Vent area and depositional mechanisms of the Upper Member of the Neapolitan Yellow Tuff (Campi Flegrei, Italy): new insights from directional fabric through image analysis

In order to provide new information about the source area and depositional mechanisms of the Upper Member of the Neapolitan Yellow Tuff (NYT), a prominent pyroclastic deposit of the Campi Flegrei Volcanic District (southern Italy), statistics on directional fabric, by means of computer-assisted image analysis on 32 rock samples, were compiled. Seventeen samples were collected along vertical direction on two selected exposures and fifteen were taken from outcrops widely distributed all around the Campi Flegrei Volcanic District. Fabric measurements within the investigated successions reveal a vertically homogeneous direction of the mean particle iso-orientation, with considerable variability in the strength of particle iso-orientation even at cm-scale. The existence of particle iso-orientation can be related to continuous sedimentation from a concentrated bedload region beneath suspension currents, producing massive or inversely graded beds by traction carpet sedimentation. The considerable vertical variability in the strength of iso-orientation is the result of very unstable flow regimes, up to the extreme condition of discrete depositional events, with a variable combination of traction carpet and/or direct suspension sedimentation. The vertical homogeneity in the mean orientation values, found in the investigated sections, may derive from the sequential deposition of laminae to thin beds, whose relatively flat upper surfaces were unable to significantly deflect the depositional system of the following currents. According to the observed homogeneous mean particle orientation values along the investigated vertical profiles, samples collected through areal distribution are considered representative of the local paleo-flow directions of the whole deposit. The mean directions of the samples collected areally show two different coherent patterns which point to the existence of two different source areas. The first, which includes all samples from the northern outcrops, appears to converge in a narrow area about 2 km NE of the town of Pozzuoli, largely in coincidence with the inferred area on the basis of the pumice fall distribution. The second, which includes samples from Capo Miseno and Posillipo areas, points to the central part of the Pozzuoli Bay, about 4 km offshore the town of Pozzuoli.     

Directional fabric measurements: an investigative approach to transport and depositional mechanisms in pyroclastic flows

Computer-assisted image analysis data of rock fabrics from two quaternary ignimbrites in the Vulsini and Cimini Volcanic Districts of Central Italy are interpreted in terms of transport and depositional mechanisms. Samples were collected vertically at m spaces up two sections through each deposit. The Orvieto–Bagnoregio ignimbrite (OBI) is a non-welded ignimbrite that shows both fluctuations in the mean particle orientation values of up to approximately ±60°, and large variations in the strength of particle iso-orientation with height. The circular frequency distributions of particle orientations are almost always anisotropic and unimodal, in line with a theoretical Von Mises distribution (the circular equivalent of a unimodal, log–normal distribution). In contrast, the welded Cimina ignimbrite (CI) shows vertical homogeneities in mean orientation values with height, and generally lower degrees of anisotropy. Such differences are interpreted as being the results of different depositional mechanisms: incremental deposition at the base of a density-stratified, partially turbulent flow for the OBI; deposition of a laminar mass flow for the CI. In the former case, during transport particles under solidus temperature are subjected to a frictional regime, particles gliding and dispersive pressures, which finally produce size-inverse grading and variable fabric development, depending on the residence time of particles at the basal shear conditions. In the latter case, elongated particles, supported in a laminar flowing viscous matrix, undergo periodic motions which tend to develop parallel-to-flow iso-orientation. Fabric data in the deposit suggest vertical constancy in the rheological properties of the flow, absence of rheological decoupling and (shearing pervasively during transport) a minor importance of plug horizons.     

Relationships between three dispersion measures used in flood frequency analysis

Three dispersion measures of a random variable, i.e., the standard deviation, the mean deviation (MD) about the mean and the second L-moment, are analyzed in terms of their properties and mutual relationships. Emphasis is placed on the MD, as it is less recognized than two other dispersion measures. The relationships between the dispersion measures are derived for distributions commonly applied in flood frequency analysis (FFA). For distributions that are unbounded, there is a distribution-dependent constant value of the ratio of dispersion measures, or equivalently of respective coefficients of variation. For two-parameter distributions that are lower-bounded, the relationship between the coefficients of variation is also distribution dependent and is not linear. For lower-bounded three-parameter distributions, the dispersion measure ratios, or equivalently the ratios of coefficients of variation, depend on the coefficient of skewness and show a strong distributional dependence. For selected distributions, the three dispersion measures are compared both in terms of the robustness to the largest samples element and the accuracy of upper quantile estimation. The MD statistics may be highly competitive to the two other dispersion measure statistics if applied in FFA for parameters estimation.     

Coupled degassing and crystallization: experimental study at continuous pressure drop, with application to volcanic bombs

 Experiments on degassing of water-saturated granite melts with a pressure drop from 100 and 450 MPa to 40 and 120 MPa, respectively, at temperatures close to feldspar liquidus (750–700  °C), were carried out to determine the modality of water exsolution and vesicle formation at the liquidus temperature. Pressure-drop rates as small as approximately 100 bar/day were used. Uniform space distributions of bubbles of exsolved water were obtained with starting glass containing a small fraction (≈0.5 vol.%) of trapped air bubbles. Volume crystallization of feldspar was observed in degassed melts supplied with seeds. Bubble size distributions (BSD) measured in granite glasses after degassing are presented. Data on vesicle characteristics (number, radius, area, elongation) were acquired on images digitized with standard software, while the reconstruction of size distributions was performed with the Schwartz-Saltikov "unfolding" procedure. Bubble size distributions of size classes in the range 5–1000 μm were acquired with proper magnification and satisfactory statistical reliability of determined number densities. The BSDs of the experimental samples are compared with the results of measurements of rapidly degassed products of Mt. Etna and Vulcano Island. Many particular features of the bubble nucleation and growth can be distinguished in an individual BSD. However, the general BSD of the whole data set, including natural ones, can be relatively well described with linear regression in bilogarithmic coordinates. The slope of this regression is approximately 2.8±0.1. This dependence is in striking contrast with distributions theoretically predicted with classical nucleation models based on homogeneous nucleation of vesicles. The theoretical distribution requires the occurrence of strong maxima that are not observed in our experimental and natural samples, thus arguing for heterogeneous nucleation mechanisms. Received: 1 October 1998 / Accepted: 25 June 1999     

Stochastic analysis of flux and head moments in a heterogeneous aquifer system

This study investigates the behavior of flux and head in a strongly heterogeneous three-dimensional aquifer system. The analyses relied on data from 520 slug tests together with 38,000 one-foot core intervals lithological data from the site of the General Separations Area in central Savannah River Site, South Carolina, USA. The skewness in the hydraulic conductivity histograms supported the geologic information for the top two aquifers, but revealed stronger clay content, than was reported for the bottom aquifer. The log-normal distribution model described adequately the hydraulic conductivity measurements for all three aquifers although, other distributions described equally well the bottom aquifer measurements. No apparent anisotropy on the horizontal plane was found for the three aquifers, but ratios of horizontal to vertical correlation lengths between 33 and 75 indicated a strong stratification at the site. Three-dimensional Monte Carlo stochastic simulations utilized a grid with larger elements than the support volume of measurements, but of sub-REV (representative elementary volume) dimensions. This necessitated, on one hand, the use of upscaled hydraulic conductivity expressions, but on the other hand did not allow for the use of anisotropic effective hydraulic conductivity expressions (Sarris and Paleologos in J Stoch Environ Res Risk Assess 18: 188–197, 2004). Flux mean and standard deviations components were evaluated on three vertical cross-sections. The mean and variance of the horizontal flux component normal to a no-flow boundary tended to zero at approximately two to three integral scales from that boundary. Close to a prescribed head boundary both the mean and variance of the horizontal flux component normal to the boundary increased from a stable value attained at a distance of about five integral scales from that boundary. The velocity field 〈q_x〉 was found to be mildly anisotropic in the top two aquifers, becoming highly anisotropic in the bottom aquifer; 〈q_y〉 was anisotropic in all three aquifers with directions of high continuity normal to those of the 〈q_x〉 field; finally, 〈q_z〉 was highly anisotropic in all three aquifers, with higher continuity along the east–west direction. The mean head field was found to be continuous, despite the high heterogeneity of the underlying hydraulic conductivity field. Directions of high continuity were in alignment with field boundaries and mean flow direction. Conditioning did not influence significantly the expected value of the flux terms, with more pronounced being the effect on the standard deviation of the flux vector components. Conditioning reduced the standard deviations of the horizontal flux components by as much as 50% in the bottom aquifer. Variability in the head cross-sections was affected only marginally, with an average 10% reduction in the respective standard deviation. Finally, the location of the conditioning data did not appear to have a significant effect on the surrounding area, with uniform reduction in standard deviations.     

Sigmoidal particle density distribution in a subplinian scoria fall deposit

A general expression to describe particle density distribution in tephra fall deposits is essential to improve fallout tephra mass determination and numerical modelling of tephra dispersion. To obtain particle density distributions in tephra fall deposits, we performed high-resolution componentry and particle density analyses on samples from the 2006 subplinian eruption of Tungurahua volcano in Ecuador. Six componentry classes, including pumice and scoria, have been identified in our sample collection. We determined the class of 300 clasts in each 0.5? fractions from ?4.5? to 3.5? and carried out water pycnometry density measurements on selected size fractions. Results indicate that the mean particle density increases with ? up to a plateau of ~2.6?g/cm³ for clasts finer than 1.5?. The density of scoria and pumice increases between ?3 and 1?, while dense particle density is sub-constant with grainsize. We show that the mean particle density ?? of the vesicular fractions is a function of grainsize i (? scale) given by a sigmoidal law: $ \mu (i)={{{K+\beta }} \left/ {{\left( {1+\alpha {e^{-ri }}} \right)}} \right.} $ , where K, ??, ?? and r are constants. These sigmoidal distributions can be used to determine accurately the load of each componentry class and should be applicable to many tephra deposits and for modelling purposes.     

Structure of the present-day velocity field of the crust in the area of the Central-Asian GPS network

The structure of the vector field of present-day horizontal velocities on the Earth’s surface is investigated with the use of the territory covered by the Central-Asian GPS network as an example. A method of identification of groups of GPS points (statistically rigid clusters), for which the rate of change in distances between them is virtually zero, is proposed and realized. Sites of the Earth’s surface (regions) containing such groups of GPS points, within the required measurement accuracy, move in the horizontal plane as two-dimensional rigid bodies. The clustering algorithm, which is based on the Student t statistics in determining the so-called statistical “sample cost,” is developed and carefully tested. The results of identification of regions take into account possible random errors in velocity measurements and do not depend on the chosen frame of reference. The method of identification of regions is sufficiently stable with respect to variations in the number of GPS sites used for clustering. Of all the sites of the Central-Asian GPS network, 323 points were selected for clustering. These sites were measured from 3 to 11 times over an 11-year interval of observations (1995–2005). The estimates of errors of velocity measurement for these sites must not exceed 1.0 mm/yr. As a result, 29 statistically rigid clusters, containing from 3 to 17 GPS sites, were identified, and the kinematic regimes of motion of regions corresponding to these clusters were determined with respect to the stable part of Eurasia. With the general direction of the translatory motion of regions toward the north, the majority of them rotate counterclockwise. Nearly one third of GPS sites do not participate in the formation of clusters; these points fall into the interregional space (IRS), which is characterized by increased strain rates. The IRS structure is partitioned into zones with four directions, of which two directions virtually coincide with directions of the principal axes of the regional strain rate tensor, and the two other directions are oriented diagonally to the principal axes. The axis of the maximum rate of the regional shortening has a north-northwestern orientation. It is in this direction (mainly along the IRS) that the crust’s contraction takes place. There is no spatial correlation between IRS zones and geological faults; however, their angular distributions and the directions of strike-slips on them are interrelated. The resulting patterns of regional motions and IRS deformations consistently reflect the dynamic action of the Indian plate on the territory under investigation.     

3D particle size distributions from 2D observations: stereology for natural applications

We have developed a general formulation for stereological analysis of particle distributions which is applicable to any particle size or size distribution (not limited to log-normal, unimodal, etc.). We have applied numerical techniques to define intersection probability distributions for any shapes (previously only known for spheres), and quantified the errors involved in using spherical coefficients for various non-spherical particles. This stereological technique is based on knowing the probability distribution of random cross-sections through various particles so that ‘small circle' cross-sections can be subtracted from an observed population to provide the 3D size distribution of particles. The results indicate that the most important parameter controlling calculated size distribution is particle aspect ratio. For a distribution of particles with a specific aspect ratio or range of aspect ratios, variations of particle form (spherical vs. cubic; rectangular vs. elliptical) do not alter the results, so the technique can be applied to a range of particle shapes. Applications can be made in a petrology, volcanology, and other fields, only a few of which can also be treated using expensive X-ray techniques.     

Statistical wind analysis for near-space applications

Statistical wind models were developed based on the existing observational wind data for near-space altitudes between 60 000 and 100 000 ft (18–30 km) above ground level (AGL) at two locations, Akon, OH, USA, and White Sands, NM, USA. These two sites are envisioned as playing a crucial role in the first flights of high-altitude airships. The analysis shown in this paper has not been previously applied to this region of the stratosphere for such an application. Standard statistics were compiled for these data such as mean, median, maximum wind speed, and standard deviation, and the data were modeled with Weibull distributions. These statistics indicated, on a yearly average, there is a lull or a “knee” in the wind between 65 000 and 72 000 ft AGL (20–22 km). From the standard statistics, trends at both locations indicated substantial seasonal variation in the mean wind speed at these heights. The yearly and monthly statistical modeling indicated that Weibull distributions were a reasonable model for the data. Forecasts and hindcasts were done by using a Weibull model based on 2004 data and comparing the model with the 2003 and 2005 data. The 2004 distribution was also a reasonable model for these years. Lastly, the Weibull distribution and cumulative function were used to predict the 50%, 95%, and 99% winds, which are directly related to the expected power requirements of a near-space station-keeping airship. These values indicated that using only the standard deviation of the mean may underestimate the operational conditions.     

Assessment of modified Anderson–Darling test statistics for the generalized extreme value and generalized logistic distributions

An important problem in frequency analysis is the selection of an appropriate probability distribution for a given sample data. This selection is generally based on goodness-of-fit tests. The goodness-of-fit method is an effective means of examining how well a sample data agrees with an assumed probability distribution as its population. However, the goodness of fit test based on empirical distribution functions gives equal weight to differences between empirical and theoretical distribution functions corresponding to all observations. To overcome this drawback, the modified Anderson–Darling test was suggested by Ahmad et al. (1988b). In this study, the critical values of the modified Anderson–Darling test statistics are revised using simulation experiments with extensions of the shape parameters for the GEV and GLO distributions, and a power study is performed to test the performance of the modified Anderson–Darling test. The results of the power study show that the modified Anderson–Darling test is more powerful than traditional tests such as the χ², Kolmogorov–Smirnov, and Cramer von Mises tests. In addition, to compare the results of these goodness-of-fit tests, the modified Anderson–Darling test is applied to the annual maximum rainfall data in Korea.     

Particle path length distributions in meandering gravel‐bed streams: results from physical models

In gravel‐bed rivers with well‐de?ned pool–bar morphology, the path length of transported bed particles must be, at least during ‘channel‐forming’ ?ows, equal to the length scale of the morphology. This is the basis for some methods for estimating bed material transport rates. However, previous data, especially from ?eld tests, are often strongly positively skewed with mean much shorter than the pool–bar spacing. One possible explanation is that positively skewed distributions occur only in channels lacking distinct pool–bar topography or only at lower discharges in pool–bar channels. A series of ?ume experiments using ?uorescent tracers was used to measure path length distributions in low‐sinuosity meandering channels to assess the relation with channel morphology and ?ow conditions. At channel‐forming ?ows, 55 to 75 per cent of the tracer grains were deposited on the ?rst point bar downstream of the point of tracer input, with 15 per cent passing beyond the ?rst bar. Path length distributions are symmetrical with mean equal to the pool–bar spacing and can be described with a Cauchy distribution. In some cases there was a secondary mode close to the point of tracer introduction; this bimodal distribution ?ts a combined gamma–Cauchy distribution. Only when discharge was reduced below the channel‐forming ?ow were frequency distributions unimodal and positively skewed with no relation to the pool–bar spacing. Thus, path length distributions become more symmetrical, and mean path length increases to coincide with pool–bar spacing, as ?ow approaches channel‐forming conditions. This is a substantial modi?cation of existing models of particle transfer in gravel‐bed rivers. Copyright © 2003 John Wiley & Sons, Ltd.     

鄱阳湖自然保护区植被生物量时空变化及水位影响

基于时间序列Moderate Resolution Imaging Spectroradiometer(MODIS)植被指数,结合野外实测生物量及水文气象等数据,分析了2001 2010年退水期鄱阳湖国家自然保护区湿地植被生物量的空间格局及其时间变化规律,并在此基础上探讨了水位变化的影响.研究结果表明:(1)湿地植被生物量密度多年均值处于0~1402 g/m²之间,除蚌湖四周生物量沿湖心向四周逐渐升高外,其他区域呈现南高北低的分布格局,且其分布与研究区高程密切相关;(2)湿地植被生物量密度多年均值在退水初期为901 g/m²,随时间变化先上升后降低;植被分布面积则随着滩地出露而逐渐增大,随后基本稳定;总生物量呈现单峰变化,在11月初达到最高值.2001 2010年研究区年均生物量变化呈现波动状态,多年均值为18.3×10⁷kg;最高出现在2006年,达到28.2×10⁷kg,2010年最低,仅有8.37×10⁷kg;(3)年均生物量与洲滩出露天数以及植被分布面积呈显著正相关,相关系数分别为0.719和0.865.水位变化为湿地生物量变化的重要影响因素.     

Testing a statistical curve‐fitting procedure for quantifying sediment populations within multi‐modal particle‐size distributions

This paper describes a series of tests designed to evaluate the capacity of a personal computer (PC) based statistical curve‐fitting program called MIX to quantify composite populations within multi‐modal particle‐size distributions. Three natural soil samples were analysed by a Coulter Multisizer, and their particle‐size distributions analysed using MIX software to identify the modes, standard deviations and proportions of their composite populations. The particle‐size distributions of the three natural soil samples were then numerically combined in equal proportions using a spreadsheet program to create synthetic particle‐size distributions of known populations. MIX was then tested on the synthetic particle‐size distributions to see if the modes and proportions it identified were similar to those modes and proportions known to characterize the synthetic particle‐size distributions. The main outcome is that MIX can very accurately describe the modal particle size and proportions of the major composite populations within a particle‐size distribution. However MIX has difficulty in identifying small populations (those contributing <10 per cent of a total particle‐size distribution), particularly when they are located in the central sections of particle‐size distributions, overlain by larger populations, or when positioned in the fine tails of distributions. Despite these minor shortcomings, MIX is a valuable tool for the examination and interpretation of particle‐size data. Copyright © 2005 John Wiley & Sons, Ltd.     

Homogeneous region delineation based on annual flood generation mechanisms

Abstract

Flood distributions can have unimodal or multimodal densities due to different flood generation mechanisms such as snowmelt and rainfall in the annual flood series. When applying nonparametric frequency analysis to annual flood data from the province of New Brunswick in Canada, unimodal, bimodal and heavy-tailed distribution shapes were found. By grouping basins with similarly-shaped densities on a geographical basis, homogeneous regions were delineated. Regional equations derived for a homogeneous region gave lower integral square errors than those of province-wide equations.     

Solar energetic particle fluences from SOHO/ERNE

We have calculated integral fluences of solar protons and helium nuclei at 19 energy thresholds between 1.6 and 90 MeV/n from the SOHO/ERNE measurements during the years 1996–2005. We have also calculated fluences of oxygen and iron in the energy range from 10 up to a few hundred MeV/n for nineteen solar energetic particle (SEP) events. These are the first results of the work aiming at a full employment of the ERNE data in investigating the fluence distributions of SEP events over the entire solar activity cycle 23 and in deriving the total dose received on-board SOHO during its mission. Some instrumental problems are identified and future developments are presented.     

On the various methods of deducing aerosol size distribution from diffusional decay measurements

Summary The value of three different methods for deducing aerosol size distribution from diffusional decay measurements — the exhaustion method proposed byPollak andMetnieks, the method byFuchs et al. and the method byNolan andScott — is investigated with numerical examples of known distributions and also by applying them to laboratory experiments. It was found that the exhaustion method and the method proposed byFuchs et al. are satisfactory for deducing the mean particle size and that none of the three methods is quite satisfactory for deducing dispersion of the distributions with higher accuracy.     

Thermomechanical analysis of long-term global modal and local deformation measurements of the Kishwaukee bridge using the bootstrap

In this paper we present a comparative analysis of global frequency and local deformation data for a large concrete bridge. The asymptotic probability distributions of the central statistics are presented, and compared with empirical bootstrap estimates. Bootstrapped distributions are calculated from reference data obtained during 1999-2000 and used to develop change-point alarm criteria for the structure, using reasonable sensitivity measures developed from FEM simulations and structural analysis. The implications of the frequency data are discussed in conjunction with the strain and displacement measurements in order to discern if the load carrying capacity of the bridge has been affected. The critical need for more advanced temperature compensation models for large structures continually in thermal disequilibrium is discussed.     

Calibration of a Par-Tec 200 laser back-scatter probe FOR in situ sizing of fluvial suspended sediment

The in situ or effective particle size distribution of fluvial suspended sediment may differ considerably from that of the chemically dispersed mineral fraction owing to flocculation. Obtaining a meaningful measure of the effective particle size distribution ideally requires that measurements should be made in situ. A rigorous assessment of the associated degree of flocculation also requires that the same measurement technique is used subsequently to establish the absolute particle size composition of the suspended sediment by analysis of the chemically dispersed mineral fraction. While few in situ measurement devices currently exist, a Par-Tec 200 laser back-scatter probe has previously been shown to be capable of making both in situ and laboratory particle size measurements of fluvial sediment. The accuracy and precision of this device is assessed in this paper. While able to distinguish relative size differences with a high degree of precision, the Par-Tec 200 performed poorly in terms of accuracy when compared with measurements made using a laser diffraction device. A calibration algorithm has been devised for the Par-Tec 200 size data, using standard sediment samples sized by means of a laser diffraction device as the reference. Application of the calibration to Par-Tec 200 measurements of heterogeneous sediment samples significantly improved the representativeness of the particle size distribution, both in terms of overall form, and the median particle size. © 1998 John Wiley & Sons, Ltd.     

Implementing quality control on a random number stream to improve a stochastic weather generator

For decades, stochastic modellers have used computerized random number generators to produce random numeric sequences fitting a specified statistical distribution. Unfortunately, none of the random number generators we tested satisfactorily produced the target distribution. The result is generated distributions whose mean even diverges from the mean used to generate them, regardless of the length of run. Non‐uniform distributions from short sequences of random numbers are a major problem in stochastic climate generation, because truly uniform distributions are required to produce the intended climate parameter distributions. In order to ensure generation of a representative climate with the stochastic weather generator CLIGEN within a 30‐year run, we tested the climate output resulting from various random number generators. The resulting distributions of climate parameters showed significant departures from the target distributions in all cases. We traced this failure back to the uniform random number generators themselves. This paper proposes a quality control approach to select only those numbers that conform to the expected distribution being retained for subsequent use. The approach is based on goodness‐of‐fit analysis applied to the random numbers generated. Normally distributed deviates are further tested with confidence interval tests on their means and standard deviations. The positive effect of the new approach on the climate characteristics generated and the subsequent deterministic process‐based hydrology and soil erosion modelling are illustrated for four climatologically diverse sites. Copyright © 2007 John Wiley & Sons, Ltd.     

Statistical Analysis of Ground Motions Estimated on the Basis of a Recipe for Strong-motion Prediction: Approach to Quantitative Evaluation of Average and Standard Deviation of Ground Motion Distribution

For seismic hazard assessment, we study the variabilities of predicted ground motion on the basis of a ??recipe for predicting strong ground motion?? and propose approximations to evaluate spatial distributions of the standard deviation for PGV, R1.0, R2.0, and R5.0 in the estimated ground motions. For strong-motion prediction, we use a finite difference method for a long period range (>1.0?s). To estimate variabilities, a Monte Carlo simulation is used and we adopt the Latin Hypercube Sampling (LHS) technique to reduce computations. In this article, we consider only aleatory variabilities in source parameters among all possible variabilities, such as those in the source parameters, the propagation characteristics and site characteristics. Model sources are assumed for dip-slip fault and strike-slip fault, and the variabilities are considered for parameters such as asperity location, rupture starting point, average asperity slip contrast, stress drop and rupture velocity. On the target site, 100 instances of PGV, R1.0, R2.0 and R5.0 data are obtained for 100 sets of parameters and an average and a standard deviation of the log normal distribution, corresponding to the variability for ground motion estimation, are statistically analyzed. For all target sites uniformly distributed in the area around the faults, the average and the standard deviation are statistically analyzed and spread to spatial maps. It is found that the spatial distributions of standard deviation values for both the dip-slip and strike-slip faults are not uniform. Approximations are attempted to develop a quantitative evaluation for spatial distributions of the standard deviation of the log normal distribution for PGV, R1.0, R2.0, and R5.0. The spatial distributions by these approximations are considered to almost reconstruct the characteristics, which are statistically analyzed by the finite difference method.