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Acoustic calibration of sodars

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1992 Meas. Sci. Technol. 3 1001

(http://iopscience.iop.org/0957-0233/3/10/012)

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Meas. Sci. Technol. 3 (1992) 1001-1007. Printed in the UK

Acoustic calibration of sodars

S D Danilovt, A E Gur'yanovt, M A Kallistratovat, I V Petenkot, S P Singalt, D R Pahwat and B S GeraS t Institute of Atmospheric Physics, Russian Academy of Sciences, Moscow, Russia t National Physical Laboratory, New Delhi, India

Received 6 August 1991, in final form 11 May 1992. accepted for publication 5 June 1992

Abstract. Measurements of calibration parameters of sodars normally require either a high mast, a tethered balloon and calm weather, or a very large anechoic chamber for precise results. In the present work, however, a simple method for the measurement of calibration parameters using a normal size anechoic chamber and a theoretical antenna directivity pattern is suggested. The method has been used for t h e two sodars belonging to the Institute of Atmospheric Physics, Moscow (Sodar LATAN-1) and the National Physical Laboratory, New Delhi.

The calibrated sodar parameters determined for the Russian Sodar LATAN-I have also been used to obtain values of t h e temperature structure parameter C: which have been found to be comDarable with the simultaneouslv measured in situ microthermal values 01 C:

1. Introduction

Sodars, among many other practical applications, allow us to measure absolute values of the temperature struc- ture parameter of turbulence, CZ, which is a very useful parameter for studying the behaviour of the optical refractive index in the atmosphere. Using the monostatic sodar C: can be found using the relationship (Little 1969):

where uLBo is the effective backscattering cross section per unit of scattering volume per unit solid angle, T is the absolute temperature of air and k is the wavenumber of the interrogating sound waves. The value of also can be found from the results of measurements of the system parameters in conjunction with the sodar equation:

where P, and P, are the transmitted and received powers respectively, c is the speed of sound, T is the pulse duration, r is the distance of the scattering volume from the sounder, S,,, is the effective area of sodar antenna, which also includes the transducer efficiencies, and L is the attenuation coefficient of the sound waves propagdt- ing in atmospheric air.

Obviously, the above needs a careful calibration of the sodar antenna system. The most direct way of acoustically calibrating a sodar is the field measurement of P, at the far field of the sodar antenna under operating voltage V,,, at the terminals of the transducer in the transmitting mode and also the measurement of the

0957-0233/92/101001 +07 504.50 0 1992 IOP Publishing Lld

sound intensity I , = Pr/S,., near the antenna (due to an additional source placed in the far field) and the input voltage Kn at the terminals of the transducer in the receiving mode. Knowing the two coefficients of trans- formation

and measuring V,,, and Kn we can find the absolute values of uls0 and C: from (2) and ( I ) respectively.

Note that such field measurements of conventional sodar antennae fitted with a loudspeaker unit and horn at the focus of the paraboloid are laborious, and require good, windless weather, high masts or tethered balloons for placing the measuring microphone and the source of sound, and some kind of turning device for directivity measurements. Such field measurements have been made by Aubry et ul (1972), Singal et al (1975) and Gerasyuk et al (1981), and it has been found that the radiation patterns d o not appear to be well defined in most of the cases due to the various limitations involved in the field experiments.

In the following, a rather simple technique is described for the acoustic calibration of a sodar trans- ducer and horn system by placing it in an anechoic chamber and using a theoretical estimate of San,. The theoretical background for such an approach is pre- sented and a procedure for measurements is described together with the results of calibration of two sodars (a designed, developed and fabricated one, LATAN-1, at the Institute of Atmospheric Physics, Russian Academy of Sciences, Moscow, and the other at the National Physical Laboratory, New Delhi, India). A comparison of the Cf values obtained by LATAN-I with the data

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of simultaneous in situ C: microthermal measurements on the meteorological mast is also given t o show accu- racy in the calibration procedure of the sodar system.

2. Theoretical background

2.1. Calculation of sound pressure in the sodar far field

The pressure field p , produced by a transducer with a horn, in the far field is normally given by the expression:

(4)

where Dh(Oh) is the dlrectiona! fzctor at the ang!e Ob between the horn axis and the direction of observation, k = w/c is the wdvenumber with angular frequency 0, rh is the distance between the centre of the horn aperture and the point of observation, V,,, is the voltage applied to the loudspeaker unit, S = 1 m is the reference distance and q,", is the output efficiency of the transducer with a horn.

The directional factor of the horn in the above expression can be described theoretically (Kinder et al 1982) as that of a piston source with an aperture radius equal to the horn aperture Rh in the form:

where J , is the Bessel function. In our case, the aperture of the horn is placed at the

focus of the parabolic reflector which will generate a parallel field in the projected zone of the reflector. The pressure factor pa", for such a plane wave can be expressed in the form:

(6) Here R is the radius of the parabolic reflector, f is the focal distance, p is the perpendicular distance measured from the axis of the paraboloid, z is the distance meas- ured along the axis of paraboloid from the focal point and A(p) is the pressure distribution over the aperture of the paraboloid.

The expression for A(p) can easily be obtained from the boundary conditions on the surface of the para- boloid. According to this expression the sum of the normal velocities in the incident (equation (4)) and the reflected (equation (6)) fields should be equal to zero at the surface. It is given by:

with 0 < p < R. i k ( r + I ) pout = &)e

In the far field of the paraboloid, the average pressure p.., at a distance r from the dish ( r = z + f ) can be written in the following way:

where D ( 0 ) is the directional factor of the acoustic antenna system of the sodar pertaining to the pressure distribution A(p) and f is the focal distance of the dish.

1002

It coincides with the directivity factor of the piston source having amplitude distribution A(p) and can be written as

I + c o s 0 loR A(dJdkp sin QP dp

(9) D(0) = Jon A(P)P dp 2

where J , is the Bessel function. It is worth mentioning that the distribution A ( p ) can

easily be measured experimentally at the level of the enclosure top edge.

On the one hand this allows us tojustify the theoreti- cal result given by equation (7) and on the other gives us an opportunity to calculate D(0) more directly than is done on the basis of the theoretical distribution A(p) .

The pressure field defined in equation (8) produced by a sodar in the far field can be thus determined if the voltage V,,,, the output efficiency K,,", and the directivity factor D(0) are known.

2.2. Relationship between transmitted and received signals

Consider now the operation of the sodar in the receiver mode. Imagine that a plane pressure wave p.e-"' is incident along the axis of the paraboloid. Then, using (7) and (8), the averaged pressure field formed over the horn aperture after reflection from the paraboloid is given by

The integral in this formula describes summation of the reflected field over the paraboloid or, more exactly, over the ring elements into which the paraboloid can be divided. &(Oh) in this in!egral appears to be due to the following two reasons. Firstly, different elements of the paraboloid send reflected signals to a given point of the horn aperture with their own phases, which makes the total pressure field at a given point smaller in amplitude in comparison with that possible at the focal point. Secondly, the horn effectively averages the press- ure field over its aperture. Indeed, the voltage developed by the loudspeaker unit is determined by the total force acting on its membrane, which is given by the integral of the pressure distribution over the aperture.

Assuming that the incident sound waves are distrib- uted over angle 0 to the axis of paraboloid, and further that waves propagating at different angles are uncorre- lated, then, instead of equation (IO) we can write the following relationship:

"

Acoustic calibration 01 sodars

Here Ip,(0)12 is the angular distribution of the squared value of the pressure of the incident sound wave on the paraboloid. It can be readily expressed through the cross section of the turbulent scattering of sound, ulsn. using eqiia:ions. (2) and (3):

where pa", can be expressed by equation (8). Further, using (7), (8) and (12), pi, can now he rewritten as:

Substituting 1 l&Iz / lKin lZ instead of Ipi.12, we obtain the result:

This expression can give us a value for the scattering coefficient u lsn , when we are able to compute the integrals in this formula. For this purpose, let us start from the calculation of D(0) which can be carried out more conveniently by substituting a model distribution of A(p) instead of real one. The simplest choice is

A(p) = A(O)[l - &p2/Rz] (15)

where E is defined from the condition that model and real distributions coincide at p = R. The correctness of this choice can be shown by moving forward a few steps to note that the discrepancy in the distribution for the two sodars, i.e. for the sodar LATAN- I (f= 0.55 m, R = 0.65 m, kR, = 3.14, c = 0.82) and for the NPL sodar ( f = 0.45 m, R = 0.6 m, kRh = 3.04, E = 0.81) is less than 20%. The model distribution A @ ) thus allows us to continue the integration in the formula for D(O). In the limit of a half width of the beam angle, the result is:

( i) (16) D(B)%I--(kRsinO)2 I - - .

Now, inserting this directivity distribution into the inte- gration J 1D(0)I4 sin B dfl and restricting it to only a half width of the beam angle, the result of integration is (taking E = 0.82):

I 8

(17) ( ~ ( c r ) ( ~ sin 0 dH x ~ __ - - R 1.3 1.571 (kR)' I - 4 6 - (kR)"

The other integration in expression (14) can also be

Denoting by b the parameter, similarly solved.

and calculating it numerically for the two sodars we find that for sodar LATAN-I, b-0.39, and for the NPL sodar, b z 0.38. Formula (14) can thus be rewritten as

This expression links the sodar parameters with the scattering coefficient and thus allows us to determine ulSn through known coefficients K~~ and K,,,, which in turn allows us to calculate C: using relationship (1).

So far the dissipation of sound in air has not been included in the calculation. However, to take it into account one may readily modify expression (19) by multiplying its right-hand side by eC4" where a is the amplitude dissipation coefficient.

3. Calibration procedure and results

Based on the above theoretical discussions, the following parameters of a transducer and horn system, namely the output efficiency tiou,, the input efficiency K~~ and the directivity distribution of the transducer, need to he known for the absolute calibration of a sodar.

To obtain the transducer output efficiency tiouc, the pressure level at a distance of 1 m from the horn aperture is measured when a voltage of 1 V is applied to the loudspeaker unit at the working frequency of the sodar. To obtain the input efficiency tiin, the voltage at the terminals of the transducer in the receiving mode is measured in an acoustic field of 1 Pa in the plane of the horn aperture. The directivity of the transducer should now be determined. We may note here that only the outer front aperture of the horn is to be considered as the source/receiver of sound. (In general, any cross section of the horn can be considered as transmitting or receiving, provided the diffraction from the wall and the outer aperture of the horn is taken into account). Since the outer aperture is the only effective source/receiver for illuminating sodar antenna. Hence during sodar operation just the centre of the outer aperture of the horn is placed at the focus of the paraboloid, making only small adjustments (close to the focal point of the paraboloid) for maximum efficiency.

Whilst measuring the input efficiency, we may further bear in mind that the size of the horn aperture is comparable to the sound wavelength. This can consider- ably disturb the incident sound field in the vicinity of the transducer, making the measurement of pressure level using the reference microphone unreliable. It is therefore necessary to first measure the voltage at the transducer terminals and use the reference microphone to determine the true pressure level. The ratio of the voltage to the pressure level gives K ~ ~ .

Standard measurement equipment for the determi- nation of the directivity of the transducer includes a revolving platform. Since the platform has large dimen- sions compared with the sound wavelength, leading to the diffraction of the sound field by the transducer placed

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S D Danilov et a/

on the platform, our measurements of directivity will not he exact. Our attempt to suppress this effect by covering the platform with an absorbing material has not led to good results. The probable reason is that one should use a very thick layer of absorbing material to achieve adequate suppression, which has not been poss- ible. It has been found that it is more convenient to place the transducer near, the edge of the platform, allowing the horn to he free, as shown in figure I . Clearly when such a scheme is used for measurements the directivity characteristics obtained should he corrected. The correction must account for the increase in distance between the horn aperture and reference microphone as the platform revolves. In addition the angle dependence of directivity diagrams has also to be corrected to the actual angle Oh instead of the angle Om, as shown in figure 2. Any other variation of this scheme, in which the loudspeaker unit is placed somewhere hetween the edge and the centre of platform, will also need correction in the manner described here.

With a view to illustrating the values of correction, the measured and corrected data are represented for the NPL sodar transducer in table I . It may be noted that the point at which the transducer was connected to the horn during measurements was placed near the edge of

Figure 1. Schematic diagram of the position of devices for measuring directivity.

/

TRANSDUCER hp /

WlTH HORN

4 \=- REFERENCE MICROPHONE

REVOLVING PLATFORM

Figure 2. Scheme lor calculating the corrected angle dependence.

the platform of rotation, the distance from this point to the reference microphone being 1 m, and that the horn length was 0.2 m.

The procedure for experimentally determining the directivity is cumbersome to repeat each time a new horn is fabricated. A theoretical model may he more convenient to use. However the horn design should be such that it offers good coincidence between the calcu- lated and the actual directivity behaviour. In figure 3 the theoretical behaviour of diagrams Dh(Oh), given by formula (5) , and the experimental curves (corrected) for transducers of sodar LATAN-I and the NPL sodar are given. It may he seen that the thenretica! and experimeE- tal curves coincide rather satisfactorily within the angles at which the paraboloids of sodars are seen from their respective foci.

In computing the above directivity behaviour, for the sodar LATAN-I an exponential horn has been used with R, = 0.085 m which at a working frequency of 2 kHz gives kRh = 3.14. The horn used for the NPL sodar is also exponential and has Rh = 0.075 m, which at a working frequency of 2.2 kHz gives k R , = 3.04. The angles at which the apertures of the reflecting para- boloids are seen from their respective foci are 61" for LATAN-1 and 67.5" for the NPL sodar.

The main parameters of the two sodars and the final

30 50 70 90 10

Angle eh

Figure 3. The comparison of the theoretical and the real directivity diagrams of loudspeaker uni t s with horns.

Table 1. Measured and corrected data for directivity determination of t h e NPL sodar source.

Measured angle 0, 0" IO" 20" 30" 40" 50" 6D" 70"

Distance d from horn aperture to microphone (m) 0.80 0.80 0.81 ' 0.83 0.86 0.88 0.92 0.95 Distance variation correction 20 log(dlO.8) 0 0 0.1 0.32 0.65 0.87 1.30 1.65

Measured level, dB 0 -0.3 -1.8 - 4.0 -6.5 -10.5 - 12.5 - 16.0 Real angle 0, 0" 12.5" 25' 37" 480 60" 70" 789

1004

Acoustic calibration 01 sodars

results of calibration thus obtained are given in table 2. It may be noted that the output efficiencies and the parameters, kR, and b, for the two sodars are more or less the same.

It may a!so be noted that the numerical coefficient in equation (19):

K = 25nb4(kR)*(kRJ4 (20) is constant for a given device. For LATAN-I it has been computed and found to be K % 10.7.

4. Comparlson of sodar (LATAN-1) and in situ microthermal measurements of C: Applying the above approach to the calibration pro- cedure, values of C: can be derived from sodar data. However, looking at the accuracy of measurements, it should be pointed out that formula (19) is approximate. Its approximate character is connected with some of the simplifications made in place of elaborate calculations. The estimations of errors made while deriving the for- mula (19) are as follows.

(1) The real directivity factor was replaced by the theoretical one. From the plots presented in figure 3 it can easily be seen that the discrepancy between the real and theoretical factor does not exceed *IS%. It can be either positive or negative depending on the horn con- struction. Note that this error may be eliminated by using the real directivity factor and implementing numerical integration starting from formula (8).

(2) The error connected. with using the model distri- bution A(p) is due to the difference between it and the distribution given by (7). A(0) and A(R) coincide for both distributions, but for an intermediate value of p the model distribution is somewhat greater leading to the possible overestimation of coefficient K given by (20) by up to 15%. This error also can be eliminated by means of a more elaborate numerical calculation with the distribution given by (7).

(3) The error due to diffraction can be roughly

Table 2. The main parameters and the results of calibration for LATAN-1 and the NPL sodar.

Parameter LATAN-1 NPL sodar

Operating frequency (Hz) 2000 2200 Horn aperture. Rh (m) 0.085 0.075 Paraboloid aperture, R (m) 0.65 0.60 Focal length. f (m) 0.55 0.45 Output efficiency K ~ " , (Pa V- ' ) 1.5 1.32 Input efficiency K,. (VPa-') 0.030 0.0176 Coefficient b a s per equation (18) 0.39 0.38 Angle with t h e paraboloid aperture from t h e focal point 61" 67"

Acoustic power, P,., (W) -6 4-10 Pulse duration, I ( 5 ) 0.05 or 0.1 0.05-0.10 Height (m) 31 to 550 40-700

Electrical power (W) 100 100

evaluated as (SF A(p)p dpl S:A(p)p dpj2 4 5%. It can lead to the overestimation of the coefficient K by up to 10% (5% in the transmitting mode and 5% in the receiving mode). To a certain extent both errors, in the distiibiition Aip) and the diffraction, can be accoilntec! for by measuring the real distribution A(p) at the top of the sodar enclosure and using it for numerical calcu- lations of D ( S ) instead of the result given by (16).

(4) The error connected with the determination of is, in fact, very small, since in its major part the non-

planar character of sound field in the vicinity of the focal plane is accounted for by (10). Only minor effects due to the special character of the field reflected by each element of the parabolic dish remain. Their influence is significantly smaller than the error due to diffraction.

,

Due to the uncertainties mentioned above, exper- imental comparison with independent in situ measure- ments of temperature structure parameter C: becomes necessary for verification of the sodar calibration procedure.

Comparison of C: values measured by sodar LATAN-I (calibrated according to the methodology described above) with those measured by microthermal probes on the meteorological mast was carried out at Mt Maydanak (2600111 above sea level, Uzbekistan, West Pamiro-Alay) in the month of August 1990, during an astronomical site testing campaign. Study of the nocturnal, optically active atmospheric turbulence over the isolated peak was the main task of the campaign (Gur'yanov et al 1992).

The microthermal measurements for comparison were carried out at heights of 33 m and 50 m. Measure- ments were made simultaneously by means of two differential fast response microthermometers, each operating with its own pair of thin wire probes, separated horizontally by 50 cm. The size of the probe was I cm in one pair and 2 cm in the other. The construction of the microthermometers and the procedure of measure- ments have been described elsewhere (Gur'yanov 1980, Gur'yanov et al 1987). The distance between the sodar antenna and the mast was 90m, the foot of the mast being located about 15 m higher than the base of the antenna. A view of the mast with the microthermometer probes and sodar antennae of LATAN-I, installed on a gentle slope, can be seen in figures 4 and 5 respectively.

Comparison of the microthermometers with their pairs of probes installed close by (figure 4), has shown an excellent agreement between the results obtained by both. For sodar-microthermal comparison, 40/20 min time averaging data was chosen, taking into account our earlier experience (Gur'yanov et al 1987). The begin- ning and the end of each session were strictly synchron- ized for both sodar and microthermal measurements.

The sodar data at the.three altitudes h, above the base of the antenna were taken for comparison with the microthermal data for each of the two different altitudes h , over the foot of the mast (h, = 33, 41 and 48 m for h,= 33 m and h,= 50, 58 and 65 m for It,= SO m) because of some uncertainty in the zero point of the

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S D Daniiov et a/

Figure 4. A view of t h e microthermometers on the mast (there were four thermometers at t h e moment of comparison).

Flgure 5. The antennas (in enclosures) of three- dimensional Doppler sodar LATAN-1 on the slope of Mt Maydanak. The data of the vertical component only were used for comparison.

height scale arising from the difference in the sodar and mast locations. The comparison of sodar data for each of the three h, altitudes with microthermal data for corresponding single h, altitude have shown an insig- nificant difference from one h, to the other, but for the 'interstitial' altitudes h, (h, = 41 m for h, = 33 m and h, = 58 m for h, = 50 m) the correlation was somewhat higher than in the other two cases. So subsequently only these pairs of altitudes were considered.

The values of the coefficients of absorption a needed for the computation of sodar C: were determined with the help of American National Standard (ANSI S1.26- 1978) and assumed to be independent of the altitude while humidity and temperature were measured at the height of 2 m near the mast.

The majority of measurements were carried out at night with only ten sessions conducted in the daytime under convective conditions. The results for the compari- son of each pair of altitudes are plotted in figure 6, while the coefficients of correlation and the geometrically

1006

Flgure 6. The results of the comparison of sodar and in situ C: measurements. Dots, night-time values; squares, day-time measurements, full lines, proportionality with nocturnal coefficient 8; broken and dotted lines, twofold and tenfold distinctions from this proportionality respectively.

averaged scaling coefficients B defined as

(21) are given in table 3. In expression (21), y = ( C : ) , is measured by microthermometers and x = (C:)s is deter- mined from sodar measurements; angle brackets denote arithmetical averaging.

In table 3, the estimates of random scattering of scale coefficients in individual sessions are also indicated (in round brackets) corresponding to multiplier/divisor IO", where U is the standard deviation of the logarithm of ratio (C:), /(C$)s. This gives the values of B in table 3 in the form:

. B = IO<IOB(Y/X)>

B = 10(1"8(Y/")) (x /+ 10"). (22)

Combining all of the nocturnal results, at the two pairs of altitudes, (see figure 6) we have obtained B =

0.70 (x /+ 1.9) and a correlation coefficient of 0.89. It may be seen that this typical 30% excess of sodar data over in situ ones is small in comparison with the great range of C: values (about four orders of magnitude at the given altitude and more than five orders of magnitude if the height dependence in the boundary layer is con- sidered). Such discrepancies cannot be explained by the errors caused by the simplified calculations considered above. The observed error may be further increased by taking into account the errors mentioned in points 2 and 3 at the beginning of the section.

Acoustic calibration of sodars

Table 3. Values of the coefficient of correlation and the geometrically averaged scaling coefficient B= (C+)-/(C+jS for microthermal and sodar data at Mt Maydanak (see text).

Microthermometer (h, = 33 m) sodar (ha = 41 m)

Microthermometer (h, = 50 m) sodar (h, = 58 m)

Coefficient of correlation 0.91 0.87 Scaling coefficient 6

nocturnal 0.67 ( x / + l . 8 ) 0.72 ( XI-2.0) daytime 1.28 ( x / - i . 9 ] 1.08 (x1-2.0)

A similar combination of daytime results gives B = 1.17 (x /-2.0). This daytime underestimation of sodar data by, typically, 17% does lie in the range of possible simplification errors roughly estimated a t the beginning of this section.

The difference between the typical nocturnal and daytime values of scaling coefficient B is apparently connected with the imperfection in the turbulence model providing the base of equation (I). Such a difference has also been reported by many authors (e.g. Neff 1978, Zhou et al 1982, Gur’yanov e t al 1987), but a conclusive explanation does not yet exist. This problem is outside of the framework of acoustic sodar calibration and is rather too complicated to be discussed here.

5. Concluslon

The results of the field comparison of sodar and in s i tu microthermal measurements of C; confirm the validity of the method of acoustic calibration of sodars proposed. However, the intensity of the nocturnal echoes turns out to be less theoretically than that observed experimen- tally. In our opinion, the main cause of the augmentation of the sodar echo-signal level a t night-time in relation to theory may be connected with the anisotropy of small-scale turbulence a t the nocturnal inversion. The direct experimental confirmation of this hypothesis will require a more accurate definition of expression ( I ) obtained in the case of isotropic turbulence using Kolmogorov’s spectrum (Tatarskii 1971).

Acknowledgments

We would like to thank E A Shurygin for programming and calculations and A N Bedulin, E V Sartory, Omkar Sharma and V K Ojha for their assistance. We would also express our thanks to the Director of the Institute of Atmospheric Physics, MOSCOW, Russia and the Direc- tor of the National Physical Laboratory, New Delhi, India for allowing us to publish this paper. Thanks

are also due to the former USSR Academy of Sciences a n d Department of Science and Technology, India for funding and for permitting us to work on the joint programme under the Indo-Soviet Long Term Collabor- ation Scheme.

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