Analysis and Optimization of Seismic Monitoring Networks with Bayesian Optimal Experiment Design

Abstract

Monitoring networks increasingly aim to assimilate data from a large number of diverse sensors covering many sensing modalities. Bayesian optimal experimental design (OED) seeks to identify data, sensor configurations, or experiments which can optimally reduce uncertainty and hence increase the performance of a monitoring network. Information theory guides OED by formulating the choice of experiment or sensor placement as an optimization problem that maximizes the expected information gain (EIG) about quantities of interest given prior knowledge and models of expected observation data. Therefore, within the context of seismo-acoustic monitoring, we can use Bayesian OED to configure sensor networks by choosing sensor locations, types, and fidelity in order to improve our ability to identify and locate seismic sources. In this work, we develop the framework necessary to use Bayesian OED to optimize a sensor network's ability to locate seismic events from arrival time data of detected seismic phases at the regional-scale. Bayesian OED requires four elements: 1) A likelihood function that describes the distribution of detection and travel time data from the sensor network, 2) A Bayesian solver that uses a prior and likelihood to identify the posterior distribution of seismic events given the data, 3) An algorithm to compute EIG about seismic events over a dataset of hypothetical prior events, 4) An optimizer that finds a sensor network which maximizes EIG. Once we have developed this framework, we explore many relevant questions to monitoring such as: how to trade off sensor fidelity and earth model uncertainty; how sensor types, number, and locations influence uncertainty; and how prior models and constraints influence sensor placement.

Figures (19)

• Figure 1: Illustration of a Bayesian inference process for seismic source location. Bayesian inference begins with a prior distribution for different earthquake locations $\theta$, shown by the contour lines on the leftmost figure. As an observer collects data, they use a likelihood function model to quantify the probability of observing that data, given that an earthquake occurs at a specific location. The observer constructs this likelihood model from physical models of seismic wave propagation, models of the sensors that detect seismic signals, and models of uncertainty (e.g. background noise modeling errors, etc.). The observer then applies Bayes' theorem to update the prior to assimilate this new information. The posterior distribution, shown by the contour lines in the rightmost image, then quantifies the probability that the seismic source has location $\theta$, given the data.
• Figure 2: Sensor locations that optimize a sensor network configuration, according to Equation \ref{['eq:optExpConfig']}. The top row displays heat maps showing the optimization surface $\mathcal{I}(S_{n+1})$, where $S_{n+1} = \{S^*_n, \mathcal{L}_{n+1}\}$, $S^*_n$ is the set of the first $n$ fixed sensor locations (white circular markers), and $\mathcal{L}_{n+1}$ is the location of the $(n+1)$th sensor (magenta circular markers) that maximizes $\mathcal{I}(S_{n+1})$. The warmer the color indicates that adding a sensor at that location is better. The bottom row displays the EIG about all events in the domain about the location of a shallow, low-magnitude seismic source at that specific latitude and longitude, i.e., it displays $\mathcal{I}(S^*_{n+1} \mid \theta' = [\mathcal{L},x,m])$ for all $\mathcal{L}$ in the domain (see Equation \ref{['eq:thetaEIG']}).
• Figure 3: Map of Transportable Array stations from December 2007 (Map from stationmap). The pink region indicates the region (Latitude $\in [ 39^oN, 43^oN ]$, Longitude $\in [ 113^oW, 107.36^oW ]$) that we use gathered sensor data that populates the parameters of the likelihood models. The orange region (Latitude $\in [ 40^oN, 42^oN ]$, Longitude $\in [ 112^oW, 109.36^oW ]$), corresponds to the monitoring region over which we build a sensor network usarray.
• Figure 4: Illustration of the 121 1D velocity models for Vp sampled from around the monitoring region in Figure \ref{['fig:ta_map']}. These representative earth models are used to estimate travel time uncertainty from earth model uncertainty.
• Figure 5: Left: A scatter plot (blue) of the estimated travel time mean $\mu \left ( \Delta, x \right )$, and estimated travel time standard deviation, $\sigma \left (\Delta, x \right )$ for various distance and depth pairs, superimposed with a fit polynomial model (red). Right: A scatter plot of predicted travel time standard deviations compared to the true value (red). A line demonstrating the performance of a perfect model is displayed in blue. The polynomial model adequately captures the bulk trend, despite some variability due to the nature of the first arriving phase.