Interpolation with Splines in Tension: A Green's Function Approach

Interpolation and gridding of data are procedures in the physical sciences and are accomplished typically using an averaging or finite difference scheme on an equidistant grid. Cubic splines are popular because of their smooth appearances; however, these functions can have undesirable oscillations between data points. Adding tension to the spline overcomes this deficiency. Here, we derive a technique for interpolation and gridding in one, two, and three dimensions using Green's functions for splines in tension and examine some of the properties of these functions. For moderate amounts of data, the Green's function technique is superior to conventional finite-difference methods because (1) both data values and directional gradients can be used to constrain the model surface, (2) noise can be suppressed easily by seeking a least-squares fit rather than exact interpolation, and (3) the model can be evaluated at arbitrary locations rather than only on a rectangular grid. We also show that the inclusion of tension greatly improves the stability of the method relative to gridding without tension. Moreover, the one-dimensional situation can be extended easily to handle parametric curve fitting in the plane and in space. Finally, we demonstrate the new method on both synthetic and real data and discuss the merits and drawbacks of the Green's function technique.     

Interpolation with Splines in Tension: A Green''s Function Approach

Interpolation and gridding of data are procedures in the physical sciences and are accomplished typically using an averaging or finite difference scheme on an equidistant grid. Cubic splines are popular because of their smooth appearances; however, these functions can have undesirable oscillations between data points. Adding tension to the spline overcomes this deficiency. Here, we derive a technique for interpolation and gridding in one, two, and three dimensions using Green's functions for splines in tension and examine some of the properties of these functions. For moderate amounts of data, the Green's function technique is superior to conventional finite-difference methods because (1) both data values and directional gradients can be used to constrain the model surface, (2) noise can be suppressed easily by seeking a least-squares fit rather than exact interpolation, and (3) the model can be evaluated at arbitrary locations rather than only on a rectangular grid. We also show that the inclusion of tension greatly improves the stability of the method relative to gridding without tension. Moreover, the one-dimensional situation can be extended easily to handle parametric curve fitting in the plane and in space. Finally, we demonstrate the new method on both synthetic and real data and discuss the merits and drawbacks of the Green's function technique.     

State,capital and spaceships: A terrestrial geography of space tourism

Over the past decade, there has been a significant structural and geographical change in space travel. Following policy, budgetary and regulatory changes in the United States, space travel may now be conducted by private companies. This privatization has also led to some geographical competition and shifts between states within the US. In this paper, I respond to MacDonald’s (2007) call for more critical geographies of outer space activity. Building from his suggestion that we look more closely at the ways in which outer space activity is constituted by “numerous familiar operations” with respect to the practices of international relations and war and to the “basic infrastructural maintenance of the state and the lives of its citizenry”, this paper explores why the US Government has allowed for private space travel and why this privatization drove some states in the US to invest heavily in such a nascent industry sector. It argues that federal and state governments both saw private space travel as a means to fulfill their “basic infrastructural maintenance” with regard to economic expansion, development and competitiveness. The paper analyses these processes through the development of space tourism. In doing so, it provides more detail and geographical context to Dickens and Ormrod’s (2007) overview of the connections between outer space and the circuits of capital. It also demonstrates the many familiar political-economic processes involved in the privatization of space travel and, as such, the possibilities for further critical geographies of space activity.     

A political cultural map of Europe. A survey approach

The existence of culture is far less contested than the ways in which one can conceptualize, theorize and measure it. Grid-group theory is a typological contribution to the field of political culture. It identifies one dimension of individuality (grid) and one dimension of social incorporation (group). Combined, these dimensions demarcate four omnipresent and interdependent cultures - hierarchy, egalitarianism, individualism and fatalism - each of which interlocks a social pattern with a type of cultural bias. The overall political cultural configuration of a country is determined by the given combination of the four cultures, each of whose size can be measured by the number of adherents. The estimated configurations of twelve European countries are obtained by analyzing two waves of the World Values Study in 1981 and 1990 (N=34,858). The results show that egalitarianism is dominant in the Nordic countries and the Netherlands, whereas fatalism is dominant in Great Britain, Ireland and southern Europe. Further analysis confirms the hypothesized correspondance between egalitarianism and Inglehart's concept of postmaterialism, both of which, in turn, are related to proenvironmental concern. This revised version was published online in August 2006 with corrections to the Cover Date.     

Distribution of Pb between sediments and pore water in Woods Lake,Adirondack State Park,New York,U.S.A.

Interstitial water samples and sediments were collected from acidified Woods Lake (pH= 5.0) to evaluate Pb profile stability and distribution between the solid and aqueous phases. A simple equation was developed to describe the distribution coefficient for Pb(Kd_Pb), based on the sample moisture content and analyte concentrations in bulk sediment and pore water. In Woods Lake sediment Kd_Pb values ranged from 6.7 × 10⁴to6.7 × 10⁵ml/g. The affinity of Pb for the solid phase was further demonstrated by inverting a sediment core and leaving it for 10 months in situ. Upon retrieval, the Pb profile was found to have remained intact over the same depth, albeit inverted, as in the surrounding sediment. Geochemical modeling using MINTEQA2 demonstrates that Pb solids that may reasonably be expected to precipitate were all undersaturated by several orders of magnitude with respect to the interstitial water, indicating that Pb was sorbed to, rather than precipitated in, Woods Lake sediment.     

Environmental problems and policies in East Central Europe: A changing agenda

Considerable environmental damage occurred during the socialist era as a consequence of the rapid rate of economic growth with low priority for the quality of life. High levels of air and water pollution impacted negatively on human health as well as the physical environment. The rural areas did not escape because of soil and forest degradation, while open pit mining and associated tipping devastated large areas of countryside especially in former Czechoslovakia and East Germany. The Chernobyl nuclear disaster proved to be a turning point since public opinion was mobilised to the point that underground movements began to lobby for improvements and environmental quality became a factor in system change in 1989. Much has been achieved over the past decade, thanks in part to expertise and financial help from abroad, and the priority currently being given to membership of the European Union means that all the requirements of the environmental 'acquis' must be met over the medium term. However, the 1990s have also introduced the sustainability agenda and current challenges are much more extensive that those of the past. While many obligations now carry the full force of law, it is also evident that sustainability is a matter for interpretation nationally and locally in the light of economic pressures and cultural values, moderated through the education process and negotiation with both statutory bodies and environmental non-governmental organisations (ENGOs). This paper therefore serves as a context for the more local studies that follow. This revised version was published online in July 2006 with corrections to the Cover Date.     

Granulites and related rocks in variscan median Europe: A dualistic interpretation

On the basis of differences in structural, petrological and radiometric features, two main clear-cut groups can be distinguished for the granulites of the Variscan median Europe.Group I comprises lenses of eclogite-granulite scattered in the internal zones of the Hercynian belt, whereas Group II includes granulitic slices related to deep-seated Alpine lineaments and xenoliths scavenged by recent volcanoes.Group I granulites belong to early thrust nappes of the Variscan orogen; they display high-pressure assemblages and are commonly associated with eclogites and garnet-peridotites. From radiometric data, this HP granulitic-eclogitic metamorphism happened around 450-400 Ma throughout median Europe. A subduction context is suggested by the low T/P gradient, the occurrence of eclogites and the contemporaneity with some blueschist-facies rocks.Group II granulites are characterized by massive occurrences, intermediate-pressure granulite-facies parageneses and the absence of eclogites. Granulitic paragneisses often display a pronounced depletion in granitophile elements and may appear as degranitized restites. Numerous basic-ultrabasic complexes occur and are interpreted as deep-seated, layered igneous intrusions emplaced during the granulitic metamorphism. Radiometric dating indicates a late Hercynian (ca. 300 Ma) age. The high T/P gradient, the occurrence of synchronous basic intrusions and the post-collision context suggest a granulitic event due to a major thermal anomaly.The Variscan cycle is thus characterized by two granulite-forming episodes. “Older granulites” of the HP type result from an early compressive stage in a crustal subduction context and “younger granulites”, from a major thermal event which originated in the upper mantle and annealed the deepest parts of the Variscan belt. A broadly similar duality in the genesis of granulitic rocks may be anticipated in other collisional belts.     

Agriculture in Eastern Europe: Communism, the transition and the future

The paper summarises the East European experience with socialist agriculture and notes that while production often failed to meet plan targets (thereby giving the impression of a sector in crisis), there was steady growth based on substantial investments in buildings, machinery, fertilisers and irrigation systems which provided food for the population at affordable prices. The increased spending power of higher salaries during the 1970s and 1980s also drew a positive response from agriculture in the context of significant reforms which, unintentionally, contributed to the demise of the communist system in 1988–1990. Transition to a market economy has cast agriculture into a state of great uncertainty through restitution and the end of price controls, combined with the disruption of trade contacts with the Former Soviet Union. Falling real wages have reduced demand while the overtures being made to the European Union (with its substantial food surpluses) suggest that reduced agricultural output may be a permanent reality. In this case the thrust of rural development will have to give more attention to the quality of the environment and the provision of alternative employments for country dwellers. It remains to be seen how far small family holdings can be maintained in the context of farm diversification and pluriactivity.     

A petrographic and fluid inclusion comparison of silver-poor and silver-rich zinc-lead ores,N.E. Washington State

Many of the zinc-lead deposits of NE Washington State are poorly known examples of Mississippi Valley Type (MVT) mineralization. This study compares inclusion fluids from the Josephine Breccia ores with the later cross-cutting sulfide-bearing quartz veins. The breccia ores are cemented mainly by open space fillings of dolomite, sphalerite, quartz, galena, jasperoid and calcite. Replacement is of minor importance. Ore and gangue deposition occurred over the range 150–250 °C with most of the temperatures less than 200 °C. The aqueous brines typically contain 17–23 equivalent weight percent NaCl with often substantial amounts of Ca and/or Mg chlorides. Homogenization temperatures do not delineate any cooling or paragenetic sequence. The cross-cutting vein quartz contains CO₂-rich inclusions with overall densities usually less than 0.7 g/cc and homogenization temperatures from 250–325 °C. Sulfur isotope analyses yield two populations with the quartz vein ores being lighter (<13 permil CDT) than the average for the conformable ores. The later veins are not remobilized MVT sulfides but represent a separate, high-silver period of mineralization.     

Factors of poultry farming adoption in Oyo State,Nigeria: A study in agency- and infrastructure-based diffusion

In this paper, use is made of the multiple regression technique in an attempt to provide explanation for the pattern of adoption of intensive poultry farming technique — an agency and infrastructure based innovation. This paper deals with two aspects of the diffusion. The first examines the factors of spatial pattern of adoption and the second, the adoption intensity. The paper then turns to consider the planning implications of these factors. In Oyo State, market factor expressed as the population size of towns, distance to agency location and from previous adopters largely account for the spatial variation in the time of adoption of ‘modern’ poultry farming technique. The intensity of adoption (poultry population) on the other hand is influenced by the number of adopters, distance to infrastructure, profitability considerations, literacy factor and the time of adoption.     

9: Processes of tectonism, magmatism and mineralization: Lessons from Europe

Metallogenic provinces in Europe range in age from the Archaean to the Neogene. Deposit types include porphyry copper and epithermal Cu–Au, volcanic-hosted massive sulphide (VMS), orogenic gold, Fe-oxide–Cu–Au, anorthosite Fe–Ti-oxide and sediment-hosted base-metal deposits. Most of them formed during short-lived magmatic events in a wide range of tectonic settings; many can be related to specific tectonic processes such as subduction, hinge retreat, accretion of island arcs, continental collision, lithosphere delamination or slab tear. In contrast, most sediment-hosted deposits in Europe evolved in extensional, continental settings over significant periods of time. In Europe, as elsewhere, ore formation is an integral part of the geodynamic evolution of the Earth's crust and mantle. Many tectonic settings create conditions conducive to the generation of water-rich magma, but the generation of ore deposits appears to be restricted to locations and short periods of change in temperature and stress, imposed by transitory plate motions. Crustal influence is evident in the strong structural controls on the location and morphology of many ore deposits in Europe. Crustal-scale fault–fracture systems, many involving strike-slip elements, have provided the fabric for major plumbing systems. Rapid uplift, as in metamorphic core complexes, and hydraulic fracturing can generate or focus magmatic–hydrothermal fluid flow that may be active for time spans significantly less than a million years. Once a hydrologically stable flow is established, ore formation is strongly dependent on the steep temperature and pressure gradients experienced by the fluid, particularly within the upper crust. In Europe, significant fracture porosity deep in the crystalline basement (1%) is not only important for magmatic–hydrothermal systems, but allows brines to circulate down through sedimentary basins and then episodically upward, expelled seismically to produce sediment-hosted base-metal deposits and Kupferschiefer copper deposits. Emerging research, stimulated by GEODE, can improve the predicting power of numerical simulations of ore-forming processes and help discover the presence of orebodies beneath barren overburden.     

Book Review: Stratigraphic Systems: Origin and Application. By Glenn S. Visher. Academic Press,San Diego, 1999

Mathematical Geosciences -