The ToFD depth equation assumes the diffractor sits on the line midway between the two probes. A tip offset across the weld, or lying ahead of or behind the current probe position along the scan axis, has a longer total path than a tip at the same depth on the centreline, so it always reads deeper than it really is.

With the transmitter and receiver index points at plus and minus S, an offset x across the weld and y along the scan direction, the total path is sqrt((S+x)^2 + y^2 + d^2) + sqrt((S-x)^2 + y^2 + d^2). Feeding half of that into the standard depth relation gives the apparent depth. The error is always positive and grows quickly for shallow flaws and large offsets, for the same reason that shallow depths are sensitive to PCS error: the arrival-time curve is nearly flat near the surface.

The y term is why point-like reflectors draw hyperbolic arcs in a D-scan. As the probes approach and pass the reflector, y falls to zero and rises again, so the apparent depth drops to the true depth at the apex and rises on either side. Always take depth readings at the apex of the arc.

ToFD gives no transverse position information at all. A signal from a diffractor well off the weld centreline is indistinguishable from a deeper one on the centreline. Where transverse position matters, use offset scans, a complementary pulse-echo or phased-array scan, or move the ToFD pair across the weld and watch how the indication depth changes.

Compression wave

Worked example

Probe centre separation	70 mm
True depth of the diffractor	20 mm
Offset across the weld	10 mm
Offset along the scan axis	0 mm
Compression velocity	5900 m/s
Half separation	35 mm
Total sound path	81.26 mm
Arrival time in material	13.77 µs
Apparent depth	20.63 mm
Depth error	0.63 mm

S = 35 mm. The two legs are sqrt((35 + 10)^2 + 20^2) = sqrt(2025 + 400) = sqrt(2425) = 49.2443 mm and sqrt((35 - 10)^2 + 20^2) = sqrt(625 + 400) = sqrt(1025) = 32.0156 mm, giving L = 81.2599 mm, that is 81.26 mm, and 81.2599/5.9 = 13.7729 us, so 13.77 us. Half of L is 40.6300 mm, so the apparent depth is sqrt(40.6300^2 - 35^2) = sqrt(1650.7933 - 1225) = sqrt(425.7933) = 20.6348 mm, that is 20.63 mm, an error of 0.63 mm.

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