Lecture 14 Transformations – 1

Learning Objectives 14.1 Why is deriving a relationship between/among data measurements the basis of generating information? How is an operation part of that process? 14.2 Describe the general character of transformations. Pick any example of a transformation and describe how a particular operation is the core of the transformation. 14.3 What is interpolation? What is inverse distance weighting in the context of interpolation?

14.1 Why is deriving a relationship between/among data measurements the basis of generating information? How is an operation part of that process?

Transformations: Background in Analytical Cartography A ‘transformational’ view of cartography as part of analysis cartography focuses on map (spatial) analysis. It emerged in the 1970’s as another perspective following the then dominant 'communication' perspective, which focused on map displays.

Transformations were first discussed as map projections (coordinate transformations), then as transformations between all spatial primitives (i.e., point, line, area, surface - PLAS) to examine relationships among PLAS.

Most of the operations discussed so far work within a single measurement framework. However, there have been suggestions for a need to transform from one framework to another. Remember: data structures implement measurement frameworks, i.e. a measurement framework is at the core of a data structure. Transformations convert one data structure to another, deriving new relationships between PLAS to discover new information about the feature(s) of interest.

Important question… how can an operation on data create (new) information? An answer… exploit relationships by generating new ways to compare dimensions (attributes, spatial and temporal) using inherent structure of geographic information.

Transformations (often) result in new geometric and/or topological relationships expressed in terms of a different data structure. As such, relationships are the foundation of spatial-temporal-attribute data structures. Understanding and using this idea is important to your success with GIS and your final project success!

Relationships are at the core of important geographic questions as the basis for GIS applications. Lecture 2 introduced critical thinking questions listed from simple to more complex then translated into an ecosystem topic. Understanding how to pose answerable questions is the basis of a successful GIS project. Examples of spatial relationships such as collocation, connectedness, adjacency, containment, and flow are among many spatial relationships. Consider various attributes in combination with spatial relationships – attribute relationships using various combination rules, e.g., dominance, contributory, interaction, as discussed in previous lectures. 14.2 Describe the general character of transformations. Pick any example of a transformation and describe how a particular operation is the core of the transformation.

Transforming data structures into other data structures provides intermediate information that often makes it easier to derive additional information.

Transformations can develop new relationships within and among dimensions of space, attributes, and time. Consider transformations among points, lines, areas, and surfaces (PLAS) as the basis for transformations. Creating a 4 x 4 matrix, PLAS x PLAS, gives us 16 generic transformations, but each matrix entry has many special data structures. Thus, dozens of transformations exist. It would take far too long to consider them all. They are all quite useful and interesting. Let us examine surface transformations to get an idea of what we are dealing with in GIS.

Surface to Surface data structure transformations provide one example. Table 9-1 depicts the surface to surface (S x S) data structure (DS) transformations, as one of the 16 entries of PLAS x PLAS matrix mentioned earlier. Assume “surface” to be the same as “field” in GIS literature. Entries in the row and column cells are the fundamental operations for the transformations. Table 9-1: Surface-Oriented Transformations on Data Structures Input DS \ Output DS Points (w.Z) Isoline DEM TIN Points (w.Z) Interpolation Interp. & Trace Interpolation Triangulation Isoline Interpolation Interp. & Trace Interpolation Triangulation* DEM Interpolation Interp. & Trace Resampling Triangulation* TIN Extraction Tracing Tracing Refinement *denotes an operation that might produce overly dense triangles without filtering

Data structures in Table 9-1 are described as follows. Points with Z values Isolated point objects with a measured height of a surface. Control within the measurement framework focuses on identifying “critical information points” as in maximum and minimum Z-value information for peak, pit, pass, valley, trough, and ridge locations The location (x,y) of the critical point is measured.

Isoline Contours of an elevation surface are established by controlling the elevation height and measuring where that height intersects the surface. Thus, the locations of points of the isoline are measured.

DEM Digital Elevation Matrix is a location controlled framework, similar to a raster. The point elevation height in a regular grid is measured. TIN A Triangulated Irregular Network is a coverage of triangles developed by connecting pairs of “critical information points” having Z-values such that triangles characterize areas with a constant slope (gradient and azimuth). The Z-value pairings are controlled such that slope is constant across a small area. The small area (formed by triangle) has its slope measured, which means heights to establish gradient and azimuth are measured. This data structure is often a ‘system’ of points with Z-values as composite measurement framework.

Cell entries in Table 9-1 are the names for operations that are the basis of transformational processes. The following is an overview description of transformation operations.

Interpolation - computes an intermediate data value based on known data values (details of kinds of interpolation are addressed below). Neighboring known data values are used to derive the unknown data value. Thus, unknown data value is commonly within the vicinity. See Spatial Interpolation examples developed using GRASS GIS – an open source GIS.

Tracing - Connect elevation points of equal height that passes through points (perhaps interpolated) to form a contour, then smooth the contour. Tracing can be as simple as linear segments one point to next. Smoothing is performed using a spline to pass a curve of a given 'tension' through all the points with appropriate continuity (smooth the contour). Constructing isolines through DEM (visualization) topsoil depth (cm). Note, “0” cm is top of (ground) surface.

Extraction – Select data from one data structure to be used in developing a second data structure. The “selective extraction” could be viewed as “information generation”, e.g. select only the ‘salient’ data such as highest or lowest elevation points.

Resampling - Select as in extraction above from one data structure to develop a second data structure, but then use that selected data to compute a new data value for the second data structure.

Triangulation – Create a set of triangles (faces of a surface having constant slope) based on X,Y,Z input points.

Refinement – Selectively prune the data structure to reduce the confounding or unnecessary information, and retain the salient information 14.3 What is interpolation? What is inverse distance weighting in the context of interpolation?

Kinds of Interpolation – interpolation is among the most popular of transformation. Many of these interpolation operations are available within the ArcGIS Geostatistical Analyst extension.

 Linear: assumes a proportional distance relationship between a particular pair of points; generalizes to three points forming a triangle (assumed to be planar). That is, estimate an elevation data value within a triangle from three known evaluation data values. RETAINS all data values.

 IDW (Inverse Distance Weighting): o Assembles a "neighborhood" of a few points, can use a fixed radius, or a target number of points (expanding and contracting to reflect density). o Computes new value from those in neighborhood, weighted so that farthest points contribute least. Weighting declines as 1/distance, 1/distance squared, etc. This is the POWER (exponent) parameter. o Will retain data values at all points (since distance goes to zero, and inverse distance goes infinite...) BUT distance within a cell is non-zero because the cell center to cell center distance is continuous o It is useful when software algorithms add "intervening opportunity" in the form of decreased weights if a point is closer to a locational barrier of some nature.

 Splines: use a numerical model based on an idealized thin spring. o Spline is programmed in GRASS using smoothing and tension parameters; examples here all are smoothed with a spline o Spline in ArcMap has "weight value" (taughtness), number of points, select either "regularized" or "tension"

 Trend surfaces: fit a polynomial of some degree (linear, parabolic, cubic, etc.) to all the points. STRONG assumption that the overall trend is more important than the particular values. Linear surface trend is a plane (flat surface). It is an average of Z’s through a X,Y,Z space; that is, the ‘average surface’.

 Kriging (optimal interpolation); based on covariance as well as value; uses decline of correlation with distance [e.g. in esri’s Geostatistical Analyst]

 Uncertainty - Information uncertainty in spatial interpolation - a paper describing various aspects of uncertainty associated with interpolation of rainfall