### Abstract

One considers an elastic halfspace with depth (x_{1}) dependent density Q and Lamé moduli (λ,m̈). Impulsive stresses τli = δ(x_{2}, x_{3})δ(t) for x_{1} = 0 are applied with displacement responses u_{i} = g_{i}(t, x_{2}, x_{3}) at x_{1} = 0 (i = 1, 2, 3). Let v_{i}(t, x_{1}) = ∫∫u_{i}dx_{2}dx_{3} (i = 1, 2 is enough) and set w(t, x_{1}) = ∫∫x_{2}u_{1}dx_{2}dx_{3}. One obtains a system of 3 differential equations for v_{1}, v_{2}, and w to which the spectral techniques of inverse scattering theory are applied as in [25]. The inverse problems for the uncoupled v_{i} can then be solved to produce 2 functions A_{1} and A_{2} involving (Q, λ, μ) as functions of “bound” variables y_{1} and y_{2} containing (Q, λ, μ), between which a relation then is determined. Analysis of the coupled equation for w then leads to a Fredholm integral equation whose solution provides an additional relation between (Q, λ, μ) from which (Q, λ, μ) can be determined as functions of x_{1}. The integral equation is reduced to a Volterra type equation by results and techniques of transmutation and then solved by a modification of standard techniques. A number of features and results of independent mathematical interest arise from the transmutation theory.

Original language | English (US) |
---|---|

Pages (from-to) | 33-73 |

Number of pages | 41 |

Journal | Mathematical Methods in the Applied Sciences |

Volume | 4 |

Issue number | 1 |

DOIs | |

State | Published - 1982 |

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*Mathematical Methods in the Applied Sciences*,

*4*(1), 33-73. https://doi.org/10.1002/mma.1670040105

**On complete recovery of geophysical data.** / Carroll, Robert; Santosa, Fadil; Payne, L.

Research output: Contribution to journal › Article

*Mathematical Methods in the Applied Sciences*, vol. 4, no. 1, pp. 33-73. https://doi.org/10.1002/mma.1670040105

}

TY - JOUR

T1 - On complete recovery of geophysical data

AU - Carroll, Robert

AU - Santosa, Fadil

AU - Payne, L.

PY - 1982

Y1 - 1982

N2 - One considers an elastic halfspace with depth (x1) dependent density Q and Lamé moduli (λ,m̈). Impulsive stresses τli = δ(x2, x3)δ(t) for x1 = 0 are applied with displacement responses ui = gi(t, x2, x3) at x1 = 0 (i = 1, 2, 3). Let vi(t, x1) = ∫∫uidx2dx3 (i = 1, 2 is enough) and set w(t, x1) = ∫∫x2u1dx2dx3. One obtains a system of 3 differential equations for v1, v2, and w to which the spectral techniques of inverse scattering theory are applied as in [25]. The inverse problems for the uncoupled vi can then be solved to produce 2 functions A1 and A2 involving (Q, λ, μ) as functions of “bound” variables y1 and y2 containing (Q, λ, μ), between which a relation then is determined. Analysis of the coupled equation for w then leads to a Fredholm integral equation whose solution provides an additional relation between (Q, λ, μ) from which (Q, λ, μ) can be determined as functions of x1. The integral equation is reduced to a Volterra type equation by results and techniques of transmutation and then solved by a modification of standard techniques. A number of features and results of independent mathematical interest arise from the transmutation theory.

AB - One considers an elastic halfspace with depth (x1) dependent density Q and Lamé moduli (λ,m̈). Impulsive stresses τli = δ(x2, x3)δ(t) for x1 = 0 are applied with displacement responses ui = gi(t, x2, x3) at x1 = 0 (i = 1, 2, 3). Let vi(t, x1) = ∫∫uidx2dx3 (i = 1, 2 is enough) and set w(t, x1) = ∫∫x2u1dx2dx3. One obtains a system of 3 differential equations for v1, v2, and w to which the spectral techniques of inverse scattering theory are applied as in [25]. The inverse problems for the uncoupled vi can then be solved to produce 2 functions A1 and A2 involving (Q, λ, μ) as functions of “bound” variables y1 and y2 containing (Q, λ, μ), between which a relation then is determined. Analysis of the coupled equation for w then leads to a Fredholm integral equation whose solution provides an additional relation between (Q, λ, μ) from which (Q, λ, μ) can be determined as functions of x1. The integral equation is reduced to a Volterra type equation by results and techniques of transmutation and then solved by a modification of standard techniques. A number of features and results of independent mathematical interest arise from the transmutation theory.

UR - http://www.scopus.com/inward/record.url?scp=84979129498&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84979129498&partnerID=8YFLogxK

U2 - 10.1002/mma.1670040105

DO - 10.1002/mma.1670040105

M3 - Article

AN - SCOPUS:84979129498

VL - 4

SP - 33

EP - 73

JO - Mathematical Methods in the Applied Sciences

JF - Mathematical Methods in the Applied Sciences

SN - 0170-4214

IS - 1

ER -