Spatial Sampling Design Research Articles

Spatial Sampling Design in the Circumpolar Active Layer Monitoring Programme

Prior to development of the Circumpolar Active Layer Monitoring (CALM) programme, little attention was paid to formal spatial sampling designs for measuring active-layer thickness (ALT). This omission made the accuracy of many data-sets questionable, in part because spatial periodicities caused by landscape features such as ice-wedge polygons and thaw lakes can significantly influence the depth of thaw. Early in the development of CALM's protocols, ALT was sampled in the continuous permafrost zone in northern Alaska and simulated by computer, to determine how ALT could be measured accurately. The simulated and field data-sets were analysed by comparing the means, variances and frequency distributions obtained using four spatial sampling designs (random, systematic, systematic random and systematic stratified unaligned). By a small margin, systematic stratified unaligned sampling provided the most accurate results. Systematic designs can, however, provide adequate estimates of the statistical moments of ALT with significant savings in cost, time and ease of implementation. Based on these results, the CALM programme recommended use of 10 x 10, 100 x 100 or 1000 x 1000 m grids, with sampling intervals of 1, 10 and 100 m, respectively. Most probed CALM sites now employ this strategy, except in terrain with unusual landscape elements or other special constraints. Copyright © 2016 John Wiley & Sons, Ltd.

Latent spatial models and sampling design for landscape genetics

We propose a spatially-explicit approach for modeling genetic variation across space and illustrate how this approach can be used to optimize spatial prediction and sampling design for landscape genetic data. We propose a multinomial data model for categorical microsatellite allele data commonly used in landscape genetic studies, and introduce a latent spatial random effect to allow for spatial correlation between genetic observations. We illustrate how modern dimension reduction approaches to spatial statistics can allow for efficient computation in landscape genetic statistical models covering large spatial domains. We apply our approach to propose a retrospective spatial sampling design for greater sage-grouse (Centrocercus urophasianus) population genetics in the western United States.

Optimal adaptive sequential spatial sampling of soil using pair-copulas

A spatial sampling design that uses pair-copulas is presented that aims to reduce prediction uncertainty by selecting additional sampling locations based on both the spatial configuration of existing locations and the values of the observations at those locations. The novelty of the approach arises in the use of pair-copulas to estimate uncertainty at unsampled locations. Spatial pair-copulas are able to more accurately capture spatial dependence compared to other types of spatial copula models. Additionally, unlike traditional kriging variance, uncertainty estimates from the pair-copula account for influence from measurement values and not just the configuration of observations. This feature is beneficial, for example, for more accurate identification of soil contamination zones where high contamination measurements are located near measurements of varying contamination. The proposed design methodology is applied to a soil contamination example from the Swiss Jura region. A partial redesign of the original sampling configuration demonstrates the potential of the proposed methodology.

Supervised Classification of the Scalar Gaussian Random Field Observations under a Deterministic Spatial Sampling Design

Given training sample, the problem of classifying a scalar Gaussian random field observation into one of two populations specified by different parametric mean models and common parametric covariance function is considered. Such problems are usually called as supervised classification or contextual classification problems. This paper concerns with classification procedures associated with Bayes Discriminant Function (BDF) under deterministic spatial sampling design. In the case of parametric uncertainty, the ML estimators of unknown parameters are plugged in the BDF. The actual risk and the approximation of the expected risk (AER) associated with aforementioned plug-in BDF are derived. This is the extension of the previous one to the case of complete parametric uncertainty, i.e., when all mean functions and covariance function parameters are unknown. Stationary geometrically anisotropic Gaussian random field with exponential covariance function sampled on regular 2-dimensional lattice is used for illustrative examples.

Multivariate geostatistical analysis of fallout radionuclides activity measured by in-situ gamma-ray spectrometry

Cs-137 and other fallout radionuclides (FRNs) have been used worldwide for more than four decades as powerful techniques to assess the magnitude and spatial pattern of soil redistribution. Cs-137, 210Pbex, and 7Be have successfully been used as tracers of soil redistribution. Their coincident use can frequently provide worthwhile information for different time scales of soil redistribution. These radionuclides can either be measured in the laboratory or in-situ. In the present study in-situ measurements using a portable HPGe detector were carried out in two loessial paired sub-catchments with fairly similar characteristics in northeast Iran. Spatial sampling design based on a minimax approach was used to determine that 60 sites were sufficient for both areas of interest. Geostatistical analysis and the linear model of co-regionalization (LMC) were applied in this study for radionuclides. The spherical model for Sample and Testifier sub-catchments with ranges of 750 m and 500 m respectively was selected as the most suitable model. The highest and the lowest non-captured variability belonged to 210Pbex and 7Be for both studied areas. The patterns of spatial variation of 7Be and Cs-137 for Sample and the patterns of spatial variation of 7Be and 210Pbex for Testifier sub-catchment were very similar. For Cs-137, global uncertainty for both sub-catchments was nearly the same, but different for other radionuclides. The distribution of local uncertainty for all radionuclides in the Testifier sub-catchment was the same. The means of spatial distance between the grid cells with the highest and the lowest uncertainty for 7Be, Cs-137, and 210Pbex, are 0.4, 1, and 7 km, respectively.

Causal modelling applied to the risk assessment of a wastewater discharge.

Bayesian networks (BNs), or causal Bayesian networks, have become quite popular in ecological risk assessment and natural resource management because of their utility as a communication and decision-support tool. Since their development in the field of artificial intelligence in the 1980s, however, Bayesian networks have evolved and merged with structural equation modelling (SEM). Unlike BNs, which are constrained to encode causal knowledge in conditional probability tables, SEMs encode this knowledge in structural equations, which is thought to be a more natural language for expressing causal information. This merger has clarified the causal content of SEMs and generalised the method such that it can now be performed using standard statistical techniques. As it was with BNs, the utility of this new generation of SEM in ecological risk assessment will need to be demonstrated with examples to foster an understanding and acceptance of the method. Here, we applied SEM to the risk assessment of a wastewater discharge to a stream, with a particular focus on the process of translating a causal diagram (conceptual model) into a statistical model which might then be used in the decision-making and evaluation stages of the risk assessment. The process of building and testing a spatial causal model is demonstrated using data from a spatial sampling design, and the implications of the resulting model are discussed in terms of the risk assessment. It is argued that a spatiotemporal causal model would have greater external validity than the spatial model, enabling broader generalisations to be made regarding the impact of a discharge, and greater value as a tool for evaluating the effects of potential treatment plant upgrades. Suggestions are made on how the causal model could be augmented to include temporal as well as spatial information, including suggestions for appropriate statistical models and analyses.

Optimal sampling for spatial prediction of functional data

This paper combines optimal spatial sampling designs with geostatistical analysis of functional data. We propose a methodology and design criteria to find the set of spatial locations that minimizes the variance of the spatial functional prediction at unsampled sites for three functional predictors: ordinary kriging, simple kriging and simple cokriging. The last one is a modification of an existing predictor that uses ordinary cokriging based on the basis coefficients. Instead, we propose to use a simple cokriging predictor with the scores resulting from a representation of the functional data with the empirical functional principal components, allowing to remove restrictions and complexity of the covariance models and constraints on the estimation procedure. The methodology is applied to a network of air quality in Bogota city, Colombia.

Comment on Article by Ferreira and Gamerman

A utility-function approach to optimal spatial sampling design is a powerful way to quantify what "optimality" means. The emphasis then should be to capture all possible contributions to utility, including scientific impact and the cost of sampling. The resulting sampling plan should contain a component of designed randomness that would allow for a non-parametric design-based analysis if model-based assumptions were in doubt. [arXiv:1509.03410]

Hidden diversity in the freshwater planktonic diatom Asterionella formosa.

Many freshwater and marine algal species are described as having cosmopolitan distributions. Whether these widely distributed morphologically similar algae also share a similar gene pool remains often unclear. In the context of island biogeography theory, stronger spatial isolation deemed typical of freshwater lakes should restrict gene flow and lead to higher genetic differentiation among lakes. Using nine microsatellite loci, we investigate the genetic diversity of a widely distributed freshwater planktonic diatom, Asterionella formosa, across different lakes in Switzerland and the Netherlands. We applied a hierarchical spatial sampling design to determine the geographical scale at which populations are structured. A subset of the isolates was additionally analysed using amplified fragment length polymorphism (AFLP) markers. Our results revealed complex and unexpected population structure in A. formosa with evidence for both restricted and moderate to high gene flow at the same time. Different genetic markers (microsatellites and AFLPs) analysed with a variety of multivariate methods consistently revealed that genetic differentiation within lakes was much stronger than among lakes, indicating the presence of cryptic species within A. formosa. We conclude that the hidden diversity found in this study is expected to have implications for the further use of A. formosa in biogeographical, conservation and ecological studies. Further research using species-level phylogenetic markers is necessary to place the observed differentiation in an evolutionary context of speciation.

Incorporating covariance estimation uncertainty in spatial sampling design for prediction with trans-Gaussian random fields

Recently, Spock and Pilz [38], demonstrated that the spatial sampling design problem for the Bayesian linear kriging predictor can be transformed to an equivalent experimental design problem for a linear regression model with stochastic regression coefficients and uncorrelated errors. The stochastic regression coefficients derive from the polar spectral approximation of the residual process. Thus, standard optimal convex experimental design theory can be used to calculate optimal spatial sampling designs. The design functionals considered in Spock and Pilz [38] did not take into account the fact that kriging is actually a plug-in predictor which uses the estimated covariance function. The resulting optimal designs were close to space-filling configurations, because the design criterion did not consider the uncertainty of the covariance function. In this paper we also assume that the covariance function is estimated, e.g., by restricted maximum likelihood (REML). We then develop a design criterion that fully takes account of the covariance uncertainty. The resulting designs are less regular and space- filling compared to those ignoring covariance uncertainty. The new designs, however, also require some closely spaced samples in order to improve the estimate of the covariance function. We also relax the assumption of Gaussian observations and assume that the data is transformed to Gaussianity by means of the Box-Cox transformation. The resulting prediction method is known as trans-Gaussian kriging. We apply the Smith and Zhu [37] approach to this kriging method and show that resulting optimal designs also depend on the available data. We illustrate our results with a data set of monthly rainfall measurements from Upper Austria.

Ex situ seed collections will benefit from considering spatial sampling design and species’ reproductive biology

Collecting seed from natural plant populations is a key tool for conservation, ecological restoration, assisted migration, studying plant mating systems, and crop breeding. Many collections rely on simple, broadly-applied rules-of-thumb for minimum sample sizes, regardless of species’ natural history, and are likely inadequate for obtaining sufficient genetic representation. There is a current paucity of information about how various logistical and biological factors influence seed sampling outcomes. Here we use simulated and empirical data to, for the first time, quantitatively evaluate the degree to which collection effectiveness is influenced by spatial arrangement, sampling intensity, and species’ reproductive biology. We clearly and quantitatively demonstrate that collections with spatially limited or biased sampling arrangements, or collections from species with high selfing or low dispersal, will need substantially more samples than are commonly recommended, or else will fail to reach the targeted genetic diversity. We also show that the marginal gain from sampling additional maternal plants will nearly always exceed that of additional seeds. Lastly we show that random sampling outperforms “easy access” and “transect” sampling by 70% and 30%, respectively. Overall, we conclude that collection guidelines tailored to particular taxa will help facilitate optimal sampling design.

Model-Based Sampling Design for Multivariate Geostatistics

The quality of inferences made from geostatistical data is affected significantly by the spatial locations, or design, of the sites that are sampled. A large body of published work exists on sampling design for univariate geostatistics, but not for multivariate geostatistics. This article considers multivariate spatial sampling design based on criteria targeted at classical co-kriging (prediction with known covariance parameters), estimation of covariance (including cross-covariance) parameters, and empirical co-kriging (prediction with estimated covariance parameters). Through a combination of analytical results and examples, we investigate the characteristics of optimal designs with respect to each criterion, addressing in particular the design’s degree of collocation. We also consider the robustness of the optimal design to the strength of spatial correlation and cross-correlation; the effects of smoothness and/or separability of the sampled process on the optimal design; the relationship between optimal designs for the multivariate problems considered here and univariate problems considered previously; and the efficiency of optimal collocated designs. One key finding is that optimal collocated designs are highly efficient in many cases. Supplementary materials are available online.

Lineage delimitation and description of nine new species of bush frogs(Anura: Raorchestes, Rhacophoridae) from the Western Ghats Escarpment.

Bush frogs of the genus Raorchestes are distributed mainly in the Western Ghats Escarpment of Peninsular India. The inventory of species in this genus is incomplete and there is ambiguity in the systematic status of species recognized by morphological criteria. To address the dual problem of taxon sampling and systematic uncertainty in bush frogs, we used a large-scale spatial sampling design, explicitly incorporating the geographic and ecological heterogeneity of the Western Ghats. We then used a hierarchical multi-criteria approach by combining mitochondrial phylogeny, genetic distance, geographic range, morphology and advertisement call to delimit bush frog lineages. Our analyses revealed the existence of a large number of new lineages with varying levels of genetic divergence. Here, we provide diagnoses and descriptions for nine lineages that exhibit divergence across multiple axes. The discovery of new lineages that exhibit high divergence across wide ranges of elevation and across the major massifs highlights the large gaps in historical sampling. These discoveries underscore the significance of addressing inadequate knowledge of species distribution, namely the "Wallacean shortfall", in addressing the problem of taxon sampling and unknown diversity in tropical hotspots. A biogeographically informed sampling and analytical approach was critical in detecting and delineating lineages in a consistent manner across the genus. Through increased taxon sampling, we were also able to discern a number of well-supported sub-clades that were either unresolved or absent in earlier phylogenetic reconstructions and identify a number of shallow divergent lineages which require further examination for assessment of their taxonomic status.

Spatial Sampling Design of the Sensor Network for Monitoring the Surface Freeze/thaw Cycles over the Heterogeneous Surface in the Heihe River Basin

Spatial Sampling Design of the Sensor Network for Monitoring the Surface Freeze/thaw Cycles over the Heterogeneous Surface in the Heihe River Basin

Trap Array Configuration Influences Estimates and Precision of Black Bear Density and Abundance

Spatial capture-recapture (SCR) models have advanced our ability to estimate population density for wide ranging animals by explicitly incorporating individual movement. Though these models are more robust to various spatial sampling designs, few studies have empirically tested different large-scale trap configurations using SCR models. We investigated how extent of trap coverage and trap spacing affects precision and accuracy of SCR parameters, implementing models using the R package secr. We tested two trapping scenarios, one spatially extensive and one intensive, using black bear (Ursus americanus) DNA data from hair snare arrays in south-central Missouri, USA. We also examined the influence that adding a second, lower barbed-wire strand to snares had on quantity and spatial distribution of detections. We simulated trapping data to test bias in density estimates of each configuration under a range of density and detection parameter values. Field data showed that using multiple arrays with intensive snare coverage produced more detections of more individuals than extensive coverage. Consequently, density and detection parameters were more precise for the intensive design. Density was estimated as 1.7 bears per 100 km2 and was 5.5 times greater than that under extensive sampling. Abundance was 279 (95% CI = 193–406) bears in the 16,812 km2 study area. Excluding detections from the lower strand resulted in the loss of 35 detections, 14 unique bears, and the largest recorded movement between snares. All simulations showed low bias for density under both configurations. Results demonstrated that in low density populations with non-uniform distribution of population density, optimizing the tradeoff among snare spacing, coverage, and sample size is of critical importance to estimating parameters with high precision and accuracy. With limited resources, allocating available traps to multiple arrays with intensive trap spacing increased the amount of information needed to inform parameters with high precision.

Spatially explicit structural equation modeling

Structural equation modeling (SEM) is a powerful statistical approach for the testing of networks of direct and indirect theoretical causal relationships in complex data sets with intercorrelated dependent and independent variables. SEM is commonly applied in ecology, but the spatial information commonly found in ecological data remains difficult to model in a SEM framework. Here we propose a simple method for spatially explicit SEM (SE‐SEM) based on the analysis of variance/covariance matrices calculated across a range of lag distances. This method provides readily interpretable plots of the change in path coefficients across scale and can be implemented using any standard SEM software package. We demonstrate the application of this method using three studies examining the relationships between environmental factors, plant community structure, nitrogen fixation, and plant competition. By design, these data sets had a spatial component, but were previously analyzed using standard SEM models. Using these data sets, we demonstrate the application of SE‐SEM to regularly spaced, irregularly spaced, and ad hoc spatial sampling designs and discuss the increased inferential capability of this approach compared with standard SEM. We provide an R package, sesem, to easily implement spatial structural equation modeling.

Spatial replicates as an alternative to temporal replicates for occupancy modelling when surveys are based on linear features of the landscape

Summary Occupancy estimates can inform biodiversity managers about the distribution of elusive species, such as the Pyrenean desman Galemys pyrenaicus, a small semi‐aquatic mammal that lives along streams. Occupancy models rely on replication within a sampling site and provide estimates of the probability of detection. However, we still do not know how occupancy and detection estimates obtained from spatial vs. temporal replications differ or the appropriateness of using one or the other when cost and logistics make one approach prohibitive. Recently, the Markovian occupancy model has been developed to analyse adjacent spatial replicates and to test for spatial dependence between them. This model has already been applied to large and highly mobile mammals using trails, but never tested for any species with linear home ranges. We compared detection and occupancy estimates obtained from both temporal and spatial sampling designs that were subsequently organized into four data configurations (sites with both spatial and temporal replicates, adjacent spatial replicates only, temporal replicates only at the segment and site scales). From that, five occupancy models with different assumptions (the standard occupancy model, the standard multiscale model, the multiscale model with Markovian process for detection, the Markovian detection model and the Markovian occupancy model) were used. We also assessed which occupancy model was the most appropriate for each data configuration to determine whether it is necessary to incorporate correlation into models. We found that the estimated detection probabilities were relatively high (≥0·58) and similar when the same model was applied to each data configuration. Spatial replication weakly underestimated occupancy. But when using this design, the Markovian occupancy model was the most supported and minimized the underestimation of occupancy, highlighting a spatial dependence between adjacent replicates. Synthesis and applications. We show that a survey based on adjacent spatial replicates for a mammal living along linear features of the landscape is a good compromise between cost and occupancy estimates, while using the Markovian occupancy model to estimate detection and occupancy. Our finding may have wider applications for the monitoring of species especially when temporal replicates are difficult or unrealistic. Spatial design, for surveys based on sign detection, could thus be applied for species with linear home ranges or when surveys are constrained by linear habitats.

Spatial sampling on streams: principles for inference on aquatic networks

AbstractFor ecological and environmental data, prior inquiries into spatial sampling designs have considered two‐dimensional domains and have shown that design optimality depends on the characteristics of the target spatial domain and intended inference. The structure and water‐driven continuity of streams prompted the development of spatial autocovariance models for stream networks. The unique properties of stream networks, and their spatial processes, warrant evaluation of sampling design characteristics in comparison with their two‐dimensional counterparts. Common inference scenarios in stream networks include spatial prediction, estimation of fixed effects parameters, and estimation of autocovariance parameters, with prediction and fixed effects estimation most commonly coupled with autocovariance parameter estimation. We consider these inference scenarios under a suite of network characteristics and stream‐network spatial processes. Our results demonstrate, for parameter estimation and prediction, the importance of collecting samples from specific network locations. Additionally, our results mirror aspects from the prior two‐dimensional sampling design inquiries, namely, the importance of collecting some samples within clusters when autocovariance parameter estimation is required. These results can be applied to help refine sample site selection for future studies and further showcase that understanding the characteristics of the targeted spatial domain is essential for sampling design planning. Published 2014. This article has been contributed to by US Government employees and their work is in the public domain in the USA.

A space-time point process model for analyzing and predicting case patterns of diarrheal disease in northwestern Ecuador

We consider modeling case-patterns under a complex spatial and longitudinal sampling design as conducted via a serial case–control study of diarrheal disease in northwestern Ecuador. We build a two-stage space-time model to understand the role of spatially and temporally referenced covariates that reflect social and natural environments in the sampled region, after accounting for unmeasured residual heterogeneities. All diarrheal case events are collected from 21 sampled communities in Esmeraldes province in Ecuador, during seven sampling cycles from 2003 to 2008. The region of interest comprises 158 communities along a river basin. Prediction of case counts at unsampled communities at a future time is of interest along with estimation of risk-related parameters. We propose a computationally feasible two-stage Bayesian approach to estimate the risk-related parameters and conduct predictive inference. We first apply the log Gaussian Cox process (LGCP), commonly used to model spatial clustering of point patterns, to accommodate temporal variation within the sampled communities. Prediction of the number of cases at unsampled communities at a future time is obtained by a disease mapping model conditional on the expected case counts from Stage I.

Spatial Sampling Design for Monitoring the Quality of Cultivated Land by Simulated Annealing

Spatial Sampling Design for Monitoring the Quality of Cultivated Land by Simulated Annealing