M. Isakson and N. Chotiros · Marcia Isakson* and Nicholas Chotiros *Corresponding author's address: Applied Research Laboratories, The University of Texas at Austin, 10000 Burnet

Proceedings of Meetings on Acoustics

Volume 19, 2013 http://acousticalsociety.org/

ICA 2013 Montreal Montreal, Canada

2 - 7 June 2013

Underwater AcousticsSession 1pUW: Seabed Scattering: Measurements and Mechanisms II

1pUW9. Rough interface acoustic scattering from layered sediments using finiteelementsMarcia Isakson* and Nicholas Chotiros

*Corresponding author's address: Applied Research Laboratories, The University of Texas at Austin, 10000 Burnet Road, Asustin,TX 78758, [email protected] Quantifying acoustic scattering from rough interfaces is important for reverberation modeling, acoustic sediment characterization, andpropagation modeling. Most models of interface scattering rely on approximations to the Helmholtz integral equation. These models generallymake such assumptions as neglecting the local angle for reflection and disregarding multiple scattering between rough layers. In this study, amixed boundary element/finite element model is used to calculate rough interface scattering from ocean sediment. The finite element method,based on the Helmholtz equation, is exact within the limits of the discretization density; reflections are calculated locally and all orders ofscattering among layers are included. Using this model, bottom loss and backscattering predictions will be calculated for several cases. [Worksupported by ONR, Ocean Acoustics.]

Published by the Acoustical Society of America through the American Institute of Physics

M. Isakson and N. Chotiros

© 2013 Acoustical Society of America [DOI: 10.1121/1.4799774]Received 16 Jan 2013; published 2 Jun 2013Proceedings of Meetings on Acoustics, Vol. 19, 070018 (2013) Page 1

INTRODUCTION Finite element (FE) scattering models offer extreme flexibility in the problem definition. FE models can include

multiple layers, gradients and custom scattering surfaces using the same amount of calculations as that of a simple model. They also offer an exact solution to a partial differential equation such as the Helmholtz equation within the limits of the discretization density.

For scattering problems, a hybrid finite element/boundary element method is often employed. In this case, the incident pressure is prescribed on a surface in a domain. The pressure and its normal derivative are calculated within the domain using finite elements. Then, the Helmholtz/Kirchhoff integral is solved to give the pressure at a point in space. This method decreases the size of the domain dramatically from using a probe for the pressure inside the domain, allowing much larger and higher frequency models.

THE FINITE ELEMENT SCATTERING MODEL

An example finite element scattering domain is shown in Fig. 1. Shown is the calculated scattered pressure

between water and sediment domains using the parameters in Tab. 1 at a grazing angle of 22 degrees. All finite element models were computed with the commercially available software, COMSOL. [1] The surface roughness between the layers is prescribed by a Von Karman spectrum given by:

W1(K )=

w1(K 2 +K0

2 )γ1/2 (1)

Here, the power spectrum, W1 , given in terms of the spatial wavenumber,K , is parameterized byw1 , the spectral strength, γ1 , the spectral exponent, andK0 , the wavenumber cut-off. These parameters are given in Tab. 1.

The pressure incident on the surface is prescribed by a Gaussian tapered plane wave given by

pinc(r )= exp{i

kinc ⋅r[1+w(r )]−(x− zcotθ)2 / g2 ) (2)

where

w(r )= [2(x− zcotθ)2 / g2−1] / (k1gsinθ)

2 (3) Here,

kinc is the incident wave vector in the top domain, k1 is its magnitude, θ is the grazing angle, and g is the

beam waist. [2] The pressure at a point far from the surface is given by the Helmholtz/Kirchhoff (HK) integral:

ps (r ) = 1

4i[ik1p(

′r ) r ⋅ ˆ′n( )r−∂ p(′r )∂ ′n

] 2π k1

e−iπ/4eik1r 2 1r

exp[ik1r ⋅ ′rr

]d ′s∫ (4) Here,

r is the vector to the receiver position and the integral is taken over the surface, s . For this study, the far field form of the HK integral is used to compute the scattered pressure, ps .


Proceedings of Meetings on Acoustics, Vol. 19, 070018 (2013) Page 2

FIGURE 1. Scattered pressure calculated on a rough interface in a finite element domain for an incident Gaussian tapered plane wave at 22 degrees grazing.

Next, the two-dimensional bistatic scattering cross section, σs , can be computed from the pressure through

normalizing the scattered intensity by the energy flux through the surface. [2]

σ2 (θ)= ps (

r ) 2 r / ( π / 2g) (5) Finally the bistatic scattering strength, SS , is given by

SS=10 log10 σ2 (6)

TABLE 1. Parameters used for fluid/fluid scattering computations. The parameters were taken from Fig. 2 of Ref. 3.

Parameters Values Water Sound Speed 1500 m/s Water Density 1000 kg/m3 Sand Sound Speed 1775 m/s Sand Attenuation 2000 kg/m3 Sand Interface Spectral Strength, w1 0.5 dB/ λ Sand Interface Wavenumber Cut-Off, KL 0.08 m 3−γ1 Sand Intergace Spectral Exponent, γ1

0.001 m-1

Frequency 1000 Hz Roughness Ratio, h /λ 0.25

PRESSURE RELEASE MODEL COMPARISONS The FE model is compared with a Kirchhoff approximation model for a pressure release, Gaussian roughness

surface as described in Fig. 4 of Ref. 2. For these calculations, an ensemble of surface roughnesses is realized using a prescribed roughness spectrum, in this case, a Gaussian spectrum following the procedure in Ref. 2. These realizations are used in both the finite element model and the Monte Carlo Kirchhoff model. The finite element scattering strength is calculated as described in the previous section. The Monte Carlo Kirchhoff results are calculated by starting with the Helmholtz scattering integral using the same incident pressure wave as the FE model. The integral equation is simplified by assuming the local scattered pressure is related to the incident pressure by



Ps (r ,Ki )≈ [1+Vww (Ki )]Pi (

r ,ki )

N̂ ⋅∇Ps (r ,Ki )≈ iN̂ ⋅

ki[1−Vww (Ki )]Pi (

r ,ki )

(7)

Here Vww (Ki ) is the global reflection coefficient for the grazing angle of the incident beam with Ki referring

to the horizontal component of the incident wavenumber vector with respect to the surface. The parameter r refers

to the position on the surface. This is similar to the derivation in Ref. 4, Appendix L with the exception of the use the global reflection coefficient. In the case of a pressure release surface, there is no difference in using a global or local reflection coefficient, but for the fluid/sediment case, there is a noticeable difference. Using this approximation, the Helmholtz integral equation is reduced to a simple integral which is numerically evaluated for each realization of the rough surface. This formulation implies three major assumptions. 1) Each local scattering event is calculated as if the scattering were from a tangent plane on the surface. There are no diffraction or multiple scattering effects. 2) The reflection coefficient does not describe local angles of the rough interface. 3) There are no shadowing effects.

The second method of employing the Kirchhoff approximation is to employ theoretical ensemble averaging over the roughness spectrum as described in Ch. 13 of Ref. 4. This leads to a closed form solution for a Gaussian spectrum. [2] This will be referred to as the “analytical Kirchhoff” model.

FIGURE 2. The bistatic scattering strength from a pressure release surface computed with finite element, Kirchhoff Monte Carlo

and analytic Kirchhoff models. The finite element, Monte Carlo Kirchhoff and analytical Kirchhoff approximation results are shown in Fig. 2

for a 1 dimensional pressure release surface. This calculation has the same parameterization as Fig. 4 of Ref. 2. The models are all calculated in two dimensions (1D surface). The red line indicates the analytic Kirchhoff model. The fluctuating blue line is the finite element model for 100 Monte Carlo realizations of the surface and the fluctuating green line is the Kirchhoff model for the same 100 realizations of the surface. Note how the finite element model captures the shadowing effect at shallow angles. These results are directly comparable to Fig. 4 of Ref. 2 with the FE model directly comparable to the full integral equation solution. This serves to validate the FE method.

WATER/SEDIMENT MODELS

In order to compare the model with measured data, a number of extensions must be employed. First, the model

must be able to compute the scattering between the water and the ocean sediment. Second, it must be able to compute the scattering from a realistic rough interface such as one described by a power law or von Karman spectrum. Lastly, the model must describe the scattering in three dimensions. Once these steps are accomplished and verified, the model may be modified to include other physics such as multiple layers, elastics, and gradients.

As an example of an extended model with two fluid domains, one describing the water and the other describing the ocean sediment, and an interface roughness described by a von Karman spectrum, consider the model described in Tab. 1. An example of the domain of this model is shown in Fig. 1.

The scattering strength at a range of angles computed via the finite element method is compared with a 2D perturbation model and a Monte Carlo Kirchhoff model. The 2D perturbation model is based on the bistatic scattering cross section from the ONR Reverberation Workshop. The bistatic scattering cross section is given by



σ1D =k13

4{a(ks ,ki )[1+Γ(ks )][1+Γ(ki )]+b(ks ,ki )[1−Γ(ks )][1−Γ(ki )]

2}P1D (ki−ks )

where

a(ks ,ki )=−(1ρ−1)cosθi cosθs +1−

κ2

ρandb(ks ,ki )= sinθs sinθi (ρ−1)withki−ks = k1(cosθi + cosθs )

(8)

In these equations, k1 is the wavenumber in the water and P1D (ki−ks ) is the 1D roughness power spectrum

evaluated at the difference between the incident wavenumber, ki and the scattered wavenumber, ks . Γ(k) is the reflection coefficient given by

Γ=ρβ1(K )−κβ2 (K )ρβ1(K )+κβ2 (K )

with

β1(K )= 1− K2 / k1

2

and

β2 (K )= 1− K2 / k2

2

(9)

Here the ratio of the wavenumbers is given by κ= k2 / k1 and the ratio of the densities is given by ρ= ρ2 /ρ1 .

The scattering strength is then computed with Eq. 6. In Fig. 3, the bistatic scattering strength is compared at four different grazing angles. In Fig. 4, the

backscattering strength is compared for a range of grazing angles. Note that in each of the bistatic scattering strength curves, perturbation theory and finite elements predict a rise in scattering strength at a scattering angle of 150 degrees. This is an effect of the critical angle at 30 degrees. The Kirchhoff results do not predict this effect because the calculation only uses the reflection coefficient of the incident wave while both perturbation theory and finite elements calculate the reflection coefficient for the local angles.

Generally, perturbation theory is considered most accurate away from the specular peak while Kirchhoff is considered most accurate near and on specular. However, these results show that perturbation theory and finite elements agree well at almost every angle except at the specular peak. For the backscattering strength, the models are almost completely in agreement with some deviation near the critical angle of 30 degrees. Because of shadowing effects, the Kirchhoff approximation is most inaccurate at shallow angles. This is especially evident in the bistatic scattering strength at 6 degrees and in the shallow angles of the backscattering strength curve.



FIGURE 3. The bistatic scattering strength calculated for four different grazing angles: (a) 6 deg, (b) 30 deg, (c) 46 deg and (d) 78 deg by finite elements, Kirchhoff Monte Carlo and 2D perturbation theory.

FIGURE 4. Backscattering strength predicted by 2D perturbation theory, finite elements and Kirchhoff Monte Carlo theory.

CONCLUSION

Results for a two-dimensional finite element model for scattering strength from a one dimensional surface were compared with Kirchhoff and perturbation theory models. The Kirchhoff model was computed both with an approximation to the Helmholtz integral equation for an ensemble of surface realizations and with a theoretically averaged analytic expression for a surface roughness described by a Gaussian roughness spectrum.

Results comparing FE and Kirchhoff theory from a Gaussian roughness spectrum on a pressure release surface revealed that while the models were in general agreement, the finite element model was able to describe shadowing effects at very low grazing angles while the Kirchhoff model was not.

(a) (b)

(c) (d)



The model was extended to a water/sediment rough interface described by a power law-like von Karman spectrum. A two fluid perturbation theory model was compared to both the finite element results and the Monte Carlo Kirchhoff approach. It was shown that both perturbation theory and finite elements captured critical angle effects which the Kirchhoff model did not predict. Furthermore, finite elements and perturbation theory agreed well at almost every angle except at the specular peak. However, there was a 2-3 dB discrepancy between the finite element results and perturbation theory for backscattering from grazing angles less than critical.

ACKNOWLEDGMENTS

The authors thank the Office of Naval Research, Ocean Acoustics and project managers, Bob Headrick and Kyle Becker for sponsoring this work.

REFERENCES

1. Comsol Multi-Physics: http://www.comsol.com/, Last Checked: 6/4/2012. 2. E. Thorsos, “The validity of the Kirchoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J.

Acoust. Soc. Am. 83, 78–92 (1988). 3. D. Jackson, R. Odom, M. Boyd, and A. Ivakin, “A GeoAcoustic Bottom Interaction Model (GABIM),” IEEE J. Ocean Eng.,

35, 603–617 (2010). 4. D. Jackson and M. Richardson. High-frequency seafloor acoustics. Springer Verlag, New York (2007).



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M. Isakson and N. Chotiros · Marcia Isakson* and Nicholas Chotiros *Corresponding author's address: Applied Research Laboratories, The University of Texas at Austin, 10000 Burnet