Feasibility study of computed vs measured high b-value(1400 s/mm2) diffusion-weighted MR images of the prostate

AIM: To evaluate the impact of computed b = 1400 s/mm2(C-b1400) vs measured b = 1400 s/mm2(M-b1400) diffusion-weighted images(DWI) on lesion detection rate, image quality and quality of lesion demarcation using a modern 3T-MR system based on a small-field-of-view sequence(sFOV). METHODS: Thirty patients(PSA: 9.5 ± 8.7 ng/mL; 68 ± 12 years) referred for magnetic resonance imaging(MRI) of the prostate were enrolled in this study. All measurements were performed on a 3T MR system.For DWI, a single-shot EPI diffusion sequence(b = 0, 100, 400, 800 s/mm2) was utilized. C-b1400 was cal-culated voxelwise from the ADC and diffusion images. Additionally, M-b1400 was acquired for evaluation and comparison. Lesion detection rate and maximum lesion diameters were obtained and compared. Image quality and quality of lesion demarcation were rated accord-ing to a 5-point Likert-type scale. Ratios of lesion-to-bladder as well as prostate-to-bladder signal intensity(SI) were calculated to estimate the signal-to-noise-ratio(SNR). RESULTS: Twenty-four lesions were detected on M-b1400 images and compared to C-b1400 images. C-b1400 detected three additional cancer suspicious lesions. Overall image quality was rated significantly better and SI ratios were significantly higher on C-b1400(2.3 ± 0.8 vs 3.1 ± 1.0, P < 0.001; 5.6 ± 1.8 vs 2.8 ± 0.9, P < 0.001). Comparison of lesion size showed no significant differences between C- and M-b1400(P = 0.22). CONCLUSION: Combination of a high b-value extrap-olation and sFOV may contribute to increase diagnostic accuracy of DWI without an increase of acquisition time, which may be useful to guide targeted prostate biopsies and to improve quality of multiparametric MRI(mMRI) especially under economical aspects in a pri-vate practice setting.

Laboratory investigation on relationship between degree of saturation, B-value and P-wave velocity

Laboratory tests were performed on Toyoura sand specimens to investigate the relationship between degree of saturation Sr, B-value and P-wave velocity Vp. Different types of pore water （de-aired water or tap water） and pore gas （air or CO2） as well as different magnitudes of back pressure were used to achieve different Sr （or B-value）. The measured relationship between B-value and Vp was not consistent with the theoretical prediction. The measurement shows that the Vp value in the specimen flushed with de-aired water is independent of B-value （or St） and is always around the one in fully saturated condition. However, the Vp value in the specimen flushed with tap water increases with B-value, but the shape of the relationship between Vp and B-value is quite different from the theoretical prediction. The possible explanation for the discrepancy between laboratory measurement and theoretical prediction lies in that the air exists in the water as air bubbles and therefore the pore fluid （air-water mixture） is heterogeneous instead of homogenous assumed in the theoretical prediction.

Continuous-cyclic variations in the b-value of the earthquake frequency-magnitude distribution

Seismicity of the Earth （M ≥ 4.5） was compiled from NEIC, IRIS and ISC catalogues and used to compute b-value based on various time windows. It is found that continuous cyclic b-variations occur on both long and short time scales, the latter being of much higher value and sometimes in excess of 0.7 of the absolute b-value. These variations occur not only yearly or monthly, but also daily. Before the occurrence of large earthquakes, b-values start increasing with variable gradients that are affected by foreshocks. In some cases, the gradient is reduced to zero or to a negative value a few days before the earthquake occurrence. In general, calculated b-values attain maxima 1 day before large earthquakes and minima soon after their occurrence. Both linear regression and maximum likelihood methods give correlatable, but variable results. It is found that an expanding time window technique from a fixed starting point is more effective in the study of b-variations. The calculated b-variations for the whole Earth, its hemispheres, quadrants and the epicentral regions of some large earthquakes are of both local and regional character, which may indicate that in such cases, the geodynamic processes acting within a certain region have a much regional effect within the Earth. The b-variations have long been known to vary with a number of local and regional factors including tectonic stresses. The results reported here indicate that geotectonic stress remains the most significant factor that controls b-variations. It is found that for earthquakes with Mw ≥ 7, an increase of about 0.20 in the b-value implies a stress increase that will result in an earthquake with a magnitude one unit higher.

Atomic Decompositions of Weak Hardy Spaces of B-Valued Martingales

The atomic decompositions of weak Hardy spaces of Banach-space-valued martingales are given. With the help of the atomic decompositions, some inequalities for B-valued martingales are established in the case 0〈r≤1. Here the results are connected closely with the p-uniform smoothness and q-uniform convexity of Banach spaces which the martingales take values in.

Spatial Variations of b-Values in the Coastal Area of Guangdong

We used earthquake catalogs recorded by Guangdong Seismological Network from 2008 to 2014 to resolve the spatial variations of b-values in the coastal area of Guangdong, particularly in three key research areas(Yangjiang, Heyuan, and offshore Nanao Island) with strong seismicity. Our results revealed that b-values exhibited significant spatial variations, and zones with low b-values could indicate the most likely seismogenic area of large earthquakes. We observed three clear low b-value patches in the offshore Nanao Island. We found a distinct high b-value peak at the depth of 11 km and two minimum peaks at about 14 and 7–8 km in the Yangjiang area. The overall b-values generally decrease with depth in the Heyuan area. The spatial variations of b-values reflect tectonic anomalies; that is, the ‘low-high-low' distribution of b-values in the offshore Nanao Island and the Yangjiang area may indicate the anomaly of the crustal structure with a weak layer. The b-values of reservoir-induced seismicity are obviously lower than that induced by tectonism. This finding indicates that the reservoir area is generally at high stress state under the condition of high pore pressure. We inferred that large earthquakes might be prone to occur at 10–12 km depth in the offshore Nanao Island, at 12–15 km depth in the Yangjiang area, and at the lower part of the seismic activity zone in the Heyuan area. Moreover, the upstream area of the Xinfengjiang reservoir is the most likely area of future large earthquakes in the Heyuan area.

The spatiotemporal variation of the b-value and its tectonic implications in North China

In this study, we adopt an improved Bayesian approach based on free-knot B-spline bases to study the spatial and temporal distribution of the b-value. Synthetic tests show that the improved Bayesian approach has a superior performance compared to the Bayesian approach as well as the widely used maximum likelihood estimation （MLE） method in fitting the real variation of b-values. We then apply the improved Bayesian approach to North China and find that the b-value has a clear relevance to seismicity. Temporal changes of b-values are also investigated in two specific areas of North China. We interpret sharp decreases in the b-values as useful messages in earthquake hazard analysis.

The b-value and fractal dimension of local seismicity around Koyna Dam(India)

Earthquakes began to occur in Koyna region （India） soon after the filling of Koyna Dam in 1962. In the present study, three datasets 1964-1993, 1993-1995, and 1996-1997 are analyzed to study the b-value and fractal dimension. The b-value is calculated using the Gutenberg- Richter relationship and fractal dimension Dcorr. using correlation integral method. The estimated b-value and Dcorr. of this region before 1993 are found to be in good agreement with previously reported studies. In the sub- sequent years after 1995, the b-value shows an increase. The estimated b-values of this region are found within the limits of global average. Also, the pattern of spatial clustering of earthquakes show increase in clustering and migration along the three zones called North-East Zone, South-East Zone （SEZ）, and Warna Seismic Zone. The earthquake events having depth ≤5 km are largely confined to SEZ. After 1993, the Dcorr. shows decrease, implying that earth- quake activity gets clustered. This seismic clustering could be helpful for earthquake forecasting.

Spatio-temporal distribution of earthquake occurrence in Eastern Himalaya and vicinity(26°N-31°N and 87°E-98°E)based on b-value and fractal dimension

This study investigates the spatial and temporal variation of fractal dimension and b-value for the eastern part of the Himalaya and adjoining area(26°N–31°N and 87°E–98°E).The analysis is carried out on the earthquake dataset of 1373 events(Mc=4.0)by sliding window technique for the period 1964 to 2020.The region is divided into three sub regions A(87°E–92°E),B(92°E–94°E)and C(94°E–98°E).The b-value computed for the region A comprising eastern Nepal is smaller compared to other two regions which infers the possible high stress and asperities in the region.High spatial fractal dimension(Dc>1.5)and low temporal fractal dimension(Dt<0.31)are computed for the regions.High spatial fractal dimension may indicate that fractures generating earthquakes are approaching a 2D structure and low temporal fractal dimension implies high clustering of earthquake’s epicenters.The b value shows a weak negative correlation with Dc for regions A and C while a weak positive correlation is observed for the region B.Based on b-value and fractal dimension,this study explains the frequency of earthquakes and heterogeneity of the seismogenic structure in this part of the Himalaya.

b-value and fractal dimension variations in Iran

Iran sits on a region with a high intrinsic level of seismic activity due to its tectonic setting.Through statistical examination of the earthquakes listed in the catalogue from International Institute of Earthquake Engineering and Seismology(IIEES),this research attempted to calculate some seismicity factors and find correlation between them.A preliminary analysis indicated changes in theb-value of the Gutenberg-Richter relationship over the study region.Thus,the study area was divided into five zones(Alborz,Zagros,Azerbaijan,Central and East)andb-value was computed for each zone.Considering faulting mechanism styles and theb-values in the region,it was found that the lowestb-values belong to the thrust events and strike-slip faulting earthquakes have intermediate values.These findings support previous studies.Furthermore,results ofb-value calculation were used for the estimation of accumulated differential stresses(σ1–σ2)over each zone.Overall,theb-value for Iran is averagely low which signifies the high stress tectonic regime in this region.Also,by having calculated fractal dimension(D)in each zone,a correlation obtained showing that in Iran region,theb-value correlates to fractal dimension byD=4.2b–2 relation which does not support Aki's(1981)speculation ofD=3b/c.

Correcting the Gutenberg–Richter b-value for effects of rounding and noise

The effects of magnitude rounding and of the presence of noise in the rounded magnitudes on the estimation of the Gutenberg-Richter b-value are explored, and the ways to correct for these effects are proposed. For typical values, b = 1 and rounding interval △M = 0.1, the rounding error is approximately -10^-3 and it can be corrected to a negligible approximately -10^-5. For the same typical values, the effect of noise can be larger, depending on the characteristics of the noise distribution; for normally distributed noise with standard deviation σ = 0.1, the correct b-value may be underestimated by a factor - 0.97.

BOUNDEDNESS OF DYADIC DERIVATIVE AND CESARO MEAN OPERATOR ON SOME B-VALUED MARTINGALE SPACES

In this article, it is proved that the maximal operator of one-dimensional dyadic derivative of dyadic integral I＊ and Cesàro mean operator σ＊ are bounded from the B-valued martingale Hardy spaces pΣα, Dα, pLα, p H#α, pKr to Lα （0 α ∞）, respectively. The facts show that it depends on the geometrical properties of the Banach space.

Joint analysis of b-value and apparent stress before the 2011 M_(W)9.0 Tohoku-Oki, Japan earthquake

Detecting tempo-spatial changes of crust stress associated with major earthquakes has implications for understanding earthquake seismogenic processes.We conducted a joint analysis of b-value and apparent stress in the source region before the March 11,2011 MW9.0 TohokuOki,Japan earthquake.Earthquakes that occurred between January 1,2000 and March 8,2011 were used to estimate bvalues,while source parameters of events with magnitudes of Ms5.0–6.9 between January 1,1997 and March 8,2011 were used to calculate the apparent stresses.Our results show that the average b-value decreased steadily from 1.26 in 2003 to 0.99 before the Tohoku-Oki mainshock.This b-value decrease coincided with an increase in the apparent stress from 0.65 MPa to 1.64 MPa.Our results reveal a clear negative correlation between the decrease in b-value and increase in apparent stress,which lasted for approximately eight years prior to the 2011 mainshock.Additionally,spatial pattern results of the relative change in b-value show that the area associated with drastic b-value decreases(25%or greater)was concentrated near the 2011 mainshock epicenter.The joint analysis of b-value and apparent stress provides a promising method for detecting anomalies that could serve as potential indicators of large earthquakes.

ATOMIC DECOMPOSITION FOR B-VALUED R.V. SEQUENCE SPACES

In this article,atomic decompositions and the duals of some B-valued r.v.se- quence spaces are investigated.The results show that it closely depends on the geometrical properties of the sequence that take values in.

Self-Organized Fractal Seismicity and b-Value of Aftershocks of the 2015 Gorkha Earthquake, Nepal

The Gorkha Earthquake that occurred on 25th April 2015 was a long anticipated, low angle thrust-faulting shallow event in Central Nepal that devastated the mountainous southern rim of the High Himalayan range. The earthquake was felt throughout central and eastern Nepal, much of the Ganges River plain in northern India, and northwestern Bangladesh, as well as in the southern parts of the Plateau of Tibet and western Bhutan. Two large aftershocks, with magnitudes 6.6 and 6.7, occurred in the region within one day of the main event, and several dozen smaller aftershocks occurred in the region during the succeeding days. In this study, we have analyzed the 350 aftershocks of the 2015 Gorkha Earthquake of Mw 7.8 to understand the spatial and temporal distribution of b-value and the fractal correlation dimension. The b-value is found to be 0.833 ± 0.035 from the Gutenberg-Richter relation by the least squares method and 0.95 ± 0.05 by the maximum likelihood method, indicating high stress bearing source zone. The spatial and temporal correlation dimension is estimated to be 1.07 ± 0.028 and 0.395 ± 0.0027 respectively. Spatial correlation dimension suggests a heterogeneous distribution of earthquake epicenters over a linear structure in space, while the temporal correlation dimension suggests clustering of aftershock activity in the time domain. The spatial variation of the b-value reveals that the b-value is high in the vicinity of the mainshock which is due to the sudden release of stress energy in the form of seismic waves. The spatial distribution of correlation dimension further confirms a linear source in the source zone as it varies from 0.8-1.0 in most of the region. We have also studied the temporal variation of b-value and correlation dimension that shows positive correlation for about first 15 days, then a negative correlation for next 45 days and after that, a positive correlation. The positive correlation suggests that the probability of large magnitude earthquakes decreases in response to increased fragmentation of the fault zone. The negative correlation means that there is a considerable probability of occurrences of large magnitude earthquakes, indicating stress release along the faults of a larger surface area [1]. The correlation coefficient between b-value and the correlation dimension is estimated to be 0.26, which shows that there is no significant relation between them.

Analysis of the b-value and Seismic Deformation in the North of China

The accuracy of b-value is limited by the uncertainty of the magnitude.In order to improvethe accuracy of the b-value,a statistic methodology was used to estimate the number of eventsbased on that there are several magnitude values for one historical event by several methods toestimate the magnitude.The b-value and seismic strain rates were calculated for the threeregions in the North of China.The method proved to be valid for all the regions,especiallyfor the data with large fluctuation.In order to study the stress state,the strain rate tensorand the principal axes of tension and compression were calculated for each of three regions bythe data of the focal mechanism,finally the stress state in each region was discussed.

Spatial Distribution of Return Period in China Based on b-value Obtained by Using a New Method

The seismicity in the territory of China, a seismotectonically complicated region, has been examined by using some complete samples of the earthquakes occurred during the last two centuries (1800 - 1999). The A-value of the Gutenberg-Richter relation was estimated by using those data samples. Taking into account the fact that the b-value is spatially more stable than the a value, the b values were calculated at the nodes of a normal grid superposing on the whole area studied and their distribution were examined. The results show that the b values increase smoothly from 0.4 to 0.93. Furthermore, keeping the b values obtained fixed, the a value distribution was also examined. In order to display more detailed information about the seismicity, smaller cell surface (10000 km2) for the calculation of the a value has chosen. The mean return period for different magnitudes was also calculated for each small cell.

Analysis on b-value Abnormality of the Minxian-Zhangxian,Gansu M_S6.6 Earthquake

In this paper,based on Gansu Digital Seismic Network data since 1990,we calculated b-value and ⊿ b-value of the neighborhood area before the 2013 Minxian-Zhangxian M_S6. 6,earthquake. The results show that the earthquake occurred at the margin of the significantly lower b-value area in southeast of Gansu and before the earthquake the neighborhood area had an obviously abnormal ⊿ b-value. From the earthquake and the spatial distribution of low b-value, the region still has the possibility of generating dangerous strong earthquakes.

Spatial Distribution Characteristic of the b-value in Dabie-Sulu and Its Adjacent Areas

The Tanlu （Tancheng-Lujiang） fault is one of the major structures in eastern China, which cuts across different blocks and controls the tectonic activity. Using the seismic data from the China Earthquake Networks Center, we investigate the spatial variations of the b-value in the Dabie-Sulu （Jiangsu-Shandong） organic belt by calculating the b-value of each grid （1~ xl~） from 1970 to 2010. The study shows that： the b-value is smaller in the Tanlu fault and its adjacent area, which might suggest that the major earthquake recurrence period is relatively long due to the lower frequency of small earthquake activity. In both sides of Tanlu fault, the b-value is higher at the edge of Sulu block and the eastern part of Dabie orogenic belt. The b values are higher in North China central orogenic belt, and small earthquake activity which occurred along the central orogenic belt is frequent. Additionally, combined with geological and geophysical study, we find that the b-value has a certain correspondence relationship with the lithologic distribution, which informs us that seismic activity features are closely related to the inhomogeneous media in the crust.

A Moment Characterization of B-Valued Generalized Functionals of White Noise

Kernel theorems are established for Bananch space-valued multilinear mappings, A moment characterization theorem for Banach space-valued generalized functionals of white noise is proved by using the above kernel theorems. A necessary and sufficient condition in terms of moments is given for sequences of Banach space-valued generalized functionals of white noise to converge strongly. The integration is also discussed of functions valued in the space of Banach space-valued generalized functionals.

Duals of Hardy Spaces of B-valued Martingales for 0

The dual of B-valued martingale Hardy space Hr^s（p） （B） with small index 0 〈 r ≤ 1, which is associated with the conditional p-variation of B-valued martingale, is characterized. In order to obtain the results, a new type of Campanato spaces for B-valued martingales is introduced and the classical technique of atomic decompositions is improved. Some results obtained here are connected closely with the p-uniform smoothness and q-uniform convexity of the underlying Banach space.