Fractals and Multifractals in the Geosciences

1. The theory of fractal geometry

1.1 Introduction to fractal geometry in nature and its difference with Euclidean geometry

1.2 Multi-fractal spectrums

1.2.1 Mono and Multi-Fractals

1.2.2 Continuous multi-fractals

1.2.3 Discrete multi-fractals

1.3 Multi-fractal distribution patterns in geochemical mineralisation

1.4 Fractal dimension and Hurst exponent

1.5 Power low frequency

1.6 Self-similarity and self-affinity


2. Introduction: characterising and mapping anomalies

2.1 Introduction

2.1.1 Population models

2.1.2 Spatial models

2.1.3 Traditional fractal and multi-fractal modelling in geochemical studies (with examples)

2.1.3.1 Number-size (N-S) fractal model

2.1.3.2 Concentration-area (C-A) fractal model

2.1.3.3 Perimeter-Area (P-A) fractal model

2.1.3.4 Concentration-distance (C-D) fractal model

2.1.3.5 spectrum-area (S-A) fractal model

2.1.3.6 Singularity model

2.1.3.7 Box-counting model

2.1.3.8 Concentration-volume (C-V) fractal model

2.1.3.9 Spectrum-volume (S-V) fractal model

2.1.3.10 Stochastic simulation methods applicable to fractal models

2.1.3.10.1 Lower-upper simulation (LUSIM)

2.1.3.10.2 Direct sequential simulation (DSSIM)

2.1.3.10.3 Sequential indicator simulation (SISIM)

2.1.3.10.4 Sequential Gaussian simulation (SGSIM)

2.1.3.10.5 Turning bands simulation (TBSIM)

2.1.3.11 Simulated-fractal models

2.1.3.11.1 Simulated size-number (SS-N) fractal model

2.1.3.11.2 Global Simulated size-number (GSS-N) fractal model

2.1.3.11.3 Co-simulated Size Number (CoSS-N) fractal model

2.2 Uncertainties in geochemical modelling

2.3 Problem statement

2.4 Aims and approach


3. Geochemical datasets

3.1 Overview

3.2 The till geochemical atlas of Sweden

3.2.1 Geology and mineralisation of Sweden

3.2.2 Quaternary glacial deposits and topography

3.2.3 Till geochemistry atlas

3.3 The soil geochemical atlas of Cyprus

3.3.1 Geology of Cyprus

3.3.2 Soil geochemistry

3.3.3 Urban geochemical map of Lemesos

3.4 General data processing

3.4.1 Log-ratio transformation


4. Novel fractal classification models

4.1 Introduction

4.2 Concentration-distance from centroids (C-DC) fractal model 304.2.1 C-DC calculations

4.3 Application of C-C and C-DC modelling: Enhancing the detection of Cu deposits in Cyprus

4.4.1 Data processing

4.4.2 Results and discussion

4.5 Application of C-C modelling: Separating anthropogenic from geogenic patterns in soils of Lemesos

4.5.1 Data processing

4.5.2 Results and discussion

4.6 Simulated fractal models

4.7 Category-based fractal modelling

4.7.1 Category-based fractal modelling based on the related bedrocks representative samples

4.7.2 Category-based fractal modelling based on the related bedrocks whole simulated samples


5. Effectiveness and uncertainty in geochemical anomaly classification models

5.1 Introduction

5.2 Comparison of model efficiencies

5. Monte Carlo simulation (MCSIM) applied to the characterised populations to quantify the thresholds’ uncertainties

5.3.1 Results and discussion

5.4 The Cyprus Soil Atlas

5.4.1 Results and discussion


6. Spatial uncertainty in categorising geological data

6.1 Introduction

6.2 Data analysis

6.2.1 Swedish till data

6.2.2 Cyprus case study


7. General discussion and conclusions

7.1 Discussion

7.1.1 Introduction

7.1.2 Performance of the developed classification models based on their systematic and spatial uncertainties

7.2 Conclusions

7.3 Future Work

7.3.1 3D modelling in depth

7.3.2 C-C and SC-SC fractal models in urban geology

7.3.3 Decision-making

7.3.4 Simulations applicable

7.3.5 Systematic uncertainty of the other classification models

7.3.6 Other multi-variate analysis approaches