Probabilistic non-linear waveform inversion method

The waveform inversion method we applied to determine the seismic moment tensor has been developed by Wéber (2005, 2006, 2009). It has already been successfully applied for estimating the full moment tensor of both local and near-regional events in the Pannonian basin (Wéber & Süle 2014, Wéber 2016a, Wéber 2016b).

The waveform inversion procedure works in the point-source approximation. A general seismic point source is described by six independent moment tensor rate functions (MTRFs) allowing the moment tensor to vary arbitrarily as a function of time. If the velocity structure and the event location are known, there is a linear connection between the seismograms and the MTRFs. Basically, the MTRFs are obtained by deconvolving the station specific Green’s functions (GFs) from the observed seismograms.

Especially in case of short epicentral distances, source mislocation can significantly bias the results of any moment tensor inversion procedure. Therefore, we should further refine the hypocentral coordinates using waveform data. Treating the hypocentral coordinates as unknown parameters, however, leads to a non-linear inverse problem. The non-linear waveform inversion procedure I have proposed to solve this problem consists of the following main steps (see the flow diagram below):

Flow diagram

Step 1
The hypocenter of the event is estimated from observed arrival times by a method that provides both the hypocentral coordinates and their uncertainty. The NonLinLoc software package or the Bayesloc multiple-event locator may be a suitable choice for this purpose.
Step 2
Using the hypocenter distribution estimated in Step 1 as a priori information and the observed waveforms as data, we perform a Bayesian inverse calculation to improve the hypocentral location. For mapping the posterior probability distribution (PPD) of the hypocenters, we apply the oct-tree importance sampling algorithm developed by Anthony Lomax and Andrew Curtis. As a result, we get a large number of points that are samples from the PPD of the hypocenter. For all these samples, the MTRFs are also calculated. Their distribution represents the uncertainty of the MTRFs due to that of the source location.
Step 3
Measurement errors and modeling errors also lead to MTRF uncertainty even for a fixed source position. To estimate the overall uncertainties of the retrieved MTRFs, we use a Monte Carlo simulation technique (Rubinstein & Kroese 2008). The goal of Monte Carlo simulation is to determine how random variation in the input data affects the uncertainty of the output. In our problem, source location and seismograms represent the input data, whereas MTRFs are the output. In the course of the simulation, we generate many new realizations of input data sets by randomly generating new hypocenters and waveforms according to their respective distributions. Then each generated input data set is inverted for MTRFs. The distribution of the obtained set of solutions approximates well the PPD of the MTRFs.

After having mapped the PPD of the source location in the previous step, generating a random hypocenter for the Monte Carlo simulation is straightforward. For generating individual realizations of noisy seismograms, we first calculate the waveform residual corresponding to the best MTRF solution obtained in Step 2. We consider this residual as a realization of the measurement and modeling errors. Convolving this error sample with a uniform white noise yields a sample of simulated error: it differs from the original sample of error but has the same amplitude spectrum. Then, we add the simulated error to the observed seismograms and obtain a new realization of waveforms which we invert for the MTRFs.
Step 4
Assuming that the focal mechanism is constant in time, the previously obtained MTRFs are decomposed into a time-invariant moment tensor and a source time function (STF). The problem is non-linear and is solved by an iterative L1 norm minimization technique (Wéber 2009). To allow only forward slip during the rupture process, we impose a positivity constraint on the STF. After the decomposition of the MTRFs, a large number of moment tensor and STF solutions are obtained that can be considered as samples from their respective PPDs. The final estimates are given by the maximum likelihood points.
Step 5
Once the moment tensors are retrieved, their principal axes are deduced. Then each moment tensor is decomposed into an isotropic (ISO) part, representing an explosive or implosive component, and into a deviatoric part, containing both the double-couple (DC) and the compensated linear vector dipole (CLVD) components. The scalar seismic moments and percentages of the DC, CLVD and ISO components are determined according to Vavrycuk (2015). Finally, the distributions of the retrieved source parameters are displayed as histogram plots.

The above described procedure can also be used when the STF is assumed to be known. In that case, Monte Carlo simulation directly results in samples from the posterior moment tensor distribution.


The reported investigation was financially supported by the Hungarian Scientific Research Fund (Nos. T042572 and K68308).


  • Rubinstein, R.Y. & Kroese, D.P., 2008: Simulation and the Monte Carlo Method, John Wiley & Sons, Hoboken, New Jersey.
  • Vavrycuk, V., 2015: Moment tensor decompositions revisited, J. Seismol. 19, 231-252.