Assumptions and the Inverse Crime

Part 0: The Forward Problem

Learning objectives

Define the inverse crime: testing an inversion on data made with its own assumptions
Explain why a matched-assumption test looks flawless yet proves nothing
See that a wrong phase leaves the data misfit untouched while ruining the model
See that a wrong bandwidth raises the misfit, so that error is catchable
State the rule: a low data misfit does not validate a model

Testing a Method Against Itself

Suppose you have written an inversion, a routine that takes a seismic trace and returns the earth reflectivity. How do you know it works? The tempting test is to feed it a synthetic whose answer you know, and check that it recovers that answer. It will, gloriously, if you build the synthetic with the very same assumptions the inversion makes. Same wavelet, same sampling, same physics. That rigged test has a name in inverse theory: the inverse crime. The recovery looks flawless, and it certifies nothing except that your forward step and your inverse step agree, which they were written to do.

Here the earth is a known reflectivity $r$ , the true data are $d = r * w_{\text{true}}$ texttrue for a true wavelet $w_{\text{true}}$ texttrue, and the inversion recovers $r_{\text{est}}$ textest by deconvolving $d$ with an assumed wavelet $w_{\text{assumed}}$ textassumed. When $w_{\text{assumed}} = w_{\text{true}}$ you commit the crime.

One Error You Cannot See, One You Can

The danger is not that the crime gives a wrong answer. It is that a matched assumption makes a good data fit look like proof, so when your real-world assumption is wrong, the same good fit reassures you falsely. The widget separates two kinds of wrong assumption, and they behave very differently.

Rotate the assumed wavelet phase and something unsettling happens: the data misfit does not move. A phase rotation is all pass, it does not change how much of the wavelet fits the data, so the trace you predict is just as good as in the crime, while the recovered reflectivity is destroyed. The data literally cannot tell you the phase was wrong. Now assume the wrong frequency instead and the misfit climbs, because a narrower wavelet cannot reproduce the trace. That error is the lucky one, the kind you catch by looking at the fit.

The Rule

Two lessons carry forward. First, to trust a method, test it on data generated with different physics than the method assumes: a finer grid, a different wavelet, added noise, or a fuller forward operator. Second, and more general, a low data misfit is not a validated model. Many earths and many assumptions can fit the same trace. Every later part of this course builds a forward operator you could commit this crime with, so knowing how to avoid it is what keeps your synthetics honest.