GEOSTATISICAL ANALYSIS

Course: Special Topics in Remote Sensing & GIS

Mirza Muhammad WaqarContact:

mirza.waqar@ist.edu.pk+92-21-34650765-79 EXT:2257

RG712

Spatial Interpolation

Spatial interpolation is the processing of using points with known values to estimate values at other points.

Spatial interpolation requires two basic inputs: Known Points Interpolation Method

The number and distribution of control points can greatly influence the accuracy of spatial interpolation. Ideally control points should be well distributed

Spatial Interpolation

Basic assumption in spatial interpolation is that the value to be estimated at a point is more influenced by a near by known point. Than those that are further away.

Control points should be well distributed. Ideal situation

Type of Spatial Interpolation

Global Interpolation: A global interpolation method uses every known point available to estimate an unknown value. To capture the general trend of the surface Take longer time to process Generate noise

Local Interpolation: A local interpolation method, on the other hand, uses a sample of known points to estimate an unknown value. To capture the local or short range variations

Type of Interpolation

Exact Interpolation: Exact interpolation predicts a value at the point location that is the same as its known value. Generate a surface that passes through the control

points.

Inexact Interpolation: Or approximate interpolation, predicts a value at the point location that differs from its known value.

Type of Interpolation

Type of Interpolation

Deterministic Interpolation method provides no assessment of errors with predicted values.

A stochastic Interpolation method, on the other hand, offers assessment of prediction error with estimated variances.

Global vs. Local

Global Interpolation Method Local Interpolation Method

Deterministic Stochastic Deterministic Stochastic

Trend Surface Regression Thiessen (exact) Kriging (Exact)

IDW (Exact)

Spline (Exact)

Trend Surface Analysis

An inexact interpolation method, trend surface analysis approximates points with known values with a polynomial equation.

The equation or the interpolator can then be used to estimate values at other points. Computationally complex model

zx,y = b0 + b1x + b2y

Regression Model

A regression model relates a dependent variable to a number of independent variables in a linear equation (an interpolator), which can then be used to prediction or estimation. Watershed Level Regression Model

Snow Water equivalent Studies Parameter-Elevation Regression on independent

Slopes Model Used to map precipitation by using climatic data and DEM

Other local regression model

Thiessen Polygon

This method don’t use an interpolator but require initial triangulation for connecting known points.

Delaunay triangulation is often used in preparing thiessen polygons. The delanuay triangulation ensure that each known point is connected to

its neighbors. Triangles are as equivalent as possible.

After triangulation, thiessen polygons can easily constructed by drawing perpendicular to the sides of triangles.

Theiessen polygons are small where known points are closer to each other and vice versa.

Thiessen Polygon

Inverse Distance Weightage (IDW)

Inverse Distance Weightage is an exact method that enforces The estimated values of a point is influence more by near by

known points than by those further apart.

A power variable control the degree of local influence. A power of 1 means a constant rate of change in values. A power of 2 or higher means the rate of change in value is higher

near a known points.

An important characteristic of IDW interpolation is that All predicted values are within the range of Max-Min values of

known points.

The Power Function (p)

Inverse Distance Weightage (IDW)

When to use IDW?

The surface calculated using IDW depends on the selection of a Power value (p) Neighborhood search strategy

IDW is an exact interpolator, where the maximum and minimum values (see diagram above) in the interpolated surface can only occur at sample points.

The output surface is sensitive to clustering and the presence of outliers. IDW assumes that the surface is being driven by the local variation, which can be captured through the neighborhood.

Kriging

Kriging is a geostatistical method for spatial interpolation.

Kirging differ from the other interpolation methods because kriging can access the quality of prediction with estimated perdition error.

Kriging

In kriging it is assumed that spatial variations of an attribute are neither random (stochastic) nor deterministic.

Instead the spatial variations may consists of three components. A spatially correlated component

Representing the variations of regionalized variable A drift representing a trend A random error term

Kiring

Kriging uses the semi-varience to measure the spatially correlated component/spatial dependence/spatial autocorrelation.

Semi-variance = 50% of variance

Range: Range of influence, the distance at which no autocorrelation exist.Nugget: Inherited error in the system at range.Sill: Leveling off of variance

Semivariogram

Semivariogram Cloud

If spatial dependence does exist in the dataset, known points that are closer to each other are expected to have small semi-variance and vice versa.

Because semivariogram cloud has all pairs of known points. A semivariogram is difficult to manage and

interpret.

Semivariogram Cloud

A process called binning is typically used to average semivariance data by distance and direction.

The first part of binning process is to group pair of sample points into lag classes.

For example, the lag size is 2000 m then the pair of points separated by less than 2000 m distance are grouped into lag class of 0-2000 m.

Similarly pair of points separated between 2000-4000 m will be grouped into lag class of 2000-4000 m and so on.

Semivariogram Cloud

Semivariogram is used as an interpolator in kriging. The semi-variogram must be fitted with a mathematical function or model.

The fitted semi-variogram can t hen be used for estimating the semivariance at any given distance.

Semivariogram Cloud

Fitting a model to a semivariogram is a difficult and often controversial task in geostatistics.

One reason for the difficulty is the number of models to choose from. Geostatistical Analyst offer 11 models.

The other reason is the lack of standardized procedure for comparing the models. Visual inspection and cross validation is the best tool to

check the strength of any model.

Semivariogram Cloud

The common models for fitting semi-variogram are Spherical Exponential

A spherical model shows a progressive decrease of spatial dependency until some distance beoyound with spatial dependences level off.

An exponential model shows less gradual pattern than a spherical model, spatial dependence decreases exponentially with increase in distance and disappear completely as the distance increase.

Types of Kriging

Ordinary Kriging Universal Kriging Co-Kriging

Ordinary Kriging

It assumes that absence of drift. It focuses on the spatially correlated

components and uses the fitted semivariogram directly for interpolation.

The general equation for estimating the Z (attribute value) at a point is

Where Z0 is the estimated valueZx is the known value at point xWx is the weight associated with point xAnd N be the number of sample points in the estimation

Universal Kiring

It assume that the spatial variation in Z-value has a drift.

A kriging method often used on data with a significant spatial trend, such as a sloping surface.

In universal kriging, the expected values of the sampled points are modeled as a polynomial trend.

Kriging is carried out on the difference between this trend and the values of the sampled points.

Co-Kriging

If two parameters are correlated then interpolation of one parameter on the base of other parameter can be done using co-kriging.

Comparison of Spatial Interpolation Method

Using same data with different interpolation methods, one can find different interpolation results.

Different predicted values can occur by using same method for different parameter values.

How can different interpolation methods be compared? Local methods are usually compared on the base of

statistical measures. Visual quality of predicted surface. Cross validation is a common technique for comparison

Cross Validation

Consider each point as unknown point and predict that point then compare the value of known point with the predicted value of the same point.

Or create subset of original data and predict the values for subset using known point and compare the predicted values with known points.

Cross Validation Procedure

Compare the interpolation method, by following procedure.1. Remove a known point from the dataset2. Use the reaming point to estimate the value at the

point previously removed3. Calculate the predicted error or estimation by

comparing the estimated value with the known value

RMS =

Standardized RMS Error

Standardize RMS Error =

A better interpolation method should produce a smaller RMS error.

A better kriging method should produce a smaller RMS error and a standardize RMS close to 1.

Validation Techniques

This technique compares the interpolation methods by 1st dividing known points into two samples. One sample for developing the model for interpolation Other sample for testing the accuracy of the model

The statistics RMS error and standardize RMS error derived from the test sample can then be used to compare the method.

Validation may not be a feasible option if the number of known points is too small to be split into two samples.

Questions & Discussion