SOURCE PARAMETERS AND 3-D ATTENUATION STRUCTURE FROM THE INVERSION
OF MICROEARTHQUAKE PULSE WIDTH DATA: METHOD AND SYNTHETIC TESTS
Aldo Zollo1 and Salvatore de Lorenzo2
1
Dipartimento di Scienze Fisiche, Università di Napoli “Federico II”, Italia
2
Dipartimento di Geologia e Geofisica, Università di Bari, Italia
Abstract
We propose a new method to determine source parameters and attenuation structure of a 3-D medium
based on first P and S rise time and total pulse width measurements from microearthquake data. The
effects of fault finiteness on seismic radiation are taken into account by assuming the rupture model for a
circular crack of Sato and Hirasawa [1973].
Ray-theory synthetic seismograms in a constant-Q anelastic medium are computed to derive a set of non
linear equations which relate the source and attenuation parameters (fault radius, orientation of the fault
plane and quality factor) to the pulse width data (half and total duration of the P and/or S waveforms).
The numerically-built relationships are used to compute the direct problem in the framework of a non
linear inversion scheme, based on the modified downhill Simplex method.
The validity and robustness of the inversion method is tested by synthetic simulations by assuming the
sources and receivers configuration of the seismic passive experiment conducted in the Campi Flegrei
caldera (Southern Italy) during the last microearthquake crisis (1982-1984).
Different heterogeneous Q models have been considered in order to assess the uncertainty and resolution
of source and attenuation parameters for the given acquisition lay-out.
The results of this simulation study indicate that first pulse width data from a local network permit to
retrieve with sufficient accuracy the heterogeneous Q structure and fault radii. A rather dense azimuthal
coverage of the sources is instead needed to recover the angles (in particular the fault strike) which
defines the fault orientation.
Introduction
Several authors have studied the problem of the joint
inversion of source parameters and attenuation
structure of the medium from microearthquake data.
Usually the inversion is performed in the frequency
domain by assuming that the theoretical far-field
displacement is described by a 4-parameters spectral
model, i.e. low frequency level, corner frequency,
high frequency decay and Q [Anderson and Hough,
1984; De Natale et al., 1987; Scherbaum, 1990,
Abercrombie, 1995]. Based on a high number of
observations, several authors assume that the high
frequency decaying spectrum can be satisfactorily
modeled in terms of an -square decaying function
[Aki, 1967, Brune, 1970; Madariaga, 1977].
A classical problem arising from these analysis is the
trade-off between the absorption factor Q parameter
and the source spectral parameters (the corner
frequency fc and the high frequency fall-off ) which
leads to an intrinsic ambiguity of the results. The
trade-off consists in the observation that seismic
spectra can be equally well fitted by assuming high Q
and (low fc /high ) or low Q and (high fc /low )
[Zollo and Iannaccone,1996]. In order to minimize
this effect, an iterative procedure consisting of two
steps (inversion of source parameters at a fixed Q
structure; inversion of Q structure with source
parameters fixed to the values previously obtained)
was proposed by Scherbaum [1990]. The trade-off is
minimized by using various recorded spectra for the
same event and by considering the attenuation
structure as a geometrical constraint.
Most of methods for the source parameter and Q
estimation neglect the effects of source directivity
produced by a finite fault. In some cases, particularly
for high frequency events (corner frequency greater
than 10 Hz) this assumption can be considered
satisfactory.
In other cases, and in particular when the distances
traveled by the waves are not too large and/or the
medium is not strongly attenuating, the shape of the
recorded waves can be primarily controlled by source
effects, also at the short wavelength scale of
microearthquake data.
A large number of observations worldwide show that
in the case of small size events (M<3), recorded at
close distances (D<10 km) and in the high frequency
range (f > 1Hz), the source finiteness and directivity
can significantly affect the shape and the amplitude of
direct P and S pulses. This is the case for
microearthquakes recorded at the Campi Flegrei
caldera (fig.1) by a temporary digital network during
the 1982-1984 seismic crisis which accompanied a
relevant ground uplift phenomenon [Courboulex,
1995].
Some experimental [Blair and Spathis, 1982] and
theoretical [Liu, 1988, de Lorenzo, 1998] results
indicate that, in these cases, the classical rise time
method for the Q estimation [Gladwin and Stacey,
1974; Kjartansson, 1979], which assume a point
source model, may be not adequate in retrieving the
attenuation parameter.
Based on these grounds, we have developed a method
for estimating source and a 3-D Q model, based on
body wave pulse-width measurements which accounts
for the fault finiteness by using the rupture model for
circular faults proposed by Sato and Hirasawa [1973]
(in the following referred as S&H). This paper
describes the theory and the synthetic tests performed
to assess the method validity and robustness. The
inversion method has been applied to P-pulse data
from a selected sample of microearthquakes recorded
at the Campi Flegrei caldera [de Lorenzo, Zollo and
Mongelli, 1999] (DZM). A detailed analysis of model
parameters uncertainty and resolution relative to the
Campi Flegrei case will be presented in this article.
The pulse width (or rise time) method
The method for Q estimation presented here is based
on the time domain formulation which derives from
the classical rise time (or pulse broadening) method
[Ricker, 1953; Gladwin and Stacey, 1974;
Kjartansson, 1979]. The rise time method assumes
that direct (P or S) pulse duration , generated by a
punctual impulsive source, increases with the distance
traveled by the wave according to the equation:
T
  0  CQ 
Q
for two reasons; first the data (the broadening of the
pulses) are not affected by subjective criterions of
phase windowing, as it happens for the spectral
technique; second, the data are less affected by
secondary arrivals due to waves diffracted by
heterogeneities in the medium and site effects.
The controversial about the applicability of the rise
time method to signals generated by sources
characterized by a finite frequency content is well
documented in literature [Blair and Spathis,1982; Liu,
1988; Wu and Lees, 1996]. The linearity among  and
t*=T/Q holds only at high cut-off frequency of the
source (>10 Hz) and implies that the method is not
adequate in retrieving attenuation from signals
generated by microearthquake with a corner
frequency less than 10 Hz [de Lorenzo, 1998] .
The method that we present in the following section,
arises from the need to account for the non linearity
between the rise time and the travel time when
seismic signal is generated by a finite dimension
seismic source. It generalizes the pulse duration
method by including a source term which is derived
from kinematics models of the seismic rupture.
The source and attenuation model
For simulating the source effect we adopted the
rupture model of Sato and Hirasawa [1973] (S&H).
S&H determined the analytical form of the P and S
far-field displacement for a growing crack spreading
outwards at a constant speed and stopping abruptly at
a circular final perimeter. The slip amplitude on the
fault surface was built in such a way to be consistent
with the Eshelby (1957)’ static solution.
In polar coordinates r0 , ,   , taking the origin of
cartesian coordinates coincident with the center of
the fault, the S&H far-field displacement of P and S

waves, at distance x from the center of the source,
can be written as:
 

 c x, t   2C FF Kv r  02 
 1 k2





x2
 1
 c x, t   2C FF Kv r  02    
 4  k k 1  k 2


(1)
where  0 is the rise time at source, T is the travel
time of the wave and Q the quality factor of the
medium.
Generally this method is retained [Liu et al., 1994;
Tonn, 1989] more reliable of the spectral techniques

 x 2 for 0 < x < 1 - k
2


(2)
where:

 for 1 - k < x < 1 +
2



vr
sin
c
v  r 
x  r t  0 
0  c 
Numerically-built
relationships
between
rise-time, pulse width and model parameters
k
(3)
 24 
K 

 7 

being r0  x , v r the constant rupture velocity
along the fault, c the P or S wave velocity in the
homogeneous medium,  the take-off angle (i.e. the
angle between the ray vector at the source and the
vector normal to the fault plane),  the applied shear
stress,  the shear modulus and CFF a constant,
depending on the considered displacement
component, which accounts for the geometrical
spreading, radiation pattern and free-surface terms.
The S&H source model predicts a 2 high
frequency fall-off spectrum at each take-off angle,
with the exception of   0 where the high frequency
decay is of the type 1 and a ratio P/S corner
frequencies greatest than 1 according to several
experimental determinations [Molnar et al., 1973].
The anelastic attenuation effect is modeled by using
the near-costant Q Azimi function [Azimi et al,
1968]:

 T 

  exp i  log
 2i 

0
 Q

(4)
where T is the travel time, Q is the quality factor
and f= 0 /2 the largest frequency in the recorded
signal (the Nyquist value).
The rise-time () and the total pulse duration ( T )
of the first P (and/or S) pulse are the observed
quantities (figure 2), related to the source and
attenuation parameter to be retrieved. The rise time is
defined as the time lag from the arrival time of the
first P-wave to the first zero crossing time and the
pulse width is defined as the time lag from the arrival
time of the signal to the second zero crossing time.
Both these quantities contain independent
informations on the rupture dimension, directivity and
the attenuation properties of the propagation medium
[Boatwright, 1984]. In the following, we have
considered as the unknown model parameters the
source radius, the dip and the strike of the fault and
the quality factor of the medium. The rupture velocity
is assumed to be known. Although we will mainly
refer to estimates of source and attenuation
parameters using the only P-wave data set, the method
can be jointly applied to P and S data as well.
In order to determine the relationships between the
observed quantities ( and T ) and source and
attenuation parameters, synthetic P-pulses have been
computed by the numerical convolution of the S&H
far-field waveform (eq.2) with the (constant-Q)
attenuation function inferred by Azimi et al. [1968]
(eq.4). The half and total width of the P pulse have
been measured, throughout an automatic procedure of
picking, on synthetic velocigrams obtained using
different values of source radius, take-off angle and
the quality factor Q in the expected range of variation
for the Campi Flegrei microearthquakes. According to
the corner frequency estimates for microearthquakes
in the Campi Flegrei area [e.g., Del Pezzo et al.,1987],
the simulations have been performed for source radii
ranging between 20 and 700 m. The attenuation
parameter T/Q was allowed to vary between 0.005
and 0.09 sec. Take off angles are varied in the
complete range 0 - 90 degrees.
A constant P-wave velocity (Vp=2.9 km/s) was
assumed in our calculation which is consistent with
the
average value
of the Campi Flegrei
tomographic model obtained by Aster and Meyer
[1988]. Furthermore, for the rupture velocity we
assumed v R =0.9 Vs, being Vs the velocity of S
waves (Vs=1.65 km/s).
In the case of a non attenuating medium (Q=),
finite source and in the far field approximation, the
rise time and pulse width are non linearly related to
the source parameters by the following expressions:


(5)
 0  0  0 sin
vR c
T0 
 0 0
 sin
vR c
(6)
We used eqs. (5) and (6) to calibrate the automatic
picking procedure of synthetic seismograms.
On the other hand, in the case of an attenuating
medium (Q) and a delta-like source function, the
relation between rise time (or pulse width) and travel
time is expected to be linear, as in eq.1, but with
different slope coefficient [Kjartansson, 1979].
Based on measurements of rise-times and pulse
durations on synthetic P-pulses for a large number of
source and attenuation models, the following set of
non linear equations has been obtained through a
non-linear regression analysis:
   0     0 , c, 
T  T0  C
T
Q
T
   0 , c, 
Q
(7)
(8)
where  0 and T0 are given respectively by (5) and
(6) and:
  0 , c,    1
0
sin   2
c
(9)
 0 , c,   1
0
sin   2
c
(10)
C  39
.  0.01
where:
(11)
 1  6.601  0.015
 2  0.946  0.015
1  0.051  0.015
 2  0.005  0.015
(12)
The eqs. (7) and (8) show the different dependence on
source parameters of the rise time and the pulse
width. In particular the slope of the pulse width as a
function of T/Q is not depending on the source time
duration. The source duration instead affects the
intercept term of the same relation.
On the other hand, the slope of the rise time as a
function of T/Q (eq. 7) depends on the source time
duration, which is controlled by rupture directivity.
This leads to a non linearity of the rise time vs. travel
time function. Fig.3 shows that eq. (7) constitute only
a first order approximation of a more complex
dependence of  on source parameters and
attenuation. This is the reason of the non critical
uncertainty on 1 ,  2 , 1 ,  2 described by eq.(12).
In the range of variation for source and attenuation
parameters that we considered for the Campi Flegrei
microearthquake data, eq.(7) leads to a satisfactory fit
of true vs. estimated rise time values (within a
7
standard deviation of 10 s).
The Inverse Method
The eq. (7) and (8) are used to formulate and to
solve the non linear inverse problem of estimating the
source parameters of a set of microearthquakes and
the spatial variation of the attenuation parameter Q.
Given N microearthquakes recorded at a network of
M receivers, the data space consists of the
measurements of rise-times  i , j and pulse widths
Ti , j (i=1,..N; j=1,..Mi , Mi being the number of
receivers that recorded the i-th event).
Concerning the source parameters space, for the
analyzed i-th event we need to estimate the radius
0,i of the S&H circular fault (we assume a fixed,
known value for rupture velocity) and the angles i,j
between the fault normal and ray vectors.
Let’s note that the angles i,j can be related to the
dip i and strike i directions of the plane
containing the circular fault for the i-th event (fig.
A1). In Appendix A the relations among these angles
are inferred for the case of straight line rays,
considered in this study. The fault dip i and strike
i angles of microearthquake i-th are therefore the
geometrical fault parameters to be inferred by the
inversion of rise-time and pulse width data.
In order to define the attenuation parameter space,
the propagation medium is subdivided into P blocks,
each of them being characterized by a different value
of the quality factor Qk (k=1,...,P). We assume that the
3D P-wave velocity structure is “a priori” known as
inferred from other active or passive seismic
investigations in the area. In the present case, first P
and S arrival times from the Campi Flegrei
microearthquake data set have been previously
inverted by Aster and Meyer [1988] to obtain a 3D
image of Vp, Vs and Vp/Vs spatial variation in the
upper 3 km of the caldera. This model is used to
compute the travel-times in the propagation medium.
The whole path attenuation term T/Q is obtained by
adding up the contribution Ti/Qi of each block crossed
by the P-wave ray:
T P Ti

Q i1 Q i
The source and attenuation model space consists of
3N+P parameters. The non linear system of equations
to be solved is therefore given by:
P T
 0 ,i  0 ,i
i , j, m

sin  i ,  i    i , j 
  i, j
vR
c
m 1 Q m
P T
 0 ,i  0 ,i
i , j, m
Ti , j 

sin  i ,  i   39
. 
vR
c
m 1 Q m
1 i  N
 i, j 
(13)
1  j  Mi
where Ti , j,m represents the travel time of the first
P-wave generated by the i-th event recorded at j-th


station passing through the m-th block, and   i,  i
is given in Appendix A.
Due to the non linearity of the system (13) we chose
to solve it by using a global optimization technique
for searching the minimum of the cost function :

N M

     Wi , j  i , jobs   i , jest
i1 j1 

2

 WTi , j Ti , jobs  Ti , jest
(14)
The choice of the weighting factors depends on the
quality of the data. In the following synthetic tests we
have used as weighting factors the same weighting
factors we determined on the real data set described in

2



DZM. A smaller weight is assigned to pulse width
data relative to the rise-time ones which is due to the
different accuracy of the two estimates from
waveform data. Pulse widths are usually more
sensitive to reading errors than the rise times, often
due to the contamination of first arrival by secondary
arrivals due to multipathing effects or to P-coda
waves. Based on this we decided to fix
WTi, j =0.5 Wi , j .
The search for the global minimum of the misfit
function was performed by using the Simplex
Downhill optimization method (SDM) [Press et. al,
1994]. SDM is a well-known nonlinear optimization
method, often used in seismology [Abercombie, 1995;
Ichinose et al., 1997]. Since the large number of
parameters to be inverted (particularly in case of the
3D heterogeneous attenuation structure) some
difficulties, related to the choice of the initial starting
model, could occur by using SDM. Some authors (e.g.
Ichinose et al, 1997) observed that a significant
exploration of the parameters space with SDM
requires the comparison of the results obtained by
using SDM with different initial starting models. We
found that this is equivalent to couple at the initial
stage the SDM with a full space searching technique
like Montecarlo or Simulated Annealing.
After a number of preliminary synthetic tests using
the acquisition lay-out of the 1984 Campi Flegrei data
set, we concluded that an efficient procedure consists
in assigning the starting models which compose the
vertexes of the initial Simplex randomly distributed
in the model space. The original optimization code
has been also modified to constrain the parameters to
be varied within an a priori selected range. The a
priori constraint on fault radii and quality factors of
each block can be roughly estimated by the
examination both of rise time and pulse width vs.
travel time data.
Using random initial models and constraints, the
computer time for convergency was strongly reduced
and a sufficiently wide exploration of the parameters
space was ensured.
The Inversion Method Applied to Synthetic Data
Several numerical tests have been performed to
study the robustness of the proposed method and the
parameter resolution assuming the source and receiver
configuration of the Campi Flegrei experiment.
In this case the considered microearthquakes
database consisted in 87 events recorded by a
minimum number of 4 stations and a maximum
number of 9 stations (see DZM).
The events have been relocated in the 3D velocity
structure determined by A&M. The same velocity
model is used in our study to compute the P-wave
travel time. Due to the relatively short range of
distances (between 3 and 10 km) and depths (95% of
hypocenters are located at depths between 0 and 3
km) we assume that ray paths can be approximated by
straight lines which greatly simplify and fasten the
computation of the attenuation term T/Q in the
inversion procedure. Moreover the longest dominant
wavelength of seismic P waves (about 0.8 km) is
reached by the minimum linear block dimensions in
which we subdivided the volume sampled by rays (1
km). The validity of the straight-line approximation
has been verified also by comparing travel time using
straight line and curved ray as in the Thurber [1983]’
tomographic method used by A&M to obtain their 3D
velocity model.
Noise-free synthetic data (P rise-times and pulse
widths) have been computed using equations (7) and
(8), by assigning, in a random way, an arbitrary value
of the fault dimensions 0 , of the dip and of the
strike  to the events selected for resolution tests.
Source dimensions were chosen in the range 0.01 km0.35 km according to the values obtained by Del
Pezzo et. al (1987). Furtherly, the noise-free synthetic
data were, modified by adding a random, gaussian
error in the range of the expected data uncertainty
(described in DZM).
Preliminary inversion tests for joint estimates of
source and attenuation parameters showed evidence
for the trade-off between Q and 0 which is already
known from the inversion of spectral data
[Boatwright, 1978; Del Pezzo et al.,1987;
Scherbaum,1990]. This led us to implement an
iterative inversion scheme which follows the
approach suggested by Scherbaum (1990).
The first step of the procedure concerns the estimate
of the source parameters for each event, assuming Q
(here denoted as the apparent Q) to be homogeneous
for the considered source to receiver travel paths. The
apparent Q can be therefore different from one event
to another, and its variation give us indication of the
anelastic heterogeneities of the volume crossed by the
rays. In the second step, the source parameters are
taken fixed to the values previously obtained and the
heterogeneous Q structure is determined by inverting
simultaneously all the rise-time and pulse width data
from the whole event data set. The inferred Q model
is used to retrieve newly the source parameters of
events. The loop is continued iteratively till a
convergency criterion is satisfied based on the
variance decrease below an arbitrary threshold value.
Resolution on source parameters
One point that we have primarily investigated is the
expected resolution on source radius and fault angle
parameters ( and ) which is critically dependent on
the azimuthal variation of rise-time and pulse width
observations (directivity effect) for small size
earthquakes.
In a first preliminary test, for the ideal case of
noise-free synthetic data, we considered a sample of
only 39 microearthquakes and considered a
homogeneous Q structure (Q=300), inverting both the
apparent Q and source parameters at the first step of
the previously described iterative procedure.
Residuals between true and inverted source
parameters are represented in fig.4 as a function of the
size of the fault. This test indicates the minimum
rupture radius which can be resolved ( 0 >30 m) for
microearthquakes in the given receiver and source
configuration and the minimum source size for which
directivity effects became significant ( 0 > 30 m). In
figg.4b-c is shown the comparison among synthetic
and inverted rise time and pulse width data, for some
of the selected events.
Synthetic tests using noisy data show in some cases
an ambiguity on fault strike determination
( est  true  180 ). This effect is observed only
in the cases of near vertical fault planes (   75 )
and cases of poor azimuthal coverage (azimuthal gap
< 50°).
Statistical Interpretation of The Results
The uncertainties on source parameters are evaluated
by a classical approach that consists in mapping
random deviates of data in the model parameter space
[see e.g Vasco et al., 1995].
Source parameters are estimated for a large number of
inversion runs carried out on different noised data sets
constructed by adding a random gaussian error to
pulse width and rise-time measurements in the range
of the expected data uncertainty (about 0.03 s on rise
time and 0.06 s on pulse width).
For each parameter an histogram which represents the
experimental distribution function can be obtained.
The histograms provide a qualitative image of
parameter resolution and uncertainty. Well
constrained parameters are expected to show a sharp
unimodal distribution function whose central moment
represents the best fitting value of that parameter.
Instead poorly constrained parameters show rather flat
or multimodal distribution function.
Based on synthetic examples, we choose to fit each
histogram with a gaussian distribution function fixing
its central moment to the average value of the
parameter, as deduced by the repeated inversions, and
by determining the best-fitting value of the variance
of the distribution, which expresses the parameter
uncertainty. The hypothesis for a nearly gaussian
distribution is therefore checked by a -square test
with a significant level of 5%.
We have tested this procedure on synthetic data,
based on 58 events extracted by the true data set, for
different cases corresponding to various assumed
heterogeneous Q structures. In figg.(5,a-b) are
reported the results for two examples. For each of
them we applied the above procedure to compute the
histograms performing 50 inversion runs.
Results of figg(5,a-b) indicate that the source radius is
the parameter better constrained by the inversion.
Referring to the orientation of the fault planes we
have obtained, by the application of the above
procedure, in all studied cases, the fault orientation
estimate is possible for about the 30% of the
considered events. Dip is usually better constrained
after the inversion than strike. Residuals on dip range
from 5° to 23° (average residual = 10° and standard
deviation of the error = 4°) with about 50% of
estimated dips. Residuals on strike range from 12° to
90° (an average residual = 45° and a standard
deviation of the error = 29°) and for about the 40% of
the events the parameter estimate was possible.
Resolution on attenuation parameters
Different resolution tests have been performed for the
retrieval of the 3D quality factor images.
Since about the 95% of considered events in Campi
Flegrei are located at depths between 0 and 3 km, we
assumed that the half-space (z > 3 km) is anelastically
homogeneous.
We considered a 3D anelastic medium (3D-case) by
subdividing the volume in blocks having sizes of
1x1x1 km 3 .
A “checkerboard test” and three “fixed geometry”
resolution tests [Sanders and Nixon, 1995, Zhao et al,
1994] are presented both for the 2D-case and 3D-case.
The term “fixed geometry” is here for attenuation
anomalies having a given shape and spatial extension.
These tests aim at assessing the significance and
reliability of the low-Q bodies retrieved from the
inversion of the true data set (see DZM).
Synthetic data were computed throughout eq. (7) and
(8) by randomly assigning source parameters to the 87
events constituting the true data-set. The synthetic
data sets were further modified by adding to each
data a random quantity in the range of the true data
uncertainties (see DZM).
The resolution on the three-dimensional Q structure
can be “a priori” roughly evaluated by computing the
hit-count maps, i.e. the schematic picture of the
number of rays crossing each cell of the investigated
volume.
The hit count map for the 87 events here considered is
shown in fig.6. We observe that the N-E part of the
area is densely sampled by rays, while elsewhere the
ray coverage is lesser. The global effect is that
resolution is expected to decrease from the center of
the investigated area toward the periphery, which is
due to the peculiar location of stations and
microearthquakes.
Checkerboard test
We subdivided the volume in a sequence of blocks
alternatively with low (Q=50) and high (Q=600)
attenuation parameters (fig.7). At the third step of
the iterative procedure, the inversion converges to the
retrieved Q structure shown in fig.8. This test
indicates that in the central part, middle depth region
of the caldera we expect a higher resolution. The
retrieved Q values are overestimated in low-Q blocks
and underestimated in high-Q blocks. This
discrepancy is generally smaller in the middle depth
(1-2 km) range than in the other depth ranges. At
shallow depths (1 - 2 km) a highly resolved image is
obtained by the inversion for the whole region with
the exception of the peripheral blocks. At large depths
(2 - 3 km) the image is less resolved and only to the
central region appear rather well constrained.
The loss of resolution with depth is a combined effect
of the increasing P-wave travel distances and of the
smaller ray sampling for deep blocks. Based on these
tests we infer that the western central part of the
caldera at depths greater than 2 km is not resolved by
the considered data set.
Fixed geometry tests
The three following tests are aimed at investigating
the resolution for detection of high-attenuation low-Q
bodies having an irregular shape and extended size.
Also we perform these tests in order to check our
ability in imaging short-wavelength anomaly pattern
and to study the degree of smearing of the resulting
images.
The reliability of the 3D Q images was studied by
assuming peculiar geometries of low-Q anomalies as
inferred by the inversion of the observed data set
(DZM).
In the first test we inverted noisy data computed by
assuming the Q model depicted in Fig. 8.
The image obtained by the inversion show that the
anomaly shape and extension is rather well resolved
but a smearing effect toward East is present. The most
relevant result of this test is the absence of significant
vertical smearing effects that lead us to consider
reliable the vertical Q variation observed from the
inversion of the real data set.
As observed in the checkerboard test, the western part
of the volume is less resolved.
Another test concerned the inversion of noised
synthetic data computed by assuming the Q model in
fig.9. The anomaly is rather well resolved, even
though an overestimate of the low-Q value in the
anomalous body is obtained. This confirms the results
of the checkerboard test with an overestimation of Q
at larger depths. No important smearing effects are
observed on the retrieved images both laterally and
vertically.
In the last test we evaluated the level of recovery of
two low-Q non adjacent bodies embedded in the
intermediate layer between 1 and 2 km depth (Fig.
10). The result of this test indicate the reliability of
the procedure of inversion in retrieving high-quality
imaging at intermediate depth, without the
introduction of significant lateral and vertical
smearing effect of the resulting Qp images.
Conclusion
The modified rise time method, presented in this
paper, arises from the need to take into account the
finiteness of the seismic source in the modeling of the
shape of the direct P and S waves, even at the
microearthquake scale. This is performed by
assuming the circular crack model proposed by Sato
and Hirasawa (1973). With respect to others time
domain modeling techniques [Liu, 1988; Wu and
Lees, 1996; de Lorenzo, 1998] the equations we
derived, other than constitute another theoretical
evidence for non linearity among rise time and travel
time, are also able to account for the directivity of
the seismic source.
We used the source and receiver configuration of the
1984 passive microearthquake experiment in Campi
Flegrei caldera (Southern Italy) to study the inversion
problem for source and 3D attenuation parameters.
Numerical simulations suggest that at the scale length
of the Campi Flegrei microearthquake the recorded
waveforms may be sensitive to the directivity of the
seismic source for fault dimensions greatest than 30 m
and an error threshold of 20-30 m, depending on data
quality.
This may indicate the primary role played by the
source on the recorded seismograms even for small
earthquake sizes.
1.
2.
3.
4.
Numerically built functions have been obtained which
constrained by the inversion of the actual data set.
relate the P rise-time and pulse width to the 5. A priori information on the average residuals on
parameters of the circular crack model (rupture time
source parameters, for the source and receivers
and fault angles). The source radius are therefore
configuration of the Campi Flegrei microearthquakes
obtained by assuming “a priori” a rupture velocity
(and the real level of noise on seismograms) (average
value (90% of shear velocity). These relations can be
residual on source radius about equal to 20 m,
easily extended to S-wave measurements and other
average residual on dip fault about equal to 10°, and
rupture velocity values.
average residual on strike fault about 45°) indicate
For data inversion we propose an iterative non linear
that, also by considering only P waves, the method is
scheme based on the Downhill Simplex optimization
able to constrain, at least in some cases, the
method. The strong trade-off among source and
orientation and the size of microearthquakes.
attenuation parameters does not allow for a joint
The uncertainties in model parameters due to the
inversion of both. We followed a scheme modified
expected errors on data indicate that the various
after Scherbaum [1990] for which at successive steps
assumptions we made (fixed Vp, Vs and Vr at source)
only source or Q parameters are inverted by fixing
should be not critical. Small
variations of this
one of the two at the value of the previous iteration.
quantities reflect in a bias on rise time and pulse
Numerical simulations confirm that this is an optimal
width data in the range of their standard deviation.
strategy and we always get convergence to a
Respect to the previous techniques of joint inversion
minimum norm set of parameters.
of source parameters and attenuation [De Natale et
The uncertainty on parameters is obtained by mapping
al., 1987; Scherbaum, 1990, Abercrombie, 1995] the
the errors on data to the model parameter space. The
method we propose is able to retrieve, other than
shape of the inferred parameter distribution can
source dimensions and quality factors also the fault
provide useful insight about the parameter resolution
plane orientation, at least in the cases in which a
and error.
sufficiently wide variation of source –receivers
The resolution tests that we carried out using noised
take-off angles is performed.
synthetic data, show that the source radius is always
well determined, while the azimuthal source coverage
and the earthquake size primary control the resolution
Appendix A
of the fault orientation angles. The largest
uncertainties concern the estimates of the fault strike.
Let’s define the angles  'i that the normal to the
The resolution studies on Q images indicate that a
fault plane forms with the vertical axis and the angle
contrast in Q of the order of 100 is detectable by the
 'i (azimuth) that the projection of the normal in the
Campi Flegrei data set, at least for regions where a
horizontal plane forms with an assigned axis (here
dense ray sampling is observed.
assumed as the WE direction).
Checkerboard and fixed geometry resolution tests
When are assigned the cartesian coordinates of the
have been performed to assess the reliability of low-Q
hypocenter (assumed known) of the i-th earthquake it
anomalous body detected by the inversion of real data
can be easily shown that the take-off angle at the j-th
(DZM).
receiver is given by:
The main results can be so summarized:
In the well-illuminated central-eastern part of the
 R i2, j  1  Ti2, j 
caldera, the actual data set is able to infer both

(a1)
i, j  arccos
lateral and vertical contrast in Q structure.
 2R i , j

The level of recovering of Q-imaging varies with

depth. It is usually higher at shallow depth, due to the
where R i , j is the length of the vector x i , j from the
higher level of ray crossing at these depth. Also the



center of the source to the receiver and Ti , j  x i , j  n i ,
number of blocks adequately recovered by the

inversion increases with decreasing depth.
being n i the normal to the fault plane.
The absence of significant vertical smearing effect in
Observing that :
fixed geometry test (figg. 12-13) is a clear indication
that, at least in the eastern part of the caldera the
Txi , j  Rsin i , j cos  i , j  sin 'i cos  'i
actual size of blocks is able to solve for vertical Q
contrasts, even if the retrieved Q imaging could be
Ty i , j  Rsin i , jsin i , j  sin 'i sin 'i
(a2)
partially overestimated, particularly at depth.
Tz i , j  R cos  i , j  cos  'i
The attenuation properties of the western,
less-illuminated part of the structure, are poorly
where  i , j and i , j specify orientation of the
straight line connecting source i and receiver j, and
that the dip  i and the strike  i of the fault plane
are related to  'i and  'i (fig.a1) from the relations:
 i   i'
 i  360   i'
(a3)
when the hypocentral coordinates are assumed
known, i , j, according to (a1), is only a function of
source parameters  i and  i , say:
 R 2i, j  1  Ti2, j 
    i ,  i 
i, j  arccos
 2R i , j

(a4)
REFERENCES
Abercrombie, R.E., Earthquake source scaling
relationship from -1 to 5 M L using seismograms
recorded at 2.5-km depth, J. Geophys. Res., 100,
24,015-24,036, 1995.
Aki, K., Scaling law of seismic spectrum, J.
Geophys. Res. 72, 1217-1232, 1967.
Anderson, J.G. and S.E. Hough, A model for the
shape of the Fourier amplitude spectrum at high
frequencies, Bull. Seism. Soc. Am., 74, 1969-1993,
1984.
Aster,R.,Meyer,R., Three-dimensional velocity
structure and hypocenter distribution in the Campi
Flegrei caldera, Italy, Tectonophysics,149, 195-218,
1988.
Azimi, S.A., Kalinin A.V., Kalinin V.V. Pivovarov,
B.L., Impulse and transient characteristic of media
with linear and quadratic absorption laws, Izvestiya,
Physics od solid Earth, 88-93, 1968.
Blair, D.P. and A. T. Spathis, Attenuation of
explosion generated pulse in rock masses, J. Geophys.
Res., 87, 3885-3892, 1982.
Boatwright, J., Detailed spectral analysis of two
small New York State earthquakes, Bull. Seism. Soc.
Am.,68, 1117-1131, 1978
Boatwright, J., Seismic estimates of stress release,
J.Geoph. Res., 89,6961-6968, 1984
Brune, J.N., Tectonic stress and the spectra of
seismic shear waves from earthquakes, J. Geophys.
Res., 75, 4997-5009, 1970.
Courbolex, F. Inversion spatio temporelle de la
source sismique a l’aide des fonctions de Green
empiriques: deconvolution par recuit simule et
application a de seismes de faible magnitude. These
de doctorat de l’Université de Paris VI (1995)
de Lorenzo, S., A model to study the bias on Q
estimates obtained by applying the rise time method
to earthquake data, in Q of the Earth, Global,
Regional and Laboratory Studies, edited by B.J.
Mitchell and B. Romanowicz, Pure and Appl.
Geophys.,153,419-438,1998.
Del Pezzo, E., De Natale, G., Martini,M., and A.
Zollo, Source parameters of microearthquakes at
Phlegraean Fields (Southern Italy) volcanic area,
Phys. Earth Planet. Inter., 47, 25-42,1987.
De Natale, G., Iannaccone, G., Martini,M. and A.
Zollo, Seismic sources and attenuation properties at
the Campi Flegrei volcanic area, Pure Appl.
Geophys., 125, 883-917, 1987.
Gladwin, M.T. and F.D. Stacey, Anelastic
degradation of acoustic pulses in rock, Phys. Earth
Planet. Inter., 8, 332-336, 1974.
Eshelby, J. D., The determination of the elastic field
of an ellipsoidal inclusion and related problems, Proc.
Roy. Soc. London, A, 241, 376-396, 1957.
Ichinose G. A., Smith, K. D. and J.G. Anderson,
Source parameters of the 15 November 1995 Border
Town, Nevada, earthquake sequence, Bull. Seism.
Soc. Am., 87, 652-667, 1997.
Kjartansson, E., Constant Q-wave propagation and
attenuation, J. Geophys. Res., 84, 4737-4748,1979.
Liu,H-P., Effect of source spectrum on seismic
attenuation measurements using the pulse broadening
method, Geophysics, 53, 1520-1526, 1988.
Liu, H.-P. Warrick, R. E. Westerlund, J. B. and E.
Kayen, In situ measurement of seismic shear-wave
absorption in the San Francisco Holocene Bay Mud
by the pulse-broadening method, Bull. Seism.
Soc.Am., 84, 62-75, 1994.
Madariaga, R., High-frequency radiation from crack
(stress drop) models of earthquake faulting, Geoph.
Journ. Royal Astron. Soc., 51, 625-651., 1977.
Molnar, P., Tucker, B. E. and J. N. Brune, Corner
frequencies of P and S waves and models of
earthquake sources, Bull. Seism. Soc. Am., 63,
2091-2104, 1973.
Press, W.H., Flannery, B.P., Teukolsky, S. A., and
W. T. Vetterling, Numerical Recipes,Cambridge
University Press, New York, 1994.
Ricker, N. H., The form and laws of propagation of
seismic wavelets, Geophysics, 18, 10-40, 1953.
Sanders, C.O. and L. D. Nixon, S wave attenuation
structure in Long Valley caldera, California, from
three component S-to-P amplitude ratio data, J.
Geophys. Res., 100, 12,395-12,404, 1995.
Sato ,T and T. Hirasawa, Body wave spectra from
propagating shear cracks, J. Phys. Earth., 21,
415,432, 1973.
Scherbaum, F.,
Combined inversion for the
three-dimensional Q structure and source parameters
using microearthquake spectra, J. Geophys. Res., 95,
12,423-12,438, 1990.
Thurber, C.H., Earthquake locations and three
dimensional crustal structure in the Coyote Lake area,
central California, J.Geophys. Res., 50, 159-170,1982
Tonn, R., Comparison of seven methods for the
computation of Q, Phys. Earth Planet. Inter., 55,
259-268, 1989.
Vasco, D.W., Johnson,L.R., and J. Pulliam, Lateral
variation in mantle structure and discontinuities
determined from P, PP, S,SS and SS-SdS travel time
residuals, J. Geoph. Res.,100, 24,037-24,060, 1995.
Wu, H. and M. Lees, Attenuation structure of Coso
geothermal area, California, from wave pulse widths,
Bull. Seism. Soc. Am., 86, 1574-1590, 1996.
Zollo, A. and G. Iannaccone, Site propagation
effects on the spectra of an SS phase recorded from a
set of microearthquakes in North Appennines, Geoph.
Res. Lett., 23, 1163-1166, 1996
Zhao, D., Hasegawa, A., and H. Kanamori, Deep
structure of the Japan subduction zones as derived
from local, regional and teleseismic events, J. Geoph.
Res., 99, 22,313-22,329, 1994.
FIGURE CAPTIONS
Fig. 1
Velocigrams generated by a local microearthquake and recorded the 3 march-1984 by the array of the University
of Wisconsin at the Campi Flegrei caldera. The time interval is equal for any seismogram (2.5 sec). It is clearly
observable the different pulse width of the direct P wave at various receivers. Because the sources-receivers
distances are very short, the different frequency content of recorded P waves may be primarily due to the
directivity of the seismic source.
Fig. 2
Schematic picture of the rise time  (the time lag between the arrival time of the signal and the first zero crossing
time) and of the total pulse width T (the time lag between the arrival time of the signal and the second zero
crossing time) on velocigrams, as considered in this study.
Fig. 3
(a) Comparison between the values of the rise time as deduced by the eq. (7) (estimated) and those measured on
synthetic seismograms (true). The scatter of the data around the best-fitting value (eq.7) (standard deviation
10 7 s) indicate that the eq.7 constitute only a first order approximation of a more complex dependence of rise
time on source parameters and attenuation.
(b) Comparison between the values of the total pulse width as deduced by the eq.8 (estimated) and those measured
on synthetic seismograms (true). The low degree of scattering of data around the best fitting value (standard
8
deviation 10 s) indicate that total pulse width is a linear function of the attenuation term t*.
Fig. 4
(a) Residuals between true and estimated source parameters (source radius and fault angles) as a function of the
source radius after the inversion of a noise-free synthetic data set realized by considering the sources-receivers
configuration corresponding to 39 earthquakes recorded at the Campi Flegrei caldera in the period march-april
1984.
Comparison between:
(b) true (dot) and estimated (open circles) rise time
(c) true (dot) and estimated (open circles) pulse width
for six events of the same data set. The directivity due to the finiteness of the rupture introduces, on the
scale-length of the Campi Flegrei caldera, a clear non-linearity of pulse width and rise time vs. distance.
Fig.5
Comparison between true and estimated source parameters after the inversion of a noisy synthetic data set (level of
noise comparable with that existing in real data). Only those parameters that can be considered resolved after the
 2 statistic test are shown.
In (5a) the source receiver configuration used consists of 58 synthetic events and the heterogeneous attenuation
structure considered in the inversion is reported in fig (7).
In (5b) the source receiver configuration used consists of 58 synthetic events and the heterogeneous attenuation
structure considered in the inversion is reported in fig (8).
Fig.6
Hit count map for the data set constituted by 87 events and considered in this study. The NE part of the area is
highly illuminated by the rays.
Fig.7
Checkerboard resolution test for the three dimensional case. On the left is shown the true attenuation structure
(low Q= 100, high Q=600), on the right the recovered Q structure.
Fig.8
3
Fixed geometry test for the three-dimensional case. It was assumed that eigth adjacent cells (1 1 1 km ), located
between 0 and 1 km of depth, are characterized by a low Q (Q=50) and the remaining structure is characterized by
a higher Q (Q=600). On the left is shown the true attenuation structure, on the right the recovered Q structure.
Fig.9
3
Fixed geometry test for the three-dimensional case. It was assumed that four adjacent cells (1 1 1 km ), located
between 2 and 3 km of depth, are characterized by a low Q (Q=100) and the remaining structure is characterized
by a higher Q (Q=600). On the left is shown the true attenuation structure, on the right the recovered Q structure.
Fig.10
Fixed geometry test for the three-dimensional case. It was assumed that two distinct blocks (1 1 1 km 3 ), located
between 1 and 2 km of depth, are characterized by a low Q (Q=100) and the remaining structure is characterized
by a higher Q (Q=650).
On the left is shown the true attenuation structure, on the right the recovered Q structure.
Fig. A1
Schematic picture of the angles defining the fault plane orientation as considered in this study.