The Discontinuous Nature of Kriging Interpolation for Digital Terrain Modeling

Kriging is a widely employed method for interpolating and estimating elevations from digital elevation data. Its place of prominence is due to its elegant theoretical foundation and its convenient practical implementation. From an interpolation point of view, kriging is equivalent to a thin-plate spline and is one species among the many in the genus of weighted inverse distance methods, albeit with attractive properties. However, from a statistical point of view, kriging is a best linear unbiased estimator and, consequently, has a place of distinction among all spatial estimators because any other linear estimator that performs as well as kriging (in the least squares sense) must be equivalent to kriging, assuming that the parameters of the semivariogram are known. Therefore, kriging is often held to be the gold standard of digital terrain model elevation estimation. However, I prove that, when used with local support, kriging creates discontinuous digital terrain models, which is to say, surfaces with “rips” and “tears” throughout them. This result is general; it is true for ordinary kriging, kriging with a trend, and other forms. A U.S. Geological Survey (USGS) digital elevation model was analyzed to characterize the distribution of the discontinuities. I show that the magnitude of the discontinuity does not depend on surface gradient but is strongly dependent on the size of the kriging neighborhood.

Impact of environmental regulations on lead concentrations in Texas ambient air and water: 1970--1990

In the last two decades, the significance of lead has been addressed in a number of environmental regulations at the national and state levels. This project investigated the environmental regulations (Clean Air Act and Amendments, 1970-1990 and Clean Water Act of 1977) and their cumulative effects on lead in ambient air and water in the state of Texas. For this purpose, historical records from the Texas Water Development Board, Texas Natural Resources Conservation Commission, and the United States Geological Survey have been assembled and analyzed for temporal and spatial trends. These trends might correspond to the phase out of lead in gasoline and other regulations.^ This study concluded that there is a significant correlation (p $\leq$.001) between environmental regulations of lead in gasoline and the concentration of lead in ambient air. Lead concentrations in ambient air have been reduced by over 90 percent in the past twenty years. An overall significant difference (p $\leq$.001) was found in mean (94, 15 respectively) lead concentrations in surface water between two time periods, one at the beginning of the twenty year period and one at the end of the study period. There has been an overall reduction of lead concentrations in surface water in Texas of approximately 84 percent. However, this reduction cannot be statistically associated with any one regulation. Groundwater data could not be analyzed for lead concentrations because of limitations of reporting data as "less than". Approximately two percent of the groundwater data was analyzed by Oneway ANOVA and no significant difference was found between the means (18, 19 respectively) of two time periods, 1977-1979 and 1988-1990. This data is consistent with the regulations having a contributory affect on declining concentrations, but other factors cannot be ruled out as having added to these declines. This study can also serve as a starting point for a more in-depth study of environmental regulations and their impact on the environment. ^

Major elements and trace ferrides in deep sea cores obtained during the Swedish Deep-Sea Expedition in 1947-1948

Samples from sediment cores collected during the Swedish Deep-Sea Expedition 1947-1948 have been analyzed in the Geochemical laboratory of the Geological Survey of Sweden. Most samples were placed at our disposal by Professor Hans Pettersson, leader of the expedition mentioned. For complementary studies, samples from the Atlantic and Indian oceans were included in our investigation and the samples placed at our disposal by Professor B. Kullenberg, Göteborg. From the Tyrrhenian Sea we got samples from Professor E. Norin, Uppsala.

Permian manganese nodules near Dillon, Montana

Concentrically ringed manganese nodules, similar in form to many found on modern ocean and sea floors, occur in a very fine grained argillaceous sandstone bed of the Permian Park City Formation near Dillon, Montana. They are enriched in many rare elements and contain us much as 2.5 percent zinc, l.3 percent nickel, and 0.22 percent cobalt. The manganese minerals are chalcophanite and todorokite. The nodules probably formed in a shallow marine oxidizing environment on the western side of the Permian sedimentary basin. The occurrence of an appreciable amount of fluorite in the bed suggests that the water was saline.

Geochemical composition of the Trincheira Complex in the southwestern Amazon craton, Brazil

We document the first-known Mesoproterozoic ophiolite from the southwestern part of the Amazon craton, corresponding to the Trincheira Complex of Calymmian age, and propose a tectonic model that explains many previously enigmatic features of the Precambrian history of this key craton, and discuss its role in the reconstruction of the Columbia supercontinent. The complex comprises extrusive rocks (fine-grained amphibolites derived from massive and pillowed basalts), mafic-ultramafic intrusive rocks, chert, banded iron formation (BIFs), pelites, psammitic and a smaller proportion of calc-silicate rocks. This sequence was deformed, metasomatized and metamorphosed during the development of the Alto Guaporé Belt, a Mesoproterozoic accretionary orogen. The rocks were deformed by a single tectonic event, which included isoclinal folding and metamorphism of the granulite-amphibolite facies. Layered magmatic structures were preserved in areas of low strain, including amygdaloidal and cumulate structures. Metamorphism was pervasive and reached temperatures of 780-853°C in mafic granulites and 680-720°C in amphibolites under an overall pressure of 6.8 kbar. The geochemical composition of the extrusive and intrusive rocks indicates that all noncumulus mafic-ultramafic rocks are tholeiitic basalts. The mafic-ultramafic rocks display moderately to strongly fractionation of light rare earth elements (LREE), near-flat heavy rare earth elements (HREE) patterns and moderate to strong negative high field strength elements (HFSE) anomalies (especially Nb), a geochemical signature typical of subduction zones. The lowest units of mafic granulites and porphyroblastic amphibolites in the Trincheira ophiolite are similar to the modern mid-ocean ridge basalt (MORB), although they locally display small Ta, Ti and Nb negative anomalies, indicating a small subduction influence. This behavior changes to an island arc tholeiites (IAT) signature in the upper units of fine-grained amphibolites and amphibole rich-amphibolites, characterized by progressive depletion in the incompatible elements and more pronounced negative Ta and Nb anomalies, as well as common Ti and Zr negative anomalies. Tectono-magmatic variation diagrams and chondrite-normalized REE and primitive mantle normalized patterns suggest a back-arc to intra-oceanic island arc tectonic regime for the eruption of these rocks. Therefore, the Trincheira ophiolite appears to have originated in an intraoceanic supra-subduction setting composed of an arc-back-arc system. Accordingly, the Trincheira Complex is a record of oceanic crust relics obducted during the collision of the Amazon craton and the Paraguá block during the Middle Mesoproterozoic. Thus, the recognition of the Trincheira ophiolite and suture significantly changes views on the evolution of the southern margin of the Amazon craton, and how it can influence the global tectonics and the reconstruction of the continents.