CSIM Full Waveform Inversion Lab

By Xin Wang, Bldg 1, 3139-WS13, 808-0386




Objective:

    Learn and explore the full waveform inversion.

Theory:

    Assume the non-linear modeling operator is A(s), s is the slowness model, a initial slowness model is s0, and the observed data is dobs, the full waveform inversion algorithm is a iterative process to find a velocity model s to minimize the misfit functional:
    f = 0.5 * || A(s) - dobs || 2
    The iterative procedures can be described as
    1. Calculate the data misfit function with the initial slowness model
      f = 0.5 * || A(s0) - dobs || 2;
    2. Apply the reverse time migration (RTM) operator LT to the data residual A(s) - dobs and get the gradient g1
    3. Find the conjugate direction for the current iteration:
      c1 = g1 + <g1, g1>/<g, g> * c0,
      where g0 is the gradient from last iteration, and c0 is the conjugate direction from last iteration;
    4. Give the trial step length alpha with, use the back tracking and hyperbolic interpolation to find the numerical step length gama * alpha
    5. Update the new slowness model:
      s1 = s0 + alpha * gama * c1;
    6. Refresh the conjugate direction, gradient and the slowness model by:
      c = c1;
      g = g1;
      s0 = s1;
    7. Repeat steps 1 to 6 until the procedures meets the designed iteration number or convergent criteria.

Procedure:

  1. Download FWI_LAB.tar.gz, extract the compressed package by tar -xvf FWI_LAB.tar.gz.

  2. Add the current directory to your MATLAB working path by addpath((genpath(pwd))) ;

  3. Go to the working directory by cd FWI_LAB;

  4. Comment lines 14 and 15 in fwi_modeling.m, and also change the command "n2s(is)" into "num2str(is)" in lines 39 and 40. Then o then run fwi_modeling to generate the synthetic data with the true cutted marmousi2 model;

  5. Comment lines 16 and 140 in fwi_debug.m. Read fwi.m or fwi_debug.m first and see how the FWI theory is implemented in the scripts;

  6. Comment lines 16 and 140 in fwi_debug.m. Run fwi_debug.m to apply FWI to the modeled data;

  7. Run fwi_plot to plot the results.

Questions:

  1. In this FWI approach, we mute the direct wave, what is the advantage and disadvantage if we include the direct wave?

  2. We limit the maximum and minimum velocity by v1(v1<vmin)=vmin;v1(v1>vmax)=vmax, why?

  3. If the starting velocity is far from the true, which results in a slow convergence, how can we speed it up?

  4. Choose another velocity model, design the acquisition geometry and the initial velocity model, run the FWI code and compare the results to see it works, if doesn't, why?

    References:

    • Seismic Inversion, Gerard T. Schuster