GEOG 318 pt 2

Derived measures on surfaces purpose

Used to compute terrain characteristics like slope relief and hydrology

Relative relief definition

Difference between highest and lowest elevation in an area

Relative relief formula

Max elevation minus min elevation

Relative relief meaning

Indicates terrain roughness and variability

Slope definition

Maximum rate of change in elevation at a point

Gradient definition

Another term for slope of a field

Slope calculation formula

Theta equals arctan of rise over run

Slope components

Rise vertical change and run horizontal distance

Slope as a vector

Has magnitude steepness and direction aspect

Slope map shows

Steepest slope direction and flow direction

Gradient representation options

Separate magnitude and direction maps or single map with arrows showing direction and magnitude

DEM slope calculation method

Uses elevation values of surrounding grid cells

X direction slope calculation

Difference between right and left neighbor divided by 2g

Y direction slope calculation

Difference between top and bottom neighbor divided by 2g

Gradient calculation

Combine x and y slopes using Pythagorean theorem

Limitations of DEM slope method

Needs dense grid ignores central point limited neighbors

Surface specific points definition

Locations where gradient equals zero

Types of surface specific points

Peaks pits saddles ridge lines flat valleys plains

Importance of surface specific points

Used in hydrology watersheds and drainage analysis

Deterministic interpolation

Assumes perfect data and no error

Stochastic interpolation

Accounts for randomness and error

Problem with deterministic interpolation

Unrealistic due to measurement error and variability

Trend surface analysis definition

Fits mathematical function to model large scale spatial trends

Trend surface purpose

Identify general patterns and remove local variation

Trend surface characteristics

Global smooth surface representing first order variation

Trend surface equation

Observed value equals trend plus residual

Linear trend surface equation

zi = beta0 + beta1xi + beta2yi

Coefficient beta0

Intercept value at origin

Coefficient beta1

Slope in x direction

Coefficient beta2

Slope in y direction

Residual definition

Difference between observed and predicted value

R squared definition

Measures how well model explains variation

R squared interpretation high

Good model fit

R squared interpretation low

Poor model fit

F test purpose

Tests if trend is statistically significant

Surface degrees of freedom

Number of coefficients minus one

Residual degrees of freedom

n minus 1 minus surface degrees of freedom

F test interpretation

If F greater than critical value reject null and model is significant

Reasons for non significant result

No trend small sample or wrong model

Trend surface limitations

Oversimplifies ignores local variation residual autocorrelation

Trend surface advantages

Simple fast exploratory method

Variogram purpose

Measure spatial autocorrelation using data

Variogram cloud definition

Plot of value differences versus distance

Number of pairs formula

n choose 2 combinations

Square root difference cloud meaning

Shows increasing difference with distance

Anisotropy definition

Spatial variation differs by direction

Isotropy definition

Same spatial variation in all directions

Lag definition

Distance interval used to group point pairs

Edge effects in variogram

Fewer pairs at large distances reduce reliability

Semivariogram definition

Plot of squared differences versus distance

Semivariance definition

Average squared difference between point pairs

Variance vs semivariance

Variance uses mean deviation semivariance uses pair differences

Experimental variogram

Empirical semivariogram with fitted model

IDW interpolation

Uses distance weighting with arbitrary parameters

Trend surface interpolation

Global smooth model

Kriging interpolation

Statistical method using spatial structure

Kriging definition

Weighted interpolation based on spatial autocorrelation

Kriging steps

Use variogram or semivariogram Fit mathematical model Calculate weights

Semivariogram nugget

Variance at zero distance due to error or micro variation

Semivariogram sill

Maximum semivariance value

Semivariogram range

Distance where spatial autocorrelation ends

High nugget meaning

High local variability

Large range meaning

Strong spatial autocorrelation

Semivariogram models

Spherical Gaussian linear hybrid

Kriging assumptions

Stationarity isotropy constant mean

Stationarity definition

Process does not change across space

First order stationarity

Constant mean

Second order stationarity

No spatial interaction

Kriging advantages

Accurate data driven uses spatial structure

Kriging limitations

Complex computationally intensive model dependent

Spatial autocorrelation definition

Measure of similarity of values across space

Positive spatial autocorrelation

Similar values cluster together

Negative spatial autocorrelation

Dissimilar values cluster together

No spatial autocorrelation

Random pattern

Why spatial autocorrelation matters

Violates independence assumption of statistics

Global measures

Single value summarizing entire area

Local measures

Measure variation at specific locations

Moran's I definition

Global measure of spatial autocorrelation

Moran's I range

Negative one to positive one

Moran's I interpretation positive

Clustering

Moran's I interpretation zero

Random

Moran's I interpretation negative

Dispersion

Spatial weights matrix

Defines neighbor relationships

Rook contiguity

Shared borders

Queen contiguity

Shared borders or corners

Moran scatterplot purpose

Visualize spatial autocorrelation

High high cluster

High values near high values

Low low cluster

Low values near low values

High low cluster

High value near low values

Low high cluster

Low value near high values

LISA definition

Local measure of spatial autocorrelation

LISA purpose

Identify clusters and outliers

LISA high high

Hotspot

LISA low low

Coldspot

LISA high low

Spatial outlier

LISA low high

Spatial outlier

Permutation test

Tests significance of spatial patterns

Multiple testing problem

Increased chance of false positives

Solution to multiple testing

Adjust significance levels

Quantity definition

What exists in a spatial dataset