Analysis of Source Spectra

The purpose of this paper is to present the fundamentally basic aspects of source spectra through the preliminary use of the one-dimensional Haskell source model which assumes a finite fault with a constant rupture velocity. Teleseismic distances are assumed, such that the far field term dominates, allowing the P and S waves emanating from the source to be effectively considered double couple point sources. The appropriate slip is modeled instantaneously such that displacement along the fault is represented by a step, ramp function. The source time function thus produces a Fourier transform of two boxcar functions with corresponding widths defined by the rise time and rupture duration [Shearer, 2009]. In the frequency domain, the product of two functions is equivalent to the convolution of two functions in the time domain. Convolution results in a pulse shape that is trapezoidal, with this pulse shape being identical at all stations but varying in length. The spectra can be divided into three regions based on the corner frequencies. The spectrum appears flat at lower frequencies (proportional to Mo). Within intermediate frequencies a ω falloff is observed, while at higher frequencies a ω falloff is expected. It is further demonstrated how upon determination of the source spectra one may utilize it to obtain radiated seismic energy, moment, and stress drop. Introduction An important aspect of the propagation of seismic waves is the source in which these waves are derived and how the release of radiated energy relates to the physical aspects of the source. Resolving such source properties is at the heart of understanding the fundamental processes that govern earthquakes. When an earthquake occurs, potential strain energy that has accumulated along faults within the Earth is released in the form of seismic waves, heat, fracturing of rock, and various other forms of work [Chandra, 1970]. Not only is that potential energy radiated as seismic waves, but it it also the inherent radiation of that energy that provides fundamental information about the mechanical aspects of the source. From the beginning of the science of seismology, seismologists have been plagued with the problem of accurately estimating this radiated seismic energy [Houston, 1990]. The difficulties arise in the need to define the seismic moment for a large dynamic range and a wide frequency band from the low frequencies beyond the corner frequencies [Prieto et al., 2006]. In an attempt to quantify radiated seismic energy (among other things), seismologists have relied heavily on a methodology known as teleseismic source spectra which allows for the characterization of the radiated seismic energy at periods from 1-50 seconds [Houston, 1990]. By quantifying the source spectra, workers are able to estimate not only the seismically radiated energy Es directly, but also moment (Mo) and stress drop (Δσ). While Es is computed from the integrated velocity spectra, the moment can be determined by examining the lower frequency limit of the displacement spectra [Shearer et al., 2006]. Stress drop, an important measure of the difference between the average state of stress on the fault plane before and after an earthquake, can be constrained through spectral corner frequencies that contain estimates on fault area [Shearer et al., 2006]. Within this paper, the fundamental theoretical aspects of source spectra will be analyzed through considering a finite fault 1D Haskell source model with a constant rupture velocity. The appropriate source time function for the Haskell model will produce a Fourier transform of two boxcar functions with corresponding widths defined by the rise time and rupture duration [Shearer, 1999]. In the frequency domain the product of two corresponding functions is equivalent to the convolution of two functions in the time domain. This convolution will effectively result in a pulse that is trapezoidal for the far field displacement. Exploration will further be made into how important characteristics of the source, such as Es, Mo, and Δσ can be determined through the constraints of the spectra. Thereafter a discussion into the uncertainties and limitations of the method will be presented. 1D Haskell Source: Finite Source As those who determine source spectra are usually interested in discovering salient source characteristics such as Es, Mo, and Δσ it is important to note the faulting parameters that primarily affect the seismic radiation: average displacement on the fault (D), the dimensions of the fault (L, W), rupture velocity (vr), and particle velocity (defined as the rate at which an individual particle on the fault travels from its initial to final position) [Lay and Wallace, 1995]. In spite of immense advances made by the seismological community in understanding the dynamics of earthquake ruptures, the most widely used models for interpreting seismic radiation are dislocation models [Madariaga, 2007]. Dislocation models simulate kinematic spreading of a displacement discontinuity along a fault plane. One of the most widely used models was introduced by Haskell [1964]. In the simplest explanation of Haskell’s model a finite fault (ribbon like) is utilized to model a uniform displacement discontinuity as it spreads out with constant rupture velocity (See Figure 3). It is assumed that the reader is clearly familiar with the notion that when modeling shear-dislocation across a fault two distinct terms are determined: the near field and the far field terms. In the respect that this paper focuses on teleseismic sources, if we consider that the distance x>>l, where l denotes the fault length, then it is most basically obvious that the far field effects would dominate. Below the derivation of the source time function will be examined for the aforementioned effects. Following Lay and Wallace (1995), without formal derivation we will begin by noting that in considering the far field P and S waves under the constraints of teleseismic distances we are effectively considering a double couple point source. Utilizing the ray coordinate system with directions l, p, φ along P, SV, and SH directions at a source as first described by Aki and Richards [1980] and the standard fault orientation parameters given by Lay and Wallace [1995] of strike (φf,) dip (δ), and rake (λ ) and source take off angles ih we can write these radiated far field terms under the predefined assumptions as the following [Equation (8.65) Lay and Wallace, 1995]: € up (r,t) = 1 4πρrα 3 ⎛