A time corrected gain curve equalises the response of the same reflector at different sound paths. Two effects have to be undone. Beam spreading makes the echo fall off as a power of range that depends on the reflector shape: a large flat back wall behaves as 1/z, a side drilled hole as 1/z^1.5 and a disc or flat bottomed hole as 1/z². In decibels that is 20·k·log10(z/z_ref) with k equal to 1, 1.5 or 2.

The second effect is material attenuation, which is linear in path length rather than logarithmic: 2·α·(z − z_ref) for the round trip. Over short paths spreading dominates; in coarse grained or austenitic material attenuation quickly becomes the larger term, and it is the one that varies most from part to part.

The sum of the two is the extra gain the TCG point needs. Add it to the reference gain and compare with what the instrument can deliver: running out of gain at the far TCG points is common on thick sections and shows up as a curve that flattens at the end. When that happens the fix is a lower frequency, a larger aperture or a different reflector depth, not more electronic gain on an already saturated receiver.

Attenuation should be measured on the part or on a block of the same material, not taken from a table. The reflector exponents are geometry and are reliable; α is metallurgy and is not.

Compression or shear

Worked example

Reference reflector sound path	25 mm
TCG point sound path	100 mm
Gain at the reference point	40 dB
Attenuation coefficient (dB/mm, one way)	0.02
Reflector type	sdh
Maximum usable instrument gain	80 dB
Wave mode	shear
Material velocity	3240 m/s
Reflector exponent k	1.5
Gain for beam spreading	18.06 dB
Gain for attenuation	3 dB
Total gain increment at this point	21.06 dB
Gain required at this TCG point	61.06 dB
Headroom remaining	18.94 dB
Time position of the TCG point	61.73 µs

For a side drilled hole k = 1.5, so spreading costs 20 × 1.5 × log10(100/25) = 30 × 0.60206 = 18.06 dB. Attenuation adds 2 × 0.02 × 75 = 3.00 dB, giving 21.06 dB above the 40 dB reference, i.e. 61.06 dB with 18.94 dB of headroom against an 80 dB ceiling. The point sits at 2 × 100/3240 = 61.73 µs.