Finite element modeling of reverberation and transmission loss

Finite element modeling of reverberation and transmission loss in shallow water waveguides with rough boundaries Marcia J. Isakson a) and Nicholas P. Chotiros Applied Research Laboratories, The University of Texas, Austin, Texas 78713-8029 (Received 5 October 2010; revised 15 November 2010; accepted 22 November 2010) A finite element model for the reverberation and propagation in a shallow water waveguide with a sandy bottom was calculated for five different environments at a center frequency of 250 Hz. The various environments included a rough water/sediment interface, a rough air/water interface, roughness at both interfaces and downward and upward refracting sound speed profiles with roughness at both interfaces. When compared to other models of reverberation such as ray theory, coupled modes, and parabolic equations, finite elements predicted higher levels of reverberation. At early times, this is due to the fathometer return, energy that is normally incident on the boundaries at zero range. At later times, the increased reverberation was due to high angle scattering paths between the two interfaces. Differences in reverberation levels among the environments indicated that scattered energy from the air/water interface is transmitted into the bottom at steep angles. This led to a large decrease in reverberation for a rough air/water interface relative to a rough water/ sediment interface. Sound speed profile effects on reverberation were minimal at this frequency range. Calculations of the scintillation index of the different environments indicated that most of the reverberation was relatively Rayleigh-like with heavier tailed distributions at longer ranges. VC 2011 Acoustical Society of America. [DOI: 10.1121/1.3531810] PACS number(s): 43.30.Gv, 43.30.Hw [DSB] Pages: 1273 1279 I. INTRODUCTION a) Author to whom correspondence should be addressed. Electronic mail: misakson@arlut.utexas.edu A predictive model of transmission loss (TL) and reverberation in shallow water waveguides is key to understanding active and passive sonar performance, self noise propagation, and acoustic communications in littoral areas. A critical factor in the modeling of shallow water waveguides is the effect of the boundaries, particularly scattering from the interfaces and transmission into the bottom. Currently, there are many methods to predict reverberation and TL in shallow water waveguides with rough boundaries. These include normal modes, 1 coupled modes, 2 parabolic equations, 3 andraybasedtheories. 4 The predicted reverberation time series of these models for a two-dimensional shallow water waveguide are in very close agreement. 5 These methods share one common approximation: energy that is scattered from the interfaces at angles close to normal is neglected. Some methods also neglect multiple scattering events. Finite element (FE) models provide a method of benchmarking the two-dimensional shallow water waveguide. As the discretization density increases, FE models approach an exact solution to the Helmholtz equation implicitly including all orders of multiple scattering. FE models have been benchmarked for scattering in two dimensions and have been found to agree closely with exact solutions such as the integral equation model. 6 The method provides a noiseless, fully customizable testbed. By comparing the FE model to other solutions, the effects of the approximations in other models can be quantified. In this paper, FE reverberation and propagation results are presented for two-dimensional waveguides with rough interface conditions. The paper is organized as follows. In Sec. II, the FE technique is introduced and details of the model are presented. Section III describes the geometric and geo-acoustic properties of the waveguide. Reverberation and propagation results for a variety of waveguides are presented and discussed in Sec. IV. Finally, conclusions are drawn in Sec. V. II. THE FE MODEL The FE method consists of solving the variational form of the Helmholtz equation. This method will only be summarized here and the interested reader is referred to Refs. 7 and 8 for a rigorous mathematical derivation. An overview of the mathematical method and application to underwater acoustics is also given in Ref. 9. A. FE modeling For this model, the water column and the sediment layer are modeled as fluids using the Helmholtz equation, r 1 þ qðxþ rpðxþ k2 pðxþ ¼0: (1) qðxþ Here q is the density, p is the pressure, x is the spatial coordinate vector, and k ¼ x/c(x) is the wavenumber. The FE method uses the variational or weak formulation of solution in which the Helmholtz equation is multiplied by a test function, /(x) and integrated. Following a derivation such as in Ref. 7, the variational expression is ð 1 r /ðxþþ qðxþ rpðxþ k2 qðxþ pðxþ/ðxþ dx ¼ 0: (2) X J. Acoust. Soc. Am. 129 (3), March 2011 0001-4966/2011/129(3)/1273/7/$30.00 VC 2011 Acoustical Society of America 1273

both the vertical and horizontal directions using the method described in Ref. 13. Waves entering the PML are only attenuated in the outgoing direction and the tangential components are unaffected. FIG. 1. (Color online) Geometry and mesh for a truncated shallow water waveguide. The mesh density increases greatly at the interfaces and near the source to resolve the solution. The model was truncated using PMLs. The test functions must form a complete, orthogonal set and must satisfy Dirichlet boundary conditions. For this problem, the piecewise quadratic functions are chosen. The problem then forms into a sparse matrix equation and mesh elements are triangular. The geometry is meshed and the problem is solved using commercially available FE modeling software. 10 B. Meshing An example of the FE mesh for a very short range problem is shown in Fig. 1. In general, seven triangular elements per wavelength were used with ten times more elements near the rough interface and the source. The interface resolution is 10 cm with one element per linear section of the interface. The sand layer depth is two acoustic wavelengths for the longest wavelength used in the Fourier synthesis, in this case 7.1 m. C. Perfectly matched layers FE domains are necessarily finite and must be truncated properly in order to reduce reflections at boundaries. The domain in this model is bounded by Berenger perfectly matched layers (PMLs) in order to simulate the Sommerfeld radiation condition at the boundaries. 11,12 The exception is the air/water interface which is modeled as a pressure release boundary. The pressure in the PML is damped in the direction perpendicular to the interface using spatial coordinates that are complexly scaled. Corner elements are damped in D. Time harmonic and time dependent solutions Since this FE model is based on the Helmholtz equation, the solutions are time harmonic. A time dependent reverberation solution may be developed through Fourier synthesis. Multiple time harmonic solutions are computed at frequencies spaced at intervals of c w /(2L) where c w is the sound speed in the water column and L is the length of the waveguide. Since the total time, T, calculated using Fourier synthesis is T ¼ 1/df, the time domain will cover the two-way travel time from the end of the waveguide. After computing the wave field at all of the necessary frequencies, the time domain solution at any point is obtained by shading the frequency response with the frequency response of the sound source and taking the inverse Fourier transform. In this case, the source frequency response is taken from the Office of Naval Research (ONR) Reverberation Workshop 5 and it is modeled as a Gaussian, 1 Q ¼ 4P o =½iH ð1þ o ðkr oþš: (3) Here Q is the frequency dependent source strength. P o is the reference pressure; in this case, 1 lpa. H o is the Hankel function of the first kind and r o is the reference radius, 1 m. In 3D, the source would be a line source. An example of a time domain solution, the entire field for a 450 m waveguide with rough boundaries, is shown in Fig. 2 at an elapsed time of about 0.11 s. In this case, the source frequency was centered at 250 Hz. The signal spectrum covers the frequency range of 210 290 Hz shaded with a Tukey window in order to provide a well-defined pulse response in the time domain. For this calculation only, the computed signal is not convolved with source spectrum. Note the fathometer returns evident close to the source. These returns are not included in most models since they correspond to energy normally incident on the boundaries. This will be evident in the reverberation time series. III. PROBLEM DEFINITION The basic problem set-up is taken from the ONR Reverberation Workshop. 5 The waveguide has a 50 m deep water FIG. 2. FE, time domain solution for a short waveguide at a time 0.11 s. The impulse response over a frequency range of 210 290 Hz is shown. Note the transmission into the sediment and the fathometer returns. 1274 J. Acoust. Soc. Am., Vol. 129, No. 3, March 2011 M. J. Isakson and N. P. Chotiros: Finite Element Reverberation Modeling

TABLE I. Waveguide parameters. Parameter Value Waveguide depth 50 m Source depth 15 m Receiver depth for reverberation 25 m Water density 1024 kg/m 3 Water sound speed 1500 m/s Water attenuation Thorp attenuation 9 Sediment density 2048 kg/m 3 Sediment sound speed 1700 m/s Sediment attenuation 0.5 db/m/khz Bottom roughness rms height, h 0.141 m Bottom roughness correlation length, l 10 m Wind speed 10 m/s column with a medium sand bottom. The sediment and water characteristics are given in Table I. The sediment/water interface roughness is characterized by a modified power law wavenumber spectrum in the following form, P 1D ¼ h 2 l pð1 þ K 2 x l2 Þ : (4) Here K is the spatial wavenumber and l is the correlation length given in Table I. The air/water interface was characterized with the Pierson/Moskowitz spectrum. 14 The actual realizations used in the models were provided by the ONR Reverberation Workshop. 15 Six waveguides are considered, as detailed in Table II. Problems 1, 2, and 3 are taken directly from the reverberation workshop. Problem 4 use sound speed profiles from other problems in the workshop and apply them to the 2D environment. IV. RESULTS TABLE II. Waveguides considered. Problem Water sound speed profile Water/sediment interface Air/water interface Control Iso-velocity Flat Flat 1 Iso-velocity Rough Flat 2 Iso-velocity Flat Rough 3 Iso-velocity Rough Rough 4 Summer profile Rough Rough Sound speed at 0 m depth: 1515 m/s Gradient: 0.3 m/s/m 5 Winter profile Sound speed at 0 m depth: 1495 m/s Gradient: 0.1 m/s/m Rough Rough FIG. 3. The reverberation as a function of time for six models for a shallow water waveguide with a rough bottom only. The first five models are taken from Ref. 5. A. Reverberation The previously published reverberation results for problem 1 from coupled modes, parabolic equations, and ray theory 5 are compared with the FE reverberation prediction in Fig. 3. In the figure, the curves are denoted by the investigator s last name. Stotts and Fromm used a ray based model, while Yang and LePage used a coupled mode solution. Lingevitch modeled with two-way rough bottom parabolic equations. Yang and Lingevitch averaged over 100 realizations while the other methods used a scattering model provided by the workshop in the time domain. Overlaid on these curves is the average of 20 realizations from the FE model. There are several differences between these models and the FE model. First, at very short times, the other models disagree with FE on the order of 10 db or more. Since FE solves the Helmholtz equation over the entire space, it also includes the energy reflected normally from the boundaries at zero range. These are colloquially known as the fathometer returns. No other model includes these high angle paths, and therefore, there is a large disagreement at short times. The more interesting difference between the FE model and the other models occurs at later times. Here the FE solution predicts much higher reverberation than the other models at particular times. For example, at about 1.6 s, the highest of the other models predict a reverberation level of 68 db while FE predicts around 61 db. This difference is very significant when looking for targets relative to a reverberant background. Since the FE model calculates the entire domain, it is difficult to determine what may be causing the differences among the models. However, all models except FE, share the common approximation of neglecting normal and nearnormal scattered energy. Although the fathometer returns are negligible after 0.4 s, there is always some scattering at normal and near-normal directions from the rough interfaces. Normal and near-normal energy between parallel boundaries tends to be multiply reflected or scattered, in a process that is often called localization. Some of the energy will be eventually scattered back to the receiver. The cumulative effect may cause the difference in reverberation results. Another hypothesis may be coherent backscattering enhancement. This occurs when the two identical paths from the source to the scatterer to the receiver constructively J. Acoust. Soc. Am., Vol. 129, No. 3, March 2011 M. J. Isakson and N. P. Chotiros: Finite Element Reverberation Modeling 1275

interfere. 16 However, at most, this enhancement leads to a 1.76 db increase in the pressure for a two-dimensional waveguide. Although this may be a factor, it does not account for all the difference among the models. Additionally, the other coherent models will include these effects. B. Waveguide comparison Although results from coupled modes, ray theory, and parabolic equations were only available for problem 1 of the workshop, FE solutions were computed for 20 realizations of every problem described in Table II. The mean reverberation for each problem is shown in Fig. 4(a) and its scintillation index is shown in Fig. 4(b). The scintillation index is defined as the standard deviation divided by the mean. There are a few interesting features to note. The flat interface case shows that the fathometer returns are negligible after 0.4 s. At later times, with the exception of problem 2, the reverberation levels of all the models are remarkably consistent. However, the reverberation from the rough bottom without a rough top is slightly higher than the other cases. The addition of the rough air/water interface may decrease the reverberation relative to the case with just a rough bottom by scattering more energy into the sediment. Energy that is scattered by the rough air/water interface will encounter the bottom at much steeper angles than energy traveling down the waveguide. This energy is much more likely to be transmitted into the bottom although some will be scattered toward the receiver. 1. Difference between problems 1 and 2 There are three effects that may contribute to the large difference in the reverberation between problems 1 and 2. First, the value of the power spectral density of the surface must be considered at the Bragg wavenumber of the source. The power spectral density, plotted in decibels referenced to 1cm 3 per cycle, for each of the interfaces is shown in Fig. 5. For 250 Hz, the Bragg wavenumber close to 0.02 cm 1 for grazing incidence. At this spatial frequency, the power spectral densities of the two surfaces are identical. Therefore, the difference cannot be attributed to a difference in the power spectral density of the roughness at the Bragg wavenumber. Another factor may be the amount of energy at each surface. In this waveguide, the modes have a node at the air/ water interface while there is not a node at the water/sediment interface since the reflection coefficient at this interface is not unity. Therefore, there is more energy to scatter at the water/sediment interface due to the modal structure. Lastly, the effects of scattering at the boundaries must be considered. Sound that is scattered from the bottom at normal or near-normal incidence will travel to the top of the waveguide be reflected without loss back toward the bottom to be scattered again. The same process from the air/water interface will encounter a very low reflection coefficient of about 8 db at the water/sediment interface. This asymmetry in reflection loss may tend to favor one boundary over another with regard to the reverberation level. FIG. 4. (a) The incoherent average of the reverberation as a function of time from the problems described in Table II. (b) The scintillation index as a function of time for all of the waveguides. FIG. 5. (Color online) Roughness power spectra for the water/sediment interface and the air/water interface. Also shown is the spatial wavenumber corresponding to the Bragg wavenumber of the central acoustic frequency. 1276 J. Acoust. Soc. Am., Vol. 129, No. 3, March 2011 M. J. Isakson and N. P. Chotiros: Finite Element Reverberation Modeling

2. Variability of reverberation In Fig. 4(b), the scintillation index for the five waveguides is plotted as a function of time. First, it is noted that after about 1 s, the scintillation index approaches one, indicating a Rayleigh distribution for the reverberation. The scintillation index for problem 1 remains below one indicating that the distribution is not fully saturated. At later times, the scintillation index for the other problems is significantly higher than one indicating non-rayleigh or heavy-tailed statistics. This is especially true for problem 3. Both of these problems appear to favor scattering from the bottom interface. C. Sources of error The FE models were analyzed to insure there are no obvious errors in the calculation and processing of the results. Two errors are considered in particular: the effects of the fathometer returns and aliasing effects in the Fourier synthesis. The analysis of the Fourier synthesis will also give an indication of the amount of residual reflection from the PMLs. 1. Fathometer returns The results from the reverberation calculated for problem 1, rough bottom only, and a flat interface are compared in Fig. 6. Although fathometer returns are a real phenomenon, it is interesting to note how they affect the solution given that other methods do not include them. It is clear in the figure that the effects from the flat surface fathometer results are insignificant past 0.4 s. 2. Aliasing in the Fourier synthesis process It is also important to quantify the effects of aliasing in the Fourier synthesis process. This type of analysis will also provide information on the reflection from the PMLs. For this analysis, three different waveguides were modeled. Each used the same roughness realization for the sediment/ FIG. 7. Reverberation as a function of time for a 4 km waveguide, 2 km waveguide, and a 2 km waveguide with 1 km of flat interfaces. water interface and had a flat air/water interface. The first waveguide was 4 km long, the second was 2 km and the third was 2 km long but the last 1 km of the bottom interface was flat. Results are shown in Fig. 7. First, note how all of the reverberation results are very consistent to the point of the end of the waveguide or the end of the roughness. The large increase in both the 2 km waveguides at 2.66 s is indicative of the aliasing of the first fathometer return. (Note that the two-way travel time to the end of this waveguide is 2.66 s.) Also, there is no evidence of reflection from the PML boundary which would be evident in the comparison of the 2 km rough bottom waveguide to the 4 km waveguide. D. Propagation Reverberation is much more sensitive to roughness than propagation. To demonstrate the effect of roughness on propagation loss, four waveguides are considered. Each is based on problem 1, but the root-mean-square (rms) roughness is increased. The first is based on the original roughness which had an rms height of 2% of the acoustic wavelength, k. Simply by scaling the bottom roughness, realizations with 5%, 10%, and 19% of the acoustic wavelength were produced. Fifty realizations of the roughness are considered and the average propagation loss is compared to that of a flat interface waveguide. The results are shown in Fig. 8. In the figure, the TL from each realization is shown in blue, while the average is shown in red. The TL from a flat interface waveguide is shown in green. As the rms roughness increases, the average TL increases relative to the flat bottom case. At a rms roughness of 0.05k there is about a 1 db difference per kilometer while for 0.19k rms height, the difference rises to about 4 db of difference per kilometer. FIG. 6. (Color online) Reverberation results from a waveguide with flat interfaces compared with that of a rough bottom interface. The flat interface fathometer returns are insignificant past 0.4 s. V. CONCLUSION Acoustic reverberation and propagation were calculated with FEs for six different shallow water waveguides. The six waveguides were (1) flat interfaces with a constant sound J. Acoust. Soc. Am., Vol. 129, No. 3, March 2011 M. J. Isakson and N. P. Chotiros: Finite Element Reverberation Modeling 1277

FIG. 8. TL as a function of range for different values of rms roughness on the bottom interface. The blue lines are individual realizations of roughness, the red line is the rough interface mean TL, and the green line is the flat interface TL. (a) h rms ¼ 0.02k; (b) h rms ¼ 0.05k; (c) h rms ¼ 0.10k; and (d) h rms ¼ 0.19k. speed, (2) rough bottom interface with a flat top interface and a constant sound speed, (3) rough top interface with a flat bottom interface and a constant sound speed, (4) rough top and bottom interfaces and a constant sound speed, (5) rough top and bottom interfaces with a summer (downward refracting) sound speed profile, and (6) rough top and bottom interface with a winter (upward refracting) sound speed profile. These results were compared with currently available models including coupled modes, parabolic equations, and ray theory for problem 1. The FE model generally predicted equal or higher reverberation levels than the other models. At very short times, this is due to the fathometer returns which are implicitly included in the FE full field calculation but neglected in the other models. At later times the enhanced reverberation is likely due to multiple high angle scattering between the boundaries. Other models which do not include normal and near-normal incidence energy on the boundaries neglect this effect. The reverberation from the different environments was fairly consistent except for problem 2 in which much of the energy was scattered from the air/water interface into the bottom. The addition of roughness at the air/water interface caused the reverberation to decrease relative to a rough bottom alone due to additional high angle energy transmitted into the sediment. The scintillation index of the reverberation indicated that the statistics were Rayleigh-like with the exception of the rough bottom only case which was not fully saturated. There were several excursions above a value of one for the scintillation index especially for the iso-velocity rough boundaries waveguide and the downward refracting sound speed profiles indicating that reverberation from the bottom is likely the cause of the non-rayleigh statistics. ACKNOWLEDGMENT The authors thank the Applied Research Laboratories at The University of Texas for the Independent Research and Development funds that initiated this work. We also thank the Office of Naval Research, Ocean Acoustics and Robert Headrick for continued funding. 1 J. Yang, D. Tang, and E. Thorsos, Reverberation due to bottom roughness using first-order perturbation theory, in Proceedings of the International Symposium on Underwater Reverberation and Clutter (2008), pp. 81 88. 1278 J. Acoust. Soc. Am., Vol. 129, No. 3, March 2011 M. J. Isakson and N. P. Chotiros: Finite Element Reverberation Modeling

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