Geodesic polygons on surfaces of revolution

This paper presents a way for computing geodesic polygons on real surfaces of revolution with partly continuous curvature and gives an application to coordinate transformations in geodesic coordinate systems. The specific formulas for an ellipsoid of revolution are briefly mentioned.     

Lithic fragments,glasses and chondrules from Luna 16 fines

Bulk compositions of igneous and microbreccia lithic fragments, glasses, and chondrules from Luna 16 fines as well as compositions of minerals in basaltic lithic fragments were determined with the electron microprobe. Igneous lithic fragments and glasses are divided into two groups, the anorthositic-noritic-troctolitic (hereafter referred to as ANT) and basaltic groups. Chondrules are always of ANT composition and microbreccia lithic fragments are divided into groups 1 and 2. The conclusions reached may be summarized as follows: (1) Luna 16 fines are more similar in composition to Apollo 11 than to Apollo 12 and 14 materials (e.g. Apollo 11 igneous lithic fragments and glasses fall into similar ANT and basaltic groups; abundant norites in Luna 16 and Apollo 11 are not KREEP as in Apollo 12 and 14; Luna 16 basaltic lithic fragments may represent high-K and low-K suites as is the case for Apollo 11; rare colorless to greenish, FeO-rich and TiO₂-poor glasses were found in both Apollo 11 and Luna 16; Luna 16 spinels are similar to Apollo 11 spinels but unlike those from Apollo 12). (2) No difference was noted in the composition of lithic fragments, glasses and chondrules from Luna 16 core tube layers A and D. (3) Microbreccia lithic fragments of group 1 originated locally by mixing of high proportions of basaltic with small proportions of ANT materials. (4) Glasses are the compositional analogs to the lithic fragments and not to the microbreccias; most glasses were produced directly from igneous rocks. (5) Glasses show partial loss of Na and K due to vaporization in the vitrification process. (6) Luna 16 chondrules have ANT but not basaltic composition. It is suggested that either liquid droplets of ANT composition are more apt to nucleate from the supercooled state; or basaltic droplets have largely been formed in small and ANT droplets in large impact events (in the latter case, probability for homogeneous and inhomogeneous nucleation is larger. (7) No evidence for ferric iron and water-bearing minerals was found. (8) Occurrence of a great variety of igneous rocks in Luna 16 samples (anorthosite, noritic anorthosite, anorthositic norite, olivine norite, troctolite, and basalt) confirm our earlier conclusion that large-scale melting or partial melting to considerable depth and extensive igneous differentiation must have occurred on the moon.     

A synthetic model for daily streamflow

This paper presents a model for synthesising daily average streamflow data that is suitable for most rivers in Great Britain. The method is based on a linear interpolation of the logorithms of 5-day average flows. The 5-day average flows are produced using N.T. Kottegoda's statistical model (Thesis, Univ. of Birmingham, 1970). The 5-day model preserves the long-term statistical characteristics of the daily data, while the short-term characteristics such as hydrograph shape are imposed by the interpolation method.

A stochastic error term is superimposed on the interpolated daily flows. This term represents the non-deterministic component of the daily time series. The analysis of the observed error terms represents an important part of this paper.

The riverflow in the Severn at Bewdley is used to demonstrate both the analysis of actual data and the generation of synthetic data. The technique is then applied to data from two other rivers with widely differing characteristics to demonstrate the range of the method.     

Petrogenetic relationships of acid and basic rocks in iceland: Sr-isotopes and rare-earth elements in late and postglacial volcanics

Strontium isotope ratios and rare-earth element abundances have been measured in acid, intermediate and basic rocks from three late to postglacial volcanic complexes, and several other postglacial basalts in Iceland. Late and postglacial basalts in Iceland have been generated from a source region which is essentially homogeneous with respect to⁸⁷Sr/⁸⁶Sr. The mean⁸⁷Sr/⁸⁶Sr ratio for the basalts analysed is 0.70328 and the range is from 0.70317 ± 6to0.70334 ± 5 (2σ).Acid rocks from the Kerlinganfjöll and Namafjall volcanic complexes have⁸⁷Sr/⁸⁶Sr ratios which are indistinguishable from analysed basalts from the same complexes. However, intermediate and acid rocks from the Torfajökull complex have significantly higher⁸⁷Sr/⁸⁶Sr ratios and could not have been derived by fractional crystallization from basaltic magmas similar to those found in the same complex. These latter rocks have most probably been produced by remelting of Tertiary gabbroic rocks in Layer 3. Most of the basalts analysed have higher total rare-earth element abundances than typical dredged ocean-ridge tholeiites, and show less light rare-earth depletion. Intermediate and acid compositions show overall higher abundances and light rare-earth enrichments. The measured rare-earth abundances are compared with abundances generated by differential partial melting of various model source regions.It is shown that both the tholeiitic and alkali basalt compositions could be generated from the same source material by different degrees of partial melting. Variable partial melting of gabbroic material may account for the rare-earth element abundances of both the rhyolitic rocks (small degrees of melting) and the intermediate rocks (more extensive melting).     

Experimental study of the stability of cordierite and garnet in pelitic compositions at high pressures and temperatures

In pelitic rocks, under conditions of low f _{O 2} and low f _{H 2 O}, the stability of the mineral pair cordierite-garnet is limited by five univariant reactions. In sequence from high pressure and low temperature to high temperature and low pressure these are: cordierite+garnet hypersthene+sillimanite+quartz, cordierite+garnet hypersthene+sapphirine+quartz, cordierite+garnet hypersthene+spinel+quartz and cordierite+garnet olivine+spinel +quartz. In this sequence of reactions the Mg/Mg+Fe²⁺ ratio of all ferro-magnesian minerals involved decreases continuously from the first reaction to the fifth. The five univariant boundaries delimit a wide P-T range over which cordierite and garnet may coexist.Two divariant equilibria in which the Mg/Mg+ Fe²⁺ ratio of the coexisting phases are uniquely determined by pressure and temperature have been studied in detail. P-T-X grids for the reactions cordierite garnet+sillimanite+quartz and cordierite+hypersthene garnet+quartz are used to obtain pressure-temperature estimates for several high grade metamorphic areas. The results suggest temperatures of formation of 700–850° C and load pressures of 5–10 kb. In rare occasions temperatures of 950–1000° C appear to have been reached during granulite metamorphism.On the basis of melting experiments in pelitic compositions it is suggested that Ca-poor garnet xenocrysts found in calc-alkaline magmas derive from admixed pelitic rocks and did not equilibrate with the calc-alkaline magma.     

The genesis of basaltic magmas

This paper reports the results of a detailed experimental investigation of fractionation of natural basaltic compositions under conditions of high pressure and high temperature. A single stage, piston-cylinder apparatus has been used in the pressure range up to 27 kb and at temperatures up to 1500° C to study the melting behaviour of several basaltic compositions. The compositions chosen are olivine-rich (20% or more normative olivine) and include olivine tholeiite (12% normative hypersthene), olivine basalt (1% normative hypersthene) alkali olivine basalt (2% normative nepheline) and picrite (3% normative hypersthene). The liquidus phases of the olivine tholeiite and olivine basalt are olivine at 1 Atmosphere, 4.5 kb and 9 kb, orthopyroxene at 13.5 and 18 kb, clinopyroxene at 22.5 kb and garnet at 27 kb. In the alkali olivine basalt composition, the liquidus phases are olivine at 1 Atmosphere and 9 kb, orthopyroxene with clinopyroxene at 13.5 kb, clinopyroxene at 18 kb and garnet at 27 kb. The sequence of appearance of phases below the liquidus has also been studied in detail. The electron probe micro-analyser has been used to make partial quantitative analyses of olivines, orthopyroxenes, clinopyroxenes and garnets which have crystallized at high pressure.These experimental and analytical results are used to determine the directions of fractionation of basaltic magmas during crystallization over a wide range of pressures. At pressures corresponding to depths of 35–70 km separation of aluminous enstatite from olivine tholeiite magma produces a direct fractionation trend from olivine tholeiites through olivine basalts to alkali olivine basalts. Co-precipitation of sub-calcic, aluminous clinopyroxene with the orthopyroxene in the more undersaturated compositions of this sequence produces derivative liquids of basanite type. Magmas of alkali olivine basalt and basanite type represent the lower temperature liquids derived by approximately 30% crystallization of olivine-rich tholeiite at 35–70 km depth. At depths of about 30 km, fractionation of olivine-rich tholeiite with separation of both olivine and low-alumina enstatite, joined at lower temperatures by sub-calcic clinopyroxene, leads to derivative liquids with relatively constant SiO₂ (48 to 50%) increasingly high Al₂O₃ (15–17%) contents and retaining olivine + hypersthene normative chemistry (5–15% normative olivine). These have the composition of typical high-alumina olivine tholeiites. The effects of low pressure fractionation may be superimposed on magma compositions derived from various depths within the mantle. These lead to divergence of the alkali olivine basalt and tholeiitic series but convergence of both the low-alumina and high-alumina tholeiites towards quartz tholeiite derivative liquids.The general problem of derivation of basaltic magmas from a mantle of peridotitic composition is discussed in some detail. Magmas are considered to be a consequence of partial melting but the composition of a magma is determined not by the depth of partial melting but by the depth at which magma segregation from residual crystals occurs. Magma generation from parental peridotite (pyrolite) at depths up to 100 km involves liquid-crystal equilibria between basaltic liquids and olivine + aluminous pyroxenes and does not involve garnet. At 35–70 km depth, basaltic liquids segregating from a pyrolite mantle will be of alkali olivine basalt type with about 20% partial melting but with increasing degrees of partial melting, liquids will change to olivine-rich tholeiite type with about 30% melting. If the depth of magma segregation is about 30 km, then magmas produced by 20–25% partial melting will be of high-alumina olivine tholeiite type, similar to the oceanic tholeiites occurring on the sea floor along the mid-oceanic ridges.Hypotheses of magma fractionation and generation by partial melting are considered in relation to the abundances and ratios of trace elements and in relation to isotopic abundance data on natural basalts. It is shown that there is a group of elements (including K, Ti, P, U, Th, Ba, Rb, Sr, Cs, Zr, Hf and the rare-earth elements) which show enrichment factors in alkali olivine basalts and in some tholeiites, which are inconsistent with simple crystal fractionation relationships between the magma types. This group of elements has been called incompatible elements referring to their inability to substitute to any appreciable extent in the major minerals of the upper mantle (olivine, aluminous pyroxenes). Because of the lack of temperature contrast between magma and wall-rock for a body of magma near to its depth of segregation in the mantle, cooling of the magma involves complementary processes of reaction with the wall-rook, including selective melting and extraction of the lowest melting fraction. The incompatible elements are probably highly concentrated in the lowest melting fraction of the pyrolite. The production of large overall enrichments in incompatible elements in a magma by reaction with and highly selective sampling of large volumes of mantle wall-rock during slow ascent of a magma is considered to be a normal, complementary process to crystal fractionation in the mantle. This process has been called wall-rock reaction. Magma generation in the mantle is rarely a simple, closed-system partial melting process and the isotopic abundances and incompatible element abundances of a basalt as observed at the earth's surface may be largely determined by the degree of reaction with the mantle or lower crustal wall-rocks and bear little relation to the abundances and ratios of the original parental mantle material (pyrolite).Occurrences of cognate xenoliths and xenocrysts in basalts are considered in relation to the experimental data on liquid-crystal equilibria at high pressure. It is inferred that the lherzolite nodules largely represent residual material after extraction of alkali olivine basalt from mantle pyrolite or pyrolite which has been selectively depleted in incompatible elements by wall-rock reaction processes. Lherzolite nodules included in tholeiitic magmas would melt to a relatively large extent and disintegrate, but would have a largely refractory character if included in alkali olivine basalt magma. Other examples of xenocrystal material in basalts are shown to be probable liquidus crystals or accumulates at high pressure from basaltic magma and provide a useful link between the experimental study and natural processes.     

Coseismic and other short-term strain changes recorded with Sacks-Evertson strainmeters in a deep mine, South Africa

Summary. During 1977 March and April, three Sacks-Evertson borehole dilatometers were installed at the ends of boreholes drilled into the sidewall of an experimental tunnel at a depth of 3.1 km in the ERPM gold mine near Johannesburg. In the following year coseismic strain changes ranging from 5 ± 10⁻¹⁰ to values exceeding 5 ± 10⁻⁶ were recorded for hundreds of mine tremors in the magnitude range -1 to 3.7 and at hypocentral distances of 50 m to about 2 km. Hypocentral coordinates and magnitudes were determined from seismograms recorded from an underground array of geophones. Amplitudes and polarities of the coseismic strain steps are generally in excellent agreement with theoretical expectations based on point-source dislocation theory; specifically, the strain steps are proportional to the seismic moment divided by the cube of hypocentral distance. At a strain level of 5 ± 10⁻⁹ or greater the tremors do not appear to be preceded by any short-term indications of instability even for tremors producing coseismic steps greater than 5 ± 10⁻⁶ and for which the strainmeters were within a source radius of the hypocentre. Continuous strain changes observed at the times when the mine excavation, at a distance of about 100 m, is extended are in good agreement with calculated changes based on the theory of elasticity. A similar calculation is consistent with post-seismic strain changes observed to follow some of the closer tremors. These post-seismic strains show a logarithmic dependence on time following the tremor and appear to be due to the interaction of a tremor with the adjacent mine excavation rather than to deformation within the actual seismic source region.     

Investigation of Image Fusion Between High—Resolution Image and Multi—Spectral Image

On the basis of a thorough understanding of the physical characteristics of remote sensing image,this paper employs the theories of wavelet transform and signal sampling to develop a new image fusion algorithm.The algorithm has been successfully applied to the image fusion of SPOT PAN and TM of Guangdong province, China The experimental results show that a perfect image fusion can be built up by using the image analytical solution and reconstruction in the image frequency domain based on the physical characteristics of the image formation.The method has demonstrated that the results of the image fusion do not change spectral characteristics of the original image.