**2.3.1 Earthquake forecasting and its verification**

Holliday et. al (2005) has based their forecast research on the association of occurrence of small earthquakes with probably future large ones. In fact, the method does not predict earthquakes, but spots regions (Hotspots regions) where they are most likely to occur in the future (about ten years).

Basically the research objective is to reduce risk areas analyzing the historical seismicity for anomalous behaviour.

The approach is based on a pattern informatics (PI) method which quantifies temporal variations in seismicity and is as follows Holliday et al, (2005):


4. The seismic intensity in box *i*, *Ii(tb, t)*, between two times *tb < t*, can then be defined as the average number of earthquakes with magnitudes greater than *Mc* that occur in the box per unit time during the specified time interval *tb to t*. Therefore, using discrete notation, we can write:

$$I\_t(t\_b, t) = \frac{1}{t - t\_b} \sum\_{t'=t\_b}^{t} N\_t(t'), \tag{1}$$

Earthquake Prediction: Analogy with Forecasting

of data with others different models.

adopted into mitigation strategies.

Philippines.(Kagan, & Jackson, 2000)

**2.3.2 Probabilistic forecasting of earthquakes** 

stochastic model and updated daily (see Table 1).

large earthquakes.

earthquakes.

preferentially in hotspots during the forecast time interval *t2 to t3*.

Models for Cyber Attacks in Internet and Computer Systems 111

change time interval (activation) or due to decreases (quiescence). The following hypothesis is taken into account: earthquakes with magnitudes larger than *Mc + 2* will occur

To evaluate the model a Relative Operating Characteristic (ROC) diagram, which can be viewed as binary forecast either to occur or not to occur, was used and presented significant results with a relative high proportion of hotspots representing locations of probably future

Although good results, the model Holliday, J.R., et. al, (2005) could be used as an input in a larger forecast system like DIFSA which would provide the communication and correlation

(Kagan, & Jackson, 2000) has developed a research with both short and long-term forecast approach and testing both with a likelihood function to 5.8-magnitude (or larger) quakes. Although the long-term approach (see Table 1), is not completely developed and is suitable to estimation of occurrence of earthquakes, it is derived from statistical, physical and intuitive arguments while the short-term forecast seismicity model is based on a specific

The research assumes that the rate density (probability per unit area and time) is proportional to a smoothed version of past seismicity and depends approximately on a negative power of the epicentral distance and linearly on magnitude of the past

The model (Kagan, & Jackson, 2000) does not use retrospective evaluation of seismic data. The parameters of long-term are evaluated on the basis of success in the forecasting of seismic activity also indicating possible earthquakes perturbations. A maximum likelihood procedure to infer optimal values are applied on short-term approach which can be

About the scientific results (Kagan, & Jackson, 2000) concluded that the research depicted a statistical relationship between successive earthquakes in a quantitative way that facilitate hypothesis testing. About the practical results the quantitative predictive assessment can be

incorporated into real-time seismic networks to provide seismic hazard estimate.

Table 1. Example of long- and short-term forecast, 1999 February 11, north of

Where the sum is performed over increments of the time series, say days.

5. In order to compare the intensities from different time intervals, it is required that they have the same statistical properties. Therfore, the seismic intensities are normalized by subtracting the mean seismic activity of all boxes and dividing by the standard deviation of the seismic activity in all boxes. The statistically normalized seismic intensity of box *i* during the time interval *tb* to *t* is then defined by

$$\dot{I}\_i(t\_b, t) = \frac{I\_i(t\_b, t) - \prec I\_i(t\_b, t)}{\sigma(t\_b, t)},\tag{2}$$

Where < *Ii(tb, t)* > is the mean intensity averaged over all the boxes and *ϭ(tb, t)* is the standard deviation of intensity over all the boxes.

6. The measure of anomalous seismicity in box *i* is the difference between the two normalized seismic intensities:

$$
\Delta I\_i(t\_b, t\_1, t\_2) = \bar{I}\_i(t\_b, t\_2) - \bar{I}\_i(t\_b, t\_1). \tag{3}
$$

7. To reduce the relative importance of random fluctuations (noise) in seismic activity, the average change in intensity is computed, *∆Ii(t0, t1, t2)* over all possible pairs of normalized intensity maps having the same change interval:

$$
\Delta I\_t(t\_0, t\_1, t\_2) = \frac{1}{t\_1 - t\_0} \sum\_{t\_b = t\_0}^{t\_1} \Delta I\_t(t\_b, t\_1, t\_2), \tag{4}
$$

Where the sum is performed over increments of the time series, which here are days.

8. The probability is defined as a future earthquake in box *i, Pi(t0, t1, t2, )*, as the square of the average intensity change:

$$P\_t(t\_0, t\_1, t\_2, \cdot) = \underline{\Delta I\_t(t\_0, t\_1, t\_2)}^\*,\tag{5}$$

9. To identify anomalous regions, it is desirable to compute the change in the probability *Pi(t0, t1, t2, )* relative to the background so that we subtract the mean probability over all boxes. This change in the probability is denoted by

$$
\Delta P\_{\ell}(t\_0, t\_1, t\_2) = P\_{\ell}(t\_0, t\_1, t\_2) - < P\_{\ell}(t\_0, t\_1, t\_2) >,\tag{6}
$$

Where *< P (t , t , t ) >* is the background probability hotspots are defined to be the regions where *∆Pi(t0, t1, t2)* is positive. In these regions, *Pi(t0, t1, t2)* is larger than the average value for all boxes (the background level). Note that since the intensities are squared in defining probabilities the hotspots may be due to either increases of seismic activity during the

4. The seismic intensity in box *i*, *Ii(tb, t)*, between two times *tb < t*, can then be defined as the average number of earthquakes with magnitudes greater than *Mc* that occur in the box per unit time during the specified time interval *tb to t*. Therefore, using discrete

5. In order to compare the intensities from different time intervals, it is required that they have the same statistical properties. Therfore, the seismic intensities are normalized by subtracting the mean seismic activity of all boxes and dividing by the standard deviation of the seismic activity in all boxes. The statistically normalized seismic

(2)

6. The measure of anomalous seismicity in box *i* is the difference between the two

7. To reduce the relative importance of random fluctuations (noise) in seismic activity, the average change in intensity is computed, *∆Ii(t0, t1, t2)* over all possible pairs of

Where the sum is performed over increments of the time series, which here are days. 8. The probability is defined as a future earthquake in box *i, Pi(t0, t1, t2, )*, as the square of

 (5) 9. To identify anomalous regions, it is desirable to compute the change in the probability *Pi(t0, t1, t2, )* relative to the background so that we subtract the mean probability over all

 (6) Where *< P (t , t , t ) >* is the background probability hotspots are defined to be the regions where *∆Pi(t0, t1, t2)* is positive. In these regions, *Pi(t0, t1, t2)* is larger than the average value for all boxes (the background level). Note that since the intensities are squared in defining probabilities the hotspots may be due to either increases of seismic activity during the

Where < *Ii(tb, t)* > is the mean intensity averaged over all the boxes and *ϭ(tb, t)* is the

Where the sum is performed over increments of the time series, say days.

intensity of box *i* during the time interval *tb* to *t* is then defined by

standard deviation of intensity over all the boxes.

normalized intensity maps having the same change interval:

boxes. This change in the probability is denoted by

normalized seismic intensities:

the average intensity change:

(1)

(3)

(4)

notation, we can write:

change time interval (activation) or due to decreases (quiescence). The following hypothesis is taken into account: earthquakes with magnitudes larger than *Mc + 2* will occur preferentially in hotspots during the forecast time interval *t2 to t3*.

To evaluate the model a Relative Operating Characteristic (ROC) diagram, which can be viewed as binary forecast either to occur or not to occur, was used and presented significant results with a relative high proportion of hotspots representing locations of probably future large earthquakes.

Although good results, the model Holliday, J.R., et. al, (2005) could be used as an input in a larger forecast system like DIFSA which would provide the communication and correlation of data with others different models.
