Confidence Limits in Double-Epoch Time-Dependent Seismic Tomography

Reliably mapping temporal variations of seismic wave speeds using local earthquakes is difficult,because the spatial coverage provided by available raysis usually sparse, time-varying,and beyond experimental control. Analyzing data from different epochs independently can lead to spurious apparent changes that are actually caused by changes in ray distributions. We circumvent this problem by inverting data sets from two epochs simultaneously, applying constraints to minimize differences between the models for different epochs as well as to optimize the fit between observations and predictions.

Results obtained using this double-epoch approach inevitably fit observed data at least as well as (and probably much better than) a single-epoch inversion can, because it allows up to twice as many adjustable parameters. This will be true even if there have been no true temporal changes, the additional free parameters merely fitting unavoidable experimental noise.

To assess the reality of apparent wave-speed changes derived by double-epoch inversions, we therefore apply variance-ratio tests ("Ftests"). Such tests require knowledge of the effective number of adjustable parameters in the inversion, which is smaller than the total number of model parameters, because of the effects of regularization and inter-epoch constraints. For approaches such as the generalized inverse or the stochastic inverse, Singular-Value Decomposition of the design-equation matrices provides an objective method for quantifying this effect.

We present examples of derived confidence levels for inferred temporal changes in wave speed, or the lack thereof, in local-earthquake tomography from Mt. Etna, Sicily, and geothermal areas in California.