Isakson Et Al: a Finite Element Model for Seafloor Roughness Scattering

A FINITE ELEMENT MODEL FOR SEAFLOOR ROUGHNESS SCATTERING

MARCIA J. ISAKSON, RAY A. YARBROUGH AND NICHOLAS P. CHOTIROS Applied Research Laboratories, The University of Texas at Austin, Austin TX 78713,USA E-mail: [email protected]

Analytic approximations based on the Helmholtz/Kirchoff integral have typically been used to estimate the contribution from scattering from the ocean bottom. However, these approximations may not be valid in many cases. In this study, analytic approximations are compared with ﬁnite element models for both canonical problems and measured data. However, analytic approximations may not be valid for natural interfaces that cannot be described by Gaussian power spectra. Finite element models, which approach exact solutions as the discretization density increases are shown to give good agreement with the approximations for canonical problems. Furthermore, ﬁnite elements are able to provide solutions for measured data in which the surface roughness does not conform to standard probability density functions. [Work sponsored by ONR Ocean Acoustics.]

1 Introduction

Seafloor roughness scattering is a major factor in ocean bottom reverberation. Previously, analytic models have been used to estimate the amount of reverberation from the ocean bottom. These models are based on the Helmholtz integral formulation and three approximations are commonly made: the Kirchoff approximation also known as the tangent plane approximation, the small slope approximation and the small roughness perturbation approximation also known as Rayleigh-Rice perturbation theory. However, these approximations typically require small to moderate interface roughness and do not generally include the effects of multiple scattering or shadowing. Finite element models can provide new insight for the seafloor scattering problem. These models approach an exact solution as discretization density increases and automat- ically include shadowing and multiple scattering. Furthermore, finite elements provide a method to calculate the scattered field without imposing any limitations on the degree of surface roughness, its probability distribution or its range dependence. Although finite elements do provide exact solutions, there are several limitations. First, the domain is necessarily finite and must be truncated. Second, the discretization is limited by computer memory. Third, finite elements compute solutions of individual rough interface realizations so many realizations must be solved to compare with a statistically based analytic theory. However, by computing many realizations, a probability density function (PDF) for the scattered field may be obtained as well as the mean solution. This scattering PDF could potentially be used in propagation modeling. In this study, finite element solutions will be compared with analytic solutions for roughness scattering problems with a goal of assessing the range of validity of the ana-

173 M. ISAKSON ET AL. lytic solutions for measured data. The finite element model will be compared to the exact integral solution and the Kirchoff and perturbation theory approximations for a variety of seafloor roughness parameters. This assessment will illustrate how finite elements can be used to bridge the gap between regions of validity for the analytic approximations. Lastly, the approximations will be compared with the finite element results for an in situ measurement of surface roughness.

2 Scattering Models Four models were used to calculate the scattering cross section for three random interfaces. These were ﬁnite elements, the exact integral solutions, the perturbation theory approximation and the Kirchoff approximation. In every model, the surface was modeled a rough pressure release surface to isolate the effects of scattering. The models are detailed below.

2.1 Finite Element Model The finite element model (FEM) was computed with a commercially available finite element code, COMSOL. The 2D scattering application mode was used for the calculations described in this paper. In the scattering application mode, the incident pressure is defined by the user. COMSOL calculates the time harmonic solution to the Helmholtz equation in the domain. The domain was truncated by Perfectly Matched Layers (PMLs) originally introduced by Berenger´ [1, 2]. Radiation conditions were applied to the outer boundaries except the bottom rough interface which was pressure release. The mesh consisted of triangular elements with Lagrange quadratic polynomial basis functions. Ten elements per wavelength were calculated. An example of the domain, solved for the scattered pressure is given in Fig. 1. For all problems in this study, the frequency was 10 kHz.

Figure 1. Domain used in FEM.

The pressure at any point outside the domain can be determined by evaluating the Helmholtz-Kirchoff integral on the surface using the pressure calculated from ﬁnite elements. The far ﬁeld Helmholtz-Kirchoff integral is given by:

∇2Ψ + k2Ψ = 0 (1)

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0 0 r 0 1 Z (~r · nˆ ) ∂p(~r ) 2 2 1 ~r · ~r p (~r) = [ik p(~r0) − ] e−iπ/4eik1|~r| exp[ik ]ds0 s 1 0 p 1 (2) 4i |~r| ∂n πk1 |~r| |~r| S Here ~r0 is the position on the surface, ~r is the receiver position, S is the rough surface and k1 is the acoustic wavenumber. For the problems addressed in this work, the first term in the integral is zero since the integration surface, S, is pressure release, p(~r0) = 0. The scattering cross section per unit angle per unit length was calculated from the intensity at the receiver position normalized by the total incident energy flux. In order to truncate the Helmholtz Kirchoff integral, a Gaussian shaded incident beam was used. This follows the procedure detailed in Ref. [3] which assessed the validity of the Kirchoff approximation. The incident Gaussian beam measured along the surface is given by Eq. (11) in this reference. The incident angle was 45o. For both the finite element solution and the integral solution, a surface of length, L, of 80λ was used with a spatial increment on 0.1λ. The incident Gaussian beam had a width of L/4.

2.2 Integral solution Scattering from a pressure release surface in 2D can be computed by solving the follow integral equation (Eq. (1) of Ref. [3]): 1 Z ∂p(r~0) (1) ~0 0 p(~r) = pinc(~r) − H0 (k1|~r − r |) 0 ds (3) 4i S ∂n (1) Here H0 is the zero-order Hankel function of the ﬁrst kind, and the integral is taken over the rough surface, S. The integral equation is solved via numerical quadrature following the method of Ref. [3] Eqs. (4) and (5). This solution is considered an exact solution for pressure release surfaces. The same incident beam used in the ﬁnite element solution is used to truncate the integral for the this solution.

2.3 Perturbation Theory The perturbation theory solution was calculated using the following equation (Eq. (37) of Ref. [3]): 3 2 2 σ = 4k sin θ sin θsW [k(cos θ + cos θs)] (4) Here σ is the scattering cross section, the scattered intensity per unit angle per unit length normalized by the incident energy ﬂux. W is the distribution function of the surface roughness. In this study, a Gaussian distribution will be used for W for perturbation theory calculations. Therefore, W is given by the following equation:

√ 2 2 W = (lh2/2 π)e−K l /4 (5)

2.4 Kirchoff theory The Kirchoff approximation was calculated using Eqs. (26)-(29) in Ref. [3]. The shadowing correction detailed in Section II.D from the same reference was also applied.

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3 Results The first step in applying FEM is to verify it against analytical approximations in the region of parameter space in which the approximations are known to be valid. The analytic approximations have been shown to be valid in different regions of the Gaussian roughness spectrum parameter space as shown in Fig. 3 reproduced from Ref. [4]. This plot shows the region of parameter space in which the approximations are valid within 1 dB. Also shown on the plot are the parameters for the three Gaussian surfaces modeled. One set of parameters is squarely in the range of validity for first order perturbation theory. Another is in the Kirchoff range. Unfortunately, surface with Gaussian roughness spectra are rarely found in nature. Shown in Fig. 2 is the measured power spectrum of a rough interface at the Experimen- tal Validation of Acoustic modeling techniques (EVA) in October 2006. These data were measured in Biodola Bay, Isola d’Elba, Italy using a laser line scan system [5]. The power spectrum represents the mean of 110 lines of measured data. Also shown in Fig. 2 is a Gaussian power spectrum that is calculated using the measured RMS height and correlation length. The third point in Fig. 3 describes the correlation length and RMS height measured at the EVA sea test. For the this third point, the Gaussian spectrum will be used to calculate the perturbation and Kirchoff approximations for comparison while he measured power spectrum is used for the finite element and integral solution approaches. All three RMS heights and correlation lengths are given in Table 1.

Figure 2. The average power spectrum for 110 measurements of 1D sea ﬂoor roughness using a laser line scan method at the EVA sea test (ﬂuctuating line). Also shown is a Gaussian spectrum using the measured correlation length and RMS height.

The scattering cross section is computed using the three analytic methods, the integral solution, perturbation theory approximation and Kirchoff approximation as well as with ﬁnite elements. The results are compared for the three cases. For each case, one hundred realizations are computed for the ﬁnite element and integral solution results.

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Figure 3. Contours for 1 dB accuracy for ﬁrst and second order perturbation theory and Kirchoff theory. Also shown are the two Gaussian spectra chosen to compare with the perturbation theory and the Kirchoff approximation. The last point (diamond) corresponds to the RMS height and correlation length of the measured surface roughness. The RMS heights and correlation lengths are given in Table 1.

Table 1. Three test cases used to compare models. Model RMS height [cm] Correlation length [cm] Case 1: Perturbation Theory Comparison 0.47 2.36 Case 2: Kirchoff Theory Comparison 3.1 13.3 Case 3: Measured Data Comparison 0.56 7.96

3.1 Case 1: Perturbation Theory Comparison

The first case compares the models in the region of parameter space that first order perturbation theory is known to be valid. The results are shown in Fig. 4. Perturbation theory gives good agreement with both the integral solution and the finite element solution except near the specular angle since it does not include coherent scattering. Kirchoff theory is only valid near specular for these parameters.

3.2 Case 2: Kirchoff Theory Comparison

For longer correlation lengths and rougher surfaces as in case 2, the Kirchoff approximation should be valid. Figure 5 shows that in this case, the Kirchoff approximation fits the integral solution and the finite element solution. The discrepancy in the backscattered angles is due to shadowing. The calculated Kirchoff approximation did not include the correction for ”illuminated” slope region developed by Wagner [6]. In this region of the parameter space, first order perturbation theory is invalid.

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Figure 4. Comparison of the the models for a rough surface with a Gaussian power spectrum with RMS height, h = 4.7 mm, and a correlation length, l = 2.36 cm. The frequency was 10 kHz. One hundred realizations were computed for the ﬁnite element and integral equation solutions.

Figure 5. Comparison of the the models for a rough surface with a Gaussian power spectrum with RMS height, h = 3.1 cm, and a correlation length, l = 13.3 cm. The frequency was 10 kHz. One hundred realizations were computed for the finite element and integral equation solutions. 3.3 Case 3: Non-Gaussian Roughness Spectrum For the case of measured roughness, the perturbation theory and Kirchoff approximation solutions were calculated using a Gaussian with the measured correlation length and RMS 178 FINITE ELEMENT INTERFACE SCATTERING MODEL height. The integral equation and finite element solutions were calculated using 100 realizations based on the measured power spectrum. The results are shown in Fig. 6. Perturbation theory, while giving a fairly good estimate near the specular angle, does not predict the scattering well at backscattered angles. This is expected since the Gaussian parameters fall outside the validity range for first order perturbation theory. The Kirchoff approximation gives a better prediction. However, it is 2-5 dB too low in the backscattered angles and more than 10 dB incorrect at forward scattered angles. Around specular, it fits the general level but fails to capture the resonant structure that both the integral solution and finite elements predict. The finite element and integral solutions are remarkably similar even after averaging 100 different realizations for each model. The structure that remains is dependent upon Bragg scattering from strong spatial wavenumbers in the measured spectrum.

Figure 6. Comparison of the the models for a rough surface. The perturbation theory and Kirchoff approximations are computed with a Gaussian power spectrum with RMS height, h = 5.6 mm, and a correlation length, l = 7.96 cm. The ﬁnite element and integral solutions were computed with one hundred realizations based on the measured power spectrum.

4 Discussion Four models of scattering were compared for three different distributions of surface roughness in two dimensions. Two of the models are considered exact solutions: finite elements and the integral solution. Two approximations were also calculated: first order perturbation theory and the Kirchoff approximation. The first case used a Gaussian roughness with a RMS height and correlation length that were within the range of validity of first order perturbation theory. For this case, the finite element solution, the integral solution and first order perturbation theory were in close agreement except at the specular angle since perturbation theory does not include coherent

179 M. ISAKSON ET AL. scattering. The Kirchoff approximation predicted the specular peak well but was not in agreement for angles far from specular. The second case used parameters for the roughness that were within the range of validity of the Kirchoff approximation. Here the finite elements, the integral solution and the Kirchoff approximation were in close agreement. There was some discrepancy in the backscattered angles which is more likely due to uncorrected shadowing in the Kirchoff approximation. The perturbation theory approach was incorrect at all angles. The third case was based on a measured power spectrum from the EVA sea test. The Kirchoff and perturbation theory approximations were calculated using a Gaussian spectrum with the calculated correlation length and RMS height. The integral solution and finite element method were modeled using 100 realizations from the measured power spectrum. Perturbation theory predicted the general levels near specular but was not in good agreement at other angles. The Kirchoff approximation predicted the general level but was 2-5 dB too low at backscattered angles and 10 dB too low at forward scattered angles. Also, both perturbation theory and the Kirchoff approximation failed to capture the resonant structure predicted by both the finite element and integral solutions. This study shows that the finite element approach is equal to the exact solution for scattering of an acoustic wave from all tested pressure release rough interfaces. This is the first step in verifying the finite element approach for rough interface scattering. The power of the finite element approach is that it is not limited to pressure release surfaces as is this form of the exact integral solution. The finite element method can be applied to fluid and elastic boundaries, and may include discrete surface scatterers, volume inclusions or range dependent properties. Furthermore, sound velocity profiles in both the sediment and water column can be included in the model. Lastly, by using the scattering application mode, the source and receiver do not have to be part of the domain. Therefore, the domain can have many fewer elements and 3D solutions are now viable with available computer resources.

Acknowledgements Thanks to the Ofﬁce of Naval Research, Ocean Acoustics and Robert Headrick for spon- soring this work.

References 1. Berenger J.P., A perfectly matched layer for the absorption of electromagnetic waves. Journal of Computational Physics 114, 18–200 (1994). 2. Berenger J.P., Perfectly matched layer for the FDTD solution of wavestructure interaction problems. IEEE Trans. Antennas Propagat. 44, 110–117 (1996). 3. Thorsos E.I., The validity of the Kirchoff approximation for rough surface scattering using a Gaussian roughness spectrum. J. Acoust. Soc. Am. 83, 78–92 (1988). 4. Jackson D.R., and Richardson M.D., High-frequency seaﬂoor acoustics. (Springer, New York 2007). 5. Chotiros N.P., Isakson M.J., Piper J.N., and Zampolli M., Seaﬂoor roughness measurement from a ROV. In Proceedings of the International Symposium on Underwater Technology 2007. Tokyo, Japan, 52-57 (2007). 6. Wagner R.J., Shadowing of randomly rough surfaces. J. Acoust. Soc. Am. 41, 138–147 (1966).

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