^{1,a)}, David Hurther

^{2}and Benjamin D. Moate

^{3}

### Abstract

Although sound has been applied to the study of sediment transport processes for a number of years, it is acknowledged that there are still problems in using the backscattered signal to measure suspended sediment parameters. In particular, when the attenuation due to the suspension becomes significant, the uncertainty associated with the variability in the scatteringcharacteristics of the sediments in suspension can lead to inversion errors which accumulate as the sound propagates through the suspension. To study this attenuation propagation problem, numerical simulations and laboratory experiments have been used to assess the impact unpredictability in the scatteringproperties of the suspension has on the acoustically derived suspended sediments parameters. The results clearly show the commonly applied iterative implicit inversion can lead to calculated sediment parameters, which become increasingly erroneous with range, as the sound propagates through the suspension. To address this problem an alternative approach to the iterative implicit formulation is investigated using a recently described dual frequency inversion. This approach is not subject to the accumulation of errors and has an explicit solution. Here the dual frequency inversion is assessed and calculated suspended sediment parameters are compared with those obtained from the iterative implicit inversion.

This work was supported by NERC UK contracts FLOCSAM and FORMOST, LEGI-OSUG France, the SANDS component of the Integrating Activity HYDRALAB III, European Commission Contract No. 022441 (RII3) and the WISE component of the Integrating Activity HYDRALAB IV, European Commission Contract No. 261520.

I. INTRODUCTION

II. BACKGROUND SCATTERING THEORY

III. PARTICLE SIZE AND CONCENTRATION PROFILES

IV. INVERSION METHODOLOGIES AND RESULTS

A. Iterative implicit inversion

B. Dual frequency inversion

C. Introduction of a size profile

D. Comparison on the inversions applied to laboratory data

E. Frequency selection and the impact of noise

V. DISCUSSION AND CONCLUSION

### Key Topics

- Suspensions
- 33.0
- Backscattering
- 29.0
- Inverse scattering
- 18.0
- Acoustical measurements
- 15.0
- Particle scattering
- 13.0

## Figures

(a) Lognormal mass distribution, *m*(*a*) (···), and particle size distribution, *n*(*a*) (–), vs particle radius *a*. *a _{m} *,

*a*

_{50}, and

*a*are the radius respectively for the mean mass, median mass and mean particle size. (b) Form functions for the intrinsic scattering characteristics of the sediments

_{c}*f*(‐‐) and the ensemble

_{io}*f*(–) with

*x*. (c) Normalized total scattering cross sections for the intrinsic scattering characteristics χ

*(‐‐) and the ensemble χ (–) with*

_{io}*x*.

*x*=

*ka*for the intrinsic scattering and

*x*=

*ka*for the ensemble scattering.

_{c}*k*is the wavenumber.

(a) Lognormal mass distribution, *m*(*a*) (···), and particle size distribution, *n*(*a*) (–), vs particle radius *a*. *a _{m} *,

*a*

_{50}, and

*a*are the radius respectively for the mean mass, median mass and mean particle size. (b) Form functions for the intrinsic scattering characteristics of the sediments

_{c}*f*(‐‐) and the ensemble

_{io}*f*(–) with

*x*. (c) Normalized total scattering cross sections for the intrinsic scattering characteristics χ

*(‐‐) and the ensemble χ (–) with*

_{io}*x*.

*x*=

*ka*for the intrinsic scattering and

*x*=

*ka*for the ensemble scattering.

_{c}*k*is the wavenumber.

Variation with *x* of (a) the ensemble form function *f* and (b) the ensemble normalized total scattering cross section χ, for β = 0.8 (‐‐), 1.0 (–), and 1.2 (···). The two vertical lines indicate the values of *x* = *ka _{c} * used in the simulated suspension inversions.

Variation with *x* of (a) the ensemble form function *f* and (b) the ensemble normalized total scattering cross section χ, for β = 0.8 (‐‐), 1.0 (–), and 1.2 (···). The two vertical lines indicate the values of *x* = *ka _{c} * used in the simulated suspension inversions.

Profiles of the variation with height above the bed, *z*, of (a) the normalized mean particle radius, *a _{c}/a_{r} *, for

*l*= 0 (–) and

*l*= 0.1 m (‐‐) and (b) the normalized mass concentration,

*C/C*.

_{r}Profiles of the variation with height above the bed, *z*, of (a) the normalized mean particle radius, *a _{c}/a_{r} *, for

*l*= 0 (–) and

*l*= 0.1 m (‐‐) and (b) the normalized mass concentration,

*C/C*.

_{r}Profiles from the iterative implicit inversion of the variation with range, *r*, of (a) normalized concentrations, *M _{s}/C_{r} *

_{,}when the particle size was range independent and known, (b) mean normalized particle size,

*a*, and (c) normalized concentration,

_{s}/a_{r}*M*, when both mean particle size and concentration were unknown. The three lines are for β = 0.8 (‐‐), 1.0 (–), and 1.2 (···), the symbols represent

_{s}/C_{r}*C*= 1.0 (○), 5.0 (□), and 20 (▵) kgm

_{r}^{−3}.

Profiles from the iterative implicit inversion of the variation with range, *r*, of (a) normalized concentrations, *M _{s}/C_{r} *

_{,}when the particle size was range independent and known, (b) mean normalized particle size,

*a*, and (c) normalized concentration,

_{s}/a_{r}*M*, when both mean particle size and concentration were unknown. The three lines are for β = 0.8 (‐‐), 1.0 (–), and 1.2 (···), the symbols represent

_{s}/C_{r}*C*= 1.0 (○), 5.0 (□), and 20 (▵) kgm

_{r}^{−3}.

(a) Profiles from the dual frequency inversion of the variation with range, *r*, of the normalized concentrations when the particle size was range independent and known. Profiles with range, *r*, from the hybrid constrained iterative implicit and dual frequency inversion of (b) the mean normalized particle size, *a _{s}/a_{r} * and (c) the normalized concentration,

*M*, when both mean particle size and concentration were unknown. The three lines are for β = 0.8 (‐‐), 1.0 (–), and 1.2 (···), the symbols represent

_{s}/C_{r}*C*= 1.0 (○), 5.0 (□), and 20 (▵) kgm

_{r}^{−3}.

(a) Profiles from the dual frequency inversion of the variation with range, *r*, of the normalized concentrations when the particle size was range independent and known. Profiles with range, *r*, from the hybrid constrained iterative implicit and dual frequency inversion of (b) the mean normalized particle size, *a _{s}/a_{r} * and (c) the normalized concentration,

*M*, when both mean particle size and concentration were unknown. The three lines are for β = 0.8 (‐‐), 1.0 (–), and 1.2 (···), the symbols represent

_{s}/C_{r}*C*= 1.0 (○), 5.0 (□), and 20 (▵) kgm

_{r}^{−3}.

Profiles of the variation with range, *r*, of the normalized concentration, *M _{s}/C_{r} *, from the constrained iterative implicit inversion when both the mean particle size and concentration were unknown. The three lines are for β = 0.8 (‐‐), 1.0 (–), and 1.2 (···), the symbols represent

*C*= 1.0 (○), 5.0 (□), and 20 (▵) kgm

_{r}^{−3}.

Profiles of the variation with range, *r*, of the normalized concentration, *M _{s}/C_{r} *, from the constrained iterative implicit inversion when both the mean particle size and concentration were unknown. The three lines are for β = 0.8 (‐‐), 1.0 (–), and 1.2 (···), the symbols represent

*C*= 1.0 (○), 5.0 (□), and 20 (▵) kgm

_{r}^{−3}.

(a) Profiles of the variation with range, *r*, of the normalized concentration, *M _{s}/C_{r} *, from the dual frequency inversion when the particle size was range dependent and known. Profiles of the variation with range, r, from the hybrid constrained iterative implicit and dual frequency inversion of; (b) the mean normalized particle size,

*a*and (c) the normalized concentration,

_{s}/a_{r}*M*, when both the mean particle size and concentration were unknown. The three lines are for β = 0.8 (‐‐), 1.0 (–), 1.2 (···) and

_{s}/C_{r}*C*= 20 kgm

_{r}^{−3}. The input profile for

*a*and

_{c}*C*are given by the symbol (×).

(a) Profiles of the variation with range, *r*, of the normalized concentration, *M _{s}/C_{r} *, from the dual frequency inversion when the particle size was range dependent and known. Profiles of the variation with range, r, from the hybrid constrained iterative implicit and dual frequency inversion of; (b) the mean normalized particle size,

*a*and (c) the normalized concentration,

_{s}/a_{r}*M*, when both the mean particle size and concentration were unknown. The three lines are for β = 0.8 (‐‐), 1.0 (–), 1.2 (···) and

_{s}/C_{r}*C*= 20 kgm

_{r}^{−3}. The input profile for

*a*and

_{c}*C*are given by the symbol (×).

Profiles of the variation with range, *r*, of the normalized concentration profiles, *M _{s}/C_{r} *, when the particle size was range independent and known. (a) Iterative implicit inversion and (b) dual frequency inversion. The lines are from simulated inversions and the open circles are from inversions using the measured backscattered signal. The symbol represent the different values for β; 0.9 (*), 0.95 (▵), 1.0 (×), 1.05 (□), 1.1 (+).

Profiles of the variation with range, *r*, of the normalized concentration profiles, *M _{s}/C_{r} *, when the particle size was range independent and known. (a) Iterative implicit inversion and (b) dual frequency inversion. The lines are from simulated inversions and the open circles are from inversions using the measured backscattered signal. The symbol represent the different values for β; 0.9 (*), 0.95 (▵), 1.0 (×), 1.05 (□), 1.1 (+).

Profiles of the variation with range, *r*, of the normalized mean particle size, *a _{s}/a_{r} * and the normalized concentration,

*M*. (a) and (b) were obtained using the iterative implicit inversion and (c) and (d) were obtained using the hybrid constrained iterative implicit dual frequency inversion. The lines are from simulated inversions and the open circles from inversions using the measured backscattered signal. The lines are for β = 0.95 (‐‐), 1.0 (–), 1.05 (···).

_{s}/C_{r}Profiles of the variation with range, *r*, of the normalized mean particle size, *a _{s}/a_{r} * and the normalized concentration,

*M*. (a) and (b) were obtained using the iterative implicit inversion and (c) and (d) were obtained using the hybrid constrained iterative implicit dual frequency inversion. The lines are from simulated inversions and the open circles from inversions using the measured backscattered signal. The lines are for β = 0.95 (‐‐), 1.0 (–), 1.05 (···).

_{s}/C_{r}Plot of the ratio of the concentration noise relative to the backscatter signal noise, σ(*m _{e} *)/σ(

*v*), with the ratio of the normalized total scattering cross sections, χ

_{e}_{1}/χ

_{2}. The solid circles were obtained using Eq. (15) and the solid line is given by (1− χ

_{1}/χ

_{2})

^{−1}.

Plot of the ratio of the concentration noise relative to the backscatter signal noise, σ(*m _{e} *)/σ(

*v*), with the ratio of the normalized total scattering cross sections, χ

_{e}_{1}/χ

_{2}. The solid circles were obtained using Eq. (15) and the solid line is given by (1− χ

_{1}/χ

_{2})

^{−1}.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content