Realistic Error Modelling for InSAR: Determination of Uncertainties in Earthquake Slip Distributions.

The major source of error in InSAR measurements results from changes in tropospheric water vapour concentrations, creating phase delays that are unrelated to ground motion. These can be distributed over distances of tens of kilometres and, if interpreted as surface deformation, can cause errors in measurement as large as 10 cm. Here we present a simple modified Monte Carlo (MC) method for determining the impact of these errors on the accuracy of model parameters derived from InSAR data. In particular, we examine the reliability of InSAR-derived earthquake slip distributions. Conventional MC bootstrap methods are often used for determining errors in model parameters derived from InSAR data. An ensemble of best-fit parameter estimates is found using different input data sets. Each of these data sets is derived from the original, but has its individual phase measurements randomly perturbed in a normal distribution about their original value using an a priori standard deviation. Errors in model parameter estimates are found from the distribution of best-fit solutions to each perturbed data set. For InSAR data, however, conventional MC fails to account for the spatial correlation of atmospheric errors between multiple sampled phase measurements. When the interferogram is sampled densely compared to the wavelength of atmospheric errors, conventional MC can grossly underestimate the errors of model parameter estimates. To produce realistic error bars for parameter estimates, the interferogram's variance-covariance matrix (VCM) must first be determined. A practical approach for this is to determine the mean covariance vs distance function (autocorrelation function), either spatially or from the interferogram's power spectrum using the Wiener-Khinchine theorem (e.g. Hanssen, 2001). This must be done using a part of the interferogram away from the deformation, or, where this is not possible, after a first-pass model has been removed. Using the covariance vs distance function, an approximate VCM can be generated for the sampled points of the interferogram. It is then possible to synthesise an ensemble of perturbed data sets that will have this VCM. These perturbed data sets can be used to produce realistic errors for model parameter estimates. We use this technique to examine the reliability of earthquake slip distributions derived from InSAR. Plots of the distribution of slip error on the fault plane are powerful tools for distinguishing those areas of the fault plane where slip patterns should be believed and those where they are spurious.