In many practical parameter estimation problems, the observation model is periodic with respect to the unknown parameters. In these cases, the appropriate estimation criterion is periodic in the parameter space, and cyclic performance bounds should be used. However, existing cyclic performance bounds do not account for the common scenario of model misspecification. The misspecified Cramér-Rao bound (MCRB) provides a lower bound on the mean-squared-error (MSE) for estimation problems under model misspecification. However, the MCRB does not provide a valid bound for periodic problems. In this paper, we close this gap by developing the cyclic MCRB, which is a lower bound on the mean cyclic error (MCE) of any Lehmann unbiased estimator under model misspecification in periodic estimation problems. Thus, it can be seen as a generalization of the cyclic CRB for cases where the assumed model (observations distribution) may be different from the true one. The proposed cyclic MCRB and the performance of the misspecified maximum likelihood (MML) estimator are compared in terms of MCE in direction-of-arrival (DOA) estimation under the misspecified assumption of white additive noise, where the true covariance is colored.

Magnitudes are common and important measures for the size of seismic events. The International Data Centre (IDC) of the Comprehensive Nuclear-Test Ban Treaty Organization estimates an event magnitude by averaging the magnitudes calculated by individual stations that detected the event, excluding outliers. This approach assumes that all station magni-tudes have the same error level and are unbiased, namely, they have no systematic errors. We show that the body-wave and surface-wave magnitudes published in the Reviewed Event Bulletin (REB) of the IDC are inconsistent with these assumptions. We thus consider a model where each station has an unknown bias and error level. Given a large collection of reported event magnitudes by a network of monitoring stations, we propose a novel approach to estimate each individual station's bias and error level. From a statistical perspec-tive, this is a challenging problem involving a huge number of variables, because in addition to the stations' biases and error levels, the event magnitudes are also unknown. Our approach is based on analyzing differences between reported magnitude values at pairs of stations, which cancels out the unknown event magnitudes and allows us to derive a sim-ple and computationally efficient algorithm. We use the estimated station biases as station correction terms and the estimated error levels to compute weights for event magnitude estimation. Using a large data set from the REB with millions of reported station magnitudes, we show that our approach yields more consistent station and event magnitudes.

In many practical parameter estimation problems, such as phase, frequency, and direction-of-arrival (DOA) estimation, the observation model is periodic with respect to the unknown parameters, and thus, the appropriate estimation criterion is periodic in the parameter space. However, iterative estimation methods, such as the Fisher-Scoring method, do not take into consideration the periodic information in order to improve the accuracy of the estimation. In this paper, we present a new iterative method - periodic Fisher-Scoring, which takes into account the signal’s periodic properties through the utilization of the cyclic Cramér-Rao bound (CRB). The cyclic CRB is a lower bound on the mean cyclic error (MCE) of unbiased estimators and, thus, is more appropriate for the derivation of the iterative method. In addition, the periodic Fisher-Scoring method uses the modulo $2 \pi$ operator at each iteration. Simulation results for DOA estimation in seismic arrays show that the proposed periodic Fisher-Scoring estimator has a lower MCE compared to the conventional Fisher-Scoring estimator. The performance improvement is more significant around the edges of the range $[-\pi,\pi]$ and under the misspecified model, i.e. under the mismatched assumption of white noise. We also show that the periodic Fisher-Scoring estimator achieves the cyclic CRB much faster than the CRB.

Manifold learning is a branch of machine learning that focuses on compactly representing complex data-sets based on their fundamental intrinsic parameters. One such method is diffusion maps, which reduces the dimension of the data while preserving its geometric structure. In this work, diffusion maps are applied to several seismic event characterization tasks. The first task is automatic earthquake-explosion discrimination, which is an essential component of nuclear test monitoring. We also use this technique to automatically identify mine explosions and aftershocks following large earthquakes. Identification of such events helps to lighten the analysts’ burden and allow for timely production of reviewed seismic bulletins. The proposed methods begin with a pre-processing stage in which a time–frequency representation is extracted from each seismogram while capturing common properties of seismic events and overcoming magnitude differences. Then, diffusion maps are used in order to construct a low-dimensional model of the original data. In this new low-dimensional space, classification analysis is carried out. The algorithm’s discrimination performance is demonstrated on several seismic data sets. For instance, using the seismograms from EIL station, we identify arrivals that were caused by explosions at the nearby Eshidiya mine in Jordan. The model provides a visualization of the data, organized by its intrinsic factors. Thus, along with the discrimination results, we provide a compact organization of the data that characterizes the activity patterns in the mine. Our results demonstrate the potential and strength of the manifold learning based approach, which may be suitable to other in other geophysics domains.

Seismic bulletins, with trustworthy phase picks, origin times, and source locations are key for regional seismic studies, such as travel-time (TT) tomography, attenuation tomography, and anisotropy studies. To lay the groundwork for such studies in Israel, we revised the seismic bulletin of Israel and the surrounding area and obtained a trustworthy TT data set. From the earthquake and explosion bulletins of the Geophysical Institute of Israel, we compiled a starting data set of about 123,000 earthquakes and explosions that occurred during the past 40 yr. After screening out the poorly recorded events, we were left with a data set of ∼38,000 well-recorded events. We then revised the remaining data set in two consecutive steps. In the first, we reviewed and updated station metadata, including changes in station metadata parameters over time. In the second step, we jointly relocated a list of selected seismic events, using the Bayesian hierarchical location software package (BayesLoc) of Myers et al. (2007) that performs joint relocation of multiple events. We observed striking dissimilarities between the spatial distributions of the newly relocated catalog and the initial locations. Although the depth distribution of the starting catalog is trimodal with peaks at 0, 5, and 10 km, the distribution in this study is unimodal, with a broad peak between 7.5 and 12.5 km. By differencing the observed arrival times and the origin times obtained through relocation with BayesLoc, we obtained a revised TT database that consists of 261,336 Pg, 132,876 Pn, 114,816 Sg, and 60,394 Sn arrivals, from a set of 30,458 jointly relocated seismic sources. We compared prerevision and postrevision TTs as a function of epicentral distance and concluded that the revised data set contains far fewer outliers and inconsistencies than the original data set. The revised TT data set may be used for seismic studies, such as TT tomography, attenuation tomography, and anisotropy studies.

For the past 40 years, the Geophysical Institute of Israel has been in charge of the recording, monitoring and relocating of local earthquakes. Due to the variety of data analysts and data sources, as well as several network upgrades, the resulting bulletin data has to be completed and homogenised, and station metadata needs to be tracked down, and sometimes corrected. For those reasons, as well as because of the lack of consensus on an accurate model for seismic velocities in the area, published source locations are often poorly constrained. We present a homogenised Israeli bulletin, including natural and man-made explosion data. We extract sets of seismic sources with location accuracy greater than 5 km (GT5), as well as GT0 explosions.We select a set of events with the highest network coverage, comprising (1) natural earthquakes, (2) man-made quarry or mine blasts, (3) GT5 earthquakes or explosions, and (4) GT0 explosions. We relocate them altogether using the BayesLoc package, a Bayesian, hierarchical, multi-event locator which produces, after source relocation, event-, station- and phase-specific correction terms. We put different a priori constraints on the different categories of seismic events, allowing poorly constrained origin parameters to improve thanks to the more accurate GT locations. BayesLoc also produces traveltime correction terms that can be used to correct systematic errors in the dataset, as well as error estimates.Eventually, we invert this homogenised local traveltime dataset in order to invert for a P-wave crustal velocity model of Israel and its surroundings. To do so, we use the Fast Marching Tomography package, which allows the representation of a wide variety of input structures (starting model and geometry of layer boundaries) and can take many different types of input data. We show preliminary inversion tests and results that are in good agreement with past local studies.This crustal model of Israel is ultimately to be used as a starting model in a larger tomographic study of the Eastern Mediterranean and Middle East region, where the Regional Seismic Travel Time approach is to be expanded, in order to improve the CTBT’s capabilities in monitoring the regional seismicity. Eventually, such a velocity model could also be used to relocate the whole earthquake catalogue more accurately, and improve the Earthquake Early Warning System currently in development in Israel.