Extension of the Variable Rotated Parabolic Equation to Problems Involving

Variable Topography

Jon M. Collis and William L. SiegmannRensselaer Polytechnic Institute

Troy, New York

Michael D. CollinsNaval Research Laboratory

Washington, D.C.

150th Meeting Acoustical Society of AmericaMinneapolis, Minnesota

17-21 October 2005

[Work supported by the Office of Naval Research]

Motivation

• Elastic sediment bottoms important for shallow water and shore propagation

• Parabolic equation method handles sloping interfaces and boundaries effectively

Mapping and Variable Rotated solutions handle variable bathymetry

Mapping solution handles some problems with variable topography

Variable Rotated solution should provide greater accuracy, especially with large slopes

• Techniques have applications in seismology, volcanology, military scenarios and meteorology

The Parabolic Equation Method

• Current implementations are efficient and accurate

• Important issues: Range-independent vs. range-dependent environments Fluid vs. elastic media

• Recent progress with elastic models: Jerzak (2001) -- existing implementation (RAMS)

improved to better handle vertical interfaces Outing (2004) -- new methods, coordinate rotation

(ROTVARS) and mapping (RAMSMAP), for variable slopes in bathymetry

Kusel (2005) -- single scattering method (SLICE) to better handle vertical elastic/elastic interfaces

The Mapping Solution• The mapping approach for treating range dependence proposed by

Collins and Dalco (JASA 107, 2000) Improved to handle multiple sediment layers by Outing (RPI

Ph.D. Thesis, 2004)

• Bathymetry mapped to become range independent Fluid-elastic boundary conditions can be satisfied accurately Ocean surface sloped Surface boundary condition easier to handle accurately than

four boundary conditions at ocean-sediment interface

• Small slopes permit dropping extra terms that appear in PE Leading order correction applied at bathymetry changes For larger slopes, solution breaks down Fields can become inaccurate beyond points of slope change

The Variable Rotated Solution

• The rotated parabolic equation proposed by Collins (JASA 87, 1990)

First PE capable of handling sloping bottoms to high (benchmark) accuracy

Capabilities extended to handle variable bathymetry, some elastic-elastic interfaces by Outing (2004)

• Rotate coordinates to align axes with constant-slope bathymetry

Solve PE in locally range-independent region Ocean surface becomes sloped, but BC easier to treat

• Connections between regions are treated with high accuracy Preserves field features, such as diffractive effects

=> Expected improved accuracy over the mapping approach!

Variable Rotation for Variable Bathymetry

• Solve in constant slope region

• March past intersection

• Interpolate/ extrapolate in crossover regions

Variable Bathymetry and Elastic Media

• In fluid layer, rotate coordinate system by applying rotation matrix

• For elastic media solve

where u, w are horizontal, vertical displacements

• Use invariance of divergence under rotation• Transform dependent variables

• Rotate, then apply

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uxw

⎛

⎝ ⎜

⎞

⎠ ⎟x+Δx

= Luxw

⎛

⎝ ⎜

⎞

⎠ ⎟x

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Δw

⎛

⎝ ⎜

⎞

⎠ ⎟= D

uxw

⎛

⎝ ⎜

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⎠ ⎟,

€

D =1

1

∂z0 1

⎛

⎝

⎜ ⎜

⎞

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€

D−1

Variable Rotation vs. Mapping• Accuracy of Variable Rotated solution should be higher than

Mapping solution No correction necessary for variable rotated solution Difficult to compare: few non-numeric benchmarks exist

RotatedMapped

Variable Rotated PE and Variable Topography

• Ocean acoustic problems with variable topography: Earthquake localization Volcanic event localization Military shore movements, blast monitoring Atmospheric noise abatement

• Boundary conditions at the air-sediment interface: Fluid sediment model => normal stress = 0 Elastic sediment model => normal, tangential stresses = 0

• Regions with variable topography interpolated/extrapolated as with bathymetry

• Technique preserves interface waves between regions with bathymetry --> topography

Scholte waves --> Rayleigh waves

Beach• Sediment layer 150 m thick, = 295 m, f = 5 Hz

• Interface wave (Scholte) transitions into surface wave (Rayleigh)

€

zs

=(2400,1200) m/s

=(3400,1700) m/s€

← (c p ,cs)

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← (c p ,cs)

Sandbar• Sediment layer 125 m thick, = 190 m, f = 25 Hz

• Transmission through sandbar near 7 km

€

zs

=(1700,800) m/s

=(2800,1400) m/s€

← (c p ,cs)

€

← (c p ,cs)

Sandbar and Beach• Sediment layer 150 m thick, = 190 m, f = 10 Hz

• Energy passes through sandbar at 8 km and into beach

€

zs

=(1700,800) m/s

=(3400,1700) m/s

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← (c p ,cs)

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← (c p ,cs)

Island• Sediment layer 150 m thick, = 195 m, f = 5 Hz

• Energy enters topography and passes through island (~ 10 km width)

€

zs

=(2400,1200) m/s

=(3400,1700) m/s

€

← (c p ,cs)

€

← (c p ,cs)

Red Beach, Camp Pendleton• Sediment layers 11 m, 63 m thick, = 30 m, f = 150 Hz

• Modal cutoff through sediment layers

€

zs

=(1650,660) m/s

=(1705,680) m/s

=(1800,720) m/s€

← (c p ,cs)

€

← (c p ,cs)

€

← (c p ,cs)

Summary & Future Direction

• Layered elastic media are relevant to sound propagation in shallow water ocean acoustics

• Variable rotated solution is highly accurate and efficient

• Method treats transitions between regions of constant slope more accurately than mapping solution

• Method handles larger slopes than mapping solution

• Topographical capabilities allow solution of a wide variety of problems not previously accessible

• Sediment layers currently constrained to follow bathymetry More capabilities if constraint removed

Thank You

• W. Jerzak and D. Outing -- for their work on earlier solution techniques upon which we based our work

• Office of Naval Research -- without their support, this research would not have been possible