Source height determination by ground effect inversion in the presence of a sound velocity gradient

Journal ofSound

SOURCE

and Vibration (1991) 145(l), 111-128

HEIGHT

INVERSION

DETERMINATION IN THE VELOCITY

K.

BY GROUND

PRESENCE

EFFECT

OF A SOUND

GRADIENT

M. LI, K. ATTENBOROUGH

AND

N. W. HEAP

Faculty of Technology, The Open University, Milton Keynes MK7 6AA, England (Received 5 July 1989, and in revised fbrm 20 April 1990) A propagation inversion method has been devised and tested that enables determination of height of a sound source which is above an acoustically soft ground. The horizontal separation between the point source and receivers is assumed to be a priori knowledge. The proposed algorithm is site adaptive since it makes use of the predicted influence of the ground on near-grazing propagation over flat, continuous ground surfaces and of the predicted influence of a stable refracting atmosphere. Stable refraction has been included in the analysis by assuming a linear sound velocity gradient in the otherwise quiescent atmosphere. This represents a first step towards an atmospheric equivalent of the matched field processing technique used in underwater acoustics. Extensive measurements have been carried out over two test sites with different ground impedance characteristics. The spectra of the acoustic signals used in the analysis were between 50 Hz and 2000 Hz and the grazing angle of each measurement varied from 5”to 8”. The algorithm has been shown to allow accurate deduction of the source height from a priori knowledge of the range where the range varies from 50 m to 100 m in one site and from 130 m to 175 m in the other site.

1.

INTRODUCTION

There are several different fields in which acoustic source height determination is important and where the approximate range of the source is known a priori. For correct modelling of the propagation of road traffic and railway noise and of the effectiveness of barriers, it is essential to have a good idea of both the acoustic source strength of the vehicles and of the height of the sources over the road or rails. In such cases although the horizontal range of the source is available (the distance to the road or rail) the precise height has to be determined acoustically. Acoustic source height determination also has military applications, both in ground-borne and airborne source identification and in aircraft tracking. In this paper a development of a scheme for source height determination, originally suggested for road traffic noise [l], is described. The method requires the source to be broadband and continuous and involves ground effect inversion. The development described here includes an allowance for a stable sound velocity gradient. In this regard it exploits two phenomena that may cause difficulty when attempting the same task by beam-forming techniques with various vertical arrays [2]. It does so, moreover, with the use of only two vertically spaced receivers, less instrumentation and rather less signal processing than would be required by beam-forming. The availability of a robust, site adaptive technique which is relatively undemanding in terms of equipment is desirable particularly for military applications. The theoretical formulation of ground effect, the ray-based method for compensating for stable refraction and the logic of the algorithm for source height determination are 111 0022-460X/91/040111

+ 18 $03.00/O

@ 1991 Academic Press Limited

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K. M. LI ET AL

discussed in the next section. In section 3 the equipment and procedures used in making measurements with stationary loudspeakers emitting broadband sound are described. Results and comparisons of geometrically measured and acoustically deduced heights at two different sites are presented in section 4. Finally the work is summarized, various limitations are discussed and potential improvements are outlined.

2. THEORY 2.1.

GROUND

EFFECT

MODEL

For the purpose of predicting near-grazing sound accepted that ground surfaces may be considered to described by a velocity potential Jlro, at a receiver point source at height H, as shown in Figure 1 may Weyl-van der Pol formula [3] as

propagation outdoors, it is widely be locally reacting. Hence, the field at height H, and separated from a be calculated by using the so-called

(1) where RI and R, are, respectively, the distances in air: the receiver and k is the wavenumber

from the source

and its mirror

k=2rf/c,

(2)

with f and c the frequency and the velocity of sound spherical wave reflection coefficient such that

in air respectively.

Q is the complex

Q=RP+(l-Rp)F(w) where

R, is the plane

wave reflection

coefficient

R, = (Z cos 0 - l)/(Z 8 being the angle of incidence surface given by

(3) given by cos ~9+ l),

and Z being the relative

normal

(4) impedance

Z=Zr+iZX. F(w)

in equation

(3) is also known

distance

as the boundary

loss factor, (6)

given by

w’=(ikR,/2)[(1/Z)+cos

01’. Receiver

Ground plane

I

Image source Figure

of the ground

(5)

F(w)=l+iGwe-“‘erfc(-iw) w being the numerical

image to

1. Schematic

source/receiver

geometry.

(7)

SOURCE

HEIGHT

DETERMINATION

113

The branch cut for calculating w is defined to be along the negative real axis and erfc ( ) is the complementary error function. The approximations inherent in equations (l), (6) and (7) have been explored elsewhere [3,4]. An empirically deduced relationship between the frequency-dependence of the relative characteristic impedance of a homogenous absorbing material and its specific flow resistivity has been used widely to provide values for Z in applying equations (1) and (7) and in fitting the theoretical predictions to measurements of excess attenuation outdoors. The real and imaginary parts of Z may be expressed as z,=

1+O*0571(fp/a)-“~754,

Z, = O.O87(fp/~)~““‘,

(839)

where p is the density of air and u is the effective flow resistivity. The qualification eflective is used here since, as argued elsewhere [5], strictly, the frequency dependent parameter is a function &I, where R is the porosity. Several instances have been found in which the empirical relationships (8) and (9) do not allow a good fit to measure excess attenuation or impedance data. Many alternative theoretical and semi-empirical models are available [6-91. One that has been found particularly useful for the work presented here is the two-parameter variable porosity model [8]. This gives normal surface impedance as Z = 0.436mf

+ i[O.436mf

+ 19*48( aelf)],

ilO)

where a, represents the effective flow resistivity at the ground surface (in this case, a, is a function of actual flow resistivity divided by porosity) and (Y, represents the effective rate of change of porosity with depth. Negative values of o, enable surface impedances to be fitted for media in which the porosity increases with depth. The use of the variable porosity model will allow tolerable fits to the measured impedances of a relatively wide range of outdoor ground surfaces, including ground surfaces in which the upper few centimeters are homogeneous [8]. However, it should be noted that the coefficients in equation (10) differ from those obtained by Attenborough [8] by a factor of two and that these discrepancies have been reported in reference [lo]. One of the differences arises from a redefinition of we. Moreover, Berry [ 111 has showed that Attenborough’s derivation of the model is incorrect as a result of the use of the wrong kind of Hankel functions which do not satisfy the Sommerfeld radiation condition at infinity. Despite this theoretical flaw, the variable porosity model in its approximate form has been found to provide satisfactory numerical agreement with the predicted impedance of the surface of a multi-layered medium in which the porosity decreases with depth in a discrete exponential manner and each layer is regarded as homogeneous, with acoustical parameters described by a four-parameter model [8]. Therefore, one should regard the variable porosity model as semi-theoretical or semi-empirical. 2.2.

RANGE

AND

HEIGHT

AMBIGUITY

Consider first the problem of deducing both range and height from an excess attenuation or level difference spectrum. The use of a vertical array of sensors (hydrophones) in a water environment for source localization has been studied widely. The main difficulty in the water environment is the existence of an ambiguity surface where the depth and range cannot be resolved uniquely. The use of matched field processing [ 121 is a possible solution. By analogy, one would expect that a similar problem will arise in the determination of the source position over land by use of an array of microphones: i.e., there will not be a unique solution for the source position. This argument may be substantiated by considering the path length difference between R, and R2,the direct and reflected path lengths.

114

K. M. Ll

ET AL.

R, and Rz are given by (see Figure 1)

It is obvious that

R,=JR'+(H,-H,)*,

R,=JR2+(H,+HJ2,

(11,12)

where R is the horizontal separation between the source and receiver. In the far field where

R z+H,+H,

(13)

one can simplify the analysis by the approximation

1/R--1/R,--I/R,, R2-RI-2HsHJR=2H,

tan A,

(14,15)

where A is the angular position of the receiver. It is seen from equation (15) that the path length difference, R2 - R,, is only dependent on A for a fixed source height and large range and does not depend on range and height separately. Subsequently, the total field potential can be rewritten as [cf. equation (l)] ikK2++

ikR, Ijlm, =~+I01

y, 1

(16) 2

where the phase angle, 4, in equation (16) is given by 4=tan-1

2[(1-F,)(FX+zXcos e)+F,(F,+z,cos e)] (F,+Z,cos e)2+(EX+zXcos e)2-(F,-1)2-F: [

1’

(17)

F,and F, in equation (17) being the real and imaginary parts of the boundary loss factor, F(w), respectively. The excess attenuation (EA.)from a point source is defined as (18) Substituting equations (15), (16) and (17) into equation (18) for a distant observer leads to E.A.=lOlog[1+2(Q]co~(2kH,tanA++)+]Q]~].

(19)

By using equation (19), one can calculate the excess attenuation as a function of frequency. In Figure 2 is shown a typical spectrum of the excess attenuation for the receiver and source at 1.2 m and 10 m, respectively, above the ground and a horizontal separation of 100 m. The variable porosity model is used for calculating the ground impedance, where

Frequency (kHz) Figure 2. Predicted excess attenuation spectrum for receiver height 1.2 m, range 100 m, source height 10 m, effective flow resistivity 2.0~ 10s MKS rayls m-’ and effective rate of change of porosity with depth IS0 m-‘.

SOURCE

HEIGHT

115

DETERMINATION

a, and (Y, are assumed to be 200 000 MKS rayls m-’ and 150 m-’ respectively. These values are typical for grassland. It is obvious that there is a series of minima and maxima in the spectrum. The first dip of the excess attenuation spectrum is known as the ground effect or impedance dip, which is largely due to the phase change on reflection at the ground surface. The location and nature of the ground effect dip is strongly affected by the angle of incidence and the acoustical nature of the ground as indicated by its normal surface impedance. The subsequent dips are due largely to the path length difference and consequent phase differences between the direct and reflected waves. These higher order dips are subsequently called path length difference (pld) dips. It is important to note the implication of equations (15) and (19): one cannot determine the range and source height uniquely from the excess attenuation spectrum irrespective of the receiver height and array size of a vertical array. It is only the angular position of the source (its elevation angle), A, which can be determined uniquely. Consequently, increasing the number of elements in the array beyond two will not result in any improvement in separate resolution of range and height. In many practical situations the range is known, but the height of the source is relatively difficult to determine. Based on the previous analysis and from a priori knowledge of the range, one can deduce the source height.

2.3. USE OF THE LEVEL DIFFERENCE SPECTRUM In the last section the properties regarding the “dips” in the excess attenuation spectrum have been discussed. Since this requires knowledge of the free field spectrum of the source, it is more convenient to use level difference instead. The level difference (L.D.) between two vertically separated microphones (L.D.) is defined as L.D. = 20 log

total sound pressure of upper receiver 1 total sound pressure of lower receiver

1 ’

(20)

The level difference spectrum has the distinctive property that it is independent of the source spectrum and level (for a continuous source). Moreover, if the lower microphone is situated on the ground the level difference spectrum corresponds to wave interference that would be seen in the excess attenuation spectrum at the upper receiver. A predicted level difference spectrum is shown in Figure 3. The result was obtained by using equation (20), where again a, and (Y, were assumed to be 200 000 MKS rayls

Frequency (kHz)

Figure effective

3. Predicted flow resistivity

level difference spectrum for receivers at 0 and 1.2 m, range 100 m, source height 10 m, 2.0~ 10’ MKS rayls m-’ and effective rate of change of porosity with depth 150 II-‘.

116

K. M. LI ETAL

m-’ and 150m-’ respectively. There are several maxima and minima in the spectrum. The first broad “dip” at low frequencies is due to the phase change of the ground-reflected sound at low frequencies. The magnitude and frequency location of this dip is closely related to the elevation angle, A (see Figure 1) and to the relevant ground parameters. Consequently, it is this part of the level difference spectrum that is used to determine the ground parameters and the angle A. In addition, the second and higher order minima may be considered to be the result of destructive interference or Lloyd’s mirror effects between the direct and ground reflected waves arriving at the upper microphone. Consider the case of sound propagation from a point source to a receiver both located above a plane, locally reacting ground surface. In the absence of meteorological effects the received signal is dependent on the relative source/receiver geometry and the acoustic influence of the ground surface, as described by equation (1). If the relative source/receiver geometry is known, as in the case of a controlled experiment, one can obtain an estimate of the impedance by comparing measured data with the theoretical predictions based on various impedance characteristics and obtaining the impedance characteristic which gives the best numerical fit between the measured and predicted data. Because of the frequency dependence of the impedance it is necessary to introduce an impedance model which describes the frequency dependence of the impedance in terms of a number of parameters, such as the one-parameter empirical model; or the two-parameter model described in section 2.1. Subsequently, the values of (T or of LY,and a, may be varied in the theoretical prediction for comparison with the measured data. The acoustical properties of the ground surfaces are then characterized by the “best fit” values of o or of a, and (Y,. These values can then be used to predict the sound level at any point above the ground surface. Such a procedure based upon various impedance models has been advocated for acoustical ground-surface characterization [14, 151. A simple extension of this fitting procedure makes some element of the source/receiver geometry a third factor to be varied in the comparison of theoretical to measured data thus allowing its determination. Moreover, we have added an additional meteorological factor, namely the linear sound velocity gradient, in our analysis. The inclusion of this meteorological factor will be discussed at length in a later section. It is essential when fitting two or more parameters in this way that the effect of changing any single parameter has a markedly different effect on the shape of the predicted level difference spectrum from the results of changing the other parameter(s). If this were not the case, it would be impossible to determine unique values for the other parameters and so characterize the ground type. The effect on the level difference spectrum of changes in the values of oe and CX,is shown in Figures 4 and 5. It can be seen that the effect of increasing a, is to increase the frequency of the peak, whereas the effect of increasing (Y, is to increase the magnitude of the ground effect dip. The proposed source height determination technique is based on measuring the level difference spectrum of the sound pressure between two vertically separated microphones, one of which is on the ground surface. This system was first suggested by Glaretas [l]. 2.4.

INCLUSION

OF A LINEAR

SOUND

VELOCITY

GRADIENT

Sound propagation over long ranges in an inhomogeneous medium has been studied for many years (for example, references [16-191). Atmospheric inhomogeneity in the form of stratification is of particular interest and is caused by temperature and/or wind gradients. The presence of these gradients implies variations of sound speed with height which causes a continuous refraction of the sound rays. It is useful to construct a simple model of the interaction between continuous refraction and ground effects.

SOURCE

-151

I

/

HEIGHT

/

117

DETERMINATION

I

1

1

0

I

/

I

I 2

Frequency

(kHz)

Figure 4. Predicted level difference spectrum for receivers at changes in effective flow resistivity (v, MKS rayls m-l). Predictions based on receivers at 0 and 1.2 m, range 100 m, source height 10 m, effective rate of change of porosity with depth 150 m-‘, with zero sound velocity gradient.

Frequency (kHz)

Figure 5. Predicted level difference spectrum for receivers at changes in effective rate of change of porosity with depth (a, m-l). Predictions based on receivers at 0 and I.2 m, range 100 m, source height IOm, effective Row resistivity 2.0~ IO5 MKS rayls m-‘, with zero sound velocity gradient.

A major difference between the effect of temperature and wind gradients is that the reciprocity law holds for the former case but not the latter. In other words, the presence of a wind gradient causes the sound field to be anisotropic such that the sound pressure is lower in the upwind direction. On the other hand, the field is exactly the same for a temperature stratified medium if the positions of the source and the receiver are interchanged. However, a wind gradient is equivalent to a temperature gradient as far as the curvature of the sound ray is concerned. The effects due to the wind and temperature gradients are additive. One may simplify the analysis by assuming the atmosphere is steady and vertically stratified. Use of the model of a vertically stratified medium allows one to use a linear sound velocity profile given by c(H) = cg( 1+ UH),

(21)

118

K. M. LI ETAL.

where a is the normalized sound velocity gradient, H is the height above ground level and subscript 0 denotes the condition at H = 0. Negative values of a are associated with upward refraction (lapse). The assumption of a linear sound velocity profile leads to a circular ray path. It is then straightforward to calculate the grazing angle and the path lengths of direct and reflected waves [ 181. The method of solution for the path lengths and grazing angle is a straightforward application of the technique described by Rudnick [ 161 and the details are not described here. However, the calculation of the phase changes of the direct and reflected waves, kR, and kR2, respectively, are complicated by the fact that k( = 27rj/c) is a function of height. Substituting equation (21) into equation (2) one can express the wavenumber as k(H)=k,/(l+aH),

(22)

where k,, is the wavenumber at H = 0. The total phase change, A, along a ray path from the source to the receiver is given by % A=

I SI

k(H) dS.

(23)

dS is the elemental arc length of the sound ray, and S, and SZ are, respectively, the arc lengths of the source and receiver measured from an arbitrarily chosen reference point. But the evaluation of the integral in equation (23) is found to be more convenient when the variable is changed from S to 4, where rj is the angle between the direction of propagation and the vertical axis (see Figure 6). Since Snell’s law states that c/sin 4 is constant along a ray, i.e., c( H)/sin 4 = co/sin C#J~ = constant,

(24)

then, with the use of equation (22) and (24), it is not difficult to show that k(H) dS = k. d&/a sin 4.

(25)

Hence equation (23) can be rewritten as

(26) where C#J~ and C&are the ray angles of the source and receiver measured from the vertical axis respectively (see Figure 6). Circular path roy I

0

R

Figure 6. An element of a curved sound ray.

SOURCE

HEIGHT

119

DETERMINATION

The total phase change given by equation (26) can be evaluated immediately to give A = (k,lo) log, [tan (&/2)/tan

(27)

(4,/2)1.

One can calculate the respective phase changes, kR, and kRz, by substituting appropriate angles (4, and &) for the direct and reflected waves into equation (27). Consequently, the field potential above an impedance plane due to a point source can be calculated by the Weyl-van der Pol formula (cf. equation (1)) but 8, R,, R2 and the phase changes (kR, and kR,) are based on circular ray paths as a result of the assumption of linear sound velocity gradient. The use of a linear sound velocity profile gives an improved estimation of the grazing angle, path lengths and phase changes. This, in turn, will give a better estimation of the total sound field due to a point source over an impedance plane. The influence of the sound velocity gradient on the level difference spectrum is shown in Figure 7. It is obvious that the effect of increasing the sound velocity gradient is to increase the frequency of the first pld dip. On the other hand it is predicted that changing the sound velocity gradient has a negligible effect on the ground effect dip.

-15

I 0

I

I

I

I

I

I

I

I

/

J 2

Frequency (kliz) Figure 7. Predicted level difference spectrum for receivers at changes in normalized sound velocity gradient (a m-‘). Predictions based on receivers at 0 and 1.2 m, range 100 m, source height 10 m, effective flow resistivity 2.0~ IO’ MKS rayls m-‘, and effective rate of change of porosity with depth 150 m-‘.

Typical results of the two prediction methods with and without ray curvature, and their comparison with data, are shown in Figures 8(a), (b). The experimental details will be discussed later in section 3. Apparently, the inclusion of sound velocity gradient in equation (1) has significantly improved the fitting of the theoretical prediction to the experimental data for a known source/receiver geometry. Further examples of the improvement in theoretical prediction by including sound velocity gradient in the calculations in the simple manner described earlier are to be found elsewhere [20]. It should be noted that A = (r/2) - 8 provided that R >>H, + Hr. We further assume in the following analysis that the sound velocity gradient is sufficiently small to cause only a single reflection. 2.5.

ALGORITHM

FOR

HEIGHT

DETERMINATION

height of the source can be determined from the angle of incidence given a priori knowledge of the range by analyzing the level difference spectrum between two vertically separated receivers. In principle given a priori knowledge of height it would be possible The

120

K.

M. Ll

ET

AL.

0

2

Frequency

(kliz)

Figure 8. Comparisons of experimental data with two prediction methods. The two parameter variable porosity model is used to calculate the ground impedance and the theoretical prediction is based on range 100 m, source height 9.39 m and (a) receivers at 0 and 1 m, (b) receivers at 0 and 2 m. Solid line, experimental data; broken line, prediction without the use of linear sound velocity gradient; chained line, prediction with the use of linear sound velocity gradient.

Frequency

(kHz)

Figure 9. Predicted level difference spectra for various source/receiver geometries with constant angular position of the source, 4 = 5.71”. The theoretical prediction is based on effective flow resistivity 2.0~ 10’ MKS rayls m-l, effective rate of change of porosity with depth 150 m-’ and receivers at 0 and 1.2 m.

SOURCE

HEIGHT

121

DETERMINATION

to calculate range, but here we concentrate on height determination. As will be discussed again later, both height and range may be determined by this procedure if more than one receiving array is used. It is indicated in Figure 9 that in the absence of a sound velocity gradient the general shapes of the level difference spectra for various source/receiver geometries are rather similar, provided that the angle of incidence and ground parameters are the same. The inclusion of a sound velocity gradient parameter gives an improved evaluation of the angle of incidence and of the direct and reflected wave path lengths. Its inclusion does not, in general, alter the position of the ground effect dip but it has a significant effect on the first pld dip. These effects have been discussed in section 2.4. Nevertheless, Figure 9 indicates that only the value of A (or 0) can be determined rather than height or range separately. As long as H, >>H,, then a priori knowledge of range ,(R) together with acoustically deduced angular position of the source (A ) and normalized sound velocity gradient (a) enables deduction of H, through H,=[Jl-(aRsinA-cash)*-sinA]/asinA.

(28)

Based on the characteristics of the level difference spectrum an algorithm has been developed which determines the source height from a priori knowledge of the horizontal separation between the source and receiver. The final outputs of the corresponding computer programs are as follows: (1) source height; (2) linear sound velocity gradient (normalized with respect to sound velocity at sea level assumed to be 344 m/s); (3) effective flow resistivity; (4) effective rate of change of porosity with depth. A flow diagram of the algorithm for a single array to enable deduction of height with a priori knowledge of range is shown in Figure 10. To summarize, the program is divided into two stages: stage I, extraction of the ground parameters and the angle of incidence from the ground effect dip; stage II, determination of the best value of sound velocity gradient from the first pld dip; the source height can then be calculated by equation (28). To limit the number of variables used in stage I, the sound velocity gradient is set initially to zero. This necessarily means that the actual range cannot be input at stage I, Input range

I

ground parameters

Fit for sound

Bandwidth need be sufficient only to include first dip in level difference soectrum

Bandwidth must include both first and second dips

Usa angle, range and velocity gradient to deduce height

Figure 10. Block diagram to show the numerical height with a priori knowledge of range.

scheme (single array algorithm)

in determining

the source

z

0

-1

Figure 1 i. The importance of the inclusion of the first path length difference dip. (a) Measured level difference spectrum with range 100 m, source height 11.7 m and receivers at 0 and 1 m. (b) Measured level difference spectrum with range 100 m, source height 10.28 m and receivers at 0 and 2 m. (c) The variation of r.m.s. error with normalized sound velocity gradient, o; theoretical predictions are based on the measured spectrum shown in Figure 1 I(a) with effective Row resistivity 2.17 x IO’ MKS rayls m-r and effective rate of change of porosity with depth 95.5 m-‘. (d) The variation of r.m.s. error with normalized sound velocity gradient, o; theoretical predictions are based on the measured spectrum shown in Figure 1 I(b) with effective flow resistivity 2.41 x 10s MKS rayls mm’ and effective rate of change of porosity with depth 39.4 m-‘.

Normaltred sound velocity gradient

1

Frequency (kl-lz)

SOURCE

HEIGHT

DETERMINATION

123

since typically the data would only be fit by a non-zero velocity gradient at this range. Consequently, to obtain estimates of ground parameters and elevation angle consistent with a near-zero velocity gradient, the initial range input is an arbitrarily large value, say 2000m. Strictly speaking, one may assume any reasonable sound velocity gradient in stage I as the ground effect dip is relatively insensitive to this parameter (see Figure 7). But the initial assumption of zero sound velocity gradient has advantages of ease and convenience in determining the ray path lengths. The ground parameters and the angle of incidence can then be deduced from the ground effect dip by using the method of “least squares” procedure for a known range. Theoretical predictions are made for individual sets of values of the ground parameters and the angle of incidence over the frequency range of interest. It is necessary only to set upper and lower bounds on the angle during the iteration process. Typically, the angle is expected to lie between 0” and 20”. At each frequency the difference between the measured and predicted level difference spectra is squared and summed to produce an overall measure of “error”. By repeating this process for hundreds of such predictions, one can obtain values of a,, (Y, and 0 which result in the smallest value of “error”. The technique is related to the “least squares” method of curve fitting. The next stage involves finding the value of sound velocity gradient which gives “best fit” data for a given range, angle of incidence and ground impedance and subject to the constraint of a single reflection for the ground-reflected component. The source height can then be determined directly from equation (28). The procedure is based on the fact that there is a shift of the first pld dip to a higher frequency for a longer range. It is therefore crucial to include not only the ground effect dip but the first pld dip in the measurement of level difference spectra. In Figure 11(a) is shown a level difference spectrum which, as a consequence of bandwidth and measurement geometry, does not include the first pld dip. Figure 11(b) features a spectrum that does include the first pld dip. Figures 11(c), (d) show the corresponding variations of r.m.s. errors with the normalized sound velocity gradient. It is obvious that a local minimum of r.m.s. error occurs only for the level difference spectrum which has included the first pld dip. Consequently there is no doubt that the first pld dip is important in choosing the best fit value of the sound velocity gradient. 3. MEASUREMENTS The algorithm for deduction of source height, the development of which was described in the last section, has been tested at two sites on data obtained by using loudspeaker sources of continuous broadband sound at heights between 2 and 30 m and vertical receiver arrays with upper microphone heights of 2 m and at horizontal ranges from 12 to 130m. The first site was the Open University sports field. An Electra-Voice Model S1202 loudspeaker was used as the sound source. It was mounted on a radar mast having a pneumatic valve system enabling the loudspeaker to be raised to a maximum height of 15 m. A modified FET amplifier and Kemo Elliptical Filter were used to amplify and filter the signal from a white noise generator. The array of receivers consisted of three vertically separated i inch condenser microphones at heights of 0, 1 and 2 m above the ground surface. The output of each microphone was connected to Briiel and Kjaer (B&K) type 2607 or 2608 measuring amplifiers. These provided the polarization voltage for the microphones and amplification of the low level signals. Two Ono-Sokki CF-900 series FFT analyzers were used in the dual channel transfer function mode and were set to make up to 64 averages during each measurement. The signal from the ground microphone

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K. M. LI ETAL

was connected to channel A of both analyzers. The output from the 2 m high microphone was connected to channel B of one analyzer and that from the 1 m high microphone was connected to channel B of the other analyzer. This enabled simultaneous measurements of the level differences to be made. Spectra from each measurement were stored on a 3.5 inch floppy disc and subsequently transferred in digital form to a VAX 8800 computer for subsequent processing. The measurements were carried out on a calm sunny day. Wind speed was less than 1 m/s and variable in direction. The second site was at the Keweenaw Research Centre (KRC), Michigan Technological University, Houghton, Michigan. An Electra-Voice loudspeaker mounted on a hydraulic lift operated from the rear of a vehicle, stationed on a 6 m high earth mound, was excited with the ampified white noise output of a Hewlett Packard 3561A spectrum analyzer. Three sound level meters (B&K) were used as receivers and were positioned at 0, 1 and 2 m heights respectively. The power spectra received at each sound level meter, from a burst of signal with approximately one minute duration, were analyzed several times to smooth estimates. From these the level differences between 0 and 1 m receivers and 0 and 2 m receivers were calculated and stored on floppy discs before subsequent transfer to the VAX mainframe computer at the Open University, U.K. Throughout these measurements the weather conditions were calm and sunny. The wind was light, being variable in strength and direction, but at no time did the speed exceed 2.0 m/s. Note, however, that this does not imply zero sound speed gradients. Indeed, the results discussed in the following section suggest the presence typically of a temperature lapse. 4. COMPARISON

OF

MEASURED

AND

PREDICTED

SOURCE

HEIGHTS

Results of testing the algorithm described in section 2 on the loudspeaker data obtained from the receivers at 0 and 2 m heights above the Open University sports field are shown in Tables l-3. Results obtained from the 0 and 2 m high receivers at KRC, Michigan are shown in Tables 4-6. In all cases the known (measured) ranges have been used as a priori information. It should be noted that in most cases the path length differences between the 0 and 1 m microphones were not great enough to capture the first pld dip (at <2 kHz). The theoretical height predictions for various ranges were satisfactory except, perhaps, those obtained from lower source height data. This relative inaccuracy in determining the lower source heights is a consequence of the assumption that H, >>H,. TABLE

1

Extraction of the bestfit groundparameters, the angularposition of the source, the normalized sound velocity gradient and source height at a known range of 50 m; the actual source height is also given in the jrst column for comparison Best fit

Measured HS (m) 5.50 5.14 4.10 4.50 4.26 4.14 3.81 3.10 2.81

, ue (MKS rayls m-‘) 2.47 x 3.83x 2.98 x 3.44 x 3.55 x 2.62 x 2.32 x 4.23 x 2.14x

10’ 10’ 10’ 10s 105 lo5 lo5 10’ 105

parameters

a, (m-‘)

A (deg)

4.30 1.07 x lo* 0.27 x lo* 1.42 x lo2 0.51 x 10’ 1.37 x lo* -4.60 x lo* 0.86 x lo2 5.34 x 10’

4.93 4.58 3.91 3.82 3.59 3.43 5.53 3.08 4.72

a (m-l) -7.38 -7.99 -7.62 -1.27 -7.41 -5.87 -1.77 -6.45 -1*00x

x x x x x x x x

1O-4 1o-4 1O-4 1o-4 1o-4 1O-4 1o-4 1O-4 lo-’

H, (m) 5.25 5.02 4.43 4.25 4.07 3.74 4.61 3.50 5.39

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HEIGHT

125

DETERMINATION

TABLE

2

As Table 1 but range 87.5 m Best fit parameters Measured H, (m)

, a, (MKS rayls m-‘) 2.24 3.41 3.58 2.84 4.91 2.39 3.95 6.37 6.89 7.54

10.33 9.74 9.38 8.97 8.74 8.20 7.97 7.43 7.20 6.43

x x x x x x x x x x

10’ lo5 lo5 10’ lo5 lo6 lo5 lo5 10’ lo5

ru, (m-l) 0.98 0.97 0.41 1.03 0.67 -2.44 1.13 0.61 -0.74 -1.33

x x x x x x x x x x

A (deg)

lo2 lo2 lo2 10’ 10’ 10’ lo? lo2 10’ lo2

TABLE

5.83 4.93 4.94 4.63 5.03 4.84 4.39 3.96 4.72 2.72

a (m-l) -3.60 -3.84 -4.62 -3.90 -3.66 -1.21 -3.95 -3.69 -4.53 -7.38

x x x x x x x x x x

1O-J 1O-J 10-j 10-j 1O-4 lo-” lo-” 10-j 10~~~ lo-”

H,(m) 10.33 9.04 9.34 8.60 9.13 7.88 X.25 7.48 8.28 7.01

3

As Table 1 but range 100 m Best fit parameters Measured

1 ce (MKS rayls m-‘)

H, (ml 11.70 11.28 11.00 10.39 10.28 9.51 9.39 8.63 8.51 7.65

2.27 x 2.10x 3,97 x 2.19 x 2.41 x 2.38 x 2.41 x 4.68 x 4.77 x 4.36 x

10’ 105 105 105 lo5 10’ 10’ lo5 10’ lo5

(Y, (m--l)

A (deg)

0.38 x 10’ 4.15 0.32 x lo2 0.95 x lo2 0.39 x lo2 0.58 x lo2 0.54 x lo2 -0.18 x lo2 0.33 x 10’ -0.65 x lo2

5.70 5.40 5.52 4.90 4.57 4.86 3.94 3.79 3.88 3.39

TABLE

a (m-‘) -3.17 -3.89 -2.85 -3.66 -4.10x -3.87 -3.58 -3.75 -3.81 -4.74

x x x x x x x x x

lo-” 1O--4 10m4 1O-4 lo-” 1O-4 1O--4 1o-4 1O-4 lo-”

H, (m! 11.59 11.52 11.11 10.42 10.06 10.46 8.68 8.50 8.69 8.30

4

As Table 1 but range 130 m Best fit parameters Measured

4 (ml 20.0 19.0 18.0 17.0 16.0 15.61

1 ue (MKS rayls m-‘) 6.30 5.63 2.35 1.32 8.50 4.65

x x x x x x

10’ lo4 lo5 10’ 10’ 10’

a, (m-l) 1.88x 4.67 x 3.76 x 4.39 x 3.21 x 2.46 x

10’ lo2 lo2 lo2 lo2 lo2

A (de& 8.75 8.52 7.56 7.38 7.01 6.46

a (m-l) 6.11 x 1.46x -5.55 x -3.45 x -2.36 x -5,97x

1O-5 lo-’ lo+ lo-‘S 1O-6 10-S

H., (ml 19.47 18.34 22.08 17.14 15.77 15.22

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TABLE

5

As Table 1 but range 160 m

Best fit parameters Measured

+

*

H, (ml

ue (MKS rayls m-‘)

a, (m-‘)

A (deg)

a (m-l)

H, (m)

19.56 18.15 16.73 15.31

3*10x 10’ 7.74 x 106 5.07 x 10’ 3.80 x 10’

-1.68x lo3 -4.04 x lo2 -3.17 x lo3 -9.65 x lo*

6.85 6.68 6.36 6.25

1.18 x 1O-4 1.01 x 1o-4 1.21 x 1o-4 1.07 x 1o-4

17.69 17.44 16.26 16.13

TABLE

6

As Table 1 but range 175 m

Best fit parameters Hs (ml

ce (MKS rayls m-‘)

20.30 19.99 18.44 16.89 16.35 13.81

1.43x106 1.50 x lo6 9.73 x lo5 2.92 x IO5 2.16 x IO5 1.06 x 10’

0*66x lo* 0.96 x lo* 2.98 x 10’ 3.27x IO* 4.19x lo2 7.69 x lo*

7.00 7.10 6.53 5.71 5.04 4.34

a (m-l)

H,(m)

7.30 x 1o-5 7.63 x 1O-5 4.83 x 10m5 2.04 x 1O-6 -5.60 x 1O-5 -5.25 x 1O-5

20.35 20.62 19.25 17.48 16.29 14.38

_

It can be seen from the tables that the values of the fitted parameters (Y,, o, and a fluctuate for a given range and various heights. This is because these parameters are allowed to vary independently in the best fit procedure in order to give the minimum letting all three parameters vary simultaneously does not “error”. Consequently, necessarily give the true value of any one. In principle, the ground parameters could be determined in advance by inverting a short range measurement. The best-fit sound velocity gradient would then be expected to vary in a more consistent manner (more indicative of the true meteorological values) by using the known ground parameters in the best fit procedure.

5. CONCLUDING REMARKS A technique for deducing source height at a known range in which the ground effect interference pattern, modified by the effects of stable refraction is exploited, has been described. The technique is based on the assumptions (1) that ground impedance may be described by a two-parameter variable porosity model, (2) that the sound velocity gradient is linear, and (3) that the range and sound velocity gradient are sufficiently small that only a single reflection need be considered. The procedure has been validated experimentally by using a fixed loudspeaker source of broadband sound at heights from 2 m to 20 m and ranges from 50 m to 175 m. The heights have been predicted acoustically to within 10% of their measured values, the accuracy being worst in percentage terms at the smallest elevation angles. This is a consequence of assumptions (2) and (3) above. The technique is least satisfactory at small grazing angles where the sound velocity gradient will be largest and non-linear. It should

SOURCE

HEIGHT DETERMINATION

127

be stressed that since the technique is based upon analysis of level difference spectra it is applicable to any broadband source. Improvements would follow from use of a more accurate propagation model based either on creeping wave analysis [21] or the Fast Field Program [22]. It should be noted that use of more than one vertically separated receiver pair, if the arrays are sufficiently distant from each other, allows triangulation for source location. This has been carried out successfully on airborne sources and will be the subject of further publication.

ACKNOWLEDGMENTS This work was carried out under MOD Procurement Executive Contract No. 2006/l (extension). The authors would like to thank the liaison officer, Dr Peter Soilleux, for many helpful discussions and his encouragement. Initial work on this project was carried out by Andrew P. Watson. The authors also wish to thank Dr Sung Lee for measurement facilities and support at KRC, Michigan, and Dr James Rogers and Dr Heather Moore for assistance with these measurements.

REFERENCES 1. C. GLARETAS

1980 Ph.D. Thesis, Pennsylvania State University. A new method for determining the acoustic impedance of the ground. 2. R. BOONE 1987 Ph.D. Thesis, Technical University of Deljt. Design and development of a synthetic acoustic antenna for highly directional sound measurements. 3. K. ATTENBOROUGH, J. M. LAWTHER and S. I. HAYEK 1980 Journal ofthe Acoustical Society ofAmerica 68, 1493-1501. Propagation from a point source above a porous half space. 4. K. ATTENBOROUGH 1982 Journal of Sound and Vibration 81,413-424. Predicted ground effect for highway noise. 5. M. E. DELANY and M. E. BAZLEY 1970 Applied Acoustics 3, 105-116. Acoustical properties of fibrous absorbent materials. 6. S. I. THOMASSON 1977 Journal of the Acoustical Society of America 61, 659-674. Sound propagation above a layer with a large refractive index. 7. R. J. DONATO 1977 Journal of the Acoustical Society of America 61, 1449-1452. Impedance models for grass-covered ground. 8. K. ATTENBOROUGH 1985 Journal of Sound and Vibration 99, 521-544. Acoustical impedance models for outdoor ground surfaces. Institute of Acoustics, September 9. J. WEMPEN 1987 One day meeting on outdoor soundpropagation, 1987, 29-36. Ground effect on long range sound propagation. 10. S. TOOMS and K. AITENBOROUGH 1988 Proceedings of the 3rd International Symposium on Long Range Sound Propagation and Coupling into the Ground, 28-30 March 1988, Mississippi, 122-142. 11. A. BERRY 1986 Private communication. Modeles d’impedance pour les sols non-homogenes. 12. R. G. FIZELL (editor) 1988 Proceedings of Workshop on Acoustic Match Field Processing, 20.-21 November 1985. Naval Research Laboratory, Washington, D.C. 13. C. F. CHIEN and W. W. SOROKA 1975 Journal qf Sound and Vibration 43, 9-20. Sound propagation along an impedance plane. 14. T. F. W. EMBLETON, J. E. PIERCY and G. A. DAIGLE 1983 Journal of the Acoustical Society ofAmerica 74, 1239-1244. Effective flow resistivity of ground surfaces determined by acoustical measurements. 15. H. M. HESS 1988 Ph.D. Thesis, The Open Uniuersity, U.K. Acoustical determination of physical properties of porous ground. 16. C. M. HARRIS (editor) 1957. Handbook of noise control. New York: McGraw-Hill. See chapter 3. 17. D. C. PRIDMORE-BROWN 1962 Journal of the Acoustical Society of America 34,438-443. Sound propagation in a temperature and wind-stratified medium. 18. T. F. W. EMBLETON, G. J. THIESSEN and J. E. PIERCY 1976 Journal of the Acoustical Society c$America 59, 278-282. Propagation in an inversion and reflections at ground.

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19. T. HIDAKA, K. KAGEYAMA and S. MASUDA 1985 Journal of the Acoustical Society oj’Japan 2, 117-125. Sound propagation in the rest atmosphere with linear sound velocity profile. 20. K. ATTENBOROUGH, N. W. HEAP and K. M. LI 1988 Proceedings ofthe Institute ofAcoustics 10,515-521. Sound velocity gradients and propagation over an impedance boundary. 21. G. A. DAIGLE, T. F. W. EMBLETON and J. E. PIERCY 1986 Journal ofrhe Acoustical Society ofAmerica 79,613-627. Propagation of sound in the presence of gradients and turbulence near the ground. 22. S. W. LEE, N. BONG, W. F. RICHARDS and R. RASPET 1986 Journal of the Acoustical Society of America 79, 628-634. Impedance formulation of the fast field program for acoustic wave propagation in the atmosphere.