Spatial Interpolation 2

Inverse Distance Weighting (IDW)

A spatial interpolation method that estimates unknown values using nearby known points, with closer points given more influence than farther points

Main principle of IDW

The weight assigned to a known point is inversely related to its distance from the unknown location

How distance affects weight in IDW

As distance increases, weight decreases; as distance decreases, weight increases

Highest-weight point in IDW

The known point closest to the unknown location has the greatest influence on the estimated value

IDW formula purpose

To calculate an estimated value at an unknown location as a weighted average of nearby known values

i in the IDW formula

The sample or known location

j in the IDW formula

The unknown location where a value is being estimated

d in the IDW formula

The distance between a known point and the unknown point

z in the IDW formula

The value at a known sample point

α (alpha) in the IDW formula

The power parameter that controls how quickly weight decreases with distance

k in the IDW formula

A normalization constant used to ensure the weights sum properly

Power parameter in IDW

A setting that controls how strongly nearby points dominate over farther points

Effect of high power (α) in IDW

Weights decrease very quickly with distance, so the nearest points dominate the interpolation and the surface becomes less smooth and more locally detailed

Effect of low power (α) in IDW

Farther points still contribute more to the estimate, creating a smoother, less locally dominated surface

Typical α values in IDW

Usually 1 or 2

Effect of increasing α on the IDW surface

The interpolated surface becomes patchier, less smooth, and more influenced by nearby points

Factors besides power that affect IDW results

Output cell size, search radius, number of neighbors, barriers, neighborhood shape, and distance metric

Output cell size in IDW

The resolution of the interpolated raster; smaller cells produce more detailed output while larger cells produce coarser output

Search radius in IDW

A setting that determines which nearby points are included in estimating each unknown location

Fixed search radius in IDW

Uses points within a constant distance around each unknown location

Variable search radius in IDW

Uses a varying area to include a specified number of nearby points for each unknown location

Barriers in IDW

Line features that limit which points can influence an estimate, so only points on the same side of the barrier are used

Examples of barriers in IDW

Ridges, rivers, coastlines, and roads

Why barriers matter in IDW

They prevent points that are close in straight-line distance but separated by a meaningful boundary from influencing each other

Effect of sample distribution on IDW

The spatial arrangement of sample points strongly changes the interpolation result even when power remains the same

Systematic sampling effect on IDW

Usually produces a more even and representative interpolation surface because the sample coverage is balanced

Random sampling effect on IDW

Can create uneven coverage and variable interpolation quality across space

Clustered sampling effect on IDW

Can produce biased or distorted surfaces because some areas are oversampled while others are undersampled

Effect of number of samples or neighbors on IDW

Using very few points creates a crude, blocky surface, while using more points generally produces a smoother and more representative surface

Shape of neighborhood in IDW

The geometric form of the local search area used to select points that influence each estimate

Circular neighborhood in IDW

Assumes isotropy, meaning influence is the same in all directions

Isotropy

The assumption that spatial influence is equal in every direction

Anisotropy

The condition where spatial influence varies by direction

Why anisotropy matters in interpolation

Some spatial processes spread more strongly in one direction than another, so a circular neighborhood may be unrealistic

Examples of anisotropic processes

Pollutant concentration influenced by wind direction or current direction

Elliptical neighborhood in IDW

A directional search area that allows interpolation to reflect anisotropic spatial influence

Effect of using an ellipse instead of a circle in IDW

It stretches the neighborhood in one preferred direction, changing which points influence the estimate and altering the resulting surface

Why IDW is a weighted average

The unknown value is calculated from a weighted combination of observed values rather than from a fitted statistical model

Range limitation of IDW predictions

IDW cannot predict values higher than the highest observed sample or lower than the lowest observed sample

Why IDW cannot create new peaks or valleys

Because it is based on averaging observed values, it cannot invent highs or lows that were never sampled

Distance metrics in IDW

Different mathematical ways of measuring distance between locations

Why IDW is local

It estimates values using nearby points within a neighborhood rather than fitting one model to the entire study area

Why IDW is exact

The interpolated surface passes through all known sample points

Why IDW is gradual

It produces smooth transitions between values rather than sudden jumps

Why IDW is deterministic

It uses mathematical rules rather than probability or statistical uncertainty models

Spline interpolation

A method that creates a smooth surface by fitting curves through known points

Origin of the word spline

A drafting tool or flexible ruler used to draw smooth curves through points

Core idea of spline interpolation

To create a smooth curve or surface that passes through known data points

Piecewise polynomial functions

Separate polynomial functions fitted to different segments of the data rather than one single polynomial for the entire dataset

Why spline uses piecewise functions

To avoid the instability and excessive wiggle that can happen when using one large high-order polynomial

What “piecewise” means in spline

The interpolation is built from multiple connected polynomial segments rather than one single equation

Why high-order polynomials can be problematic

They can oscillate too much and produce unrealistic wiggles or instability across the surface

Most commonly used spline order

Cubic spline, or 3rd-order spline

Cubic spline

A spline made from connected 3rd-order polynomial segments that balances smoothness and flexibility

Main advantage of cubic spline

It creates smooth curves without becoming as unstable as higher-order polynomials

Impact of an outlier on spline

An outlier can pull the curve or surface strongly and distort the interpolation nearby

Why spline is often considered local

It applies the interpolation algorithm repeatedly over piecewise segments or local sections rather than fitting one global surface to the whole dataset

Why spline is often considered exact

The surface passes through all known sample points

Why spline is gradual

It produces smooth, continuous changes rather than abrupt jumps

Why spline is deterministic

It uses mathematical functions rather than statistical uncertainty models

Approximate spline

A smoothing version of spline that does not necessarily pass exactly through every observed point

Difference between exact spline and approximate spline

Exact spline honors all observed values, while approximate spline smooths the surface and may not pass exactly through every point

Main difference between IDW and spline

IDW estimates values using distance-weighted averages of nearby points, while spline fits smooth mathematical curves or surfaces through points

Main similarity between IDW and spline

Both are commonly treated as local, deterministic interpolation methods that often produce gradual surfaces

Main limitation of spline

It can be strongly distorted by outliers and may behave unrealistically if the data contain problematic values