Normalised cross-correlation: numerical conditioning

deepwave.misfits.normalised_cross_correlation() documents what it computes and what its zero-norm branch returns. This page records the numerical-conditioning device that sits underneath that definition, and why the definition is written the way it is.

Internal max-abs rescaling

Before the dot product and the norms are computed, each trace is divided, independently, by its own maximum absolute value. That scale factor is detached: it is purely a numerical-conditioning device, not part of the mathematical NCC formula, so no gradient flows through it.

The rescaling leaves the result mathematically unchanged, because NCC is exactly scale-invariant in each argument separately (\(\mathrm{NCC}(ap, bt) = \mathrm{NCC}(p, t)\) for any \(a, b > 0\)). What it buys is that every intermediate quantity is \(O(1)\) regardless of the traces’ raw amplitude, so the sum of squares behind the L2 norm cannot underflow. Computing the norm directly from raw samples would suffer exactly that underflow at small-but-nonzero amplitudes, and ordinary seismic data is routinely far below unit scale – particle-velocity or strain-rate traces of \(10^{-8}\) or smaller are unremarkable. With the rescaling, such data is normalised and compared exactly like unit-scale data, all the way down to the point where a trace’s own peak amplitude underflows to exactly 0 in the working dtype.

Why the zero branch is a branch, not an epsilon

The guard on a zero denominator is a genuine branch on the value 0.0, not a rounded-down small-denominator threshold, so it does not fire for small-but-nonzero amplitudes. It applies only when at least one of pred/target is an exactly all-zero trace, whose L2 norm is exactly 0.0 rather than merely small; the rescaling above is what makes that a safe test to write, since it removes the underflow that would otherwise turn a small-but-real trace into a zero one.

In the implementation, torch.where evaluates both branches, so a placeholder denominator of 1 is substituted wherever the true denominator is 0, purely to avoid a 0 / 0 division. That branch’s result is then discarded by the outer torch.where, and its gradient contribution vanishes because the dot product is also 0 wherever the denominator is 0 (a zero-scale trace is exactly the zero vector, so its dot product with anything is 0).

Why no mean is subtracted

Not subtracting a mean is a deliberate choice, and the reason is stated in the function’s own docstring: it keeps the misfit sensitive to any DC offset, which is physically meaningful for wave data, since a non-zero mean is itself a mismatch worth penalising.

Mean-centring would not, in fact, cost the exact per-trace amplitude-scale invariance that makes this misfit useful for FWI – that property survives either way. The argument for omitting it is the narrower one that it is an unneeded operation in the formula, and the formula is easier to reason about, and cheaper, without it.