Inversion of levelling data: how important is error treatment?

Abstract

S U M M A R YEven 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 nonmeasurement 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 χ 2 (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 χ 2 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.

Paired deformation sources of the Campi Flegrei caldera (Italy) required by recent (1980-2010) deformation history

“…The uplift covariance matrix is not diagonal and cannot be computed exactly because of lack of information on original measurements and the adjustment procedure. We use a nondiagonal covariance matrix obtained by combining the nondiagonal covariance matrix due to leveling measurement errors (, proportional to , L being section length in km) and the diagonal covariance matrix due to nonmeasurement errors, under the simple hypothesis of uncorrelated nonmeasurement errors on benchmark uplifts (0.01 m, estimated as in Amoruso and Crescentini []).…”

Modelling deformation due to a pressurized ellipsoidal cavity, with reference to the Campi Flegrei caldera, Italy

“…Accurate inversion of real data requires the use of trade‐off curves to balance the different data sets [e.g., Amoruso et al , 2008], information criteria to select the best model [e.g., Amoruso and Crescentini , 2007], and a more realistic representation of the medium; here we show that these criteria are not sufficient when inverting superficial displacements using a general moment tensor when a vertically flattened ellipsoidal source is involved. This may be the case for CF, and may be true as well for other calderas.…”

A horizontal crack in a layered structure satisfies deformation for the 2004–2006 uplift of Campi Flegrei

“…Data inversion leads to minimizing a cost function which measures the disagreement between model and observations for different model parameters. We use two different cost functions, namely the mean squared deviation of residuals (chi‐square fitting, , appropriate for normally distributed errors) and the mean absolute deviation of residuals (, appropriate for two‐sided‐exponentially distributed errors and commonly used for robust fitting) [e.g., Amoruso et al , 2002]: where j = 1, …, M indicates different data sets, w j is the weight of each data set in the cost function, i = 1, …, N j indicates rotated independent data ( x i ) in each data set [ Amoruso and Crescentini , 2007], f i ( a ) is model prediction of x i , and σ i is uncertainty of x i . Here we use w j = 1 for each data set.…”