTL effect. We have modified and implemented new algorithms in our GPS software, GINS 7.1. GPS OTL constituents are estimated based on 1-day batch solutions. We compare the observed GPS OTL constituents of M₂, S₂, N₂ and K₁ waves with the selected ocean tide models on global and regional grids. Large phase-lag and amplitude discrepancies over 20° and 1.5 cm in the vertical direction in the semi-diurnal band of M₂ between predictions and GPS/models are detected in the Bay of Mont St-Michel. From a least squares spectral analysis of the GPS time-series, significant harmonic peaks in the integer multiples of the orbital periods of the GPS satellites are observed, indicating the existence of multipath effects in the GPS OTL constituents. The GPS OTL observations agree best with FES2004, NEA2004, GOT00.2 and CSR4.0 tide models.     

The presence, characteristics and earthquake implications of the St. Lawrence fault zone within and near Lake Ontario (Canada–USA)

Questions persist concerning the earthquake potential of the populous and industrial Lake Ontario (Canada–USA) area. Pertinent to those questions is whether the major fault zone that extends along the St. Lawrence River valley, herein named the St. Lawrence fault zone, continues upstream along the St. Lawrence River valley at least as far as Lake Ontario or terminates near Cornwall (Ontario, Canada)–Massena (NY, USA). New geological studies uncovered paleotectonic bedrock faults that are parallel to, and lie within, the projection of that northeast-oriented fault zone between Cornwall and northeastern Lake Ontario, suggesting that the fault zone continues into Lake Ontario. The aforementioned bedrock faults range from meters to tens of kilometers in length and display kinematically incompatible displacements, implying that the fault zone was periodically reactivated in the study area. Beneath Lake Ontario the Hamilton–Presqu'ile fault lines up with the St. Lawrence fault zone and projects to the southwest where it coincides with the Dundas Valley (Ontario, Canada). The Dundas Valley extends landward from beneath the western end of the lake and is marked by a vertical stratigraphic displacement across its width. The alignment of the Hamilton–Presqu'ile fault with the St. Lawrence fault zone strongly suggests that the latter crosses the entire length of Lake Ontario and continues along the Dundas Valley.The Rochester Basin, an east–northeast-trending linear trough in the southeastern corner of Lake Ontario, lies along the southern part of the St. Lawrence fault zone. Submarine dives in May 1997 revealed inclined layers of glaciolacustrine clay along two different scarps within the basin. The inclined layers strike parallel to the long dimension of the basin, and dip about 20° to the north–northwest suggesting that they are the result of rigid-body rotation consequent upon post-glacial faulting. Those post-glacial faults are growth faults as demonstrated by the consistently greater thickness, unit-by-unit, of unconsolidated sediments on the downthrown (northwest) side of the faults relative to their counterparts on the upthrown (southeast) side. Underneath the western part of Lake Ontario is a monoclinal warp that displaces the glacial and post-glacial sediments, and the underlying bedrock–sediment interface. Because of the post-glacial growth faults and the monoclinal warp the St. Lawrence fault zone is inferred to be tectonically active beneath Lake Ontario. Furthermore, within the lake it crosses at least five major faults and fault zones and coexists with other neotectonic structures. Those attributes, combined with the large earthquakes associated with the St. Lawrence fault zone well to the northeast of Lake Ontario, suggest that the seismic risk in the area surrounding and including Lake Ontario is likely much greater than previously believed.     

Horizontal and vertical components of earthquake ground motions at liquefiable sites

Field observations on ground motions from recent earthquakes imply that current knowledge is limited with regard to relating vertical and horizontal motions at liquefiable sites. This paper describes a study with the purpose of clarifying this emerging issue to some extent. A series of numerical analyses is carried out on a liquefiable soil deposit with a verified, fully coupled, nonlinear procedure. It is shown that the transformation of vertical motions in the deposit differs considerably from the transformation of horizontal motions. Both the amplitude and frequency content of the horizontal motions are strongly dependent on the shaking level or the associated nonlinear soil behavior. The transfer function for vertical motions is however likely to be independent of the intensity of input motions; no reduction in the amplitude occurs even in the case of strong shaking. The results are shown to be in consistence with the laboratory observations on shaking table tests and recent field observations that less nonlinearity exists for vertical motions. It is also shown that the possibility exists for using information on spectral ratios between the horizontal and vertical surface motions to quickly identify in situ soil behavior and liquefaction that are not readily covered by conventional field or laboratory experimentation procedures.