Empirical Green’s Function Research Articles

Improving ambient noise crosscorrelations using a polarization-based azimuth filter

Seismic interferometry is widely used to image and monitor earth structures at different scales by extracting an empirical Green’s function (EGF) from the crosscorrelation of ambient noise under the assumption of diffuse wavefields or energy equipartitioning. However, EGFs may be incorrectly estimated and lead to a misunderstanding of subsurface structures because the distribution of ambient noise sources is neither isotropic nor stationary. To extract reliable EGFs from the nonideal ambient noise data, we develop a polarization-based azimuth filter (PAF) to attenuate the ambient noise energy of nonstationary sources to improve the quality of crosscorrelation functions. The PAF is a time-frequency domain azimuth filter for 3C seismic data, which uses time-frequency polarization analysis to measure the azimuths of ambient noise signals and then filters the 3C data to obtain the signals from desired directions. Based on the stationary-phase theory, we use the PAF to extract the ambient noise signals from stationary-phase zones, which improves the quality of the crosscorrelation function and eliminates the requirement for long observations. A synthetic test and two short-time engineering examples demonstrate that uneven source distributions might cause serious spurious signals in EGFs, and we demonstrate the improvement in EGFs using our method. In addition, the PAF may allow for the complete separation of the fundamental and higher modes of Rayleigh waves, which uncovers more information implicit in the ambient noise data.

CSRM‐1.0: A China Seismological Reference Model

AbstractHigh‐resolution seismic model is crucial for advancing our understandings on geological processes and enhancing seismic hazard mitigation programs. We construct a high‐resolution China Seismological Reference Model (CSRM‐1.0) in the top 100 km of the crust and uppermost mantle in continental China following a top‐down construction process. The employed seismic constraints include P‐wave polarization angle from tele‐seismic event, short‐period Rayleigh wave ellipticity from ambient noise, long‐period Rayleigh wave ellipticity from earthquake data, receiver function, empirical Green's function from ambient noise, Rayleigh wave phase/group velocity dispersion curves from regional earthquakes, and Pn‐wave travel time extracted from seismic data of 4,435 stations. CSRM‐1.0 has a spatial crustal resolution of ∼60 km beneath the north‐south seismic belt and trans‐North China orogen regions and ∼120 km beneath the rest of continental China, and a spatial mantle resolution of ∼300 km. CSRM‐1.0 exhibits prominent velocity heterogeneities in the crust and uppermost mantle and an eastward thinning of the crust, geographically correlating with geological settings. CSRM‐1.0 improvements include accurate estimation of shallow seismic structure, increased spatial resolution and improved model accuracy. Crustal composition inferred from CSRM‐1.0 exhibits a general transition from a felsic upper crust to a mafic lower crust. Mafic rocks in the lower crust are found predominantly along inter‐block boundaries and sporadically within the interiors of blocks, likely resulted from preferential inter‐block intrusions of magmas related to various oceanic plate subductions and the Emeishan mantle plume. This study contributes seismic constraints and CSRM‐1.0 to the CSRM product center (http://chinageorefmodel.org) as a backbone open‐access geophysical cyberinfrastructure.

Frequency‐Bessel Transform Method for Multimodal Dispersion Measurement of Surface Waves From Distributed Acoustic Sensing Data

AbstractThe array‐based frequency‐Bessel transform method has been demonstrated to effectively extract dispersion curves of higher‐mode surface waves from the empirical Green's functions (EGFs) of displacement fields reconstructed by ambient noise interferometry. Distributed acoustic sensing (DAS), a novel dense array observation technique, has been widely implemented in surface wave imaging to estimate subsurface velocity structure in practice. However, there is still no clear understanding in theory about how to accurately extract surface‐wave dispersion curves directly from DAS strain (or strain rate) data. To address this, we extend the frequency‐Bessel transform method by deriving Green’s functions (GFs) for horizontal strain fields, making it applicable to DAS data. First, we test its performance using synthetic GFs and verify the correctness of extracted dispersion spectrograms with theoretical results. Then, we apply it to three field DAS ambient‐noise data sets, two recorded on land and one in the seabed. The reliability and advantages of the method are confirmed by comparing results with the widely used phase shift method. The results demonstrate that our extended frequency‐Bessel transform method is reliable and can provide more abundant and higher‐quality dispersion information of surface waves. Moreover, our method is also adaptable for active‐source DAS data with simple modifications to the derived transform formulas. We also find that the gauge length in the DAS system significantly impacts the polarity and value of extracted dispersion energy. Overall, our study provides a theoretical framework and practical tool for multimodal surface wave dispersion measurement using DAS data.

An EGF technique to infer the source parameters of a circular crack growing at a variable rupture velocity

Circular crack models with a constant rupture velocity struggle to effectively model both the amplitude and duration of first P-wave pulses generated by small magnitude seismic events. Assuming a constant rupture velocity is unphysical, necessitating a deceleration phase in the rupture velocity to uphold the causality of the healing process. Moreover, a comprehensive failure model might encompass an initial nucleation phase, typically characterized by an increase of the initial rupture velocity. Studies have demonstrated that quasi-dynamic circular crack models featuring variable rupture velocities can accurately model the shape of the observed first P-wave pulse. Based on these principles, an Empirical Green’s function (EGF) approach was previously formulated to estimate the source parameters of small magnitude earthquakes, called MAIN. In addition to determine the source radius and stress drop, this method also enables the inference of the temporal evolution of rupture velocity. However, this method encounters difficulties when the noise-to-signal ratio in the recordings of smaller earthquakes used as EGF exceeds 5%, a common situation when employing regional-scale recordings of small-magnitude earthquakes as EGF. Through synthetic tests, we demonstrated that, in such instances, the problem of this technique is that the alignment between the onset of P waves of EGF and MAIN is not rightly recovered after the initial inversion step. Consequently, a novel inversion method has been developed to address this issue, enabling the identification of the optimal alignment of P-wave arrivals in EGF and MAIN across all stations. A Bayesian statistical approach is proposed to meticulously investigate the solutions of model parameters and their correlations. Using the new technique on a small magnitude earthquake (ML = 3.3) occurred in Central Italy enabled us to identify the most likely rupture models and examine the issue of correlation among model parameters. Application of Occam’s Razor Principle suggests that, for the investigated event, a circular crack model should be favored over a heterogeneous rupture model.

Azimuthal crustal variations and their implications on the seismic impulse response in the Valley of Mexico

AbstractPrevious studies have suggested prominent variations in the seismic wave behavior at the 5 s period when traveling across the Valley of Mexico, associating them with the crustal structure and contributing to the anomalous seismic wave patterns observed each time an earthquake hits Mexico City. This article confirms the variations observed at 0.2 Hz by analyzing the Green tensor diagonal retrieved from empirical Green functions (EGF) calculated using seismic noise data cross-correlations of the vertical and horizontal components. We observe time and phase shifts between the east and north EGF components and show that they can be explained by the crustal structure from the surface up to 20 km depth; we also observe that the time and phase shifts are more significant if the distance between the source and the station increases. Additionally, the article presents an updated version of the velocity model from receiver functions and dispersion curves (VMRFDC v2.0) for the crustal structure under the Valley of Mexico. To validate this model, we compare the EGFs with synthetic Green functions determined numerically. To do so, we adaptatively meshed this model using an iterative algorithm to numerically simulate the impulse response up to 0.5 Hz. Finally, the comparisons between noise and synthetic EGF showed that the VMRFDC v2.0 model explains the time shifts and phase differences at 0.2 Hz, previously observed by independent studies, suggesting it correctly characterizes the crustal structure anomalies beneath the Valley of Mexico.

Application of empirical green function method for synthesis of response spectrum compatible time histories

Application of empirical green function method for synthesis of response spectrum compatible time histories

Seismic Imaging of Shallow Depths using Ambient Noise Tomography in the Hyderabad Region, Eastern Dharwar Craton, India

ABSTRACT Hyderabad and its surrounding region are situated within Eastern Dharwar Craton, primarily composed of Archean granite. We use ambient noise tomography to obtain the 3-D shallow subsurface shear seismic velocity structure of 150 x 200 km2 area around Hyderabad. We used continuous waveforms that were recorded by a digital seismic network with 10 three-component broadband seismographs from November 2020 to December 2021. Using the vertical component of waveforms, the Empirical Green’s functions (EGFs) have been estimated, and the group velocity dispersion curves in the time period range of 1.0–10 s have been extracted from these. We inverted these dispersion curves to estimate 3-D variation of shear wave velocity by using direct surface wave tomographic method. The results suggest that the subsurface velocity structure is heterogeneous. The shear wave velocity varies from 3.4 to 3.7 km/s in the shallow subsurface. The limited seismicity in this area is mostly confined to the boundary between high and low-velocity changes.

Ambient Seismic Noise Tomography Within the Floridan Aquifer System, Santa Fe River‐Sink Rise, Florida, US

AbstractThis study utilized ambient seismic noise tomography to investigate correlations of hydrogeological systems with heterogeneous porosity with the internal velocity structure of karst aquifers using the Floridan Aquifer System (FAS) as an example. We conducted a seismic signal analysis with 30 days of vertical component seismic data acquired within the Santa Fe River Sink‐Rise system (Sink‐Rise System), using power spectral density estimates and frequency‐wavenumber analysis to identify any noise source biases. Empirical Green's functions were derived through phase cross‐correlations and phase‐weighted stacking. The functions allowed extraction of group velocity dispersion measurements, which were inverted to generate 2D tomographic images at various periods. Tomographic images show multiple layers with varying group velocities that aligned with known hydrogeological features of the FAS, including its base, low‐porosity and low‐permeability dolostone semiconfining units characterized by elevated velocities, and zones of elevated porosity from karstification with reduced seismic velocities. Our results show how ambient seismic noise can be used to evaluate aquifer physical properties, identify unknown features, and identify the location and continuity of aquifer units.

Empirical Green’s function analysis of some induced earthquake pairs from the Groningen gas field

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.

A Source and Ground Motion Study of the Veracruz Earthquakes of 29 October 2009 (Mw5.7) and 4 August 2021 (Mw4.8): Evidence of Strong Azimuthal Variation of Attenuation

We study two moderate earthquakes that occurred offshore the State of Veracruz, in the southwestern Gulf of Mexico, on 29 October 2009 (Mw 5.7) and 4 August 2021 (Mw 4.8). The former was located near the town of Alvarado and latter near the city of Veracruz. The events were well recorded by accelerographs and seismographs at local and regional distances. W-phase regional centroid moment tensor inversion reveals that they had reverse-faulting mechanism, similar to several other earthquakes in the southwestern Gulf of Mexico. Of the seven focal mechanisms now available along the southwestern margin, two are strike slip and the rest are of thrust type, suggesting a heterogeneous stress regime. We take advantage of local and regional recordings produced by these two earthquakes to study the characteristics of the ground motion. Source spectra computed at each station separately (without correcting for the site effect), assuming a reasonable geometrical spreading and Q = 141f 0.63, show remarkably high variability due to difference in path and local site effects. The geometric mean apparent source spectrum (source spectrum including site effects) of both earthquakes may be modeled by an ω2 -Brune source model with a stress drop, Δσ, of 40 MPa. These source spectra, along with the application of stochastic method, yield peak ground acceleration (PGA) and velocity (PGV) as a function of distance in general agreement with the observations. Of greater practical importance is the ground motion at sedimentary sites in the city of Veracruz and at the Laguna Verde Nuclear Power Plant (LVNPP) site, especially from a postulated Mw 6.5 earthquake which is a reasonable scenario event for the region. For the city of Veracruz and LVNPP we estimate site effect with respect to the ω2 -Brune source with Δσ of 2 MPa. There is some support for this Δσ. We apply both stochastic and empirical Green’s functions (EGF) techniques in the estimation of the ground motion. The recording of the 2021 earthquake is taken as the EGF, and Δσ of the EGF and the target event are assumed to be the same and equal to 2 MPa. The predicted PGA and PGV at sedimentary sites in the city of Veracruz and LVNPP above the hypocenter (depth = 20 km) from the postulated Mw 6.5 earthquake are 0.2 g and 10 cm/s and 0.18 g and 3 cm/s, respectively. These results are preliminary as they are based on several assumptions.

Rayleigh wave attenuation tomography based on ambient noise interferometry: methods and an application to Northeast China

SUMMARY Although ambient noise interferometry has been extensively utilized for seismic velocity tomography, its application in retrieving attenuation remains limited. This study presents a comprehensive workflow for extracting Rayleigh wave amplitude and attenuation from ambient noise, which consists of three phases: (1) retrieval of empirical Green's functions (EGFs), (2) selection and correction of amplitude measurements and (3) inversion of attenuation, site amplification and noise intensity terms. Throughout these processes, an ‘asynchronous’ temporal flattening method is used to generate high-quality EGFs while preserving relative amplitudes between stations. Additionally, a novel ‘t-symmetry’ criterion is proposed for data selection along with the signal-to-noise ratio. Furthermore, 2-D sensitivity kernels are utilized to estimate the focusing/defocusing effect, which is then corrected in amplitude measurements. These procedures are designed to deliver reliable attenuation measurements while maintaining flexibility and automation. To validate the effectiveness of the proposed noise-based attenuation tomography approach, we apply it to a linear array, NCISP-6, located in NE China. The obtained results correlate reasonably well with known geological structures. Specifically, at short periods, high attenuation anomalies delineate the location of major sedimentary basins and faults; while at longer periods, a notable rapid increase of attenuation is observed beneath the Moho discontinuity. Given that attenuation measurements are more sensitive to porosity, defect concentration, temperature, melt and volatile ratio than seismic velocities, noise-based attenuation tomography provides important additional constraints for exploring the crustal and upper mantle structures.

Radioactive source localization employing resistive electrode array (REA) detector.

Objective.In this feasibility study, we explore an application of a Resistive Electrode Array (REA) for localization of a radioactive point source. The inverse problem posed by multichannel REA detection is studied from mathematical perspective and involves the questions of the minimal configuration of the conductive leads that can achieve this goal. The basic configuration consists of a circularly shaped REA with four opposite electrical lead-pairs at its perimeter.Approach.A robust mathematical reconstruction method for a 3D radioactive source relative to the REA is presented. The characteristic empirical Green's function for the detector response of the REA is determined by numerically solving Laplace equations with appropriate boundary conditions. Based on this model, Monte Carlo simulations of the inverse problem with Gaussian noise are performed and the overall accuracy of the localization is investigated.Main results.The results show a 3D error distribution of localization which is uniform in the (x,y)-plane of the REA and strongly correlated in the orthogonalz-axis. The overall accuracy decreases with higher distance of the source to the detector which is intuitive due to approximate flux dependence following the inverse square law. Further, a saturation in accuracy regarding the number of electrical leads and a linear dependence of the reconstruction error on the measurement noise level are observed.Significance.A broad range of REA detector configurations and their characteristics are investigated by this study for radioactive source localization allowing diverse practical applications with detector diameters ranging from millimeters to meters.

Accurate Magnitude and Stress Drop Using the Spectral Ratios Method Applied to Distributed Acoustic Sensing

AbstractThe reliable estimation of earthquake magnitude and stress drop are key in seismology. The novel technology of distributed acoustic sensing (DAS) holds great promise for source parameter inversion owing to the measurements' high spatial density. In this study, I demonstrate the robustness of DAS for magnitude and stress drop estimation using the empirical Green's function deconvolution method. This method is applied to nine co‐located earthquakes recorded in Israel following the 2023 Turkey earthquakes. Spectral ratios were stacked along the fiber, and fitted with a relative Boatwright source spectral model. Excellent fits were obtained even for similar sized earthquakes. Stable seismic moments and stress drops were calculated assuming that the moment of one earthquake is known. DAS derived estimates were found to be more stable and reliable than those obtained using a dense accelerometer network. The results demonstrate the great potential of DAS for source studies.

Review of source models for reproducing and predicting strong motions by empirical Green's function method

AbstractShort‐period ground motions from earthquakes are calculated by semiempirical methods such as the empirical Green's function method or the stochastic Green's function method. In this review paper, I summarized source models developed for reproducing and predicting ground motions in the theoretical methods at first, and then reviewed the source models applied to reproducing and predicting ground motions in the empirical Green's function method in these about 40 years. Finally, I showed some issues to be solved about source modeling in order to predict accurate short‐period motions in the near‐fault regions.

Broadband source model of the 2021 MW 7.1 Fukushima‐ken Oki earthquake in Japan based on seafloor and onshore strong‐motion records

AbstractThe 2021 MW 7.1 Fukushima‐ken Oki earthquake was the largest down‐dip compressional intraslab earthquake since August 2016, which is when S‐net records from the seafloor started to be provided. In this study, the empirical Green's function method was used with strong‐motion records from S‐net on the seafloor and KiK‐net onshore to estimate the broadband (0.2–10 Hz) source model of this earthquake. Three strong‐motion generation areas (SMGAs) with very high stress drops were estimated around the edge of the fault. One SMGA with a stress drop of 125 MPa was located in the oceanic crust in the shallow part of the Pacific Plate. The other two SMGAs with stress drops of 313 and 188 MPa were located in the deep part where previous source models without S‐net estimated a small slip. The two deep SMGAs increased the contribution of S‐net to more than that of KiK‐net. The short‐period spectral level was as high as that of the 2011 MW 7.1 Miyagi‐ken Oki earthquake, which was also a down‐dip compressional intraslab earthquake. The estimated broadband source model simulated the horizontal and vertical motions well at stations within 200 km of the hypocenter, except for surface waves at a few S‐net stations.

A Comprehensive Stress Drop Map From Trench to Depth in the Northern Chilean Subduction Zone

AbstractWe compute stress drops for earthquakes in Northern Chile recorded between 2007 and 2021. By applying two analysis techniques, (a) the spectral ratio (SR) method and (b) the spectral decomposition (SDC) method, a stress drop map for the subduction zone consisting of 51,510 stress drop values is produced. We build an extended set of empirical Green’s functions (EGF) for the SR method by systematic template matching. Outputs are used to compare with results from the SDC approach, where we apply cell‐wise obtained global EGF's to compensate for the structural heterogeneity of the subduction zone. We find a good consistency of results of the two methods. The increased spatial coverage and quantity of stress drop estimates from the SDC method facilitate a consistent stress drop mapping of the different seismotectonic domains. Albeit only small differences of median stress drop, strike‐perpendicular depth sections clearly reveal systematic variations, with earthquakes at different seismotectonic locations exhibiting distinct values. In particular, interface seismicity is characterized by the lowest observed median value, whereas upper plate earthquakes show noticeably higher stress drop values. Intermediate depth earthquakes show comparatively high average stress drop and a rather strong depth‐dependent increase of median stress drop. Additionally, we observe spatio‐temporal variability of stress drops related to the occurrence of the two megathrust earthquakes in the study region. The presented study is the first coherent large scale 3D stress drop mapping of the Northern Chilean subduction zone. It provides an important component for further detailed analysis of the physics of earthquake ruptures.

High-resolution seismic velocity structure using induced inter-event interferometry at the geothermal sites, Geysers, Eastern California

The possibility of retrieving Rayleigh wave empirical Green's functions (EGFs) using virtual seismometers provides an opportunity to calculate high-resolution velocity models without a set of dense stations and/or expensive seismic imaging/monitoring. The high-resolution tomographic map can give an overview of the fracturing within the saturated rock. Theoretically, the cross-correlations of earthquakes contain significant information about the EGFs between event-pairs. In this work, we developed an application of virtual seismometers on a very small scale in superficial layers where anthropogenic seismicity occurred. To test this method's capabilities, we applied it to investigate the NW-part of The Geysers (TG) geothermal field. We used 1276 micro-earthquakes, occurred in a cuboid with the horizontal side of ∼1 × ∼2 km and the vertical edge at a depth from 0.9 to 2.2 km, to obtain the high-resolution (100 × 100 m) velocity structure. A group of strict criteria was imposed separately for single event waveform and event pairs to stabilize the tomographic results. After retrieving the inter-event EGF signal and calculating the Rayleigh wave group velocity dispersion curves, we obtained the group velocity maps for each horizontal layer. The tomographic maps as a function of depth indicated a low-velocity anomaly related to existing inactive faults, topography variation of the top of the steam zone, and low-velocity anomalies possibly stimulated by fluid injection. The topography of the top of the steam varies from ∼1.0 km to 1.3 km of maps, and also anomalies connected with fluid migration appear at the depth of injection wells (1.5–1.9 km).

Ambient seismic noise tomography of the Suwannee suture zone using cross-coherence interferometry and double beamforming

SUMMARY We use continuous data from more than 200 vertical-component broad-band seismic stations for the years 2012 and 2013 to model the crustal and lithospheric structure of the southeastern United States (SEUS). Seismic interferometry via cross-coherence is used to retrieve the surface wave Empirical Green's function (EGF) between station pairs. We mitigate the problem of non-stationary sources contributing to the extracted EGF by applying double beamforming to pairs of subarrays of stations. The recovered Rayleigh waves are used to compute group velocity dispersion maps which are then inverted to find shear wave velocity profiles using a Markov Chain Monte Carlo technique. EGFs are computed for more than 80 000 station pairs after filtering to the period band 7–60 s. Results reveal tectonic features in the SEUS that support recent claims that the northernmost extent of the Suwannee suture is located near the boundary of the Carolinia and Charleston terranes and that the Charleston and Suwannee terranes are not seismologically distinct within the resolution limits of the data set.

Recognition of the causative fault of the 2017 MW 4.9 Malard (Tehran, Iran) earthquake from directivity analysis of the recorded ground motions

On December 20, 2017, a light shallow earthquake (Mw 4.9) with a purely strike-slip mechanism occurred on a hidden unknown fault, 30 km west of the city of Tehran. The purpose of this study is to determine the causative fault plane based on the directivity effect that was observed during this shallow crustal depth earthquake, referred to as the Malard earthquake. This was achieved by using data from 96 seismic and accelerometer stations, and employing a variety of methods.For the analysis of the directivity effect, we employed methods including empirical Green's function deconvolution in the frequency domain, inversion of corrected ground motions based on empirical models, and comparison of relative peak ground acceleration and velocity between the mainshock and the largest aftershock. All approaches indicate that the Malard earthquake occurred on a previously unknown fault with a strike of 71°, a steep southward dip, and a confirmed left-lateral strike-slip mechanism, as evidenced by the analysis of aftershock distribution. The earthquake exhibited a strong directivity effect, with rupture propagating unilaterally from the hypocenter to the southwest at a velocity of 2.5 km/s.In addition to enhancing our understanding of active earthquake sources in the densely populated areas of Iran, the findings of this study will also contribute to future earthquake risk assessments in the Tehran region. However, it's important to note that Tehran was situated in the backward zone of the Malard earthquake rupture, where the least ground motion was recorded. Future earthquakes on this fault may exhibit different propagation patterns, toward the city. The determination of a preferred propagation direction can only be achieved probabilistically through the study of a substantial number of earthquakes in the region. The absence of recorded ground motion data will lead to significant uncertainty in predicting the propagation direction of future earthquakes, which must be taken into consideration during hazard analysis.

Estimation of Uniform Risk Spectra Suitable for the Seismic Design of Structures

The aim of this paper is to present a performance-based method to estimate uniform risk spectra (URS) for the seismic design and assessment of structures. These spectra, computed with the proposed methodology, provide the lateral capacity (in terms of spectral acceleration) that should be given to a structure, characterized by a reference single degree of freedom system, to achieve a predetermined exceedance rate of economic loss. This procedure involves the seismic hazard assessment necessary to define a seismic design level consistent with the accepted loss value, using a large enough number of synthetic seismic records of several magnitudes, which were obtained by means of an improved empirical Green function method. The statistics of the expected losses of a reference single degree of freedom system are obtained using Monte Carlo simulation, considering the seismic demand and the lateral strength of the structure as random variables. The method is divided into two main stages: (1) definition of the seismic hazard at the site of interest and (2) the probabilistic analysis of the seismic performance in terms of an economical loss ratio of nonlinear SDOF. To illustrate the proposed methodology and, subsequently, to validate it, a URS was computed for a site located in the Mexico City lake-bed zone, and its use in the design of three reinforced concrete frames is shown. The results show that the proposed spectra provide a sufficient approximation between the seismic risk level considered in the seismic design and that of the designed structure. It is concluded that the proposed procedure is a significant improvement over others considered in the literature and a useful research tool for the further development of risk-based earthquake engineering.