ALBERT

Description: SUMMARY Investigating the mechanisms of small seismic sources usually consists of three steps: determining the moment tensor of the source; decomposing the moment tensor into parameters that can be interpreted in terms of physical mechanisms and displaying those parameters. This paper concerns the second and third steps. Two existing methods—the Riedesel-Jordan and Hudson-Pearce-Rogers parameters and displays—are reviewed, compared and contrasted, and advantages and disadvantages of the two methods are discussed. One disadvantage is that neither method takes into consideration the effect of anisotropy on the interpretation. In microseisms, anisotropy can be important. A new procedure based on the biaxial decomposition of the potency tensor is introduced which explicitly allows for anisotropy and interprets the moment tensor in terms of an isotropic pressure change and a displacement discontinuity on a fault. It is shown that this interpretation is always possible for any moment tensor whatever the anisotropy. To compare the pressure change with the displacement discontinuity, it is useful to be able to determine the volume change from the pressure source in any medium. This depends on the embedded bulk modulus, which differs from the normal bulk modulus. The embedded modulus in isotropic media is well known and the equivalent anisotropic result is derived in this paper. Interpreting a seismic source in terms of the volume change due to a pressure change and a displacement discontinuity on a fault allows a simple 3-D graphical glyph to be used to display the interpretation.

Description: This note points out that the statement in ‘A new moment-tensor decomposition for seismic events in anisotropic media’ by Chapman and Leaney that ‘this interpretation is always possible for any moment tensor whatever the anisotropy’ needs qualification. The statement is true for isotropic and weakly anisotropic media, but the decomposition is not possible for all anisotropies, and for some anisotropies and moment tensors is possible but non-unique. However, it is possible for most expected anisotropies. Conditions when the statement is invalid are discussed, and examples of realistic anisotropy are given for which the statement is valid.

Description: Finite-difference simulations are an important tool for studying elastic and acoustic wave propagation, but remain computationally challenging for elastic waves in three dimensions. Computations for acoustic waves are significantly simpler as they require less memory and operations per grid cell, and more significantly can be performed with coarser grids, both in space and time. In this paper, we present a procedure for correcting acoustic simulations for some of the effects of elasticity, at a cost considerably less than full elastic simulations. Two models are considered: the full elastic model and an equivalent acoustic model with the same P velocity and density. In this paper, although the basic theory is presented for anisotropic elasticity, the specific examples are for an isotropic model. The simulations are performed using the finite-difference method, but the basic method could be applied to other numerical techniques. A simulation in the acoustic model is performed and treated as an approximate solution of the wave propagation in the elastic model. As the acoustic solution is known, the error to the elastic wave equations can be calculated. If extra sources equal to this error were introduced into the elastic model, then the acoustic solution would be an exact solution of the elastic wave equations. Instead, the negative of these sources is introduced into a second acoustic simulation that is used to correct the first acoustic simulation. The corrected acoustic simulation contains some of the effects of elasticity without the full cost of an elastic simulation. It does not contain any shear waves, but amplitudes of reflected P waves are approximately corrected. We expect the corrected acoustic solution to be useful in regions of space and time around a P -wave source, but to deteriorate in some regions, for example, wider angles, and later in time, or after propagation through many interfaces. In this paper, we outline the theory of the correction method, and present results for simulations in a 2-D model with a plane interface. Reflections from a plane interface are simple enough that an analytic analysis is possible, and for plane waves, we give the correction to the acoustic reflection and transmission coefficients. Finally, finite-difference calculations for plane waves are used to confirm the analytic results. Results for wave propagation in more complicated, realistic models will be presented elsewhere.