Quality measures for soil surveys by lognormal kriging

Abstract

If we know the variogram of a random variable then we can compute the prediction error variances (kriging variances) for kriged estimates of the variable at unsampled sites from sampling grids of different design and density. In this way the kriging variance is a useful pre-survey measure of the quality of statistical predictions, which can be used to design sampling schemes to achieve target quality requirements at minimal cost. However, many soil properties are lognormally distributed, and must be transformed to logarithms before geostatistical analysis. The predicted values on the log scale are then back-transformed. It is possible to compute the prediction error variance for a prediction by this lognormal kriging procedure. However, it does not depend only on the variogram of the variable and the sampling configuration, but also on the conditional mean of the prediction. We therefore cannot use the kriging variance directly as a pre-survey measure of quality for geostatistical surveys of lognormal variables. In this paper we present an alternative. First we show how the limits of a prediction interval for a variable predicted by lognormal kriging can be expressed as dimensionless quantities, proportions of the unknown median of the conditional distribution. This scaled prediction interval can be used as a presurvey quality measure since it depends only on the sampling configuration and the variogram of the log-transformed variable. Second, we show how a similar scaled prediction interval can be computed for the median value of a lognormal variable across a block, in the case of block kriging. This approach is then illustrated using variograms of lognormally distributed data on concentration of elements in the soils of a part of eastern England.

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“…In the literature, there are several theoretical studies of LOK such as Journel and Huijbregts [20] and Dowd [25], and some useful application of LOK in geology such as Gilbert and Simpson [26], Mc Grath and Zhangb [27], Paul and Cressie [28] and Lark and Lapworth [29].…”

Optimization of the sampling scheme for maps of physical and chemical properties estimated by kriging

“…It is important to observe that both scale and non-scale variograms have the same kriging values (Vieira et al, 1997). Therefore, the kriging variance can be used as a tool to evaluate the quality of sampling schemes (Groenigen, 2000;Lark & Lapworth, 2012).…”

The offset correlation, a novel quality measure for planning geochemical surveys of the soil by kriging

“…Di et al, 1989;Ruffo et al, 2005). The methodology 54 has been extended to cover prediction by cokriging (McBratney and Webster, 1983), cases 55 where the mean is not assumed to be stationary (Brus and Heuvelink, 2007) and where 56 the variable is log-normal (Lark and Lapworth, 2012) and to account for uncertainty in 57 the variogram model Lark, 2006, 2007;Zhu and Stein, 2006).…”