Popperian Inversion

How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth?

Sherlock Holmes (Doyle, 1890)

Although the falsification principle has been around before, it is now connected with the name of Sir Karl Popper, who established it as basic principle of scientific progress (Popper, 1934). This principle states that empirical sciences cannot positively identify the truth (no, I'm not going to discuss the meaning of "truth" here), but can only sort out what is wrong, leaving the remainder as possibly true. It is meaningless to say that one non-falsified hypothesis is "more likely" than another one, since the available data just does not allow for a further distinction of non-falsified hypotheses. Hypotheses are right or wrong, not likely. This is important to remember, especially in the case of geophysical modeling, where models simplify the geological structure very much: usually, all models are wrong because of these simplifications, and it is stupid nonsense to say that one wrong model is more likely than another wrong model.The ultimate goal of any geophysical inversion process must be to find all solutions that are not falsified by data, since any of them could be the true solution.

In consequence, all hypotheses (hypocenters in the case of earthquake location) that are not falsified must be kept in mind until new data comes along that allows for further falsifications. The often favored "best fitting solution" often just explains measurement errors by model structure and is therefore not really a better approximation to the truth.

The LOCSMITH software strictly follows the falsification approach: it does not attempt to find the true solution, but confines itself on the distinction between falsified and non-falsified candidate solutions. To do so, it tests many candidate hypocenters a for possible falsification by observations.

If a forward computation using a candidate 4D hypocenter h(x,y,z,t)

The compatible solutions establish a set H={h(x,y,z,t) | h not falsified} which is the solution set of the location problem.

The goal of the location process is to produce a description of this solution set.

If earthquake location were a linear inversion problem, the parameter space defined by x,y,z,t would be a vector space, and the solution set is simply a subspace of this vector space - and it is also a vector space. Ideally, it is a zero dimensional space, i.e. a point, and the solution is unique. If the problem is underdetermined, the solution set becomes a higher-dimensional space, i.e. a line, a plane or a volume, and the solution is non-unique.

Unfortunately, earthquake location is a nonlinear inversion problem, so the solution set does not necessarily form a vector space. In general, the solution set of a non-linear inversion problem might be any kind of potato-shaped volume, or a even a collection of disconnected potato-shaped volumes, or - wort case scenario - a dust of isolated points. That's why nonlinear inversion is so difficult.

The only way to properly solve the generic nonlinear inversion problem is therefore the systematic testing of many candidate solutions - the exhaustive grid search. LOCSMITH is an implementation of such a grid search, but tries to filter our solution-free parts of the grid very early by applying a deterministic importance sampling.

An additional complication - additional to the limitations of empirical science and the non-linearity of the actual problem - is that observational data is always incomplete and erroneous. Incompleteness of data makes the problem underdetermined in the sense that with incomplete data, one will never obtain a unique solution. Errors can make the problem underdetermined in the sense that the introduction of observational errors can "blur" a unique solution into an potato. Errors can also shift the solution to a incorrect position (bias). All these effects must be considered in the definition of the method (difficult with biases).


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