Interpolation

S

this is a useful material for knowing interpolation in arc GIS.

32 ArcUser July–September 2004 www.esri.com
By Colin Childs, ESRI Education Services
Editorʼs note: In addition to supplying tools for spatial analysis (i.e.,
for modeling suitability, distance, or hydrology), the ArcGIS 9 Spatial
Analyst extension provides tools for spatial data analysis that apply sta-
tistical theory and techniques to the modeling of spatially referenced data.
Elevation, temperature, and contamination concentrations are types of
data that can be represented by surfaces. Each raster cell represents a mea-
surement such as a cellʼs relationship to a fixed point or specific concentra-
tion level. Because obtaining values for each cell in a raster is typically
not practical, sample points are used to derive the intervening values using
the interpolation tools in ArcGIS Spatial Analyst. This article provides an
introduction to the interpolation methods used by these tools.
GIS is all about spatial data and the tools for managing, compiling,
and analyzing that data. ArcGIS Spatial Analyst extension provides a
toolset for analyzing and modeling spatial data. A set of sample points
representing changes in landscape, population, or environment can be
used to visualize the continuity and variability of observed data across a
surface through the use of interpolation tools. These changes can be ex-
trapolated across geographic space. The morphology and characteristics
of these changes can be described. The ability to create surfaces from
sample data makes interpolation both powerful and useful.
Understanding Surfaces
Before discussing different interpolation techniques, the differences in
the methods used for surface representation need to be discussed. Each
representation is useful for specific situations. This discussion concen-
trates on the creation of a surface in the native ArcGIS grid format.
A grid representation of a surface is considered to be a functional sur-
face because for any given x,y location, it stores only a single Z value as
opposed to multiple Z values. Functional surfaces are continuous because
an x,y location has one and only one Z value regardless of the direction
from which the x,y point is approached. Functional surfaces are 2.5-di-
mensional surfaces, not true three-dimensional surfaces.
Functional surfaces can be used to represent terrestrial surfaces that
depict the earthʼs surface, statistical surfaces that describe demographic
and other types of data, and mathematical surfaces that are based on arith-
metic expressions. Surface representation in its simplest form is done
by storing x,y and Z values that define the location of a sample and the
change characteristic represented by the Z value.
Contours or isolines are used to define a common characteristic along
a line. Technically, contours join locations of equal value to each other.
In the case of a contour line representing height, it is a line drawn on a
map that connects points of equal elevation above a datum that usually
represents mean sea level.
A triangulated irregular network (TIN) is a vector data structure used
to store and display surface models. A TIN partitions geographic space
using a set of irregularly spaced data points, each of which has x-, y-,
and Z-values. These points are connected by edges that form contiguous,
nonoverlapping triangles and create a continuous surface that represents
the terrain.
A grid is a spatial data structure that defines space as an array of cells
of equal size that are arranged in rows and columns. In the case of a grid
that represents a surface, each cell contains an attribute value that repre-
sents a change in Z value. The location of the cell in geographic space is
obtained from its position relative to the gridʼs origin.
Interpolation Methods
To create a surface grid inArcGIS, the SpatialAnalyst extension employs
one of several interpolation tools. Interpolation is a procedure used to
predict the values of cells at locations that lack sampled points. It is based
on the principle of spatial autocorrelation or spatial dependence, which
measures degree of relationship/dependence between near and distant
objects.
Spatial autocorrelation determines if values are interrelated. If values
are interrelated, it determines if there is a spatial pattern. This correlation
is used to measure
n Similarity of objects within an area
n The degree to which a spatial phenomenon is correlated to itself in
space
n The level of interdependence between the variables
n Nature and strength of the interdependence
Different interpolation methods will almost always produce different results.
Points
Methods of Surface Representation
Contours
TINs Grids
Interpolation is the proce-
dure used to predict cell
values for locations that
lack sampled points.
Interpolating Surfaces
in ArcGIS Spatial Analyst
1.1 1.2 1.3 1.5 1.6 1.7 1.8
21
Interpolated Rainfall
Known Rainfall Values
1 Mile
0 0.2 0.5 0.7 1
Developer’s Corner
www.esri.com ArcUser July–September 2004 33
There are two categories of interpolation techniques: deterministic
and geostatistical. Deterministic interpolation techniques create surfaces
based on measured points or mathematical formulas. Methods such as
Inverse Distance Weight (IDW) are based on the extent of similarity of
cells while methods such as Trend fit a smooth surface defined by a math-
ematical function. Geostatistical interpolation techniques such as Kriging
are based on statistics and are used for more advanced prediction surface
modeling that also includes some measure of the certainty or accuracy of
predictions.
The characteristics of an interpolated surface can be controlled by
limiting the input points used in the calculation of output cell values.
This can be done by limiting the number of points sampled or the area
from which sampled points are taken. Specifying the maximum number
of points to be sampled will return the points closest to the output cell
location until the maximum number is reached. Alternatively, specifying
a fixed radius in map units will select only input points within the radius
distance from the center of the output cell unless there are not enough
points within that radius.
Maximum Number
The characteristics of an interpolated surface can be controlled
by limiting the input points used. Points can be limited by
specifying a maximum number of points, which will select the
nearest points until that maximum number of points is reached.
Alternatively, a radius can be specified in map units.
Fixed Radius
With Barrier
Without Barrier
Barriers, which reflect the presence of fault lines, cliffs,
streams, and other features that create linear discontinuity in
surfaces, also control how surfaces are generated. IDW and
Kriging support the use of barriers.
Many interpolation tools incorporate barriers that define and control
surface behavior in terms of smoothness and continuity. Barriers are
needed because sometimes interpolation operations should not be per-
formed across features, such as fault lines, levees, cliffs, and streams, that
create a linear discontinuity in the surface. By using barriers, changes in
the behavior of the surface can be described and enforced.
Available Interpolation Options
ArcGIS Spatial Analyst at version 9 offers several interpolation tools for
generating surface grids from point data. PointInterp, Natural Neighbors,
and Trend methods and the Topo to Raster command have been added to
the IDW, Spline, and Kriging interpolation methods that were available in
ArcGIS 8.3 Spatial Analyst. Natural Neighbor and Trend methods were
available in ArcGIS 8.3 3D Analyst, although Trend was only accessible
programmatically. Each method uses a different approach for determin-
ing the output cell values. The most appropriate method will depend on
the distribution of sample points and the phenomenon being studied.
The characteristics of an interpolated surface can be
controlled by limiting the input points used in the
calculation of output cell values.
Continued on page 34
Interpolating Surfaces in ArcGIS Spatial Analyst
Continued from page 33
34 ArcUser July–September 2004 www.esri.com
IDW
The IDW function should be used when the set of points is dense enough
to capture the extent of local surface variation needed for analysis. IDW
determines cell values using a linear-weighted combination set of sample
points. The weight assigned is a function of the distance of an input point
from the output cell location. The greater the distance, the less influence
the cell has on the output value.
Kriging
A powerful statistical interpolation method used for diverse applications
such as health sciences, geochemistry, and pollution modeling, Kriging
assumes that the distance or direction between sample points reflects a
spatial correlation that can be used to explain variation in the surface.
It fits a function to a specified number of points or all points within a
specified radius to determine the output value for each location. Kriging
is most appropriate when a spatially correlated distance or directional
bias in the data is known and is often used for applications in soil science
and geology.
The predicted values are derived from the measure of relationship in
samples using sophisticated weighted average technique. It uses a search
radius that can be fixed or variable. The generated cell values can exceed
value range of samples, and the surface does not pass through samples.
There are several types of Kriging. Ordinary Kriging, the most com-
mon method, assumes that there is no constant mean for the data over an
area mean (i.e., no trend). Universal Kriging does assume that an overrid-
ing trend exists in the data and that it can be modeled.
Inverse Distance Weighted
Spline
Spline estimates values using a mathematical function that minimizes
overall surface curvature. This results in a smooth surface that passes
exactly through the input points. Conceptually, it is like bending a sheet
of rubber so that it passes through the points while minimizing the total
curvature of the surface. It can predict ridges and valleys in the data and
is the best method for representing the smoothly varying surfaces of phe-
nomena such as temperature.
There are two variations of spline—regularized and tension. A regu-
larized spline incorporates the first derivative (slope), second derivative
(rate of change in slope), and third derivative (rate of change in the sec-
ond derivative) into its minimization calculations. Although a tension
spline uses only first and second derivatives, it includes more points in
the spline calculations, which usually creates smoother surfaces but in-
creases computation time.
Spline
Kriging
PointInterp
A method that is similar to IDW, the PointInterp function allows more
control over the sampling neighborhood. The influence of a particular
sample on the interpolated grid cell value depends on whether the sample
point is in the cellʼs neighborhood and how far from the cell being inter-
polated it is located. Points outside the neighborhood have no influence.
PointInterp
Developer’s Corner
www.esri.com ArcUser July–September 2004 35
The weighted value of points inside the neighborhood is calculated us-
ing an inverse distance weighted interpolation or inverse exponential
distance interpolation. This method interpolates a raster using point fea-
tures but allows for different types of neighborhoods. Neighborhoods can
have shapes such as circles, rectangles, irregular polygons, annuluses, or
wedges.
Natural Neighbor
Natural neighbor interpolation has many positive features, can be used for
both interpolation and extrapolation, and generally works well with clus-
tered scatter points. Another weighted-average method, the basic equa-
tion used in natural neighbor interpolation is identical to the one used in
IDW interpolation. This method can efficiently handle large input point
datasets. When using the Natural Neighbor method, local coordinates de-
fine the amount of influence any scatter point will have on output cells.
Natural Neighbor
Trend
Trend
Trend is a statistical method that finds the surface that fits the sample
points using a least-squares regression fit. It fits one polynomial equation
to the entire surface. This results in a surface that minimizes surface vari-
ance in relation to the input values. The surface is constructed so that for
every input point, the total of the differences between the actual values
and the estimated values (i.e., the variance) will be as small as possible. It
is an inexact interpolator, and the resulting surface rarely passes through
the input points. However, this method detects trends in the sample data
and is similar to natural phenomena that typically vary smoothly.
Topo to Raster
By interpolating elevation values for a raster, the Topo to Raster method
imposes constraints that ensure a hydrologically correct digital elevation
model that contains a connected drainage structure and correctly repre-
sents ridges and streams from input contour data. It uses an iterative fi-
nite difference interpolation technique that optimizes the computational
efficiency of local interpolation without losing the surface continuity of
global interpolation. It was specifically designed to work intelligently
with contour inputs.
IDW and Spline are two deterministic methods that create surfaces
from samples based on the extent of similarity or degree of smooth-
ing. However, while a spline surface passes exactly through each
sample point, an IDW will pass through none of the points. Kriging is
a geostatistical method that uses a powerful statistical technique for
predicting values derived from the measure of relationship in samples
and employs sophisticated weighted average techniques.
Conclusion
This brief introduction supplies a framework for exploring the interpola-
tion tools available in ArcGIS 9 Spatial Analyst. A more robust and com-
prehensive arsenal of tools for data exploration and spatial data analysis
is available from theArcGIS 9 GeostatisticalAnalyst extension. More in-
formation on how to perform spatial data analysis and applications for the
tools in ArcGIS 9 Spatial Analyst may be found in the product manual,
Using ArcGIS Spatial Analyst. Papers that provide background informa-
tion on interpolation methods and their application in various industries
and disciplines can be found at www.esri.com/geostatisticalanalyst.
Each of the interpolation methods available in the
ArcGIS Spatial Analyst extension uses a different
approach for determining output cells. Choose a
method based on the distribution of sample points
and the phenomenon being studied.
Krige
Spline
IDW
Distance
Z-value

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Interpolation

  • 1. 32 ArcUser July–September 2004 www.esri.com By Colin Childs, ESRI Education Services Editorʼs note: In addition to supplying tools for spatial analysis (i.e., for modeling suitability, distance, or hydrology), the ArcGIS 9 Spatial Analyst extension provides tools for spatial data analysis that apply sta- tistical theory and techniques to the modeling of spatially referenced data. Elevation, temperature, and contamination concentrations are types of data that can be represented by surfaces. Each raster cell represents a mea- surement such as a cellʼs relationship to a fixed point or specific concentra- tion level. Because obtaining values for each cell in a raster is typically not practical, sample points are used to derive the intervening values using the interpolation tools in ArcGIS Spatial Analyst. This article provides an introduction to the interpolation methods used by these tools. GIS is all about spatial data and the tools for managing, compiling, and analyzing that data. ArcGIS Spatial Analyst extension provides a toolset for analyzing and modeling spatial data. A set of sample points representing changes in landscape, population, or environment can be used to visualize the continuity and variability of observed data across a surface through the use of interpolation tools. These changes can be ex- trapolated across geographic space. The morphology and characteristics of these changes can be described. The ability to create surfaces from sample data makes interpolation both powerful and useful. Understanding Surfaces Before discussing different interpolation techniques, the differences in the methods used for surface representation need to be discussed. Each representation is useful for specific situations. This discussion concen- trates on the creation of a surface in the native ArcGIS grid format. A grid representation of a surface is considered to be a functional sur- face because for any given x,y location, it stores only a single Z value as opposed to multiple Z values. Functional surfaces are continuous because an x,y location has one and only one Z value regardless of the direction from which the x,y point is approached. Functional surfaces are 2.5-di- mensional surfaces, not true three-dimensional surfaces. Functional surfaces can be used to represent terrestrial surfaces that depict the earthʼs surface, statistical surfaces that describe demographic and other types of data, and mathematical surfaces that are based on arith- metic expressions. Surface representation in its simplest form is done by storing x,y and Z values that define the location of a sample and the change characteristic represented by the Z value. Contours or isolines are used to define a common characteristic along a line. Technically, contours join locations of equal value to each other. In the case of a contour line representing height, it is a line drawn on a map that connects points of equal elevation above a datum that usually represents mean sea level. A triangulated irregular network (TIN) is a vector data structure used to store and display surface models. A TIN partitions geographic space using a set of irregularly spaced data points, each of which has x-, y-, and Z-values. These points are connected by edges that form contiguous, nonoverlapping triangles and create a continuous surface that represents the terrain. A grid is a spatial data structure that defines space as an array of cells of equal size that are arranged in rows and columns. In the case of a grid that represents a surface, each cell contains an attribute value that repre- sents a change in Z value. The location of the cell in geographic space is obtained from its position relative to the gridʼs origin. Interpolation Methods To create a surface grid inArcGIS, the SpatialAnalyst extension employs one of several interpolation tools. Interpolation is a procedure used to predict the values of cells at locations that lack sampled points. It is based on the principle of spatial autocorrelation or spatial dependence, which measures degree of relationship/dependence between near and distant objects. Spatial autocorrelation determines if values are interrelated. If values are interrelated, it determines if there is a spatial pattern. This correlation is used to measure n Similarity of objects within an area n The degree to which a spatial phenomenon is correlated to itself in space n The level of interdependence between the variables n Nature and strength of the interdependence Different interpolation methods will almost always produce different results. Points Methods of Surface Representation Contours TINs Grids Interpolation is the proce- dure used to predict cell values for locations that lack sampled points. Interpolating Surfaces in ArcGIS Spatial Analyst 1.1 1.2 1.3 1.5 1.6 1.7 1.8 21 Interpolated Rainfall Known Rainfall Values 1 Mile 0 0.2 0.5 0.7 1
  • 2. Developer’s Corner www.esri.com ArcUser July–September 2004 33 There are two categories of interpolation techniques: deterministic and geostatistical. Deterministic interpolation techniques create surfaces based on measured points or mathematical formulas. Methods such as Inverse Distance Weight (IDW) are based on the extent of similarity of cells while methods such as Trend fit a smooth surface defined by a math- ematical function. Geostatistical interpolation techniques such as Kriging are based on statistics and are used for more advanced prediction surface modeling that also includes some measure of the certainty or accuracy of predictions. The characteristics of an interpolated surface can be controlled by limiting the input points used in the calculation of output cell values. This can be done by limiting the number of points sampled or the area from which sampled points are taken. Specifying the maximum number of points to be sampled will return the points closest to the output cell location until the maximum number is reached. Alternatively, specifying a fixed radius in map units will select only input points within the radius distance from the center of the output cell unless there are not enough points within that radius. Maximum Number The characteristics of an interpolated surface can be controlled by limiting the input points used. Points can be limited by specifying a maximum number of points, which will select the nearest points until that maximum number of points is reached. Alternatively, a radius can be specified in map units. Fixed Radius With Barrier Without Barrier Barriers, which reflect the presence of fault lines, cliffs, streams, and other features that create linear discontinuity in surfaces, also control how surfaces are generated. IDW and Kriging support the use of barriers. Many interpolation tools incorporate barriers that define and control surface behavior in terms of smoothness and continuity. Barriers are needed because sometimes interpolation operations should not be per- formed across features, such as fault lines, levees, cliffs, and streams, that create a linear discontinuity in the surface. By using barriers, changes in the behavior of the surface can be described and enforced. Available Interpolation Options ArcGIS Spatial Analyst at version 9 offers several interpolation tools for generating surface grids from point data. PointInterp, Natural Neighbors, and Trend methods and the Topo to Raster command have been added to the IDW, Spline, and Kriging interpolation methods that were available in ArcGIS 8.3 Spatial Analyst. Natural Neighbor and Trend methods were available in ArcGIS 8.3 3D Analyst, although Trend was only accessible programmatically. Each method uses a different approach for determin- ing the output cell values. The most appropriate method will depend on the distribution of sample points and the phenomenon being studied. The characteristics of an interpolated surface can be controlled by limiting the input points used in the calculation of output cell values. Continued on page 34
  • 3. Interpolating Surfaces in ArcGIS Spatial Analyst Continued from page 33 34 ArcUser July–September 2004 www.esri.com IDW The IDW function should be used when the set of points is dense enough to capture the extent of local surface variation needed for analysis. IDW determines cell values using a linear-weighted combination set of sample points. The weight assigned is a function of the distance of an input point from the output cell location. The greater the distance, the less influence the cell has on the output value. Kriging A powerful statistical interpolation method used for diverse applications such as health sciences, geochemistry, and pollution modeling, Kriging assumes that the distance or direction between sample points reflects a spatial correlation that can be used to explain variation in the surface. It fits a function to a specified number of points or all points within a specified radius to determine the output value for each location. Kriging is most appropriate when a spatially correlated distance or directional bias in the data is known and is often used for applications in soil science and geology. The predicted values are derived from the measure of relationship in samples using sophisticated weighted average technique. It uses a search radius that can be fixed or variable. The generated cell values can exceed value range of samples, and the surface does not pass through samples. There are several types of Kriging. Ordinary Kriging, the most com- mon method, assumes that there is no constant mean for the data over an area mean (i.e., no trend). Universal Kriging does assume that an overrid- ing trend exists in the data and that it can be modeled. Inverse Distance Weighted Spline Spline estimates values using a mathematical function that minimizes overall surface curvature. This results in a smooth surface that passes exactly through the input points. Conceptually, it is like bending a sheet of rubber so that it passes through the points while minimizing the total curvature of the surface. It can predict ridges and valleys in the data and is the best method for representing the smoothly varying surfaces of phe- nomena such as temperature. There are two variations of spline—regularized and tension. A regu- larized spline incorporates the first derivative (slope), second derivative (rate of change in slope), and third derivative (rate of change in the sec- ond derivative) into its minimization calculations. Although a tension spline uses only first and second derivatives, it includes more points in the spline calculations, which usually creates smoother surfaces but in- creases computation time. Spline Kriging PointInterp A method that is similar to IDW, the PointInterp function allows more control over the sampling neighborhood. The influence of a particular sample on the interpolated grid cell value depends on whether the sample point is in the cellʼs neighborhood and how far from the cell being inter- polated it is located. Points outside the neighborhood have no influence. PointInterp
  • 4. Developer’s Corner www.esri.com ArcUser July–September 2004 35 The weighted value of points inside the neighborhood is calculated us- ing an inverse distance weighted interpolation or inverse exponential distance interpolation. This method interpolates a raster using point fea- tures but allows for different types of neighborhoods. Neighborhoods can have shapes such as circles, rectangles, irregular polygons, annuluses, or wedges. Natural Neighbor Natural neighbor interpolation has many positive features, can be used for both interpolation and extrapolation, and generally works well with clus- tered scatter points. Another weighted-average method, the basic equa- tion used in natural neighbor interpolation is identical to the one used in IDW interpolation. This method can efficiently handle large input point datasets. When using the Natural Neighbor method, local coordinates de- fine the amount of influence any scatter point will have on output cells. Natural Neighbor Trend Trend Trend is a statistical method that finds the surface that fits the sample points using a least-squares regression fit. It fits one polynomial equation to the entire surface. This results in a surface that minimizes surface vari- ance in relation to the input values. The surface is constructed so that for every input point, the total of the differences between the actual values and the estimated values (i.e., the variance) will be as small as possible. It is an inexact interpolator, and the resulting surface rarely passes through the input points. However, this method detects trends in the sample data and is similar to natural phenomena that typically vary smoothly. Topo to Raster By interpolating elevation values for a raster, the Topo to Raster method imposes constraints that ensure a hydrologically correct digital elevation model that contains a connected drainage structure and correctly repre- sents ridges and streams from input contour data. It uses an iterative fi- nite difference interpolation technique that optimizes the computational efficiency of local interpolation without losing the surface continuity of global interpolation. It was specifically designed to work intelligently with contour inputs. IDW and Spline are two deterministic methods that create surfaces from samples based on the extent of similarity or degree of smooth- ing. However, while a spline surface passes exactly through each sample point, an IDW will pass through none of the points. Kriging is a geostatistical method that uses a powerful statistical technique for predicting values derived from the measure of relationship in samples and employs sophisticated weighted average techniques. Conclusion This brief introduction supplies a framework for exploring the interpola- tion tools available in ArcGIS 9 Spatial Analyst. A more robust and com- prehensive arsenal of tools for data exploration and spatial data analysis is available from theArcGIS 9 GeostatisticalAnalyst extension. More in- formation on how to perform spatial data analysis and applications for the tools in ArcGIS 9 Spatial Analyst may be found in the product manual, Using ArcGIS Spatial Analyst. Papers that provide background informa- tion on interpolation methods and their application in various industries and disciplines can be found at www.esri.com/geostatisticalanalyst. Each of the interpolation methods available in the ArcGIS Spatial Analyst extension uses a different approach for determining output cells. Choose a method based on the distribution of sample points and the phenomenon being studied. Krige Spline IDW Distance Z-value