Transformation in GIS

A geographical transformation is a mathematical operation that converts the coordinates of a point in one geographic coordinate system to the coordinates of the same point in another geographic coordinate system.

Since geographic coordinate systems contain datums that are based on spheroids, a geographic transformation also changes the underlying spheroid. There are several methods, which have different levels of accuracy and ranges, for transforming between datums.

A geographic transformation always converts geographic (latitude–longitude) coordinates. Some methods convert the geographic coordinates to geocentric (X,Y,Z) coordinates, transform the X,Y,Z coordinates, and convert the new values back to geographic coordinates

Affine transformation is a geographic transformation that scales, rotates, skews, and/or translates images or coordinates between any two Euclidean spaces. It is commonly used in GIS to transform maps between coordinate systems.

In an affine transformation, parallel lines remain parallel, the mid-point of a line segment remains a mid-point and all points on a straight line remain on a straight line.

Geometric transformation is the process of using a set of control points and transformation equations to register a digitised map, satellite image, or an air photo to a projected coordination system.

Geometric transformation converts a newly digitised map into projected coordinates by a process called map-to-map transformation. A remotely sensed image is converted to projected coordinates using image-to-map transformation. This is also called georeferencing. 

Different methods have been proposed for transformation from one coordinate system to another. Each method is differentiated based on the geometric property it preserves and the changes it allows. Transformation results in either:

• Changes in position and direction
• Uniform change of scale or
• Changes in size and shape

Below are listed the various transformations and their effect on a rectangular object:

1. Equiarea transformation permits rotation of rectangle and preserves its shape and size.
2. Similarity transformation permits rotation of rectangle and preserves its shape but not the size.
3. Affine transformation allows angular distortion but preserves parallelism of lines
4. Projective transformation allows both angular and length distortions and thus allows the rectangle to be transformed into an irregular quadrilateral.

Generally, Affine transformations are used for map-to-map or image-to-map transformations and projective transformation is used for aerial photographs with relief displacement.

Spatial Analysis in GIS

SPATIAL ANALYSIS IN GIS

GIS is differentiated from other information systems due to its spatial analysis functions.
Spatial analysis functions are used to answer questions about the real world using  GIS databases as a model of the real world.
Spatial analysis techniques are used to create an image of reality that can be easily understood.
Basic spatial analysis can be performed at various levels:

1. Sorting data in attribute tables for presentation
2. Performing arithmetic, boolean and statistical operation on attribute tables
3. Compiling new data based on original and derived attributes or based on geographic relationships.
4. Within each level operations may be logical, arithmetic, geometric, statistical or a combination of any of these four types.

The two fundamental functions of GIS are:

1. Generation of maps and
2. Generation of tabular reports 

Spatial analysis requires logical connections between attribute data and map features. Spatial analysis builds operational procedures on spatial relationship between map features. 

Attribute query involves selecting information by the use of logical questions. When no spatial information is required to ask a question, the query is considered an attribute query. 

A spatial query involves selecting features based on spatial relationships. The answer to such queries can be obtained by a hard-copy map or by using a GIS.

Basic GIS analysis involves attribute queries and spatial queries. Complicated analysis require a series of GIS operations involving multiple attribute and spatial queries, alteration of original data and generation of new data sets.

An effective spatial analysis uses the best available methods appropriately for different types of attribute queries, spatial queries and data alteration. 

The design of the analysis depends on the purpose of the study.

The use of GIS to inquire geographic features and retrieve associated attribute information is called identification. This process generates new set of maps by query and analysis. Spatial analysis helps make the new information clearer. GIS operational procedure and analytical tasks that are suited for spatial analysis are discussed below:

1. Single layer operations are procedures that correspond to queries and alterations of data that operate on a single data layer. For example, creating a buffer zone (silence zone) around all schools in a city is a single layer operation
2. Multi layer operations are used for manipulation of spatial data on multiple data layers. For example, the overlay of two input data layers produces a map of combined polygons.
3. Topological overlays: These are multi layer operations that allow combining features from different layers to form a new map and give new information and features that were not present in the individual maps.
4. Point pattern analysis deals with examination and evaluation of spatial patterns and processes of point features.
5. Network analysis: It is designed specifically for line features organized in connected features and typically applies to transportation problems and network analysis. For example: school bus routing, walking distance, bus stop optimization, etc
6. Surface analysis deals with the spatial distribution of surface information in a three dimensional structure.
7. Grid analysis involves processing of spatial data in a regularly spaced form. 
8. Fuzzy spatial analysis is based on fuzzy set theory. Fuzzy set theory is a generalization of boolean algebra where zones of gradual transition are used to divide classes instead of crisp boundaries. Fuzzy algebra offers various other methods to combine different sets of data for landslide zonation map preparation. Fuzzy logic can also be used to handle mapping errors or uncertainty.

Geostatistical tools for spatial analysis

Geostatistics studies the spatial variability of regionalized values. Tools to characterise spatial variability are:

1. Spatial auto-correlation function and
2. Variogram

Spatial auto-correlation examines the correlation of a random process with itself in space. Examples of such phenomena are: 

                                      -Total amount of rainfall

                                      -Toxic element concentration

                                      -Elevation at triangulated points, etc

The spatial auto-correlation function depicted as a graph is called a spatial auto-correlogram and this gives an insight into the spatial behaviour of the phenomena under study.

A variogram is calculated from the variance of pairs or points at different separation.

Spatial analysis is a vital part of GIS and can be used for many applications like:

1. Site suitability
2. Natural resource management
3. Environmental Disaster Management

Spatial Analysis is the heart or core of GIS because it includes transformations, manipulations and methods that can be applied to geographic data to support decisions, reveal patterns and anomalies not immediately obvious and add value.

Spatial analysis is a set of methods whose results change when locations of the objects being analysed or the frame used to analyse them changes.

Spatial analysis can be:

1. Inductive: Examining empirical evidence and searching for patterns that might support new theories or general principles
2. Deductive: Focussing on testing of known theories against data
3. Normative: Using spatial analysis to develop new or better designs

Analysis can also be carried out on attribute tables of a GIS by plotting one variable against the other as a scatterplot and examining the dependence of one variable on one or more independent variables

Regression analysis can be used to find the simplest relationship and multiple regression can be used to understand the effects of multiple independent variables.

Changing relationship between variables with space is called spatial heterogeneity.

One of the most powerful features of a GIS is the ability to join tables based on common geographic location.

The point-in-polygon operation is used to determine if a point lies inside or outside a polygon.

The polygon overlay is similar to the point-in-polygon operation.

Overlay in raster is very simple. The attributes of each cell are combined according to a set of rules. 

The ability to calculate and manipulate distances forms the basis of spatial analysis.

Distance along a route (represented by a poly-line) is calculated by adding the lengths of each segment of the poly-line.

Since poly-lines short-cut corners, the length of a poly-line is shorter than the length of the object it represents leading to slight discrepancy.

Buffering builds new objects by identifying all areas that are within certain specified distance of the original object.

Point patterns can be identified as clustered, dispersed or random.

There are SIX CATEGORIES of spatial analysis:

1. Queries and reasoning
2. Measurements
3. Transformations
4. Descriptive summaries
5. Optimization and
6. Hypothesis testing

Queries and reasoning are the most basic analysis operations where GIS is used to answer simple questions. No changes occur in the database and no new data are produced.

Measurements involve measurement of simple properties of objects such as length, area or shape and relationship between pair of objects such as distance or direction.

Transformations are simple methods of spatial analysis that change data-sets by combining them or comparing them to obtain new data-sets and finally new ideas. Transformations use simple arithmetic, geometric or logical rules. They include operations that convert raster data to vector data and vice versa. They may create fields from collections of objects or detect collection of objects in fields.

Descriptive summaries attempt to capture the essence of a data-set in one or two numbers. 

Optimization techniques are normative in nature and are designed to select ideal locations for objects given specific constraints. They are widely used in market research, package delivery industry, etc.

Hypothesis testing focusses on reasoning from the results of a limited sample to make genaralizations about an entire population. Hypothesis testing is the basis for inferential statistics  and forms the core of statistical analysis.

Spatial analysis can be done by overlay analysis by overlaying land use and flood zone to determine the residential parcels inside a flood zone area. This data can be used by insurance companies to target their insurance sales.

Farmers can use interpolation to examine soil samples from a farm area.

Shop owners can establish their stores based on location (distance and density) analysis

Data types in spatial analysis

The three types of data used to characterize problems of spatial analysis are:

1. Events or point patterns: Examples of this type are: crime spots, disease occurrences, localization of vegetal species, etc.
2. Continuous surfaces: Examples of this type are: geological maps, topographical maps, ecological maps, etc.
3. Areas with counts: Examples of this type are: population surveys, health statistics, etc that are demarcated by closed polygons (postal zones, municipalities, etc)

Problems of spatial analysis deal with environmental and socioeconomic data. 

Basic concepts of spatial analysis

1. Spatial dependency is an important concept to understand and analyse a spatial phenomena. "Everything is related to everything else, but near things are more related than distant things" -Waldo Tobler (First law of Geography)
2. The computational expression of the concept of spatial dependence is spatial autocorrelation.
3. Statistical inference for spatial data: An important consequence of spatial dependence is that statistical inferences on this type of data will not be as efficient as in the case of independent samples of the same size.

Spatial interpolation is the process of manipulating spatial information to extract new information and meaning from original data. GIS provides spatial analysis tools for calculating feature statistics and carrying out geoprocessing activities as data interpolation.

The two widely used interpolation methods are:

1. Inverse Distance Weighting (IDW) and
2. Triangular Irregular Networks (TIN)

Other interpolation methods are:

• Regularized Splines with Tension (RST)
• Kriging or Trend Surface Interpolation

Connectivity functions

CONNECTIVITY FUNCTIONS

Connectivity functions examine relationships between objects in terms of adjacency and relative-effective-distance. Operators associated with connectivity include:
-Network analysis
-Diffusion models
-Cellular automata and
-Agent based models

The two connectivity functions widely used in GIS are:
-Contiguity and
-Spreading

Contiguity analysis exploits the topological relationships between objects. It helps determine if two objects are adjacent or if they share a node. Contiguity analysis also helps determine a pattern of spread.
Grouping vector data is used to reclassify a map and dissolve polygon boundaries. Grouping is also done on raster data but is less elegant than grouping operation on vector data.

Contiguity is used to measure shortest and longest straight line distances across and area and to identify areas of terrain with specified size and shape characteristics.

Proximity Functions. The simple distance between features. Four parameters are used to measure proximity are listed below
1. target locations.
2. unit of measurement.
3. a function to calculate proximity and
4. the area to be analyzed.

A common type of proximity analysis is the buffer zone.

Network Functions
A network is a set of interconnected linear features that form a pattern or framework. City Streets, Power Transmission Lines, and Airline Service Routes are examples.
There are three principal types of GIS Analysis performed by Networking.
1. Prediction of loading on the network itself (prediction of flood crests),
2. Rate optimentation (emergency routing of ambulances), and
3. Resource allocation (zones for servicing rescue areas)

Networks analysis entails four components.
1. set of resources (goods to be delivered)
2. one or more locations where the resources are located (several warehouses where the goods are located)
3. an objective to deliver the resources to a set of destinations (customer locational data base) and
4. Set of constraints that places limits on how the objective can be met (is it economically feasible to deliver goods from one point to another)

Spread Functions
Spread functions help determine the "BEST" way to go from point A to point B

Seek or Stream Functions
Seek or Stream Functions refer to a function that is directed outward in a step-by-step manner using a specified decision rule. This function can be used to evaluate erosion potential.

Intervisibility functions
This function is a graphic depiction of the area that can be seen from the specified target areas. Intervisibility functions rely on digital elevation data to define the surrounding topography.

Classification of GIS

A GIS application can be classified into the following types:

1. Four dimensional GIS
2. Multimedia or hypermedia GIS
3. Web GIS and
4. Virtual Reality GIS

The above types of GIS are briefly discussed below:

1. Four dimensional GIS are designed to handle three dimensions of space and one dimension of time. The spatio-temporal representations can handle only two dimensions of space and one dimension of time. 
2. Multimedia/hypermedia GIS allow the user to access a range of georeferenced multimedia data by selecting resources from a georeferenced image map base. A map serving as the primary index to multimedia data in a multimedia geo-representation is called a hypermap. Multimedia and virtual geo-representations can be stored either in extended relational databases, object databases or in application specific data stores.
3. Web GIS: Widespread access to the internet coupled with the use of web browsers and the explosion of geographic information has made it possible to develop new forms of multimedia geo-representations on the web. Many geomatic web solutions are web-based and are rapidly overtaking desktop GIS with the future trends following the same direction.
4. Virtual Reality GIS: Virtual Reality GIS have been developed to allow the creation, manipulation and exploration of geo-referenced virtual environments. For example, use of Virtual Reality Markup Language (VRML) to experiment with different scenarios. Virtual Reality GIS can also be web-based. An example of application of Virtual Reality GIS is 3D simulation for planning in various scenarios.

Analysis functions in GIS

GIS ANALYSIS FUNCTIONS

GIS analysis functions fall into four categories:

1. Retrieval/Classification/Measurement functions
2. Overlay functions
3. Neighbourhood functions and
4. Connectivity functions

• Retrieval functions basically involve a selective search

Classification/Reclassification functions involve two operations:

1. Identifying a set of features as belonging to a group and
2. Defining patterns

Measurement functions measure distances, lengths, perimeters and areas

A selective search is an example of a retrieval function. It involves selection of attributes based on graphic selection tools used to select areas in the map displayed.

Overlay functions could be:

1. Arithmetic 
1. Addition
2. Subtraction
3. Multiplication
4. Division
2. Logical
1. Used to find where specific conditions occur (and, or, >,< etc)

Vector methods are good for sparse data sets while raster methods are easier for grid calculations

Neighbourhood functions

The basic functions that fall under this domain are:

1. Average
2. Diversity
3. Minimum/Maximum and
4. Total

The parameters that need to be defined to operate these functions are:

1. Target locations
2. Specification of neighbourhood
3. Function to be performed on neighbourhood elements
4. Search operation is one of the most common neighbourhood function
5. Neighbourhood function on a vector model is a specialised search function while on a raster model, polygons are on a separate layer and points and lines are on a separate layer.
6. Theissen polygon operation

Cartographic modeling by GIS analysis - procedure with an example

CARTOGRAPHIC MODELING

Cartographic modeling involves the use of basic GIS functions in a logical sequence to solve complex spatial problems.

It was developed to model:

Statement of conditions or assumptions

1. Land-use planning alternatives and
2. Applications that require integrated analysis of multiple geographically distributed factors

• The term was coined by Dana Tomlin in 1983.
• Cartographic modeling lies completely under the raster domain. 
• The nature of analysis is purely additive or subtractive and this complements the values assigned to the raster format of the data
• The digitised data is layered and these layers are combined to construct constraint maps that can be analysed with reference to any specific geographic problem to arrive at the best alternative.

A cartographic model has the ability to form a logical sequence. The process of cartographic modeling is characterised by working backward to insure that all data that will be needed are identified. This helps to avoid collecting data that will not be needed. The process insures that any judgements to be made are explicitly identified. Hence, subjective judgements are an integral part of cartographic modeling.

Cartographic modeling is a common way of expressing and organising the methods by which spatial variables and spatial operations are selected and used to develop an analytical solution within a GIS.

Cartographic modeling is based on the concept of data layers, operations and procedures. Cartographic modeling capabilities are found in most GIS software. 

Modeling is a logical or mathematical formulation that attempts to simulate some aspect of the real world.

The five steps involved in cartographic modeling are listed below and elaborated subsequently:

1. Statement of problem or objectives
2. Statement of conditions or assumptions
3. Methodology
4. Implementation and
5. Evaluation

Statement of problem involves dividing the problem into sub-problems. The objective provides a direction and a clear end to the activity. It helps by the possible routes to solving for the objectives

Statement of conditions or assumptions includes the conditions of the problem. For example: current state, background or case history of the problem. Assumptions in the model define the limitations of the analysis. An assumption of most models is that the processes of the past will continue in the future.

Methodology involves:

1. Assembly of sub-models into a model, which can be sub-divided into:
1. Identification of sub-problems (analogous to the concept of divide and conquer)
2. Development of sub-models that address the sub-problems
3. Development of a strategy for integrating the sub-models
4. Developing a flowchart that shows the parts in the context of the whole
2. Identification of:
1. Data sets needed
2. Spatial operations
3. Non-spatial operations and
4. Interaction of spatial and non-spatial data

Implementation involves:

Implementing the model using the analytic tools available in GIS. It also involves implementation of techniques to circumvent the limitations of the GIS system.

Evaluation

This involves testing the effectiveness of the model. If the model does not conform to expectations,  its assumptions and components should be re-examined and adjusted where necessary. The above procedures should be performed in an iterative fashion until the objectives are achieved.

EXAMPLE

Problem: The municipal corporation of a city would like to measure the environmental equity as compared to the siting of waste transfer stations

Restating the problem: Is one particular income class bearing the burden of waste transfer stations ?

Conditions and assumptions: 

1. Impact of waste transfer sites on surrounding communities is negligible beyond 500m
2. Within 500m the effect is uniform
3. Income classes are distributed evenly throughout a census tract

Methodology:

• Data sets needed
• income by census tract
• location of waste transfer sites
• Spatial operations
• points in polygon
• buffer
• overlay
• Operations on attributes
• select
• reclass
• calculate area estimates and
• generate statistics

Implementation:

• Outline for flow of implementation
• select only waste transfer sites in the city
• generate a 500m buffer around these sites
• select income data from census data and reclass into three income classes: low, medium and high
• add a field to hold the original area of each census tract prior to polygon intersection
• recalculate the income classes based on percentage of census tract left in the intersected polygons. Use the original area field that was brought along in the intersection
• calculate the totals for each of three generated income classes for the entire city
• generate pie charts for the number in each income class for:
•  the entire city and 
• affected areas
• create a map showing output

Evaluation

• The results obtained should be evaluated against the methodology used to test the validity of the model.
• The model should accurately represent the process being modeled.
• A statistical analyses that includes both qualitative and quantitative observations should be performed.
• Based on the above listed criteria, changes to improve the model should be documented and the modeling should be repeated.

Raster data analysis

RASTER DATA ANALYSIS

• Raster data analysis is based on cells and rasters
• Raster data analysis can be performed at the level of individual cells, or a group of cells, or cells within an entire raster
• The type of cell value is an important aspect of raster data analysis
• Various types of data are stored in raster format
• Raster data analysis is software specific-raster data. Hence in order to use Digital Elevation Models(DEMs), Satellite images and other raster data in data analysis, they must first be processed and imported to software-specific raster data
• The core of raster data analysis comprises of local operations that are cell-by-cell operations.
• A local operation can create new raster from a single input raster or multiple input raster.
• Converting a floating point raster to an integer raster is a simple local operation
• Converting a slope raster measured in percent to one measured in degrees is also a local operation
• Reclassification is a local operation that creates a new raster by classification. Reclassification is also called as recoding or transforming
• Local operations with multiple rasters are also known as compositing, overlaying or superimposing maps
• Map algebra is defined as local operations with multiple input rasters.
• Local operations compute an output raster dataset where the output value at each location is a function of the value associated with that location on one or more raster datasets
• Focal or Neighbourhood functions produce an output raster dataset in which output value at each location is a function of the value at a location and the value of cells in a specified neighbourhood around that location
• Zonal functions compute an output raster dataset where the output value for each location depends on the value of the cell at the location and the association that the location has within a cartographic zone.
• Global functions compute an output raster dataset in which the output value at each cell location is potentially a function of all the cells in the input raster datasets.
• The two types of global functions are:
• Euclidian distance and
• Weighted distance
• The raster calculator provides powerful tools in algebra syntax to perform mathematical calculations using operators and functions. It can also be used to set-up selection queries or type spatial analyst function syntax. Inputs can be grid datasets or raster layers, shape files, coverages, tables, constants and numbers.
• The raster calculator provides four groups of mathematical functions: Logarithmic, Arithmetic, Trigonometric and Exponential.
• The raster calculator allows boolean, relational and arithmetic operators
• Raster data analysis also involves:
• Terrain analysis
• Contour
• Slope
• Aspect
• Hillshade
• Viewshed
• Cut/Fill
• Hydrologic analysis
• Basin
• Fill
• Flow accumulation
• Flow direction
• Flow length
• Sink
• Snap pour point
• Snap pour
• Stream link
• Stream order
• Stream to feature
• Stream shape
• Watershed
• Reclassification is replacing input cell values with new output cell values. The reasons to reclassify data are listed below:
• To replace values based on new information
• To group certain values together
• To reclassify values to a common scale
• To set specific cells to "NoData" or set "NoData" cells to a value