Reducing the influence of microphone errors on in- situ ground impedance measurements

Reducing the influence of microphone errors on in- situ ground impedance measurements Roland Kruse, Sophie Sauerzapf Oldenburg University, Inst. of Physics, 6111 Oldenburg, Germany Abstract The transfer function method is a procedure to determine the surface impedance of grounds in-situ. Disadvantageously, it is very sensitive to errors in the magnitude and phase of the measured transfer function at frequencies below about 500 Hz. One source of these errors is the differences in the frequency responses and calibration of the two microphones used. Two methods routinely used for impedance tube measurements are employed to reduce these errors: One microphone is subsequently placed at the two measurement positions or the measurement is done with two microphones in normal and switched position. The two microphone / switched positions technique leads to credible impedance estimates down to 100 Hz on the investigated ground of medium impedance. The one microphone / switched positions technique, on the other hand, shows a comparable performance only for small source-microphone distances, low wind speed and background noise. Keywords: Ground impedance; In-situ impedance measurement PACS 43.58.Bh

Introduction The acoustical properties of outdoor grounds have a large influence on sound propagation over short to medium ranges when the effect of meteorological conditions is still small. For a proper description of the ground, at least the surface impedance needs to be known. It may be determined by a ground impedance model [1,] and an in-situ measurement. For higher frequencies and not too small angles of incidence, simple impulse-echo methods [3,4] which rely on the separation of direct and reflected pulse in the time domain and assume plane waves, are appropriate. For the low frequency range, such a plane wave approximation is insufficient as the spherical nature of the wave has to be taken into account. By use of a sound field model [5] and measuring the sound pressure simultaneously at two positions, the surface impedance can be estimated by comparing the transfer function measured between the two positions with the one predicted by the sound field model ( sound field matching ). The only currently standardised method for this purpose, ANSI S1.18 [6], achieves the matching by comparing the magnitude of the transfer function ( level difference spectrum ) with pre-calculated level difference spectra ( templates ) [7]. Templates are provided for a number of absorber parameter combinations, for two absorber models, and three geometries. Best fit is determined by choosing the absorber parameter(s) which give the lowest sum of squared differences between 1/3- octave averages of measured and tabulated values; the surface impedance follows from the absorber parameters. The template method is robust and does not require sophisticated mathematical procedures. However, it is limited to the specific ground impedance model the templates are derived from. This limitation can be overcome by direct deduction of the surface impedance from the transfer function ( numerical inversion ), a process currently under discussion to become an ANSI standard 1. Unfortunately, at low frequencies, this method is sensitive to errors in the measurement geometry and, to an even higher degree, to errors in the measured transfer function [8]. One source of errors in the transfer function is the microphones which can have slightly different frequency responses especially if not using a (phase) matched pair. This problem is not specific to in-situ measurements but also applies to impedance tube measurements [9]. Two solutions for it are commonly used: On the one hand, the difference between the microphones can be assessed by measuring the transfer function with the microphones being in normal and switched position [10]. On the other hand, it is possible to measure with just one microphone which is subsequently placed at the two positions [11]. Both procedures assume that the sound field is stationary. In this article, these two techniques are applied to the in-situ ground impedance measurement with the transfer function method to investigate if they are beneficial in outdoor situations. All impedances presented are normalized to the impedance of air and an exp(+iωt) time dependence is assumed. Experimental procedure Measurement set-up Measurements were done on a grass covered soccer field with two geometries (on two different days) as listed in table 1. The measurement set-up is shown in fig.1. For geometry B, the sound source was a 10 cm broadband loudspeaker in a (13 cm)³ closed cabinet. The temperature was 0 C and the wind speed < m/s. The ground was dry. For geometry A (comparable to geometry A from ANSI S1.18 but accounting for the chosen microphone support allowing only heights that are multiples of five centimetres), a 5 cm woofer was used to ensure a sufficiently high sound pressure despite the increased sourcemicrophone distance. The temperature was 18 C with wind speeds of 3 to 4 m/s. The ground was wet. Some background noise from construction works was present. 1 Workgroup 0, ANSI S1

The emitted signal was pink (pseudo-random) 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 with windscreens were used. Four (geometry B) or five (geometry A ) measurements at different locations were done. At each location, the following transfer functions were recorded with a B&K PULSE 11 measurement system ( 6400 Hz, f = 8 Hz, 150 averages with 50% overlap): One-microphone technique: The single microphone was first placed in the upper position and the transfer function (H upper ) in relation to the input signal was measured. Then, the measurement was repeated with the microphone in the lower position (H lower ). The transfer function T 1 between the upper and lower microphone position is T 1 = H upper / H lower. Switched microphone technique: Two microphones were used and the transfer function (T normal ) between them measured in normal position. Then, the positions of the microphones were switched and the measurement repeated (T switched ). The transfer function T corrected for differences between the microphones is given by eq.1, P 1/ denoting the sound pressure spectrum at location 1/ and S 1/ the sensitivity of microphone 1/. T = T P S P S 1 1 1 1 normal Tswitched = = P S P S1 P1 P (1) The transfer functions T 1, T, T normal and T switched from the four / five measurement locations were averaged before calculating the surface impedance as suggested in [1]. Sound field model and impedance deduction For the calculation of the surface impedance from the transfer functions, a widespread model for the reflection of spherical waves on a locally reacting impedance plane was used [5]. The velocity potential in the height d above the surface can be described by eq. - 4. ( d ) e = r 1 1 + [ R + (1 R e ) F] r ikr ikr p p φ () with the boundary loss function λ ² F = 1 i π λe erfc( iλ) = 1 i π λw( λ) (3) r 1 and r are the lengths of the direct and reflected path; R p is the reflection coefficient for plane waves. k denotes the wave number in air, erfc the complementary error function and w the Faddeeva or complex error function [13]. The numerical distance λ is defined in eq.4 representing a simplification of the definition [14]. It is a function of the angle of incidence θ and the normalized surface admittance β. 1 λ = ikr (sin( θ ) + β ) (4) The transfer function T model is defined as the ratio of the velocity potential at the upper microphone position divided by the potential at the lower microphone position (eq.5). T mod el φ( d, upper) = (5) φ( d, lower) The surface impedance Z was calculated from the measured transfer function T and geometry by the Newton-Raphson algorithm which finds the zero of the function (T model - T) and therefore gains the surface impedance Z which minimizes the difference between the measured transfer function and the transfer function predicted by the model. This very

efficient method has been described in [15]. The calculation was started at the lowest frequency point (50 Hz) with a seed value for the iteration of β = 0.05. The obtained solution is used as seed value for the next higher frequency point. While the iteration may, in general, converge on a wrong solution (local minimum), such a behaviour was not observed even if the initial seed value was varied. For all calculations, Matlab R007a was used. The Matlab code can be found on the author s webpage. For calculation of the complex error function w, the routine Faddeeva_ from the Matpack 3 library was used because Matlab s erfc function does not support complex arguments. Results The four measured / calculated transfer functions for geometry B are presented in fig.. At frequencies below about 1.5 khz, the differences between them appear to be very low. The maximum differences between the transfer functions, at 00 Hz, are 0.1 db and 0.15. Even in the higher frequency range the differences stay fairly small, with the result of the measurement with just one microphone deviating most from the other measurements. Fig.3 shows the corresponding surface impedance estimates. In this case, the results agree well for frequencies above 500 Hz. Below 400 Hz, there are large differences between the measurements done with the microphones in normal and in switched position; the measurement in normal position indicates a lower impedance than the same measurement with the microphone positions switched. The results obtained using the one microphone technique (T 1 ) and the corrected two microphone transfer function (T ) are located between the other two results, but are not equivalent to a simple average. The one-microphone results are not as smooth as the ones obtained using two microphones. The impedances deduced from measurements with geometry A are shown in fig.4. Again, there are considerable differences between the measurements in normal and switched position below 400 Hz, and only the corrected transfer function T leads to an impedance which increases with decreasing frequency down to approx. 80 Hz, as expected for a rigidporous medium. The one microphone technique, on the other hand, indicates a highly varying impedance below 500 Hz. During the measurements with geometry A it was also noticed that the coherence between the signals from two microphones was clearly higher (near unity for 50 Hz < f < khz) than the coherence between the input and a single microphone signal. Discussion The results primarily show the weakness of the transfer function method: Though the measured transfer functions show only very small differences at low frequencies, these have a large impact on the deduced impedance. The larger differences above 1.5 khz have no significant effect. Especially noteworthy is the difference between the results with normal and switched microphone positions. Because both measurements have been done at the same locations immediately one after another, it is unlikely that actual changes in the set-up or ground impedance are the reason for the deviations. The likeliest cause is differences in the microphones frequency responses. This assumption is supported by the fact that, at least for geometry B, the one-microphone technique T 1 and the corrected transfer function T both are not affected by the microphone errors - yield similar results. The performance of the one microphone technique depends on the external conditions. For small source-microphone distances (geometry B) and low wind speeds and background noise, it offered results comparable to the switched microphone technique. However, under less favourable conditions (wind speeds of 3-4 m/s and background noise), no credible results could be obtained below 500 Hz. The drawback of this method is that the spectra at the two positions are not measured at the same instant. Changes in the propagation conditions, resulting from atmospheric turbulence, and varying background noise levels will http://www.physik.uni-oldenburg.de/aku/ 3 http://www.matpack.de

cause fluctuations in the transfer function - in relation to the input signal - and consequently influence the calculated transfer function T 1 between the two positions. Although the effect of turbulence is expected to be small for frequencies below 500 Hz [16], even these minor changes of the level difference can have a huge effect on the predicted impedance. To complicate matters further, the effect of turbulence on (the coherence within) a sound field increases with increasing distance between two positions. This agrees well with the observed higher coherence between the two microphone signals (distance 0 cm) compared to the coherence between input and single microphone signal (distance 1 m). Conclusion For the purpose of outdoor impedance measurements, the two-microphone/ switched position technique commonly used for impedance tube measurements offers significant advantages compared to using a pair of microphones in just one position. Credible results can be expected down to 100 Hz for geometry A and not too hard grounds. Geometry B displayed a poorer low frequency performance but better results for higher frequencies. In this context one should remember that errors in the transfer function are not the only source of errors in the deduced impedance, inaccuracies in the geometry can have a similar effect. The degree of this effect will depend on the specific geometry. The same holds true for uncertainties in the level difference. As a consequence, the improvement to be expected by using the switched microphone technique will vary, a geometry which is largely insensitive to such uncertainties will barely benefit from it. The use of a single microphone is the less favourable method as it is more sensitive to changing propagation conditions and background noise. It can only be recommended for small source-microphone distances, low wind speed and background noise. A general problem of in-situ measurements remains: It is sometimes difficult to decide which measurement result is correct because the true impedance is unknown. The impedance measured with geometry B (fig.3) offers a good example: while the results obtained with the microphones in normal position lack the expected increase in the real part of the impedance below 500 Hz, the results from the measurements with the microphone positions switched can t be rejected without additional information. Fortunately, in this case, the impedance obtained by using geometry A (fig.4) is obviously wrong for frequencies smaller than 300 Hz except for the impedance following from the corrected transfer function T. However, one may not always have data for different geometries. Thus, the use of the switched microphone technique can be recommended to get an impression of the accuracy of one s data by comparing normal and switched results. Acknowledgements The authors would like to thank the members of the ANSI S1 workgroup 0 for their support and helpful suggestions.

References 1. Attenborough K et al. Predicting outdoor sound. Routledge Chapman & Hall (006). Attenborough K. Ground Parameter Information for Propagation Modeling. Journal of the Acoustical Society of America (199); 9(1): 418-47 3. Mommertz E. Angle-Dependent In-situ Measurement of Reflection Coefficients Using a Subtraction Technique. Applied Acoustics (1995); 46: 51-63 4. 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 (003) 5. Nobile MA, Hayek SI. Acoustic propagation over an impedance plane. Journal of the Acoustical Society of America (1985); 78(4): 135-1336 6. ANSI S1.18 American National Standard Template Method for Ground Impedance. Acoustical Society of America (1999) 7. Embleton, T. F. W., J. E. Piercy, et al. (1983). "Effective flow resistivity of ground surfaces determined by acoustical measurements." Journal of the Acoustical Society of America 74(4): 139-144. 8. Kruse R, Mellert V (007). Effect and minimization of errors in in-situ ground impedance measurements. Applied Acoustics; doi:10.1016/j.apacoust.007.05.010 9. ISO 10534-. Determination of sound absorption coefficient and impedance in impedance tubes - Part : Transfer-function method. International Organization for Standardization (1998) 10. Chung JY, Blaser DA. Transfer function method of measuring in-duct acoustic properties. I. Theory. Journal of the Acoustical Society of America (1980); 63(3): 907-913 11. Chu WT. Single-microphone method for certain applications of the sound intensity technique. Journal of Sound and Vibration (1985); 101(3): 443-445 1. Kruse R. Application of the two-microphone method for in-situ ground impedance measurements. Acta Acustica united with Acustica (007); 93(5): 837-84 13. Abramowitz M, Stegun I. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (1965). This book is copyright free and readily available on the internet. 14. Ingard U. On the Reflection of a Spherical Wave from an Impedance Plane. Journal of the Acoustical Society of America (1951); 3(3): 39-335 15. Taherzadeh S, Attenborough K. Deduction of ground impedance from measurements of excess attenuation spectra. Journal of the Acoustical Society of America (1999); 105(3): 039-04 16. Clifford SF, Lataitis RJ. Turbulence effects on acoustic wave propagation over a smooth surface. Journal of the Acoustical Society of America (1983); 73(5): 1545-1550

Tables Geometry: A B Lower microphone height [h rl ] 5 cm 5 cm Upper microphone height [h ru ] 45 cm 0 cm Source height [h s ] 35 cm 0 cm Source-receiver distance [R] 1.75 m 1.0 m Tab. 1: Measurement geometries for the transfer function method. Figures R hs Loudspeaker θ Ground hru hrl Upper mic. Lower mic. Fig. 1: Measurement set-up for the transfer function method.

10-0.5 Mangitude of transfer function [db] 0-10 -0.3-0.35-0.4 00 00 -.6 -.8-3 -3. Magnitude Phase 100 50 0 Phase of transfer function [deg] 00 300 500 1000 000 3000 Frequency [Hz] Fig. : Transfer functions measured with two different techniques over a dry soccer field. Geometry B from ANSI S1.18; Average of four measurements at different locations. T normal (-), T switched (--), T ( ) and T 1 (.-.-) (one-microphone). 00 Hz region (190 10 Hz) magnified.

40 30 Im(Z) Re(Z) 0 10 0-10 -0-30 -40 00 300 500 1000 000 3000 Frequency [Hz] Fig. 3: Estimated surface impedance Z of a dry soccer field measured with two different techniques. Geometry B from ANSI S1.18; Average of four measurements at different locations. Calculated from T normal (-), T switched (--), T ( ) and T 1 (.-.-) (one-microphone)

0 15 Im(Z) Re(Z) 10 5 0-5 -10-15 -0 100 00 500 1000 3000 Frequency [Hz] Fig. 4: Estimated surface impedance Z of a wet soccer field measured with two different techniques. Geometry A (table 1); Average of five measurements at different locations. Calculated from T normal (-), T switched (--), T ( ) and T 1 (.-.-) (one-microphone)