Model Construction:
interpolation techniques
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Introduction
• There is no ideal DEM.
– DEM generation techniques can not capture the full complexity of a surface.
– There is always a sampling problem and a representation problem.
• Continuous Surface Discrete Surface Continuous ⇒ ⇒ Surface.
Surface Characteristics
• Functional Surfaces:
– Store a single z value for any given X,Y location.
– Represent continuous surfaces.
– Referred to as 2.5 dimensional surfaces.
• Solid Models:
– True 3 dimensional models capable of storing multiple Z values for any given (X, Y) location.
– Capable of representing discontinuous surfaces.
– Examples: Machine parts, highway structures, buildings.
Surface Characteristics
• Surface Continuity:
– Continuous Surface: If you approach a given X, Y location on a functional surface from any direction, you will get the same Z value.
– Solid models are capable of storing more than one Z value for a given (X, Y) location (discontinuous
surfaces).
Surface Characteristics
• Surface Smoothness:
– In addition to being continuous, a smooth surface has the additional property that regardless of the direction from which you approach a given point on the surface, the surface normal is constant.
– Geologically young terrain have sharp ridges and valleys. In contrast, older terrain are smoothed by weathering forces.
Surface Smoothness
Non-Smooth Surface Smooth Surface
REMEMBER
• Our objectives.
– Reality (continuous surface) digital/discrete ⇒ representation.
• Established through a sampling process.
• Discussed in chapter 4.
– Digital representation Reality (at least our best guess of reality).
• Established through interpolation process.
Interpolation Vs. Extrapolation
• Interpolation:
– The process of estimating the values of an attribute (e.g., elevation) at internal unsampled sites using measurements made at reference points.
– The interpolation point lies within the range defined by the reference points.
• Extrapolation:
– The process of predicting the values of an attribute (e.g., elevation) at external unsampled sites using measurements made at reference points.
– The extrapolation point lies outside the range defined by the reference points.
Interpolation Vs. Extrapolation
•Reference Points.
•Interpolation Point.
•Extrapolation Point.
Measured Height Derived Height
Interpolation: Where and Why?
• The data we have do not cover the domain of interest completely.
• The discretized surface has a different level of resolution, cell size, or orientation from that required
• A continuous surface is currently being represented by a data model that is different from required.
• Need elevation Z at single unsampled points.
• Need elevation Z of a rectangular grid from irregularly sampled points: gridding.
• Need to compute locations X,Y of points along contours: contour interpolation.
• Densification or coarsening of rectangular grids: resampling.
Interpolation: Remarks
• There is no ‘best’ interpolation algorithm that is clearly superior to all other interpolation methodologies.
• The quality of the resulting DTM is determined by:
– The distribution and accuracy of the original data/referncepoints.
– The adequacy of the underlying interpolation model.
• Most important criteria for selecting an interpolation method include among others:
– Desired accuracy.
– Involved computational effort.
– Possibility of taking structural features into account.
– Adaptability to the varying characteristics of the terrain.
Classification of Interpolation Method
• Global Fit: Calculate a single function
describing a surface that covers the entire map area.
• Local Fit: Estimate the surface at
interpolation points using only a selection
of the nearest data/reference points.
Global Interpolation
Trend Surface Analysis (TSA)
• A global fit interpolation method, which has the following characteristics:
– Elevations are approximated by a polynomial expansion.
– The coefficients of the polynomial function are determined through a least squares adjustment.
– Each original observation is considered to be the sum of a deterministic polynomial function of the
geographic/planimetric coordinates plus a random error.
– Because of the least-squares fitting procedure, no other polynomial equation of the same degree can provide a better approximation of the data.
TSA: Concept
• Observed data points are assumed to be the superposition of two components:
– Regional/trend: Low frequency component of the surface (borrowed from Fourier Analysis).
– Local fluctuations: High frequency component of the surface.
• The optimum trend, or linear function, must
minimize the squared sum of deviations (local
fluctuations) between the original surface and its
trend.
Trend Surfaces in 3-D
First Order Trend Surface
• A first order linear trend surface equation has the form:
•Question: How can we determine the coefficients e(x,y) so that e is minimised?
First Order Trend Surface
• Observation Equations:
•Normal Equations:
First Order Trend Surface
• Normal Equations:
Normal Equations
Trend Surface Analysis
• Second Order Polynomial Trend Surface:
•Third Order Polynomial Trend Surface:
Trend Surface Analysis
• Fourth Order Polynomial Trend Surface:
Trend Surface Analysis
• Fifth Order Polynomial Surface:
Order Polynomial Surface:
TSA: Example
• Given: irregularly distributed set of reference points
– Fit a first, second and third order trend surfaces through the given points.
– Calculate the estimated elevations and the corresponding residuals for the given reference points.
– Calculate the percentage of goodness of fit of the regression.
TSA: Example (Original Surface)
TSA: Example (Original Surface)
TSA: (Trend Surface, 2nd Order)
TSA: (Original/Trend Surfaces , 2nd
Order)
TSA: (Residual Plot , 2nd Order)
TSA: Example (Coefficients)
TSA: Remarks
• The coefficients of the design matrix (A) are
proportional to the planimetric coordinates of the reference points.
• When dealing with large numbers, numerical instability is expected.
• Solution:
– Option (1): Shift the coordinate system to the centroid of the area of interest.
– Option (2): Normalize the coordinates to the range (0:1).
TSA: Remarks
• CentroidBased Computation:
–Shift the coordinates:
– Caution: After the fitting make sure to shift the coordinates back to the original coordinate system.
TSA: Remarks
TSA: Remark
•The coefficients are different.
•The numerical stability is not good when using the raw coordinates.
TSA: Precautions
• There must be adequate data control.
– – The number of observations should be much greater than the number of coefficients.
• The spacing between the observation points is important.
– It can affect the size and resolution of features seen.
• Clustering can cause problems or bias.
– We are assuming equal weight for all the measured elevations.
• The distribution can affect the shape of the surface.
• There are problems near boundaries.
– The surface can "blow-up" at the corners.
– For this reason, it is important to have a buffer around the area of concern.
TSA: Advantages
• Unique surface generated.
• Easy to program.
• Same surface estimated regardless of orientation of reference geographic.
• Calculation time for low order surfaces is low.
• If the raw data are statistical in nature, with
perhaps more than one value at an observation
point, trend surfaces provide statistically optimal
estimates of a linear model that describes their
spatial distribution.
TSA: Disadvantages
• Statistical assumptions of the model are rarely met in practice.
• Local anomalies can not be seen on contour maps of low order polynomials (but can be seen on
contour maps of residuals).
• The surfaces are highly susceptible to edge effects.
• Difficulty to describe a physical meaning to
complex and high order polynomial.
Problems with Trend Surfaces
• Polynomial surface is too simple in comparison to most natural surfaces.
• Difficulty in extrapolating beyond the area of data control.
– Prone to the generation of seriously exaggerated estimates.
• Computational difficulties may be encountered if a very high order polynomial trend surface is fitted.
• The matrix solution may become unstable, or
rounding errors may result in erroneous trend
surface coefficients.
