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Illustration of wave propagation effects that occur from different portions of a fault as an earthquake ruptures. Each EGF accounts for a different travel path through the heterogeneous geology. There are several limitations to EGFs. The selection of an EGF for use in source studies follows strict criteria that are not always possible to fulfill. Thus, a suitable EGF function may not be available, limiting the number of earthquakes that can be analyzed. The bandwidth of the EGF for which enough signal above noise is available is another limiting factor in source studies. For very small EGF earthquakes or noisy surface stations, the available bandwidth may not be enough to perform the source analysis, even though there is sufficient signal from the earthquake being analyzed. Limited instrument bandwidth, another potentially serious limitation, may bias the source parameters. However, the EGF method profits from having a good distribution of recording stations, providing a good azimuthal coverage so that source directivity effects may be accounted for. There are also a number of limitations in applying EGF methods to ground-motion modeling: 1. In virtually all practical applications, there are insufficient number of EGFs to provide an impulse response for all portions of a fault rupture to be modeled. 2. EGFs cannot accurately model variations in focal mechanism solutions. 3. Noise levels in recordings limit their usable frequency band (usually between 0.2 Hz to 25.0 Hz or narrower in seismically active regions such as the Western U.S). 4. A distribution of stations with good azimuthal coverage is necessary to ascertain the locations and source parameters of the small earthquakes that would provide EGFs.
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... the last couple of decades, empirical Green’s functions [EGFs] have been increasingly used in earthquake source studies, crustal attenuation studies, strong ground-motion prediction, finite rupture modeling, and site-response studies. Theoretically, Green's functions are the impulse response of the medium, and EGFs are recordings used to provide this impulse response. In this chapter, we review the theoretical and observational basis for identifying and using small earthquakes as empirical Green's functions—and their application. We generally refer to analyses that use EGFs as “the EGF method”. The seismic record of an earthquake contains information on the earthquake source, the path the seismic waves propagated through, the site response of the geology beneath the recording site, and the response of the instrument that recorded the ground motion. In seismology, we are interested in isolating either source, path, or site information, depending on the study focus. The instrument response is usually known and easy to remove from recordings, but the same is not true for the other factors. Seismic waves propagate in the earth in a complex way. They are reflected and refracted at interfaces between rocks with different properties, attenuated throughout the path between the earthquake source and the station by scattering and anelastic effects, subjected to energy focusing and defocussing due to lateral changes in the refractive properties of the rock, and amplified and highly attenuated near the recording station as waves pass through unconsolidated near-surface material. These propagation complexities are not well captured by crustal models, which provide the basis for calculating Green's functions, especially for high frequencies (> 1 Hz). At higher frequencies, wave propagation is very sensitive to small crustal heterogeneities, which are generally not well known; at low frequencies (< 1 Hz), wave propagation can be modeled fairly accurately. EGFs can be used instead of mathematical calculations to more accurately represent seismic wave propagation in the geologically heterogeneous crust. The EGF method is the best available method because it empirically corrects for unknown path and site effects, for which a short wavelength resolution is needed. However, true EGFs contain the source rupture process of the small earthquakes in the recorded seismograms. No earthquake has a true impulsive source. Therefore, one must be careful using EGFs. There has not been uniformity in defining EGF’s. Recordings of earthquakes with magnitudes less than 1.0 to 7.0 have been used as EGFs. Here, we apply a strict mathematical definition to EGFs and discuss different uses of EGFs with respect to this definition. We define EGFs as recordings of sources that satisfy impulsive-point dislocation criteria within the frequency band the EGFs are used. The word “satisfy” is included in the definition of EGFs because there is no true impulsive point source in nature. Therefore, the definition of an EGF can be dependent upon the frequency range of interest. Sources in nature may be explosions, rock bursts, earthquakes, or any impulsive source. Under certain constraints, EGFs can be used as point sources. The impulsive point source definition is consistent with the mathematical definition of an elastodynamic Green’s function, which is the response at a particular location to a uni-directional, unit-impulsive, point source at another location. EGFs were initially used in attenuation and earthquake-source studies (e.g., Bakun and Bufe, 1975, and references within). Bakun and Bufe noted that for small earthquakes (<M4), for which corner frequencies are within the frequency band most affected by attenuation and site effects, deconvolution could isolate source and propagation-path effects. Initially, the method was applied in the frequency domain by a spectral ratio method. The underlying reasoning for doing this was that common station recordings of closely located earthquakes shared the same propagation path and could be used to form “ratios of spectra,” and thus “cancel the common propagation path effects” (Bakum and Bufe, 1975). Subsequent studies further developed EGF methods. Frankel (1982), Mueller (1985), Hough et al., (1991), Abercrombie and Rice (2005), and Viegas et al. (2010) performed analysis of EGFs in the frequency domain to obtain source parameters of larger earthquakes and attenuation prosperities of the medium. Frankel and Kanamori (1983), Frankel et al. (1986), Mori and Frankel (1990), Mori et al. (2003) performed similar analysis in the time domain. Hough (1997), Hough et al. (1999), Prejean and Ellsworth (2001), Ide et al. (2003), Prieto et al. (2004), Shearer et al. (2006) analyzed multiple earthquakes simultaneously to obtain source properties and kappa (near site attenuation). Mayeda et al. (2007) and Viegas (2009) used source spectra derived from coda waves instead of direct waves to obtain source properities. Hartzell (1978) and Wu (1978) first suggested using EGFs to calculate strong ground motion. Using small earthquakes to provide EGF’s for synthesizing larger earthquakes is very practical; small earthquakes occur hundreds of times more frequently than larger earthquakes and EGFs can be readily obtained in a short period of time before a large earthquake occurs. Hartzell and Wu suggested using EGFs as the Green’s function in the representation relation along with synthetic rupture processes for calculating (synthesizing) the resulting ground motion. That is, the fault of the large earthquake is represented as a summation of subfaults, or elemental point sources, for which EGFs are available. Figure 1 dramatizes the wave propagation effects that occur from different portions of a fault, so that as an earthquake ruptures, EGFs account for the different travel paths through the heterogeneous geology. Hutchings (1991) and Hartzell et al. (1999) designed particular rupture models to represent what actual earthquake might do. This followed Boatwright (1981), who designed quasi-dynamic sources. Guatteri et al. (2003) further developed source modeling by including actual dynamic rupture process in the calculations. Irikura (1984) used a relatively large earthquake as an EGF and modified it to represent the strong ground motion from an even larger earthquake (usually 1 or 2 magnitude units higher). In this approach the EGF was not isolated as an impulsive point source, and the larger earthquake was made up of fairly large sub-events. Joyner and Boore (1986) examined the statistics of how to add up sub-events to create a larger earthquake. Frankel (1991) proposed a fractal summation of scaled EGFs to represent the source process as a statistical process. Tumarkin (1994) and Abrahamson and Bolt (1997) modified recordings of relatively large earthquakes to fit a target spectra of an even larger earthquake. Somerville et al. (1991) used the recordings near a large earthquake as an empirical source function and calculated the wave propagation effects. All these source models have different implications for the synthesis process, and are discussed below. We also examine validation procedures to verify the usefulness of methods. 5. Good-quality recordings of seismograms have to be captured at the locations of interest for ground-motion synthesis. In this chapter, we describe the theoretical and observational basis for identifying and using small earthquakes as EGFs and review some of the typically used methods in earthquake source and strong-ground-motion studies. We also discuss the advances in the scientific knowledge of earthquake sources made possible by EGFs. Finally, we discuss using the application of the EGF method (in strong-ground-motion prediction) to probabilistic seismic-hazard analysis. The representation relation (Aki and Richards, 2002) is the fundamental, elastodynamic, mathematical description of an earthquake and the resulting ground motion. It is expressed ...
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Citations
... The review article by Hutchings and Viegas (2012) gives an overview of the development and applications of the empirical Green's function method. Here we give a concise account of the EGF method in the form in which we have applied it to the Groningen data. ...
... To understand our results we have built simple kinematic source models which represent the larger event as a composite of multiple slip patches (such an approach is discussed by Hutchings and Viegas 2012). The motivation was to develop a conceptual framework for interpreting the field data, provide expressions for the RSTF duration as a function of station azimuth, and enable us to generate synthetic seismograms on which to validate the data processing. ...
We have applied the empirical Green’s function (EGF) method to 53 pairs of earthquakes, with magnitudes ranging from M = 0.4 to M = 3.4, induced by gas production from the Groningen field in the Netherlands. For a subset of the events processed, we find that the relative source time functions obtained by the EGF deconvolution show clear indications of a horizontal component of rupture propagation. The earthquake monitoring network used has dense azimuthal coverage for nearly all events such that wavelet duration times can be picked as a function of source-station azimuth and inverted using the usual Doppler broadening model to estimate rupture propagation strike, distance, and velocity. Average slip velocities have also been estimated and found to be in agreement with typical published values. We have used synthetic data, from both a simple convolutional model of the seismogram and more sophisticated finite difference rupture simulations, to validate our data processing workflow and develop kinematic models which can explain the observed characteristics of the field data. Using a measure based on the L1-norm to discriminate results of differing quality, we find that the highest quality results show very good alignment of the rupture propagation with directions of the detailed fault map, obtained from the full-field 3D seismic data. The dip direction rupture extents were estimated from the horizontal rupture propagation distances and catalogue magnitudes showing that, for all but the largest magnitude event (the M = 3.4 event of 8th January 2018), the dip-direction extent is sufficiently small to be contained wholly within the reservoir.
... The EGF approach is usually the only method available for retrieving the MRFs of small events (M w < 5). However, there are the risks of noise in the EGF, differences in path effects and focal mechanisms between target and EGF events (Hutchings and Viegas, 2012), and frequency-band limitations. Instability in deconvolution and the effects of reflected and/or converted waves owing to slight differences in locations and focal mechanisms can inadvertently make the MRFs appear more complex than they really are. ...
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... The general idea of the spectral ratio method (SR) is to use for each target earthquake a smaller event with similar location and focal mechanism as an empirical Green's function (EGF) (Hutchings & Viegas, 2012). By spectral division between target and EGF seismograms the path and site terms which are assumed to be identical are removed (cf. ...
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... Spectral ratio approaches employ empirical Green's functions (EGFs) to eliminate path and site terms in the observed seismogram spectrum and to isolate the event source term (e.g. Hutchings & Viegas, 2012). While this makes them more robust against systematic errors introduced by over-or under-correcting for attenuation structures and radiation pattern, they require the existence of a suitable nearby EGF event which, in practise, is a rather strong limitation. ...
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... It should be mentioned that the effects of liquefaction and soil nonlinearity caused by shear modulus or damping ratio are largely not included in the EGFs. These effects need to be added within a post-processing step (Aki and Richards 2002;Hutchings and Viegas 2012). Different approaches have been developed in the time or frequency domain to account for soil nonlinearity (e.g., Hashash et al. 2010;Seyhan and Stewart 2014). ...
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... Mayorga (2010), Courboulex et al. (2010) and Comino et al. (2012) have used smaller events as empirical Green's functions to simulate strong ground motion, and in some cases historical earthquakes occurred in the analyzed zones. However, Hutchings & Viegas (2012) analyzed the main limitations of the use of smaller earthquakes as Green's empirical functions, among them: in virtually all practical applications, there is insufficient number of EGFs to provide, an impulse response for all portions of a fault rupture to be modeled; empirical Green´s functions (EGFs) cannot accurately model variations in focal mechanism solutions and a distribution of stations with good azimuthal coverage is necessary to ascertain the locations and source parameters of the small earthquakes that would provide EGFs. ...
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... Expected values of the pseudo-velocity and displacement spectra using the LS7 scenario and rise-time in the range t r = 2-10 s at the Tocopilla station in the NS-and EW-direction are shown in Fig. 11. Because this methodology does not capture the high frequencies accurately (Hutchings and Viegas 2012), the spectra are analyzed only for fundamental system periods T ≥ 1 s. The figure shows that the effect of the rise-time on the spectral response values is significant and that as slip rise-time increases, spectral responses decrease. ...
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Three other rupture models based on previous studies of interplate locking for north Chile and capable of generating Mw\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_w$$\end{document} 8.3–8.6 earthquakes with an estimated maximum slip of 9.2 m, were incorporated in the analyses. Also, four additional scenarios with moment magnitudes in the range Mw\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_w$$\end{document} 8.6–8.9 were generated by concatenating these physical scenarios into larger rupture areas within the north segment. Using these scenarios, synthetic ground motions were built at four observation sites: Pisagua, Iquique, Tocopilla, and Calama. Response sensitivity was studied for three key rupture parameters: mean rupture velocity, slip rise-time, and rupture directivity. Responses selected were peak ground displacement (PGD), spectral pseudo-velocities, Sv\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_v$$\end{document}, and spectral displacements, Sd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_d$$\end{document}. First and second order variations of PGD, Sv\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_v$$\end{document}, and Sd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_d$$\end{document} relative to the source parameters were computed and used together with a Taylor series expansion to propagate uncertainty into the responses as a function of vr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_r$$\end{document} and rise-time tr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_r$$\end{document}. To study the effect of rupture directivity, three different foci locations were considered for each scenario: north, south, and at the centroid of the slip model. Response PGD values show no clear trends with rupture velocity, vr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_r$$\end{document}; however, the variability increases as the system period increases. The effect of the slip rise-time is significant, and as tr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_r$$\end{document} increases, the spectral responses tend to decrease, suggesting that shorter slip rise-times lead to higher seismic demands in long period structures. The results obtained for the directivity analysis suggest that two factors control the expected waveforms and spectral responses: first, the direction of the rupture relative to the location of each site, and the hypocentral distance.
... Mayorga (2010), Courboulex et al. (2010) and Comino et al. (2012) have used smaller events as empirical Green's functions to simulate strong ground motion, and in some cases historical earthquakes occurred in the analyzed zones. However, Hutchings & Viegas (2012) analyzed the main limitations of the use of smaller earthquakes as Green's empirical functions, among them: in virtually all practical applications, there is insufficient number of EGFs to provide, an impulse response for all portions of a fault rupture to be modeled; empirical Green´s functions (EGFs) cannot accurately model variations in focal mechanism solutions and a distribution of stations with good azimuthal coverage is necessary to ascertain the locations and source parameters of the small earthquakes that would provide EGFs. ...
A procedure to obtain a suitable set of accelerograms for vulnerability and risk assessment of buildings and bridges in Cuba is presented. Scaled signals of small or moderate earthquakes in the Oriente fault system area were analyzed. This area was chosen because it is the one where the strongest earthquakes of the southeastern region of the country take place. Signals obtained and their parameters for the engineering analysis are shown (peak ground acceleration, velocity and displacement, Arias Intensity and Housner Intensity), the latter two parameters are related to the earthquake intensity. The maximum amplitudes of the new signals are in period intervals of 0.10 sec. - 0.60 sec. These values are consistent with the periods manifested by the most common structures in the city of Santiago de Cuba. Consequently, by performing linear and non-linear dynamic analysis based on these accelerograms, it is assessed a seismic action that, because of its frequency, influences the appearance of damages in such structures.
