Spherical Slepian functions and the polar gap in geodesy

The estimation of potential fields such as the gravitational or magnetic potential at the surface of a spherical planet from noisy observations taken at an altitude over an incomplete portion of the globe is a classic example of an ill-posed inverse problem. We show that this potential-field estimation problem has deep-seated connections to Slepian's spatiospectral localization problem which seeks bandlimited spherical functions whose energy is optimally concentrated in some closed portion of the unit sphere. This allows us to formulate an alternative solution to the traditional damped least-squares spherical harmonic approach in geodesy, whereby the source field is now expanded in a truncated Slepian function basis set. We discuss the relative performance of both methods with regard to standard statistical measures such as bias, variance and mean squared error, and pay special attention to the algorithmic efficiency of computing the Slepian functions on the region complementary to the axisymmetric polar gap characteristic of satellite surveys. The ease, speed, and accuracy of our method make the use of spherical Slepian functions in earth and planetary geodesy practical.     

Multifractal modeling and spatial statistics

In general, the multifractal model provides more information about measurements on spatial objects than a fractal model. It also results in mathematical equations for the covariance function and semivariogram in spatial statistics which are determined primarily by the second-order mass exponent. However, these equations can be approximated by power-law relations which are comparable directly to equations based on fractal modeling. The multifractal approach is used to describe the underlying spatial structure of De Wijs 's example of zinc values from a sphalerite-bearing quartz vein near Pulacayo, Bolivia. It is shown that these data are multifractal instead of fractal, and that the second-order mass exponent (=0.979±0.011 for the example) can be used in spatial statistical analysis.     

Appreciation of Contributions by Georges Matheron and John Tukey to a Mineral-Resources Research Project

Georges Matheron (1930–2000) and John Tukey (1915–2000) were among the most prominent mathematical statisticians of the 20th century. Both men produced numerous important new theoretical and practical results. This personal appreciation of their work concentrates on contributions to mineral-resources research and describes their influence on my work in mineral-resource evaluation studies at the Geological Survey of Canada (1966–1983).     

Multifractal modeling and spatial point processes

The multifractal model can be applied to spatial point processes. It provides new, approximately power-law type, expressions for their second-order intensity and K (r) functions. The box-counting and cluster dimensions are different but mutually interrelated according to multifractal theory. This approach is used to describe the underlying spatial structure of gold mineral occurrences in the Iskut River area, northwestern British Columbia. The box-counting and cluster dimensions for the example are estimated to be 1.335±0.077 and 1.219±0.037, respectively. The relatively strong clustering of the gold deposits is reflected by the fact that both values are considerably less than the corresponding Euclidean dimension (=2).     

Closed Form Solutions of the Two-Dimensional Turning Bands Equation

The turning bands method generates realizations of isotropic Gaussian random fields by means of appropriately summed line processes. For two-dimensional simulations the relation between the isotropic correlation function of the random field and the correlation function to be simulated along the lines is given by an integral equation of Abel type. We present closed form solutions of this integral equation for almost all two-dimensional correlation models encountered in practice and discuss their numerical implementation. As an additional benefit, our tables and illustrations serve as a concise guide to correlation models useful in geostatistics.     

The topography of Iapetus' leading side

We have used Cassini stereo images to study the topography of Iapetus' leading side. A terrain model derived at resolutions of 4-8 km reveals that Iapetus has substantial topography with heights in the range of −10 km to +13 km, much more than observed on the other middle-sized satellites of Saturn so far. Most of the topography is older than 4 Ga [Neukum, G., Wagner, R., Denk, T., Porco, C.C., 2005. Lunar Planet. Sci. XXXVI. Abstract 2034] which implies that Iapetus must have had a thick lithosphere early in its history to support this topography. Models of lithospheric deflection by topographic loads provide an estimate of the required elastic thickness in the range of 50-100 km. Iapetus' prominent equatorial ridge [Porco, C.C., and 34 colleagues, 2005. Science 307, 1237-1242] reaches widths of 70 km and heights of up to 13 km from their base within the modeled area. The morphology of the ridge suggests an endogenous origin rather than a formation by collisional accretion of a ring remnant [Ip, W.-H., 2006. Geophys. Res. Lett. 33, doi:10.1029/2005GL025386. L16203]. The transition from simple to complex central peak craters on Iapetus occurs at diameters of 11±3 km. The central peaks have pronounced conical shapes with flanking slopes of typically 11° and heights that can rise above the surrounding plains. Crater depths seem to be systematically lower on Iapetus than on similarly sized Rhea, which if true, may be related to more pronounced crater-wall slumping (which widens the craters) on Iapetus than on Rhea. There are seven large impact basins with complex morphologies including central peak massifs and terraced walls, the largest one reaches 800 km in diameter and has rim topography of up to 10 km. Generally, no rings are observed with the basins consistent with a thick lithosphere but still thin enough to allow for viscous relaxation of the basin floors, which is inferred from crater depth-to-diameter measurements. In particular, a 400-km basin shows up-domed floor topography which is suggestive of viscous relaxation. A model of complex crater formation with a viscoplastic (Bingham) rheology [Melosh, H.J., 1989. Impact Cratering. Oxford Univ. Press, New York] of the impact-shocked icy material provides an estimate of the effective cohesion/viscosity at . The local distribution of bright and dark material on the surface of Iapetus is largely controlled by topography and consistent with the dark material being a sublimation lag deposit originating from a bright icy substrate mixed with the dark components, but frost deposits are possible as well.     

The Correlation Bias for Two-Dimensional Simulations by Turning Bands

The turning bands method (TBM) generates realizations of isotropic Gaussian random fields by summing contributions from line processes. We consider two-dimensional simulations and study the correlation bias attributable to the use of only a finite number L of lines. Our analytical and numerical results confirm that the maximal bias is of order 1/L, and that L = 64 lines suffice for excellent covariance reproduction. The notorious banding observed in simulations with an insufficient number of lines is a related but different phenomenon and depends strongly on the choice of the line simulation technique. Clear-cut recommendations for the number of lines necessary to avoid the effect can only be based on practical experience with the specific code at hand.     

Evolution of anthropogenic and biomass burning emissions of air pollutants at global and regional scales during the 1980�C2010 period

Several different inventories of global and regional anthropogenic and biomass burning emissions are assessed for the 1980?C2010 period. The species considered in this study are carbon monoxide, nitrogen oxides, sulfur dioxide and black carbon. The inventories considered include the ACCMIP historical emissions developed in support of the simulations for the IPCC AR5 assessment. Emissions for 2005 and 2010 from the Representative Concentration Pathways (RCPs) are also included. Large discrepancies between the global and regional emissions are identified, which shows that there is still no consensus on the best estimates for surface emissions of atmospheric compounds. At the global scale, anthropogenic emissions of CO, NO_x and SO₂ show the best agreement for most years, although agreement does not necessarily mean that uncertainty is low. The agreement is low for BC emissions, particularly in the period prior to 2000. The best consensus is for NO_x emissions for all periods and all regions, except for China, where emissions in 1980 and 1990 need to be better defined. Emissions of CO need better quantification in the USA and India for all periods; in Central Europe, the evolution of emissions during the past two decades needs to be better determined. The agreement between the different SO₂ emissions datasets is rather good for the USA, but better quantification is needed elsewhere, particularly for Central Europe, India and China. The comparisons performed in this study show that the use of RCP8.5 for the extension of the ACCMIP inventory beyond 2000 is reasonable, until more global or regional estimates become available. Concerning biomass burning emissions, most inventories agree within 50?C80%, depending on the year and season. The large differences between biomass burning inventories are due to differences in the estimates of burned areas from the different available products, as well as in the amount of biomass burned.