DGS / AVG equivalent reflector size

DGS — Distance, Gain, Size, called AVG in German — replaces a set of drilled reference blocks with a calculation. Instead of comparing an indication with a machined reflector at a similar range, the response is compared with a plane back wall, and theory supplies the rest: how far the echo from a small disc reflector falls below the back-wall echo at the same distance.

In the far field the on-axis pressure of a circular probe decays as S/(λz). A small disc of area S_r re-radiates like a piston of that area, which makes the disc-to-back-wall ratio 2 S_r/(λ z), or π d²/(2 λ z) for a disc of diameter d. Rearranged, that gives the equivalent reflector size: the diameter of the flat, circular, perfectly-oriented disc that would produce the echo actually observed.

The relation is a far-field one, so the sound path must be beyond about three near-field lengths, N = D²/(4λ). Inside the near field the on-axis pressure oscillates through maxima and minima and no simple amplitude law applies, which is why DGS diagrams are drawn against normalised distance A = z/N and normalised size G = d/D, and why probes are chosen so the region of interest lies past the last maximum.

ERS is a sizing convention, not a flaw size. A real crack is rough, tilted and partly transparent, and almost always returns less than a disc of the same area, so ERS under-states planar flaws and over-states nothing. Codes that accept DGS say so explicitly and require the probe's own DGS diagram, a back-wall reference on sound material, and a transfer correction for surface and attenuation differences.

λ = c / f
near field N = D_eff² / (4 λ)
normalised distance A = z / N
disc echo relative to back wall at the same range:  ΔV = 20 log₁₀( π d² / (2 λ z) )
so  d_ERS = √( 2 λ z · 10^(ΔV/20) / π )
ΔV = [20 log₁₀(H/H_ref) − (g_ind − g_bw)] − [−20 log₁₀(z_f/z_bw) − 2α(z_f − z_bw)] + ΔV_transfer
H = screen height %, g = instrument gain in dB — not the A and G of the DGS diagram