G. M. Philip

Triangle based interpolation

Triangle based interpolation is introduced by an outline of two classical planar interpolation methods, viz. linear triangular facets and proximal polygons. These are shown to have opposite local bias. By applying cross products of triangles to obtain local gradients, a method designated “slant-top proximal polygon interpolation” is introduced that is intermediate between linear facets and polygonal interpolation in its local bias. This surface is not continuous, but, by extending and weighting the gradient planes, a C1surface can be obtained. The gradients also allow a roughness index to be calculated for each data point in the set. This index is used to control the shape of a blending function that provides a weighted combination of the gradient planes and linear interpolation. This results in a curvilinear, C1,interpolation of the data set that is bounded by the linear interpolation and the weighted gradient planes and is tangent to the slant-top interpolation at the data points. These procedures may be applied to data with two, three, or four independent variables.

Matheronian geostatistics—Quo vadis?

Components of geostatistical estimation, developed as a method for ore deposit assessment, are discussed in detail. The assumption that spatial observations can be treated as a stochastic process is judged to be an inappropriate model for natural data. Problems of semivariogram formulation are reviewed, and this method is considered to be inadequate for estimating the function being sought. Characteristics of bivariate interpolation are summarized, highlighting kriging limitations as an interpolation method. Limitations are similar to those of inverse distance weighted observations interpolation. Attention is drawn to the local bias of kriging and misplaced claims that it is an “optimal” interpolation method. The so-called “estimation variance,” interpreted as providing confidence limits for estimation of mining blocks, is shown to be meaningless as an index of local variation. The claim that geostatistics constitutes a “new science” is examined in detail. Such novelties as exist in the method are shown to transgress accepted principles of scientific inference. Stochastic modeling in general is discussed, and purposes of the approach emphasized. For the purpose of detailed quantitative assessment it can provide only prediction qualified by hypothesis at best. Such an approach should play no part in ore deposit assessment where the need is for local detailed inventories; these can only be achieved properly through local deterministic methods, where prediction is purely deductive.

Australian Tertiary schizasterid echinoids

The taxonomy of schizasterid spatangoids is discussed and the term is restricted to include only Schizaster-like echinoids. Within the genus Schizaster Agassiz the following morphological subgenera are recognized: Paraster Pomel, Dipneustes Arnaud and Ova Gray. Seven species are described from Palaeocene to Miocene rocks of Australia. New species are S. (Paraster) carinatus, S. (Paraster) tatei, S. (Schizaster) halli and S. (Dipneustes) fosteri. Schizaster and its various morphotypes are thought to have evolved from a Linthia-like root stock through progressive morphological changes in test shape, form of ambulacra and adoral test morphology. Such changes are interpreted as being adaptations which allowed the sea urchins to occupy various levels of substrate in different sediment types. Morphotypes appear to have evolved iteratively, perhaps due to variation in the onset of sexual maturity.

Notocarpos garratti gen. et sp. nov., a new Silurian mitrate carpoid from Victoria

Notocarpos garratti gen. et sp. nov. is described from the middle Ludlovian Humevale Formation of the Clonbinane district, Victoria. It is compared with similar anomalocystitid carpoids and is found to resemble most closely Allanicytidium flemingi Caster & Gill 1968 from the Early Devonian Reefton Beds of New Zealand. N. garratti provides evidence that anomalocystitids rested with the flattened thecal surface against the sea floor (i.e., an orientation opposite to that proposed by Jefferies, 1968). It is further suggested that the stele was adapted to provide a rearward mode of locomotion.

A method for assessing local variation among scattered measurements

Automatic triangulation of scattered locations permits analysis of local variation in a dependent variable through calculation of a roughness index. This is approached by treating triangles of the triangulation (including the dependent variable) as vectorial structures, and accumulating at each data point the vector sum of the cluster of triangles surrounding it. The roughness index is defined as the complement of the ratio of the area of a triangle cluster to the area of component triangles as projected onto a gradient plane defined by their vector sum. The roughness index provides a measure of consistency of data values relative to surrounding observations and can be interpreted as a local index of reliability of interpolation.

Algebraic dispersion fields on ternary diagrams

Previously published dispersion fields on ternary diagrams have been constructed variously, and their derivations have not been well-specified. Here an explanation of their bases is provided through an algebraic method for calculating two related forms, designated thesilhouette dispersion field and thegirth dispersion field. Such dispersion estimates can be made more precise by specifying the percentage of samples that fall within the field. Because such fields represent a mechanistic rather than a probabilistic approach, their use in comparison of sample sets must be viewed with caution.

Tertiary brissid echinoids of Southern Australia

Nine species of brissid echinoids are described from Eocene to Miocene strata of Victoria, South Australia and the southern part of Western Australia. Redescribed are Gillechinus cudmorei Fell 1964a; Brissopsis tatei Hall 1907; Brissopsis praeluzonica Fell 1964b; Cyclaster archeri (Tenison Woods 1867); Granobrissoides australiae (Cotteau 1889); Meoma tuberculata Hutton 1873 and Spatagobrissus laubei (Duncan 1877). Brissus fosteri and Brissopsis australis are described as new. Brissopsis praeluzonica is described from outside New Zealand for the first time. Micraster brevistella Laube 1869; Hemiaster posita Hutton 1873; Cyclaster morgani Cotteau 1889; and Cyclaster lycoperdon Bittner 1892 are all placed in synonymy with Cyclaster archeri. Granobrissoides Lambert 1920 is reinstated to generic status and G. australiae described in detail for the first time. The late Middle Miocene Eupatagus laubei Duncan 1877 is placed in Spatagobrissus, a genus hitherto known only from the living South African species S. mi...

A provisional phylogeny for some reconstructed Late Devonian polygnathid conodont apparatuses

New collections of Late Devonian age, from the Canning Basin, Western Australia, permit an examination of the evolutionary origin of stratigraphically important palmatolepan elements in terms of reconstructed apparatuses. The apparatuses conform to the polygnathid plan of Klapper & Philip (1971, 1972) and are described in such terms. The reconstruction of Mesotaxis Klapper & Philip (based on Polygnathus asymmetricus) is emended to an apparatus with a lippertiform N element and a ramiform symmetry transition series. Early forms of Palmatodella (here used to include all apparatuses with a palmatolepan P element) possess a similar lippertiform N element, a palmatodellan O element, and a symmetry transition series clearly derived from that of Mesotaxis. In Palmatodella triangularis the N element is lost and is not represented in younger Fammenian species of Palmatodella. It is considered premature to attempt to relate the new reconstructions to a detailed generic taxonomy.

Neighborhood discontinuities in bivariate interpolation of scattered observations

Because of the need for computational efficiency, bivariate interpolation methods applied to scattered observations often involve two stages. Initially the variable is estimated at regular grid nodes using a running subset of data (usually of fixed number). This, however, will produce discontinuities in the interpolated surface. Thus a second stage, curvilinear interpolation technique, is applied to estimated values to smooth out the effect of discontinuities. Such problems can be overcome efficiently in processing large data sets by interpolating over natural neighbor subsets. Interpolation procedures that generate discontinuities in the interpolated surface are inappropriate for geological applications, where dislocations due to structural complications may be present.

Late Silurian or Early Devonian corals from the continental slope off southern New South Wales

Latest Silurian or Early Devonian tabulate corals have been recovered in a dredge haul from submarine bedrock at a water depth of 2620 – 2155 m on the continental slope off southern New South Wales 45 km east of Tuross Heads from the submerged and unexplored part of the eastern Lachlan Orogen. The corals have been identified as latest Silurian to Early Devonian taxa, Favosites gothlandicus and Squameofavosites squamuliferus forma nitidus. The presence of shallow water deposits containing these forms indicates that a crustal thickening event or events took place near the end of Silurian or early in Devonian time in the eastern Lachlan Orogen east of the present coastline. This occurred well after the end of subduction in Early Silurian time which is marked by the welding of the Cambrian — Late Ordovician Wagonga Group accretionary wedge or collided terrain on to the eastern margin of the orogen and significantly earlier than the emplacement of the Moruya Suite granites in Middle Devonian time.