# Optimal Transmission of Interface Vibration Wavelets—A Simulation of Seabed Seismic Responses

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## Abstract

**:**

## 1. Introduction

## 2. The Properties of Infinite Half-Space Models

## 3. A New Half-Space Physical Model to Simulate Saturated Sediments

_{r}, of the material shear speed thereafter. This contrasts with analyses of power-law profiles, including that of Chapman and Godin [22], where the shear waves, absent in the fluid, start with a wave speed of zero which increases with depth as a power law, favouring a special case of a half-power. When used with a constant density, this gives a linear increase in shear modulus, but still has a discontinuity of the gradient at the interface. In contrast, the model described here has a step change in shear speed at the seabed. We then find characteristic wavelets which propagate well, primarily influenced by sediment properties close to the seabed.

_{s}, plus two independent mechanical properties, for example, the shear wave speed V

_{s}and compressional wave speed V

_{p}. Other options include the Young’s modulus E and Poisson ratio ν. The fluid half-space only needs its density ρ

_{w}and the speed of water pressure waves, V

_{w}, defined. The speed of the interface waves for the Rayleigh model is somewhat less than V

_{s}, which, in turn, is less than V

_{p}. For the special case when ν = 0.25 (a “Poisson solid”), the interface wave travels at ≈0.92 V

_{s}. The same value has been chosen for this model, but only as a limit case at infinite depth. It is also reported for rocks such as granite, and was used by Zhu et al. [20] for concrete.

_{r}of V

_{s}with depth, d. V

_{p}also increases linearly, determined by the choice of ν = 0.25 at infinite depth. V

_{s}and V

_{p}now refer to the interface values which also control the values at depth d in the sediment:

## 4. The FE Model Simulates the Propagation of a Transient Ripple across the Interface

_{s}, at the interface was 128 m/s. A small shear speed gradient g

_{r}of 1 (m/s)/m, or 1 s

^{−1}, was set for this model run. The compressive speed at the interface, V

_{p}, was 1520 m/s. The solid had a uniform density of 2250 kg/m

^{3}. The water had a density of 1000 kg/m

^{3}and wave speed of 1500 m/s.

## 5. Communications Would Be Distorted by Strong Filtration Imposed by the Sediment

## 6. The Structure of the Wavelet as a Combination of Hermite Polynomials

_{k}), also described as a “Mexican Hat” [25], is used here to describe amplitude versus time t. Constants K, peak time t

_{0}, and zero-to-zero width τ, can be adjusted to fit data from the FE analysis.

_{s}= 128 m/s, τ = 40 ms, and g

_{r}= 4 s

^{−1}. It used node 2162 at 222 m radius, chosen to provide a symmetric hump form. Additional plots show the Ricker form (red) and an improved mathematical form (dotted orange), representing a Morphing Mexican Hat shape (MMH), the function shown in Equation (4), which is a combination of two terms from the Hermite polynomial sequence.

_{k}), the FE data shows additional zero crossings which are provided by the fourth-order Hermite polynomial, the second derivative of the Ricker function, R

_{k}′′. A combination of terms provided a better fit with extra zero crossings. Other combinations of terms may prove useful in the analysis of more complex seismic wavelets where more zero crossings are created as the energy is dispersed both spatially and in time.

## 7. The Morphing Nature of the Wavelet Structure

_{r}≈ 4 s

^{−1}) from a “hump” to a “dip” and back, with the symmetric hump being periodically inverted via antisymmetric forms. This dynamic wavelet structure is described here as a “Morphing Mexican Hat”, or MMH function (Figure 3 and Equation (4)), which is a combination of two derivative functions R

_{k}and R

_{k}″ of the scaled and offset time T (Equation (1)).

^{2}, yielding T

^{2}= 0.725 or 8.275, and T = ±0.8515 (inner zeroes) or T = ±2.877 (outer zeroes). Once this fit to the zeroes was found, the intervening shape fitted the data well, to provide a better understanding and prediction of this special interface wave and its morphing process.

## 8. The FE Forcing Pulse Drivers

_{0}± 4τ when the residual amplitude is less than K/10

^{12}.

## 9. Modeling Comparisons with FE (PAFEC) and Wavenumber Integration (OASES)

^{3}. The compressional wave speed is 1520 m/s and shear wave speed 128 m/s in the sediment.

## 10. The Retention of Wavelet Energy Is Key to the Optimal Transmission

_{s}= 128 m/s at the interface and g

_{r}= 4 s

^{−1}.

## 11. Stability of the Wavelet Shape

## 12. The Shear Speed Gradient Controls the Morphing Frequency

_{r}= 4 s

^{−1}(or (m/s)/m) the morphing cycle takes 0.67 s, a frequency of 1.488 Hz, with humps occurring every ~83 m. When the gradient is reduced to 2 s

^{−1}, the morphing frequency is reduced to 0.748 Hz. For the second run, the overall range was doubled, using larger elements and a slower transient step rate. However, the vertical mesh density was retained, thus checking that the results were not sensitive to the element proportions. Over this range, the morphing frequency, f

_{m}, is then given by f

_{m}≈ 0.37 g

_{r}.

_{r}, has units of s

^{−1}. Checks made using other gradients show in Table 1 that this trend is only approximately linear.

## 13. Wavelets in a Hamilton Profile FE Model Are Also Seen to Morph Regularly

## 14. The Variation of Group Velocity with Wavelet Width

## 15. Reduction of the Wavelet Peak with Time and Distance

## 16. Comparison with Other Seismic Surface Waves

## 17. Matching the FE Outward Displacement Data to the Ricker Mathematics

## 18. Presenting the Data as a Hodograph—The Orthogonal Motion

## 19. Using Hodographs to Observe the Morphing Process

## 20. The Morphing Mathematics

_{1}, y

_{1}to x

_{2}, y

_{2}is given as a function of the Euler angle, θ, (Wertz [40]).

## 21. Wavelet Morphing in a Moving Frame Provides Sinusoidal Vibration Functions

## 22. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**A view of a graded sedimentary solid from above. This shows a vibration wavelet travelling outward from the central point of excitation. Note the small advanced ripples and a lack of any disturbance behind the peak of the largest, slowest mode.

**Figure 2.**This display shows the region around node 2143, an element junction at 217 m radius from the model centre to the left. Results for the solid displacements are vector magnitudes, colour coded up to 80 μm, using a similar colour display used in Figure 1, where blue shows a low value. The deformation is exaggerated for clarity. The water pressures are bi-polar to ±200 Pa with zero deviation from normal shown as green.

**Figure 3.**Finite element (FE) time series data, showing vertical (upward) displacement versus two mathematical fit functions.

**Figure 7.**The wavelet changes shape with time and distance. Here, data from two nodes at different distances (1680 at 50.75 m, 9002 at 966.75 m) are shown offset in time and scaled by a cylindrical spreading factor of √ (966.75/50.75) in amplitude.

**Figure 8.**Times and positions for successive humps–compare gradients of g

_{r}= 4 (m/s)/m (

**plot a**) and 2 (m/s)/m (

**plot b**).

**Figure 12.**The horizontal outward displacement data associated with the Figure 3 vertical data.

**Figure 14.**The vertical and horizontal motions of different nodes as the wavelet passes. These nodes are at ranges 243 m, 264 m, 285 and 306 m in plots (

**a**) to (

**d**).

**Figure 15.**Four successive hodographs show the changes during the morphing process at ranges 243 m, 284 m, 285 and 306 m in plots (

**a**) to (

**d**).

Shear Speed Gradient (s^{−1}) | 1 | 2 | 4 | 8 | 16 | 32 |
---|---|---|---|---|---|---|

Morphing frequency (Hz) | 0.335 | 0.748 | 1.488 | 2.912 | 5.682 | 10.91 |

Pulse Width Tau ms | Transit Speed ms^{−1} | Z/z Width #1 ms | Z/z Width #2 ms | |
---|---|---|---|---|

1 | 16 | 117.89 | 15.4 | 16.6 |

2 | 20 | 117.74 | 18.8 | 20.4 |

3 | 25 | 117.5 | 23 | 24 |

4 | 32 | 117.38 | 29 | 29 |

5 | 40 | 117.39 | 36 | 36 |

6 | 50 | 117.60 | 45 | 45 |

7 | 64 | 117.92 | 56.3 | 58 |

8 | 80 | 118.79 | 70.4 | 74 |

9 | 100 | 119.47 | 86 | 88 |

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**MDPI and ACS Style**

Hazelwood, R.A.; Macey, P.C.; Robinson, S.P.; Wang, L.S.
Optimal Transmission of Interface Vibration Wavelets—A Simulation of Seabed Seismic Responses. *J. Mar. Sci. Eng.* **2018**, *6*, 61.
https://doi.org/10.3390/jmse6020061

**AMA Style**

Hazelwood RA, Macey PC, Robinson SP, Wang LS.
Optimal Transmission of Interface Vibration Wavelets—A Simulation of Seabed Seismic Responses. *Journal of Marine Science and Engineering*. 2018; 6(2):61.
https://doi.org/10.3390/jmse6020061

**Chicago/Turabian Style**

Hazelwood, Richard A., Patrick C. Macey, Stephen P. Robinson, and Lian S. Wang.
2018. "Optimal Transmission of Interface Vibration Wavelets—A Simulation of Seabed Seismic Responses" *Journal of Marine Science and Engineering* 6, no. 2: 61.
https://doi.org/10.3390/jmse6020061