GEOG 463 Final

surfaces also have a height value of Z
- Z values constitute a statistical representation of the magnitude of the feature
Surface representation
Surfaces can be represented using contour or isolines, points, TINs and rasters
Most surface analysis in GIS is done on raster or TIN data

A sample should be representative, ie
Cover the entire area
Be large enough to be statistically representative
Be unbiased and randomly chosen
Be practically/ economically possible to collect

Spatial sampling- process of selecting points (of an attribute) from a continuous field
Statistical samples allow the inference of population characteristics froma subset (that is cheap and easy to get)
Interpolated surfaces are depended (in part) on the sampling scheme used to create them

simple random sampling
Systematic sampling
Systematic sampling with local random allocation
Contour sampling/adaptive sampling

If you have a high density sample points, local variation is captured- appropriate fine detail studies
If the oppositie, you will copy regions varitaion/trends at expense of local varibaility - appropriate for large detail studies
Density of sampling points must be chosen to reflect the scale of variation of phenomenon you are measuring

Conventional statistics does not consider spatial variation
Determine the ‘nature’ of spatial variations- and quantify the → spatial auto correlation
Interpolate data based on knowledge of spatial variations- rather than guessing

Exact: interpolated surface passes through ALL original data points (reproduces data values)
Inexact- does NOT pass through all data points (does not reproduce original data values)
Global: estimated value is a function of all the values in the data set (usually based on statistics- variance and regression)
Local: estimated values is a function of a subset of values within the vicinity of the unknown value (more consistent with toblers law)

Entails fitting mathematical models through points to generate a smooth surface
Spatial coordinates (x and y) are independent variables, and the attribute value is teh dependent variable

Global method that uses all the data points
1. Fit a polynomial function to the points, so that the attribute is estimated from its position in x and y
2. Use least-squares method to estimate the coefficients (smae principle as multiple regression)

Improving the fit
1. Use second or higher order polynomials to improve the fit
2. For higher order polynomials add more terms
Better fit to the points, but
Large fluctuations between and outside the points
More data points needed

Trend interpolation results in a smooth surface that represents gradual trends in the surface over an area of interest. This type of interpolation can be used for:

Fitting a surface to the sample points when the surface varies gradually from geion to region
Examining or removing the effects of long-range or global trends. In such circumstances the technique is otehn referred to as trend surface analysis

Trend interpolation creates a gradually varying surface using low-order polynomials that describe a physical process

Superficially easy to understand
Difficult to attribute physical meaning to higher order trend surface (over modled)
Sensitive to outliers
Inexact - does not pass through points

Attribute of interest very sparsely sampled
Identity a secondary attribute that is very densely sampled
Predict the values at un-sampled locations for the sparse attribute based on a regression model (co-incident points) developed between the two

Thiessen polygons - exact
Inverse distance - exact
Splines - exact

Created by
1. Subdividing lines joining nearest neighbour points
2. Drawing perpendicular bisectors through these lines
3. Assembling polygon edges using these bisectors

Divide a region based on the spatial configuration of the data points
Assume homogeneity of the values within polygons boundaries
Most common use is to establish area territories for a set of points
Not very appropriate for elevation- abrupt change

Combine idea of proximity from Thiessen with the gradual change of trend surface
Main idea- value of an attribute at an un-sampled location is inversely proportional to its distance from the estimated value
1. For every point to be estimated, the distance to every known data point is determined
2. Inverse, inverse square, or inverse cubed is computed (value of r, the power parameter)
3.Each known point is multiplied by its inverse square distance term and values summed
4. Interpolated value= sum of above divided by sum of inverse square distance

Good:
Can alter search criteria at will
Is frequently used (though this is not necessarily a good thing)

Bad:
sensitive to outliers
Sensitive to clustering producing weird patterns
Has no built in method to test quality of predictions
Inverse distance is somewhat arbitrary in choice of exponents (r-value)

Fits a piece-wise polynomial function to a neighbourhood of sample points
Typically produces a smooth surface

Good
Can be used to smooth surfaces where artefacts (sharp boundaries) appear
Aesthetically pleasing surfaces
Good for analysis of surface geometery/topology (deruvatuves can be computed)

Bad
No easy way to evaluate efficacy of splines
Represents unrealistically smooth version of reality

If a variable is distributed in space, it is “regionalized”
Regionalized Variables take on values according to spatial location.
Given a variable z, measured at a location i , the variability in z can be broken down into three components: structural, random and error variance

“…The variogram is a quantitative descriptive statistic that can be graphically represented in a manner which characterizes the spatial dependence of a data set…”
…it describes the spatial correlation of a variable…
Our interpolation variable is treated as a random variable
A random variable is simply a variable for which the exact value is unknown
All we can know is the likelihood of obtaining different values based on repetitive sampling

The geostatistical measure that describes the rate of change of the regionalized variable is known as the semivariance. The Semi-Variogram (also called just Variogram in literature) plots the SemiVariance on the Y-Axis versus the lag-distance on the X-Axis.

The SemiVariance is used for:
1.Descriptive analysis: where the spatial structure of the data is investigated using the semivariogram
2. Predictive applications: where the semivariogram is fitted to a theoretical model, parameterized, and used to predict the regionalized variable at other non-measured points (kriging).

Semi-variance is analytically related to spatial covariance and is more commonly used with interpolation

A semivariogram is a plot of the structure function that, like autocorrelation, describes the relationship between measurements taken some distance apart. Semivariograms define the range or distance over which spatial dependence exists.

The nugget is the semivariance at a distance 0.0, (the y –intercept)

The sill is the value at which the semivariogram levels off (its asymptotic value)

The range is the distance at which the semivariogram levels off (the spatial extent of structure in the data)
The distance at which the curve reaches the sill. This defines the maximum distance at which there is any spatial dependence

Step 1: Measure ALL possible distances between ALL points (no repetition)
Step 2: Calculate average distance →
~ Lag distance (should be smaller than range)
Step 3: Calculate semi-variance
Step 4: Plot the variogram, and analyze characteristics (sill, nugget, range)

If a variable exhibits different ranges in different directions, then there is a geometric anisotropy

Expresses the difference in semivariance in different directions.

SemiVariograms are computed in two perpendicular directions (a Semivariogram surface), and the anisotropy can be described as the ratio of the major and minor axes of an ellipsoid (isolines of semivariance)

If the nugget variance dominates the local variation (=experimental semivariogam does not show tendency to decrease as h → 0
- Interpolation would be meaningless
- Use overall mean from all sample points in the region of interest.
A noisy variogram, in which the experimentally derived semivariances are scattered suggests:
- That too few samples were used in the computation of the variogram (at least 50-100 sample points per variogram)
- You should get more sample points, if possible

Kriging – an interpolation method that uses spatial autocorrelation (semivariogram) to optimize the weights of an interpolator.

Weights for interpolation are derived from a variogram and thus in a non-arbitrary manner (unlike inverse distance).

Map of Kriging Variance is automatically produced and gives you information about the reliability of your interpolated map over an area.

Kriging is an exact interpolator – original sample values are reproduced in output maps.
Kriging is a local interpolator – predictions are made based on sampled values in a neigbhourhood defined by a search radius.

Step 1. Evaluate semivariogram
Step 2. Measure the shortest and longest distance between sample pairs → to determine number of lags and lag spacing
Step 3: Calculate variance of your data → can be used as an indication of the sill when modelling the variogram (not needed)
Step 4: Verify if there is a global trend in your data. Data should be detrended. After detrending, the variogram is computed on the residuals. Interpolation is done on residuals. Finally the trend is added to the interpolated residuals.
Step 5: Verify if there is Anisotrophy. If present, you must calculate the anisotropical variogram (indicate angle of largest range).
Step 6: You now have the variogram model that Kriging will use for determination of the weights of sample points.

For Kriging, the search radius should not be larger than the range on the corresponding variogram of the data
The range is the limit of spatial autocorrelation in the sample data
Any other lags beyond the range are ineffective for developing the weights for local interpolation

Without independent evaluation data:
Visually – does the surface look natural – are there visual artefacts?
Validation with independent data
Cross Validation-“Leave-one-out” principle.
Validation statistics
-Mean error
-Mean absolute error
-Root mean squared error (RMSE): measures the variance of the fit

Non-linear Kriging – if your sample data are severely skewed, you must transform your data to a normal distribution, then kriging, then transform the result back – the most frequent is the log-normal transform
Anisotropic Kriging – when you take into consideration range anisotropy (calculated from directional variograms)
Stratified Kriging – classify area into meaningful subareas and develop separate variograms and kriging for each sub-area separately
Universal Kriging – used when severe trends are present – fit a trend surface through your samples - subtract the values of the trend surface from your sample attribute values – kriging these residuals – after kriging, add the trend back to your kriging residual map
Co-kriging – one variable (e.g. rainfall) is sparsely sampled, the other variable (e.g. a vegetation index from satellite data) is densely sampled – you know the two vary in the same manner in space (e.g. high NDVI, high rainfall)
The variogram developed for the densely sampled data will improve the quality of your interpolation for the sparsely sampled data set for kriging
The result is a really good rainfall map
Block Kriging – used to coarsen the resolution of your output interpolated surface
For example, if your soil moisture observations are taken at a spatial resolution of 3 cm, you can estimate the average values over 5 m
Universal/Ordinary kriging
Ordinary Kriging is the most general and widely used of the kriging methods and is the default. It assumes the constant mean is unknown. This is a reasonable assumption unless there is a scientific reason to reject it.

Suitability modeling
Distance modeling- what is the best path to build a connection between point A and B
Hydrologic modeling- in which direction will the water flow
Surface modeling - long-term average groundwater depth (m below surface)

Terminology
DEM: refers to a regular array of pixels/cells of elevation values. In practice, it coils be used to refer to a DTM or DSM
DTM: refers to bare terrain models (including elevations of geographic features such as rivers, ridges, break lines, etc)
DSM: refers to digital models including features above the surface of the earth (ie buildings)
Importance of DEMs for surface analysis
Exact terrain/surface parameters: slope, aspect, hydrology, characteristics

Compare the 4 rise values
Keep highest riseun
Calculate slope in percentage
Slope is calculated in the steepest direction of elevation change
Four nearest method for smooth terrain
Rough terrain- All neighbours are included by weighted differently

Orientation of a pixel is derived from the slope
Aspect identifies the down slope direction of the maximum rate of change in value from each cell to its neighbors

Hydrologic functions on DEM
To model the topographic form of a drainage basin
Determining the drainage network and associated drainage divides
Estimating slopes for understanding drainage patterns and processes
Watershed and stream grids

Watershed: mapped as all grid cells that drain to an outlet

Catchments: For every stream segment, there is a corresponding catchment

World is too complex and detailed
The data structure or models used by GIS to represent the world are not discriminating or flexible enough
We make decisions (how to categorize data, how to define zones) that are not always fully informed or justified and are always biased
It is impossible to make a perfect representation of the world, so uncertainty is inevitable
Uncertainty degrades the quality of a spatial representation

1. Lineage
Concerned with historical and compilation aspects of the data such as: source, content, data capture specification, geographic coverage, compilation method, etc
2. Positional accuracy
Information about how closely coordinate values of map features match their true location
3. Attribute accuracy
Information on the error in the values of attribute data elements
4. Logical consistency
5. Completeness

Accuracy: “the degree to which a map (position measurements) or database (attribute data) matches true or accepted values”
Precision: refers to the consistency and exactness of the measurements:
1. The ability of a measurement (position attribute) to be consistently reproduced”
And
2. The number of significant digits to which a value (position, attribute) has been reliably measured

Error: general term that encompasses both imprecision and inaccuracy
Main types: positional or attribute
Positional error:
GPS is not error free

Track and document errors:
Visual (eg topology)
Use error analysis (sensitivity analysis, error propagation) to find the impacts of input uncertainty on output
The concepts introduced earlier should be documented for each data layer
METADATA!

Commonly, a cell is not purely one thing or another, but might contain some x, some y, and maybe some z within its area
These impure cells are termed mixed pixels or “mixels”
the raster structure may distort the shape of spatial objects
All detail about variation is lost
Socioeconomic data are often best represented as points

For various reasons data are usually aggregated and reported at zonal level
This distorts data in 2 ways
- Spatially inappropriate representation (polygons instead of points)
- Forced data into zones whose boundaries may not respect natural distribution patterns

Visual inspection during data editing and cleaning
Attributes can be checked by using annotation, line colors and patterns
Statistical analysis identify extreme values of attributes

Monte carlo simulation- used in overlays (error simulation)
Sensitivity analysis

Multiple versions of the input map are simulated based on the known error characteristics
Repeats 100s of runs of the spatial model to produce multiple output maps
Maps can be aggregated to obtain a spatial map of uncertainty

Sensitivity analysis investigates how a model (numerical, conceptual, or a complex integration of both) responds to changes in the information provided to it as input
The main goal of sensitivity analysis is to identify the parts of the model that are critical and those that are not