This repository was created to perform the estimation of variogram and correlation models, which are essential for many engineering uses. In particular, variogram and correlation models form the backbone for Kriging, by which values can be estimated at unknown locations. There are several available packages (see References), whilst some are thorough and fairly robust, they are largely niche in most cases and not tailored to specific use cases without additional modifications. In this regard, this repository focuses mainly on engineering applications such as:

1. Spatial Correlation Models (SCMs) for parameter and residual estimation (e.g., Vs30, $\delta W$)
2. Kriging Estimates of values at unknown locations
3. Sequential Gaussian Simulation (SGS) for statistically robust estimates at unknown locations

Refer to Theory for getting a quick overview of the process for estimating semivariance and correlation whereby models are developed. Theory also provides an overview of the available kernels and some of the underlying assumptions required to conduct robust geostatistical analyses.

1. Clone the repository:

   git clone https://github.com/AaronJR474/VarioCorreKrigE.git
   cd VarioCorreKrigE

2. Install the required dependencies:

   pip install -r requirements.txt

   AND Optional for Bayesian Markov Chain Monte Carlo (MCMC) analysis:

   pip install numpyro >= 0.18.0

pip install git+https://github.com/AaronJR474/VarioCorreKrigE.git

OR Optional for Bayesian Markov Chain Monte Carlo (MCMC) analysis:

pip install "git+https://github.com/AaronJR474/VarioCorreKrigE.git#egg=VarioCorreKrigE[bayesmcmc]"

The package is broken down into three main functions as follows:

1. variofit.py: Estimation of Semivariance and fitting of variogram models (including bootstrapping: "pair" or "point" resampling)

   from VarioCorreKrigE.variofit import variofit, variofitmulti
   from VarioCorreKrigE.variofit import VARIOGRAM_MODELS

   

2. correfit.py: Estimation of correlation coefficients and fitting of correlation models (including bootstrapping: "pair" or "point" resampling)

   from VarioCorreKrigE.correfit import correfit, correfitmulti
   from VarioCorreKrigE.correfit import CORRELATION_MODELS

   

3. Estimation of cross-variograms and cross-correlograms

   from VarioCorreKrigE.variofit import crossvariofit, multicrossvariofit
   from VarioCorreKrigE.correfit import crosscorrefit, multicrosscorrefit

   

4. skrig.py: Estimation of Simple Kriging with predefined or custom models (including Sequential Gaussian Simulation)

   from VarioCorreKrigE.skrig import simple_kriging, simple_kriging_custom_corr, sgs_simple_kriging, sgs_simple_kriging_custom_corr
   from VarioCorreKrigE.correfit import CORRELATION_MODELS
   from VarioCorreKrigE.variofit import VARIOGRAM_MODELS

   

5. okrig.py: Estimation of Ordinary Kriging with predefined or custom models

   from VarioCorreKrigE.skrig import ordinary_kriging, ordinary_kriging_custom_corr
   from VarioCorreKrigE.correfit import CORRELATION_MODELS
   from VarioCorreKrigE.variofit import VARIOGRAM_MODELS

6. cckrig.py: Estimation of Cokriging (Simple and Ordinary Kriging variants) as well as Co-located kriging using Markov 1 and 2 Models

   from VarioCorreKrigE.cckrig import simple_cokriging, ordinary_cokriging, SCCK_MM1, SCCK_MM2, OCCK_MM1, OCCK_MM2, ICCK_MM1, ICCK_MM2, OICCK_MM1, OICCK_MM2

   

7. bayes_krig.py: Empirical Bayesian Kriging (EBK) using either bootstrapped variograms/correlograms or posterior from Bayesian MCMC (see: VarioCorreKrige_bayesmcmc_example.ipynb)

   from VarioCorreKrigE.bayes_krig import bayes_krig

   

In the folder Examples, there are detailed examples for each of the mentioned functions, including the ability to utilize custom correlation/variogram models not currently within the package. The examples also demonstrate the creation of custom covariance matrices and corresponding pairwise distances for a user-tailored Kriging analysis.

Future versions will be extended to include:

• Directional/Anisotropic variograms/correlograms
• Linear Model of Coregionalization fitting toolbox
• Universal Kriging
• Bayesian Updating for Combining Conditional Distributions

Myers, Donald. (1997). Multivariate geostatistics By Hans Wackernagel. Mathematical Geology. 29. 307-310. 10.1007/BF02769635.

Pyrcz, M.J., 2024, Applied Geostatistics in Python: a Hands-on Guide with GeostatsPy [e-book]. Zenodo. doi:10.5281/zenodo.15169133

Deutsch, J., 2015–2025, Geostatistics Lessons [online]. Available at: https://geostatisticslessons.com/.

Mirko Mälicke, Egil Möller, Helge David Schneider, & Sebastian Müller. (2021, May 28). mmaelicke/scikit-gstat: A scipy flavoured geostatistical variogram analysis toolbox (Version v0.6.0). Zenodo. http://doi.org/10.5281/zenodo.4835779

Müller, S., Schüler, L., Zech, A., and Heße, F.: GSTools v1.3: a toolbox for geostatistical modelling in Python, Geosci. Model Dev., 15, 3161–3182, https://doi.org/10.5194/gmd-15-3161-2022, 2022.

This project is provided “as is” without any warranties or guarantees of any kind. Use of this code is at your own risk. The authors and contributors are not liable for any damages, data loss, or issues arising from the use, modification, or distribution of this software.

This repository may contain experimental, incomplete, or evolving features. Nothing here should be considered production‑ready unless explicitly stated.

You are responsible for reviewing and complying with all relevant licenses, data‑use agreements, and third‑party dependencies referenced in this project.