ToFD depth resolution against depth

Differentiating the ToFD depth relation shows how much arrival-time change a millimetre of depth actually produces: dt/dd = 2*d / (c*sqrt(S^2 + d^2)). Inverting it gives the depth uncertainty produced by a timing uncertainty, delta_d = delta_t * c * sqrt(S^2 + d^2) / (2*d).

The gradient goes to zero at the scanning surface, so near-surface depth uncertainty is unbounded: at a 70 mm PCS a diffractor at 1 mm and one at 3 mm differ by under 40 nanoseconds of arrival time. Deep in the wall the gradient tends to 2/c and the uncertainty tends to delta_t*c/2, which for steel and a 20 nanosecond timing uncertainty is about 0.06 mm. This asymmetry is why ToFD sizes mid-wall and lower-wall flaws far better than shallow ones, and why through-wall height of a deep flaw is more trustworthy than the absolute depth of a shallow one.

The timing uncertainty to enter is not only the digitiser step. It combines the digitisation interval, the operator repeatability in placing a cursor on a diffracted signal (typically a fraction of a cycle), and any jitter in the trigger. Using one digitisation interval alone gives an optimistic answer.

This figure is only the timing contribution. Errors in the PCS, in the assumed velocity, in probe alignment and in flaw position off the centreline all add to it, and near the surface the PCS term normally dominates. Quote depth accuracy from measurements on a reference block containing targets at known depths, and use this calculation to understand where the errors come from.

Timing uncertainty is not the only clock-related limit. The transmitted pulse has a finite duration tau = n_cycles/f, and two diffracted signals closer in time than that duration merge into one. Feeding tau through the same gradient gives the minimum resolvable separation of two tips at a given depth, h_min = tau*c*sqrt(S^2 + d^2)/(2*d). With a 70 mm PCS and a well damped two-cycle 5 MHz pulse that is about 3.6 mm at 12 mm depth: a flaw of smaller through-wall height there shows a single merged signal and cannot be sized from its tips at all, however fine the digitiser.

S = PCS / 2
dt/dd = 2 × d / (c × √(S² + d²))
Δd = Δt × c × √(S² + d²) / (2 × d)
τ = n_cycles / f
h_min = τ × c × √(S² + d²) / (2 × d)