1. Map Interpolation

2. Introduction 
One of the strength of Geographical Information Systems (GIS) is the performance of spatial analysis on geographic features (points, lines or polygons).These features are defined as follows:-
Points: Spatial features with a precise location that can be described by a single coordinate pair,
Line: A spatial feature with a precise location described by a series of coordinate pairs.
Polygons: Spatial features that are areas or zones enclosed by precisely defined boundaries. The boundaries of a polygon are formed from one or more lines. These geographic features are representations of real world phenomenon and are described by attributes collected together in a specific site.

3. The issue of geographically constrained data interpolation can be described logically for data collected within specific sites in an area of interest (AOI), can be extended spatially to sites where no sample collection has taken place. These non sampled spaces that occur during any measurement or sampling process can cover a wide intervals, separated in space and time, and may possibly represent the desired results, but only in a very coarse level. This output will raise the concern that conducting discrete measurements points based on an x, y coordinates, sampled either in a regular or irregular grid sampling points and getting high quality results with spatial continuity of the information is a challenging problem which can be dealt with interpolation.

4. Defining Interpolation
Interpolation is process by which a surface is created, usually a raster dataset, through the input of data collected at a number of sample points and those computed points. A simple example is to reallocate data from sample points to surface by an interpolating model, for instance using Inverse weighted Distance (IDW) or Linear Interpolating functions.
On the other hand the computed interpolation process is used to insert many new values in between key to obtain a smoother result. In a normal circumstance the higher the density of measured sites the lower the sparsity effect is.

5. Figure 1. Area of Interest and Discrete Measurement Points
Figure 2. The AOI and Labeled Measurement Sites

6. The purpose of the process is to reduce any coarse result due to the sparsity effect defined by number of sites per area AOI, (see figure 1). The area of interest encompasses sampling points and serves as a barrier to control the extent of the interpolated surface and determine the spatial coverage within a distance from the features enclosed in (see figure 2) and eventually the anticipated spatial analysis results. Furthermore, the area represented by a polygon feature and the points in should be within the same domain and posses identical Georefrencing parameters so that there will not be any cascading or locational displacement effect.

7. Each sampling or observation point is described by data of geographic tuple,(a) location parameters which bears the x, y coordinates and elevation (with specific unit). Associated to this are the aspatial attributes (tabular descriptive data that GIS links to map features) which depict the entities’ autochthonous or allocthonous characteristics of the site; or any measurement results (tabular data) from laboratory works, etc.
In this particular case which (to be demonstrated later in this article, the Pollution entity is described by measured heavy metals distribution interpolated surfaces.

8. Geodatabase for Interpolated data
The working mapping scale and the point interval may vary, and always there is sparsity in distribution. Depending on the sparsity factor, the relationship and trend of the attribute values (example Pollution and Sediments and association with Location entity) can be inter-related and eventually interpreted. A data model is required to achieve the goal by associating each entity in an interpolation process to visualize the relationship.
The data model will be mandatory in order to describe and define properly the data types of each attribute in the system. Non typed attribute can generate ambiguous results error during computation which leads to question the information integrity of the system.

9. The following data model which is described by the data structure is designed to store the data to be interpolated. Referential keys between the geographic feature (Location) table and the attribute tables (Sediments and Pollution) are defined to keep the integrity of information being interpolated.
Definition of the data type of each attribute in each entity can be executed using the SQL’s DDL and the variables are stored within a Geodatabase. This specific example database is created within sde: sqlserver.

10. Physical Creation of Interpolated Variables
CTREATE TABLE Location (
LocNo INTEGER primary key,
LonDD DOUBLE NOT NULL,
LATDD DOUBLE NOT NULL);
CREATE TABLE Pollution (
PolNo INTEGER primary key,
LocNo INTEGER REFERENCES Location (LocNO),
mercury_ppm DOUBLE,
Cadimium_ppm DOUBLE);
CREATE TABLE Sed (
SedNo INTEGER primary key,
ClayP DOUBLE,
SiltP DOUBLE,
sandP DOUBLE);
The highlighted attributes will be incorporated in the interpolation process and the relationship in between will be explained by means of surfaces generated using GIS.

11. Spatial Relationship: Feature vs. Tabular Data
The feature table (Location) which bears the coordinates and is the master table to be converted to a point feature. The Pollution and Sediment tables are the attribute data which describe the specific location in terms of pollution content and sediments grain sizes distribution. With the applying of GIS functions the data stored in the later two will is appended to the feature table for the purpose of spatial data interpolation.
Figure 3. The Data Model and Implementation
Any interpolated information can be interpreted in relation to each other. Example any high value of Cd or Hg corresponding to high value clay will be attributed to the affinity of the heavy metal-clay particles in the environment.
Likewise zones that show with high interpolated values can be retrieved usin any spatial query related to the defined structure from within the Geodatabase. For instance finding the interpolated geographic surface which contains the values representing greater that 2*standard deviation. The spatial information is retrieved from the integrated and relationally constrained entitles within the system. The data stored within the Geodatabase, partially displayed below.

12. Tabular Data for Interpolation
Table 1: The Data tables

13. Role of Interpolation
What makes having detailed mapping information too expensive is visiting every location in the study area or area of interest (AOI) to measure the height, magnitude, or concentration of a phenomenon. The alternative is interpolations which can be conducted by devising methodologies of measuring the phenomenon in a set of strategically dispersed sample locations, so that the predicted values can be assigned to all other non sampled locations and generate the interpolated surface. 
During the phase of spatial analysis we face another problem, related to the variegated attributive nature of the geographic features. For example we have two observable fact: temperature and precipitation (average and agglomerated for a month respectively), given the two not in a coinciding grid. This will cause cascading effects on the mapping interpolation. To perform the interaction computations responsive, we should transform the data to a common grid or common coordinate system, by processes of Georefrencing. The maps incorporated to illustrate the process of interpolation have been projected using relevant datum and ellipsoid applicable in the area of interest.
The role of interpolation is wide in scope and deep usage. The later depends on how we utilized the data such as the Temperature, soil parameters, ground water characteristics, pollution sources, and vegetation data. In the stage of data production we can calculate the values of a particular phenomenon in predefined spots using interpolation procedures. For example if we want to deliver the data in a regular grid, but the samples are measured in scattered points we have to calculate the values of grid points from the samples using interpolation procedures.

14. Variations in Interpolation processes
Surface interpolation functions create a continuous surface from discrete set of measured points through the input of data collected at a number of sample points. The continuous surface representation is in a raster dataset format. Raster is composed of a matrix of pixels. Each pixel is identified by its centroid; which is composed of the x, y and xyz coordinates in 2D and 3D representation respectively.
Interpolation of specific value associated to the coordinates can be executed on both dimensions. There are several forms of interpolation, each which treats the data differently, depending on the properties of the data set. In comparing interpolation methods, the first consideration should be whether or not the source data will change (exact or approximate).

15. Linear Interpolation
There are different ways to derive prediction by interpolator function (linear, IDW, spline, kriging) of a location; each method is referred to as a model and in each model there are different assumptions made inherently related to the data being interpolated.
Certain models are more applicable for specific data—for example, one model may account for local variation better than another.
Example:
(a) Linear Interpolation of geometric object This can be explained by linear interpolation between two known points, where the value y at x is computed by linear interpolation. If the two known points are given by the coordinates and the linear interpolant is straight line between these points. For a value x in the interval, the value y along the straight line is given from the equation see eqn 1.
(b) LP Linear is often used to fill the gaps in a table and the value that required to be interpolated is calculated for generating an interpolated graphical display. In the following example six stations, the elevation value for station 2 is missed. A specific z value along the line can be computed (interpolated) to find the approximate z (m) values in between stations, for instance, see table 2 and Equation 1 and figure 3

16. Table 2: Data Table Figure 4. Linear computed (A: Before; B: After)
The Z value can be computed to display the coherence of the values of measurement using an interpolation function. In this case the missed value will be calculated using the function
Figure 4. Linear computed (A: Before; B: After)
Eqn 1: Linear Tabular Equation
The result of (z2) will be 20. Inserting the value into the cell and drawing the graph will produce the graph figure 3(a), showing the corresponding coordinate vs. elevation relationship with a Smooth slop. The function can be incorporated into a lookup table to retrieve any data missed from and to generate complete interpolated surface.

17. Line feature and Interpolated Points
Interpolating points over linear features or vector requires precaution by considering the geographic reality of the points being interpolated over line feature. For instance, when plotting vector data, it might be possible to connect each point with a straight line segment. This will raise a question about the true knowledge of the region between known points. For instance data consisting of points
Figure 5. Interpolation Accuracy
along a coastline might be sparse; using the assumption when plotting vector data requires that points be connected with straight line segments, essentially indicates a lack of knowledge about conditions between the measured points. In the absence of other knowledge, filling in data based on linear interpolation can be false and misleading (see figure 5).
The degree of error can be minimized by increasing observation point’s density along the specified profile of measurement; see example illustrated bellow.

18. Automated Interpolation
Figure 6. Rasterized Vector (A: vector, B: interpolated raster)
Another type interpolation that generates an interpolated object is rasterizing. When objects are rasterized into two-dimensional images from the corner of two points (vertices), see figure, 6 all the pixels between these two points are filled in by an internal interpolation algorithm, which determines their color and other properties. All the initial characteristics of the vector features are changed to Raster format. The spatial interpolated information becomes continuous rather than discrete. The pixel properties can be resymbolized and depending the on the original vector composition the variegated hues are displayed to reflect the continuous surface spatial information (see Figure 6).

19. Generating Interpolating Geographic Features
Geographic data is modeled in one of three ways: as a collection of features in Vector_format, grid of cells data in raster format (spatial_index), or set of triangulated points model surface in (irregular triangulated network) TIN format. A triangulated irregular network (TIN) is a digital data structure used in a geographic information system (GIS) for the representation of a surface. A TIN comprises a triangular network of vertices, known as mass points, with associated coordinates in three dimensions connected by edges to form a is a collection of plane figures that fills the plane with no overlaps and no gaps triangular tessellation. Data having a discrete location with a defined shape and boundary is modeled using the vector format and the data is represented by features stored in feature classes. A feature can have one of these types of geometries: point, multipoint, Polyline, or polygon.
The geometry is composed of two-dimensional (x, y) or three-dimensional (x, y and Z) geographic coordinates. Figure 7 shows the point sampling type, based on an x, y and z values. The z value in this can be an elevation at that specific site, or the attribute value that is being measured or describing the phenomenon. Specific for this example, the illustration is for the heavy metals concentration measured in each station. The second layer is a digital raster map, a 3 band, 30m resolution, with a SPR psad-56. The display is meant to depict on how the discrete sampled points result can be classified using statistical tools and thematically display them on the map. The colored size variation indicates the magnitude of the concentration.
Geographic feature classes can be interpolated using interpolating tools, for instance the Inverse_distance_weighting, (IDW) method is applied to produce geographically interpolated features applicable for spatial data interpretations.

20. The Inverse Distance Weight (IDW)
Inverse Distance weighting is a method for multivariate interpolation (spatial interpolation on functions of more than one variable), a process of assigning values to unknown points by using values from usually scattered set of known points.It can be described by the following formula Equation 1: The 
IDW
is a simple IDW weighting function, in this case the x denotes an interpolated (arbitrary) point, xk is an interpolating (known) point, d is a given distance (metric operator) from the known point xk to the unknown point x, N is the total number of known points used in interpolation and p is a positive real number, called the power parameter. Here, the weight decreases as distance increases from the interpolated points. Greater values of p assign greater influence to values closest to the interpolated point. For 0 < p < 1 u(x) has smooth peaks over the interpolated points xk, while as p > 1 the peaks become sharp. The choice of value for p is therefore a function of the degree of smoothing desired in the interpolation, the density and distribution of samples being interpolated, and the maximum distance over which an individual sample is allowed to influence the surrounding ones. Furthermore, in straight line distance between X and Xk, the distance between the interpolated and interpolating can be verified by considering the coordinates on the sites of the sites by :

21. IDW interpolation explicitly implements the assumption that things that are close to one another are more alike than those which are farther apart. Interpolators occur embedded in well developed GIS applications. To predict a value for any unmeasured location, i.e., within a specified AOI; IDW will use the measured values surrounding the prediction location assuming that each measured point has a local influence and diminishes with distance. The surface calculated using IDW depends on the selection of a power value (p) neighborhood search strategy, measured values, and the distance in between estimated. Based on the following parameters The weight function varies from a value of unity at the scatter point to a value approaching zero as the distance from the scatter point increases. The weight functions are normalized so that the weights sum to unity.
P: the weighting power, it defines the rate at which weights fall off with distances (d),the distance between the interpolated and sample distances. In most interpolating function (example IDW) the value of P ranges between 2 and 1.5.
IDW is an exact interpolator, where the maximum and minimum values in the interpolated surface can only occur at sample points. Further more it 
assumes that the interpolated surface being driven by the local variation and neighborhood influences.

22. Inverse distance weighted methods are based on the assumption that the interpolating surface should be influenced most by the nearby points and less by the more distant points.The interpolating surface is a weighted average of the scatter points and the weight assigned to each scatter point diminishes as the distance from the interpolation point to the scatter point increases. The distance d is calculated using the formula: Where (x,y) are the coordinates of the interpolation point and (xi,yi) are the coordinates of each scatter point. The weight function varies from a value of unity at the scatter point to a value approaching zero as the distance from the scatter point increases.
Figure 7. IDW-Interpolated Values of Clay Percent Concentrate
Figure 8. Contour-Interpolated Values of Mercury Concentration
The interpolated Mercury distributions from the discrete Measurement points are reprocessed to generate a contoured surface (ArcGIS ESRI 2009). The narrow spaces between the contours show the higher distribution of the feature being spatially analyzed. The interval between two contours is one meter, the base being 0m. A total of 20 contour lines are generated for this specific illustration. However, decreasing the interval will increase the total number of contour numbers. 

23. Synopsis
The fundamental principles of Geospatial data states that things that are closer tends to be more alike than those which are far apart. This is true if we consider and have adequate knowledge on the properties (physical, chemical) of the data that are being interpolated and also on the environmental factors that influence the original constituent of the data, for instance, are these data autochthonous or vv allocthonous?
What about the issue on the lateral and vertical continuity, etc ? These are questions that require meticulous digestion before providing the interpolated surface for the making of a decision. When trying to build the elevation surface, you can assume that the sample values closest to the prediction location will be similar.
However, the amount of the sample must also be taken into account. One fact remain intact is that, in normal environment, as one moves farther away from the prediction location, the influence of the points will decrease. Considering a point too far away may actually be detrimental because the point there may be located in an area that is dramatically different from the prediction location.

24. Figure 9. Overlay of Interpolated Surfaces
One solution is to consider enough points to give a good prediction and acquire adequate information on the underlying composition. Considering the distance relationship, the values of the closer points are usually weighted more heavily than those farther away and the role of interpolation is vital at this point. Inverse Distance Weighted (IDW) interpolation functions based on a local method; that means that the unknown values, can be determined based upon points in a designated local neighborhood measured value which is the known point. For IDW the assumption is made that the closer neighbors are a better estimator of the unknown value than the farthest neighbors, and therefore are weighted according to this inverse distance relationship (see equation 3).
A combination of interpolated surfaces, spatially interpreted, defined on a common spatial reference and resolution is depicted (see figure 9) to visualize the output of interpolating results on the perspectives of mapping interpolation.

25. Understanding raster interpolation
Interpolation predicts values for cells in a raster from a limited number of sample data points. It can be used to predict unknown values for any geographic point data: elevation, rainfall, chemical concentrations, noise levels, and so on.  The illustration on the left shows a point dataset of known rainfall-level values. The illustration on the right shows a raster interpolated from these points. Unknown values are predicted with a mathematical formula that uses the values of nearby known points.

26. Why interpolate to raster?
The assumption that makes interpolation a viable option is that spatially distributed objects are spatially correlated; in other words, things that are close together tend to have similar characteristics. For instance, if it is raining on one side of the street, you can predict with a high level of confidence that it is also raining on the other side of the street. You would be less certain if it was raining across town and less confident still about the state of the weather in the next county. Using the previous analogy, it is easy to see that the values of points close to sampled points are more likely to be similar than those that are farther apart. This is the basis of interpolation. A typical use for point interpolation is to create an elevation surface from a set of sample measurements. 
In the following illustration, each symbol in the point layer represents a location where the elevation has been measured. By interpolating, the values between these input points will be predicted.