Pedometry

1.1. Why did the need for pedometry arise?

Traditional soil science, despite all its achievements, encountered a fundamental limitation: the soil cover, like any natural system, exhibits enormous spatial and temporal variability. As early as 1941, the founder of modern genetic soil science, Hans Jenny, formulated his famous equation (Jenny, 1941):

where

• S – soil properties,
• cl – climate,
• o – organisms,
• r – relief,
• p – parent material,
• t – time.

This equation shows that soil is a function of multiple factors, each of which varies in space. Consequently, the soil itself cannot be uniform.

Nevertheless, for decades, soil studies – especially applied ones – often ignored this fundamental heterogeneity. The reasons were both practical (limited resources for sampling and analysis) and conceptual (the habit of thinking in terms of soil “types” and “classes”).

The situation began to change in the 1960s–70s, when three developments came to the fore:

1. Geostatistics – methods originally developed in mining for estimating mineral reserves proved ideally suited for analysing spatial variability in soils (Matheron, 1963). Key works by Burgess and Webster (1980) showed that soil properties exhibit spatial autocorrelation – that is, nearby points are more similar to each other than distant ones.

2. Information technology – the advent of Geographic Information Systems (GIS), Global Positioning Systems (GPS), and remote sensing methods provided the technical foundation for collecting and processing vast amounts of spatially referenced data.

3. Precision agriculture – the practical needs of agriculture for differentiated resource management required detailed spatial information about soils at the individual field level (Wollenhaupt et al., 1994).

1.2. Defining characteristics of pedometry

Thus, pedometry can be defined through the following key features:

First, it is a quantitative approach. Pedometry deals with numbers, not qualitative descriptions. Soil carbon content is expressed as a percentage or grams per kilogram, not as “high” or “low”. Boundaries between soil classes are determined statistically, not intuitively.

Second, it is a spatially‑oriented approach. Every measurement has precise coordinates. This allows analysis not only of the values of properties but also of their relative positions, which fundamentally changes the logic of data interpretation.

Third, it is a probabilistic approach. Pedometry never gives a single “correct” value at a point – it provides an estimate and a measure of its uncertainty. This is crucial for making informed decisions.

1.3. The place of pedometry in the system of soil sciences

Pedometry does not abolish or replace classical soil science. Rather, it is its instrumental extension and development (McBratney et al., 2000). Within the system of soil sciences, pedometry lies at the intersection of several disciplines:

• Soil physics – provides methods for measuring and modelling physical properties.
• Soil chemistry – gives an understanding of chemical processes that determine soil properties.
• Soil biology – reveals the role of living organisms in forming soil properties.
• Mathematical statistics – provides tools for data analysis and uncertainty assessment.
• Geoinformatics – supplies tools for working with spatial data.

The key difference between pedometry and traditional approaches lies in the recognition that the soil cover represents a continuous spatial process, not a mosaic of discrete objects (White, 2006). This means that between any two points in a field there exists a gradient of properties that can be studied and modelled.

It is precisely this fundamental property of soil that makes it meaningless to try to characterize an entire field with a single sample – and it is precisely to deal with this continuous variability that pedometry was created. In the following sections, we will examine how spatial heterogeneity manifests itself, how to measure it, and how to use the obtained information for agronomic decision‑making.

2.1. Soil as a continuous medium

Unlike discrete objects such as individual trees or stones, the soil cover forms a continuum – a continuous medium where properties change gradually from point to point. Between two adjacent points in a field there is no sharp boundary: humus content, particle‑size distribution, and acidity change smoothly, though at different rates.

This property was noted by the founders of soil science. However, for a long time soil scientists sought to simplify this continuity by creating classifications and distinguishing soil “types” – as if soil were akin to plant or animal species. In practice, the boundaries between soil types in the field are almost always blurred, and within one “type” properties can vary several‑fold (White, 2006).

Pedometry starts from the premise that soil is a regionalized variable. The term, introduced by Georges Matheron, means that the value of a property at a given point depends on the values at neighbouring points, but this dependence weakens with distance (Wendroth et al., 2012). It is precisely this property – spatial autocorrelation – that makes interpolation and mapping of soil properties possible, and its absence would render any predictions meaningless.

2.2. Causes of spatial heterogeneity

Why is soil so variable? The answer is provided by the same Jenny equation (Jenny, 1941):

Each of the soil‑forming factors is itself spatially heterogeneous:

• Climate (cl) – at the macroscale determines zonal soil types, but even within one field it can vary due to microclimatic effects (e.g., northern slopes are cooler than southern ones, and depressions accumulate cold air) (Weil, 2017).

• Relief (r) – perhaps the main source of variability at the local level. A change in elevation of just a few metres affects moisture regime, erosion processes, and the redistribution of fine particles and organic matter. This is why catenas – sequences of soils along a slope – are so typical of many landscapes (Weil, 2017).

• Parent material (p) – can change even within a single field, especially in areas with glacial deposits or at contacts between different geological formations. Figure 19.1 in Weil (2017) shows how even within 50 metres, soils on sandy and clayey deposits can differ dramatically.

• Organisms (o) – including vegetation, soil fauna and microorganisms, as well as humans. Even a single tree can create a unique “micro‑soil halo” around itself through litter, root exudates, and microclimate modification. Human activity – fertilization, liming, tillage – creates patchiness that can persist for decades (Weil, 2017).

• Time (t) – soils of different ages on the same territory will have different degrees of development. For example, fresh alluvial deposits in a river floodplain are young soils (Entisols), while on a neighbouring terrace more developed soils (Alfisols or Ultisols) may occur.

Thus, soil heterogeneity is not accidental or an artefact of measurements, but a regular consequence of the diversity of soil‑forming factors (White, 2006).

2.3. Scales of heterogeneity

Spatial variability of soils manifests at all levels. Three main scales are usually distinguished (Weil, 2017; White, 2006):

Micro‑ and small‑scale variability (millimetres to metres) – associated with local effects: worm and root channels, structural units, individual aggregates, patches of organic matter. This variability often accounts for a significant portion of total variance, but in practice it is difficult to account for in agronomic decisions. It appears as the so‑called “nugget effect” (discussed in the section on the variogram). An illustrative example: infiltration measurements using small (5–10 cm diameter) and large (30–50 cm) rings give completely different estimates of variability – because small rings may hit or miss a macropore, while large rings always include several macropores (see Fig. 19.2 in Weil, 2017).

Mesoscale variability (metres to hundreds of metres) – covers a single field, a group of fields, or a small catchment. The main factor here is relief. It is at this level that agronomists and farmers work. This is where catenas are observed: on the hilltop – well‑drained, more acidic soil with a thin humus horizon; on the slope – eroded, lighter in colour; at the foot – with signs of gleying, darker, with higher organic matter content due to material washed down from above (Weil, 2017). This is the variability that precision agriculture attempts to account for.

Large‑scale variability (kilometres to continents) – associated with climatic zones, large geological structures, and vegetation formations. At this level, soil zones are distinguished (Arctic, taiga, chernozem, desert, etc.). For agronomy, this is important for choosing adapted varieties, irrigation, and farming systems, but at the field level this variability is less critical because it is “averaged” within a single farm.

2.4. Structured and unstructured variability

From the perspective of pedometry, it is important to distinguish two types of variability (Wendroth et al., 2012):

• Structured (systematic) – when properties change regularly in space, for example from the hilltop to the foot. This change can be described by a mathematical function (trend). Such variability is predictable and can be modelled.

• Unstructured (random) – when neighbouring points may differ greatly without apparent regularity. This may be due to microheterogeneity, measurement errors, or random factors. Such variability cannot be predicted from neighbouring points – it appears as “noise”.

A real field is always a mixture of structured and random variability. The task of pedometry is to separate one from the other and to estimate the proportion of each component. Only then can we determine whether interpolation is worthwhile and how often samples should be taken.

2.5. Why does one sample not work?

Now we can clearly answer the key question of this lecture. A single composite sample from the entire field gives only the arithmetic mean value of a property. But this mean hides the enormous variability within the field. Imagine taking the arithmetic mean of the numbers 1 and 100 – we get 50.5, but this number does not characterize either the first or the second value. Likewise, an averaged sample from a field where pH is 5.0 in one part and 7.5 in another will give a pH of about 6.0–6.5, which does not correspond to the real need of any part.

Moreover, even if the field appears visually uniform (colour, relief, vegetation), its soil properties can differ several‑fold – and these differences will determine yield, fertilizer efficiency, and the need for reclamation (Weil, 2017; White, 2006). Therefore, agronomic recommendations based on a single averaged sample will be suboptimal for most points in the field: somewhere we apply excess fertilizer (leading to losses and pollution), somewhere insufficient (yield loss).

Pedometry offers a different approach: measure properties at different points, identify patterns of their spatial distribution, and use these patterns for differentiated management. But first, we need to understand how the distance between points is related to the similarity of their properties. This is the subject of the next section – on spatial autocorrelation.

3.1. What is scale in soil research

In traditional cartography, scale is defined as the ratio of length on the map to the corresponding length on the ground. However, in digital soil mapping and pedometry, this term is increasingly replaced by the concepts of “resolution” and “spacing” (McBratney et al., 2003; White, 2006).

• Resolution – the size of the smallest element (pixel or cell) for which a property value is determined. For example, a resolution of 30 metres means that one cell covers an area of 30 × 30 m.
• Spacing – the distance between neighbouring observation points. In a regular grid, spacing equals the distance between grid nodes.
• Extent – the total area covered by the study.

These three parameters – resolution, spacing, and extent – determine which scales of variability can be detected in a study and which remain “hidden”. This is fundamentally important: if we survey a field with a spacing of 50 metres, we will never know how properties change over 5 metres. And if we study only one field, we cannot say anything about regional patterns.

3.2. Levels of scale in soil research

In soil science, five main scale levels are commonly distinguished (McBratney et al., 2003; Weil, 2017) (see also Table 37.1 in McBratney et al., 2003):

Level D1 – local (less than 20 m)

• Resolution: less than 5 × 5 m
• Extent: up to 50 × 50 km
• Corresponds to scale 1:5,000 and larger
• Examples: precision agriculture, horticulture, nurseries, experimental plots. Microrelief, local patches, and the influence of individual plants are studied here.

Level D2 – detailed (20–200 m)

• Resolution: 5–20 m
• Extent: 500 m – 200 km
• Corresponds to scale 1:5,000 – 1:20,000
• Examples: detailed soil surveys of farms, irrigation projects, land reclamation. Catenas and changes in soils according to relief elements become visible.

Level D3 – medium (cadastral) (200 m – 2 km)

• Resolution: 20–200 m
• Extent: 2–2000 km
• Corresponds to scale 1:20,000 – 1:200,000
• This is the main level for soil surveys at scales 1:50,000 – 1:100,000. Soil associations and large landscape units are distinguished here.

Level D4 – regional (2–20 km)

• Resolution: 200 m – 2 km
• Extent: 20–20,000 km
• Corresponds to scale 1:200,000 – 1:2,000,000
• Examples: regional soil maps, land‑use planning, resource assessment.

Level D5 – continental / global (more than 20 km)

• Resolution: greater than 2 km
• Extent: more than 200 × 200 km
• Corresponds to scales smaller than 1:2,000,000
• Examples: FAO World Soil Map, global climate models, assessment of carbon stocks on a planetary scale.

3.3. Change of dominant factors at different scales

When moving from one scale level to another, not only the quantitative characteristics change – the factors themselves that determine soil variability change (McBratney et al., 2003; Weil, 2017):

At the local level (D1–D2), relief (r) and organisms (o), especially humans, become the main factors. Micro‑highs and micro‑lows, old furrows, manure storage sites, and the canopies of individual trees – all create the “mosaic” that the agronomist sees within a single field. Detailed measurements and spatial statistics are important here.

At the medium level (D3), differences in parent material (p) and soil age (t) are added to relief. For example, within one farm, soils on loess, till, and alluvium may be found – and each of these parent materials will give its own soil type. Here, knowledge of geology and geomorphology is needed in addition to measurements.

At the regional level (D4), climate (cl) becomes decisive. It is at this level that soil zones are distinguished: podzolic soils of the taiga, chernozems of the steppes, sierozems of deserts. Climate also influences vegetation, which acts as an indicator of soil conditions.

At the global level (D5), combinations of major climatic belts, tectonic structures, and biogeographic regions begin to operate. Here we speak of soil orders (Soil Taxonomy) or reference soil groups (WRB) – i.e., the most general categories.

3.4. The problem of transferring data between scales

One of the most important consequences of scale dependence is the non‑transferability of patterns from one scale to another. What works at the field level does not necessarily work at the regional level, and vice versa (Wendroth et al., 2012).

Wendroth et al. (2012) provide a striking example: clay content and soil electrical resistivity were measured along a 440‑metre transect. When measurements were taken with a spacing of 40 metres, the correlation between the two properties was weak (r = -0.505). When the spacing was reduced to 5 metres, the correlation coefficient increased to -0.669. But – what is even more important – the parameters of the linear regression changed: the slope and intercept were different for different spacings (see Table 10.1 in Wendroth et al., 2012).

This means that if we build a regression model on data with a 40‑m spacing and then apply it to predict properties on a 5‑m grid, we will get incorrect results. The model is “scale‑dependent”. That is why empirical patterns cannot simply be extrapolated from one scale to another – spatial structure at each level must be taken into account.

3.5. From field to continent: practical significance

Why does an agronomist need to understand scale? Because the type of decisions made depends on the scale:

• At the field level (D1–D2), we decide on differentiated fertilization, liming, and sowing. Detailed soil property maps with a resolution of 5–20 m are needed here. This is where precision agriculture and pedometry in its “applied” form operate.

• At the farm or district level (D3), we plan crop rotations, reclamation measures, and estimate overall fertilizer requirements. A resolution of 20–200 m and work with soil associations are sufficient here.

• At the regional level (D4), land management, environmental monitoring, and resource assessment issues are addressed. Soil maps at scales 1:200,000 – 1:1,000,000 are quite adequate here.

• At the global level (D5), we assess carbon fluxes, climate change, and food security. Here, a resolution of 1 km is already an achievement (the global soil database SoilGrids works at 1 km resolution but aims for 100 m) (Weil, 2017).

In other words: for each scale – its own data set, its own methods, and its own questions. Attempting to use global soil map data to manage a specific field is as absurd as trying to draw a world map from data from a single field.

3.6. Scale and sampling density

Scale directly determines sampling density – the number of observations per unit area (White, 2006). For detailed surveys (1:5,000–1:10,000), 4–5 observations per hectare are recommended. For reconnaissance surveys (1:100,000 and smaller), one observation per several square kilometres may be sufficient.

But sampling density is not an arbitrary number. It must be sufficient to “see” the spatial structure of the property. If the sampling spacing is greater than the range of spatial autocorrelation, we will not see any structure – all measurements will appear random (Wendroth et al., 2012). This is the subject of the next section – on spatial autocorrelation.

4.1. What is spatial autocorrelation

Spatial autocorrelation is a statistical measure of how similar the values of a given property are at nearby locations (Wendroth et al., 2012). In short: nearby points are more similar than distant ones.

Imagine soil organic carbon content along a slope. At the top of the hill there is little – erosion removes the top layer. Slightly lower down the slope there is more, at the foot even more, in a wetland depression maximum. If we measure carbon at points 2 metres apart, their values will be close. If we compare a point at the top and a point at the foot, the difference will be huge. This is a manifestation of spatial autocorrelation: similarity decreases with increasing distance.

Formally, autocorrelation is described through autocovariance – the expected product of deviations of two observations from the mean, separated by distance h (Wendroth et al., 2012):

And the normalized form – the autocorrelation function *r(h)* – shows the strength of the relationship between values as a function of distance h (Wendroth et al., 2012):

Do not be intimidated by the formulas: the essence is simple. At h = 0 (zero distance – comparing a point with itself), autocorrelation is 1 – perfect correlation. As h increases, autocorrelation generally decreases and eventually becomes close to zero – the relationship disappears. The distance at which autocorrelation drops to zero is called the range of autocorrelation or simply the range. Beyond this distance, values are statistically independent.

4.2. Why autocorrelation matters for soil studies

Knowledge of the existence and nature of spatial autocorrelation is crucial for all soil research. Here is why:

First, autocorrelation determines whether interpolation makes sense. If autocorrelation is absent (r(h) ≈ 0 for all h > 0), then properties change randomly, and predicting values at points between measurements is impossible. This is the classical situation assumed by traditional statistical analysis, but for soils it is rare. In practice, soil properties almost always have spatial structure – it just needs to be properly identified (Wendroth et al., 2012).

Second, autocorrelation determines sampling density and scheme. If the range of autocorrelation is, say, 20 metres, then sampling with a 50‑metre spacing will not reveal the structure – we will have a set of virtually independent points, and any maps will be only a rough approximation. Conversely, if we know the range, we can choose a spacing that allows us to “see” the structure. Ideally, the spacing should be 2–5 times smaller than the range.

Third, autocorrelation allows us to assess the uncertainty of predictions. If we know the spatial structure, we can say not only “in this point humus content is about 3%” but also “with 90% probability it lies between 2.7% and 3.3%”. Without knowing the structure, any uncertainty estimate would be arbitrary.

4.3. How autocorrelation manifests: an example with soil moisture

Wendroth et al. (2012) provide an instructive example demonstrating how spacing affects the detection of autocorrelation. Soil moisture at a depth of 40–60 cm was measured along a 440‑metre transect in an agricultural landscape in north‑eastern Germany.

When samples were taken with a spacing of 40 metres, the autocorrelation function r(h) dropped practically to zero already at h > 0 (see Fig. 10.1a,b in Wendroth et al., 2012). This meant that at this spacing, spatial structure was not detectable – moisture changes appeared random. Any interpolation would have been unjustified.

When the spacing was reduced to 20 metres, the picture did not change fundamentally – autocorrelation was still absent (Fig. 10.1c,d). It seemed that soil moisture had no spatial structure at all.

But when the spacing was reduced to 10 metres, the situation changed dramatically (Fig. 10.1e,f). A clear structure appeared: autocorrelation was high at 10 m and gradually decreased with distance. Finally, at a spacing of 5 metres, the structure was fully revealed (Fig. 10.1g,h) – autocorrelation clearly decreased with distance, reaching zero at about 30–40 metres.

Conclusion: autocorrelation was always present in the data. It was simply not “visible” at too coarse a spacing – it was masked by apparent noise. This example shows why we cannot simply choose a sampling spacing “by eye” – we need to know or estimate the nature of the spatial structure.

4.4. Autocorrelation and cross‑correlation

In addition to autocorrelation of a single variable, pedometry also makes use of cross‑correlation – a measure of the relationship between two different properties measured at points separated by distance h (Wendroth et al., 2012).

The formula for cross‑correlation is analogous to the autocorrelation one, but instead of one variable A, two are taken – A and B:

Cross‑correlation answers the question: if we know clay content at point x, how well does it predict electrical resistivity at a neighbouring point x+h? This is important when one property is easy and cheap to measure (e.g., electrical conductivity) and another is labour‑intensive (e.g., particle‑size distribution). If there is cross‑correlation between them, we can use the “cheap” measurements to predict the “expensive” ones.

In the same study (Wendroth et al., 2012), cross‑correlation between clay content and soil electrical resistivity was measured. At 40‑m spacing, cross‑correlation was absent (Fig. 10.2c) – the two properties appeared independent. But at 5‑m spacing, cross‑correlation became clearly visible (Fig. 10.2d): clay content at a point was related to resistivity at a neighbouring point up to 5–10 metres. Moreover, the regression relationship between the properties itself changed with spacing (see Table 10.1 in Wendroth et al., 2012) – slope and intercept were different for different scales.

This has enormous practical significance. For example, many pedotransfer functions – empirical equations that predict complex properties (water retention, hydraulic conductivity) from simple ones (particle‑size distribution, organic matter) – are often built on data collected at different spacings and scales. If the scale dependence of cross‑correlation is not taken into account, such functions may give systematic errors when applied at another scale.

4.5. Autocorrelation and traditional statistics

It is important to understand the fundamental difference between traditional statistics and an approach that accounts for autocorrelation. Traditional methods (analysis of variance, regression, significance tests) assume independence of observations. This means that each measurement is considered completely independent of others – as if we randomly select points across the field.

But, as we have seen, soil properties are autocorrelated. Nearby points are not independent – they are similar. This violates the basic assumption of traditional statistics. The result: we may find statistically significant differences where there are none (Type I error) or miss real differences (Type II error). Moreover, estimates of variance and standard errors will be biased (Wendroth et al., 2012; White, 2006).

That is why special methods – geostatistics, which account for spatial autocorrelation – are needed for analysing soil data. And that is why random sampling, which works well in many other fields, is often suboptimal in soil science.

4.6. What if autocorrelation is absent?

It may also happen that spatial autocorrelation is indeed not detected – even with a sufficiently fine spacing. This can mean several things (Wendroth et al., 2012):

1. True randomness – the property really varies chaotically at all scales. This is rare but possible for some properties, especially those strongly dependent on random events (e.g., distribution of individual stones or large pores).

2. Incorrect scale choice – structure exists but at a different scale. For example, at the 5‑m scale there is no structure, but it appears at 50 m or vice versa. This requires multi‑scale analysis.

3. Measurement errors – noise in measurements can mask the real structure. This is especially common with weak signals (e.g., low concentrations of pollutants).

4. “Nugget effect” – variability at scales smaller than the sampling spacing. This is a very common situation, and we will discuss it in the next section on the variogram.

If autocorrelation is absent, interpolation (e.g., kriging) becomes unjustified. In that case, the best we can do is use the mean value over the whole field or switch to a finer measurement grid. But that is a matter of practical compromise between accuracy and cost.

5.1. From autocorrelation to variogram

The autocorrelation function r(h), discussed in the previous section, shows the strength of the relationship between values as a function of distance h. The closer r(h) is to 1, the stronger the similarity; the closer to 0, the weaker. This is intuitive and a good way to see structure. But autocorrelation has a drawback: it is “normalized”, i.e., its values always lie between -1 and 1. We see the strength of the relationship, but not the magnitude of variability in the original units of measurement.

The variogram solves this problem. It shows not the strength of the relationship, but the mean squared difference between values at points separated by distance h (Wendroth et al., 2012). If values are close – the difference is small; if far – large. Unlike autocorrelation, the variogram is expressed in the squares of the units in which the property is measured.

The formula for the semivariogram is as follows (Wendroth et al., 2012; White, 2006):

Here γ(h) is the semivariogram (often simply called “variogram”); N(h) is the number of point pairs separated by distance h; the sum is taken over all such pairs. The division by 2 makes the variogram comparable to ordinary variance. But the main point is not the formula but its meaning: the variogram increases with distance h because points become less similar, and the difference between them on average increases.

5.2. What a variogram looks like and what it shows

A typical variogram has three key elements (Wendroth et al., 2012; White, 2006) (see also Fig. B14.3.1 in White, 2006):

1. Nugget – the value of the variogram at h = 0. Logically, at zero distance, the difference between a point and itself should be zero. But in practice, when the variogram is extrapolated to zero, a positive value is often obtained. This is the nugget. It reflects variability at scales smaller than our minimum sampling spacing, plus measurement errors. If the nugget is large, it means that a significant part of the variability occurs over very short distances – for example, at the scale of individual structural units or worm channels. This is the “micro‑variability” that we cannot see at our sampling spacing.

2. Sill – the value to which the variogram levels off at large distances. When the distance becomes large enough, points cease to be similar, and the mean difference between them reaches a maximum. This sill is often close to the overall variance of the property in the entire sample.

3. Range – the distance at which the variogram reaches the sill. Beyond the range, values are statistically independent – autocorrelation disappears. Within the range, values are similar, and interpolation makes sense.

Thus, the variogram gives us three crucial parameters: nugget (micro‑variability), sill (total variance), and range (radius of autocorrelation). These determine how we need to sample and whether we can build maps.

5.3. Practical example: soil moisture

Let us return to the familiar example from Wendroth et al. (2012) of soil moisture along a 440‑metre transect. For each spacing – 40 m, 20 m, 10 m, 5 m – a variogram was computed.

At 40‑metre spacing, the variogram looked like “pure nugget” – no increase with distance, immediately a sill. This meant that all variability occurred at scales smaller than 40 m. We could not “see” structure because the spacing was too coarse.

At 20‑metre spacing, the situation did not change – the variogram again showed only nugget. Structure was still hidden.

At 10‑metre spacing, a weak increase appeared, but it was uncertain – there were too few points for calculation, and the variogram was noisy.

At 5‑metre spacing, the structure became clearly visible: the variogram rose from the nugget to the sill, reaching it at about 30–40 metres. The range was about 30–40 metres. This meant that soil moisture correlated at distances up to 30–40 m, and then the relationship disappeared.

It is important to note: the range is not a “correct” radius but an empirical characteristic that depends on the scale of investigation, soil type, season, and other factors. It will differ under different conditions.

5.4. The problem of the nugget and what lies behind it

The nugget is not just a statistical artefact. It has a physical meaning (Wendroth et al., 2012; White, 2006).

First, it is micro‑variability – variability of properties at distances smaller than our sampling spacing. In soil, this may be related to:

• Individual structural units (aggregates)
• Worm and root channels
• Patches of organic matter
• Local accumulations of mineral particles
• Micro‑highs and micro‑lows of the relief

This micro‑variability can account for a significant portion of total variance – sometimes up to 50% or more. It cannot be accounted for at the usual sampling spacing, but it needs to be known to assess prediction uncertainty.

Second, the nugget includes measurement errors – both laboratory and field. If we take two samples at the same point and get different values, this also contributes to the nugget.

How to distinguish between the two? Wendroth et al. (2012) describe a simple but effective method: at some points, take not one but two or three adjacent samples (so‑called “duplicates” or “nests”) at distances of several centimetres or metres apart. Then compute the variance between these duplicates – this will be an estimate of the “true” micro‑variability plus measurement error. If it is substantially less than the nugget from the variogram, then the nugget is mainly due to variability at scales between the duplicates and the sampling spacing. If it is comparable to the nugget – then the main cause is measurement errors.

In their example with soil moisture in Kentucky (Fig. 10.3c and 10.4c in Wendroth et al., 2012), duplicates (samples taken a few centimetres apart) showed very low variance – significantly lower than the variogram nugget. This meant that the main part of the nugget was due not to measurement errors but to real variability at scales from a few centimetres to 5 metres. In other words, the soil is indeed very variable over short distances – and this must be accounted for when interpreting data.

5.5. Variogram models

An experimental variogram is a set of points calculated from real data for different distances h. But for use in calculations (e.g., for kriging), we need to “smooth” these points by fitting a mathematical function – a variogram model (Wendroth et al., 2012; White, 2006).

The most common models:

• Spherical – the most popular. It rises from the nugget to the sill, reaching the sill at the range. It describes many soil properties well.
• Exponential – rises quickly at first, then slowly approaches the sill, but theoretically reaches it only at infinity. In practice, the range is defined as the distance at which 95% of the sill is reached.
• Gaussian – has an S‑shape, slow growth near zero, then acceleration and levelling off. Used for properties with very smooth spatial structure.
• Power models – have no sill, grow indefinitely. Used for properties with a trend (e.g., humus content increasing along a transect).

Choosing a model is not just a mathematical exercise. It should reflect the physical nature of the process. For example, the spherical model assumes that the range is a real physical limit of correlation. The exponential model assumes that correlation decays gradually and never completely disappears.

5.6. Variogram and anisotropy

So far, we have discussed variograms assuming that variability is the same in all directions. This is called isotropy. But in practice, soil properties often have different variability in different directions – this is anisotropy (Wendroth et al., 2012).

For example, on a slope, organic matter content may change rapidly down the slope (from top to bottom) and slowly across the slope (along contours). Or on a field where tillage was done in one direction, properties may have a “striped” structure.

In such cases, not one but several variograms are constructed – for different directions (e.g., 0°, 45°, 90°, 135°). If they differ, we speak of geometric anisotropy – different ranges in different directions. Or of zonal anisotropy – different nugget or sill.

Accounting for anisotropy is important for correct interpolation: if we do not take into account that correlation is stronger in one direction than in another, we will get distorted maps.

5.7. Variogram and sampling density

From all of the above, a key practical conclusion follows: the variogram is a tool for choosing the optimal sampling density (Wendroth et al., 2012; White, 2006).

If we know the range of autocorrelation, we can choose a sampling spacing such that:

• Spacing is smaller than the range – otherwise we will not see the structure (as in the example with 40‑m spacing when range was 30–40 m).
• Spacing is sufficiently small to reliably estimate the nugget and the initial rise of the variogram.
• Spacing is not excessively small – this increases costs without a proportional gain in accuracy.

It is usually recommended that the spacing be about 1/3–1/5 of the range. In the example above, with a range of 30–40 m, the optimal spacing would be 6–10 m – which was confirmed in practice (structure appeared at 10 m and became clear at 5 m).

In addition, the variogram allows estimating the required number of samples to achieve a given accuracy. The larger the nugget (micro‑variability), the more samples are needed to “average out” this noise. The larger the range, the less frequently samples can be taken.

It is important to understand: the variogram is not a property of the soil itself, but a property of the sample (Wendroth et al., 2012). If we change the extent (study area) or the spacing, the variogram parameters will change. Therefore, the variogram is always constructed for a specific study and cannot be mechanically transferred to another site or scale.

6.1. Why sampling density matters

Recall the example of soil moisture from Wendroth et al. (2012). At a sampling spacing of 40 m, the researchers found no spatial structure – the data appeared as random noise. At a spacing of 5 m, the structure became clearly visible. The same data, but at different sampling densities, gave fundamentally different conclusions about the presence or absence of spatial pattern.

This is not just an academic curiosity. Incorrect sampling density leads to serious consequences:

• Too sparse a network – we do not see the real structure; properties appear random. Any interpolation becomes unjustified; all decisions are based on averages that do not reflect the true diversity of the field.
• Too dense a network – we waste money on sampling and analysis, gaining only a marginal increase in accuracy. Cost‑effectiveness drops.

The optimal density is one that allows reliable estimation of the spatial structure (variogram) and then using it for interpolation with acceptable accuracy.

6.2. Factors determining the required sampling density

Several factors influence sampling density (Wendroth et al., 2012; White, 2006; Weil, 2017):

1. Nature of spatial variability of the property. If the range of autocorrelation is large (e.g., 100 m), spacing can be relatively coarse (20–30 m). If the range is small (e.g., 10 m), spacing must be fine (2–5 m). Nugget variance also matters: the larger it is, the more samples are needed to “cover” micro‑variability.

2. Purpose of the study. For agronomic management at the field level (precision agriculture), high density is needed – up to 4–5 samples per hectare for detailed maps (White, 2006). For regional resource assessment, 1 sample per several square kilometres may suffice.

3. Budget and available resources. A denser network requires more time, labour, and laboratory analyses. Often the optimal density is determined not only by scientific but also by economic considerations.

4. Measurement method. If the property can be measured quickly and cheaply (e.g., electrical conductivity or spectral reflectance), density can be higher than for labour‑intensive analyses (particle‑size distribution, humus content).

5. Size of the study area. The larger the area, the lower the relative density (samples per hectare), but the extent (overall range) must be sufficient to cover all major landscape types.

6.3. How to determine optimal density: the role of the variogram

The key tool for choosing density is a preliminary pilot study with a finer grid than intended. A variogram is built from these data, and the nugget, sill, and range are estimated. Then, knowing the range, we can choose a spacing that is 3–5 times smaller than the range (Wendroth et al., 2012).

For example, if the pilot study gives a range of 60 m, the optimal spacing for the main network is 12–20 m. If the range is 20 m – spacing 4–7 m.

But a pilot study incurs additional costs. Other approaches can also be used:

• Using literature data – for similar soil‑climate conditions, approximate range values from published works can be used. However, these may differ greatly, so this approach is less reliable.
• Expert estimates – an experienced soil scientist can infer the nature of variability based on relief, soil maps, and visual observations. But this is only an approximation.

Ideally, if resources permit, a detailed (fine) sampling is first carried out on a small area (a “reference” site), a variogram is built, and then the main network for the entire territory is designed based on its parameters. This approach minimizes overall costs without sacrificing accuracy.

6.4. Strategies for locating sampling points

In addition to density, the location of sampling points is also important. Several strategies exist (White, 2006; Wendroth et al., 2012):

Systematic (regular) grid – points are placed at equal distances from each other, forming a rectangular or triangular network. This is the most common approach for detailed surveys. It provides uniform coverage and simplifies geostatistical analysis. A disadvantage is that it may miss local anomalies if they fall between grid nodes.

Random sampling – points are chosen at random. This method is good for estimating means and variances but poor for revealing spatial structure and mapping. Traditional statistics is based on random sampling, but for soil studies it is often suboptimal (Wendroth et al., 2012).

Stratified random sampling – the area is divided into homogeneous zones (strata), and points are chosen randomly within each stratum. This combines the advantages of random sampling with consideration of spatial heterogeneity.

Targeted (purposive) sampling – points are chosen where changes are expected (e.g., at soil boundary contours, at the foot of slopes). This method is effective for identifying boundaries but requires prior knowledge of the landscape. In traditional free survey, soil scientists work exactly this way (White, 2006).

Adaptive sampling – an initial sparse grid is taken, and then additional points are added in areas of high variability. This saves resources by concentrating efforts on the most “informative” areas.

In recent years, the conditioned Latin Hypercube Sampling (cLHS) method has gained popularity, which allows selecting a limited number of points that maximally cover the variation space of all available covariates (relief, vegetation, remote sensing data) (Minasny & McBratney, 2006, cited in Weil, 2017). This is particularly useful for digital soil mapping, where we want the training set to represent the full diversity of landscape conditions.

6.5. Relationship between density and map scale

There is a direct relationship between sampling density and the scale of the final map (White, 2006). It is generally accepted that for reliable depiction of a soil boundary on a map, at least 4–5 observations per square centimetre of map are needed. Based on this, the required density for a given scale can be calculated.

For example, at a scale of 1:10,000, 1 cm² of map corresponds to 1 hectare on the ground. If 4 observations per cm² are needed, density should be 4 samples per hectare. For a scale of 1:25,000, 1 cm² corresponds to 6.25 ha, and 4 samples per cm² give a density of about 0.64 samples per hectare – i.e., one sample per 1.5 ha. For a scale of 1:50,000, density will be even lower.

Thus, the smaller the scale (the larger the area represented on one sheet), the fewer samples per unit area are needed. However, we lose detail – small heterogeneities will not be shown.

6.6. Errors due to insufficient density

If sampling density is insufficient, typical errors occur (Wendroth et al., 2012):

1. Missing spatial structure – the variogram shows only nugget, and we erroneously conclude that the property is random. In fact, structure exists but is not resolvable at the given spacing.

2. Bias in estimates – interpolation gives incorrect values, especially in areas with sharp gradients. The map becomes “smoothed”, losing real contrasts.

3. Incorrect uncertainty estimation – we either underestimate variability (if we ignore structure) or overestimate it (if we try to interpolate without structure). In either case, predictions become unreliable.

4. Wrong agronomic decisions – if we do not see that pH is 5.0 in one part of the field and 7.5 in another, we may apply lime uniformly, leading to over‑liming in some areas and deficiency in others.

6.7. Sampling density in precision agriculture

In precision agriculture, sampling density is especially critical. The goal is to create detailed maps of soil properties (pH, nutrient content, humus, particle‑size distribution) with a resolution that allows differentiated management of agronomic operations (Wollenhaupt et al., 1994; Weil, 2017).

Typical recommendations for fields of moderate size (10–100 ha) are a grid of 1–2 samples per hectare (spacing 50–100 m). However, as we have seen, the actual required density depends on the range of autocorrelation. In some cases, even 1 sample per hectare may be excessive (if the range is large), while in others it may be insufficient.

That is why pedometry offers not rigid standards but a flexible approach: first estimate the spatial structure (pilot study, literature data), then choose the optimal spacing and point placement scheme. This saves resources without losing accuracy.

In addition, precision agriculture increasingly uses remote and proximal sensing methods (electrical conductivity, spectroscopy) – they provide very dense measurement networks but require calibration with a limited number of laboratory analyses. Pedometry allows combining these heterogeneous data (cokriging, regression kriging) to obtain high‑quality maps at minimal cost for “expensive” analyses.

6.8. Practical recommendations

Let us summarize practical recommendations for choosing sampling density (Wendroth et al., 2012; White, 2006):

1. Always start with a pilot study – at least 30–50 points on a small area to estimate the variogram and range. This will pay off many times over when designing the main network.

2. Choose the spacing of the main network to be no more than 1/3–1/5 of the range. If range is 60 m – spacing 12–20 m; if 30 m – spacing 6–10 m.

3. Use stratified or adaptive schemes to ensure representativeness of all landscape elements.

4. For detailed precision agriculture maps – density 1–4 samples per hectare, but adjust according to the variogram.

5. For reconnaissance surveys – density can be much lower, but it is important that coverage is sufficient to identify major soil associations.

6. If resources are limited, it is better to reduce the number of samples but place them optimally (e.g., using Latin hypercube) than to take many samples without regard to structure.

7. Always document the spacing, scheme, and justification for the chosen density – this will allow future interpretation and comparison with other studies.

7.1. What is precision agriculture and what is its problem?

Precision agriculture (site‑specific farming) is a management system for agronomic operations that accounts for the spatial heterogeneity of the soil cover and crops within a single field (Weil, 2017). Its goal is to apply resources (fertilizers, crop protection agents, seeds, water) not uniformly across the field but differentially, according to the actual needs of each location.

The idea seems simple and obvious. However, in practice it encounters a fundamental obstacle: we do not know how soil properties vary across the field until we measure them at a sufficient number of points. Without this knowledge, any differentiated management becomes guesswork.

The traditional approach – a composite sample from the whole field – assumes that the field is uniform. But we already know that this is not so. In practice, even within one field, there may coexist areas with pH from 5.0 to 7.5, humus content from 1% to 4%, and different availability of phosphorus and potassium (Weil, 2017). An averaged fertilizer recommendation will be suboptimal for most points: somewhere we apply excess (losses, pollution), somewhere insufficient (yield loss).

7.2. How pedometry supports precision agriculture

Pedometry provides precision agriculture with a scientific basis for decision‑making. Let us examine the key stages of this process.

7.3. Practical example: differentiated fertilization

Consider a concrete example. A 100‑ha field where fertilizers were previously applied unevenly (more in some places, less in others). Traditional approach: take one composite sample, obtain an “average” phosphorus content, and recommend a uniform fertilizer rate for the whole field.

Pedometric approach (Wollenhaupt et al., 1994; Weil, 2017):

1. Conduct a pilot sampling (e.g., 30 points).

2. Build a variogram for available phosphorus.

3. Estimate the range (say, 40 m).

4. Choose a network with a spacing of 10–15 m for the main sampling (total 50–70 points).

5. Build a phosphorus map using kriging.

6. Based on the map, create recommendations: areas with high content – no phosphorus; medium – 30 kg/ha; low – 60 kg/ha.

7. Apply fertilizers differentially, using equipment with GPS and variable rate application.

Result: fertilizer savings on high‑content areas, yield increase on low‑content areas, reduced phosphorus losses to the environment, and higher overall efficiency.

Without pedometry (without knowledge of spatial structure and without maps), such differentiation would be impossible – we would not know where the content is high and where it is low.

7.4. Economic and environmental aspects

Pedometry in precision agriculture has not only agronomic but also economic and environmental significance.

Economic benefits:

• Reduced costs for fertilizers and crop protection by applying them only where needed (savings can be 10–30%).
• Increased yield on problematic areas through targeted elimination of limiting factors.
• Improved product quality (more uniform ripening, less lodging).
• Ability to make informed decisions about land purchase or lease based on knowledge of its spatial heterogeneity.

Environmental benefits:

• Reduced risk of nutrient leaching (especially nitrates) into groundwater by eliminating over‑application.
• Reduced greenhouse gas emissions (N₂O) from excess nitrogen fertilizers.
• Conservation of soil biota biodiversity by reducing chemical loading.
• More efficient use of water resources under differentiated irrigation.

Thus, pedometry is not just an academic discipline. It is a tool for sustainable development of agriculture, combining economic efficiency with environmental responsibility (Weil, 2017).

7.5. Limitations and prospects

It is important to understand that pedometry and precision agriculture are not a panacea. They have limitations:

• Cost of implementation – equipment, software, training, analyses require significant investment. For small farms, they may be prohibitive.
• Complexity of interpretation – spatial models give probabilistic estimates, not absolute truths. Qualified personnel are needed for their use.
• Temporal variability – soil properties change not only in space but also in time. Maps need to be updated.
• Not all properties are equally “well” interpolable – some properties have weak autocorrelation and are difficult to map.

However, the prospects for development are enormous. Advances in remote sensing, UAVs, on‑the‑go sensors, artificial intelligence, and machine learning are making pedometry increasingly accessible and effective (Weil, 2017). Systems are already emerging that allow creating maps of soil properties in real time, right during field work. And at the heart of all these systems are the very principles we have discussed in this lecture: accounting for spatial variability, autocorrelation, variogram, optimal sampling density.

Conclusion

We have travelled the full path: from realising that soil is heterogeneous and cannot be characterized by one sample, through quantitative description of this heterogeneity (autocorrelation, variogram), to practical recommendations on sampling and, finally, to applying this knowledge in precision agriculture.

Key takeaways from the lecture:

1. The soil cover is continuous and variable at all scales – from millimetres to continents. This is a consequence of the diversity of soil‑forming factors.

2. A single composite sample gives only an averaged value that does not reflect the real diversity of the field and cannot serve as a basis for optimal agronomic decisions.

3. Spatial autocorrelation is a fundamental property of soil data. Nearby points are similar, distant ones are not. This allows interpolation between measurement points.

4. The variogram is the main tool of pedometry. It shows how quickly similarity decays with distance and allows estimating the range, nugget (micro‑variability), and sill (total variance).

5. Sampling density must be determined based on the variogram. The optimal spacing is 1/3–1/5 of the range. Without this, we either fail to see structure or waste resources.

6. Precision agriculture is impossible without pedometry. It provides the scientific foundation for differentiated resource management, enabling the creation of property maps, accounting for uncertainty, integrating heterogeneous data, and optimising costs.

Pedometry is not just a set of statistical methods. It is a new way of thinking about soil as a continuous, variable, but knowable system. And this way of thinking is becoming increasingly relevant in the era of digital agriculture, where data are the main resource and science is the key to sustainable development.