Kriging with Measurement Error

The user will find this kriging option in the Interpolate / Estimation / (Co-)Kriging... window, Special Kriging Options...button.

A slight modification of the theory makes it possible to take into account variable measurement errors at data points, provided the variances of these errors are known.

Suppose that, instead of Zɑ we are given Zɑ+eɑ where eɑ is a random error satisfying the following conditions:

Then the kriging estimator of Z can be written and the variance becomes:

Then the kriging system of Z0 remains the same except that Vɑ is now added to the diagonal terms Kɑɑ; no change occurs in the right-hand side of the kriging system.

These data error variances Vɑ are related, though not identical, to the nugget effect.

Let us first recall the definition of the nugget effect.

By definition, the nugget effect refers to a discontinuity of the variogram or the covariance at zero distance. Mathematically, it means that the field Z(x) is not continuous in the mean square sense. The origin of the terminology "nugget effect" is as follows.

Gold ore is often discovered in the form of nuggets, i.e. pebbles of pure gold disseminated in a sterile matrix. Consequently, the ore grade varies discontinuously from inside to outside the nugget. It has been found convenient to retain the term "nugget effect" even if this is due to causes other than actual nuggets.

Generally, discontinuity of the variogram is only apparent. If we could investigate structures at a smaller scale, we would see that Z(x) is in fact continuous but with a range much smaller than the nearest distance between data points. This is the reason why one could conveniently replace this nugget effect by a transition scheme (say a spherical variogram) with a very short range.

But the "nugget effect" (as used in the modeling phase) can also be due to another factor: the measurement error. In this case, the discontinuity is real and is due to errors of the type eɑ. This time, the discontinuity remains whatever the size of the structure investigation. If the same type of measurement error is attributed to all data, the estimate is the same whether:

  • you do not use any nugget effect in your model and you provide the same Vɑ for each data, or
  • you define a nugget effect component in your model whose sill C is precisely equal to Vɑ.

Unlike the estimate itself, the kriging variance differs depending on which option is chosen. Indeed, the measurement error Vɑ is considered as an artefact and is not a part of the phenomenon of interest. Therefore, a kriging with a variance of measurement error equal for each data and no nugget effect in the model will lead to smaller kriging variances than the estimation with a nugget component equal to Vɑ.

The use of data error variances Vɑ really makes sense when the data is of different qualities. Many situations may occur. For example, the data may come from several surveys: old ones and new ones. Or the measurement techniques may be different: depth measured at wells or by seismic, porosities from cores or from log interpretation, etc ...

In such cases error variances may be computed separately for each sub-population and, if we are lucky, the better quality data will allow identification of the underlying structure (possibly including a nugget effect component), while the variogram attached to the poorer quality data will show the same previous structure incremented by a nugget effect corresponding to the specific measurement error variance Vɑ.

In other cases, it could be possible to evaluate directly the precision of each measurement and derive Vɑ : if we are told that the absolute error on Z is ΔZ, by reference to Gaussian errors we may consider that, ΔZ = 2σ and take: Vɑ = (ΔZ / 2)2.

Another use of this technique, is in the post processing of the macro kriging where we calculate "equivalent samples" with measurement error variances. These variances are in fact calculated from a fitted model depending on the number of initial samples inside pseudo blocks.