🗺️Geospatial Engineering Unit 8 – Spatial Analysis & Geostatistics

Key Concepts and Definitions

• Spatial analysis involves examining the geographic patterns, relationships, and interactions among features, objects, or phenomena in space
• Geostatistics focuses on the study and analysis of spatial data using statistical methods that consider the spatial context and dependencies
• Spatial data represents information about the location, shape, and attributes of geographic features or objects
• Spatial autocorrelation measures the degree to which spatial features are correlated with themselves across space (Tobler's First Law of Geography)
• Interpolation estimates unknown values at unsampled locations based on known values at sampled locations
• Kriging is a geostatistical interpolation method that uses a weighted average of neighboring samples to estimate unknown values
• Spatial regression models incorporate spatial dependencies and autocorrelation into traditional regression analysis
• Variograms quantify the spatial variability and structure of a dataset by measuring the dissimilarity between pairs of observations as a function of their separation distance

Spatial Data Types and Structures

• Vector data represents discrete features as points, lines, or polygons with associated attributes
  • Points are used for features with a single coordinate (cities, landmarks)
  • Lines represent features with a length but no width (roads, rivers)
  • Polygons represent features with a closed boundary and an interior (buildings, land parcels)
• Raster data represents continuous surfaces or fields using a grid of cells or pixels with assigned values
  • Each cell contains a value representing a specific attribute or measurement (elevation, temperature)
  • Raster data is commonly used for satellite imagery, digital elevation models, and thematic maps
• Geodatabases are specialized databases designed to store, manage, and manipulate spatial data
• Spatial data can be organized using different coordinate reference systems (CRS) to define the location and projection of features on the Earth's surface
• Metadata provides essential information about spatial datasets, including their origin, quality, accuracy, and intended use

Exploratory Spatial Data Analysis

• Exploratory spatial data analysis (ESDA) involves visualizing and summarizing spatial patterns, trends, and relationships in the data
• Choropleth maps use color or shading to represent the intensity or magnitude of a variable across different geographic areas
• Spatial clustering methods identify groups of similar or dissimilar features based on their spatial proximity and attribute values
  • Hot spot analysis (Getis-Ord Gi*) identifies statistically significant spatial clusters of high or low values
  • Cluster and outlier analysis (Anselin Local Moran's I) identifies spatial clusters, outliers, and patterns of spatial association
• Spatial outliers are observations that exhibit unusual or extreme values compared to their neighboring features
• Spatial data can be explored using various statistical measures, such as the mean, median, standard deviation, and quartiles, to understand the distribution and central tendency of the data
• Spatial data mining techniques discover hidden patterns, associations, and relationships in large and complex spatial datasets

Spatial Autocorrelation

• Spatial autocorrelation refers to the presence of systematic spatial variation in a variable, where nearby locations tend to have similar values
• Positive spatial autocorrelation indicates that similar values tend to cluster together in space (high values near high values, low values near low values)
• Negative spatial autocorrelation indicates that dissimilar values tend to cluster together in space (high values near low values, low values near high values)
• Global measures of spatial autocorrelation, such as Moran's I and Geary's C, quantify the overall degree of spatial clustering or dispersion in a dataset
• Local indicators of spatial association (LISA) identify the presence and significance of local spatial clusters or outliers
• The modifiable areal unit problem (MAUP) arises when the results of spatial analysis are sensitive to the scale and aggregation of the spatial units used
• Spatial weights matrices define the spatial relationships or connectivity between features based on criteria such as contiguity, distance, or k-nearest neighbors

Geostatistical Methods

• Geostatistical methods model the spatial variability and uncertainty of a continuous variable using probabilistic models
• Variogram analysis quantifies the spatial dependence and structure of a variable by measuring the dissimilarity between pairs of observations as a function of their separation distance
  • Empirical variograms are constructed from the observed data by plotting the average squared differences between pairs of observations against their separation distances
  • Theoretical variogram models (spherical, exponential, Gaussian) are fitted to the empirical variogram to characterize the spatial structure and provide input for interpolation
• Kriging is a geostatistical interpolation method that estimates unknown values at unsampled locations using a weighted average of neighboring observations
  • Ordinary kriging assumes a constant but unknown mean and relies on the spatial structure captured by the variogram
  • Universal kriging incorporates a trend or drift in the mean value across the study area
  • Cokriging incorporates additional correlated variables to improve the estimation accuracy
• Geostatistical simulation generates multiple realizations of a spatial variable that honor the observed data and the spatial structure while quantifying the uncertainty
• Cross-validation assesses the accuracy and reliability of geostatistical models by iteratively removing each observation and predicting its value using the remaining data

Interpolation Techniques

• Interpolation estimates unknown values at unsampled locations based on known values at sampled locations
• Deterministic interpolation methods create surfaces based on mathematical functions without considering the spatial structure or uncertainty
  • Inverse distance weighting (IDW) estimates values based on a weighted average of nearby observations, with weights decreasing as the distance increases
  • Spline interpolation fits a smooth surface that passes exactly through the observed data points while minimizing the overall curvature
  • Trend surface analysis fits a polynomial surface to the observed data to capture global trends or patterns
• Geostatistical interpolation methods, such as kriging, incorporate the spatial structure and variability of the data to provide optimal estimates and quantify the associated uncertainty
• The choice of interpolation method depends on the nature of the data, the desired properties of the interpolated surface, and the assumptions about the underlying spatial process
• Interpolation accuracy can be assessed using cross-validation techniques, such as leave-one-out or k-fold cross-validation, to compare the predicted values with the observed values
• Anisotropy refers to the directional dependence of spatial variability, where the spatial structure varies with the orientation or direction in space

Spatial Regression Models

• Spatial regression models incorporate spatial dependencies and autocorrelation into traditional regression analysis to account for the spatial structure in the data
• Spatial lag models (SLM) include a spatially lagged dependent variable as an explanatory variable to capture the influence of neighboring observations on the response variable
• Spatial error models (SEM) incorporate a spatially correlated error term to account for the spatial autocorrelation in the residuals
• Geographically weighted regression (GWR) allows the regression coefficients to vary spatially, capturing local variations in the relationships between variables
  • GWR estimates separate regression equations for each location using a spatial kernel to weight the neighboring observations
  • The bandwidth of the spatial kernel determines the extent of spatial influence and can be fixed or adaptive
• Spatial autoregressive models (SAR) combine both spatial lag and spatial error components to capture the spatial dependencies in the dependent variable and the error term
• Model selection techniques, such as Akaike Information Criterion (AIC) or Bayesian Information Criterion (BIC), help choose the most appropriate spatial regression model based on the trade-off between model fit and complexity
• Spatial regression models can be used for prediction, hypothesis testing, and understanding the spatial patterns and relationships in the data

Applications and Case Studies

• Spatial analysis and geostatistics find applications in various domains, including environmental science, public health, urban planning, and natural resource management
• Environmental applications include mapping and monitoring air pollution, water quality, soil contamination, and ecological patterns
  • Geostatistical methods can be used to interpolate pollutant concentrations, identify hotspots, and assess the spatial extent of environmental hazards
  • Spatial regression models can investigate the relationships between environmental variables and socioeconomic factors or land use patterns
• Public health applications involve analyzing the spatial distribution of diseases, identifying risk factors, and planning interventions
  • Spatial cluster analysis can detect disease outbreaks or areas with elevated disease risk
  • Spatial interpolation can estimate the prevalence or incidence of diseases at unsampled locations
• Urban planning applications include analyzing land use patterns, transportation networks, and urban growth
  • Spatial regression models can examine the factors influencing property values, crime rates, or accessibility to services
  • Spatial optimization techniques can support decision-making in facility location, resource allocation, or infrastructure planning
• Natural resource management applications involve mapping and assessing the distribution and abundance of resources, such as minerals, forests, or water
  • Geostatistical methods can estimate the spatial variability of resource attributes, such as ore grades or forest biomass
  • Spatial decision support systems can integrate spatial analysis and geostatistics to guide sustainable resource extraction and conservation efforts