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 approxi- mations are commonly made: the Kirchoff approximation also known as the tangent plane approximation, the small slope approximation and the small roughness perturbation ap- proximation also known as Rayleigh-Rice perturbation theory. However, these approxima- tions 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 inter- face 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 measure- ment 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 approxi- mation 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 ele- ment 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 ele- ments 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 ele- ments. The far field Helmholtz-Kirchoff integral is given by: ∇2Ψ + k2Ψ = 0 (1) 174 FINITE ELEMENT INTERFACE SCATTERING MODEL 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 follow- ing 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 shadow- ing correction detailed in Section II.D from the same reference was also applied. 175 M. ISAKSON ET AL. 3 Results The first step in applying FEM is to verify it against analytical approximations in the re- gion 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 corre- lation 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 mea- sured 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. 176 FINITE ELEMENT INTERFACE SCATTERING MODEL 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.