Computer Analysis for Acoustic Sensing of Multilayer Sediment
Authors: Magnuson Allen H. , Stewart Gary K. , Newman A.K. 
Published: October 2008

ABSTRACT
An analytical model is proposed for application to the remote acoustic sensing of sediments on the continental shelf. A Green's function is used as a model for the acoustic response-in the liquid layer overlying the sediment. The sediment stratification is represented as a series of discrete layers, each layer being a homogeneous viscoelastic solid. A recurrence relation is developed to eliminate potentials in the intermediate solid layers. After applying boundary conditions at the liquid-solid interface, the acoustic response is expressed as an improper integral. The recurrence relation reduces the multilayer problem to a form ideal for solution on the digital computer. The computer analysis of the problem is discussed and compared to previously obtained analytical solutions for the single-layer case.

INTRODUCTION
The problem of remote acoustic classification and identification of sediments on the continental shelf has become increasingly important due to increased interest in the coastal zone. Presently, most acoustic sounding for sediments is done using seismic profiling where the intensity of the return is indicated as a function of pulse return time. This technique is valuable in obtaining a qualitative understanding of the sub bottom; however, more detailed quantitative information is usually required for underwater construction and commercial dredging for sand and gravel. In addition, it is difficult to interpret multiple return signals which imply sub layering in the sediment.
Recent sounding and coring data taken as part of the joint UNH-Raytheon project in Narragansett Bay in Massachusetts Bay shows that commercially important sand and gravel deposits lie on a first layer 10 to 15 feet deep, below which lies finer clay sediment. Therefore, knowledge of the thickness of the first layer is important from a commercial standpoint.
This paper takes into account the multiple layering of the sub bottom using a realistic three-dimensional model for the coupled acoustic-viscoelastic field shown in Figure 1. Earlier investigators, notably Thomsonand Haskellconsidered only the two-dimensional case for dissipationless solids. Hamilton in Ref.concludes that unconsolidated underwater sediments can be modeled as elastic solids. Recently, Hamiltonhas presented data indicating that the sediments behave as a lightly damped elastic solid.
The analysis of this problem is described in detail by Magnuson in Ref. (5). The three-layer problem is discussed by Stewart in Ref. (6).

ANALYTICAL APPROACH
We model the acoustic response as a Green's function. The Green's function G(w) for the dissipationless liquid satisfies the following equation in the frequency domain:(Mathematical equation available in full paper) where k =w/c is the wave number in the fluid, r is the field point, r is the location of the source andthe Dirac delta function. The right-hand side of eq. (1) is a monopole point source excitation representing an acoustic transducer, and H(W) is the Fourier transform of the time dependence of the source. The displacement field and the pressure field p are related to the Green's function as follows:(Mathematical equation available in full paper)