Post on 13-Feb-2021
Moment Tensor Resolvability: Application to Southwest Iberia1
2
Jǐŕı Zahradńık1 and Susana Custódio23
1Charles University in Prague, Faculty of Mathematics and Physics, Czech Republic4
2University of Coimbra, Centre for Geophysics – Geophysical Institute, Portugal5
6
August 1, 20117
8
Jǐŕı Zahradńık (corresponding author):9
Charles University in Prague, Faculty of Mathematics and Physics, V Holesovickach 2, 18010
00 Prague 8, Czech Republic, Czech Republic, jiri.zahradnik@mff.cuni.cz11
1
Abstract12
We present a method to assess the uncertainty of earthquake focal mechanisms13
based on the standard theory of linear inverse problems. We start by computing the14
uncertainty of the moment tensor, M. We then map the uncertainty of M into uncer-15
tainties of strike, dip, and rake. The inputs are: source and station locations, crustal16
model, frequency band of interest, and an estimate of data error. The output is a17
6D error ellipsoid, which shows the uncertainty of the individual parameters of M.18
We focus on the double-couple (DC) part of M. The method is applicable when data19
(waveform) are available, as well as when no waveforms are available. The latter is20
particularly useful for network design. One of the most interesting applications that we21
show are maps of DC resolvability, which are computed without waveforms. Examples22
are shown for earthquakes in Southwest Europe. We find that the resolvability de-23
pends critically on source depth. Shallow DC sources (10 km) are theoretically better24
resolved than deeper sources (40 and 60 km). The DC resolvability of the 40-km deep25
event improves considerably when the Portuguese network is supplemented by stations26
in Spain and Morocco. Adding a few OBS stations would improve the resolvability.27
However, a similar improvement in resolvability can be achieved with a densification28
of land stations. A dense land network is able to resolve M well in spite of the large29
azimuthal gap, which spans ∼ 200o.30
2
1 Introduction31
Earthquake models can be divided into two broad categories: point- and finite-sources.32
Finite-fault models contain information about the spatial evolution of slip on the fault.33
Point-source models simplify the earthquake as slip that occurs on a single point in space.34
Point-source models are useful to describe the geometry of faulting as well as non-shear35
components of the source, even for small and moderate-size earthquakes. These parameters36
are quite useful, e.g. to understand the tectonics of a given region. The most general and37
widely-used description of a seismic point source is the moment tensor (M), a 2nd-rank,38
three-by-three, symmetric tensor [Gilbert , 1971; Aki and Richards , 2002]. M describes any39
source of deformation in an elastic medium in terms of force-couples. The moment tensor40
of shear/tectonic earthquakes represents a simple double couple of forces. In this case,41
M contains information both about the magnitude of the earthquake and the geometry of42
faulting. The strike, dip and rake angles can be derived from M.43
At present, seismologists use ground-motion waveforms to infer M on a routine basis.44
The problem of studying the seismic source given observed ground motion is an inverse45
problem (the corresponding direct problem is to compute the ground motion caused by a46
seismic source). The result of any inverse problem has limited reliability. The inversion of M47
is now common practice, however the errors of M are rarely reported. The assessment of the48
reliability of M is particularly important when the dataset is less-than-perfect, e.g., when49
only a few stations are available, when a large azimuthal gap exists, or when the signal-50
to-noise (S/N) ratio is low in the frequency band of interest. Why are the uncertainties51
3
of M not routinely reported? We can identify two main reasons: (i) It is relatively easy52
to compute error estimates for the individual components of M, because they are linearly53
related to the waveforms (for a fixed depth and origin time). However, the more intuitive54
faulting parameters – strike, dip and rake angles – are not linearly related to the waveforms,55
so their uncertainties are more difficult to estimate. (ii) The evaluation of the absolute errors56
of M requires an assessment of data error, which in principle cannot be obtained. A third57
reason may be suggested: Users of M (or strike, dip and rake) are often more interested in a58
final result than in a lengthy/extensive discussion of error. However, using the results of an59
inversion without thinking about its errors is dangerous: A solution to an inverse problem60
can always be found; however the solution will be meaningless if the errors are too large.61
This paper focuses on the uncertainty of M. We address both issues stated above: (i) We62
show how to compute the errors of M and how to translate them into errors of strike, dip63
and rake; (ii) We discuss data errors, point ways to estimate their magnitude, and indicate64
applications for which an actual assessment of data error is not needed. Finally, the method65
that we present is simple and fast; thus it can be implemented for routine inversions of M.66
The resolution of the moment tensors retrieved from waveform inversions has been a67
concern to seismologists for long [e.g. Riedesel and Jordan, 1989; Vasco, 1990]. Nevertheless,68
the issue remains a topic of active research. Riedesel and Jordan [1989] showed how to use69
the covariance of M (a 4th-rank tensor) to graphically visualize the uncertainty of moment70
tensors. However their method is based on perturbation theory, hence only applicable to71
small parameter variations [Vasco, 1990]. Vasco [1990] studied the errors in M arising72
from noise in data and imperfect Green’s functions. He proposed a method to map data73
4
errors into bounds of three invariants of M: the trace (which corresponds to the isotropic74
component of the source), the determinant, and the sum of the diagonal minors of the75
determinant. More recently Yagi and Fukahata [2011] proposed an inversion algorithm that76
takes into account the uncertainty of the Green’s functions (their method concerned a finite-77
fault inversion). They computed the covariance matrix from the misfit between observed and78
synthetic waveforms. Then they rigorously included the covariance in the inverse problem79
to iteratively improve the estimate of the model parameters. Vavryčuk [2007]; Bukchin80
et al. [2010] showed that trade-offs between individual components of the moment tensor81
impose limits on the resolvability of M, even for error-free data and Green’s functions (GFs).82
Such trade-offs are caused by symmetries in the seismic wave field and/or source-station83
configuration. In such situations, different earthquakes (as defined by M) generate indistinct84
waveforms at the stations where data is recorded.85
A popular approach to assess the robustness of M is to find a family of acceptable86
solutions rather than just one best-fitting solution. In this case, the family itself is used to87
define the confidence intervals of the model parameters. For example, Š́ılený [1998] generated88
families of acceptable Ms using a genetic algorithm. Some authors repeat their inversions89
with data contaminated with artificial noise; e.g. Hingee et al. [2011] considered noise as high90
as 100% of the data amplitude. Others repeat the inversions in artificially perturbed crustal91
models [Š́ılený , 2004; Wéber , 2006; Vavryčuk , 2007]. Ford et al. [2009] discussed inaccurate92
crustal models in a very practical way: in terms of temporal shifts between observed and93
synthetic waveforms. Ford et al. [2010] proposed the network sensitivity solution (NSS), a94
way to assess the reliability of the non-DC components of M. The NSS is obtained through95
5
a grid search over the whole space of model parameters. Misfits (or equivalently, variance96
reductions) are computed for all possible source models. The results are displayed using a97
source type plot, which allows the easy identification of shear vs volumetric components of98
M [Hudson et al., 1989]. All methods above provide families of acceptable solutions without99
specifying absolute confidence intervals. In fact, absolute confidence intervals can only be100
computed given an independent assessment of data error (which cannot be obtained). The101
studies described above show that defining a range of acceptable solutions is useful, even if102
just in a relative sense (i.e., without determining absolute probabilities for specific confidence103
levels).104
In this paper we present a simple approach: We use the well-established theory of linear105
inverse problems to map data errors into errors of the model parameters [e.g. Menke, 1989;106
Press et al., 1992; Parker , 1994]. Thus we obtain an estimate of the uncertainty of M. We107
then map the uncertainty of M into uncertainties of strike, dip, and rake. Our approach108
is similar to that of Vasco [1990] in that we numerically map data errors into errors of109
the model parameters. Another method that shares similarities with ours is the NSS of110
Ford et al. [2010]. The NSS extensively explores the parameter space in order to image111
the uncertainty of M. However, the NSS requires the repeated computation of synthetic112
ground-motion and variance reduction (or misfit). Our method does not require the actual113
computation of synthetic ground-motion, it is simply based on the analysis of the Green’s114
functions. Also, our method allows an easy assessment of the resolvability of pure double115
couple (DC) sources. The uncertainty of pure DC sources is not visualized in the NSS116
because all pure DC solutions are plotted on a single point of the source-type plot.117
6
The uncertainty analysis presented in this paper uses the full moment tensor. However,118
we focus on the DC part of M, leaving the resolvability of non-DC components for a separate119
study. The uncertainty of M, as presented in this paper, is useful in two distinct situations:120
(i) Waveforms are available, they have been inverted to find M, and the resulting source121
parameters need “error bars”; (ii) No waveforms are available, but we are still interested122
in the resolvability of a given focal mechanism. The latter is typical of network design or123
experiment planning. The tools that we present in this paper are quite simple and based on124
standard theory; nevertheless they provide a fast and computationally cheap way to assess125
the resolvability of M, both with and without data.126
Examples of the assessment of DC uncertainty are shown for earthquakes in Southwest127
Iberia. First, we study the resolvability of a reference earthquake located offshore. We then128
investigate how the resolvability of DC solutions varies with earthquake depth, frequency129
range, crustal structure, station configuration, and data error. We also present maps of130
resolvability for different network configurations. Some of the questions that we answer are:131
Is it enough to use the Portuguese network in order to resolve DC parameters for offshore132
earthquakes? Or do we need to add stations from neighbor countries? Or do we even need133
to add ocean bottom seismometers (OBSs), thus improving the azimuthal coverage? This134
last question has practical implications, because the deployment of OBS stations is more135
expensive and difficult than that of traditional land stations.136
7
2 Method137
In this section we describe the linear inverse problem of finding the moment tensor M given138
observed ground motion. We then show how to compute the resolvability of the inferred139
model parameters. We also indicate how these values can be translated into uncertainties140
of strike, dip, and rake. Intentionally, part of the theory is presented in an Appendix; both141
because the theory behind the computations is standard and because moving the theory to142
the Appendix allows a more intuitive interpretation of the concepts.143
2.1 The Moment Tensor144
Ground motion is linearly related to the moment tensor:145
di(t) =6∑
j=1
Gij(t)mj (1)
In the symbolic equation above di(t) are the displacement waveforms (each one recorded146
on a different station/component i), Gij(t) is the matrix of Green’s functions (which we147
compute based on crustal structure, source location, station location, and frequency range),148
and mj are the unknown parameters which fully determine M. Suppose that we have N149
waveforms to use in the inversion, such that i = 1, . . . , N . The number of model parameters150
to determine is six, equal to the number of independent components of M. Then d(t) is a151
column vector of size N × 1, G(t) is a matrix N × 6, and m is a column vector of size 6× 1.152
The inversion is formally over-determined when N À 6.153
The six parameters mj may be the actual moment tensor components. More conve-154
8
niently, mj may also be the coefficients of elementary focal mechanisms, which in turn are155
linear combinations of the independent components of M [Kikuchi and Kanamori , 1991;156
Zahradńık et al., 2005, 2008a,b]. In this case, the columns of G are the so-called elementary157
seismograms, which correspond to the six elementary moment tensors M1, . . . ,M6. The158
moment tensor M is constructed as a linear combination of Mi:159
M = m1M1 + . . . + m6M6 (2)
In our analysis we assume that the time dependence of M is known (e.g. the source160
time function is a delta function). We further assume that the earthquake origin time and161
location are known. Finding M, or solving (1) and (2), is then a linear problem.162
2.2 Best Solution vs Range of Acceptable Solutions163
The goal of the inverse problem is to find a set of model parameters (mj), which generates164
synthetic ground motion that agrees with the data. The solution to this problem is found165
by minimizing the misfit between observed data (di) and synthetics (si). Synthetics are166
given by the right-hand side of (1), such that s = Gm. We can proceed in two ways:167
(a) We can compute a single “best” (minimum-misfit) solution, e.g. through least-squares168
minimization, LSQ; (b) We can use a global minimization method, e.g grid search, to obtain a169
set of acceptable solutions with misfits ranging from a minimum to a threshold value. Option170
(a) is attractive when we look for a single solution for a genuinely linear problem, like the171
inversion of M. However, we favor option (b) as it provides a distribution of acceptable172
parameters (e.g. strike, dip, and rake angles), or a distribution of the deviations from their173
9
optimal values. Option (b) allows us to learn about non-optimal solutions, which still match174
the data well.175
Let us start with case (a): We want to find a single “best” solution. The least-squares176
procedure minimizes a misfit e, which is given by the sum of the squared residuals:177
e =∑
i
(di − si)2 (3)
The summation is performed over all stations, components and times. In the Appendix178
we outline two different ways of obtaining the LSQ solution (using normal equations or179
singular value decomposition, SVD). An obvious drawback of this approach is that if the180
Green’s functions are inaccurate or the observed waveforms are noisy, their perfect match by181
synthetic seismograms is misleading. Therefore, a better strategy is to introduce a weighted182
misfit that takes into account the data variance, σ2i , corresponding to the standard deviation,183
σi, of data i:184
χ2 =∑
i
(di − si
σi
)2(4)
Note that only statistical errors are taken into account in σi (i.e., errors that would vanish185
if we could repeat one observation enough times and then compute an average). Systematic186
errors (those that would not disappear with repetition and average) are not taken into187
account in σi. Let us assume that all the waveforms are affected by the same data variance188
σ2d. (We make this assumption for the sake of simplicity. An analogous derivation using189
different σi for each data piece is straightforward.) In this case σi = σd for all waveforms i,190
and (4) simplifies to:191
10
χ2 =
∑i(di − si)2
σ2d(5)
The set of model parameters that generates the lowest χ2 is the optimal solution, which192
we will name the χ2min-solution.193
Now let us consider case (b): We are interested not only in the optimal solution (χ2min-194
solution), but also in a family of reasonable solutions. We define the class of acceptable195
solutions as those for which the weighted misfit has a maximum value of χ2th, where the196
subscript th stands for threshold.197
χ2th = χ2min + ∆χ
2 (6)
∆χ2 is the tolerated increase in misfit with respect to the minimum-misfit solution. We198
can then define the family of acceptable solutions as those for which, e.g., ∆χ2 < 1:199
∆χ2 = χ2th − χ2min < 1 (7)200
(d− sth)2σ2d
− (d− smin)2
σ2d< 1 (8)
201
e2th < e2min + σ
2d (9)
We can easily see that the solutions for which ∆χ2 < 1 are those whose misfits (e2th)202
are larger than the minimum misfit (e2min) by less than the data variance (σ2d). The number203
of solutions within ∆χ2 < 1 depends explicitly on data error. More (or less) solutions are204
accepted when σ2d is larger (or smaller).205
11
In actual moment-tensor inverse problems, it is common practice to maximize the variance206
reduction, VR, instead of minimizing the misfit e.207
VR = 1−∑
i(di − si)2∑i d
2i
(10)
We can also express ∆χ2=1 in terms of VR, using (5), (7) and (10):208
∑i
d2i (VRmin − VRth) = σ2d (11)
or209
VRth = VRmin − σ2d∑i d
2i
(12)
The equations above demonstrate that choosing a set of acceptable solutions via prescrip-210
tion of a variance reduction threshold, VRth, is in fact equivalent to assigning a value to the211
data variance, σ2d. For example, accepting only the optimal solution (i.e. VRth = VRmin) is212
equivalent to assuming that the data is error-free (σd = 0). Equation (12) also shows that a213
good numerical value for the variance reduction (i.e., VR close to VRmin) is meaningless if214
σ2d is underestimated.215
In this section we introduced the concept of an ensemble of acceptable solutions based216
on data error. This concept is very natural if we have a dataset, and various solutions match217
the data almost as well as the optimal solution. Another important point of this section is218
that we must take into account data error in order to compute meaningful uncertainties of219
the model parameters. The next section will focus on the determination of the ensemble of220
acceptable solutions according to the χ2 criterium.221
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2.3 Resolvability of the Moment Tensor222
Above we explored a situation in which we had a dataset with data variance σ2d, and wanted223
to find a family of acceptable solutions of M. The same idea can be applied to a situation224
where no data are available, but we are still interested in the uncertainty of a given focal225
mechanism. For example, we want to know how a given network resolves the moment tensor226
of offshore vs onshore earthquakes. Or we want to know how the resolvability of a deep227
event compares with that of a shallow event. Or we want to estimate the error of a solution228
reported by an agency, but we do not have the actual data. Or we plan a future deployment,229
and want to know what station configuration is best. In the situations above we need to230
evaluate the uncertainty (or resolvability) of a given focal mechanism without using data.231
The family of acceptable solutions (∆χ2 < ∆χ2threshold) can be defined by prescribing σ2d,232
exactly as explained in the previous section. However, if we have no data, how can we233
find a family of acceptable solutions? The theory of linear inverse problems has a simple234
answer for such question: Errors of the model parameters are related to data errors through235
the Green’s functions matrix G. Therefore, we can mathematically analyze a given source-236
station configuration, prescribe σ2d, and compute the uncertainty of a given focal mechanism.237
The data itself is actually not needed to compute parameter uncertainties.238
In moment tensor inversions, we need to determine six model parameters, m1, ...,m6239
(Equations (1) and (2)). Thus the theoretical uncertainty region is six-dimensional (6D)240
[e.g. Press et al., 1992]. In linear problems, the surfaces of constant L2 misfit (e.g. constant241
e, χ2, ∆χ2, or VR) are ellipsoids in the 6D model space. The use of an L2 formulation242
13
implicitly assumes that the data errors are Gaussian (i.e. normally distributed). The shape243
of the 6D region will, in general, be more complex if data errors are non-Gaussian or if the244
problem is non-linear. The reader is probably familiar with the error ellipsoids used for245
earthquake locations. A few striking differences between the error ellipsoids of earthquake246
locations and those of moment tensors are worth mentioning: The main difference is that the247
location of earthquakes is a strongly non-linear problem; thus the location error ellipsoids248
are of little interest (they are valid only for the linearized problem). The proper assessment249
of location errors requires the computation of non-elliptical 3D error volumes [e.g. Lomax ,250
2005]. On the other hand, the error ellipsoids of M cannot be fully visualized because they251
are six-dimensional. Thus, and although available in closed mathematical form, the error252
ellipsoid of M has to be transformed into more meaningful quantities in order to become253
useful.254
Throughout this paper we will present uncertainty regions (or error ellipsoids) rather255
than confidence regions (as strictly defined in a statistical sense). The error ellipsoid can256
be “translated” into a confidence region (e.g., a 90% confidence region) only if data errors257
are normally distributed and their exact absolute value is known. We refer the reader to258
section 15.6 of Press et al. [1992] for more details on how to compute ellipsoids of different259
confidence levels. Next we briefly explain how to compute the error ellipsoid of M; the260
Appendix contains a full-detail description of the algorithm.261
The errors of the model parameters σ(m) depend exclusively on the Green’s functions G262
and on the data variance σ2d. In fact, the shape and orientation in model space of the error263
ellipsoid is computed from the eigenvalues and eigenvectors of the matrix GTG (Figure 1;264
14
see the Appendix for more details). The data variance σ2d scales the size of the ellipsoid. In265
order to compute the error of the model parameters σ(m) we need the following information:266
location of the earthquake (epicenter and depth), station coordinates, crustal model, and267
frequency band (all these parameters enter the computation of G); and an estimate of268
data error σd. We use the discrete wavenumber method, which takes as input 1D layered269
models of velocity-density-attenuation, in order to compute Green’s functions [Bouchon,270
1981; Coutant , 1989]. Data error is extensively discussed in the next section.271
The error ellipsoid is centered at a point mref of the parameter space. In the situation272
where we have data, mref corresponds to the optimal solution (∆χ2=0). If we do not have273
actual data, mref is the focal mechanism (or M) whose resolvability we wish to assess. Let274
us consider the ellipsoid defined by ∆χ2 ≤ 1. This choice of ∆χ2 threshold is arbitrary:275
We will not use ∆χ2 to assign a probability to a specific confidence level. The family of276
acceptable solutions inside the ellipsoid, including its surface, is defined as:277
macceptable = mref + ∆m (13)
In the equation above, macceptable is an acceptable solution and ∆m is a radius vector from278
the center of the ellipsoid to a point in the surface of the ellipsoid. In order to numerically279
find the family of acceptable solutions, we simply perform a 6D grid search around mref280
and find the solutions for which ∆χ2 ≤ 1. Thus we determine which δmj (variations of mj281
around mref ) fall inside the ellipsoid for all j = 1, . . . , 6. Or in other words, we determine282
the amount by which we can vary the source parameters, δmj, without leaving the ∆χ2 ≤ 1283
region. Because the inverse problem is linear in Mi, the same 6D ellipsoid (set of points284
15
∆m) is valid for investigating the uncertainty of any mref . That is, we can use the same285
error ellipsoid to study the resolvability of an earthquake with any focal mechanism (notice286
that the actual knowledge of M is not required to compute the error ellipsoid). This method287
of exploring the parameter space is considerably faster than other sensitivity studies which288
need to use seismograms and compute misfits.289
The 6D error ellipsoid contains information about the uncertainty of the full moment290
tensor, including trade-offs between individual components of M. We then have to translate291
these uncertainties into meaningful quantities. We will focus on the double-couple (DC)292
component of M, as described by the strike, dip and rake angles. The relation between M293
and strike, dip, and rake is non-linear. Therefore in order to compute a family of acceptable294
solutions in terms of strike, dip and rake, we must choose a source mechanism mref . The295
family of acceptable DC solutions can be displayed using nodal lines. In addition, we use296
the “Kagan’s angle” [Kagan, 1991], hereafter K-angle, in order to characterize the family297
of aceptable solutions. K-angle is the smallest rotation between two orthogonal bases de-298
scribing different DCs (e.g. the smallest angle between the P, T, and B bases of two DCs;299
or the smallest angle between the slip vectors of two DCs). In our application, the K-angle300
corresponds to the smallest amount by which we have to rotate each acceptable solution in301
order to get mref . Although less intuitive than strike, dip, and rake, K-angle conveniently302
quantifies the deviation of the family of aceptable solutions from mref using one single value.303
The K-angle is especially useful to plot maps of resolvability, which show the geographical304
variation of DC resolvability (in this case we need one single parameter do characterize the305
resolvability) – see section 3.8.306
16
2.4 Data Error307
A key parameter in the estimation of the uncertainty of M is data error. In principle, one can308
never know the exact value of data error. In this section we discuss several ways to estimate309
σd. Let us start by considering the situation when we do have a dataset to perform the310
inversion. Vasco [1990] proposed that the value of the data error be the maximum between311
the noise background level (σn) and the misfit between observed and synthetic waveforms:312
σd = max (σn, |d− s|) (14)
The noise term σn generally produces values for data error which are on the order of 1%313
– 10% of the data amplitude (noisier data are normally left out of the inversion). The term314
arising from the misfit between data and synthetics (|d− s|) can lead to considerably higher315
estimates of σd.316
Let us now consider the case in which we do not have data, but still want to assess the317
resolvability of M. In this case we can estimate the order of magnitude of expectable data318
error from empirical arguments. Let the data error, as defined by its standard deviation σd,319
be related to data amplitude by:320
σd = C.G.M0 (15)
Where C is a constant, G is the peak displacement of the medium in response to a unit-321
moment source dislocation (in the studied source-station configuration and frequency band),322
and M0 is the seismic moment of the earthquake whose resolvability we wish to assess. Note323
17
that G.M0 is an estimate of the data amplitude. Thus, if C = 1, σd is of the same order of324
magnitude as (synthetic) waveform data. What is a realistic estimate of C?325
Equation (11) shows that σ2d is related to the data power∑
i d2i through the variance326
reduction difference VRmin−VRth. Consider the following thought experiment: We inverted327
a dataset, found the strike, dip, and rake angles through grid search, and defined a family of328
acceptable solutions by prescribing a variance reduction threshold VRth. Specifically, assume329
that VRmin = 0.7 and we chose VRth = 0.6. According to (11), accepting such values for330
the variance reduction is equivalent to estimating the magnitude of data variance:331
σ2d = (0.7− 0.6)∑
i
d2i (16)
Recall that the summation of the data di is made for all stations, components and times.332
Assume that we have 7 stations, and 3 waveforms were recorded at each station. Further333
assume that the dominant wave-group in the ground displacement is a 10-second triangle334
with peak value P. Then:335
σ2d = 0.1∑
i d2i
= 0.1 ∗ 3 ∗ 7 ∗ 102P 2
= 10.5 ∗ P 2
(17)
and336
σd = 3.2 ∗ P (18)
Thus σd is proportional to the data amplitude (σd ∝ P ), in good agreement with (15):337
σd ∝ G.M0. In fact, (18) is equivalent to (15) if C = 3.2. Choosing a family of acceptable338
18
solutions via prescription of VRth close to VRmin is equivalent to prescribing a value for data339
error of the same order of magnitude as the data itself. Note that in the example above data340
error was even larger than the data amplitude (C > 1). The exact value of C depends on the341
exact value of VRth, on the number of waveforms N , and on the source time function. This342
example shows that choosing σ2d lower than the data amplitude is probably too optimistic.343
One more example: Consider identical waveforms shifted with respect to each other by344
τ . For simplification, assume that the identical waveforms are harmonic and their period is345
T . The mismatch ² between the two waveforms is then given by:346
² = d(t)− d(t− τ) ≈ ḋ(t)τ = d(t)2πT
τ (19)
If the difference between waveforms, ², is taken as a representation of the data error σd,347
then σd is proportional to data itself d(t). In this case, C = (2π/T )τ . Data error is smaller348
than the data amplitude (C < 1) only for small time shifts, τ < T/2π. Although extremely349
simple, this example is very relevant for inversions of M. In fact, time shifts between the350
dominant quasi-periodic wavegroup of data and synthetics are common in moment tensor351
inversions. In opposition, the amplitude of the observed and synthetic waves is normally352
well matched. The time-shifts between data and synthetics are due to inaccurate crustal353
models or mislocations of the earthquake hypocenter. Some practitioners introduce artificial354
time shifts to align the observed and synthetic seismograms prior to the inversion. In case355
τ > T/2π, such alignment corresponds to declaring C > 1, i.e. that data error is larger than356
the data itself (e.g. Ford et al. [2009] considered time shifts up to half-cycle T/2).357
The example above demonstrate that realistic estimates of the data error, mainly due to358
19
inexact crustal models, must be relatively large and comparable to data amplitude. Thus,359
an appropriate choice is C ≥ 1.360
3 Applications to Earthquakes in Southwest Iberia361
We now present examples of the assessment of DC resolvability for earthquakes on the362
active plate boundary between Africa (Nubia) and Eurasia, west of Portugal [e.g. Buforn363
et al., 1988; Serpelloni et al., 2007]. The region is characterized by events of various focal364
mechanisms [Buforn et al., 1988, 1995; Borges et al., 2001; Stich et al., 2003; Buforn et al.,365
2004; Stich et al., 2006, 2010], that occur at depths down to 60 km [Grimison and Chen, 1986;366
Engdahl et al., 1998; Stich et al., 2010; Geissler et al., 2010]. Figure 2 shows a map of the367
studied region, displaying both the seismicity and the seismic network. In the applications368
below we study the resolvability of the DC component of M without using actual waveforms.369
In the remainder of the text we will simply use the expression “resolvability” or “resolvability370
of DC” to refer to the resolvability of the DC component of M. We investigate the effect of371
earthquake depth, focal mechanism, station configuration, crustal structure and frequency372
band on the resolvability of a DC. All the results are summarized in Table 1. We also assess373
the sensibility of the results to our choice of data error σd. Finally, we compute maps of374
resolvability for different station networks. The cases studied below were designed to mimic375
real situations.376
20
3.1 Reference Setup377
We will start by studying the resolvability of a DC source for a reference configuration as de-378
termined by earthquake location (epicenter and depth), focal mechanism, crustal structure,379
frequency band, station network, and data error. We will then change these parameters380
one by one to investigate their effect on DC resolvability. The reference configuration is381
defined as follows: The station network is composed of seven land stations located in Portu-382
gal, Spain, and Morocco (hereafter named IB network). These seven stations are: PMAFR,383
PFVI, PNCL and PBDV in Portugal; SFS in Spain; and RTC and NKM in Morocco. Many384
more broadband stations are currently installed in Iberia and northern Africa (Figure 2).385
We chose to use a limited set of stations in order to show a simple application of the method.386
The analysis with a reduced set of stations is also instructive to learn about the resolvability387
of DCs in less favorable situations, in which only a few stations are available. The refer-388
ence earthquake is located offshore, on the Horseshoe Abyssal Plain (HAP), at 35.90oN and389
10.31oW (Figure 2). This location was chosen after the epicenter of a MW6 earthquake that390
occurred on February 12, 2007 [Stich et al., 2007]. The reference epicenter is located in a re-391
gion of frequent seismic activity [Geissler et al., 2010]. The reference source depth is 40 km,392
also chosen after the February 12, 2007, earthquake. Earthquakes occur frequently at depths393
of 40–60 km below the HAP [e.g. Geissler et al., 2010]. The crustal structure is described394
by the 1D layered model proposed by Stich et al. [2003] for the Hercynian basement, Meso-395
zoic platforms and mixed propagation paths. The reference frequency band is 0.025 to 0.08396
Hz. The reference focal mechanism mref is described by scalar moment M0 = 1 ∗ 1016 Nm,397
21
and strike, dip, and rake angles equal to 128o, 46o, and 138o, respectively (or equivalently398
250o, 61o, and 52o). We assume that data error σd is equal to 10−5 m. This value is ob-399
tained using Eq. (15): We take the seismic moment of the February 12, 2007, earthquake400
(M0 = 1 ∗ 1016 Nm), compute G (∼ 10−21 m/(Nm)), and set C = 1.401
We start by determining the ∆χ2 < 1 error ellipsoid for the reference configuration402
(described above), according to the Appendix. Once we have the error ellipsoid, we find the403
corresponding acceptable values of the faulting angles – strike, dip and rake: 1) We take all404
the (discrete) points inside the ellipsoid, macceptable (13); 2) We convert macceptable into actual405
moment tensors Macceptable (2); and 3) We convert Macceptable into strike, dip and rake using406
the standard relations between M and faulting angles. This simple transformation allows407
an intuitive visualization of the whole family of acceptable solutions. Figure 3 (a–c) shows408
the faulting angles plotted as a function of the misfit ∆χ2. The left- and right-hand edges409
of the plots correspond to the center (∆χ2 = 0) and surface (∆χ2 = 1) of the ellipsoid,410
respectively. The two strips in each panel correspond to the two DC conjugate solutions411
(two possible faulting planes). Note that we find solutions close to the reference values of412
strike, dip, and rake, even on the surface of the ellipsoid (we will see further ahead that413
they do not occur simultaneously though). A qualitative analysis of Figure 3 (a–c) already414
yields interesting results: For example, the fault plane which strikes 250o is better resolved415
than that which strikes 128o. Figure 3(d) shows the same information as panels 3(a)–(c)416
displayed in a different format: The deviations of strike, dip and rake from the reference417
solution are now quantified with the K-angle. The K-angle summarizes in one single value418
the deviation of a given solution from the reference solution. The histograms of Figure 3 (e–h)419
22
show the statistical distribution of strike, dip, rake, and K-angle. Note that the histograms420
of the strike, dip and rake angles are peaked at their reference values (one peak for each421
conjugate solution). However K-angle is not peaked at zero, as would be expected if the422
most frequently-occurring solution were mref . The Appendix explains that most points423
inside the error ellipsoid are located on, or very close to, the ellipsoid surface. This sampling424
of the error ellipsoid results from the regular sampling of the 6D parameter space. On the425
surface of the ellipsoid, the reference values occur individually but not simultaneously. In426
fact, the reference values occur simultaneously only at the center of the ellipsoid.427
For the reference configuration, the mean value of the K-angle is Kmean = 14o, its max-428
imum value is Kmax = 34o, and its standard deviation is σ(K) = 6o. These values are not429
meaningful in an absolute sense because they depend on our order-of-magnitude estimate430
of data error. However they are useful in a relative sense, as they allow us to compare the431
resolvability of a given focal mechanism in different configurations of the inverse problem.432
We will mostly use Kmean throughout the paper in order to compare the DC resolvability433
in different situations. Kmean measures the deviation of the acceptable solutions (mostly434
located on the surface of the ∆χ2 = 1 ellipsoid) from mref .435
3.2 Source Depth436
We will now repeat the analysis above using two different hypocenter depths: a shallower437
10-km source and a deeper 60-km source. Figure 4 shows that the uncertainty of the focal438
mechanism increases dramatically with source depth. The mathematical explanation for439
such decrease is that different source-station configurations have different Green’s functions440
23
(matrix G); hence different singular vectors and singular values, and different error ellipsoids.441
The physical explanation is not as straightforward; nevertheless it is understandable that442
deeper sources produce a simpler wave field, with weaker Lg waves. Thus seismograms from443
deep sources carry less, or lower-amplitude, information about the source. Note that the444
excellent resolvability of the shallow source (10 km) assumes that we know the exact Green’s445
functions. The uncertainty of the DC solution for a 10-km deep source is underestimated by446
our analysis if the shallow crustal structure is less well known than its deeper counterpart.447
3.3 Station Network448
We now test the resolvability of DC parameters using different station networks. The earth-449
quakes that happen offshore SW Iberia are normally recorded by stations in Iberia and450
northern Africa; however a large azimuthal gap does exist, which hinders the studies of seis-451
mic sources. In order to improve our knowledge of the local earthquake activity one might452
think of deploying a permanent ocean bottom seismometer (OBS) network. It is interesting453
to investigate how the resolvability of M would improve using such OBS network. Thus we454
tested the following stations configurations (Figure 2): (i) Network IB (the already-described455
reference configuration) with 7 land stations distributed in Portugal, Spain and Morocco;456
(ii) Network PT, composed of only four stations in Portugal; (iii) Network IB+OBS, com-457
posed of the IB land stations plus four OBS stations; and (iv) Network IBdense, composed458
of the IB network plus 23 land stations along the coastline of Portugal, Spain and Mo-459
rocco, totalizing 30 stations . The location of the four OBS stations were chosen as the true460
locations of temporary OBS stations [Geissler et al., 2010]. Figure 5 shows that, for the461
24
reference network IB, the family of acceptable solutions has a mean deviation from the true462
focal mechanism Kmean = 14o. Kmean increases to 22
o when we use the reduced land network463
PT, corresponding to a strong decrease in the resolvability of DCs. The resolvability is in464
turn dramatically improved when we add four OBS stations to the IB network, with Kmean465
dropping to 6o. A similar improvement in the resolvability is achieved with a densification of466
the land network (Kmean = 8o for network IBdense). Notice that the resolvability of the focal467
mechanism improves using IBdense, in spite of the large azimuthal gap. The resolvability of468
a 10-km deep event using the IB network is similar to that of a 40-km event using IB+OBS469
or IBdense; in both cases the resolvability of the DC is good. On the other hand, the resolv-470
ability is very poor for a 60-km deep event using network IB, equalling the resolvability of a471
40-km deep event using network PT.472
3.4 Frequency Range473
So far we worked with waveforms in the 0.025 – 0.08 Hz frequency range. If we increase474
the low-frequency limit from 0.025 Hz to 0.05 Hz (i.e. if we remove the low frequencies475
from the inversion), the resolvability deteriorates, with Kmean rising to 25o. This is exactly476
what happens in real inversions of weak events, where the low frequencies cannot be used477
due to their low S/N ratio. On the contrary, if we use higher frequencies (increase the478
upper frequency limit from 0.08 Hz to 0.2 Hz) the resolvability improves slightly, with Kmean479
dropping to 13o and Kmax dropping to 26o. However, the improvement in the resolvability of480
the DC based on high frequencies is of little usage: In practice, the existing crustal models481
do not provide realistic Green’s functions at near-regional distances up to 0.2 Hz.482
25
3.5 Crustal Structure483
We can formally test different crustal models while keeping the source-station configuration484
and frequency range unchanged. In this situation we alter the matrix G, thus changing485
the error ellipsoid. We tested three different crustal structures: (i) The regional model of486
Stich et al. [2003], used in the reference setup; (ii) The global model PREM [Dziewonski487
and Anderson, 1981]; (iii) A homogeneous, fast, low-attenuation, mantle-like, homogeneous488
half-space, with VP = 8.1 km/s, VS = 4.5 km/s, QP = 1340, and QS = 600; and (iv) A ho-489
mogeneous, slow, high-attenuation, crust-like, homogeneous half-space, with VP = 5.8 km/s,490
VS = 3.2 km/s, QP = 300, and QS = 150. Kmean for each of the models above is 14o, 14o,491
31o, and 16o, respectively. It is interesting to notice that the resolvability of the DC is very492
similar whether the true crustal structure resembles PREM, the regional model of Stich et al.493
[2003], or a crust-like homogeneous half-space. However, the resolvability decreases strongly494
if the medium is fast, in spite of low-attenuation (case iii).495
The results stated above concern the resolvability of a DC mechanism if the true crust is496
like that of each studied model. These results must not be confused with a test of sensibility497
of the Green’s functions, where one would assess the effect of an erroneous crustal structure498
on the source parameters. It is legitimate to question how the focal mechanism changes if the499
same waveforms are inverted with various crustal models (see references in the Introduction).500
The analysis proposed in this paper cannot address such question. When we compare crustal501
models (i)–(iv), we simply investigate the focal-mechanism uncertainty if the true crust502
resembles A, or if the true crust resembles B. The proposed method does not show how the503
26
mechanism changes if the true crust A is erroneously represented by model B.504
3.6 Focal Mechanism505
A given source-station configuration provides different K-angle distributions for different506
reference focal mechanisms, due to the previously mentioned non-linearity of the inverse507
problem with respect to faulting angles. We tested the DC resolvability of four typical focal508
mechanisms in the study region (Figure 2): (i) strike, dip, and rake angles equal to 128o,509
46o, 138o (reference setup); (ii) 52o, 57o, 99o; (iii) 204o, 68o, −16o; (iv) 316o, 35o, −170o.510
The mean K-angles obtained for each case are 14o, 14o, 16o, and 16o, respectively. These511
results indicate that Kmean, and therefore the DC resolvability, does not vary significantly512
with focal mechanism.513
3.7 Data Variance514
All the calculations above assumed a constant value for the data variance σ2d. In particular,515
we assumed σd = 10−5 m, which we obtained by setting C = 1 in (15) (see sections 2.4516
and 3.1). When we evaluate (16), using synthetic data in order to compute di, we obtain517
σd = 1.4 ∗ 10−5; i.e. a similar value, C=1.4.518
Data variance plays a key role in our analysis, controling the size of the error ellipsoid519
(see the Appendix). By prescribing a constant value for σ2d we are able to compare the520
uncertainty of DCs for different configurations of the inverse problem. Figure 6 shows the521
sensitivity of K-angle to data variance σ2d. As expected, an increase in σ2d leads to an increase522
in K-angle. K-angle is very sensitive to σ2d in the range 10−10 to 10−8 m2.523
27
3.8 Maps of Resolvability524
The last application that we present are maps of DC resolvability. In order to plot maps525
of resolvability we perform the analysis presented above for different source epicenters. We526
start by choosing a station network (as defined by station locations), crustal model, fre-527
quency range, earthquake depth, focal mechanism, and data variance. We then compute528
the uncertainty of strike, dip, rake, and finally K-angle for several epicenters on a regular529
grid (of source latitudes and longitudes). The maps of resolvability can be easily computed530
and allow an immediate visualization of the geographical variation of DC resolvability; these531
maps are a useful tool to plan seismic networks or experiments.532
Figure 7 shows the geographical variation of K-angle (which indicates the resolvability533
of the DC) for three different station configurations: IB, IB+OBS, and IBdense. The pa-534
rameters used to compute the maps are those of the reference setup. The IB network is able535
to resolve the DC parameters for onshore earthquakes better than for offshore earthquakes.536
The addition of OBS stations (IB+OBS) allows an improvement of the resolvability of off-537
shore sources. The dense land network, IBdense, is able to resolve DC parameters of offshore538
earthquakes as well as the network IB+OBS. IBdense further improves the resolvability of539
onshore earthquakes in comparison to IB and IB+OBS.540
4 Discussion and Conclusions541
In this paper we presented a method, based on the standard theory of inverse linear problems,542
to compute the uncertainty of the focal mechanisms inferred from the inversion of M. The543
28
input parameters are the source location (epicenter and depth), stations locations, crustal544
model, frequency band of interest, and estimate of data error. The output is a 6D error545
ellipsoid, which contains a family of acceptable solutions in the space of model parameters546
mj. This 6D ellipsoid shows the deviation of the acceptable focal mechanisms from the547
optimal (or reference) solution. The larger the data variance σ2d, the larger the variability548
of the aceptable focal mechanisms. We restricted our analysis to the uncertainty of the549
double-couple component of M, although the 6D error ellipsoid contains information about550
the uncertainty of all the components of M. We mapped the uncertainties of the model551
parameters mj into uncertainties of the faulting angles strike, dip and rake. The results are552
useful to assess, for example, if one faulting angle (e.g. strike) is better resolved than the553
other (e.g. dip), or which of the two possible faulting planes is better resolved. We used554
K-angle to quantify with a single parameter the deviation of the acceptable DC solutions555
from the reference solution.556
The resolvability of DC sources is determined by data error and by the eigenvectors557
and eigenvalues of the underlying inverse problem. These eigenvectors and eigenvalues are558
computed from the Green’s functions matrix G, thus tightly related to the source-station559
configuration. Some sets of eigenvectors and eigenvalues allow an adequate retrieval of the560
moment tensor M, while some others do not. Large eigenvalues correspond to robust source561
parameters. Small eigenvalues are associated with poorly-resolved parameters. The problem562
is very similar to that of linear finite-fault inversions, although it uses a significantly smaller563
number of model parameters [Page et al., 2009; Custódio et al., 2009; Zahradńık and Gallovič,564
2010; Gallovič and Zahradńık , 2011].565
29
The method presented in this paper is applicable to linear problems only. The inversion566
of M is a linear problem assuming that the source location, origin time, and source time567
function are known. The algorithm can be generalized to the non-linear case, where we568
search for the source position and origin time, in addition to M. This generalization is the569
focus of on-going work and will be presented in a separate paper. A non-linear approach570
will allow the study of the trade-off between depth and non-DC components of M. The571
uncertainty of the non-DC components can then be correctly assessed.572
A proper assessment of data error σd is required in order to determine the uncertainty of573
DC sources. We showed that realistic data errors must be of the same order of magnitude of574
the data itself, mostly due to inaccurate crustal models (and respective Green’s functions).575
Our method allows an estimation of the uncertainty of DC parameters if data is available.576
In this case we process the available data, solve the inverse problem, and obtain a minimum-577
misfit “best” solution. Then we prescribe a value for data error and estimate the uncertainty578
of each faulting angle. The method can also be applied when no data (waveforms) are579
available, which is particularly useful for network design. In this paper we present several580
examples for which an exact knowledge of data errors is not needed. We simply choose a581
reasonable value for σ2d, keep it fixed, and study the comparative resolvability of DC sources582
in different scenarios. In other words, we assess the resolvability of the focal mechanism in583
a relative sense.584
In particular, we assessed the relative effect of source epicenter, source depth, stations585
locations, crustal structure, frequency range, and data error on the resolvability of DCs. We586
focused on the resolvability of earthquakes in Southwest Europe. A few stations were selected587
30
to represent the seismic networks of Portugal, Spain, and Morocco. We also considered a588
few OBS sites in the Atlantic Ocean. The analysis was performed in the frequency range589
below 0.1 Hz, which is typical for the near-regional inversions of M. Table 1 summarizes the590
results.591
We find that, for a same network, the resolvability depends critically on source depth.592
Shallow DC sources (10 km) are better resolved than deeper sources (40 and 60 km) by593
a factor of two. Our analysis indicates that the focal mechanism of a shallow offshore594
earthquake can be well resolved even by a small subset of land stations in Portugal (in595
spite of the large azimuthal gap). These results are supported by unpublished inversions of596
different-depth earthquakes [Krizova-Cervinkova, 2008]. The author repeated single-station597
waveform inversions in a near-regional network of 10 stations in Greece for the shallow598
Trichonis Lake earthquake [Kiratzi et al., 2008] and for the intermediate-depth Leonidio599
earthquake [Zahradńık et al., 2008c]. The focal mechanisms inferred from single-station600
inversions of the shallow event were all almost identical. However, all single-station inversions601
of the intermediate-depth event failed.602
Our uncertainty analysis also indicates that, if more stations are available, the resolv-603
ability of the 40 to 60-km deep event can be as good as that of the shallow event using604
less stations. The resolvability of DCs improves considerably when the Portuguese network605
(PT) is supplemented by stations in Spain and Morocco (IB). Further improvement can be606
achieved by adding a few OBS stations offshore. Indeed, the theoretical DC resolvability of607
a 40-km deep event using IB+OBS is almost as good as the resolvability of the 10-km deep608
event using only IB network. The improvement of the DC resolvability due to additional609
31
OBS stations is partly due to the decrease of the azimuthal gap. Yet we showed that a610
similar improvement in resolvability can be achieved with a densification of land stations611
(IBdense), where the azimuthal gap remains as large as ∼ 200o. The resolvability of DC612
sources does not depend only on azimuthal coverage. A good azimuthal coverage is useful,613
but not always strictly necessary.614
Finally, a word on the use of OBS waveforms in regional earthquake inversions. Within615
project NEAREST, a temporary deployment of broadband OBS (BB-OBS) stations recorded616
earthquakes offshore SW Iberia (Figure 2). Geissler et al. [2010] used BB-OBS first-motion617
polarities in order to determine improved focal mechanisms of weak offshore events. The618
use of BB-OBS waveforms of regional earthquakes is challenging due to noise and various619
disturbances in the data, e.g. tilts [Crawford and Webb, 2000]. We attempted to use the620
NEAREST BB-OBS waveforms in a moment-tensor inversion: We started by taking the621
waveforms from a single MW4.5 event – the largest event in the dataset – recorded at five622
three-component OBS stations. We then tried to remove the tilt-like disturbances from the623
horizontal components according to Zahradńık and Plešinger [2005, 2010]. Almost no signal624
was left in the frequency band of interest for near-regional waveform inversion after the625
successful removal of the tilt-like disturbances. The use of BB-OBS waveforms of regional626
earthquakes might be more successful for smaller source-station distances (
http://seismo.geology.upatras.gr/isola/].632
Data and Resources633
The software ISOLA was used to prepare the matrices for the computations of resolvability.634
Green functions in ISOLA are computed using the AXITRA code of Coutant [1989]. GMT635
[Wessel and Smith, 1998] and Matlab were used for figure plotting.636
Acknowledgments637
We thank Frantǐsek Gallovič, Efthimios Sokos and Daniel Stich for valuable discussions and638
support. Partial funding was obtained from the grant agencies in the Czech Republic: GACR639
210/11/0854 and MSM 0021620860. This research was supported by a Marie Curie Inter-640
national Reintegration Grant within the 7th European Community Framework Programme641
(PIRG03–GA–2008–230922).642
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40
Jǐŕı Zahradńık780
Charles University in Prague, Faculty of Mathematics and Physics,781
V Holesovickach 2, 180 00 Prague 8, Czech Republic, Czech Republic.782
jiri.zahradnik@mff.cuni.cz783
784
Susana Custódio785
Instituto Geof́ısico da Universidade de Coimbra,786
Av. Dr. Dias da Silva, 3000-134 Coimbra, Portugal.787
susanacustodio@dct.uc.pt788
789
41
Table 1: Summary of K-angles for different configurations of the inverse problem. The
underlined parameters are those which are different from the reference setup. Lower K-
values indicate better resolved DC mechanisms.
Source Station Frequency Crustal Focal Kmean Kmax σ(K)
Depth Network Range Structure Mechanism (o) (o) (o)
(km) (Hz) Strike/Dip/Rake (o)
40 IB 0.025 – 0.08 Regional 1D 128 / 46 / 138 14 34 6
10 IB 0.025 – 0.08 Regional 1D 128 / 46 / 138 6 14 2
60 IB 0.025 – 0.08 Regional 1D 128 / 46 / 138 22 91 10
40 PT 0.025 – 0.08 Regional 1D 128 / 46 / 138 22 92 11
40 IBdense 0.025 – 0.08 Regional 1D 128 / 46 / 138 8 19 3
40 IB+OBS 0.025 – 0.08 Regional 1D 128 / 46 / 138 6 15 2
40 IB 0.05 – 0.08 Regional 1D 128 / 46 / 138 25 82 11
40 IB 0.025 – 0.2 Regional 1D 128 / 46 / 138 13 26 5
40 IB 0.025 – 0.08 PREM 128 / 46 / 138 14 35 6
40 IB 0.025 – 0.08 Homog. Fast 128 / 46 / 138 31 91 14
40 IB 0.025 – 0.08 Homog. Slow 128 / 46 / 138 16 35 6
40 IB 0.025 – 0.08 Regional 1D 52 / 57 / 99 14 38 6
40 IB 0.025 – 0.08 Regional 1D 204 / 68 / -16 16 40 6
40 IB 0.025 – 0.08 Regional 1D 316 / 35 / -170 16 32 6
42
Figure Captions790
Figure 1. The shape and orientation of the error ellipsoid is determined by the singular791
vectors V(i) and singular values wi. The singular vectors V(i) determine the direction of792
the ellipsoid axes, i.e. they determine the orientation of the ellipsoid in the 6D parameter793
space. The singular values determine the size of the ellipsoid axes, thus the shape of the794
ellipsoid. Data error, σd, scales the size of the whole ellipsoid. The surface of the ellipsoid is795
characterized by ∆χ2 = constant. See the Appendix for more details. Figure adapted from796
Press et al. [1992].797
798
Figure 2. Map of the seismicity and seismic network in southwest Iberia. The red star indi-799
cates the reference earthquake epicenter. Network PT – squares; Network IB – squares and800
hexagons; Network IB+OBS – squares, hexagons and circles; Network IBdense – diamonds;801
Small solid triangles – permanent stations; Small white triangles – temporary stations (cur-802
rent or past). Small gray dots show the background seismicity in the instrumental record803
(post-1960) [Martins and Mendes-Victor , 1990; Pena et al., in press; Carrilho et al., 2004;804
Preliminary Seismic Information, 2010]. Larger dots show the locations of MW > 3.5 earth-805
quakes in the period 2007 – 2010. The four different focal mechanisms studied are plotted806
in the upper left corner, with their strike, dip, and rake values.807
808
Figure 3. (a–c) Strike, dip and rake angles for the family of acceptable solutions vs misfit,809
∆χ2, for the reference setup: The source-station geometry of network IB is shown in Fig-810
43
ure 2; source depth is 40 km; strike, dip, rake angles are 128o, 46o, 138o, equivalent to 250o,811
61o, 52o. (d) K-angle corresponding to the focal mechanisms shown in panels (a–c). (e–h)812
Histograms of strike, dip, rake and K-angle.813
814
Figure 4. (a–c) Histograms of K-angle indicating the resolvability of a DC source for three815
different source depths: 10, 40 and 60 km. The mean (and maximum) values of the K-angle816
are 6o (14o), 14o (34o), and 22o (91o), respectively. Dashed lines indicate the average K-817
angle. (d–f) The same results plotted using the more familiar nodal lines. Note the excellent818
resolvability of the shallow source (10 km).819
820
Figure 5. Resolvability of the 40-km deep DC using four different station networks (Fig-821
ure 2): (a) PT network (squares), (b) IB network (squares and hexagons), (c) IB+OBS822
network (squares, hexagons and circles), (d) IBdense network (diamonds). The resolvability823
of this offshore DC source is very similar whether we use IB+OBS or IBdense.824
825
Figure 6. Variation of the mean of K-angle, Kmean, and standard deviation, σ(K), with826
data variance σ2d.827
828
Figure 7. Maps of K-angle indicating the resolvability of a DC source using different station829
networks: (a) IB; (b) IB+OBS; (c) IBdense. Dark cells indicate better resolved DC sources.830
The stations that compose each network are shown by white triangles. This map concerns831
earthquake sources at depths of 40 km. Using OBSs in addition to the land IB network832
44
improves the resolvability of the DC source considerably, particularly for offshore sources.833
The dense land network, IBdense, provides a similar improvement of the DC resolvability of834
offshore earthquakes, and further improves the resolvability of onshore earthquakes.835
45
Figures836
Figure 1
m1
m2 V(1)
V(2)
Leng
th 1
/w 1
Length 1/w2
Δχ2 =c
onsta
nt
46
Figure 2
−12˚
−12˚
−10˚
−10˚
−8˚
−8˚
−6˚
−6˚
−4˚
−4˚
34˚ 34˚
36˚ 36˚
38˚ 38˚
40˚ 40˚
42˚ 42˚
44˚ 44˚
(316˚, 35˚, -170˚)
(128˚, 46˚, 138˚)
(52˚, 57˚, 99˚)
(204˚, 68˚, -16˚)
47
Figure 3
(a)
0 0.2 0.4 0.6 0.8 10
50
100
150
200
250
300
350
Str
ike
(o)
Δχ2
(b)
0 0.2 0.4 0.6 0.8 10
10
20
30
40
50
60
70
80
90
Dip
(o)
Δχ2
48
Figure 3
(c)
0 0.2 0.4 0.6 0.8 1
−150
−100
−50
0
50
100
150R
ake
(o)
Δχ2
(d)
0 0.2 0.4 0.6 0.8 10
10
20
30
40
50
60
70
80
90
K-a
ng
le (
o)
Δχ2
49
Figure 3
(e)
0 50 100 150 200 250 300 3500
2000
4000
6000
8000
10000
12000
Strike (o)
Fre
quency (
Num
ber
of S
olu
tions)
(f)
0 10 20 30 40 50 60 70 80 900
1000
2000
3000
4000
5000
6000
7000
8000
Dip (o)
Fre
quency (
Num
ber
of S
olu
tions)
50
Figure 3
(g)
0 20 40 60 80 100 120 140 160 1800
500
1000
1500
2000
2500
3000
3500
4000
Rake (o)
Fre
qu
en
cy (
Nu
mb
er
of
So
lutio
ns)
(h)
0 10 20 30 40 50 600
500
1000
1500
2000
2500
3000
3500
4000
K-angle (o)
Fre
quency (
Num
ber
of S
olu
tions)
51
Figure 4
(a)
0 10 20 30 40 50 600
1000
2000
3000
4000
5000
6000
7000
8000
9000Depth = 10 km
K-angle (o)
Fre
quency
(N
um
ber
of S
olu
tions) K-mean = 6˚
(b)
0 10 20 30 40 50 600
500
1000
1500
2000
2500
3000
3500
4000Depth = 40 km
K-angle (o)
Fre
quency
(N
um
ber
of S
olu
tions)
K-mean = 14˚
52
Figure 4
(c)
0 10 20 30 40 50 600
500
1000
1500
2000
2500Depth = 60 km
K-angle (o)
Fre
quency
(N
um
ber
of S
olu
tions)
K-mean = 22˚
(d)
53
Figure 4
(e)
(f)
54
Figure 5
(a)
0 10 20 30 40 50 600
200
400
600
800
1000
1200
1400
1600
K-angle (o)
Fre
quency
(N
um
ber
of S
olu
tions)
PT
K-mean = 22˚
(b)
0 10 20 30 40 50 600
500
1000
1500
2000
2500
3000
3500
4000IB
K-angle (o)
Fre
quency
(N
um
ber
of S
olu
tions)
K-mean = 14˚
55
Figure 5
(c)
0 10 20 30 40 50 600
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
K-angle (o)
Fre
quency
(N
um
ber
of S
olu
tions)
IB+OBS
K-mean = 6˚
(d)
0 10 20 30 40 50 600
1000
2000
3000
4000
5000
6000
7000
K-angle (o)
Fre
qu
en
cy (
Nu
mb
er
of
So
lutio
ns)
IBdense
K-mean = 8˚
56
Figure 6
0
10
20
30
40
50
60
70
Data Variance σd2 (m2)
K−
angle
(˚)
Kmeanσ(K)
10 −10 10 −8 10 −6 10 −4 10 −2 10 0
57
Figure 7
(a)
14˚W 12˚W 10˚W 8˚W 6˚W 4˚W
32˚N
34˚N
36˚N
38˚N
40˚N
42˚N
0 10 20 30 40
Longitude Latit
ude
K-angle
IB
58
Figure 7
(b)
14˚W 12˚W 10˚W 8˚W 6˚W 4˚W
32˚N
34˚N
36˚N
38˚N
40˚N
42˚N
0 10 20 30 40
Longitude
Latit
ude
K-angle
IB+OBS
59
Figure 7
(c)
14˚W 12˚W 10˚W 8˚W 6˚W 4˚W
32˚N
34˚N
36˚N
38˚N
40˚N
42˚N
0 10 20 30 40
Longitude
Latit
ude
K-angle
IBdense
60
Appendix837
This appendix focuses on the theory and algorithm employed to construct the 6D error838
ellipsoid of M. We underscore that this analysis is standard and general for linear inverse839
problems. Further details on the method can be found, e.g., in Menke [1989]; Press et al.840
[1992]; Parker [1994].841
Recall Equation (1) in the main text, which states the linear relation between ground842
motion (d) and moment tensor parameters (m), through the Green’s functions (G):843
d = Gm (A-1)
In order to take into account data error, we divide both sides of (A-1) by data standard844
deviation σd:845
d̃i =diσd
(A-2)
G̃ij =Gijσd
(A-3)
Equation (A-1) can then be re-written as:846
d̃ = G̃m (A-4)
G̃ is the so-called design matrix. Solving (A-4) in order to find m is equivalent to847
minimizing χ2 (Equation (5) in the main text). The general least-squares (LSQ) solution of848
61
(A-4) can be obtained with the aid of normal equations. In this case the solution is given849
by:850
mmin = (G̃T G̃)−1G̃T d̃ (A-5)
Alternatively, the solution can be obtained by means of singular value decomposition851
(SVD) of the matrix G̃:852
G̃ = U.W(VT ) (A-6)
Matrices U and V are orthonormal, and matrix W is diagonal. The columns of U and V853
are the left- and right-singular vectors of G̃, respectively. Hereafter we use “singular vectors”854
to designate the columns of matrix V, which we represent by V(i) The diagonal elements855
of matrix W are the singular values wi of G̃ . The model parameters mj obtained through856
SVD are then:857
mmin =6∑
i=1
(U(i).d̃
wi
)V(i) (A-7)
Where the sum runs over i = 1, . . . , 6 model parameters. The parameter variances and858
covariances can be expressed in a very transparent form, using the singular vectors V(i) and859
singular values wi:860
σ2(mj) =6∑
i=1
(Vjiwi
)2(A-8a)
861
Cov(mj,mk) =6∑
i=1
(VjiVkiw2i
)(A-8b)
62
Within the scope of this paper we will not use the parameter co-variances. The co-862
variance matrix contains explicit information about the trade-off between every pair of model863
parameters. The information about parameter trade-off is also contained in the error ellip-864
soid. The error ellipsoid of M can be obtained from the singular vectors (V(i)) and singular865
values (wi) of G̃ (Figure 1 in the main text). The orthonormal singular vectors V(i) define866
the direction of the ellipsoid principal axes. The length of the ellipsoid axes are inversely867
proportional to the singular values wi. Thus, small singular values correspond to long axes of868
the ellipsoid (poorly resolved parameters). The most robust parameters are those associated869
with large singular values wi. The shape and orientation of the error ellipsoid is fully deter-870
mined by V(i) and wi. The actual size of the ellipsoid is determined by the data variance871
σ2d (which enters the computation via singular values wi) and choice of ∆χ2 threshold. The872
surface of the ellipsoid, defined as a surface of constant ∆χ2, can then be written as [Press873
et al., 1992]:874
∆χ2 = w21(V(1).δm)2 + . . . + w26(V(6).δm)
2 (A-9)
where δm is the vector between the center of the ellipsoid and a point in the parameter875
space.876
Once we know the singular vectors V(i) and singular values wi we can easily determine877
numerically the error ellipsoid: We set the size of the ellipsoid by choosing the limiting ∆χ2878
value, hereafter ∆χ2 = 1. Then we find the model parameters δmj which are within the879
ellipsoid 0 ≤ ∆χ2 ≤ 1. In practice, we discretize δm1, δm2, . . . , δm6, and grid-search the880
6D parameter space within the limits given by ±σ(m1), . . . ,±σ(m6), calculated fom (A-8a).88163
Note that the surface ∆χ2 = 1 is chosen arbitrarily, not associated with a specific confidence882
level. The grid search is fast, requiring only a few seconds on a laptop (it can be relatively883
coarse, e.g. 11 points for each axis, six nested loops, totalizing 116 numerical evaluations of884
(A-9)).885
Figure A-1 shows selected 2D cross-sections of ∆χ2 = 1 error ellipsoids along axes m1 and886
m2. The center of the ellipsoid (∆χ2 = 0) is the reference solution mref . The discrete points887
inside the ellipsoid are the family of acceptable solutions, macceptable = mref + δm. Note888
that the projection of the ellipsoid onto the individual axes equal the individual parameter889
standard deviations σ(mi). This is a consequence of the adopted choice of limiting ∆χ2 = 1.890
The ellipsoid cross-sections allow a visualization of the well-known trade-offs between model891
parameters. However the practical interest of this visualization is limited, because: 1) we892
have to consider all six dimensions; and 2) the model parameters do not have an intuitive893
meaning. More meaningful are the uncertainties of the faulting angles (strike, dip, and rake894
– shown in the main text), which result from the interaction of all six model parameters.895
Although we used SVD in this derivation, in practice we do not need to decompose G̃.896
In fact, we only need to compute the eigenvalues and eigenvectors of GTG. This shortcut897
further contributes to the numerical efficiency of the method. The parameter variances898
σ2(mj) are actually the diagonal elements of G̃T G̃−1 , and σ2(mj)/σ2d are the diagonal899
elements of GTG−1. The singular vectors V(i) of G̃ are simply the eigenvectors of GTG.900
And the singular values of G̃ can be calculated from the eigenvalues λi of GTG:901
wi =√
λi/σ2d (A-10)
64
and, therefore:902
∆χ2 =λ1(V(1).δm)
2 + . . . + λ6(V(6).δm)2
σ2d(A-11)
If we now recall recall equations (7), (8), (11) (in the main text) and (A-11), we find that903
the ∆χ2 = 1 surface can be expressed in several equivalent ways:904
σ2d = (d− sth)2 − (d− smin)2
=∑
i d2i (VRmin − VRth)
= λ1(V(1).δm)2 + . . . + λ6(V(6).δm)
2
(A-12)
Equation (A-12) is an interesting by-product of our analysis: It demonstrates that chosing905
a family of acceptable solutions via prescription of a variance-reduction threshold (in case we906
have data) is equivalent to prescribing a value for data error σd. Moreover, σd is explicitly907
related to the eigenvectors V (i) and λi, which we can compute without data. This equivalence908
is not often recognized.909
Let us next summarize the computational algorithm. Consider first that waveforms are910
available:911
1. Matrix G is formed; matrix GTG is computed.912
2. Column vector d is formed.913
3. The LSQ solution of (A-4), given by (A-5), provides the optimum solution mmin.914
4. The LSQ solution is adopted as the reference solution mref = mmin.915
5. The singular vectors V(i) are computed as eigenvectors of GTG, and the singular values916
wi are computed from the eigenvalues λi of GTG according to (A-10).917
65
6. The standard deviations of the model parameters σ(m) are obtained from (A-8).918
7. A 6D grid search is performed to obtain the discrete points δm satisfying 0 ≤ ∆χ2 ≤ 1919
(A-9).920
8. The set of model parameters macceptable given by macceptable = mref + δm is the family921
of acceptable solutions.922
If waveform data are not available, and only the uncertainty analysis is to be made, we limit923
the analysis to steps 1 and 5 – 8. In this case, mref is chosen arbitrarily. If we are interested924
in the resolvability of a specific focal mechanism, we can easily compute the coefficients mj925
from strike, dip, rake and scalar moment. We then proceed as indicated above.926
A final issue deserves attention: How does the uniform sampling of the 6D parameter927
space relate to the sampling of the misfit ∆χ2? Figure A-2 shows the distribution of all values928
of ∆χ2 found while grid-searching the 6D volume. The misfit ∆χ2 is sufficiently smo