Moment Tensor Resolvability: Application to Southwest...

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, [email protected]

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

12

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

References643

Aki, K., and P. G. Richards (2002), Quantitative Seismology, University Science Books.644

Borges, J. F., A. J. S. Fitas, M. Bezzeghoud, and P. Teves-Costa (2001), Seismotectonics of645

Portugal and its adjacent Atlantic area, Tectonophysics, 331, 373–387, doi:10.1016/S0040-646

1951(00)00291-2.647

33

Bouchon, M. (1981), A simple method to calculate Green’s functions for elastic layered648

media, Bull. Seism. Soc. Am., 71 (4), 959.649

Buforn, E., A. Ud́ıas, and M. A. Colombas (1988), Seismicity, source mechanisms and tecton-650

ics of the Azores-Gibraltar plate boundary, Tectonophysics, 152, 89–118, doi:10.1016/0040-651

1951(88)90031-5.652

Buforn, E., C. Sanz de Galdeano, and A. Ud́ıas (1995), Seismotectonics of the ibero-653

maghrebian region, Tectonophysics, 248 (3-4), 247 – 261, doi:DOI: 10.1016/0040-654

1951(94)00276-F, focal Mechanism and Seismotectonics.655

Buforn, E., M. Bezzeghoud, A. Ud́ıas, and C. Pro (2004), Seismic Sources on the Iberia-656

African Plate Boundary and their Tectonic Implications, Pure and Applied Geophysics,657

161, 623–646, doi:10.1007/s00024-003-2466-1.658

Bukchin, B., E. Clévédé, and A. Mostinskiy (2010), Uncertainty of moment tensor determi-659

nation from surface wave analysis for shallow earthquakes, J. Seismol., 14 (3), 601–614,660

doi:10.1007/s10950-009-9185-8.661

Carrilho, F., A. Pena, J. Nunes, and M. L. Senos (2004), Catálogo Śısmico Instrumental662

1970–2000, Tech. rep., Instituto de Meteorologia, ISBN: 972-9083-12-6, Depsito legal No:663

221 955/05.664

Coutant, O. (1989), Program of numerical simulation AXITRA, Tech. rep., LGIT, Grenoble.665

Crawford, W. C., and S. C. Webb (2000), Identifying and removing tilt noise from low-666

34

frequency (

Gilbert, F. (1971), Excitation of the normal modes of the Earth by earthquake sources,687

Geophys. J. Int., 22, 223–226, doi:10.1111/j.1365-246X.1971.tb03593.x.688

Grimison, N. L., and W.-P. Chen (1986), The Azores-Gibraltar plate boundary: Focal mech-689

anisms, depths of earthquakes, and their tectonic implications, Journal of Geophysical690

Research, 91, 2029–2048, doi:10.1029/JB091iB02p02029.691

Hingee, M., H. Tkalčić, A. Fichtner, and M. Sambridge (2011), Seismic moment tensor692

inversion using a 3-D structural model: applications for the Australian region, Geophys.693

J. Int., 184, 949, doi:10.1111/j.1365-246X.2010.04897.x.694

Hudson, J. A., R. G. Pearce, and R. M. Rogers (1989), Source type plot for inversion of the695

moment tensor, J. Geophys. Res., 94, 765, doi:10.1029/JB094iB01p00765.696

Kagan, Y. Y. (1991), 3-D rotation of double-couple earthquake sources, Geophys. J. Int.,697

106, 709, doi:10.1111/j.1365-246X.1991.tb06343.x.698

Kikuchi, M., and H. Kanamori (1991), Inversion of complex body waves–III, Bull. Seism.699

Soc. Am.700

Kiratzi, A., et al. (2008), The April 2007 earthquake swarm near Lake Trichonis and701

implications for active tectonics in western Greece, Tectonophysics, 452, 51, doi:702

10.1016/j.tecto.2008.02.009.703

Krizova-Cervinkova (2008), Moment inversion of earthquakes in Greece by ISOLA method,704

Master’s thesis, Charles University in Prague, (unpublished, in Czech).705

36

Lomax, A. (2005), A Reanalysis of the Hypocentral Location and Related Observations706

for the Great 1906 California Earthquake, Bull. Seism. Soc. Am., 95 (3), 861–877, doi:707

10.1785/0120040141.708

Martins, I., and L. A. Mendes-Victor (1990), Contribuição para o estudo da sismicidade de709

Portugal Continental., Tech. Rep. 18, IGIDL, Universidade de Lisboa.710

Menke, W. (1989), Geophysical data analysis: Discrete inverse theory, Academic Press.711

Page, M. T., S. Custódio, R. J. Archuleta, and J. M. Carlson (2009), Constraining earthquake712

source inversions with GPS data: 1. Resolution-based removal of artifacts, J. Geophys.713

Res., 114, 01,314, doi:10.1029/2007JB005449.714

Parker, R. L. (1994), Geophysical inverse theory, Princeton University Press.715

Pena, A., C. Nunes, and F. Carrilho (in press), Catálogo Śısmico Instrumental 1961–1969,716

Tech. rep., Instituto de Meteorologia.717

Preliminary Seismic Information (2010), Boletins Sismológicos Mensais, Tech. rep., Instituto718

de Meteorologia, http://www.meteo.pt/en/publicacoes/tecnico-cientif/noIM/boletins/.719

Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery (1992), Numerical720

recipes in FORTRAN. The art of scientific computing, Cambridge University Press, 2nd721

ed.722

Riedesel, M. A., and T. H. Jordan (1989), Display and assessment of seismic moment tensors,723

Bull. Seism. Soc. Am., 79 (1), 85–100.724

37

Serpelloni, E., G. Vannucci, S. Pondrelli, A. Argnani, G. Casula, M. Anzidei, P. Baldi,725

and P. Gasperini (2007), Kinematics of the Western Africa-Eurasia Plate Boundary From726

Focal Mechanisms and GPS Data, Geophysical Journal International, 169 (3), 1180–1200,727

doi:DOI: 10.1111/j.1365-246X.2007.03367.x.728

Š́ılený, J. (1998), Earthquake source parameters and their confidence regions by a ge-729

netic algorithm with a ‘memory’, Geophys. J. Int., 134, 228–242, doi:10.1046/j.1365-730

246x.1998.00549.x.731

Š́ılený, J. (2004), Regional moment tensor uncertainty due to mismodeling of the crust,732

Tectonophysics, 383, 133–147, doi:10.1016/j.tecto.2003.12.007.733

Sokos, E. N., and J. Zahradńık (2008), ISOLA a Fortran code and a Matlab GUI to perform734

multiple-point source inversion of seismic data, Computers & Geosciences, 34, 967, doi:735

10.1016/j.cageo.2007.07.005.736

Stich, D., C. J. Ammon, and J. Morales (2003), Moment tensor solutions for small and737

moderate earthquakes in the Ibero-Maghreb region, J. Geophys. Res., 108 (B3), 2148,738

doi:10.1029/2002JB002057.739

Stich, D., E. Serpelloni, F. de Lis Mancilla, and J. Morales (2006), Kinematics of the Iberia740

Maghreb plate contact from seismic moment tensors and GPS observations, Tectono-741

physics, 426, 295–317, doi:10.1016/j.tecto.2006.08.004.742

Stich, D., F. d. L. Mancilla, S. Pondrelli, and J. Morales (2007), Source analysis of the743

38

February 12th 2007, Mw 6.0 Horseshoe earthquake: Implications for the 1755 Lisbon744

earthquake, Geophysical Research Letters, 34, L12,308, doi:10.1029/2007GL030012.745

Stich, D., R. Martn, and J. Morales (2010), Moment tensor inversion for iberia-746

maghreb earthquakes 2005-2008, Tectonophysics, 483 (3–4), 390 – 398, doi:DOI:747

10.1016/j.tecto.2009.11.006.748

Vasco, D. W. (1990), Moment-tensor invariants: Searching for non-double-couple earth-749

quakes, Bull. Seism. Soc. Am., 80 (2), 354–371.750

Vavryčuk, V. (2007), On the retrieval of moment tensors from borehole data, Geophys.751

Prosp., 55, 381, doi:10.1111/j.1365-2478.2007.00624.x.752

Wéber, Z. (2006), Probabilistic local waveform inversion for moment tensor and hypocentral753

location, Geophys. J. Int., 165, 607–621, doi:10.1111/j.1365-246X.2006.02934.x.754

Wessel, P., and W. H. F. Smith (1998), New, improved version of Generic Mapping Tools755

released, EOS Trans. Amer. Geophys. U., 79 (47), 579.756

Yagi, Y., and Y. Fukahata (2011), Introduction of uncertainty of Green’s function into757

waveform inversion for seismic source processes, Geophys. J. Int.758

Zahradńık, J., and F. Gallovič (2010), Toward understanding slip inversion uncertainty and759

artifacts, J. Geophys. Res., 115 (B9), B09,310, doi:10.1029/2010JB007414.760

Zahradńık, J., and A. Plešinger (2005), Long-period pulses in broadband records of near761

earthquakes, Bull. Seism. Soc. Am., 95 (5), 1928–1939, doi:10.1785/0120040210.762

39

Zahradńık, J., and A. Plešinger (2010), Toward understanding subtle instrumentation effects763

associated with weak seismic events in the near field, Bull. Seism. Soc. Am., 100 (1), 59–73,764

doi:10.1785/0120090087.765

Zahradńık, J., A. Serpetsidaki, E. Sokos, and G.-A. Tselentis (2005), Iterative deconvolution766

of regional waveforms and a double-event interpretation of the 2003 Lefkada earthquake,767

Greece, Bull. Seism. Soc. Am., 95 (1), 159–172, doi:10.1785/0120040035.768

Zahradńık, J., J. Jansky, and V. Plicka (2008a), Detailed waveform inversion for moment769

tensors of M∼4 events: Examples from the Corinth Gulf, Greece, Bull. Seism. Soc. Am.,770

98 (6), 2756–2771, doi:10.1785/0120080124.771

Zahradńık, J., E. Sokos, G.-A. Tselentis, and N. Martakis (2008b), Non-double-couple772

mechanism of moderate earthquakes near Zakynthos, Greece, April 2006; explana-773

tion in terms of complexity, Geophysical Prospecting, 56, 341–356, doi:10.1111/j.1365-774

2478.2007.00671.x.775

Zahradńık, J., F. Gallovič, E. Sokos, A. Serpetsidaki, and A. Tselentis (2008c), Quick776

fault-plane identification by a geometrical method: Application to the MW 6.2 Leoni-777

dio earthquake, 6 January 2008, Greece, Seismol. Res. Lett., 79 (5), 653–662, doi:778

10.1785/gssrl.79.5.653.779

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

[email protected]

784

Susana Custódio785

Instituto Geof́ısico da Universidade de Coimbra,786

Av. Dr. Dias da Silva, 3000-134 Coimbra, Portugal.787

[email protected]

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

Moment Tensor Resolvability: Application to Southwest...

Documents

Transcript of Moment Tensor Resolvability: Application to Southwest...