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Seismic moment

The seismic moment is a fundamental physical quantity used by seismologists to quantify the size of an earthquake, defined as the product of the shear modulus (rigidity) of the surrounding rock, the area of the fault surface that ruptures, and the average displacement (slip) along that fault.^[1] Mathematically expressed as M_0 = \mu A D, where \mu is the shear modulus (typically around 30 GPa for crustal rocks), A is the rupture area, and D is the average slip, it represents the total inelastic deformation associated with the earthquake source and is measured in units of Newton-meters (N·m).^[1] This scalar value provides a direct measure of the earthquake's potency, independent of distance or local geology, making it superior to earlier magnitude scales like the Richter scale for assessing large events.^[2] Introduced in modern form by Keiiti Aki in 1966 through analysis of seismic wave spectra, the concept of seismic moment evolved from earlier ideas of earthquake mechanics and became central to earthquake science.^[3] In 1979, Thomas C. Hanks and Hiroo Kanamori formalized its use in the moment magnitude scale (M_w), defined as M_w = \frac{2}{3} \log_{10} M_0 - 10.7 (for M_0 in dyne·cm) or equivalently M_w = \frac{2}{3} \log_{10} M_0 - 6.0 (for N·m), which has since replaced other scales as the standard for global earthquake reporting due to its consistency across all sizes and depths.^[4] Seismic moment is typically estimated from the low-frequency amplitudes of seismic waves via moment tensor inversion or from geodetic observations like GPS and InSAR, allowing for rapid assessment during and after events.^[5] For example, the 1960 Chile earthquake had a seismic moment of approximately $2.5 \times 10^{23} N·m, corresponding to M_w 9.5, illustrating its role in characterizing the largest recorded seismic releases.^[6] Unlike energy-based measures, seismic moment emphasizes the fault mechanics, aiding in hazard modeling, source physics studies, and comparisons of tectonic strain accumulation over time.^[7]

Definition and Physical Interpretation

Basic Concept

Seismic moment serves as a fundamental measure of an earthquake's size, quantifying the overall strength of the seismic source in terms of the total inelastic deformation during fault rupture. Physically, it represents the product of the Earth's crustal rigidity, the surface area of the fault that slips, and the average displacement along that fault, providing a direct indicator of the scale of deformation at depth. This approach captures the intrinsic "strength" of the earthquake independent of observational biases in wave propagation.^[2] In contrast to intensity scales, which assess the local effects of ground shaking and vary by location, seismic moment focuses exclusively on source characteristics such as fault dimensions and slip, offering a consistent, physics-based evaluation unaffected by distance or site conditions. A striking example is the 1960 Valdivia earthquake in Chile, the largest instrumentally recorded event, which released an immense seismic moment equivalent to a rupture spanning approximately 1,000 kilometers along the subduction zone with average slips of 20-40 meters, underscoring how this measure reveals the vast scale of megathrust faulting.^[8]

Key Physical Components

The shear modulus, denoted as μ, represents the rock's resistance to shear deformation, quantifying the stiffness of the material under applied shear stress.^[9] In the Earth's crust, typical values for μ range around 30 GPa for upper crustal rocks, such as granitic or sedimentary formations.^[9] This parameter varies significantly with depth, increasing from approximately 20–30 GPa in the shallow crust to 40–60 GPa or higher in the lower crust due to elevated pressures and temperatures that enhance rock rigidity.^[10] Additionally, μ differs by rock type; for instance, more competent igneous rocks like basalt exhibit higher values than fractured or porous sedimentary rocks.^[11] The fault area, A, refers to the surface area over which the earthquake rupture propagates, encompassing both the length along the fault strike and the down-dip width.^[12] This area is influenced by tectonic setting, with plate boundary faults—such as those in subduction zones—often supporting larger ruptures spanning hundreds of kilometers due to sustained stress accumulation along extensive plate interfaces.^[12] In contrast, intraplate faults, occurring within stable continental interiors, typically involve smaller areas, limited by localized stress concentrations and the absence of prolonged plate motion, resulting in ruptures rarely exceeding tens of kilometers.^[12] Average slip, D, describes the mean displacement across the fault plane during rupture, representing the relative movement of rock masses on either side.^[13] Measuring D poses challenges due to its spatial heterogeneity—slip often varies patchily along the fault, with maximum values at the hypocenter and tapering elsewhere—requiring integration of field observations, geodetic data, and seismic inversions for reliable averages.^[14] This parameter relates to stress drop, the difference in shear stress before and after rupture, as higher slips generally correspond to greater stress releases, though actual values depend on fault friction and rock properties.^[13] These components—shear modulus, fault area, and average slip—interact to quantify the total inelastic deformation associated with the earthquake, forming the integrated product that defines the seismic moment.^[15] For example, a large fault area with modest slip, as in some subduction zone events, can yield substantial seismic potency comparable to a smaller area with extreme slip, such as in intraplate ruptures, highlighting how variations in one component can compensate for others to influence overall seismic potency.^[12]

Mathematical Formulation

Scalar Seismic Moment

The scalar seismic moment M_0 provides a fundamental measure of an earthquake's physical size by quantifying the irreversible deformation associated with fault slip in the Earth's crust. It originates from the elastic rebound theory, which describes how tectonic forces gradually deform rocks elastically across a fault until accumulated strain energy is abruptly released through rupture, generating seismic waves. This theory posits that the potential energy stored prior to rupture is proportional to the shear stress times the displacement, leading to a source strength metric that scales with the fault's geometric and material properties. The formula for the scalar seismic moment derives from the representation theorem in elastodynamics, where the seismic source is modeled as a distribution of double-couple forces equivalent to fault slip in an elastic medium. For a simple finite fault, integrating the moment density over the rupture surface yields the scalar form M_0 = \mu A D, where \mu is the shear modulus of the surrounding rock (typically $2 \times 10^{10} to $4 \times 10^{10} Pa for crustal rocks, representing resistance to shear deformation in N/m² or dyne/cm²), A is the area of the fault surface that ruptures (in m² or cm²), and D is the average relative displacement (slip) across the fault (in m or cm). This expression captures the total "strength" of the source as the product of rigidity, rupture scale, and slip amplitude, directly linking to the released strain energy under the assumption of quasi-static rebound. In the International System of Units (SI), M_0 is expressed in newton-meters (N·m), which has the dimension of energy (equivalent to joules), reflecting the physical work done during slip. Historically, the centimeter-gram-second (CGS) system used dyne-centimeters (dyne·cm), with a conversion factor of $1 N·m = 10^7 dyne·cm to accommodate smaller computational scales in early seismology. Given the enormous values for major earthquakes (often exceeding $10^{18} N·m), M_0 is frequently handled logarithmically in analyses, though the scalar itself remains a linear measure of source potency. The scalar approximation relies on several simplifying assumptions: the medium is linearly elastic and isotropic (no directional variations in material properties), the slip is uniform across a planar fault surface (neglecting irregular rupture geometries or variable slip patches), and the fault is embedded in an infinite homogeneous half-space. These idealizations facilitate quantification but may introduce errors for complex events. For example, consider a hypothetical moderate earthquake with \mu = 3 \times 10^{10} Pa, A = 100 km² (= 10^8 m²), and D = 1 m. Substituting into the formula gives M_0 = (3 \times 10^{10}) \times (10^8) \times 1 = 3 \times 10^{18} N·m (or $3 \times 10^{25} dyne·cm), illustrating the scale for a typical crustal event.

Moment Tensor

The seismic moment tensor, denoted as M_{ij}, is a 3×3 symmetric matrix that encapsulates the directional characteristics of an earthquake source by representing it as a superposition of equivalent force couples acting on the fault plane.^[16] This tensor has six independent components due to its symmetry: the three diagonal elements (M_{11}, M_{22}, M_{33}) correspond to potential volume changes like expansions or contractions, while the three off-diagonal elements (M_{12} = M_{21}, M_{13} = M_{31}, M_{23} = M_{32}) describe shear deformations, including strike-slip and dip-slip motions.^[17] The formalism was introduced by Gilbert in 1970 to model the excitation of Earth's normal modes by seismic sources. The scalar seismic moment M_0, which quantifies the overall size of the earthquake, is derived from the moment tensor using its Frobenius norm:

M_0 = \sqrt{\frac{1}{2} \sum_{i,j=1}^3 M_{ij}^2},

where the factor of $1/\sqrt{2} normalizes for the double-couple component, ensuring consistency with the physical potency of the source.^[18] This relation treats M_0 as a scalar summary of the tensor's magnitude, independent of its orientation.^[19] In most tectonic earthquakes, the moment tensor is constrained to a double-couple model, which represents pure shear faulting across a planar dislocation without any isotropic (explosive or implosive) component that would imply volume change.^[20] The double-couple corresponds to the traceless deviatoric part of the tensor, where the sum of the diagonal elements is zero, modeling the opposing forces from two perpendicular pairs of tractions on the fault—equivalent to the relative slip of rigid blocks along a shear plane.^[21] This excludes non-shear sources like volcanic explosions, focusing on faulting mechanisms dominant in natural seismicity.^[22] Focal mechanisms, derived from the eigenvalues and eigenvectors of the moment tensor via inversion of seismic waveforms, are commonly visualized as stereographic "beach ball" projections on a lower-hemisphere plot.^[20] These diagrams delineate compressional (shaded) and dilatational (unshaded) quadrants based on the principal stress axes; for a thrust fault, the beach ball typically shows a pattern with a compressional quadrant in the direction of hanging-wall up-dip motion over the footwall, indicating reverse slip on a low-angle plane.^[20]

Estimation Methods

Seismogram-Based Calculation

Seismic moment is commonly estimated from seismograms recorded by global or regional seismic networks, leveraging the recorded waveforms to infer source properties through spectral analysis or full-waveform inversion techniques. These methods rely on long-period body waves (P and S) and surface waves, which carry information about the overall energy release without being overly sensitive to high-frequency details that may be distorted by propagation effects. The primary advantage of seismogram-based approaches is their ability to provide rapid, remote assessments of earthquake source parameters for events worldwide, enabling consistent cataloging of seismic moments for magnitudes typically above Mw 5.0. One foundational technique for estimating the scalar seismic moment, M0, involves low-frequency spectral leveling, which examines the displacement or velocity spectrum of seismograms at frequencies below the source corner frequency. In this method, the spectrum exhibits a flat plateau at low frequencies, corresponding to the moment-rate spectrum level, where the amplitude is proportional to the seismic moment scaled by source radiation pattern, hypocentral distance, and medium properties. This plateau is derived from long-period waves that are less affected by source finiteness, allowing M0 to be isolated by fitting the observed spectral level after correcting for geometric spreading and anelastic attenuation. The approach stems from the omega-squared source model, which predicts a constant low-frequency asymptote for far-field displacement spectra.^[23] For more comprehensive source characterization, broadband waveform inversion techniques are employed to retrieve the full moment tensor from seismograms spanning a wide frequency band, typically 0.003–0.1 Hz for global events. These methods invert long-period body and mantle surface waves simultaneously to solve for the centroid location, source time function, and six-component moment tensor, yielding both the scalar moment and fault orientation. A prominent implementation is the Global Centroid Moment Tensor (GCMT) project, which has systematically applied this inversion since 1976, processing data from the World-Wide Standardized Seismograph Network and its successors to produce over 25,000 solutions as of 2025. The inversion minimizes waveform misfit using least-squares optimization, assuming a point-source approximation valid for most moderate-to-large earthquakes.^[24]^[25] Recent advances include machine learning methods, such as convolutional neural networks trained on global seismic records, which improve rapid magnitude and moment estimation, particularly when fine-tuned for specific regions or events.^[26] Processing seismograms for these estimations requires frequency-dependent corrections to account for propagation and recording effects that could bias the source spectrum. Attenuation, modeled as an exponential decay with quality factor Q(f) that varies with frequency, must be removed to recover the unattenuated source signal, particularly for intermediate frequencies where Q increases roughly linearly. Site effects, including local amplification due to shallow velocity structure, introduce frequency-selective gains that are corrected using empirical transfer functions or borehole data. Instrument response, which convolves the true ground motion with the seismometer's frequency-dependent sensitivity, is deconvolved via digital filters to obtain broadband displacement records essential for low-frequency analysis. These corrections ensure that the inferred moment reflects the true source release rather than path or site distortions.^[27]^[28] Uncertainties in seismogram-based moment estimates arise from inherent trade-offs among source parameters, such as depth, focal mechanism, and scalar moment, exacerbated by the limited resolution of long-period data. For shallow crustal events, waveform inversions like those in the GCMT catalog often exhibit depth-moment ambiguity, where a deeper isotropic source can mimic a shallower double-couple mechanism, leading to moment uncertainties of up to 0.2 log units. Mechanism trade-offs, particularly for strike-slip versus dip-slip orientations, further propagate errors, with catalog-wide analyses showing variance in principal axes exceeding 20 degrees for some events. These issues are evident in the GCMT dataset since 1976, where post-hoc validations against geodetic data reveal systematic biases in thrust fault depths for subduction zones, underscoring the need for multi-dataset integration to refine estimates.^[29]^[30]

Geodetic and Field Estimation

Geodetic methods, such as Global Positioning System (GPS) and Interferometric Synthetic Aperture Radar (InSAR), provide direct measurements of surface deformation to estimate seismic moment by inverting co-seismic displacements for fault slip distributions.^[31] These techniques capture horizontal and vertical offsets at GPS stations and line-of-sight displacements across broad areas via InSAR, respectively, which are modeled using elastic dislocation theory to determine rupture area A and average slip D.^[32] The scalar seismic moment M_0 is then calculated as M_0 = \mu A D, where \mu is the crustal rigidity modulus, typically assumed to be around 30 GPa for the upper crust.^[32] Paleoseismology employs field techniques like trenching across fault zones and mapping offset geomorphic features, such as stream channels or alluvial fans, to quantify slip for prehistoric earthquakes where instrumental records are unavailable. Trenches expose stratigraphic evidence of past ruptures, allowing measurement of single-event displacements, while offset features provide average slip per event when correlated with dated layers.^[33] These slip values, combined with estimates of rupture length from scarp mapping, yield A and D for computing M_0 via the same rigidity-based formula, often calibrated against empirical displacement-magnitude relations to infer moment magnitudes for long-term seismic hazard assessment. Strong-motion accelerometers deployed in the near-field capture high-frequency ground accelerations, which are integrated to displacements and inverted to resolve detailed, heterogeneous slip distributions, particularly the high-frequency components that contribute to total moment release.^[34] These instruments, often collocated with GPS for baseline corrections, enable estimation of near-fault slip variability that seismic waves alone may undersample, allowing integration of high-frequency slip into the overall M_0 calculation by partitioning the rupture model.^[34] For the 2011 Tohoku earthquake (M_w 9.0), geodetic data from over 1,200 GPS sites and seafloor observations initially refined and upwardly revised seismic estimates of moment release, yielding a total M_0 of approximately $4.0 \times 10^{22} N·m with maximum slips exceeding 60 m over a 500 km rupture length.^[35] This adjustment, driven by InSAR and GPS inversions showing deeper and larger slips than early teleseismic models, highlighted the value of geodetic constraints in resolving megathrust events.^[36]

Relation to Magnitude Scales

Moment Magnitude Scale

The moment magnitude scale, denoted as M_w, is a logarithmic measure of earthquake size directly derived from the scalar seismic moment M_0, providing a physically grounded quantification of the total energy released during an event. This scale addresses limitations in earlier magnitude systems by basing its values on the actual mechanics of fault rupture rather than empirical recordings that can saturate for large earthquakes.^[37] Developed by Thomas C. Hanks and Hiroo Kanamori in 1979, the scale was introduced to extend reliable magnitude estimates to very large earthquakes, where traditional scales like the local magnitude M_L or surface-wave magnitude M_s underestimate sizes due to instrumental saturation.^[37] The formula is given by

M_w = \frac{2}{3} \log_{10} M_0 - 6.0,

where M_0 is expressed in newton-meters (N·m); the factor of $2/3 ensures the scale's logarithmic nature aligns with the approximate cubic relationship between seismic moment and radiated energy, while the constant -6.0 normalizes values to approximate equivalence with the Richter scale for moderate events around magnitude 6. Key properties of the M_w scale include its open-ended nature, with no theoretical upper limit, allowing accurate assessment of "great" earthquakes; its firm physical basis in fault parameters like area, slip, and rigidity; and its consistency across the full spectrum of earthquake sizes, from small to enormous, without the saturation issues of amplitude-based scales.^[37] For instance, the 1960 Valdivia earthquake in Chile, with M_w 9.5, released a seismic moment approximately 250 times greater than that of the 1906 San Francisco earthquake (M_w 7.9), illustrating how M_0 quantifies the vastly larger total energy release in subduction zone events compared to strike-slip ruptures.^[38] This direct tie to M_0 ensures M_w remains a standard for comparing global seismic events.^[37]

Comparisons with Other Magnitudes

The Richter scale, or local magnitude (ML), measures earthquake size based on the maximum amplitude of seismic waves recorded by seismographs, primarily calibrated for shallow Southern California events. However, it saturates for large earthquakes, underestimating magnitudes above approximately Mw 8 because it does not account for the total energy release from extended fault ruptures. In contrast, the seismic moment (M0) quantifies the total potency of the source by integrating fault area, slip, and rigidity, providing a more accurate measure of overall energy without saturation limits, as demonstrated in events like the 1960 Chile earthquake where ML underestimated the size compared to later moment-based assessments. Body-wave magnitude (mb) relies on the amplitude of high-frequency P-waves, while surface-wave magnitude (Ms) uses longer-period surface waves, both of which are frequency-dependent and perform inconsistently for deep or very large earthquakes. For instance, mb tends to underestimate deep-focus events due to attenuation of short waves, and Ms becomes unreliable for shallow great earthquakes where surface waves are less prominent. The moment magnitude (Mw), derived from M0, overcomes these issues by incorporating the full seismic wave spectrum and source mechanics, ensuring consistency across event types and depths. The energy magnitude (Me) estimates the logarithm of radiated seismic energy (ER), focusing on the dynamic aspects of wave propagation rather than static fault properties. While Me correlates with observed shaking intensity, it can diverge from M0-based measures because seismic moment emphasizes source rigidity and displacement (potency), not the variable fraction of energy that radiates as seismic waves. This distinction is evident in tsunamigenic events where high M0 values indicate substantial seafloor displacement, even if radiated energy appears lower relative to Me. Empirical relations often approximate conversions between scales, such as Mw ≈ ML for small-to-moderate earthquakes (typically below magnitude 6) in tectonically similar regions, derived from regression analyses of global datasets. Logarithmic formulas like Mb = 0.91(Mw) + 0.98 have been established for body waves, but discrepancies arise in complex scenarios, such as the 2004 Sumatra-Andaman earthquake, where initial Ms estimates reached only 8.5 while Mw exceeded 9.1 due to the event's immense rupture length and slow slip. These relations underscore seismic moment's superiority as a fundamental physical parameter for unifying magnitude scales across diverse earthquake characteristics.

Historical Development

Early Concepts

The foundations of seismic moment concepts trace back to early 20th-century understandings of earthquake mechanics, building on 19th-century principles of rock elasticity. In 1910, Harry Fielding Reid proposed the elastic rebound theory following the 1906 San Francisco earthquake, explaining how tectonic stresses accumulate gradually across faults, deforming rocks elastically until the accumulated strain exceeds frictional resistance, leading to sudden slip and energy release. This model linked stress buildup directly to fault displacement, providing a physical basis for quantifying earthquake size through parameters like fault area and slip, though quantitative measures remained undeveloped at the time.^[39] The limitations of early magnitude scales became evident with the 1960 Valdivia earthquake in Chile, the largest instrumentally recorded event at moment magnitude 9.5.^[40] Traditional scales like the Richter local magnitude (ML) and surface-wave magnitude (Ms) saturated for such great events, assigning it an Ms of only 8.5 based on short-period recordings, which underestimated the total energy release and fault dimensions.^[41] This discrepancy highlighted the need for measures insensitive to distance and frequency, spurring interest in source parameters derived from long-period seismic waves that better capture the overall fault rupture. In the pre-tensor era of the 1950s and 1960s, seismologists began deriving scalar estimates of earthquake source strength from long-period surface waves, such as Rayleigh and Love waves, which reflect low-frequency fault motion without saturation issues.^[42] These efforts focused on integrating wave spectra to infer products of rigidity, fault area, and average slip, laying groundwork for formal seismic moment calculations. By 1966, Keiiti Aki advanced this approach in his analysis of the Niigata earthquake, introducing the scalar seismic moment as the product of shear modulus, fault area, and displacement, estimated directly from G-wave (long-period Rayleigh) spectra.^[43] Freeman Gilbert built on this in 1971, incorporating the moment concept into models of earthquake excitation of Earth's normal modes, emphasizing its role in source spectra for global wave propagation.^[44]

Modern Formulation

The modern formulation of seismic moment emerged in the late 1970s with the formal definition of the moment magnitude scale, M_w = \frac{2}{3} \log_{10} M_0 - 10.7, where M_0 is the scalar seismic moment in dyne-cm, providing a standardized, non-saturating measure of earthquake size directly tied to physical source properties. This work by Hanks and Kanamori also emphasized the use of the full moment tensor to represent the orientation and geometry of faulting, moving beyond scalar approximations to capture the complete seismic source mechanism.^[4] Building on this, the Global Centroid Moment Tensor (GCMT) project initiated routine inversions of long-period body and surface waves starting from events in 1977, enabling systematic global catalogs of moment tensors and scalar moments for moderate-to-large earthquakes. The foundational method for these inversions, which retrieves the centroid location, orientation, and M_0 from waveform data, was detailed by Dziewonski et al., allowing for consistent estimation across diverse tectonic settings.^[45] The project's first comprehensive application to 201 events in 1981 demonstrated its efficacy for catalog-wide M_0 determinations, with scalar moments typically ranging from $10^{24} to $10^{29} dyne-cm. Refinements in the 1990s leveraged expanding broadband seismograph networks, such as the Global Seismographic Network, to enhance resolution of moment tensors across wider frequency bands, improving accuracy for regional and complex sources. These advances facilitated the adoption of finite-fault models, which distribute slip over extended rupture areas rather than point sources, particularly for large or multifaceted events; for instance, the 1992 Landers earthquake (M_w 7.3) was modeled with a total M_0 of approximately $6.0 \times 10^{26} dyne-cm across multiple segments, revealing spatiotemporal rupture propagation.^[46]^[47] Current standards, as outlined in IASPEI guidelines, require reporting M_0 in the International System of Units (N·m) alongside the moment tensor decomposition, ensuring interoperability in global catalogs and emphasizing uncertainties for very large events. Updates in the 2010s addressed challenges with great earthquakes, such as revisions to the 2011 Tohoku M_w 9.1 event, where initial point-source estimates of M_0 \approx 3 \times 10^{29} dyne-cm were refined upward to $5.3 \times 10^{29} dyne-cm using finite-fault inversions incorporating tsunami and geodetic data, highlighting the need for multi-dataset integration.^[48]

Applications and Significance

In Earthquake Assessment

Seismic moment plays a crucial role in sizing earthquakes during rapid response efforts, enabling the determination of rupture area and released energy. The seismic moment M_0 is related to the fault area A and average slip D by M_0 = \mu A D, where \mu is the shear modulus, allowing seismologists to infer rupture dimensions when combined with fault geometry assumptions. For instance, the U.S. Geological Survey's National Earthquake Information Center (NEIC) utilizes moment magnitude M_w, derived directly from M_0, to assess earthquake size within 20 minutes for events above magnitude thresholds, facilitating timely alerts and preliminary impact evaluations.^[49] This approach provides a more reliable measure of total energy release compared to local magnitudes, especially for large events, supporting immediate situational awareness.^[50] Stress drop estimation from seismic moment offers insights into the dynamic properties of an earthquake source, aiding aftershock forecasting. The average stress drop \Delta \sigma can be approximated as \Delta \sigma \approx \mu \frac{D}{W}, where W is the fault width, derived from the relation between M_0, slip, and fault dimensions; this quantifies the shear stress change during rupture.^[51] Higher stress drops indicate more efficient energy release and are associated with reduced aftershock productivity for earthquakes of similar magnitude, as they leave less residual stress to trigger secondary events.^[52] Such estimates, often refined using spectral analysis of seismograms, inform models of aftershock sequences by linking source physics to seismicity rates.^[53] The seismic moment tensor further enables assessment of rupture directivity, distinguishing unilateral from bilateral propagation and its influence on ground shaking. Unilateral ruptures propagate predominantly in one direction from the hypocenter, amplifying peak ground velocities in the forward direction due to constructive interference of seismic waves, while bilateral ruptures spread symmetrically with less directional bias.^[54] By inverting near-field ground motion data with the moment tensor, directivity can be quantified rapidly, revealing effects such as up to twofold shaking amplification along the rupture path, which is critical for evaluating localized damage potential. A notable case is the 2010 Haiti earthquake (M_w 7.0), where finite-fault inversions using seismic moment constrained the slip distribution to two patches at approximately 14 km depth, with maximum slips of 5 m and limited slip in the upper 6 km, consistent with the absence of surface rupture.^[55] This analysis highlighted the absence of surface rupture and concentrated energy release near Port-au-Prince, explaining the intense near-field shaking despite the event's moderate size; the resulting models informed damage simulations by incorporating Coulomb stress changes (0.1–0.2 MPa) on adjacent faults, enhancing post-event recovery planning and hazard reassessments.^[56]

In Seismic Hazard Analysis

Seismic moment plays a central role in probabilistic seismic hazard analysis (PSHA) by informing the frequency-magnitude distributions of earthquakes, which are essential for generating hazard maps. The Gutenberg-Richter law, adapted to describe the distribution of seismic moments, posits a power-law relationship where the frequency of events decreases with increasing moment, allowing modelers to estimate the likelihood of large-magnitude earthquakes over long periods. This adaptation enables the calculation of recurrence intervals for specific moment thresholds, which are integrated into PSHA frameworks to delineate zones of varying hazard levels, such as those used in national building codes.^[57]^[58] In ground motion prediction, seismic moment serves as a key input to empirical ground motion prediction equations (GMPEs), which scale the intensity of shaking based on source characteristics. GMPEs typically incorporate moment magnitude—a logarithmic measure derived directly from seismic moment—to predict parameters like peak ground acceleration or spectral acceleration at a site, accounting for distance, site conditions, and fault type. For instance, in subduction zone settings, higher seismic moments correspond to greater energy release, leading to amplified shaking intensities that are quantified through these models to inform seismic design spectra. This scaling ensures that hazard assessments reflect the physical size and potency of potential ruptures.^[59] The integration of seismic moment into plate tectonics models further enhances long-term hazard evaluation by estimating moment release rates along tectonic boundaries. Globally, subduction zones account for the majority of seismic moment release, with an average rate of approximately $10^{22} N·m per year, derived from geodetic measurements of plate convergence and historical seismicity catalogs. These rates help quantify the seismic budget of regions like the Pacific Ring of Fire, where accumulated strain not released aseismically builds toward future large events, guiding the prioritization of monitoring and mitigation efforts in high-risk areas.^[60]^[61] Emerging applications leverage automated forecasting systems to assess aftershock hazards from early aftershock patterns, such as predicting magnitudes and durations. For example, following the 2023 Turkey-Syria earthquakes, the AFCAST system facilitated rapid assessments of aftershock hazards by analyzing sequences shortly after the mainshocks, enabling timely adjustments to evacuation and infrastructure response strategies in affected regions.^[62]

References

[1]
[PDF] earthquake magnitude, intensity, energy, power law relations and ...
Hanks and Kanamori (1979) proposed the moment magnitude scale by : Mw = 2/3 log Mo - 10.7. (6) where Mo is seismic moment of the earthquake in dyne cm. The ...
[2]
Moment magnitude, Richter scale | U.S. Geological Survey
Moment magnitude gives the most reliable estimate of earthquake size. Moment is a physical quantity proportional to the slip on the fault multiplied by the ...
[3]
Seismic Moment and Local Magnitude Scales in Ridgecrest ...
Apr 15, 2025 · (2011) found M = 0.594 * M L + 0.985 ⁠, both for small earthquakes in Switzerland.Missing: formula | Show results with:formula
[4]
A moment magnitude scale - Hanks - 1979 - AGU Journals - Wiley
May 10, 1979 · The nearly coincident forms of the relations between seismic moment M0 and the magnitudes ML, MS, and Mw imply a moment magnitude scale M ...
[5]
Magnitude Types | U.S. Geological Survey - USGS.gov
MW = 2/3 * (log10(MO) - 16.1), where MO is the seismic moment. Note this is also unit-dependent; the formula above is for moment in dyne-cm.
[6]
Magnitude Explained: Moment Magnitude vs. Richter Scale - IRIS
The seismic moment defines how much force is needed to generate the recorded waves. That information is plugged into the moment magnitude scale to give us the ...
[7]
[PDF] BULLETIN OF THE EartuquaKE - RESEARCH INSTITUTE
Estimation of earthquake moment, released energy, and stress-strain drop from the G wave spectrum. By Keiiti AKI,. Earthquake Research Institute. (Read July 20, ...
[8]
Source Estimate for the 1960 Chile Earthquake From Joint Inversion ...
Feb 11, 2019 · The seismic moment was 0.9–1 × 1023 Nm. Fujii and Satake (2013) jointly used tsunami and geodetic data to estimate the slip distribution. They ...
[9]
Seattle Fault Rupture - USGS WR CMG
Jan 20, 2004 · For all cases, a Poisson ratio of 0.25 is used. For the homogeneous case, a constant shear modulus of 30 GPa is used.
[10]
Implications for slip to the trench duri - GeoScienceWorld
Jun 14, 2018 · Earthquake data have been used to estimate shear modulus variations with depth (Dziewonski and Ander- ... ative to crustal rocks (shear modulus ~ ...
[11]
Crustal rheology and faulting at strike-slip plate boundaries
Mar 10, 2000 · Since the shear modulus /a is fixed (rl/2-2=r/1/r/2), T(k) depends only on the viscosity contrast between the two layers and on the thickness of ...Missing: variability | Show results with:variability
[12]
Scaling differences between large interplate and intraplate ...
Mar 3, 2017 · This implies that intraplate faults have a higher frictional strength than do plate boundaries, and hence that faults are velocity or slip ...
[13]
Relation between average slip and average stress drop for ...
Mar 3, 2017 · Relations between the average slip and average stress drop for simple rectangular faults of different length to width ratios are determined ...
[14]
Resolution and uncertainties in estimates of earthquake stress drop ...
Mar 15, 2021 · Our models and understanding of the dynamics of earthquake rupture are based largely on estimates of earthquake source parameters, such as stress drop and ...
[15]
Variation of Proportionality Between Stress Drop and Slip, With ...
Feb 21, 2023 · Earthquake stress drop Δσ is related to fault slip via , where μ, D, and Lc denote shear modulus, average slip, and fault dimension.Introduction · Methods · Results · Discussion
[16]
A geometric setting for moment tensors - Oxford Academic
Computationally a moment tensor is a 3 × 3 symmetric matrix that describes an earthquake source as a sum of three force couples.Missing: 3x3 | Show results with:3x3
[17]
[PDF] Information Sheet IS 3.9 1 - Topic Moment tensor inversion and ...
The moment tensor (3x3 matrix) is symmetric, i.e. it has six independent components. The diagonal elements represent linear vector dipoles, the off diagonal ...
[18]
new moment-tensor decomposition for seismic events in anisotropic ...
A new procedure based on the biaxial decomposition of the potency tensor is introduced which explicitly allows for anisotropy and interprets the moment tensor.
[19]
[PDF] A Probabilistic Bayesian Framework for the Seismic Moment Tensor
Sep 30, 2022 · 2.2 Random Moment Tensor Sampling. The seismic MT is a 3x3 matrix consists of nine force couples that represent the equivalent body forces ...
[20]
Focal Mechanisms... or "Beachballs" | U.S. Geological Survey
Example of a focal mechanism for a normal fault (left) and a thrust fault (right). A sketch of the cross-section for the normal fault is under the focal ...
[21]
https://mxrap.com/theory/moment-tensor-guide/
[22]
Probabilistic Moment Tensor Inversion for Hydrocarbon‐Induced ...
Aug 11, 2020 · The moment tensor is decomposed according to the Frobenius norm (Silver and Jordan, 1982). Details on the optimization setup and complete ...
[23]
TESLA, A Tool for Automatic Earthquake Low‐Frequency Spectral ...
Jul 3, 2023 · Moreover, the low‐frequency spectral level can be used to estimate the seismic moment of the earthquakes and moment magnitude. TESLA first ...
[24]
Global Centroid Moment Tensor Project
Development and implementation of improved methods for the quantification of earthquake source characteristics on a global scale. How to cite the CMT catalog:.CMT catalog web search · CMT Catalog Files · CMT Catalog Citation...Missing: estimation | Show results with:estimation
[25]
Global seismicity of 1981: centroid-moment tensor solutions for 542 ...
Dziewonski et al., 1981. A.M. Dziewonski, T.-A. Chou, J.H. Woodhouse ... Systematic analysis of the Mediterranean region using the Centroid-Moment Tensor method.
[26]
Frequency-Dependent Attenuation of P Waves and Estimation of ...
Dec 1, 2008 · After correcting the spectra for site attenuation characteristics, the earthquake source parameters have been determined by using both P- and S- ...
[27]
Frequency‐Dependent Attenuation of P and S Waves in Southern ...
Jun 26, 2018 · Abstract. Accurate models of crustal attenuation structure are important for simulating seismic wavefields at high frequencies (f > 1 Hz).
[28]
Improvements and Heterogeneities of the Global Centroid Moment ...
Oct 11, 2024 · The data were obtained from the Global Centroid Moment Tensor (Global CMT) project, from the files available at www.globalcmt.org/CMTfiles.html.
[29]
Uncertainty estimations for moment tensor inversions: the issue of ...
The table summarizes the moment tensor source parameters obtained by using TDMT procedure with the five 1-D wave speed models tested in our paper, and CMT3D ...
[30]
A joint analysis of GPS motions and InSAR to infer the coseismic ...
Coseismic GPS and InSAR data have been utilized to model the slip distribution of earthquakes assuming a linear elastic behaviour of the deformation field ( ...
[31]
Geodetic slip model of the 2011 M9.0 Tohoku earthquake
Sep 1, 2011 · [2] The March 11, 2011 M9.0 Tohoku Earthquake resulted from dip slip on the Japan trench accommodating ∼8 cm/yr convergence between the Pacific ...Missing: revision upward
[32]
Variability in magnitude of paleoearthquakes revealed by trenching ...
Oct 27, 2015 · Our study shows that moderate-magnitude events can be preserved in the stratigraphy and exposed by paleoseismic trenching under ideal conditions ...
[33]
Recovering Seismic Displacements through Combined Use of 1-Hz ...
Mar 9, 2017 · The use of near-field strong ground motion accelerograms for both understanding the seismic source and for improving the response of ...
[34]
The 2011 Magnitude 9.0 Tohoku-Oki Earthquake - Science
May 19, 2011 · Geodetic observations of crustal ... seismic moment, and static stress drop (fig. S4). Estimates of moment magnitude range from 8.8 to 9.2.
[35]
Coseismic slip distribution of the 2011 off the Pacific Coast of ...
Jul 26, 2012 · Slip of the 2011 Tohoku earthquake is estimated from land and seafloor geodetic data Maximum slip (85 m) released strain energy accumulated ...
[36]
https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2012JB009186
[37]
What was the magnitude? - Earthquake Hazards Program
The magnitude of 1906 has been redetermined in two recent studies that used rather different approaches: M = 7.7.
[38]
Reid's Elastic Rebound Theory - Earthquake Hazards Program
This gradual accumulation and release of stress and strain is now referred to as the "elastic rebound theory" of earthquakes.Missing: paper | Show results with:paper
[39]
M 9.5 - 1960 Great Chilean Earthquake (Valdivia Earthquake)
M 9.5 - 1960 Great Chilean Earthquake (Valdivia Earthquake) · 1960-05-22 19:11:20 (UTC) · 38.143°S 73.407°W · 25.0 km depth.Regional Information · Impact Summary · Interactive Map · ShakeMapMissing: moment | Show results with:moment
[40]
How big was that quake? - SERC (Carleton)
Oct 10, 2007 · On the Richter scale, the Alaska quake was "bigger" having a MS of 8.6 compared to the MS of the Chilean quake of 8.5. However, in measuring the ...
[41]
[PDF] 4.02 Seismic Source Theory - Elsevier
The physical origin of the double couple model was established in the early 1960s, thanks to the observational work of numerous seismologists and the crucial ...
[42]
Determination of earthquake source parameters from waveform data ...
Determination of earthquake source parameters from waveform data for studies of global and regional seismicity. A. M. Dziewonski,. A. M. ...
[43]
Toward real-time estimation of regional moment tensors
Mar 3, 2017 · Recent advances in broadband station coverage, continuous telemetry systems, moment-tensor procedures, and computer data-processing methods ...
[44]
Spatial and temporal distribution of slip for the 1992 Landers ...
Mar 3, 2017 · Our fault parameterization involves a variable-slip, multiple-segment, finite-fault model which treats the diverse data sets in a self- ...
[45]
[PDF] 1 SUMMARY OF MAGNITUDE WORKING GROUP ... - IASPEI
Mar 27, 2013 · The IASPEI standard procedures address the measurement of amplitudes and periods from digital data for use in calculating the generic magnitude ...
[46]
National Earthquake Information Center (NEIC) - USGS.gov
Oct 11, 2018 · First, the NEIC determines, as rapidly and as accurately as possible, the location and size of all significant earthquakes that occur worldwide.
[47]
Rapid calculation of a Centroid Moment Tensor and waveheight ...
Sep 27, 2011 · The rCMT system's main purpose is to calculate very rapid reliable moment magnitude estimates and mechanisms for earthquakes greater than 7.0 ...
[48]
[PDF] Earthquake stress drop estimates: What are they telling us?
Landers aftershocks have strong along-strike variations in stress drop with possible correlation to slip models. • Hard to resolve any temporal changes.
[49]
Regional and stress drop effects on aftershock productivity of large ...
Oct 8, 2016 · Higher stress-drop megathrust earthquakes produce fewer aftershocks than lower stress-drop events of similar magnitude Productivity ...
[50]
Stress‐Drop Estimates and Source Scaling of the 2011 Mineral ...
Sep 27, 2017 · The estimated stress drops for 46 aftershocks (⁠ 0.71–4.13) range from to 50 MPa with a median value of 3 MPa.Missing: forecasting | Show results with:forecasting
[51]
[PDF] Near-Real-Time Estimates on Earthquake Rupture Directivity Using ...
Aug 8, 2018 · Here we propose a method to estimate the directivity using real-time interpolated shaking maps of near-field (<25 km) ground motion data ...
[52]
[PDF] Coseismic Slip Distribution of the 2010 M 7.0 Haiti Earthquake and ...
These tests illustrate that the lack of shallow slip in our preferred model is robust. ... Coulomb Stress Trans- fer Model for the January 12 2010 Mw 7.0 Haiti ...
[53]
[PDF] Overview of the 2010 Haiti Earthquake
The 12 January 2010 Mw 7.0 earthquake in the Republic of Haiti caused an estimated 300,000 deaths, displaced more than a million people, and damaged.
[54]
[PDF] Introduction to Probabilistic Seismic Hazard Analysis
This limited magnitude distribution is termed a bounded. Gutenberg-Richter recurrence law. ... earthquake magnitudes with a Gutenberg-Richter distribution.
[55]
Seismic moment distribution revisited: II. Moment conservation ...
In this paper, we try to estimate the upper bound of the seismic moment–frequency relation, using the moment conservation principle. Quantitative plate ...Missing: adapted | Show results with:adapted
[56]
[PDF] Ground-motion prediction equations 1964–2010
This report summarizes all empirical ground-motion prediction equations (GMPEs), to esti- mate earthquake peak ground acceleration (PGA) and ...
[57]
Global Plate Motions and Earthquake Cycle Effects - AGU Journals
Jun 14, 2018 · The GBM moment accumulation rate, 1.09 × 1022 N·m/year, is equivalent to a MW=8.66 earthquake per year, while the average yearly release as ...
[58]
Comparison of seismic moment release rates along different types of ...
In this paper, we will assume that μ = 50 GPa for subduction zones and 30 GPa for ridge-transforms. Estimates of Westimate are highly variable, even for ...Summary · Introduction · Data and Methods · Results
[59]
Automated Assessment of Hazards of Aftershocks of the M w 7.8 ...
Nov 28, 2023 · The paper analyzes the AFCAST automated assessment of the aftershock hazards caused by the Mw 7.8 earthquake in Turkey of February 6, 2023 (the ...