Skip to main content

Gaussian Process Regression Reviewed in the Context of Inverse Theory

Abstract

We review Gaussian process regression (GPR) and analyze it in the context of Inverse Theory—the collection of techniques used in geophysics (among other fields) to understand the structure of data analysis problems and the quality of their solutions. By viewing GPR as a special case of generalized least squares (least squares with prior information), we derive expressions for a variety of standard Inverse Theory quantities, including the data and model resolution matrices, the importance (influence) vector, and the gradient of the solution with respect to a parameter. We study the impulse response in the one-dimensional continuum limit and provide formulas for its area and width. Finally, we demonstrate how the importance vector can be used to design an optimum GPR experiment, through a process we call importance winnowing.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Data Availability

This paper contains no data.

References

Download references

Acknowledgements

This research was partially supported by the National Science Foundation under Grant EAR 20-02352.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to William Menke.

Ethics declarations

Conflict of interest

The authors declare no conflicts of interest with respect to this work.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Menke, W., Creel, R. Gaussian Process Regression Reviewed in the Context of Inverse Theory. Surv Geophys 42, 473–503 (2021). https://doi.org/10.1007/s10712-021-09640-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10712-021-09640-w

Keywords

  • Gaussian process regression
  • Geostatistics
  • Importance
  • Influence
  • Interpolation
  • Inverse Theory
  • Least Squares
  • Resolution