Regular Variations of Intensity of the Westward Auroral Electrojet according to the Geomagnetic <Emphasis Type="Italic">AL</Emphasis> Index

Diurnal and seasonal variations of the geomagnetic AL index are studied. It is found in disturbed days that the mode of AL diurnal variation depends on the angle between the Sun–Earth line prolongation in the direction towards the magnetotail and the plane of geomagnetic equator; on quiet days, AL depends on the angle of attack between the geomagnetic axis and the Earth–Sun line. Seasonal AL variations are characterized by annual variations with summer maximum and semiannual variations with equinoctial maxima. It is shown that the semiannual AL variations can be described by a simplified model of plasma convection in the magnetotail based on a plasma electron cooling mechanism.     

<Emphasis Type="Italic">Q</Emphasis><Subscript>c</Subscript> and <Emphasis Type="Italic">Q</Emphasis><Subscript>S</Subscript> wave attenuation of South African earthquakes

Quality factor Q, which describes the attenuation of seismic waves with distance, was determined for South Africa using data recorded by the South African National Seismograph Network. Because of an objective paucity of seismicity in South Africa and modernisation of the seismograph network only in 2007, I carried out a coda wave decay analysis on only 13 tectonic earthquakes and 7 mine-related events for the magnitude range 3.6?≤?M _L ?≤?4.4. Up to five seismograph stations were utilised to determine Q _c for frequencies at 2, 4, 8 and 16 Hz resulting in 84 individual measurements. The constants Q ₀ and α were determined for the attenuation relation Q _c(f)?=?Q ₀ f ^α . The result was Q ₀?=?396?±?29 and α?=?0.72?±?0.04 for a lapse time of 1.9*(t _s???t ₀) (time from origin time t ₀ to the start of coda analysis window is 1.9 times the S-travel time, t _s) and a coda window length of 80 s. This lapse time and coda window length were found to fit the most individual frequencies for a signal-to-noise ratio of at least 3 and a minimum absolute correlation coefficient for the envelope of 0.5. For a positive correlation coefficient, the envelope amplitude increases with time and Q _c was not calculated. The derived Q _c was verified using the spectral ratio method on a smaller data set consisting of nine earthquakes and one mine-related event recorded by up to four seismograph stations. Since the spectral ratio method requires absolute amplitudes in its calculations, site response tests were performed to select four appropriate stations without soil amplification and/or signal distortion. The result obtained for Q _S was Q ₀?=?391?±?130 and α?=?0.60?±?0.16, which agrees well with the coda Q _c result.     

Attenuation coefficients of Rayleigh and <Emphasis Type="Italic">Lg</Emphasis> waves

Analysis of the frequency dependence of the attenuation coefficient leads to significant changes in interpretation of seismic attenuation data. Here, several published surface-wave attenuation studies are revisited from a uniform viewpoint of the temporal attenuation coefficient, denoted by χ. Theoretically, χ( f) is expected to be linear in frequency, with a generally non-zero intercept γ?=?χ(0) related to the variations of geometrical spreading, and slope dχ/df = π/Q _e caused by the effective attenuation of the medium. This phenomenological model allows a simple classification of χ( f) dependences as combinations of linear segments within several frequency bands. Such linear patterns are indeed observed for Rayleigh waves at 500–100-s and 100–10-s periods, and also for Lg from ~2 s to ~1.5 Hz. The Lg χ( f) branch overlaps with similar linear branches of body, Pn, and coda waves, which were described earlier and extend to ~100 Hz. For surface waves shorter than ~100 s, γ values recorded in areas of stable and active tectonics are separated by the levels of \(\gamma _{D} \approx 0.2 \times 10^{-3}\) s^???1 (for Rayleigh waves) and 8 ×10^???3 s^???1 (for Lg). The recently recognized discrepancy between the values of Q measured from long-period surface waves and normal-mode oscillations could also be explained by a slight positive bias in the geometrical spreading of surface waves. Similarly to the apparent χ, the corresponding linear variation with frequency is inferred for the intrinsic attenuation coefficient, χ _i , which combines the effects of geometrical spreading and dissipation within the medium. Frequency-dependent rheological or scattering Q is not required for explaining any of the attenuation observations considered in this study. The often-interpreted increase of Q with frequency may be apparent and caused by using the Q-based model of attenuation and following preferred Q( f) dependences while ignoring the true χ( f) trends within the individual frequency bands.     

On the Ultra Long Propagation of Felt Ground Motion Due to the <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">w</Emphasis></Subscript> 8.3 Deep-Focus Sea-of-Okhotsk Earthquake of May 24, 2013

This study uses macroseismic data and wave equations to solve the problem of ultra long propagation of felt ground motion (over 9000 km from the epicenter) due to the Sea-of-Okhotsk earthquake. We show that the principal mechanism of this phenomenon could be excitation of a previously unknown standing radial wave as a mode of the Earth’s free oscillations, ₀S₀, due to the superposition of an incident and a reflected spherical P wave in the epicentral area of the Sea-of-Okhotsk earthquake. The standing wave generates slowly attenuating P waves that travel over the earth’s surface that act as carrying waves; when superposed on these, direct body waves acquire the ability to travel over great distances. We show previously unknown parameters of the radial mode ₀S₀ for the initial phase of earth deformation due to the large deep-focus earthquake. We used data on the Sea-of-Okhotsk and Bolivian earthquakes to show that large deep-focus earthquakes can excite free oscillations of the Earth that are not only recorded by instrumental means, but are also felt by people, with the amplification of the macroseismic effect being directly related to the phenomenon of resonance for multistory buildings.     

Mechanism of formation of <Emphasis Type="Italic">Q</Emphasis>-disturbances in the <Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript> region of the equatorial ionosphere

Using model simulations, the morphological picture (revealed earlier) of the disturbances in the F ₂ region of the equatorial ionosphere under quiet geomagnetic conditions (Q-disturbances) is interpreted. It is shown that the observed variations in the velocity of the vertical E × B plasma drift, related to the zonal E _y component of the electric field, are responsible for the formation of Q-disturbances. The plasma recombination at altitudes of the lower part of the F ₂ region and the dependence of the rate of this process on heliogeophysical conditions compose the mechanism of Q-disturbance formation at night. The daytime positive Q-disturbances are caused exclusively by a decrease in the upward E × B drift, and this type of disturbances could be related to the known phenomenon of counter electrojet. Possible causes of formation of the daytime negative Q-disturbances are discussed.     

Diurnal and Longitudinal Variations of the Structure of an Equatorial Anomaly During Equinoxes According to Intercosmos-19 Satellite Data

Longitudinal and local time variations in the structure of the equatorial anomaly under high solar activity in the equinox are considered according to the Intercosmos-19 topside sounding data. It is shown that the anomaly begins to form at 0800 LT, when the southern crest is formed. The development of the equatorial anomaly is associated with well-known variations in the equatorial ionosphere: a change in the direction of the electric field from the west to the east, which causes vertical plasma drift W (directed upward) and the fountain effect. At 1000 LT, both anomaly crests appear, but they become completely symmetrical only by 1400 LT. The average position of the crests increases from I = 20° at 1000 LT to I = 28° at 1400 LT. The position of the crests is quite strong, sometimes up to 15°, varies with longitude. The foF2 value above the equator and the equatorial anomaly intensity (EAI) at 1200–1400 LT vary with the longitude according to changes in the vertical plasma drift velocity W. At this time, four harmonics are observed in the longitudinal variations of W, foF2, and EAI. The equatorial anomaly intensity increases to the maximum 1.5–2 h after the evening burst in the vertical plasma drift velocity. Longitudinal variations of foF2 for 2000–2200 LT are also associated with corresponding variations in the vertical plasma drift velocity. The equatorial anomaly intensity decreases after the maximum at 2000 LT and the crests decrease in size and shift towards the equator, but the anomaly is well developed at midnight. On the contrary, after midnight, foF2 maxima in the region of the anomaly crests are farther from the equator, but this is obviously associated with the action of the neutral wind. At 0200 LT, in contrast to the morning hours, only the northern crest of the anomaly is clearly pronounced. Thus, in the case of high solar activity during the equinoxes, a well-defined equatorial anomaly is observed from 1000 to 2400 LT. It reaches the maximum at 2000 LT.     

Behavior of Parameters of Enhancements in the Nighttime Electron Concentration in the Ionospheric <Emphasis Type="Italic">F</Emphasis>2 Layer

We have analyzed the behavior of the F2 layer parameters during nighttime periods of enhanced electron concentration by the results of vertical sounding of the ionosphere carried out with five-minute periodicity in Almaty (76°55′ E, 43°15′ N) in 2001–2012. The results are obtained within the frameworks of the unified concept of different types of ionospheric plasma disturbances manifested as variations in the height and half-thickness of the layer accompanied by an increase and decrease of N _m F2 at the moments of maximum compression and expansion of the layer. A good correlation is found between height h _Am , which corresponds to the maximum increase, and layer peak height h _m F, while h _Am is always less than h _m F. The difference between h _Am and h _m F linearly increases with increasing h _m F. Whereas the difference is ~38 km for h _m F = 280 km, it is ~54 km for h _m F = 380 km. Additionally, the correlation is good between the increase in the electron concentration in the layer maximum ΔN _m and the maximum enhancement at the fixed height ΔN; the electron concentration enhancement in the layer maximum is about two to three times lower than its maximum enhancement at the fixed height.     

Estimating seismic moments and Lg <Emphasis Type="Italic">Q</Emphasis> using Lg spectra

A multi-event and multi-station inverse method is presented in the paper to simultaneously estimate the seismic moments (M ₀) and source corner frequencies (f _c) of several Jiashi (Xinjiang, China) earthquakes, as well as the apparent Lg Q models for the paths from Jiashi to eight seismic stations (WMQ, AAK, TLG, MAKZ, KUR, VOS, ZRN and CHK) in Central Asia. The resultant seismic moments correlate well with the M ₀ values obtained by Harvard University using the centroid moment tensor (CMT) inversion and the surface-wave magnitudes as well. After the correction by a typical value of average radiation coefficient for regional SV waves, the M ₀ values from Lg spectral inversion are still close to the corresponding values obtained from CMT inversion. The obtained apparent Q _0Lg values (Lg Q at 1 Hz) are consistent with the tectonic features of corresponding propagation paths. The Q _0Lg values are 351±87, 349±86 and 300±27 for the paths from Jiashi to AAK, TLG and MAKZ, respectively. They are smaller than Q _0Lg values for the paths to KUR, VOS, ZRN and CHK, which are 553±72, 569±58, 550±57 and 603±65, respectively. These results agree with the condition that the paths to AAK, TLG and MAKZ mainly propagate through the mountainous Tianshan area where relatively strong seismic activities and large variations of topography are exhibited, while the paths to KUR, VOS, ZRN and CHK mainly propagate through the stable area of Kazak platform. The Q _0Lg value for the path to WMQ is 462±56. This is also in agreement with the condition that the path to WMQ is basically along the border area between Tianshan Mountain and Tarim Basin, and along this path the variations of topography and crustal thickness are moderate in comparison with that along the path to MAKZ.     

Peculiarities of Rayleigh wave displacement in a transversely isotropic half space

We consider a transversely isotropic medium with vertical axis of symmetry (VTI). Rayleigh wave displacement components in a homogeneous VTI medium contain exp(±kr_jz), where z is the vertical coordinate, k is the wave number, and j?=?1, 2; r_j may be considered as depth-decay factor. In a VTI medium, r_j can be a real or imaginary as in an isotropic medium, or it can be a complex depending on the elastic parameters of the VTI medium; if complex, r₁ and r₂ are complex conjugates. In a homogeneous VTI half space, Rayleigh wave displacement is significantly modified with a phase shift when r_j changes from real to complex with variation of VTI parameters; at the transition, the displacement becomes zero when r₁?=?r₂. In a liquid layer over a VTI half space, Rayleigh waves are dispersive. Here, also Rayleigh wave displacement significantly modified with a phase shift when r_j changes from real to complex with a decrease of period. At very low period, phase velocity of Rayleigh waves becomes less than P-wave velocity in the liquid layer giving rise to Scholte waves (interface waves). The amplitudes of Scholte waves with a VTI half space are found to be significantly larger than those with an isotropic half space.     

Effect of zonal E × B plasma drift on electron density in the low-latitude ionospheric <Emphasis Type="Italic">F</Emphasis> region at a solar activity maximum near vernal equinox

The variations in the density of the ionospheric F2 layer maximum (NmF2) under the action of the zonal plasma drift perpendicularly to the magnetic (B) and electric (E) fields in the direction geomagnetic west-geomagnetic east have been studied using the three-dimensional nonstationary theoretical model of electron and ion densities (N _e and N _i ) and temperatures (T _e and T _i ) in the low-latitude and midlatitude ionospheric F region and plasmasphere. The method of numerical calculations of N _e , N _i , T _e , and T _i , including the advantages of the Lagrangian and Eulerian methods, is used in the model. A dipole approximation of the geomagnetic field (B), taking into account the non-coincidence of the geographic and geomagnetic poles and differences between the positions of the Earth’s and geomagnetic dipole centers, is accepted in the calculations. The calculated NmF2 and altitudes of the F2 layer maximum (hmF2) have been compared with these quantities measured at 16 low-latitude ionospheric sounding stations during the geomagnetically quiet period October 11–12, 1958. This comparison made it possible to correct the input model parameters: the NRLMSISE-00 model [O], the meridional component of the neutral wind velocity according to the HWW90 model, and the meridional component of the equatorial plasma drift due to the electric field specified by the empirical model. It has been indicated that the effect of the zonal E × B plasma drift on NmF2 can be neglected under daytime conditions and changes in NmF2 and hmF2 under the action of this drift are insignificant under nighttime conditions north of 25° and south of ?26° geomagnetic latitude. The effect of the zonal E × B plasma drift on NmF2 and hmF2 is most substantial in the nightside ionosphere approximately from ?20° to 20° geomagnetic latitude, and the neglect of this drift results in an up to 2.4-fold underestimation of NmF2. The found dependence of the effect of the zonal E × B plasma drift on NmF2 and hmF2 on geomagnetic latitude is related to the longitudinal asymmetry of B, asymmetry of the neutral wind about the geomagnetic equator, and changes in the meridional E × B plasma drift at a change in geomagnetic longitude.     

Dependence of geomagnetic disturbances on extreme values of the solar wind <Emphasis Type="Italic">E</Emphasis><Subscript><Emphasis Type="Italic">y</Emphasis></Subscript> component

The dependence of the zonal geomagnetic indices (AE, Ap, Kp, Kn, and Dst) on the solar wind parameters (the electric field E _y component, dynamic pressure P _d and IMF irregularity σB) has been studied for two types of events: magnetic clouds and high-speed streams. Based on the empirical relationships, it has been established that the AE, Ap, Kp, and Kn indices are directly proportional to the E _y value at E _y < 12 mV m^?1 and are inversely proportional to this value at E _y > 12 mV m^?1 for the first-type events. On the contrary, the dependence of Dst on E _y is monotonous nonlinear. A linear dependence of all geomagnetic indices on E _y is typical of the second-type events. It has been indicated that the specific features of geoeffectiveness of magnetic clouds and high-speed solar wind streams are caused by the dependence of the electric field potential across the polar cap on the electric field, solar wind dynamic pressure, and IMF fluctuations.     

Dynamics of the midlatitude ionospheric <Emphasis Type="Italic">F</Emphasis> region at sunrise

The behavior of the F2 layer at sunrise has been studied based on vertical-incidence ionospheric sounding data in Almaty (76°55′E, 43°15′N). Records with small amplitudes of electron density background fluctuations were selected in order to exactly estimate the onsets of a pronounced increase in the electron density at different altitudes. It has been indicated that the electron density growth rate is a function of altitude; in this case, the growth rate at the F2 layer maximum is much lower than such values at fixed altitudes of ～30–55 km below the layer maximum. The solar zenith angle (χ) and the blanketing layer thickness (h ₀) at the beginning of a pronounced increase in the electron density at altitude h are linearly related to the h value, and these quantities vary within ～90° < χ < 100° and 180 km < h ₀ < 260 km, respectively.     

Region-specific <Emphasis Type="Italic">S</Emphasis>-wave attenuation for earthquakes in northwestern Iran

In this study, continuous wavelet transform is applied to estimate the frequency-dependent quality factor of shear waves, Q _S , in northwestern Iran. The dataset used in this study includes velocigrams of more than 50 events with magnitudes between 4.0 and 6.5, which have occurred in the study area. The CWT-based method shows a high-resolution technique for the estimation of S-wave frequency-dependent attenuation. The quality factor values are determined in the form of a power law as Q _S (f)?=?(147?±?16)f ^0.71?±?0.02 and (126?±?12)f ^0.73?±?0.02 for vertical and horizontal components, respectively, where f is between 0.9 and 12 Hz. Furthermore, in order to verify the reliability of the suggested Q _S estimator method, an additional test is performed by using accelerograms of Ahar-Varzaghan dual earthquakes on August 11, 2012, of moment magnitudes 6.4 and 6.3 and their aftershocks. Results indicate that the estimated Q _S values from CWT-based method are not very sensitive to the numbers and types of waveforms used (velocity or acceleration).     

Interaction of geometry and mechanical property of trapezoidal sedimentary basins with incident <Emphasis Type="Italic">SH</Emphasis> waves

This paper investigated the effects of basin geometry and material property on the response of 2D trapezoidal sediment-filled basin to incident plane SH waves. Ten basin configurations with different geometries were developed, and then their seismic responses to both Ricker wavelets and seismic records were simulated by using an explicit finite difference scheme. The definition of deep/shallow basin, the precondition for the observation of prominent surface waves and the influential area of edge effects of the shallow basin were discussed quantitatively in this study. The followings were concluded: in the common velocity contrast range (v _s1/v _s2 < 10), the fundamental frequency a basin with W/H > 3.0 can be estimated approximately by 1D theory. The complexity of peak ground acceleration distribution pattern, the width of the most affected section as well as the amplitude of ground motion in the Edge Region increase with incident frequency. Prominent surface waves can only be observed when the incident wavelength is shorter than the critical wavelength λ _c . The interaction between incident wave and basin dynamic property plays a dominant role on the peak ground acceleration amplitude while the interaction between incident wave and geometry plays a more significant role on the peak ground acceleration distribution. For very shallow basin, different areas along the basin width are affected to different extents. Only a limited area close to the basin edge is influenced significantly. It is more feasible to propose spectral aggravation factor for different surface zones respectively than a uniform constant as a tool to calibrate the 1D-based design spectrum so as to take the basin effects into account.     

An adaptive modal pushover analysis procedure (<Emphasis Type="Italic">VMPA</Emphasis>-<Emphasis Type="Italic">A</Emphasis>) for buildings subjected to bi-directional ground motions

A new modal pushover analysis procedure (VMPA-A) is developed and implemented in MATLAB code for three-dimensional buildings subjected to bidirectional ground motions. VMPA-A uses stepwise force patterns to represent changes in the dynamic characteristics because of the accumulated structural damages. The hybrid-spectrum concept is introduced to account for the bidirectional ground motion effects. Due to enactments of the equal displacement rule and the secant stiffness-based linearization process, nonlinear analysis is performed for specific displacement targets without stipulation of full modal capacity curves for each mode. Horizontal components of an earthquake record are considered simultaneously, and the consistency between the force and displacement vectors for each mode is provided. These are the main advantages of the proposed procedure against modal pushover analysis (MPA). An existing 21-story reinforced concrete building is analyzed to exemplify VMPA-A. The response parameters such as displacements, story drifts, internal forces, strains, etc. are discussed by comparing the results of VMPA-A with nonlinear time history analyses, which is accepted as the “exact solution”. Though consistent demand estimations are obtained for story drifts, displacements and deformations, some conservative results are obtained for story shears.     

Signatures of Alfvenic Field-Line Resonance in the Behavior of Preonset Auroral Arcs

The evolution of preonset auroral arcs before full-scale auroral poleward expansion (the time T₀ indicates the expansion onset) is studied based on ground-based optical observations filtered by the gradient method. In one of the three events studied in detail, the preonset arc exhibits periodic poleward excursions ~10 min before T₀. The excursions extend over 1° in latitude, being repeated with a period of 2.5 min (frequency 6.7 mHz), and can be explained by the theory of classical (i.e., linear nondispersive) Alfvénic fieldline resonance (FLR), which is proposed to form and evolve at the location of subsequent substorm initiation. In two other events, the preonset arc evolves somewhat differently. Having appeared 15–20 min before T₀, the arc brightens and develops a fine structure in the transverse direction, with new arcs detaching and propagating away from it. Such signatures may indicate a nonlinear dispersive FLR that periodically produces soliton-like structures propagating across and away from the resonance layer. The involved nonlinearity has a ponderomotive nature. The dispersive effects become significant if, as a result of fine structuring, perturbations are produced on the scales of order of the electron inertial length or ion gyroradius.     

Spatial and time variations in critical frequencies of the ionospheric <Emphasis Type="Italic">F</Emphasis> region above the zone of equatorial earthquake preparation

Spatial and time variations in the critical frequencies foF2 before the strong earthquake of August 15, 1963, with the magnitude M = 7.75 are analyzed. The epicenter of the earthquake was located in the vicinity of the magnetic equator in the American longitudinal sector. The data of the topside ionosonde on board Alouette-1 and of the series of ground-based ionosondes has been used for this purpose. The ground-based ionosondes made it possible to detect an insignificant anomalous decrease in foF2 within the zone of earthquake preparation a few days prior to the earthquake. This result confirms the conclusion drawn earlier on the basis of the satellite data. The modification of the ionosphere at the F-region level is more evident in the satellite than in the ground-based data. It is also noted that the character of the time variations in foF2 a day before the earthquake is similar to the so called “quiet time” Q-disturbances in the ionosphere, when the electron concentration at the F region maximum differs from the median values by more than 20% under undisturbed geophysical conditions.     

Magnitude <Emphasis Type="Italic">M</Emphasis><Subscript>w</Subscript> in metropolitan France

The recent seismicity catalogue of metropolitan France Sismicité Instrumentale de l’Hexagone (SI-Hex) covers the period 1962–2009. It is the outcome of a multipartner project conducted between 2010 and 2013. In this catalogue, moment magnitudes (M _w) are mainly determined from short-period velocimetric records, the same records as those used by the Laboratoire de Détection Géophysique (LDG) for issuing local magnitudes (M _L) since 1962. Two distinct procedures are used, whether M _L-LDG is larger or smaller than 4. For M _L-LDG >4, M _w is computed by fitting the coda-wave amplitude on the raw records. Station corrections and regional properties of coda-wave attenuation are taken into account in the computations. For M _L-LDG ≤4, M _w is converted from M _L-LDG through linear regression rules. In the smallest magnitude range M _L-LDG <3.1, special attention is paid to the non-unity slope of the relation between the local magnitudes and M _w. All M _w determined during the SI-Hex project is calibrated according to reference M _w of recent events. As for some small events, no M _L-LDG has been determined; local magnitudes issued by other French networks or LDG duration magnitude (M _D) are first converted into M _L-LDG before applying the conversion rules. This paper shows how the different sources of information and the different magnitude ranges are combined in order to determine an unbiased set of M _w for the whole 38,027 events of the catalogue.     

A real <Emphasis Type="Italic">P</Emphasis>-wave and its dependence on the presence of gas

Attenuation of seismic compression waves leads to the real existence of a fast P₁ wave in rocks which are fully saturated with dropping fluid and a slow P₂ wave in the rocks containing gas in their pores. This accounts for the seismic blanking zones below the gas horizons for the P₁ waves. Oscillations of gaseous inclusions ensure the energy transfer to the dominant frequencies which are different for the cases of passive seismic (few Hz) and active source seismic (10–20 Hz). The intervals of dominant frequencies are determined from the negative attenuation of these low-frequency waves. According to the observations and the suggested equation, random noise amplifies the signal at these frequencies. Thus, the P₂ waves at the dominant frequency of the active source seismics are applicable for elaborating on the details of the saturation of the production layer by hydrocarbons. The relation to the AVO method (Amplitude Variation with Offset) and dilatancy effect during the preparation of an earthquake is noted.     

Sampling uncertainties and source <Emphasis Type="Italic">b</Emphasis> likelihood for the Gutenberg-Richter <Emphasis Type="Italic">b</Emphasis> value from the Aki-Utsu method

The Aki-Utsu method of Gutenberg-Richter (G-R) b value estimation is often misapplied so that estimations not using the G-R histogram are often meaningless because they are not based on adequate samples. We propose a method to estimate the likelihood Pr(b?b _m , N, M ₁, M ₂) that an observed b _m estimate, based on a sample of N magnitudes within an [M ₁????≤?ΔM/2,?M ₂?+?ΔM/2) range, where ΔM?=?0.1 is the usual rounding applied to magnitudes, is due to a “true” source b value, b, and use these likelihoods to estimate source b ranges corresponding to various confidence levels. As an example of application of the method, we estimate the b values before and after the occurrence of a 7.4-magnitude earthquake in the Mexican subduction zone, and find a difference of 0.82 between them with 100% confidence that the b values are different.