=============================== VIP

Variational Inversion Package, which implements variational methods for geophysical imaging problems, including seismic travel time tomography and full waveform inversion.

Authors

• Xin Zhang x.zhang2@ed.ac.uk

Requirements

Cython, Dask, H5py

Install

In the VIP folder, run

.. code-block:: sh

sh setup.sh

This builds up the VIP package, but does not install the package into your Python environment. As a result, to use the package you need to tell python where the package is. For example, when running scripts, do

.. code-block:: python

PYTHONPATH=/your/VIP/path python vip_example.py

See examples in tests folder. Instead you may want to install the package,

.. code-block:: sh

sh setup.sh install

This will install the package into your Python environment, after which the package can be used directly in your scripts.

Variational Inversion

This package implements three different variational inference methods: ADVI, SVGD, and stochastic SVGD (sSVGD). To use them,

.. code-block:: python

from vip.pyvi.svgd import SVGD, sSVGD
from vip.pyvi.advi import ADVI

All methods require a function that takes model parameters as input and calculates the gradients of logarithm posterior pdf function w.r.t parameters. For example,

.. code-block:: python

def dlnprob(theta):
    # theta has a shape of (num_of_particles, num_of_parameters)
    # some calculation of theta
    return logp, grad, None

where logp is the logarithm of (unnormalized) posterior pdf value (or negative misfit value), grad is the gradient. The third output is used to return other auxiliary variables (e.g., a mask array), and can be safely ignored. Thereafter,

.. code-block:: python

svgd = SVGD(dlnprob, kernel='rbf')

This creates a SVGD method which uses radial basis function kernel. To sample the posterior pdf,

.. code-block:: python

losses, x = svgd.sample(x0, n_iter=1000, stepsize=0.01, optimizer='sgd')

where x0 and x are variables containing starting and final particles with a shape of (num_of_particles, num_of_parameters) respectively. This will run the SVGD algorithm for 1,000 iterations using stochastic gradient descent (sgd) algorithm. Supported optimization algorithms include sgd, adagrad, adadelta and adam. To use sSVGD algorithm,

.. code-block:: python

ssvgd = sSVGD(dlnprob, kernel='rbf')
losses, x = ssvgd.sample(x0, n_iter=2000, stepsize=0.01, burn_in=1000)

This will sample the posterior using sSVGD method for 2,000 iterations with a burn-in period of 1,000. To use ADVI,

.. code-block:: python

advi = ADVI(dlnprob, kernel='meanfield')
losses, phi = advi.sample(n_iter=2000, stepsize=0.01, optimizer='adam')

This runs mean-field ADVI for 2,000 iterations using the adam optimization algorithm. The vector phi contains the mean (first half) and the logarithm of the standard deviation (second half) of the final Gaussian distribution. To use fullrank ADVI, set kernel to "fullrank". In this case, assume the number of parameters is n, the first n elements of phi are the mean, and the rest n^2 elements are the Cholesky decomposition (L) of the covariance matrix.

Examples

• For a complete 2D Full-waveform inversion example, please see the example in tests/fwi2d.
• For a complete 2D travel time tomography example, please see the example in tests/tomo2d.
• For an example implementation of 3D Full-waveform inversion, please see the example in tests/fwi3d. Note that this requires users to provide an external 3D FWI code to calculate misfit values and gradients. See details in forward/fwi3d.

References

• Zhang, X., & Curtis, A. (2024). VIP-Variational Inversion Package with example implementations of Bayesian tomographic imaging. Seismica, 3(1).
• Zhang, X., & Curtis, A. (2020). Seismic tomography using variational inference methods. Journal of Geophysical Research: Solid Earth, 125(4), e2019JB018589.
• Zhang, X., Nawaz, M. A., Zhao, X., & Curtis, A. (2021). An introduction to variational inference in geophysical inverse problems. In Advances in Geophysics (Vol. 62, pp. 73-140). Elsevier.
• Zhang, X., Lomas, A., Zhou, M., Zheng, Y., & Curtis, A. (2023). 3-D Bayesian variational full waveform inversion. Geophysical Journal International, 234(1), 546-561.