Interpolation using a generalized Green's function for a spherical surface spline in tension

A variety of methods exist for interpolating Cartesian or spherical surface data onto an equidistant lattice in a procedure known as gridding. Methods based on Green's functions are particularly simple to implement. In such methods, the Green's function for the gridding operator is determined and the resulting gridding solution is composed of the superposition of contributions from each data constraint, weighted by the Green's function evaluated for all output–input point separations. The Green's function method allows for considerable flexibility, such as complete freedom in specifying where the solution will be evaluated (it does not have to be on a lattice) and the ability to include both surface heights and surface gradients as data constraints. Green's function solutions for Cartesian data in 1-, 2- and 3-D spaces are well known, as is the dilogarithm solution for minimum curvature spline on a spherical surface. Here, the spherical surface case is extended to include tension and the new generalized Green's function is derived. It is shown that the new function reduces to the dilogarithm solution in the limit of zero tension. Properties of the new function are examined and the new gridding method is implemented in Matlab® and demonstrated on three geophysical data sets.     

Deposits of depletive high-density turbidity currents: a flume analogue of bed geometry, structure and texture

Flume experiments were performed to study the flow properties and depositional characteristics of high‐density turbidity currents that were depletive and quasi‐steady to waning for periods of several tens of seconds. Such currents may serve as an analogue for rapidly expanding flows at the mouth of submarine channels. The turbidity currents carried up to 35 vol.% of fine‐grained natural sand, very fine sand‐sized glass beads or coarse silt‐sized glass beads. Data analysis focused on: (1) depositional processes related to flow expansion; (2) geometry of sediment bodies generated by the depletive flows; (3) vertical and horizontal sequences of sedimentary structures within the sediment bodies; and (4) spatial trends in grain‐size distribution within the deposits. The experimental turbidity currents formed distinct fan‐shaped sediment bodies within a wide basin. Most fans consisted of a proximal channel‐levee system connected in the downstream direction to a lobe. This basic geometry was independent of flow density, flow velocity, flow volume and sediment type, in spite of the fact that the turbidity currents of relatively high density were different from those of relatively low density in that they exhibited two‐layer flow, with a low‐density turbulent layer moving on top of a dense layer with visibly suppressed large‐scale turbulence. Yet, the geometry of individual morphological elements appeared to relate closely to initial flow conditions and grain size of suspended sediment. Notably, the fans changed from circular to elongate, and lobe and levee thickness increased with increasing grain size and flow velocity. Erosion was confined to the proximal part of the leveed channel. Erosive capacity increased with increasing flow velocity, but appeared to be constant for turbidity currents of different grain size and similar density. Structureless sediment filled the channel during the waning stages of the turbidity currents laden with fine sand. The adjacent levee sands were laminated. The massive character of the channel fills is attributed to rapid settling of suspension load and associated suppression of tractional transport. Sediment bypassing prevailed in fan channels composed of very fine sand and coarse silt, because channel floors remained fully exposed until the end of the experiments. Lobe deposits, formed by the fine sand‐laden, high‐density turbidity currents, contained massive sand in the central part grading to plane parallel‐laminated sand towards the fringes. The depletive flows produced a radial decrease in mean grain size in the lobe deposits of all fans. Vertical trends in grain size comprised inverse‐to‐normal grading in the levees and in the thickest part of the lobes, and normal grading in the channel and fringes of the fine sandy fans. The inverse grading is attributed to a process involving headward‐directed transport of relatively fine‐grained and low‐concentrated fluid at the level of the velocity maximum of the turbidity current. The normal grading is inferred to denote the waning stage of turbidity‐current transport.     

Are Workers Attracted to Social Interaction Opportunities? A Study of Face-to-Face Contact Opportunities by Occupation and Industry

The spatial structure of a region is known to affect the degree of face-to-face interaction opportunities for a city’s residents. These interaction opportunities are important building blocks in aspects of economic production. To date, though, there is scant empirical evidence linking interaction opportunities to worker locations. In this article, using a spatial measure of social interaction potential (SIP), we seek to discover whether, and by how much, opportunities for interaction differ at home and work locations for workers within different industry and occupation groups in U.S. metropolitan areas. Based on the time-geographic concept of joint accessibility, SIP is sensitive to population and employment densities, as well as travel times associated with worker commutes in a region. We compare SIP at the census tract level of geography both within and between Metropolitan Statistical Areas (MSAs) nationally and test SIP distributions by occupation and industrial categories using nonparametric Kruskal–Wallis and Dunn tests. The study finds that several categories of higher skill and creative workers live and work in higher SIP areas. These findings provide evidence in support of theories of knowledge creation that rely on spontaneous face-to-face interaction and also indicate the effect of lifestyle preferences in location choices for highly skilled and arts workers.     

Analysis of Observed and Predicted Tsunami Travel Times for the Pacific and Indian Oceans

I have examined over 1500 historical tsunami travel-time records for 127 tsunamigenic earthquakes that occurred in the Pacific and Indian Oceans. After subjecting the observations to simple tests to rule out gross errors I compare the remaining reports to simple travel-time predictions using Huygens method and the long-wave approximation, thus simulating the calculations that typically take place in a tsunami warning situation. In general, I find a high correspondence between predicted and reported travel times however, significant departures exist. Some outliers imply significantly slower propagation speeds than predicted; many of these are clearly the consequences of observers not being able to detect the (possibly weak?) first arrivals. Other outliers imply excessively long predicted travel times. These outliers reflect peculiar geometric and bathymetric conditions that are poorly represented in global bathymetric grids, leading to longer propagation paths and consequently increased travel times. Analysis of Δt, the difference between observed and predicted travel time, yields a mean Δt of 19 minutes with a standard deviation of 131 minutes. Robust statistics, being less sensitive to outliers, yield a median Δt of just 18 seconds and a median absolute deviation of 33 minutes. Care is needed to process bathymetry to avoid excessive travel-time delays in shallow areas. I also show that a 2×2 arc minute grid yields better results that a 5×5 arc minute grid; the latter in general yielding slightly slower propagation predictions. The largest remaining source of error appears to be the inadequacy of the point-source approximation to the finite tsunami-generating area.     

Constraining explosive volcanism: subjective choices during estimates of eruption magnitude

When estimating the magnitude of explosive eruptions from their deposits, individuals make three sets of critical choices with respect to input data: the spacing of sampling sites, the selection of contour intervals to constrain the field measurements, and the hand contouring of thickness/isomass data, respectively. Volcanologists make subjective calls, as there are no accepted published protocols and few accounts of how these choices will impact estimates of eruption magnitude. Here, for the first time, we took a set of unpublished thickness measurements from the 1959 Kīlauea Iki pyroclastic fall deposit and asked 101 volcanologists worldwide to hand contour the data. First, there were surprisingly consistent volume estimates across maps with three different sampling densities. Second, the variability in volume calculations imparted by individuals’ choices of contours is also surprisingly low and lies between s?=?5 and 8 %. Third, volume estimation is insensitive to the extent to which different individuals “smooth” the raw data in constructing contour lines. Finally, large uncertainty is associated with the construction of the thinnest isopachs, which is likely to underestimate the actual trend of deposit thinning. The net result is that researchers can have considerable confidence in using volume or dispersal data from multiple authors and different deposits for comparative studies. These insights should help volcanologists around the world to optimize design and execution of field-based studies to characterize accurately the volume of pyroclastic deposits.     

The spatial and temporal distribution of marine geophysical surveys

We examine how bathymetric mapping coverage varies with distance from the coastline, here a proxy for the effort involved in collecting the data. Distances to the nearest coastline were evaluated on a 1′ × 1′ global grid. We evaluate the density of marine survey track lines, which falls off with increasing distance from the coastline and drops off precipitously for the most remote regions. Bathymetric coverage shows a marked asymmetry between the southern and northern hemispheres, the latter having a factor of 2–4 denser coverage. We find a rapid decrease in data acquisition for previously unexplored regions beginning in 1973–1975. This rate change may reflect a transition from serendipitous exploration to more targeted investigations as the plate tectonics hypothesis became accepted, but it could also reflect the 1970s oil shocks. Coverage of the seafloor varies logarithmically with mapping resolution. At 0.5° resolution, only ∼60% of the seafloor has been mapped; the 50% mark was reached in 1979 and coverage of unexplored seafloor has since been less rapid. For comparison, at 1′ resolution less than 10% of the seafloor has been mapped. Given rising fuel costs we predict the most remote areas will see a decline in future surveys. Better coordination of exploration among agencies and nations could mitigate this concern and improve global coverage, as could future altimetric mapping dedicated to bathymetric prediction.     

Flexure across a continent–ocean fracture zone: the northern Falkland/Malvinas Plateau, South Atlantic

 Bathymetry, satellite-derived gravity, and interpreted seismic reflection data across the northern Falkland/Malvinas Plateau fossil continent–ocean transform rim may record the degree of mechanical coupling across the boundary after ridge–transform intersection time. The rim comprises a broad microcontinental block in the east and a continental marginal fracture ridge 50–100 km wide elsewhere. Free-air gravity anomalies tentatively suggest that the fracture ridge is locked against oceanic elastic lithosphere both to the north (Argentine Basin) and south (Central Falkland Basin). Received: 18 January 1996 / Revision received: 25 March 1995