INTRODUCTION
Summary Nonlinear optimization methods (or inversion) were investigated for analyzing synthetic microseismic arrival times. Two direct search techniques, the genetic algorithm and pattern search, were used to find the layered-earth velocity values from P-wave arrival times from a simulated perforation shot. For locating microseismic hypocenters, the gradient-based Levenberg-Marquardt algorithm was used to invert reduced arrival times from borehole and surface receiver arrays. Both categories of nonlinear optimization method, direct search and gradient-based, were effective for inverting arrival times to the required model parameters. Our experience suggests that the direct search methods, in particular pattern search, are simpler and faster in this application, i.e., inverting microseismic arrival time data to obtain layer either velocities or hypocenter coordinates.
Unknown model parameters in fitting geophysical survey results can be found by minimizing the misfit between observed and calculated arrival times using non-linear optimization schemes (generally called inversion techniques by geophysicists). The misfir or objective function to be minimized must be parameterized by an input vector of the variables to be found. Optimization techniques fall in two categories: gradient-based, and direct-search. An example of gradient-based techniques is the Levenberg-Marquardt algorithm (Levenberg, 1944; Marquardt, 1967). Examples of direct search techniques are the pattern search method and the genetics algorithm. These algorithms are available in the utility program optimtool which is bundled in the MATLAB (2009) Optimization Toolbox. Gradient-based optimization methods such as the Levenberg-Marquardt (LM) algorithm often can be trapped in local minima when the objective function is a complex nonlinear equation involving many variables. The more variables there are, the greater the likelihood for the existence of local minima, saddle points, or long narrow data valleys. Using a gradient method to find the global minimum in the objective function for such cases tends to be problematic. Alternatives to gradient based methods exist in the form of sophisticated global search techniques such as the genetic algorithm (GA) or pattern search (PS). These direct search methods are described in the literature; see, for example, Whitley (1997), and Kolda et al. (2003). We tested these nonlinear optimization techniques for their effectiveness in solving two problems related to microseismic monitoring. The first problem is estimating the velocity values in an earth model knowing the location of a perforation shot source and the arrival times at a receiver array. The second problem is locating the microseismic hypocenter knowing the arrival times at the receiver array as well as the velocity model. We performed the tests using synthetic arrival times calculated by raytracing through a horizontally-layered velocity model. The left panel on Figure 1 is a section view showing the layered-earth velocity model with a microseismic source in a treatment well and an array of geophones in a vertical observation well. The geometry in cylindrical coordinates has azimuthal symmetry about the observation well or the microseismic source. Assuming P-wave velocities, Snell’s Law ray-tracing from the source to the geophone array gives a set of first-arrival times as a function of depth in the observation well.