Omori-Utsu Parameters

## 1. Introduction

This chapiter is dedicated to the analysis of some aftershock sequences occurred in Algeria-Morocco region, namely Al Hoceima earthquakes of May 26, 1994 (Mw6.0) and February 24, 2004 (Mw6.1) which occurred in northern Morocco, the October 10, 1980 El Asnam earthquake (Mw7.3), the May 21, 2003 Zemouri earthquake (Mw6.9) and March 26 2006 Laalam earthquake (Mw5.2) in northern Algeria.

Aftershock sequence is usually attributed to the strain energy not released by the mainshock. Statistical properties of aftershocks have been extensively studied for long time. Most of them dealt with the distribution of aftershock in time, space and magnitude domain. Several authors have noted the importnace of systematic investigation of aftershock sequences to earthquake prediction and a number of statistical models have been proposed to describe seismicity characters in time, space and magnitude (Utsu, 1961; Utsu et al., 1995; Guo and Ogata, 1997; Ogata, 1999). It is also well known that aftershock sequence offer a rich source of information about Earth's crust and can provide and understanding of the mechanism of earthquakes, because tens of thousands of earthquakes can occur over a short period in small area. The tectonic setting and the mode of faulting are factors others than fault surface properties that might control the behavior of the sequence (Kisslinger and Jones, 1991; Tahir et al., 2012). It is widely accepted that aftershock sequence presents unique opportunity to study the physics of earthquakes. In the same time importnat questions concern the fundamental origin of some widely applicable scaling laws, as the Gutenberg-Richter frequency-magnitude relationship, the modified Omori law or the Omori-Utsu law for aftershock decay and Bath's law for the difference between the magnitude of the largest aftershock and mainshock.

The frequency-magnitude distribution (Gutenberg and Richter, 1944), firstly examinated, describes the relation between the frequency of occurence and magnitude of earthquake

where,

The temporal decay of aftershocks is important task because it contains information about seismogenic process and physical condition in the source region. It is well known that the occurrence rate of aftershock sequences in time is empirically well described by the Omori-Utsu law as proposed by Utsu (Utsu, 1969). The power law decay represented by the Omori-Utsu relation is an example of temporal self-similarity of the earthquake source process. The variability of the parameter

A model describing the temporal behavior of cumulative moment release in aftershock sequences is analyzed as alternative approach with respect to the Omori-Utsu law. In this model the static fatigue is assumed to be the principal explanation of the aftershock temporal behavior. Following Marcellini (1997), the different aftershock sequences are analyzed using this model. The most important result shows that globally the model explain and fit correctly the data.

Modified Bath's law is analyzed for each aftershock sequence, it allows us to derive some important properties related to the energy partionning. This approach is used to derive the energy dropped during the mainshock.

The fractal dimension

## 2. Tectonic sketch and seismicity overview

In this section we give a large overview of the tectonic skech and the seismicity of the studied region (Pelaéz et al., 2012). The Morocco-Algeria region, namely the Maghrebian region (Fig. 1) occupies the NW part of the African (Nubia) Plate in what is referred to its continental crust. Its oceanic crust continues till the area of the Azores Islands. To the North it is immediately situated the Eurasian Plate, although between the Gibraltar Arc and the South of Italy an intermediate complex domain is intercalated. This domain is formed by some oceanic basins, as is the Algero-Provençal Basin and the Thyrrehnian Basin, and by a former region, presently disintegrated and now forming the Betic-Rifean Internal Zone, the Kabylias (in Algeria), the Peloritani Mountains (Sicily) and the Calabrian area in Italy (AlKaPeCa domain). This area underwent from the early Miocene a northwards subduction of Africa, then opening the small oceanic basins quoted, accompanied by the disintegration of the AlKaPeCa domain.

Presently, the convergence between the Nubia Plate and Iberia has an approximate NNW-SSE direction, with values of the order of 3 to 5 cm/year, according the places. This compression is accompanied, at least in the area of the Gibraltar Arc (in the Alboran Sea) by a noticeable ENE-WSW tension, in some cases even more important than the compression. For this reason, in the Alboran area, some extensional movements can be important The Maghrebian region is also a complex area in which the Saharan Shield affected by the Pan-African Orogeny (Precambrian to early Cambrian) is in contact with the Atlasic Mountains of mainly Alpine age (Fig. 1). To the North of the Atlas is situated the Moroccan Meseta, to the West, and the High Plateaus in Algeria, which to the North contact with the Rif and Tell mountains, typically Alpine chains. The Saharan Shield forms part of the Precambrian areas of Africa, clearly cratonized and generally not affected by later important deformations. In fact, in the Maghrebian area it corresponds to a clearly stable area. In Morocco, the so called Antiatlas, corresponds to a Precambrian and, mainly, Paleozoic area, making a tectonic transition between the shield and the Atlas. The Atlasic Mountains correspond to an intracontinental chain. To the West, in Morocco, the High Atlas reach the coast in the Agadir area and continues to the Northeast and East, passing, although with lesser heights, to the Saharan Atlas, which cross Algeria and reach the central part of Tunis. They can be considered as aulacogens bordering the northern part of the Saharan shield. To the North, the Middle Atlas in Morocco has a different direction, NE-SW, separating the Moroccan Meseta and the High Plateaus, both forming by Paleozoic rocks, although with a Mesozoic and Tertiary cover, well developed in some areas. On the whole, the Atlasic Mountains has been tectonically unstable from the Triassic times, and along the Alpine orogeny suffered important deformations and, more recently, also important volcanism, reaching the Quaternary. The Rif and Tell thrust southwards the Moroccan Meseta and the High Plateaus, and even in some places part of the Atlasic Mountains. They are formed by sedimentary External zones (only slightly affected by metamorphism in some Moroccan places) and by Internal zones. Mostly Internal zones (divided in several tectonic complexes) are affected by alpine metamorphism, moreover the existence of previous Paleozoic and even older deformations. In any case, their present structure has being formed during the Alpine Orogeny. They appear mainly to the E of Tetuan, in Morocco, and in the Kabylias, in Algeria. These Alpine chains have being structured from the Cretaceous to the Oligocene-early Miocene. Later, were formed numerous Neogene basins, clearly cutting in many cases previous structures. In this time, particularly from the late Miocene to the present, a near N-S compression provoked the existence of strike-slip faults (NE-SW, sinistral, and NW-SE, dextral), moreover reverse fault, many of which has N70ºE to E-W direction. In many cases, the cited strike-slip faults moved mainly as normal faults, releasing by this way the regional tension, practically perpendicular to the compression.

In this study, we analyze the aftershock sequences triggered by some important and damaging earthquakes which occurred in the Morocco-Algeria region. The seismic activity in this region is mainly characterized by moderate to destructive magnitude events. It has been the site of numereous historical earthquakes, and was therefore subject to extensive damage and several thousands of casualties in the past. Among the largest recorded earthquake in Morocco, the February 29, 1960 Agadir (Morocco) earthquake (Ms6.0), the May 26, 1994 and February 24, 2004 Al Hoceima earthquakes with Mw6.0 and Mw6.1 respectively (Pelàez et al., 2007). In northern Algeria, earthquakes up to magnitude Ms 7.3 have been recorded, namely the October 10, 1980 El Asnam earthquake (Ms 7.3) (Ouyed et al.,1981) and May 21, 2003 Zemouri earthquake (Mw6.9) earthquake, (Hamdache et al., 2004). It is important to point out that these two events were the most damaging events occurred in northern Algeria and, the region of EL Asnam experienced in the past damaging earthquake on 09 September 1954 (Hamdache et al., 2010). The original epicentral database and the arrival time readings of the different aftershock sequences used, were obtained from the Spanish National Geographical Institut (IGN) earthquake catalog. For the sequences triggered by the May 21, 2003 Zemouri earthquake and by March 20, 2006 Laalam event, we used the data recorded by portable seismological stations network monitored by CRAAG. It is well known that the process to identify aftershock is related to the seismicity declustering process, a crucial step in separating an erthquake catalog into foreshock, aftershock and mainshock. This process is widely used in seismology, in particular for seismic hazard assessment and in earthquake prediction model. There are sevceral declustering algorithms that have been proposed. Up to now, most users have applied either the algorithm of Gardner and Knopoff (1974) or Reasenberg (1985). Gardner and Knopoff (1974) introduced a procedure for identifying aftershocks within seismicity catalog using inter-event distances in time and space. They also provided specific space-time distances as a function of mainshock magnitude to identify aftershocks. This method is known as a window method and is one of the most used. Reasenberg's algorithm (1985) allows tolink up aftershock triggering within an earthquake cluster: if A is the mainshock of B, and B is the mainshock of C, then all A, B and C are considered to belong to one common cluster. When defining a cluster, only the largest earthquake is finaly kept to be the cluster's mainshock. In this study we have used the single-link cluster algorithm (Frohlich and Davis, 1990). We first identify large magnitude within the database. Earthquakes occurring within a certain distance

where

The May 26, 1994 Al Hoceima earthquake (Mw 6.0) took place at 12 km depth, the focal mechanism indicates the presence of a main set of sinistral fault with a N-S trend, which may involve several parallel surface (Calvert et al., 1997). Analysis of the aftershock sequence highlighted the presence of NNE-SSW distribution of the seismicity (El Alami et al., 1998). The sequence used covers a period of about one year from the mainshock, including 318 events with magnitude ranged from 2.0 to 6.0. The February 24, 2004 Al Hoceima earthquake (Mw 6.1) took place at a depth of 10 to 14 km. The focal mechanism suggests that the active nodal plane corresponds to a sinistral strike-slip fault oriented N11 N and dipping 70 toward the E (Stich et al., 2005). The aftershocks were aligned preferently in NNE-SSW, in the same way as one of the nodal planes of the focal mechanism. As pointed by Galindo-Zaldivar et al., (2009) the main stresses for the two aftershock sequences trigged by the May 26, 1994 and February 24, 2004 events have a trend with NW-SE compression (P-axis) and orthogonal NE-SW extension (T-axis) compatible with the present convergence of the Africa and Eurasia plate. The aftershock sequence includes 1233 events with magnitude ranged between 0.6 to 6.1. The time span of the sequence is about 1 year.

The October 10, 1980 El Asnam earthquake (Mw 7.3) is one of the most important and most damaging event occurred in northern Algeria. This event has taken place on the Oued-Fodda reversse fault. The later is segmented into three segments, ruptured along 26 km. This fault is located in the Cheliff high seismogenic Quaternary bassin, considered as very active. It is important to point out that almost all the seismicity in northern Algeria is located around the Plio-Quaternary intermountains active bassins (Meghraoui et al., 1996). The aftershock sequence used include the 130 most important magnitude events, ranged between 2.4 and 7.3. In the same way, the May 21, 2003 earthquake (Mw 6.9) has been located in the NE continuation of the south reversse fault system of the Quaternary Mitidja bassin (Maouche et al., 2011). The aftershock sequence we used include 1555 magnitude events, ranged between 0.9 to 6.9 and recorded during the first 40 days from the main shock (Hamdache et al., 2004). The March 20, 2006 Laalam earthquake (Mw 5.2), was located in the Babor chain, in the 'Petite Kabylie' south of Bejaia city. This chain belongs to the Tell Atlas, which is a portion of the Alpine belt in northern Africa. As pointed out by Beldjoudi et al., (2009) the region is affected by several faults. The regional seismicity analysis shows that the Babors chain seems to belongs to a "transition zone" between a large belt of reverse faulting along the western and central part of northern Algeria and a more distributed zone where deformation is mainly accomodated through strike-slip faulting (Beldjoudi et al., 2009). The aftershock sequence include 111 of the best recorded events with more than 54 with

## 3. Magnitude-frequency relationship

The frequency-size distribution (Gutenberg and Richter, 1944) describes the relation between the frequency of occurrence and the magnitude of earthquake

the statistic

The first step in the analysis of the Gutenberg-Richter law, is the determination of the threshold magnitude of completness. It is defined as the lowest magnitude for which all the events are reliably detected (Rydelek and Sacks, 1989). There are many approaches to the estimation of

The maximum curvature (MAXC) method (Weimer and Wyss, 2000) defines the completeness magnitude as the magnitude for which the first derivative of the frequency magnitude curve is maximum (being the maximum of the non-cumulative frequency-magnitude distribution). The goodness of fit (GFT) method (Weimer and Wyss, 2000) compares the observed frequency-magnitude distribution with a synthetic distribution, and the goodness of fit is calculated as the absolute difference of the number of earthquake in each magnitude bins between the observed and synthetic distribution. The synthetic distribution is calculated using *a* and *b*-values estimated from the observed dataset by increasing the cutoff magnitude. The completness magnitude, is given by the magnitude for which 90% of the data are fitted by a straight line. The entire magnitude range (EMR) method was developed by Ogata and Katsura (1993) and modified later by Woessner and Weimer (2005). The maximum likelihood estimation method is used to estimate the power law G-R law parameters *a* and *b*. The same method is applied to estimate the mean and standard deviation of the Normal distribution considered for the incomplete part of the frequency-magnitude distribution. *μ* and *σ* are the magnitude at which

The frequency-magnitude distribution as shown previously is defined as,(Gutenberg and Richter, 1954)

where,

where

where

where

It is straightforward to see that for

(Tinti and Mulargia, 1987; Gutorp and Hopkins, 1986). It has been observed, however, that for a difference of about 3.0 in magnitude between

For the two aftershock sequences triggered by the mainshocks occurred around Al Hoceima city in Morocco on 1994 and 2004, the threshold magnitude

In Fig.2, the frequency-magnitude relation for the 1994 and 2004 aftershocks series of Al Hoceima (Morocco) are displayed. Based on maximum curvature procedure (MAXC), the magnitude of completeness was taken equal to 2.8 for the 1994 aftershock seqiuence and 3.4 for the 2004 sequence. It is important to point out that the changing point procedure (Amores, 2007) gives the same results. Using these threshold magnitudes, we derive the

For the sequence of October 10,1980 El Asnam earthquake and March 20, 2006 Laalam earthque, we have used the maximum curvature procedure to derive the threshold magnitude, we obtained

## 4. Decay rate of aftershock activity — Omori-Utsu law

The third studied scaling law is related to the decay rate of aftershock activity. It is well know that the decay rate of aftershock activity with time is governed by the modified Omori law or Omori-Utsu law (Utsu et al. 1995),

In this study aftershock sequences are modeled using point process defined by the following conditional intensity,

with,

where,

The log likelihood is then given by;

with

it follows that the maximum likelihood estimate MLE

Following, Ogata (1983), the maximum estimation of the parameters are obtained by using the Davidon-Fletcher-Powell optimization algorithm (Press et al. 1986, pages 277, 308) applied to equation 14. The standard deviation of the estimated parameters through the maximum likelihood procedure are derived using the inverse of the Fisher information matrix

the results obtained using this procedure for the aftershock sequences are shown on Fig. 4.

It is often observed that a sequence of aftershocks contains secondary aftershocks, which are aftershocks of a major aftershock (Utsu, 1970). Secondary aftershock are typically detected as changing-point in the activity rate of the sequence using Akaike information criteria (AIC) (Akaike, 1974). In this study, the changing-point in the activity rate of the sequence is detected by using the plot of cumulative number of events vs time from the mainshock. Assuming one secondary aftershock occurred at time

and the cumulative function is given by;

a model with smaller value of

The Omori-Utsu parameters obtained for each aftershock sequence for magnitude above the threshold magnitude,

Omori-Utsu parameters for | |||||

Al Hoceima 1994 | 0.76 | 0.0040 | 12.47 | ||

Al Hoceima 2004 | 0.85 | 0.034 | 47.38 | ||

El Asnam 1980 | 0.84 | 0.025 | 4.48 | ||

Zemouri 2003 | 1.13 | 0.311 | 107.53 | ||

Laalam 2006 | 0.69 | 0.014 | 10.27 |

As pointed previously, it is often observed that a sequence of aftershocks contains secondary aftershocks, aftershocks of a major aftershock (Utsu, 1970). If a secondary aftershock equence starts at time

denoting,

The analysis of the aftershock sequence of Al Hoceima 2004, give us that a suspecious point of aftershock activity change is given by

of non-negative conditional intensity function produces a 1-1 transformation of the time from

from the graphs of the residual process we can deduce, as shown previously using the

Akaike Information Criteria | ||||||||||

Model 1 | Model 2 | Model 3 | Model 4 | B. Model | ||||||

Al Hoceima 1994 | -639.1429 | -609.7559 | -609.276 | -633.7404 | Model 1 | |||||

AlHoceima 2004 | -3616.8231 | -36020.7106 | -3609.2007 | -3610.6351 | Model 1 | |||||

El Asnam 1980 | -47.1127 | -32.118 | -31.2841 | -42.9912 | Model 1 | |||||

Zemouri 2003 | -8973.1703 | -9244.9968 | -9342.0748 | -9353.1883 | Model 4 | |||||

Laalam 2006 | -293.5383 | -274.8197 | -275.9812 | -288.698 | Model 1 |

Following figure 7.6 displays for El Asnam 1980, Zemouri 2003 and Laalam 2006 aftershock sequences the adjustment of the data with the appropriate model, as deduce from the Table2

The temporal aftershock decay is also analyzed using the approach introduced by Marcellini (1997). This approach describes the temporal behavior of the cumulative seismic moment released in aftershock sequences. It is an alternative approach to the Omori-Utsu law, previously analyzed. Static fatigue is assumed to be the principal explanation of the aftershock temporal behavior. Under the condition that the main shock causes a redistribution of stress, the initial stress condition of the afterhock sequence at main shock origin time

where

where

Following Marcellini (1997), we consider the aftershock zone as characterized by barriers that breaks after a given elapsed time proportional to the stress intensity factor

to test if the static model fatigue holds as represented by equation (22), two conditions must be checked:

(a) The validity of equation 22, which is adjusted to different aftershock data and plotted on Figure 8. The same figure shows the estimates

The confidence limite of

(b) The validity of the definition of the constants

Taking into account the obtained coefficient of correlation

## 5. Energy partitioning

The other scaling law exalined in this study is the modified Bath law. In its original form, Bath law states that the difference

Extensive studies about the statistical variability of

where,

combining this last relation with the empirical relation of the energy dissipated

we derive the partionning of the energy, in the following way. It is wellestablished that the magnitude distribution of aftershocks clearly exhibits a near-universal scaling relative to the mainshock magnitude. To explore this relation for our aftershock sequences, we will determine the ratio of the total energy radiated by the afytershock sequence to the seismic energy radiated by the mainshock. The energy radiated during an earthquake is related empirically to its moment magnitude

with,

This relation is applied directly to describe the link between the energy radiated by the mainshock

following Shcherbakov et al.(2004a), the total energy radiated during the aftershock sequence

by combining the former equations, Eq. 33 and 34, we obtain

in addition, different version of the above equation, Eq 35 is obtained if we use the equation giving

taking into account the modified Bath's law, we obtain

the ratio of the total radiated energy by the aftershocks

assuming that all earthquakes have the same seismic efficiency, which means that the ratio of the radiated energy to the total drop is stored as elastic energy is also the ratio of the drop in the stored elastic energy due to the aftershocks to the drop in the stored elastic energy due to the mainshock. Finally, the following relation is derived

Using the Eq. 39 for the studied aftershock sequences, the results are shown on the following table.

Ratio of the elastic energy released | ||||||

Al Hoceima 1994 | 1.07 | 1.10 | 0.05 | |||

Al Hoceima 2004 | 1.13 | 0.50 | 0.35 | |||

El Asnam 1980 | 0.82 | 1.20 | 0.02 | |||

Zemouri 2003 | 1.10 | 0.82 | 0.14 | |||

Laalam 2006 | 0.99 | 1.70 | 0.01 |

From the obtained results shown on Table 3, we deduce the percentage of the total energy radiated during the mainshock. From this point of view, 95 % of the total energy has been radiated during the Al Hoceima 1994 mainshock and 65% during Al Hoceima 2004 mainshock. On the other side the El Asnam 1980 main shock radiated about 98% of the total energy, 2% has been radiated by the aftershock sequence, but we shouldpoint out that these results depend directly on the quality of data used and it is clear that whatever the sequence of aftershocks used, it is still incomplete, especially at the beginning just after the occurrence of the main shock. For the aftershock sequence triggered by the El Asnam earthquake of 1980, it seems that the sequence used is truncated due to the delay in the implementation of seismological network just after the main shock. Nevertheles, the results shown on Table 3, gives a large overview on the ratio of the total energy radiated by the main shock and by the aftershocks. Thus, 86 % of the total energy has been radiated during the main shock of Zemouri 2003 and 99% during Laalam 2006 mainshock.

## 6. Spatial aftershock distribution

It is well known that seismicity is a classical example of a complex phenomenon that can be quantified using fractal theory (Turcotte, 1997). In particular, fault networks and epicenter distributions have fractal properties (Goltz, 1998). Thus, a natural way to analyze the spatial distribution of seismicity is to determine the fractal dimension

where

where

In practice, however, for large values of

where

The results obtained are close to 2.0, which allow us to deduce that the spatial distribution of the epicenters tends to be uniform on the plane.

The ratio of the slip on the primary fault to the total slip over the fault system is given by (Khattri, 1995)

where

The obtained results are shown on Table 3, we deduce that during Al Hoceima earthquake of 1994, 62% of the total slip accomodates the primary rupture, 60 % during Al Hoceima 2004 earthquake. During the El Asnam earthquake of 10 October 1980, 59% of the total slip accomodates the primary fault segment, during Zemouri earthquake of 2003 this ratio has been estimate to 57% and 45% during Laalam earthqiuake of 2006. It is important to point out that in each case the remainder of the slip is distributed over the secondary rupture.

Fractale dimension and ratio of the slip | ||||||

D2 ± σD2 | Range | Sp/S | ||||

Al Hoceima 1994 | 1.60 | 1.79 - 12.56 | 0.62 | |||

Al Hoceima 2004 | 1.67 | 2.16 - 17.27 | 0.60 | |||

El Asnam 1980 | 1.70 | 12.90 - 31.35 | 0.59 | |||

Zemouri 2003 | 1.79 | 1.60 - 10.00 | 0.57 | |||

Laalam 2006 | 2.13 | 1.57 - 8.90 | 0.45 |

In this section we attempt to analysis the inter-event distance distribution of probability. We use a non-parametric approach to analysis the density of probability of the inter-event distances, especially the kernel density estimation, this methodlogy is clearly presented in Silverman (1986). We used it in the following way. Given a sample of

where

in Eq. 44,

considere a sample of

The last term of the right member of the last equality is independnat of the parameter

where,

futhermore, the score function could be written in the following form

with,

assuming that the minimum of

we observe that in the case of the Gaussian kernel, the kernel

the parameter

the parameter

where

The test

**Theorem** (Silverman, 1981)

The kernel density estimate *k* bumps if and only

The test

**Proof.** The proof of this theorem is given in details in Silverman (1981).

Under the null hypothesis, samples can be simulated from the kernel density estimate by using Efron formula,

where *n* simulated from the sample

if

# is the sign of the number of element in the set. In practice, we apply the series of tests

The Fig. 11(a) gives the estimated density obtained using the rule of thumb. The cross validation optimal smoothing method gives a bandwidth parameter

For the aftershock sequence triggered by the 21 May 2003 Zemouri earthquake (Mw 6.9), the obtained results are shown on Fig. 13.

The estimated density using the cross validation optimal smoothing has been obtained with a bandwidth parameter

## 7. Conclusion

Aftershock sequences in Algeria-Morocco region have been analyzed in order to estimate and derive with accuracy the parameters of the most important scaling laws in statistical seismology. For each aftershock sequence the threshold complteness magnitude

This study is a first attemp to perform analysis of aftershock sequneces triggered by main events in the Algeria-Morocco region.