Solving an Ill-Posed Problem

Douglas Oldenburg
University of British Columbia
Lindsey Heagy
University of British Columbia

The above work shows that the inverse problem is nonunique and ill-conditioned. This leads to the statement that “the inverse problem is ill-posed”. Any framework must handle these fundamental elements.

The major weapon for handling nonuniqueness is to add more information and find a model that satisfies the geophysical data as well as this a priori information. The simplified flow chart for the inversion is now shown in .

The prior information takes many forms and throughout this module we will have opportunity to illustrate all of them. When thinking about a general geophysical inversion where we attempt to find a distribution of a physical property that can be translated into geology, we might have:

  • General characteristics of the model - Is the geology expected to be smooth, discontinuous or compact?
  • Geologic constraints - Knowledge about faults, overburden, rock units
  • Reference model
  • Bounds for the physical property
  • Information from other geophysical surveys
  • Expected rock units and physical property measurements.

The second issue of importance is the ill-conditioning of the inverse problem. The fundamental ill-conditioning is often automatically ameliorated by adding additional information into the problem. For instance, when adding a smallest model component into a Tikhonov inversion. However, there will always be ill-conditioning resulting from over-fitting noisy data and this needs to be addressed.

There are two general approaches to solving the inverse problem. The first is a probabilistic approach that rests on Bayesian analysis.

1.6.1. Bayesian (Probabilistic)

Use Bayes’ Theorem

P(mdobs)P(dobsm)P(m)P(m|d^{obs})\propto P(d^{obs}|m)P(m)

where P(m)P(m) is the prior information about mm, P(dobsm)P(d^{obs}|m) is the probability about the data errors (likelihood), and P(mdobs)P(m|d^{obs}) is the posterior probability for the model.

There are two pathways for analysis. The full Bayesian solution seeks to characterize
P(mdobs)P(m|d^{obs}) and draw inferences from that distribution. The second is to find a particular solution that maximizes P(mdobs)P(m|d^{obs}). This is call the maximum a posteriori (MAP) estimate.

Abundant resources for inversion using a Bayesian philosophy exist, but a classic is that of . We will have occasion towards the latter part of this module to work with a Bayesian formulation but most of our work will use the Tikhonov method.

1.6.2. Tikhonov (Deterministic)

In this approach the goal is to find a single “best” solution by solving an optimization problem:

Minimize  ϕ=ϕd+βϕmsubject to  ml<m<mu,\begin{aligned}&\text{Minimize}~~\phi=\phi_d+\beta\phi_m\\ &\text{subject to}~~m_l<m<m_u,\end{aligned}

where ϕd\phi_d is the data misfit, ϕm\phi_m is the regularization (model norm), β\beta is the trade-off parameter, and ml,mum_l, m_u are lower and upper bounds on the model. The constraints on the model from the geophysical information are embedded in the data misfit while the a priori information is encoded into the model norm and bounds.

  1. Tarantola, A. (2005). Inverse Problem Theory and Methods for Model Parameter Estimation. Society for Industrial. 10.1137/1.9780898717921