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Spatiotemporal Deconvolution of Hydrophone Response for Linear and Nonlinear Beams—Part II: Experimental Validation

K. Wear,Anant Shah,C. Baker

Published 2022 in IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control

ABSTRACT

This article reports experimental validation for spatiotemporal deconvolution methods and simple empirical formulas to correct pressure and beamwidth measurements for spatial averaging across a hydrophone sensitive element. The method was validated using linear and nonlinear beams transmitted by seven single-element spherically focusing transducers (2–10 MHz; <inline-formula> <tex-math notation="LaTeX">${F}$ </tex-math></inline-formula>/#: 1–3) and measured with five hydrophones (sensitive element diameters <inline-formula> <tex-math notation="LaTeX">${d}_{g}$ </tex-math></inline-formula>: 85–1000 <inline-formula> <tex-math notation="LaTeX">$\mu \text{m}$ </tex-math></inline-formula>), resulting in 35 transducer/hydrophone combinations. Exponential functions, exp(<inline-formula> <tex-math notation="LaTeX">$-\alpha {x}$ </tex-math></inline-formula>), where <inline-formula> <tex-math notation="LaTeX">${x} = {d}_{g}$ </tex-math></inline-formula>/(<inline-formula> <tex-math notation="LaTeX">$\lambda _{{1}}{F}$ </tex-math></inline-formula>/#) and <inline-formula> <tex-math notation="LaTeX">$\lambda _{{1}}$ </tex-math></inline-formula> is the fundamental wavelength, were used to model focal pressure ratios <inline-formula> <tex-math notation="LaTeX">${p}'/{p}$ </tex-math></inline-formula> (where <inline-formula> <tex-math notation="LaTeX">${p}'$ </tex-math></inline-formula> is the measured value subjected to spatial averaging and <inline-formula> <tex-math notation="LaTeX">${p}$ </tex-math></inline-formula> is the true axial value that would be obtained with a hypothetical point hydrophone). Spatiotemporal deconvolution reduced <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> (followed by root mean squared difference between data and fit) from 0.29–0.30 (7%) to 0.01 (8%) (linear signals) and from 0.29–0.40 (8%) to 0.04 (14%) (nonlinear signals), indicating successful spatial averaging correction. Linear functions, <italic>Cx</italic> + 1, were used to model FWHM’/FWHM, where FWHM is full-width half-maximum. Spatiotemporal deconvolution reduced <inline-formula> <tex-math notation="LaTeX">${C}$ </tex-math></inline-formula> from 9% (4%) to −0.6% (1%) (linear signals) and from 30% (10%) to 6% (5%) (nonlinear signals), indicating successful spatial averaging correction. Spatiotemporal deconvolution resulted in significant improvement in accuracy even when the hydrophone geometrical sensitive element diameter exceeded the beam FWHM. Responsible reporting of hydrophone-based pressure measurements should always acknowledge spatial averaging considerations.

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