Journal of Seismology

, Volume 23, Issue 5, pp 1017–1030 | Cite as

An inverse problem in seismology: derivation of the seismic source parameters from P and S seismic waves

  • Bogdan Felix ApostolEmail author
Original Article


This paper presents the solution of an inverse problem in Seismology, which aims at deriving the seismic source parameters from P and S seismic waves. In particular, the paper gives the deduction of the seismic-moment tensor. The problem is tackled in this paper under three particular circumstances. First, we use the amplitude of the far-field (P and S) seismic waves as input data. We use the analytical expression of the seismic waves in a homogeneous isotropic body with a seismic-moment source of tensorial forces, the source being localized both in space and time. We assume that the position of the seismic source is known. The far-field waves provide three equations for the six unknown parameters of the general tensor of the seismic moment, such that the system of equations is under-determined. Second, the Kostrov vectorial (dyadic) representation of the seismic moment for a shear faulting is used. This representation relates the seismic moment to the focal displacement in the fault and the orientation of the fault (moment-displacement relation); it reduces the seismic moment to four unknown parameters. Third, the fourth missing equation is derived from the energy conservation and the covariance condition. The four equations derived here are solved and the seismic moment is determined, as well as other parameters of the seismic source, like focal volume, focal slip, fault orientation, and duration of the seismic activity in the source. It turns out that the seismic moment is traceless, its magnitude is of the order of the elastic energy stored in the focal region (as expected), and the solution is governed by the unit quadratic form associated with the seismic-moment tensor (related to the magnitude of the longitudinal displacement in the P wave). A useful picture of the seismic moment is the conic represented by the associated quadratic form, which is a hyperbola (seismic hyperbola). This hyperbola provides an image for the focal region: its asymptotes are oriented along the focal displacement and the normal to the fault. Also, the special case of an isotropic seismic moment is presented. Numerical examples are provided for this procedure, and the limitations are discussed.


Seismic source Inverse problem Seismic waves Seismic moment Elasticity Seismic hyperbola 



The author is indebted to L. C. Cune, the colleagues in the Institute for Earth’s Physics, Magurele-Bucharest, for many enlightening discussions, and to the members of the Laboratory of Theoretical Physics at Magurele-Bucharest for many useful discussions and a throughout checking of this work. The author is also indebted to the anonymous editor and reviewers for many helpful suggestions for improving this paper.

Funding information

This work was partially supported by the Romanian Government Research Grant No. PN16-35-01-07/2016 and No. PN18-15-01-01/2018.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Institute for Earth’s PhysicsMagurele-BucharestRomania

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