Uncertainty Analysis

What is uncertainty analysis?

Uncertainty analysis, or uncertainty assessment, quantifies the uncertainty (or equivalent, confidence) of the model features recovered from geophysical inversions. For example, in an inverted 3D density model, we might have recovered a cell with a density contrast of 0.07 g/cc. When it comes to interpretation and decision-making, a relevant question is how confident we are in this particular value, which amounts to asking: how much uncertainty is associated with this value? Is it 0.07+/-0.01 g/cc, or 0.07+/-0.1 g/cc? Obviously, they represent two very different levels of uncertainties, and have different implications for subsequent geological interpretation. In many cases, uncertainty analysis can be understood as the determination of error bars or confidence intervals.

Or, the inversion might show that one cell has a larger conductivity value than another cell, which may be a critical piece of evidence for showing more part melt in one area than another. But before making that observation or conclusion, one must be very confident in what he or she sees in the inverted resistivity model. In other words, he or she must answer the following question: how certain it is that the former model element has a larger conductivity than the latter? Is it 20% certain or 80% certain? This is also the type of questions that uncertainty analysis tries to answer.

Another example comes from the realm of magnetization inversion where a set of magnetic data is inverted for a distribution of magnetization vectors consisting of magnitudes and directions. The recovered magnetization directions sometimes can be tied to geological timing and inform the formation or tectonic history of the geological features of interest. Assuming a large Koenigsberger ratio (Q ratio), a low inclination indicates that the magnetic rocks in the study area were magnetized close to the paleo-equator. It also helps determine the paleomagnetic pole position. Therefore, quantifying the uncertainty of the recovered magnetization directions becomes critical as it directly influences the confidence and reliability of all the subsequent interpretations.

These are just a few examples from my research work. There are many other examples that speak to the importance of uncertainty analysis in literature. In a word, uncertainty analysis connects geophysical inversions and geological interpretations, and determines how geophysical inversion results should be properly interpreted so that the final interpretations are statistically valid and scientifically sound. Any geological interpretations based on geophysical inversions without uncertainty analysis is doubtful, or at least, less convincing.

Basic approach

Uncertainty of geophysical inversion is closely related to its non-uniqueness, i.e., the existence of many (in many cases, infinite) models that fit the geophysical measurements equally well. After all, if there is only one model that can reproduce the measurements, the uncertainty of the inverse solution is apparently zero. Note that, a model of zero uncertainty does not directly translate to 100% accuracy. Or in general, smaller uncertainty does not mean larger accuracy.

There is a vast body of literature out there on the topic of uncertainty analysis. Researchers have developed many different approaches to quantifying uncertainty. And the technical details differ. But the basic idea is largely the same, that is, to try to find as many equally valid models as possible. (Equally valid models simply mean the models that fit the observed data equally well.) This requires some kind of strategies to explore the model space. Then, by examining the variabilities of these models, we can develop some practical insights into the uncertainty.

In the Bayesian approach, this is achieved by drawing many samples from the posterior probability density function (which is simply the product of prior probability density function and the likelihood). The degree to which these samples differ from each other reflects the underlying uncertainty. The biggest challenge comes from the fact that drawing samples from a very high dimensional PDF is computationally prohibitive. This is the fundamental reason why Bayesian approach is largely limited to 1D and 2D inverse problems. At the forefront of research efforts in the realm of Bayesian inversion is the development of computationally efficient strategies that would make 3D Bayesian inversions computationally feasible.

In the deterministic approach, inversion is accomplished by minimizing an objective function. The specifics of an objective function determine the characteristics of an inverted model. Varying the objective function would result in an inverted model that looks different. Therefore, to generate a larger sequence of equally valid models, one has to somehow develop many objective functions, each of which would result in a model with unique features. A collection of many such models would allow us to explore the model space and develop an understanding of the uncertainties of inverted features.

Some of my work on uncertainty analysis

Below I give an example of the recent work that my PhD student, Xiaolong Wei, and I have done on quantifying uncertainties in the deterministic inversion framework. As explained above, the basic idea of our work is to generate a large number of equally valid models by adjusting the tunable parameters in the objective function. Figure 1 summarizes the 30 different models that we have obtained by inverting the same set of gravity gradiometry data but with different objective functions. These 30 different models all fit the observed Gzz data equally well but each of them has a unique set of features. The variabilities of the anomalous features in these models reflect the underlying uncertainties. Therefore, by quantifying these variabilities, we have successfully obtained a measure of the uncertainties of the inverse problem. Many statistical methods are available to quantify the variabilities, the simplest being the standard deviations.

We believe that it is better to present all 30 models to geologists for geological interpretations, instead of just picking one 'best' model based on some geophysicist's personal preference. The consistent features in the 30 models should be interpreted with more confidence. The variable features, on the other hand, should be interpreted with care. And geological insights and decision making based on these variable features should take their uncertainties into account.

My student Xiaolong and I are also working on 3D Bayesian inversions. Our goal is to develop a set of numerical methods and computational frameworks that enable us to perform uncertainty analysis of 3D inversions efficiently in both deterministic and Bayesian frameworks.