# 1 Back to index

This a just a paragraph to put the link to the site index.

# 2 Objectives and methods

The objective of this site is to explain how we compute the level II and III data of ICOS sites which correspond to a set of variables for the whole site and for sp1 sites.

Here we present the theory of the stocks calculation and the design-based (DB) estimates of the different statistical parameters needed.

## 2.1 Definitions and method to compute carbon and nitrogen stocks at a sp1 site

The different components of a soil sample could be represented schematically as in figure 1.

The major part of the carbon and nitrogen stock are within the fine soil part so that the soil stock is approximated by the C or N stock contained in the fine fraction of the soil (FE). In addition to the fine fraction, soil also contains some rock fragments (RF, > 2 mm) that can considerably contribute to the total mass of the sample and hence the soil bulk density (BD), but not to the soil C or N stock. Moreover, following Mehler et al., (2014), living root fragments (> 2 mm) are not considered to be part of soil organic C and N stocks, but as part of the plant biomass.

As detailed in Poeplau et al. (2017), the soil stocks of C and N for each Sp1 sample are quantified as the C and N contents of fine soil (as measured by the LAS laboratory), multiplied by the mass of fine soil per sample volume and multiplied by the thickness of the sampled layer. Some approaches classically implemented use the bulk density (BD) instead of the mass of fine soil per sample volume and evaluate the mass of fine soil by accounting for the volumetric rock fraction (see Poeplau et al. 2017 for a review of the different methods).

In the ICOS project, a certain volume of soil is sampled, dried and weighed. After sieving, the mass of rock and root fragments is separated from the fine soil, and thus there is a separation into fine and coarse soil fractions (rock fragments, roots). Then, the OC content is determined in a sieved fine soil sample (usually < 2 mm). This procedure allows a quantification of the C or N stock (CStock and NStock in g m^{-2}) in the fine soil fraction of a given sample using the following equations:

\[\begin{equation} CStock = \frac{mass_{fine soil} } {Volume_{sample} } thickness_{sample} * C_{content} \end{equation}\]

\[\begin{equation} NStock = \frac{mass_{fine soil} } {Volume_{sample} } thickness_{sample} * N_{content} \end{equation}\]

where \(mass_{fine soil}\) is the dry mass of fine soil in the sample (in \(kg\)), \(Volume_{sample}\) is the volume of the sample (in \(m^{3}\)), \(thickness_{sample}\) is the sample layer thickness (in m), and \(C_{content}\) and \(N_{content}\) are the carbon and nitrogen contents as analysed by the laboratory (in \(g kg^{-1}\)).

In the following, for convenience, we define the fine soil stock (\(FSS\)) of a sample as:

\[\begin{equation} FSS = \frac{mass_{fine soil} } {Volume_{sample} } thickness_{sample} \end{equation}\]

Because of the difficulties to account of the volume of large stones, the ICOS current protocol does not include the specific situations. It is evaluated on a case per case basis.

## 2.2 Design based estimates of the statistical parameters (mean, total and uncertainty)

In a Design-based approach, the locations of sampling units within
the design are chosen randomly and not based on convenience or prior
information (purposive sampling). In ICOS a computer assisted approach
is used to determine the coordinates of the locations using a random
number generated by an ad-hoc procedure (Brus et al, 2019). The strength
of this type of sampling designs is that the probability of selecting
the sampling units is known and can be used for statistical inference.
This type of design is best suited for estimating global quantities such
as means and totals (de Gruitjer et al., 20). A wide range of
probability sampling designs are ready to use. Icos estimates rely on a
** stratified simple random sampling**.

With a stratified simple random sampling the target area is divided into sub-areas, called strata. Within every sub-area, or stratum, simple random sampling is then applied. With simple random sampling, all sampling locations are selected with equal probability and independently from each other.

When compared to a simple random sampling, a stratified random sampling leads generally to a smaller sampling variance of the estimated global quantities for the same number of sampling points (sample size), or smaller sample size for the same sampling variance of the estimated mean. Stratified random sampling is thus more efficient than simple random sampling. This is achieved by forming strata that are all as homogeneous as possible regarding the soil property in which we are interested.

In ICOS the strata are designed to be of equal size and compact in
space. We used the `spCosa`

R package to define the strata.
In each stratum, 2 sp1 sites are randomly sampled.

## 2.3 Definitions

In the following the sp2 number is \(k\), the layer number is \(l\) and the sp1 number is \(i\). The following variables are used to compute the stocks. The indexes and exponents indicate wheter a variable is measured at an sp1 or sp2 site and to which layer it belongs to.

- layer \(l\),
- sp1 site \(i\),
- sp2 site \(k\),
- \(SosmVolume^{l}_{ik}\) is the volume of the sample in \(cm^{3}\),
- \(sosmW30E^l_{ik}\) is the dry mass of fine soil (dried at 30°C) of the sample in \(g\),
- \(residWater^l_ik\) is the residual water content of the sample in %
- \(Ep_i^l\) is the layer thickness in \(cm\),
- \(sosmW105S^{l}_{ik}\) is the dry mass of rocks fragments (dried at 105°C) in \(g\),
- \(sosmW70R^{l}_{ik}\) is the dry mass of coarse plants fragments (dried at 70°C) in \(g\),
- \(FSS_{ik}^l\) is the fine soil stock in \(g.cm^{−2}\) ,
- \(RF_{ik}^l\) is the rocks fraction (no units, from 0 to 1),
- \(z^l_i\) is the carbon (or nitrogen) content in \(gC.kg^{-1}\) dry soil (or \(gN.kg^{-1}\) dry soil),
- \(t^l\) is the soil carbon (or nitrogen) stock in \(t.ha^{−1}\),
- \(BD_{ik}^l\) is the bulk density for in \(g.cm^{−3}\).

For the Design based estimates of the statistical parameters:

- \(H\) is the number of strata,
- \(n_h\) is the number of sp1 sites per stratum,
- \(n_{sp2}\) is the number of sp2 sites per sp1 site,
- \(N\) is the total number of sp1 samples where \(N = \sum_{h=1}^H n_h\),
- \(t^l_i\) is the carbon (or nitrogen) stock in \(g.m^{-2}\) or in \(kg.m^{-2}\),
- \(A_h\) is the area of the stratum \(h\) (in \(m^2\)).

# 3 Estimation of the the soil carbon and nitrogen stocks for each layer \(l\) and each sp1 site

To compute the stock \(t_i^l\) for the layer \(l\) at the sp1 site \(i\), we first calculate the fine soil stock (\(FSS_{ik}^l\)) from the sp2 samples \(k\).

## 3.1 Fine soil soil stock (\(FSS_{ik}^l\))

The fine soil stock (\(FSS_{ik}^l\)) is computed for each layer \(l\) and sp2 site \(k\) as:

\[\begin{equation} FSS^l_{ik} = \frac{mass_{FineSoil}} { Volume_{sample} } Ep^l = \frac{(sosmW30E^l_{ik}) * (100 - residWater^l_ik)/100}{SosmVolume^{l}_{ik}} Ep^{l}_{ik} \end{equation}\]

It is then averaged over the sp2 site as follow:

\[\begin{equation} \overline{FSS}_i^l = \frac{1}{n_{sp2}} \sum_{k=1}^{n_{sp2}} FSS^l_{ik} \end{equation}\]

## 3.2 Carbon and Nitrogen stock in the fine soil fraction of the samples

We then compute the stock at each layer \(l\) and sp1 point \(i\), \(t_i^l\)

\[\begin{equation} t_{_i}^l = 10 \frac{ \overline{FSS}_i^l}{1000}z_{i}^l \end{equation}\]

## 3.3 Bulk density

The BD and rock fraction measured from the sp2 samples \(k\) are also important properties that are useful for soil description and modelling purposes. The BD is calculated from the combination of the fine soil fraction, rock and root fragments (figure 1) as follow for each layer \(l\) and sp2 site \(k\):

\[\begin{equation} BD^{l}_{ik} = \frac{ ( sosmW30E^{l}_{ik} (100 - residWater^l_{ik} /100) ) + sosmW105S^{l}_{ik} + sosmW70R^{l}_{ik}) } {SosmVolume^{l}_{ik}} \end{equation}\]

It is then averaged over the sp2 site as follow:

\[\begin{equation} \overline{BD}_i^l = \frac{1}{n_{sp2}} \sum_{k=1}^{n_{sp2}} BD^l_{ik} \end{equation}\]

## 3.4 Rock Fraction

The rock fraction is computed as follow for the layer \(l\) of sp2 site \(k\):

\[\begin{equation} RF^l_{ik} = \frac{sosmW105S_{ik}} {sosmW105S_{ik}+sosmW30E_{ik} } \end{equation}\]

It is then averaged over the sp2 site as follow:

\[\begin{equation} \overline{RF}_i^l = \frac{1}{n_{sp2}} \sum_{k=1}^{n_{sp2}} RF^l_{ik} \end{equation}\]

Before 2020 and for a few ICOS sites, the rock fragments were dried at 30°C only. In these conditions, the \(RF^l_{ik}\) was computed in the following way:

\[\begin{equation} RF^l_{ik} = \frac{sosmW30S_{sp2}} {sosmW30S_{sp2}+sosmW30E_{sp2} } \end{equation}\]

# 4 The theory to compute design based estimates of the soil properties

## 4.1 Design based estimates of the mean and its sampling variance

### 4.1.1 Design based estimates of the mean

The design-based estimator of the mean of a variable \(x\) in a stratified simple random sampling is:

\[\begin{equation} \hat{ \overline{x} } = \sum_{h=1}^{H} w_h \hat{\overline{x} }_{h} \end{equation}\]

where \(w_h = A_h/\sum_{h}A_h\) and \(\hat{\overline{x} }_{h}\) is the estimated mean for each stratum \(h\) using the simple random sampling estimator:

\[\begin{equation} \hat{\overline{x}}_{h} =\frac1{n_h} \sum_{i=1}^{n_h}x_i \end{equation}\]

With for equal area compact strata, which is the case in ICOS as they were designed for that purpose, \(w_h\) simplifies to:

\[\begin{equation} w_h = A_h/\sum_{h}A_h = 1/H \end{equation}\]

where \(H\) is the number of strata.

Under these conditions, the estimator of the mean becomes:

\[\begin{equation}
\hat{ \overline{x} } = \sum_{h=1}^{H} \frac1{n_h} \hat{\overline{x}
}_{h}
\end{equation}\]

As the number of observations per stratum is the same (\(n_h = 2\)), the estimator of the mean for \(\hat{\overline{x}}\) can be simplified as follows:

\[\begin{equation} \hat{ \overline{x} } = \frac1N \sum_{i=1}^{N} x_i \end{equation}\]

### 4.1.2 Design based estimates of the sampling variance of the mean

The sampling variance of the estimated mean of a variable \(x\) is estimated by:

\[\begin{equation} \hat V (\hat{ \overline{x} } ) = \sum_{h=1}^{H} w_h^2 \frac{ \hat{ S^2_h} (\hat{ \overline{x} } ) }{n_h} \end{equation}\]

where \(\hat{ S^2_h} (\hat{ \overline{x} })\) is the variance of \(x\) in the stratum \(h\), which is given by:

\[\begin{equation} \hat{ S^2_h} (\hat{ \overline{x} })= \frac{1}{n_h-1}\sum_{i=1}^{n_h} ( x_{hi} -\hat{\overline{x}}_h )^2 \end{equation}\]

Similary to what was done for the mean, we know that \(w_h = \frac1H\) for all strata \(h\) for equal area compact strata. The sampling variance could then be reduced to:

\[\begin{equation} \hat V (\hat{ \overline{x} } ) = (\frac1H)^2 \sum_{h=1}^{H} \frac{ \hat{ S^2_h} (\hat{ \overline{x} } ) }{n_h} \end{equation}\]

Here, we need to know the id of the stratum for every \(x_i\) to compute the sampling variance of the strata

## 4.2 Design based estimates of the total and its sampling variance

The same simplification holds for the total than for the mean as the area of the strata are the same.

### 4.2.1 Design based estimates of the total

The design-based estimator of the total of a variable \(x\) for area \(|A|\) in a stratified simple random sampling with compact strata is:

\[\begin{equation} \hat{ T } (x) = \frac{|A|}N \sum_{i=1}^{N} x_i \end{equation}\]

### 4.2.2 Design based estimates of the sampling variance of the total

The sampling variance of the estimated mean of a variable \(x\) is estimated by:

\[\begin{equation} \hat V (\hat{ T }(x) ) = (\frac{|A|}H)^2 \sum_{h=1}^{H} \frac{ \hat{ S^2_h} ( x ) }{n_h} \end{equation}\]

Here, we need to know the id of the stratum for every \(x_i\) to compute the sampling variance of the strata

## 4.3 Design based estimates of the spatial variance

The spatial variance of a variable \(x\) between locations in the area represents the variablility of the target value between locations.

An unbiased estimator of the spatial variance \(S^2(x)\) is

\[\begin{equation} \hat{S^2(x)} = \hat{\overline{x^2}} - (\hat{\overline x})^2 + \hat V(\hat{\overline x} ) \end{equation}\]

where \(\hat{\overline{x^2}}\) denote the estimated mean of (\(x^2\)), where the mean is taken as when calculating \(\hat{\overline x}\).

## 4.4 Representing the uncertainty of the estimated statical estimates

The uncertainty about the estimated mean could be expressed using not merely a single number, but a confidence interval: the bounds of the 90%-confidence interval are given by:

\[\begin{equation} \hat{x} \pm Student.t_{1-\alpha/2}^{(n-1)}.\sqrt{\hat V (\hat{x}) } \end{equation}\]

With \(\alpha = 0.10\), and where \[Student.t^{(n-1)}_{1-\alpha/2}\] is the \((1-\alpha/2)\) quantile of the Student \(t\) distribution with \((n-1)\) degrees of freedom. \(n\) is the number of sampling observations to compute the mean, eg \(N = n_h*H\).

## 4.5 ICOS level II and III data

to produce the different estimates for level II and III data, one need to replace \(x\) in the previous equation per the carbone and nitrogen stock or content.

For example, the carbon content of the layer \(l\) for sp1 site \(i\) is noted \(z_i^l\):

\[\begin{equation} \hat{ \overline{z^l} } = \frac 1 N \sum_{i=1}^{N} z_i^l \end{equation}\]

# 5 Changing the units of the stocks

To express the stock of C or N in \(t.ha^{-1}\), we simply need to multiply \(t_{_i}^l\) by 10.

To express the stock of C or N in \(g.m^{-2}\), we simply need to multiply \(t_{_i}^l\) by 1000.

# 6 References

Brus, D.J., 2019. Sampling for digital soil mapping: A tutorial supported by R scripts. Geoderma 338, 464–480. https://doi.org/10.1016/j.geoderma.2018.07.036

de Gruijter, J., Brus, D.J., Bierkens, M.F.P., Knotters, M., 2006. Sampling for Natural Resource Monitoring. Springer-Verlag.

Mehler, K., Sch?ning, I., & Berli, M. (2014). The Importance of Rock Fragment Density for the Calculation of Soil Bulk Density and Soil Organic Carbon Stocks Soil Physics. Soil Science Society of America Journal, 78, 1186-1191. https://doi.org/10.2136/sssaj2013.11.0480

Poeplau, C., Vos, C., & Don, A. (2017). Soil organic carbon stocks are systematically overestimated by misuse of the parameters bulk density and rock fragment content. SOIL, 3(1). https://doi.org/10.5194/soil-3-61-2017