Spatiotemporal Deconvolution of Hydrophone Response for Linear and Nonlinear Beams I: Theory, Spatial Averaging Correction Formulas, and Criteria for Sensitive Element Size

Abstract

This article reports spatiotemporal deconvolution methods and simple empirical formulas to correct pressure and beamwidth measurements for spatial averaging across a hydrophone sensitive element. Readers who are uninterested in hydrophone theory may proceed directly to Appendix A for an easy method to estimate spatial averaging correction factors. Hydrophones were modeled as angular spectrum filters. Simulations modeled 9 circular transducers (1–10 MHz; F/1.4–F/3.2) driven at 6 power levels and measured with 8 hydrophones (432 beam/hydrophone combinations). For example, the model predicts that if a 200-μm membrane hydrophone measures a moderately nonlinear 5-MHz beam from an F/1 transducer, spatial-averaging correction factors are 33% (peak compressional pressure or p_c), 18% (peak rarefactional pressure or p_r), and 18% (full width half maximum or FWHM). Theoretical and experimental estimates of spatial-averaging correction factors agreed to within 5% for linear and moderately nonlinear signals. Criteria for maximum appropriate hydrophone sensitive element size as functions of experimental parameters were derived. Unlike the oft-cited International Electro-technical Commission (IEC) criterion, the new criteria were derived for focusing rather than planar transducers and can accommodate nonlinear signals in addition to linear signals. Responsible reporting of hydrophone-based pressure and beamwidth measurements should always acknowledge spatial averaging considerations.

I. INTRODUCTION

A. Pressure and Beamwidth Measurements

The primary goal of this article and a companion article [1] is to report spatiotemporal deconvolution to correct for effects of hydrophone frequency-dependent sensitivity and finite sensitive element size on pressure and beamwidth measurements. Accurate pressure and beamwidth measurements are essential for documenting experiments and for establishing safety and effectiveness of medical devices.

In this article, a transverse-scan-based spatial deconvolution method is developed and compared with an analytic inverse filter spatial deconvolution method [2–4] that does not require a transverse scan. While both methods predict the effects of spatial averaging on pressure measurements, only the scan-based method predicts the effects of spatial averaging on beamwidth measurements. The analytic inverse filter method is included in IEC 62127–1 Ed. 2 (to be published in 2022).

A second goal of this article is to provide simple empirical formulas to predict dependences of spatial averaging errors in p_c, p_r, and FWHM on sensitive element size, frequency, F/#, and beam nonlinearity. These formulas can help researchers choose appropriate hydrophones for future measurements or calculate appropriate correction factors for past measurements. These formulas are convenient because they do not require a transverse scan or the considerable signal processing involved for scan-based spatial deconvolution.

Spatiotemporal deconvolution may be divided into two steps: temporal deconvolution and spatial deconvolution.

B. Temporal Deconvolution

The easiest way to convert hydrophone output voltage to acoustic pressure is to divide the voltage waveform by a scalar sensitivity factor (e.g., in units of V/MPa). However, this method is inaccurate when the hydrophone sensitivity varies substantially with frequency across the signal spectrum. In these cases, temporal deconvolution can significantly improve accuracy [5–19].

Temporal deconvolution is particularly helpful for nonlinear signals. Nonlinear signals have significant spectral content at harmonics (integer multiples) of the fundamental frequency due to nonlinear propagation. Harmonic content increases with pressure level and propagation distance [20]. Nonlinear signals are commonly found in acoustic radiation force impulse (ARFI) imaging [21–25], harmonic imaging [26], pulsed Doppler measurements [27], and high intensity therapeutic ultrasound (HITU) [3, 9, 10, 28–31].

C. Spatial Deconvolution

A pioneering approach for spatial deconvolution (a. k. a. “spatial averaging correction”) involved on-axis and off-axis hydrophone measurements that can be used to compute a frequency-independent scale factor for linear pressure waves [32, 33]. Subsequent approaches accommodated linear and nonlinear waves. Some of these employed numerical methods to model spatial averaging [34–38]. Another used an analytic filter [2], which has been validated for membrane, needle, and fiber-optic hydrophones measuring nonlinear signals in diagnostic and HITU pressure ranges [3, 4, 39]. While most spatial averaging correction methods assume circular transducers, one approach has been validated for nonlinear signals transmitted by linear and phased arrays [40–42].

One transverse-scan-based spatial deconvolution method demonstrated improved transverse resolution for narrowband, 2.25-MHz, linear signals detected using receivers with surface areas that were large (63 mm²) compared with most hydrophones (< 0.8 mm²) [43–45]. The method did not consider frequency-dependence of effective receiver size, which has been shown to significantly impact spatial averaging effects for modern hydrophones with geometrical sensitive element diameters of 85–1000 μm [2, 4, 39].

The present article and a companion article [1] report validation of a spatiotemporal deconvolution method for linear and nonlinear signals using transducers (1–10 MHz; F/1–F/3) and hydrophone sensitive element diameters (85–1000 μm) that are relevant to modern ultrasound measurements. The transducers include models designed for HITU.

Fig. 1 illustrates the frequency-dependent relationship between harmonic focal spot size (i.e., FWHM) and hydrophone effective sensitive element diameter d_eff(f) generated using previously validated models [2, 4, 46]. As harmonic frequency increases, harmonic component beamwidth decreases, thereby increasing the potential for spatial averaging.

Fig. 1.

FWHM at harmonic frequencies for Sonic Concepts H101 transducer (3.3 MHz, F/1) (black asterisks). Also shown are frequency-dependent effective sensitive element diameters d_eff(f) for five membrane hydrophones with geometrical sensitive element diameters d_g of 200 through 1000 μm (dashed lines).

D. Outline of this Article

Section II presents the hydrophone transverse-scan-based spatiotemporal response model and the analytic filter model [2] that will be used for comparison.

Section III describes the simulation, models for hydrophone frequency-dependent effective sensitive element size, and signal processing steps for modeling the hydrophone response.

Section IV shows dependences of spatial averaging errors on hydrophone sensitive element size, frequency, F/#, and nonlinearity.

Section V provides empirical parametric models for spatial averaging corrections for pressure and FWHM as functions of experimental parameters. Formulas are derived for maximum appropriate hydrophone sensitive element size as a function of experimental parameters.

Section VI offers concluding remarks.

Appendix A summarizes a simple method for estimating spatial averaging correction factors.

Appendix B describes a method for quantifying the nonlinearity of a pressure wave.

II. Theory

A. Hydrophone Response Model

For a circularly symmetric transducer, the hydrophone input pressure pulse and the hydrophone output voltage at the measurement plane may be denoted by p(t, r) and u(t, r), where t is time and r is radial distance from the beam propagation axis. Applying spatial and temporal Fourier transformation yields P(f, ρ) and U(f, ρ), where f is temporal frequency and ρ is spatial frequency.

The hydrophone output voltage may be represented in the spatiotemporal transform domain as

$$U\left(f,\rho \right)=M_L\left(f\right)D_{\rho }\left[a_{eff}\left(f\right),\rho \right]P\left(f,\rho \right)$$ (1)

M_L(f) is the loaded sensitivity of the hydrophone. D_ρ[a_eff(f), ρ] describes hydrophone directional response, where a_eff(f) is the frequency-dependent effective sensitive element radius [46–48].

P(f, ρ) is closely related to the angular spectrum of the incident pressure wave. The difference is that it is expressed as a function of spatial frequency ρ instead of angle θ. Spatial frequency and angle are related by [2, 49]

$$\theta ={cos}^{-1}\left[\sqrt{1-{\left(\lambda \rho \right)}^2}\right]$$ (2)

and may be simplified to ρ = (sin θ)/λ, where λ is wavelength.

Hydrophone directivity is typically expressed as a function of plane-wave incident angle θ. For a circularly symmetric hydrophone, it may be defined as

$$D_{\theta }\left[a_{eff}\left(f\right),k,\theta \right]=\frac{2J_1\left[k{a}_{eff}\left(f\right)sin\theta \right]}{k{a}_{eff}\left(f\right)sin\theta }$$ (3)

where k=2π/λ and J₁() is a Bessel function of the first kind, order 1. Eq. (3) has been used to model a circular piston in a rigid planar baffle, with a_eff(f) set equal to the geometrical sensitive element radius (a_g) at all frequencies [50], If a_eff(f) is obtained by fitting (3) to frequency-dependent directivity models or measurements, then (3) can be used to accommodate a variety of hydrophone designs. Empirical models for a_eff(f) have been obtained for membrane [32, 46, 51, 52], needle [2, 47, 51], and reflectance-type fiber-optic (RTFO) [2, 48] hydrophones.

For spatiotemporal deconvolution, it is convenient to express directivity as a function of ρ.

$$D_{\rho }\left[a_{eff}\left(f\right),\rho \right]=\frac{2J_1\left[2\pi \rho a_{eff}\left(f\right)\right]}{2\pi \rho a_{eff}\left(f\right)}$$ (4)

Spatiotemporal deconvolution may be accomplished using

$$P\left(f,\rho \right)\approx \frac{U\left(f,\rho \right)}{M_L\left(f\right)D_{\rho }\left[a_{eff}\left(f\right),\rho \right]}{L}_f\left(f\right)L_{\rho }(f,\rho )$$ (5)

where U(f, ρ) is the spatiotemporal transform of a transverse scan u(t, r), and L_f (f) and L_ρ (f, ρ) are filters to protect from inaccuracies at frequencies where signal-to-noise-ratio (SNR) may be low or either M_L(f) or D_ρ[a_eff(f), ρ] may fall to values far below their maximum levels. Butterworth [3, 4] and Wiener [14] low-pass filters have been used for L_f (f) for sensitivity deconvolution. In the companion article, a Wiener filter is used for L_ρ (f, ρ) [1]. Pressure in the spatiotemporal domain, p(t, r), may be recovered by applying inverse temporal and spatial Fourier Transforms to P(f, ρ).

B. Focal Plane Pressure Field (and Spatial Fourier Transform) for a Spherically Focusing Transducer

Since the convolution formula (1) will be tested for circular, spherically focusing piston transducers, it will be useful to derive theoretical forms for the corresponding focal-plane pressure field and spatial Fourier Transform to compare with results obtained numerically (present article) and experimentally (companion article [1]).

The source is assumed to have surface pressure p₀, radius a_s, and radius of curvature R. The transducer is driven at fundamental frequency f₁, which corresponds to wavelength λ₁ where λ₁=c/f₁ and c = speed of sound. In the low amplitude limit, in which propagation is linear (i.e., little harmonic generation), the radial pressure distribution for the fundamental component at the focal plane is given by [53]

$$p\left(f_1,r\right)=-i{p}_0\frac{\pi a_s^2}{{\lambda }_{1}R}exp(i{k}_{1}R)exp\left(i{k}_1\frac{r^2}{2R}\right)jinc\left(\frac{a_{s}r}{{\lambda }_{1}R}\right)$$ (6)

where k₁=2π/λ₁ and jinc() is defined by [54]

$$jinc\left(\alpha \right)=besinc\left(\alpha \right)=2\frac{J_1\left(2\pi \alpha \right)}{2\pi \alpha }$$ (7)

Eq. (6) is similar in magnitude to the far field diffraction pattern from a circular, planar source [54, 55].

In the present article and the companion article [1], driving voltages are allowed to be sufficiently high to generate significant harmonic activity. However, it has been demonstrated that (6) is still a good approximation for the fundamental component at diagnostic [2, 4, 39] and even high-intensity therapeutic [3] levels. In the present article, the validity of (6) is investigated by evaluating the dependence of the FWHM of the fundamental component of the beam at the focal plane as a function of nonlinearity. (The spatial deconvolution method described in Sec. II-A does not rely on this assumption, however.)

The two-dimensional spatial Fourier transform of a circularly symmetric function is given by the Hankel or Fourier-Bessel transform [54].

$$G\left(\rho \right)=2\pi \underset{0}{\overset{\mathrm{\infty }}{\int }}rg\left(r\right)J_0\left(2\pi r\rho \right)dr$$ (8)

where J₀() is a Bessel function of the first kind, order 0. In order to take the Hankel transform of (6), it is useful to note that, for typical medical ultrasound pressure waves, the radial variation of the quadratic phase factor, exp[ik₁r²/(2R)], is gradual compared with the radial variation of the jinc[a_sr/(λ₁R)] factor, as has been shown previously (see [2], sec. II-E, (18) and (19)). Therefore, the quadratic phase factor may be ignored here.

The Hankel transform of jinc(br) (where b is a scale factor) is [54, 56]

$$H\left\{jinc\left(br\right)\right\}=\frac{1}{\pi b^2}circ\left(\frac{\rho }{b}\right)$$ (9)

where circ(x) =1 (x< 1); ½ (x=1); 0 (x>1) [54]. Therefore, the magnitude of the spatial Fourier Transform of (6) is approximately given by

$$|P\left(f_1,\rho \right)|\approx p_0{\lambda }_{1}Rcirc\left(\frac{{\lambda }_{1}R\rho }{a_s}\right)$$ (10)

Eq. (10) implies that the spatial Fourier Transform has a sharp cutoff spatial frequency at ρ_cutoff=a_s/(λ₁R). This is tested numerically in the present paper and experimentally in the companion article [1].

C. Analytic Filter Method for Estimating Spatial Averaging

The spatiotemporal deconvolution analysis just presented will be compared with an inverse analytic filter approach that has been validated for membrane, needle, and fiber-optic hydrophones measuring nonlinear signals in diagnostic and HITU pressure ranges [3, 4, 39]. The analytic spatial averaging filter S_p(f) for a nonlinear tone burst is given by [2]

$$S_p\left(n{f}_1\right)=\frac{1}{{\Omega }_n^2}\left[1-e^{-{\Omega }_n^2}\right],$$ (11)

where

$${\Omega }_n^2=\frac{a_{eff}^2\left(n{f}_1\right)}{2{\sigma }_n^2},$$ (12)

f=nf₁, n is the harmonic number, and the magnitude of the radial profile of each harmonic component of the nonlinear beam is given by |w_n(r)|=exp(−r²/2σ_n²) for r values extending from 0 to the outer edge of the hydrophone effective sensitive element [2]. Eq. (11) is a low-pass filter that depends on the ratio of the frequency-dependent hydrophone effective sensitive element area to the frequency-dependent beam −6dB area. The fundamental component Gaussian beam width parameter is given by σ₁≈1.93R/(k₁a_s) [2]. The harmonic beam width parameters are given by σ_n=σ₁/n^q, where q is approximately 0.8 at diagnostic pressure levels (around 5 MPa) [2, 4, 39] and approximately 0.6 at HITU pressure levels (up to 49 MPa) [3]. The FWHM of each harmonic component is ${\sigma }_{n}2\sqrt{2ln2}\approx 2.35{\sigma }_n$[2].

III. Methods

A. KZK Simulation

Simulations used HIFU Simulator Ver 1.2 software [57] to solve the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation as described previously [2]. Simulation enabled investigation of hydrophone spatial averaging effects over wider ranges of nonlinearity than could be achieved experimentally in the companion article [1]. Experimental pressures are limited by the need to reduce the risk of cavitation and to protect hydrophones from damage. The KZK equation is theoretically valid up to angles of about 15–20° from the propagation axis (approximately equivalent to F/# > 1.5) [58, 59]. The parameters for water used in the simulations were sound speed c=1480 m/s, density ρ_w=1 g/cm³, absorption at 1 MHz = 0.217 dB/m, exponent of absorption vs. frequency = 2, and nonlinear coefficient β=3.5. One hundred harmonics were used in each simulation. The grid size was 40 points per fundamental wavelength in the axial dimension and 50 points per fundamental wavelength in the radial dimension.

B. Transducers

Table I lists parameters for the single-element, circular, spherically focusing transducers modelled in the simulations. They included Sonic Concepts (Bothell, WA) H151, H196, and H249, which are designed for HITU applications. The other parameters correspond to transducers manufactured by Blatek (State College, PA), and Panametrics (Waltham, MA) (V308, V311 and V381). The driving frequencies (1–10 MHz) and the F/#s (F/1.4–F/3.2) were chosen to encompass a wide range of common applications. Many transducers in Table I were chosen to match transducers for which experimental data were acquired and reported in the companion article [1].

TABLE I.

Transducer Parameters For Simulation

Manufacturer	Model	Driving Frequency (MHz)	Diameter (mm)	Focal Length (mm)	F/#
Sonic Concepts	H151	1.10	64	100	1.6
Sonic Concepts	H249	1.35	50	125	2.5
Sonic Concepts	H151	3.41	64	100	1.6
Sonic Concepts	H196	3.41	40	100	2.5
Panametrics	V381	3.50	19	38	2.0
Blatek	Custom	3.50	64	89	1.4
Sonic Concepts	H249	4.19	50	125	2.5
Panametrics	V308	5.00	19	38	2.0
Panametrics	V311	10.00	12	38	3.2

C. Hydrophones, Sensitivities and Directivities

The first set of simulations assumed membrane hydrophones with nominal geometrical sensitive element diameters of 200, 500, 600, and 1000 μm, corresponding to the four hydrophones used in the companion experimental article [1].

Modeling directivity as a filter requires knowledge of D_ρ[a_eff(f), ρ] at the frequencies that contribute significantly to the signal (e.g., fundamental and harmonics). Since this set of frequencies might not correspond exactly to the set of frequencies at which directivity was measured, it is useful to have parametric models that can be used to interpolate or extrapolate a_eff(f) for any arbitrary set of frequencies.

Frequency-dependent effective sensitive element radii a_eff(f) for the membrane hydrophones were derived from directivity measurements at multiple frequencies as described previously [4, 46]. When directivity measurements are unavailable, generic empirical formulas are useful for modeling a_eff(f). For membrane hydrophones, an empirical model has been validated: [a_eff(f)−a_g]/a_g=C/(ka_g)^m where a_g is the nominal geometrical sensitive element radius, and C and m are fitting parameters, with C=1.89 and m=1.36 [46].

The second set of simulations assumed needle / RTFO hydrophones with nominal geometrical sensitive element diameters of 200, 500, 600, and 1000 μm. For this set, frequency-dependent effective sensitive element radii were predicted using a previously validated “rigid piston” model for needle and RTFO hydrophones [47, 48, 60]. The model gives the effective radius a_eff(f) by [a_eff(f)−a_g]/a_g=Aexp(−Bka_g), where A=1.85 and B=1.05 [2].

As stated already, another source of distortion for hydrophone measurements of broadband pressure signals is frequency-dependent sensitivity. The effects of frequency-dependent sensitivity have previously been considered and can be countered by inverse filtering [6, 7, 11–13, 17, 18, 61]. The simultaneous effects of frequency-dependent sensitivity and spatial convolution can be managed by applying two separate inverse filters independently, as shown in (5) [4, 42]. To isolate the effects of hydrophone spatial averaging in the simulation, only distortion due to spatial convolution was considered, with the understanding that in practice both forms of distortion occur and can be corrected independently. In the companion article [1], deconvolutions for both frequency-dependent sensitivity and spatial response are performed.

D. Spatial Convolution

The signal model in (1) requires P(f, ρ), which may be computed via spatiotemporal Fourier transformation of the transverse pressure distribution in the focal plane, p(t, r). Therefore, for each source transducer / transmit-power combination, a digitized transverse focal-plane pressure matrix p(jΔt, mΔr) was generated by HIFU Simulator, where Δt was the temporal sampling interval, Δr was the transverse spatial separation between adjacent scan lines, and j and m were integers. Fig. 2 shows an example of a simulated focal waveform.

Fig. 2.

Simulated focal waveform incident upon hydrophone for Sonic Concepts H151 transducer.

The pressure matrix p(jΔt, mΔr) was Fourier transformed along the time dimension using a Fast Fourier Transform (FFT) to yield a spectral pressure matrix P’(jΔf, mΔr) (where the prime denotes that a temporal transform but not a spatial transform has been performed). Fig. 3 shows an example of a focal spectral magnitude. Because of the harmonic structure of the spectra, the spectral matrix could be contracted into a much smaller harmonic matrix P_H’(nf₁, mΔr) where n is the harmonic number, f₁ is the fundamental frequency, and spectral strengths at harmonic frequencies nf₁ were found by integrating peaks over frequency intervals nf₁±f₁/10. Contraction from the spectral matrix P’(jΔf, mΔr) to the harmonic matrix P_H’(nf₁, mΔr) significantly reduced the number of spatial Fourier transforms required in subsequent steps and therefore significantly reduced computation time. This is consistent with previous models that represented nonlinear pressure signals as sums of harmonics [41, 62, 63].

Fig. 3.

Simulated focal spectral magnitude incident upon hydrophone for Sonic Concepts H151 transducer.

The harmonic matrix P_H’(nf₁, mΔr) was Hankel transformed along the spatial dimension to give the spatiotemporal transform P_H(nf₁, sΔρ) where Δρ=ρ_cutoff/10 was the spatial frequency sampling interval, s was an integer that ranged from 0 to 100 (i.e., ρ_max=10ρ_cutoff) and ρ_cutoff=a_s/(λ₁R) (see Sec. II-B). (The 2D Fourier transform of a circularly symmetric function is equivalent to a Hankel transform or a Fourier-Bessel transform [64]).

For each transducer / transmit-power / hydrophone combination, U_H(nf₁, sΔρ) was obtained from (1) by computing the element-by-element product of D_ρH[a_eff(nf₁), sΔρ] (for that hydrophone) and P_H(nf₁, sΔρ) (for that transducer / transmit-power combination). As stated previously, M_L(f) was assumed to be one for all frequencies. In other words, it was assumed that in practice spectra would be sensitivity deconvolved independently as has been described previously [4, 6, 7, 11–13, 17, 18, 65].

U_H(nf₁, sΔρ) was then inverse transformed in the spatial frequency dimension (another Hankel transform), giving U_H’(nf₁, mΔr), a matrix of transverse beam scans for the fundamental and harmonic frequencies. U_H’(nf₁, mΔr), could be summed over all harmonic frequencies (the first index) to yield the total broadband, focal-plane, transverse beam scan.

E. Waveform Nonlinearity Indexes

Waveform nonlinearity was characterized by the non-linear propagation parameter 𝜎_m [66], local distortion parameter 𝜎_q [33, 67], and the spectral index SI [68]. According to IEC 62127–1, 𝜎_q < 0.5 corresponds to “little” nonlinear distortion while 𝜎_q ≥ 0.5 corresponds to “considerable” nonlinear distortion [33]. SI is the fraction of the power spectrum contained in harmonics of the fundamental frequency [68]. SI, 𝜎_m, and 𝜎_q have been found to be relatively robust, compared with other nonlinearity metrics, for describing transfer of energy from an ultrasound beam fundamental frequency to higher harmonics [67].

Each simulation was repeated for a range of input powers to generate values of 𝜎_q of approximately 0.5, 1, 2, 3, 4, and 5. This corresponded to values of 𝜎_m up to approximately 4 and values of SI up to approximately 50%. In addition, all signals could be low-pass filtered to isolate the fundamental signal to simulate linear signals with 𝜎_q ≈ 𝜎_m ≈ SI ≈ 0.

IV. Results

A. Beam Distortion by Hydrophone in Spatial Domain and Spatial Frequency Domain

Fig. 4 shows simulated focal-plane transverse beam cross sections transmitted by the Blatek transducer and measured with an ideal point hydrophone (d_g = 0 μm) and four membrane hydrophones with d_g = 200, 500, 600, and 1000 μm. Due to spatial convolution, the measured peak compressional pressure decreases as d_g increases.

Fig. 4.

Simulated focal-plane transverse beam cross sections for 3.5 MHz, F/1.4 transducer measured with ideal point hydrophone (d_g = 0 μm) and four membrane hydrophones with d_g = 200, 500, 600, and 1000 μm.

Fig. 5 shows the first two harmonic components (at frequencies f₁ and 2f₁) of the spatial Fourier Transform from a simulated transverse scan for the H151 transducer, before and after convolution with the transfer function of the d_g=500 μm hydrophone. The simulated spatial Fourier Transform for the fundamental frequency (f₁) is close to the theoretically expected function, circ(λ₁Rρ/a_s) from (10). Spatial convolution (i.e., multiplication by directivity in spatial-frequency domain) is a low-pass filter that preferentially passes lower spatial frequency components.

Fig. 5.

Spatial Fourier Transform from transverse scan for the H151 transducer. The first two harmonic components are shown before and after scan-based spatial convolution with the transfer function of the d_g=500μm hydrophone. The theoretical function for the fundamental frequency (f₁) is circ(λ₁Rρ/a_s) from (10).

B. Effects of Spatial Averaging on Pressure Measurements

Fig.s 6 (membrane hydrophones) and 7 (needle / fiber optic hydrophones) show relative errors due to hydrophone spatial averaging for p_c (left column), p_r (middle column) and FWHM of p_c in the focal plane (right column) as functions of d_g/(λ₁F/#). The abscissa d_g/(λ₁F/#) is an index of the ratio of the hydrophone geometrical sensitive element diameter d_g to the fundamental component focal spot size. Recall that the FWHM of the fundamental component is 1.41λ₁F/# [2, 69]. Linear signals (σ_q = 0) are shown in top rows of Fig.s 6 and 7, and three levels of nonlinearity (σ_q > 0) are shown in lower rows.

Fig. 6.

Relative errors due to hydrophone spatial averaging in focal peak compressional pressure p_c (left column), focal peak rarefactional pressure p_r (middle column) and FWHM (right column) measured with membrane hydrophones plotted as functions of d_g/(λ₁F/#) for linear signals (top row) and three levels of nonlinearity (other rows). AF-SC: analytic filter spatial convolution. SB-SC: scan-based spatial convolution. See Sec. IV-B.

Fig. 7.

Relative errors due to hydrophone spatial averaging in focal peak compressional pressure p_c (left column), focal peak rarefactional pressure p_r (middle column) and FWHM (right column) measured with needle / fiber optic hydrophones plotted as functions of d_g/(λ₁F/#) for linear signals (top row) and three levels of nonlinearity (other rows). AF-SC: analytic filter spatial convolution. SB-SC: scan-based spatial convolution. See Sec. IV-B.

Relative errors in focal pressure magnitudes were computed as (p’−p)/p (where p’ = the measured value subjected to spatial averaging and p = the true axial value that would be obtained with a hypothetical point hydrophone).

Pressure data were modeled using exponential functions, p’/p = exp(−αx), and Gaussian functions, p’/p = exp(−βx²), where x=d_g/(λ₁F/#). In each case, the model with the lower root-mean-squared difference (RMSD) between simulated data and functional fit was chosen (see Sec. V-B).

Figs. 6 and 7 show excellent agreement between analytic filter spatial convolution (AF-SC) and scan-based spatial convolution (SB-SC), suggesting approximate equivalence of the two methods.

Fig. 8 shows the pressure error parameter α_c for p_c for membrane hydrophones, computed by AF-SC and SB-SC, as functions of local distortion parameter σ_q. The AF-SC and SB-SC methods exhibit excellent agreement, reinforcing the validity of both methods. The dependence of α_c on σ_q is linear up to σ_q = 5. The data are consistent with experimental measurements at σ_q = 0 and σ_q = 1 taken from the companion article [1].

Fig. 8.

Relative error parameters α_c for focal peak compressional pressure magnitude. σ_q: nonlinear distortion parameter. SIM: simulation. EXPT: experiment. AF-SC: analytic filter spatial convolution. SB-SC: scan-based spatial convolution. M: membrane hydrophone.

Table II shows data from Fig. 8 with values for σ_q, σ_m and SI. Table II also shows previously reported values for the pressure error parameter α_c for p_c simulated using the AF-SC method for membrane hydrophones [4]. The present and previous values for α_c are similar even though the two simulations assumed different sets of source transducers. Table II also shows experimental values from the companion article [1]. For experimental data, ground truth pressure values were estimated from measurements performed with a high-resolution hydrophone (d_g=85 μm) that had been corrected using spatiotemporal deconvolution [1].

TABLE II.

Pressure Error Parameter Alpha For Membrane Hydrophones

Pressure				Compressional				Rarefactional
Data				Simulation	Simulation	Previous Simulation	Expt	Simulation	Simulation	Previous Simulation	Expt
Method				AF-SC	SB-SC	AF-SC		AF-SC	SB-SC	AF-SC
Linearity	σq	σm	SI	αc (RMSD)	αc (RMSD)	αc (RMSD)	αc (RMSD)	αr (RMSD)	αr (RMSD)	αr (RMSD)	αr (RMSD)
Linear	0	0	0	0.24 (3%)	0.25 (4%)		0.29 (7%)	0.24 (3%)	0.25 (4%)		0.30 (7%)
Nonlinear	0.5	0.4	4%	0.31 (3%)	0.33 (4%)	0.34 (4%)		0.20 (3%)	0.22 (3%)	0.25 (4%)
Nonlinear	1.0	0.8	12%	0.40 (3%)	0.43 (4%)	0.43 (4%)	0.40 (8%)	0.20 (3%)	0.21 (4%)	0.23 (4%)	0.29 (6%)
Nonlinear	2.0	1.6	29%	0.65 (5%)	0.65 (5%)	0.68 (5%)		0.20 (3%)	0.21 (4%)	0.23 (4%)
Nonlinear	3.0	2.2	35%	0.76 (4%)	0.72 (5%)			0.20 (3%)	0.23 (5%)
Nonlinear	4.0	3.1	45%	0.87 (4%)	0.84 (5%)			0.19 (3%)	0.24 (5%)
Nonlinear	5.0	3.9	49%	0.95 (4%)	1.01 (4%)	1.02 (5%)		0.18 (3%)	0.23 (3%)	0.23 (4%)

Exponentially decaying model functions, exp(-αx), were fit to focal pressure magnitude ratios p’/p (where p’ = measured value subjected to spatial averaging and p = true axial value that would be obtained with a hypothetical point hydrophone), where x = d_g / (λ₁F/#). Nonlinearity indexes are σ_q (local distortion parameter), σ_m (nonlinear propagation parameter), and SI (spectral index). RMSD: root-mean-squared difference between data and functional fits. AF-SC: analytic filter spatial convolution. SB-SC: scan-based spatial convolution. Previous simulation data are from Wear et al., IEEE Trans. Ultrason. Ferro., Freq. Control., 67(12), 2674–2691, 2020.

Fig. 9 shows the pressure error parameter α_r for p_r for membrane hydrophones, computed with both AF-SC and SB-SC methods, as functions of local distortion parameter σ_q. The AF-SC and SB-SC methods exhibit excellent agreement, reinforcing the validity of both methods. The parameter α_r is nearly independent of σ_q up to σ_q = 5. This is because, as can be seen in Fig. 2, increases in harmonic content due to increases in σ_q are manifested more in the compressional component (which varies rapidly with time) and less in the rarefactional component (which varies slowly with time) [70]. The data are consistent with experimental measurements at σ_q = 0 and σ_q = 1 taken from the companion article [1]. By comparing Figs. 8 and 9, it may be seen that at σ_q = 0, α_c ≈ α_r. This is because at σ_q = 0, the wave is linear and therefore p_c ≈ p_r.

Fig. 9.

Relative error parameters α_r for focal peak rarefactional pressure magnitude. σ_q: nonlinear distortion parameter. SIM: simulation. EXPT: experiment. AF-SC: analytic filter spatial convolution. SB-SC: scan-based spatial convolution. M: membrane hydrophone.

Fig.s 10 and 11 (and Table III) are like Figs. 8 and 9 (and Table II), except that they were generated assuming needle / RTFO hydrophones instead of membrane hydrophones. Similar trends are apparent.

Fig. 10.

Relative error parameters α_c and β_c for focal peak compressional pressure magnitude. σ_q: nonlinear distortion parameter. SIM: simulation. AF-SC: analytic filter spatial convolution. SB-SC: scan-based spatial convolution. N/FO: needle or RTFO hydrophone.

Fig. 11.

Relative error parameters β_r for focal peak rarefactional pressure magnitude. σ_q: nonlinear distortion parameter. SIM: simulation. AF-SC: analytic filter spatial convolution. SB-SC: scan-based spatial convolution. N/FO: needle or RTFO hydrophone.

TABLE III.

Pressure Error Parameters Alpha and Beta For Needle Hydrophones

Pressure				Compressional			Rarefactional
Data				Simulation	Simulation	Previous Simulation	Simulation	Simulation	Previous Simulation
Method				AF-SC	SB-SC	AF-SC	AF-SC	SB-SC	AF-SC
Linearity	σq	σm	SI	βc (RMSD)	βc (RMSD)	βc (RMSD)	βr (RMSD)	βr (RMSD)	βr (RMSD)
Linear	0	0	0	0.16 (1%)	0.16 (1%)		0.16 (1%)	0.16 (1%)
Nonlinear	0.5	0.4	4%	0.21 (2%)	0.23 (1%)	0.24 (3%)	0.13 (1%)	0.14 (1%)	0.13 (1%)
Nonlinear	1.0	0.8	12%	0.29 (4%)	0.32 (3%)	0.30 (5%)	0.12 (1%)	0.13 (1%)	0.13 (1%)
				αc (RMSD)	αc (RMSD)		βr (RMSD)	βr (RMSD)
Nonlinear	2.0	1.6	29%	0.52 (4%)	0.52 (5%)		0.12 (1%)	0.13 (1%)	0.12 (1%)
Nonlinear	3.0	2.2	35%	0.63 (3%)	0.58 (7%)		0.12 (1%)	0.15 (3%)
Nonlinear	4.0	3.1	45%	0.71 (2%)	0.67 (6%)		0.12 (1%)	0.16 (4%)
Nonlinear	5.0	3.9	49%	0.78 (2%)	0.80 (6%)		0.12 (1%)	0.15 (3%)	0.13 (2%)

Exponentially decaying and Gaussian model functions, exp(-αx) (p_c for σ_q ≥2) or exp(-βx²) (all other pressures), were fit to focal pressure magnitude ratios p’/p (where p’ = measured value subjected to spatial averaging and p = true axial value that would be obtained with a hypothetical point hydrophone), where x = d_g / (λ₁F/#). Nonlinearity indexes are σ_q (local distortion parameter), σ_m (nonlinear propagation parameter), and SI (spectral index). RMSD: root-mean-squared difference between data and functional fits. AF-SC: analytic filter spatial convolution. SB-SC: scan-based spatial convolution. Previous simulation data are from Wear et al., IEEE Trans. Ultrason. Ferro., Freq. Control., 67(12), 2674–2691, 2020.

C. Effects of Spatial Averaging on FWHM Measurements

Relative errors in FWHM were modeled using linear model functions, FWHM’ / FWHM = 1 + Cx. Fig. 12 shows the FWHM error parameter C for membrane and needle / RTFO hydrophones, computed with SB-SC, as functions of local distortion parameter σ_q. The dependence of C on σ_q is linear up to σ_q = 5 for both types of hydrophones. The values of C for membrane hydrophones are very consistent with experimentally-derived values taken from the companion article [1].

Fig. 12.

Error parameter C for FWHM as a function of nonlinear distortion parameter σ_q. SIM: simulation. SB-SC: scan-based spatial convolution. EXPT: experiment. M: membrane hydrophone. N/FO: needle or RTFO hydrophone.

Table IV shows the data from Fig. 12 in numerical form. Table IV also includes values for σ_m and SI.

TABLE IV.

FWHM Error Parameter C For Membrane and Needle Hydrophones

Hydrophone Type				Membrane	Membrane	Needle
Data				Simulation	Expt	Simulation
Method				SB-SC	SB-SC	SB-SC
	σq	σm	SI	C (RMSD)	C (RMSD)	C (RMSD)
Linear	0	0	0	5% (2%)	9% (4%)	4% (3%)
Nonlinear	0.5	0.4	4%	16% (4%)		14% (5%)
Nonlinear	1.0	0.8	12%	30% (5%)	30% (10%)	27% (6%)
Nonlinear	2.0	1.6	29%	62% (9%)		57% (10%)
Nonlinear	3.0	2.2	35%	74% (10%)		67% (12%)
Nonlinear	4.0	3.1	45%	100% (13%)		89% (14%)
Nonlinear	5.0	3.9	49%	120% (20%)		109% (20%)

Linear model functions Cx were used to fit to relative errors in FWHM, where x = d_g / (λ₁F/#). Nonlinearity indexes are σ_q (local distortion parameter), σ_m (nonlinear propagation parameter), and SI (spectral index). RMSD: root-mean-squared difference between data and functional fits. SB-SC: scan-based spatial convolution.

V. Discussion

A. Comparison of Two Spatial Averaging Models

Two methods (AF-SC and SB-SC) were compared for modeling the effects of hydrophone spatial averaging on pressure measurements for linear and nonlinear signals. The AF method has the advantage that it can be performed on axial pressure measurements, without the need for a transverse scan. The analytic filter (11) can be computed based on transducer and hydrophone properties. Spatial averaging effects may be suppressed by inverse filtering the hydrophone output with the analytic filter. Although the SB method requires a transverse scan, it has the advantage that it can also assess the effects of spatial averaging on beam FWHM.

The analytic filter method makes assumptions about the signal. First, it models the beam as a superposition of gaussian beams with radial profile magnitudes of harmonic components given by |w_n(r)|=exp(−r²/2σ_n²) for r values extending from 0 to the outer edge of the hydrophone effective sensitive element [2]. This approximation has been validated for needle, membrane, and fiber-optic hydrophones at diagnostic and HITU pressure levels [3, 4, 39]. It is validated again in the companion article [1].

Second, the analytic filter model assumes that the FWHM of the fundamental component equals the linear-propagation value. Fig. 13 shows simulated FWHM of the fundamental component as a function of σ_q for nine simulated source transducers. The theoretical FWHM (1.41λ₁F/#) are shown by the horizontal lines. For σ_q ≤ 6, the fundamental FWHM is approximately equal to the linear-propagation value, even though σ_q = 6 corresponds to significant nonlinearity. For example, HITU signals at frequencies of 1.5–4 MHz and pressure levels of 16–49 MPa have been reported to have σ_q values ranging from 3 to 4 [3].

Fig. 13.

Simulated FWHM of fundamental component as a function of nonlinear distortion parameter σ_q for nine transducers. The theoretical FWHM is 1.41 λ₁ F/#.

Based on the simulation results of the present article and the experimental results of the companion article [1], the two methods give very similar results for pressure measurements performed using membrane, needle, and RTFO hydrophones at frequencies ranging from 1–10 MHz, F/# ranging from 1.4–3.2, and nonlinear distortion ranging from none (σ_q = 0) to severe (σ_q = 5), as shown in Figs. 6 and 7, and Tables II and III.

B. Simple Empirical Formulas to Correct for Finite-Sensitive-Element Effects

Figs. 6 and 7 show that spatial averaging errors are largely determined by a small set of experimental parameters: hydrophone type, d_g, λ₁, F/#, and σ_q. (Of these, the only one that requires some effort to estimate is σ_q—see Appendix B.) Therefore, simple empirical formulas obtained from the analysis presented here may be used to estimate spatial averaging errors when the experimenter knows these parameters.

The best models (lowest RMSD between model and simulation) between measured pressures p’ and spatial-averaging-corrected pressures p were as follows. For membrane hydrophones (σ_q ≤ 5) and needle / RTFO hydrophones (2 ≤ σ_q ≤ 5),

$$p_c\approx p_c^{\mathrm{\prime }}\mathrm{e}\mathrm{x}\mathrm{p}\left({\alpha }_{c}x\right)$$ (13)

where x=d_g/(λ₁F/#). For needle / RTFO hydrophones (σ_q ≤ 2),

$$p_c\approx p_c^{\mathrm{\prime }}\mathrm{e}\mathrm{x}\mathrm{p}\left({\beta }_c{x}^2\right)$$ (14)

For membrane hydrophones (σ_q ≤ 5),

$$p_r\approx p_r^{\mathrm{\prime }}\mathrm{e}\mathrm{x}\mathrm{p}\left({\alpha }_{r}x\right)$$ (15)

For needle / RTFO hydrophones (σ_q ≤ 5),

$$p_r\approx p_r^{\mathrm{\prime }}\mathrm{e}\mathrm{x}\mathrm{p}\left({\beta }_r{x}^2\right)$$ (16)

Expressions for α_c, α_r, β_c, and β_r may be found in Table V.

TABLE V.

Best Models for Spatial Averaging Errors

Hydrophone	Parameter	σq range	model	RMSD
Membrane	p c	σq ≤ 5	αc≈0.27+0.15σq	0.04
Membrane	p r	σq ≤ 5	αr≈0.24	0.03
Membrane	C	σq ≤ 5	C≈(5+27σq) %	6%
Needle / fiber optic	p c	0 ≤ σq ≤ 2	βc≈0.15+0.17σq	0.01
Needle / fiber optic	p c	2 ≤ σq ≤ 5	αc≈0.34+0.09σq	0.02
Needle / fiber optic	p r	σq ≤ 5	βr≈0.14	0.02
Needle / fiber optic	C	σq ≤ 5	C≈(3+23σq) %	5%

Exponentially decaying and Gaussian model functions, exp(-αx) or exp(-βx²), were fit to focal pressure magnitude ratios p’/p (where p’ = measured value subjected to spatial averaging and p = true axial value that would be obtained with a hypothetical point hydrophone), where x = d_g / (λ₁F/#). Linear model functions Cx were used to fit to relative errors in FWHM. The local distortion parameter is σ_q. RMSD: root-mean-squared difference between data and functional fits.

For all hydrophones,

$$FWHM\approx \frac{FWH{M}^{\mathrm{\prime }}}{Cx+1}$$ (17)

Figs. 8–12 and Table V show the dependences of α, β, and C on σ_q.

C. Criteria for Maximum Hydrophone Sensitive Element Size when Spatiotemporal Deconvolution is not Performed

To choose an appropriate hydrophone for a measurement, the relative error models may be used to solve for criteria for maximum appropriate hydrophone sensitive element diameters as functions of experimental parameters for cases when spatiotemporal deconvolution is not performed. The relative errors in pressures and FWHM are given by

$${\epsilon }_p=\frac{p^{\mathrm{\prime }}-p}{p}=\exp \left(-\alpha x\right)-1or\exp \left(-\beta x^2\right)-1$$ (18)

$${\epsilon }_{FWHM}=\frac{{FWHM}^{\mathrm{\prime }}-FWHM}{FWHM}=Cx$$ (19)

where, as shown in Table V, α, and β assume different values for compressional and rarefactional pressures, and x=d_g/(λ₁F/#). First, maximum acceptable relative error levels, ε_p and ε_FWHM may be chosen (e.g., −0.1 and 0.1 respectively, corresponding to −10% and 10%). For pressure parameters with exponential spatial averaging error functions,

$$d_g\le -\frac{{\lambda }_{1}F/\#}{\alpha }\mathrm{l}\mathrm{n}({\epsilon }_p+1)$$ (20)

For pressure parameters with Gaussian spatial averaging error functions,

$$d_g\le \frac{{\lambda }_{1}F/\#}{\sqrt{\beta }}\sqrt{-\mathrm{l}\mathrm{n}({\epsilon }_p+1)}$$ (21)

For FWHM,

$$d_g\le \frac{{\lambda }_{1}F/\#}{C}{\epsilon }_{FWHM}$$ (22)

where formulas for α, β, and C are given in Table V.

These criteria have advantages compared with a hydrophone maximum sensitive element size criterion from IEC 62127–1 [33, 71]. First, unlike the IEC criterion, (20)–(22) were derived based on focused transducers instead of planar transducers. Although IEC 62127–1 states that its criterion can also be used for focused transducers, it is not optimal for focused transducers since it was derived for planar transducers. Second, unlike the IEC criterion, (20)–(22) allow for nonlinear ultrasound and therefore can accommodate nonlinear modes such as pulsed Doppler, harmonic imaging, ARFI and HITU. (To be fair, nonlinear ultrasound is far more common today than it was in 1985, when the IEC criterion was derived.) Third, unlike the IEC criterion, (20)–(22) provide separate criteria for p_c, p_r, and FWHM, which allows the investigator to tailor the criterion to a particular measurement task. For example, if cavitation probability is the main concern, then p_r is the main parameter of interest. Fourth, (20) – (22) are expressed in terms of the geometrical sensitive element diameter d_g, which is often more readily accessible than the effective sensitive element diameter d_eff (on which the IEC criterion is based).

VI. Conclusion

Two methods have been shown to be equivalent for predicting errors in pressure measurements due to hydrophone spatial averaging over a broad range of experimental parameters: an analytic filter method and a scan-based method. The analytic filter method has the advantage of not requiring a transverse scan. The scan-based method has the advantage of predicting errors in beam FWHM in addition to predicting errors in pressures.

The analysis shows that hydrophone spatial averaging errors can be significant for many common ultrasound experiments. For example, for an F/1 transducer transmitting a moderately nonlinear beam (σ_q ≈ 1) at 5 MHz and measured with a 200-μm membrane hydrophone, correction factors for peak compressional pressure, peak rarefactional pressure, and FWHM are 33%, 18%, and 18% respectively.

Simple empirical formulas have been obtained for prediction of spatial averaging errors in pressure and FWHM as functions of experimental parameters. Criteria for maximum acceptable hydrophone sensitive element size as functions of experimental parameters have been derived. These criteria are useful for informing choice of an appropriate hydrophone for future experiments.

Using the spatial averaging correction formulas, accurate measurements of focal pressures and FWHM are possible even when the hydrophone geometrical sensitive element diameter exceeds the beam fundamental component focal spot size. This provides investigators with increased flexibility of hydrophone choice, which can be beneficial when selection of hydrophones in a laboratory is limited or when larger sensitive element size is desired for increased sensitivity and increased SNR.

Responsible reporting of hydrophone-based pressure measurements should always acknowledge spatial averaging considerations.

Biography

Keith A. Wear (Senior Member, IEEE) received the B.A. degree in applied physics from the University of California at San Diego, San Diego, CA, USA, and the M.S. and Ph.D. degrees in applied physics, with a Ph.D. minor in electrical engineering, from Stanford University, Stanford, CA.

He was a Postdoctoral Research Fellow with Physics Department, Washington University, St. Louis, MO, USA. He is currently a Research Physicist with the U.S. Food and Drug Administration, Silver Spring, MD, USA.

Dr. Wear is a Fellow of the Acoustical Society of America, the American Institute for Medical and Biological Engineering, and the American Institute of Ultrasound in Medicine (AIUM). He received the 2019 AIUM Joseph H. Holmes Basic Science Pioneer Award. He served as Associate Editor-in-Chief for IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL (IEEE-TUFFC) (2019–2021). He has served as Associate Editor of IEEE-TUFFC (2002–2021), Journal of the Acoustical Society of America (2012-present), and Ultrasonic Imaging (2013-present). He was the Technical Program Chair of the 2008 IEEE International Ultrasonics Symposium (IUS), Beijing, China. He was the General Program Chair of the 2017 IEEE IUS, Washington, DC, USA. He has served as chair of the AIUM Technical Standards Committee (2014–2016), AIUM Bioeffects Committee (2021–2023), AIUM Basic Science and Instrumentation Community (2004–2006 and 2014–2016), and AIUM Therapeutic Ultrasound Community (2013–2015). He is Chair of the American Association of Physicists in Medicine Task Group 333 on Magnetic Resonance Guided Focused Ultrasound Quality Assurance. His research interests include hydrophone measurement methodology, high intensity therapeutic ultrasound, photoacoustics, and quantitative ultrasound.

Appendix A: Simple Method for Estimating Spatial Averaging Correction Factors

First, compute x=d_g/(λ₁F/#) from the hydrophone geometrical sensitive element diameter d_g, fundamental wavelength λ₁, and transducer f-number F/#.

Second, for nonlinear signals, if x ≤1, use (B5) or (B6) from Appendix B to estimate the local distortion parameter σ_q from focal pressure measurements p_c’ and p_r’. (If x > 1, see Appendix B.) For linear signals (e.g., p_c ≈ p_r), σ_q ≈ 0. (Note that correction factors for p_r’ do not require estimates of σ_q.)

Third, use formulas in Table V to compute α_c or β_c, α_r or β_r, and C.

Fourth, use (13)–(17) to compute spatial-averaging correction factors to convert measured values p_c,’ p_r’, and FWHM’ into spatial-averaging corrected values p_c, p_r, and FWHM.

Alternatively, radiofrequency data may be inverse filtered with the analytic filter (11) described in Sec. II-C. If measurements for a_eff(f) for the hydrophone (e.g., derived from directivity measurements) are unavailable, then generic versions for a_eff(f) given in Sec. III-C may be used. The inverse analytic filter (IAF) method corrects pressure (but not FWHM) measurements for spatial averaging. The IAF method is included in IEC 62127–1 Ed. 2 (to be published in 2022).

This methodology may also be relevant to capsule hydrophones. There is anecdotal evidence that a_eff(f) is asymptotically (ka_g > 0.7) similar for capsule and needle hydrophones (see Fig. 2 in [42]).

Appendix B: How to Estimate σ_q from Focal Pressure Measurements

The models in Table V for p_c and FWHM (but not the models for p_r) require knowledge of the local distortion parameter, σ_q [33, 66, 67, 72],

$${\sigma }_q=z{p}_m\frac{2\pi f_{awf}\beta }{{\rho }_w{c}^3}\frac{1}{\sqrt{F_a}}$$ (B1)

where z is the axial propagation distance, p_m=(p_c+p_r)/2, f_awf is the acoustic working frequency (f_awf ≈ f₁), β is the nonlinearity parameter (β=1+B/2A=3.5 for pure water at 20°C), ρ_w is the density of water (997 kg/m³), F_a is the local area factor,

$$F_a=\sqrt{\frac{0.69A_{SAeff}}{A_{b,-6dB}}},$$ (B2)

A_SAeff is the effective source aperture area, and A_b,−6dB is the beam area included within the half maximum pressure contour. A_SAeff may be estimated from nominal aperture dimensions or by fitting a theoretical diffraction pattern to a spatially deconvolved transverse hydrophone scan [1].

As shown in Fig. 14, an empirical model for A_b,−6dB(σ_q) found by simulation for the transducers in Table I is

$$A_{b,-6dB}({\sigma }_q)\approx \pi {\left(0.7{\lambda }_{1}F/\#\right)}^{2}exp(-0.12{\sigma }_q)$$ (B3)

Fig. 14.

Simulated ratio of nonlinear to linear beam −6 dB cross-sectional areas. The linear beam −6 dB cross-sectional area is π(0.7λ₁F/#)². Data are based on KZK simulation for transducers in Table I. The exponential decay function found by least squares regression is exp(−0.12σ_q). The RMSD between simulated data and exponential model is 6%.

The RMSD between simulated data and exponential model (B3) is 6%. Substituting (B3) into (B2) gives

$$F_a\approx \sqrt{\frac{0.69A_{SAeff}}{\pi {\left(0.7{\lambda }_{1}F/\#\right)}^2}}\mathrm{e}\mathrm{x}\mathrm{p}(0.06{\sigma }_q)$$ (B4)

Eq. (15) or (16) (which do not depend on σ_q—see Table V) may be used to estimate p_r. A system of two equations, (13) or (14) (which do depend on σ_q because α_c and β_c depend on σ_q—see Table V) and (B1) can be solved for p_c and σ_q numerically by looping over a plausible range of guesses for σ_q (e.g., 0 to 6 in increments of 0.1; computation time = 89 μs on Intel® i7–8665U/1.9GHz/32 GB RAM) and selecting the value of σ_q that achieves the closest equality for (B1).

If d_g/(λ₁F/#) ≤ 1 and σ_q < 5 (a common situation), approximations may be made to provide rapid, useful estimates of σ_q. First, exponentials in (13) – (16) may be approximated by exp(τ)≈1+τ. Second, exp(−0.12σ_q) in (B3) may be approximated by one. While the first approximation leads to underestimation of σ_q, the second approximation leads to overestimation of σ_q, so the two tend to cancel each other. For membrane hydrophones,

$${\sigma }_q\approx \frac{K(1+0.25x)(p_c^{\mathrm{\prime }}+p_r^{\mathrm{\prime }})}{1-0.15{Kp}_c^{\mathrm{\prime }}x}$$ (B5)

For needle / RTFO hydrophones,

$${\sigma }_q\approx \frac{K(1+0.15x^2)(p_c^{\mathrm{\prime }}+p_r^{\mathrm{\prime }})}{1-0.17{Kp}_c^{\mathrm{\prime }}{x}^2}$$ (B6)

where

$$K=\frac{z}{2}\frac{2\pi f_{awf}\beta }{{\rho }_w{c}^3}\frac{1}{\sqrt{F_{a0}}}$$ (B7)

and

$$F_{a0}=\sqrt{\frac{0.69A_{SAeff}}{\pi {\left(0.7{\lambda }_{1}F/\#\right)}^2}}$$ (B8)

Fig. 15 shows accuracies of the numerical solution of (13) and (B1) (for all simulated data) and the approximate formula (B5) (for the subset of simulated data for which d_g/(λ₁F/#) ≤1) for estimating σ_q for the transducers in Table I measured with membrane hydrophones (d_g=200, 500, 600, 1000 μm). For the numerical method, the average RMSD in σ_q estimates was 13% for both membrane and needle / RTFO hydrophones. For the approximate formulas (B5) and (B6) (for the subset for which d_g/(λ₁F/#) ≤ 1), the average RMSD in σ_q estimates was 10% for both membrane and needle / RTFO hydrophones. Eq. (B5) is validated experimentally in the companion article (see Sec. III-F in [1]).

Fig. 15.

Estimated σ_q vs true σ_q for membrane hydrophones for KZK simulations for transducers in Table I. Results are shown for the numerical solution of (13) and (B1) for all.simulations and the approximate formula (B5) for the subset for which x=d_g/(λ₁F/#)≤1.