Theoretical aspects of seismic waves and sources Massimo Cocco INGV Earthquakes produce effects to the environment & the society Damages are produced by severe ground shaking due to seismic waves and local site amplifications and by the vulnerability of buildings and human constructions. This course aims to introduce the fundamental equations underlying the main theoretical and numerical approaches used in seismology. The main goal is to provide basic knowledge on mathematical and physical background on quantitative seismology Its ambitions is to contribute in educating a new generation of engineering seismologists to collaborate with geophysicists and seismologists

Earthquakes do occur in specific areas and are associated with the seismic wave emission caused by the rupture propagation on a fault plane

Earthquakes & dislocations

Earthquakes modify the environment INGV

Surface breakage Kokoxili earthquake Mw 7.9 (Qinghai Province, China)

Chi-Chi earthquake: recorded seismograms gal cm

The representation theorem The equation of motion the displacement field due to both the body forces throughout a volume V and the boundary conditions on the surface S Green functions

In seismology it is more useful to consider S as a surface including two adjacent surfaces internal to V. This specific form of the representation theorem is necessary to analyze an earthquake as the displacement’s discontinuity (i.e., slip) on a buried fault. The common practice to achieve this goal is to develop body-force equivalents (Aki & Richards, 2002, pp.38-58) for simple shear across a fault surface demonstrating that different systems of forces can be equivalent to exactly the given displacement discontinuity (slip). The surface  is chosen to include two adjacent surfaces (     ) internal to volume V.  -  + V  -  + V n S Reciprocity of Green functions Aki & Richards, 2002, eq.3.1)

KINEMATIC REPRESENTATION is the coseismic slip is the discontinuity in dynamic tractions.

equivalent surface distribution of body forces there exists an equivalent surface distribution of body forces, which produces the same effects of a physical discontinuity of displacement across the fault surface KINEMATIC TRACTIONS slip on fault Landslide or Explosion (Mt. St. Helens) add 3 for isotropic explosion

Moment density tensor This term is the n-component of the displacement field at caused by a couples of forces at and is the strength of the (p,q) couple

Moment Tensors STRIKE SLIP FAULT TENSILE CRACK is the scalar moment tensor

MOMENT TENSOR Cosesimic slip Displacement field for a point source Scalar Seismic Moment The nine components of moment tensor and the nine force couples.

The Green Functions Point dislocation source For many purposes in seismology equivalent body forces acting at a point are an adequate source model for ground displacement observed during an earthquake. However, the equivalent body forces associated with fault slip are double couple and not a unidirectional force. Equation of motion Equation of motion for a Green function (9)

Further property equation of motion with zero initial conditions if the source is now extended throughout a volume Equation of Motion Source Solution which has the important property that the field at x and t is caused by the source activity in the element dV at x at the retarded time

The Lamé’s potentials the representation of displacement field in terms of the Helmholtz potentials. Equation of motion The common solution for the elastodynamic Green function in a homogeneous, elastic unbounded medium can be derived by solving the equation of motion with a body force which is applied in x the direction with amplitude, that is

Homogeneous, elastic unbounded medium with unidirectional body force the displacement field using these Lamé potentials using We can now generalize this formula by changing the subscript 1 to j, the result represents the displacement caused by a point force in the x j -direction; introducing the direction cosines

Green function definition the definition of Green function for a homogenous, isotropic unbounded medium Scales as r -2 Scales as r -1

Far-field P-wave This seismic wave attenuates as r -1 it propagates at a velocity The displacement waveform is proportional to the applied force at the retarded time The direction of the associated displacement at is parallel to the direction from the source, since The far-field P-wave is longitudinal (called sometimes radial) and its direction of particle motion is in the direction of propagation

Far-field S-wave This seismic wave attenuates as r -1 it propagates at a velocity The displacement waveform is proportional to the applied force at the retarded time The direction of the associated displacement at is perpendicular to the direction from the source, since The far-field S-wave is transversal and its direction of particle motion is perpendicular to the direction of propagation

Far-field radiation pattern radiation patterns showing the directions and the amplitudes of P and S motions in the far-field caused by a point force in the x j -direction Radiation patterns for P- and S-waves in the far field for point force in the x j -direction

Near-field term The near-filed term of the displacement field Both the P- and S-waves contribute to the near field term, because it contains contributions from the gradient of  and the curl of . It is neither irrotational (having zero curl) nor solenoidal (having zero divergence) and it is not possible to decompose this contribution in P and S waves. Ground velocity Ground displacement

The double-couple solution double-couple solution in an infinite, homogeneous isotropic medium. Radiation Pattern moment rate function NF IT FF

radiated displacement field from any moment tensor the application to seismic waves requires specializing the solution to cases where the moment tensor arises from a shear dislocation. In such cases, slip lies on the fault plane

Radiation Pattern for a double couple source According to the equivalent body forces representation, the physical discontinuity of displacement (that is, slip) is by definition represented by a distribution of double-couple forces. This implies that there exists an auxiliary plane perpendicular to the fault plane The next step is to transform the solution for the displacement field radiated by a shear dislocation from its Cartesian coordinates form into a spherical polar coordinate framework radial and 2 transverse directions, respectively

displacement field and seismic moment

Far Field radiation patterns for P and S waves far-field the displacement Radiation patterns for a fault geometry shown in (a). (b) displays the radiation pattern for P-waves showing amplitude (left) and direction (right). (c) shows the same for S-waves