# Time-only Molchan test for the tectonic waves (time-dependent)

Minimum magnitude Mc = 6 and Mc = 6.5

Molchan test is a standard test used by the Collaboratory for the Study of Earthquake Predictability (CSEP) for the evaluation of earthquake forecasts. Molchan test is alarm-based and evaluates whether a given alarm function has a skill, that is if it chooses better alarm sets than a random function.

The Molchan diagram shows the fraction of space-time occupied by alarms (x-axis) versus the miss-rate (y-axis). In the time-only Molchan test, the x-axis shows the fraction of time occupied by alarms.

According to the Omega-Theory the time dependent shear traction model is built from the time- invariant reference model (average seismic potential, Ps(r)) in the following way:

S(r,t) = (Ps(r) + δPs(r,t)) x W(r,t); the complete model “MES total N[1]”.

Here, δPs(r,t) = 0.5 dN(r,t) is the dynamic increase of the seismic potential due to an anomalously increased rate of events dN(r,t) in the hot spots (aftershocks, foreshocks, or swarms), and W(r,t) is the effect of the tectonic waves.

## Step 1: Elimination of hot spots (graphs above)

To show the effect of the tectonic waves, the effect of the hot spots (δPs(r,t) term) should be eliminated by slightly randomizing the epicenters of the earthquakes up to dε degrees. The graphs above illustrate such tests for the reference model (average seismic potential, left graphs) and the dynamic seismic potential (right graphs). The results show that randomizing the epicenters of the earthquakes leads to the “negative” skill of the reference model with the area skill score below 0.5. The results for the dynamic seismic potential show that this model has no skill for dε = 2.3 degrees for M5.5+ earthquakes and dε = 1.5 for M6.5+ earthquake. The area skill score (af) is close to 0.5.

## Step 2: Test of non-randomness of the alarm function (left graphs below)

The tectonic waves produce seismic states (Žalohar, 2018), which are only accurate up to a time error of dt. Seismic states are related to geometric sequences of earthquakes (the Omega-sequences), however, natural Omega-sequences are not perfectly geometric. Forecasting earthquakes based on such natural Omega-sequences is thus necessarily confronted with limited accuracy dt. Therefore, the QEFS generates 7-days compilation charts (forecasts) where dt = 3.5. This means that earthquakes are considered as successful forecasts if they happen within the time window ± dt around the alarms. This time window should also be considered in the Molchan test, however, applying dt in this test artificially increases the skill even for the random noise. Therefore, the Molchan test for the tectonic waves must include the non-randomness test by randomizing the times of earthquakes. In this case, it is expected that the alarm function ( = shear traction) has no skill for the time-randomized earthquakes, which is confirmed in the left graphs below. The area skill score (af) is close to 0.5. We used dε = 2.3 for M5.5+ earthquakes and dε = 1.5 for M6.5+ earthquakes. We also used dt = 4.

This test shows that the shear traction has little to do with random earthquakes and is thus non-random.

## Step 3: Test of the effect of the tectonic waves (right graphs below)

After the Molchan tests in Step 1 and Step 2 described above showed no skill related to hot spots and random earthquakes, the Molchan test can be performed for the alarm function ( = shear traction) and for the non-randomized (original) times of earthquakes. Now, it is expected that some gain in skill should be detected, which is confirmed in the right graphs below. The gain of skill is not related to hot spots and random earthquakes, but to the tectonic waves directly.

**References:**

- Molchan, G.M., 1990.
*Strategies in strong earthquake prediction*. Physics of the Earth and Planetary Interiors 61, 84-98. - Molchan, G.M., 1991.
*Structure of optimal strategies in earthquake prediction*. Tectonophysics 193, 267-276. - Molchan, G.M., Kagan, Y.Y., 1992.
*Earthquake Prediction and Its Optimization*. Journal of Geophysical Research 97/B4, 4823-4838. - Molchan G.M., 2012.
*On the testing of seismicity models*. Acta Geophysica 60, 624-637. - Zechar, D.J., Jordan, T.H., 2007.
*Testing alarm-based earthquake predictions*. Geophysical Journal International, 172(2), 715-724. - Žalohar, J., 2018.
*The Omega-Theory; A New Physics of Earthquakes*. Elsevier, 558 pp.