Graph theory for analyzing pair-wise data: application to geophysical model parameters estimated from interferometric synthetic aperture radar data at Okmok volcano, Alaska

Abstract

Graph theory is useful for analyzing timedependent model parameters estimated from interferometric synthetic aperture radar (InSAR) data in the temporal domain. Plotting acquisition dates (epochs) as vertices and pair-wise interferometric combinations as edges defines an incidence graph. The edge-vertex incidence matrix and the normalized edge Laplacian matrix are factors in the covariance matrix for the pair-wise data. Using empirical measures of residual scatter in the pair-wise observations, we estimate the relative variance at each epoch by inverting the covariance of the pair-wise data. We evaluate the rank deficiency of the corresponding least-squares problem via the edge-vertex incidence matrix. We implement our method in a MATLAB software package called GraphTreeTA available on GitHub (https://github.com/feigl/gipht). We apply temporal adjustment to the data set described in Lu et al. (Geophys Res Solid Earth 110, 2005) at Okmok volcano, Alaska, which erupted most recently in 1997 and 2008. The data set contains 44 differential volumetric changes and uncertainties estimated from interferograms between 1997 and 2004. Estimates show that approximately half of the magma volume lost during the 1997 eruption was recovered by the summer of 2003. Between June 2002 and September 2003, the estimated rate of volumetric increase is (6.2 ± 0.6)×10 6 m 3 /year. Our preferred model provides a reasonable fit that is compatible with viscoelastic relaxation in the five years following the 1997 Electronic supplementary material The online version of this article (

Geodetic Measurements and Numerical Models of Deformation at Coso Geothermal Field, California, USA, 2004–2016

“…The pairs from Environmental Satellite (Envisat) are selected using a minimum spanning tree (MST) algorithm according to orbital separation (e.g., [26]). The same data set was used in a previous deformation modeling study at Coso [27].…”

“…To analyze this trend, we perform time-series analysis using temporal adjustment on the set of volume changes ∆V i,j estimated from the individual InSAR pairs spanning the time intervals from t i to t j . This procedure converts the pair-wise volume changes ∆V i,j for individual interferometric pairs into a cumulative value of volume change at each point t i and t j in time [26]. As described previously, this approach implicitly assumes that the temporal dependence and spatial dependence of the deformation field are separable functions [26,39].…”

“…This procedure converts the pair-wise volume changes ∆V i,j for individual interferometric pairs into a cumulative value of volume change at each point t i and t j in time [26]. As described previously, this approach implicitly assumes that the temporal dependence and spatial dependence of the deformation field are separable functions [26,39]. Accordingly, we write the vector displacement field u as…”

Spatio–Temporal Analysis of Deformation at San Emidio Geothermal Field, Nevada, USA Between 1992 and 2010

“…To determine any temporal trends in the deformation, we performed time-series analysis on the estimated volume changes derived from the spatial deformation modeling. To handle the pair-wise nature of these volume changes, we used a graph-theoretic approach to the procedure known as temporal adjustment, which converts a series of individual, pair-wise volume changes ∆V i,j into cumulative volume change at individual points t i and t j in time [26,31]. We considered several different parameterizations for the temporal function f (t).…”

“…Time series at San Emidio, showing cumulative volume change from temporal adjustment of volume change rates estimated from InSAR data spanning 1992 to 2010. Black lines show the modeled volume change with 68% confidence intervals (dashed lines) as estimated by temporal adjustment with a single-rate temporal function[31]. Red segments indicate measurements of observed volume change derived from individual interferometric pairs.…”