-P-r 2 rle- )Sin dtd 1 de

(3)

r is the pulse duration, c is the sound speed in water, and a, is the attenuation due to the

suspension. Only first order multiple scattering has been included, there is therefore an

expected upper limit to the applicability of the expression of the order of a few tens of kgm3.

Assuming rc<

-K, K, -M2((r)
r
(4)




K,(r)-
p ,



K-Pr c16 r,2J,(kaSinO) 4 dO
K- P Jr [ Sin 6 dS
( 16 Jo katSinO I









K, contains the sediment information, and K, represents the system parameters. J, is the first

order Bessel function and describes the directivity for the usual disc transceiver employed,

a, is the transceiver radius and 0 is the angle subtended to the acoustic axis.



The parameter required is the suspended load, and this is obtained from the backscattered

pressure, therefore rearranging, the concentration is given by


M- (+)
K,(r)K,
(5)
An interesting problem with equation (5) is that to evaluate it, a, needs to be known,

however this is a function of M(r) the unknown. This requires a procedure where

sequentially M(r) is evaluated and a numerical integration conducted to evaluate a, and this

calculation repeated step-wise down through the water column. The process has the problem

of feeding back inaccuracies into the calculation which can accumulate, and is analogous to

positive feedback. This is particularly acute at high concentrations and the further ranges

when substantial deviations from the actual profile may be computed from the acoustic data.

Also a degree of imprecision in a, will not result in a similar error in M(r), but may be

substantially larger depending on the concentration and range. However, the corollary to this

is that for low values of a, the problem is well defined.



2.1 Form Function

The backscattered form function needs to be known if equation (4) and (5) are to be

evaluated. Most recent data',3 show that a non-dimensional universal expression of the form

close to,













Y-O1+K x2
'1


Where



CO-(1-v e (x-x)/t'"2)(1 v 2e -[(x-x)/ri21



2
Kf,-y P



YK-=(K-K/o yp-3(p s-p )/(2p,+p)


can be used to describe the backscattering by sedimentary particles.


. ,



-r
0I





In






CM

-V


I I I 1 I i i ,


. V VI I I I


2 3 5 6 789
10


2 3 '1 5 6 789
101


ka

Fig 1. Form function


10-1


2 3 1 5


. .









x=ka,, K and p are the compressibility and density respectively and subscripts s and o refer

to the suspension and water. vP=0.37, 7,=0.5, x1=1.4, and v,=0.28, r,_=2.2, x,=2.8. For

non-cohesive sedimentary material a reasonable value for Kf=1.14. The curve for the

backscattered form function, obtained using equation (6) is shown in Fig 1, where it can be

seen to compare well with recent observations.



2.2 Attenuation

The attenuation due to absorption by the water, and scattering and absorption by the

sediment, is introduced separately. The water absorption can be described by,


ac-(55.9-2.37.T + 0.0477.T2 -0.000384.T3).10-3 j (7)

f is the frequency in Mega-Hertz, T is the temperature in degrees Celsius, and a, is the

attenuation in Nepers m1. This is the absorption for fresh water, however, for frequencies

above 1MHz it is almost identical to sea water. The attenuation due to sediment scattering

a, is given by the two components, that owing to scattering, a,, and absorption, aO2.

as-C1+Ca2 (8)

The component due to scattering can be expressed as


al-(N/2)a, (9)

N is the number of particles per unit volume and o, is the total scattering cross-section. al

can to be written as,


ai <0 > (10)
8npj










M has replaced N.


oC





CO
CD



O
I,


(o


O


56789 2 3 4 56789 2 3 4 5
100 101

kas





Fig 2 Normalised total scattering cross section


It is convenient to normalise at by twice the particle's geometric cross-section, X=t/2ra,2,

this is the normalised total scattering cross section, and re-expressing gives


3M
a1 =
4p ,


Where


f'ap(as)daf 'aXp(as) da,
-
f"a'p(a)da
0 S


(11)


CD 00










(4/3)K4 (12)
(12)
[1+x2+(4/3)Kax4]



K,-(y +y,/3)/6

Using usual values for compressibility and density gives K,=0.18. The form of the

expression for X is compared with recently obtained data' in Fig 2.





To account for the absorption due to the suspended load the following expression was

employed4


Mk(o-1)2 s (13)
a2 2P, s 2+(G+8)2


where


a-p/po



6- 11+---
2 (2 )


9 1 1]
s--- 1+--
4 4p P

3=(kc/2v)"2, and v is the cinematic viscosity. This expression is valid for ka, < <1, which

is the region where attenuation due to scattering becomes negligible and viscous absorption

begins to dominate. It is useful to write the attenuation in terms of a sediment attenuation

constant which has unit of Nepers m' (kgm"3)'. The attenuation constant is then just

multiplied by the range and concentration to obtain the attenuation coefficient.
















or








01
-t;



o



i,
oi
x /
01,
CO I






o















CO. 80. 120. 160.
0










as




















These are give by r'=c,/M and can be can be written as
S

































k(o-1)2 S (14b)
as









The values for and 2 are shown in Fig 3. This shows the dominance in sediment

scattering for the larger particles, and the increasing importance of particle attenuation for

the finer material. In general the attenuation is a function of range due to the varying

suspended load with height above the bed. The attenuation coefficient due to the suspension

evaluated up to a particular range is therefore given by


ai r ( M(r) dr (15)
r 0

i=1,2.



2.3 Amplitude statistics.

The backscattered signal profile from a homogeneous suspension of constant concentration

is not uniform but variable due to the random phase of the returns from the scatterers. The

outcome of the supposition of random phase statistics uniformly distributed over 2r leads to

a Rayleigh probability distribution for the amplitude distribution which is given by,


p(A) 2A e -( 2^l (16)






A is the amplitude of the backscattered pressure and is the mean square amplitude.

Fig 4 shows an example of the statistics of the backscattered echo.For a Rayleigh distribution

the standard deviation of the signal is approximately half the root-mean-square amplitude,

therefore to obtain 5% accuracy in we need to obtain 100 independent estimates,

this will equate to a 10% precision in M. The importance of the statistical nature of the

backscattered signal is that it may compromise the spacial and temporal resolution available.










0


4.. 4











1.0o 20.0 40.0 60.0 80.0 100.0
P (Pa)



Fig 4 Comparison of a Rayleigh distribution with the measured data.



A main asset of the acoustic backscatter approach is its potential to provide high temporal

and spatial resolution measurements of suspended sediments. However, incorrect inferences

regarding suspended load will result if inaccurate account of the signal fluctuation due to the

random phasing of the acoustic returns is not properly considered. Therefore there is the

need to be aware of the degree of accuracy required if authentic variations in the suspended

load are to be ascertained.



2.4 Simplified expressions

The above gives a description of the interaction of sound with marine sediments. In a number

of circumstances simplified results can be obtained and these are considered now.



a. Rayleigh scattering: Long wavelength limit ka, < <0.5

In this case it can be seen from equations (6) and (12) that f = Kfx2 and x = (4/3)Kx4. The









concentration is therefore given by


-M(r)- I r2e4r(.1 (17)



Similarly the sediment attenuation constant can be written as


-1-- k43 (18)
P,

Both of which are simple expressions to evaluate. It can be seen that both M(r) and are

strongly dependent on the particle size, therefore any error in this parameter will play an

important role in the accuracy of the sediment concentration.



b. Optical scattering: Short wavelength limit ka,> >5

In this case f 1 and X = 1 which leads to


M(r)- -S2 r 9
SKt J

and

3
Cl-1 (20)
4 p,

Again these are readily evaluated functions. In this case the dependence on a, is linear and

therefore the estimate of concentration is not as sensitive to errors in the particle size as in

the Rayleigh case.



However, there is still the problem that to evaluate M(r) we need to know a, which is

dependent on the unknown M(r). Certainly if the assumption can be made that a,= 0.0 the

problem is considerably simpler and the result are more reliable.

14













3.OBSERVATIONS AND THEORY


To examine the correctness of the theoretical description comparisons are made with data

collected both from laboratory experiments and observations taken in an in estuarine

environment. These measurements were taken using a 3MHz system designed to examine

non-cohesive sediment processes in the bottom lm of the benthic boundary layer. The

diameter of the material was approximately 180tim. The 3MHz data set provides observations

which can be rigorously compared with the theoretical predictions.



3.1 Laboratory data

Figures 5a-c show laboratory results taken in a sediment tower using sediments recovered

from the estuary. The tower generated a uniform homogeneous suspension over

approximately 1.5m. The concentration in the laboratory was obtained using pumped

sampling. The variation in the range dependence of the backscattered root-mean-square

pressure, < P,, >, for a number concentration levels is shown in figure 5a. In all cases there

is seen to be a steady fall off in pressure with range arising from spherical spreading and

attenuation due to absorption by the water and the sediment scattering. It can be seen that as

the suspended concentration level rises and the range from the transceiver increases the effect

of the suspension attenuation becomes particularly significant, with the highest concentration

giving the lowest signal levels beyond about a metre. Two theoretical curves, calculated

using equation (4), are compared with the data; the dashed lines were determined with the

sediment attenuation neglected, and the solid lines with account taken of the sediment













attenuation.

10 a




C 102




V 101 L
a-3
S'0..28 kgm-3
0.064

S222
1 0-
0.0 30.0 60.0 90.0 120.0 150.0
r cm


103 b
-10 7.5 cm



10

CL
A\ 525





92.5

10 -
10-2 101 100 101
M-kgm3



Cb










'o ____ i_____ _____ i___ l--------_------------------- ---



0. 10. 20. 30. L0. 50. 60. 70. 80. 90. 100.
r (cm)

Fig 5 a) Range dependence of the backscattered pressure for different suspended load. b)


Variation of backscattered pressure with mass concentration at selected ranges. c) Ratio of


acoustic concentration to pumped sample value.









It is readily seen that the attenuation due to the presence of the suspension needs to be

accounted for at the higher concentrations and further ranges. When account is taken of a,

the agreement between theory and observations is seen to be good over the complete distance

covered and for concentration levels encompassing close to two orders of magnitude. Figure

5b shows the variation of backscattered signal with concentration at ranges, 7.5, 22.5, 52.5

and 92.5cm from the transceiver. The curves were obtained by averaging the rms pressure

amplitude over +5cm at each of the ranges. Two curves are shown again and employ the

same nomenclature as figure 5a. For low concentrations and close ranges there is seen to be

a linear dependence of on VM, however, as the integrated effect of the attenuation

accumulates there is a departure from the linear curve. It can be seen that for the further

ranges from the transceiver for increasing concentrations, the backscattered pressure actually

begins to decrease. The effect of the attenuation more than offsets the signal increase due

to the linear dependence on VM and begins to dominate the backscatter response. Good

agreement is again seen between observations and predictions. This non-linear response of

upon VM with range and concentration demonstrates the difficulty in trying to

obtain an empirical calibration for an acoustic backscatter device. Comparison of the

variation with range of the ratio of the acoustic estimate of the concentration, MA, calculated

using equation (5), to the measured pumped sample value, Mp, are shown in Fig 5c. The

results were taken using a concentration of 2Kgm". Ideally the value of this ratio should be

unity and constant with range. Three scenarios are presented (ignore the dashed lines). The

centre result was computed by evaluating a, using the measured concentration, while the

upper and lower curves were obtained with no input from the measured concentration. In the

marine environment a, cannot be input to the calculation since the concentration is the value

unknown. Therefore the step-wise procedure discussed earlier was employed to calculate









M(r) and a, sequentially, and feed back these values to progressively compute the suspended

concentration profile. It can be seen that a small variation of K, or Kt, in this case an 8%,

changes the concentration from the upper line to the lower line. This results in a change of

concentration, at a range of close to Im, of over two orders of magnitude. This is a

relatively extreme result, however, it does demonstrate the possible difficulties that can be

encountered when suspended loads are high, and attenuation by the sediments dominates the

range dependence of the backscattered signal.



3.2 Marine data

The estuarine measurements were carried out during a Spring tidal period over three

consecutive floods. The bed consisted of fine sand and the area was dominated by sandwaves

which typically have a wavelength of 15-20m with a trough-to-crest height of the order of

0.8m. Measurements of the suspended load were taken above the bed at 0.1, 0.2, 0.4 and

0.8m using pumped sampling. The acoustic concentrations values were computed using

equation (5). The acoustic estimates of concentration were obtained by averaging the acoustic

data over the same period as the pumped sample data, this was 60s. The results of these

acoustic estimates of the suspended sediment concentration over three consecutive floods are

compared with the pumped sample data in Fig 6. The comparison of the concentration

estimates obtained using acoustic backscattering with the estuarine pump sample data

collected over the three consecutive floods show good agreement. The gradient of the

experimental data is close to unity and positioned about the line MA=Mp.






























Ex 0

0 A X
A (o




;+



10- 10- 10-1 100 ltO
Mp (Kgmn3)






6. Acoustic concentration versus pump sampled data.




The similarity of the pumped sampled data and the acoustic estimates of concentration

provide confidence in the accuracy of the acoustic technique. Th6e acoustic data can now be

employed to analyse the details of sediment transport. These results are shown below in Fig

7. Fig 7a shows the full concentration field, and Fig 7b shows values with the mean

concentration subtracted and only values above the mean plotted. The latter show the high









concentration events. These data cover a period of 80s, and illustrate the variability of the

suspended load on a time scale of the order of seconds. It is expected that by identifying the

relationship between these sediment structures, and the turbulent tidal flow, that a detailed

understanding of sediment transport process can be developed.


Fig 7. a) Measured suspended concentration over 80s. b) Fluctations from the mean over the

same period as a).









4.ABS SYSTEM's


4.1 Scattering regime of the ABS

For the 1-5 MHz ABS systems currently being employed, it is useful to ascertain the

scattering regime into which the data usually fall. Most studies have been concerned with

non-cohesive sediments with particle sizes ranging from a,= 50-150/m, this covers the

ka,= 0.2-3. This regime is primarily in an intermediate region above Rayleigh scattering and

below optical scattering. Therefore in only a limited number of circumstances will the

approximations in equations (17)-(20) be valid, and typically equations (6) and (12) will need

to be employed.



For the attenuation three contributions are present; the absorption by the water, the scattering

from to the sediments and the absorption due to the sediments. Values at 3MHz are chosen

to establish the relative importance. Using equation (7), and assuming a temperature of

14 o C, the attenuation is 0.28 Nepers m-' which for a range of operation of Im gives a

signal level reduction of 5 dB or close to 40%. To estimate the attenuation due to the

suspended sediment a particle size of a,=100gm is used. From data comparable to that

shown in Fig 3 it can be shown that if the sediment size is in the non-cohesive regime

scattering dominates. For a uniform concentration of 0.5 Kgm3-, aci=0.28 Nepers and

a2=0.0025 Nepers m-1. The viscous absorption due to the particles is therefore negligible,

and the sediment concentration is comparable to the water absorption. Therefore typically for

scattering by non-cohesives at Mega-Hertz frequencies, attenuation due to scattering plays

an important role in interpreting the data.









4.2 Calibration of an ABS

Ideally to evaluate equation (5) all the parameter on the RHS the equation should be known.

In general it would be useful to evaluate IK, however, this is often not practicable, since it

requires measurements of the transducer beam pattern and absolute pressure measurements.

Therefore it has to be accepted that in general K, will be treated as a scaling constant in the

system. This presents no problems as long as the system parameters remain constant. To

obtain K,, requires sediment samples comparable to the suspended sediments at the marine

site to be employed in a laboratory calibration. This will then effectively provide, IK*K, and

the value for, ,s. If it can be assumed that the particle size remains constant through the

water column in the marine environment then these parameters are constant. If a, is varying

with height above the bed the interpretation of the acoustic data becomes more difficult, and

it is then necessary to know the temporal and spatial variation of the particle size to interpret

the backscatter data. This would be difficult to obtain, however, a useful approximation

would be to measure the mean particle size profile with height above the bed and employ this

to interpret the backscatter ABS data. Therefore a basic calibration would be obtained using

in-situ samples at different heights above the bed over the measurement period at a particular

site. Half the data can then used to calibrate the acoustic data and the rest is used to test the

calibration. This calibration can then be applied to the whole acoustic data set and known

accuracies can be compute from the calibration test. Given sediment attenuations are not to

great, M(r) <0.1Kgm-3, and the variance in particle size at a given height above the bed is

reasonably low, a(a,)<20%, between in-situ samples, then the acoustic inversion should

valid. This is underpinned by the acknowledgement that has been obtained from

about 100 backscattered profiles. This number is required to negate the variability in

backscattered signal amplitude due to the statistical distribution of the amplitude, owing to









the phasing of the echo from each of the individual particles insonified.









CONCLUSIONS


An examination of the application of acoustic scattering for measuring suspended sediment

concentration has been presented. A theoretical description of the interaction of sound with

suspensions has been tendered, and this compares favourably with laboratory observations.

In the marine environment time averaged acoustic concentration predictions were seen to be

in good agreement with pumped sample data taken in an estuary.



To obtain accurate results requires, significant data averaging of the amplitude of the

backscattered signal to overcome the statistical distribution of the backscattered echo,

estimates of the particle size distribution with height above the bed and careful evaluation of

Kt*K, and a,. There is also the inversion problem which is typically manifest at high

concentrations and the further ranges from the transceiver. However, the gains in the

description of the suspension, which can be attained using the acoustic approach, does

provide the impetus for resolving some of the acoustic problems, and obtaining a detailed

picture of suspension processes.



Finally the use of acoustics for measuring suspended sediments is an expanding and ongoing

process. The theory is moderately well developed, although more so for non-cohesive than

cohesive sediments. Further studies are required in the latter. Once confidence has developed

in the acoustic approach, it should provide a significant advance in a our ability to monitor

and predict sediment movement.











References

1 A.E. Hay. Sound scattering from a particle-laden, turbulent jet. J. Acoust. Soc. Am.

90(4) 2055-2074 1991.

2 P.D. Thorne and S.C. Campbell. Backscattering by a suspension of spheres. J.

Acoust. Soc Am. 92(2) Ptl 978-985. 1992.

3 P. D. Thorne, K. Waters, and T. J. Brudner. Scattering of sound by irregularly

shaped particles. ARL report Uni Texas at Austin. ARL-TR-92-23

3 R. J. Urick. The absorption of sound in suspensions of irregular particles. J. Acoust.

Soc Am. 20(3) 283-289. 1948.