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FiSH: MATLAB Tools to Turn Fault Data into Seismic-Hazard Models by B. Pace, F. Visini, and L. Peruzza
ABSTRACT Faults have been increasingly integrated into seismic-hazard assessments. We have developed a package of MATLAB (http://www.mathworks.com/products/matlab, last accessed January 2016) tools (called FiSH), designed to help seismic-hazard modelers analyze fault data. These tools enable the derivation of expected earthquake rates, given common fault data, and allow researchers to test the consistency between the magnitude–frequency distributions (MFDs) assigned to a fault and some available observations. The basic assumption of FiSH is that the geometric and kinematic features of a fault are the expression of its seismogenic potential. Three tools have been designed to integrate the variable levels of information available: (1) the first tool allows users to convert fault geometry and slip rates into a global budget of the seismic moment released in a given time frame, taking uncertainties into account; (2) the second tool computes the recurrence parameters and associated uncertainties from historical and/or paleoseismological data; and (3) the third tool outputs time-independent or time-dependent earthquake rates for different MFD models. We present a test case to illustrate the capabilities of FiSH, on the Paganica normal fault in central Italy that ruptured during the 2009 L’Aquila earthquake sequence (mainshock M w 6.3). The source codes are open, and we encourage users to handle the scripts, communicate with us regarding bugs, and/or suggest further improvements. Our intent is to distribute these tools in order to help researchers to pinpoint potential inconsistencies and obtain reliable fault-based seismic-hazard evaluations.
INTRODUCTION Probabilistic seismic hazard assessment (PSHA) has been used extensively by the scientific community over the last two decades and has led to the distribution of many seismic-hazard doi: 10.1785/0220150189
analysis tools (e.g., Bender and Perkins, 1987; Ordaz et al., 2013; Pagani et al., 2014). The majority of these tools have been conceived to customize the representation of the results of a given PSHA model in terms of maps, hazard curves and spectra, and disaggregation analyses (e.g., for Italy, http:// esse1‑gis.mi.ingv.it/s1_en.php, last accessed December 2015; Europe, http://www.efehr.org:8080/jetspeed/portal/hazard. psml, last accessed December 2015; and the United States of America, http://earthquake.usgs.gov/hazards/apps/, last accessed December 2015). Several applications have focused on the visualization and/or analysis of individual seismic-hazard components, such as the ground-motion prediction equation or the magnitude–frequency distribution (MFD) of seismic sources (e.g., Wiemer, 2001; Field et al., 2003; http://www.opensha. org/apps, last accessed December 2015). Tools that are solely dedicated to converting geological field data on faults into seismic-hazard model components (such as source geometries and earthquake rupture forecasts) are rare. In general, fault source modeling uses simplistic rules based on the size and slip rate of mapped faults to estimate seismicity rates by adopting a priori MFDs (of the Gutenberg–Richter type, following the characteristic earthquake model, or by hybrid formulations). The practice of using fault sources is growing in popularity, including in regions with moderate seismic activity, such as the European countries. In these areas, fault identification may be affected by similarly large uncertainties in the historical and instrumental seismic histories of more active areas that have not been inhabited for long period time. Certain studies have effectively applied a time-dependent perspective to combine historical and instrumental seismic data with geological and paleoseismological information, partially compensating for a lack of information (e.g., in Italy, Pace et al., 2006, and Peruzza et al., 2011; in New Zealand, Stirling et al., 2012; in California, Field et al., 2014, 2015). FiSH is a graphical user interface (GUI)-supported tool that is designed to help seismic-hazard modelers analyze fault data. FiSH is a package of MATLAB routines that is designed to evaluate expected earthquake rates, given certain fault data
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constraints, and to test the consistency between the MFD that is assigned to a fault and any available observations. The FiSH output can be used as input in other software packages that use faults to evaluate seismic hazards, particularly the software packages OpenQuake (Pagani et al., 2014), which was developed for the Global Earthquake Model, and CRISIS2008 (Ordaz et al., 2013). The initial concept of FiSH originated in the early 2000s because of the encountered difficulties with using faults in PSHA for central Italy (Barchi et al., 2000; Peruzza and Pace, 2002; Pace et al., 2006). Since then, new theoretical formulations have been developed (Peruzza et al., 2010), and the first MATLAB routines were released in 2013 through an Italian project (Italian Presidenza del Consiglio dei Ministri—Dipartimento della Protezione Civile [INGV-DPC], Project S22012, in Pace et al., 2013). We tested and implemented these tools over the past two years, and this article represents the official release of FiSH v.1.02. We illustrate the capabilities of FiSH with a test case that involves the Paganica fault, the causative fault of the L’Aquila earthquake in 2009 that led to 309 casualties (Boncio et al., 2010). The example shows that we can derive time-independent and time-dependent earthquake rates that are mutually consistent and hopefully more realistic than what would be obtained with simplistic rules by considering (1) field geological data on fault geometry and slip rates, (2) earthquake series from historical and paleoseismological studies, and (3) the year of the last known major earthquake to have occurred on the fault.
GENERAL FEATURES The basic assumption of FiSH is that the geometric and kinematic features of a fault are the expression of its seismogenic potential. Defining the seismogenic sources that underlie the use of FiSH is important. Taking advantage of the segmentation model defined by Boncio et al. (2004), a “seismogenic master fault” in this work is a major structure that can be considered substantially continuous at depth for several kilometers. Seismogenic master faults are separated from each other by firstorder structural or geometrical complexities (Fig. 1). For central Italy, Boncio et al. (2004) defined these complexities as the following barriers: (1) 3–4 km fault gaps among aligned structures; (2) sharp bends or intersections with cross structures that are 4–9 km long (often transfer faults) and oriented at nearly right angles to the intersected faults; and (3) overlapping or underlapping en echelon arrangements with separations between 2 and 5 km faults. More recently, Field et al. (2015) used a list of plausibility criteria to determine which subsections of a fault can interact to produce an earthquake, including distance and azimuthal changes between fault sections, cumulative azimuthal and rake changes, and the Coulomb criterion. Based on this framework, a seismogenic source is a fault that can break entirely during an earthquake. As shown in Figure 1, the shape and size of the seismogenic source must be evaluated in three dimensions by defining the depth extent of the fault, the dip angle, and the length along strike. These parameters describe 2
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(a)
(b)
Length surface fault plane projection
dip-angle
Thickness of the seismogenic layer
fault plane
▴ Figure 1. (a) Schematic block diagram that illustrates the concept of seismogenic master faults (major fault structures that are substantially continuous at depth) and the major complexities that separate them. (b) Three-dimensional sketch of a seismogenic master fault that shows the main geometrical parameters that are required by FiSH: the along-strike length of the source, the dip-angle, and the thickness of the seismogenic layer. the geometry of a seismogenic source in FiSH. The fault length and seismogenic depth are assumed to be known parameters when using our tools; similarly, the crustal properties (e.g., shear modulus and strain drop) may be derived from specific studies or the general literature. FiSH is written in MATLAB (v.R2011b), a Mathworks (http://www.mathworks.com, last accessed December 2015) commercial software package widely used by geosciences researchers. A MATLAB license is mandatory to run FiSH, but, due to FiSH’s GUI, no knowledge of the MATLAB language is required to use the tools. The source codes are available at http://fish-code.com (last accessed January 2016), and we welcome implementations or user feedback. FiSH has been tested using several operating systems (Macintosh, Unix, and PC), and it should run on all platforms that support MATLAB. Three tools (workflow in Fig. 2), designed to integrate variable levels of information that is available for each fault, have been developed. The first tool allows users to convert common fault data, such as geometry and slip rates, into a global budget of the seismic moment that was released in a given time frame, taking uncertainties into account. The second tool computes the recurrence parameters (RP) and associated uncertainties from historical and/or paleoseismological data. The third tool March/April 2016
INPUT
historical data, paleoseismological data
OUTPUT
maximum magnitude + σ, mean recurrence time + CV, elapsed time, M0 rate
mean recurrence time +CV, scale/shape parameters
GR parameters (Mt, b-value)
OUTPUT
OUTPUT
scale relationship code, geometry, slip rate, shear modulus, strain drop (observed magnitude, last eq time)
INPUT
INPUT
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expected earthquake rates, probabilities
Input for seismic hazard software (e.g., OpenQuake)
▴ Figure 2. Workflow that illustrates how the FiSH routines integrate the different levels of information that is available for a fault. The dotted arrows and box indicate the optional inputs that must be handled by the user. outputs time-independent or time-dependent earthquake rates for different MFD models. In particular, the codes that are currently provided with FiSH accomplish the following tasks: 1. Moment budget (MB) uses fault data, such as the style of faulting, geometry, and slip rate variability, to calculate the global budget of the seismic moment rate that is permitted by the structure based on predefined size–magnitude relationships. We calculate MB using two values: the maximum magnitude M max and the associated mean recurrence time T mean , including their uncertainties in terms of the standard deviation of a Gaussian distribution for M max and the coefficient of variation (CV, standard deviation of the recurrence times over their mean) for T mean. 2. RP uses the time series of past major earthquake occurrences (historical events assigned to a fault and/or paleoseismological data that include the uncertainty in dates). Using a Monte Carlo approach, RP calculates the distributions of recurrence times of major events on that fault and their variability. 3. Activity rates (AR) use the output from MB to calculate the Poisson and time-dependent (if the elapsed time from the last earthquake is available) probabilities of earthquake occurrences and the expected annual rates based on different predefined MFD models. Currently, FiSH supports
four types of MFDs: a single magnitude value, a characteristic distribution with a Gaussian bell that is symmetric around a central magnitude, a classical Gutenberg–Richter type (Gutenberg and Richter, 1944) and a truncated Gutenberg–Richter type (Ordaz, 1999; Kagan, 2002).
TOOLS BACKGROUND In this section, we describe the FiSH routines and the formulas they use. MB Tool MB combines standard data that were collected for faults or are assumed to be known, such as the size and rheological properties, with empirical size–magnitude relationships and allows for the treatment of their uncertainties. The input list that is defined by the user consists of the following: (1) the fault name, (2) a scale relationship code, (3) the length along strike (kilometers), (4) the dip angle (degrees), (5) the thickness of the seismogenic layer (kilometers), (6–7) the minimum and maximum slip rate (mm=yr), (8) the maximum observed magnitude (M w , if any), (9) the standard deviation in magnitude of the observed event (if any), (10) the date of the last event (year, if any), (11) the shear modulus, and (12) the strain drop. The first 10 input data are mandatory, whereas the last two are op-
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tion (1) to avoid the occasional rounding errors that can occur in the formula that was originally published by Hanks and Kanamori (1979). 2. A value of magnitude (M ASP , ASP for aspect ratio) is computed by modifying the along-strike dimension if it exceeds the length that is predicted by the aspect ratio relationships, which were derived by Peruzza and Pace (2002) from a slightly modified Wells and Coppersmith (1994) dataset, followed by the application of equation (1). 3. Two magnitude values that depend on the choice of the scale relationship are calculated. For example, if the user chooses to use Wells and Coppersmith (1994), the two magnitudes are given by empirical relationships that were established for either the maximum subsurface fault rupture length (MRLD) or maximum rupture area (MRA). 4. A value that corresponds to the maximum observed magnitude (M obs ), if available in the input file, is calculated.
tional and can be set homogeneously for all the faults in the GUI or entered into the input file (only the significand; exponents are set to 10−5 for strain drop and 1010 Pa for the shear modulus); fields 8–10 must be set as NaN (not a number) if they are unknown. The scale relationship code (see information in the MB interface in the “Scale Relationship” format button) allows the program to use the proper empirical relationship to transform the fault size into the maximum magnitude, that is, the magnitude that corresponds to the complete rupture of the entire fault plane. Currently, four relationships for tectonic and volcanic contexts have been implemented (Wells and Coppersmith, 1994, and Leonard, 2010, for tectonic contexts; Villamor et al., 2001, and Azzaro et al., 2015, for volcanic contexts); the scale relationship code assigns the chosen relationship(s) and the associated set of coefficients to the fault. The user has to prepare an input text file (Fig. 3) with fields that are separated by blanks or commas and contain one row for each fault to be analyzed. Next, MB prompts the user for the output filename and for other choices; the shear modulus μ and strain drop k (defined as the displacement-to-length ratio, D=L) are values that are set as homogeneous by default because they are commonly assumed to be constant in a seismotectonic region (Scholz, 1990). However, these values can be also set differently for each fault (e.g., for the sensitivity testing of their impact on the hazard input). These values are present in the scalar seismic moment (M 0 ) formula as follows:
Because all the empirical relationships and observations are affected by uncertainties, MB is designed to take these factors into account and return a maximum magnitude value and a standard deviation. The uncertainties in the empirical scaling relationship are taken from the studies of Wells and Coppersmith (1994), Villamor et al. (2001), Peruzza and Pace (2002), Leonard (2010), and Azzaro et al. (2015). Currently, the uncertainty in MM 0 is fixed and set to 0.3, whereas the uncertainty in M obs is assigned by the user. To combine the maximum magnitudes, MB draws a probability curve for each magnitude estimate by assuming a normal distribution. The user can define the number of standard deviations (σ) for truncating the normal distribution of magnitudes at both sides (left empty for unbounded distributions). MB successively sums the probability density curves and fits the summed curve to a normal distribution to obtain the mean of the maximum magnitude M max and its standard deviation σM max . MB also allows the introduction of weights to create the probability density curves of the magnitudes. Therefore, M max represents an evaluation of the maximum rupture that is allowed by the fault geometry and the rheological properties. An output figure of this process is given
1
M 0 μDLW μkL2 W ;
EQ-TARGET;temp:intralink-;df1;40;421
in which L, W , and D are, respectively, the along-strike fault length, the down-dip width, and the average displacement of a rectangular source (Fig. 1). For each fault, depending on the input choice, up to five M max values (and their errors) are computed: 1. A magnitude value MM 0 is directly computed by applying the standard formula M w = 2/3 (log M 0 − 9:1), which was accepted by International Association of Seismology and Physics of the Earth’s Interior (IASPEI, 2005), to equa-
(a)
Name
ScR
Length
Dip
Seis_thick
SRmin
SRmax
Mobs
σ Mobs
Last_eq_time
Input file format
Paganica
WC94-N
20
50
15
0.6
0.8
6.3
0.1
2009
(b)
Id
Mmax
σ Mmax
Tmean
CV
Telap
Output file format
1
6.5
0.2
922
0.71
6
Mo
rate
7.678 × 1015
▴
Name Paganica
Figure 3. Example of input/output files for moment budget (MB) model. (a) Input file formats: ScR, scale relationship code (see the Scale-Relationship button in the MB interface); length, fault length along strike (kilometers); dip, dip angle; Seis_thick, thickness of the seismogenic layer (kilometer); SR min , minimum value of slip rate (mm= yr); SR max , maximum value of slip rate (mm= yr); M obs , maximum observed magnitude (M w ); σM obs , standard deviation in the magnitude of the observed event; and Last_eq_time date of the last assigned full-rupture earthquake. (b) Output file formats: id, identification number; M max , maximum magnitude value; σM max , maximum magnitude standard deviation; T mean , mean recurrence time (year) of M max , CV, coefficient of variation of T mean , which is defined as the ratio between the standard deviation of T mean and T mean ; T elap , time elapsed (year) since the last assigned earthquake; M o rate , seismic moment rate (N · m · yr−1 ); and name, name of the fault.
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3.5
Probability density function
adopted by IASPEI (2005) for the relationship between the moment magnitude and seismic moment (in N·m). To take into account the uncertainties in M max and the slip rates and to explore how these uncertainties affect the variability in T mean , we introduced formal error propagation following Peruzza et al. (2010). The general formula for error propagation is given by a Taylor series:
Paganica
4
3
2.5
EQ-TARGET;temp:intralink-;df3;323;661
2
F x; y; z; … ≈ F x; ¯ y¯; z; ¯ …
1.5
1
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
7
Magnitude
Figure 4. Example of MB graphic output for the Paganica fault (described in the Paganica Fault Case Study section), which shows the combination of magnitude estimates from empirical relationships and observations, both of which are affected by uncertainties. In this example, four magnitudes are estimated, and MB draws a probability curve for each magnitude by following normal distributions that are symmetrically distributed around the central values. The magnitudes are defined as follows: MM 0 is from the standard formula (IASPEI, 2005); maximum subsurface fault length (MRLD) and maximum rupture area (MRA) are from the empirical relationship of Wells and Coppersmith (1994) for length and area; and M obs is the largest observed moment magnitude (6.3 in this case). The black dashed line represents the summed probability density curve, the vertical black line represents the central value of the Gaussian fit of the summed probability density curve (M max ), and its standard deviation (σM max ) is given by the horizontal black dashed line. The input values that were used to obtain this figure are provided in Figure 3a.
for each fault (an example that is described later is given in Fig. 4), so the user can easily compare the distributions and the impact of different magnitudes on the final value. To obtain the mean recurrence time of M max , hereafter referred as T mean , MB uses the criterion of “segment seismic moment conservation” that was proposed by Field et al. (1999), which divides the seismic moment that corresponds to M max by the moment rate given a slip rate T mean
1 101:5M max 9:1 ; Char Rate μV LW
σ 2F
EQ-TARGET;temp:intralink-;df4;323;578
▴
EQ-TARGET;temp:intralink-;df2;52;200
∂F z − z ¯ …; ∂x
3
and the variance is given by
0.5
0 6
∂F ∂F x − x ¯ y − y¯ ∂x ∂x
2
in which T mean is the mean recurrence time in years, Char Rate is the annual mean rate of occurrence, M max is the previously computed mean maximum magnitude, μ is the shear modulus, V is the average long-term slip rate (obtained by averaging the minimum and maximum slip rates defined by the user in the input file), and L and W are the geometrical parameters of the fault. The coefficients 1.5 and 9.1 are the standard values that are
2 2 2 ∂F ∂F ∂F 2 2 ≈ σx σy σ 2z …: ∂x ∂y ∂z
4
The variance is applied to the T mean obtained from equation (2) using partial derivatives of the slip rate V and magnitude M max . The variance of T mean becomes EQ-TARGET;temp:intralink-;df5;323;485
σ 2T mean
9:11:5M max 1:5 log10 2 10 ≈ dM 2max μV LW 109:11:5M max 2 2 − dV ; μV 2 LW
5
in which dM max and dV are generic small variations in the magnitude and slip rate. Here, they are substituted with the standard deviation σM (previously described) and σV for the analysis of error propagation. The parameter σV is assumed to be half of the user-defined minimum and maximum slip rates (given in the input file). The output of MB consists of a figure with curves of the magnitude distributions, as well as the derived mean and standard deviation (e.g., Fig. 4). The output also includes a file (Fig. 3b) that contains the following information for each fault: the identification number, the mean value of M max and its standard deviation, T mean (yr), the CV for T mean, the time elapsed (yr) since the last earthquake (if given), the seismic moment rate (N · m · yr−1 , defined as the ratio between the seismic moment that corresponds to M max and T mean ), and the name of the fault. This output can be used as the input for the AR tool. RP Tool Recent work has focused on deriving statistical parameters, such as the mean and standard deviation of the recurrence times, from paleoseismological or historical earthquake time series by considering the uncertainties in dating (e.g., Parsons, 2008). The user should thus check them with the parameters that are derived from the moment budget to enlighten the potential limitations that are associated with paleoseismological and historical datasets, for example, incomplete catalogs or uncertain magnitudes of the causative events.
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The RP tool is designed to take these uncertainties into account and show how they can influence the desired parameters. Starting from an input file that contains the youngest and oldest years of occurrence for each event in the series, RP produces n simulations of the earthquake catalog (hereafter referred to as “paleoeqs-simulations”) using a uniform distribution for the occurrences within its window of uncertainty. This simplistic view does not consider the most recent developments in age modeling by paleoseismologists (e.g., the OxCal program, Bronk Ramsey and Lee [2013], available at http:// c14.arch.ox.ac.uk/embed.php?File=oxcal.html#program, last accessed December 2015) but, conversely, enables the users to handle the literature data, which are usually provided as rough time intervals. Future implementations of FiSH could introduce arbitrary probability density distributions. For each paleoeqs-simulation, RP estimates the arithmetic mean of the recurrence times and its standard deviation (we call this case “arithmetic”). RP also fits each paleoeqs-simulation using three probability functions, assuming that the events follow a Brownian passage time (BPT), a Weibull, and a Poisson distribution. For the BPT distribution (Matthews et al., 2002), the probability density function is r 2 mean T mean t−T 2T mean α2 t ; P t e 2πα2 t 3
6
EQ-TARGET;temp:intralink-;df6;40;481
in which T mean is the mean recurrence time, α is the aperiodicity of the distribution, and t is the time in years. RP returns two values, T mean and α. For the Weibull distribution (Patel et al., 1976), the probability density function is b t b−1 P t exp −t=ab ; a a
7
EQ-TARGET;temp:intralink-;df7;40;375
in which b > 0 is the shape parameter and a > 0 is the scale parameter of the distribution. The T mean and variance (σ) of the Weibull distribution are provided by T mean aΓ1 b−1 ;
8
σ 2 a2 Γ1 2b−1 − Γ1 b−1 2 :
9
EQ-TARGET;temp:intralink-;df8;311;697
EQ-TARGET;temp:intralink-;df9;311;652
RP returns the values a, b, T mean , and CV (CV σ=T mean ).
In a Poisson distribution of the earthquakes, the times between events are described by an exponential distribution, for which the probability density function is
10
Pt λe−λt ;
EQ-TARGET;temp:intralink-;df10;311;579
in which λ is the inverse of the mean recurrence time (T mean ) obtained by RP. Because this fit is performed for each of the paleoeqssimulations, n output parameters are obtained. These values are shown as contour graphs (for the arithmetic, BPT, and Weibull distributions) and histograms of T mean (for the Poisson distribution), see an example in Figure 6. The input required by the RP is an index file (with a .txt extension, see Fig. 5) that lists the names of the files that contain the earthquake time series for each fault. These files (with a .paleo extension) should be prepared with one row for each event and two columns for the earliest and latest years of earthquake occurrences, thus accounting for the dating uncertainties; for historical events, the same year has to be repeated. Once the input file is uploaded, RP prompts the user to set the number of simulations n and the output name. The user can define a seed number in the GUI to control the random generator so that the user can reproduce the same random simulation, and the calculations are replicable using the same seed number. This structure enables the
(a) Input file format (filename.txt)
(b) Input file format (Paganica.paleo)
Paganica
Oldest 2009 1461 890 –760 –2900
Youngest 2009 1461 1150 670 –760
(c) Output file format
ARITH:
CV = 0.65 Tmean = 1013
BPT param:
alpha = 0.50 Tmean = 799
WBL param:
a = 1233 b = 1.27
WBL param (CV,Tmean)
▴
CV = 0.53 Tmean = 1021
POIS param:
Tmean = 1013
Paganica
Figure 5. Example of input/output files for recurrence parameters (RP). (a) Input file formats (filename.txt): index file that lists the names of the site or fault where the data were collected. The name must correspond to the filename (in this example, Paganica) where the data are stored. Multiple names (and therefore many faults or alternative age determinations, with one case per row) are allowed within a single run. (b) Input file formats (filename.paleo): The youngest and oldest years of occurrence of each recognized event are listed in two columns. The filename must correspond to the name in filename.txt. (c) Output file formats: The output parameters are calculated using simple sample statistics (the arithmetic mean of recurrence times and its standard deviation, provided in ARITH) or by assuming that the recurrences follow a given distribution (Brownian passage time, BPT param.; Weibull, WBL param.; exponential (Poisson), POIS param.).
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users to treat, for example, only some of the faults that were previously characterized in MB or to check the changes from different age modeling/interpretations on the same fault trench. The outputs consist of a figure (e.g., Fig. 6) that includes sample data, contour graphs of the paleoeqs-simulations according to the probability distributions and histogram for each row of the index file, and a summary file (Fig. 5) with the statistical parameters of all the rows. The pairs of T mean and CV that are obtained by RP can be entered into the AR tool depending on the user’s need.
AR is designed to return annual earthquake rates. The user first enters the standard output of the MB tool or a new file from
the RP results that contains the following for each row/fault: (1) the identification number, (2) M max , (3) its standard deviation σM max , (4) T mean , (5) CV, (6) the seismic moment rate, (7) the elapsed time from the last assigned earthquake (if given), and (8) the name of the fault. All the fields must be filled, but not all the inputs will necessarily be used; the basic information for modeling Gutenberg–Richter (GR) MFDs, that is, the minimum magnitude and b-value, is handled separately in a second input file. We hope that this choice will allow a user to easily perform several tests using a common input. Once the user has added the input file(s), AR prompts the user to name the output file, choose the MFD, set an observation window and magnitude bin, and flag the figure options. The standard output consists of two text files with the
(a)
(b)
AR Tool
Paganica −arithmetic
1400
4
1200
300
1000
200
800
100
Tmean
Eq#
Paganica −data 5
3 2
1 −4000
600 −2000
0
2000
4000
0
0.5
1
Range of occurrence (year)
(c)
1.5
CV
(d)
Paganica −POIS distribution
Paganica −BPT distribution 1400
1500
800 1200
Tmean
Count
1000
500
600 1000
400
800
0 600
(e)
800
1000 Tmean
1200
600 0.2
1400
200 0.4
0.6
0.8
1
1.2
Alpha
Paganica −WBL distribution
Paganica −WBL(Tmean,CV) distribution
(f) 1400
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400
1000
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800
200
600
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0
1
2 b
3
Tmean
a
1200
300 1000
200
800
4
600 0.2
100 0.4
0.6
0.8
1
1.2
CV
▴ Figure 6. Example of RP graphic output for the Paganica fault (described in the Paganica Fault Case Study section), which shows (a) the used occurrence data, (b, d–f) the contour plots that display the isolines of the pairs of parameters that were evaluated by the three two-parameter methods, with the level of isolines indicated in the colored scale bar, and (c) the histogram of the average recurrence time from the Poisson distribution. The input values for this figure are provided in Figure 5. Seismological Research Letters
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(a)
Id
Mmax
Input file format
1
6.5
(b) Output file format (rates)
Id
Mmaxlb
1
6.3
Id
Mmaxlb
σ Mmax 0.2
Tmean
CV
Telap
922
0.71
6
Bin 0.1
Mo
Name
rate
7.678 × 1015
Paganica
Rates -04
1.5000 × 10
2.1824 × 10
-04
Name -04
2.4730 × 10
2.1824 × 10
-04
1.5000 × 10
-04
Paganica
(c) Output file format (probability)
1
6.3
Window 50
Probability 4.8 × 10
Name
-02
Paganica
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Figure 7. Example of input/output files for activity rates (AR). (a) Input file formats: id, identification number; M max , maximum magnitude value; σM max , maximum magnitude standard deviation; T mean , mean recurrence time of M max (year), CV, coefficient of variation of T mean , which is defined as the ratio between σT mean and T mean ; T elap , time elapsed (year) since the last assigned earthquake; M o rate , seismic moment rate (N · m · yr−1 ); and name, name of the fault. (b) Output file formats (rates): id, identification number; M max lb , lower bound of M max (M max − σM max ); bin, magnitude bin width; rates, earthquake annual rates (five numbers in this example) from M max lb to M max ub M max σM max per bin of magnitude; and name, name of the fault. In this example, the rates refer to a characteristic Gaussian model; see the Characteristic Gaussian Model (Poisson) section for details. (c) Output file formats (probability): id, identification number; M max lb , lower bound of M max M max − σM max ; window, probability window (year); probability, probability of an earthquake with a magnitude greater than or equal to M max lb in the next probability window; and name, name of the fault. In this example, the probability refers to a characteristic Gaussian model; see the Characteristic Gaussian Model (Poisson) section for details. EQ-TARGET;temp:intralink-;df11;311;489
annual rates and probabilities in the time window that was selected in the GUI (e.g., Fig. 7); additional graphics are plotted for each row/fault (e.g., Fig. 8). AR balances the annual earthquake rates over the range of magnitudes for the MFD with the moment rate. The details of this process for the various choices of MFDs are explained below. Single-Value Model (Poisson) In the single-value Poisson model, the MFD collapses into a single value that is given by (M max , 1=T mean ) in the input file. Similar to the MB tool, the seismic moment rate is given by the seismic moment of M max (using the standard conversion formula) divided by the T mean . If this value differs from the sixth value that is listed in the AR input file (e.g., the MB output has been modified according to the RP results or by imposing the seismic moment rate, for example, by geodetic considerations), a warning message appears. If the option that forces the use of the input seismic moment rate is flagged, a new T mean is computed; a warning message informs the user if this new recurrence time differs with respect to the original second and third input values. The same flow chart applies to the other choices that are described hereafter; thus, the code serves to check inconsistencies between geometrical and energetic formulations and to perform sensitivity testing on the seismicity rates. Single-Value Model (BPT) In the single-value BPT model, AR also uses the CV and the elapsed time of the input file. As in the previous case, the MFD collapses into a single value that is given by the M max of the input file, but its annual rate is given in terms of a time-dependent probability that is assigned to the fault using the BPT formula (equation 6). The conditional probability (Pjelap) that an event occurs during the next ΔT yr, given an elapsed time T elap since the last event, is defined as follows: 8
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PT elap ≤ T ≤ T elap ΔT jT > T elap
PT elap ≤ T ≤ T elap ΔT : 1 − P0 ≤ T ≤ T elap
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In this case, the T elap is a mandatory input, ΔT is the observation window that is set in the GUI, and the CV of the mean recurrence time is the α value in equation (6). MB returns a CV value, which is defined as the ratio between σT mean and T mean , but the user can assign different values, for example, after analyzing the date sequence with RP. Similar to the previous (Poisson) case, a check on the seismic moment rate is performed. Thus, using the simplification that was proposed by Wu et al. (1995), a fictitious recurrence time T fict for M max is computed by solving the equivalence of the probabilities as follows: T fict
EQ-TARGET;temp:intralink-;df12;311;315
−ΔT ; LN 1 − Pjelap
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in which Pjelap, which is the conditional probability from the BPT model (equation 11), is set equal to the probability of a Poisson process given a selected observation period ΔT . The inverse of T fict is used to obtain the rate of occurrence at M max . The output files for each fault are similar to the previous case; the annual rates and probabilities do not refer to any time window, but only to the ΔT yr after the starting time, which is used to compute the elapsed time; forward retrospective testing can be performed at any starting time by modifying only the T elap parameter in the input file. If the figure option is set in the GUI, an additional plot that represents the Poisson and BPT probabilities versus the elapsed time is given (see the example in Fig. 8, which is described in more detail in the next section). Single-Value Model (User-Defined Probability) In the most generic case, AR can use probabilities that are defined by the user: with this option, the input file must also March/April 2016
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Characteristic Gaussian Model (BPT) Similar to the previous case and the Single Value Model (UserDefined Probability) section, AR can consider the BPT model to be valid under the assumption that an earthquake with a magnitude in the range M max lb –M max ub allows the renewal process to begin again. Thus, the probability of equation (6) must be representative of an earthquake with a magnitude in the range M max lb –M max ub . Therefore, T mean from equation (6) is the inverse of the cumulative annual rate of earthquakes with M w ≥ M max lb , and the fictitious recurrence time T fict by equation (12) is computed similarly to that in the Single Value Model (User-Defined Probability) section. The output is the same as in the Characteristic Gaussian Model (Poisson) section.
Paganica 0.08 BPT POIS P|elap T
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P of occurrence in50 years
M>=Mmaxlb
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0 0
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Figure 8. Example of AR graphic output that shows the hazard functions and probabilities of the occurrence of an earthquake with M w between 6.3 and 6.7. The BPT curve (black curve) and the time-dependent probability of occurrence are computed for the next 50 yr (P\elap, red circle), given an elapsed time of 6 yr. The time-independent model (POIS, black dashed line) refers to a period of 50 yr. The mean recurrence time of a magnitude that is equal or greater than the lower bound of the M max (6.3 in this case) is also provided (T M≥M max lb , blue vertical line).
contain an additional column with the probability of the full rupture of the fault in the selected time window. A fictitious recurrence time T fict for M max is now computed by solving the equivalence of the probabilities in equation (10), where Pjelap is the user-defined probability in the input file. Incorrect results may derive from the uncontrolled utilization of the ΔT window in the GUI. The inverse of T fict is used to obtain the rate of occurrence at M max . This choice can be used, for example, if different probability density functions from BPT (e.g., Weibull, which is characterized by the RP tool) are used to model time dependency. Characteristic Gaussian Model (Poisson) In addition to the Single-Value Model (Poisson) section, AR can use M max with its standard deviation (σM max ). The MFD is provided by a Gaussian curve that is centered on the maximum magnitude in the input file and ranges from a lower bound M max lb M max − σM max to an upper bound M max ub M max σM max with a step that must be specified by the user in the GUI. The seismic moment rate is thus partitioned and balanced over this range of magnitudes. One output file lists M max lb in the user-defined bin width and the annual rates for all the magnitude bins; the other lists the Poisson probability that at least one earthquake will occur with a magnitude greater than or equal to M max lb . The particular format of the first output file is dictated by the potential need for an output that can be easily used as input for the OpenQuake (Pagani et al., 2014) and CRISIS2008 (Ordaz et al., 2013) software when fault sources are used.
Characteristic Gaussian Model (User-Defined Probability) If the input file contains an additional column with the probability that is assigned to the full rupture, AR can compute seismicity rates and probabilities for the magnitude bins of a Gaussian bell. The logical flowchart is the same as in the previous case. Classical GR The identification number is used by AR when modeling seismicity rates with a GR distribution. An additional input file that lists the identification number, the lower threshold magnitude of the distribution (M t ), and the b-value of the Gutenberg–Richter distribution is required. In this case, the distribution is truncated at both ends by M t and M max , as EQ-TARGET;temp:intralink-;df13;323;409
log 10 N M a − bM
with M t ≤ M ≤ M max ;
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in which N M is the number of earthquakes with a magnitude greater than or equal to M t and less than or equal to M max , and a is the logarithm of the number of earthquakes with M ≥ 0. AR computes a to balance the total seismic moment rate (given by summing the rates of moment magnitudes for the bins of magnitudes with a user-defined step) with the seismic moment rate that was obtained by the pair M max and T mean . The output consists of one file with seismicity rates that lists the M t , the user-defined bin, and the annual rates for each magnitude bin. Given the well-known problems of decumulation, the use of this formulation has been progressively abandoned in PSHA. Truncated GR This case is similar to the previous one, and the additional input file is the same as in the classical GR case. The distribution is truncated at both ends by M t and M max , but the truncated GR is characterized by a smooth transition to M max instead of the abrupt drop of the classical GR model. Following Kagan (2002), the cumulative density function of the truncated GR model is ϕM
EQ-TARGET;temp:intralink-;df14;323;123
M t =Mβ − M t =M u β 1 − M t =M u β
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in which M u is evaluated by shifting the M max of a magnitude bin to ensure a nonzero rate of earthquakes at M max , and β is 2=3b; this equivalence is handled within the code, and the input file must define the b-value. As in classical GR, the seismic moment rate is partitioned and balanced over this range of magnitudes with a step that must be specified by the user in the GUI. The output is the same as in the previous case.
PAGANICA FAULT CASE STUDY The example shown below is intended to illustrate the capabilities of FiSH. The case study is taken from published articles, because presenting new data and/or interpretations is beyond the scope of this article. The Paganica normal fault in central Italy ruptured during the 2009 L’Aquila earthquake sequence, causing an M w 6.3 earthquake on 6 April. The geometrical and slip rate parameters that comprise the main input for FiSH are mainly taken from Pace et al. (2006, 2014), whereas the paleoseismological data are from Cinti et al. (2011). In the following, we describe in detail the results obtained using the three FiSH tools. MB Example Following Peruzza et al. (2011), the last earthquake that was assigned to the fault is the 2009 L’Aquila event (M w 6.3), which resulted in an elapsed time of 6 yr in 2015, and the uncertainty in the magnitude was fixed to 0.1. This variability is what can be assigned to the instrumental estimate of M w by full seismic moment tensor inversion. The geometric parameters of the fault (a length along strike of 20 km, a dip angle of 50°, and a seismogenic thickness of 14 km) and slip rates (ranging from 0.6 to 0:8 mm=yr) were taken from previous geologic studies by Boncio et al. (2004) and Roberts et al. (2004), both of which were revised by Pace et al. (2010). The input file for MB is shown in Figure 3. The header in the input file is mandatory, and values must be given in the same order. To run MB, the user must define the path to the main folder in the MATLAB window and type “mb” into the command line of MATLAB to initiate an empty MB session. The MB GUI will open, allowing the user to load the input file, name the output file, and set the strain drop, the shear modulus, and eventually the number of standard deviations (σ) for truncating the normal distribution of magnitudes at both sides. The GUI window also contains information on the main features of the code. MB accepts as many rows as the user needs, and the output will result in a text file that contains a row for each fault and as many figures (in .EPS format) as faults. These figures show the magnitude probability density functions, their sum, and the resulting mean and the standard deviation. In this example, the use of the relationship of Wells and Coppersmith (1994) for normal faults (scale relationship code WC94N), which uses the geometrical and kinematic parameters of the fault, suggests that the full rupture of the source is given M max 6:5, which is associated with a T mean of 922 yr. The text input and output files and output figure are shown in Figures 3 and 4, respectively. In this case, the use of 1σ 10
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truncation slightly moves M max toward the maximum observed value (6.4), but a contemporaneous decrease in T mean (653 yr) keeps the seismic moment rate almost fixed (from 7:678 × 1015 to 7:675 × 1015 N · m · yr−1 ). RP Example The RP tool accepts the dates of past events (historical or paleoearthquakes) as the input. An example input file, prepared using paleoseismological data from the Paganica fault (Cinti et al., 2011), is shown in Figure 5. The series contains five earthquakes, of which the most recent is the A.D. 2009 earthquake. Three events occurred between A.D. 1461 and 760 B.C., and the oldest one occurred between 2900 B.C. and 760 B.C. Note that the header in the input file is mandatory, and values must be given in the same order. If no uncertainty is entered for an event, as is the case for the two most recent historical earthquakes, the youngest and oldest ages are the same. To run RP, set the path and type “rp” into the command line of MATLAB to start a new session. The RP GUI allows the user to load an input file and set the name of the output file and the number of simulations (in the Paganica example, we used 10,000 simulated paleoearthquake series). The RP output figure, which is shown in Figure 6, contains the selected occurrence data (Figs. 5b and 6a), the number of simulations, the contour plots of the three two-parameter methods (Fig. 6b,d–f ), and a histogram of the average recurrence time from the Poisson distribution (Fig. 6c). With 10,000 paleoearthquake simulations, Figure 6b shows a quite large area of (T mean , CV) pairs, with two peaks with the maximum frequency (more than 300 simulations) at values of ∼800 − 0:55 and ∼1000 yrs (0.65), respectively. Figure 6c shows the histogram of T mean , which has a range of 700– 1200 yr, assuming a Poisson distribution. Figure 6d shows that the two peaks are more numerous (> 600 pairs of T mean and CV) and somewhat more clearly separated (with values of ∼800 [0.5] and 1200 yr [0.8], respectively) when using a BPT distribution to model the recurrence of the earthquake series, which suggests the two most likely recurrence behaviors of the Paganica fault. Finally, the results from the Weibull model (Fig. 6e and f ) are given in terms of a and b coefficients (equation 7) and equivalent values of T mean and CV (equations 8 and 9). The highest frequency of a and b pairs (> 600 pairs) falls between 1200 and 1400 yr for a and between 1.3 and 1.5 for b and splits into two peaks whose highest frequency (> 300 pairs) T mean − CV results are between ∼800 and 1000 (0.4–0.6) and ∼1150 yr (0.7). The output also includes a text file that indicates the mode of pair(s) of parameters that are evaluated by each of the three methodsarithmetic, BPT, and Weibull and the mode of the mean recurrence time(s) that are obtained by assuming a Poisson distribution of events. These results are shown in Figure 8. AR Example The structure of the input file for AR can be the same as the output file of MB (Fig. 7); conversely, the user can modify or edit a new, similar input file. To run AR , set the path and type March/April 2016
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Comparing Rates In Figure 9, we compare the cumulative rates of occurrence that were computed by the four chosen MFDs with the rates that were calculated by RP, which only used paleoseismological data (Fig. 5). Because RP does not include the magnitude of the events, the rate of occurrence (i.e., the inverse of the mean recurrence time, which is computed using the arithmetic method in Figure 5c) is plotted as a horizontal line in Figure 9. The curves that were obtained from AR share the same seismic moment rate from the input file, 7:678 × 1015 N · m · yr−1 (Fig. 3b). Interestingly, the moment rate in an ideal situation, in which the fault rupture is dictated by single-value behavior and the paloeseismological data are complete, should be the same as that from MB. In contrast, assuming complete paleoseismological data and that the fault ruptures in the greatest earthquakes have magnitudes that follow a Gaussian distribution, the horizontal line and the characteristic Gaussian (GaussCH) model should intersect at the lower bound of M max (6.3 in this case), as should the GR and truncated-GR cases. In Figure 9, the RP-derived rate intersects three of the AR-derived rates (the characteristic Gaussian, GR, and truncated GR) and lies below the single-value model. We suggest two possible interpretations according to the limitations of the input data: (1) the paleoseismological data are com-
Truncated GR Classical GR Gauss CH 10–2
Annual cumulative rates
“ar” into the command line of MATLAB to start a new AR session. The AR GUI prompts the user to load the input files, set the name of the output file, choose an MFD, define the probability window and the magnitude bin width, and select if the user wants the output to include a figure (one for each row of the input file) that shows hazard functions and the probability of occurrence from the BPT and Poisson models. The second input file is required only for MFDs that follow GR or truncated-GR behavior; this second file contains the b-value and M t parameters, which were described in the Classical GR and Truncated GR sections. The GUI window also contains information on the main features of the code. The header in the input file is mandatory, and values must be given in the same order. AR accepts as many rows as the user needs and will return a file containing a row for each fault and as many figures (in .EPS format) as input rows. In the Paganica example, we use the output file from MB (Figs. 3b and 7a) to run AR with four possible MFD choices: SingleValue, GaussCH, ClassicalGR, and TruncatedGR. Figure 8 shows the hazard functions for the time-independent model (i.e., a Poisson distribution of earthquakes in time) and the time-dependent model (i.e., a BPT distribution). The MFD is set to GaussCH, with a central value of M w 6.5 and a standard deviation of 0.2, as listed in the input file (Fig. 7a). Because 6 yr have elapsed since the last earthquake occurred in 2009, the time-dependent probability of the occurrence of an event with 6:3 ≤ M w ≤ 6:7 over the next 50 yr is ∼0:00001% (Pjelap red circle in Fig. 8). The Poisson probability of an earthquake with 6:3 ≤ M w ≤ 6:7 in 50 yr is about 5% (Figs. 7c and 8). Together with the probability of occurrence, AR returns the annual earthquake rates for the given magnitude bins in the form of incremental rates, as shown in Figure 7b.
SingleValue rate from RP
10–3
10–4 5.5
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5.8
5.9
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6.4 6.5
6.6 6.7
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Figure 9. Comparison of the magnitude–frequency distributions for the Paganica fault (described in the Paganica Fault Case Study section), which were obtained using the single-value Poisson model (solid blue circle in the figure), characteristic Gaussian model (black line), and classical and truncated Gutenberg–Richter models (red and blue lines, respectively). These models are computed by AR with the input in Figure 7. The horizontal black dashed line represents the inverse of the RP-derived T mean considering the arithmetic case (mean of the recurrence times of all the simulations), as shown in Figures 5c and 6b (ARITH results).
plete for events with magnitudes in the range 6.3–6.7, if the fault has characteristic Gaussian behavior, and (2) the paleoseismological data are not complete for events with magnitudes lower than 6.1–6.2, if the fault has GR or truncated GR behavior, or the paleoseismological data are complete, and the slip rates overestimate the seismic moment that is released by the fault. From a seismic-hazard perspective, paleoseismologically inferred rates of occurrence should be cross-checked with geologically derived long-term slip rates and/or with rates of occurrence that are derived from other approaches (such as geodetic rates that are computed for seismogenic faults). Our intent is to distribute tools that can help researchers to pinpoint potential inconsistencies and obtain more reliable fault-based seismic-hazard evaluations.
FINAL REMARKS FiSH is available at https://sites.google.com/site/francvisini/ (last accessed December 2015) and should be run in MATLAB R2011b. The compressed file also contains a collection of sample data files (inputs and outputs). Although tested and designed to work on any MATLAB-supported platform, it is possible that some users will experience problems while running various functions. The source codes are open, and we encourage users to handle the scripts, communicate with us regarding bugs, and/or suggest further improvements. FiSH is intended to be useful for seismic-hazard practitioners, but it must be used under the general principle “trash in–trash out.” The use of incomplete earth-
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quake series and/or earthquake associations in RP or unrealistic geometries in MB can be only partially detected by the code. In our experience, the output figures help the user to detect input data problems. Although they have not been scheduled yet, new releases of FiSH are likely. These updates will largely be driven by research interests, and new features will be documented on the same website.
DATA AND RESOURCES FiSH is written in MATLAB (v.R2011b), a Mathworks software (http://www.mathworks.com, last accessed December 2015) and is available at http://fish-code.com (last accessed January 2016). All data used in the examples for moment budget (MB Example), recurrence parameter (RP Example), and activity rate (AR Example) came from published sources listed in the references.
ACKNOWLEDGMENTS This study has benefitted from partial funding that was provided by the Italian Presidenza del Consiglio dei Ministri Dipartimento della Protezione Civile (INGV-DPC), Project S2-2012 and V3-2014, and the Ministero dell’Istruzione, dell’Università e della Ricerca (MIUR) funded project "Futuro in Ricerca" (FIRB) Abruzzo (Code RBAP10ZC8K_006). This article does not necessarily represent DPC’s official opinion and policies. We thank the reviewers for their accurate reviews and useful suggestions.
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B. Pace DiSPUTer Università “G. d’Annunzio” Campus Universitario di Madonna delle Piane Via dei Vestini 30 66013 Chieti Scalo, Italy
[email protected]
F. Visini Istituto Nazionale di Geofisica e Vulcanologia Via dell’Arcivescovado 8 67100 L’Aquila, Italy
[email protected]
L. Peruzza OGS (Istituto Nazionale di Oceanografia e Geofisica Sperimentale) Borgo Grotta Gigante 42/C 34010 Sgonico, Trieste, Italy
Seismological Research Letters
[email protected]
Published Online 13 January 2016
Volume 87, Number 2A
March/April 2016
13
