Mathematical Geosciences

, Volume 51, Issue 1, pp 29–51 | Cite as

Entropy-Based Weighting in One-Dimensional Multiple Errors Analysis of Geological Contacts to Model Geological Structure

  • Weisheng HouEmail author
  • Chanjie Cui
  • Liang Yang
  • Qiaochu Yang
  • Keith Clarke


In each step of geological modeling, errors have an impact on measurements and workflow processes and, so, have consequences that challenge accurate three-dimensional geological modeling. In the context of classical error theory, for now, only spatial positional error is considered, acknowledging that temporal, attribute, and ontological errors—and many others—are part of the complete error budget. Existing methods usually assumed that a single error distribution (Gaussian) exists across all kinds of spatial data. Yet, across, and even within, different kinds of raw data (such as borehole logs, user-defined geological sections, and geological maps), different types of positional error distributions may exist. Most statistical methods make a priori assumptions about error distributions that impact their explanatory power. Consequently, analyzing errors in multi-source and conflated data for geological modeling remains a grand challenge in geological modeling. In this study, a novel approach is presented regarding the analysis of one-dimensional multiple errors in the raw data used for model geological structures. The analysis is based on the relationship between spatial error distributions and different geological attributes. By assuming that the contact points of a geological subsurface are decided by the geological attributes related to both sides of the subsurface, this assumption means that the spatial error of geological contacts can be transferred into specific probabilities of all the related geological attributes at each three-dimensional point, which is termed the “geological attribute probability”. Both a normal distribution and a continuous uniform distribution were transferred into geological attribute probabilities, allowing different kinds of spatial error distributions to be summed directly after the transformation. On cross-points with multiple raw data with errors that follow different kinds of distributions, an entropy-based weight was given to each type of data to calculate the final probabilities. The weighting value at each point in space is decided by the related geological attribute probabilities. In a test application that accounted for the best estimates of geological contacts, the experimental results showed the following: (1) for line segments, the band shape of geological attribute probabilities matched that of existing error models; and (2) the geological attribute probabilities directly show the error distribution and are an effective way of describing multiple error distributions among the input data.


Multiple error distribution Entropy-based weight Geological attribute probability 



The authors greatly appreciate the valuable comments of the anonymous reviewers in the refinements of this manuscript. This study was substantially supported by the NSFC Program (41472300, 41502068, and 41102207), the Fundamental Research Funds for the Central Universities (12lgpy19, 15lgjc45), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (RFDP) (20100171120001). The research was performed while the first author was a visiting research scholar at the Department of Geography, University of California, Santa Barbara, and the department’s support is gratefully acknowledged.


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Copyright information

© International Association for Mathematical Geosciences 2018

Authors and Affiliations

  • Weisheng Hou
    • 1
    • 2
    Email author
  • Chanjie Cui
    • 1
    • 2
  • Liang Yang
    • 1
    • 3
  • Qiaochu Yang
    • 1
    • 2
  • Keith Clarke
    • 4
  1. 1.School of Earth Sciences and EngineeringSun Yat-sen UniversityGuangzhouChina
  2. 2.Guangdong Provincial Key Laboratory of Mineral Resource Exploration and Geological ProcessesGuangzhouChina
  3. 3.Geological SciencesStanford UniversityStanfordUSA
  4. 4.Department of Geography, 1720 Ellison HallUniversity of California, Santa BarbaraSanta BarbaraUSA

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