Inverse problems are at the heart of geoscience. But I only hear hardcore geophysicists talk about them. Maybe this is because they're hard problems to solve, requiring mathematical rigour and computational clout. But the language is useful, and the realization that some problems are just damn hard — unsolvable, even — is actually kind of liberating. 

Forwards first

Before worrying about inverse problems, it helps to understand what a forward problem is. A forward problem starts with plenty of inputs, and asks for a straightforward, algorithmic, computable output. For example:

• What is 4 × 5?
• Given a depositional environment, what sedimentological features do we expect?
• Given an impedance log and a wavelet, compute a synthetic seismogram.

These problems are solved by deductive reasoning, and have outcomes that are no less certain than the inputs.

Can you do it backwards?

You can guess what an inverse problem looks like. Computing 4 × 5 was pretty easy, even for a geophysicist, but it's not only difficult to do it backwards, it's impossible:

20 = what × what

You can solve it easily enough, but solutions are, to use the jargon, non-unique: 2 × 10, 7.2 × 1.666..., 6.3662 × π — you get the idea. One way to deal with such under-determined systems of equations is to know about, or guess, some constraints. For example, perhaps our system — our model — only includes integers. That narrows it down to three solutions. If we also know that the integers are less than 10, there can be only one solution.

Non-uniqueness is a characteristic of ill-posed problems. Ill-posedness is a dead giveaway of an inverse problem. Proposed by Jacques Hadamard, the concept is the opposite of well-posedness, which has three criteria:

• A solution exists.
• The solution is unique.
• The solution is well-conditioned, which means it doesn't change disproportionately when the input changes. 

Notice the way the example problem was presented: one equation, two unknowns. There is already a priori knowledge about the system: there are two numbers, and the operator is multiplication. In geoscience, since the earth is not a computer, we depend on such knowledge about the nature of the system — what the variables are, how they interact, etc. We are always working with a model of nature.

Tomorrow, I'll look at some specific examples of inverse problems, and Evan will continue the conversation next week.