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General inversion theory

What does a scientist do? A scientist observes, explains, predicts, and publishes. Scientists ultimately observe nature. The fundamental insight that nature's laws are universal, allows a scientist to simplify complex nature into simpler experiments. To qualify as scientific, the experiments and observations have to be reproducible by other interested scientists.

The scientist then employs a scientific theory to explain the observations. Such theories tend to be mathematical laws that combine a set of parameters that describe nature's state to the observed quantities.

From all the models that might explain an observations, a scientist chooses the most parsimonious model. In many cases, a scientist has to choose among several computational models. Quantum versus Newtonian mechanics, pixel versus graphical object parameterization, or L1 versus L2 norm are classic cases of competing mathematical models. Additionally, practical considerations, such as computational cost and implementation time, may influence a scientist's choice.

A scientific theory is usually used in two ways: forward and backwards. Most of the time actually both ways, simultaneously. To test a scientific theory, a scientist predicts an experiment's observation from the experimentally controlled state of the system of interest. If the best fitting predicted data of a mathematical model does not fit the observed data, then the scientist needs to find an improved theory. Such a replacement usually involves a better approximation of the underlying physics (an acoustic wave equation is replaced by an elastic one, or quantum effects are taken into account). Interestingly the improved models usually extend the traditional ones. The traditional models are not wrong but remain useful approximations of the improved theory in the traditional cases.

Observations and a scientific model of nature can also be used to deduce the state of nature under observation. Once a scientist decides on a computational model numerical analysis devises methods to approximate the optimal estimate of the model parameters from the given observations. Interestingly, the forward simulation is usually simpler than its inversion: a case of scrambling of information or, for the more metaphysically inclined, an application of the second law of thermo dynamics.

Furthermore, a computational model suggests an experiment that resolves parameters of the model optimally.

Seismic. In general, I believe inversion continues to be a fruitful source of innovation in geophysics. Inversion theory distinguishes model formulations. Inversion theory guides the design of optimal acquisition geometry. Finally, inversion leads to an optimal approximation of model parameters. I expect succesfull traditional processing approaches to reappear as individual approximative steps of corresponding inversion formulations.

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Next: Integration of experiments Up: Wave propagation in an Previous: Wyllie's model
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