Inversion is an important tool in the interpretation of geophysical data. The process aids in the reconstruction of rock property distributions from measurements of their physical responses. This is achieved by a structured search into a parameter (rock properties) space through the use of physical forward modeling.
In the early stages of development, such searches were performed manually by a human operator making adjustments to some guess of the geological setting. The search would proceed in a trial and error fashion to match measured data and reconstruct a reasonable geological model. Much research in the last two decades has been concentrated on automating this process with the use of sophisticated inversion procedures.
Automation appears to eliminate most of the subjective judgement involved in repeated forward modeling by removing any input from the operator. In fact, the subjectivity is not removed, it is simply hidden because the presence of (purely subjective) assumptions is still crucial for the successful outcome of automatic inversion procedures.
The inherent non-uniqueness underlying all geophysical inverse problems results in additional information being necessary to select a single solution among the ensemble of infinite rock property distributions. Such information may be provided in the form of:
- Specific starting model for the inverse run
- Specific parameters restricting the search to predetermined geometrical shapes
- Extra mathematical requirement for the solution.
If this information is tied up in the inner working of a black box inversion algorithm, the average user may be completely unaware that he or she is influencing the result.
In recent years, fast computers allowed the development of sophisticated forward modeling of geodynamic processes. Plate tectonics, faulting and folding, mantle convection, and fluid flows could be treated in an inverse process, much like seismic/potential field problems. The potential of such applications would be tremendous, given that reconstructing initial geological configurations from their geological responses is very much what geology is about. It is an implicit inverse problem tackled on a daily basis in every geologist's brain.
At present, geodynamic modeling is almost exclusively confined to the forward modeling stage. The physical intuition of the modeler is very important in this field because the data are usually extremely sparse and may only provide constraints on some integral property of the system. The quality of a solution is often judged accordingly to its resemblance to patterns seen in the field, to the fact that it does not contradict basic geological principles, or simply to the modeler's expectations. Fit-to-data can be used as a further criterion when available.
For the time being, subjectivity, knowledge, experience, and intuition still play a major role in geophysical/geological modeling. The implementation of fully automated systems for data analysis and artificial intelligence approaches have been attempted with only limited success. It is still the geoscientist, not the tool, that discovers mineral and oil deposits.
Recently, research in artificial intelligence developed systems to support artistic creativity (Takagi 1998). They have been used, for example, in graphic design and music composition. The systems take advantage of fast computation to generate a suite of images or music sequences. Then, an artist looks at the different images or listens to pieces of music and ranks them according to his/her own tastes. An inversion strategy takes into account such judgement in a formal mathematical way to generate a new set of images and/or music sequences, iteratively converging towards the artist's tastes and/or inspiration.
We propose the extension of such techniques to geophysical and geological applications, in which subjective judgement is necessary either to discriminate between ambiguous solutions or to evaluate geological models in the absence of sufficient constraints. In doing so, we present the first step in the development of a system for interactive inversion of geophysical/geological processes. The system offers three main useful features:
- Allows a more systematic application of forward modeling codes.
- Provides a formal role for relevant geological experience and knowledge in inversion that often is extremely difficult to translate into mathematically rigorous constraints.
- May suggest valid solutions falling outside the range of original expectation, by facilitating a "brainstorm-like" process between the geoscientist and the inversion procedure.
The purpose of interactive inversion is to allow the user to direct the parameter space search according to his/her subjective judgement. In order to do so, the traditional numerical measure of data mismatch is replaced by the user's evaluation.
Humans find it hard to express subjective judgement with absolute values, while they generally find it much easier to compare different instances of the same process and rank them accordingly to certain criteria. Interactive inversion works by producing different possible solutions and presenting them to the user for judgement and ranking. Genetic inversion works by optimizing an ensemble of solutions.
Genetic algorithms are a search method suitable for the global optimization of irregular multimodal functions. Starting with a set of initial solutions, these algorithms progressively modify the solution set by mimicking the evolutionary behavior of biological systems, until an acceptable result is achieved.
Genetic algorithms (GAs) are able to solve non-linear, non-local optimization problems, without the need for curvature information and without the need for derivatives. This feature is particularly important for our application, since no derivative information for the subjective judgement is available.
In a GA, a potential solution is represented in the form of a chromosome. Each parameter to be determined can be interpreted as a gene, and the concatenation of the parameters resembles the chain in a chromosome. Early GAs represented each gene in binary form, but further research showed that a straightforward representation as real numbers can be more effective in high dimensional spaces.
A GA works by applying three basic operators, corresponding to the biological processes of selection crossover and mutation, to a population (ensemble) of chromosomes.
Formally, the modifications to a GA required to work interactively are minimal. Once a set of chromosomes is generated, it is fed to a forward code. Then a set of outputs (in either images of animations) is produced. The images are visualized and the user ranks them according to his or her subjective judgement.
The ranking is input to the GA, which uses it to produce the next generation of chromosomes. Notice that, since ranking is implicitly used in linear normalization selection, effectively no formal algorithmic change in the code is imposed by replacing a measure of fitness with the subjective evaluation.
The model to be run required the reactivation of a sequence of three existing extensional basins during a compression phase of deformation. The difficult part of this computation was to obtain a mechanically self-consistent initial condition for the reactivation (three extensional faults). Trial-and-error modeling of an extended, layered system found a suitable starting point after numerous iterations, which took 90% of the time for the entire modeling exercise (in this case two weeks).
It was interesting to know if the interactive inversion could produce the desired initial condition more rapidly than the trial and error approach. A rough sketch of the desired result was given to an operator who had not participated in an earlier trial-and-error exercise together with a suitable model with a number of free parameters.
The main feature of the section is the three extensional faults, with approximately regular spacing. The aim of the analysis is to deduce what kind of stress and material properties can allow the formation of such faulting pattern.
The interactive inversion needed eight GA generations to find the target section. The human time required for such results was just a few minutes, which is to rank eight GA generations, and two hours of computer time for each generation to produce the animations. The computer time could be greatly reduced by implementing the procedure in parallel. Each model is entirely independent of the other.
There is an additional benefit to the interactive inversion. The process of selecting the best model also maps out much of the local parameter space. An analysis of the parameters for the better models reveals which are important for the desired behaviour and which are not. Further, since the GA provides models closely related to the best one, the influence of the controlling parameters on the outcome can be discerned.
From the exact placement of the brittle structures, this is the only model that is useful, however, geologically similar results are C and D, which also have nearly-periodic brittle structures in the uppermost layer. Analysis of the parameters for these models shows that the models C, D, G are identical except for the upper layer thickness - clearly the spacing of the brittle structures is controlled most strongly by this parameter. We learn, therefore, not just how to produce a particular model, but also the controlling physical variables, which we can use in related models.
The main feature of the method is that it offers a way to include geological knowledge and experience into the inverse process. This is extremely hard to achieve in a rigorous mathematical formalism. As mentioned above, traditional inverse techniques include constraints, which can be specified easily in purely mathematical terms.
These assumptions are generated from mathematical convenience in the first instance, and geological relevance in the second. As a result they rarely allow the proper modeling of complex geology. This is largely at the core of the difficult communication often experienced between geophysicists and geologists which results in a lack of common ground between the two camps.
The technique we have developed tries to build a bridge between the two views by providing an obvious and simple way to include geological knowledge into the inversion with no need for complex mathematical modeling. It allows the user to watch the inversion as it proceeds through the parameter space and so removes much of the black box nature of the automatic inversion. Insights that may be obtained range from a better understanding of the parameter space, to a better understanding of the range of variability of geological processes.
This is very important, since often experience and strong preconceptions may hinder alternative geological interpretations. The stochastic sampling of the solution space typical of GAs may develop unexpected solutions or alternative geological scenarios. In a typical automatic black box inverse run, such alternative solutions would be most likely discarded and lost.
This offers an opportunity for "brainstorming" between inversion and user. Such brainstorming could go one step further because there is no reason to limit the subjective judgement to a single user. Different geologists might work on the same inversion, or even better, geologists and geophysicists could cooperate in the inversion providing two different classes of knowledge and experience to the problem.
While the interactive approach is a necessity for processes where only a subjective ranking exists, the methodology may also be useful in traditional inversions where ambiguities are present. In this case, the judgement of the experienced user is required to discriminate between models, which have the same objective fit to the data, but might be quite distinct in terms of geological probability. Thus, the numerical mismatch would guarantee data fitting and the subjective judgement would disregard "un-geological" scenarios.
In geoscience, measurements rarely supply sufficient constraints on a problem to allow for a unique and stable solution from inversion. Additional external constraints are used in these cases, but are often constructed more for mathematical convenience than for strict geological appropriateness. This is because it is often hard to code analytical or numerical geological information.
We have presented a simple way an inverse run can be driven entirely by subjective judgement from users with reasonable geological knowledge and experience. While this approach does not remove non-uniqueness from the solution, it allows the reconstruction of solutions satisfying basic criteria of geological reliability.
The method has proven to be successful in both synthetic and real applications, and compares well to traditional numerical approaches. It requires only minimal time and effort from the user. We believe this technique can greatly widen the range of geological problems admissible for inversion and can be used for many applications in geosciences, either alone or in conjunction with traditional numerical techniques.
Takagi, H., 1998, Interactive evolutionary computation: system optimization based on human selective evaluation; IEEE Int. Conf. on Intelligent Engineering Systems, 1-6, Vienna, Austria, Sept 1998.
Editor's Note: This is a synopsis of an article that will appear in an upcoming issue of Geophysics. Dr. Boschetti has organized a workshop "Future directions in the analysis of potential field data: inversion, signal processing, interpretation" for the Australian SEG Conference 2001, in Brisbane, Australia, in August, 2001. He can be contacted at Tel: ++61 8 9284 8421.