Introduction to Inverse Theory
Measuring
In many areas of life we measure things, even without knowing, we measure our heart rate, the speed of our car, the amount of money in bank, the number of days to our next holiday. As humans, we love to measure. But what is typically of interest is not necessarily what we measure, the observations, but rather what infers about the underlying characteristics of what gave rise to those measurements. For instance, we might measure the size of a house and how many bedrooms and bathrooms it has. The purpose is not necessarily to know this information but to infer from it how good that house is, with the basic assumption here being that a bigger house is a better house, which is typically borne out by house prices. Not all problems are as simple, and indeed there is a lot more that goes into the value of a house than simply the number of rooms! Take, for example, reconstructing the internal structure of the earth from seismological observations. This spectrum of difficulty in problems has an underlying commonality though; that there is some reasoning or physics as to why those observations exists. For instance; house price is related to house size, or the physics behind the acoustic wave equation for the previous examples. As scientists have endeavored to determine, for thousands of years, this underlying order of the universe (and "things" in general); we tend to have a fairly good grasp of the physics that give rise to the observations. If we don't we can always approximate this or employ mathematical models to estimate the physics or reasoning.
Measurements are cool, but we're really interested in what gave rise to them
We are thus left with some observations and some approximation of the reasoning or physics that gave rise to
these observations but not the thing we really want; the conditions that gave rise to those observations. In
most circumstances, understanding these conditions is the end goal. For instance, we may observation some
amount of Xrays at various detectors but it is the reconstructed computerized tomography (CT) image (which
gave rise to the variation in amplitude of energy received at each receiver) that is important. It isn't the
seismograph responses that are useful but what they can tell us about the epicenter of the earthquake that
gave rise to the seismic activity. Beyond this, by knowing the conditions that gave rise to the current
observations, we can predict into the future (with varying degrees of accuracy) also.
An approach to determining these conditions, typically called model variables, from the observations (also
called data, measurements, and other such names) and knowledge the physics or reasoning that lead to the
observations is called inversion, but you will also see it referred to as optimization. Due to it being a
key aspect of operational research then this may also get confused with inversion (operational research
would also include other areas such as time series analysis and statistical modelling).
Relation to data science and modelling
In many problems in data science, it concerns itself with finding an approximation of that physics/reasoning
that maps between some input model data, and some output decision; does this photo (input) contain a cat
(yes/no, the result). Neural networks, for instance, do this by finding approximating that physics/reasoning
with some complex network of weights and simple transforms which are to be determined. By contrast,
inversion takes the output and a best guess of teh physics/reasoning that gave rise to those outputs to
determine what the initial conditions were.
In this respect, we can see some problems as having three parts; an input, some mappings or transformations
which represent the physics or reasoning, and some output. The missing component determines what approach to
take to; if you are missing knowledge of the mapping then use data science, if you don't know the initial
conditions or input then use inversion, and if you don't know the results then you can forward model the
inputs through the transformation to predict the output.
