The Impact of Biased Sampling on the Estimation of the Semivariogram Within Fractured Media Containing Multiple Fracture Sets

Monte Carlo simulation was used to examine the error (statistical bias) introduced in estimating a sample semivariogram through application of oriented sampling patterns to variables which are correlated with fracture orientation. Sample semivariograms of the directional components of the water velocity were used to illustrate that oriented sampling schemes can provide biased data sets which result in error in the estimation of the semivariogram, particularly in the estimation of the sill (or variance). Three sampling patterns were used to analyze directional semivariograms of the components of the fluid velocity: sampling along lines parallel to the mean regional hydraulic gradient, sampling among lines perpendicular to the mean regional hydraulic gradient, and sampling along fracture segments. The first two sampling patterns were shown to introduce substantial error in the sills of the velocity variograms. It is argued that this error is due to the combination of unequal sampling of fractures with different orientations (i.e., sampling bias) and the systematic variation in the magnitude of the velocity components with orientation of the fracture. As a consequence, it is suggested that correction factors developed to correct fracture frequency statistics need to be extended to improve estimation of spatial moments of variables which are correlated with fracture orientation.     

Influence of borehole quantity and distribution on lithology field simulation

This paper aims to study the influence of the number and distribution of drill samples on the simulation accuracy of the lithology field. This research mainly applies the variation function method in geo-statistics, and determines important indicators such as the variation, and then the lithology field is simulated by sequence index simulation. It is shown that(1) simulation error decreases with the increase of sampling density;(2) at the scale and complexity of this study, when the sampling density reaches 40/km^2, the average error of the lithology field simulation can be less than 2.0%;(3) in the study mode of examples, the simulation results of random sampling in the whole region are the most ideal, with an average error of 5.4%. The average error of the simulation results of the centralized sampling is about 10 times that of the random sampling method;(4) known from the influence analysis of the degree of study sample unevenness influence on the imitation results, under the same sample size, the simulation error decreases with the increase of the most adjacent index. When the nearest index reaches 1, the simulation error will be less than 6%, and the error variable range is within 3%.     

Influence of borehole quantity and distribution on lithology field simulation

This paper aims to study the influence of the number and distribution of drill samples on the simulation accuracy of the lithology field. This research mainly applies the variation function method in geo-statistics, and determines important indicators such as the variation, and then the lithology field is simulated by sequence index simulation. It is shown that (1) simulation error decreases with the increase of sampling density; (2) at the scale and complexity of this study, when the sampling density reaches 40 /km2, the average error of the lithology field simulation can be less than 2.0%; (3) in the study mode of examples, the simulation results of random sampling in the whole region are the most ideal, with an average error of 5.4%. The average error of the simulation results of the centralized sampling is about 10 times that of the random sampling method; (4) known from the influence analysis of the degree of study sample unevenness influence on the imitation results, under the same sample size, the simulation error decreases with the increase of the most adjacent index. When the nearest index reaches 1, the simulation error will be less than 6%, and the error variable range is within 3%.     

The effect of drift on the experimental semivariogram

When nonlinear drift is present, the nature of the bias in the experimental semivariogram estimator of the semivariogram function is determined by the extent and density of the sampling as well as by the drift function itself. The bias caused by drift may affect the interpretation of the experimental semivariogram over its entire range.     

Lognormality in the observed size distribution of oil and gas pools as a consequence of sampling bias

Economic filtration has been offered as an explanation of the observed lognormality in the size distribution of discovered oil and gas deposits. The result leads to the conclusion that one cannot impute the shape of the underlying parent distribution from the observed discoveries size distribution. The fact that the largest pools tend to be discovered early in the exploration history of an area of interest suggests the existence of an inherent sampling bias in the discovery process. The bias is influenced by the levels of geologic knowledge and technological sophistication. Furthermore, the existence of the bias leads to lognormality in the observed discoveries size distribution of oil and gas pools. A discovery process model explicitly incorporating the notion of sampling bias was applied to a series of Weibull parent frequency size distributions. The selected parent distributions are of a class suggested in the literature as more reflective of nature's size distribution and have empirical support. The distribution of discoveries resulting from the application of the model to the chosen parent size distributions were tested for lognormality using a chi-squared test. Lognormality was found to be an acceptable model of the discoveries size distribution over a wide range of resource exhaustion measures. When combined with the notion of economic filtration, sampling bias leads to the conclusion that one should not expect the lognormal distribution to accurately represent the underlying parent size distribution of oil and gas deposits.     

偏极性现象研究及其地球动力学意义

对于１０８年或更长的地史时期而言,地磁场表现出偏极性特征。统计表明,偏极性变化与磁极倒转频率、长期变、古强度变化等古地磁现象以及地球自转速率变化、真极移和海底扩张等地球动力学现象均有不同程度的相关关系。偏极性呈现周期性变化,其中６４０～５３３Ｍａ的周期对应于王鸿祯等（１９９７）所提出的联合古陆周期。笔者认为,Ｍａｒｕｙａｍａ（１９９４）的地幔羽模型可能有助于说明岩石圈板块通过全地幔对流的方式介入核幔边界条件,进而影响或控制了偏极性的长周期变化。     

浓度梯度引起的固相速度测量误差

用粒子示踪技术测量明渠挟沙水流中固相颗粒的速度时,流速梯度和浓度梯度的存在会引起测量误差.选用典型的垂线流速与浓度分布公式对时均速度测量误差进行分析,得出了误差的解析表达式.对该解析式的理论分析表明,测量误差恒小于零,即速度测量值比实际值偏小.误差的大小与采样窗口的相对位置、尺寸及颗粒的悬浮指数有关.对误差大小随自变量...     

Probabilistic modelling of joint orientation

Observed frequencies of joint orientations are subject to error due to sampling bias. This error should be corrected before statistical inference is made on the distribution of orientation. Corrections (weighting functions) are developed for sampling bias in orientation for finite joints of different sizes and shapes intersecting rectangular exposures. Chi-square goodness-of-fit procedures available for hemispherical normal and bivariate normal distributions are modified to make them applicable for both raw data and data corrected for sampling bias. The aforementioned corrections and procedures were applied to a joint orientation cluster to study the effect of (a) joint orientation, (b) joint size and (c) joint shape, on the statistical distribution of the orientation. The influences of all these aforementioned factors were found to be significant. However, at present, joint sizes and shapes are not measured in field joint surveys. Therefore, it is suggested to make attempts to obtain joint sizes and shapes in field joint mapping surveys. Since the currently available probability distributions are not adequate to represent all joint orientation distributions, it is suggested either to look for new probability distributions or to develop procedures to use empirical distributions in modelling orientation distributions.     

Quantifying sample bias in clast fabric measurements

Sampling elongate clasts that protrude from a planar face for clast macro‐fabric analysis introduces a bias into reconstructed fabrics because clasts aligned perpendicular to the sampling face are over‐represented relative to those aligned parallel to the face. This study develops a probability‐based mathematical analysis to quantify sample bias for a variety of clast shapes and population fabrics, including isotropic, clustered and girdled fabric styles. Bias is expressed in terms of sample eigenvalues and eigenvectors relative to those of the parent population. Results indicate that sampling always has the effect of artificially drawing fabrics towards perpendicularity to the sampling face relative to the populations from which they are drawn. This rotation generally has the effect of artificially strengthening population fabrics, by up to 30% in the case of very weakly clustered or girdled populations. However, fabric strengthening is by no means universal and sampling alters different parent populations in different, sometimes complex, ways: the analysis in this paper identifies situations where sampling can strengthen or weaken parent fabrics, where it can rotate parent fabrics (by up to 90° in the case of a very weak population fabric), and where it can even change the style of a population fabric. For example, near‐isotropic population fabrics can appear clustered; weakly clustered and weakly girdled population fabrics can appear isotropic; weakly clustered population fabrics can appear weakly girdled; and girdled population fabrics can appear clustered. Overall, this analysis indicates that weakly orientated population fabrics are most susceptible to bias. Thus, a weakly clustered population fabric aligned parallel to a sample face is particularly susceptible to major sample bias in both fabric strength (artificially overestimated) and orientation (artificially rotated perpendicular to the face). Bias can be reduced either by sampling all the clasts from a cube‐shaped block of sediment excavated from the face, or by sampling equal numbers of clasts from the three orthogonal faces of the same sedimentary unit.     

Analysis of a new surface-sampling technique for unconsolidated sand populations

A new surface-sampling technique for unconsolidated sands on aeolian dunes is introduced and analyzed in terms of an elementary mathematical model. The sampling bias inherent to the technique is modeled using a simple geometric argument. This shows that samples drawn from 2D slices of a 3D population are naturally biased toward larger sand grains, with the n^th moment of the surface (2D) distribution proportional to the (n+1)^th moment of the bulk (3D) distribution. Laboratory tests of the sampling technique allow an empirical relation for the sampling bias to be determined, and comparison shows the model and empirical relations to be in close agreement for the particular population distributions considered. Possible corrections to the model, as suggested by the empirical analysis, are also discussed.     

Nonstationarity of the Mean and Unbiased Variogram Estimation: Extension of the Weighted Least-Squares Method

When concerned with spatial data, it is not unusual to observe a nonstationarity of the mean. This nonstationarity may be modeled through linear models and the fitting of variograms or covariance functions performed on residuals. Although it usually is accepted by authors that a bias is present if residuals are used, its importance is rarely assessed. In this paper, an expression of the variogram and the covariance function is developed to determine the expected bias. It is shown that the magnitude of the bias depends on the sampling configuration, the importance of the dependence between observations, the number of parameters used to model the mean, and the number of data. The applications of the expression are twofold. The first one is to evaluate a priori the importance of the bias which is expected when a residuals-based variogram model is used for a given configuration and a hypothetical data dependence. The second one is to extend the weighted least-squares method to fit the variogram and to obtain an unbiased estimate of the variogram. Two case studies show that the bias can be negligible or larger than 20%. The residual-based sample variogram underestimates the total variance of the process but the nugget variance may be overestimated.     

空间分辨率与取样方式对DEM流域特征提取的影响

随着数字水文的兴起和分布式水文模型研究的发展, 利用DEM提取水文特征, 进而进行水文模拟的方法越来越广泛地为水文学者所采用. 空间分辨率的改变与DEM重新取样方式对水文模拟都会产生重要影响. 采取不同取样方法获得多种尺度的DEM, 对不同分辨率下的流域特征值进行了统计分析与比较, 引入熵的概念度量不同分辨率的DEM包含的信息量, 以及不同取样方式对信息量的影响. 并计算了以50 m DEM所包含的信息量为基准, 在不同的信息损失下所要求的最低分辨率.     

Reversals of the geomagnetic field,magnetostratigraphy, and relative magnitude of paleosecular variation in the phanerozoic

The percentage of normal and reversed magnetization in land-based paleomagnetic studies of Phanerozoic rocks (0 to ? 570 m.y.) have been compiled in order to determine the long-term variation in polarity bias of the geomagnetic field. Where possible the results are compared with the record from marine magnetic anomalies. Only rarely is there an even balance between normal and reversed polarity. During the past 350 m.y. two quiet intervals can be recognized when few reversals occurred, the Cretaceous (KN about ? 81 to ? 110 m.y.) and Permo-Carboniferous (PCR about ? 227 to ? 313 m.y.). Less firmly established are two other quiet intervals, one in the Jurassic (JN about ? 145 to ? 165 m.y.), and one in the Triassic (TRN about ? 205 to ? 220 m.y.). Between these quiet intervals there are disturbed intervals when reversals were comparatively frequent. From ? 680 to ? 350 m.y. the paleomagnetic record is inadequate to delineate a succession of quiet and disturbed intervals although one is probably present. Maximum entropy spectral analysis reveals three periodicities, a dominant one at about 300 m.y. and others, less well-defined, at 113 and 57 m.y. The variations in polarity bias are compared with the paleosecular variation, and it is shown that the magnitude of the paleosecular variation is greater in disturbed than in quiet intervals. This indicates that the magnitude of paleosecular variation and polarity bias are governed by variations in the balance between non-dipole and dipole components of the field, and that these variations probably had their origin in processes near the core—mantle interface. The correspondence between the dominant periods of 300 m.y. and plate tectonics is noted and a causal relationship suggested.     

Sea level in the Mediterranean Sea: seasonal adjustment and trend extraction within the framework of SSA

The sea level change is a crucial indicator of our climate. The spatial sampling offered by satellite altimetry and its continuity during the past years are the major assets to provide an improved vision of the Mediterranean sea level changes. In this paper, an automatic signal extraction approach, based on Singular Spectrum Analysis (SSA), is utilized for analysis and seasonal adjustment of the Mediterranean Sea level series. This automatic approach enables us to overcome the difficulties of visual identification of trend constituents that sometimes we encounter when using the conventional SSA method. The results indicate that the Mediterranean mean sea level is dominated by several harmonic components. The annual signal is particularly strong and almost covers 73.62 % of the original sea level series variation whiles its amplitude is about 15 cm. The extracted trend also indicates that the Mediterranean main sea level has significantly been raised during the period 1993–2012 by 2.44?±?0.4 mm yr^?1. As an important consequence, considering the current situation, if this trend continues, the Mediterranean Sea level will be raised about 22 cm by the end of this century, which makes a dramatic effect on several issues such as land, flora, fauna, and people activities established along the Mediterranean coastlines.     

Design-based versus model-based sampling strategies: Comment on R. J. Barnes' “bounding the required sample size for geologic site characterization”

Two fundamentally different sources of randomness exist on which design and inference in spatial sampling can be based: (a) variation that would occur on resampling the same spatial population with other sampling configurations generated by the same design, and (b) variation occurring on sampling other populations, hypothetically generated by the same spatial model, using the same sampling configuration. The former leads to the design-based approach, which uses classical sampling theory; the latter leads to the model-based approach and uses geostatistical theory. Failure to recognize these two sources of randomness causes misunderstanding about dependence of variables and the role of randomization in sampling, unwarranted narrowing down the choice of sampling strategies to those that are model-based, and abuse in simulation experiments. This is exemplified in Barnes' publication on the required sample size for geologic site characterization by nonparametric tolerance intervals. A basic design-based strategy like Simple Random Sampling is shown to require smaller sample sizes than the model-based strategy advocated by Barnes. In addition, Simple Random Sampling is completely robust against model errors and less complicated.     

平原区中大比例尺土壤地球化学调查采样方法--浙江省平湖市试验研究

文章以浙江省平湖市为典型三角洲平原区,开展了多种尺度、不同采样方法的对比试验。结果表明,田块内土壤元素分布较为均匀,小范围内元素的空间分异性较小,而田块之间土壤元素含量差异性明显增加;Hg,Cd等典型污染元素空间分异性较强,而人为污染扰动较弱的元素的空间分布较为均一;不同采样方法取得的区域性资料,其统计值接近,空间分布模式也相似,但有些元素仍有较明显差异。因此,中大比例尺地球化学调查时,应根据调查研究的目标任务,充分考虑地块分布、土壤类型和土地利用方式,选择合理的采样点。土壤样品可采取多坑点采集、多子样组合的方式,以保证样品的代表性,同时减少分析测试工作量。     

Statistical analysis of the effective stress method and modifications for prediction of ultimate bond strength of soil nails

Uncertainty in the predicted ultimate pullout strength of soil nails can be significant due to the complexity of nail–soil interactions, inherent variability in soil properties and the effects of nail installation. The paper first presents a statistical evaluation of the accuracy of ultimate bond strength of soil nails using the effective stress method (ESM) equation that has been adopted in Hong Kong. A total of 113 ultimate nail capacity data points from field pullout tests were collected from the literature and used to estimate the accuracy of the current ESM. Based on the available data, the current ESM default pullout model is found to be excessively conservative (on average) by at least a factor of three. The spread in prediction accuracy measured by the coefficient of variation (COV) of bias is in the range of 36–43 % after removing anomalous test data. Here, bias is the ratio of measured to predicted pullout load capacity. In addition, the accuracy of the current ESM equation for prediction of nail bond strength is shown to be dependent on the magnitude of predicted ultimate bond strength and magnitude of nominal vertical effective stress which is undesirable. The paper examines four candidate-modified bond strength equations with empirical coefficients that have been back-fitted to measured bond strengths to improve the overall accuracy of the equation and to reduce or remove the undesirable dependencies noted above. One equation with an empirically corrected stress-dependent term is judged to be the best candidate model based on the mean of bias values, spread (COV) of bias values, lack of dependencies and simplicity. Finally, the relative contributions of random variability in soil shear strength to measurement bias in bond strength (prediction accuracy) for each soil type are computed for the best bond strength model. Analysis of the contribution of soil shear strength to prediction accuracy showed that the combination of variability due to factors other than soil shear strength was greater than the variability in soil shear strength alone, where the latter is defined by the soil secant friction coefficient.     

Determining the spatial scale of variation in soil radon concentration

Increasing concern about possible links between emissions of radon and certain types of malignant disease has led to local and regional surveys to measure radon concentrations in the soil and in dwellings. The spatial scale at which radon varies is largely unknown, and so efficient sampling schemes and methods of mapping cannot be selected. To determine the spatial scale of radon variation in the English Midlands we measured radon in the soil using solid-state nuclear track detection in three areas of different geological complexity. In two of them we used an unbalanced multistage sampling scheme with seven stages of nesting: the Hereford survey covered distances from 10 m to 7.5 km, and the Buxton survey distances from 1 m to 3.75 km. The results from the nested surveys suggested that geology exerts a strong control on the variation. Finally radon was measured every 20 m along a 2-km long transect which crossed several lithologies close to Nottingham. The soil radon values changed in an erratic way along the transect. The sample variogram of radon has a substantial nugget variance, suggesting that much of the variation occurred for distances less than the sampling interval. The structure at the longer scale seems to be controlled by the underlying geology. These results have implications for designing further surveys and for selecting a method of mapping. Stratification based on lithology might be the only feasible solution to sampling, estimating and mapping radon concentrations over large areas. Where the locally erratic component of variation is large, estimation by kriging, for example, would confer little additional benefit compared with that by classification.