Proceedings of Meetings on Acoustics

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1 Proceedings of Meetings on Acoustics Volume 19, ICA 2013 Montreal Montreal, Canada 2-7 June 2013 Underwater Acoustics Session 2pUWb: Arctic Acoustics and Applications 2pUWb8. A parabolic equation for under ice propagation Adam M. Metzler*, Jon M. Collis and William L. Siegmann *Corresponding author's address: ARL - Environmental Sciences Group, The University of Texas at Austin, PO Box 29, Austin, Texas 78613, ametzler@arlut.utexas.edu Parabolic equation methods are useful to accurately and efficiently model propagation in the ocean for range-dependent environments. Most methods treat environments where the ocean surface is a flat perfect reflector. For some problems the ocean surface characterized as a random rough scattering water-air boundary produces significant effects on the overall propagation. A rough surface parabolic equation has been developed [A. P. Rosenberg, J. Acoust. Soc. Am. 105, 144 (1999] that extends the split-step Padé approach in the RAM implementation, and others have extended split-step Fourier methods. For an upper ice surface, elasticity should be incorporated into the rough scattering boundary. In this paper, a parabolic equation is presented that captures effects of seismo-acoustic interactions and scattering from a rough surface modeled as rigid as opposed to pressure release. Particular attention is given to upward-refracting sound speed profiles, typically found in Arctic environments, which encourage interactions with an ice surface. The scattering technique can be extended to environments with fluid/ice interfaces. [Work supported by ARL:IR&D] Published by the Acoustical Society of America through the American Institute of Physics 2013 Acoustical Society of America [DOI: / ] Received 22 Jan 2013; published 2 Jun 2013 Proceedings of Meetings on Acoustics, Vol. 19, 0051 (2013 Page 1

2 INTRODUCTION Developments in underwater acoustic modeling for the Arctic have been limited due to the complicated nature of the polar extreme. In Arctic regions, the sound speed minimum occurs at or near the ice-covered surface. The upward refracting sound speed profile causes any long-range propagation to repeatedly interact with the ice cover. At low frequencies it is believed that the rough underside of the ice has a negligible effect on propagation [1]. As frequency increase, the jagged underside of the ice cover, and the scattering that it causes, becomes increasingly more important for accurate propagation modeling. The present discussion is motivated by a general renewed interest in the Arctic, in particular for acoustical applications. In addition to scattering concerns, there are also scenarios in which the ice cover may vanish. The thrust of this work will be to introduce a parabolic equation solution that is capable of accurately simulating propagation in range-dependent Arctic environments. Due to the rough underside of the ice, a rough surface scattering parabolic equation [2] may be used to treat the fluid/ice interface. However, since the ice is modeled as an elastic layer, a rigid boundary condition may be more appropriate than a pressure-release condition. ARCTIC PARABOLIC EQUATION The canonical environment that we consider in this work is depicted in Fig. 1, which shows an elastic layer (representing an ice cover overlying a fluid layer overlying an elastic basement. Although Fig. 1 suggests an environment with no range dependence, the parabolic equation solution presented in this section can handle range-dependent environments. For Arctic environments, range dependence is considered in the form of an upward refracting sound speed profile, variable bathymetry, and variable ice thickness. FIGURE 1: Example layered Arctic environment. The elastic parabolic equation is derived from the elastic equations of motion [3] in terms of Proceedings of Meetings on Acoustics, Vol. 19, 0051 (2013 Page 2

3 the (u x, w dependent variables, where u x is the horizontal derivative of the horizontal displacement and w is the vertical displacement [4, 5]. In this form the equations are ( λ + 2μ 2 u x x 2 μ 2 w x ( μ u x ( (λ w + 2μ + ρω 2 u x + λ 3 w ( μ 2 w x 2 = 0, (1 x ρω 2 w + μ u x + (λu x = 0, (2 where λ and μ are Lamé constants related to the wave speeds by ( λ = ρ c 2 p 2c2 s, μ = ρc 2 s, (3 where c p and c s are the compressional and shear speeds in the sediment. Attenuation is incorporated into the system by assuming that both sound speeds are complex, with compressional and shear attenuations, β p and β s, being included in the imaginary part of the sound speeds. Equations (1 and (2 can be written as a system of equations of the form ( (L 2 x 2 + M ux w = 0, (4 where L and M are matrices containing parameters and operators that vary in depth only. Equation (4 can be factored into a product of incoming and outgoing operators, and since in most ocean acoustic propagation scenarios outgoing energy dominates incoming energy, the parabolic equation equation is x ( ux w = i ( L 1 M ( 1/2 ux w. (5 Numerical solution of Eq. (5 is achieved through an approximation of the operator ( L 1 M 1/2 with a rational-linear Padé series, yielding a split-step Padé solution [6]. For the Arctic environment depicted in Fig. 1, continuity conditions must be explicitly enforced at the fluid/solid interfaces that occur at both the fluid/ice and fluid/sediment boundaries. These interface conditions are { } 1 ρω 2 (λδ = {w} E, (6 F { {λδ} F = λu x + ( λ + 2μ w }, (7 E { 0 = (λu x + ( } (λ w + 2μ + ρω 2 w, (8 E which correspond to continuity of vertical displacement, continuity of normal stress, and vanishing of tangential stress respectively. In Eqs. (6 to (8 the subscripts F and E represent the fluid and elastic sides of the interface respectively, and Δ = u/ x + w/, the dilatation in the fluid. Range dependence due to either bathymetric changes or changes in ice cover thickness are treated by applying a stair-step approximation where the environment is modeled as a series of range-independent regions [7]. This procedure introduces artificial vertical interfaces at the boundary of two adjacent range-independent regions. An approximate energy conservation condition is applied at these vertical interfaces to accurately propagate the solution between range-independent regions [8]. Proceedings of Meetings on Acoustics, Vol. 19, 0051 (2013 Page 3

4 TABLE 1: Geoacoustical parameters for the ice and sediment for the example environment considered. c p (m/s c s (m/s ρ (g/cm 3 β p (db/λ β s (db/λ Ice Sediment PRELIMINARY RESULTS Consider initially that the overlying ice layer is penetrable and may terminate or initiate at specific ranges. The interface conditions of Eqs. (6 to (8 have been incorporated into a parabolic equation solution and results are presented in this section. Propagation examples are considered for the region depicted in Fig. 2a for a source placed at a latitude of N and a longitude of 150 W. For this example environment, a source is placed 100 m into the ocean and a receiver is tracked at a depth of 150 m. Geoacoustical parameters for both the ice and sediment are provided in Table 1 and the upward refracting sound speed profile taken from the month of January is shown in Fig. 2b. The bathymetry is nearly constant throughout range, sloping downwards from 30 m to 4100 m over approximately 150 km. (A (B FIGURE 2: (a Bathymetric plot of the Arctic. The red dot represents the location of the source for the example environment considered with propagation moving in the direction of the arrow. (b Sound speed profile in the fluid for the example environment. Initially, consider the environment described in Table 1 and Fig. 2b, but as range independent. Two cases are examined, one without any ice cover and one with a 10 m thick ice cover throughout the domain. A 30 Hz point source is located 100 m below the sea surface. Transmission loss results for these two environments are compared in Fig. 3. Contour plots for the first 250 m in depth are shown in Fig. 3a for the environment with no ice cover and in Fig. 3b for the environment with 10 m thick ice cover. When ice is present, less energy is propagated in the water column, probably as a result of absorption into the elastic ice layer. These differences are illustrated further in Fig. 3c which compares transmission loss in both environments at a receiver depth of 150 m. As range increases the differences between the curves increases, further suggesting that the differences are a result of absorption loss due to the presence of the ice cover. Consider now the range-dependent case with variable bathymetry and treat the fluid/ice Proceedings of Meetings on Acoustics, Vol. 19, 0051 (2013 Page 4

5 Depth (m Depth (m (A (B Loss (db re 1m (C FIGURE 3: Results for the range-independent example. Transmission loss contour plots for an environment with (a no ice and (b a 10 m thick ice layer. The increase in transmission loss for the 10 m thick case is due to absorption into the ice layer. (c Transmission loss curves for the environment with no ice (red and the environment with 10 m thick ice cover (black at a receiver depth of 150 m. interface as rough. The rough interface is created by assuming a starting thickness and using a random walk procedure to create roughness along the interface. An example is shown in Fig. 4a. The black curve represents the random fluid/ice interface by starting at a depth of 10 m and using 500 random walks over the 150 km range, or a new point every 300 m. For a frequency of 30 Hz and using the sound speed profile in Fig. 2b, this results in a new point approximately every six wavelengths. The red curve represents the average thickness of the ice using the random surface. Transmission loss curves at a receiver depth of 150 m for both the random fluid/ice interface environment and the average fluid/ice interface environment is shown in Fig. 4b. The two curves are in very good agreement throughout the entire range suggesting that at this frequency and for this amount of random points, the roughness of the underside of the ice does not significantly affect the propagation in the water column. Figure 4c depicts transmission loss for similar environments, but now with a random fluid/ice interface obtained by using 3000 points over the 150 km, or approximately a new point every wavelength. Using this interface (not shown and its corresponding average ice thickness results in less agreement between the two environments. This suggests that the roughness of the underside of the ice affects the propagation if these effects are on the order of a wavelength. Moreover, results were unobtainable for fluid/ice interfaces created by this random walk procedure for higher frequencies, suggesting that scattering methods must be investigated to accurately produce propagation solutions for environments with rough fluid/ice interfaces. SCATTERING TREATMENTS IN PARABOLIC EQUATION SOLUTIONS Consider the overlying ice layer as a thin layer, i.e. the layer thickness is considerably less than that of an acoustic wavelength. In order to maintain computational efficiency when computing solutions to problems of this type, variable grid spacing may be employed to allow for refined gridding within the ice layer as opposed to more coarse gridding, which may be Proceedings of Meetings on Acoustics, Vol. 19, 0051 (2013 Page 5

6 Depth (m (A Loss (db re 1m (B Loss (db re 1m (C FIGURE 4: Results for the ice roughness example. (a Location of the fluid/ice interface as created by a random walk using 500 points over the 150 km range (black and the average of the random walk (red. Transmission loss results at a receiver depth of 150 m for (b the environment using the fluid/ice interface in (a, and (c an environment created using a random walk with 3000 points over the 150 km range. applicable in the water column and elastic basement [9]. It is in this manner that the parabolic equation solution of Rosenberg proposed to treat rough surface scattering [2]. While this solution may allow for refined gridding near a rough sea surface, for example, it is not clear that this approach is applicable to scattering from rough elastic interface, such as from ice ridges. An alternative to this, for scattering from an overlayer of ice, would be to simply treat the interface as rigid. Mathematically, this would amount to applying a Neumann boundary condition on the surface interface when there is an ice cover. This would remove the need for an additional surface condition at the air interface. As this condition would be applied in the fluid, it would be of the form p(x, z = 0 = 0, (9 where p = λδ and Δ is the dilatation defined in Sec. 2. The boundary condition in Eq. (9, along with the variable grid spacing, is expected to be the most accurate and efficient approach to incorporate under-ice scattering into the Arctic parabolic equation. SUMMARY An accurate and efficient parabolic equation has been developed for propagation in ice-covered Arctic environments. The parabolic equation can handle range dependence in both bathymetric features and terminating and reappearing ice. For weak under-ice roughness (when compared to acoustic wavelength, it appears that the roughness does not have a significant effect on the other all propagation. However, for higher frequencies or more finer and drastic roughness, scattering techniques must be used for accurate solutions. A Neumann boundary condition will allow for a more accurate model for scattering since the underside of the ice is expected to be rigid as opposed to a pressure-release interface. Proceedings of Meetings on Acoustics, Vol. 19, 0051 (2013 Page 6

7 ACKNOWLEDGMENTS Work of the first author supported by the Internal Research and Development program of the Applied Research Laboratories at The University of Texas at Austin. REFERENCES [1] O. I. Diachok, Effects of sea-ice ridges on sound propagation in the arctic ocean, J. Acoust. Soc. Am. 59, (1976. [2] A. P. Rosenberg, A new rough surface parabolic equation program for computing low-frequency acoustic forward scattering from the ocean surface, J. Acoust. Soc. Am. 105, (1999. [3] H. Kolsky, Stress Waves in Solids (Dover, New York (1963. [4] A. J. Fredricks, W. L. Siegmann, and M. D. Collins, A parabolic equation for anisotropic media, Wave Motion 31, (2000. [5] W. Jerzak, W. L. Siegmann, and M. D. Collins, Modeling Rayleigh and Stonely waves and other interface and boundary effects with the parabolic equation, J. Acoust. Soc. Am. 117, (2005. [6] M. D. Collins, A split-step Padé solution for the parabolic equation method, J. Acoust. Soc. Am. 93, (1993. [7] F. B. Jensen and W. A. Kuperman, Sound propagation in a wedge-shaped ocean with a penetrable bottom, J. Acoust. Soc. Am., (19. [8] M. D. Collins and E. K. Westwood, A higher-order emergy conserving parabolic equation for range-dependent ocean depth, sound speed, and density, J. Acoust. Soc. Am. 89, (1991. [9] J. M. Collis, Low-frequency seismo-acoustic propagation near thin and low shear speed ocean sediment layers, Proc. of Meetings on Acoust. 8, 0001 (2009. Proceedings of Meetings on Acoustics, Vol. 19, 0051 (2013 Page 7