Variable resolution block kriging using a hierarchical spatial data structure

Abstract

Kriging is an optimal method of spatial interpolation that produces an error for each interpolated value. Block kriging is a form of kriging that computes averaged estimates over blocks (areas or volumes) within the interpolation space. If this space is sampled sparsely, and divided into blocks of a constant size, a variable estimation error is obtained for each block, with blocks near to sample points having smaller errors than blocks farther away. An alternative strategy for sparsely sampled spaces is to vary the sizes of blocks in such away that a block's interpolated value is just sufficiently different from that of an adjacent block given the errors on both blocks. This has the advantage of increasing spatial resolution in many regions, and conversely reducing it in others where maintaining a constant size of block is unjustified (hence achieving data compression). Such a variable subdivision of space can be achieved by regular recursive decomposition using a hierarchical data structure. An implementation of this alternative strategy employing a split-and-merge algorithm operating on a hierarchical data structure is discussed. The technique is illustrated using an oceanographic example involving the interpolation of satellite sea surface temperature data. Consideration is given to the problem of error propagation when combining variable resolution interpolated fields in GIS modelling operations.     

Qualitative GIS and the visualization of narrative activity space data

Qualitative activity space data, i.e. qualitative data associated with the routine locations and activities of individuals, are recognized as increasingly useful by researchers in the social and health sciences for investigating the influence of environment on human behavior. However, there has been little research on techniques for exploring qualitative activity space data. This research illustrates the theoretical principles of combining qualitative and quantitative data and methodologies within the context of geographic information systems (GIS), using visualization as the means of inquiry. Through the use of a prototype implementation of a visualization system for qualitative activity space data and its application in a case study of urban youth we show how these theoretical methodological principles are realized in applied research. The visualization system uses a variety of visual variables to simultaneously depict multiple qualitative and quantitative attributes of individuals' activity spaces. The visualization is applied to explore the activity spaces of a sample of urban youth participating in a study on the geographic and social contexts of adolescent substance use. Examples demonstrate how the visualization can be used to explore individual activity spaces to generate hypotheses, investigate statistical outliers, and explore activity space patterns among subject subgroups.     

Silica-rich deposits and hydrated minerals at Gusev Crater, Mars: Vis-NIR spectral characterization and regional mapping

The Mars Exploration Rover (MER) Spirit has discovered surprisingly high concentrations of amorphous silica in soil and nodular outcrops in the Inner Basin of the Columbia Hills. In Pancam multispectral observations, we find that an absorption feature at the longest Pancam wavelength (1009 nm) appears to be characteristic of these silica-rich materials; however, spectral analyses of amorphous silica suggest that the ∼1009 nm spectral feature is not a direct reflection of their silica-rich nature. Based on comparisons with spectral databases, we hypothesize that the presence of H₂O or OH, either free (as water ice), adsorbed or bound in a mineral structure, is responsible for the spectral feature observed by Pancam. The Gertrude Weise soil, which is nearly pure opaline silica, may have adsorbed water cold-trapped on mineral grains. The origin of the ∼1009 nm Pancam feature observed in the silica-rich nodular outcrops may result from the presence of additional hydrated minerals (specific sulfates, halides, chlorides, sodium silicates, carbonates or borates). Using the ∼1009 nm feature with other spectral parameters as a “hydration signature” we have mapped the occurrence of hydrated materials along the extent of Spirit’s traverse across the Columbia Hills from West Spur to Home Plate (sols 155-1696). We have also mapped this hydration signature across large panoramic images to understand the regional distribution of materials that are spectrally similar to the silica-rich soil and nodular outcrops. Our results suggest that hydrated materials are common in the Columbia Hills.     

New Cassini RADAR results for Saturn’s icy satellites

Cassini radar tracks on Saturn’s icy satellites through the end of the Prime Mission in 2008 have increased the number of radar albedo estimates from 10 (Ostro et al., 2006) to 73. The measurements sample diverse subradar locations (and for Dione, Rhea, and Iapetus almost always use beamwidths less than half the target angular diameters), thereby constraining the satellites’ global radar albedo distributions. The echoes result predominantly from volume scattering, and their strength is thus strongly sensitive to ice purity and regolith maturity. The combination of the Cassini data set and Arecibo 13-cm observations of Enceladus, Tethys, Dione, Rhea (Black et al., 2007), and Iapetus (Black et al., 2004) discloses an unexpectedly complex pattern of 13-to-2-cm wavelength dependence. The 13-cm albedos are generally smaller than 2-cm albedos and lack the correlation seen between 2-cm and optical geometric albedos. Enceladus and Iapetus are the most interesting cases. We infer from hemispheric albedo variations that the E-ring has a prominent effect on the 13-cm radar “lightcurve”. The uppermost trailing-side regolith is too fresh for meteoroid bombardment to have developed larger-scale heterogeneities that would be necessary to elevate the 13-cm radar albedo, whereas all of Enceladus is clean and mature enough for the 2-cm albedo to be uniformly high. For, Iapetus, the 2-cm albedo is strongly correlated with optical albedo: low for the optically dark, leading-side material and high for the optically bright, trailing-side material. However, Iapetus’ 13-cm albedo values show no significant albedo dichotomy and are several times lower than 2-cm values, being indistinguishable from the weighted mean of 13-cm albedos for main-belt asteroids, 0.15 ± 0.10. The leading side’s optically dark contaminant must be present to depths of at least one to several decimeters, so 2-cm albedos can mimic the optical dichotomy; however, it does not have to extend any deeper than that. The fact that both hemispheres of Iapetus look Asteroid-like at 13 cm means that coherent backscattering itself is not nearly as effective as it is at 2 cm. Since Iapetus’ entire surface is mature regolith, the wavelength dependence must involve composition, not structure. Either the composition is a function of depth everywhere (with electrical loss much greater at depths greater than a decimeter or two), or the intrinsic electrical loss of some pervasive constituent is much higher at 13 cm than at 2 cm. Ammonia is a candidate for such a contaminant. If ammonia’s electrical properties do not depend on frequency, and if ammonia is globally much less abundant within the upper one or two decimeters than at greater depths, then coherent backscattering would effectively be shut down at 13 cm, explaining the Asteroid-like 13-cm albedo.     

The equatorial shape and gravity field of Mercury from MESSENGER flybys 1 and 2

On 14 January and 6 October 2008 the MESSENGER spacecraft passed within 200 km of the surface of Mercury. These flybys by MESSENGER provided the first observations of Mercury from a spacecraft since the Mariner 10 flybys in 1974 and 1975. Data from the Mercury Laser Altimeter (MLA) provided new information on the equatorial shape of Mercury, and Doppler tracking of the spacecraft through the flybys provided new data on the planet’s gravity field. The MLA passes were on opposite hemispheres of the planet and span collectively ∼40% of the equatorial circumference. The mean elevation of topography observed during flyby 1, in the longitude range 0-90°E, is greater than that seen during flyby 2 in the longitude range 180-270°E, indicating an offset between centers of mass and figure having a magnitude and phase in general agreement with topography determined by Earth-based radar. Both MLA profiles are characterized by slopes of ∼0.015° downward to the east, which is consistent with a long-wavelength equatorial shape defined by a best-fitting ellipse. The Doppler tracking data show sensitivity to the gravitational structure of Mercury. The equatorial ellipticity of the gravitational field, C_2,2, is well determined and correlates with the equatorial shape. The S_2,2 coefficient is ∼0, as would be expected if Mercury’s coordinate system, defined by its rotational state, is aligned along its principal axes of inertia. The recovered value of the polar flattening of the gravitational potential, J₂, is considerably lower in magnitude than the value obtained from Mariner 10 tracking, a result that is problematic for internal structure models. This parameter is not as well constrained as the equatorial ellipticity because the flyby trajectories were nearly in the planet’s equatorial plane. The residuals from the Doppler tracking data suggest the possibility of mascons on Mercury, but flyby observations are of insufficient resolution for confident recovery. For a range of assumptions on degree of compensation and crustal and mantle densities, the allowable crustal thickness is consistent with the upper limit of about 100 km estimated from the inferred depth of faulting beneath a prominent lobate scarp, an assumed ductile flow law for crustal material, and the condition that temperature at the base of the crust does not exceed the solidus temperature. The MESSENGER value of C_2,2 has allowed an improved estimate of the ratio of the polar moment of inertia of the mantle and crust to the full polar moment (C_m/C), a refinement that strengthens the conclusion that Mercury has at present a fluid outer core.     

Progressive formation of modern drumlins at Múlajökull,Iceland: stratigraphical and morphological evidence

The drumlin field at Múlajökull, Iceland, is considered to be an active field in that partly and fully ice‐covered drumlins are being shaped by the current glacier regime. We test the hypothesis that the drumlins form by a combination of erosion and deposition during successive surge cycles. We mapped and measured 143 drumlins and studied their stratigraphy in four exposures. All exposures reveal several till units where the youngest till commonly truncates older tills on the drumlin flanks and proximal slope. Drumlins inside a 1992 moraine are relatively long and narrow whereas drumlins outside the moraine are wider and shorter. A conceptual model suggests that radial crevasses create spatial heterogeneity in normal stress on the bed so that deposition is favoured beneath crevasses and erosion in adjacent areas. Consequently, the crevasse pattern of the glacier controls the location of proto‐drumlins. A feedback mechanism leads to continued crevassing and increased sedimentation at the location of the proto‐drumlins. The drumlin relief and elongation ratio increases as the glacier erodes the sides and drapes a new till over the landform through successive surges. Our observations of this only known active drumlin field may have implications for the formation and morphological evolution of Pleistocene drumlin fields with similar composition, and our model may be tested on modern drumlins that may become exposed upon future ice retreat.     

The effect of Mn on mineral stability in metapelites revisited: new a–x relations for manganese‐bearing minerals

The a–x relations recently presented in White et al. ( 2014 , Journal of Metamorphic Geology, 32, 261–286) are extended to include MnO. This provides a set of internally consistent a–x relations for metapelitic rocks in the MnO–Na₂O–CaO–K₂O–FeO–MgO–Al₂O₃–SiO₂–H₂O–TiO₂–O₂ (MnNCKFMASHTO) system. The mixing parameters for the Mn‐bearing minerals were estimated using the micro‐? approach of Powell et al. ( 2014 , Journal of Metamorphic Geology, 32, 245–260). Then the Mn‐end‐member thermodynamic properties were calibrated using a database of co‐existing minerals involving literature data from rocks and from experiments on natural materials. Mn‐end‐members were calibrated for orthopyroxene, cordierite, staurolite, chloritoid, chlorite, biotite, ilmenite and hematite, assuming known properties for the garnet end‐member spessartine. The addition of MnO to phase diagram calculations results in a marked expansion of the stability of garnet‐bearing assemblages. At greenschist facies conditions garnet stability is extended down temperature. At amphibolite facies conditions, the garnet‐in boundary shifts to lower pressure. While the addition of MnO greatly influences the stability of garnet, it has relatively little effect on the stability of other common metapelitic minerals, with the resultant diagrams being topologically very similar to those calculated without MnO. Furthermore, the addition of MnO in the amounts measured in most metapelites has only a small effect on the mode of garnet, with calculated garnet modes remaining smaller than 1% in the P–T range outside its predicted Mn‐free P–T range.     

An electron‐optical study of melt‐related microstructures in granulite facies rocks from the Torngat Orogen

Cordierite–quartz and plagioclase–quartz intergrowths in a paragneiss from northern Labrador (the Tasiuyak Gneiss) were studied using SEM, STEM and TEM. The gneiss experienced granulite facies conditions and partial melting during both regional and, subsequently, during contact metamorphism. The microstructures examined all results from the contact metamorphism. Cordierite–quartz intergrowths occur on coarse and fine scales. The former sometimes exist as a ‘geometric’ intergrowth in which the interface between cordierite and quartz appears planar at the resolution of the optical microscope and SEM. The latter exists in several microstructural variants. Plagioclase is present as a minor component of the intergrowth in some examples of both the coarse and fine intergrowth. Grain boundaries in cordierite–quartz intergrowths are occupied by amorphous material or a mixture of amorphous material and chlorite. Cordierite and quartz are terminated by crystal faces in contact with amorphous material. Chlorite is sometimes found on cordierite surfaces and penetrating into cordierite grains along defects. Quartz contains (former) fluid inclusions 10–20 nm in maximum dimension. The presence of planar interfaces between cordierite and the amorphous phase is reminiscent of those between crystals and glass in volcanic rocks, but in the absence of compelling evidence that the amorphous material represents former melt, it is interpreted as a reaction product of cordierite. Plagioclase–quartz intergrowths occur in a number of microstructural variants and are commonly associated with cordierite–quartz intergrowths. The plagioclase–quartz intergrowths display simple, non‐planar interfaces between plagioclase and quartz. Quartz contains (former) fluid inclusions of dimensions similar to those observed in cordierite–quartz intergrowths. The boundary between quartz and enclosing K‐feldspar is cuspate, with quartz cusps penetrating a few tens of nanometres into K‐feldspar, commonly along defects in K‐feldspar and sometimes with very low dihedral angles at their tips. This cuspate microstructure is interpreted as melt pseudomorphs. The plagioclase–quartz intergrowths share some features with myrmekite, but differ in some respects: the composition of the plagioclase (An₃₇Ab₆₂Or₁–An₃₈Ab₆₁Or₁); the association with cordierite–quartz intergrowths; and microstructures that are atypical of myrmekite (e.g. quartz vermicules shared with cordierite–quartz intergrowths). It is inferred that the plagioclase–quartz intergrowths may have formed from, or in the presence of, melt. Inferred melt‐related microstructures preserved on the nanometre scale suggest that melt on grain boundaries was more pervasive than is evident from light optical and SEM observations.