Crucial to the development of earthquake forecasting schemes is the manifestation of spatiotemporal correlations between earthquakes as highlighted, for example, by the notion of aftershocks. Here, we present an analysis of the statistical relation between subsequent magnitudes of a recently proposed self-similar aftershock rates model of seismicity, whose main distinguishing feature is that of interdependence between trigger and triggered events in terms of a time-varying frequency–magnitude distribution. By means of a particular statistical measure, we study the level of magnitude correlations under specific types of time conditioning, explain their provenance within the model framework and show that the type of null model chosen in the analysis plays a pivotal role in the type and strength of observed correlations. Specifically, we show that while the variations in the magnitude distribution can give rise to large trivial correlations between subsequent magnitudes, the non-trivial magnitude correlations are rather minimal. Simulations mimicking southern California (SC) show that these non-trivial correlations cannot be observed at the

An outstanding question in earthquake dynamics is how reliably one is able
to

It is through the aforementioned constitutive statistical models
that forecasting of seismicity is often implemented

We first give a brief overview of the SSAR model (Sect.

The SSAR model recasts the standard Omori–Utsu rate equation into
a self-similar version. A distinguishing feature of the rate equation
in the SSAR model is that it only depends on the difference between
mother–daughter events making it self-similar (e.g., the rates
of a magnitude 3 mother and magnitude 2 daughter event are the same
as those of a magnitude 5 mother and a magnitude 4 daughter event); the scaling relation along with the relationship amongst its exponents was explored in the context of the rates for all events above a magnitude

Plot of Eq. (

Analogously to the ETAS model, the full SSAR model is given by a time-varying seismic rate (also called the conditional intensity or stochastic intensity), which takes on the following form:

Since the SSAR model was tested using a catalog from SC

Time–magnitude plot for a realization of the SSAR model with a lower magnitude threshold of

In this section we aim to answer the question of what is the type and strength of the effective magnitude correlations in the SSAR model (for an analysis of magnitude correlations in the SC catalog, see

Our study of magnitude correlations, similar to methods used in

Another important aspect in our analysis is how we randomly choose the magnitudes

For all three types of conditioning one can state the following. If the quantity

In Fig.

Magnitude correlations of the unconditioned SSAR-SC catalog for

To test whether magnitude correlation becomes stronger if one considers pairs of events that are related, we now focus on the magnitude correlation analysis for

Magnitude correlations for the SSAR-SC catalog with

In contrast, when estimating

As discussed above, the underlying difference between sub-catalog and full-catalog randomizing for

While in our model simulations we can readily identify mother–daughter pairs, i.e., the ground truth is known, this is not the case for field data. Thus, for such catalogs – including the SC catalog – one would need to infer mother–daughter pairs, i.e., decluster the catalog first

Magnitude correlations for the SSAR-SC catalog with

To clarify the reason for the difference between the

Ratio of total number of events satisfying

The aforementioned maximization comes with a trade-off; although a higher

Through a particular statistical measure (Sect.

Finally, the significantly higher strength of the trivial correlations compared to the non-trivial correlations is the main outcome of our analysis. Thus, when it comes to improving earthquake forecasting efforts our analysis leads us to believe that looking at the time variations in the frequency–magnitude distribution could perhaps be a more fruitful approach then focusing on non-trivial correlations. Using the SSAR model instead of the ETAS model in existing forecasting frameworks would be one way to utilize this. This remains a challenge for the future.

Python code, synthetic catalog data and plot data can be found in

The supplement related to this article is available online at:

AFZM prepared the paper with corresponding discussions, edits, contributions and modifications from JD. AFZM developed the model code, statistical analysis code and created the figures.

The authors declare that they have no conflict of interest.

Andres F. Zambrano Moreno would like to thank Jordi Baro for providing ETAS C++ code which helped greatly in understanding the type of coding that would be required for the creation of the SSAR code and for helpful consultations, Mohammed Yaghoobi for helpful discussions on the interpretation of the magnitude correlation plots, and Ayush Mandawal for lending an ear and participating in general dialogues on the topic.

This research has been partially supported by NSERC through a Discovery Grant to Jörn Davidsen.

This paper was edited by Ilya Zaliapin and reviewed by Robert Shcherbakov and one anonymous referee.