Phaedon C. Kyriakidis

Geostatistical Space–Time Models: A Review

Geostatistical space–time models are used increasingly for addressing environmental problems, such as monitoring acid deposition or global warming, and forecasting precipitation or stream flow. Each discipline approaches the problem of joint space–time modeling from its own perspective, a fact leading to a significant amount of overlapping models and, possibly, confusion. This paper attempts an annotated survey of models proposed in the literature, stating contributions and pinpointing shortcomings. Stochastic models that extend spatial statistics (geostatistics) to include the additional time dimension are presented with a common notation to facilitate comparison. Two conceptual viewpoints are distinguished: (1) approaches involving a single spatiotemporal random function model, and (2) approaches involving vectors of space random functions or vectors of time series. Links between these two viewpoints are then revealed; advantages and shortcomings are highlighted. Inference from space–time data is revisited, and assessment of joint space–time uncertainty via stochastic imaging is suggested.

Error in a USGS 30-meter digital elevation model and its impact on terrain modeling.

Calculations based on US Geological Survey (USGS) digital elevation models (DEMs) inherit any errors associated with that particular representation of topography. We investigated the potential impact of error in a USGS 30 m DEM on terrain analysis over 27 km 2 . The difference in elevation between 2652 differential Global Positioning Systems measurements and USGS 30-m DEM derived elevations provided the comparative error dataset. Analysis of this comparative error data suggested that although the global (average) error is small, local error values can be large, and also spatially correlated. Stochastic conditional simulation was used to generate multiple realizations of the DEM error surface that reproduce the error measurements at their original locations and sample statistics such as the histogram and semivariogram model. The differences between these alternative error surfaces provide a model of uncertainty for the unknown DEM error spatial distribution. These DEM errors had a significant impact on terrain attributes which compound elevation values of many grid cells (e.g. slope, wetness index, etc.). A case study using terrain modeling demonstrates that the result of error propagation is most dramatic in valley bottoms and along streamlines. q 2000 Elsevier Science B.V. All rights reserved.

A Geostatistical Framework for Area-to-Point Spatial Interpolation

The spatial prediction of point values from areal data of the same attribute is addressed within the general geostatistical framework of change of support; the term support refers to the domain informed by each datum or unknown value. It is demonstrated that the proposed geostatistical framework can explicitly and consistently account for the support differences between the available areal data and the sought-after point predictions. In particular, it is proved that appropriate modeling of all area-to-area and area-to-point covariances required by the geostatistical framework yields coherent (mass-preserving or pycnophylactic) predictions. In other words, the areal average (or areal total) of point predictions within any arbitrary area informed by an areal-average (or areal-total) datum is equal to that particular datum. In addition, the proposed geostatistical framework offers the unique advantage of providing a measure of the reliability (standard error) of each point prediction. It is also demonstrated that several existing approaches for area-to-point interpolation can be viewed within this geostatistical framework. More precisely, it is shown that (i) the choropleth map case corresponds to the geostatistical solution under the assumption of spatial independence at the point support level; (ii) several forms of kernel smoothing can be regarded as alternative (albeit sometimes incoherent) implementations of the geostatistical approach; and (iii) Tobler’s smooth pycnophylactic interpolation, on a quasi-infinite domain without non-negativity constraints, corresponds to the geostatistical solution when the semivariogram model adopted at the point support level is identified to the free-space Green’s functions (linear in 1-D or logarithmic in 2-D) of Poisson’s partial differential equation. In lieu of a formal case study, several 1-D examples are given to illustrate pertinent concepts.

Climate variability has a stabilizing effect on the coexistence of prairie grasses

How expected increases in climate variability will affect species diversity depends on the role of such variability in regulating the coexistence of competing species. Despite theory linking temporal environmental fluctuations with the maintenance of diversity, the importance of climate variability for stabilizing coexistence remains unknown because of a lack of appropriate long-term observations. Here, we analyze three decades of demographic data from a Kansas prairie to demonstrate that interannual climate variability promotes the coexistence of three common grass species. Specifically, we show that (i) the dynamics of the three species satisfy all requirements of “storage effect” theory based on recruitment variability with overlapping generations, (ii) climate variables are correlated with interannual variation in species performance, and (iii) temporal variability increases low-density growth rates, buffering these species against competitive exclusion. Given that environmental fluctuations are ubiquitous in natural systems, our results suggest that coexistence based on the storage effect may be underappreciated and could provide an important alternative to recent neutral theories of diversity. Field evidence for positive effects of variability on coexistence also emphasizes the need to consider changes in both climate means and variances when forecasting the effects of global change on species diversity.

Geostatistics for conflation and accuracy assessment of digital elevation models

A geostatistical methodology is proposed for integrating elevation estimates derived from digital elevation models (DEMs) and elevation measurements of higher accuracy, e.g., elevation spot heights. The sparse elevation measurements (hard data) and the abundant DEM-reported elevations (soft data) are employed for modeling the unknown higher accuracy (reference) elevation surface in a way that properly reflects the relative reliability of the two sources of information. Stochastic conditional simulation is performed for generating alternative, equiprobable images (numerical models) of the unknown reference elevation surface using both hard and soft data. These numerical models reproduce the hard elevation data at their measurement locations, and a set of auto and crosscovariance models quantifying spatial correlation between data of the two sources of information at various spatial scales. From this set of alternative representations of the reference elevation, the probability that the unknown reference va...

Geostatistical Mapping of Precipitation from Rain Gauge Data Using Atmospheric and Terrain Characteristics

Abstract A geostatistical framework for integrating lower-atmosphere state variables and terrain characteristics into the spatial interpolation of rainfall is presented. Lower-atmosphere state variables considered are specific humidity and wind, derived from an assimilated data product from the National Centers for Environmental Prediction and the National Center for Atmospheric Research (NCEP–NCAR reanalysis). These variables, along with terrain elevation and its gradient from a 1-km-resolution digital elevation model, are used for constructing additional rainfall predictors, such as the amount of moisture subject to orographic lifting; these latter predictors quantify the interaction of lower-atmosphere characteristics with local terrain. A “first-guess” field of precipitation estimates is constructed via a multiple regression model using collocated rain gauge observations and rainfall predictors. The final map of rainfall estimates is derived by adding to this initial field a field of spatially interpo...

Geostatistical Solutions for Super-Resolution Land Cover Mapping

Super-resolution land cover mapping aims at producing fine spatial resolution maps of land cover classes from a set of coarse-resolution class fractions derived from satellite information via, for example, spectral unmixing procedures. Based on a prior model of spatial structure or texture that encodes the expected patterns of classes at the fine (target) resolution, this paper presents a sequential simulation framework for generating alternative super-resolution maps of class labels that are consistent with the coarse class fractions. Two modes of encapsulating the prior structural information are investigated-one uses a set of indicator variogram models, and the other uses training images. A case study illustrates that both approaches lead to super-resolution class maps that exhibit a variety of spatial patterns ranging from simple to complex. Using four different examples, it is demonstrated that the structural model controls the patterns seen on the super-resolution maps, even for cases where the coarse fraction data are highly constraining.

A geostatistical approach for mapping thematic classification accuracy and evaluating the impact of inaccurate spatial data on ecological model predictions

Spatial information in the form of geographical information system coverages and remotely sensed imagery is increasingly used in ecological modeling. Examples include maps of land cover type from which ecologically relevant properties, such as biomass or leaf area index, are derived. Spatial information, however, is not error-free: acquisition and processing errors, as well as the complexity of the physical processes involved, make remotely sensed data imperfect measurements of ecological attributes. It is therefore important to first assess the accuracy of the spatial information being used and then evaluate the impact of such inaccurate information on ecological model predictions. In this paper, the role of geostatistics for mapping thematic classification accuracy through integration of abundant image-derived (soft) and sparse higher accuracy (hard) class labels is presented. Such assessment leads to local indices of map quality, which can be used for guiding additional ground surveys. Stochastic simulation is proposed for generating multiple alternative realizations (maps) of the spatial distribution of the higher accuracy class labels over the study area. All simulated realizations are consistent with the available pieces of information (hard and soft labels) up to their validated level of accuracy. The simulated alternative class label representations can be used for assessing joint spatial accuracy, i.e., classification accuracy regarding entire spatial features read from the thematic map. Such realizations can also serve as input parameters to spatially explicit ecological models; the resulting distribution of ecological responses provides a model of uncertainty regarding the ecological model prediction. A case study illustrates the generation of alternative land cover maps for a Landsat Thematic Mapper (TM) subscene, and the subsequent construction of local map quality indices. Simulated land cover maps are then input into a biogeochemical model for assessing uncertainty regarding net primary production (NPP).

Population-density estimation using regression and area-to-point residual kriging

Census population data are associated with several analytical and cartographic problems. Regression models using remote‐sensing covariates have been examined to estimate urban population density, but the performance may not be satisfactory. This paper describes a kriging‐based areal interpolation method, namely area‐to‐point residual kriging, which can be used to disaggregate the residuals remaining from regression. Compared with conventional cokriging, the area‐to‐point residual kriging is much simpler in that only a semivariogram model for the point residuals is required, as opposed to a set of auto‐ and cross‐semivariogram models involving the dependent variable and all the covariates. In addition, area‐to‐point residual kriging explicitly accounts for any scale differences between source data and target values. The method is illustrated by disaggregating population from census units to the land‐use zones within them. Comparative results for regression with and without area‐to‐point residual kriging show that area‐to‐point residual kriging can substantially improve interpolation accuracy.

Spatial and temporal variability in the R-5 infiltration data set: Déjà vu and rainfall-runoff simulations

This paper is a continuation of the event-based rainfall-runoff model evaluation study reported by Loague and Freeze [1985[. Here we reevaluate the performance of a quasi-physically based rainfall-runoff model for three large events from the well-known R-5 catchment. Five different statistical criteria are used to quantitatively judge model performance. Temporal variability in the large R-5 infiltration data set [Loague and Gander, 1990] is filtered by working in terms of permeability. The transformed data set is reanalyzed via geostatistical methods to model the spatial distribution of permeability across the R-5 catchment. We present new estimates of the spatial distribution of infiltration that are in turn used in our rainfall-runoff simulations with the Horton rainfall-runoff model. The new rainfall-runoff simulations, complicated by reinfiltration impacts at the smaller scales of characterization, indicate that the near-surface hydrologic response of the R-5 catchment is most probably dominated by a combination of the Horton and Dunne overland flow mechanisms.