226 IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 22, NO. 2, APRIL 1997Benchmarks for Validating Range-DependentSeismo-Acoustic Propagation CodesJoo Thiam Goh, Member, IEEE, Henrik Schmidt, Peter Gerstoft, and Woojae SeongAbstract—The availability of fast and relatively low-cost com-puting power has resulted in radical changes to the role ofseismo-acoustic modeling. With the increase in the number ofmodels available, there is the inevitable question of how canone go about validating all these numerical schemes. Recently,the issue of establishing reference solutions for range-dependentocean acoustic problems was addressed within the AcousticalSociety of America. This has resulted in a set of well-definedbenchmarks for range-dependent fluid problems. However, todate, there is no consistent set of benchmarks for the range-dependent seismo-acoustic codes. In this paper, we present acollection of problems intended for general use by the modelingcommunity for validation of new computational schemes. Anumber of new seismo-acoustic codes are applied to producereference solutions for these benchmarks.Index Terms—Benchmarking, elastic, modeling.I. INTRODUCTIONIN OCEAN acoustics, the recent shift in emphasis fromdeep to shallow water and littoral environments has ledto a significant effort in developing environmental acousticsmodels incorporating improved treatment of the dominantphenomenon in such environments—the bottom interaction.The shallow-water environment is an extremely complicatedwaveguide bounded above by a rough sea surface and belowby an inhomogeneous, multilayered elastic sea bed. Further,the acoustic properties of the water column are affected by theclose proximity to the atmosphere, giving rise to a significantspatial and temporal variability. The elastic sea bed addsanother degree of complication and it is only recently thatmodelers have been able to account for its effect, to somedegree. The existence of seismic interface waves, inhomo-geneous waves, and headwaves, and interference of multiplyreflected waves are all important phenomena, and the energycarried by seismic waves is not negligible compared to thewaterborne field.The most general approaches to modeling seismo-acousticbottom interaction are the direct numerical solutions to theManuscript received June 12, 1996; revised December 17, 1996. This workwas supported in part by the Office of Naval Research, in part by the HighLatitude Dynamics and the Ocean Acoustics programs.J. T. Goh is with the Defence Science Organization, Singapore 118230,Republic of Singapore.H. Schmidt is with the Department of Ocean Engineering, MassachusettsInstitute of Technology, Cambridge, MA 02142 USA.P. Gerstoft is with SACLANT Undersea Research Centre, 19138 La Spezia,Italy.W. Seong is with the Department of Ocean Engineering, Seoul NationalUniversity, Seoul 151-742, Korea.Publisher Item Identifier S 0364-9059(97)03400-6.elastic and fluid wave equations. Thus, fluid-elastic interac-tion problems have been handled using both finite-differencemethods (FDM’s) [1], and finite-element methods (FEM’s) [2].However, since these methods rely on spatial and temporaldiscretizations which are small compared to the wavelengths,they are normally restricted to modeling short-range propaga-tion and scattering. Even with today’s computers, the use ofFDM and FEM to problems involving ranges of hundreds orthousands of wavelengths, characteristic of ocean acoustics, isprohibitive.The parabolic equation (PE) algorithm today is withoutdoubt the most popular and versatile approach to modelingrange-dependent ocean waveguides. However, in trying toextend the PE theory to elastic medium, two main problemsarise. Firstly, the field is described by a vector (displacement)rather than a scalar. Secondly, two different wave speeds existin a solid and in a heterogeneous media or at boundaries,there is continuous conversion from one wave type to another.Furthermore, elastic bottoms support a wide spectrum ofpropagation angles. Therefore, even though several PE modelshave been proposed for wave propagation in elastic media[3]–[9], only a few of these models were actually implemented.Notable implementations include those of Wetton and Brooke[7] and Collins [8], [9]. Thus, for the most part, the parabolictheories for elastic waves have not been adequately testednumerically, particularly in two-way formulations. In additionto being limited to weak range dependence, a major drawbackof the PE as well as the discrete methods is the fact that thesolutions are not as easily interpreted physically. Thus, themodal structure of the field can only be determined throughpostprocessing [10].For range-independent seismo-acoustic propagation model-ing, SAFARI [11] is in widespread use for providing exactreference solutions. Since SAFARI is based on integral trans-forms of the wave equation, it is not directly applicable torange-dependent problems. To overcome this inherent lim-itation of spectral approaches, Lu and Felsen [12] derivedan adiabatic transformation of the wavenumber integrals forweakly range-dependent problems. However, their methodworks well only for cases where the wave field is largelydominated by discrete modes [13]. Its extension to the elasticcase is also nontrivial—if at all possible.Recently, two new modeling approaches were developedfor solving the elastic wave equation in range-dependentenvironments. Both divide the environment into horizontallystratified sectors, coupled along vertical interfaces. Anothercommon feature is the use of wavenumber integration for0364–9059/97$10.00 © 1997 IEEEAuthorized licensed use limited to: IEEE Xplore. Downloaded on November 22, 2008 at 23:08 from IEEE Xplore. Restrictions apply.GOH et al.: VALIDATING RANGE-DEPENDENT SEISMO-ACOUSTIC PROPAGATION CODES 227generating the Green’s functions for the sectors. The maindifference is the handling of the coupling of seismic energyat the vertical interfaces between these range-independent sec-tors. One, in principle, exact method—the so-called spectralsuper-element approach—solves the coupled integral equa-tion using a high-order panel-boundary-element formulation[14], [15]. The other approximate approach solves the re-flection/transmission problem locally at a discrete numberof depths, yielding a distribution of virtual panel sources[16]. Both methods use standard wavenumber integration tocompute the forward and backward scattered field within thesectors. The testing