Short Range Propagation Through an Inhomogeneous Medium

N . G. Pace' W. J . Woo8 G. M . A . Jones' 'University of Bath, Claverton Down, Bath, UK

'Defence Research Agency, Portland, Dorset, UK

Ahstmct - Inhomogeneity is a characteristic of ev- ery real medium, and two types are observed, regu- lar and random. Regular inhomogeneities are pro- duced by spatial variation of the mean character- istics of a medium, for example, velocity variation with temperature, dep th and salinity. Random in- homogeneities are produced by deviation from the mean values. In this paper a medium with random inhomogeneities is modelled and ray diagrams illus- t ra te the effect of the magnitude and scale size of the inhomogeneities on the propagation of sound. The sound speed of the medium varies with both range and dep th and thus may be described by cor- relation lengths in both dimensions. The results presented relate to an existing extreme experimen- tal situation in the propagation range is 170m with horizontal correlation length of 20m, vertical cor- relation length Zm, and standard deviation of the local sound speed from the local mean of 0.5m/s. The modelling results are compared with experi- mental results.

I. INTRODUCTION

A theoretical overview of propagation in a random medium is given in [l]. In the following section ex- pressions for medium characteristics which depend on the correlation function are presented. Here a brief de- scription of the correlation function for refractive index is given.

Fluctuations in the refractive index, T I , are described by p(x, y, z , t ) . The correlation function of a process stationary in time and spatially homogeneous is given by

where overbar means averaging with respect to time t , and x = 2 2 - x1,y = y2 - y1 and z = 22 - 21.

achieves its maximum at z = y = z = 0, where

The correlation coefficient N = Nl2/7, behaves such that as distance between points is increased, cor- relation coefficient decreases from its maximum value of 1 and becomes small (< 1) at a distance, a (scale of inhomogeneities).

In section 111 a crude derivation of the focusing factor in terms of the number of rays which fall in an interval is given. From models of the inhomogeneous medium a prediction of intensity fluctuation with depth at a fixed range is given. These are then compared with data from an actual experiment with similar configuration.

- NI1 = p2.

Table I Data measured on circumference of' 300m radius circle. Where a is the horizontal correlation length, and N11 is the mean square fluctuation of refractive index.

ata set

R210603 I 40.50 I 0.0034

R220603 I 213.00 I 0.0063

R300603 254.75 0.0080

11. THEORETICAL PREDICTIONS In a medium with random inhomogeneities the index of refraction is a random function of the co-ordinates and time. The origin of the propagation variabilities are thought to be mainly thermal microstructures su- perimposed on temperature gradients.

An experimental study of temperature inhomo- geneities in the ocean is given in [2] using a fastact- ing thermometer. At depths of 30 to 60 m the mean temperature fluctuations amounted to 0.04 deg C, with a - 60 cm as the mean size of the inhomogeneities. This corresponds to small fractional changes in the in- dex of refraction; the mean square fluctuation of the acoustic index of refraction is equal to 2 = 5 x lo-'.

Experimental data was collected on Loch Goil in March 1993, and around 2,300 readings were taken around a circum of radius 300 m. Correlation lengths less than 1 m would not be seen. Ten consecutive sets of data collected are shown in table 1 , where the mean square fluctuation of the acoustic index of refraction is about 5 x and the mean size of the inhomo- geneities is between 40 and 290 m.

The estimates of the mean square fluctuation of re- fractive index given in table 1 is far greater than those

III- I 72 0-7803-2056-5 1994 IE€E

observed by [2] where they were making observations on a fine scale, and therefore we would expect the inho- mogeneities of size greater than 40 m to have a greater effect on propagation than inhomogeneities which are less than 1 m in size.

A . Fluctuations of the Transit T i m e

As conditions in the medium change, the time taken for a ray to go a given distance changes; moreover, the ray tube is deformed, which leads to intensity fluctu- ations. Let us consider the problem of transit time fluctuations and intensity fluctuations, assuming that the deviations from their initial direction (taken along the x axis) is small.

Takingt to be the time taken to go the distance L say, and t to be the mean transit time (time of travel in the homogeneous medium), then At = t - 1 and from [ 11 the mean square transit time fluctuation is

where CO is the mean velocity, that is the underlying velocity. The correlation coefficient N(x, y, z ) differs from zero in the region where x has values of order a . Since the inclination of the ray is small, we have y << a and z << a in the region where the values of the correlation coefficient are appreciable, therefore y and z have been set to zero.

We can immediately find the phase fluctuation @ = w A t :

00

5 = 2 7 k a L 4 N ( z , 0,O) dx (3)

where k is the wave number.

B. Fluctuations of the Intensi ty

In [l] an expression for the change of intensity de- termined by the change of cross section of the ray tube gives

log - = - 1 ( L - x)V2p dx.

Replacing the intensity ratio by the ratio of squared amplitudes, gives the the mean square amplitude fluc- tuations as:

L

(4) Z 10

. \ 2 . .m -- (log e-- = $ 2 L 3 J o [V2V2NIZ=, dx. ( 5 )

Setting N = exp (-(zz + z2)/a2], as the function rep- resenting the inhomogeneous medium, we obtain

(7)

111. AN ESTIMATE OF SOUND INTENSITY Following is the derivation of an expression for inten- sity. Many approximations are made in the derivation but is useful as a robust approximation to leading or- der. The difficulty of calculating intensity for the inho- mogeneous medium is that it is unstable, since if two rays separated by a small angle leaves the source then a t the range r where we want to measure the intensity the rays may not have remained close together, which is a usual requirement for the applicability of the ex- pressions of the formulae for calculating the intensity. Another short fall in the estimation of the intensity in the inhomogeneous medium is that two rays which set off from the source not necessarily close together may cross over a t the range r thus producing a point of high intensity.

A derivation of an expression for the focusing factor is given in [3] and is given as

where x1 is the angle the ray leaves the source, and x is the grazing angle at the receiver at range r . A corresponding expression which considers the variation in the ray tube in the z direction (depth) rather than the r direction (range) gives

(9)

If the angles x1 and x are assumed to be small, then cos , cos x N 1, so that the expression for the focusing factor can be further simplified to

where 6x1 is the small angle between the two rays which leave the source, and 62 is the vertical distance between the two rays a t the range r . Finally, can as- sume that r6x1 = 620, where 620 is the distance be- tween two rays a t the range r in the homogeneous medium when they set off from the source at an an- gle 6x1 apart.

Hence

gives an expression for the focusing factor as a ratio of the distance between two rays in the homogeneous and inhomogeneous medium.

To get a more useful expression, a t the range r can take a depth interval 6d (6d = 0.5m was used in Fig. la,b,c and 2a,b,c), and seek a relation between the fo- cusing factor in that depth interval at a specific depth, to the number of rays which fall in that depth interval.

Taking rays to leave the source at an angle 68 apart; No is the number of rays which falls in this depth in- terval in the homogeneous case, and Ni is the number of rays which falls in this depth interval in the inhomo- geneous medium represented by the i th example of the range dependent medium. We have generated data to

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represent the random medium, and due to the random nature of the data 100 examples are needed for results to become convergent.

Assuming the N, rays are uniformly distributed over the interval 6d, the spacing between the rays are 620

and 6z;, and are given by:

Hence from the expression for the focusing factor we get the expression

For the depth interval 6d, we have 100 estimates f, of the focusing factor f . By taking 100 different data sets to represent the inhomogeneous medium, the mean and standard deviation of the set f;(i = 1,100) is suf- ficiently stable.

We are interested in the mean 7, and the standard deviation f s D of the estimates fi(i = 1,100). If the estimates f; are then assumed to be Normally dis- tributed, then most of the estimates fi will fall in the interval (7 - fSD, 7 + f so ) .

A dB value for the measured intensities is 10 log I / l o , and the related expression in terms of the number of rays which falls into a depth interval is lOlogN/No, where 7 is the average over the 100 examples of the medium. In Fig. la,b,c and 2a,b,c plots are given of lolog?, lolog(f' + f s D ) and lolog(? - f s ~ ) with depth, which corresponds to the band of expected level of signal with depth. If the value of 7 - f s D is less than zero, then one cannot take the log of it, but this actually corresponds to a dB level of minus infinity, and these points are omitted from the graphs.

With a constant initial velocity profile and the stan- dard deviation of the data representing the medium being 0.1 m/s and at a range of 170m, Fig. l a shows that in the depth interval between 27.5 and 31.5 m the departure from spherical spreading is significant. Let us define spherical spreading to be such that the dB value of the standard deviation about the mean is less than 2. This means up to 4dB difference between 10 log(f + fSD) and 10 log(? - fSD).

When the standard deviation of the data represent- ing the medium is 0.25 m/s, departure from spherical spreading is predicted in the depth interval between 24.5 and 35 m , as shown in Fig. lb . For a standard deviation of 0.5 m/s for the medium, Fig. IC shows a significant departure from spherical spreading in the depth interval between 22 and 38.5 m. The tailing off at depths less than 18 m and greater than 40 m is due to the edge of the cone of rays taken. The cone taken in the examples is -4.49 degrees to 4 degrees.

With an initial velocity profile taken at Loch Goil on 3 February 1994 at a range of 170m, and the stan- dard deviation of the data representing the medium is 0.1 m/s, Fig. 2a shows a significant departure from spherical spreading in the depth interval between 28 and 31.5 m. One notices that there is about 1 dB dif- ference for predicted signal level above a depth of 30 in compared to the signal level below a depth of 30 m. When the standard deviation of the data representing

~

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the medium is 0.25 m/s, Fig. 2b shows departure from spherical spreading is significant in the depth interval between 25 and 34.5 m. Again a 1 dB difference in predicted signal level on either side of the 30 m mark. For a standard deviation of 0.5 m/s of the range depen- dent medium, Fig. 2c shows departure from spherical spreading is significant in the depth interval between 21 and 37m. This time the signal level on either side of the 30m mark varies by 1-2 dB.

At a trial in February in Loch Goil, dB values were calculated related to the peak voltage of the signal re- ceived from a transmitted pulse. The quantity which gives the dB value is 20 log Vd, where V is the voltage measured at the receiver and d the horizontal distance between the transmitter and receiver. The transmitter was at a depth of 30 m and one of the receivers was on axis at a depth of 30m, and a second receiver took readings at depths of 22,30,38 and 42 m consecutively. Hence we have a comparison of readings taken at the same time but at different depths.

A relation is required between the intensity (or the number of rays which fall in a depth interval) and the voltage measured at a receiver. The voltage V is pro- portional to the pressure p , so that V 0: p . Intensity is related to the pressure squared, so that

I oc v 2 .

If Vo is the voltage at the receiver in a homogeneous medium,

1 d2

v,2 oc Io 0: -,

and 20logVd differs from 2OlogV/Vo by a constant. Hence the two quantities 10 log V2d2 and 10 log N/No should only differ by a constant with depth.

So as to give a measure of the fluctuation at the re- ceiver at various depths, Fig. 3, 4 and 5 give the 1dB variation on the measured voltage squared. The val- ues displayed is the dB difference between 10 l o g ( v r + ( V 2 ) s , ) and l O l o g ( T - (V2)s,). Fig. 3 shows t,he variation on the voltage measured when the receivers were placed at depths of 30 m and 22 m. Fig. 4 shows the variation when the receivers were placed at depths of 30 m (on axis) and 38 m, and Fig. 5 shows the va.ri- ation when the receivers were placed at depths of 30 m and 42 m.

The variation off axis is less than 6 dB for all the sets of data collected, whereas the variation on axis (at 30 m) is greater than 8 dB on 5 data sets. Hence a more stable signal is expected off axis. The variation off axis is of the same order as that shown in Fig. 2c, which represents the intensity predicted at a range of 170 m by rays which leave the source at a depth of 30 m and travels through a medium with an underlying velocbty which w a s measured in February at Loch Goil and an inhomogeneous medium with a standard deviation of 0.5 m/s.

IV. CONCLUSION A model of the sea has been developed which repre- sents a real situation in Loch God. Since the model is generated by random numbers, we found that the data was convergent when we averaged over a hundred realisations.

Fig. la

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Fig. 1. Intensity fluctuation at a range of 170 m for a medium with constant initial velocity profile. The inhomogeneous medium has a standard deviation of a) 0.1 m/s, b) 0.25 m/s, C) 0.5 m/s. The reference intensity is that for sperical spreading.

111- 175

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Fig. 2. Intensity fluctuation at a range of 170 m for a medium with an initial velocity profile which was mea- sured a t Loch Goil in February. The inhomogeneous medium has a standard deviation of a) 0.1 m/s, b) 0.25 m/s, c) 0.5 m/s. The reference intensity is that for sperical spreading.

111- 176

A 14 12 10

B 8 .

A

--__ Receiver at 22m

A Receiver at 30m

Fig. 3. Voltage variation measured by receivers a t a range of 170 m from the source and at depths of 22 m and 30 m. Signals transmitted were of 3 frequencies: 3 kHz, 30 kHz and 70 kHz.

10

8

6

4

2

0

%

A

Receiver at 38m

A Receiver at 30m

' A A

A A m

A I A m = . -

0 2 4 6 8 10

Data set

Fig. 4. Voltage variation measured by receivers a t a range of 170 m from the source and a t depths of 30 m and 38 m. Signals transmitted were of 3 frequencies: 3 kHz, 30 kHz and 70 kHz.

A Remver at 30m

10

A A

Data set

'The method used for measuring the intensity is rea- sonably robust, but assumes that all rays in a depth interval are of the same phase. An expression for the focusing factor is currently being looked at to incorpo- rate the phase.

From the results obtained by the theoretical model, and the data collected at Loch Goil, we conclude that readings taken on axis varies significantly from spher- ical spreading. Depending on the size of the inho- mogeneities (standard deviation of the inhomogeneous medium data), spherical spreading is approached off axis. The larger the standard deviation of the inhomc- geneous medium data, then the further off axis readings are taken so that spherical spreading is approached.

This phenomena can be visualised as rays which leave the source at a small angle gets refracted within a patch which corresponds to correlation lengths in the r direction of 20 m and in the t direction of 2 m. Rays which leave the source at'larger angles do not get to- tally refracted within a patch of the inhomogeneous medium, and roughly follow the same path as if it were in a homogeneous medium.

ACKNOWLEDGEMENT This work has been carried out with the support of DRA(Port1and) under Grant No. 21 12/094/CSM.

REFERENCES [l] L. A. Chernov, W a v e Propagation i n a Random

Medium, translation, McGraw-Hill Book Com- pany, New York, 1960.

[2] L. J.Liebermann, "Effect of Temperature Inho- mogeneities in the Ocean on the Propagation of Sound," it J . Acoust. Soc. Am., vol. 23, pp.563- 570,1951.

[3] L. M. Brekhovskikh, and Y. P. Lysanov, Fun- damentals of Ocean Acoustics, Springer-Verlag, 1982, pp.38-42.

Fig. 5. Voltage variation measured by receivers a t a range of 170 m from the source and a t depths of 30 m and 42 m. Signals transrnitted were of 3 frequencies: 3 kHz, 30 kHz and 70 kIIz.

111- 177