# Graph Theory for Analyzing Pair-wise Data: Application to Interferometric Synthetic Aperture Radar Data

Graph theory is useful for estimating time-dependent model parameters via weighted least-squares using interferometric synthetic aperture radar (InSAR) data. 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 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. (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) x 10^6 m^3/yr. Our preferred model provides a reasonable fit that is compatible with viscoelastic relaxation in the five years following the 1997 eruption. Although we demonstrate the approach using volumetric rates of change, our formulation in terms of incidence graphs applies to any quantity derived from pair-wise differences, such as wrapped phase or wrapped residuals.

Date of final oral examination: 05/19/2016

This thesis is approved by the following members of the Final Oral Committee: Kurt L. Feigl, Professor, Geoscience

Michael Cardiff, Assistant Professor, Geoscience

Clifford H. Thurber, Vilas Distinguished Professor, Geoscience

#### Citation Formats

`University of Wisconsin. (2016). Graph Theory for Analyzing Pair-wise Data: Application to Interferometric Synthetic Aperture Radar Data [data set]. Retrieved from https://gdr.openei.org/submissions/1075.`

`Reinisch, Elena. `*Graph Theory for Analyzing Pair-wise Data: Application to Interferometric Synthetic Aperture Radar Data*. United States: N.p., 28 Jul, 2016. Web. https://gdr.openei.org/submissions/1075.

`Reinisch, Elena. `*Graph Theory for Analyzing Pair-wise Data: Application to Interferometric Synthetic Aperture Radar Data*. United States. https://gdr.openei.org/submissions/1075

`Reinisch, Elena. 2016. "Graph Theory for Analyzing Pair-wise Data: Application to Interferometric Synthetic Aperture Radar Data". United States. https://gdr.openei.org/submissions/1075.`

`@div{oedi_1075, title = {Graph Theory for Analyzing Pair-wise Data: Application to Interferometric Synthetic Aperture Radar Data}, author = {Reinisch, Elena.}, abstractNote = {Graph theory is useful for estimating time-dependent model parameters via weighted least-squares using interferometric synthetic aperture radar (InSAR) data. 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 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. (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) x 10^6 m^3/yr. Our preferred model provides a reasonable fit that is compatible with viscoelastic relaxation in the five years following the 1997 eruption. Although we demonstrate the approach using volumetric rates of change, our formulation in terms of incidence graphs applies to any quantity derived from pair-wise differences, such as wrapped phase or wrapped residuals.`

Date of final oral examination: 05/19/2016

This thesis is approved by the following members of the Final Oral Committee: Kurt L. Feigl, Professor, Geoscience

Michael Cardiff, Assistant Professor, Geoscience

Clifford H. Thurber, Vilas Distinguished Professor, Geoscience}, doi = {}, url = {https://gdr.openei.org/submissions/1075}, journal = {}, number = , volume = , place = {United States}, year = {2016}, month = {07}}

#### Details

Data from **Jul 28, 2016**

Last updated
**Jan 27, 2020**

Submitted
**Jul 13, 2018**

#### Organization

University of Wisconsin

#### Contact

Kurt Feigl

608.262.8960

#### Authors

#### Keywords

energy, PoroTomo, InSAR, time series, temporal adjustment, graph theory, poroelastic tomography, thesis, paper, model, parameters, modeling, time-dependent, time-varying, weighted least-squares, inversion, radar, interferometric, synthetic aperture, laplacian, covariance, matrix, MatLab, GraphTreeTA, Alaska, AK, Okmok, volcano, magma, volume, method, implementation, application, viscous, flow, viscoelastic relaxation, remote sensing#### DOE Project Details

Project Name
**PoroTomo Project**

Project Lead
**Elisabet Metcalfe**

Project Number
**EE0006760**