Reply to “Standardized vs. customary ordinary cokriging…” by A. Papritz

doi:10.1016/j.geoderma.2008.04.008

Geoderma

Volume 146, Issues 1–2, 31 July 2008, Pages 397-399

https://doi.org/10.1016/j.geoderma.2008.04.008 Get rights and content

Introduction

Papritz (2008) has raised two main objections to our earlier study (Bishop and Lark, 2006):

(1)
We report predictions from the Papritz and Flühler (1994) cokriging estimator (PCK) for treatment responses
(2)
We use standardised ordinary cokriging (Goovaerts, 1997) (ECK) in general, and more specifically we recommend ECK on the basis of simulation studies. Papritz (2008) concludes that we err either
- (i)
  by assuming that the means of the treatment responses are equal, or
- (ii)
  not assuming the means to be equal but incorrectly using the ECK kriging variance estimator.

The rationale behind our original study (Bishop and Lark, 2006) was to find the best cokriging estimator for estimation of both treatment responses and the contrasts between them. We expected ordinary cokriging (DCK) to give us the best estimates (in the least-squares sense) of a target variable (e.g. the response to one treatment), given data on this variable and coregionalised ones (the other treatments). The estimator proposed by Papritz and Flühler (1994) (PCK) was expected to give us the best estimate of a contrast between the variables (such as their difference). One option would be use to DCK to estimate treatment responses, and PCK to estimate the contrasts, but we would prefer to have a coherent set of estimates (where the local estimate of any contrast is equal to the contrast between the local estimates of the responses).

For this reason we evaluated DCK, PCK and ordinary kriging in simulation studies. We also considered standardised ordinary cokriging (ECK) as described by Goovaerts (1997), in which each variable is first standardised to a common mean and then cokriging weights are found which sum to one over all the coregionalised variables. The basic characteristics of the cokriging estimators are summarised in Table 1.

While, as Papritz (2008) points out, the estimates of the treatment effects from the PCK estimator (by comparison to the contrasts) will not be optimal and should not be used, we thought that these results were worth reporting to emphasise this message for practitioners who would be equally interested in the contrast between treatment responses as well as their individual values.

We accept Papritz's general objections to ECK on theoretical grounds and our published findings are rightly questioned given that we only reported the theoretical kriging variances for our simulation study. Based on these we found ECK to be the best estimator and proceeded to recommend that in the paper and in future papers (Bishop and Lark, 2007).

However, our recommendation to use ECK was based on a more detailed simulation study which is described below. It was basis of the original submission to Geoderma but was simplified on recommendations by the editor and reviewers. The simulation results do support our recommendation, as we will show, an interesting result given the theory Papritz (2008) outlines.

Section snippets

The linear model of coregionalisation

A method similar to that described by Lark (2002) was used to generate two-dimensional simulations of two random functions using a LMCR. A LMCR for 2 variables, Z₁ and Z₂, treats them as a linear combination of a common set of independent underlying random functions. For example, the simulated random function Z₁(x) was obtained with the equation: $Z_{1} (x) = \sum_{k = 1}^{2} \sum_{j = 1}^{2} a_{1, j}^{k} y_{j}^{k} (x)$ where y_j^k(x) is a random function where (μ = 0, σ² = 1), j = 1 or 2, and k is an index indicating the semivariogram structure, in

Results

The α′Σα term inflated the kriging variances by 0.0001–1% (e.g. a typical result was that kriging variance = 0.334 for the ECK estimator, and α′Σα = 0.00014) depending on the actual prediction location. Therefore, for our simulation study there was a negligible difference between the kriging variances using the SCK and ECK estimator.

Regardless of the LMCR–experimental design combination, kriging method or whether the response or contrast was being predicted, the mean values across all realisations

Discussion and conclusions

As Papritz (2008) stated it would be strange to assume equal means when using ECK to estimate the responses to different treatments as in an experiments we are actually attempting to ascertain whether there is a difference between treatments. We did not assume this and used the samples means when using the ECK estimator. The theoretical kriging variances indicate that the variances should decrease in the order SCK > DCK > ECK (Papritz, 2008). Contrary to this our simulation results showed that the

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