Inversion of levelling data: how important is error treatment?

Abstract

Even if proper treatment of error statistics is potentially essential for the reliability of experimental data inversion, a critical evaluation of its effects on levelling data inversion is still lacking. In this paper, we consider the complete covariance matrix for levelling measurements, obtained by combining the covariance matrix due to measurement errors and the covariance matrix due to non-measurement errors, under the simple hypothesis of uncorrelated non-measurement errors on bench mark vertical displacements. The complete covariance matrix is reduced to diagonal form by means of a rotation matrix; the same rotation transforms the data to independent form. The eigenvalues of the complete covariance matrix give the uncertainties of the transformed independent data. This procedure can be used also with non-normal distributions of errors, in which case misfit functions other than χ2 (e.g. the mean absolute deviation) are minimized. Here we focus on two test cases (the 1989 Loma Prieta earthquake and the 1908 Messina earthquake) inverting both real data and synthetics. The inversion of synthetic data sets does not evidence any systematic dependence of retrieved parameter values on the covariance matrix. Most retrieved fault parameter values are close to what used in the forward model, whatever covariance matrix is used. As a consequence, large discrepancies among results obtained using covariance matrices including different combinations of measurement and non-measurement errors when inverting measured and synthetic data sets would possibly indicate the need for further investigations. While measurement errors can be a priori evaluated, it is difficult to estimate non-measurement errors. Our synthetic tests using a uniform-slip rectangular fault in a homogeneous elastic half-space show that, if measurement errors have been correctly evaluated, average non-measurement errors can be estimated by choosing their weight inside the covariance matrix so that the ratio between the total square residual for the χ2 best-fitting model and the number of degrees of freedom is about one. This result holds even if non-measurement errors are different from bench mark to bench mark and/or not normally distributed. In case of more complex models (e.g. one distributed-slip fault) it is difficult to estimate the actual number of degrees of freedom. Our synthetic tests show that this difficulty can be surmounted by using the Akaike Information Criterion, which actually decreases when turning from a uniform-slip inverse model to a distributed-slip inverse model only if the correct amount of non-measurement errors is included in the covariance matrix.