EMWTINV is a Python tool for performing Hybrid Bayesian Inversion (HBI) for hydrogeophysical data interpretation.

Hybrid Bayesian Inversion (HBI) is a Bayesian extension of Coupled Hydrogeophysical Inversion (CHI) that enables efficient estimation of hydrological properties from geophysical data. It can integrates petrophysical relationships and hydrological models with significant uncertainty in calibration parameters.

HBI decomposes the subsurface model into two components:

• Groundwater Model:

  • Predictable using petrophysical and hydrological modeling
  • Characterized by posterior probability density functions (PDFs)

• Background Model:

  • Not predictable from petrophysical or hydrological models
  • Represented by deterministic resistivity structures (1D/2D/3D)
  • Its uncertainty is not characterized

This decomposition allows computationally expensive Bayesian inference to be applied only to the groundwater component (which is lower-dimensional), while the background model is handled using conventional least-squares inversion.

To jointly estimate:

• The posterior PDFs of the Groundwater Model
• The maximum likelihood estimate (MLE) of the Background Model

HBI applies the Expectation-Maximization (EM) algorithm, where each iteration executes the following sequence:

• E-step: Bayesian inversion of the groundwater model using Metropolis-Hastings sampling.
• M-step: Least-squares inversion of the background model.

EMWTINV is executed via the emwtinv_multi.py script. Example applications are located in the /Script directory.

• /Source: Source code
• /Script: Example scripts for applying EMWTINV
• /Utils: Tools for plotting and analyzing results

Before running, place the following files in the same directory as emwtinv_multi.py:

Defines input data, geophysical method, E-step and M-step parameters.

📁 Geophysical Data

• Data_list: Path to the observation file with geophysical data.
• Method: Geophysical method to be used. Options include:
  • DC2D – Electrical Resistivity Tomography (2D)
  • DC1D – Electrical Vertical Sounding
  • TEM1D – Transient Electromagnetics (1D)
  • MT1D – Magnetotellurics (1D, not tested)
  • GRAV2D – Gravity (2D, not tested)

🔧 M-STEP Parameters (Traditional Inversion of Background Model)

• externalmodel: Initial model flag (True/False)
• external_constant_model: Use a halfspace model as initial model (True/False)
• externalmodel_ref: Reference model flag (True/False)
• model_file: Path to the initial background model file (if externalmodel is True)
• model_mesh: Mesh file used for the initial background model
• modelref_file: Path to the reference model used during M-step regularization
• modelref_mesh: Mesh file used for the reference model
• invert_initial_bkmodel: Use traditional inversion to estimate the initial background model (True/False)
• reference_value: Value used for halfspace model when external_constant_model is True
• sub_lmd: Not used
• emmax_iter: Number of HBI iterations

🧮 SimPEG Regularization Parameters for Initial Background Model

• invparameters0.Collingrate: Number of iterations to reduce beta
• invparameters0.Beta: Trade-off factor
• invparameters0.Minbeta: Minimum beta
• invparameters0.Collingfactor: Factor to reduce beta
• invparameters0.alpha_x/y/z/s: Regularization weights for gradients and model smallness
• invparameters0.chifact: Misfit target

🧮 SimPEG Regularization Parameters for Background Model (M-Step)

• invparameters.Collingrate: Number of iterations to reduce beta
• invparameters.Beta: Trade-off factor
• invparameters.Minbeta: Minimum beta
• invparameters.Collingfactor: Factor to reduce beta
• invparameters.alpha_x/y/z/s: Regularization weights for gradients and model smallness
• invparameters.chifact: Misfit target

🔄 E-STEP Parameters (Bayesian Inversion of Groundwater Model)

• estep_niter: Number of groundwater model samples per MPI thread
• estep_beta: Not used
• estep_beta_factor: Not used
• estep_prop: Model generator type. Options:
  • REF – 2D unconfined aquifer constrained by water table (Zwt)
  • 2D – 2D unconfined aquifer (no Zwt)
  • 1D – 1D unconfined aquifer
• vmin: Not used
• vmax: Not used

Defines the prior distributions and bounds for petrophysical parameters used to generate groundwater models.

Example for 2D unconfined aquifer with ERT.

Parameter Description

• Zmin Minimum elevation of the water table at the center of the model
• Zmax Maximum elevation of the water table at the center of the model
• dZ Discretization step for water table elevation
• log_sigma_min Minimum log bulk resistivity
• log_sigma_max Maximum log bulk resistivity
• beta_min Minimum CK-relationship correlation factor (β)
• beta_max Maximum CK-relationship correlation factor (β)
• por_min Minimum porosity
• por_max Maximum porosity
• m1, m2 Range of Archie’s cementation factor (m)
• logA1, logA2 Range of log aquifer resistivity (log Ω·m) for Archie’s Law
• sigma0 Not used
• sigma0_var Not used
• magic_lmd Not used

python emwtinv_multi.py

mpirun -n <number_of_processes> python -m emwtinv_multi.py

Note: Parallel computing is used only during the E-step for Bayesian sampling. The total number of samples will be:

number_of_processes × samples_per_process

Parallelization is implemented via the mpi4py library. On Windows, mpi4py supports only single-process execution.

• SimPEG: Geophysical simulation and inversion

  pip install simpeg

• mpi4py: MPI for parallel Bayesian sampling

  pip install mpi4py

  ⚠️ Full parallel execution is supported only on Linux

• NumPy / SciPy: Core scientific libraries

  pip install numpy scipy

1. Clone the repository

git clone https://github.com/your-username/emwtinv.git
cd emwtinv

2. Install the required Python libraries (see above)

3. Explore the following directories:

   • /Source: Core source code
   • /Script: Example applications
   • /Utils: Visualization and result analysis tools

MIT License