REMOTE CHARACTERIZATION OF RANDOM SCATTERER DISTRIBUTIONS IN STRATIFIED MARINE ENVIRONMENT
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REMOTE CHARACTERIZATION OF RANDOM SCATTERER
DISTRIBUTIONS IN STRATIFIED MARINE ENVIRONMENT
Anatoliy N. Ivakin
Applied Physics Laboratory, University of Washington
1013 NE 40th Street, Seattle WA 98105, USA
Fax: 206-543-6785, email: aniv@uw.edu
Abstract: Discrete scatterers, such as particles and objects of various kinds, are common
in marine environment. Natural examples are particles in suspended sediment, shells,
rocks, oil droplets, bubbles (either in the sediment or in water column). Effects of these
objects, their size, shape, material properties, spatial variability, on underwater acoustic
propagation and scattering can be significant. Quantification of these effects requires a
relevant parameterization of scattering objects to provide necessary inputs to acoustic
models. Such models then can be used for development of various algorithms and
techniques for remote characterization of marine environment. In this paper, we describe
a simple, physics-based model, which provides a relationship between the scattering
intensity and statistical characteristics of randomly distributed, either in water column or
in the seabed, arbitrary sized and shaped discrete objects. This model is rather general
and able to predict scattering in environment having arbitrary stratification. Its first
version has a primary application to analysis of bottom scattering, and is named
GAMBID, Geo-Acoustic Model of Bottom Interaction and Discrete scattering. This model
can be considered as a supplement to GABIM, a recently published model, which treats
only continuous heterogeneity of the sediment. The GAMBID-model is applicable to both
types of scatterers, continuous and discrete, arbitrarily distributed in any part of stratified
marine environment, seabed or water column. The scattering kernel is given by the local
volume scattering coefficient, which is defined in two different ways. For continuous
heterogeneity, it is defined by a spectral function of heterogeneity, and, for discrete
scatterers, by their size/shape distributions. The model is applied to analysis of scattering
from inclusions in stratified sand/mud sediments with shell inclusions, and model/data
comparisons are presented.
Keywords: Inclusions, particle size and shape distributions, acoustic scattering 6/7/2011
8:25 PM .1. INTRODUCTION
Discrete scatterers, such as particles and objects of various kinds, are common in
marine environment. Examples are suspended sediment particles, shells, rocks, hydrate
inclusions in the seabed, oil droplets, fish, shelled animals, gas voids and large bubbles
(either in the sediment or in water column), and many others. Effects of these objects, their
size, shape, internal structure, material properties, spatial (and in many cases temporal)
variability on acoustic propagation and scattering can be significant. Quantification of
these effects requires a relevant parameterization of these objects to provide necessary
inputs to acoustic models. Such models then can be used for development of various
algorithms and techniques for remote characterization of marine environment.
In this paper, we describe a simple, physics-based model, which provides a relationship
between the scattering intensity and statistical characteristics of randomly distributed,
either in water column or in the seabed, arbitrary sized and shaped discrete objects. This
model is rather general and able to predict scattering in environment having arbitrary
stratification. Its first application has been for analysis of the SAX04 geoacoustic data set
[1], and for this reason it was (preliminary) named GAMBID, Geo-Acoustic Model of
Bottom Interaction and Discrete scattering. First results of its utilizing for the SAX04 data
analysis were presented in [2]. This model can be considered as a supplement to GABIM
[3], which, as is now, treats only the case of continuous heterogeneity of the sediment.
The general model is comprised of two parts. First part is its propagation kernel, which
describes the two-way acoustic propagation (from source to scattering point and then to
receiver) in an arbitrarily stratified environment. It requires environmental inputs in terms
of the depth-profiles of acoustic parameters of the medium (density, sound speed and
attenuation). The second part is the scattering kernel, given by the effective volume
scattering coefficient, defined locally, in any part of the environment, either in the seabed
or in water column. It requires inputs in terms of the scattering cross-sections for
individual objects defined as functions of their size and shape, and their statistical
size/shape distributions.
In Section 2 of this paper, the general approach is outlined and the expression for the
scattered intensity is obtained. The approach is applicable to both types of heterogeneity,
continuous and discrete. In Section 3, expressions for the individual scattering functions
and the volume scattering coefficient are obtained. In Section 4, the model is applied to
analysis of acoustic backscatter from inclusions in the SAX04 sediment using the
GAMBID, the model is discussed, and model/data comparisons are presented.
2. APPROACH
The marine environment is spatially heterogeneous, so that essential for acoustics
parameters are randomly fluctuating around some background. The background however,
in most cases, is also spatially dependent (or at least stratified, i.e. depth-dependent).
Stratification is known to significantly complicate the problem of sound propagation
underwater. Also, and even in greater extent, it makes more difficult the problem of
scattering occurring due to random fluctuations (heterogeneity) if this environment. Here,
we consider a possibility to significantly simplify accounting for the effect of
stratification, using an approach similar to described in [4] for scattering from
inhomogeneities in stratified sediments. This is generalized, so that it can be used in anypart of marine environment, seabed or water column, with continuous or discrete
heterogeneity. It describes the inhomegeneous medium in terms of its spatially fluctuating
compressibility and density, ~ and ~ , and can be outlined as follows.
The field in such medium is convenient to describe not in terms of acoustic pressure,
p , but using a wave function ~ ~
~ p / ~ , where a fixed density parameter is
introduced for convenience (to keep same dimensions). The wave function obeys a more
~
simple (than e.g. in [5]) equation of a standard Helmholtz form, ( Q)~ 0 , with
~ ~
Q Q(r ) defined, at given frequency and position vector r , as follows
~
Q 2~~ ~1 / 2 ~ 1 / 2 (1)
Consider also a reference medium (or an “unperturbed” background) with compressibility
and density, and , with the corresponding function Q(r ) . Perturbations of the
~
medium are given by the heterogeneity function Q Q . If perturbations are relatively
small (which, however is not a necessary requirement here), then the heterogeneity
function is linear with respect to fluctuations of the medium parameters.
In the first order, the scattered intensity in such medium is given by the expression
2
s (r1 ) M V (r ) (r )G(r , r1 ) d 3 r
2
(2)
with M V (r ) being the volume scattering coefficient, or the scattering cross section per unit
volume, defined as follows
M V (r ) ( / 2) (q, r ) (3)
where is the local 3D (energy) spectrum of heterogeneity, and q ( 1 ) is the
local scattering vector, defined through the phases of the wave function, exp(i ) ,
and corresponding Green function, G G exp(i1 ) , both for the unperturbed medium.
The simplicity of Eq.(2) results from assumption that the spectrum of heterogeneity is a
smooth enough function, allowing neglecting bistatic scattering effects by ignoring
difference in wave vectors appearing in the stratified environment with both down-going
and up-going waves [2].
Transition back to the pressure function in (2) is simple, being equivalent to
replacement of and G by corresponding pressure and Green function (which both are
continuous) concurrently with replacement of M V by its “effective” value given by the
expression M Veff ( / ) 2 M V . This transition can be useful e.g. in the presence of
interfaces, where the reference medium parameters have discontinuities.
Eq. (2) can be applied to any part of marine environment, e.g. heterogeneous sea-water
column or near-sea-surface layer, where compressibility may fluctuate significantly due to
spatial variations of bubble concentration. Also, it can be used for description of volume
scattering due to randomly distributed discrete fluctuations of various kind. In this case,
the heterogeneity function becomes a sum of separate contributions, describing individual
discrete scatterers. If they are sparse enough (inclusion-type), their positions can be
considered random and mutually uncorrelated. Assuming summation of scatteringintensities from different inclusions, we again obtain (2), but with the volume scattering
coefficient defined as follows
M V v (a) v (a) da CV v (4)
where v / v is the individual scattering cross section of the inclusion divided by its
volume, and averaged over other parameters, such as shape and orientation,
v(a)N (a)
v (a) is the particle volume-size distribution function, with N defined as
Va
the average number of inclusions with given size within small intervals (a, a a) , in the
volume element V , and CV v (a)da 1 is the total volume concentration of
inclusions. The inequality must hold due to the inclusion sparseness assumption.
The size-parameter in Eq. (4) for non-spherical particles requires a definition, which
should be given in the “acoustically relevant” and complete manner that is as a parameter
(or a vector-parameter, i.e. a set of parameters) whose determining is sufficient for the
description of the particle scattering cross-section, . In common analysis, the size-
parameter is defined by the particle volume, e.g., as the radius of the sphere with the same
volume (the equivalent radius) through the relation v(a) (4 / 3)a 3 , or the equivalent
diameter D 2a (frequently called “true size”). However, from acoustics stand-point, it is
not sufficient, because, as shown in the following section, to provide a prediction of the
scattering-cross section for such particles, such definition should be accompanied by an
appropriate parameterization of the particle shape.
3. SCATTERING COEFFICIENTS
Consider now the scattering cross section for the inclusions, assuming that they can
be particles of arbitrary size, shape, and orientation. First, consider separately the cases of
particles small and large comparing to the wavelength, corresponding to low and high
frequencies, and then the case of intermediate sizes or frequencies.
For small particles, if they are randomly oriented and shaped, the average (over the
particle orientation) scattering cross section can be presented as follows
o k 4 v 2 Ro
2
4 2 (5)
where k / c and c () 1 / 2 are the wave number and sound speed in surrounding
“effective” fluid (which is considered as the reference medium), parameter Ro is
dimensionless, controlled by contrast of material properties, practically independent from
the particle shape, and only slightly dependent from the angle of scattering in all backward
hemisphere, which is important because the possibility of ignoring the bistatic effect is
also one of conditions for validity of the general expression (2). For solid inclusions (such
as suspended particles) we have Ro 1 , for gas bubbles in the sediment or in water
Ro ~ / 1 , and for fluid-like inclusions (such as oil droplets in water), we have
R 2 1 , with ( ~ c~) /( c) 1 being the relative contrast of impedance.
o c c
Combining (4) and (5), one obtains a simple low-frequency expression for the volumescattering coefficient in a cloud of Rayleigh particles, small (compared with the
wavelength), randomly oriented and arbitrarily shaped scatterers:
Ro2
M V CV k4 v (6)
16 2
where v is the particle average volume.
Scattering from non-spherical particles, whose dimensions (at least one of them) are not
small in comparison with the wave length, is more complicated and generally less
understood, than in the case of small particles. Here we take an empirical and somewhat
heuristic approach, based on comparison of existing solutions for backscatter from various
large particles of simple shape (such as the disk, spheroid, cone, cuboid, and some
others), showing that the angular patterns, being extremely complicated and sensitive to
the body orientation, become much simpler if averaged over a range of orientations. At
very high frequencies results of such averaging can be approximated by a single (the same
for different shapes) simple equation
S R /(16 ) (7)
2
Here S is the total surface area of the scattering particle, and the parameter R is the
reflection coefficient at normal incidence, defined by material contrast of the particle and
practically independent from the particle shape. For inclusions with large contrast, such as
for solid particles and gas voids, we have R 1 , and for low-contrast fluid-like inclusions
R c / 2 . Like in the case of small particles, for typical values of material parameters
and scattering angles within the backward hemisphere, the bistatic effects can be ignored.
We also use a dimensionless shape-parameter q S / S o (known also as Weston’s
ratio), with S o D 2 being the surface area of the sphere with the same volume (for
brevity, equivalent surface area). Because the sphere has the minimal surface area among
possible shapes with fixed volume, the inequality holds: q 1 , so that for spherical (and
only spherical) particles q 1 . It can be considered as a scattering enhancement factor due
to “non-sphericity” of scatterer. Using (4) and (7) results in a simple high frequency
expression (or the geometry-acoustics, frequency-independent limit) for the volume
backscattering coefficient in a cloud of particles larger than the wavelength,
2
R
M V CV S /v (8)
16
where S / v 6 q / D is the average specific surface area of the inclusions (which are
arbitrarily sized and shaped).
Having Eqs.(5) and (7) as limiting cases of low and high frequencies, we introduce a
“bridge” approximation for intermediate frequencies, or a smoothing function of the form
o
1 /
, 0 (9)
where is free parameter, which can be chosen from comparison with known solutions at
the intermediate frequencies, to describe, with a reasonable accuracy, the scattering cross-
section for a wide range of particle sizes, from small to large compared to the wavelength.
Alternatively, numerical solutions can be exploited based on the T-matrix method,
applicable for bodies of arbitrary shapes, but more time- and labor- consuming.4. SCATTERING FROM INCLUSIONS IN A STRATIFIED SEABED
The approach described in this paper was used for developing a model (GAMBID) for
prediction of the backscatter intensity from a stratified seabed with inclusions. All
necessary inputs and ground truth for GAMBID can be provided by particle analysis of
sediment cores and samples. The particle size distribution in this model is comprised of
two parts, central and coarse. The information about the central part is most common in
seafloor databases and usually provides statistics of particles comprising the sediment
matrix. The major parameter of this statistics is the particle mean size. Analysis of
sediment cores provides the depth-dependence of the mean size, which, through its known
empirical relationships [6] with the sediment acoustic parameters (density, sound speed
and attenuation), gives their depth-dependencies as well. This completely defines the
propagation kernel of GAMBID, allowing calculation of the depth-dependent pressure
functions in the sediment, as required in Eq. (2).
Analysis of the coarse part of size distribution is not yet common and much less
comprehensive, although only this can provide necessary input parameters for particles
significantly larger than the mean size, which can be considered as sparse inclusions in the
sediment matrix. Such analysis should be given in terms of the equivalent size and
accompanied by evaluation of the particle surface-based shape factor (usually related to
the particle size). This completely defines the scattering kernel of GAMBID, allowing
calculation of the volume scattering coefficient, based on knowledge of particle individual
scattering functions, and their size and shape distributions as given by Eqs. (4-9).
The described model was applied to analysis of the SAX04 geoacoustic and
environmental data set [1,2] and used for calculating the propagation and scattering in the
complicated, stratified (mud-to-sand) sediment with inclusions of three types, located in
different layers of the continuously stratified sediment. First type is carbonate shells
uniformly distributed in the sediment basement, with medium sand matrix. The basement
was covered by a transition mud-to-sand layer (about 3 cm thick), resulted from
redistribution of sediment (comprised of mud, sand, and shell particles) after a strong
weather event, see for more detail [1,2]. In lower part of this layer, at depths where
sediment became dense enough to support heavier particles, shells have settled (centered
between 2 and 2.5 cm depth), considered as second type of inclusions. The higher part of
the transition layer (at 0-2 cm depths) was a rather uniform mixture of mud and medium
sand, in the very top of which (at about 1-3 mm depths) was a thin venire layer of coarse
quartz sand particles, considered as inclusions of third type.
In Fig.1a, the frequency dependence of individual scattering functions, were calculated
using Eqs. (5,7,9), and results are shown in Fig.1b in terms of the reduced scattering cross-
section, 4 / D 2 , for the three different types of inclusions in the SAX04 sediment. These
cross-sections were exploited then for calculation of the volume scattering coefficient
using Eq.(4) with size/shape distributions obtained from the SAX04 sediment core
analysis for inclusions (coarse sand and shells), see [2], and results are given in Fig.1b.
The depth-dependent pressure functions were calculated according to depth-profiles of
the density, sound speed and attenuation in the stratified SAX04 sediment, and results are
shown in Fig.2. Combined with results given in Fig. 1b, they are used for calculating the
backscatter intensity using Eq.(2), given in terms of the seabed scattering strength. The
results are shown in Fig.3, where comparisons with the SAX04 acoustic backscatter data
are given as well, showing that all three types of inclusions are important (at different
frequencies) for explaining the measured backscatter.Two main conclusions result from this analysis. First is that the seabed scattering
strength is significantly affected by the sediment stratification. Second is that the
scattering strength shows sensitivity to the inclusion size, shape, and depth distributions.
Therefore, the physics-based model presented here (GAMBID), can be used as a
foundation for development of new techniques for seabed characterization using measured
dependencies of backscatter intensity from the frequency and scattering angles.
shell in mud (r) or in sand (b); sand in mud(g) shells: D=0.7-20mm, Q=1-3.8mm, Cv=0.003; sand in mud: D=0.7mm, Q=1, Cv=0.03
0 0
D=0.7mm, Q=1, =-1
D=5mm, Q=3, =-1
Effective Volume Scattering Coefficient, dB/m
-5 D=5mm, Q=1, =-1 -10
Reduced Scattering Cross-Section, dB
D=5mm, Q=3, =-1
D=5mm, Q=3, =-10
-10 -20
-15 -30
-20 -40
-25 -50
-30 -60
1 2 3 1 2 3
10 10 10 10 10 10
Frequency, kHz Frequency, kHz
(a) (b)
Fig. 1: Frequency dependence of (a) the individual reduced scattering cross-section, and
(b) the volume scattering coefficient, for three different types of SAX04 inclusions.
h = 0.03m, =[1.4,2.016], c=[1.48,1.742], =[0.005,0.01], N=553, f = [70 400] kHz
0 0
-0.005 = 20 -0.005 = 25
-0.01 -0.01
-0.015 -0.015
-0.02 -0.02
depth, m
depth, m
-0.025 -0.025
-0.03 -0.03
-0.035 -0.035
-0.04 70 kHz -0.04
-0.045 400 kHz -0.045
-0.05 -0.05
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Field Magnitude Field Magnitude
0 0
-0.005 = 30 -0.005 = 35
-0.01 -0.01
-0.015 -0.015
-0.02 -0.02
depth, m
depth, m
-0.025 -0.025
-0.03 -0.03
-0.035 -0.035
-0.04 -0.04
-0.045 -0.045
-0.05 -0.05
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Field Magnitude Field Magnitude
Fig.2: Depth dependent pressure magnitude at various grazing angles and frequencies.-10 -10
-15 -15
-20 = 20 -20 = 25
-25 -25
Scattering Strength, dB
Scattering Strength, dB
-30 -30
-35 -35
-40 -40
-45 -45
-50 -50
-55 -55
-60 -60
1 2 3 1 2 3
10 10 10 10 10 10
Frequency, kHz Frequency, kHz
shells in sand/mud
-10 shells in sand -10
-15 sand in mud -15
Williams' data
-20 -20 = 35
-25 -25
Scattering Strength, dB
Scattering Strength, dB
= 30
-30 -30
-35 -35
-40 -40
-45 -45
-50 -50
-55 -55
-60 -60
1 2 3 1 2 3
10 10 10 10 10 10
Frequency, kHz Frequency, kHz
Fig.3: Frequency dependences of the bottom backscattering strength for three types of
inclusions, at different grazing angles, 20, 25, 30, and 35 degrees, versus SAX04 data [1].
5. ACKNOWLEDGEMENT
This work was supported by the US Office of Naval Research.
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