Modeling of strong motion generation area of the Uttarkashi earthquake using modified semiempirical approach

The semiempirical approach based on envelope summation method given by Midorikawa (Tectonophysics 218:287–295, 1993) has been modified in this paper for modeling of strong motion generation areas (SMGAs). Horizontal components of strong ground motion have been simulated using modifications in the semiempirical approach given by Joshi et al. (Nat Hazard 71:587–609, 2014). Various modifications in the technique account for finite rupture source, layering of earth, componentwise division of energy and frequency-dependent radiation pattern. In this paper, SMGAs of the Uttarkashi earthquake have been modeled. Two different isolated wave packets in the recorded accelerogram have been identified from recorded ground motion, which accounts for two different SMGAs in the entire rupture plane. The approximate locations of SMGAs within the rupture plane were estimated using spatio-temporal variation of 77 aftershocks. Source parameters of each SMGA were calculated from theoretical and observed source displacement spectra computed from two different wave packets in the record. The final model of rupture plane responsible for the Uttarkashi earthquake consists of two SMGAs, and the same has been used to simulate horizontal components of acceleration records at different station using modified semiempirical technique. Comparison of the observed and simulated acceleration records in terms of root mean square error confirms the suitability of the final source model for the Uttarkashi earthquake.     

Scenarios of Seismic Risk in the United Arab Emirates,an Approximate Estimate

We estimate the losses due to 10 scenario earthquakes in 150 settlements of the United Arab Emirates (UAE). For southern Iran, we use four source zones and the maximum magnitudes in them as determined by GSHAP (7.2 M 8.1). For six local scenario earthquakes, we use the range 5.5 M 6.5, place the sources mainly on mapped faults and vary the distance to major cities from 10 to 60 km. In the test case of the Masafi earthquake (M5, 11 March 2002), the method and data bank we use yield the correct results, suggesting that our approach to the problem is valid for the UAE. The sources in Iran are expected to cause only minor damage, except for an M8.1 earthquake in the Makran region. For such an event we expect some deaths, several hundred injured and a loss of 3–6% of the value to the building stock in the northeastern UAE, including Oman. The losses for local scenarios with epicenters in the unpopulated areas of the UAE and for scenarios with M < 5.8 are estimated to be minor. Because the two major mapped faults run through several of the large cities, scenarios with short epicentral distances from cities have to be considered. Scenarios with M6 near cities lead to estimates of about 1000 ± 500 deaths, and several thousand injured. Most buildings are expected to be damaged to a moderate degree and the loss to buildings is estimated around 1/4of their value. If the magnitude should reach 6.5, the losses to humans and to building value could be staggering. These estimates are approximate because: (1) there exists no local seismograph network that could map active faults by locating microseismicity; (2) there exist no historically old buildings that could serve as tests for effects due to strong ground motion in the past; (3) there exist no microzonation of the subsurface properties in this region of unconsolidated building ground; (4) there exist no detailed inventory of building fragility. Nevertheless, our conclusion that there exists a substantial seismic risk in the UAE is reliable, because our method yields accurate results in the cases of earthquakes with known losses during the last several decades in the Middle East.     

Estimation of earthquake source parameters in the Kachchh seismic zone,Gujarat, India,using three component S-wave spectra

Earthquake source parameters and crustal \(Q_{0}\) values for the 138 selected local events of (\(\hbox {M}_{\mathrm{w}}{:}2.5{-}4.4\)) the 2001 Bhuj earthquake sequence have been computed through inversion modelling of S-waves from three-component broadband seismometer data. SEISAN software has been used to locate the identified local earthquakes, which were recorded at least three or more stations of the Kachchh seismological network. Three component spectra of S-wave are being inverted by using the Levenberg–Marquardt non-linear inversion technique, wherein the inversion scheme is formulated based on \(\omega ^{2}\) source model. SAC Software (seismic analysis code) is being utilized for calculating three-component displacement and velocity spectra of S-wave. The displacement spectra are used for estimating corner frequency (in Hz) and long period spectral level (in nm-s). These two parameters play a key role in estimating earthquake source parameters. The crustal \({Q}_{0}\) values have been computed simultaneously for each component of three-component broadband seismograph. The estimated seismic moment (\(M_{0}\)) and source radius (r) using S-wave spectra range from 7.03E+12 to 5.36E+15 N-m and 178.56 to 565.21 m, respectively. The corner frequencies for S-wave vary from 3.025 to 7.425 Hz. We also estimated the radiated energy (\(E_{S}\)) using velocity spectra, which is varying from 2.76E+06 to 4.07E+11 Joules. The estimated apparent stress drop and static stress drop values range from 0.01 to 2.56 and 0.53 to 36.79 MPa, respectively. Our study also reveals that estimated \(Q_{0}\) values vary from 119.0 to 7229.5, with an average \(Q_{0}\) value of 701. Another important parameter, by which the earthquake rupture process can be recognized, is Zuniga parameter. It suggests that most of the Kachchh events follow the frictional overshoot model. Our estimated static stress drop values are higher than the apparent stress drop values. And the stress drop values are quite larger for intraplate earthquakes than the interplate earthquakes.     

Bayesian estimation of seismic hazard for two sites in Switzerland

Seismic hazard analysis is based on data and models, which both are imprecise and uncertain. Especially the interpretation of historical information into earthquake parameters, e.g. earthquake size and location, yields ambiguous and imprecise data. Models based on probability distributions have been developed in order to quantify and represent these uncertainties. Nevertheless, the majority of the procedures applied in seismic hazard assessment do not take into account these uncertainties, nor do they show the variance of the results. Therefore, a procedure based on Bayesian statistics was developed to estimate return periods for different ground motion intensities (MSK scale).Bayesian techniques provide a mathematical model to estimate the distribution of random variables in presence of uncertainties. The developed method estimates the probability distribution of the number of occurrences in a Poisson process described by the parameter . The input data are the historical occurrences of intensities for a particular site, represented by a discrete probability distribution for each earthquake. The calculation of these historical occurrences requires a careful preparation of all input parameters, i.e. a modelling of their uncertainties. The obtained results show that the variance of the recurrence rate is smaller in regions with higher seismic activity than in less active regions. It can also be demonstrated that long return periods cannot be estimated with confidence, because the time period of observation is too short. This indicates that the long return periods obtained by seismic source methods only reflects the delineated seismic sources and the chosen earthquake size distribution law.     

Ground motion estimation during the Kashmir earthquake of 8th October 2005

In this article, analytical methods have been used to estimate ground motion during the 8 October 2005, Kashmir earthquake. Peak ground acceleration (PGA) values at several stations in the epicentral region have been estimated by empirical analytical source mechanism models. As an alternate analysis, PGA estimates have also been obtained using the stochastic finite fault seismological model. The estimated PGAs are compared with that obtained from damage values. A PGA contour map in the near-source region is provided. It is found that very near to the epicenter, PGA would have reached more than 1 g. It is demonstrated that empirical analytical models can be effectively used to estimate ground motion due to rupture of active faults.     

Comparison of earthquake source characteristics in the Kachchh Rift Basin and Saurashtra horst,Deccan Volcanic Province,western India

Seismic source parameters of small to moderate sized intraplate earthquakes that occurred during 2002–2009 in the tectonic blocks of Kachchh Rift Basin (KRB) and the Saurashtra Horst (SH), in the stable continental region of western peninsular India, are studied through spectral analysis of shear waves. The data of aftershock sequence of the 2001 Bhuj earthquake (\(M_{w}\) 7.7) in the KRB and the 2007 Talala earthquake (\(M_{w}\) 5.0) in the SH are used for this study. In the SH, the seismic moment (\(M_{o})\), corner frequency \((f_{c})\), stress drop (\(\varDelta \sigma \)) and source radius (r) vary from \(7.8\times 10^{11}\) to \(4.0\times \)10\(^{16}\) N-m, 1.0–8.9 Hz, 4.8–10.2 MPa and 195–1480 m, respectively. While in the KRB, these parameters vary from \(M_{o} \sim 1.24 \,\times \, 10^{11}\) to \(4.1 \times 10^{16}\) N-m, \(f_{c }\sim \) 1.6 to 13.1 Hz, \(\varDelta \sigma \sim 0.06\) to 16.62 MPa and \(r \sim 100\) to 840 m. The kappa (K) value in the KRB (0.025–0.03) is slightly larger than that in the SH region (0.02), probably due to thick sedimentary layers. The estimated stress drops of earthquakes in the KRB are relatively higher than those in SH, due to large crustal stress concentration associated with mafic/ultramafic rocks at the hypocentral depths. The results also suggest that the stress drop value of intraplate earthquakes is larger than the interplate earthquakes. In addition, it is observed that the strike-slip events in the SH have lower stress drops, compared to the thrust and strike-slip events.     

Site response of the Ganges basin inferred from re-evaluated macroseismic observations from the 1897 Shillong, 1905 Kangra,and 1934 Nepal earthquakes

We analyze previously published geodetic data and intensity values for the M _s = 8.1 Shillong (1897), M _s = 7.8 Kangra (1905), and M _s = 8.2 Nepal/Bihar (1934) earthquakes to investigate the rupture zones of these earthquakes as well as the amplification of ground motions throughout the Punjab, Ganges and Brahmaputra valleys. For each earthquake we subtract the observed MSK intensities from a synthetic intensity derived from an inferred planar rupture model of the earthquake, combined with an attenuation function derived from instrumentally recorded earthquakes. The resulting residuals are contoured to identify regions of anomalous intensity caused primarily by local site effects. Observations indicative of liquefaction are treated separately from other indications of shaking severity lest they inflate inferred residual shaking estimates. Despite this precaution we find that intensites are 1–3 units higher near the major rivers, as well as at the edges of the Ganges basin. We find evidence for a post-critical Moho reflection from the 1897 and 1905 earthquakes that raises intensities 1–2 units at distances of the order of 150 km from the rupture zone, and we find that the 1905 earthquake triggered a substantial subsequent earthquake at Dehra Dun, at a distance of approximately 150 km. Four or more M = 8 earthquakes are apparently overdue in the region based on seismic moment summation in the past 500 years. Results from the current study permit anticipated intensities in these future earthquakes to be refined to incorporate site effects derived from dense macroseismic data.     

Destructive near-fault strong ground motion from the 2016 Kumamoto prefecture,Japan, M7.3 earthquake

The sequence of the 2016 Kumamoto, Japan, earthquake, which included an initial M6.5 foreshock on April 14, followed by a larger M7.3 mainshock on April 16, and subsequently occurred high aftershock activity, caused significant damage in Kumamoto and neighboring regions. The near-field strong motion record by strong motion network (K-NET and KiK-net) and the intensity meter network demonstrated clearly the characteristics of the strong ground motion developed by the shallow (H = 12 km), inland earthquake comprising short-time duration (<15–20 s) but large (>1G) ground accelerations. The velocity response spectra of the near-fault motion at Mashiki and Nishihara showed large levels (>300–550 cm/s) in the short-period range (T = 1–2 s), several times larger than that of the near-field record of the destructive 1995 Kobe earthquake (M7.3) and that of the 2004 Mid-Niigata earthquake (M6.8). This period corresponds to the collapse vulnerability of Japanese wooden-frame houses, and is the major cause of severe damage during the Kumamoto earthquake. The response spectra also showed extremely large levels (>240–340 cm/s) in the long-period (T > 3 s) band, which is potentially disastrous for high-rise buildings, large oil storage tanks, etc. to have longer resonant period. Such long-period motion was, for the most parts, developed by the static displacement of the fault movement rather than by the seismic waves radiating from the source fault. Thus, the extreme near-fault long-period motion was hazardous only close to the fault but it attenuated very rapidly away from the fault.     

Energetic and spatial characterization of seismicity in the Algeria–Morocco region

We estimate the energetic and spatial characteristics of seismicity in the Algeria–Morocco region using a variety of seismic and statistical parameters, as a first step in a detailed investigation of regional seismic hazard. We divide the region into five seismotectonic regions, comprising the most important tectonic domains in the studied area: the Moroccan Meseta, the Rif, the Tell, the High Plateau, and the Atlas. Characteristic seismic hazard parameters, including the Gutenberg–Richter b-value, mean seismic activity rate, and maximum possible earthquake magnitude, were computed using an extension of the Aki–Utsu procedure for incomplete earthquake catalogs for each domain, based on recent earthquake catalogs compiled for northern Morocco and northern Algeria. Gutenberg–Richter b-values for each zone were initially estimated using the approach of Weichert (Bull Seismol Soc Am 70:1337–1346, 1980): the estimated b-values are 1.04 ± 0.04, 0.93 ± 0.10, 0.72 ± 0.03, 0.87 ± 0.02, and 0.77 ± 0.02 for the Atlas, Meseta, High Plateau, Rif, and Tell seismogenic zones, respectively. The fractal dimension D ₂ was also estimated for each zone. From the ratio D ₂/b, it appears that the Tell and Rif zones, with ratios of 2.09 and 2.12, respectively, have the highest potential earthquake hazard in the region. The Gutenberg–Richter relationship analysis allows us to derive that in the Tell and Rif, the number of earthquake with magnitude above Mw 4.0, since 1925 normalized to decade and to square cell with 100-km sides is equal to 2.6 and 1.91, respectively. This study provides the first detailed information about the potential seismicity of these large domains, including maximum regional magnitudes, characteristics of spatial clustering, and distribution of seismic energy release.

     

Seismic hazard analysis for critical infrastructures in California

California is in a highly seismically active region, and structures must be designed and constructed to withstand earthquakes. Seismic hazard analysis to estimate realistic earthquake ground motions and surface fault rupture offsets is done for various mitigation measures. The best policy is to avoid constructing structures crossing seismogenic faults. Because earthquake timings are unpredictable within our current understanding, the best method is time-invariant deterministic seismic hazard analysis (DHSA) to assess effects from the largest single earthquake called Maximum Credible Earthquake (MCEs) expected from seismogenic faults. Time-dependent hazard estimates such as those arrived at through probabilistic seismic hazard analysis (PSHA) are inherently unreliable. Hazard analyses based on MCEs have been in continuous use for the design and construction of highways and bridges in California for over 30 years.

This paper presents an alternative to other methods of analysis, e.g., Abrahamson (2000) [Abrahamson, N.A., 2000. State of the practice of seismic hazard evaluation. Melbourne: proceedings of GeoEng, 2000].     

Energetic and spatial characterization of seismicity in the Algeria–Morocco region

We estimate the energetic and spatial characteristics of seismicity in the Algeria–Morocco region using a variety of seismic and statistical parameters, as a first step in a detailed investigation of regional seismic hazard. We divide the region into five seismotectonic regions, comprising the most important tectonic domains in the studied area: the Moroccan Meseta, the Rif, the Tell, the High Plateau, and the Atlas. Characteristic seismic hazard parameters, including the Gutenberg–Richter b-value, mean seismic activity rate, and maximum possible earthquake magnitude, were computed using an extension of the Aki–Utsu procedure for incomplete earthquake catalogs for each domain, based on recent earthquake catalogs compiled for northern Morocco and northern Algeria. Gutenberg–Richter b-values for each zone were initially estimated using the approach of Weichert (Bull Seismol Soc Am 70:1337–1346, 1980): the estimated b-values are 1.04 ± 0.04, 0.93 ± 0.10, 0.72 ± 0.03, 0.87 ± 0.02, and 0.77 ± 0.02 for the Atlas, Meseta, High Plateau, Rif, and Tell seismogenic zones, respectively. The fractal dimension D ₂ was also estimated for each zone. From the ratio D ₂/b, it appears that the Tell and Rif zones, with ratios of 2.09 and 2.12, respectively, have the highest potential earthquake hazard in the region. The Gutenberg–Richter relationship analysis allows us to derive that in the Tell and Rif, the number of earthquake with magnitude above Mw 4.0, since 1925 normalized to decade and to square cell with 100-km sides is equal to 2.6 and 1.91, respectively. This study provides the first detailed information about the potential seismicity of these large domains, including maximum regional magnitudes, characteristics of spatial clustering, and distribution of seismic energy release.     

March 11, 2011 M 9.0 Tohoku earthquake in Japan: Tectonic setting of source,macroseismic, seismological,and geodynamic manifestations

The results of interpretation of seismological, geological, geophysical, geodetic, and macroseismic data on the source zone of the catastrophic Tohoku earthquake with M = 8.8–9.0 (from different estimates), which occurred March 11, 2011 off the eastern coast of Honshu Island, are reported. Consideration of the seismotectonic features of the Western Pacific; the distribution of epicenters and hypocenters of the main shock, fore- and aftershocks; the solution of focal mechanisms of the strongest shocks; and the data on directions of lateral and vertical displacement of the island surface makes it possible to contour the source region, reconstruct the structure of the source in the subsurface, and estimate the deformation of the lithosphere resulted from this great seismic event.     

Seismic hazard scenario and attenuation model of the Garhwal Himalaya using near-field synthesis from weak motion seismometry

In this paper, we present a seismic hazard scenario for the Garhwal region of the north-western Himalayan range, in terms of the horizontal Peak Ground Acceleration. The scenario earthquake of moment magnitude M _w 8.5 has a 10% exceedance probability over the next 50 years. These estimates, the first for the region, were calculated through a stepwise process based on:

• An estimation of the Maximum Credible Earthquake from the seismicity of the region and Global Seismic Hazard Assessment Program considerations, and
• four seismotectonic parameters abstracted from near field weak-motion data recorded at five stations installed in Chamoli District of the Garhwal region in the aftermath of the 1999 Chamoli earthquake. The latter include
• The frequency dependent power law for the shear wave quality factor, Q _S
• the site amplification at each station using horizontal-to-vertical-spectral ratio and generalized inversion technique
• source parameters of various events recorded by the array and application of the resulting relations between the scalar seismic moment M ₀ (dyne-cm) and moment magnitude M _w and the corner frequency, ? _c (Hz) and moment magnitude M _w to simulate spectral acceleration due to higher magnitude events corresponding to the estimated Maximum Credible Earthquake, and
• regional and site specific local spectral attenuation relations at different geometrically central frequencies in the low, moderate and high frequency bands.

     

Disaster prevention,disaster preparedness and local community resilience within the context of disaster risk management in Cameroon

This paper presents a simulation of three components of near-field ground shaking recorded during the main shock at three stations of the September 16, 1978, Tabas (M _w = 7.4), Iran, earthquake, close to the causative fault. A hybrid method composed of a discrete wavenumber method developed by Bouchon (Bouchon in Bull Seismol Soc Am 71:959–971, 1981; Cotton and Coutant in Geophys J Int 128:676–688, 1997) and a stochastic finite-fault modeling based on a dynamic corner frequency proposed by Motazedian and Atkinson (Bull Seismol Soc Am 95:995–1010, 2005), modified by Assatourians and Atkinson (Bull Seismol Soc Am 97:935–1949, 2007), is used for generating the seismograms at low (0.1–1.0 Hz) and high frequencies (1.0–20.0 Hz), respectively. The results are validated by comparing the simulated peak acceleration, peak velocity, peak displacement, Arias intensity, the integral of velocity squared, Fourier spectrum and acceleration response spectrum on a frequency-by-frequency basis, the shape of the normalized integrals of acceleration and velocity squared, and the cross-correlation with the observed time-series data. Each characteristic is compared on a scale from 0 to 10, with 10 being perfect agreement. Also, the results are validated by comparing the simulated ground motions with the modified Mercalli intensity observations reported by reconnaissance teams and showed reasonable agreement. The results of the present study imply that the damage distribution pattern of the 1978 Tabas earthquake can be explained by the source directivity effect.     

Monitoring active fault creep as a tool in seismic hazard mitigation. Insights from creepmeter study at Chihshang,Taiwan

In 1998 we installed five creepmeters across the Chihshang Fault, the active plate suture in eastern Taiwan. Daily creepmeter data indicated decreasing creeping rate from 1999 to 2003, suggesting increasing seismic hazard. The fault was ruptured by the Chengkung earthquake ($\mathit{Mw}=6.6$) on 10 December 2003. Through extrapolation of our earlier creep data of 1986–1991 and 1992–1997, we evaluate the minimum deficit in aseismic creep shortening as 106 or 46 mm (respectively) before this earthquake. The near-surface co-seismic shortening was limited, but the total shortening resulting from the earthquake, including post-seismic creep, was about 97 mm. This suggests that near the surface most of the detectable deficit has been absorbed by this earthquake and subsequent creep. We thus point out that creepmeter installation and monitoring bring a powerful tool in seismic hazard mitigation. To cite this article: J.-C. Lee et al., C. R. Geoscience 337 (2005).     

Effect of frequency-dependent radiation pattern in the strong motion simulation of the 2011 Tohoku earthquake,Japan, using modified semi-empirical method

We perform a strong ground motion simulation using a modified semi-empirical technique (Midorikawa in Tectonophysics 218:287–295, 1993), with frequency-dependent radiation pattern model. Joshi et al. (Nat Hazards 71:587–609, 2014) have modified the semi-empirical technique to incorporate the modeling of strong motion generation areas (SMGAs). A frequency-dependent radiation pattern model is applied to simulate high-frequency ground motion more precisely. Identified SMGAs (Kurahashi and Irikura in Earth Planets Space 63:571–576, 2011) of the 2011 off the Pacific coast of Tohoku earthquake (M _w  = 9.0) were modeled using this modified technique. We analyzed the effect of changing seismic moment values of SMGAs on the simulated acceleration time series. Final selection of the moment values of SMGAs is based on the root-mean-square error (RMSE) of waveform comparison. Records are simulated for both frequency-dependent and constant radiation pattern function. Simulated records for both cases are compared with observed records in terms of peak ground acceleration, peak ground velocity and pseudo-acceleration response spectra at different stations. Comparison of simulated and observed records in terms of RMSE suggests that the method is capable of simulating record, which matches in a wide frequency range for this earthquake and bears realistic appearance in terms of shape and strong motion parameters. The results confirm the efficacy and suitability of rupture model defined by five SMGAs for the developed modified technique.     

Computation of Probabilistic Seismic Hazard for the GHSAP Test Area ‘Caucasus’

As a result of work carried out during the first two stages of the Global Seismic Hazard Assessment Program (GSHAP) for the Test Area Caucasus, a uniform earthquake catalogue was compiled and a Seismic Source Zones Model was designed. At the final stage of the program, the computation of seismic hazard was done by different methods.The results of a computation done using the Probabilistic Seismic Hazard Assessment methodology, as well as primary intermediate steps and preparatory work are given in the present paper. Peak horizontal ground acceleration is chosen as the parameter representing seismic hazard. Final computer calculations were done with the SEISRISK III program. The two final Seismic Hazard maps for different return periods are presented. The work was carried out at the National Survey for Seismic Protection of the Republic of Armenia.     

Quality control in geochemistry from a comparison of four central tendency and five dispersion estimators and example of a geochemical reference material

Data quality control in geochemistry constitutes a fundamental problem that is still to be solved from the application of statistics and computation. We used refined Monte Carlo simulations of 10,000 replications and 190 independent experiments for sample sizes of 5 to 100. Statistical contaminations of 1 to 4 observations were used to compare 9 statistical parameters (4 central tendency—mean, median, trimean, and Gastwirth mean, and 5 dispersion estimates—standard deviation, median absolute deviation, S _n , Q _n , and \( {\widehat{\sigma}}_n \)). The presence of discordant observations in the data arrays rendered the outlier-based and robust parameters to disagree with each other. However, when the mean and standard deviation (outlier-based parameters) were estimated from censored data arrays obtained after the identification and separation of outlying observations, they generally provided a better estimate of the population than the robust estimates obtained from the original data arrays. This inference is contrary to the general belief, and therefore, reasons for the better performance of the outlier-based methods as compared to the robust methods are suggested. However, when all parameters were estimated from censored arrays and appropriate precise and accurate correction factors put forth in this work were applied, all of them became fully consistent, i.e., the mean agreed with the median, trimean and Gastwirth mean, and the standard deviation with the median absolute deviation, S _n , Q _n , and \( {\widehat{\sigma}}_n \). An example of inter-laboratory chemical data for a Hawaiian reference material BHVO-1 included sample sizes from 5 to 100, which showed that small samples of up to 20 provide inconsistent estimates, whereas larger samples of 20–100, especially >40, were more appropriate for estimating statistical parameters through robust or outlier-based methods. Although all statistical estimators provided consistent results, our simulation study shows that it is better to use the censored sample mean and population standard deviation as the best estimates.     

Non-uniform model for the synchrotron radiation of Sgr A* and other low-luminosity galactic nuclei

A model for non-uniform source of synchrotron radiation with a power-law radial distribution of the magnetic field and relativistic-electron density along one-or two-sided jets is described. Non-relativistic jets with both constant cross sections (collimated jets) and cross sections that are proportional to distance (conical jets) are considered. Formulas that can be used to determine source parameters from the spectral index, source size, and index of the relativistic-electron energy spectrum based on multi-frequency observations are obtained. In the case of a conical jet, these formulas coincide with the analogous formulas for a spherical source obtained by A.P. Marscher. Relations that can be used to estimate the magnetic-field strength from the brightness temperature in the self-absorbed region are also obtained. As examples, the inhomogeneous-source model is applied to the compact radio sources at the centers of the Milky Way, Sgr A*, and the low-luminosity galactic nuclei M81* and M87*, which are associated with supermassive black holes. The inner radius of the radiation region is determined. For Sgr A*, this distance turns out to be comparable to the gravitational radius, smaller than the radius of the last stable orbit for a non-rotating black hole, and consistent with the radius of the last stable orbit expected for a rotating black hole. The inner radii in M81* and M87* are ～15 R _S , an order of magnitude larger than for Sgr A*. Estimates of the magnetic field at the inner radius are 400 G for M81*, 0.65–5.3 kG for Sgr A*, and 20–100 kG for M87*. These magnetic fields and the Blandford-Znajek model for the radiation of a rotating black hole are used to estimate the rotational speed of the black holes, which are in agreement with the characteristic variability time scales for these three objects. However, the accuracy of these estimates is modest, and is limited primarily by the accuracy of interferometric measurements at millimeter wavelengths.