A systematic study on effects of calibration-waveguide geometry and least-squares formulation on ear-probe source calibrations

Abstract

Measuring ear-canal absorbance and compensating for effects of the ear-canal acoustics on otoacoustic-emission measurements using an ear probe rely on accurately determining its acoustic source parameters. Using pressure measurements made in several rigid waveguides and models of their input impedances, a conventional calibration method estimates the ear-probe Thévenin-equivalent source parameters via a least-squares fit to an over-determined system of equations. Such a calibration procedure involves critical considerations on the geometry and number of utilized calibration waveguides. This paper studies the effects of calibration-waveguide geometry on achieving accurate ear-probe calibrations and measurements by systematically varying the lengths, length ratios, radii, and number of waveguides. For calibration-waveguide lengths in the range of 10–60 mm, accurate calibrations were generally obtained with absorbance measurement errors of approximately 0.02. Longer waveguides resulted in calibration errors, mainly due to coincident resonance frequencies among waveguides in the presence of mismatches between their assumed and actual geometries. The accuracy of calibrations was independent of the calibration-waveguide radius, except for an increased sensitivity of wider waveguides to noise. Finally, it is demonstrated how reformulating the over-determined system of equations to return the least-squares reflectance source parameters substantially reduces calibration and measurement errors.

I. INTRODUCTION

Ear-canal acoustic impedance, reflectance, and absorbance are conveniently measured using an ear probe inserted into and sealed to the ear canal and require a preliminary calibration to obtain the ear-probe acoustic source parameters. These quantities are useful for several hearing-diagnostic applications, including middle-ear diagnostics [e.g., Piskorski et al. (1999) and Keefe et al. (2000)] and compensating for the effects of ear-canal acoustics on otoacoustic-emission measurements [e.g., Souza et al. (2014) and Charaziak and Shera (2017)]. The research literature is dominated by one underlying ear-probe source calibration method and formulation—proposed originally by Allen (1986)—often referred to as the multi-tube calibration method. With this method, pressure measurements made using the ear probe in several cylindrical rigid waveguides of different geometry are related to transmission-line models of their input impedances by the ear-probe Thévenin-equivalent source parameters. Allen (1986) solved this over-determined system of linear equations for the least-squares Thévenin-equivalent source parameters of the ear probe.

Calibrating an ear probe using the multi-tube method involves critical considerations on the number of used waveguides and their geometry. Various reports have utilized vastly different calibration-waveguide lengths, e.g., 12–20 mm (Nørgaard et al., 2017b), 12.1–30.7 mm (Voss and Allen, 1994), 16.4–67.4 mm (Lewis and Neely, 2015), 18.5–83 mm (Lewis, 2018; Lewis et al., 2009; McCreery et al., 2009; Scheperle et al., 2011; Scheperle and Hajicek, 2020; Scheperle et al., 2008), 24.3–73.3 mm (Groon et al., 2015), 28–70 mm (Charaziak and Shera, 2017), 233–500 mm (Keefe et al., 1992), 268–684 mm (Keefe et al., 2000), 475–954 mm (Margolis et al., 1999), and 904–1830 mm (Keefe, 2020). For measurements in adult ears, most reports have used 4–6 waveguides with cross-sectional area close to the average adult ear canal (radii of 3.5–4 mm) which minimizes measurement errors when evanescent modes are not accounted for during calibration and measurements (Nørgaard et al., 2018b). The effects of varying calibration-waveguide geometry and number on achieving accurate ear-probe source parameters have not been systematically studied in the literature.

In this paper, we study the influence of variations in calibration-waveguide geometry and number on obtaining accurate acoustic Thévenin-equivalent calibrations of an ear probe. The lengths, length ratios, and radii of 3D-printed calibration waveguides, and the number of calibration waveguides included in the least-squares solution are systematically varied within bounds of what is practically feasible. Calibration errors are quantified in terms of the accuracy in measuring the absorbance of a standardized occluded-ear simulator, estimating the sound pressure at the reference microphone terminating the ear simulator as the integrated pressure (Lewis et al., 2009), and predicting the emitted pressure (Charaziak and Shera, 2017) of a simulated otoacoustic-emission response when operating the ear-simulator reference microphone as an electro-static speaker. We further demonstrate how reformulating the over-determined system of equations according to Keefe (2020) to return the least-squares ear-probe reflectance source parameters substantially reduces calibration and measurement errors.

II. BACKGROUND

By representing the ear probe as an acoustic Thévenin-equivalent circuit with its two unknown source parameters, the source pressure $P_{\mathrm{s}}$ and source impedance $Z_{\mathrm{s}}$, Allen (1986) related the pressures measured by the ear-probe microphone P_n in the nth of N total acoustic loads to their respective modeled input impedances Z_n,

$$Z_n{P}_{\mathrm{s}}-P_n{Z}_{\mathrm{s}}=P_n{Z}_n.$$ (1)

When the number of loads N > 2, the system of equations becomes over-determined and the least-squares source parameters $P_{\mathrm{s}}$ and $Z_{\mathrm{s}}$ minimize the error

$${ϵ}_{\text{ls}}=\sum _{n=1}^N{\left|Z_n{P}_{\mathrm{s}}-P_n{Z}_{\mathrm{s}}-P_n{Z}_n\right|}^2.$$ (2)

Writing the system of equations in matrix form, $P_{\mathrm{s}}$ and $Z_{\mathrm{s}}$ can be calculated from

$$\left[\begin{array}{c}{P}_{\mathrm{s}}\\ Z_{\mathrm{s}}\end{array}\right]={\left[\begin{array}{cc}{Z}_1& -P_1\\ ⋮& ⋮\\ Z_N& -P_N\end{array}\right]}^{+}\left[\begin{array}{c}{Z}_1{P}_1\\ ⋮\\ Z_N{P}_N\end{array}\right],$$ (3)

where the plus superscript denotes the Moore-Penrose pseudo-inverse matrix.

Previous reports employing the multi-tube calibration method use rigid cylindrical waveguides as the acoustic loads. The pressures P_n are measured with the ear probe inserted into each waveguide n using an ear tip, and their input impedances Z_n are calculated using a plane-wave transmission-line model,

$$Z_n=Z_0\mathrm{\hspace{0.17em}coth\hspace{0.17em}}\gamma l_n.$$ (4)

The complex characteristic impedance Z₀ and propagation constant γ, including thermo-viscous losses, can be calculated according to Keefe (1984) based on the waveguide radius r. In a two-step process, the lengths l_n are first estimated using iterative methods that minimize some error in the least-squares fit. These estimated lengths l_n are then used to calculate the final least-squares source parameters. This multi-tube calibration method and formulation proposed by Allen (1986) remains the most frequently used in the research literature.

The transmission-line model underlying Eq. (4) assumes plane-wave propagation, however, higher-order evanescent modes are elicited in the area discontinuity between the probe tube (which individually connects the internal transducers in the ear-probe body to the ear canal) and the calibration waveguides (Karal, 1953; Keefe and Benade, 1981; Brass and Locke, 1997; Fletcher et al., 2005). Failing to account for these modes during the conventional length-estimation procedure introduces errors and a dependence of the ear-probe source parameters on the calibration-waveguide radius (Nørgaard et al., 2018b). Nørgaard et al. (2017b) removed this radius dependence and corresponding calibration errors from the source parameters by placing the ear probe (without an ear tip) at a well-defined location in the calibration waveguides using a custom ear-probe adapter. They defined the lengths l_n and instead estimated the inertances L_n approximating the contribution from evanescent modes, using a similar two-step process,

$$Z_n=Z_0\mathrm{\hspace{0.17em}coth\hspace{0.17em}}\gamma l_n+j\omega L_n,$$ (5)

where j is the unit imaginary number and ω is the angular frequency.

Different formulations exist for calculating the least-squares ear-probe source parameters and the specific formulation chosen affects the sensitivity of the least-squares solution to various phenomena. However, the formulation of Allen (1986) [Eq. (1)] is used almost exclusively in the research literature. Due to standing waves, P_n and Z_n vary by orders of magnitude across frequency with large values near resonance frequencies and low values near anti-resonance frequencies. The location of these resonance and anti-resonance frequencies depends chiefly on the lengths of the calibration waveguides. Furthermore, P_n are affected by noise near anti-resonance frequencies because the sound pressure drops to low levels, especially for shorter and wider waveguides which exert less thermo-viscous losses. Two aspects are relevant to consider regarding the conventional formulation of Eq. (1) and the resulting least-squares source parameters. First, P_n acts linearly on each term in which it occurs. This causes measurement noise present at anti-resonance frequencies for individual waveguides to be conveniently suppressed in the least-squares solution [Eq. (3)]. Second, the least-squares solution minimizes the squared absolute error in $P_n{Z}_n$ [the right-hand side of Eq. (1)], and Z_n occurs in the first term on the left-hand side of Eq. (1). This causes individual waveguides to excessively bias the least-squares solution at their resonance frequencies where both P_n and Z_n are large. Furthermore, small geometrical errors in the calibration waveguides—e.g., physical deviations from the dimensions used in the transmission-line model—result in absolute errors in P_n and Z_n that are much greater near resonance frequencies.

An alternative calibration method determines the ear-probe reflectance source parameters, the source incident pressure P₀ and source reflectance ${\Gamma }_{\mathrm{s}}$, based on measurements in a semi-anechoic waveguide (Keefe, 1997; Keefe and Simmons, 2003). These reflectance source parameters provide a characterization of the ear probe which is mathematically equivalent to acoustic Thévenin- or Norton-equivalent models. Keefe (2020) combined the direct measurement of the incident pressure P₀ in a semi-anechoic waveguide with measurements in rigid waveguides to provide a unique source incident pressure P₀ and a least-squares source reflectance ${\Gamma }_{\mathrm{s}}$ in a so-called “known-P₀” calibration. Keefe (2020) further reformulated Eq. (1) in terms of the unknown P₀ and ${\Gamma }_{\mathrm{s}}$ in an “unknown-P₀” calibration,

$$(1+{\Gamma }_n)P_0+P_n{\Gamma }_n{\Gamma }_{\mathrm{s}}=P_n,$$ (6)

where the reflectance of each calibration waveguide [given Z_n of Eq. (4)]

$${\Gamma }_n=\frac{Z_n-Z_0}{Z_n+Z_0}=e^{-2\gamma l_n}.$$ (7)

The least-squares reflectance source parameters now minimize the error

$${ϵ}_{\text{ls}}=\sum _{n=1}^N{\left|(1+{\Gamma }_n)P_0+P_n{\Gamma }_n{\Gamma }_{\mathrm{s}}-P_n\right|}^2,$$ (8)

and can be calculated using the matrix form

$$\left[\begin{array}{c}{P}_0\\ {\Gamma }_{\mathrm{s}}\end{array}\right]={\left[\begin{array}{cc}1+{\Gamma }_1& P_1{\Gamma }_1\\ ⋮& ⋮\\ 1+{\Gamma }_N& P_N{\Gamma }_N\end{array}\right]}^{+}\left[\begin{array}{c}{P}_1\\ ⋮\\ P_N\end{array}\right].$$ (9)

That is, the incident pressure P₀ is not measured in an anechoic waveguide, but derived as part of the least-squares solution in the unknown-P₀ formulation of Keefe (2020) in Eq. (6). While this formulation in the context of the Keefe (2020) paper appears to mainly serve the immediate purpose of returning the least-squares reflectance source parameters—as opposed to the Thévenin-equivalent source parameters when using the formulation of Allen (1986) in Eq. (1)—it also has the advantage of normalizing least-squares errors among the waveguides and the bias of individual waveguides on the least-squares source parameters because $\left|{\Gamma }_n\right|\in [0,1]$. That is, with such reformulation, the least-squares source parameters are less biased by individual waveguides at their resonance frequencies. At the same time, P_n still acts linearly on each term in which it occurs whereby the least-squares reflectance source parameters are similarly affected by noise at anti-resonance frequencies.

III. METHODS

A. Evaluation calibration waveguides

We studied the influence of calibration-waveguide geometry and number on acoustic source calibrations of an ear probe using 3D-printed cylindrical evaluation calibration waveguides of varying geometry with a construction principle identical to Nørgaard et al. (2017b). The calibration waveguides were manufactured from Accura Extreme White 200 (3D Systems, Inc., Rock Hill, South Carolina), an ABS-like material (acrylonitrile butadiene styrene). The specified relative tolerance was ±10 μm per mm of printed material (0.1%). That is, when the ear probe is inserted into a calibration waveguide, the tip of its probe tube is flush with the waveguide reference input plane [see Nørgaard et al. (2017b), Fig. 2], and the geometrical dimensions of the acoustic enclosure in front of the ear probe is well-defined within the bounds of the relative tolerance of the 3D-printed material and the absolute tolerance of the custom ear-probe adapter. As described in the following, we took a number of precautions to reduce the number of variable parameters in calibration-waveguide dimensions to something that is practically realizable using the 3D-printing technology at hand. Note that, with our 3D-printed waveguides, we were unable to specify the roughness of the inner walls, however inspecting the layers from the 3D-printing process with a pointed object, the wall appeared consistently smooth. Non-smooth walls may affect the acoustic thermo-viscous boundary layer and introduce excess damping of that predicted by a lossy uniform transmission-line model [Eq. (4)].

Three sets of evaluation calibration waveguides with radius r_i of the ith set were constructed with $r_1=2$ mm, $r_2=4$ mm, and $r_3=5$ mm. $r_1=2$ mm represents the smallest calibration-waveguide radius reported in the literature (Nørgaard et al., 2017b), and is slightly larger than the typical radii of probe tubes (1.65 mm for our utilized ear probe). As previously mentioned, $r_2=4$ mm corresponds approximately to adult ear canals and has been most frequently used in the research literature. Finally, $r_3=5$ mm was included to represent a diameter larger than adult ear canals, but not so large that that higher-order modes propagate at audio frequencies [20.1 kHz for $r_3=5$ mm, see Blackstock (2000)].

For each set of waveguides, the length l_m of the mth of M 3D-printed waveguides was defined using the following expression:

$$l_m=l_1{\varphi }_l^{m-1}.$$ (10)

Setting $l_1=5$ mm, the length ratio ${\varphi }_l=1.17$, and $m=1,2,\dots ,M$ with M = 24, a total of 3 × 24 = 72 evaluation waveguides were constructed with lengths ranging from 5 to 185 mm. The length ratio ${\varphi }_l$ ensures that length differences between sequential waveguides are identical on a logarithmic scale and that resonance frequencies among the evaluation calibration waveguides are spaced equidistantly on a logarithmic frequency axis. ${\varphi }_l=1.17$ was chosen in order to, as much as possible, avoid coincident resonance frequencies for waveguides with sequential lengths l_m. Note that, because the utilized 3D-printing technique specifies a relative tolerance, the relative tolerance on variations in calibration-waveguide resonance frequencies is identical (0.1%) for all waveguides.

B. Ear-probe evaluation calibrations

Based on the measured pressures $P_{i,m}$ with the ear-probe inserted into each m of M evaluation calibration waveguides of the ith set (i.e., of different diameters) and their modeled input impedances $Z_{i,m}$, we calculated the Thévenin-equivalent source pressures $P_{\mathrm{s},i,n,N,\Delta n}$ and source impedances $Z_{\mathrm{s},i,n,N,\Delta n}$ using the formulation of Allen (1986) based on all possible combinations of N waveguides, starting with the nth, using every $\Delta n$ th waveguide,

$$\left[\begin{array}{c}{P}_{\mathrm{s},i,n,N,\Delta n}\\ Z_{\mathrm{s},i,n,N,\Delta n}\end{array}\right]={\left[\begin{array}{cc}{Z}_{i,n}& -P_{i,n}\\ Z_{i,n+\Delta n}& -P_{i,n+\Delta n}\\ Z_{i,n+2\Delta n}& -P_{i,n+2\Delta n}\\ ⋮& ⋮\\ Z_{i,n+(N-1)\Delta n}& -P_{i,n+(N-1)\Delta n}\end{array}\right]}^{+}\left[\begin{array}{c}{Z}_{i,n}{P}_{i,n}\\ Z_{i,n+\Delta n}{P}_{i,n+\Delta n}\\ Z_{i,n+2\Delta n}{P}_{i,n+2\Delta n}\\ ⋮\\ Z_{i,n+(N-1)\Delta n}{P}_{i,n+(N-1)\Delta n}\end{array}\right].$$ (11)

That is, to further reduce the complexity of our analyses, we only consider combinations of evaluation calibration waveguides with sequential lengths l_m in $\Delta n$ steps of identical radii r_i. Note that m refers to a specific waveguide in each set i consisting of a total of M waveguides, whereas n refers to the shortest waveguide used for a specific calibration using a total of N waveguides. We also calculated the source incident pressure $P_{0,i,n,N,\Delta n}$ and the source reflectance ${\Gamma }_{\mathrm{s},i,n,N,\Delta n}$ using the unknown-P₀ formulation of Keefe (2020) and the modeled reflectances of the calibration waveguides ${\Gamma }_{i,m}$,

$$\left[\begin{array}{c}{P}_{0,i,n,N,\Delta n}\\ {\Gamma }_{\mathrm{s},i,n,N,\Delta n}\end{array}\right]={\left[\begin{array}{cc}1+{\Gamma }_{i,n}& P_{i,n}{\Gamma }_{i,n}\\ 1+{\Gamma }_{i,n+\Delta n}& P_{i,n+\Delta n}{\Gamma }_{i,n+\Delta n}\\ 1+{\Gamma }_{i,n+2\Delta n}& P_{i,n+2\Delta n}{\Gamma }_{i,n+2\Delta n}\\ ⋮& ⋮\\ 1+{\Gamma }_{i,n+(N-1)\Delta n}& P_{i,n+(N-1)\Delta n}{\Gamma }_{i,n+(N-1)\Delta n}\end{array}\right]}^{+}\left[\begin{array}{c}{P}_{i,n}\\ P_{i,n+\Delta n}\\ P_{i,n+2\Delta n}\\ ⋮\\ P_{i,n+(N-1)\Delta n}\end{array}\right].$$ (12)

To enable comparison of the two formulations in our analyses and using identical equations in the following, reflectance source parameters were converted to corresponding Thévenin-equivalent source parameters,

$$Z_{\mathrm{s},i,n,N,\Delta n}=Z_{0,i}\frac{1+{\Gamma }_{\mathrm{s},i,n,N,\Delta n}}{1-{\Gamma }_{\mathrm{s},i,n,N,\Delta n}}\hspace{1em}\text{and}\hspace{1em}{P}_{\mathrm{s},i,n,N,\Delta n}=P_{0,i,n,N,\Delta n}\frac{Z_{\mathrm{s},i,n,N,\Delta n}+Z_{0,i}}{Z_{\mathrm{s},i,n,N,\Delta n}},$$ (13)

where $Z_{0,i}$ is the complex characteristic impedance of the ith set of evaluation calibration waveguides.

Here, we consider $\Delta n=1$, 2, and 3, resulting in effective length ratios of ${\varphi }_l^{\Delta n}=1.17$, 1.37, and 1.60, respectively, in our evaluation calibrations. These ${\varphi }_l^{\Delta n}$ do not directly represent those of calibration-waveguide lengths utilized in previous reports for which the length ratios are not constant across calibration waveguides. We evaluated $P_{\mathrm{s},i,n,N,\Delta n}$ and $Z_{\mathrm{s},i,n,N,\Delta n}$ for $N=2,3,\dots ,8$ with all possible combinations of i and n, as described in Sec. III A, for each N, resulting in 1008 sets of $P_{\mathrm{s},i,n,N,\Delta n}$ and $Z_{\mathrm{s},i,n,N,\Delta n}$ for each of the two different considered formulations (Allen, 1986; Keefe, 2020).

The analytical input impedances $Z_{i,m}$ of the waveguides were calculated using a lossy transmission-line model. We accounted for evanescent modes by estimating the inertance $L_{i,m}$ and additional losses due to larger flow gradients near the probe-tube aperture by estimating the loss factor ${\zeta }_{i,m}$ in each waveguide (Nørgaard et al., 2017b),

$$Z_{i,m}=Z_{0,i}\coth {\gamma }_i{l}_m+j\omega L_{i,m}+\sqrt{\omega }{\zeta }_{i,m}.$$ (14)

Here, γ_i is the propagation constant of the ith set of waveguides. The waveguide reflectances were then calculated as

$${\Gamma }_{i,m}=\frac{Z_{i,m}-Z_{0,i}}{Z_{i,m}+Z_{0,i}}\ne e^{-2{\gamma }_i{l}_m}.$$ (15)

$L_{i,m}$ and ${\zeta }_{i,m}$ were estimated by employing the two-step procedure described by Nørgaard et al. (2017b) utilizing the formulation of Allen (1986) with N = 4 waveguides at a time and $\Delta n=1$, i.e., for n = 1, 5, 9, 13, 17, and 21 for waveguides of each radius r_i. However, using different n, N, or $\Delta n$ for this two-step estimation procedure resulted in practically identical $L_{i,m}$ and ${\zeta }_{i,m}$, and, thus, identical $Z_{i,m}$.

C. Ear-simulator measurement

The ear probe was inserted into a uniform occluded-ear simulator according to IEC 60318‐4 (2010) using an ear tip and a perpendicular insertion relative to the cross-sectional plane with an approximate residual ear-canal length of 25 mm. We measured a single ear-probe pressure $P_{\text{ec}}$ in the occluded-ear simulator and used it to calculate the ear-canal evaluation impedances $Z_{\text{ec},i,n,N,\Delta n}$ using each calibrated set of $P_{\mathrm{s},i,n,N,\Delta n}$ and $Z_{\mathrm{s},i,n,N,\Delta n}$,

$$Z_{\text{ec},i,n,N,\Delta n}=Z_{\mathrm{s},i,n,N,\Delta n}\frac{P_{\text{ec}}}{P_{\mathrm{s},i,n,N,\Delta n}-P_{\text{ec}}}.$$ (16)

Next, we estimated the plane-wave ear-canal evaluation impedances $Z_{\text{ec},\text{pl},i,n,N,\Delta n}$ by subtracting an ear-canal evanescent-modes inertance $L_{\text{ec}}$ (Nørgaard et al., 2017a),

$$Z_{\text{ec},\text{pl},i,n,N,\Delta n}=Z_{\text{ec},i,n,N,\Delta n}-j\omega L_{\text{ec}}.$$ (17)

The ear-canal evaluation reflectances ${\Gamma }_{\text{ec},i,n,N,\Delta n}$ were then calculated,

$${\Gamma }_{\text{ec},i,n,N,\Delta n}=\frac{Z_{\text{ec},\text{pl},i,n,N,\Delta n}-R_{0,\text{ec}}}{Z_{\text{ec},\text{pl},i,n,N,\Delta n}+R_{0,\text{ec}}},$$ (18)

where the ear-canal lossless characteristic impedance $R_{0,\text{ec}}=\rho c/A_{\text{ec}}$ with air density ρ, speed of sound c, and ear-canal cross-sectional area $A_{\text{ec}}$. $R_{0,\text{ec}}$ and $L_{\text{ec}}$ were estimated using Nørgaard et al. (2017a) from the ear-simulator measurement based on a reference calibration, as described in Sec. III D. In this way, the estimation of $R_{0,\text{ec}}$ and $L_{\text{ec}}$ is independent of effects of calibration errors on the evaluation source parameters $P_{\mathrm{s},i,n,N,\Delta n}$ and $Z_{\mathrm{s},i,n,N,\Delta n}$—that we aim to quantify—on individual ${\Gamma }_{\text{ec},i,n,N,\Delta n}$. The ear-canal evaluation (power) absorbances are

$$A_{\text{ec},i,n,N,\Delta n}=1-|{\Gamma }_{\text{ec},i,n,N,\Delta n}{|}^2.$$ (19)

Based on the measured ear-canal pressure $P_{\text{ec}}$ and the ear-canal evaluation reflectances ${\Gamma }_{\text{ec},i,n,N,\Delta n}$, we further calculated the evaluation integrated pressures $P_{\text{int},i,n,N,\Delta n}$ (Lewis et al., 2009),

$$P_{\text{int},i,n,N,\Delta n}=\left|P_{\text{ec}}{H}_{\text{em},i,n,N,\Delta n}\right|H_{\text{int},i,n,N,\Delta n},$$ (20)

with the evaluation total-to-integrated-pressure transfer functions

$$H_{\text{int},i,n,N,\Delta n}=\left|\frac{1}{1+{\Gamma }_{\text{ec},i,n,N,\Delta n}}\right|+\left|\frac{{\Gamma }_{\text{ec},i,n,N,\Delta n}}{1+{\Gamma }_{\text{ec},i,n,N,\Delta n}}\right|,$$ (21)

and the evanescent-modes-to-plane-wave pressure transfer functions (Nørgaard et al., 2017a)

$$H_{\text{em},i,n,N,\Delta n}=\frac{Z_{\text{ec},\text{pl},i,n,N,\Delta n}}{Z_{\text{ec},i,n,N,\Delta n}}.$$ (22)

The evaluation integrated pressures $P_{\text{int},i,n,N,\Delta n}$ approximate the pressure at the tympanic membrane and were compared to the pressure measured by the reference microphone $P_{\text{mic}}$ terminating the occluded-ear simulator.

Finally, we simulated an otoacoustic emission in the ear simulator by operating the reference microphone at the coupler termination as an electro-static speaker injecting the volume flow $U_{\text{tm}}$ (Nørgaard et al., 2019a). The total pressure $P_{\text{spl}}$ at the ear probe in response to this simulated otoacoustic emission was measured and used to calculate the evaluation emitted pressures (Charaziak and Shera, 2017)

$$P_{\text{epl},i,n,N,\Delta n}=P_{\text{spl}}{H}_{\text{epl},i,n,N,\Delta n},$$ (23)

with the evaluation total-to-emitted-pressure transfer functions

$$H_{\text{epl},i,n,N,\Delta n}=\frac{1-{\Gamma }_{\text{ec},i,n,N,\Delta n}\Gamma {\prime }_{\mathrm{s},i,n,N,\Delta n}}{T(1+\Gamma {\prime }_{\mathrm{s},i,n,N,\Delta n})}.$$ (24)

Note that the ear-probe source reflectances in the occluded-ear simulator

$$\Gamma {\prime }_{\mathrm{s},i,n,N,\Delta n}=\frac{Z_{\mathrm{s},i,n,N,\Delta n}-R_{0,\text{ec}}}{Z_{\mathrm{s},i,n,N,\Delta n}+R_{0,\text{ec}}}\ne {\Gamma }_{\mathrm{s},i,n,N,\Delta n},$$ (25)

differ from ${\Gamma }_{\mathrm{s},i,n,N,\Delta n}$ [Eq. (12)] because they characterize the ear probe in the specific assumed lossless ear canal. Disregarding here the emitted-pressure phase, we set T = 1 [see Charaziak and Shera (2017)].

D. Reference calibration and measurement

To establish a reference ear-canal absorbance $A_{\text{ec},\text{ref}}$ for assessing the accuracy of evaluation ear-canal absorbances $A_{\text{ec},i,n,N,\Delta n}$ using calibrations based on different combinations of evaluation calibration waveguides, a reference calibration of the ear probe was conducted according to Nørgaard et al. (2017b) using their proposed steel calibration waveguides with radius r = 2 mm and lengths l = 12, 14.5, 17.5, and 20 mm (logarithmic-mean length of 15.7 mm) to obtain a set of reference Thévenin-equivalent source parameters $P_{\mathrm{s},\text{ref}}$ and $Z_{\mathrm{s},\text{ref}}$. The reference ear-canal impedance $Z_{\text{ec},\text{ref}}$ and plane-wave impedance $Z_{\text{ec},\text{pl},\text{ref}}$, reflectance ${\Gamma }_{\text{ec},\text{ref}}$, absorbance $A_{\text{ec},\text{ref}}$, integrated pressure $P_{\text{int},\text{ref}}$, and emitted pressure $P_{\text{epl},\text{ref}}$ were then calculated using Eqs. (16)–(25) based on $P_{\mathrm{s},\text{ref}}$ and $Z_{\mathrm{s},\text{ref}}$. $L_{\text{ec}}$ and $R_{0,\text{ec}}$ were estimated from $Z_{\text{ec},\text{ref}}$ according to Nørgaard et al. (2017a) and further used for calculating the evaluation ear-canal plane-wave impedances $Z_{\text{ec},\text{pl},i,n,N,\Delta n}$ [Eq. (17)] and reflectances ${\Gamma }_{\text{ec},i,n,N,\Delta n}$ [Eq. (18)], respectively, as described in Sec. III C.

Nørgaard et al. (2019a) demonstrated that the combination of calibration (Nørgaard et al., 2017b) and measurement (Nørgaard et al., 2017a) methods used as reference here produces accurate integrated pressures, and reflectances and emitted pressures independent of insertion depth into a uniform occluded-ear simulator when the ear probe is inserted perpendicular to the ear-simulator cross-sectional plane. Nørgaard et al. (2020) further reproduced this reflectance of the occluded-ear simulator using an alternative measurement technique based on a two-microphone probe by approximating the acoustic flow as the finite-difference pressure gradient (Seybert and Ross, 1977). This alternative technique does not require the ear-canal cross-sectional area $A_{\text{ec}}$ in the calculation of ear-canal reflectance. We thus assert that our reference reflectance ${\Gamma }_{\text{ec},\text{ref}}$, absorbance $A_{\text{ec},\text{ref}}$, and emitted pressure $P_{\text{epl},\text{ref}}$ in the occluded-ear simulator are accurate.

Rather than observing calibration errors as a function of frequency, we consider various maximum errors ${\text{max}}_fϵ$ across different frequency ranges f. The maximum absolute absorbance measurement errors were calculated as

$$\text{max}{ϵ}_{A_{\text{ec},i,n,N,\Delta n}}={\text{max}}_f|A_{\text{ec},i,n,N,\Delta n}-A_{\text{ec},\text{ref}}|.$$ (26)

The largest absolute deviations in dB between evaluation integrated $P_{\text{int},i,n,N,\Delta n}$ and reference-microphone $P_{\text{mic}}$ pressures were calculated as

$$\text{max}{ϵ}_{P_{\text{int},i,n,N,\Delta n}}={\text{max}}_f\left|20\hspace{0.17em}{\text{log}}_{10}\frac{\left|P_{\text{mic}}\right|}{P_{\text{int},i,n,N,\Delta n}}\right|,$$ (27)

and, similarly, the maximum deviations between evaluation emitted pressures $P_{\text{epl},i,n,N,\Delta n}$ and the reference emitted pressure $P_{\text{epl},\text{ref}}$,

$$\text{max}{ϵ}_{P_{\text{epl},i,n,N,\Delta n}}={\text{max}}_f\left|20\hspace{0.17em}{\text{log}}_{10}\frac{\left|P_{\text{epl},i,n,N,\Delta n}\right|}{\left|P_{\text{epl},\text{ref}}\right|}\right|.$$ (28)

Maximum errors are plotted as a function of the logarithmic-mean lengths $l_{\text{avg},n,N,\Delta n}$ of the utilized calibration waveguides,

$$l_{\text{avg},n,N,\Delta n}=\text{exp}\hspace{0.17em}\sum _{k=0}^{N-1}\frac{\hspace{0.17em}\text{log}\hspace{0.17em}{l}_{n+k\Delta n}}{N},$$ (29)

where k is an integer index variable used for the summation.

E. Measurements and equipment

All our reported measurements were performed using a FireFace UC sound card (RME Audio, Haimhausen, Germany), controlled through custom-written matlab (The MathWorks, Inc., Natick, Massachusetts) software and the third-party utility Playrec.¹ We used a custom Titan-based ear probe (Interacoustics A/S, Middelfart, Denmark), modified for improved high-frequency performance and reduced internal crosstalk. This ear probe utilizes a probe tube with flush speaker and microphone tubes. Acoustic ear-probe pressure responses in the evaluation and reference calibration waveguides, and the occluded-ear simulator were measured by supplying a wideband acoustic click stimulus to the ear-probe speaker at a sampling rate of 44.1 kHz. The stimulus was frequency-normalized to produce a flat zero-phase electrical spectrum on the terminals of the ear-probe microphone with the ear probe inserted into an anechoic waveguide of radius r = 4 mm. The electrical frequency normalization applied to the stimulus ensured that the dominant noise source—i.e., electrical noise from the ear-probe microphone—had an approximately constant influence across frequency. Stimuli were presented at the beginning of blocks in lengths of 4096 samples (approximately 92.9 ms in duration) enabling capturing the entire transient response of the longest evaluation calibration waveguides. Thus, the spectral resolution of our analyses when calculating the discrete Fourier transform is 10.8 Hz. For each measurement, 256 blocks were recorded and synchronously averaged to reduce noise. To ensure a similar influence of noise across all measurements, a simple artifact-rejection algorithm was employed that discarded two sequential blocks if the total RMS voltage of the difference between the blocks, divided by $\sqrt{2}$, exceeded –85 dB V (corresponding to 39 dB SPL, assuming a frequency-independent microphone sensitivity of –30 dB V/Pa). In that case, the stimulus was repeated until 256 artifact-free blocks had been recorded and included in the synchronous average. We calibrated the ear-probe microphone for its complex frequency-dependent acoustic sensitivity using the method and coupler described by Nørgaard et al. (2018a). A Type 4157 occluded-ear simulator (Brüel & Kjær Sound & Vibration A/S, Nærum, Denmark) according to IEC 60318‐4 (2010) with a radius of r = 3.75 mm was used with a 3D-printed part attached that uniformly extended its nominal length (i.e., total length minus probe insertion) to approximately 25 mm. The ear probe was inserted into the ear simulator using a green, 9-mm, mushroom-shaped silicone-rubber ear tip (Sanibel Supply, Middelfart, Denmark). The frequency-independent flow from the reference microphone operating as an electro-static speaker $U_{\text{tm}}$ was estimated at low frequencies as described by Nørgaard et al. (2019a).

F. Supplementary materials

In the supplementary materials to this paper,² we include the reported results in the matlab data file data.mat. Executing the script plotData.m analyzes the data and generates the figures reported in Sec. IV. Relevant individual lines in the script are cross-referenced against corresponding equations in this paper. The script was tested using matlab version R2021b, requiring no extra toolboxes. Note that uncommenting the lines defining the length of rolling-average smoothing windows requires the matlab Signal Processing Toolbox. We invite interested readers to apply their own or custom calibration methods, e.g., two-step procedures for acoustically estimating waveguide lengths, to the data to visualize calibration and measurement errors.

IV. RESULTS

A. Reference measurement and distributions of evaluation measurements

Figure 1(a) shows the measured reference absorbance $A_{\text{ec},\text{ref}}$ of the occluded-ear simulator, and Figs. 1(b), 1(c) show 2D histograms of all 1008 evaluation absorbances $A_{\text{ec},i,n,N,\Delta n}$ [Eq. (19)] at individual bin frequencies on the x axis and an absorbance bin size of 0.01 on the y axis using calibrations based on the considered formulations of Allen (1986) [Eq. (11)] and Keefe (2020) [Eq. (12)], respectively. Notice the logarithmic colorbar of the 2D histograms. The majority of measurement points across all $A_{\text{ec},i,n,N,\Delta n}$ coincide with $A_{\text{ec},\text{ref}}$, but a large number of individual measurement points are affected by substantial errors. $A_{\text{ec},\text{ref}}$ and the yellow regions in the $A_{\text{ec},i,n,N,\Delta n}$ histograms correspond to reflectance magnitudes practically identical to those reported by Nørgaard et al. (2020) (their Fig. 3), who compared reflectance measurements of the occluded-ear simulator using identical calibration and measurement methods to an alternative measurement technique using the two-microphone probe. It is evident how the formulation of Keefe (2020) results in a slight decrease in the spread of measurement errors, especially visible at low frequencies and otherwise mainly when comparing Figs. 1(b), 1(c) on top of each other. Figures 1(d)–1(f) show quantities identical to Figs. 1(a)–1(c), but with absorbances averaged into one-third-octave bands with center frequencies ranging from 125 Hz to 16 kHz. Visually, this averaging process seems to reduce the incidence of measurement errors when comparing Figs. 1(b), 1(c) to Figs. 1(e), 1(f), but some errors remain. Similar to Figs. 1(b), 1(c), the formulation of Keefe (2020) seems to reduce the incidence of measurement errors in $A_{\text{ec},i,n,N,\Delta n}$.

FIG. 1.

(Color online) The reference (a),(d) absorbances $A_{\text{ec},\text{ref}}$, (g),(j) integrated-pressure errors $P_{\text{mic}}/P_{\text{int}}$, and (m),(p) normalized emitted pressures $P_{\text{epl},\text{ref}}/U_{\text{tm}}{R}_{0,\text{ec}}$, and 2D histograms (note the logarithmic colorbar) of the evaluation (b),(c),(e),(f) absorbances $A_{\text{ec},i,n,N,\Delta n}$ [Eq. (19)], (h),(i),(k),(l) integrated-pressure errors $P_{\text{mic}}/P_{\text{int},i,n,N,\Delta n}$ [see Eq. (20)], and (n),(o),(q),(r) normalized emitted pressures $P_{\text{epl},i,n,N,\Delta n}/U_{\text{tm}}{R}_{0,\text{ec}}$ [see Eq. (23)] in the occluded-ear simulator, [(a)–(c), (g)–(i), (m)–(o)] at individual bin frequencies, (d)–(f) with the absorbances averaged into one-third-octave bands, and at individual bin frequencies with the total-to-integrated-pressure $H_{\text{int},i,n,N,\Delta n}$ [Eq. (21)] and total-to-emitted-pressure $H_{\text{epl},i,n,N,\Delta n}$ [Eq. (24)] transfer functions smoothed using (j)–(l) 79-point and (p)–(r) 149-point, respectively, Blackman-window moving averages, using evaluation calibrations based on the formulations of (b),(e),(h),(k),(n),(q) Allen (1986) [Eq. (11)] and (c),(f),(i),(l),(o),(r) Keefe (2020) [Eq. (12)].

Figure 1(g) shows the ratio of the pressure recorded at the reference microphone to the reference integrated pressure $P_{\text{mic}}/P_{\text{int},\text{ref}}$, and Figs. 1(h), 1(i) show 2D histograms of the ratios to the evaluation integrated pressures $P_{\text{mic}}/P_{\text{int},i,n,N,\Delta n}$ [see Eqs. (20) and (21)] at individual bin frequencies on the x axis and a bin size of 0.5 dB on the y axis using evaluation calibrations based on the formulations of Allen (1986) and Keefe (2020), respectively. These ratios then represent the stimulus errors were the integrated pressures $P_{\text{int},\text{ref}}$ or $P_{\text{int},i,n,N,\Delta n}$ used for in-situ calibration of stimuli. $P_{\text{int},\text{ref}}$ is within ±1 dB of $P_{\text{mic}}$ up and 16 kHz, and the majority of measurement points in $P_{\text{int},i,n,N,\Delta n}$ coincide with $P_{\text{int},\text{ref}}$, but substantial errors can be observed at several remaining measurement points across $P_{\text{int},i,n,N,\Delta n}$. The slight deviation of $P_{\text{int},\text{ref}}$ and $P_{\text{int},i,n,N,\Delta n}$ from $P_{\text{mic}}$ above 16 kHz is a result of the dust protector in the occluded-ear simulator which causes an absorption of sound near and above 20 kHz. It is evident that large measurement errors in the reflectances ${\Gamma }_{\text{ec},i,n,N,\Delta n}$ at low frequencies do not introduce errors into $P_{\text{int},i,n,N,\Delta n}$, however, at higher frequencies, corresponding errors in the ear-canal reflectance start to impact the integrated pressure. Figures 1(j)–1(l) show quantities identical to Figs. 1(g)–1(i) with the total-to-integrated-pressure transfer functions $H_{\text{int},\text{ref}}$ and $H_{\text{int},i,n,N,\Delta n}$ [Eq. (21)] smoothed using a 79-point Blackman-window moving average across frequency. The length of this window was chosen to avoid dominant errors in the integrated pressures $P_{\text{int},\text{ref}}$ and $P_{\text{int},i,n,N,\Delta n}$ due to smoothing near ear-canal anti-resonance frequencies, which are represented in $H_{\text{int},\text{ref}}$ and $H_{\text{int},i,n,N,\Delta n}$. The effect of the smoothing is visible at 11 kHz in Figs. 1(j)–1(l). Comparing Figs. 1(h), 1(i) to Figs. 1(k), 1(l), errors in $P_{\text{int},i,n,N,\Delta n}$ have, to some degree, been reduced from the smoothing process, but not entirely. Both without and with smoothing, the improvements using Keefe (2020) over Allen (1986) are hardly evident unless comparing these figures on top of each other.

Last, Fig. 1(m) shows the reference emitted pressure normalized by the acoustic flow injected at the tympanic membrane and the ear-canal characteristic impedance $P_{\text{epl},\text{ref}}/U_{\text{tm}}{R}_{0,\text{ec}}$, and Figs. 1(n), 1(o) show 2D histograms of the normalized evaluation emitted pressures $P_{\text{epl},i,n,N,\Delta n}/U_{\text{tm}}{R}_{0,\text{ec}}$ [see Eqs. (23) and (24)] at individual bin frequencies with with a y-axis bin size of 0.5 dB using evaluation calibrations based on the formulations of Allen (1986) and Keefe (2020), respectively. The emitted pressures normalized by $U_{\text{tm}}{R}_{0,\text{ec}}$ deviate from unity (0 dB) to the degree that the simulated middle ear of the occluded-ear simulator does not operate as an ideal flow source (Nørgaard et al., 2019a) and are within –2 and 0 dB up to 16 kHz. Again, most measurement points in $P_{\text{epl},i,n,N,\Delta n}$ coincide with $P_{\text{epl},\text{ref}}$, but substantial errors are also present. Figures 1(p)–1(r) show quantities identical to Figs. 1(m)–1(o) with the total-to-emitted-pressure transfer functions $H_{\text{epl},\text{ref}}$ and $H_{\text{epl},i,n,N,\Delta n}$ [Eq. (24)] smoothed using a 149-point Blackman-window moving average across frequency. The length of this window was chosen to avoid dominant errors in integrated pressures $P_{\text{epl},\text{ref}}$ and $P_{\text{epl},i,n,N,\Delta n}$ due to smoothing near ear-canal resonance frequencies, which are represented in $H_{\text{epl},\text{ref}}$ and $H_{\text{epl},i,n,N,\Delta n}$. When comparing Figs. 1(n), 1(o) to Figs. 1(q), 1(r), the smoothing seems to drastically decrease the incidence of errors in $P_{\text{epl},i,n,N,\Delta n}$. Note that the smoothing introduces low-frequency errors into $P_{\text{epl},\text{ref}}$ and $P_{\text{epl},i,n,N,\Delta n}$ because of the inverse-frequency behavior in $H_{\text{epl},\text{ref}}$ and $H_{\text{epl},i,n,N,\Delta n}$ resulting from the compliance of the ear canal and middle ear. Similarly here, the merits of using Keefe (2020) over Allen (1986) are hardly evident unless a direct comparison is made.

B. Measurement errors using Allen (1986)

1. Measurement errors up to 8 kHz

Figure 2 shows maximum absorbance ${ϵ}_{A_{\text{ec},i,n,N,\Delta n}}$ [Eq. (26)], integrated-pressure ${ϵ}_{P_{\text{int},i,n,N,\Delta n}}$ [Eq. (27)], and emitted-pressure ${ϵ}_{P_{\text{epl},i,n,N,\Delta n}}$ [Eq. (28)] errors as a function of the logarithmic-mean length $l_{\text{avg},n,N,\Delta n}$ [Eq. (29)] of the utilized evaluation calibration waveguides starting with the nth of N in steps of $\Delta n$ waveguides using evaluation calibrations based on the formulation of Allen (1986) across a frequency range f of 0.25–8 kHz. Each subfigure column shows results for each of the considered evaluation-calibration-waveguide radii r_i.

FIG. 2.

(Color online) The maximum (a)–(f) absorbance ${ϵ}_{A_{\text{ec},i,n,N,\Delta n}}$ [Eq. (26)], (g)–(l) integrated-pressure ${ϵ}_{P_{\text{int},i,n,N,\Delta n}}$ [Eq. (27)], and (m)–(r) emitted-pressure ${ϵ}_{P_{\text{epl},i,n,N,\Delta n}}$ [Eq. (28)] errors in the occluded-ear simulator over the frequency range f of 0.25–8 kHz as a function of the logarithmic-mean length $l_{\text{avg},n,N,\Delta n}$ [Eq. (29)] of the utilized evaluation calibration waveguides of radii (a),(d),(g),(j),(m),(p) $r_1=2$ mm, (b),(e),(h),(k),(n),(q) $r_2=4$ mm, and (c),(f),(i),(l),(o),(r) $r_3=5$ mm, using the waveguide step sizes (solid lines) $\Delta n=1$, (dashed lines) $\Delta n=2$, and (dotted lines) $\Delta n=3$ using evaluation calibrations based on the formulation of Allen (1986) [Eq. (11)]. Results are shown [(a)–(c), (g)–(i), (m)–(o)] for individual bin frequencies without smoothing, (d)–(f) with absorbances averaged into one-third octave bands, and for individual bin frequencies with the total-to-integrated-pressure $H_{\text{int},i,n,N,\Delta n}$ [Eq. (21)] and the total-to-emitted-pressure $H_{\text{epl},i,n,N,\Delta n}$ [Eq. (21)] transfer functions smoothed using (j)–(l) 79-point and (p)–(r) 149-point, respectively, Blackman-window moving averages.

Figures 2(a)–2(c) show $\text{max}{ϵ}_{A_{\text{ec},i,n,N,\Delta n}}$ for individual bin frequencies. For all radii r_i, it is evident how shorter calibration-waveguide lengths generally perform poorly, mainly as a result of noise; for shorter waveguides, reduced thermo-viscous losses affecting forward- and reverse-propagating waves undergoing a round trip cause larger anti-resonance frequency dips and more noise in their proximity. For $\Delta n=1$, this effect is more pronounced because anti-resonance frequencies are more closely spaced in frequency, and for $\Delta n=2$ and 3, this influence of noise with shorter waveguides is therefore reduced. Additionally, wider waveguides exhibit further reduced thermo-viscous losses, evident from increased measurement errors in Fig. 2(c). Finally, absolute tolerances in the mechanical probe adapter introduce larger relative tolerances for shorter compared to longer waveguides. For longer waveguides, substantial absorbance errors start to occur with logarithmic-mean lengths above approximately 60 mm. These increasing errors are mainly a result of coincident resonance frequencies among the utilized evaluation calibration waveguides in the presence of small geometrical errors. As discussed in Sec. II, the absolute change in $P_n{Z}_n$ due to geometrical errors is much greater at resonance frequencies compared to anti-resonance frequencies. Thus, when two or more resonance frequencies coincide in the presence of geometrical errors, these waveguides with large errors in $P_n{Z}_n$ bias the least-squares solution simultaneously, resulting in calibration errors. This effect of coincident resonance frequencies seems more pronounced for $\Delta n=2$ and 3, compared to $\Delta n=1$, presumably because these resonances start to coincide at a lower frequency. Furthermore, the effect of coinciding resonance frequencies is less pronounced for smaller radii because of the increased thermo-viscous damping and, thus, a smaller relative change in $P_n{Z}_n$ given some length error. For longer waveguides, noise also starts to affect the calibrations because anti-resonance frequencies occur at lower frequencies and noise is usually inversely proportional to frequency. For all radii, similar absorbance measurement errors are observed using logarithmic-mean lengths $l_{\text{avg},i,n,N,\Delta n}$ around 20 mm. The evaluation calibration waveguides with $r_1=2$ mm seem to provide accurate calibrations within a wide range of $l_{\text{avg},i,n,N,\Delta n}$ from approximately 10–60 mm with absorbance measurement errors as low as 0.02. Note that such small errors might as well be a result of small errors in the reference absorbance $A_{\text{ec},\text{ref}}$. For larger radii, this range narrows down with increasing radius. N = 4 waveguides seems to constitute the point of diminishing returns where adding additional calibration waveguides in many cases does not qualitatively improve the result of the absorbance measurement. A two-load N = 2 calibration appears unfeasible in most cases with seemingly random errors occurring across different waveguide dimensions and numbers. However, in some cases, the two-load calibration does provide accurate results. Figures 2(d)–2(f) show $\text{max}{ϵ}_{A_{\text{ec},i,n,N,\Delta n}}$ with absorbances $A_{\text{ec},\text{ref}}$ and $A_{\text{ec},i,n,N,\Delta n}$ averaged into one-third-octave bands. While the smoothing results in a lower $\text{max}{ϵ}_{A_{\text{ec},i,n,N,\Delta n}}$ across the combinations of calibration waveguides, many of the above described issues persist, and one certainly needs to consider the dimensions of the calibration waveguides.

Figures 2(g)–2(i) show $\text{max}{ϵ}_{P_{\text{int},i,n,N,\Delta n}}$ at individual bin frequencies. From these results, it is evident how even large absorbance measurement errors have little to no impact on the calibration of stimulus levels, and choosing calibration-waveguide geometries that provide good performance for this purpose is less critical. Still, a slightly improved performance is observed using the $r_1=2$ mm evaluation calibration waveguides compared to $r_2=4$ mm and $r_3=5$ mm. In most cases, $N\ge 4$ provides accuracy within 1 dB, except for calibration waveguides with logarithmic-mean lengths larger than approximately 100 mm. Note that the integrated pressure mainly seeks to compensate for the effects of quarter-wave anti-resonance frequencies and, with the nominal ear-canal length of 25 mm in the occluded-ear simulator, only a single anti-resonance frequency is present at approximately 3.5 kHz within the current considered frequency range of 0.25–8 kHz. Figures 2(j)–2(l) show $\text{max}{ϵ}_{P_{\text{int},i,n,N,\Delta n}}$ at individual bin frequencies with only the total-to-integrated-pressure transfer functions $H_{\text{int},i,n,N,\Delta n}$ [Eq. (21)] smoothed using the 79-point Blackman-window moving average. It is evident that the smoothing process in most cases has a minimal effect on the already low stimulus errors.

Figures 2(m)–2(o) show $\text{max}{ϵ}_{P_{\text{epl},i,n,N,\Delta n}}$ at individual bin frequencies. It is evident that the effects of calibration errors on the emitted pressure are larger compared to the integrated pressure. Still, most combinations of calibration-waveguide geometry result in emitted-pressure errors smaller than 1 dB. Note that the emitted pressure seeks to compensate for the amplification of an otoacoustic-emission response near half-wave resonance frequencies, for which the first one occurs at approximately 7 kHz in this case. Figures 2(p)–2(r) show $\text{max}{ϵ}_{P_{\text{epl},i,n,N,\Delta n}}$ at individual bin frequencies with only the total-to-integrated-pressure transfer functions $H_{\text{epl},i,n,N,\Delta n}$ [Eq. (21)] smoothed using the 149-point Blackman-window moving average (no smoothing was applied to the reference total-to-emitted-pressure transfer function $H_{\text{epl},\text{ref}}$). For the specific results in the occluded-ear simulator, it is evident that the smoothing reduces deviations in the emitted-pressure error, which is now within 1 dB in most cases.

2. Measurement errors up to 16 kHz

Figure 3 shows quantities and conditions identical to Fig. 2, as described in Sec. IV B 1, but in the frequency range f of 0.25–16 kHz. We deliberately chose the upper limit of this frequency range due to the deviation between the reference $P_{\text{int},\text{ref}}$ and evaluation $P_{\text{int},i,n,N,\Delta n}$ integrated pressures from the reference-microphone pressure $P_{\text{mic}}$ toward 20 kHz [see Figs. 1(m)–1(r)]. For the absorbances in Figs. 3(a)–3(c), maximum errors are expectedly generally larger when including an additional octave in the frequency spectrum. Errors due to coincident resonance frequencies start occurring at a lower logarithmic-mean length of approximately 45 mm. Furthermore, the raw calibrations in the $r_3=5$ mm waveguides are all affected by rather large errors. This seems to be a result of an increased sensitivity to noise at higher frequencies when fewer waveguides are included in the calibration because their anti-resonance frequencies occur above 8 kHz. For the one-third-octave-band-averaged absorbances in Figs. 3(d)–3(f), errors are now down to a low level, however, note that these results only include three more bands compared to the results in Figs. 2(d)–2(f). Conclusions similar to those above for the narrow frequency range can be drawn from the integrated-pressure errors in Figs. 3(g)–3(i). $r_1=2$ mm provides the lowest errors, followed by $r_2=4$ mm and then $r_3=5$ mm. The smoothing process utilized in Figs. 3(j)–3(l) provides only little improvement. Finally, similar observations are made for the emitted-pressure errors in Figs. 3(m)–3(o), where the smoothing procedure in Figs. 3(p)–3(r) provides a more substantial improvement.

FIG. 3.

(Color online) Quantities and conditions identical to Fig. 2, but showing respective maximum errors across the frequency range f of 0.25–16 kHz.

C. Measurement errors using Keefe (2020)

Figures 4 and 5 show quantities and conditions identical to those in Figs. 2 and 3, respectively, but using the formulation of Keefe (2020) [Eq. (12)]. While most of the various observations described in Sec. III B using the formulation of Allen (1986) are equally descriptive for these results, it is evident that maximum absorbance, integrated-pressure, and emitted-pressure errors have been significantly reduced across the majority of considered combinations of evaluation calibration waveguides and that a wider range of calibration-waveguide lengths provide accurate calibrations. This improvement is mostly visible for waveguide combinations with logarithmic-mean lengths shorter than 10 mm and longer than 60 mm that previously resulted in larger errors, and particularly for $r_2=4$ mm and even more for $r_3=5$ mm. Presumably, this is because P_m and Z_m are increasingly sensitive to geometrical errors at their resonance frequencies with increasing radius and the formulation of Keefe (2020) largely mitigates the problematic aspects of these errors. In the region of logarithmic-mean lengths between 10 and 60 mm, calibration errors are already low using the formulation of Allen (1986), and no significant improvement in using the formulation of Keefe (2020) can be observed. The two-load calibrations N=2 are, naturally, exactly identical to those in Figs. 2 and 3.

FIG. 4.

(Color online) Quantities and conditions identical to Fig. 2, but using evaluation calibrations based on the formulation of Keefe (2020) [Eq. (12)].

FIG. 5.

(Color online) Quantities and conditions identical to Fig. 2, but using evaluation calibrations based on the formulation of Keefe (2020) [Eq. (12)] and showing respective maximum errors across the frequency range f of 0.25–16 kHz.

V. DISCUSSION

By necessity, our study and analyses included only a subset of possible combinations of calibration-waveguide geometry. While our length ratio ${\varphi }_l$ [Eq. (10)] is constant across our utilized waveguide sets, the integer step variable $\Delta n$ allows us to further consider effective length ratios of ${\varphi }_l^{\Delta n}$ in our evaluation calibrations. Our utilized waveguide lengths, length ratios, radii, and number of waveguides encompass those utilized in some other studies employing the multi-tube calibration method, e.g., Voss and Allen (1994), Scheperle et al. (2008), McCreery et al. (2009), Lewis et al. (2009), Scheperle et al. (2011), Groon et al. (2015), Lewis and Neely (2015), Charaziak and Shera (2017), Nørgaard et al. (2017b), Lewis (2018), and Scheperle and Hajicek (2020). To reduce complexity of our data analyses, and due to manufacturing limitations associated with 3D printing, we have not considered longer waveguides such as those used by Keefe et al. (1992), Margolis et al. (1999), Keefe et al. (2000), and Keefe (2020).

The results of this study clearly show how our longer waveguides perform poorly for ear-probe source calibrations mainly due to coincident resonance frequencies resulting from their increased spectral density. The results further demonstrate how reformulating the system of equations in terms of the waveguide reflectances and the reflectance source parameters (Keefe, 2020) to a large degree mitigates this issue. However, our results cannot shed light on the performance of calibrations using waveguides longer than those considered in this study [e.g., Keefe et al. (1992), Margolis et al. (1999), Keefe et al. (2000), and Keefe (2020)]. Acoustically, such longer waveguides differ substantially with their increased thermo-viscous losses and less pronounced resonance peaks. These characteristics might result in a smaller impact of geometrical errors at coincident resonance frequencies, despite their increased spectral density, and a lower sensitivity to noise.

Our results illustrate how octave-band averaging or smoothing in many cases can adequately mitigate calibration errors. However, with considerately chosen calibration-waveguide geometry, it is possible to obtain reliable ear-probe source calibrations, based on which measurements do not require smoothing to provide accurate results at all individual bin frequencies, even at high frequencies. Of the waveguide combinations utilized in this study, this was generally achieved for logarithmic-mean length of 10–60 mm, using 3 or more waveguides, and using radii of 2 and 4 mm. We find the ability to maintain accurate data in a format as raw as possible preferable to avoid requiring such smoothing procedures.

It is worth noting that some previous studies utilized ear probes with protruded microphone tubes [e.g., Keefe et al. (1992), Voss and Allen (1994), and Huang et al. (2000)], which seeks to minimize the influence of higher-order evanescent modes emanating from the speaker-tube aperture on the microphone-tube aperture. Huang et al. (2000) calibrated an ear probe with a protruded microphone tube using waveguides of varying radii and noted a dependence of the ear-probe source parameters on the radius resulting from the varying residual space between the probe-tube speaker and microphone apertures. They also demonstrated that, in the case of a protruded microphone tube, any mismatch between the radii of the calibration waveguides and the ear canal introduces considerable measurement errors due to the difference in residual space between the tube apertures during calibration and measurement. Presumably as a result of this dependency, later studies and commercial probe systems seem to prefer ear probes with flush probe tubes. However, when evanescent modes are not accounted for during calibration, this still introduces a dependency on the calibration-waveguide radius (Nørgaard et al., 2017b). Our results demonstrate that, when evanescent modes are adequately accounted for, the calibration-waveguide cross-sectional area does not need to match that of an adult ear canal. Instead, disregarding errors due to noise with increasing diameter, we obtained virtually identical calibrations using our various considered calibration-waveguide radii.

In our study, calibration errors near coincident resonance frequencies occur due to mechanical tolerances in our calibration waveguides because we define their geometry in our formulations. In cases where the ear probe is inserted into each waveguide using an ear tip and the calibration-waveguide lengths are estimated acoustically, geometrical errors instead occur due to the effects of the coupling between the ear probe and the waveguide on this length estimation, e.g., evanescent modes (Nørgaard et al., 2017b) and oblique ear-probe insertions (Nørgaard et al., 2019b). When the ear probe is inserted into a single waveguide and its length is varied using, e.g., a movable piston (Iseberg et al., 2015), length-estimation errors are identical for all waveguides and not detected in the error of the least-squares fit, but the source parameters still depend on the calibration-waveguide radius (Nørgaard et al., 2018b). When the ear probe is manually inserted into each waveguide, small differences in the coupling introduce length-error differences among the waveguides which may result in calibration errors at coincident resonance frequencies.

Our study has demonstrated that our calibration waveguides with logarithmic-mean lengths ranging from approximately 10 to 60 mm result in more accurate ear-probe source calibrations, opposed to waveguides shorter than 10 mm and longer than 60 mm, as a result of their lower spectral density of resonance frequencies. These dimensions also seem preferable to longer waveguides from manufacturing and practical points of view. Waveguides shorter than 10 mm seem undesirable because they are more susceptible to noise and suffer excessively from absolute tolerances related to the placement of the ear probe. We were surprised that our two-load N = 2 calibrations, in which the system of two equations with two unknowns returns a unique solution for the source parameters, in a few cases provide accurate measurements. However, despite its practical merits, we find such two-load calibration undesirable because it offers no immediate means of assessing the quality of a given calibration. Conversely, for a calibration with N > 2 loads, the calibration error can conveniently be quantified in terms of a non-dimensional error (Keefe et al., 1992), or the ability of the least-squares source parameters in estimating the acoustic pressures (Scheperle et al., 2011) or the input impedances (Nørgaard et al., 2017b) of the calibration waveguides at each bin frequency. These variables are able to identify various types of calibration errors, e.g., due to not properly inserting the ear probe into a calibration waveguide. In this study, we have not considered anechoic- and semi-anechoic-waveguide calibrations where the ear-probe source incident pressure is uniquely determined (Keefe, 1997, 2020; Keefe and Simmons, 2003). Similar to the two-load calibration, such methods rely on the ability to accurately determine the unique source incident pressure and do not share the practical merits of short rigid waveguides.

Ideally, measurements of ear-canal acoustic impedance and derived quantities using a source-calibrated ear probe are independent of the type of utilized ear probe. However, calibration errors due to any mismatch between the calibration-waveguide model and true acoustic impedances giving rise to measurement errors depend on the characteristics of the ear probe, more specifically, its acoustic source impedance. Strictly speaking, the results reported in this study are therefore specific for our utilized ear probe. However, because typical ear probes consist of a number of transducers housed in the ear-probe body and connected to the acoustic load using individual tubes in the probe tube with an outer radius similar to our utilized ear probe (1.65 mm), their source impedances are similar and generally larger than the characteristic impedance of our utilized calibration waveguides. Therefore, we believe that our results are still valid as general guidance for selecting appropriate dimensions of calibration waveguides.

Strictly speaking, our reference measurement cannot be considered the ground truth because the reflectance of the occluded-ear simulator remains unknown. Nørgaard et al. (2020) merely demonstrated that their two utilized measurement techniques using an ear probe and a two-microphone probe result in identical reflectances of the occluded-ear simulator, but its true reflectance characteristic remains unknown. Note that IEC 60318‐4 (2010) only specifies the characteristics of the occluded-ear simulator up to 10 kHz, thus, its similarity with an adult ear above this frequency is uncertain. However, we do not consider the specific utilized ear simulator a crucial part of our study and similar results would likely have been obtained using a different type of occluded-ear simulator. Furthermore, because our study involved repeatedly manually inserting the ear probe into calibration waveguides, couplers, and ear simulators, it cannot be ruled out that small physical changes to the ear probe occurred during the course of the measurement sequence, gradually altering its true source parameters. Still, the alignment between the reference absorbance $A_{\text{ec},\text{ref}}$ and the majority of evaluation-absorbance $A_{\text{ec},i,n,N,\Delta n}$ measurement points, and the remaining derived quantities in Fig. 1 is quite striking. This gives us confidence that our approach is adequate for quantifying measurement errors in ear-canal absorbance and reflectance, and in accounting for the effects of the ear-canal acoustics on otoacoustic-emission measurements.

VI. CONCLUSION

Calibrating an ear probe to determine its acoustic source parameters using the multi-tube calibration method and rigid calibration waveguides involves critical considerations on calibration-waveguide geometry and number. In this study, we have investigated the influence of calibration-waveguide geometry on obtaining accurate calibrations by systematically varying the lengths, length ratios, radii, and number of calibrations waveguides. Calibration errors were quantified by observing the maximum errors introduced into measurements of ear-canal absorbance, estimating the sound pressure at the tympanic membrane, and calculating the emitted pressure in an occluded-ear simulator over frequency ranges up to 8 and 16 kHz. These measurements were referenced against calibration and measurement methods previously shown to provide accurate results in the occluded-ear simulator. We found that shorter waveguides with logarithmic-mean lengths in the range of 10 to 60 mm generally produced accurate calibrations. These calibrations were independent of the calibration-waveguide radius and length ratio, thus eliminating the need to match radii of the calibration waveguides with the ear canal. However, calibrations using wider waveguides tended to be more sensitive to noise. Increased calibration errors were observed using our utilized calibration waveguides longer than 60 mm due to coincident resonance frequencies among the waveguides, and using our waveguides shorter than 10 mm due to a larger influence of noise at anti-resonance frequencies. However, conclusions cannot be drawn for waveguides longer than 185 mm, the longest used in our study. Finally, reformulating the over-determined system of equations in terms of the reflectance source parameters (Keefe, 2020), as opposed to the Thévenin-equivalent source parameters (Allen, 1986), significantly reduced calibration and measurement errors across the majority of considered combinations of calibration-waveguide geometry.