Poeplau et al. (2017) recently outlined the systematic overestimation of soil organic carbon (SOC) stocks due to incorrect application of bulk density and rock fragment content in calculation of SOC stocks. Unfortunately, the method they propose to rectify this is associated with a greater error (due to assumption of rock density, extra calculation steps and propagation of errors) than the simpler mass-balance-derived equation for SOC stock calculations, outlined below. Using a mass balance approach to C stocks we define

where C_stock is the amount of carbon stored in a given soil area (kg m⁻²) and depth, d (cm); mass proportion_C is the carbon content of the whole soil (g kg⁻¹) and ρ is the bulk density of the whole soil (g cm⁻³).

Using a mass balance approach on the mass proportion of C in the whole soil, we obtain

where C_content,fine is the mass proportion of C in the fine soil fraction (g kg⁻¹), mass proportion_fine is the mass proportion of the fine soil to the whole soil sample (g kg⁻¹) and C_{content,  coarse} is the mass proportion of C in the coarse soil fraction (g kg⁻¹), and mass proportion_coarse is the mass proportion of the coarse soil to the whole soil sample (g kg⁻¹), generally referred to as the rock content. C_{content,  coarse} is assumed to be negligible (i.e. = 0) in all methods, so the Eq. (2) simplifies to

The mass proportion_fine is

Substituting Eq. (6) into Eq. (3) we obtain

Substituting Eq. (7) into Eq. (1) we obtain

This looks similar to Eq. (5) in Poeplau et al. (2017). However, they use the volumetric proportion, not the mass proportion of rock fragments, which is mathematically incorrect. They also state that their Eq. (6) “resembles” Eq. (8). However, their M4 is actually a more convoluted and obtuse equivalent to the commonly known and applied Eq. (8) (Ellert and Bettany, 1995; Goidts et al., 2009; Mikha et al., 2013; Orgill et al., 2013). This can be shown by combining Eqs. (3) and (6) from Poeplau et al. (2017), which illustrates that the inclusion of rock density to calculate SOC stocks is unnecessary and redundant.

Equation (8) is also mathematically equivalent to calculations according to Eqs. (7) and (8) in Poeplau et al. (2017). However, the recommended use of the mass of fine fraction for the calculations by Poeplau et al. (2017) also has a greater potential error than using the mass proportion of rocks according to Eq. (8). The advantage of using the rock mass to correct the stocks is that rocks are (nearly) entirely conserved during sieving, whereas fine soil mass is lost as dust during sieving, increasing uncertainty in the calculations. In contrast, M4 (Eqs. 3 and 6) of Poeplau et al. (2017) requires an estimation of rock density (they recommend assuming a rock density of 2.63 g cm⁻³) to calculate the bulk density of the fine soil sample as well as to adjust for rock content. Rock density depends on parent material, with basalts having higher densities than granites, so this assumption increases error and uncertainty (Hazelton and Murphy, 2016).

Unfortunately, the additional calculations required in M4 also increase the uncertainty of the estimate due to error propagation. This can be illustrated by calculating the error terms of both equations. The squared relative error of Eq. (8) is

With mass proportion${}_{\mathrm{rock}}=\frac{{\mathrm{mass}}_{\mathrm{rock}}}{{\mathrm{mass}}_{\mathrm{sample}}}$ and ${\mathit{\rho }}_{\mathrm{sample}}=\frac{{\mathrm{mass}}_{\mathrm{sample}}}{{\mathrm{volume}}_{\mathrm{sample}}}$ we obtain

The squared relative error of M4 in Poeplau et al. (2017) is

Using the Eq. (3) in Poeplau et al. (2017) for ρ_fine and with volume proportion${}_{\mathrm{rock}}=\frac{{\mathrm{volume}}_{\mathrm{rock}}}{{\mathrm{volume}}_{\mathrm{sample}}}$ we obtain

With ${\mathit{\rho }}_{\mathrm{rock}}=\frac{{\mathrm{mass}}_{\mathrm{rock}}}{{\mathrm{volume}}_{\mathrm{rock}}}$ the squared relative error of M4 in Poeplau et al. (2017) is therefore

As can be seen, M4 has more sources of error than Eq. (8). M4 is therefore statistically inferior and should be avoided. This is in line with applying the law of parsimony (Occam's razor) to the problem of SOC stock calculations, which states that when presented with competing answers to a problem, one should choose the one which makes the fewest assumptions. Calculations according to their M4 are therefore less precise due to extra sources of error (derived from either analytical or assumed rock density as well as error propagation). As such, using Eq. (8) above, based on the C content of the fine soil, mass proportion of rocks and bulk density in the whole sample will yield the most precise estimate of C stocks.

With regards to eliminating the depth, d, from the calculations (Eq. 9 in Poeplau et al., 2017, suggested by Wendt and Hauser, 2013), it would appear that the error of this method is lower still. However, this is deceptive because the error associated with sampling a specific depth remains, so the mathematical simplification does not eliminate the error term.

Of key concern – and not addressed here – is the calculation of SOC stocks in stony soils, as here an accurate estimation of rock content is highly difficult. Estimating rock content from the profile face is highly error prone because 2-D surface areas are not representative of irregular 3-D structures, such as rocks. Therefore, estimating rock content from the profile face is not volumetric. Taking larger volumes of sample in very large cores to determine the bulk density of the whole soil would help to alleviate this issue, but this would be associated with more field and laboratory work. A systematic study into this issue, similar to the systematic evaluation of sources of error when upscaling to SOC analyses to landscape stocks (Goidts et al., 2009), could help to resolve the issue.

In summary, Poeplau et al. (2017) have clearly demonstrated the need to adjust for coarse fragments > 2 mm in SOC stock calculations. Unfortunately, their recommendation has added some confusion to the correct method of calculation of SOC stocks via the introduction of unfamiliar formulas. Whilst mathematically correct, their formulas are associated with larger errors than the standard equation, so they present no clear advantage. As such, we recommend the use of Eq. (8) for SOC stock calculations.