Solid-phase mass balance approaches for estimating rock weathering rates rely on calculating the difference between the amount of a mobile element or cation, j (e.g., Mg, Ca, Na, K), present in the rock before and after weathering has occurred. For this calculation it must be assumed that the frame of reference is the same for measurements taken before and after weathering—i.e., that the same amount of rock is accounted for at both time steps (Reershemius et al., 2023). Given that feedstock application and mixing are unlikely to be perfectly homogeneous at the field scale, and that soil erosion can move material after application of feedstock, it is problematic to assume that a spatially uniform amount of feedstock mass applied in any given location can be perfectly resolved by sampling alone. Alternatively, the amount of rock accounted for in a sample can be estimated by referencing to the amount of a detrital trace element, i, measured in the sample, provided that this element is immobile during weathering (Brimhall and Dietrich, 1987; Chadwick et al., 1990, 1999; Brimhall et al., 1991; Kurtz et al., 2000; White et al., 2001; Anderson et al., 2002; Riebe et al., 2003; Tabor et al., 2004; Sheldon and Tabor, 2009; Brantley and Lebedeva, 2011; Fisher et al., 2017; Lipp et al., 2021; Kantola et al., 2023; Reershemius and Suhrhoff, 2023; Reershemius et al., 2023).

In order to facilitate the use of an immobile tracer i to calculate the amount of feedstock initially present in a sample, the addition of a feedstock to soil can be represented by a two-endmember mixing model between soil baseline (c_s) and feedstock (c_f) (Reershemius et al., 2023; see also Supplementary Figure S36). A mixture of composition c_m between soil and feedstock will fall along a mixing line in i vs. j space (Supplementary Figures S1, S36). Weathering will result in loss of j but not i from this mixture, and therefore the composition of the same mixture after weathering, c_n, will plot below the mixing line (Supplementary Figures S1, S36). Note that the compositional change due to weathering will likely be different for different cations j, as the mineral phases in rocks within which these cations are held may have different susceptibilities to weathering, and incongruent weathering or precipitation of secondary phases will result in preferential reincorporation of some cations j into solid phases. Assuming that feedstock dissolution has a negligible effect on [i] (but see section 4.4), the composition of c_m in i vs. j space can be reconstructed from [i] of c_n. The difference in [j] between c_m and c_n can then be used to calculate a sample-specific feedstock cation loss, which is directly proportional to dissolution.

This study builds upon the two-endmember mixing framework described above (Reershemius et al., 2023) to examine the impact of uncertainty in the compositions of a mixture post-weathering, c_n, and the soil (_s) and feedstock (_f) endmembers, c_s and c_f, on the calculation of dissolution using solid-phase mass balance (Supplementary Figures S1, S36). The derivations are laid out in more detail in the Supplementary material.

We first derive the uncertainty, e, for the mobile cation (j) concentration in a mixture of the initial soil and feedstock endmembers: (1) e m j = e s j + ( e f j − e s j ) ∗ [ j ] m − [ j ] s [ j ] f − [ j ] s

Note that in these equations, uncertainties for feedstock, soil, or mixture are in concentration units, and for the soil and feedstock are calculated by multiplying concentrations with assumed % uncertainty for e (see section 2.3). For the purposes of this framework, we do not distinguish between analytical and sampling uncertainty, but present both as a aggregated uncertainty term.

Assuming that at field deployment scale no change in composition is resolvable if it does not exceed aggregated analytical and sampling uncertainty (see sections 4.2 and 4.4 for a discussion of this assumption), the limit of resolvability for Δ[j] due to dissolution is given by the expression: (2) e n j + e m j < [ j ] m − [ j ] n

And therefore, substituting Eq. 1 into Eq. 2: (3) e n j + e s j + ( e f j − e s j ) ∗ [ j ] m − [ j ] s [ j ] f − [ j ] s < [ j ] m − [ j ] n

Next, the relation of a (mass ratio of feedstock in a mixture based on the applied mass of feedstock) and b (the fraction of feedstock dissolved) to changes in mobile cation concentration is defined as follows: (4) a ∗ b ∗ [ j ] f = [ j ] m − [ j ] n

Combining Eqs. 3 and 4 gives an expression for the minimum resolvable a and b given uncertainty, which is solved with respect to either variable: (5) a ∗ b ∗ [ j ] f > e n j + e m j

Substituting and rearranging gives the following expressions (see Supplementary material for full derivation): (6a) a > 2 ∗ e s j ∗ j s j s ∗ j f + e s j ∗ j f ∗ b − e s j ∗ j f + e f j ∗ j s + 2 ∗ e s j ∗ j s (6b) b > 2 ∗ ( 1 − a ) a ∗ e s j ∗ [ j ] s + e s j ∗ [ j ] f + e f j ∗ [ j ] s [ j ] s ∗ [ j ] f + e s j ∗ [ j ] f

In order to compare between different detrital elemental tracers i in this mass-balance framework, we define a metric, φ, which denotes whether a signal from feedstock application is sufficient to overcome uncertainty, eⁱ, in the concentration of the immobile tracer i. Here φ is defined as: (7) φ = | Δ [ i ] m − Δ [ i ] s | e m i + e s i

Thus, φ > 1 where feedstock addition is resolvable above uncertainty, and φ < 1 where it is not. Where i is a single element, this can be rewritten as follows (see Supplementary material for full derivation): (8) φ = | a ∗ [ i ] f + ( 1 − a ) ∗ [ i ] s − [ i ] s | 2 e s i + ( e f i − e s i ) ∗ a

We can also employ the same framework to derive an expression for φ for a situation in which the ratio of multiple tracers ( i α i β ) is used (see Supplementary material). Note that in this framework mixing in the shallow soil sampling horizon is assumed to be homogeneous. However, this does not imply that homogeneous mixing is necessarily required to resolve signals from EW deployments using mass-balance, but rather that in considering the role of uncertainty in resolving EW it is most instructive to consider perfect mixing of feedstock and soil in deployments on average, as feedstock may be over-or under-sampled spatially in real settings.

We use the expressions above to calculate values of applied feedstock mass a, dissolution b, and resolvability φ for a range of feedstocks and background soil compositions, given different scenarios for feedstock application a, feedstock dissolution b, and aggregated sampling and analytical uncertainty e. To allow comparison across a wide range of soil types and geographies, we use data for five feedstocks as well as soil compositions from across the coterminous US.

In the analysis presented here, we limit our selection of feedstocks to mafic and ultramafic rocks, which have been widely proposed for EW (Schuiling and Krijgsman, 2006; Hartmann and Kempe, 2008; Köhler et al., 2010; Hartmann et al., 2013; Taylor et al., 2016; Edwards et al., 2017; Beerling et al., 2018, 2020; Andrews and Taylor, 2019; Lewis et al., 2021). These lithologies contain the highest concentrations of weatherable cations of all abundant silicate rocks, and therefore have the highest CDR potential per mass deployed (see Supplementary Figure S34). These rock types also contain many trace elements in relative abundance compared with other silicate rocks, carbonates, and background soils (see Figure 1). This is useful when considering which detrital trace elements could be used as representative immobile tracer(s) i for feedstock addition in a mass-balance framework. We use specific compositions that are representative of feedstocks used in previous EW trials (Lewis et al., 2021; Kantola et al., 2023), as well as global average values for common rock types. Individual feedstocks considered here are Pioneer Valley Basalt (PVB), Blue Ridge Basalt (BRB), and Almklovdalen Olivine (AO) (see also Figures 1, 2; Supplementary Figures S2–S4, and Supplementary Datasheet 1). Elemental concentrations of these feedstocks were measured as laid out in detail previously (Reershemius et al., 2023), and are consistent with published values where available (BRB; Lewis et al., 2021). We stress that for this study, these serve purely as representative feedstocks for illustrative purposes. We also use mean compositions of basalt and peridotite from the GEOROC database (Lehnert et al., 2000) in our analysis to compare to the individual marketed feedstocks (see Figure 1). Only samples in the GEOROC database described as “whole rock” with data for all elements of interest are used. For the mean basalt composition, we restrict samples to those with “Rock Name” designated as “basalt”; those samples designated as “subaerial”; and those collected in the contiguous USA to be more representative of realistic source material for EW feedstocks, (limiting the range of longitude and latitude in the search criteria to those of the extremities of the contiguous USA; 24.52° to 49.38°N, and 66.9° to 124.7°W; n = 1,557). For the mean peridotite composition, samples were allowed to be globally distributed and were restricted to those with “Rock Name” designated as “peridotite,” “lherzolite,” “dunite,” “harzburgite,” or “wehrlite” (n = 257).

These feedstock data are combined with soil data from a USGS dataset for soils of the coterminous US, utilizing the data for the top 5 cm of soil (Smith et al., 2013). The data are ideal for capturing large-scale soil variability because they are spatially balanced without adhering to a strict grid-based system (4,857 sample sites; 1 per 1,600 km²; see Figure 3; Supplementary Figures S5–S7 for maps of elemental and mineralogical concentrations) (Stevens and Olsen, 2000, 2003, 2004). Soil density data, relevant for the calculation of mass mixing ratios (a) based on feedstock application amounts in t ha⁻¹, were obtained from the National Soil Survey Geographic (gNATSGO) Database and the Gridded Soil Survey Geographic (gSSURGO) Databases (Boiko et al., 2021). The density data were interpolated using an inverse distance weighted interpolation algorithm (IDW) in ArcMap and extracted from the resulting layer for each sample site contained in the geochemical database (while capping data to the mean ± σ of all density data).

Using these data, a, b, and φ were calculated for each sample site using soil geochemical data and assuming a range of applied feedstock mass (a = 10, 25, 50, or 100 t ha⁻¹ total application); feedstock dissolution (b = 5, 15, or 30%); and aggregated sampling and analytical uncertainty (e = 1, 2, 5, 10, or 25%). Given that analytical uncertainty can be optimized to be <2% using isotope dilution, the lower end of these % uncertainty values represent cases where sampling uncertainty is neglected or has been reduced to the same order as analytical uncertainty (see discussion in sections 4.1, 4.2 and 4.4). Because of the greater ease of sampling from stockpiles, feedstock uncertainty was always assumed to be 1%. Although this is optimistic, the impact of the assumed feedstock uncertainty on parameters assessed here is low due to the low proportion of feedstock in soil-sampling mixtures at the feedstock application amounts considered here (generally <10% feedstock in soil-feedstock mixtures at application amounts < ~150 t ha⁻¹; see also Supplementary section S1.1 and Supplementary Figure S1). a and b were calculated in all cases using elements j (Mg + Ca + Na), excluding K which is a minor constituent in all feedstock compositions and is a major constituent of NPK fertilizers used in agriculture. φ was calculated separately for those elements i for which concentration data was present in the USGS soils database used: Ti, Al, Nb, Th, Y, Cr, and Ni. Additionally, φ was calculated for a subset of immobile element ratios i α i β (see above): Ti/Th, Al/Th, Y/Th, Ti/Nb, Y/Nb, Cr/Ti, Cr/Th, Ni/Ti, and Ni/Th. The data from individual sites were interpolated (IDW) in ArcMap. Resulting maps utilize “scientific color maps” (Crameri et al., 2020).

The framework we present here allows for the selection of arbitrary values for feedstock application amount (a), dissolution (b), and aggregated uncertainty on measurements (e). The usefulness of this discussion therefore depends on the selection of sensible criteria. Parameter values explored here were chosen to be realistic for early EW deployment scenarios.

Masses of applied feedstock (a), were chosen such that even the highest a values included here may be exceeded after 2–3 years of actual EW deployment. Academic enhanced weathering field trials commonly apply >40 t basalt ha⁻¹ per year (Larkin et al., 2022; Kantola et al., 2023; Beerling et al., 2024). Models estimating global CDR potential of EW have also used this application rate (Beerling et al., 2020, 2023; Kantzas et al., 2022), or alternatively cumulative addition of 150 t ha⁻¹ (Strefler et al., 2018). The lower application amounts are representative of basalt application as done purely for fertilization purposes (Swoboda et al., 2022), and are consistent with application amounts chosen in a modeling study focusing on the biological dimension of CDR from EW which uses 10, 30, and 50 t ha⁻¹ (Goll et al., 2021). However, application rates above 20 t ha⁻¹ will be a strong deviation from liming application rates (e.g., Mallarino et al., 2013).

The goal of EW is near complete dissolution of feedstock; achieving close to the full CDR potential of a feedstock will be important for EW to be cost effective. However, there are significant uncertainties regarding the timescales over which extensive dissolution will occur across settings. In practice, it has been difficult to relate field to lab or modeling based weathering rates for natural silicates (White and Brantley, 2003; Andrews and Taylor, 2019; Calabrese et al., 2022) because the former tend to be slower due to, e.g., surface passivation in soils through secondary precipitates and biofilms or fluctuating moisture content (Daval et al., 2018; Calabrese et al., 2022). In addition, factors such as grain size and, relatedly, specific surface area, likely exert a dominant influence on the rate in EW deployments (Strefler et al., 2018; Rinder and von Hagke, 2021; Amann et al., 2022; Rijnders et al., 2023; Vanderkloot and Ryan, 2023). As a result, it is difficult to predict a representative average value for b in EW deployments; and a trival portion of potential feedstocks have so far been tested. In mesocosm and field EW experiments, the fraction of applied feedstock that was weathered ranged for example from ~5.5% after 100 days (Vienne et al., 2022), 10% after 235 days (Reershemius et al., 2023) to ~22% after ~2.5 years average basalt residence time (Beerling et al., 2024). The values of 5 to 30% assessed here may therefore be seen as representative for a few months to years post-application, but will likely be exceeded over longer time horizons.

Our framework is agnostic as to the source of uncertainty e. It can therefore be seen as a aggregated measure of both analytical uncertainty and sampling uncertainty resulting from soil heterogeneity (also see section 4.2). Elemental concentrations of solid-phase samples can be measured using a variety of different methods, including X-ray fluorescence spectroscopy (XRF), inductively coupled plasma optical emission spectroscopy (ICP-OES), and ICP mass spectrometry (ICP-MS). Global analytical error for these methods varies, from <1% using isotope dilution to calculate concentrations using ICP-MS measurements, to >5% for typical XRF measurements (Reershemius et al., 2023). Hence, uncertainty deriving from spatial variability in background soil composition and soil + basalt mixture composition at the field scale are likely to be the larger source of uncertainty in real deployments. The scale of this uncertainty may also vary for different elements i and j depending on site-specific conditions. Importantly, uncertainty on an individual measurement of a sample will be much greater than the uncertainty of a mean calculated from many samples, when considering composition at the field scale. For example, in an EW field trial conducted in the US Corn Belt, Beerling et al. (2024) found that standard deviation of elemental concentration across pre-deployment soils was 7% for [Ti], 11% for [Ca], 13% for [Mg], 20% for [Al], 21% for [Na], and 25% for Th; however, standard error of the mean—an estimate of the variability of the sample mean—was 2% for [Ti], 3% for [Ca], 3% for [Mg], 5% for [Al], 5% for [Na], and 6% for Th. Regardless, it is clear that sampling methods will play a large role in determining uncertainty, and further systematic interrogation of this issue in the context of EW quantification and MRV is of critical importance. In our analysis, the smallest values chosen for e may be conceptualized as accounting for only analytical uncertainty, or for cases where sampling methods are successful in significantly limiting uncertainty that is a result of spatial variability, while larger e values (10–25%) may be seen as being more representative of aggregate uncertainty including both analytical and sampling uncertainties pertaining to individual measurements.

Differences in concentration of mobile (j) as well as immobile elements (or their ratios; i) between soil and feedstock are essential for detecting EW feedstock dissolution and addition, respectively. For a given feedstock, variability in EW signal resolvability is therefore driven by concentration gradients of relevant elements in soils (Figures 4–7). In terms of detecting feedstock dissolution (i.e., a and b; Figures 4, 5), areas where soil mobile cation concentrations are particularly low are ideal. These tend to be acidic soils in the southeastern US (compare with Figure 3D; Supplementary Figure S5). Importantly, this also implies that such soil-based methods may work better in sub-tropical and tropical soils, which are typically depleted in mobile cations due to intense chemical weathering (Brady and Weil, 2016), and which have been proposed as particularly promising sites for EW (e.g., Edwards et al., 2017). Soils enriched in j, where detecting feedstock dissolution is more challenging, are generally those with higher pH and therefore higher clay mineral and in some cases carbonate content (Figures 3C–F). These soil compositions should generally require less remediation of soil acidity than other arable lands that are depleted in j, and might therefore be less obvious targets for EW deployment from a purely agronomic perspective.

Similarly, the detection of feedstock addition (i.e., φ) also depends on the difference between soil and feedstock composition with respect to chosen immobile element tracers. Some connections to bedrock geology can be drawn here—for example Ti as a proxy for feedstock addition does not work well in the northwestern US (Figure 7), presumably because a higher prevalence of mafic volcanic rocks as protolith (Figure 3C) contributes to higher soil Ti concentrations. Interestingly, land use type does not have a clear impact on calculated φ parameters (Supplementary Figure S35). Spatial patterns of φ are more variable than for a and b. Concentrations of different immobile tracers i do not necessarily covary in soils, meaning that the immobile tracer i with the lowest threshold values of a and e on average may not be the best tracer to use in all sites. This variability is demonstrated in Figure 8, which shows that the immobile tracer i that results in the highest resolvability φ of a feedstock application varies greatly by geographic location (i.e., soil composition) and by choice of feedstock (maps for BRB, PVB, and AO can be found in the supplement together with larger maps for mean basalt and mean peridotite; Supplementary Figure S32).

In some cases, low resolvability of feedstock addition as a result of small Δ[i] may be overcome by comparing the ratio of immobile tracers in soils with that of feedstock, i.e., Δ( [ i ] α [ i ] β ) where i^α is a tracer i that is more enriched in feedstock than in soil (e.g., Ti in most mafic rocks), and i^β is a tracer i that is more enriched in soil than feedstock (e.g., Th for most mafic rocks). As a result, immobile element ratios outperform single elements in many locations for PVB, which generally has lower immobile element concentrations (see Supplementary Figure S32; best φ for PVB; and Supplementary Figures S23, S24). However, while this measure amplifies the compositional difference between feedstock and soil, it also changes the form of the mixing line, such that it may not necessarily be beneficial to the resolvability of a signal for basalt addition in all cases (see Figure 2). An additional benefit of this approach is that if the concentrations of immobile tracers i^α and i^β are correlated, using a ratio of trace elements may help reduce sampling uncertainty because the variability of [ i ] α [ i ] β in soils should be lower than variability of either i^α and i^β individually. As soil heterogeneity may be the main lever on aggregated uncertainties (e), this effect may be crucial in some cases for the detection of EW signals and should be considered in addition to mere optimization of φ values for a given location. A caveat for this approach is that the ratio of two detrital elements will change if one is more mobile than the other during weathering—i.e., if a larger relative amount of i^α is lost from the solid phase than i^β, or vice versa. If employing ratios of detrital tracers, this difference must be accounted for.

Crucially, the analysis presented here demonstrates the importance of successfully handling soil heterogeneity with appropriate sampling protocols and/or statistical resampling techniques, because otherwise spatial variability in soil elemental concentration at the field scale will be a major source of uncertainty relative to analytical uncertainty (see section 4.1). However, as noted above, the uncertainty of mean elemental concentration (i.e., the standard error of the mean) is significantly lower than that of individual measurements; and will decrease as the number of samples used to calculate a mean increases.

Broadly, our results suggest that depending on the quality of sampling and the size of analytical uncertainty, soil-based mass-balance approaches will often be useful for tracking EW in major agricultural regions of the US. Given that EW field trials usually apply large amounts of basalt each year (e.g., >40 t ha⁻¹) (Andrews and Taylor, 2019; Larkin et al., 2022; Kantola et al., 2023), feedstock addition and dissolution would be detectable even when considering aggregated analytical and sampling error of 10% (see Figures 4–6). However, these high rates may not be practical for many farmers and likely foster lower weathering efficiencies (for example due to clay formation). Similarly, feedstock addition is detectable in the Corn Belt using Ti for mean basalt and Ni or Cr for peridotite (Figures 7, 8; see section 4.4 for discussion of elemental immobility). However, if field-scale heterogeneity cannot be addressed adequately by sampling protocols (i.e., e ≥ 25%), addition and dissolution of feedstocks during EW deployments will be difficult to resolve. This is also the case for lower masses of total feedstock application than is commonly practiced in field trials to date—which, again, may be more realistic for deployments on farms. Nevertheless, it is important for EW practictioners to have the ability to assess resolvability of an EW deployment given their specific field conditions (see section 4.5).

There is considerable overlap in the compositions of US top soils and potential EW feedstocks (see Figures 1, 2). However, Δ[i] and Δ[j] between feedstock and soil are key factors in determining the resolvability of a signal for feedstock application and dissolution, respectively. Given that most stakeholders (e.g., EW suppliers and farmers) are much more flexible with regards to feedstock composition than soil, the choice of feedstock and immobile tracer i must be informed by the specific soil composition of the site where soil-based mass-balance is being utilized.

In this context, maximizing Δ[j] through feedstock selection may already be in the interest of the EW supplier, because the concentration of mobile elements in feedstocks linearly scales with their CDR potential (Renforth, 2012, 2019; Supplementary Figure S34). Feedstocks should then be chosen that are enriched in a detrital element i that has low background concentrations in local soils. It should be noted here that not all of the “immobile” elements assessed here are immobile during weathering in all settings. To avoid underestimating CDR (see section 4.4), it is therefore in the interest of the stakeholder to ensure that the chosen tracer i is largely immobile during weathering on the relevant timescales. If this condition is not met, properly accounting for feedstock addition may require longer sediment cores (e.g., 1 m—to test for migration of i to below the ploughed layer) or testing effluent waters and harvested biomass.

The impact that feedstock choice may have on the resolvability of EW signals through soil-based mass-balance approaches is demonstrated by the difference in φ values between BRB and PVB (Figure 7; Supplementary Figures S19, S20). This is especially pronounced when comparing mafic to ultramafic feedstocks. Mafic rocks such as basalts generally have concentrations of some lithophile elements that can be used as immobile tracers, including Ti, Al, and Y, that are high relative to soils (Figure 1). By contrast, ultramafic rocks are depleted in these elements relative to soils and mafic rocks. As a result, there is a smaller selection of potential tracers i that may be used to track application of ultramafic rocks in EW deployments (e.g., Cr, Ni)—though given the much higher concentrations of these elements in peridotites than in soils resolvability φ for these elements in peridotite feedstocks is much greater (see also Supplementary Figures S30, S31). If EW suppliers are flexible with regards to feedstock type (basalt vs. peridotite), switching to peridotite may therefore be a viable strategy for soils where high concentration of lithophile elements may prevent detection of feedstock addition for basalt.

In any case, if soil-based mass balance approaches are to be widely used for the purpose of calculating EW rates in field deployments, care must be taken that feedstock application amount, feedstock choice, analytical methodology, and choice of immobile tracer are all selected with consideration of site-specific soil properties. Efficacy of soil-based mass balance approaches can be greatly improved by choosing a suitable feedstock. However, there may not be an optimal feedstock for all soils, particularly if concentrations of major and detrital elements in soils are high. In this context it is important to note that resolvability in this framework only refers to the efficacy of soil-based mass-balance approaches, not to the question of whether weathering and CDR occur. In other words, a given scenario below the resolvability threshold only means that soil-based mass-balance approaches will be inadequate for quantifying CDR in these settings, but CDR may still be occurring through EW.

Provided soils and feedstocks are matched in an effective way, and soil sampling is effective at constraining and minimizing uncertainty due to spatial heterogeneity in background soil and soil+basalt mixtures at the field scale, soil-based mass-balance approaches have clear potential for use in MRV of EW. However, there are some limitations to this class of MRV approaches that are important to consider.

One clear result of the feasibility assessment presented here is that soil heterogeneity needs to be addressed adequately for soil-based mass balances approaches to be able to quantify EW. At the largest aggregated sampling and analytical uncertainty assessed here (e = 25%), olivine weathering in most agricultural areas of the US requires very large masses of feedstock (a ≥ 100 t ha⁻¹) and extensive dissolution (b ≥ 40%) in order to be resolvable, while basalt dissolution will not be detectable even at 100 t ha⁻¹ (see Supplementary Figures S8–S17). At higher aggregated sampling and analytical uncertatinty and 50 t basalt application ha⁻¹, none of the detrital elements tested here would be reliable for assessing feedstock addition (Supplementary Figures S18–S24). These findings have fundamental implications for the utility of soil-based approaches. It is paramount for EW studies wishing to base MRV on soil-based approaches to successfully handle soil heterogeneity, an aspect not addressed in previous mesocosm experiments (Reershemius et al., 2023), or rigorously tested in large-scale field trials to date (e.g., Kantola et al., 2023; Beerling et al., 2024). Moving forward, EW deployments should prioritize testing and optimization of sampling protocols to capture in-field heterogeneity at multiple scales, which could potentially drive down the uncertainty derived from sampling a spatially heterogeneous background. Importantly, if the central tendency of soil elemental concentration at the field scale can be robustly and representatively determined through sampling (i.e., determining mean concentration with a small standard error of the mean), then spatial variability is not necessarily a barrier to resolvability. There is a clear need to test this hypothesis with additional field trials.

The effect of enrichment on the concentration of detrital elements i with dissolution of the feedstock is also an important issue that is not incorporated into the mass-balance frameworks presented here or elsewhere (Reershemius and Suhrhoff, 2023; Reershemius et al., 2023). However, a correction for feedstock dissolution can be easily applied that accounts for this concentration effect after loss of j has been calculated, given the ratio of mass soil:feedstock and the relative concentrations of i in soil and feedstock (Reershemius et al., 2023). As such, ignoring this concentration effect does not change the outcomes of our analyses. Furthermore, this concentration will cause the post-weathering soil+feedstock mixture to fall further away from the initial mixing line (see Supplementary Figure S36). Not accounting for it in this framework is therefore a conservative approach when estimating resolvability of EW signals where i in feedstock is greater than in soil.

There are some proposed EW feedstocks—notably wollastonite (Haque et al., 2019, 2020b,c; Paulo et al., 2021; Jariwala et al., 2022; Wood et al., 2022; Feng and Hicks, 2023)—that do not have significant concentrations of trace elements that can be used for the purposes of tracking feedstock application. The same is true for using carbonate feedstocks (Knapp and Tipper, 2022; Zhang et al., 2022). In these cases, using a detrital tracer i to track weathering rates is not workable. Addition of a feedstock that is more depleted in i than the soil to which it is deployed will result in the soil + feedstock mixture becoming more dilute in i. If this dilution effect overcame limitations of analytical and sampling uncertainty, an amount of feedstock application could be calculated as presented here. However, subsequent dissolution of this feedstock will result in an increase in the concentration of i such that the composition of soil + feedstock after weathering tracks the mixing line between the feedstock and soil (Supplementary Figure S36). This means that a signal for feedstock addition and subsequent weathering is indistinguishable from a signal for lower feedstock addition on its own, which defeats the purpose of using a detrital trace element to track weathering rates. Given this consideration, it should be noted that some potential detrital (or semi-immobile, detrital-like) elements i (Th, Zr, and REEs such as Pr, Nd, Sm—see Figures 1, 2; Supplementary Figure S3) on their own are unsuitable for this purpose for almost all mafic and ultramafic rocks. Moreover, despite apparent resolvability φ > 1 for AO when using Ti as a detrital tracer at high application a and low uncertainty e, it should be noted that given [Ti] is greater in most soils than in AO, the dissolution effect outlined above means that this result is erroneous (see Supplementary Figure S22). When performing these analyses, cases should therefore be excluded where [i]_s > [i]_f, which is also reflected in the online tool we present in section 4.5.

Furthermore, this framework is agnostic to potential mobility in certain detrital trace elements i during weathering. One practical measure of elemental (im-)mobility during weathering on a global scale is the ratio of world average river water concentration over the abundance of in the continental crust (Gaillardet et al., 2014). This measure indicates that Ti and Al may be among the most immobile elements. However, Ni, and Cr are more than 10 times more mobile. Given that parent material concentration rather than soil/leaching condition is the primary control on Ni and Cr concentrations in natural soils developing on both ultramafic (Kierczak et al., 2021) and basaltic (Suhrhoff, 2022) bedrocks, the assumption of immobility may be warranted for most soils to some degree. However, both Ni and Cr show higher mobility at low soil pH (Alloway, 2012). Specific caution should therefore be placed on the immobility of these tracers in acidic soils, though this also means that EW itself may positively impact Ni and Cr immobility. Furthermore, Cr mobility can also be promoted by elevated Cl-and F-concentrations (Seigneur and Constantinou, 1995; Alloway, 2012) as well as the presence of fulvic and organic colloids through formation of soluble Cr(III) complexes (James, 1996; Alloway, 2012). Mobility of Cr is further complicated through its redox sensitivity as Cr(VI) is generally more mobile than Cr(III) (Alloway, 2012). The oxidation of Cr(III) to Cr(VI) may be promoted by the presence of organic colloids (James, 1996; Alloway, 2012) and presence of Mn oxides in soils (Bartlett and James, 1979; James and Bartlett, 1988; Alloway, 2012). Importantly in the context of EW, elevated soil Cr(VI) concentrations have also been reported following P fertilization (Becquer et al., 2003; Alloway, 2012). Hence, the assumption of immobility should be a particular focus of EW employments in highly acidic soils, soils high in labile organic matter, and high P-fertilization regimes. Crucially, if detrital elements are lost from soils, this will cause underestimation of feedstock addition and of the fraction of weathering that has occurred. This is because the loss will not affect measured mobile cation concentrations but will reduce the amount of initial feedstock assumed to be present. Having a partially immobile tracer therefore means foregoing a part of earned CDR credits, but should not introduce the risk of exaggerating CDR.

Finally, some of the detrital elements discussed here—primarily Ni and Cr—may be toxic if available to plants and animals at elevated concentrations in soils. This has received extensive attention elsewhere in the context of EW (Haque et al., 2020a; Suhrhoff, 2022; Dupla et al., 2023; Vink and Knops, 2023) and is not the focus of this study. For these feedstocks and detrital elements, the need to quantify feedstock addition through the presence of a detrital element and the wish to avoid environmental risks lead to inherently contradictory feedstock choices, requiring a compromise between both. In this context we wish to stress that for these elements, the premise should not be to maximize φ as much as possible through the selection of high concentration feedstocks. Rather, assuming that other detrital elements are not viable tracers for feedstock addition, elemental concentrations in feedstocks should only be as high as necessary to detect feedstock addition (say, φ > 2), but as low as possible to avoid environmental risks.

A key point we wish to emphasize is that although we present this framework in the context of a US case study, our approach can be applied to any dataset of soil composition, at any scale, and that the framework outlined here can be used as a tool for EW practitioners to use soil composition data from their fields as a guide for site-specific best practice in EW deployment. Moving forward, it will be particularly interesting to test the utility of soil-based approaches in tropical soils. These are also regions that may exhibit the potential for high weathering rates as a result of local climate (Edwards et al., 2017; Beerling et al., 2020), as well as the strongest potential for EW-driven fertilization effects (Goll et al., 2021). Tropical soils have very different background concentrations of mobile and immobile elements. Because mobile cations are typically depleted in such soils (Brady and Weil, 2016), soil-based approaches may be particularly sensitive in detecting dissolution of EW feedstock. Furthermore, because the ability of soil-based approaches to resolve EW signals scales directly with the concentration difference between soil and feedstock, the uncertainty imposed by sampling of soil heterogeneity may be lower in relation to the overall EW signal compared to temperate soils of high (mobile) cation concentrations. On the other hand, depending on site-specific conditions, tropical soils can have elevated immobile element concentrations or elevated mobility of immobile elements (e.g., Cornu et al., 1999; Du et al., 2012; Jiang et al., 2018), such that accurate detection of feedstock addition might be more challenging. In principle, although the utility of this framework will vary from site to site, the high potential for detection of both EW feedstock dissolution and addition in the southeastern US (Figures 4–8) is promising for soil-based approaches in tropical settings.

In order to facilitate practitioner use of site-specific data for individual deployments, we present this framework in a user-friendly online tool (Figure 9). This tool allows users to provide data on soil composition, expected aggregated analytical and sampling uncertainty, as well as different potential feedstock compositions, and reports information on the required mass of feedstock addition a,¹ relative feedstock dissolution b,² and resolvability of feedstock addition based on a single detrital tracer i³ and ratios of two detrital tracers i⁴ required to resolve EW feedstock application and dissolution using solid phase mass-balance. This tool is meant to help practitioners plan EW deployments where weathering rates and related CDR can be robustly monitored and verified. Additionally, this tool can also be used to vet reported weathering rates calculated using soil-based mass balance in EW deployments, given information on the use of tracers, feedstocks, and soil compositions, and could be used as part of efforts to standardize oversight in the growing market for carbon credits using EW.