Sonar systems for imaging continental margins rely on non-specular scattering from muds and sands, which can be modeled as fluid saturated, porous solids with low shear moduli. To study the physical mechanisms responsible for forward and back scattering in these media it is useful to have a forward modeling technique which applies to rough and heterogeneous porous bottoms. A numerical scattering chamber using the time-domain finite difference method applied to the range dependent Biot equations can be used to study the scattering of low-grazing angle pulse beams from fluid-saturated porous media. For example, for a flat sea floor over a homogeneous porous half-space, converted shear waves and converted compressional "slow" waves are shown to be excited in the sub-bottom even when the grazing angle is below the critical angle for compressional "fast" waves. Any scattering element near the seafloor will act as a secondary point source by Huygen's principle and when excited by an acoustic wave will have the potential to generate a family of interface and body waves of "fast" compressional, "slow" compressional and shear type. Funded by The Office of Naval Research.
Importance of Porous Media
- sands and muds in marine sedimentary environments (detecting mines on and in the seafloor by acoustic backscatter)
- petroleum reservoirs (distinguishing gas from oil in seismic records, estimating permeability from geoacoustic measurements)
- free gas under gas hydrate deposits
- porous and fractured igneous rocks in layer 2 and upper layer 3 of the oceanic crust
- hydrothermal vents
- rock-magma interaction under ridges or volcanoes
Limitations of Ray Theory
- Ray theory is a common and intuitive model for describing the propagation of high frequency sound in homogeneous and smoothly varying media
- It fails to adequately predict the behavior of waves when the medium parameters vary over length scales comparable to seismic wavelengths (for example: diffractions over seamounts, diffractions around axial magma chambers, back-scattering strengths in swath bathymetry and side-scan sonar imagery, wavefront healing, etc)
- So it is important to have a full wave method to study acoustic and elastic wave behavior in range dependent porous media.
The wave equation for a heterogeneous, non-uniform porosity, poro-elastic medium
(1)Note that just about all other wave equations in Biot's porous media papers, including all of the familiar equations, assume uniform porosity and homogeneous material. In our finite difference approach we rely on gradients of elastic parameters (and porosity) to compute the effects of interfaces. We do not introduce boundary conditions explicitly. So it is important to have the correct equations for heterogeneous, non-uniform porosity material.
Model Parameters
| Water | |
| Compressional Velocity | 1500m/s |
| Density | 1000 kg/m3 |
| Low Shear Velocity Medium | |
| Compressional Velocity | 3700m/s |
| Shear Velocity | 140m/s |
| Density | 2650 kg/m3 |
| High Shear Velocity Medium | |
| Compressional Velocity | 3000m/s |
| Shear Velocity | 1730m/s |
| Density | 1700 kg/m3 |
Poro-elastic medium - Sand
Porosity |
f |
0.47 |
|
Viscosity |
h |
10-6 kg/(m-s) |
|
Permeability |
k |
10-10 m2 |
|
Fluid Density |
r f |
1000kg/m3 |
|
Solid Grain Density |
r s |
2650kg/m3 |
|
Grain Bulk Modulus |
Ks |
3.6x1010 Pa |
|
Fluid Bulk Modulus |
Kf |
2.25x109 Pa |
|
Dry Skeleton Bulk Modulus |
KB |
4.36x107 Pa |
|
Dry Skeleton Shear Modulus |
N |
2.61x107 Pa |
|
Figure 1: This snapshot shows a Gaussian pulse-beam at 15° grazing angle in a homogeneous medium with water properties. It was taken 50 Periods after the initiation of the beam. Only a single "ray" is represented by this beam.
Figure 2: The same beam as in Figure 1 insonifies the interface between water and a "high shear velocity" medium such as basalt. In this case the shear velocity of the bottom is faster than the compressional velocity of the water. The 15° grazing angle is sub-critical for both compressional and shear waves in the bottom and there is total internal reflection. Compressional and shear wave components of the evanescent wave below the interface (the direct wave root) can be observed.
Figure 3: This figure is the same as Figure 2 but considers the interface between water and a "low shear velocity" medium such as mud or sand. In this case the shear velocity of the bottom is slower than the compressional velocity of the water. The 15° grazing angle is sub-critical for compressional waves but it is super criticial for shear waves. The compressional wave is still evanescent in the bottom but the shear wave is a propagating shear body wave.
Figure 4: This figure is the same as Figures 2 and 3 but considers the interface between water and a "porous" medium such as a sandstone. The shear velocity of the bottom is slower than the compressional velocity of the water. In addition to the shear body wave in the bottom there is now a compressional "slow wave" which is also a body wave.
Figure 5: If a heterogeneity is introduced at the interface, in this case a facet that is one wavelength high, scattering into body and interface waves occurs. Even though the "ray" is incident on the facet at normal incidence there is no "reflected" or "transmitted" wave at the facet. Instead the facet acts as a secondary point source (Huygen's Principle). Since the secondary point source is near the interface, both forward and backward propagating Stoneley waves are excited. There is also the complete family of compressional and shear body waves, including head waves, that would be expected for a point source on the interface.
Figure 6: When the facet heterogeneity is placed on the interface with a "low shear velocity" medium much less energy goes into the Stoneley waves. A diffracted shear body wave can be observed, but most of the energy is in the diffracted compressional wave.
Figure 7: When the facet heterogeneity is placed on the interface with a porous bottom a "diffracted "slow wave" is generated. This is the first demonstration of full wave scattering at an interface between a fluid and a porous medium. Note that the diffracted "fast" compressional wave is much stronger for a porous medium than for the high and low shear velocity media (Figures 5 and 6).Advantages of Time Domain Finite Differences (TDFD)
All rigidity effects (in addition to the compressibility effects) including interface waves at free and fluid-solid boundaries are included.
All mode conversions between compressional (fast and slow), shear and interface waves are considered.
Interface roughness and sub-bottom heterogeneities, with length scales on the order of acoustic wavelengths, can be studied with the same methodology.
Models can consist of arbitrary combinations of acoustic, elastic, anelastic and poro-elastic media.
Wavefront snapshots, in addition to time series at arbitrary locations within the model, provide useful insights into the propagation and scattering of sound in complex media.
TDFD methods can be applied to Gaussian pulse-beams, a wave approximation to rays.
All multiple interactions between scatterers are considered.
Problems are scaled to wavelengths and periods so that results are applicable to a wide range of frequencies.
"Take-Away" Messages
We have a methodology for quantitatively studying wave propagation and scattering in complex, range dependent media including combinations of acoustic, elastic, anelastic and poro-elastic materials.
Facets (cliffs) that are a wavelength high act more like point scatterers than planar interfaces.
Not everything that is perpendicular to a wave front is a ray.
In porous bottoms there is a strong fast compressional wave generated by scattering at a seafloor heterogeneity. This is an important mechanism for getting energy into the bottom at non-specular angles.
In porous bottoms there are weak shear and weak slow-compressional waves in the sediments.
In soft and porous bottoms interface (Stoneley) waves are not a significant product of seafloor scattering. This contrasts with hard bottoms where interface waves are a major propagation mechanism for ambient noise.
References:
Biot, M.A. (1962). "Mechanics of deformation and acoustic propagation in porous media," J. Appl. Phys. 33, 1482-1498.
Chotiros, N.P. (1995). "Biot model of sound propagation in water-saturated sand," J. Acoust. Soc. Am. 97, 199-214.
Stephen, R.A. (1997). "Time domain finite difference methods for range dependent Biot media," in High Frequency Acoustics in Shallow Water, edited by N.G. Pace, E. Pouliquen, O. Bergam, and A.P. Lyons, (SACLANTCEN Conference Proceedings Series, La Spezia, Italy), CP-45, pp. 501-508.
Stoll, R.D., and Kan, T.K. (1981). "Reflection of acoustic waves at a water-sediment interface," J. Acoust. Soc. Am. 70, 149-156.
