VIP -- Variational Inversion Package with example implementations of Bayesian tomographic imaging

TL;DR

This paper introduces VIP, a Python package that brings variational inference to geophysical inverse problems by implementing ADVI, SVGD, and sSVGD and validating them with 2D travel-time tomography and 2D full-waveform inversion. It provides a scalable framework with forward models, adjoint-based gradients, and unconstrained transforms to approximate the posterior $p(\mathbf{m}|\mathbf{d}_{obs})$ efficiently, offering a practical alternative to traditional MCMC. The study demonstrates that variational methods can deliver accurate posterior means and uncertainties faster than MH-McMC, while highlighting the trade-offs: ADVI can be faster but biased in some regions; SVGD and especially sSVGD better capture complex posteriors at higher computational cost. VIP’s modular design, HPC-ready implementation, and test examples aim to broaden access to Bayesian inversion for a wide range of geophysical problems, encouraging community contributions.

Abstract

Bayesian inference has become an important tool to solve inverse problems and to quantify uncertainties in their solutions. Variational inference is a method that provides probabilistic, Bayesian solutions efficiently by using optimization. In this study we present a Python Variational Inversion Package (VIP), to solve inverse problems using variational inference methods. The package includes automatic differential variational inference (ADVI), Stein variational gradient descent (SVGD) and stochastic SVGD (sSVGD), and provides implementations of 2D travel time tomography and 2D full waveform inversion including test examples and solutions. Users can solve their own problems by supplying an appropriate forward function and a gradient calculation code. In addition, the package provides a scalable implementation which can be deployed easily on a desktop machine or using modern high performance computational facilities. The examples demonstrate that VIP is an efficient, scalable, extensible and user-friendly package, and can be used to solve a wide range of low or high dimensional inverse problems in practice.

Figures (6)

• Figure 1: Code structure of VIP. Each rectangle represents a folder or file in the package. Users can implement their own forward functions similarly to the way this is implemented in examples tomo and fwi2d.
• Figure 2: (a) Locations of seismometers (blue triangles) around British Isles used in this study. (b) Terrane boundaries in the British Isles from galetti2017transdimensional. Abbreviations are as follows: OIT, Outer Isles Thrust; GGF, Great Glen Fault; HBF, Highland Boundary Fault; SUF, Southern Uplands Fault; WBF, Welsh Borderland Fault System.
• Figure 3: Mean (top row) and standard deviation (bottom row) maps of group velocity at 10 s period obtained using ADVI, SVGD, sSVGD and MH-McMC respectively. White triangles denote locations of seismometers. Black numbers are referred to in the main text.
• Figure 4: (a) The true model used in the full waveform inversion example. 10 sources are located at the depth of 20 m (red stars) and 200 receivers (not shown) are equally spaced at the depth of 360 m on the seabed. (b) The prior distribution of seismic velocity, which is set to be a Uniform distribution with an interval of 2 km/s at each depth. An additional lower bound of 1.5 km/s is also impose to the velocity to ensure that the rock velocity is higher than the velocity in water. (c) An example particle generated from the prior distribution.
• Figure 5: The mean (top row) and standard deviation (bottom row) obtained using SVGD (left panel) and sSVGD (right panel), respectively. Black dashed lines denote well log locations referred to in the main text.