Effects of Surface Geology on Seismic Motion

We introduce a new conserved quantity, Normalized Energy Density (NED), alternative to the conventional definition of energy for a layered structure in a 2D SH problem. NED is defined by the average of power of a half transfer function multiplied by the impedance, and the conservation across the material interface is analytically proved for a two-layered case. For three, four, and tenlayered cases, the conservation is examined by applying the Monte Carlo simulation method, and then NED is supposed to be conserved through the layers. However, the conservation is not guaranteed when the material has damping. We perform another Monte Carlo simulation in order to identify the effect on NED of the damping property by applying the conventional Haskell matrix method with the damping coefficient and the apparent quality factor Qa. NED decreases as the damping coefficient increases, and also almost the same characteristics are obtained by the apparent quality factor. The magnitude of damping properties corresponds to the decreases of NED observed in the top layer, whereas the NEDs vary widely. We introduce another damping property, T/Q, considering a travel time from the basement to the free surface, and its variation becomes smaller. We also propose the analytical relation of NED v.s. T/Q to give almost the upper boundary of the possible solution. The correlation curve enables us to identify the damping property directly from NED. INTRODUCTION Conserved quantities, such as mass, momentum and energy, in elasto-dynamic problems are the fundamental variables when analyzing wave propagation in a continuous medium. In addition, the balance principles associated with these quantities, e.g., the balance of mass and the balance of momentum, govern the deformation within the framework of Newtonian mechanics. The balance of energy is one of the principles used to quantify the seismic energy radiated from an earthquake source. Radiation energy E is theoretically defined as the total energy transmitted through a certain surface. When a particular region, e.g. a seismic fault, generates all of the energy, the integration on the arbitrary surface surrounding the region is theoretically conserved even for a general heterogeneous medium. The representation has already been introduced in Love (1927). The energy of seismic events was first applied by Richter (1935) in order to measure the size of earthquakes by using the local magnitude scale, although it was not exactly equal to the definition of the energy. Afterward, Kanamori (1977) proposed the use of moment magnitude, defined from the seismic moment that is related to the energy release during the events, whose energy is different from the radiation energy. A detailed discussion on radiation energy is introduced in Kostrov and Das (1988), Fukuyama (2005), and Abercrombie et al. (2006).