Using graph clustering to locate sources within a dense sensor array

Abstract

We develop a model-free technique to identify weak sources within dense sensor arrays using graph clustering. No knowledge about the propagation medium is needed except that signal strengths decay to insignificant levels within a scale that is shorter than the aperture. We then reinterpret the spatial coherence matrix of a wave field as a matrix whose support is a connectivity matrix of a graph with sensors as vertices. In a dense network, well-separated sources induce clusters in this graph. The geographic spread of these clusters can serve to localize the sources. The support of the covariance matrix is estimated from limited-time data using a hypothesis test with a robust phase-only coherence test statistic combined with a physical distance criterion. The latter criterion ensures graph sparsity and thus prevents clusters from forming by chance. We verify the approach and quantify its reliability on a simulated dataset. The method is then applied to data from a dense 5200 element geophone array that blanketed 7 km ×10 km of the city of Long Beach (CA). The analysis exposes a helicopter traversing the array and oil production facilities.

Machine Learning in Seismology: Turning Data into Insights

“…For example, the use of probabilistic graphical models and graph theory in seismology has become increasingly prevalent. The deployment of large-N arrays (Karplus and Schmandt, 2018) provides one such opportunity, in which weak event signals can be extracted using graph clustering (Riahi and Gerstoft, 2017) or similarity theory (Li, Peng, et al, 2018). Separately, Trugman and Shearer (2017) use graph theory and hierarchical cluster analysis to obtain high-precision earthquake location estimates using differential travel times from pairs of earthquakes observed at a set of common stations.…”

Graph clustering for localization within a sensor array

“…, u k N ] T be the vertex indicator vector of U k (u k i = 1 if v k i 2 U k and 0 otherwise). The connectivity matrix of G is then (see [15]):…”

“…Let I(v) be the support indicator function of a vector or matrix v. The lack of overlap of the g k and the supportindicator function properties (see [15]) allow us to write the support of the sum in (13) as We now define E ij = I( ) as the connectivity matrix of a graph G with N vertices (sensors), i.e. there is an edge between vertices i and j if E ij = I(C) ij = 1.…”

“…Such clusters can be found through an eigenvalue decomposition of a matrix (the graph Laplacian) that is derived from the connectivity matrix [14]. For more details on the approach see the full paper [15].…”

Graph clustering for localization within a sensor array

“…Let I(v) be the support indicator function of a vector or matrix v. The lack of overlap of the g k and the supportindicator function properties (see [15]) allow us to write the support of the sum in (13) as We now define E ij = I( ) as the connectivity matrix of a graph G with N vertices (sensors), i.e. there is an edge between vertices i and j if E ij = I(C) ij = 1.…”