Transcript WFM 6202: Remote Sensing and GIS in Water Management © Dr.

WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful Islam
WFM 6202: Remote Sensing and GIS in Water
Management
[Part-B: Geographic Information System (GIS)]
Lecture-6: Interpolation
Akm Saiful Islam
Institute of Water and Flood Management (IWFM)
Bangladesh University of Engineering and Technology (BUET)
December, 2008
WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful Islam
Principle of Interpolation
 Interpolation is the procedure of estimating
the value of properties at unsampled points or
areas using a limited number of sampled
observations.
WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful Islam
Interpolation Techniques
• 1. Pointwise interpolation
• 1(a) Thiessen polygon
• 1(b) Weighted Average
• 2. Interpolation by curve fitting
– 2.1 Exact interpolation
• 2. 1(a). Nearest neighbor
• 2. 1.(b) Linear interpolation
• 2. 1(c) Cubic interpolation
– 2.2 Approximate interpolation
• 2.2(a) Moving Average
• 2.2(b) B-spline
• 2.2(c) Curve Fitting by Least Square Method
 3. Interpolation by surface fitting
– 3.1 Regular grid
• 3.1(a) Bilinear Interpolation
• 3.1(b) Bicubic Interpolation
– 3.2 Random points
• 3.2(a) TIN
WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful Islam
1. Pointwise Interpolation
 Pointwise interpolation is used in case the sampled
points are not densely located with a limited influence
or continuity in surrounding observations, for example
climate observations such as rainfall and
temperature, or ground water level measurements at
wells.
WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful Islam
1(a) Thiessen Polygons
 Thiessen polygons can be generated using distance
operator which creates the polygon boundaries as the
intersections of radial expansions from the observation
points.
 This method is also known as Voronoi tessellation.
WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful Islam
1(b) Weighted Average
 A window of circular shape with the radius of
dmax is drawn at a point to be interpolated, so
as to involve six to eight surrounding
observed points.
WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful Islam
2. Interpolation by Curve Fitting
 the principle of curve fitting respectively to
interpolate the value at an unsampled point
using surrounding sampled points.
WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful Islam
2. Curve Fitting
Curve fitting is an important type of interpolation in
many applications of. Curve fitting is divided into two
categories.
 2.1 exact interpolation : a fitted curve passes through all
given points.
 2.2 approximate interpolation : a fitted curve does not
always pass through all given points.
WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful Islam
2.1 Exact interpolation
There are three methods:
2.1(a) nearest neighbor : the same value as that
of the observation is given within the proximal
distance
WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful Islam
2.1 Exact interpolation
2.1(b) linear interpolation: a piecewise
linear function is applied between two
adjacent points.
WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful Islam
2.1 Exact interpolation
• 2.1(c) cubic interpolation : a third order
polynomial is applied between two adjacent points
under the condition that the first and second order
differentials should be continuous.
WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful Islam
2.2. Approximate Interpolation
There are three methods;
2.2(a) Moving Average: a window with a range of -d to
+d is set to average the observation within the region
WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful Islam
2.2 Approximate Interpolation
2.2(b) B-Spline: a cubic curve is
determined by using four adjacent
observations
WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful Islam
2.2 Approximate Interpolation
2.2(c) Curve Fitting by Least Square Method.
Least square method (sometimes called regression model) is a
statistical approach to estimate an expected value or function with the
highest probability from the observations with random errors. The
highest probability is replaced by minimizing the sum of square of
residuals in the least square method.
Equation
Y  ax  b
Slope
a
 ( Xi  X )(Y  Y )
 ( Xi  X )
intercept
b  Y  aX
2
WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful Islam
3. Interpolation by Surface Fitting
 the principle of surface fitting respectively to
interpolate the value at an unsampled point
using surrounding sampled points.
WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful Islam
3. Surface Fitting
• Surface fitting is widely used for interpolation of points on
continuous surfaces such as digital elevation model
(DEM), geoid, climate model (rainfall, temperature,
pressure etc.) and so on.
• Surface fitting is classified into two categories:
– 3.1 surface fitting for regular grid and
– 3.2 surface fitting for random points.
• 3.1 Surface Fitting for Regular Grid
Following two methods are commonly used.
3.1(a) Bilinear Interpolation
3.1(b) Bicubic Interpolation
WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful Islam
3.1 Surface Fitting for Regular Grid
3.1(a) Bilinear Interpolation
Bilinear function is used to interpolate z using the
following formula with respect to normalized
coordinates (u, v) of the original coordinates (x, y)
WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful Islam
3.1Surface Fitting for Regular Grid
3.1(b) Bicubic Interpolation
Third order polynomial is used to fit a continuous
surface using 4 x 4 = 16 adjacent points.
WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful Islam
3.2 Surface Fitting for random
Points
3.2. (a) Triangular network called as Triangulated
Irregular Network (TIN) is applied
WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful Islam
Compare Interpolation methods
• Thiessen polygons are Used for service area analysis of public
facilities such as hospitals. Originally proposed to estimate aerial
averages precipitation in 1985.
• Inverse Distance Weighted can be a good way to take a first look
at an interpolated surface. However, there is no assessment of
prediction errors. Accuracy depends on the selection of a power
value and the neighborhood search strategy. A smaller (6) actually
produce better estimations than a larger number (12).
• Thin-plate Splines (applies to surface) are recommended for
smooth, continuous surfaces such as elevation and water table. Also
used for interpolating mean rainfall surface and land demand
surface.
WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful Islam
Geo-statistical method- Kriging
•
•
•
•
Kriging is a geostatistical method for spatial
interpolation.
It can assess the quality of prediction with estimated
prediction errors.
It uses statistical models that allow a variety of map
outputs including predictions, prediction standard
errors, probability, etc.
Semivariogram can be fitted as:
1. Ordinary Kriging models:
Spherical, Circular, Exponential,
Gaussian and Linear.
1. Universal Kriging models:
Linear with Linear drift, and
Linear with Quadratic drift
WFM 6202: Remote Sensing and GIS in Water Management © Dr. Akm Saiful Islam
Semivariogram
• The semivariogram functions
quantifies the assumption that
things nearby tend to be more
similar than things that are
farther apart. Semivariogram
measures the strength of
statistical correlation as a
function of distance.
• Semivariance:
Y(h) = ½ [(Z(xi) - Z(xj)]2
• Covarience = Sill – Y(h)