yongpeng25 / topopt-fwi

Full Waveform Inversion (FWI) appoach based on Topology Optimizatin (TO) for the salt reconstruction problem.

TopOptFWI

Full Waveform Inversion (FWI) approach based on Topology Optimization (TO) for the salt reconstruction problem.

Problem Description

• The wave propagation problem in the frequency domain is solved using a Finite Element (FE) model.
• Squared slowness model is split into two parts: background and salt structure.
• The Solid Isotropic Material with Penalization (SIMP) model is used to interpolate the squared slowness.
• A Helmholtz-type filter is used to smooth both control variables and gradient.
• A Heaviside projection is used to ease the identification of sharp interfaces with strong velocity contrast.
• Gradient calculation with respect to the control variables is aided by the dolfin-adjoint package.
• Control variables are updated using an L-BFGS-B algorithm.

Dependencies and Installation

This code is based on Python 3, relies on the installation in a Linux environment, and requires the installation of the following packages:

• FEniCS 2019.1.0
• dolfin-adjoint 2019.1.0
• numpy 1.22.0
• matplotlib 3.5.1
• scipy 1.7.3

Example

To run the example from the terminal:

python3 example.py 

Configuring the problem

The "input_data" class defines the problem to be solved:

class input_data:
    """Parameters to configure the FWI algorithm"""
    def __init__(self):
        # General Configuration ------------------------------------------------- #
              
        self.tofwi = True               # True = TO-based FWI
                                        # False = Traditional FWI
                                    
        self.example = 'b'              # Synthetic true model ('a','b','c' or 'd')
        
        self.true_salt_vel = 4.5        # Salt velocity for the true model (km/s)
        
        self.inverse_crime = True       # True = data from constant density model
                                        # False = data from variable density model
                                    
        self.r_s = 0.5                  # Filter radius for gradient smoothing (km)

        self.n_processes = 1            # Number of processes (int)
        
        # Initial Model --------------------------------------------------------- #   
        
        self.initial_slope = 0.8333     # Slope for the background velocity
                                        # (Exact slope is 0.8333)
        
        self.estimated_salt_vel = 4.5   # Salt velocity estimate (km/s)
        
        # TO-based FWI Configuration -------------------------------------------- #
        
        # (These variables have no influence on the inversion if the traditional 
        # FWI approach was chosen).
        
        self.update_background = False  # True = reconstruct background velocity
                                        # False = fixed background velocity

        self.r_a = 0.1                  # Filter radius for variable smoothing (km)

        self.q = 3                      # Penalty exponent for the interpolation 

        self.projection = True          # True = Heaviside projection active
                                        # False = Heaviside projection disabled
        # ----------------------------------------------------------------------- #

Solving the Inverse Problem

Importing the main module and the "input_data" class:

import TopOptFWI 
from input_data import input_data

Preprocessing step (create Mesh, Spaces, Absorbing layer, etc):

prep = TopOptFWI.PreProcessing(TopOptFWI.model_data(),input_data())
prep.initialize_model()

Solving the Forward Problem (to generate the observed data):

obs_data = TopOptFWI.ForwardProblem(prep)

Initializing Optimization Variables:

ai,bi,mi,opts,bds = TopOptFWI.InitializeOpt(prep)

Solving the Inverse Problem:

ai,bi,mi,outer_iter = TopOptFWI.Inversion(prep,obs_data,ai,bi,mi,opts,bds)

Saving Solutions and Evaluating Errors:

TopOptFWI.PlotVelocity([ai,bi,mi],prep,outer_iter)
TopOptFWI.EvalErrors([ai,bi,mi],prep,obs_data)

Results

True Model:

Synthetic model taken from Kadu et al. 2016 with a background velocity varying linearly with depth and salt bodies with constant velocity of 4.5 km/s. There are 21 sources with 15 Hz Ricker signature placed on top of the model and 101 receivers at 50 m depth. Frequency band between 2.5 and 3.5 Hz with a spacing of 0.125 Hz.

Initial Model:

The initial model is a velocity model with the exact linear background.

Reconstructed Model:

Solution considering the TO-based FWI with fixed background.

Other synthetic models

Synthetic models taken from Kadu et al. 2016 with a background velocity varying linearly with depth and salt bodies with constant velocity of 4.5 km/s. A different true model can be used by changing the instance variable "example" in the "input_data" class.

input_data.example = 'a' input_data.example = 'c'

input_data.example = 'b' input_data.example = 'd'

Citation

If our work is useful for your research, please consider citing:

@article{gonccalves2023salt,
  title={Salt reconstruction in full-waveform inversion using topology optimization techniques},
  author={Gon{\c{c}}alves, JF and Silva, ECN},
  journal={Geophysical Journal International},
  volume={234},
  number={2},
  pages={1484--1504},
  year={2023},
  publisher={Oxford University Press},
  doi={10.1093/gji/ggad150},
  url={https://doi.org/10.1093/gji/ggad150},
}

@article{gonccalves2020identification,
  title={Identification problem of acoustic media in the frequency domain based on the topology optimization method},
  author={Gon{\c{c}}alves, Juliano F and Moreira, Joao BD and Salas, Ruben A and Ghorbani, Mohammad M and Rubio, Wilfredo M and Silva, Em{\'\i}lio CN},
  journal={Structural and Multidisciplinary Optimization},
  volume={62},
  number={3},
  pages={1041--1059},
  year={2020},
  publisher={Springer},
  doi={10.1007/s00158-020-02638-9},
  url={https://doi.org/10.1007/s00158-020-02638-9},
}

Contact

If you have any questions, please feel free to message the authors: Juliano F. Gonçalves ([email protected]) or Emílio C.N. Silva ([email protected]).