Spatial Sampling Scheme Research Articles

Optimal spatial sampling schemes for environmental surveys

A practical problem in spatial statistics is that of constructing spatial sampling designs for environmental monitoring network. This paper presents a fractal-based criterion for the construction of coverage designs to optimize the location of sampling points. The algorithm does not depend on the covariance structure of the process and provides desirable results for situations in which a poor prior knowledge is available. The statistical characteristics of the method are explored by a simulation study while a design exercise concerning the Pescara area monitoring network is used to demonstrate potential designs under realistic assumptions.

Asymptotic distributions of M-estimators in a spatial regression model under some fixed and stochastic spatial sampling designs

In this paper, we consider M-estimators of the regression parameter in a spatial multiple linear regression model. We establish consistency and asymptotic normality of the M-estimators when the data-sites are generated by a class of deterministic as well as a class of stochastic spatial sampling schemes. Under the deterministic sampling schemes, the data-sites are located on a regular grid but may have aninfill component. On the other hand, under the stochastic sampling schemes, locations of the data-sites are given by the realizations of a collection of independent random vectors and thus, are irregularly spaced. It is shown that scaling constants of different orders are needed for asymptotic normality under different spatial sampling schemes considered here. Further, in the stochastic case, the asymptotic covariance matrix is shown to depend on the spatial sampling density associated with the stochastic design. Results are established for M-estimators corresponding to certain non-smooth score functions including Huber’s ψ-function and the sign functions (corresponding to the sample quantiles).

The dynamics of land‐cover change in western Honduras: exploring spatial and temporal complexity

AbstractThis paper presents an econometric analysis of land‐cover change in western Honduras. Ground‐truthed satellite image analysis indicates that between 1987 and 1996, net forest regrowth occurred in the 1015 km2study region. While some forest regrowth can be attributed to a 1987 ban on logging, the area of forest regrowth greatly exceeds that of previously clear‐cut areas. Further, new area was also deforested between 1987 and 1996. Thus, the observed land‐cover changes most likely represent a complex mosaic of changing land‐use patterns across time and space. Using satellite imagery from 1987, 1991 and 1996, we estimate a series of models, including binary probit models for each date, and a random‐effects probit model using panel techniques. We also experiment with spatial sampling schemes designed to reduce residual spatial autocorrelation, and qualitatively compare the impact of spatial sampling on model accuracy. Lastly, we find that changes in relative prices, infrastructure improvement, and topography are all significantly related to changing land‐cover patterns.

The dynamics of land-cover change in western Honduras: exploring spatial and temporal complexity

This paper presents an econometric analysis of land-cover change in western Honduras. Ground-truthed satellite image analysis indicates that between 1987 and 1996, net forest regrowth occurred in the 1015 km 2 study region. While some forest regrowth can be attributed to a 1987 ban on logging, the area of forest regrowth greatly exceeds that of previously clear-cut areas. Further, new area was also deforested between 1987 and 1996. Thus, the observed land-cover changes most likely represent a complex mosaic of changing land-use patterns across time and space. Using satellite imagery from 1987, 1991 and 1996, we estimate a series of models, including binary probit models for each date, and a random-effects probit model using panel techniques. We also experiment with spatial sampling schemes designed to reduce residual spatial autocorrelation, and qualitatively compare the impact of spatial sampling on model accuracy. Lastly, we find that changes in relative prices, infrastructure improvement, and topography are all significantly related to changing land-cover patterns.

Reduction of soil water spatial sampling density using scaled semivariograms and simulated annealing

Spatial sampling density influences the reliability and feasibility of environmental studies. Optimizing spatial sampling schemes is important, particularly when multiple observations must be repeated over extended periods. The goal of this study was to develop a sampling density reduction method for a network of 57 soil water content (SWC) measurement locations in an 8-ha microwatershed, given observations taken at five different dates. We sought a subset of 10 points that would best predict (via spatial interpolation) the SWC at the remaining locations. Three observation dates (calibration set) were used to develop the method, and the remaining two (validation set) were used to test it. Calibration date semivariograms were coalesced into one scaled semivariogram, used with ordinary kriging to predict SWC outside the chosen subset. We defined four different scenarios by combining two simulated annealing algorithms, Sacks and Schiller (S&S) and Spatial Simulated Annealing (SSA), and two fitness functions, one based on scaled kriging variance (SKV), and the other based on actual mean squared prediction error (scaled mean squared error, SMSE). We searched for an optimal subset in each calibration scenario; each was then used to predict SWC throughout the microwatershed on the validation dates. The results were compared with those of regular grids and randomly generated patterns. Temporal stability was tested by analyzing deviations between individual and field average observations and using Spearman's rank correlation. The S&S and SSA algorithms performed similarly well, although SSA converged better. The SKV-based scenarios had lower SKV in the calibration and validation sets than the SMSE-based scenarios, the regular grids, and the random patterns. However, the SMSE-based scenarios produced an optimal subset having minimal SMSE over both data sets vs. the other methods. This subset produced mostly low relative errors: for January 25, 1993, 50% of the predicted points fell within ±5%, 82% within ±10%, and 6.5% fell outside 15%; for December 23, 1993, 35% were within ±5%, 72% within ±10%, and 23% fell beyond 15%. However, kriging assumptions were violated on December 23. The SMSE-based scenarios predicted validation set SWC better than the SKV scenarios because they included microwatershed locations that did not obey the stationarity assumptions of kriging but were temporally stable and captured the full range of SWC variation. Modifying the proposed method to perform kriging with a physically based trend model will further improve its predictive accuracy.

Bias Correction and Bootstrap Methods for a Spatial Sampling Scheme

Motivated by sampling problems in forestry and related fields, we suggest a spatial sampling scheme for estimating the intensity of a point process. The technique is related to the 'wandering quarter' method. In applications where the cost of identifying random points is high relative to the cost of taking measurements, for example when identification involves travelling within a large region, our approach has significant advantages over more traditional approaches such as T-square sampling. When the point process is Poisson we suggest a simple bias correction for a 'naive' estimator of intensity, and also discuss a more complex estimator based on maximum likelihood. A technique for pivoting, founded on a fourth-root transformation, is proposed and shown to yield second-order accuracy when applied to construct bootstrap confidence intervals for intensity. Bootstrap methods for correcting edge effects and for addressing non-Poisson point-process models are also suggested.

The influence of variogram parameters on optimal sampling schemes for mapping by kriging

Using spatial simulated annealing (SSA), spatial sampling schemes can be optimised for minimal kriging variance. Two optimisation criteria are presented in this paper. The first criterion minimises the average kriging variance, the second the maximum kriging variance. In a simple case with 23 observations, performances of a sampling scheme obtained with SSA were compared with performances of a triangular grid. SSA reduced the average kriging variance from 40.64 to 39.99 [unit] 2. The maximum kriging variance was reduced from 86.83 to 53.36 [unit] 2. Starting with a preliminary, irregularly data set of 100 observations, an additional sampling scheme of 10 observations was optimised. This reduced the average kriging variance from 21.62 to 15.83 [unit] 2. The maximum kriging variance was reduced from 70.22 to 34.60 [unit] 2. As the kriging variance is considerably influenced by variogram parameters, their influence on the optimised sampling schemes was investigated. A Gaussian variogram produced a different sampling scheme than an exponential variogram with the same nugget, sill and (effective) range. Exponential, spherical and linear variograms without nugget resulted in irregular similar sampling schemes. A very short range resulted in irregular sampling schemes, with observations separated by distances larger than twice the range. For a spherical variogram, the magnitude of the relative nugget effect did not affect the sampling schemes, except for very high values (0.75).

Constrained optimisation of soil sampling for minimisation of the kriging variance

This paper introduces the extended Spatial Simulated Annealing (SSA) method to optimise spatial sampling schemes for obtaining the minimal kriging variance. Sampling schemes are optimised at the point level. Boundaries and previous observations can be taken into account. This procedure extends ordinary SSA which focuses entirely on variogram estimation and even distribution of observations over the area. We applied it to texture and phosphate content on a river terrace in Thailand. First, sampling was conducted for estimation of the variogram using ordinary SSA. The variograms thus obtained were used in extended SSA, yielding a reduction of the mean kriging variance of the sand percentage from 28.2 to 23.7(%) 2. The variograms were used subsequently in a geomorphologically similar area. Optimised sampling schemes for anisotropic variables differed considerably from those for isotropic ones. The schemes were especially efficient in reducing high values of the kriging variance near boundaries of the area.

Constrained Optimization of Spatial Sampling using Continuous Simulated Annealing

AbstractSpatial sampling is an important issue in environmental studies because the sample configuration influences both costs and effectiveness of a survey. Practical sampling constraints and available preinformation can help to optimize the sampling scheme. In this paper, spatial simulated annealing (SSA) is presented as a method to optimize spatial environmental sampling schemes. Sampling schemes are optimized at the point‐level, taking into account sampling constraints and preliminary observations. Two optimization criteria have been used. The first optimizes even spreading of the points over a region, whereas the second optimizes variogram estimation using a proposed criterion from the literature. For several examples it is shown that SSA is superior to conventional methods of designing sampling schemes. Improvements up to 30% occur for the first criterion, and an almost complete solution is found for the second criterion. Spatial simulated annealing is especially useful in studies with many sampling constraints. It is flexible in implementing additional, quantitative criteria.

Sampling errors in anchovy egg abundance estimates using the PAIROVET net

A replicate haul experiment was carried out using the vertically towed PAIROVET net in an area of high abundance of anchovy eggs in the Bay of Biscay. The main aim of the experiment was to compare the performance of ships of different size working in the same area. In addition, we tested the performance of the PAIROVET net and the deployment methodology used to sample anchovy eggs for biomass assessment purposes. We concluded that under the experimental conditions experi- enced in this cruise, different ships yield similar performances of the vertically towed PAIROVET net if the methodology is kept similar. The ANOVA suggests that the systematic central sampling (SCS) scheme currently in use for the Bay of Biscay anchovy biomass estimations (one station out at 3 miles) adequately represents the surrounding area along the transect line. The among-station component of variance within the unit sampling area of the current SCS turned out to be comparable and, in some cases, smaller than the within-station variance. Consequently, the spatial design of the SCS adopted during routine surveys appears to be a reliable strategy for sampling anchovy eggs in areas of high egg abundance, as in this experiment. In addition, it ensures fairly independent data units. In the area of the experiment, the most disperse egg stages (like 2- and 3-day-old eggs) show a larger within- station component of variance than the spatial component of variance within the rectangles of the current SCS. This suggests that for these stages, the sampling can still be improved cheaply by increas- ing the precision of egg abundance estimates per station with larger sample sizes, without changing the spatial sampling scheme or the number of stations. However, the benefits of this strategy will hardly affect the precision of the 1-day-old egg estimates because of their patchy distribution.

Maximum likelihood estimation under a spatial sampling scheme

It is shown that the maximum likelihood estimator in a model used in the statistical analysis of computer experiments is asymptotically efficient.

Application of soil‐vegetation correlation to optimal resolution mapping of solonetzic rangeland

An optimal resolution surveying technique was developed and tested in the solonetzic grassland of Hortobagy, Hungary. Indication of soil properties by vegetation is utilized to create a sampling design by applying a satellite image of the pilot area and to estimate soil properties on the basis of the statistical correlation between vegetation and soil data. The spatial sampling scheme was designed in accordance with the local heterogeneity of the satellite image. In order to study the predictability of the vegetation categories in terms of soil chemical properties, discriminant analysis was used with pH, electrical conductivity, and sodium activity at three depths, used as grouping variables. Based on the taxonomy of vegetation types, about two‐thirds of the cases can be classified and correlated with the soil chemical properties.

Reorganization and diversification of signals in vision

Extensive processing and reorganization of information are required as retinally generated signals flow centrally. Insight into some principles of the reorganization and transformations that occur in the early stages of the visual pathway, prior to the higher stages of pattern recognition, is provided through an analysis of receptive fields and cell counts. The rate of information flow is reduced approximately 100-fold by a retinal position-dependent (inhomogeneous) spatial sampling scheme. This principle of specialization, or nonuniform processing, is further elaborated in the retino-cortical mapping. In the central fovea, where the retinal spatial sampling rate is the highest and processing function is uniform, there are about 4000 striate cortical neurons receiving information from each cone. This number, which provides a measure of function multiplicity, drops to about 200 over the range of the near periphery. In the peripheral field beyond eight degrees, where information is compressed at the retina, the functional multiplicity stays approximately constant at 200. As it seems that no two cells perform exactly the same operation, it is concluded that the striate cortex performs many simultaneous functional mappings.

Optimal Randomization for Experiments in which Autocorrelation is Present

For a linear or spatial ordering of the experimental units, systematic methods of applying the treatments to the experimental units minimize the variance of linear contrasts of treatment means. In the linear ordering the optimal randomization scheme for the application of k treatments is to apply a treatment to every kth experimental unit, the equivalent of systematic sampling in the sampling literature. In the spatial ordering the optimal randomization schemes are equivalent to Quenouille's (1949) systematic aligned or unaligned spatial sampling schemes.

Optimal randomization for experiments in which autocorrelation is present

SUMMARY For a linear or spatial ordering of the experimental units, systematic methods of applying the treatments to the experimental units minimize the variance of linear contrasts of treatment means. In the linear ordering the optimal randomization scheme for the application of k treatments is to apply a treatment to every kth experimental unit, the equivalent of systematic sampling in the sampling literature. In the spatial ordering the optimal randomization schemes are equivalent to Quenouille's (1949) systematic aligned or unaligned spatial sampling schemes.

Effect of spatial sampling on an acoustical image reconstruction

It is often difficult and unpractical, in the acoustic imaging area, to insert a focusing lens between the observed object and the ultrasonic receiver. Furthermore, the focal length must, in the B scan mode, be varied very fast in order to keep the line sources lying at different depths well focused. A spatial-sampling scheme may be used for circumventing these problems by taking off the possibly complex processing unit from the element contacting the observed object, without additional complexity or degradation in the flexibility and usefulness of the apparatus. The sampling is done using a receiving transducer array to convert the acoustic signals into electrical ones, which may be amplified before being applied to another reemitting transducer array: an acoustical field identical to the original one is so reconstructed. The sampling operation will, however, alter the image quality by introducing parasitic images. The dependence of the perturbations upon the system parameters is studied in this paper and criteria are derived for limiting them.

Quadrat Analysis of Urban Dispersion: 4. Spatial Sampling

This paper is the concluding part of a four-part essay on quadrat analysis. It shows how quadrat analysis of spatial dispersion may be linked with Jessen's (1970) method of probability sampling with marginal constraints to produce more efficient spatial sampling schemes.