Skip to main content

Maximum Entropy Method

  • Living reference work entry
  • First Online:
Encyclopedia of Mathematical Geosciences


The principle of maximum entropy states that the most suitable probability model for a given system maximizes the Shannon entropy subject to the constraints imposed by the data and – if available – other prior knowledge of the system. The maximum entropy distribution is the most general probability distribution function conditionally on the constraints. In the geosciences, the principle of maximum entropy is mainly used in two ways: (1) in the maximum entropy method (MEM) for the parametric estimation of the power spectrum and (2) for constructing joint probability models suitable for spatial and spatiotemporal datasets.


The concept of entropy was introduced in thermodynamics by the German physicist Rudolf Clausius in the nineteenth century. Clausius used entropy to measure the thermal energy of a machine per unit temperature which cannot be used to generate useful work. The Austrian physicist Ludwig Boltzmann used entropy in statistical mechanics to quantify the ran...

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Institutional subscriptions



maximum entropy method


maximum entropy


Bayesian Maximum entropy


  • Burg JP (1972) The relationship between maximum entropy spectra and maximum likelihood spectra. Geophysics 37(2):375–376

    Article  Google Scholar 

  • Christakos G (1990) A Bayesian/maximum-entropy view to the spatial estimation problem. Math Geol 22(7):763–777

    Article  Google Scholar 

  • Christakos G (2000) Modern spatiotemporal geostatistics, International Association for Mathematical Geology Studies in mathematical geology, vol 6. Oxford University Press, Oxford

    Google Scholar 

  • Hristopulos DT (2020) Random fields for spatial data modeling: a primer for scientists and engineers. Springer Netherlands, Dordrecht

    Book  Google Scholar 

  • Jaynes ET (1957a) Information theory and statistical mechanics. I. Phys Rev 106(4):620–630

    Article  Google Scholar 

  • Jaynes ET (1957b) Information theory and statistical mechanics. II. Phys Rev 108(2):171–190

    Article  Google Scholar 

  • Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27(3):379–423

    Article  Google Scholar 

  • Skilling J (2013) Maximum entropy and Bayesian methods: Cambridge, England, 1988, vol 36. Springer Science & Business Media, Cham

    Google Scholar 

  • Skilling J, Bryan R (1984) Maximum entropy image reconstruction-general algorithm. Mon Not R Astron Soc 211:111–124

    Article  Google Scholar 

  • Ulrych TJ, Bishop TN (1975) Maximum entropy spectral analysis and autoregressive decomposition. Rev Geophys 13(1):183–200

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Dionissios T. Hristopulos .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Hristopulos, D.T., Varouchakis, E.A. (2021). Maximum Entropy Method. In: Daya Sagar, B., Cheng, Q., McKinley, J., Agterberg, F. (eds) Encyclopedia of Mathematical Geosciences. Encyclopedia of Earth Sciences Series. Springer, Cham.

Download citation

  • DOI:

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-26050-7

  • Online ISBN: 978-3-030-26050-7

  • eBook Packages: Springer Reference Earth and Environm. ScienceReference Module Physical and Materials ScienceReference Module Earth and Environmental Sciences

Publish with us

Policies and ethics