Probabilistic earthquake location with non-linear, global search methods allows the use of 3D models and produces comprehensive uncertainty and resolution information represented by a probability density function over the unknown hypocentral parameters We describe a probabilistic earthquake location methodology and introduce an efficient Metropolis-Gibbs, non-linear, global sampling algorithm to obtain such locations Using synthetic travel times generated in a 3D model, we examine the locations and uncertainties given by an exhaustive grid-search and the Metropolis-Gibbs sampler using 3D and layered velocity models, and by a iterative, linear method in the layered model We also investigate the relation of average station residuals to known static delays in the travel times, and the quality of the recovery of known focal mechanisms With the 3D model and exact data, the location probability density functions obtained with the Metropolis-Gibbs method are nearly identical to those of the slower but exhaustive grid-search The location PDFs can be large and irregular outside of a station network even for the case of exact data With location in the 3D model and static shifts added to the data, there are systematic biases in the event locations Locations using the layered model show that both linear and global methods give systematic biases in the event locations and that the error volumes do not include the “true” location — absolute event locations and errors are not recovered The iterative, linear location method can fail for locations near sharp contrasts in velocity and outside of a network Metropolis-Gibbs is a practical method to obtain complete, probabilistic locations for large numbers of events and for location in 3D models It is only about 10 times slower than linearized methods but is stable for cases where linearized methods fail The exhaustive grid-search method is about 1000 times slower than linearized methods but is useful for location of smaller number of events and to obtain accurate images of location probability density functions that may be highly-irregular

Probabilistic Earthquake Location in 3D and Layered Models

Multigluon cross sections and 5-jet production at hadron colliders

We discuss a novel strategy for training neural networks using sequential Monte Carlo algorithms and propose a new hybrid gradient descent / sampling importance resampling algorithm (HySIR). In terms of computational time and accuracy, the hybrid SIR is a clear improvement over conventional sequential Monte Carlo techniques. The new algorithm may be viewed as a global optimization strategy that allows us to learn the probability distributions of the network weights and outputs in a sequential framework. It is well suited to applications involving on-line, nonlinear, and nongaussian signal processing. We show how the new algorithm outperforms extended Kalman filter training on several problems. In particular, we address the problem of pricing option contracts, traded in financial markets. In this context, we are able to estimate the one-step-ahead probability density functions of the options prices.

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Sequential Monte Carlo Methods to Train Neural Network Models

Arrival time The time of the first measurable energy of a seismic phase on a seismogram. Centroid The coordinates of the spatial or temporal average of some characteristic of an earthquake such as surface shaking intensity or moment release. Data space If the data are described by a vector d, then the data spaceD is the set of all possible values of d. Direct search A search or inversion technique that does not explicitly use derivatives. Usually involves repeated application of a forward problem. Earthquake early warning The goal of earthquake early warning is to estimate the shaking hazard of a large earthquake at a nearby population center or other critical site before destructive S and surface waves have reached the site. This requires that useful probabilistic constraint on the location and size of an earthquake is obtained very rapidly. Earthquake location An earthquake location specifies a spatial position and time of occurrence for an earthquake. The location may refer to the earthquake hypocenter and corresponding origin time, a mean or centroid of some spatial or temporal characteristic of the earthquake, or another property of the earthquake that can be spatially and temporally localized. This term also refers to the process of locating an earthquake. Epicenter The point on the Earth’s surface directly above a hypocenter. Error A specified variation in the value assumed by a variable. See also uncertainty. Forward problem A mathematical expression that relates the parameters defining a physical system (model space) to the data (data space) produced by a physical process acting on the system.

C. L. Mallows

Papers

Monte Carlo Methods.