Estimating temporal change in soil monitoring: II. Sampling from simulated fields

Design-based and model-based methods of estimating temporal change of soil properties over a finite area have been compared. Two large fields of auto- and cross-correlated data were simulated, each representing the spatial distribution of a variable at one time. The fields were then sampled repeatedly. The means of stratified and systematic random samples and geostatistical global estimates were used to infer the mean difference between the fields. All estimators were unbiased, but their variances differed. Pairing the positions on the two occasions increased the precision of the design–based estimates. Systematic sampling was slightly more precise than stratified sampling. Kriging was less precise than both because some of the sample information must be used to estimate the variograms at short lags. Neither balanced differences nor the normal formula for simple random sampling predicted the estimation variances of small (n< 50) systematic samples accurately. For larger samples the method of balanced differences performed well. If the spatial variation is unknown in advance and only small samples can be taken then stratified random sampling with two observations per stratum is the preferred design. It resulted in the best combination of precision and accuracy in predicting the sampling error.     

Two-stage sequential sampling: A neighborhood-free adaptive sampling procedure

Designing an efficient sampling scheme for a rare and clustered population is a challenging area of research. Adaptive cluster sampling, which has been shown to be viable for such a population, is based on sampling a neighborhood of units around a unit that meets a specified condition. However, the edge units produced by sampling neighborhoods have proven to limit the efficiency and applicability of adaptive cluster sampling. We propose a sampling design that is adaptive in the sense that the final sample depends on observed values, but it avoids the use of neighborhoods and the sampling of edge units. Unbiased estimators of population total and its variance are derived using Murthy’s estimator. The modified two-stage sampling design is easy to implement and can be applied to a wider range of populations than adaptive cluster sampling. We evaluate the proposed sampling design by simulating sampling of two real biological populations and an artificial population for which the variable of interest took the value either 0 or 1 (e.g., indicating presence and absence of a rare event). We show that the proposed sampling design is more efficient than conventional sampling in nearly all cases. The approach used to derive estimators (Murthy’s estimator) opens the door for unbiased estimators to be found for similar sequential sampling designs.     

Bayesian Calibration of Blue Crab (<Emphasis Type="Italic">Callinectes sapidus</Emphasis>) Abundance Indices Based on Probability Surveys

Abundance and standard error estimates in surveys of fishery resources typically employ classical design-based approaches, ignoring the influences of non-design factors such as varying catchability. We developed a Bayesian approach for estimating abundance and associated errors in a fishery survey by incorporating sampling and non-sampling variabilities. First, a zero-inflated spatial model was used to quantify variance components due to non-sampling factors; second, the model was used to calibrate the estimated abundance index and its variance using pseudo empirical likelihood. The approach was applied to a winter dredge survey conducted to estimate the abundance of blue crabs (Callinectes sapidus) in the Chesapeake Bay. We explored the properties of the calibration estimators through a limited simulation study. The variance estimator calibrated on posterior sample performed well, and the mean estimator had comparable performance to design-based approach with slightly higher bias and lower (about 15% reduction) mean squared error. The results suggest that application of this approach can improve estimation of abundance indices using data from design-based fishery surveys.     

Sampling and bulking strategies for estimating soil properties in small regions

Estimates of mean values of soil properties within small rectangular blocks of land can be obtained by kriging provided the semi-variogram is known. This paper describes optimal rectangular grid sampling configurations whereby estimation variances can be minimized. For linear semi-variograms square blocks are best estimated by sampling at the nodes of a centrally placed grid with its interval equal to the block side divided by the square root of the sample size. For spherical semi-variograms the same configuration is almost optimal. The estimation variance of a bulked sample can be identical with that of a kriged estimate where the semi-variogram is linear and equal portions of soil are taken from each node on the optimally configured grid and provided the soil property is additive. For spherical semi-variograms the above is approximately true. Comparisons with estimates that take no account of known spatial dependence show that the true variances can be much less than those apparent using classical theory, and the necessary sampling effort much less. Within block-variances are often needed for planning, and an appendix gives two-dimensional auxiliary functions from which they can be calculated for linear and spherical semi-variograms.     

Non-homogeneity of variance components from spatially nested sampling of the soil

Spatially nested sampling and the associated nested analysis of variance by spatial scale is a well-established methodology for the exploratory investigation of soil variation over multiple, disparate scales. The variance components that can be estimated this way can be accumulated to approximate the variogram. This allows us to identify the important scales of variation, and the general form of the spatial dependence, in order to plan more detailed sampling by design-based or model-based methods. Implicit in the standard analyses of nested sample data is the assumption of homogeneity in the variance, i.e. that all variations from sub-station means at some scale represent a random variable of uniform variance. If this assumption fails then the comparable assumption of stationarity in the variance, which is an important assumption in geostatistics, will also be implausible. However, data from nested sampling may be analysed with a linear mixed model in which the variance components are parameters which can be estimated by residual maximum likelihood (REML). Within this framework it is possible to propose an alternative variance parameterization in which the variance depends on some auxiliary variable, and so is not generally homogeneous. In this paper we demonstrate this approach, using data from nested sampling of chemical and biogeochemical soil properties across a region in central England, and use land use as our auxiliary variable to model non-homogeneous variance components. We show how the REML analysis allows us to make inferences about the need for a non-homogeneous model. Variances of soil pH and cation exchange capacity at different scales differ between these land uses, but a homogeneous variance model is preferable to such non-homogeneous models for the variance of soil urease activity at standard concentrations of urea.     

Comparison of survey estimates of the finite population variance

The Environmental Monitoring and Assessment Program (EMAP) of the U.S. Environmental Protection Agency has conducted several probability surveys of aquatic resources. Such surveys usually have unequal probability of including population elements in the sample. The Northeast lakes survey, which motivated this study of variance estimation, was such a survey. We examine ten estimators for the finite population variance using a Monte Carlo factorial experiment that considers three population characteristics. The results show that the correlation between the inclusion probabilities and the response is the most important factor that differentiates the estimators. Under conditions of low correlation (approximately <0.4), a common feature in environmental surveys, the sample variance is best, elsewhere, two ratio estimators, one based on consistency and the Horvitz-Thompson Theorem (HT) and the other based on the Yates-Grundy form, behave similarly and best.     

空间自相关性对冬小麦种植面积空间抽样效率的影响

空间抽样是实现区域农作物面积高效估算的重要手段,农作物分布受自然条件等因素影响普遍存在空间自相关性,但以往针对空间相关性对农作物面积抽样效率的影响研究明显不足。该研究选取安徽省凤台县为研究区,通过2017年4月4景GF-1全色多光谱影像(Panchromatic and Multispectral, PMS)与Google Earth高空间分辨率影像相结合提取研究区冬小麦。设计10种抽样单元尺度、3种抽样外推方法、2种相对允许误差和5种样本布局方式,构建多种冬小麦面积空间抽样方案。利用全局莫兰指数(global Moran’s index)评价不种尺度下抽样单元内冬小麦面积比的空间自相关强度,分析空间自相关性对冬小麦面积抽样效率(抽样误差、样本容量和空间布局)的影响。研究结果表明,抽样单元内冬小麦面积比的空间自相关强度随单元尺度的增大而减小,全局莫兰指数相应地由0.75降至0.50。无论在何种尺度下抽样单元内冬小麦面积比都呈显著的空间正相关性;抽样外推冬小麦面积总体的误差随空间自相关强度的减小呈先减小后明显增大的趋势。在10种抽样单元尺度中,当抽样单元尺度为2000m且抽样比为5%时,无论采用何种抽样方法外推总体的误差均为最小(简单随机抽样、系统和分层抽样外推总体的相对误差分别为17.94%、9.48%和1.82%);当相对允许误差设计为5%时,简单随机抽样外推总体所需样本容量随空间自相关强度的降低从660降至56。而分层抽样的样本容量不受空间自相关性的影响;5种样本布局方式中,采用分层随机抽样方式外推冬小麦面积总体的平均相对误差、平均变异系数和均方根误差最小,分别为1.82%、3.19%和0.11×108 m2。该研究可为有空间自相关存在下的农作物面积空间抽样方案合理设计提供参考依据。     

冬小麦种植面积空间抽样效率影响因子分析

基于遥感与抽样的农作物种植面积测量方法结合了遥感和抽样理论的优势,已经成为农作物种植面积测量中有着广泛应用前景的测量方法。以格网为单元,进行分层空间抽样,分析在二值图像的情况下,抽样格网大小、分层层数对抽样精度、抽样精度方差、抽样比的影响;将二值图像分类结果定义为作物区,随机混入不同丰度10%,20%,……,100%的冬小麦,在不同冬小麦丰度（即不同的分类误差）的前提下,分析抽样格网大小、分层层数、分类误差对抽样精度、抽样比的影响,确定最优分层定义为6层,在分类误差小于40%（即冬小麦丰度大于60%）的前提下,可以有效地进行空间抽样推算区域冬小麦种植面积,为农作物种植面积测量空间抽样方案的优化提供理论基础。     

作物种植面积空间对地抽样方法设计

传统的粮食作物种植面积估算一般采用目录抽样方法，由于缺乏现实、有效的先验知识，抽样过程受精度和效率制约。本文利用地理信息系统、遥感和全球定位系统技术，结合传统的随机、系统和分层抽样方法，设计三种新的粮食作物种植面积空间对地抽样方案，并在此基础上开展试验对比研究，分析它们在抽样精度、最少样本量和稳定性方面的差异。结果表明，本文所设计的空间分层抽样方法所需样本量较小，且具有较高的估算精度和稳定性，可以用于大范围农作物种植面积监测。     

Robust estimation of the pseudo cross‐variogram for cokriging soil properties

The pseudo cross‐variogram can be used for cokriging two or more soil properties when few or none of the sampling locations have values recorded for all of them. The usual estimator of the pseudo cross‐variogram is susceptible to the effects of extreme data (outliers). This will lead to overestimation of the error variance of predictions obtained by cokriging. A solution to this problem is to use robust estimators of the pseudo cross‐variogram, and three such estimators are proposed in this paper. The robust estimators were demonstrated on simulated data in the presence of different numbers of outlying data drawn from different contaminating distributions. The robust estimators were less sensitive to the outliers than the non‐robust one, but they had larger variances. Outliers tend to obscure the spatial structure of the cross‐correlation of the simulated variables as described by the non‐robust estimator. The several estimators of the pseudo cross‐variogram were applied to a multitemporal data set on soil water content. Since these were obtained non‐destructively, direct measurements of temporal change can be made. A prediction subset of the data was subsampled as if obtained by destructive analysis and the remainder used for validation. Estimators of the auto‐variogram and pseudo cross‐variogram were applied to the prediction data, then used to predict the change in water content at the validation sites by cokriging. The estimation variances of these predictions were best calculated with a robustly estimated model of coregionalization, although the validation set was too small to conclude that the non‐robust estimators were unsuitable in this instance.     

Estimating temporal change in soil monitoring: I. Statistical theory

Detecting small temporal change of spatially varying soil properties demands precise estimation. Design– and model–based methods are compared for estimating temporal change of soil properties over finite areas. Analytical expressions for the estimators and their variances arc derived for the two approaches, and formulae for the expectations of the variances under the random–process model are developed. Among the randomized designs simple, stratified, and systematic random sampling using the arithmetic mean as estimator have been studied. Pairing the sampling positions on the different occasions increases the precision of design–based estimation if the observations are positively cross–correlated. The relative precisions of the means of stratified and systematic samples depends on the spatial correlation. Neither is more precise than the other in all circumstances. The stratified design provides an unbiased estimator for the sampling error, which is not available from systematic samples. Theoretically, the geostatistical global estimator is more precise than the estimates derived from any of the classical designs when many realizations arc repeatedly sampled at random. In practice, with only a single realization of the process, this is no longer relevant. Moreover, errors in estimating the variograms add to the total error of the method. It seems that only by sampling from large auto–correlated random fields can the precisions of the methods be compared in practice.     

The Use of Calibration Weighting for Variance Estimation Under Systematic Sampling: Applications to Forest Cover Assessment

The purpose of this note is to propose a variance estimator under non-measurable designs that exploits the existence of an auxiliary variable well correlated with the survey variable of interest. Under non-measurable designs, the Sen–Yates–Grundy variance estimator generates a downward bias that can be reduced using a calibration weighting based on the auxiliary variable. Conditions of approximate unbiasedness for the resulting calibration estimator are given. The application to systematic sampling is considered. The proposal proves to be effective for estimating the variance of the forest cover estimator in remote sensing-based surveys, owing to the strong correlation between the reference data, available from a systematic sample, and the satellite map data, available for the whole population and hence exploited as an auxiliary variable. Supplementary materials accompanying this paper appear online.     

Using regression models in design‐based estimation of spatial means of soil properties

The precision of design‐based sampling strategies can be increased by using regression models at the estimation stage. A general regression estimator is given that can be used for a wide variety of models and any well‐defined sampling design. It equals the π estimator plus an adjustment term that accounts for the differences between the π estimators for the spatial means of the auxiliary variables and the true spatial means of these variables. The regression estimator and ratio estimator follow from certain assumptions on the model and the sampling design. These are compared with the π estimator in two case studies. In one study a bivariate field of linearly related variables was simulated and repeatedly sampled by Simple Random Sampling without replacement and sample sizes 10, 25, 50, 100 and 200. For all sample sizes the ratio of the standard error of the simple regression estimator to that of the π estimator was approximately 55%. The bias of the simple regression estimator was negligibly small. The confidence interval estimators were valid for all sample sizes except for n = 10. Also the ratio estimator was approximately unbiased, and the confidence interval estimators were valid for all sample sizes, even for n = 10. This is remarkable because the ratio estimator assumes that the intercept of the regression line is 0 which was incorrect for the simulated field. On the other hand, only approximately 55% of the potential gain was achieved because the model was inappropriate. In a second study the spatial means of the Mean Highest Watertable of map units were estimated by Stratified Simple Random Sampling and the combined (multiple) regression estimator. The NAP elevation, the local elevation, the Easting and the Northing were used as auxiliary variables. For all map units except one the combined (multiple) regression estimator was more precise than the π estimator. The ratio of the standard errors varied from 0.36 to 1.04. The domain for which the regression estimator was less precise than the π estimator showed strong variation between strata. For this domain it was more efficient to group the strata into two groups and to fit simple models for these groups separately.     

Estimating Distance Sampling Detection Functions When Distances Are Measured With Errors

Distance sampling methods assume that distances are known but in practice there are often errors in measuring them. These can have substantial impact on the bias and precision of distance sampling estimators. In this paper we develop methods that accommodate both systematic and stochastic measurement errors. We use the methods to estimate detection probability in two surveys with substantial measurement error. The first is a shipboard line transect survey in the North Sea in which information on measurement error comes from photographically measured distances to a subset of detections. The second is an aerial cue-counting survey off Iceland in which information on measurement error comes from pairs of independently estimated distances to a subset of detections. Different methods are required for measurement error estimation in the two cases. We investigate by simulation the properties of the new estimators and compare them to conventional estimators. They are found to perform better than conventional estimators in the presence of measurement error, more so in the case of cue-counting and point transect estimators than line transect estimators. An appendix on the asymptotic distributions of conditional and full likelihood estimators is available online.     

The components of variation associated with sampling soil for the measurement of major and trace nutrients in grazed fields in S.E. Scotland

Abstract. The laboratory, spatial and temporal components of variation associated with sampling soil for the measurement of pH, organic matter and extractable P, K, Mg, S, Cu, and Co were studied over two years using soil samples from 15 farms in S.E. Scotland. On each farm a selected field was divided into 4–8 sectors, and sampled three times each year, in June, August and October, by bulking 25 cores taken in a 'W' pattern. Analysis of variance showed that inter-field variation was greater than that between sampling dates for most of the properties measured. Restricted Maximum Likelihood Estimation showed that for all elements except K and S the variation between fields was greater than that within a field. Temporal variation was usually smaller than spatial, but K and Co showed similarly small temporal and spatial variations. Variation associated with laboratory procedures was much less than either spatial or temporal variation except for S, most of the total observed variation of which resulted from laboratory error. It is suggested that the most cost effective field sampling technique is to split a field into sectors, sample each individually and analyse a bulked sample made up from the sectors.     

A New Method for Computing Reliable Sample Size for Prescribing Soil Test Based Nutrient Management Interventions

ABSTRACT

Conventional sampling schemes for soil test guided nutrient management do not duly consider spatial variability. Fisher’s least significant difference (LSD) classical technique is sometimes manipulated for computing minimum sample size. However, it does not consider spatial dependence and relies on sample variance. Here, we present a new LSD-based robust method that uses semivariogram sill as a variance surrogate and then explore through sensitivity analysis novel alternative measurement units to reduce sample size rendered large by spatial variability. For differentiating crop response based categories, 273–22,320 samples were required for primary nutrients. Required sample size for detecting desired critical shifts in micronutrient status varied from 16–28,854. Changing to millimole units for potassium (K) and iron (Fe) further reduced sample size significantly. Thus, LSD-based technique can be made robust by using geostatistical techniques. Conventional measuring units in highly variable plant nutrients can be replaced with more practicable and economical units.     

植被异质性样区真实性检验的优化采样策略

遥感反演植被产品的真实性检验是推动其在农业领域应用水平提升的重要保证,其中异质性植被样区的优化采样设计是真实性检验地面测量过程中的关键技术。该研究以遥感影像作为先验知识,通过K-means聚类分层选取初始样点,利用空间模拟退火算法规划最优采样方案,并采用同期地面实测数据进行检验。研究结果表明,空间模拟退火算法在样点与总体空间变异性的一致性、插值面的精度、插值点和实测点的相关性3个方面都明显优于传统采样方案,2块样区优化后的采样方案插值面与影像面的均方根误差分别为3.1026和2.9627,插值点与实测点的皮尔逊相关系数分别为0.601和0.757,表明空间模拟退火算法可以为真实性检验地面试验提供可靠的优化采样策略。     

Designing sampling schemes for effect monitoring of nutrient leaching from agricultural soils

A general methodology for designing sampling schemes for monitoring is illustrated with a case study aimed at estimating the temporal change of the spatial mean P concentration in the topsoil of an agricultural field after implementation of the remediation measure. A before‐after control‐impact (BACI) sample‐pattern is proposed, with stratified random sampling as a spatial sampling design. The strata are formed as compact blocks of equal area, so that the sample locations cover the field very well. Composite sampling, where the aliquots of a composite come from different strata, is proposed in order to save laboratory costs. The numbers of composites and aliquots per composite are optimized for testing the hypothesis that the mean P concentration didn’t change or has increased. Initially, this is done for a known variogram, temporal correlation, variance of laboratory measurement error, initial mean P concentration, and time needed for fieldwork. The optimal sample size to achieve a power of 0.90 at a 10% decrease of the mean P concentration is six composites of six aliquots each. Next, the effect of uncertainty about these model parameters on the optimal sample size and on the power of the test for a fixed sample size is analyzed. This analysis showed that, to obtain a probability of 95% that the power ≥ 0.90, the sample size must be increased to 7 composites of 10 aliquots each.     

Using a Master Sample to Integrate Stream Monitoring Programs

The need for aquatic resource condition surveys at scales that are too extensive to census has increased in recent years. Statistically designed sample surveys are intended to meet this need. Simple or stratified random sampling or systematic survey designs are often used to obtain a representative set of sites for data collection. However, such designs have limitations when applied to spatially distributed natural resources, like stream networks. Stevens and Olsen proposed a design that overcomes the key limitations of simple, stratified random or systematic designs by selecting a spatially balanced sample. The outcome of a spatially balanced sample is an ordered list of sampling locations with spatial distribution that balances the advantages of simple or stratified random samples or systematic samples. This approach can be used to select a sample of sites for particular studies to meet specific objectives. This approach can also be used to select a “master sample” from which subsamples can be drawn for particular needs. At the same time, these individual samples can be incorporated into a broader design that facilitates integrated monitoring and data sharing.     

华北山地6种天然次生林土壤氮素的空间异质性特征

 利用地统计学理论和方法研究华北山地次生林区6种天然次生林土壤氮素的空间异质性特征。结果表明:1)次生阔叶林土壤总氮质量分数相对较高（3 1004 500 mg/kg）,而针叶林相对较低（9001 300 mg/kg）,各森林类型土壤中有效NH4+-N质量分数均高于有效NO3--N,形成以NH4+-N占优势的氮营养生境;2)针阔混交林中,土壤全氮的变异强度最大,变异的空间相关性较差（随机性变异占总变异的42.7%）,针叶林中,全氮空间变异强度相对较弱,但以自相关变异为主（结构方差比为72.2%81.0%）,呈现弱的斑块分布特征;3)阔叶林中,NH4+-N具有很强的空间自相关变异,NO3--N异质性程度相对较弱,针叶林中,NH4+-N变异强度较小,而NO3--N空间变异却相对明显;4)不同森林类型对土壤全氮及各有效氮形态的空间异质性特征有影响;5)植被种类、植被组成、植被多样性等因素的差异及由此导致的树种空间分布格局是影响总氮量及氮矿化,进而导致氮素不同形态在林分间甚至林分内不同空间样点间异质性形成的重要原因。