Acoustic tomography by Hamiltonian methods including the adiabatic approximation

Wunsch, Carl

doi:10.1029/rg025i001p00041

Cited by 19 publications

(6 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This analogy can also be used for the second-order traveltime. As noted by Wunsch (1987) and Farra and Madariaga (1987), one can consider the traveltime in ray tracing problems as the Lagrangian of an equivalent problem in classical mechanics. (The basic connection is of course the variational principle of the action respectively the traveltime.)…”

Section: R Snieder and M Sambridgementioning

confidence: 99%

“…The equations of dynamic ray tracing can be cast in a Hamiltonian form. Perturbation methods for Hamiltonian systems are well developed, and a number of authors have used this to develop a perturbation theory for ray tracing problems (Chapman 1985;Wunsch 1987;Farra & Madariaga 1987;Farra, Virieux & Madariaga 1989;Virieux 1991). As an alternative to Hamiltonian perturbation theory, one can also apply perturbation theory directly to the equation of kinematic ray tracing (Moore 1991).…”

Section: Introductionmentioning

confidence: 99%

“…Another ambiguity in setting up a perturbation treatment is the choice of the independent parameter and the employed coordinate system. Wunsch (1987) uses the horizontal distance of a ray as independent parameter and solves for the depth of a ray. Farra & Madariaga (1987) use the distance along the unperturbed ray as independent parameter and employ ray coordinates.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Ray perturbation theory for traveltimes and ray paths in 3-D heterogeneous media

Snieder

Sambridge

1992

Geophysical Journal International

View full text Add to dashboard Cite

S U M M A R YBoth paraxial ray tracing and two-point ray tracing are powerful tools for solving wave propagation problems. When a slowness model is mildly perturbed from a reference model, one can use perturbation theory for the determination of the ray positions and the traveltimes. An extension of Fermat's theorem is presented, which states that the traveltime is stationary with respect to the perturbations in the ray position provided that the endpoints of the ray are perturbed along the wavefront of the unperturbed ray. It is shown that when the ray perturbation satisfies this condition the second-order traveltime perturbation can be computed from the first-order ray perturbation. A perturbation analysis of the equation of kinematic ray tracing leads to a simple second-order differential equation for the ray deflection expressed in ray coordinates. This constitutes a perturbation method based on a Lagrangian formulation, and leads to a first-order expression for the ray deflection and a second-order expression for the traveltime perturbation. This is of relevance to non-linear traveltime tomography because it leads to an efficient method for evaluating the lowest order ray deflection and the non-linear effect this has on the traveltimes. The theory is applicable both to two-point ray tracing and to the determination of paraxial rays. The derivations in this paper are completely self-contained. All expressions, including the transformation to ray coordinates, are derived from first principles. In this way one obtains insight in the approximations that are actually made. A scale analysis leads to dimensionless numbers that give an indication whether the theory is applicable to a specific problem. For the special case of a layered reference medium the final equations are particularly simple. Plane discontinuities in the reference model and the slowness perturbation are incorporated in the theory. The final expressions for the ray deflection and the traveltime perturbation can be implemented numerically in a simple way. It is indicated how applications to very large-scale problems can be achieved. Several examples, including the propagation of waves through a quasi-random model of the earth's mantle illustrate the theory.

show abstract

Section: R Snieder and M Sambridgementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Ray perturbation theory for traveltimes and ray paths in 3-D heterogeneous media

Snieder

Sambridge

1992

Geophysical Journal International

View full text Add to dashboard Cite

show abstract

“…In the past quadrennium, the analytical theory was considerably simplified by adopting Hamiltonians for calculating the rays, without changing the basic conclusions of quadratic dependence and warm bias. Wunsch (1987) brought Hamiltonian techniques to bear on the ray forward problem for long‐range (many turning point) acoustic transmissions. He introduced the adiabatic approximation for range‐dependent ray calculations, simplifying the algebra for sufficiently smooth perturbations.…”

Section: Theorymentioning

confidence: 99%

A Review of Ocean Acoustic Tomography: 1987–1990

Worcester

Cornuelle

Spindel

1991

Reviews of Geophysics

View full text Add to dashboard Cite

Research in ocean acoustic tomography during the last quadrennium has resulted in substantial progress in understanding the capabilities and limitations of the technique. Theoretical studies and numerical simulations have led to greater understanding of the oceanographic information available in a vertical slice from acoustic transmissions between a single pair of instruments, of the horizontal geometries required to map the ocean mesoscale field with specified precision, and of the properties of tomographic reconstructions of the two‐dimensional vector field of current. Simultaneously, the instrumentation used in tomographic experiments has been significantly improved, increasing the precision of the measurements and making gyre and basin scale experiments feasible between moored instruments. Experimental efforts to test the accuracy with which the ocean temperature and current fields can be measured acoustically have now demonstrated that tomographic techniques provide measurements with oceanographically useful precision up to ranges of about 1000 km. Such demonstrations are difficult due to the incompatibility between point measurements and the spatial averages provided by tomographic techniques. The experiments have also yielded significant information on the character of acoustic propagation at long range in the ocean. Experimental precision is now adequate to distinguish between competing algorithms for sound speed as a function of temperature, salinity, and depth. Finally, and most importantly, emphasis has shifted to use of the technique for studying the ocean, rather than on understanding the capabilities and limitations of the technique. Two major experiments, the Greenland Sea Tomography Experiment and the Gulf Stream Extension Tomography Experiment, both conducted during 1988–89, were devoted to improving our understanding of ocean dynamics, although results are not yet available. There is increased emphasis on exploiting the integrating nature of acoustic transmissions to study gyre and global scale temperature variability, phenomena difficult to study in any other way.

show abstract

“…We extend and complete the work of Virieux (1989). These techniques have recently been applied in the problem of oceanographic tomography (Miller 1986;Wunsch 1987). Because the J .…”

Section: Introductionmentioning

confidence: 99%

Ray Tracing On A Heterogeneous Sphere By Lie Series

Virieux

Ekström

1991

Geophysical Journal International

View full text Add to dashboard Cite

We propose a method for fast analytical ray tracing on a heterogeneous sphere for surface waves. We first select the specific coordinates of orbital motion which have action/angle properties. We then apply the Lie perturbation approach and, when the square of the slowness is expanded in spherical harmonics, we obtain an analytical formula for the perturbed parameters of the ray. These expressions are sensitive to both the odd and even parts of the expansion. Traveltimes are computed by perturbation, while geometrical spreading is estimated numerically between two nearby perturbed rays. For the 'Gulf of Alaska' earthquake of November 1987, the analytical ray follows the same deviations with respect to the great circle as the numerical one, when we use the phase velocity model of Montagner & Tanimoto (1990) at period of 167 s. The agreement is excellent for traveltime computations. When the numerical ray tracing predicts a focus/defocusing effect, the perturbed ray tracing gives the same trend. Moreover, variations of the shooting angles between trains can be as high as 20" which might modify the radiation pattern seen by the station for different trains. When the perturbed ray deviates too strongly, a reinitialization technique will guarantee a given accuracy. This reinitialization, which is not required for long periods (>150 s), is probably necessary at shorter periods.

show abstract

Acoustic tomography by Hamiltonian methods including the adiabatic approximation

Cited by 19 publications

References 22 publications

Ray perturbation theory for traveltimes and ray paths in 3-D heterogeneous media

Ray perturbation theory for traveltimes and ray paths in 3-D heterogeneous media

A Review of Ocean Acoustic Tomography: 1987–1990

Ray Tracing On A Heterogeneous Sphere By Lie Series

Contact Info

Product

Resources

About