ToFD depth error from PCS error

The ToFD depth calculation uses the half separation S directly, so any error in the probe centre separation feeds straight into every depth reported. If the true separation is larger than the value entered by an amount delta, the sound path to a flaw at true depth d is longer than the assumed geometry allows for, and the instrument reports the flaw as deeper than it is: d_apparent = sqrt(d^2 + S*delta + delta^2/4).

Differentiating that at small delta gives a sensitivity of S/(2*d) millimetres of depth error per millimetre of PCS error, which grows without limit as the flaw approaches the scanning surface. With a 70 mm PCS a flaw at 10 mm depth picks up roughly 1.75 mm of depth error for every 1 mm of PCS error; the same 1 mm error at 40 mm depth is worth less than 0.5 mm. Errors of that size are easy to make: index points that are not where the drawing says they are, wedge wear, probe holders that shift on a curved surface, or a PCS measured between wedge fronts instead of between index points.

Two things guard against it. Measure and record the PCS actually achieved between index points, and check it against the lateral wave and backwall arrival times on known-thickness material before and after every scan. A PCS error shows up immediately as a lateral wave that does not arrive when it should, which is why the lateral wave is worth protecting in the gate.

Note that the sign is one-directional in the sense that matters: because the error enters as a squared path term, a PCS larger than assumed always reads deep, and a PCS smaller than assumed always reads shallow. Through-wall height is less affected than absolute depth, because much of the error is common to both tip readings.

S = PCS / 2
sensitivity ≈ S / (2 × d)   [mm depth error per mm PCS error]
d_apparent = √( d² + S × δ + δ² / 4 )
error = d_apparent − d
δ for a target error e: δ = −2S + 2 × √( S² + 2 × d × e + e² )