Joint inversion

What is inversion?

Geophysicists collect data usually on or above the surface of the Earth in order to understand the subsurface structures, compositions and dynamics. An important component of interpreting geophysical data is to create a good image of the Earth through a mathematical process called 'inversion'. Mathematically, geophysical inversion works in much the same way as medical imaging.

However, inversion of geophysical data is highly non-unique and ill-posed. Non-uniqueness means that there are an infinite number of models that can fit the measured data (to their error levels). Ill-posedness means that inversion is an unstable process, where small perturbations in the measured data (e.g., measurement errors and noise) could result in large variations of the resulting models. Therefore, ever since geophysicists started using inversion in their work, many research efforts have been devoted to address these two problems.

In general, there are two ways of dealing with the above-mentioned problems.

One is to impose as much prior information/constraints as possible to an inversion so that the resulting model not only reproduces the measured geophysical data but also complies with all the prior information/constraints (e.g., the measured dip, specified ranges of inverted values, etc). In literature, this type of inversions is called 'geologically constrained inversion'. The greatest challenge for this type of inversion arises from the less-recognized fact that incorporating prior geological information into inversion is mathematically very challenging. Sometimes, it may seem deceptively simple. For example, assuming that from density measurements on rock samples, we know that the density values that the rocks in our study area can only assume two values: 2.45 g/cc and 2.57 g/cc. Now, the questions is, how can we constrain our gravity inversion in such a way that, after inversion is completed, the recovered densities only show these two values? This idea dates back to at least 1990s (even earlier). However, in the geophysical community, a good solution did not exist until 2010s. One of my contributions to this area of study is to solve this problem by combining the classical Tikhonov regularized inversion formalism with fuzzy c-means clustering.

What is joint inversion?

A second strategy is to invert multiple types of geophysical data in a unified framework so that the resulting models are mutually consistent and satisfy all the geophysical measurements. The basic assumption or philosophy is that a model that is consistent with multiple geophysical data types is more trustworthy than a model consistent with only one type of geophysical data. A good example that employs the same idea (but does not fall into the category of joint inversion) is the gravity modeling that is typically done in oil and gas industry with seismic constraints. Gravity modeling without seismic constraints is simply meaningless when it comes to hydrocarbon exploration.

An essential component of joint inversion is to design a coupling strategy so that the exchange of either structural or physical information (or both) can happen between, say, a density model and a velocity model. Without such a coupling strategy, no exchange of information would happen. Then, it reduces to separate inversions where inversion of gravity data is completely independent of inversion of, say, seismic data.

There are generally two types of coupling strategies.

One strategy is to promote structural similarity between two different physical property models. The basic assumption is that, the geology in a study area is unique, and thus, we expect a certain degree of structural similarity between, say, density and velocity models. The research questions for this structure-based coupling strategy are: (1) how to define structures? and (2) how to quantify the similarity (or equivalently, differences) between two models? Different answers to these two questions would result in different structural similarity measures and joint inversion algorithms. The most popular structural similarity measure so far has been the cross-gradients proposed by Gallardo and Meju in 2003. The cross-gradients method uses gradient (i.e., first order derivative) to define structures, and uses the cross product of the two model gradients as a measure of structural similarity at any location in a model.

Another coupling strategy is based on the observation that physical properties are usually correlated. For example, many lab measurements have shown that there is a linear relationship between density and velocity. It is, therefore, natural and logical to expect inversions to be able to reproduce such a linear relationship among the inverted density and velocity values. The basic idea for the second type of coupling strategy is to assume some kind of relationships existing between physical properties and to constrain inversion in such a way that these relationships are reproduced among inverted values.

In literature, joint inversion based on structural similarity measures is typically termed structure-based joint inversion, and joint inversion based on physical property correlations is termed petrophysics-based joint inversion.

The petrophysics-based joint inversion has been around for a long time. Joint inversion of seismic and gravity has been typically done using the petrophysics-based approach because there is usually a well-established analytical relationship between density and velocity.

What are the challenges?

As far as I'm concerned, several issues remain to be addressed when it comes to petrophysics-based joint inversion. One is the multi-modality of physical property relationships. That is, in a study area, there exists more than one relationship between two physical properties. For example, the crossplot of measured resistivity and velocity values (i.e., the black dots) in the following figure from Moorkamp et al. (2013) clearly shows two major clustering features, one encircled by the blue circle and corresponding to the salt and the other exhibiting a quasi-quadratic trend and representing the background sediments. The bimodal nature of the physical property distribution is a truthful reflection of the subsurface geology in the region. However, it also makes it difficult to incorporate such multi-modal petrophysical data into inversion, because (1) no analytical relationship exists between velocity and resistivity in such a situation, and (2) we simply do not know which one of the multiple relationships should be used for a specific region.

Another issue comes from qualitative information about physical properties. For example, we might have some qualitative knowledge such as Rock A is denser and more conductive than Rock B. This type of prior knowledge is of greater availability. It does not require rock property measurements. Instead, it could come from expert knowledge or from a textbook. However, because of its qualitative nature, there has not been any proven approach to incorporating such qualitative information into petrophysics-based joint inversion.

Some of my work on joint inversion

One of my contributions to the field of joint inversion is that I have developed a coupling strategy that allows multi-modal petrophysical data to be incorporated into joint inversion. This coupling strategy is based on fuzzy c-means clustering. It encourages the jointly inverted values to show a certain number of clusters at certain locations. Both the number of the clusters and their locations are determined by the prior petrophysical data.

Joint inversion of gravity and seismic traveltime for salt imaging

The following movies show how this coupling strategy, when applied to joint inversion of gravity and seismic traveltimes, is able to push inversion to honor and reproduce the multi-modality of prior petrophysical data. More details can be found in Sun and Li (2016).

This movie summarizes the jointly inverted density model from each iteration. The inverted density model from the first 5 or 6 iterations looks much like the model that one would obtain from a separate inversion of gravity data. It is smooth and does not contain many detailed structures of the salt body. But from Iteration 7, we start to see a lot of details being added to the model. The dipping flank is much better imaged. The final density model from joint inversion is a much better representation of the salt body than the separately inverted density model, and is only possible from joint inversion.

This movie shows the jointly inverted velocity model from each iteration. The inverted velocity model after 3 iterations looks much like the one from a separate inversion of seismic traveltimes. The triangle-shaped top part is largely missing. And the dipping flank is only partially images without much detailed structural information. But from Iteration 4, we start to see (1) the top and dipping flank are being much improved, (2) geometrical details of the salt body are recovered, and (3) the velocity values in the background and within the salt body are becoming more and more uniform and closer to the true values. Again, all these improvements are a consequence of the additional information brought by gravity and petrophysical data into the joint inversion. Such additional information positively influences the construction of the velocity model through the coupling term (which serves to exchange information between different sources of data) in the joint inversion.

This movie shows how the jointly inverted density and velocity values (i.e., the yellow dots) behave iteratively in a crossplot, as compared to the true physical property values (i.e., the red dots). The jointly inverted values in the first two or three iterations show a high degree of scattering. But starting from Iteration 4, we can clearly see that the yellow dots (i.e., the inverted values) start to move toward one of the two red dots (i.e., the true values), as if they were being drawn by an attraction force from the red dots. This means that our clustering constraint is coming into play. As the iterative inversion process goes on, the yellow dots keep moving closer to the red dots. Eventually, the yellow dots show very similar clustering patterns to the red dots (i.e., they show two clusters and their average locations are close to the red dots). That is, our joint inversion is able to reproduce the multi-modal petrophysical data (and of course, the gravity and seismic traveltime data as well).

Joint inversion of magnetic and induced polarization data for sulfide exploration

I have also applied the same methodology to the exploration of sulfide deposit where I have jointly inverted magnetic and induced polarization (IP) data with a priori petrophysical data as constraints. A detailed report of this work can be found in Sun and Li (2014). Below I show the final results in a movie and two figures.

The four yellow dots in this movie represent the true chargeability and susceptibility values. They correspond to the background, the stock, the alteration zone and the sulfide deposit. The red dots are the recovered chargeability and susceptibility values from joint inversion of magnetic and IP data. As the iteration goes on, we observe that the red dots are moving toward one of the four dots, thanks to the clustering constraint. In the end, the red dots (i..e, the recovered values) exhibit four clusters well separated from each other. Their locations are also highly consistent with the true physical property values as marked by the red dots.

Geology interpretation based on jointly inverted models can then be done more easily and reliably because (1) in spatial domain, they contain structurally consistent features, and (2) in parameter domain, they display clustering features that are highly consistent with a priori petrophysical data. The right panel in Figure 2 shows the geology differentiation result from joint clustering inversion. The four geological units are identified at correct locations with reasonable shapes. The geological model in Figure 2 integrates information from magnetic data, IP data and petrophysical data, and guides further exploration efforts.

Current status and road ahead

Joint inversion has been an active area of research in the past 15 years or so. But unfortunately, it has not been widely used in both industry and academia. There are a few reasons for it. First, it is mathematically difficult. Designing a proper coupling strategy that can be implemented elegantly via mathematics requires some in-depth mathematical knowledge. Secondly, it is complicated. It often requires major modifications of existing separate inversion codes, which is usually a massive undertaking. Thirdly, successful applications to field data do exist but not many. The added value of joint inversion, therefore, has not been well proven when it comes to real-world applications. In other words, in most cases, separate inversion results might just result in very similar understanding of geology and lead to similar decisions. More case studies are needed to really convince people that there is something that only joint inversion can provide.

In addition, several challenges still remain to be addressed. For example, how to incorporate qualitative and incomplete petrophysical data into joint inversion? How to overcome the local minima problem with the cross-gradients method? Therefore, in addition to more and better case studies, we also need to develop new methods and algorithms.

Thirdly, so far there has not been any open source numerical framework that allows researchers to do all sorts of numerical experiments without having to coding everything from scratch. This has slowed down progress, and needs to be changed by motivated community volunteers with a good understanding of joint inversion methods and an excellent set of coding skills.

In summary, I believe case studies, algorithmic development and infrastructure construction are all needed in order to maximize the use and value of joint inversion. I think we will see more work along these three parallel (yet not independent) lines in future.

A note for readers

Thanks for reading it through! The above text is a very high-level summary of my understanding of joint inversion. I have left out many technical details as well as many of my thoughts on joint inversion. If you have any thoughts or ideas that you want to share with me, or if you are inspired and want to let me know, please feel free to contact me.