## 1 Introduction
### Sound Speed Models of the Sub-Bottom Layers  
Gradients in the geoacoustic properties of sediments are commonly observed in coring data and in bottom sound-speed inversions [25,37]. They are important to the physics of the scattering process because they control both how the sub-bottom is illuminated [38] and how the scattered field propagates back to the receivers [39]. These gradients in the sub-bottom are usually not smooth and have rather rapid, fluttered fluctuations. However, in seismic work, where the wavelengths of the radiated signals are considerably larger than these fluctuations, the overall representation by analytic functions is of practical value [40].  
The variation of the sound speed with sub-bottom depth can be described using various models of differing complexity. For general acoustic problems and common geoacoustic parameters of the seafloor, one can use a homogeneous liquid half-space representation of the bottom. The implementation of a liquid homogeneous half-space bottom model showed its particular applicability for gas-rich lake-floor sediments [3,7]. In this case, the seafloor sound speed was particularly low (\(\sim\)350 m s\({}^{-1}\)), and the reflection coefficient was found to be exceptionally high (\(|V|~{}\sim~{}0.8\)), due to the significant concentration of gas bubbles. The liquid half-space bottom model has been successfully applied to sound-speed estimations for thin (\(<\)1 m thick), gas-saturated layers.  
The simplest representation of a multilayered bottom is a piecewise-constant model with a discrete number of horizontal layers, each having a constant sound speed [41]. More complex models, which are associated with the compaction of sediments deposited along ocean margins, assume that the sound speed varies continuously with depth. The classical instantaneous linear velocity model \(c_{b}(z)=c_{0}+Kz\) and the classical exponential model \(c_{b}(z)=c_{0}\exp\left(\frac{K_{0}z}{c_{0}}\right)\)[42] are generally accepted and often confirmed by measurements of thin elastic sediment sections [43]. Here \(c_{0}\) is the topmost seafloor sound speed, \(K\) is a constant vertical sound speed gradient in the linear model, and \(K_{a}\) is the gradient at the top interface in the exponential model. However, these models increase towards infinity with the increasing thickness of the sub-bottom layer. For large thicknesses, these models no longer represent any physical compaction of the sub-bottom layer [44]. Moreover, in the exponential model, the gradient increases with depth, which contradicts the expected behavior [44,45]. There are other models that represent the bottom velocity structure (e.g., the parabolic model [40] or the linear decrease of slowness [46]), each having its advantages and limitations. [PERSON] and [PERSON] [44] suggested an asymptotically bounded, exponential-velocity model. Their model better represents the vertical sound-speed variations in compacted sediments, especially in the case of thick layers. For such sediments (e.g., as found in Gulf of Mexico) the gradient is higher in the upper layer and gradually decreases with depth. In practice, seismic velocity is affected by sediment composition, pore shape, pore pressure, pore-fluid saturation and confining pressure, all of which are partly associated with the compaction, and by temperature. These factors may cause local anomalies with respect to the general trend of increased velocity with depth [44].