Spatial Interpolation III

ENVE 424/524

Spatial Interpolation

We can estimate the value of a variable at a location based on the values of that variable at neighboring locations.

The trick is defining the weight

point monitoring

data

continuous surface of

estimates (map)

spatial estimation method

zi is the estimated value at location i

n is the number of data pointszj is the value at data point j

wij is the weight assigned to data point j

Kriging Steps

1) Describe the spatial variation in the sample data

2) Summarize the spatial variation by a mathematical function

3) Use this model to determine the interpolation weights

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Variations on Kriging

Universal (kriging with a trend)

Probability

SimpleUses a known (at least assumed), constant mean across domain. Usually used by subtracting point data values from mean, interpolating the residuals, and then adding back to mean.

Uses a modeled trend across domain. The trend is subtracted from the point data values and the residuals interpolated.

Indicator krigingPoint data are transformed to indicator variables (usually binary). For example, a threshold value is defined and data points that are below the threshold are assigned a value of 1 and those above the threshold are assigned a value of 0. The indicator values are interpolated and the resulting surface shows the probabilities (0-1) of exceeding (or being below) the threshold.

Similar to indicator kriging but also incorporates the difference between a data pint value and the defined threshold. Using this “proximity” to threshold information can result in more accurate probabilities.

Block Kriging

Used when estimates are required not at points but an average value is required within a prescribed local area.

The estimate from block kriging using four points (Figure a) is equal to averaging the estimates from kriging at each of the four points individually.

Kriging with secondary informationKriging can incorporate secondary information to aid the estimation process when the primary variable is sparsely sampled.

Secondary information that is physically related to the primary variable and measured at a high spatial density than the primary variable can aid in “filling in” spatial gaps during interpolation.

(hard information)

(soft information)

Examples: Primary Secondary

rainfall amount digital elevation model

particulate matter concentrations visibility measurements

zinc concentrations

seasonal rainfall (1970-1980) seasonal rainfall (1990-2000)

cadmium concentrations

Kriging with an external drift (KED)

A secondary variable related to the primary variable can be used to model the "drift", or trend, of the primary variable in the kriging estimation.

In areas without primary sampling sites, the secondary variable aids the estimate through a function relating the secondary to primary variable.

As the estimation location gets farther from available primary samples, the estimate approaches the drift value.

The secondary variable must vary smoothly in space and must be known at all estimation locations as well as all primary sampling locations. The function relating the secondary and primary variable must be linear.

Ozone measurements in Paris Air quality model grid

&

Ozone concentrations using KED

Cokriging

Cokriging is an extension of ordinary kriging that uses a high spatial resolution secondary variable network to improve the estimation of the primary variable. Like ordinary kriging, a variogram model is developed for the primary variable.

A second variogram model is developed for the secondary variableA third variogram model is generated from the cross correlation of the primary and secondary variables.

The secondary data is "transformed" to the scale of the primary variable in order to derive a cross variogram. The secondary data are then treated identically to the primary data during spatial estimation.

V Variogram U Variogram UV Cross-Variogram

All U and V points are paired, binned, and semi-variance calculatedV = Primary Data U = Secondary Data

Cokriging Example

Cokriging does not improve estimates much if

-- all the variogram models are very similar in shape

-- the secondary information is does not have a higher density than the primary information

Kriging Error Variance

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Kriging Error

Factors influencing interpolation error:

1) Number of nearby samples

2) Proximity of nearby samples

3) Spatial arrangement of samples

4) Nature of phenomenon being estimated

Remove site from network (red circle)Obtain estimate at removed site location.

Interpolation Error = Estimated Value – Measured Value

Cross Validation

Cross Validation, a “jackknife technique”, provides a measure of interpolation error

Updates

ArcGIS now available for class use on the computer in Urbauer 319A

Plan is to have it available on two other machines in Cupples II first floor.

Reading for error and uncertainty lectures

Focus on Chapters 6 and 15 in Geographic Information Systems and Science, Longley et. Al

Optional: Chapter 9 in Principles of Geographical Information Systems, Burrough and McDonnell

Review for Exam

Exam will cover material from lecture 1 through today’s lecture

Exam questions will be based on problem sets, lecture slides, lab exercises, and the reading.

The exam questions will be a mix of:

-- multiple choice

-- match

-- short answer

-- equation-based calculations

Types of Attributes

• Nominal – Simply identifies or labels an entity so that it can be distinguished from another. e.g. sensor ID, building name (Lopata House vs. Lopata Hall)

– Cannot be manipulated using mathematical operations. However, frequency distributions are meaningful.

• Ordinal – Values based on an order or ranking, e.g. agricultural potential classes

– Cannot be manipulated using mathematical operations. However, frequency distributions are meaningful.

• Interval – Differences between entities are defined using fixed equal units, e.g. Celsius temperature

– Can be manipulated using addition and subtraction

• Ratio - Differences between entities can be defined using ratios, e.g. distance

– Can be manipulated using multiplication and division

• Cyclic - differences between entities depend on direction, e.g. wind direction

A common approach to classifying attributes is based on their level of measurement