We propose a mathematical framework for formulation of biostratigraphic methods based on two order relations ≤^__??_ and <<^_??_ on a set whose members are intervals of a totally ordered set (A, ≤) defined by: [a, b] ≤^_??_ [a′, b′] iffa≤a′and b≤b′, and [a, b] <<^_??_ [a′, b′] iffb≤a′or [a, b] = [a′, b′] .Let Σ₀ be a set whose members are individuals of animals and plants which lived within an intervalT, and S₀be a partition ofΣ₀. An element ofS₀is called here“an individual class”.We define a range of an individual class σ, denoted by τ (σ), by union of life intervals of all individuals in σ. This framework enables us to formulate easily several basic concepts. A combination of individual classes which lived together at a timetis given bys (t) = {α|t∈τ (α) } .A range in which the same combination of individual classes lived together is expressed by an equivalence class relative to an equivalence relation-such thatp-qiffs (p) =s (q) .It is possible to order individual classes based on a relation κ such that α κ β iff τ (α) ≤^_??_τ (β) . Further we discuss the details of time classification based on a setSwhich satisfies three assumptions (1) Sis a subset ofS₀such that τ (σ) is an interval ofTfor all σ∈S, (2) (S, κ) is a total ordering, and (3) s (t) is a non-empty set for allt∈T. The results suggest that the proposed mathematical framework is useful to formulate basic concepts of biostratigraphy.

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To observe and to analyze the internal structures of various kinds of rock samples, a 3-dimensional CT (Computerized Tomography) image was reconstructed from the data obtained by an industry X-ray CT scanner with high intensity X-ray beam. FBP (Filtered BackProjection) method was used for the reconstruction of the CT image from the X-ray intensity data. Spatial resolution and size of the image were 0.2×0.2×1 mm and 1024×1024×320 pixels (voxels), respectively. CT value (X-ray liner absorption coefficient for the object) was stored as a 16 bits integer on each pixel in the image. Density calibration of the CT value in the image was performed using the standard samples with known densities. New methods were applied to the calibration to correct the effect of low CT values in the edge region on the mean CT value of each sample. Using the result of the calibration, density of an object in the CT image was determined from the CT value with decision error of about 0.3 g/cm³.

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In order to develop efficient algorithms for computer processing of geological data, we need to establish a mathematical framework which integrates a variety of geological methods. This paper presents a mathematical model composed of three sets B, Z and F whose members are cards, numbers and symbols, respectively, relations between sets and several axioms. This is a preliminary study to formulate a fundamental framework of biostratigraphic methods constructed on the basic principles of geology; the law of original horizontality, the law of original lateral continuity, the law of superposition and the law of faunal succession. In addition to three axioms A₁, A₂ and A₃ similar to those which Shiono and Wadatsumi (1992) proposed as properties of layering strata, an axiom A₄ is assumed to introduce a property of fossils or the law of faunal succession into the theoretical system.
It is confirmed that A₁, A₂ and A₃ are formally equivalent to Shiono and Wadatsumi (1992) 's axioms through a fact that they provide similar theorems.From A₄, we derived several inference rules on order relations between cards and/or groups of cards, which are interpreted to formulate biostratigraphic procedures.Further, we obtained an important result that if A₁, A₂, A₃ and A₄ hold true on B, then properties equivalent to A₁, A₂ and A₃ hold true also on a quotient set B/D, where D is an equivalence relation such that aDb iff a and b are marked with the same symbol.Thus the present model, in spite of its simplicity, reflects fundamental properties of strata baring fossils and provides a clue to systematize mathematical bases of biostratigraphy.

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Artificial neural network (ANN) is a computational model of biological nervous system. The major advantage of this system is that it can process the data having nonlinear behavior and classify appropriately complex and noisy image. The most widely used technique for modifying the connection weights in ANN is termed error back-propagation, which minimizes the error between the output from the network and the desired output using the steepest descent method.However, an ideal modifying rule should also consider the characteristics of spatial correlation of data.
We propose a new modifying rule of the connection weights of network, ESV criteria, which considers both error and semivariogram. The semivariogram expresses the spatial correlation of data by a relationship between the distance of data pair and difference of their values. Typical interpolation method using the semivariogram is Kriging, a weighting average method. The proposed method based on the ESV criteria is called Neural Kriging (NK) .
In order to evaluate the effectiveness of NK, it was applied to a reconstruction problem of a defined data-distribution model from randomly selected discrete data. Two kinds of model, isotropie spherical model and orthogonal anisotropic Gaussian model, were examined. It was revealed that the interpolation errors of NK are smaller than those of ANN and Ordinary Kriging for both models. Therefore, NK is regarded as an interpolation method of high accuracy which can be used for randomly distributed data with any structure of spatial correlation.
Furthermore, NK was applied to the distribution analysis of temperature and pH of water sampled from hot springs and geothermal investigation wells in the Hohi area, northeastern Kyushu, Japan. The result made it clear that the trend of their distributions is harmony with thedirection of principal fracture system in this area.

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