Semivariogram Range Research Articles

Crustal Structure Beneath Mexico City From Joint Inversion of Receiver Functions and Dispersion Curves

AbstractOver the years, it has been clear to seismologists that the crustal structure beneath Mexico City is highly complex. This fact may influence the seismic waves' behavior when traveling through it. Consequently, many studies have been done to understand this crustal structure. This article presents a high‐resolution S‐wave velocity model obtained from joint inversion of radial receiver functions and group‐velocity dispersion curves beneath Mexico City (VMRFDC). This model includes azimuthal variations observed with receiver functions, which may be interpreted as crustal discontinuities. Receiver functions were retrieved from velocity and acceleration records, and dispersion curves were collected from a previous tomographic study. We performed a geostatistical analysis that revealed a horizontal spatial correlation of the crustal structure for up to 7.5 km. This distance was obtained from the S‐wave velocity omnidirectional semivariogram range per depth. Using ordinary kriging, we utilized this spatial correlation to estimate S‐wave velocities per depth and introduced two novel techniques: a resolution test for kriging and a geostatistical‐dependent smoothing technique. Finally, we compare our results with previous works, making our model consistent with geological and geophysical features reported in the literature.

Sampling density to detect spatial dependence of potassium, calcium and magnesium in sandy soils

The sampling grid density for georeferenced soil collection must be large enough to allow the identification of the spatial dependence of attributes with representative accuracy of the cultivated area, but not large enough to make fertility mapping unfeasible. The objective of this study was to define, from the evaluation of geostatistical parameters obtained from a super dense soil sampling, an efficient grid for detecting the spatial dependence of potassium (K+), calcium (Ca2+), and magnesium (Mg2+) in a sandy soil. The experiment was conducted in a 3.2 hectare annatto crop (Bixa orellana L.), in 2017. The geostatistical grid consisted of 31 points per hectare, totaling 101 georeferenced points in an 18x18 m spacing. Soil was sampled at the depths of 0-0.20 m and 0.20-0.40 m. A strong spatial dependence was found for all soil attributes in both depths, while the semivariograms fitted to the spherical model with good coefficients of determination (R²) indicating a spatial correlation between the attributes. The range of spatial dependence was close to 100 m for all attributes in both layers. In sandy soils, an efficient sampling grid to detect the spatial dependence of K+, Ca2+ and Mg2+ must consider a semivariogram range of approximately 100 meters.

Gaining Insight from Semi-Variograms into Machine Learning Performance of Rock Domains at a Copper Mine

Machine learning (ML) is increasingly being leveraged by the mining industry to understand how rock properties vary at a mine site. In previously published work, the rock type, granodiorite, was predicted with high accuracy by the random forest (RF) ML method at the Erdenet copper mine in Mongolia. As a result of the optimistic results (86% overall success rate), this paper extended the research to determine if ML would be successful in modeling rock domains. Rock domains are groups of rocks that occur together. There were two additional goals. One was to determine if the variograms could predict or help understand how ML methods would perform on the data. The second was to determine if 2D modeling would perform well given the disseminated nature of the deposit. ML methods, multilayer perceptron (MLP), k-nearest neighborhood (KNN) and RF, were applied to model six rock domains, D0–D5, in 2D and 3D. Modeling performance was poor in 2D. Prediction performance accuracy was high in 3D for the domains D1 (92–94%), D2 (94–96%) and D4 (85–98%). Note that the domains D1 and D2 together constituted about 80% of the samples. Conclusions drawn in this paper are based on the results of 3D modeling since 2D modeling performance was poor. Prediction performance appeared to depend on two factors. It was better for a domain when the domain was not a minuscule proportion of the sample. It was also better for domains whose indicator semi-variogram (ISV) range was high. For example, though D4 only contributed 15% of the samples, the range was high. MLP did not perform as well as KNN and RF, with RF performing the best. The hyperparameters of KNN and RF suggested that performance was best when only a small number of samples were used to make a prediction. One overall summary conclusion is that the two most important domains, D1 and D2, could be predicted with high accuracy using ML. The second summary conclusion is that semi-variograms can provide insight into ML performance.

Simulation of yield and water balance using WHCNS and APSIM combined with geostatistics across a heterogeneous field

Estimating water balance is the foundation of improving water productivity (WP) and managing crop production efficiently in fields with significant spatial variability of soil properties. Agricultural system models are useful tools to simulate crop yield, WP, and water balance; however, they are rarely focused on geostatistical characteristics on the spatial scale. If agricultural system models are used spatially, it is important to consider whether the geostatistical characteristics of the simulations are similar to those based on measured data. In this study, two process-based models, the widely-used Agricultural Production Systems sIMulator (APSIM) and the newly-developed soil Water Heat Carbon Nitrogen Simulator (WHCNS), were used to simulate crop yield, water balance, and WP in a 54-ha field with spatially variable soil properties. Performance of the simulations was good for both models; however, the simulation accuracy of WHCNS was higher than for APSIM. Geostatistical characteristics of measured maize yield and final soil water storage (SWSf) were different from the simulated data. The fitted semivariogram models of simulated yield and SWSf had a higher semivariogram range and lower random variation than that of measured data. The fitted semivariogram models and geostatistical characteristics of simulated water balance also varied between the two models. Although the agricultural system models simulated the spatial distribution of variables efficiently, their spatial structure was changed in comparison with the spatial structure of measured data. This would affect the interpolation precision of spatial distribution maps. More work is required on the robustness and prediction accuracy of both models for their implementation in a spatial way.

Investigating the potential of Sentinel-1 to detect varying spatial heterogeneity in pasture cover in grasslands

ABSTRACT Selective grazing by livestock may be indicative of a site’s grass species diversity and depending on the grazing intensity; this may or may not promote further diversity. However, the detection of sites with spatial heterogeneity in pasture cover as a manifestation of selective grazing has not yet been investigated using satellite remote sensing. Thus, this study was conducted to address the question; can Sentinel-1 detect spatial heterogeneity induced by livestock grazing in grassy fields? Since Synthetic Aperture Radar (SAR) imaging is noted to be sensitive to vegetation architectural arrangement, this study used Sentinel-1 C-band SAR to detect spatial heterogeneity created by selective livestock grazing. The study examined a range of semivariogram, grey-level co-occurrence matrix (GLCM), and eigenvector-eigenvalue polarimetric decomposition features. The coefficient of variation estimates of the GLCM contrast feature consistently produced the strongest correlation (R 2 = 0.71) with Lloyd’s Patchiness Index and semivariogram sill while the polarimetric scattering entropy (range estimates) produced a significant linear correlation with semivariogram sill (R 2 = 0.55, p < 0.05). Inferably, the GLCM contrast and polarimetric scattering entropy can predict spatial heterogeneity in a grazing environment. This is the first time polarimetric scattering entropy estimated from Sentinel-1 has been used for the detection of spatial heterogeneity in a grazing landscape, which makes this study different from past similar studies. Nonetheless, we recommend the testing of this parameter (polarimetric scattering entropy) with a multitemporal SAR data and encourage future studies to investigate the potential of Sentinel-1 for the detection of spatial distances between grass clumps.

The impact of groundwater depth on the spatial variance of vegetation index in the Ordos Plateau, China: A semivariogram analysis

As important sensitive feedback of ecosystems, spatial distribution and patterns of vegetation can remarkably reflect eco-environmental conditions in arid and semiarid areas, where groundwater plays a significant role. The impact of groundwater depth (GD) on the spatial variance in remote-sensing vegetation index is highlighted in this study over the Ordos Plateau, China. The 250 m resolution mean annual Enhanced Vegetation Index (EVI) data are analyzed and compared to the same resolution data of altitude, air temperature, aridity index and GD. A linear regression analysis indicates a significant weak correlation between the EVI and the climatic and topographic controls at the regional scale. A semivariogram analysis is introduced to quantify the GD-dependent variance in the residual EVI (WEVI) filtered from this linear regression. The pixel-to-pixel dependency of WEVI on GD is nonlinear and significant only when GD is smaller than 10 m. The semivariance of WEVI increases with the lag distance and slightly fluctuates around the sill when the distance is sufficiently larger than 5 km. The sill of WEVI variance in the situation of GD ≤ 5 m is significantly higher than that in deeper groundwater situations. The semivariogram range of WEVI in the middle zone of the study area is lowest due to the lowest semivariogram range of GD. The nugget effect in the WEVI variance is weak but positively related to the nugget of GD variance when groundwater is shallow. High sensitivity of the vegetation cover to the shallow groundwater should be considered in the regulation of groundwater resources for ecological benefits.

English

Micronutrients are important for crop growth, production and their deficiency and toxicity affect crop yield. However, the up dated information about their status and spatial distribution in Ethiopian soils is scarce. Therefore, fertilizer recommendation for crops in the country has until recently focused on nitrogen and phosphorus macronutrients only. But many studies have revealed the deficiency of some micronutrients in soils of different parts of Ethiopia. To narrow this gap, this study was conducted in Kedida Gamela, Kecha Bira and Damboya districts of Kambata Tambaro (KT) Zone, Southern Ethiopia, through assessing and mapping the status and spatial distribution of micronutrients. The micronutrients were extracted by using Mehlich-III multi-nutrient extraction method and their concentrations were measured by using Inductively Coupled Plasma Optical Emission Spectrometer (ICP-OES). The fertility maps and predication were prepared by co-Kriging method using Arc map 10.0 tools and the status of Melich-III extractable iron (Fe), Zinc (Zn), boron (B), copper (Cu) and molybdenum (Mo) were indicated on the map. The extracted Fe ranged from 50.04 to 209.72 mg/kg, 60.08 to 240 mg/kg and 46.84 to 412.23 mg/kg in Kedida Gamela, Kecha Bira and Damboya districts soils, respectively. Zn ranged from 1.3 to 28 mg/kg, 0.9 to 47 mg/kg and 1.0 to 39 mg/kg for Kedida Gamela, Kecha Bira and Damboya districts, respectively. The calculated manganese activity index (MnAI) indicated that Mn is in excess and even toxic. B ranged from 0.02 to 1.83 mg/kg, 0.4 to 1.44 mg/kg and 0.06 to 2.03 mg/kg in soils of Kedida Gamela, Kecha Bira and Damboya woredas, respectively indicating that most of soils were deficient in B. Cu ranged from 0.6 to 2.9 mg/kg, 0.5-1.44 mg/kg and 0.8 to 3.4 mg/kg in Kedida Gamela, Kecha Bira and Damboya districts indicating its status fall between low and optimum category. Mo ranged from 2.21 to 18.71 mg/kg in the soils of the study area indicating that all soils were sufficient in Mo content. The means of all micronutrients except B showed significant differences among districts and showed moderate spatial dependences. The range of semivariogram for all studied micronutrients was greater than the average sampling distance indicating that it was adequate enough to catch spatial variability of them. In order to strengthen this result, plant sample analysis and calibration of micronutrients with plant response are recommended. Key words: Boron, Copper, Iron, Kriging, Manganese, Mehlich-III, Molybdenum, Zinc and spatial dependency.

Spatial interpolation quality assessment for soil sensor transect datasets

Near-ground geophysical soil sensors provide valuable information for precision agriculture applications. Indeed, their readings can be used as proxy for many soil parameters. On-the-go soil sensor surveys are, typically, carried out intensively (e.g., every 2m) over many parallel transects. Two types of soil sensors measurements are considered in this paper: apparent electrical conductivity (4 fields in California, USA) and reflectance (1 field in Italy). Two types of spatial interpolations are carried out, universal kriging (model-based) and inverse distance weighting (deterministic). Interpolation quality assessment is usually carried out using leave-one-out (loo) resampling. We show that loo resampling on transect sampling datasets returns overly-optimistic, low interpolation errors, because the left-out data point has values very close to that of its neighbors in the training dataset. This bias in the map quality assessment can be reduced by removing the closest neighbors of the validation observation from the training dataset, in a (spatial) h-block (SHB) fashion. The results indicate that, for soil sensor data acquired along parallel transects: (i) the SHB resampling is a useful tool to test the performance of interpolation techniques and (ii) the optimal (i.e., rendering the same errors of un-sampled locations between transects) SHB threshold distance (h.dist) for neighbor-exclusion is proportional to the semi-variogram range and partial sill. This procedure provides research scientists with an improved means of understanding the error of soil maps made by interpolating soil sensor measurements.

How are landscape complexity and vegetation structure related across an agricultural frontier in the subtropical Chaco, NW Argentina?

Human-driven alteration of the Chaco strongly affects ecological patterns and associated processes at all spatial scales. To understand these modifications, sufficient methods for describing and quantifying high levels of landscape complexity caused by human activities in the region are urgently needed. Most methods involve the use of passive remote sensors, which capture complexity in only two dimensions (2D). A common 2D approach has been to calculate landscape metrics, such as Shannon's Landscape Diversity Index. But, it is not clear what aspects of three dimensional (3D) vegetation structure are being captured by these metrics. 3D structure is known to be as important as or more important than 2D structure in determining landscape patterns of biodiversity of many groups of organisms. In addition, studies have used a limited number of coarsely defined land-cover classes to calculate metrics. Our question was: how is vegetation structure related to remote sensing attributes in an agricultural frontier in the subtropical dry Chaco, NW Argentina? A secondary question was to explore the relationships between traditional landscape metrics and the semivariogram, a geostatistical tool used to describe 2D complexity. We described landscape complexity from the panchromatic QuickBird band and measured vegetation structure in 22-1 ha plots across an agricultural frontier in the subtropical dry Chaco, northern Argentina. A total of 2683 individual trees in 51 plant species and 21 families were measured in the field and 25,665 points were recorded to estimate foliage height diversity. Four landscape complexity groups were identified by a two-way cluster analysis using the 2D metrics. Four vegetation variables differed significantly among the 2D complexity groups: the standard deviation of the Enhanced Vegetation Index, the coefficient of variation of density per transect (CV density), mean tree diameter (DBH), and foliage height diversity (FHD). Largest patch index and semivariogram range were negatively related to CV density, mean DBH and FHD, while semivariogram sill, mean shape index, landscape shape index and number of patches were positively related to all three vegetation variables. Landscape metrics were not related to tree species diversity or density as previously shown, probably as a result of structural similarity among the dominant tree species in the Chaco biome.

Comparison of the EM38 and EM38-MK2 electromagnetic induction-based sensors for spatial soil analysis at field scale

Electromagnetic induction (EMI) sensors are frequently used for the assessment of soil properties, particularly the clay content, soil salinity, water content and other soil characteristics. In this work, we compared two of these sensors, EM38 and EM38-MK2, with regards to their performance in (i) point-wise measurements with direct juxtaposition of the ECa readings, (ii) determining the soil texture and thickness of Quaternary sediments, (iii) the interpolation procedure (with semivariograms) of ECa and (iv) visual inspection maps of ECa and soil texture. Measurements were performed on a research area representing highly variable site conditions (sandy, gravelly, silty and clayey Tertiary and Quaternary sediments). The point-wise juxtaposition delivered no significant differences between the ECa values of both sensors. The determination of the soil texture and the interpolation results, however, exhibited distinct gradations between the two sensors. The regression analyses indicated a primarily higher R2 with the EM38 in comparison to the EM38-MK2. Furthermore, the EM38 provided in both modes lower semivariogram range values. This result leads to the conclusion that EM38 delivers more small-scale area-wide ECa distributions.

NDVI spatial pattern and the potential fragility of mixed forested areas in volcanic lake watersheds

Upland forested areas of watersheds undergo changes due to many factors including ecological succession, natural disturbances and human activity. The rate of natural and man-induced ecological changes in these landscapes is a function of the structural and functional characteristics of the component ecosystems. Analyzing spatial patterns and detecting fragile areas are thus crucial for making previsions about the chance and rate of disturbance propagation within and between the ecosystems. In this study we have tested the hypothesis of occurrence of a relationship between the extent of temporal change and spatial heterogeneity of mixed forested areas in the watershed of two Italian volcanic lakes by using remotely sensed data. Landsat images were acquired in summer 1987, 1992 and 2000, when Nature Reserves were established, and the temporal variation in the Normalized Difference Vegetation Index (NDVI) was determined by the change detection analysis. To analyze the spatial variability of NDVI, semivariograms were calculated using data from five randomly chosen forested areas (10km2-wide) per watershed. Results show that NDVI varied greatly across the two study sites and most of the variation was spatially structured. NDVI varied also over time. A linear positive relationship was observed between the number of pixels changing between dates and the semivariogram range, as the maximum distance of spatial dependence estimated from the starting NDVI image. Spatial homogeneity of NDVI is thus suggested as an indicator of intrinsic fragility (i.e. susceptibility to change) of mixed forests and the semivariogram range as a rapid estimator that can be considered by forest managers and agencies.

Response of Wild Blueberry Yield to Spatial Variability of Soil Properties

Crop management practices within wild blueberry fields are implemented uniformly, with inadequate attention being given to substantial variations in soil properties and fruit yield. These variations emphasize the need for precise and site-specific crop management based on proper characterization and quantification of spatial soil variability. The objective of this research was to characterize and quantify the spatial patterns of variability in soil properties and fruit yield. A grid pattern of sampling points was established at each experimental site to collect soil and fruit yield samples during 2009 to 2010 at North River and Carmal sites, Nova Scotia, Canada. The soil samples were collected from 0- to 15-cm depth at each grid point. These soil samples were analyzed for soil organic matter, pH, texture, electrical conductivity, and inorganic nitrogen (N). The volumetric moisture content (&thetas;v) was recorded at each grid point using time domain reflectometry. Ground conductivity readings were also recorded using dual EM conductivity meter at same selected grid points. Fruit yield was mapped using calibrated digital color photography during the crop year. The coefficient of variation of the soil properties and fruit yield suggested moderate to high variability (coefficient of variation >15%) except for soil pH. The results of correlation analysis indicated that the values for inorganic N, soil organic matter, electrical conductivity, clay, and &thetas;v were generally higher in high-yielding areas and vice versa. The trend was opposite for sand and silt content, which were higher in low-yielding areas. Most of the soil properties had a large spatial variation with semivariogram range of 20 to 50 m and varied greatly within field. Kriged maps of soil properties and fruit yield also showed substantial variation within field. Characterization of spatial soil variability in wild blueberry fields would assist in planning future soil sampling in new fields showing soil and/or yield variability. The selection of soil sampling should be based on the sampling interval suggested by the variogram range (<20 m). Proper soil sampling can help in identifying yield-limiting soil properties and develop prescription maps for site-specific nutrient management to ameliorate unproductive areas and reduce environmental contamination.

Spatial Variability of Soil Physical Properties Affecting Florida Citrus Production

Spatial variability of citrus yield is very common in Florida, but very little has been done to identify the responsible soil factors. Our objective was to characterize variations in soil physical properties in sandy soils where citrus is produced in Florida. A citrus grove was divided into five productivity zones defined as "very poor," "poor," "medium," "good," and "very good" based on tree canopy volume. Six random soil samples were collected from each zone at depths of 0 to 15, 15 to 30, 30 to 45, and 45 to 60 cm. Samples were analyzed for soil color, particle size and sand size distribution, and water retention parameters. Although the coefficients of variation (1%-60%) were large for color, texture, and water retention, the values were generally higher in areas with large trees and vice versa. The trend was opposite for soil lightness and sand content, which were lower in areas of greater growth. Most of the soil properties had a large spatial variation with semivariogram range of 100 to 250 m and varied greatly with increased productivity. Differences between the productivity zones were even more prominent at greater soil depth. Characterization and quantification of spatial soil variability in citrus groves would assist in planning future soil sampling in new fields showing soil and/or tree growth variability. The selection of soil sampling sites should be based on the sampling interval suggested by the variogram range (<75 m) and include the depth (45-60 cm) influencing productivity.

Laser scanning of fine scale pattern along a hydrological gradient in a peatland ecosystem

Lowland ombrotrophic (rain-fed) peatlands are a declining ecological resource in Europe. Peatlands display characteristic patterns in vegetation and surface topography, linked to ecological function, hydrology, biodiversity and carbon sequestration. Laser scanning provides a means of precisely measuring vegetation pattern in peatlands, and thus holds promise as a tool for monitoring peatland condition. Terrestrial laser scanning (TLS) was used for measurement of vegetation pattern along an eco-hydrological gradient at a UK peatland (Wedholme Flow, Cumbria) at fine grain sizes (<1 cm spatial resolution over 10 m spatial extent). Seven sites were investigated—each showed varying water table and ecological characteristics. TLS data were analysed using semi-variogram analysis which enabled the scale of spatial dependence in vegetation structures to be measured. In addition ecological, hydrological and positional surveys were conducted to elucidate interpretation of spatial patterns. Results show that TLS was able to rapidly measure vegetation patterns associated with eco-hydrological condition classes. Intact sites with hummock-hollow topography showed an isotropic pattern with a grain size or length-scale of 1 m or less (indicated by semi-variogram range). Degraded sites with high shrub cover showed increased sill variance values at larger range distances—typically around 3–4 m. The work presented shows the advantages of TLS methodologies for rapid measurement of 3-D vegetation canopy structure and surface microtopography, at fine spatial scales, in short vegetation. The paper considers how these approaches may be extended to monitoring peatland structure over larger spatial extents from airborne LiDAR systems.

A semi-variogram approach for estimating stems per hectare in Eucalyptus grandis plantations using discrete-return lidar height data

Estimating stems per hectare (SPHA) for a given forest area from high spatial resolution remotely sensed data usually follows the identification of individual trees. A common method of tree identification is through local maxima filtering, which in the context of a lidar canopy height model (CHM), seeks to locate the highest value within a specified neighbourhood of pixels. Hence, specifying an appropriate window size is a critical consideration. This study investigated the potential of the semi-variogram range towards defining an average window size for a given plot within Eucalyptus species plantations. The analysis also included comparisons of CHMs with three pixel sizes (spatial resolutions) (0.2 m, 0.5 m, and 1 m) at lidar point density of 5 points/m 2 and three lidar point densities (1 point/m 2, 3 points/m 2, and 5 points/m 2). These variations were introduced to study the effect of interpolated height surface resolution and lidar point density, respectively, on the identification of trees. Semi-variogram analysis yielded range values that varied distinctly with spatial resolution and point density. Computation of SPHA based on the semi-variogram range values resulted in overall accuracies of 73%, 56%, and 41% for 0.2 m, 0.5 m, and 1 m resolutions, respectively. A comparative approach, that defines window size based on pre-determined tree spacing, yielded corresponding accuracies of 82%, 82%, and 68% at the respective CHM resolutions. Point density comparisons based on interpolated CHM of 0.2 m resolution and the semi-variogram approach resulted in similar results between 5 points/m 2 (73%) and 3 points/m 2 (70%), whereas 1 point/m 2 returned the lowest accuracy (56%). Similar trends with superior accuracies were observed using the pre-determined tree spacing approach from the same resolution CHM: 82% (5 points/m 2), 80% (3 points/m 2), and 74% (1 point/m 2). While all estimates were negatively biased, the CHM with a 0.2 m spatial resolution at a point density of 3 points/m 2 resulted in a reasonable level of accuracy, negating the need for high density (>3 points/m 2) lidar surveys for this purpose. It was concluded that the semi-variogram approach showed promise for estimation of SPHA, particularly due to its independence from a priori knowledge regarding the tree stocking of the plantation.

Determining the spatial scale for analysing mobile measurements of air pollution

When dealing with spatial data or modelling in a geographical context, identifying an appropriate scale for analysis is a critical precursor; however, it is difficult to determine due to limited availability of data at an adequate spatial resolution. This paper describes a mobile monitoring method to collect spatially representative measurements of woodsmoke particulates in support of spatial modelling. A geostatistical technique is described to characterize the spatial scale of woodsmoke particulates collected for 19 evenings over two heating seasons in Victoria, British Columbia, Canada. Semivariograms were applied to 20 data sets (19 evenings and a combined data set) to characterize the appropriate spatial-analysis scale as defined by the semivariogram range, the maximum distance of spatial dependence. Typically, the semivariogram range occurred at 2673 m. This method can be used to identify an optimal sampling interval for woodsmoke data collection, to define the neighbourhood size for performing spatial analyses, and to produce robust model variables and parameters by characterizing the degree of spatial autocorrelation in the data set.

Reliability of sampling designs for spatial snow surveys

Spatial patterns are an inherent property of most naturally occurring materials at a large range of scales. To describe spatial patterns in the field, several observations are made according to a certain sampling design. The spatial structure can be described by the semivariogram range, and nugget and sill variances. We test how reliably seven sampling designs estimate these parameters for simulated spatial fields with predefined spatial structures using a Monte Carlo approach. Five designs have been used previously in the field for snow cover sampling, whereas two designs with semi-random sampling locations have not been used in the field. The designs include 84–159 sampling locations covering small mountain slopes typical of snow avalanche terrain. The results from the simulations show that all designs: (a) give reasonably unbiased estimates of the semivariogram parameters when averaged over many simulations, and (b) show considerable spread in the semivariogram parameter estimates, causing large uncertainty in the semivariogram estimates. Our results suggest that any comparisons of the estimated semivariogram parameters made with the sampling designs will be associated with large uncertainties. To remedy this, we suggest that optimal sampling designs for sampling slope scale snow cover parameters must include more sampling locations and a stratified randomized sampling design in the future.

Why did Sudicky [1986] find an exponential‐like spatial correlation structure for hydraulic conductivity at the Borden research site?

The exponential‐like spatial correlation structure for hydraulic conductivity that Sudicky (1986) found at the Borden research site arises from hierarchical sedimentary architecture. The sediments can be described in terms of a hierarchy of stratal unit types wherein larger‐scale unit types (hierarchical level II) are composed of smaller‐scale unit types (hierarchical level I). There is a hierarchy of permeability populations corresponding to this stratal hierarchy. The shape and effective range of the composite semivariogram is mostly defined by how the proportion of lag transitions crossing these unit types grows as a function of lag distance. Lag transitions across level I units but within the same level II unit type and transitions across level II unit types contribute about equally in defining the composite sample semivariogram in the vertical direction. These two types of cross transitions have shorter and longer length scales, respectively. The composite semivariogram reflects additive contributions from the two length scales. The proportion of cross‐transition lags grows with an exponential‐like curve because the unit types at either level reoccur with a relatively high variability in length. The shape and range of the semivariogram are modeled from knowing only the univariate statistics for length among the unit types, and the sill is modeled from knowing univariate statistics for hydraulic conductivity populations among the unit types. Understanding the relationship between the semivariogram structure and these quantifiable attributes of the hierarchical stratal architecture helps in reducing the equivocal aspects of choosing a semivariogram model. It also helps in identifying bias in sample semivariograms and in assessing stationarity of second‐order spatial bivariate statistics.

Differences between Soil Ammonium and Nitrate Spatial Pattern in Six Plant Communities. Simulated Effect on Plant Populations

Geostatistical descriptions of soil heterogeneity patterns for plant communities are abundant in literature, however no effort has been done to compare spatial structure of different nutrients, and to look for a common pattern in different plant communities. Furthermore, there is no information on the consequences of different soil spatial patterns on resource availability for plant populations. Conditional simulations on the spatial distribution of soil NH4-N and NO3-N were carried out in order to study the effect of contrasted patch sizes on N availability for individuals with increasing root system size. The semivariogram range (an indication of patch size) for soil NH4-N and NO3-N in six plant communities was found to be very variable, and was higher for soil NH4-N than for soil NO3-N in each community. A positive correlation was observed between organic matter and NH4-N spatial ranges in the six plant communities, but not between NO3-N and NH4-N. Probabilities of finding a high soil N concentration within simulated plant populations depended on N patch size and root system size. Thus, a population taking up NH4-N (higher spatial range values) would be more heterogeneous (i.e. individuals will have differing probabilities of finding a high soil N concentration) than the same population taking up NO3-N. Likewise, a seedling population taking up NH4-N or NO3-N would be more heterogeneous than a large tree population in the same area, where individuals would have similar probability of finding a high soil N concentration. These results showed that the spatial patch size of limiting resources has important consequences at the population level, since it determines the probability of finding a favorable site, and therefore differing performances of individuals within a population.

Soil Ammonium vs. Nitrate Spatial Pattern in Six Plant Communities: Simulated Effect on Plant Populations

Geostatistical descriptions of soil heterogeneity patterns for plant communities are abundant in literature, however there is no information on the consequences of different soil spatial patterns on resource availability for plant populations. Conditional simulations on the spatial distribution of soil NH4–N and NO3–N were carried out in order to study the effect of contrasted patch sizes on nitrogen availability for individuals with increasing root system size. The semivariogram range (an indication of patch size) for soil NH4–N and NO3–N in six plant communities was found to be very variable, and was higher for soil NH4–N than for soil NO3–N in each community. A positive correlation was observed between organic matter and NH4–N spatial ranges in the six plant communities, but not between NO3–N and NH4–N. Probabilities of finding a high soil N concentration within simulated plant populations depended on N patch size and root system size. Thus, a population taking up NH4–N (higher spatial range values) would be more heterogeneous (i.e. Individuals will have differing probabilities of finding a high soil N concentration) than the same population taking up NO3–N. Likewise, a seedling population taking up NH4–N or NO3–N would be more heterogeneous than a large tree population in the same area, where individuals would have similar probability of finding a high soil N concentration. These results showed that the spatial patch size of limiting resources has important consequences at the population level, since it determines the probability of finding a favourable site and therefore differing performances of individuals within a population.