Geophys. J. Int. (2006) 166, 322–338 doi: 10.1111/j.1365-246X.2006.02998.xG

JISei

smol

ogy

Fault plane orientations of small earthquakes of the 1997Umbria-Marche (Italy) seismic sequence from P-wavepolarities and rise times

Marilena Filippucci,1,2 Salvatore de Lorenzo1,2 and Enzo Boschi31Dipartimento di Geologia e Geofisica, Universita di Bari, Bari, Italy. E-mail: marilena.filippucci@geo.uniba.it2Centro Interdipartimentale per la Valutazione e Mitigazione del Rischio Sismico e Vulcanico, Universita di Bari, Bari, Italy3Istituto Nazionale di Geofisica e Vulcanologia, Roma, Italy

Accepted 2006 March 10. Received 2006 March 10; in original form 2005 February 21

S U M M A R YA two-step technique has been developed with the aim of retrieving the fault plane orientationof a small-magnitude earthquake. The technique uses a set of non-linear equations, whichrelate the rise times of first P waves to source parameters (source dimension L, dip δ and strikeφ of the fault) and intrinsic Qp. At the first step of the technique, the inversion of P polaritiesprovides two fault plane orientations for each focal mechanism solution. At the second step,the inversion of rise times is carried out to retrieve L and Qp by fixing δ and φ to the valuesof the fault plane orientation inferred at the first step. A robust analysis, based on the randomdeviates technique, allows us to evaluate if one fault plane exists which systematically betterfits data and can be considered the ‘true’ fault plane. A parameter is introduced to quantify thelevel of resolution of the ‘true’ fault plane. A probabilistic approach, based on the assumptionthat errors on data are Gaussian distributed, allows us to a posteriori validate or reject the‘true’ fault plane. The technique has been applied to 47 small earthquakes (1.3 < ML < 3.9)occurred below the town of Sellano, during the 1997 Umbria-Marche (Central Italy) seismiccrisis. The strike of the inferred fault planes is generally along the Apennine direction. Faultplane solutions can be mainly subdivided into two groups: a first group of SW-dipping faultsand a second group of NE-dipping faults. These results agree with the present day knowledgeof the fault systems in the area. An average Qp equal to 354 ± 63 has been estimated, inagreement with previous attenuation studies.

Key words: circular crack, directivity, fault plane solution, non-linear inversion, polarities,rise time.

1 I N T RO D U C T I O N

The seismic activity of Central Apennines is well documented in

the Italian historical earthquake catalogue (Boschi et al. 2000). Con-

cerning the more recent activity, an important seismic crisis occurred

in the Umbria-Marche region in 1997. The sequence was charac-

terized by nine strong shocks, with magnitude higher than M w =5, and more than two thousand aftershocks (Amato et al. 1998).

The focal mechanisms for the main shocks indicate normal faulting

on NW–SE striking fault planes with tension axes oriented in the

range 40◦–60◦, roughly perpendicular to the strike of the Apennines

(Ekstrom et al. 1998). Two of the nine largest earthquakes occurred

on 12th and 14th October, near the town of Sellano, with magnitude

respectively M w = 5.2 and M w = 5.6 (Fig. 1). During the crisis,

the seismicity migrated from north towards south in the Apennine

direction and the activity was concentrated on a 40 km area elon-

gated in the NW–SE direction, mainly shallower than 9 km. A dense

seismological network, installed in the epicentral area by several

institutions, recorded a high number of high-quality data (Michelini

et al. 2000) with a high coverage of the azimuths (Fig. 1).

While several investigations have focused on the moderate (4 <

ML < 5) and high (ML > 5) magnitude earthquakes of the sequence

(Olivieri & Ekstrom 1999; Pino & Mazza 2000; Capuano et al.2000; Morelli et al. 2000, among the others), in only a few stud-

ies the source properties of the small earthquakes occurred in this

area have been analysed (Bindi et al. 2001). A high fraction of the

energy radiated during the unusually long Umbria-Marche seismic

sequence was released by the aftershocks, with a total extent of the

aftershock zone (40 km) larger than the cumulative length of the

individual ruptures of the largest shocks (25–30 km) (Amato et al.1998). For this reason it is of fundamental importance any attempt

to address the role played by these small seismic ruptures.

Classically, the focal mechanism of a small earthquake is in-

ferred from the non-linear inversion of P polarities. In the case

322 C© 2006 The Authors

Journal compilation C© 2006 RAS

Fault plane orientations of earthquakes of 1997 Umbria-Marche seismic sequence 323

Figure 1. Epicentres (grey circles) of the earthquakes recorded during the 1997 Umbria-Marche seismic crisis and recording stations (modified from Govoni

et al. 1999). The focal mechanisms of the two mainshocks occurred in Sellano (Ekstrom et al. 1998) are shown.

of poor sampling of the focal sphere, the focal mechanism can

be poorly constrained and the inversion can give rise to multiple

solutions.

More refined techniques, as those based on the seismic moment

tensor inversion (e.g. Trifu et al. 2000, and references therein) have

been also used to detail the seismic source properties of the earth-

quakes.

In the last twenty years, many efforts have been made to reduce

the uncertainty in the retrieved focal mechanisms and to overcome

the problem of multiple or weak solutions, adding S-wave informa-

tion (Zollo & Bernard 1991; Nakamura 2002; among the others).

Moreover, many attempts have been made to determine which of

the two nodal planes of the focal mechanism is the true fault plane.

In particular, many authors (e.g. Sato & Hirasawa 1973; Mori 1996;

Courboulex et al. 1997; de Lorenzo & Zollo 2003; Warren & Shearer

2006) tried to constrain the fault plane orientation by modelling the

directivity source effect, that is, the variation of the waveform am-

plitude and/or pulse width of first P and/or S waves with varying the

source to receiver angle.

In this article we selected 47 small-magnitude earthquakes oc-

curred in Sellano during the 1997 Umbria-Marche seismic crisis.

After the calculation of the focal mechanisms of these events, we

tried to determine which of the two nodal planes of the focal mech-

anism is the ‘true’ fault plane. To this end, we used a recently

developed technique (de Lorenzo et al. 2004) which inverts the

P pulse broadenings and modified it to account for a priori con-

straints on the fault plane orientation coming from the inversion of

P polarities.

2 R I S E T I M E I N V E R S I O N T E C H N I Q U E

2.1 Theoretical relationships

Starting from the pioneering works of Ben Menahem (1961) and

O’Neill & Healy (1973), several studies have focused on the

estimate of source and Q parameters by modelling the rise times

of body waves generated by low magnitude earthquakes. On a ve-

locity seismogram, the rise time τ of a wave can be defined as the

time interval between the onset of the wave and its first zero cross-

ing time (Fig. 2). It is known that the rise time of first P and/or Sbody waves depends on the angle from which the source sees the re-

ceiver. This phenomenon, known as the directivity source effect, has

been theoretically explained in terms of both the spatial finiteness

of the seismic source and its finite rupture velocity (e.g. Madariaga

1976). It has been demonstrated (Boatwright 1984) that the direc-

tivity source effect depend also on the shape of the fractured area

(unilateral, circular or mixed). In this article we assume, as source

model, a circular crack which ruptures with constant rupture veloc-

ity and instantaneously stops when the final perimeter of radius L is

reached (Sato & Hirasawa 1973). Under these assumptions and in

the far-field range (Fraunhofer approximation), the source rise time

τ 0 (i.e. the rise time in an ideal elastic medium) obeys the following

equation (e.g. Zollo & de Lorenzo 2001):

τ0 = L

Vr

(1 − Vr

csin θ

), (1)

where Vr is the rupture velocity, c is the body wave velocity at the

source and θ is the takeoff angle (the angle between the normal �n

C© 2006 The Authors, GJI, 166, 322–338

Journal compilation C© 2006 RAS

324 M. Filippucci, S. de Lorenzo and E. Boschi

τV

elo

city

Time

Figure 2. The rise time of a first P wave on a velocity seismogram.

to the source plane and the tangent �t to the ray leaving the source)

(Fig. 3b). When θ = 0, the source rise time assumes its maximum

value (τ0 = LVr

), which represents also the duration of the slipping

on the fault. θ is function of dip δ and strike φ of the fault and of

the source to receiver distance, according to the equation (Zollo &

de Lorenzo 2001):

θ (δ, φ) = arccos

{x2 + 1 − S2

2S

}, (2)

where x is the length of the vector �x =→

C P (C is the centre of the

source; P is the point on the Earth’s surface reached by the tangent

N

E

Z

n

φ

’φ

δ

’δ

N

E

Z

n θ

Ray

leavi

ng th

e sour

ce

(a) (b)

(c)

tC

P

C

γ

α

x

E

N

Z

S

W Ra

y le

avi

ng

th

e s

ou

rce

t

Figure 3. The angles defining the fault plane orientation. The grey rectangle indicates the fault plane, that is, the plane containing the circular crack. (a)

Geometrical relation between the unit normal n to the fault plane, the dip δ and the strike φ of the fault plane and the angles δ′ and φ′ given in eq. (5). (b) The

takeoff angle θ between the unit normal n to the fault plane and the tangent �t to the ray leaving the source. c) Geometrical relation between x, γ and α given in

eq. (4).

to the ray in C) (Fig. 3) and:

�S = �x − �n. (3)

In Cartesian coordinates, �S is given by:

Sx = x sin γ cos α − sin δ′ cos φ′,

Sy = x sin γ sin α − sin δ′ sin φ′,

Sz = x cos γ − cos δ′,

(4)

where γ and α specify the orientation of the vector �x (Fig. 3c). δ′

and φ′ are related to δ and φ (Fig. 3a) by the equations:

δ = δ′

φ = 360 − φ′.(5)

As an effect of the seismic absorption along the ray-path de-

scribed by T /Qp (where T is the traveltime of the wave and Qp the

intrinsic quality factor of the P waves), an additional broadening

of the first P pulse occurs. It has been shown (de Lorenzo 1998)

that this additional broadening depends in turn on the source. In de

Lorenzo et al. (2004) the non-linear equation relating the rise time

to model parameters (L , δ, φ, Qp) has been found. It is:

τ = τ0 + η(L , c, δ, φ)T

Q p+ λ(L , c, δ, φ), (6)

C© 2006 The Authors, GJI, 166, 322–338

Journal compilation C© 2006 RAS

Fault plane orientations of earthquakes of 1997 Umbria-Marche seismic sequence 325

where:

η(L , c, δ, φ) = η1

L

csin θ (δ, φ) + η2, (7)

λ(L , c, δ, φ) = λ1

L

csin θ (δ, φ) + λ2, (8)

being η1, η2, λ1 and λ2 constants which depend on the body wave

velocity at the source.

2.2 Inversion technique

We propose an improvement of the inversion method of Zollo &

de Lorenzo (2001) to account for a priori information on dip and

strike. With respect to the previous study, the imposition of (δ, φ)

allows us to reduce the number of unknown model parameters from

four (L, δ, φ and Qp) to only two (L and Qp). We remark that the

estimated Qp is an apparent Qp, that is, the geometrical average of

Qp along the ray paths available for the event.

The technique we developed consists of two steps:

At the first step, focal mechanisms are computed from the

inversion of P polarities using the FPFIT code (Reasenberg &

Oppheneimer 1985), providing the values of (δ i , φ i ) (1 ≤ i ≤ Np)

for each of the Np nodal planes.

At the second step, by imposing δ = δ i and φ = φ i (1 ≤ i ≤ Np),

Np inversions of rise times are carried out, providing Np estimates

of L and Qp.

The inversions are performed by using the Simplex Downhill

method (Nelder & Mead 1965) to search for the absolute minimum

of the L2-norm weighted misfit function between observed and es-

timated rise times, with weights equal to the inverse of the variance

on data.

The results of the Np inversions are then compared to determine

which of the Np fault planes produces the best fit of the observed

rise times. This plane will be considered the most probable true fault

plane and it will be indicated with the index 1. If FPFIT does not

generate multiple solutions, the index 2 will indicate the conjugate

plane. If multiple solutions are inferred from the inversion of Ppolarities, the index 2 will indicate the next most probable fault

plane solution.

We use the random deviates technique (Vasco & Johnson 1998) to

evaluate how the errors on data map in the model parameter space.

To this aim, we build N rand different data sets adding to each datum

a random quantity. This random quantity is selected inside the range

of error affecting each datum. Our data are represented by rise times

τ obs, hypocentre locations, angles between rays and the vertical axis

γ , azimuths α and takeoff angles θ (Fig. 3b and c).

We define standard deviation of the inversion for a given fault

plane the quantity:

σ =√∑N

j=1

(τ obs

j − τ teoj

)2

N − 1, (9)

where N is the number of rise times available for the event. For a

given event, the fault plane 1 will be considered really constrained

by the rise times only when the standard deviation of the inversion σ i1

for the fault plane 1 is systematically (i.e. for 1 ≤ i ≤ N rand) smaller

than the standard deviation σ i2 of the inversion for the fault plane 2.

Considering the standard deviation reduction of the ith inversion:

�σ i = σ i2 − σ i

1

σ i2

1 ≤ i ≤ Nrand. (10)

The fault plane 1 will be considered the most probable when:

�σ i ≥ 0 ∀i : 1 ≤ i ≤ Nrand. (11)

For this event L and Qp are estimated by averaging the results of

the N rand inversions using the fault plane 1.

3 DATA S E L E C T I O N

We have initially considered more than sixty low-magnitude earth-

quakes, recorded during the 1997 Umbria-Marche seismic crisis

(Govoni et al. 1999). The network included 15 temporary and 8

permanent stations. Ten of these stations (blue triangles in Fig. 1)

consist of four Lennartz LE-3D/5s (flat velocity response between

0.2 and 40 Hz) and six Lennartz LE-3D/1s (flat velocity response

between 1 and 40 Hz) seismometers equipped with MarsLite data

loggers recording on 230 MByte optical disks. Sensors have three

component (vertical, north-south and east-west) and these stations

recorded in continuous mode at 125 samples per sec. Permanent

stations are managed by the RSM (Osservatorio Geofisico Speri-

mentale di Macerata) and RESIL (Regione Umbria) and recorded

in continuous mode at 62.5 samples per second. These stations con-

sist of Mark L4C-3D seismometers (flat response between 1 and

40 Hz) equipped with MARS88/FD data loggers.

The earthquakes (1.3 ≤ ML ≤ 3.9) considered in this study have

been localized inside of a square of 0.1◦ × 0.1◦ around the Sellano

town (Fig. 4). Foci depth varies from 2.7 to 7.0 km, with the greatest

concentration of events between 3 and 4 km. The inferred foci are

affected by an error less than 300 m on horizontal coordinates and

less than 1 km on the vertical coordinate.

The application of the methods based on pulse widths requires a

careful selection of the P waveforms. First of all, we have to account

for the waveform complexity. Since the theoretical model is based

on the assumption of a simple unipolar source, it should be applied

to those events whose waveforms indicate a simple rupture process.

Alternatively, in the case of source heterogeneities, the inversion

result could refer to the first stage of a multiple rupture process

(Boatwright 1984). This observation has been recently validated by

Fisher (2005), which showed that the multiple arrivals following the

first P wave can be modelled, in some cases, in terms of a multiple

rupture process at the source.

LONGITUDE

LA

TIT

UD

E

12°45’ 13°05’

42°50’

43°05’

Figure 4. Horizontal projection of the seismic rays considered in this study.

The epicentres of the studied events are localized inside a grid of 0.1◦ ×0.1◦ around the town of Sellano. Recording stations are named and repre-

sented by triangles.

C© 2006 The Authors, GJI, 166, 322–338

Journal compilation C© 2006 RAS

326 M. Filippucci, S. de Lorenzo and E. Boschi

EVENT DATE: 26/10/1997 ORIGIN TIME: 17:47

TESTING UNIPOLARITY COMPARING V AND V PLOTS2

STATION TREV

Vel

oci

ty(c

ou

nt/

Vo

lt*

s)

Time ( s)

(a)

Vel

oci

ty(c

ount/

Volt

*s)

Dis

pla

cem

ent

(count/

Volt

)

Time ( s)

(b)

Vel

oci

ty(c

ount/

Volt

*s)

Vel

oci

ty(c

ount

/Volt

*s

)

2

22

2

Time ( s)

(c)

Figure 5. (a) Vertical component of a velocity seismogram recorded at the station TREV. (b) Comparison between the selected first P velocity waveform in

the grey band in (a) and the integrated displacement field. (c) Comparison between the selected first arrival P velocity waveform in the grey band in (a) and the

square of the velocity field. The dashed line is the onset measured on the v2 seismogram.

A simple way to test the degree of complexity of a seismic rupture

consists of integrating the P wave velocity waveform v(t) to obtain

the displacement field. In the case of a simple rupture process the

displacement field would not significantly differ from an unipolar

shape. Since the procedure of picking can be affected by a more or

less pronounced degree of subjectivity, the comparison between v(t)and v2(t) plots (Boatwright 1980) can be useful to better determine

the onset of the P phase. As an example, Fig. 5a shows a vertical

velocity seismogram. The first P wave is characterized by both a

sharp onset and a unipolar displacement waveform (Fig. 5b). In this

case, the onset can be easily detected on both v(t) and v2(t) (Fig. 5c).

Fig. 6(a) shows another vertical velocity seismogram. Even in this

case, the integrated displacement waveform is simple and unipolar

in shape (Fig. 6b). However, in this case, the onset of the P wave is

not very clear on v(t) and the comparison with v2(t) is very useful

to better constrain the onset of the wave (Fig. 6c).

Following Deichmann (1997), we discarded the data affected by a

clear multipathing effect during the first half-cycle of the wave. This

effect is visible as a sharp discontinuity during the first half-cycle of

the wave (Fig. 7) and can be due to the presence of thin layers below

the recording site, where a fraction of the seismic energy remains

trapped and is subjected to multiple reflections (de Lorenzo & Zollo

2003).

In a preliminary analysis we compared the amplitude on the three

components of motion and verified that the dominant motion of

the first P waves is along the vertical. This is in agreement with

the significant Vp velocity increase with depth (e.g. Cattaneo et al.2000; Michelini et al. 2000). For this reason, rise times have been

measured on the vertical component of the velocity seismograms.

We did not perform the correction for the instrumental response

because the filtering operated in the deconvolution can generate

artificial signals (Mulargia & Geller 2003, and references therein).

C© 2006 The Authors, GJI, 166, 322–338

Journal compilation C© 2006 RAS

Fault plane orientations of earthquakes of 1997 Umbria-Marche seismic sequence 327

EVENT DATE: 19/10/1997 ORIGIN TIME: 11:03

STATION CASC

Vel

oci

ty(c

ount/

Volt

s)

Time ( s)

Z component(a)

(c)

Vel

oci

ty(c

ou

nt/

Vo

lt*

s)V

elo

city

(co

un

t/V

olt

*s

)

2

22

2

Time ( s)Time ( s)

Vel

oci

ty(c

ou

nt/

Vo

lt*

s)D

isp

lace

men

t(c

ou

nt/

Vo

lt)

(b)

TESTING UNIPOLARITY COMPARING V AND V PLOTS2

Figure 6. (a) Vertical component of a velocity seismogram recorded at the station CASC. (b) Comparison between the selected first P velocity waveform in

the grey band in (a) and the integrated displacement field. (c) Comparison between the selected first P velocity waveform in the grey band in (a) and the square

of the velocity field. The dashed line is the onset measured on the v2 seismogram.

Since the minimum frequency of pulses is about 5 Hz, where the im-

pulse response of the instruments is always flat, rise times should be

unaffected by instrumental effects. Moreover, pulse width methods

do not use amplitude information.

Secondary arrival

Time (s)

Vel

oci

ty(c

ount/

Volt

s)

Figure 7. A sharp discontinuity on the first half-cycle of a P wave, which

could be caused by multipathing effects.

Rise time and its standard deviation have been measured taking

into account the effect of the noise N(t) on the vertical velocity

seismogram v(t) (Fig. 8). Three estimates of the rise time have been

carried out. The first is based on the assumption that the onset of the

P wave has been rightly picked at the time T 0, which corresponds

to a first zero crossing at time T 1 and consequently a rise time equal

to T 1 − T 0 (Fig. 8). The other two measurements take into account

the effect of the noise. To this end, we measured the average level of

noise 〈|N |〉 in L1 norm in a time window TW = 0.5 s which precedes

the onset of the P wave (Fig. 8), using the relation:

〈|N |〉 = 1

TW

∫ TW

0

|N (t)| dt . (12)

The intersections of the lines v = ± 〈|N |〉 with v(t) allow us to

determine a minimum T A1 − T A

0 and a maximum T B1 − T B

0 estimate

of the rise time (Fig. 8). These three estimates are then used to

compute the average value of τ obs and its standard deviation �τ obs.

C© 2006 The Authors, GJI, 166, 322–338

Journal compilation C© 2006 RAS

328 M. Filippucci, S. de Lorenzo and E. Boschi

<|N|>=3

< >τ =82+8 ms-

(a)

T0

T0

A

T0

B

..

.

(d)

T1

T1

A

T1

B

...

(e)

(b)

v(t)

(co

un

t/V

olt

s)|N

(t)|

(co

un

t/V

olt

s)

Time (s)

Time (s)

(c)

v(t)

(co

un

t/V

olt

s)

Time (s)

0.5 s

v(t)

(co

un

t/V

olt

s)

v(t)

(co

un

t/V

olt

s)

Time (s) Time (s)

Figure 8. Measurement of rise time on the first P wave. (a) a velocity seismogram v(t) recorded at station ETZ; the square encloses a time window of 0.5 s of

noise N(t) before the estimated onset of the P wave. (b) The absolute value |N(t)| of N(t) used to compute the average value 〈|N |〉. (c) Intersections of the lines

v = ±〈|N |〉 with the first P wave. (d) Intersection points T A0 and T B

0 of the rising part of P wave with the lines v = ±〈|N |〉. (e) Intersection points T A1 and T B

1

of the descending part of P wave with the lines v = ±〈|N |〉.

4 R E S U LT S

In order to perform the inversions of data, seismic rays have been

traced in the layered 1-D velocity model (Table 1) inferred by

Cattaneo et al. (2000) together with the localization of the events.

The ray-tracing algorithm is described in Lee & Stewart (1981).

Table 1. The 1-D velocity model used in this paper (Cattaneo

et al. 2000).

Depth of the top of the layer (km) Vp (km s−1)

1 4.9

2 5.3

3 5.7

4 6.1

5 6.2

6 6.3

4.1 First step: Inversion of P polarities

For the studied events, the relative source to receiver position and

the used velocity model give rise to a high azimuthal coverage

of the focal sphere but to a poor exploration of the angles be-

tween seismic rays and the vertical axis (Fig. 9). For this reason,

the focal mechanism solutions are not always well constrained by

P polarity data. Furthermore, data errors, unmodelled refractions

(Reasenberg & Oppheneimer 1985) and scattering from hetero-

geneities (Nakamura 2002) can be the cause of discrepant polarities

and, consequently, of multiple solutions, as we inferred, for example,

for the events #39 and #29 in Fig. 9.

The results of FPFIT inversions are summarized in Table 2. Focal

mechanisms for all the selected events are shown in Fig. 10. It is

worth noting that a great variety of solutions is inferred. The domi-

nant focal mechanism is normal faulting. In many cases, solutions

show a dominant strike-slip component.

C© 2006 The Authors, GJI, 166, 322–338

Journal compilation C© 2006 RAS

Fault plane orientations of earthquakes of 1997 Umbria-Marche seismic sequence 329

( ) ( )

( )( )

Figure 9. Lower hemisphere focal projection of P polarities of two earth-

quakes (#29 and #39) considered in this study. The poor exploration of the

angles formed by the rays with the vertical axis can cause multiple solutions.

(a) and (b) show the two possible focal solutions for the event #39; (c) and

(d) show the two possible focal solutions for the event #29.

4.2 Second step: Inversion of rise times

According to the random deviates approach, described in Sec-

tion 2.2, for each studied event N rand = 100 noisy data sets have

been built, randomly varying either rise times and hypocentre loca-

tions. Rise time was let vary by adding to its average value a random

quantity selected in the range of its standard deviation. Hypocentres

were let vary inside a range (a cube of 1 km3) that is greater than

the estimated error on coordinates, to take into account not only the

error on the localizations but also possible 3-D effects. The conse-

quence of varying the hypocentre locations is that γ , α and θ (eqs

2, 3 and 4) vary as well.

For a given event, N rand inversions are then carried out for each

of the Np nodal planes. We define the average standard deviation of

the inversion for the fault plane 1 the quantity:

〈σ1〉 = 1

Nrand

Nrand∑i=1

σ i1, (13)

and, similarly, 〈σ 2〉 will be the standard deviation of the inversion

for the fault plane 2. The average standard deviation reduction from

the fault plane 1 to the fault plane 2 is defined as the quantity:

〈�σ 〉 = 1

Nrand

Nrand∑i=1

�σ i , (14)

where �σ i is given in eq. (10).

As an example, in Fig. 11 the inversion scheme is summarized for

the event #1. For this event, first of all we select the fault plane 1 by

observing that 〈σ 1〉 < 〈σ 2〉 (Figs 11a and b). In addition, since the

criterion (11) has been satisfied, the fault plane 1 can be considered

the ‘true’ fault plane, with an average standard deviation reduction

〈�σ 〉 of about 50 per cent (Fig. 11c).

As described by eq. (6), rise times contain information on both

source and attenuation effects. To evaluate the directivity source

effect, as described in eq. (1), rise times must be corrected for the

attenuation term. Once L and Qp are determined, the source rise

times (observed τ obs0 and estimated τ teo

0 ) can be computed from the

rise times (observed τ obs and estimated τ teo) rearranging eq. (6) in

the following way:⎧⎨⎩τ obs

0,i, j = τ obsi, j − η(L j , c j , δ, φ) Ti

Q j− λ(L j , c j , δ, φ),

τ teo0,i, j = τ teo

i, j − η(L j , c j , δ, φ) TiQ j

− λ(L j , c j , δ, φ),1 ≤ i ≤ N j ,

(15)

where Nj is the number of the rise times available for the jth event.

Observed and estimated source rise times can be plotted versus

the takeoff angle to visualize either the quality of the matching and

the range of the takeoff angle which is covered by the data (Fig. 11d).

The effect of adding a noise to both rise times and hypocentre

locations is that the error on observed source rise time�τ obs0 (vertical

red bar in Fig. 11d) is greater than the error it would have considering

only the standard deviation on rise time �τ obs, as can be computed

by using the error propagation formula on eq. (15). In addition, the

variation of hypocentre location gives rise to a variation of the takeoff

angle (horizontal red bar in Fig. 11d), for a given receiver, that

depends on both the source to receiver position and the orientation of

the fault plane. This is the reason why horizontal red bars (Fig. 11d)

vary from a station to another one.

The N rand inversions cause the estimated source rise time to span

an interval, represented by vertical black bar in Fig. 11(d), whose

central value lies on the theoretical τ 0 versus θ trend, described by

eq. (1) and represented by the continuous line in Fig. 11(d).

The inversion results for some of the 47 events are shown in

Fig. 12, as plots of �σ versus the number of inversion. These plots

allow us to choose the events for which the fault plane orientation

can be considered constrained by the rise times, that is, the events

for which the criterion (11) is satisfied. In this way 24 events were

selected and listed in Table 3, together with the values of (L , Qp,

δ1, φ1). �L and �Qp in Table 3 represent the standard deviation on

L and Qp respectively, as computed by averaging the results of the

N rand inversions for the fault plane 1 of a given event. The number

of rise times available for each event N , the average relative error on

observed source rise times 〈�τobs0

τobs0

〉 and on observed rise times 〈�τobs

τobs 〉are also listed in Table 3. The comparison among τ obs

0 and τ teo0 versus

θ , for these 24 events, is shown in Fig. 13.

4.3 Fault plane resolution

The analysis hitherto carried out has allowed us to select the 24

events for which the fault plane orientation can be considered

constrained by the rise times. We need now to quantify the de-

gree of resolution of the retrieved fault plane solutions. Indeed, the

matching in Fig. 13 is helpful for a first qualitative visualization of

the resolution but, alone, cannot be enough to definitively assess it.

In building a parameter R which quantifies the level of resolution of

the ‘true’ fault plane of a given event, the following points have to

be considered:

(1) R depends on the exploration of the takeoff angle. If we in-

dicate with �θ = θ max − θ min the range of takeoff angles cov-

ered by data, R will increase with increasing �θ . It must be noted

that �θ depends on how the fault plane is oriented with respect

to the source to receivers layout, according to eqs (2), (3), (4) and

(5).(2) R will increase with increasing the number of rise times N

available for the event.(3) R depends on the global quality of the observed source rise

times of each event, which can be quantified by the average relative

error on observed source rise times, given by 〈�τobs0

τobs0

〉.

C© 2006 The Authors, GJI, 166, 322–338

Journal compilation C© 2006 RAS

330 M. Filippucci, S. de Lorenzo and E. Boschi

Table 2. Results of the inversion of P polarities. ID represents the identification number of the event (letters B,C,D indicate multiple

solutions); ML is the local magnitude of the events; fault plane solutions are indicated with (δ, φ); errors on dip and strike are indicated

respectively with �δ and �φ; F is the L1 norm misfit function. The dark grey cells indicate the fault plane solutions constrained by the

inversion of rise times.

ID Origin time ML δ1(◦) φ1(◦) δ2(◦) φ2(◦) �φ (◦) �δ (◦) F(y-m-d-h-min)

1 9711011946 2.4 60 300 50 60 0 0 0.15

2 9711012017 2.7 55 90 75 350 5 10 0.15

2B 9711012017 2.7 45 160 45 355 0 8 0.19

3 9711012343 2.4 60 300 50 60 0 0 0.23

3B 9711012343 2.4 60 20 30 165 10 5 0.15

4 9710261747 2.4 80 185 20 300 13 5 0.01

5 9711020521 3.2 60 280 40 135 3 8 0.02

6 9711020745 2.6 80 285 55 20 8 13 0.17

6B 9711020745 2.6 75 230 20 70 5 5 0.17

7 9711021028 2.9 40 100 65 220 5 20 0.15

8 9710181637 2.4 85 105 80 15 10 30 0

9 9710181728 2.8 50 175 50 35 8 10 0.08

10 9710181733 2.2 35 145 60 348 10 13 0

11 9710182048 2.2 60 158 40 15 5 8 0.21

11B 9710182048 2.2 90 60 25 330 10 3 0.21

12 9710190046 2 90 90 75 180 28 23 0

13 9710190205 1.6 75 32 25 150 20 33 0

14 9710290515 2 85 140 10 255 3 8 0

15 9710190421 1.8 75 115 80 23 18 53 0

16 9710190557 2.4 40 10 55 165 10 13 0

17 9710190759 2.6 60 75 45 305 13 8 0

17B 9710190759 2.6 20 120 75 332 10 5 0

18 9710190852 2.9 75 120 20 300 10 3 0

19 9710190859 3 80 55 70 322 10 35 0

20 9710190912 3 30 160 60 340 10 10 0.17

21 9710190956 3 60 125 30 295 3 10 0

22 9710191103 2.6 50 50 60 292 13 30 0.17

22B 9710191103 2.6 50 160 40 325 0 0 0.17

23 9710191515 1.6 85 135 15 250 3 8 0

24 9710200145 2.2 40 165 50 0 0 0 0.13

25 9710200609 3.9 65 320 70 60 8 20 0

26 9710200610 3.2 40 15 55 170 10 10 0

27 9710200618 2.4 75 310 50 50 8 15 0.14

27B 9710200618 2.4 60 140 80 235 0 23 0.14

27C 9710200618 2.4 35 135 60 0 3 3 0.14

28 9710200618 2.5 75 210 50 110 18 48 0

29 9710200619 3.4 60 310 40 85 8 5 0.15

29B 9710200619 3.4 85 45 10 295 5 10 0.09

30 9710200654 2.1 5 20 85 100 18 10 0.13

30B 9710200654 2.1 85 190 5 80 20 8 0.13

31 9710291925 2.2 70 15 20 220 13 0 0

32 9710200732 2.4 25 155 70 300 8 8 0

33 9710200746 2.4 80 60 15 195 20 5 0

33B 9710200746 2.4 90 100 20 190 10 5 0

34 9710201028 3.3 70 115 20 295 8 10 0.07

35 9710201345 2.4 70 90 80 355 10 28 0

36 9710201353 2.4 90 20 80 290 8 35 0

37 9710202117 2 50 150 65 270 5 10 0.15

38 9710211059 2.4 80 270 65 5 3 20 0.18

38B 9710211059 2.4 45 145 70 260 5 15 0.18

39 9710211141 2.5 60 115 45 355 8 10 0

39B 9710211141 2.5 70 175 55 280 8 10 0

40 9710211316 2.3 80 270 60 5 3 15 0.18

40B 9710211316 2.3 50 150 60 270 5 10 0.09

41 9710230140 2.5 50 310 50 175 18 10 0

41B 9710230140 2.5 25 315 75 190 23 8 0

41C 9710230140 2.5 80 255 15 120 10 8 0.07

42 9710230318 3 70 35 80 130 3 10 0.19

C© 2006 The Authors, GJI, 166, 322–338

Journal compilation C© 2006 RAS

Fault plane orientations of earthquakes of 1997 Umbria-Marche seismic sequence 331

Table 2. (Continued.)

ID Origin time ML δ1(◦) φ1(◦) δ2(◦) φ2(◦) �φ (◦) �δ (◦) F(y-m-d-h-min)

42B 9710230318 3 80 35 50 130 5 10 0.19

42C 9710230318 3 15 150 75 320 5 3 0.19

43 9710240611 2.1 50 55 50 285 8 10 0.02

43B 9710240611 2.1 10 15 80 225 8 8 0.02

43C 9710240611 2.1 50 20 50 160 3 5 0.02

43D 9710240611 2.1 70 150 25 280 8 10 0.01

44 9710241448 2.1 30 310 80 205 23 5 0.11

45 9710241600 2.9 90 345 85 75 5 20 0.14

46 9710241746 2.6 55 315 75 215 3 10 0.09

47 9710252202 2.5 80 330 10 180 10 0 0

Figure 10. Plot of the focal mechanism solutions of the forty seven events considered in this study and summarized in Table 3. The number on the top-left of

the focal sphere is the identification number of the event.

(4) R depends on the adherence of the rise times predicted by the

model (L , Qp, δ1, φ1) to data, which is identified by 〈σ 1〉.(5) R depends on the average standard deviation reduction 〈�σ 〉.

The higher 〈�σ 〉 is, the greater the difference between the two fault

planes in reproducing the observed rise times will be.

To take into account all the above-mentioned points, we quantify

the level of resolution using the parameter R defined as:

R = N�ϑ

π/2log10

〈�σ 〉⟨�τobs

0

τobs0

⟩〈σ1〉

. (16)

�θ

π/2, 〈σ1〉, 〈�σ 〉 and R are listed in Table 3.

From eq. (16) it follows that R has not an upper bound. Therefore,

we choose to quantify the resolution R of the events with respect to

its maximum value Rmax, corresponding to the best resolved event

(event#1), by the ratio R/Rmax (Table 3).

4.4 ‘A posteriori’ validation of the results

If we assume that the errors �τ obsi on rise times are Gaussian dis-

tributed, the quantity:

N∑i=1

(τ obs

i − τ teoi

)2(�τ obs

i

)2(17)

will represent a χ 2 variable (Press et al. 1989) with a number of

degrees of freedom equal to N − k − 1, where N is the number

of rise time data and k = 2 the number of parameters (L and Qp)

inferred from the inversion of rise times. Under this assumption, we

can then perform, in the dip-strike plane, a χ 2 test for a given level

C© 2006 The Authors, GJI, 166, 322–338

Journal compilation C© 2006 RAS

332 M. Filippucci, S. de Lorenzo and E. Boschi

0 20 40 60 80 100

0.006

0.008

0.010

0.012

0.014

0.016

σ1(s

)

Number of inversion

0 20 40 60 80 100

0.012

0.016

0.020

0.024

2σ

(s)

Number of inversion

1

0 20 40 60 80 100

0.20

0.30

0.40

0.50

0.60

0.70

Δσ

20 40 60 80 100

0.03

0.05

0.07

0.09

0.11

0.02

0.04

0.06

0.08

0.10

0.12

2

(a) (b)

(c) (d)

Number of inversion

So

urc

erise

tim

eτ

0(s

)

Takeoff angle (°)θ

Figure 11. The inversion of rise times for the event #1. At the top, the focal mechanism of the event is shown. (a) The plot of the standard deviation versus the

number of inversion for the fault plane 1; the solid line represents 〈σ 1〉. (b) The plot of the standard deviation versus the number of inversion for the fault plane 2;

the solid line represents 〈σ 2〉. (c) Plot of the standard deviation reduction between the two planes versus the number of inversion. The line represents 〈�σ 〉given in eq. (14). (d) Matching between theoretical and observed source rise times versus the takeoff angle, for the fault plane 1; vertical red bar represents the

error �τ obs0 on the observed source rise time; vertical black bar represents the range �τ teo

0 of the theoretical source rise times explored in the N rand inversions;

horizontal red bar represents the range of the takeoff angles explored in the N rand inversions; the solid line is the theoretical trend of τ 0 versus θ (eq. 1).

of significance αχ (e.g. Cramer 1946) in order to accept or reject

the following null hypothesis:

H 0: the point (δ, φ) is the ‘true’ fault plane orientation.

The χ2 test has been performed for each event in Table 3. In the

dip-strike plane, a grid-search method with a grid step of 1◦ × 1◦ has

been implemented to find the points where H 0 can be accepted for

a given level of significance αχ . Four levels of significance (αχ =25, 10, 5 and 1 per cent) have been successively taken into account.

Results are shown in Fig. 14, where grey areas represent regions of

acceptance of H 0 and black dots are the FPFIT fault plane solutions.

When the black dot corresponding to the fault plane 1 lies inside the

grey area and the others are outside, the fault plane 1 will represent

the true fault plane at a level of significance αχ . It is worth noting that

the extension of the regions of acceptability of the null hypothesis

vary from an event to another one, indicating that the set of rise

times of each event is differently sensitive to the directivity source

effect. The events resolved for the highest level of significance of

25 per cent (#1, #18, #25, #47) are also characterized by a very small

extension of the grey areas, which indicates a very well-resolved

directivity source effect. In other cases (e.g. events #9, #44, #46)

grey areas are very wide, indicating that the set of available rise

times is less sensitive to variations of the fault plane orientation.

C© 2006 The Authors, GJI, 166, 322–338

Journal compilation C© 2006 RAS

Fault plane orientations of earthquakes of 1997 Umbria-Marche seismic sequence 333

0 20 40 60 80 100

0.00

0.05

0.10

0.15

0.20

0.25

Event#5

Event#7

0 20 40 60 80 100

-0.04

-0.02

0.00

0.02

0.04

0.06

Δσ

Event#45

0 20 40 60 80 100

0.20

0.30

0.40

0.50

0.60

0.70

0 20 40 60 80 100

-0.20

-0.10

0.00

0.10

0.20

0.30

Event#35

Number of inversion

Δσ

Number of inversion

Δσ Δσ

Number of inversion Number of inversion

a

b

Figure 12. Same as in Fig. 11(c) for some of the studied events. (a) Events for which the criterion (11) is satisfied. (b) Events for which the criterion (11) is

not satisfied.

It is worth noting that this analysis forced us to discard two events

(#23, #24), which were previously selected by using the criterion

(11).

5 D I S C U S S I O N A N D C O N C L U S I O N S

Despite the accuracy we used in collecting the data set, as discussed

in Section 3, we cannot be a priori sure that the secondary wave field

(scatterers and multiple reflections) does not affect the estimated

rise times. For the resolved 22 events, we can hypothesize that the

bias in measuring rise times, due to coda energy, is not so high to

hide the directivity source effect. For the remaining 25 unresolved

events, the bias in rise times, due to coda energy, can represent one

of the difficulties in retrieving the fault plane orientation. It has

to be noted that, if the coda energy can affect the first half-cycle of

P wave, it will affect more critically the second half-cycle of P wave.

This observation is supported by the failure of another attempt we

made, jointly inverting rise times τ and pulse widths �T , where �Tis measured as the time interval between the P wave onset and its

second zero crossing time (de Lorenzo et al. 2004). Indeed, using τ

and �T , we were unable to obtain stable and unique solutions, that

is, solutions which satisfy the criterion (11).

The main limitation of the present technique is represented by the

assumption of a fixed average rupture velocity (Vr = 0.9Vs) on which

the theoretical eq. (6) has been built (de Lorenzo et al. 2004). As

shown in de Lorenzo & Zollo (2003), this assumption could affect

the source radius estimated from the inversion of rise times. For this

reason the present estimates of L have to be considered dependent

on this assumption.

It is well known (de Lorenzo 1998) that methods using rise times

of first P waves represent the most reliable techniques to infer the

intrinsic quality factor Qp. This is because only the very beginning

part of the wave-field, that usually is relatively free of complicated

propagation effects, is used (Liu et al. 1994). Qp for the resolved

events (Table 3) ranges from a minimum of 143 to a maximum of

493, with an average value 〈Qp〉 = 354 ± 63. This result validates

a previous intrinsic Qp estimate in the area (Qp = 310), obtained

applying a pulse width technique on one event (de Lorenzo & Zollo

2003).

In the same area, a study of Q coda attenuation has been per-

formed and the dependence of the anelastic attenuation parameters

on frequency has been determined to be (Castro et al. 2000):

Qc( f ) = 77 f 0.6 for the western part of the Apennine chain,

and:

Qc( f ) = 55 f 0.8 for the eastern part.

C© 2006 The Authors, GJI, 166, 322–338

Journal compilation C© 2006 RAS

334 M. Filippucci, S. de Lorenzo and E. Boschi

#3#1 #2

#8

20 40 60 80 100

-0.02

0.02

0.06

0.10

-0.04

0.00

0.04

0.08

20 40 60 80 100

0.03

0.05

0.07

0.09

0.11

0.02

0.04

0.06

0.08

0.10

0.12

20 40 60 80

-0.02

0.02

0.06

0.10

-0.04

0.00

0.04

0.08

0.12

20 30 40 50 60 70 80 90

-0.02

0.02

0.06

0.10

0.14

0.18

-0.04

0.00

0.04

0.08

0.12

0.16

0.20

#5

0 10 20 30 40 50 60 70 80 90

-0.02

0.02

0.06

0.10

0.14

0.18

-0.04

0.00

0.04

0.08

0.12

0.16

0.20

#6

0 10 20 30 40 50 60 70 80

-0.02

0.02

0.06

0.10

0.14

0.18

-0.04

0.00

0.04

0.08

0.12

0.16

0.20

#9

30 40 50 60 70 80 90

0.02

0.06

0.10

0.00

0.04

0.08

0.12

#11

30 40 50 60 70 80 90

-0.02

0.02

0.06

0.10

-0.04

0.00

0.04

0.08

0.12

10 20 30 40 50 60 70 80 90

0.02

0.06

0.10

0.14

0.00

0.04

0.08

0.12

#34

10 20 30 40 50 60 70 80 90

-0.02

0.02

0.06

0.10

0.14

0.18

-0.04

0.00

0.04

0.08

0.12

0.16

0.20

#18

10 20 30 40 50 60 70 80 90

0.02

0.06

0.10

0.14

0.00

0.04

0.08

0.12

0.16

# 2 1

10 20 30 40 50 60 70 80 90 100

-0.02

0.02

0.06

0.10

-0.04

0.00

0.04

0.08

0.12

#23

50 60 70 80 90

-0.02

0.02

0.06

-0.04

0.00

0.04

0.08

#24

10 20 30 40 50 60 70 80 90

0.02

0.06

0.10

0.14

0.18

0.00

0.04

0.08

0.12

0.16

0.20

30 40 50 60 70 80 90

-0.02

0.02

0.06

0.10

0.14

0.18

-0.04

0.00

0.04

0.08

0.12

0.16

0.20

# 2 9# 2 5 #37

40 50 60 70 80 90

0.02

0.06

0.00

0.04

40 50 60 70 80 90 100

-0.02

0.02

0.06

0.10

-0.04

0.00

0.04

0.08

0.12

#38

30 40 50 60 70 80 90

-0.02

0.02

0.06

0.10

0.14

-0.04

0.00

0.04

0.08

0.12

0.16

#39

30 40 50 60 70 80 90

-0.02

0.02

0.06

0.10

0.14

-0.04

0.00

0.04

0.08

0.12

0.16

#40

10 20 30 40 50 60 70 80 90 100

0.02

0.06

0.10

0.14

0.00

0.04

0.08

0.12

0.16

#41

40 50 60 70 80 90 100

0.02

0.06

0.00

0.04

# 44

0 10 20 30 40 50 60 70 80 90

0.02

0.06

0.10

0.14

0.00

0.04

0.08

0.12

0.16

#45 #46

10 20 30 40 50 60 70 80 90 100

0.02

0.06

0.10

0.14

0.00

0.04

0.08

0.12

10 20 30 40 50 60 70 80 90

0.02

0.06

0.10

0.14

0.00

0.04

0.08

0.12

0.16

# 47

Sou

rce

rise

tim

eτ 0

(s)

Take off angle θ (°)

Figure 13. Same as in Fig. 11(d) for all the events of Table 3.

C© 2006 The Authors, GJI, 166, 322–338

Journal compilation C© 2006 RAS

Fault plane orientations of earthquakes of 1997 Umbria-Marche seismic sequence 335

Table 3. Results of the inversions of rise times. ID is the identification number of the event. N is the number of rise time data available for the event. δ1(◦)

and φ1(◦) are dip and strike of the fault plane 1. L is the source radius; Qp is the apparent Qp of the event. �L and �Qp are the standard deviations on L and

Qp . 〈σ 1〉 is the average standard deviation of the inversion for the fault plane 1. 〈�σ 〉 is the average value of standard deviation reduction defined in eq. (14).�θπ/2 is the covered range of the takeoff angles normalized to π /2. The average relative error on observed source rise times is 〈 �τobs

0

τobs0

〉. The average relative error

on observed rise times is 〈 �τobs

τobs 〉.R is the resolution; R/Rmax is the resolution normalized to its maximum value. The dark grey lines indicate the two events

which were rejected after the χ2-test.

ID N δ1(◦) φ1(◦) L (m) �L(m) Q �Q 〈σ 1〉 (ms) 〈�σ 〉 �θπ/2 〈 �τobs

0

τobs0

〉 〈 �τobs

τobs 〉 R RRmax

1 10 60 300 340 40 424 86 7.8 0.48 0.73 0.15 0.08 19.0 1

2 10 55 90 343 36 460 72 13.1 0.285 0.59 0.49 0.15 9.6 0.51

3 9 60 300 262 64 217 47 11.2 0.24 0.69 0.28 0.12 11.7 0.62

5 13 60 280 359 40 365 66 26.5 0.08 0.60 0.37 0.14 7.1 0.37

6 12 80 285 284 64 176 35 25.8 0.175 0.83 0.49 0.16 11.4 0.60

8 8 85 105 286 68 198 60 29.3 0.172 0.71 0.54 0.17 5.9 0.31

9 11 50 175 255 52 236 38 12.8 0.085 0.50 0.34 0.13 7.1 0.38

11 11 60 158 221 46 408 90 13.6 0.075 0.56 0.42 0.12 6.9 0.36

18 12 75 120 282 80 178 43 10.1 0.085 0.64 0.25 0.11 11.8 0.62

21 10 60 125 381 72 412 98 15.3 0.13 0.63 0.30 0.11 9.1 0.48

23 11 85 135 214 54 171 33 19.8 0.04 0.60 0.56 0.16 3.7 0.19

24 8 40 165 178 58 199 42 13.0 0.02 0.34 0.61 0.11 1.1 0.06

25 10 65 320 443 72 363 97 17.4 0.37 0.76 0.24 0.15 14.9 0.78

29 12 60 310 379 42 457 70 24.1 0.125 0.41 0.37 0.14 5.7 0.30

34 14 70 115 369 52 351 84 27.8 0.135 0.61 0.49 0.13 8.5 0.45

37 8 50 150 187 20 492 35 5.4 0.2 0.43 0.18 0.11 8.0 0.42

38 12 45 145 273 24 437 89 13.7 0.112 0.47 0.32 0.12 8.0 0.42

39 11 60 115 358 24 450 33 21.6 0.07 0.55 0.40 0.15 5.5 0.29

40 11 50 150 280 38 433 85 16.4 0.095 0.47 0.35 0.15 6.2 0.33

41 11 50 310 293 24 461 79 14.2 0.095 0.64 0.29 0.13 9.6 0.51

44 8 30 310 149 48 245 58 6.9 0.265 0.37 0.33 0.13 6.2 0.32

45 7 90 345 441 54 460 62 12.6 0.47 0.69 0.20 0.12 10.9 0.57

46 10 55 315 271 48 409 82 13.0 0.16 0.70 0.24 0.11 12.0 0.63

47 11 80 330 353 30 460 50 11.1 0.325 0.72 0.21 0.10 17.0 0.89

Owing to the scattering effects included in the coda of signals,

Qc and Qp estimates are not directly comparable. However, if we

consider the average frequency content of the pulses considered in

this study ( f ∼ 10 Hz), Qc will vary from 306 (in the western part)

to 347 (in the eastern part), covering a range of values that is about

the same range covered by Qp. This indicates that, for the average

frequency content of rise times considered in this study, the effect of

scattering on observed rise times has to be considered less important

than the intrinsic absorption. Therefore, even if the observed first Pwaves are, in some cases, characterized by a highly energetic P coda

train, this result may a posteriori indicate that coda energy doesn’t

critically affect the first half-cycle of the P wave, at least for the

resolved events.

Concerning the inferred focal mechanisms, we have to observe

that, as an effect of the marked increase of velocity with depth due

to the velocity model, the polarities tend to be projected towards

the periphery of the focal sphere (Fig. 9). The effect of the veloc-

ity model on the fault plane solutions was tested re-computing the

focal mechanisms for the simplest case of a homogeneous veloc-

ity model. The use of a homogeneous velocity model produces a

lower focusing effect of points on the periphery of the focal sphere

but only slight variations of the solutions (maximum differences

of the order of 5◦–10◦ for both dip and strike). Furthermore, the

adopted velocity model (Cattaneo et al. 2000) does not take into

account lateral heterogeneities of the elastic parameters. The main

feature of the 3-D tomographic images of Vp (Michelini et al. 2000)

is represented by smooth undulations of elastic discontinuities. The

maximum amplitudes of these discontinuities is in the range of a

few hundred meters, which is comparable with, when not less than,

the error on hypocentre location. Assuming that hypocentres have in

the 3-D model the same position than in the 1-D model, a maximum

difference in the takeoff angles of about 3◦–4◦ would be estimated.

This difference cannot critically affect the retrieved focal solutions,

being of the same magnitude order or less than the range of the take-

off angles covered in the inversions (horizontal red bars of Fig. 13).

Let us consider now the inferred fault plane orientations. By us-

ing the criterion (11), we operated a first selection of the fault planes

listed in Table 2. In this way we selected 24 events, listed in Table 3,

for which the fault plane 1 can be considered the ‘true’ fault plane.

To visualize the goodness of the fit, we compared observed and

estimated source rise times versus the takeoff angle (Fig. 13). It is

worth noting that, to simulate possible 3-D path effects, the hypocen-

tre locations were let vary inside a range (1 km3), greater than or

comparable with the variation in hypocentre localizations from 1-D

to 3-D inferred by Chiaraluce et al. (2003) using a refined technique

of hypocentre localization (Waldhauser & Ellsworth 2000). In some

cases, for instance for the events #1, #3, #18 and #47, both a good

matching of the model to data and a good sampling of takeoff angles

is inferred (Fig. 13). In other cases, as for the events #24,#37 and

#44, the quality of the fit is poor and only a little sampling of the

takeoff angles is obtained (Fig. 13). As we have carefully examined

in Section (4.3), the evaluation of the level of resolution of the in-

ferred fault planes is a difficult task. Indeed, the resolution depends

on several factors, such as the number of available data, the range

covered by the takeoff angles, the quality of data, the adherence

of the model to data and the standard deviation reduction given in

eq. (9). We introduced the parameter R to quantitatively and jointly

account for all these variables (eq. 16). Looking at the R values

C© 2006 The Authors, GJI, 166, 322–338

Journal compilation C© 2006 RAS

336 M. Filippucci, S. de Lorenzo and E. Boschi

EVENT#1 EVENT#18 EVENT#2

EVENT#37 EVENT#5 EVENT#6 EVENT#8

EVENT#25 EVENT#47 EVENT#3

=5%αχ

=1%αχ =1%αχ =1%αχ

=25%αχ =25%αχ =25%αχ =5%αχ=25%αχ0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

=5%αχ

0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

EVENT#9

0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

=1%αχ

EVENT#11

=1%αχ

0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

EVENT#21

=1%αχ

0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

=1%αχ

EVENT#29

0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

=1%αχ

EVENT#34

0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

EVENT#38

=1%αχ

0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

EVENT#40

=1%αχ

0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

EVENT#39

=1%αχ0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

EVENT#41

=1%αχ

0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

EVENT#44

=1%αχ

0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

EVENT#45

=1%αχ

0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

EVENT#46

=1%αχ

0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

EVENT#23

=1%αχ

0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

EVENT#24

=1%αχ

0

60

120

180

240

300

360

0 30 60 90Dip (°)

Str

ike

(°)

Figure 14. Plot in the δ − φ plane of the (δ, φ) values which satisfy the H 0 hypothesis (grey areas), for the events of Table 3. Black dots represent the FPFIT

fault plane solutions. On the top of each map, the FPFIT fault plane solutions are plotted in the focal sphere: solid line in the beach ball represents the ‘true’

fault plane; hatched lines the remaining planes.

(Table 3), we can observe that the best resolved events are the events

#1 and #47, while the worst resolved events are #23 and #24. Gen-

erally, a good agreement is found between the value of R and the

quality of the fit (Fig. 13).

Although R can quantify the level of resolution, it cannot help

us to solve this fundamental problem: is the selected fault plane

really constrained by the inversion? This is the reason why we tested

the uniqueness of the retrieved fault planes using a probabilistic

approach based on the χ2 test. With this approach, the two events

with the least level of resolution R (#23 and #24) have been found

to be ambiguous (Fig. 14) and have been rejected; for the remaining

22 events, the fault plane can be considered the true fault plane for alevel of significance αχ , reported in Fig. 14. For the highest level ofsignificance (αχ = 25 per cent) only the fault plane of the events #1,

#18, #25, #47 was constrained; for the level of significance αχ =5 per cent the fault plane of the events #2, #3, #37 was constrained;

C© 2006 The Authors, GJI, 166, 322–338

Journal compilation C© 2006 RAS

Fault plane orientations of earthquakes of 1997 Umbria-Marche seismic sequence 337

1 5

8 921

25

34 37

44

45

46 47

2

3 6

11

29

3839 40

41

18

NE-dipping

SW-dippingfaults

a

b

1

3

5

6

8

9

11

18

21

25

29

34

37

38

39

40

41

4445

46

47

SW-dipping

NE-dippingc

500 m

500 m

N

Figure 15. (a) Orientation of the twenty two ‘true’ fault planes. On the top the SW-dipping faults are shown; on the centre the NE-dipping faults are shown;

on the bottom one strike-slip and one S-dipping faults are shown. (b) the inferred Apennine faults (SW-dipping; NE-dipping and vertical) are grouped in three

focal spheres. (c) Epicentre locations of the SW-dipping and of the NE-dipping faults.

for the remaining 15 events the fault plane was constrained for the

level of significance αχ = 1 per cent. The χ 2 test was also useful to

evaluate the sensitivity of rise times to the directivity source effect.

For instance, even if the fault plane solution of the events #6, #9,

#11, #21, #44, #46 has been uniquely constrained, the wideness of

the grey area in the dip-strike plane indicates that a great variation

of fault plane orientation corresponds to a small variation of rise

times.

The ‘true’ fault plane solutions are shown in Fig. 15(a). We un-

derline that the majority of the inferred fault planes (21 of 22) strikes

along a direction that is roughly parallel to the strike of the Apen-

nine chain, as was found for the eight mainshocks of the sequence

(e.g. Olivieri & Ekstrom 1999; Pino & Mazza 2000). With the ex-

ception of only one solution, the retrieved fault planes can be mainly

grouped in two classes: 10 faults dipping in SW direction and the

other 10 dipping in NE direction (Fig. 15b). Several seismologi-

cal studies found out the existence of SW-dipping fault systems in

the Umbria-Marche area. In particular, Chiaraluce et al. (2003) in-

ferred that the location of the aftershocks of the whole sequence

mainly identifies a complex pattern of small segments activated by

the moderate magnitude earthquakes; the authors found out that

the seismicity occurs mainly on planar SW-dipping fault planes.

Moreover, in a detailed geological and seismological investigation,

Boncio & Lavecchia (2000) attributed the seismicity, in the area of

Sellano, to the M. Civitella-Preci SW-dipping fault. It is worth noting

that the epicentres of the SW-dipping faults inferred in this study

are roughly aligned along the same direction (∼330◦) (Fig. 15c)

of the Civitella-Preci fault. Moreover, Boncio & Lavecchia (2000)

stated that the activated SW-dipping faults terminate, at depth, in

the common NE-dipping normal detachment Altotiberina fault. In

this frame, the inferred NE-dipping solutions can be related to the

seismic activity along the Altotiberina.

The results of this study indicate that the directivity source effect

on rise times, when clearly recognizable, can be very helpful to

solve, at least in some cases, the ambiguity between the true and the

auxiliary fault plane.

A C K N O W L E D G M E N T S

We are very grateful to P. Capuano and C. Trifu for their careful re-

view of the manuscript. The editor T. Dahm is particularly thanked

for suggestions which allowed us to improve the technique of inver-

sion of rise times. Two anonymous reviewers are acknowledged for

their useful suggestions and comments. Many thanks to Ann Brown

for the corrections to the English text.

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