Fitting matrix-valued variogram models by simultaneous diagonalization (Part I: Theory)

Suppose that ¯(x₁),...,¯Z(x_n). are observations of vector-valued random function ¯(x). In the isotropic situation, the sample variogram γ^*(h) for a given lag h is $$\bar \gamma ^ * (h) = \frac{1}{{2N(h)}}\mathop \sum \limits_{s(h)} (\overline Z (x_1 ) - \overline Z (x_1 )) \overline {(Z} (x_1 ) - \overline Z (x_1 ))^T $$ where s(h) is a set of paired points with distance h and N(h) is the number of pairs in s(h).. For a selection of lags h₁, h₂, .... h_k such that N (h₁) > O. we obtain a ktuple of (semi) positive definite matrices $\bar \gamma ^ * (h_{ 1} ),. . . ., \bar \gamma ^ * (h_{ k} )$ . We want to determine an orthonormal matrix B which simultaneously diagonalizes the $\bar \gamma ^ * (h_{ 1} ),. . . ., \bar \gamma ^ * (h_{ k} )$ or nearly diagonalizes them in the sense that the sum of squares of offdiagonal elements is small compared to the sum of squares of diagonal elements. If such a B exists, we linearly transform $\overline Z (x)$ by $\overline Y (x) = B\overline Z (x)$ . Then, the resulting vector function $\overline Y (x)$ has less spatial correlation among its components than $\overline Z (x)$ does. The components of $\overline Y (x)$ with little contribution to the variogram structure may be dropped, and small crossvariograms fitted by straightlines. Variogram models obtained by this scheme preserve the negative definiteness property of variograms (in the matrix-valued function sense). A simplified analysis and computation in cokriging can be carried out. The principles of this scheme arc presented in this paper.     

Discrete element modelling of the influence of cover strength on basement-involved fault-propagation folding

A discrete element model is used to investigate the influence of sedimentary cover strength on the development of basement-involved fault-propagation folds. We find that uniformly weak cover best promotes the development of classical, trishear-like fault-related folds showing marked anticlinal thinning and synclinal thickening, with cover dips increasing downwards towards the fault tip. Uniformly strong cover results in more rounded fold forms with only minor hinge thickening/thinning and significant basement fault-propagation into the sedimentary cover. Heterogeneous, layered, cover sequences with marked differences in strength promote the development of more complex and variable fold forms, with a close juxtaposition of brittle and macroscopically ductile features, which diverge from the predictions of simple kinematic models. In these structures the upper layers are often poor indicators of deeper structure. In addition, we find that in layered cover sequences fault-propagation into the cover is a complex process and is strongly buffered by the weaker cover units.     

Constraints of Housing Age and Migration on Residential Mobility

This paper investigates the role of housing age in constraining residential mobility, measured as the percent of households that have moved into their homes in the past 15 months. The leading explanation for why mobility rates differ so much among regions of the United States has been the overall level of growth. The present analysis shows that the growth effect operates through both the newness of population (migration) and the newness of housing available for occupancy by all local residents. The posited explanation for this housing age effect is that progressively older units contain increasingly settled occupants, yielding fewer opportunities for in-movers in areas with older housing. It is empirically demonstrated that households in older housing have lower likelihood of recent mobility even after controlling for age, tenure, migration status, and state location of residence. The analysis reveals the temporal interdependency of mobility, migration, person age, and housing age.     

A new form of the cokriging equations

Myers developed a matrix form of the cokriging equations, but one that entails the solution of a large system of linear equations. Large systems are troublesome because of memory requirements and a general increase in the matrix condition number. We transform Myers’s system into a set of smaller systems, whose solution gives the classical kriging results, and provides simultaneously a nested set of lower dimensional cokriging results. In the course of developing the new formulation we make an interesting link to the Cauchy-Schwarz condition for the invertibility of a system, and another to a simple situation of coregionalization. In addition, we proceed from these new equations to a linear approximation to the cokriging results in the event that the crossvariograms are small, allowing one to take advantage of a recent results of Xie and others which proceeds by diagonalizing the variogram matrix function over the lag classes.     

Tree-Ring Based Reconstructions of Precipitation for the Southern Canadian Cordillera

This paper reconstructs precipitation variability in the southern Canadian Cordillera over the past 3–400 years using dendroclimatologicaltechniques. Fifty-three total ring-width (RW) chronologies, 28 earlywood (EW) and 28 latewood (LW) chronologies were developed from open-grown, low-elevation stands of Pseudotsuga menziesii (Douglas-fir) and Pinusponderosa (ponderosa pine) across the southern Canadian Cordillera. RW, EW and LW chronologies from both species were used to develop 13 annual (prior July to current June) precipitation reconstructions across the region. The reconstructions range in length from 165 to 688 years, pass verification tests and capture 39–64% of the variancein the instrumental record. Coincident, prolonged intervals of dry conditions are estimated for the years: 1717–1732, 1839–1859, 1917–1941 and1968–1979. Shorter dry intervals are identified between 1581–1586, 1626–1630,1641–1653 1701–1708, 1756–1761, 1768–1772, 1793–1800,1868–1875, 1889–1897 and 1985–1989. The historic drought of the 1920–1930s was the longest but not the most intense across this area in the last 300 years. Wet conditions occur in the majority of reconstructions for the years: 1689–1700, 1750–55,1778–1789, 1800–1830, 1880–1890, 1898–1916 and 1942–1960. Thesedata, in conjunction with data from adjacent areas, are used to provide the first maps of decadal precipitation anomalies for the region between 1700 and 1990.     

High-pressure experimental growth of diamond using C–K2CO3–KCl as an analogue for Cl-bearing carbonate fluid

High-pressure, high-temperature diamond growth experiments have been conducted in the system C–K₂CO₃–KCl at 1050–1420 °C, 7.0–7.7 GPa. KCl is of interest because of the strong effect of halogens on the phase relations of carbonate-rich systems [Geophys. Res. Lett. 30 (2003) 1022] and because of the occurrence of KCl coexisting with alkali silicate–carbonate fluids in natural-coated diamond [Geochim. Cosmochim. Acta 64 (2000) 717]. We have used system C–K₂CO₃–KCl as an analogue for these mantle fluids in diamond growth experiments. The presence of KCl reduces the potassium carbonate liquidus to ≤1000 °C at 7.7 GPa, allowing it to act as a solvent catalyst for diamond growth at temperatures below the continental geotherm. This is a reduction on the minimum diamond growth temperature reported in the alkali-carbonate–C–O–H system [Lithos 60 (2002) 145]. Diamond growth using carbonate solvent catalysts is characterised by a relatively long induction period. However, the addition of KCl also reduced the period for diamond growth in carbonate to 5 min; no such induction period appears to be necessary. It is suggested that KCl destabilises carbonate, allowing greater solubility and diffusion of carbon.     

Further examples of ventifacts and unusual patterned ground from the Falkland Islands, South Atlantic

Abstract Examples of ventifacts and forms of unusual patterned ground that differ in situation, characteristics and/or origin to types reported previously are described from the Falkland Islands, South Atlantic. The ventifacts, represented by clasts projecting from an eroding tillite shore platform, have been grooved by the passage of windblown sand. The patterned ground forms comprise stone polygons on clay loam regolith, stone polygons on peat, peat‐filled desiccation cracks on peat, and clusters of on‐edge clasts (stone packings) on a gravel beach. Aeolian transport of particles is considered responsible for the infill of the polygonal cracks. The clusters of on‐edge clasts are thought to have developed as a result of wave action, in particular backwash and/or undertow. Although these features occupy small surface areas they provide useful information about local geomorphic processes and testify to the marked diversity of such features in nature.     

Discrete-element modelling of detachment folding

A two‐dimensional, discrete‐element modelling technique is used to investigate the initiation and growth of detachment folds in sedimentary rocks above a weak décollement level. The model depicts the sedimentary rocks as an assemblage of spheres that obey Newton's equations of motion and that interact with elastic forces under the influence of gravity. Faulting or fracturing between neighbouring elements is represented by a transition from repulsive–attractive forces to solely repulsive forces. The sedimentary sequence is mechanically heterogeneous, consisting of intercalated layers of markedly different strengths and thicknesses. The interlayering of weak and strong layers within the sedimentary rocks promotes the localization of flexural flow deformation within the weak layers. Even with simple displacement boundary conditions, and straightforward interlayering of weak and strong layers, the structural geometries that develop are complex, with a combination of box, lift‐off and disharmonic detachment fold styles forming above the décollement. In detail, it is found that the modelled folds grow by both limb rotation and limb lengthening. The combination of these two mechanisms results in uplift patterns above the folds that are difficult, or misleading, to interpret in terms of simple kinematic models. Comparison of modelling results with natural examples and with kinematic models highlights the complexities of structural interpretation in such settings.