Detection Is Not Identifiability: An Observability Theory for Earthquake Source Inference from InSAR

Summary Interferometric Synthetic Aperture Radar (InSAR) provides spatially dense measurements of coseismic deformation, but detection of a deformation signal does not imply that its generating earthquake source is well constrained or that heterogeneous fault slip is spatially recoverable. This study develops an information-theoretic framework that separates three levels of earthquake-source inference: statistical detection, local source-parameter identifiability and finite-fault recoverability. Detection is quantified using a whitened known-template statistic; local identifiability is characterised through the eigenspectrum of a prior-scaled Fisher information matrix, from which an effective identifiable dimension, information gain and invariant weak-subspace diagnostics are derived; and finite-fault recoverability is quantified using a prior-conditioned slip-information operator and an effective recoverable slip dimension. Controlled elastic-dislocation experiments vary earthquake magnitude, centroid depth, fault dip, viewing geometry, spatial covariance, nuisance components, data loss, spatial sampling and complementary observations. To isolate fault geometry, the principal benchmarks use common strike, rake and rupture scaling and vary only dip between ${30}^ \circ $, ${55}^ \circ $ and ${85}^ \circ $. At 10 km centroid depth, the ${P}_{{\rm{det}}} = 0.5$ boundary precedes the illustrative ${d}_{{\rm{eff}}} = 5$ level by 0.600, 0.533 and 0.436 magnitude units for the three dips, respectively; this ordering persists wherever both transitions lie within the analysed magnitude–depth domain. Complementary line-of-sight geometries and additional geodetic observations increase information when they constrain source directions that remain weak in the existing data, whereas correlated noise, selective removal of informative observations and spatial decimation substantially reduce local identifiability. Finite-fault experiments show that numerical model dimension and information-supported dimensionality can diverge strongly: at 10 km depth, increasing the discretisation from 12 to 160 slip patches raises the prior-conditioned effective recoverable dimension only from 5.88 to 6.56, while at 20 km depth it changes from 2.86 to 2.94. The resulting magnitude–depth regime maps demonstrate that InSAR earthquake-source observability is task dependent: detection, local source identifiability and finite-fault recoverability quantify different properties of the same source–observation system and should therefore be evaluated separately.