D.J. Brus

Sampling for natural resource monitoring

The book presents the statistical knowledge and methodology of sampling and data analysis useful for spatial inventory and monitoring of natural resources. The authors omitted all theory not essential for applications or for basic understanding. This presentation is broader than standard statistical texts, as the authors pay much attention to how statistical methodology can be employed and embedded in real-life spatial inventory and monitoring projects. Thus they discuss in detail how efficient sampling schemes and monitoring systems can be designed in view of the aims and constraints of the project.

Random sampling or geostatistical modelling? Choosing between design-based and model-based sampling strategies for soil (with discussion)

Abstract Classical sampling theory has been repeatedly identified with classical statistics which assumes that data are identically and independently distributed. This explains the switch of many soil scientists from design-based sampling strategies, based on classical sampling theory, to the model-based approach, which is based on geostatistics. However, in design-based sampling, independence has a different meaning and is determined by the sampling design, whereas in the model-based approach it is determined by the postulated model for the process studied. Design-based strategies are therefore also valid in areas with autocorrelation. Design-based and model-based estimates of spatial means are compared in a simulation study on the basis of the design-based quality criteria. The simulated field consists of four homogeneous units that are realizations of models with different means, variances and variograms. Performance is compared for two sample sizes (140 and 1520) and two block sizes (8 × 6.4 km 2 , 1.6 × 1.6 km 2 ). The two strategies are Stratified Simple Random Sampling combined with the Horvitz-Thompson estimator ( STSI , t HT ), and Systematic Sampling combined with the block kriging predictor ( SY , t OK ). Point estimates of spatial means by ( SY , t OK ) were more accurate in all cases except the global mean (8 × 6.4 km 2 block) estimated from the small sample. In interval estimates on the other hand, p -coverages were in general better with the design-based strategy, except when the number of sample points in the block was small. Factors that determine the effectiveness and efficiency of the two approaches are the type of request, the interest in objective estimates, the need for separate unique estimates of the estimation variance for all points or subregions, the interest in valid and accurate estimates of the estimation or prediction variance, the quality of the model, the autocorrelation between observation and prediction points, and the sample size. These factors will be assembled in a decision-tree that can be helpful in choosing between the two approaches. Models can also be used in the design-based approach. They describe the population itself, whereas in the model-based approach they describe the data generating processes. Errors in such models result in less accurate estimates, but the estimated accuracy is still valid.

Spacial prediction and mapping of continuous soil classes

Abstract Some problems associated with the use of discrete classes in soil mapping are discussed and a new approach to the production of soil class maps which attempts to circumvent such difficulties is presented. The method involves prediction of k +1 continuous classes resulting from a fuzzy k -means with extragrades grouping procedure, onto a fine grid. The prediction of individual class memberships is optimized by using ordinary kriging of log-ratio transformed memberships with a non-linear back transformation. The resulting k +1 rastered maps can be manipulated in various ways to produce isogram, choropleth or chorochromatic maps. In addition to potentially providing more soil information to the user this procedure also has implications in cartography and for geographic information systems. An example is given from a 6 km × 8 km area in the eastern part of the Netherlands.

The performance of spatial interpolation methods and choropleth maps to estimate properties at points: A soil survey case study

A study was designed to compare the performance of six spatial interpolation methods to estimate soil properties at unvisited points. These methods were global mean, moving average, nearest neighbour, inverse squared distance, Laplacian smoothing splines and ordinary point kriging. These methods were also applied in combination with a choropleth map (soil map) by stratifying the area. The soil properties estimated were thickness of A1 horizon, maximum areic mass of phosphate adsorbed by soil, mean highest water table and mean lowest water table. The performance of the methods was measured by estimating the spatial means of the squared and absolute error (quality criteria not conditional on the sample of test points) by a stratified simple random sample of test points. The mean squared error was very large in proportion to the spatial variation over the total area for all methods and properties. Differences between methods were small. In general, no statistically significant stratification or weighting effects were found. The effect of weighting plus stratification was usually not significant either. Overall, weighting with inverse squared distance was as satisfactory as weighting by ordinary point kriging. However, the latter was superior near data points. Also, when combined with soil map stratification, kriging was more reliable in the sense that it estimated all properties well. Estimates obtained using the means of six soil map units were better, although not significantly, than those obtained from unstratified kriging and as good as kriging within three map units.

An R package for spatial coverage sampling and random sampling from compact geographical strata by k-means

Both for mapping and for estimating spatial means of an environmental variable, the accuracy of the result will usually be increased by dispersing the sample locations so that they cover the study area as uniformly as possible. We developed a new R package for designing spatial coverage samples for mapping, and for random sampling from compact geographical strata for estimating spatial means. The mean squared shortest distance (MSSD) was chosen as objective function, which can be minimized by k-means clustering. Two k-means algorithms are described, one for unequal area and one for equal area partitioning. The R package is illustrated with three examples: (1) subsampling of square and circular sampling plots commonly used in surveys of soil, vegetation, forest, etc.; (2) sampling of agricultural fields for soil testing; and (3) infill sampling of climate stations for mainland Australia and Tasmania. The algorithms give satisfactory results within reasonable computing time.

A sampling scheme for estimating the mean extractable phosphorus concentration of fields for environmental regulation

Abstract A soil sampling scheme for estimating the mean extractable P concentration of fields is designed to be used as a tool for environmental regulation of the application rates of manure. The field to be sampled, is split up into geographically compact blocks of equal area that are used as strata. From each stratum one sampling point is selected by Simple Random Sampling. These samples are bulked into one composite for the field. The geographical stratification is performed by restricted least-squares clustering of raster cells using the coordinates of the midpoints as classification variables and the within-group sum of squares as the minimisation criterion. Using a variance model and a cost model, the numbers of sample points and laboratory analyses are optimised simultaneously, given a maximum allowed variance of the total error (sampling error plus measurement error). To predict the sampling variance, variograms have been estimated for 16 fields differing in land-use, soil parent material and phosphate level. A pooled relative variogram was used to predict the sampling variance for various sample sizes (5 to 50), field-areas (1 to 10 ha) and phosphate levels (for grassland 20 to 80 mg P 2 O 5 extracted in ammonimum lactate per 100 g soil, for arable land 20 to 80 mg P 2 O 5 extracted in water per 1 dm 3 soil). The cost model consists of three components: (i) fieldwork cost; (ii) field equipment cost and, (iii) laboratory cost. For the 16 fields, the predicted sampling variance of the Stratified Sampling design is 0.8 to 0.4 times the predicted variance of Simple Random Sampling if 40 points were sampled. To estimate the mean extractable P concentration with a total variance ≤9, replicate measurement of the composite only pays if the mean extractable P concentration of the field exceeds 40 to 50. This critical phosphate level increases with the maximum allowed variance of the total error.

Predictions of spatially averaged cadmium contents in rice grains in the Fuyang valley, P.R. China.

Soils in the Fuyang valley (Zhejiang province, southeast China) have been contaminated by heavy metals. Since rice (Oryza sativa L.) is the dominant crop in the valley and because of its tendency to accumulate Cd in its grains, assessment of the human health risk resulting from consumption of locally produced rice is needed. In this study, we used a regression model to predict the average Cd content in rice grains for paddy fields. The multiple linear model for log(Cd) content in rice grains with log(HNO(3)-Cd), pH, log(clay), and log(soil organic matter, SOM) as predictors performed much better (R(2)(adj) = 66.1%) than the model with log(CaCl(2)-Cd) as a single predictor (R(2)(adj) = 28.1%). This can be explained by the sensitivity of CaCl(2)-extracted Cd for changes in redox potential and as a result of the drying of the soil samples in the laboratory. Consequently, the multiple linear model was used to predict the average Cd contents in rice grains for paddy fields, and to estimate the probability that the FAO/WHO standard of 0.2 mg kg(-1) will be exceeded. Eleven blocks had a probability smaller than 10% of exceeding this standard (safe blocks). If a lognormal distribution is assumed, 35 blocks had a probability larger than 90% (blocks at risk). Hence, risk reduction measures should be undertaken for the blocks at risk. For 27 blocks the probability was between 10 and 90%. For these blocks the uncertainty should be reduced via improvement of the regression model and/or increasing the number of sample locations within blocks.

A structured approach to designing soil survey schemes with prediction of sampling error from variograms

Abstract There is a growing need for soil survey information with quantified accuracy, which can be met by applying appropriate statistical methods. For any survey a scheme should be designed specifying how and where data are to be recorded in the field, and how they are to be analysed statistically. Three formal levels are distinguished as the basis for a knowledge-based system for designing soil survey schemes. The first level depicts the role of the design in a survey project, the second distinguishes different steps in the design, and the third defines a framework of the basic concepts. Choosing a sampling design and comparing the accuracy of possible designs are essential in our support system. The accuracy of results from a scheme is defined as the mean squared error due to sampling. A computational method to predict the sampling error from prior information in the form of variograms is given and is illustrated with a case study.

Effect of the sampling design of ground control points on the geometric correction of remotely sensed imagery

The acquirement of ground control points (GCPs) is a basic and important step in the geometric correction of remotely sensed imagery. In particular, the spatial distribution of GCPs may affect the accuracy and quality of image correction. In this paper, both a simulation experiment and actual-image analyses are carried out to investigate the effect of the sampling design for selecting GCPs on the geometric correction of remotely sensed imagery. Sampling designs compared are simple random sampling, spatial coverage sampling, and universal kriging model-based sampling. The experiments indicate that the sampling design of GCPs strongly affects the accuracy of the geometric correction. The universal kriging model-based sampling design considers the spatial autocovariance of regression residuals and yields the most accurate correction. This method is highly recommended as a new GCPs sampling design method for geometric correction of remotely sensed imagery.

Estimation of non-ergodic variograms and their sampling variance by design-based sampling strategies

Design-based sampling strategies based on classical sampling theory offer unprecedented potentials for estimation of non-ergodic variograms. Unbiased and uncorrelated estimates of the semivariance at the selected lags and of its sampling variance can be simply obtained. These estimates are robust against deviations from an assumed spatial autocorrelation model. The same holds for the variogram model parameters and their sampling (co)variances. Moreover, an objective measure for lack of fit of the fitted model can simply be derived. The estimators for two basic sampling designs, simple random sampling and stratified simple random sampling of pairs of points, are presented. The first has been tested in real world for estimating the non-ergodic variograms of three soil properties. The parameters of variogram models and their sampling (co)variances were estimated with 72 pairs of points distributed over six lags.