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1 Geostatistics and Precision Agriculture

The previous chapter describes how precision agriculture can be used to improve farm management to achieve economic and environmental benefits. Short-range differences in soil attributes mean that spatially differentiated management can create economic or environmental benefits. Effective precision agriculture requires accurate soil mapping at subfield scales so that management practices can be modified. Improvements in farming technology, for instance, GPS-controlled farm equipment, decrease the difficulty and cost associated with spatially differentiated management. This improves the ease of implementation and makes high-resolution soil maps more valuable.

A key challenge for geostatistics in precision agriculture is the detection of soil variability at important subfield scales and the systematic incorporation of this variability into accurate field scale soil maps.

2 Soil Survey, the Variogram and Kriging

Soil attributes are typically difficult and expensive to observe. As a result, soil attribute maps made for the purpose of precision agriculture are typically created from point observations which represent a small proportion of the area to be mapped. Estimates of soil attributes in unknown areas are based on the observations and an expectation of the regions between them. This process of predicting attributes in unobserved areas is known as kriging (explained in more detail in Chap. 10). Assumptions about spatial variability are typically derived from the variogram, which links spatial separation distance to expectations about variability. Chapters 10 and 11 explain the variogram, its uses and the different methods of calculating the variogram in more detail.

The variogram is sometimes called the ‘cornerstone of geostatistics’. Accurate estimation of the variogram is critical to the production of accurate soil maps. Because of the hidden nature of most soil attributes, we can usually only directly observe a small proportion of an area of interest. In Chap. 10, the distinction between the experimental or empirical variogram and the model variogram is described in some detail. The empirical variogram plots the average variance against separation distance for a number of distinct lags. The model variogram uses the information from the empirical variogram to estimate the expected variability at all lags. The purpose of the model variogram is to estimate the true underlying spatial variation at a level of detail that allows useful predictions.

Interpolation of results into unobserved points depends on the spatial structure estimated by the variogram. The closer the estimated variogram to the underlying spatial structure, the more accurate the subsequent interpolation. In general, a variogram computed from samples with finer spacing and more observations will estimate the underlying spatial structure more accurately than a variogram computed from samples with coarser spacing. Finer spacing allows the detection of spatial structure across more scales. The extent to which this is true will depend on the interaction of the spacing with the underlying spatial structure. For example, if there is no spatial relationship between points more than 5 m apart, then decreasing spacing from 50 m of separation distance to 10 m will not improve the variogram. We pause here to explain some key components of the variogram and how they are affected by survey design.

3 Key Components of the Variogram

While the variogram can take many forms (see Chap. 10, Sect. 10.1.1, for detail on some commonly used models), there are three components which are typically considered the most important indicators of spatial structure. Shown in the stylised diagram in Fig. 21.1 and subsequently described, these components are the nugget, the sill and the range. The estimation of each of these parameters depends strongly on the sampling design.

Fig. 21.1
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Stylised diagram, showing the three most important indicators of spatial structure, the nugget, the sill and the range

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3.1 Nugget

In principle the nugget effect captures nonspatial variation: measurement error; random variation. However, in practice the nugget will also capture spatial variation that occurs at scales less than the smallest sampling interval. If the sampling interval is wider than an important scale of spatial variation, then this will increase the nugget. Aliquotting (pooling of samples) will decrease the nugget. A larger support (area over which the sample is taken) will also decrease the nugget.

3.2 Total Sill

The total sill is defined as the maximum variability that can be expected for a particular soil property. Beyond a certain separation distance (the range, see below), the expected variability will not increase past the value of the total sill (or in some cases will increase only very slowly and slightly past the sill). Only bounded variogram models have a sill. The total sill is more likely to be affected by the number of samples and the wider separation distance than the minimum sampling spacing. When modelling spatial variability at a field scale, it is unlikely that the maximum variability of the soil will be reached. However, unbounded models are rarely fit. In the context of precision agriculture, we can think of the total sill as being the maximum variability within this particular context or a local maximum. If we extended the sampling to a regional level, it is likely we would reach another magnitude of variability.

3.3 Partial Sill

The distance between the nugget and the sill is known as the partial sill. The partial sill is the component of variation that can be spatially attributed. In Fig. 21.1, the semivariance increases linearly with distance at short separation distances. As the separation distance increases, the rate of increase in the semivariance decreases, before eventually reaching a plateau as the semivariance reaches the total sill. This pattern is commonly found in variogram models (see Chap. 10 for more detail on the functional forms). Accurately determining the change in variability between the nugget and the sill requires sufficiently dense sampling at appropriate scales of variability.

3.4 Range

Like the lag and the sill, the range estimated by the theoretical variogram will be strongly affected by the sampling design. As the lag increases, the confidence intervals widen (Oliver and Webster 2014) which can make it difficult to accurately fit a variogram at lags approaching the full extent. Precision agriculture surveys are typically conducted over areas from a few hectares to a few hundred hectares. This precludes them from capturing landscape or continental scale variability. Despite this, the majority of variogram fits from precision agriculture studies are bounded. The modelled range and sill can be thought of as the ‘local sill’ associated with the ‘local range’ associated with the particular extent and spacing of that survey. It is very likely that if the extent of the survey was increased, another degree of variability would be found. It is critical to consider the potential effect of both the extent and the spacing on the range when considering precision agriculture studies and how their findings may inform your own work.

4 Soil Survey Design: Capturing Spatial Variability

The task of ensuring that a soil survey captures the necessary scales of variability is not a trivial one. The expense of a soil survey, and the commonly destructive nature of soil sampling, creates a pressure to reduce the number of sampling points as much as possible. However, if sampling is insufficient or too sparse (relative to the underlying spatial structure), it will compromise the variogram and thus the accuracy of the maps which are calculated from it. It is noted in Chap. 11 that one of the chief difficulties associated with the design of a soil surveyfor estimating a variogram is that the underlying spatial structure is unknown at the time the survey is designed. If the underlying spatial structure of a soil attribute is known, we can design a soil survey sufficiently to create a map at a particular level of detail.

Where budget allows it, the best practice is to undertake a preliminary soil survey to estimate important scales of variation before a more comprehensive soil survey is designed. This survey should be nested in its design to increase the chances of capturing important scales of variability (Pettitt and McBratney 1993; Webster and Oliver 2001). However, a preliminary survey will rarely be economically feasible. If it is not possible to conduct a preliminary survey, a soil survey design is likely to benefit from the consultation of alternative sources of information about soil variability, such as existing literature or covariates.

Budget or practical pressures may be sufficient to impede the collection of sufficient data to reliably calculate an empirical variogram. A stable variogram calculated from classical geostatistical methods (as described in Chap. 10) requires around 100 observations. More modern methods (as described in Chap. 11) typically require around 50. It is not possible to estimate spatial variability at distances less than the minimum separation distance. In cases where there are few or widely spread soil observations, alternative sources of information may be useful for the calculation of a variogram for kriging.

The expense of gathering soil observations creates a need for cheaper sources of information about soil variability, either to assist in the planning of a soil survey or to use in the process of kriging itself. Many authors have identified sources and strategies for the production of this information.

5 Variograms from the Precision Agriculture Literature

Variograms calculated for the same soil attribute may be a useful source of information. However, variograms calculated for the same property can vary significantly for a number of reasons that should be carefully considered. As outlined in Chap. 12, parent material, soil type, land use and climate will all have an effect on soil spatial variability. Consideration of these factors will be important in the selection of a variogram. Unfortunately, knowledge about soil spatial variability does not extend to the quantification of which of these factors are most important for determining spatial variability of different soil attributes.

The underlying variability of soil attributes might be different for the reasons mentioned above. In addition, the methods used to detect soil variability might create differences in the shape of the variogram. Different projects may focus on detecting variability at different scales for management or budget reasons. Even if the soil type is similar between two studies, the spatial variability might not be measured in a way that provides useful information.

It is also very important that the statistical methods used are assessed critically, before results are used or duplicated. There are a number of variograms included in Tables 21.1, 21.2, 21.3, 21.4, 21.5, 21.6, 21.7 and 21.8

Table 21.1 Compilation of key properties for clay variograms
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Table 21.1 (continued)
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Table 3 Table 21.1
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Table 21.2 Compilation of key properties for sand variograms
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Table 5 Table 21.2
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Table 21.3 Compilation of key properties for pH variograms
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Table 7 Table 21.3
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Table 21.3 (continued)
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Table 21.4 Compilation of key properties for carbon variograms
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Table 21.4 (continued)
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Table 11 Table 21.4
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Table 21.4 (continued)
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Table 21.5 Compilation of key properties for available nitrogen variograms
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Table 21.5 (continued)
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Table 21.6 Compilation of key properties for total nitrogen variograms
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Table 21.7 Compilation of key properties for phosphorus variograms
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Table 17 Table 21.7
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Table 21.7 (continued)
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Table 21.8 Compilation of key properties for potassium variograms
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Table 21.8 (continued)
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Table 21.8
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Table 21.8 (continued)
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which appear to use insufficient sample numbers for variogram estimation. There are several variograms which appear to assign spatial structure at a magnitude that appears to be meaningless relative to the units (i.e. total sill of less than 1% for the soil texture fraction). Footnote 1 These results have been included for completeness, but we wish to draw the readers’ attention to the fact that some of the variograms included in this collection may have issues associated with them.

We describe below some key trends we have noted in the compilation of soil variograms and include a graphical summary of the variograms we have compiled. Variograms from McBratney and Pringle (1999) are included as well as those we have collected from the intervening period. Summary details and references for each variogram are given in Tables 21.1, 21.2, 21.3, 21.4, 21.5, 21.6, 21.7 and 21.8. We encourage the reader to consult each source directly for more detail about the sampling design and process.

Variograms for the same soil attribute from existing literature can be a useful source of information about field scale soil. It is important to exercise caution when consulting this literature, as variograms have been created from different soil types, for different purposes and possibly with important methodological limitations.

5.1 Field Scale Soil Variograms: Key Trends

5.1.1 Variogram Forms

Across all soil properties, a number of functional forms were fit to variograms. For each soil property, at least one study found no spatial structure (i.e. pure nugget) to be the best fit. This suggests that either the spatial structure occurs at scales finer than those surveyed or that the variability in the property in question is less than the experimental error. Spherical and exponential variograms were commonly used. Some papers described the functional form as ‘experimental’. These models were fit with an exponential model.

5.1.2 Total Sill

Between studies the total sill (nugget plus partial sill) changes by several orders of magnitude for each property. As expected, this variability is the least pronounced for bounded properties (pH, OM %, sand % and clay %) and much more pronounced for micronutrients, which vary by around three to five orders of magnitude.

Variability in total sill is similar to that observed by McBratney and Pringle (1999). Inspection of the summary tables (Tables 21.1, 21.2, 21.3, 21.4, 21.5, 21.6, 21.7 and 21.8) indicates that for the majority of soil properties, the range in the total sill is similar for the variograms collected by McBratney and Pringle (1999) and the more recently collected properties. The maximum variability reached within 1 km has remained within an order of magnitude for all properties. For soil pH, organic carbon and potassium, the maximum variability found in the more recent literature search is two to three times greater than the maximum variability found in the literature reviewed by McBratney and Pringle (1999). The other properties have a very similar maximum.

5.1.3 Nugget

Like McBratney and Pringle (1999), we find wide variability (several orders of magnitude) in the nugget parameter. McBratney and Pringle (1999) suggest that this variability is largely due to the strong effect of sampling design on the nugget. A variogram can only model the spatial structure that is detectable by the sampling design. In general, the wider the spacing, the more spatial variability will be attributed to the nugget component of the model. If the survey spacing is wider than the spatial structure, the variogram will appear as a pure nugget model. A wider support and the use of aliquotting will reduce the ‘noise’ in the data and decrease the nugget. It has been often proposed, and is quite likely, that the majority of soil properties would have more than one layer of soil structure. The differences in estimated nugget likely reflect both the underlying differences in spatial structure at the field scale and the capacity of different survey designs to capture this variability.

5.1.4 Range

Like the lag and the sill, the range estimated by the theoretical variogram will be strongly affected by the sampling design. As the lag increases, the confidence interval around the variance increases (Oliver and Webster 2014). Precision agriculture surveys are typically conducted over areas from a few hectares to a few hundred hectares which limits the extent of spatial variability they can capture. Despite this, the majority of variograms fit from precision agriculture studies are bounded. The modelled range and sill can be thought of as the ‘local sill’ and ‘local range’ associated with the particular extent and spacing of that survey. It is very likely that if the extent of the survey was increased, another degree of variability would be found. It is critical to consider the potential effect of both the extent and the spacing on the range when considering precision agriculture studies and how their findings may inform your own work.

5.2 Field Scale Soil Variograms: Methodological Differences

5.2.1 Survey Design

Another substantial difference between the survey designs was whether or not aliquott ing was used. This is particularly significant when comparing these spatial studies because some studies model the range at distances that other studies were combining soil at. This is typically done for samples taken within 1 m of each other. This practice is likely to reduce the nugget (or white noise) and also to reduce any short-term spatial trends which might be occurring.

There is significant variation in the survey design which may influence the mapping of spatial variability. Nested designs are better placed to capture spatial trends across a variety of scales than designs with even spacing; however, because of the additional costs associated with these, they are less common.

5.2.2 Model Fitting Process

Oliver and Webster (2014) wrote an explanatory piece of work, describing the best methods for soil scientists to model variograms for kriging. They also described common mistakes made by soil scientists when calculating variograms. The majority of papers we assess do not follow all of Oliver and Webster’s recommendations for reporting methods. This makes it difficult to assess how well a fitted variogram captures underlying spatial variability. Few papers report summary statistics for a variety of models, and few papers present variogram clouds to illustrate the utility of the fit. This does not necessarily mean that the fitted models are not accurate, but it does make it difficult to assess the model.

Another point worth considering is the possibility that trends are being overfitted. Perhaps some of the models presented in this chapter would have been better represented by a nugget. These issues around model quality are not new, but a degree of caution is required when interpreting the results.

The more recent literature has included studies which have found much lower values for the total sill for several soil attributes. For clay the lowest value found for the total sill is two orders of magnitude lower than the lowest value reported by Petit and McBratney. Sand is one order of magnitude lower. Some modelled variograms occur over a very tiny range of variability relative to the magnitude of the property they are measuring. Whether it is necessary or feasible to model a spatial structure of less than 1% for soil texture properties is questionable.

5.2.3 Measurement Methods

The properties we have included here are commonly measured soil properties with known agronomic implications. However, measurement of these properties is rarely simple or consistent. Differences in measurement methods and differences in which component of the property is being measured will influence both the shape and magnitude of the variogram.

pH is an extremely commonly measured property. However, within the studies we have assessed, there are differences in the solution, the dilution rate and the equipment used to measure the pH. This problem becomes more complex when considering more difficult-to-measure properties such as potassium and phosphorus. Different studies have used different extractants to target different fractions for these nutrients.

The variogram is affected by the distribution of the property it is being calculated for. The variograms calculated for potassium and phosphorus show the greatest differences in total sill. We expect that this is because the target of the measurements varies as well as the measurement method used.

Some articles were found which estimated total carbon or inorganic carbon. However, there were relatively few such studies, so we have not included them here. We have included studies which measured organic matter as a proxy for organic carbon. We converted these using the van Bemmelen factor (1.724). Pribyl (2010) illustrates that an accurate conversion factor for different soils can vary from 1.4 to 2.5. Error in the conversion will be small relative to the overall spread of the variograms.

5.3 Field Scale Soil Variograms: A Compilation

Figures 21.2, 21.3, 21.4, 21.5, 21.6, 21.7, 21.8. and 21.9. provide a visual compilation of field scale variograms for each of the soil properties initially examined by McBratney and Pringle (1999). We include both the original variograms used by McBratney and Pringle and variograms published since then. We only include variograms which were calculated from untransformed data and which were based on physical observations (i.e. not from remotely observed data). The black bold lines represent the average variogram (Sect. 21.7). Tables 21.1, 21.2, 21.3, 21.4, 21.5, 21.6, 21.7 and 21.8 correspond to each figure and include reference details and key parameters for each variogram. Because of the wide range of values of the source variograms, the scales used in the figures cannot include all of them, and some low sill variograms have not been included. We include the details in the tables for completeness, but suspect that they are unlikely to provide useful information. The figures affected and the number of source variograms not included are pH 1, carbon 3, total nitrogen 1 and potassium 2.

Fig. 21.2
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Compilation of field scale variograms for clay. The bold black line represents the average variogram. Summary details and references for each variogram (a–ag) are given in Table 21.1

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Fig. 21.3
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Compilation of field scale variograms for sand. The bold black line represents the average variogram. Summary details and references for each variogram (a–r) are given in Table 21.2

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Fig. 21.4
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Compilation of field scale variograms for pH. The bold black line represents the average variogram. Summary details and references for each variogram (a–ag) are given in Table 21.3

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Fig. 21.5
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Compilation of field scale variograms for carbon The bold black line represents the average variogram. Summary details and references for each variogram (a–al) are given in Table 21.4

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Fig. 21.6
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Compilation of field scale variograms for available nitrogen. The bold black line represents the average variogram. Summary details and references for each variogram (a–m) are given in Table 21.5

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Fig. 21.7
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Compilation of field scale variograms for total nitrogen. The bold black line represents the average variogram. Summary details and references for each variogram (a–h) are given in Table 21.6

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Fig. 21.8.
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Compilation of field scale variograms for phosphorus. The bold black line represents the average variogram. Summary details and references for each variogram (a–ab) are given in Table 21.7

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Fig. 21.9.
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Compilation of field scale variograms for potassium. The bold black line represents the average variogram. Summary details and references for each variogram (a–ad) are given in Table 21.8

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6 Estimation of the Variogram from Proportional Variograms

McBratney and Pringle (1999) observe a strong relationship between mean squared and variance for several soil properties and develop a method for estimating a ‘proportional variogram’. This method has the advantage of capturing the much higher levels of variability that tend to occur when the mean values of the property are extreme. Similar to McBratney and Pringle (1999), we find that some soil properties (phosphorus, nitrogen, potassium and carbon) appear to have a strong linear relationship between the mean and the standard deviation. This could imply that the calculation of a proportional variogram would be useful for these properties. However, closer interrogation reveals that this relationship is largely driven by high leverage points. It is not possible to fit a robust curve to link the mean and standard deviation. Proportional variograms are based on the relationship between the mean and the variance. Because we cannot be confident about this relationship, it is not prudent to calculate proportional variograms.

We advise against the use of proportional variograms as an estimate for variability and as such do not update McBratney and Pringle’s (1999) estimates of proportional variograms.

7 Estimation of the Variogram from Average Variograms

McBratney and Pringle (1999) calculate average variograms for seven soil properties. These average variograms are calculated from the sample of variograms they compile. The fourth root transform of each approximated by the spherical model is taken and then backtransformed. Exponential or spherical models are fitted.

McBratney and Pringle (1999) note the broad spread of variability in the variograms they collected and suggest that this might make the ‘average’ variograms less useful. Despite this, they suggest that the average variogram is a useful starting point where no other information is available. Kerry and Oliver (2003) find evidence that average variograms can be useful for prediction when parent material and soil forming factors are similar, but emphasise that they do not expect a global average variogram to provide much useful information.

We do not believe a global average provides useful information for predictive purposes. However, as McBratney and Pringle (1999) suggested, the average variogram does provide a useful reference for those interested in soil variability. We produce average variograms for illustrative purposes (by averaging the fourth root transform of each variogram at finely spaced intervals and then plotting the backtransformed values), but we do not fit these with a functional form.

As suggested by McBratney and Pringle (1999) and illustrated by Kerry and Oliver (2004), the concept of the average variogram has the most use for prediction when it is calculated from a select subset of existing variograms. The diversity of climate regimes, soil types and land use for which field scale variograms have been calculated means that discretion is essential in the selection process. Differences in sampling regime, soil measurement protocols and geostatistical methods add another layer of complexity that needs to be navigated in appropriate selection. We discuss these issues further in the next section.

We advise against the use of the ‘global average’ variogram as an estimate of local soil variability. Instead, where suitable variograms are available, an average of variograms with similar conditions is taken. Discretion and expert knowledge will need to be used in this selection process. The process outlined by McBratney and Pringle (1999) for calculation of an average variogram can be followed.

8 Ancillary Information

Kerry and Oliver have written several papers investigating the potential of using cheaper more densely available ancillary information to supplement expensive and sometimes sparse soil survey data. In 2004, they compared the spatial structure of a number of ancillary data sources to the spatial structure of a number of fixed soil properties. They found that variograms calculated from aerial colour photographs of 3.4 m ‘sampling density’ can estimate range with sufficient accuracy to helpfully guide sampling density of the soil survey. Kerry and Oliver (2008) paper extends the use of ancillary information to kriging. They suggest that the primary requirement for using data is that the ancillary variogram has a similar sill-to-nugget ratio to the property being studied.

Where it is available, ancillary information can be a useful source of information. Care must be taken in the selection of appropriate ancillary information. It may be useful to consult a range of ancillary variables.

9 Expert Knowledge

Truong et al. (2013) propose the use of expert knowledge as a means to estimate the variogram when there are not enough observations to calculate a reliable variogram using geostatistics. They point to an increasing realisation from other disciplines that experts’ knowledge provides a useful source of information that can be incorporated into statistical models. Truong et al. (2013) also suggest that expert knowledge may be useful when there is no data available or even when the available data for some reason are unreliable or unsuitable.

Truong et al. (2013) propose a strict methodology for eliciting knowledge from experts in order to construct a variogram. Their methods are designed to avoid bias. This process is still in the prototype stage. Currently, those seeking to supplement data with expert knowledge will not be able to avoid some bias. However, in many cases, subjective expert knowledge may be the best available option.

Even when there are data available, a degree of subjectivity will be required to assess the usefulness and representativeness of these data. Where possible, it will obviously be preferred that these subjective decisions are informed by those with expertise in the area of interest (geographical or topical).

We strongly encourage the use of expert knowledge in the selection of appropriate datasets for the modelling of spatial variability. Where datasets are unavailable or deemed inappropriate, it may be necessary to rely entirely on expert knowledge to estimate the variogram. Eventually, it may be possible to elicit expert knowledge using a formal process such as that described by Truong et al. (2013).

10 Quick Variograms

There may be situations when ancillary information, variograms from literature or even expert knowledge are unavailable or unreliable. In these cases, we would like to propose the following sampling approach that can be used to estimate a rough variogram at very low cost. The method proposed will necessarily be imprecise, but is a better alternative than not having any information. We anticipate that this method would be particularly useful in cases where alternative sources of information are available but unreliable.

We suggest sampling in eight locations (four widely spaced points, each with a closely spaced pair as per the diagram shown in Fig. 21.10). Obviously, the sampling design will be affected by the shape of the field. We advise that sampling occurs as close to the boundaries as possible while avoiding the edge effect.

Fig. 21.10
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Sampling approach for estimating a ‘rough’ variogram. Here, it is shown that sampling is recommended in eight locations (four widely spaced points, each with a closely spaced pair, i.e. A-B, C-D, E-F and G-H)

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In Fig. 21.10, one can calculate four bin sizes.

  • Close spacing (proxy for nugget): four pairs A-B, C-D, E-F, G-H

  • Maximum spacing (proxy for sill): eight pairs (A-G, A-H, B-G, B-H, C-E, C-F, D-E, D-F)

  • Intermediate spacing 1: eight pairs A-C, A-D, B-C, B-D, E-G, E-H, F-G, F-H

  • Intermediate spacing 2: eight pairs A-E, A-F, B-E, B-F, C-G, C-H, D-G, D-H

The close spacing (the close pairs) can be used as a proxy for the nugget, and the maximum spacing (the diagonals) can be used as a proxy for the total sill.

  • If the nugget and the sill are similar, we can assume that the appropriate model is the nugget model.

  • If the nugget and the sill are different, we will need to estimate the range and select a model for the variability.

There is no obvious proxy for the range that can be calculated from a small number of data points.

The two intermediate spacings may be useful to indicate where the range should occur. If they are similar to the total sill, then the range should be less than the intermediate spacings. If they are smaller than the total sill, then the range should be greater than the intermediate spacings.

We suggest that the intermediate spacings be used to determine the limits of where the range could occur. The range should then be taken as the halfway point between the limits.

For example, if the smallest intermediate bin has a variance similar to the sill, then we should set the range to be equal to the halfway point between the nugget and the smaller intermediate bin. If both of the intermediate bins have variances smaller than the sill, we should set the range to be equal to halfway between the total sill and the intermediate bin.

Modelling a range larger than the maximum separation distance is always unlikely in variograms, because of the tendency of models to break down beyond half of the field extent. We do not think that unbounded models (i.e. ranges greater than the maximum separation distance) are necessary to consider here.

Quick variograms are not able to provide a precise estimation of soil variability. Their primary and important advantage is that they are calculated from data for the property of interest in the location of interest. Quick variograms provide a useful source of information when either (i) there are no other sources of information available or (ii) for checking alternative sources of information against a reference point.

11 Recommendations

  • Where it is economically and practically feasible, the best practice for estimating variograms is to conduct a preliminary survey and then a comprehensive field survey at the spatial scales of interest.

  • Where resources for field survey are limited, it is desirable to find a variogram for the same property which has been calculated for a similar soil type. This variogram may be useful as a source of information for survey design. It may also be possible to use this variogram for kriging. Due to the large variability in soil variograms calculated for each soil property, it is critical to use discretion in this selection process.

  • If more than one existing variogram from a similar soil type is identified, the information from both should be used. An averaging process may be a useful way to combine this information. Alternatively, they could be used separately to provide a range of predictions.

  • Ancillary information (such as that from aerial photographs) should be considered to estimate variograms for soil properties. It can be used for survey planning and for kriging. Care must be taken to ensure that these variograms have a similar nugget-to-sill ratio to the property of interest. If available, it is preferred to use information from soil survey over ancillary information.

  • If it is not possible to find a variogram for a similar area, other options are available. For example, expert knowledge could be consulted. Consultation of expert knowledge may occur in a formal process-based manner or in a more informal way.

  • Quick field surveys, with limited sampling, may provide a cost-effective way to estimate rough variograms where other information is unavailable. These methods may also be useful when it is desirable to supplement information with observations directly from the field.

  • Variograms are useful for designing detailed sampling surveys for agricultural and environmental purposes.