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Effect and minimization of errors in in situ ground impedance measurements
Available online at www.sciencedirect.com
Applied Acoustics 69 (2008) 884–890 www.elsevier.com/locate/apacoust
Effect and minimization of errors in in situ ground impedance measurements Roland Kruse *, Volker Mellert Oldenburg University, Institute of Physics, 26111 Oldenburg, Germany Received 19 February 2007; received in revised form 18 May 2007; accepted 27 May 2007 Available online 19 July 2007
Abstract The transfer function method is a procedure to measure the surface impedance of grounds in situ. In this article, the influence of measurement errors on the predicted surface impedance is investigated numerically. Even small errors in the range of accuracy of common measurement equipment can lead to significant errors in the impedance. This is especially true for errors in the transfer function at frequencies below about 500 Hz and for highly reflecting grounds. To a lesser degree, errors in the measurement geometry contribute to the uncertainty of the estimated impedance. To minimize these effects, an improved geometry is suggested for the frequency range 100– 400 Hz significantly reducing the average error. However, even with this optimized geometry the average error for high impedance grounds, like compacted silt, in this frequency range will be around 50%. Therefore, the use of the transfer function method cannot be recommended in this case unless the requirements in accuracy are very low for a specific application or particular favourable measurement conditions are given. Ó 2007 Elsevier Ltd. All rights reserved. PACS: 43.58.Bh Keywords: Ground impedance; In situ impedance measurement
1. Introduction The surface impedance of outdoor grounds is an important parameter for the prediction of the sound propagation, especially for small distances when the effect of meteorological factors is often low compared to the ground effect. Generally, the measurement of the ground impedance can only be done using in situ methods because small samples for use in the impedance tube [1] do not correctly represent the properties of the extended and sometimes layered/inhomogeneous outdoor ground. Commonly used for in situ impedance measurements – both in room acoustics and for outdoor use – are impulse–echo (‘‘Adrienne’’) methods [2,3] and the two*
Corresponding author. Tel.: +49 441 798 5255; fax: +49 441 798 3698. E-mail address: [email protected] (R. Kruse).
0003-682X/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2007.05.010
microphone (transfer function) technique [4, ‘‘Iterative Procedure’’]. While the impulse–echo methods, because of their simple theoretical background, are easy the implement, they rely on a plane wave like reflection and are therefore not adequate for low frequencies and small or even grazing incidence angles. The two-microphone method, on the other hand, is commonly used in combination with a sound field model taking the spherical nature of the waves at low frequencies into account and is therefore, theoretically, well suited for that purpose. A variant of this method is used in the American standard ANSI S1.18 [5]. In this standard, the magnitude of the transfer function between two microphones situated above the ground is measured using one of three predefined geometries. The user will then (visually) compare this ‘‘level difference spectrum’’ with spectra provided in the standard and choose the one that fits best over the whole frequency range. These spectra (‘‘templates’’) were derived using a
R. Kruse, V. Mellert / Applied Acoustics 69 (2008) 884–890
sound field model and a surface impedance from either a one or a two-parameter absorber model. Thereby, the user will obtain absorber parameters by which the surface impedance can easily be calculated. Currently, a new ANSI standard is proposed (working group S1/WG 20) to allow a direct deduction of the ground impedance without the use of templates. This procedure, while requiring a sophisticated data evaluation, would have the advantage of being independent of any absorber model. The use of the transfer function method in such a standard raises concerns about the robustness of the method, because the method may become more widespread and may be used by persons less experienced in this area of acoustics. The influence of selected measurement errors on the predicted surface impedance is investigated in this paper. Small errors in the measurement geometry can hardly be avoided. Distance measurements, e.g. with a tape measure, have limited accuracy and the ground surface is not well defined. Another source of errors is the fact that the model requires the knowledge of the position of the acoustical centre of the loudspeaker which can be different from its geometrical centre [6]. Errors may not only occur in the geometry but also in the measured transfer function T. Phase errors arise from the fact that standard microphones are not phase-matched. Errors in the magnitude of T can arise from slight differences in calibration and frequency response of the microphones. Both errors may also follow from unwanted reflections, e.g. from the measurement equipment or other reflecting objects nearby as well as from meteorological influences (wind and temperature) causing spatial or temporal changes in the speed of sound. The necessary measurement precision is derived from the error predictions. Subsequently, an optimization is done to select the best measurement geometry for a given frequency range and surface impedance. All impedances presented are normalized to the impedance of air and assume an exp(ixt) time dependence.
885 R=1m Upper mic. hru = 20 cm
Loudspeaker hs = 20 cm
Lower mic. hrl = 5 cm
θ Ground
Fig. 1. Measurement set-up for the transfer function method. Geometry B from ANSI S1.18.
T ¼
pðupperÞ /ðupperÞ ¼ pðlowerÞ /ðlowerÞ
ð1Þ
Five measurements at different locations were done. Geometry B from ANSI S1.18 was used as it showed the best overall performance on this type of ground: Source height (hs) = 20 cm, upper microphone height (hru) = 20 cm, lower microphone height (hrl) = 5 cm, source–receiver distance (R) = 1 m. 2.2. Sound field model and impedance deduction A widespread model for the reflection of spherical waves on a locally reacting impedance plane [7] was used for the calculation of the surface impedance from the two-microphone transfer function. This model assumes an exp(ixt) time dependence. The velocity potential / is a function of the lengths of the direct path r1 and reflected path r2, the wave number k, the angle of incidence h and the normalized surface admittance b. It is defined by /¼
eikr1 eikr2 þ ½Rp þ ð1 Rp ÞF r1 r2
ð2Þ
with pffiffiffi 2 F ¼ 1 þ i pkek erfcðikÞ
ð3Þ
2. Experimental procedure
Rp is the reflection coefficient for plane waves (Eq. (4)).
2.1. Measurement set-up
Rp ¼
Measurements of the two-microphone transfer function were done on a (grass-covered) soccer field. The temperature was 27 °C and the wind speed <1 m/s. The background noise was below 45 dB(A). The measurement set-up is shown in Fig. 1. The sound source was a 10 cm loudspeaker in a (13 cm)3 closed cabinet. The emitted signal was pink noise with a level of not less than 80 dB(A) at the microphone positions to ensure the signal was well above the background noise. B&K 4189 microphones (1/200 , IEC 651 Type 1) with windscreens were used. The transfer function T as defined in Eq. (1) was determined using B&K PULSE 10, p representing the sound pressure at the two-microphone positions. It is equal to the ratio of the velocity potentials /:
The numerical distance k is defined in Eq. (5) representing a common simplification of the definition by [8]: rffiffiffiffiffiffiffiffiffiffiffi 1 k¼ ikr2 ðsinðhÞ þ bÞ ð5Þ 2
sinðhÞ b sinðhÞ þ b
ð4Þ
The surface impedance Z was calculated from the measured transfer function T (Eq. (1)) and geometry by the Newton–Raphson algorithm which finds the zero of the function (predicted T observed T) and therefore gains the surface impedance Z which minimizes the difference between the observed transfer function and the transfer function predicted by the model. This very efficient method has been described in [9]. The calculation was started at the lowest frequency point (100 Hz) with a seed value for the
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R. Kruse, V. Mellert / Applied Acoustics 69 (2008) 884–890
iteration of b = 0.05. The obtained solution is used as seed value for the next higher frequency point. While the iteration may, in general, sometimes converge on a ‘‘wrong’’ solution (local minima), such a behaviour was not observed even if the initial seed value was varied. For all calculations, Matlab R2006b was used. The Matlab code can be found on the authors webpage. 1 2.3. Effect of errors To determine the influence of errors in the geometry or transfer function on the estimated surface impedance, a two-parameter model (Eqs. (6) and (7)) [5] was (least square) fitted to the impedance using data from 400 to 4000 Hz, averaged over 1/3rd octave bands, to obtain an estimate Z0 of the impedance at low frequencies. rffiffiffiffiffi 1 re ð6Þ ReðZ 0 =q0 c0 Þ ¼ pffiffiffiffiffiffiffiffiffiffi pcq0 f rffiffiffiffiffi 1 r e c 0 ae ImðZ 0 =q0 c0 Þ ¼ pffiffiffiffiffiffiffiffiffiffi þ ð7Þ 8pcf pcq0 f q0 and cc are the density and speed of sound in air, c the ratio of specific heats, re the effective flow resistivity and ae a parameter representing an effective rate of change of porosity with depth. Now, the transfer function T was calculated from this impedance using geometry B (or geometry B with slight errors). T was then slightly changed in amplitude (±0.2 dB) or phase (±0.5°) and the surface impedance calculated. The predicted impedance with measurement errors is then compared to the nominal impedance Z0. These error estimated were chosen to account mainly for the differences in the microphone frequency responses which, in a free sound field, were determined to be 0.1 dB, respectively, 0.5°, on average. Additional 0.1 dB were added to allow for other (external) influences. 2.4. Geometry optimization To find an optimized geometry for the lower frequency range, which is least sensitive to measurement errors, the following procedure was used: (1) Bounds for the geometry were chosen to allow for an easy set-up and keeping the distances small to minimize the influence of meteorological factors. (2) The achievable precision of the measurement (respectively the measurement errors) was defined. (3) The magnitude of the estimated surface impedance jZj was calculated for all geometries on a 5 cm grid. (4) Because of the possible nonlinear relation between the errors and their effect on Z, it is not sufficient to calculate the error in Z only for the maximum error estimate. The calculation was done for four val1
http://www.physik.uni-oldenburg.de/aku/.
ues within the error range, e.g. for the error in jTj, estimated to be ±0.2 dB, the values 0.2, 0.1, 0.1, 0.2 dB were chosen. The resulting error Ex was defined as the standard deviation of jZj for the four error values, calculated at each frequency point with the predictand and jZ0j. (5) To allow for an efficient calculation, the effects of the errors were considered to be independent except for the magnitude and phase of the transfer function. This reduces the number of calculations for each frequency point and geometry to 32 instead of 4096 needed if all error combinations were considered. (6) Step 3 was repeated for three surface impedances Z0 (low, medium, high) from two-parameter models [5]: Pine forest floor (re = 7.5 kPa s/m2, ae = 16 m1), soccer field (re = 200 kPa s/m2, ae = 40 m1) and compacted silt (re = 4000 kPa s/m2, ae = 115 m1). The calculation was done for the frequency range 100–400 Hz divided into 1/3rd octave steps, the number of frequency points N thereby being seven. (7) In analogy to the Gaussian error propagation law, the (relative) average total error ATE for each geometry was defined as the geometrical sum of the single errors Ex – divided by the ‘‘true’’ impedance Z0 – averaged over the frequency range f. 1 X ATE ¼ N f
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E2T þ E2hru þ E2hrl þ E2hs þ E2R jZ 0 j
ð8Þ
The optimal geometry – observing the boundary conditions in Table 1 – shall be the geometry with the lowest ATE. 3. Results The estimated surface impedance of the soccer field (average of five measurements) is shown in Fig. 2. While the course of the impedance is rather smooth and does agree well with impedance models for grass-covered ground [10] for frequencies above about 400 Hz, the decrease in impedance at lower frequencies does not at all agree with models for porous absorbers.
Table 1 Boundary conditions for the geometry optimization
Transfer function Lower microphone height Distance upper–lower mic. Source height Source–receiver distance
Range (m)
Error estimate
– 0.05–1 0.1–1 0.2–1.5 1–3
±0.2 dB, ±0.5° ±2 cm ±2 cm ±2 cm ±5 cm
Ranges of the geometry and estimates of the achievable measurement precision.
R. Kruse, V. Mellert / Applied Acoustics 69 (2008) 884–890 20
887
30
15 20 10 10
Impedance
Impedance
5 0 -5 -10
0
-10
-15 -20 -20 -25 100
300
500
1000
2000
-30 100
4000
300
500 1000 Frequency [Hz]
Frequency [Hz]
Fig. 2. Predicted surface impedance of a grass-covered soccer field. Real part (—), imaginary part (– –), fitted two-parameter model ( ).
2000
4000
Fig. 4. Effect of errors in the microphone position on the estimated surface impedance for geometry B. Correct impedance (—), upper microphone 2 cm too high (– –), lower microphone 2 cm too high ( ).
3.1. Effect of errors The fitted two-parameter model on which the error analysis is based is shown in Fig. 2. The obtained absorber parameters re = 200 kPa s/m2 and ae = 40 m1 are comparable to the values re = 183 kPa s/m2 and ae = 40 m1 which are stated for lawn in the ANSI standard. In Fig. 3, the effect of an error of 0.2 dB, respectively, 0.5° in the transfer function is shown. Even small errors in T cause high errors in the estimated surface impedance at low frequencies. It should also be noted that the magnitude of the error does not necessarily increase continuously with decreasing frequency. In Fig. 4, the effect of an error in the microphone positions is presented. The effect is much smaller than the effect
of an error in T, but errors in the upper microphone position can still cause significant deviations. Not shown but also observed is the fact that errors in the source height and source–receiver distance have only a minor effect on the surface impedance estimate (for geometry B and the assumed impedance of the soccer field). The large effect of transfer function errors will now be further analysed. In Figs. 5 and 6, the relative error in the real and imaginary part of the impedance is shown in relation to magnitude and phase errors of T. The effect is not only large but also asymmetric: For most of the error combinations the real part of the impedance will be too low and the imaginary part too high. Furthermore, there’s a strong interaction between the effect of magnitude and phase errors.
30
20
Impedance
10
0
-10
-20
-30
-40 100
300
500 1000 Frequency [Hz]
2000
4000
Fig. 3. Effect of errors in the transfer function on the estimated surface impedance for geometry B. Correct impedance (—), transfer function increased by 0.2 dB (– –), phase error of +0.5° ( ).
0.2
1
1
5 0.
2 2
0.2
1
0.15 0.1 0
0.5 0.5
0.2
0
0 -0.05
-0 .2
0.2
0.5
0
0.2
-0 .2
0
-0 .5
-0.2
-0.1 -0.15
2 1
1
0.2
0.05
0 -0.2
-0 .5
-0.5
-0.5 -1
Error in magnitude of the transfer function [dB]
0.5
0.25
-0.2
-1
-0.25 -1
-0.5
0
0.5
1
Error in phase of the transfer function [deg]
Fig. 5. Effect of errors in the magnitude and phase of the transfer function on the real part of the predicted surface impedance for geometry B at 250 Hz. Relative error (1 = 100%).
R. Kruse, V. Mellert / Applied Acoustics 69 (2008) 884–890
0.25
2
1 1
0.5
0. 2
0
3.2. Geometry optimization 0.5
.5 -0
0.2
-0 .2
0.05
1
2 0.51 0.2 -0 .20 -1 -2
0.5
0.1
-1
0 -0.05
0
-0 .5
-0.1
0
-0.2
-0 .2
-0.5
-0.15 0
-1
-0.2 0.2
-2
-0.25 -1
0.2
-0.5
0
0.5
1
Error in phase of the transfer function [deg]
Fig. 6. Effect of errors in the magnitude and phase of the transfer function on the imaginary part of the predicted surface impedance for geometry B at 250 Hz. Relative error (1 = 100%).
1.5
Average total error
only true for the transfer function but also for the geometry.
1
0.2 0.15
0. 2
Error in magnitude of the transfer function [dB]
888
1
The results of the optimization of the geometry for the frequency range 100–400 Hz for three different surface impedances can be found in Table 2. For all impedances, the optimum values R = 3 m, hs = 0.5 m and hrl = 5 cm are the same, while the upper microphone height varies between 1 m and 0.8 m. The effect of this optimization on the average total error ATE is shown in Fig. 7. In comparison with the predefined geometries from the ANSI standard, the average error can be reduced significantly for all three surface impedances to about half of the value of the best of these geometries. However, for the high surface impedance the ATE is still 53%. In Fig. 8, the effect of an error of 0.2 dB, respectively, 0.5° in the transfer function is shown, comparable to Fig. 3 for geometry B. The error is significantly reduced, especially in the imaginary part.
Table 2 Results of the geometry optimization for the frequency range 100–400 Hz for three different surface impedances
0.5
0
Z_low
Z_medium
Z_high
Lower microphone height (cm) Upper microphone height (m) Source height (m) Source–receiver distance (m)
Fig. 7. Average total error (relative) in the predicted surface impedance for three surface impedances and geometries A–C [5] as well as the optimized geometry. Geometry A ( ), B ( ), C ( ), optimized geometry ( ).
Low impedance
Medium impedance
High impedance
5
5
5
1
0.95
0.8
0.5 3
0.5 3
0.5 3
40 30
jT j 0:03 dB; phaseðT Þ 0:2 ; hru 1 cm; hrl 2 cm; hs 2:5 cm; R 5 cm
20
Impedance
From Figs. 5 and 6 as well as likewise diagrams2 for the combinations hru, hrl and hs, R, the following estimates for the necessary measurement precision (geometry B, 250 Hz) are derived, if the error is to be smaller than 20% in both real and imaginary part and no interactions between the three error combinations are considered:
10 0 -10 -20 -30
A likewise analysis revealed that for high surface impedances or lower frequencies the necessary precision becomes even higher. In case of high surface impedances this is not 2
Matlab code for generating these diagrams can be found on the author’s website.
-40 100
300
500 1000 Frequency [Hz]
2000
4000
Fig. 8. Effect of errors in the transfer function on the estimated surface impedance for the optimized geometry. Correct impedance (—), transfer function increased by 0.2 dB (– –), phase error of +0.5° ( ).
R. Kruse, V. Mellert / Applied Acoustics 69 (2008) 884–890
889
4. Discussion
5. Conclusion
Taken into account the results of the error analysis, it seems that the often reported decrease of the predicted surface impedance at low frequencies is the result of measurement uncertainties. The investigated soccer field was large, flat and homogeneous, it therefore resembles one of the best measurement location one can hope to encounter during ground impedance measurements. On the other hand, small errors in the transfer function result in deviations from the expected surface impedance as observed in the measurement, with the real part being too low and the imaginary part too high. To a lesser degree, this is also true for errors in the geometry, especially the upper microphone height. To obtain acceptable results at low frequencies with geometry B, which is well suited for frequencies above 400 Hz and ground with medium impedance, the necessary measurement precision for an error less than 20% is already high at 250 Hz and can only be achieved with very great care and high quality equipment including phase-matched microphones. Fortunately, the effect of errors in the geometry on the impedance is less asymmetric compared to the effect of errors in the transfer function and can therefore be reduced by averaging multiple measurements on the same ground including repositioning of source and receivers. The optimized geometry for the frequency range 100– 400 Hz does feature a significant advantage over the predefined geometries with respect to its error sensitivity, but the average error for high impedance surfaces is still high. The very high error for geometry C in this case does agree with the standards recommendation to use this geometry only for very soft grounds. It should be kept in mind that the necessary measurement precision itself depends on how accurate the surface impedance needs to be measured and therefore on the specific application. In general, the effect of a given percental error in the impedance is highest near the first ground dip, the frequency range in which the presence of the ground leads to a high attenuation compared to the free field situation. The position of the ground dip depends on the impedance and geometry. As an example, the ground dip above the soccer field with source and receiver at 1 m height and the distance between them being 100 m is located around 750 Hz. In this case, an error of 50% in the surface impedance leads to an underestimation of the sound pressure at 400 Hz by 17 dB, while at 1 kHz the sound pressure is overestimated by only 3 dB. Below 400 Hz, the error decreases and reaches 1.3 dB at 100 Hz. For higher propagation distances, the error does decrease slower with decreasing frequency. Thus, the frequency range below 1 kHz should receive particular attention regarding errors in the predicted surface impedance.
The two-microphone method is an established procedure for the in situ measurement of the surface impedance. A disadvantage is that it is sensitive to errors both in the measured transfer function and, to a minor degree, to errors in the geometry. The effect of these errors depends on the measurement geometry and the surface impedance. For low impedance surfaces, it is possible to measure with acceptable accuracy at frequencies >100 Hz even with the predefined geometries from the ANSI standard. Unfortunately, such highly absorbing materials rarely occur in nature with the important exception of snow. For materials with higher flow resistivity, the use of the predefined geometries cannot be recommended for frequencies below about 500 Hz. The optimization procedure, on the other hand, leads to a geometry which provides a significant higher error tolerance and should enable the researcher to obtain reasonable results down to 100 Hz. However, for materials with a very high surface impedance like compacted silt or asphalt, even the optimized geometry cannot guarantee low errors. Taken in mind that the proposed source–receiver distance is 3 m and therefore the maximum allowed distance for the optimization it is possible that better geometries with larger distances exist. Larger distances, however, would be more affected by meteorological factors and require large surfaces. Therefore, the use of the two-microphone method is not recommended in such situations unless the requirements in accuracy are low for a specific application or very favourable measurement conditions (large, flat surfaces, no wind) and high quality equipment including phase- matched microphones are available. Acknowledgements The authors thank their student Ping Rong for his helpful comments on the data processing and interpretation and his valuable Matlab programming. References [1] ISO 10534-2. Determination of sound absorption coefficient and impedance in impedance tubes – Part 2: transfer-function method. International Organization for Standardization; 1998. [2] Mommertz E. Angle-dependent in situ measurement of reflection coefficients using a subtraction technique. Appl Acoust 1995;46: 251–63. [3] CEN/TS 1793-5. Road traffic noise reducing devices – test method for determining the acoustic performance – intrinsic characteristics – in situ values of sound reflection and airborne sound insulation. European Committee for Standardization; 2003. [4] Allard JF, Champoux Y. In situ two-microphone technique for the measurement of the acoustic surface impedance of materials. Noise Contr Eng J 1989;32(1):15–23. [5] ANSI S1.18-1999 (R2004). Template method for ground impedance. American National Standards Institute; 2004. [6] Fuhs S, Ho¨ldrich R, Tomberger G. Validierung des Entfernungsgesetzes und Korrektur der Gruppenlaufzeit und des akustischen
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Zentrums des Lautsprechers im Adrienne-Verfahren. DAGA 2006. Braunschweig. [7] Nobile MA, Hayek SI. Acoustic propagation over an impedance plane. J Acoust Soc Am 1985;78(4):1325–36. [8] Ingard U. On the reflection of a spherical wave from an impedance plane. J Acoust Soc Am 1951;23(3):329–35.
[9] Taherzadeh S, Attenborough K. Deduction of ground impedance from measurements of excess attenuation spectra. J Acoust Soc Am 1999;105(3):2039–42. [10] Donato RJ. Impedance models for grass-covered ground. J Acoust Soc Am 1977;61(3):1449–52.
Applied Acoustics 69 (2008) 884–890 www.elsevier.com/locate/apacoust
Effect and minimization of errors in in situ ground impedance measurements Roland Kruse *, Volker Mellert Oldenburg University, Institute of Physics, 26111 Oldenburg, Germany Received 19 February 2007; received in revised form 18 May 2007; accepted 27 May 2007 Available online 19 July 2007
Abstract The transfer function method is a procedure to measure the surface impedance of grounds in situ. In this article, the influence of measurement errors on the predicted surface impedance is investigated numerically. Even small errors in the range of accuracy of common measurement equipment can lead to significant errors in the impedance. This is especially true for errors in the transfer function at frequencies below about 500 Hz and for highly reflecting grounds. To a lesser degree, errors in the measurement geometry contribute to the uncertainty of the estimated impedance. To minimize these effects, an improved geometry is suggested for the frequency range 100– 400 Hz significantly reducing the average error. However, even with this optimized geometry the average error for high impedance grounds, like compacted silt, in this frequency range will be around 50%. Therefore, the use of the transfer function method cannot be recommended in this case unless the requirements in accuracy are very low for a specific application or particular favourable measurement conditions are given. Ó 2007 Elsevier Ltd. All rights reserved. PACS: 43.58.Bh Keywords: Ground impedance; In situ impedance measurement
1. Introduction The surface impedance of outdoor grounds is an important parameter for the prediction of the sound propagation, especially for small distances when the effect of meteorological factors is often low compared to the ground effect. Generally, the measurement of the ground impedance can only be done using in situ methods because small samples for use in the impedance tube [1] do not correctly represent the properties of the extended and sometimes layered/inhomogeneous outdoor ground. Commonly used for in situ impedance measurements – both in room acoustics and for outdoor use – are impulse–echo (‘‘Adrienne’’) methods [2,3] and the two*
Corresponding author. Tel.: +49 441 798 5255; fax: +49 441 798 3698. E-mail address: [email protected] (R. Kruse).
0003-682X/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2007.05.010
microphone (transfer function) technique [4, ‘‘Iterative Procedure’’]. While the impulse–echo methods, because of their simple theoretical background, are easy the implement, they rely on a plane wave like reflection and are therefore not adequate for low frequencies and small or even grazing incidence angles. The two-microphone method, on the other hand, is commonly used in combination with a sound field model taking the spherical nature of the waves at low frequencies into account and is therefore, theoretically, well suited for that purpose. A variant of this method is used in the American standard ANSI S1.18 [5]. In this standard, the magnitude of the transfer function between two microphones situated above the ground is measured using one of three predefined geometries. The user will then (visually) compare this ‘‘level difference spectrum’’ with spectra provided in the standard and choose the one that fits best over the whole frequency range. These spectra (‘‘templates’’) were derived using a
R. Kruse, V. Mellert / Applied Acoustics 69 (2008) 884–890
sound field model and a surface impedance from either a one or a two-parameter absorber model. Thereby, the user will obtain absorber parameters by which the surface impedance can easily be calculated. Currently, a new ANSI standard is proposed (working group S1/WG 20) to allow a direct deduction of the ground impedance without the use of templates. This procedure, while requiring a sophisticated data evaluation, would have the advantage of being independent of any absorber model. The use of the transfer function method in such a standard raises concerns about the robustness of the method, because the method may become more widespread and may be used by persons less experienced in this area of acoustics. The influence of selected measurement errors on the predicted surface impedance is investigated in this paper. Small errors in the measurement geometry can hardly be avoided. Distance measurements, e.g. with a tape measure, have limited accuracy and the ground surface is not well defined. Another source of errors is the fact that the model requires the knowledge of the position of the acoustical centre of the loudspeaker which can be different from its geometrical centre [6]. Errors may not only occur in the geometry but also in the measured transfer function T. Phase errors arise from the fact that standard microphones are not phase-matched. Errors in the magnitude of T can arise from slight differences in calibration and frequency response of the microphones. Both errors may also follow from unwanted reflections, e.g. from the measurement equipment or other reflecting objects nearby as well as from meteorological influences (wind and temperature) causing spatial or temporal changes in the speed of sound. The necessary measurement precision is derived from the error predictions. Subsequently, an optimization is done to select the best measurement geometry for a given frequency range and surface impedance. All impedances presented are normalized to the impedance of air and assume an exp(ixt) time dependence.
885 R=1m Upper mic. hru = 20 cm
Loudspeaker hs = 20 cm
Lower mic. hrl = 5 cm
θ Ground
Fig. 1. Measurement set-up for the transfer function method. Geometry B from ANSI S1.18.
T ¼
pðupperÞ /ðupperÞ ¼ pðlowerÞ /ðlowerÞ
ð1Þ
Five measurements at different locations were done. Geometry B from ANSI S1.18 was used as it showed the best overall performance on this type of ground: Source height (hs) = 20 cm, upper microphone height (hru) = 20 cm, lower microphone height (hrl) = 5 cm, source–receiver distance (R) = 1 m. 2.2. Sound field model and impedance deduction A widespread model for the reflection of spherical waves on a locally reacting impedance plane [7] was used for the calculation of the surface impedance from the two-microphone transfer function. This model assumes an exp(ixt) time dependence. The velocity potential / is a function of the lengths of the direct path r1 and reflected path r2, the wave number k, the angle of incidence h and the normalized surface admittance b. It is defined by /¼
eikr1 eikr2 þ ½Rp þ ð1 Rp ÞF r1 r2
ð2Þ
with pffiffiffi 2 F ¼ 1 þ i pkek erfcðikÞ
ð3Þ
2. Experimental procedure
Rp is the reflection coefficient for plane waves (Eq. (4)).
2.1. Measurement set-up
Rp ¼
Measurements of the two-microphone transfer function were done on a (grass-covered) soccer field. The temperature was 27 °C and the wind speed <1 m/s. The background noise was below 45 dB(A). The measurement set-up is shown in Fig. 1. The sound source was a 10 cm loudspeaker in a (13 cm)3 closed cabinet. The emitted signal was pink noise with a level of not less than 80 dB(A) at the microphone positions to ensure the signal was well above the background noise. B&K 4189 microphones (1/200 , IEC 651 Type 1) with windscreens were used. The transfer function T as defined in Eq. (1) was determined using B&K PULSE 10, p representing the sound pressure at the two-microphone positions. It is equal to the ratio of the velocity potentials /:
The numerical distance k is defined in Eq. (5) representing a common simplification of the definition by [8]: rffiffiffiffiffiffiffiffiffiffiffi 1 k¼ ikr2 ðsinðhÞ þ bÞ ð5Þ 2
sinðhÞ b sinðhÞ þ b
ð4Þ
The surface impedance Z was calculated from the measured transfer function T (Eq. (1)) and geometry by the Newton–Raphson algorithm which finds the zero of the function (predicted T observed T) and therefore gains the surface impedance Z which minimizes the difference between the observed transfer function and the transfer function predicted by the model. This very efficient method has been described in [9]. The calculation was started at the lowest frequency point (100 Hz) with a seed value for the
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iteration of b = 0.05. The obtained solution is used as seed value for the next higher frequency point. While the iteration may, in general, sometimes converge on a ‘‘wrong’’ solution (local minima), such a behaviour was not observed even if the initial seed value was varied. For all calculations, Matlab R2006b was used. The Matlab code can be found on the authors webpage. 1 2.3. Effect of errors To determine the influence of errors in the geometry or transfer function on the estimated surface impedance, a two-parameter model (Eqs. (6) and (7)) [5] was (least square) fitted to the impedance using data from 400 to 4000 Hz, averaged over 1/3rd octave bands, to obtain an estimate Z0 of the impedance at low frequencies. rffiffiffiffiffi 1 re ð6Þ ReðZ 0 =q0 c0 Þ ¼ pffiffiffiffiffiffiffiffiffiffi pcq0 f rffiffiffiffiffi 1 r e c 0 ae ImðZ 0 =q0 c0 Þ ¼ pffiffiffiffiffiffiffiffiffiffi þ ð7Þ 8pcf pcq0 f q0 and cc are the density and speed of sound in air, c the ratio of specific heats, re the effective flow resistivity and ae a parameter representing an effective rate of change of porosity with depth. Now, the transfer function T was calculated from this impedance using geometry B (or geometry B with slight errors). T was then slightly changed in amplitude (±0.2 dB) or phase (±0.5°) and the surface impedance calculated. The predicted impedance with measurement errors is then compared to the nominal impedance Z0. These error estimated were chosen to account mainly for the differences in the microphone frequency responses which, in a free sound field, were determined to be 0.1 dB, respectively, 0.5°, on average. Additional 0.1 dB were added to allow for other (external) influences. 2.4. Geometry optimization To find an optimized geometry for the lower frequency range, which is least sensitive to measurement errors, the following procedure was used: (1) Bounds for the geometry were chosen to allow for an easy set-up and keeping the distances small to minimize the influence of meteorological factors. (2) The achievable precision of the measurement (respectively the measurement errors) was defined. (3) The magnitude of the estimated surface impedance jZj was calculated for all geometries on a 5 cm grid. (4) Because of the possible nonlinear relation between the errors and their effect on Z, it is not sufficient to calculate the error in Z only for the maximum error estimate. The calculation was done for four val1
http://www.physik.uni-oldenburg.de/aku/.
ues within the error range, e.g. for the error in jTj, estimated to be ±0.2 dB, the values 0.2, 0.1, 0.1, 0.2 dB were chosen. The resulting error Ex was defined as the standard deviation of jZj for the four error values, calculated at each frequency point with the predictand and jZ0j. (5) To allow for an efficient calculation, the effects of the errors were considered to be independent except for the magnitude and phase of the transfer function. This reduces the number of calculations for each frequency point and geometry to 32 instead of 4096 needed if all error combinations were considered. (6) Step 3 was repeated for three surface impedances Z0 (low, medium, high) from two-parameter models [5]: Pine forest floor (re = 7.5 kPa s/m2, ae = 16 m1), soccer field (re = 200 kPa s/m2, ae = 40 m1) and compacted silt (re = 4000 kPa s/m2, ae = 115 m1). The calculation was done for the frequency range 100–400 Hz divided into 1/3rd octave steps, the number of frequency points N thereby being seven. (7) In analogy to the Gaussian error propagation law, the (relative) average total error ATE for each geometry was defined as the geometrical sum of the single errors Ex – divided by the ‘‘true’’ impedance Z0 – averaged over the frequency range f. 1 X ATE ¼ N f
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E2T þ E2hru þ E2hrl þ E2hs þ E2R jZ 0 j
ð8Þ
The optimal geometry – observing the boundary conditions in Table 1 – shall be the geometry with the lowest ATE. 3. Results The estimated surface impedance of the soccer field (average of five measurements) is shown in Fig. 2. While the course of the impedance is rather smooth and does agree well with impedance models for grass-covered ground [10] for frequencies above about 400 Hz, the decrease in impedance at lower frequencies does not at all agree with models for porous absorbers.
Table 1 Boundary conditions for the geometry optimization
Transfer function Lower microphone height Distance upper–lower mic. Source height Source–receiver distance
Range (m)
Error estimate
– 0.05–1 0.1–1 0.2–1.5 1–3
±0.2 dB, ±0.5° ±2 cm ±2 cm ±2 cm ±5 cm
Ranges of the geometry and estimates of the achievable measurement precision.
R. Kruse, V. Mellert / Applied Acoustics 69 (2008) 884–890 20
887
30
15 20 10 10
Impedance
Impedance
5 0 -5 -10
0
-10
-15 -20 -20 -25 100
300
500
1000
2000
-30 100
4000
300
500 1000 Frequency [Hz]
Frequency [Hz]
Fig. 2. Predicted surface impedance of a grass-covered soccer field. Real part (—), imaginary part (– –), fitted two-parameter model ( ).
2000
4000
Fig. 4. Effect of errors in the microphone position on the estimated surface impedance for geometry B. Correct impedance (—), upper microphone 2 cm too high (– –), lower microphone 2 cm too high ( ).
3.1. Effect of errors The fitted two-parameter model on which the error analysis is based is shown in Fig. 2. The obtained absorber parameters re = 200 kPa s/m2 and ae = 40 m1 are comparable to the values re = 183 kPa s/m2 and ae = 40 m1 which are stated for lawn in the ANSI standard. In Fig. 3, the effect of an error of 0.2 dB, respectively, 0.5° in the transfer function is shown. Even small errors in T cause high errors in the estimated surface impedance at low frequencies. It should also be noted that the magnitude of the error does not necessarily increase continuously with decreasing frequency. In Fig. 4, the effect of an error in the microphone positions is presented. The effect is much smaller than the effect
of an error in T, but errors in the upper microphone position can still cause significant deviations. Not shown but also observed is the fact that errors in the source height and source–receiver distance have only a minor effect on the surface impedance estimate (for geometry B and the assumed impedance of the soccer field). The large effect of transfer function errors will now be further analysed. In Figs. 5 and 6, the relative error in the real and imaginary part of the impedance is shown in relation to magnitude and phase errors of T. The effect is not only large but also asymmetric: For most of the error combinations the real part of the impedance will be too low and the imaginary part too high. Furthermore, there’s a strong interaction between the effect of magnitude and phase errors.
30
20
Impedance
10
0
-10
-20
-30
-40 100
300
500 1000 Frequency [Hz]
2000
4000
Fig. 3. Effect of errors in the transfer function on the estimated surface impedance for geometry B. Correct impedance (—), transfer function increased by 0.2 dB (– –), phase error of +0.5° ( ).
0.2
1
1
5 0.
2 2
0.2
1
0.15 0.1 0
0.5 0.5
0.2
0
0 -0.05
-0 .2
0.2
0.5
0
0.2
-0 .2
0
-0 .5
-0.2
-0.1 -0.15
2 1
1
0.2
0.05
0 -0.2
-0 .5
-0.5
-0.5 -1
Error in magnitude of the transfer function [dB]
0.5
0.25
-0.2
-1
-0.25 -1
-0.5
0
0.5
1
Error in phase of the transfer function [deg]
Fig. 5. Effect of errors in the magnitude and phase of the transfer function on the real part of the predicted surface impedance for geometry B at 250 Hz. Relative error (1 = 100%).
R. Kruse, V. Mellert / Applied Acoustics 69 (2008) 884–890
0.25
2
1 1
0.5
0. 2
0
3.2. Geometry optimization 0.5
.5 -0
0.2
-0 .2
0.05
1
2 0.51 0.2 -0 .20 -1 -2
0.5
0.1
-1
0 -0.05
0
-0 .5
-0.1
0
-0.2
-0 .2
-0.5
-0.15 0
-1
-0.2 0.2
-2
-0.25 -1
0.2
-0.5
0
0.5
1
Error in phase of the transfer function [deg]
Fig. 6. Effect of errors in the magnitude and phase of the transfer function on the imaginary part of the predicted surface impedance for geometry B at 250 Hz. Relative error (1 = 100%).
1.5
Average total error
only true for the transfer function but also for the geometry.
1
0.2 0.15
0. 2
Error in magnitude of the transfer function [dB]
888
1
The results of the optimization of the geometry for the frequency range 100–400 Hz for three different surface impedances can be found in Table 2. For all impedances, the optimum values R = 3 m, hs = 0.5 m and hrl = 5 cm are the same, while the upper microphone height varies between 1 m and 0.8 m. The effect of this optimization on the average total error ATE is shown in Fig. 7. In comparison with the predefined geometries from the ANSI standard, the average error can be reduced significantly for all three surface impedances to about half of the value of the best of these geometries. However, for the high surface impedance the ATE is still 53%. In Fig. 8, the effect of an error of 0.2 dB, respectively, 0.5° in the transfer function is shown, comparable to Fig. 3 for geometry B. The error is significantly reduced, especially in the imaginary part.
Table 2 Results of the geometry optimization for the frequency range 100–400 Hz for three different surface impedances
0.5
0
Z_low
Z_medium
Z_high
Lower microphone height (cm) Upper microphone height (m) Source height (m) Source–receiver distance (m)
Fig. 7. Average total error (relative) in the predicted surface impedance for three surface impedances and geometries A–C [5] as well as the optimized geometry. Geometry A ( ), B ( ), C ( ), optimized geometry ( ).
Low impedance
Medium impedance
High impedance
5
5
5
1
0.95
0.8
0.5 3
0.5 3
0.5 3
40 30
jT j 0:03 dB; phaseðT Þ 0:2 ; hru 1 cm; hrl 2 cm; hs 2:5 cm; R 5 cm
20
Impedance
From Figs. 5 and 6 as well as likewise diagrams2 for the combinations hru, hrl and hs, R, the following estimates for the necessary measurement precision (geometry B, 250 Hz) are derived, if the error is to be smaller than 20% in both real and imaginary part and no interactions between the three error combinations are considered:
10 0 -10 -20 -30
A likewise analysis revealed that for high surface impedances or lower frequencies the necessary precision becomes even higher. In case of high surface impedances this is not 2
Matlab code for generating these diagrams can be found on the author’s website.
-40 100
300
500 1000 Frequency [Hz]
2000
4000
Fig. 8. Effect of errors in the transfer function on the estimated surface impedance for the optimized geometry. Correct impedance (—), transfer function increased by 0.2 dB (– –), phase error of +0.5° ( ).
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4. Discussion
5. Conclusion
Taken into account the results of the error analysis, it seems that the often reported decrease of the predicted surface impedance at low frequencies is the result of measurement uncertainties. The investigated soccer field was large, flat and homogeneous, it therefore resembles one of the best measurement location one can hope to encounter during ground impedance measurements. On the other hand, small errors in the transfer function result in deviations from the expected surface impedance as observed in the measurement, with the real part being too low and the imaginary part too high. To a lesser degree, this is also true for errors in the geometry, especially the upper microphone height. To obtain acceptable results at low frequencies with geometry B, which is well suited for frequencies above 400 Hz and ground with medium impedance, the necessary measurement precision for an error less than 20% is already high at 250 Hz and can only be achieved with very great care and high quality equipment including phase-matched microphones. Fortunately, the effect of errors in the geometry on the impedance is less asymmetric compared to the effect of errors in the transfer function and can therefore be reduced by averaging multiple measurements on the same ground including repositioning of source and receivers. The optimized geometry for the frequency range 100– 400 Hz does feature a significant advantage over the predefined geometries with respect to its error sensitivity, but the average error for high impedance surfaces is still high. The very high error for geometry C in this case does agree with the standards recommendation to use this geometry only for very soft grounds. It should be kept in mind that the necessary measurement precision itself depends on how accurate the surface impedance needs to be measured and therefore on the specific application. In general, the effect of a given percental error in the impedance is highest near the first ground dip, the frequency range in which the presence of the ground leads to a high attenuation compared to the free field situation. The position of the ground dip depends on the impedance and geometry. As an example, the ground dip above the soccer field with source and receiver at 1 m height and the distance between them being 100 m is located around 750 Hz. In this case, an error of 50% in the surface impedance leads to an underestimation of the sound pressure at 400 Hz by 17 dB, while at 1 kHz the sound pressure is overestimated by only 3 dB. Below 400 Hz, the error decreases and reaches 1.3 dB at 100 Hz. For higher propagation distances, the error does decrease slower with decreasing frequency. Thus, the frequency range below 1 kHz should receive particular attention regarding errors in the predicted surface impedance.
The two-microphone method is an established procedure for the in situ measurement of the surface impedance. A disadvantage is that it is sensitive to errors both in the measured transfer function and, to a minor degree, to errors in the geometry. The effect of these errors depends on the measurement geometry and the surface impedance. For low impedance surfaces, it is possible to measure with acceptable accuracy at frequencies >100 Hz even with the predefined geometries from the ANSI standard. Unfortunately, such highly absorbing materials rarely occur in nature with the important exception of snow. For materials with higher flow resistivity, the use of the predefined geometries cannot be recommended for frequencies below about 500 Hz. The optimization procedure, on the other hand, leads to a geometry which provides a significant higher error tolerance and should enable the researcher to obtain reasonable results down to 100 Hz. However, for materials with a very high surface impedance like compacted silt or asphalt, even the optimized geometry cannot guarantee low errors. Taken in mind that the proposed source–receiver distance is 3 m and therefore the maximum allowed distance for the optimization it is possible that better geometries with larger distances exist. Larger distances, however, would be more affected by meteorological factors and require large surfaces. Therefore, the use of the two-microphone method is not recommended in such situations unless the requirements in accuracy are low for a specific application or very favourable measurement conditions (large, flat surfaces, no wind) and high quality equipment including phase- matched microphones are available. Acknowledgements The authors thank their student Ping Rong for his helpful comments on the data processing and interpretation and his valuable Matlab programming. References [1] ISO 10534-2. Determination of sound absorption coefficient and impedance in impedance tubes – Part 2: transfer-function method. International Organization for Standardization; 1998. [2] Mommertz E. Angle-dependent in situ measurement of reflection coefficients using a subtraction technique. Appl Acoust 1995;46: 251–63. [3] CEN/TS 1793-5. Road traffic noise reducing devices – test method for determining the acoustic performance – intrinsic characteristics – in situ values of sound reflection and airborne sound insulation. European Committee for Standardization; 2003. [4] Allard JF, Champoux Y. In situ two-microphone technique for the measurement of the acoustic surface impedance of materials. Noise Contr Eng J 1989;32(1):15–23. [5] ANSI S1.18-1999 (R2004). Template method for ground impedance. American National Standards Institute; 2004. [6] Fuhs S, Ho¨ldrich R, Tomberger G. Validierung des Entfernungsgesetzes und Korrektur der Gruppenlaufzeit und des akustischen
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Zentrums des Lautsprechers im Adrienne-Verfahren. DAGA 2006. Braunschweig. [7] Nobile MA, Hayek SI. Acoustic propagation over an impedance plane. J Acoust Soc Am 1985;78(4):1325–36. [8] Ingard U. On the reflection of a spherical wave from an impedance plane. J Acoust Soc Am 1951;23(3):329–35.
[9] Taherzadeh S, Attenborough K. Deduction of ground impedance from measurements of excess attenuation spectra. J Acoust Soc Am 1999;105(3):2039–42. [10] Donato RJ. Impedance models for grass-covered ground. J Acoust Soc Am 1977;61(3):1449–52.