Adaptive Simulated Annealing Velocity Modeling for Rayleigh Wave Dispersion

Inversion of Rayleigh-wave dispersion curve is realized using the adaptive simulated annealing (ASA) method. Instead of using uniform probability distributions, ASA uses the Ingber generation probability distribution. This distribution justifies an exponential temperature annealing schedule in perturbing models and guarantees the convergence of the algorithm. The exponential decrease of temperature leads to a quick finding of the global minimum. Tests on both real and synthetic Rayleigh dispersion data sets indicate that our ASA optimization is accurate. Introduction Shallow shear velocity structure is very important for seismic design of engineered structures and facilities. Recent building codes in both Canada (National Building Code of Canada) and the United States (BSSC 2000) are closely reliant on measurements of shallow shear-velocity structure at building sites. Seismic hazard evaluation of a region, comprehensive earthquake preparedness, and development of the national seismic hazard map also benefit from the local shallow shear velocity structure at large numbers of sites. Several techniques have been used to successfully estimate the dispersion curve of the surface waves contained in microtremors (e.g., Horike 1985; Liu et al. 2000). The refraction microtremor (ReMi) technique (Louie 2001) has been widely applied both commercially (through SeisOpt ReMiTM from Optim Inc.) and in research (Scott et al. 2004; Stephenson et al. 2005) to produce reliable dispersion curves. Commercial ReMi software yields a local 1-D shearvelocity structure by manually forward-modeling a dispersion curve picked from ReMi data. The modeling, however, has to be done by several practitioners to avoid human bias. An accurate and automatic inversion method is needed to quickly generate shallow shear-wave velocity profiles without bias. Adaptive simulated annealing (ASA) optimization is an effective alternative. Unlike linear inversion, simulated annealing (SA) is a directed Monte Carlo inversion method for 1 Grad. Research Asst., Nevada Seismological Lab., Univ. of Nevada, Reno, NV 89557; donghong@seismo.unr.edu 2 Vice President, Optim Inc., Reno, NV; satish@optimsoftware.com 3 Assoc. Professor, Nevada Seismological Lab., University of Nevada, Reno, NV 89557 finding the global minimum of a non-linear error function. Luke et al. (2003) showed that linear inversion yields excellent dispersion results for simple profiles; however, for more complex profiles multiple solutions with equally good data fits are possible. SA has been successfully used in many geophysical inverse problems (Pullammanappallil and Louie 1993, 1994; Sen and Stoffa 1996; Martinez et al. 2000; Beaty et al. 2002) since the work of Metropolis et al. (1953) and of Kirkpatrick et al. (1983). However, SA is quite time-consuming in finding an optimal fit, and it is difficult to fine tune to specific problems, relative to other fitting techniques (Ingber 1993). Instead of using a uniform distribution to perturb models, ASA uses the Ingber generation probability distribution, which then justifies an exponential temperature-annealing schedule. The exponential decrease guarantees the convergence of the algorithm, leading to quick convergence at the global minimum (Ingber 1993). Inverse theory The concept of the simulated annealing (SA) is based on the manner in which liquids freeze or metals re-crystallize in the process of annealing. In an annealing process a melt, initially at high temperature T and disordered, is slowly cooled so that the system at any time is approximately in thermodynamic equilibrium. As cooling proceeds, the system becomes more ordered and approaches a ground state at T=0. Hence the process can be thought of as an adiabatic approach to the lowest energy state. If the initial temperature of the system is too low or cooling is done insufficiently slowly the system may become quenched forming defects or freezing out in metastable states (ie. trapped in a local minimum energy state). Metropolis et. al. (1953) introduced the SA algorithm for simulating the evolution of a solid in a heat bath to thermal equilibrium. His original scheme was that an initial state of a thermodynamic system was chosen at energy E and temperature T. Holding T constant the initial configuration is perturbed and the change in energy ∆E is computed. If the change in energy is negative the new configuration is accepted. If the change in energy is positive it is accepted with a probability given by the Boltzmann factor exp(-∆E/T). This processes is then repeated sufficient times to give good sampling statistics for the current temperature, and then the temperature is decremented and the entire process repeated until a frozen state is achieved. By analogy the generalization of this Monte Carlo approach to inversion of dispersion curve is straight forward. The current state of the thermodynamic system is analogous to the calculated phase velocities based on the current model m which involves the Pand S-wave velocities, thickness, and density of each layer. The energy of the thermodynamic system is analogous to the least-square error E of the calculated and observed phase velocities, defined as

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