Helmut Schaeben

Technische Universität Bergakademie Freiberg

Even though the descriptive definition of orientation is the same in both settings, the explicitnotation of a crystallographic orientation as (3 3) matrix in terms of Euler angles featuredby the popular MATLAB toolbox MTEX differs by an inversion from the quasi-standard notation datedback to the early days of quantitative texture analysis champione...

To be a quantitative and testable tectonic model, plate tectonics requires spherical geometry and spherical kinematics in terms of finite rotations conveniently parametrized by their angle and axis and described by unit quaternions. In treatises on “Plate Tectonics” infinitesimal, instantaneous, and finite rotations, absolute and relative rotations...

For phase transformations within polycrystalline materials, the connection between the crystal orientations of parent grains and those of child grains is usually expressed in terms of (theoretical or measured) orientation relationships. This paper introduces a new approach to various problems associated with orientation relationships: (i) estimatio...

To be a quantitative and testable tectonic model, plate tectonics requires spherical geometry and spherical kinematics in terms of finite rotations conveniently parametrized by their angle and axis and described by unit quaternions. In treatises on ’Plate Tectonics’ infinitesimal, instantaneous, and finite rotations, absolute and relative rotations...

The orientation of a crystal can be described by a rotation that transforms it into some standard position. Because crystals tend to be symmetrical, this rotation is known only up to multiplication by an element of the symmetry group. The first part of this paper summarises the embedding approach to studying crystallographic orientations. The secon...

Regression or regression-like models are often employed in mineral prospectivity modeling, i.e., for the targeting of resources, either based on 2D map images or 3D geomodels both in raster mode or based on spatial point processes. Machine learning techniques like artificial neural networks are often applied and give decent results in the predictio...

Revisiting a spiral design for X-ray pole figure measurementsand a symbolic definition of a cumulative crystallographic orientation distribution a one-dimensional deterministic approximately uniform sequential design is appliedto evaluate and cumulate a given orientation density function resulting in a properly definedcumulative crystallographic or...

Regression models are often employed in prospectivity modeling for the targeting of resources. Logistic regression has a well understood statistical foundation and uses an explicit model from which knowledge can be gained about the underlying phenomenon. In this paper, a model selection procedure based on logistic regression enhanced with nonlinear...

Regression models are often employed in prospectivity modeling for the targeting of resources. Logistic regression has a well understood statistical foundation and uses an explicit model from which knowledge can be gained about the underlying phenomenon. In this paper, a model selection procedure based on logistic regression enhanced with nonlinear...

As analytical tools evolve, more geochemical data are obtained so that effective and robust tools are demanded for capturing detailed information of underlying geological processes such as alteration, mineralization and weathering. The data are formed by a number of elements, whose concentrations are interdependent due to the fact that their total...

Regression or regression‐like models are often employed in potential modeling, i.e., for the targeting of resources, either based on 2D map images or 3D geomodels both in raster mode or based on spatial point processes. Recently, machine learning techniques such as artificial neural networks have gained popularity also in potential modeling. Using...

Currently tailings deposits have become new resources that are challenging and valuable to exploit. To properly exploit them, we require a 3D spatial characterization of their mineral content. In a natural deposit, this is achieved by sampling at several locations and applying geostatistics to estimate block values. Certain characteristics of taili...

The majority of popular methods of prospectivity modeling refer to the raster mode of digital two-dimensional map images or three-dimensional geomodels, thus requiring a user decision concerning the spatial resolution right from the beginning, imposing a constraint on all subsequent computations . In contrast to geostatistics and its variogram or c...

Geological interpretation plays a crucial role in every phase of subsurface characterization from exploration to exploitation, e.g. of an oil reservoir or a mineral deposit. In general, the distribution of physical properties is controlled by the architecture of geological objects. Therefore, defining it becomes the initial step of geological model...

Technical and economic evolution of the mineral industry resulted in a new view of mining tailings. Formerly tailings are considered not valuable but now they become new resources that have promising economic values. The spatial estimation of mineral distribution is essential for optimally exploiting tailings, but this faces several issues such as...

While tests for pairwise conditional independence of random variables have been devised, testing joint conditional independence of several random variables seems to be a challenge in general. Restriction to categorical random variables implies in particular that their common distribution may initially be thought of as contingency table, and then in...

To be useful in numerical simulations of e.g. deformation processes, EBSD datasets of crystallographic orientations have to be downsized by several orders of magnitude yet preserving the orientation density function approximately. The objective is either to preserve the overall shape of the initially kernel estimated orientation density function an...

The orientation of a rigid object can be described by a rotation that transforms it into a standard position. For a symmetrical object the rotation is known only up to multiplication by an element of the symmetry group. Such ambiguous rotations arise in biomechanics, crystallography and seismology. We develop methods for analyzing data of this form...

The objective of prospectivity modeling is prediction of the conditional probability of the presence T = 1 or absence T = 0 of a target T given favorable or prohibitive predictors B, or construction of a two classes 0,1 classification of T. A special case of logistic regression called weights-of-evidence (WofE) is geologists’ favorite method of pro...

This paper presents the background for the calculation of anisotropic piezoelectric properties of single crystals and the graphical display of the results in two or three dimensions, and the calculation of the aggregate properties from constituent crystals and the texture of the aggregate in a coherent manner. The texture data can be obtained from...

A task of the EU funded project ProMine “Nano-particle products from new mineral resources in Europe” coordinated by the Geological Survey of Finland (GTK) is spatial geomodelling of mineralized belts and development of spatio-temporal geomodels. An essential prerequisite is a genuine data model to manage geoscience data and geoscience models effic...

This contribution is an attempt to present a self-contained and comprehensive survey of the mathematics and physics of the material behavior of rocks in terms of texture and anisotropy. Being generally multiphase and polycrystalline, where each single crystallite is anisotropic with respect to its physical properties, texture, i.e., the statistical...

Logistic regression is a classical linear model for logit-transformed conditional probabilities of a binary target variable. It recovers the true conditional probabilities if the joint distribution of predictors and the target is of log-linear form. Weights-of-evidence is an ordinary logistic regression with parameters equal to the differences of t...

In plate tectonics each Euler pole has to be estimated from geological field data , where is a point on the sphere, , and is an axis in the tangent plane at . The data can be viewed as regression data, in which the positions are values of a predictor on , while the corresponding tangential axes are responses in the tangent plane to . A regression m...

Despite a missing definition of equivalence of mathematical models or methods by Zhang et al. (Math Geosci, 2013), an “equivalence” (Zhang et al., Math Geosci, 2013, p. 6,7,8,14) of modified weights-of-evidence (Agterberg, Nat Resour Res 20:95–101, 2011) and logistic regression does not generally exist. Its alleged proof is based on a previously co...

Many different versions of geoscience data concurrently exist in a database for different geological paradigms, source data, and authors. The aim of this study is to manage these versions in a database management system. Our data include geological surfaces, which are triangulated meshes in this study. Unlike revision/version/source control systems...

This paper presents the background for the calculation of various numbers that can be used to characterize crystal-preferred orientation (CPO), also known as texture in materials science, for large datasets using the combined scripting possibilities of MTEX and MatLabw. The paper is focused on three aspects in particular: the strength of CPO repres...

New light is shed on mathematical methods of potential modeling from the point of view of Markov random fields. In particular, weights-of-evidence and logistic regression models are discussed in terms of graphical models possessing Markov properties, where the notion of conditional independence is essential, and will be related to log-linear models...

40Ar/39Ar dating of young (<1 Ma) and/or low potassium samples requires degassing of large amounts of material either by a laser or a furnace. A furnace excels in precise and reproducible temperature control and in providing a homogeneous temperature distribution even for relatively large samples amounts. For most systems, however, the degassed min...

Weights-of-evidence is the special case of logistic regression, if the predictor variables are conditionally independent indicator variables given the target variable. In this case the contrasts of the weights of evidence are identical to the parameters of the corresponding logistic regression model. If the modeling assumption of conditional indepe...

A model for spatially and temporally indexed multi-dimensional geoscience data has been developed by first embedding a combinatorial topological model in terms of G-Maps in the domain Rm×TimeRm×Time(m∈N)(m∈N), and then converting it into an object-relational model which can easily be implemented in an object-relational database system.Geoscience ob...

Digital geological applications have become 3D up to 4D modelling of the underground. The modellers are working very heterogeneously in terms of its applied software systems. On the other hand the 3D/4D modelling of the subsurface has become part of the geological surveys all around the world. This implies a wide spread group of users working in di...

Mathematical methods of potential modeling are considered from the point of view of generalized linear models and Markov stochastic graphs. In particular, weights-of-evidence and logistic regression models will be discussed in terms of stochastic graphs possessing a Markov property, where the notion of conditional independence is essential, and wil...

GST - Geosciences in space and time is being developed and implemented as hub to facilitate the exchange of spatially and temporally indexed multi-dimensional geoscience data and corresponding geomodels amongst partners. It originates from TUBAF's contribution to the EU project "ProMine" and its perspective extensions are TUBAF's contribution to th...

Various attempts are known to turn the “catalogue” of mineral deposit models compiled by Cox and Singer (1986) operational, and have initiated activities called “potential mapping”, “potential modeling”, or “targeting”. The common ultimate objective is to estimate the probability for a given location that a mineralization of a given type occurred....

Pole figure measurements with an X‐ray texture goniometer equipped with a point detector are rather time consuming: depending on the angular resolution to be recorded, of the order of several hours per pole figure. Conventionally, the pole hemisphere is scanned along latitudinal small circles according to a regular grid of constant step sizes in bo...

ProMine is a European Union (EU) co-funded project, focused on “Innovative concepts and processes for strategic mineral supply and for new high added value mineral-based products”. It started in 2009, for a four years period, and involves 27 partners from 11 countries. A first major result from the project is the release of the collaboratively deve...

A new measurement strategy is introduced to determine very pronounced crystallographic textures of polycrystalline materials most accurately and efficiently. It features optimized adaptively refined successive X-ray pole figure measurements. To optimize both the total experimental time as well as the angular resolution, and thus the accuracy of pol...

3D models offer reliable and realistic reproduction of structures of the underground. They are used in many applications: e. g. reservoir modelling, prospective, numeric simulations, extractions planning etc. While working on complex projects various software are used for the creation of the model, the prediction of properties and final simulation....

1 This paper presents the theoretical background for the calculation of physical 2 properties of an aggregate from constituent crystal properties and the texture of the 3 aggregate in a coherent manner. Emphasis is placed on the important tensor proper-4 ties of 2 nd and 4 th rank with applications in rock deformation, structural geology, geo-5 dyn...

In crystallography, a probability density function f(g) defined on the group SO(3) of rotations is referred to as orientation density function; the analysis of preferred crystallographic orientation is referred to as texture analysis. In nondestrcutive practical applications it is common practice to measure diffraction pole figures $ \tilde P(h,r)...

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This contribution is an attempt to present a self-contained and comprehensive survey of the mathematics and physics of material behavior of rocks in terms of texture and anisotropy. Being generally multi-phase and poly-crystalline, where each single crystallite is anisotropic with respect to its physical properties, texture, i.e., the statistical a...

While crystallography conventionally presumes that a single crystal carries a unique crystallographic orientation, modern experimental techniques reveal that a single crystal may exhibit an orientation distribution. However, this distribution is largely concentrated; it is extremely concentrated when compared with orientation distributions of polyc...

The MATLAB™ toolbox MTEX provides a unique way to represent, analyse and interpret crystallographic preferred orientation, i.e. texture, based on integral ("pole figure") or individual orientation ("EBSD") measurements. In particular, MTEX comprises functions to import, analyse and visualize diffraction pole figure data as well as EBSD data, to est...

The generalized totally geodesic Radon transform associates the mean values over spherical tori to a function f defined on 3⊂ℍ, where the elements of 3 are considered as quaternions representing rotations. It is introduced into the analysis of crystallographic preferred orientation and identified with the probability density function corresponding...

Patterns of crystallographic preferred orientation are referred to as texture. The specific subject of texture analysis is the experimental determination and interpretation of the statistical distribution of orientations of crystals within a specimen of polycrystalline material, which could be metals or rocks. The objective is to relate an observed...

A novel algorithm for ODF (orientation density function) estimation from diffraction pole figures is presented which is especially well suited for sharp textures and high-resolution pole figures measured with respect to arbitrarily scattered specimen directions, e.g. by area detectors. The estimated ODF is computed as the solution of a minimization...

The inversion of the one-dimensional Radon transform on the rotation group SO(3) is an ill-posed inverse problem which applies to x-ray tomography with polycrystalline materials. This paper presents a novel approach to the numerical inversion of the one-dimensional Radon transform on SO(3). Based on a Fourier slice theorem the discrete inverse Rado...

There are a number of deformation mechanisms that involve mixing of particles of different phases. For example, in cataclastic flow, particles are fragmented and displaced past each other, in diffusion creep, grains of one phase nucleate and grow between grains of other phases. The resulting mixtures may form "random", "clustered" or "anticlustered...

A novel control of a texture goniometer, which depends on the texture being measured itself, is suggested. In particular, it is suggested that the obsolete control with constant step sizes in both angles is replaced by an adaptive successive refinement of an initial coarse uniform grid to a locally refined grid, where the progressive refinement cor...

This communication demonstrates a sharp inequality between the L^{2}-norm and the entropy of probability density functions. This inequality is applied to texture analysis, and the relationship between the entropy and the texture index of an orientation density function is characterized. More precisely, the orientation space is shown to allow for te...

Texture analysis is used here as short term for analysis of crystallographic preferred orientation. Its major mathematical objective is the determination of a reasonable orientation probability density function and corresponding crystallographic axes probability density functions from experimentally accessible diffracted radiation intensity data. S...

1. Abstract Due to a novel method of pole intensity to orientation density inversion applying radial basis functions and an implementation employing fast Fourier transform algorithms for , 2 S 2 2 × S S , and SO(3) it is now possible to process intensity data arbitrarily scattered on the pole sphere and successively compute numerical approximations...

Resolving the inverse spherical Radon transform problem R-1 : Rf 7→ f for an orientation probability density function f : SO(3) 7→ R+ is equivalent to solving the corresponding Darboux-type (ultrahyperbolic) differential equation (Δh - Δr)Rf = 0 governing the spherical Radon transform Rf. The general solution of the latter is known both in terms of...

3–dimensional (3d) automatic crystal orientation mapping (ACOM) allows to sam-ple parallel 2d section raster images which represent the orientation of crystal grains. The fabric, i.e. the partition of the specimen into 3d crystal grains, can be approximated given its intersections with parallel planes provided by series of 2–dimensional raster orie...

In texture analysis intensities of a diffraction experiment are recorded as experimentally accessible pole figures which are modeled as a "pole figure projection" of an orientation probability density function. The recovery of the orientation density function by means of inversion of the pole figure projection has been a major issue ever since its...

The representations of both orientation and pole density functions in terms of spherical wavelets are exposed and their properties and relationship are discussed. The main idea of wavelet analysis is to obtain a multiscale representation of the functions or the data which allows localization in space and frequency. This mainstream idea of modern da...

The radial basis function method is applied to reconstruct the orientation density function and the;pole density function from any arbitrary set of X-ray intensities. The mathematical foundations are given as well as numerical results.

Pole point plots are generalized to augmented pole point plots visualizing a second crystallographic direction subjected to a (random) orientation. Then ω-sections of the space of crystallographic orientations generalizing both γ- as well as σ-sections are introduced as augmented pole point plots ordered according to the orientation distance betwee...

This communication compiles propositions concerning the spherical geometry of rotations when represented by unit quaternions. The propositions are thought to establish a two-way correspondence between geometrical objects like circle, sphere, and torus spanned by real unit quaternions representing rotations and geometrical objects constituted by dir...

Spherical geometry of quaternions representing rotations is employed to provide a unique correspondence between distinguished cases of the Bingham distribution on the 3-dimensional sphere S3 of rotations and a classification of ideal textures, i.e. patterns of preferred crystallographic orientations. It is shown that the Bingham distribution can re...

We consider a one-dimensional Radon transform on the group SO(3) which is motivated by texture goniometry. In particular we will derive several inversion formulae and compare them with the inversion of the one-dimensional spherical Radon transform on the three dimensional unit sphere for even functions. Comment: 20 pages

The generalized spherical Radon transform associates the mean values over spherical tori to a function $f$ defined on $\mathbb{S}^3 \subset \mathbb{H}$, where the elements of $\mathbb{S}^3$ are considered as quaternions representing rotations. It is introduced into the analysis of crystallographic preferred orientation and identified with the proba...

This communication compiles propositions concerning the spherical geometry of rotations when represented by unit quaternions. The propositions are thought to establish a two-way correspondence between geometrical objects in the space of real unit quaternions representing rotations and geometrical objects constituted by directions in the three-dimen...

(g f with known mean 1. The measured diffraction intensities P(h,r) are mean values of ) (g f along geodesic fibers } : { ) , ( r gh g r h B = = or corresponding tube shaped blocks in the domain of the random field. We propose a kind of block kriging to interpolate to unmeasured P(h,r). The method surpasses other methods in simplicity and generalit...

This communication compiles propositions concerning the spherical geometry of rotations when represented by unit quaternions. The propositions are thought to establish a two-way correspondence between geometrical objects like circle, sphere, and torus in the 4 dimensional space of real unit quaternions representing rotations and geometrical objects...

In our contribution, we present the definitions of the spherical, generalized spherical and dual spherical Radon transforms as they haven proven useful in applied analysis of preferred crystallographic orientation. We mention several inverse formulae originating in material sciences, mathematical tomography, and integral geometry. Then we focus on...

Spherical geometry of quaternions is employed to characterize the Bingham distribution on the 3-dimensional sphere \mathbbS3 Ì \mathbbR4 {\mathbb{S}}^3 \subset {\mathbb{R}}^4 as being uniquely composed of a bipolar, a circular and a spherical component. A new parametrization of its dispersion parameters provides a classification of patterns of c...

Since the domain of crystallographic orientations is three-dimensional and spherical, insightful visualization of them or visualization of related probability density functions requires (i) exploitation of the effect of a given orientation on the crystallographic axes, (ii) consideration of spherical means of the orientation probability density fun...

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The objective of this contribution is to emphasize the fundamental role of spherical harmonics in constructive approximation on the sphere in general and in texture analysis in particular. The specific purpose is to present some methods of texture analysis and pole-to-orientation probability density inversion in a unifying approach, i.e. to show th...

Nach der Bestandsaufnahme durch G. Teege [1] im August-Heft des Informatik-Spektrums möchten wir mit diesem Beitrag die Perspektiven geowissenschaftlicher Informationssysteme skizzieren.

Since the domain of crystallographic orientations is three-dimensional and spherical, their in- sightful rendering or rendering of related probability density functions requires (i) to exploit the eect of a given orientation on crystallographic axes, (ii) to consider spherical means of the orientation probability density function with respect to lo...

The general objective of this communication is to present the basics of mathematical texture analysis as part of integral geometry involving spherical analogues of the X-ray and Radon transform, and in particular to clarify its relationship with mathematical tomography. Copyright © 2002 John Wiley & Sons, Ltd.

The problem to predict a rotation (orientation) from corresponding geocoded data is discussed and a general solution by virtue of embedding the group of rotations in a real vector space is presented. It is referred to as kriging in embedding spaces as developed in part I of this contribution, and basically the same arguments apply and lead to equiv...

The problem to predict a direction, axis, or orientation (rotation) from corresponding geocoded data is discussed and a general solution by virtue of embedding a sphere/hemisphere in a real vector space is presented. Its explicit justification in terms of mathematical assumptions concerning stationarity/homogeneity and isotropy is included. The dat...

The crystal size distribution model that is now in frequent use in the geosciences was first introduced by [Marsh (1988)][1] and by [Cashman and Marsh (1988)][2]. In a recent paper, however, [Pan (2001)][3] questioned the validity of this model and concluded that there is an inherited correlation of