1 meuse (cont)

1. “There is clear evidence of anisotropy in the variograms”

What this means

Directional variograms (e.g. 0°, 45°, 90°, 135°) are not the same

The rate at which semivariance increases depends on direction

👉 Therefore, spatial dependence is direction-dependent, not isotropic.

2. “The highest level of spatial correlation is shown in the 45° variogram”

Key idea

Highest spatial correlation ⇔ lowest semivariance for a given distance

Or equivalently: largest range

So this tells us:

Along 45°, observations remain similar over longer distances

This is the major axis of anisotropy

Plain-language interpretation

The process varies most smoothly along the 45° direction.

This often aligns with a physical or structural direction (wind, river, slope, coastline, etc.).

3. “When the sill appears to be different in different directions, this suggests a trend in the data”

⚠️ This is a very important point.

Reminder: what the sill represents

The sill ≈ total variance of a stationary process

Under second-order stationarity:

The sill should be the same in all directions

So if sills differ by direction…

It usually means:

The assumption of stationarity is violated

There is a large-scale trend (drift) in the data.

Intuition

A trend inflates semivariance more in some directions than others

This can look like anisotropy, but it’s actually non-stationarity

Not all apparent anisotropy is “true” anisotropy — some of it is trend.

4. Why this is a problem

Geostatistical models assume:

Constant mean (or trend removed)

Direction-invariant variance

If you ignore the trend:

The variogram tries to explain large-scale structure

You get direction-dependent sills, which are hard to model properly.

5. Ways to deal with this situation

Now the three solutions make sense.

1. Fit a nested semivariogram with an infinite range in one direction

What this means

Add a component that never reaches a sill in one direction

Infinite range ≈ captures a trend-like structure

Interpretation

Small-scale variation → finite-range variogram

Large-scale trend → infinite-range component.

When used

When trend is directional and you don’t want to model it explicitly

Common in classical geostatistics

2. Fit a semivariogram along one direction with no trend

What this means

Choose the direction where the data appear most stationary

Often the direction perpendicular to the trend

Interpretation

“If I analyze only this direction, the trend effect is minimal.”

Limitations

You lose information from other directions

Mainly a diagnostic or pragmatic solution

3. Fit a trend to the data, then fit a semivariogram to the residuals ✅ (most principled)

What this means

Model the mean structure:

Linear trend

Polynomial surface

Covariates (elevation, latitude, etc.)

Compute residuals

Fit a variogram to the residuals

Why this works

Residuals are closer to stationary

Directional sills should now be similar

Remaining anisotropy reflects true spatial dependence

This is often the preferred modern approach.

1.1 Key takeaway

Directional variograms indicate clear anisotropy, with the strongest spatial correlation occurring along the 45° direction. However, the presence of direction-dependent sills suggests non-stationarity due to an underlying trend in the data. To address this, the trend may be explicitly modeled and removed prior to variogram estimation, or alternatively, a nested semivariogram with an infinite-range component may be employed to capture large-scale variation. Variogram modeling of the residuals then allows valid inference under stationarity assumptions.

Different ranges → anisotropy

Different sills → likely trend

Always ask:

“Is this true anisotropy, or trend masquerading as anisotropy?”