Improved calculation of warming-equivalent emissions for short-lived climate pollutants

Anthropogenic global warming at a given time is largely determined by the cumulative total emissions (or stock) of long-lived climate pollutants (LLCPs), predominantly carbon dioxide (CO2), and the emission rates (or flow) of short-lived climate pollutants (SLCPs) immediately prior to that time. Under the United Nations Framework Convention on Climate Change (UNFCCC), reporting of greenhouse gas emissions has been standardised in terms of CO2-equivalent (CO2-e) emissions using Global Warming Potentials (GWP) over 100-years, but the conventional usage of GWP does not adequately capture the different behaviours of LLCPs and SLCPs, or their impact on global mean surface temperature. An alternative usage of GWP, denoted GWP*, overcomes this problem by equating an increase in the emission rate of an SLCP with a one-off “pulse” emission of CO2. We show that this approach, while an improvement on the conventional usage, slightly underestimates the impact of recent increases in SLCP emissions on current rates of warming because the climate does not respond instantaneously to radiative forcing. We resolve this with a modification of the GWP* definition, which incorporates a term for each of the short-timescale and long-timescale climate responses to changes in radiative forcing. The amended version allows “CO2-warming-equivalent” (CO2-we) emissions to be calculated directly from reported emissions. Thus SLCPs can be incorporated directly into carbon budgets consistent with long-term temperature goals, because every unit of CO2-we emitted generates approximately the same amount of warming, whether it is emitted as a SLCP or a LLCP. This is not the case for conventionally derived CO2-e.

(SLCPs). The atmospheric lifetime and radiative impacts of different GHGs differ dramatically. Acknowledgement of this reality led to the widescale adoption of the GWP100 methodology. GWP100 equates emissions using a scaling factor -CO2-e. GHGs are assigned a GHG equivalency, then that number is used to determine the emissions' potential impact. Following GWP100, a pound of methane equates to 25 pounds of CO2. Thus, methane is calculated as 25CO2e. However, this simplified scaling factor fails to recognize the amount of time emissions remain in the atmosphere -an equally important factor in determining potential atmospheric impact. The GWP* methodology seeks to remedy this oversight. 1 Anthropogenic warming estimations are largely determined by the cumulative total emissions of LLCPs and the emission rates of SLCPs. GWP* equates an increase in the emissions rate of an SLCP with a single "pulse" emission of CO2, and thus considers not only the initial intensity of GHGs, but also the amount of time that they remain in the atmosphere. This approach is a significant improvement on the conventional GWP100 methodology. Further, the GWP* methodology modifies the conventional GWP definition to consider CO2 warming equivalents (CO2-we) rather than CO2-e. Following GWP*, SLCPs can be incorporated directly into carbon budgets consistent with long-term temperature goals, because every unit of CO2-we emitted generates approximately the same amount of warming, whether it is emitted as a SLCP or a LLCP. This is not the case for conventionally derived CO2-e measurements.

Greater Recognition of Grassland Carbon Sinks
NCBA is pleased with the Agency's effort to recognize existing GHG emission offsets. As the Agency noted in the Draft Inventory, carbon sinks account for a 20% offset of agricultural GHG emissionssignificantly reducing the net impact of the industry. NCBA encourages the bolstering of this section generally, so that regulated stakeholders and consumers alike can assess the net impact of GHG emitters. Going forward, NCBA urges EPA to specifically consider the environmental benefit of managed grazing, a conservation practice implemented by ranchers across the country. It is well-known that rotational grazing leads to increased carbon sequestration. 2 Globally, if soil organic carbon in agricultural lands and grasslands increase 10% over the course of the 21st century, carbon dioxide concentrations in the atmosphere could be reduced by 110 ppm. 3 NCBA appreciates the opportunity to comment on the Agency's Draft Inventory and looks forward to continued engagement on this important issue.

INTRODUCTION
Comprehensive climate policies must appraise a range of greenhouse gases and aerosols, which can differ significantly in their radiative efficiencies and atmospheric lifespans, and hence the nature of their climate impacts. 1 To reflect this, different climate pollutants are often expressed using a common emission metric. Emissions reporting under the United Nations Framework Convention on Climate Change (UNFCCC) now requires the use of 100-year Global Warming Potential (GWP 100 ) to account for all gases as carbon dioxide equivalent (CO 2 -e) quantities. Despite its prevalence in the UNFCCC and national climate policies, GWP has received criticism, [2][3][4] not least that it cannot be used to appraise temperature-related goals, 5 and other equivalence metrics have been proposed. [6][7][8][9] Indeed, Shine 3 notes that strong caveats were in place when GWP was introduced in the Intergovernmental Panel on Climate Change's First Assessment Report 10 : "It must be stressed that there is no universally accepted methodology for combining all the relevant factors into a single [metric]… A simple approach [i.e., the GWP] has been adopted here to illustrate the difficulties inherent in the concept." Working Group 1 of the Fifth Assessment Report, AR5, did not recommend any metric and emphasised that the choice of metric depends on the specific goal of the climate policy. In AR4, however, the GWPs were the recommended metric to compare the effects of long-lived greenhouse gases, 11 and AR5 values of GWP 100 have now been adopted for emissions reporting (see the textual proposal from 12 December 2018 on the transparency framework for action and support referred to in Article 13 of the Paris Agreement: https:// unfccc.int/process/bodies/subsidiary-bodies/ad-hoc-workinggroup-on-the-paris-agreement-apa/information-on-apa-agendaitem-5).
The temperature response to emissions is ambiguous under GWP 1,12,13 and this ambiguity is particularly relevant in the context of the Paris Agreement, given its stated aim of 'holding the increase in the global average temperature well below 2°C above pre-industrial levels and pursuing efforts to limit the temperature increase to 1.5°C.' Beyond the reference to a balance of emissions by sources and removals by sinks well before the end of the century, neither the means by which this is to be achieved nor the metrics used to assess progress are explicitly stated. 14 Tanaka and O'Neill 15 demonstrate that net-zero aggregate CO 2 -e emissions based on GWP 100 (which is often assumed to be the definition of the balance of sources and sinks described in the Paris Agreement) are not essential to limit warming to 1.5°C. Wigley 16 posits that the balance of sources and sinks in Article 4.1 of the Paris Agreement is scientifically inconsistent with the temperature goals in Article 2.1. These papers show how moving from the temperature goals articulated in the Paris Agreement to emissions targets and profiles is not something that is currently well-handled by conventional carbon accounting; they also show that the area is receiving renewed scrutiny as countries, firms and sectoral bodies try to work out mitigation strategies of their own. This paper demonstrates a method that unambiguously links aggregated greenhouse gas emissions with their warming outcomes on decade to century timescales, allowing short-lived climate pollutants (SLCPs) to be brought into a carbon budget framework. 17 It is designed to be useful for informing policies that specifically aim to limit global warming, as is required under the Paris Agreement. This method builds on the revised usage of GWP, denoted GWP*, proposed in Allen et al., 12,18 building on Shine, et al. 6 Specifically, we address a shortcoming in the originally proposed definition of GWP*, in that it did not account for the delayed temperature response to past increases in SLCP emissions, bringing aggregate emissions into closer agreement with both CO 2 -forcing-equivalent emissions 19 and the temperature response.

RESULTS
A revised definition of GWP* A new usage of GWPs, denoted GWP*, allows emissions of shortlived and long-lived climate pollutants (SLCP & LLCPs) to be more consistently expressed within a single metric by equating a change in the emission rate of an SLCP as equivalent to a single emissions pulse of a long-lived pollutant. As originally defined in Allen, et al., 18 a step-change in emission rate of an SLCP (ΔE SLCP tonnes per year) is equivalent to a one-off pulse emission of ΔE SLCP × GWP H × H tonnes of CO 2 , where GWP H is the conventional Global Warming Potential relative to CO 2 , integrated over a timehorizon H years. Emissions of LLCPs, defined here as those having an atmospheric lifetime longer than H, will still behave as a cumulative pollutant within time-horizon H, and therefore equivalent emissions for LLCPs are derived simply by multiplying those emissions by GWP H . This rate-based equivalence for SLCPs overcomes the problems inherent in GWP (or any pulse-based metric) in not adequately distinguishing their largely non-cumulative behaviour. However, although a sustained SLCP emissions rate will result in a stable atmospheric concentration and hence maintain the same level of forcing, some additional long-term warming will occur while the climate system is still equilibrating to past increases in SLCP emissions. Note this is not a cumulative impact of emissions mirroring that of CO 2 : it is, rather, the delayed response associated with equilibration to a past increase in forcing. After a sufficiently long period of constant emissions (on the order of centuries), SLCP-induced warming will stabilise, whereas CO 2 -induced warming continues to increase as long as CO 2 emissions remain above zero. After CO 2 emissions reach zero, ongoing thermal adjustment in surface temperature is largely balanced by ocean uptake of CO 2 , 20 at least in the absence of strong Earth System feedbacks. 21 The multi-century component of the thermal response of the climate system, together with carbon cycle feedbacks, 22 act to prolong the warming impact of SLCP emissions. 23 As noted in Allen, et al., 12 this can be incorporated 'by including a small contribution that scales with time-integrated [SLCP] emissions.' This component was not pursued in Allen, et al. 12 for simplicity and because the contribution of this multi-century adjustment to past increases in SLCP emissions is small compared to the impact of current changes in SLCP emissions under the scenarios considered in that paper (see their Fig. 2 Fig. 1 to this paper). Nevertheless, it may be significant for individual countries whose SLCP emissions have increased within the past half-century or so and are now approximately stable.

and supplementary
Thus, we propose a re-definition of GWP* to incorporate both timescales, as well as providing a theoretical justification below, we here adjust this empirically to produce the best fit between cumulative CO 2 -warming-equivalent (CO 2 -we) emissions and resultant warming (see below and Methods for full details). Calculated using this re-defined GWP*, CO 2 -we emissions of an SLCP in a given year are defined: where GWP H is the conventional global warming potential for a given SLCP over time-horizon H, ΔE SLCP the change in SLCP emission rate over the preceding Δt years, E SLCP the SLCP emissions for that year, and r and s the weights assigned to the rate and stock contributions, respectively. The only difference between this formulation and that of Allen et al. 12,18 is that they used r = 1 and s = 0. Including the time period Δt spreads the CO 2 -we pulse corresponding to a change in SLCP emission rate over Δt. Allen et al. 12 suggest at least 20 years, which has the effect of reducing the volatility in CO 2 -we emissions and improving the correspondence with temperature response. The first ("rate") term on the right-hand side, r ΔESLCP Δt H GWP H , represents the response to the changing SLCP emission rates. The second ("stock") term, s × E SLCP × GWP H , is added to represent the longterm equilibration to past increases in forcing, which can be approximated by a small term scaling with cumulative SLCP emissions. In other words, the rate term approximates the shorttimescale climate response to a change in radiative forcing; the stock term approximates the long-timescale equilibration which occurs even when there is constant radiative forcing.
The exact values of r and s will depend on the precise timescales of the climate response to radiative forcing, on how long ago the increase in SLCP emissions occurred, and on carbon cycle feedbacks, all of which are uncertain and scenariodependent. Constraining r + s = 1 ensures that total CO 2 -we emissions over 100 years corresponding to a steady SLCP emission starting in year 1 are the same as total CO 2 -e emissions would be, consistent with the original derivation of GWP* presented in Allen, et al. 18 The ratio s/(rH) corresponds to the fractional rate of decline of SLCP emissions for these to be considered equivalent to a zero rate of CO 2 -we emissions and hence not cause further warming. This depends on the SLCP and details of the scenario in question. Our objective is a simple and reliable indicator of the relationship between emissions and recent and near-term future warming associated with the largest non-CO 2 climate drivers, so we estimate r and s using a multiple linear regression onto the response to methane emissions in commonly used scenarios (representative concentration pathways, RCPs), focusing on the time period 1900-2100.
The method is applicable to other SLCPs, but the optimal values of r and s could be different for each SLCP dependent on past emissions. As many short-lived industrial gases have only started being released in recent decades, the warming responses to these gases is likely distinct to those for methane, with a greater emphasis on the immediate effects of changing emission rates, and thus not necessarily reflecting the same r and s values as derived for methane here. That said, GWP*, like any metric, depends on strong assumptions of linearity, so the additional precision may not be worth the additional complexity. The application of GWP* to specific other gases is beyond the scope of this paper but warrants further investigation, given their impacts are substantial and potentially growing. For example, the radiative forcing from total halocarbons (long and short-lived) is less than half that from methane 1 but may grow with increasing demand for air conditioning.
Empirical estimation of flow and stock contributions The global mean surface temperature (GMST) responses (ΔT) to methane radiative forcing from the RCP2.6, 4.5 and 6 scenarios taken from the AR5 database, 24 were derived using the default configuration of the FaIR simple climate model. 25 Methane was M. Cain et al.
chosen as it has the largest impact on radiative forcing of all the SLCPs, and these scenarios were used as they represent pathways approximately in line with current policies, as well as a more ambitious pathway (RCP 2.6). Multiple linear regression was then used to find values of r and s to best fit the relationship between cumulative emissions of CO 2 -we and ΔT (using the RCP emissions from the AR5 database), constrained such that r + s = 1. Using this approach, r = 0.75 and s = 0.25 are the mean values based on the three RCPs and are found to provide a good fit for all three (see methods section below).
To demonstrate the improved warming-equivalence of GWP*, cumulative emissions (left axes) are shown alongside the corresponding warming in Fig. 1. The change in GMST calculated from the methane radiative forcing is shown as a time series (black dashed line, right axis) for RCP 2.6 (upper), RCP 4.5 (middle) and RCP 6 (lower). The cumulative CO 2 -e emissions calculated using GWP 100 (cyan) show that there is no agreement between CO 2 -e emissions and warming when methane emissions are stable or in decline. CO 2 -we emissions calculated with GWP*, especially when both stock and flow properties are included (purple), show a clear improvement with the cumulative emissions matching the temperature response.
Orange lines show cumulative CO 2 -we emissions retaining only the flow term i.e., setting r = 1, s = 0 in equation 1, equivalent to the definition in ref., 12 which scales with smoothed annual emissions (dotted line, inner left axis). GWP* defined by only the change in rate of methane emissions (i.e., as originally defined) overestimates the cooling that would occur under decreasing methane emissions because it fails to take into account the century-timescale response to earlier methane emission increases. Figure 2 shows cumulative CO 2 -e and CO 2 -we emissions of methane plotted against the modelled temperature response, relative to 1900 and up to 2100, for the RCPs. The linear relationship between cumulative CO 2 emissions and warming is the basis for the carbon budget concept, which describes how much CO 2 can be emitted before any given threshold of global mean warming is reached. Previous studies 26,27 have computed CO 2 budgets conditioned on specific scenarios for non-CO 2 forcing. GWP* allows non-CO 2 forcings to be included in the carbon budget itself, as it describes a linear relationship between cumulative CO 2 -we emissions of SLCPs and warming, as shown in purple in Fig. 2.
Under GWP 100 (cyan) this relationship breaks down completely when SLCP emissions start to decline. Unphysically, GWP as traditionally used implies declining methane emission rates still contribute to increasing cumulative CO 2 -e, when they are in fact causing cooling. Hence, the negative gradient towards the end of the cyan scenarios in Fig. 2. If only the flow properties of methane are considered (orange), reducing methane emission rates are now equivalent to a negative CO 2 emission, so the line 'turns back' on itself as cumulative emissions decline alongside reducing temperatures. Although much closer to a linear relationship than GWP 100 , in this exclusively rate-based version there is now more CO 2 -we removal than would be expected to explain a given amount of cooling, if we consider that the truly equivalent relationship should mimic that of CO 2 emissions and temperature response, which is approximately linear. 28 This is because the ratebased formulation alone does not account for the ongoing warming from the thermal equilibration to past methane emission increases. Incorporating a term that accounts for this behaviour, as proposed above (purple lines), largely overcomes the problem, providing a close match to a linear relationship for the scenarios , which is the approximate slope of Fig. 2 for the GWP* flow and stock (purple) Fig. 2 Cumulative methane emissions (1900-2100) from the historical period plus RCP 2.6 (solid), RCP 4.5 (dashed) and RCP 6 (dotted) converted to CO 2 -e emissions using GWP 100 (cyan), GWP* using only the flow properties (orange), GWP* for both flow and stock properties using r = 0.75 and s = 0.25 (purple), against modelled warming response to the methane radiative forcing from the scenario database. TCRE values of 1.5 (shallowest gradient), 2.0, and 2.5 K per trillion tonnes Carbon equivalent (or 0.41, 0.55 and 0.68 K per trillion tonnes of CO 2 -we) are shown by grey lines tested here. The relationship shown here corresponds to a Transient Climate Response to cumulative carbon Emissions (TCRE, or the amount of warming per unit carbon emitted, shown by the grey lines in Fig. 2) of about 1.8°C per TtC during the historical period and slightly less in the RCP projections. The slope changes because there is a time lag in the climate's response, therefore negative emissions do not immediately reverse the effects of the same amount of emissions. This effect was reported by Zickfeld, et al. 29 for CO 2 removals. The TCRE depends directly on the prescribed TCR of the FaIR model used to generate the temperature: but, crucially, would be similar for all non-CO 2 forcings provided their different efficacies are taken into account using effective radiative forcing. 1 Adding a small correction to account for the fact that the climate system takes time to equilibrate to higher forcing permits a physically plausible interpretation of "equivalence" in the calculation of carbon budgets. This improves the accuracy of the temperature outcome of the equivalence metric, however it should be noted that this is still an application of GWP 100 and therefore does not capture everything that a climate model includes. Collins, et al. 9 investigate the impact of methane on the remaining carbon budget using an intermediate complexity climate-carbon model, and note that their results show a close correspondence to the GWP* metric as proposed in Allen, et al. 18 Physical interpretation and justification We can illustrate the physical interpretation of r and s values by considering some more idealised scenarios. Setting the left-hand side of equation 1 (CO 2 -we emissions) to zero, we are able to calculate the methane trend required to be equivalent to no further CO 2 emissions: ΔE SLCP /Δt = −[s/(rH)]E SLCP , which is required to generate a radiative forcing pathway that will approximately stabilise temperatures over the time period Δt. Hence with r = 0.75, s = 0.25 and H = 100 years, [s/(rH)] = 0.3% is the rate at which methane emissions need to decline to give stable methaneinduced warming. This makes zero CO 2 -we emissions under GWP* consistent with stable temperatures, matching the temperature response to zero CO 2 emissions.
The definition of CO 2 -we using GWP* is independent of SLCP lifetime (assumed to be much shorter than H), but it does depend on the SLCP forcing history: if temperatures are close to equilibrium following a very gradual forcing increase over many centuries, a near-zero decline rate (near constant SLCP emissions) would be consistent with no further warming. Faster rates of decline would be required to maintain no further warming following a rapid increase in SLCP forcing, because the climate system would be further from equilibrium. Here we have based the coefficients and therefore the rate of decline on a combination of historical (1900 onwards) and future scenario emissions to encompass climate response in the near future to emissions over the last century.
We have used an empirical method to find a definition of GWP* that preserves the link between an emission and the warming it generates in the medium term up to 2100. The physical interpretation of equation 1 is that the flow term (with coefficient r) represents the fast climate response to a change in radiative forcing, generated by the atmospheric and ocean mixed-layer response. 30 The timescale of this response is about 4 years here. 31 The stock term (with coefficient s) represents the slower timescale climate response to a change in radiative forcing, due to the deep ocean response. This effect means that the climate responds slowly to past changes in radiative forcing, and is why the climate is currently far from equilibrium. We have approximated this response by treating a quarter of the climate response to a SLCP as "cumulative". The timescale for this response is uncertain, 32 and is of the order a few centuries, as discussed below.
The exponential decline of 0.3% per year corresponds to a time constant of about 300 years, consistent with the equilibration timescale of the climate system. This timescale is largely governed by deep ocean adjustment to relatively recent forcing increases identified by Geoffroy, et al. 32 (multi-model mean of 290 years, with a standard deviation of 107 years). If the equilibration timescale of the climate system were shorter, then s would be lower. If the deep ocean response timescale were the same as the atmosphere and mixed ocean timescale (about 4 years), then r would be 1, s would be 0, and the definition of GWP* from Allen, et al. 12 would not change.
The rate at which global methane emissions need to decline to reduce the rate of methane-induced warming to zero has not been explored systematically with complex models and would depend on the details of its atmospheric chemistry. An indication of the rate of decline of emissions of a generic SLCP required to stabilise SLCP-induced warming following a linear increase in SLCP emissions over a multi-decade period can be provided by considering the generic response to the following widely studied scenario: a linear increase in forcing to the equivalent of a CO 2 doubling over 70 years, followed by constant forcing. If the thermal response of the climate system is characterised by a short (sub-decadal, atmosphere and mixed layer ocean) adjustment time, d 1 , and a long (multi-century, deep ocean) adjustment time, d 2 ≫ 70 years, temperatures after year 70 adjust exponentially from their value at year 70 (the Transient Climate Response, or TCR) to their long-term equilibrium value (the Equilibrium Climate Sensitivity, or ECS) with an adjustment timescale d 2 . Hence, warming would rise at a fractional rate of (ECS−TCR)/(d 2 × TCR) per year in the decades immediately after forcing is held constant (see supplementary Fig. 2). On multi-decade timescales longer than the SLCP lifetime, SLCP emissions would need to fall at the same fractional rate to yield no further warming, since the rate of SLCP emissions scales with SLCP-induced forcing and the temperature response is linear in forcing.
For representative values, (ECS = 2.75°C, TCR = 1.6°C, d 2 = 239 years, after Millar, et al. 31 ) this indicates a decline rate, (ECS−TCR)/ (d 2 × TCR), of 0.3% per year, corresponding to a time-scale of 333 years (the inverse of 0.3%/year). This is consistent with our estimates of r = 0.75 and s = 0.25 for H = 100 years, which give a time-scale (r/s) × H of 300 years. This indicates the approximate rate of decline of methane emissions required for no further warming. However, this timescale depends on the multi-century response of the climate system and carbon cycle feedbacks, all of which are poorly constrained by available observations and modelling: targeted experiments varying methane emission growth and decline rates would give a more precise indication.
The relationship between stable or declining methane emissions is shown in Fig. 3 for a range of model parameters in the simple climate model, FaIR (assuming a constant methane lifetime). Different colours show simulations with a 1-sigma range of d 2 values from Geoffroy, et al., 32 and for a range of realised warming fractions (the ratio of TCR:ECS) based on that in the CMIP5 ensemble. 33 The dashed lines show the scenario in which methane emissions are kept stable for 130 years after a 70-year ramp up to approximately present day emission rates. For all parameter combinations, constant methane emissions cause a continued warming. The solid lines show scenarios which reduce the emissions by a fractional rate of (ECS−TCR)/(d 2 × TCR) per year to compensate for the slow climate response. As predicted, these give stable temperatures over the decades following the emissions peak. Decline rates range from between 0.06% (TCR:1.6°C, ECS:2.0°C, d 2 :397.0 years) and 0.55% (TCR:1.6°C, ECS:3.2°C d 2 :183.0 years). Note that these values have been calculated based on a simple climate model that emulates the response from complex climate models. There remains considerable uncertainty in how the real climate would evolve, for example through feedbacks that are not yet included in climate models, which may not be fully reflected in this range of estimates.
While methane emissions may thus appear to have a cumulative impact on global temperature, this is better interpreted as the delayed response to relatively recent methane emissions increases. Constant anthropogenic methane emissions, if maintained indefinitely, clearly have no further warming impact (being indistinguishable from constant natural emissions). This apparent cumulative impact is important, and captures the potential benefits of early methane mitigation 9 not apparent through a solely rate-based equivalence, but is only about 25% (s) of the impact indicated by GWP 100 and closer to that indicated by the 100-year Global Temperature-change Potential (GTP) including carbon cycle feedbacks. This should not, however, be interpreted as simple support for a lower metric value than GWP 100 : most scenarios and policy interventions involve changes in methane emission rates outside the range zero to −0.3%/year, in which case the first term on the RHS of equation 1 (neglected by conventional metrics) dominates.

DISCUSSION
We have demonstrated how it is possible to represent both the short-lived nature of methane and the long-timescale adjustment of the climate system in a single metric, GWP*. This metric allows SLCP emissions to be converted to CO 2 -equivalent emissions and preserve an unambiguous link to global warming, which we have therefore termed CO 2 -warming-equivalent. Just as many different methods have been proposed for the calculation of CO 2equivalent emissions, so there are multiple ways of calculating CO 2 -warming-equivalent emissions. Some, like GWP*, rely heavily on linearization; others, like CO 2 -forcing-equivalent emissions 19 or explicit modelling of the temperature response, 9 have a stronger physical justification and are likely to be more accurate for specific applications, at the cost of simplicity and generality. While complete disambiguation requires all details to be specified, the aim of CO 2 -we emissions is clear: to calculate the CO 2 emission pathway that would yield the same global temperature change on all relevant timescales as that caused by the time-history of some non-CO 2 climate forcer. Given this objective, appropriate methodological decisions should always yield broadly similar CO 2 -we pathways for the same climate forcer, in stark contrast to CO 2 -e emissions, for which legitimate methodological decisions such as the choice of time-horizon can change results by over an order of magnitude. Conventional pulse-based metrics treat SLCPs like methane as a stock pollutant only, thereby neglecting the rapid climate response to changes in SLCP emission rates, which dominate the temperature response while emission rates are changing.
Single-number metrics like GWP typically overestimate the cumulative effects of SLCPs; but there is some apparent cumulative impact of SLCP emissions, which arises not because they accumulate in the atmosphere, but because a component of the climate system's response to past forcing increases is characterised by a slow equilibration timescale. Based on historical emissions and RCPs 2.6, 4.5 and 6, GWP* is found to best represent the temperature impacts of methane emissions by modifying the definition in Allen, et al. 12 to weight the flow response (impact of changing methane emission rates) by 0.75 and the stock response (equilibration of the climate system to past methane emission increases) by 0.25.
The benefits of GWP* are most apparent when SLCP emission rates are declining, as this is when CO 2 -e emissions derived from conventional GWP 100 would indicate a temperature response of the wrong sign (further warming instead of cooling). Under the Paris Agreement, nations have agreed to limit global warming to well below 2°C, and to pursue efforts to limit it to 1.5°C. Using GWP* to calculate CO 2 -we emissions can therefore be useful in linking emissions scenarios with temperature goals. It uses GWP 100 in a novel way, and is thus consistent with current requirements that countries use this metric in emissions accounting. GWP* allows the contributions of all climate forcing agents to be aggregated to reach a global total cumulative CO 2 -we emission, which can then be multiplied by the TCRE to give an estimate of resultant warming over any given time period: where TCRE is the Transient Climate Response to cumulative carbon Emissions, ∑CO 2 −we is the cumulative short-lived GHG emissions aggregated using GWP* and ∑CO 2 −e is the cumulative long-lived GHG emissions aggregated using GWP 100 . This method provides a simple and transparent mechanism by which to estimate whether countries are on track to meet the Paris Agreement goals in the global stocktake. It also allows SLCPs and cumulative gases to continue to be included together in reporting mitigation ambitions, maintaining fungibility while improving environmental integrity.

Method to derive r and s
Data from the RCP database has been used to investigate how methane emissions relate to warming, and how different emission metrics represent that warming. The RCP database contains emissions rates and radiative forcings for different greenhouse gases from 1765 to 2100 for the four RCP scenarios (RCP2.6, RCP4.5, RCP6 and RCP8.5). Here, the representation of methane based on GWP 100 and GWP* is considered.
The methane emissions time series for each of the RCP scenarios are converted to CO 2 -e emissions timeseries using a GWP 100 of 28. 1 The temperature response to the methane radiative forcing (from the RCP database) is calculated using the FaIR model, 25 with a factor of 1.65 applied to account for the secondary effects of ozone and stratospheric water vapour as recommended in Myhre, et al. 1 Our analysis is consistent with assumptions in Myhre, et al. 1 and therefore does not include more recent findings, for example updated radiative forcings from Etminan, et al. 34 In all scenarios, the warming trend does not correspond closely to the cumulative CO 2 -e emissions calculated using GWP 100 .
The coefficients to weight the long and short term effects are found for each scenario using a linear regression model of the equation ΔT = a. C GWP* + b.C GWP100 , where C GWP100 is the cumulative CO 2 -e emission of methane defined conventionally using GWP 100 , and C GWP* is the cumulative CO 2 -e emission of methane defined using GWP* from Allen, et al. 12 with Δt = 20 years. Coefficients a and b were found by ordinary least squares multiple linear regression using the statsmodels package in python (http://www.statsmodels.org), and then the normalised coefficients were set by defining r = a/(a + b) and s = b/(a + b), such that r + s = 1.
When radiative forcing is put into the FaIR model, a temperature response is calculated using an impulse response function as described in Smith, et al., 25 with 2 response timescales (239 and 4.1 years as default) and a default TCR and ECS of 1.6 and 2.75 K. The radiative forcings from the IPCC AR5 database used to drive this model have been calculated using the model MAGICC, which includes carbon cycle feedbacks. Note that this model does not include less well understood earth system feedbacks, such as permafrost feedbacks, which could act to accelerate warming. Therefore, the method is most suited for use in more ambitious mitigation scenarios where warming is slower, and abrupt positive feedbacks less likely to be triggered. Table 1 shows the values for r and s that are generated using the above method, for historical data from 1900 and RCP 2.6, 4.5 and 6 data to 2100. The mean and standard deviation of these values are used in this work for r and s. The time period 1900 to 2100 is chosen as it represents the recent historical increase in methane emissions, as well as capturing three possible futures.

DATA AVAILABILITY
The RCP datasets analysed during the current study are available at http://www.iiasa. ac.at/web-apps/tnt/RcpDb.

CODE AVAILABILITY
The code used to calculate r and s, CO 2 -we emissions and produce Figs 1 and 2 is available at https://gitlab.ouce.ox.ac.uk/OMP_climate_pollutants/co2-warmingequivalence.

Introduction
Ruminants, particularly beef cattle, are perceived by many as a problem since they are a source of greenhouse gas (GHG) due to the methane produced by rumen fermentation [1]. However, it is premature to decide on appropriate management actions or policies until full ecosystem analyses have outlined net emissions by considering all emissions compared to carbon sequestration associated with different options in the beef production chain [2]. Since the major portion of the beef production chain involves animals grazing on perennial pastures, an important initial step would be to gather GHG emissions and carbon (C) sequestration data to determine net emissions using life cycle assessment (LCA) for different grazing strategies on perennial pastures.
For beef cattle production, management practices in different regions vary greatly in terms of stocking rate, mean cow size, calving season, primary forage types and fertilizer use [3]. Therefore, an ideal LCA model is one that is regionally specific. Our LCA modeling is applicable to the cow-calf only production phase in the Southern Great Plains (SGP) region of USA, where one third of US cow-calf only farms are located [4], and no known net-emission LCA study regarding cow-calf production on perennial pastures has been conducted.
Traditional ranching in the South Central U.S. has generally been based on continuous yearlong grazing practices. Rotational grazing, also known as multi-paddock grazing (MP), has been recommended since the mid-20th century as an important tool to adaptively manage grazing land ecosystems for the purpose of sustaining productivity and improving animal management. Under rotational grazing management, one paddock is grazed at a time while the other paddocks recover. There is published and anecdotal evidence from producers that, if applied appropriately to produce most advantageous results, rotational grazing can lead to improved forage and livestock production [5][6][7][8].
Simulation modeling also indicates that there tend to be larger profit margins and restoration of ecological condition with rotational grazing compared to traditional grazing [9,10]. However, few studies have been conducted to compare GHG emissions and C sequestration relations among different grazing strategies.
In this study we calculate the carbon footprints for cow-calf farmers under continuous and rotational grazing strategies using life cycle assessment (LCA) modeling, which is a standard assessment of the environmental impacts associated with a wide range of agricultural systems using a "cradle-to-grave" approach. Compared to the sector approach which only includes emissions from direct farm activities, LCA also includes indirect emissions generated by farm inputs and pre-chain activities.
Previous LCA studies on beef production have consistently reported that the cow-calf phase contributes the most emissions to the overall beef production system [11]. However, they generally omitted carbon sequestration, which has great potential to mitigate GHG emissions for cow-calf production as C sequestration exceeds emissions when animals feed solely by grazing perennial pastures [5,12]. In addition, net emissions are rarely analyzed on the same farm to estimate the GHG balance [13] and changes in C stock resulting from different grazing management practices are generally not known [14]. As grazing management practices have impacts on both GHG emissions and carbon sequestration, it is important to consider C sequestration in conjunction with GHG emissions on the same ranch to provide an objective evaluation of the GHG mitigation potential of advanced grazing management strategies.
In our study we considered both GHG emissions and C sequestration to calculate net GHG emissions for cow-calf farms under different grazing strategies. Based on Teague et al. [15], three grazing management alternatives on neighboring commercial ranches in three proximate counties in north Texas tall grass prairie are considered, including: (1) continuous grazing with light stocking (LC), representing the best-case scenario for continuous grazing; (2) traditional heavily stocked continuous grazing (HC), representing the most commonly used grazing management; and (3) adaptively managed and stocked rotational grazing, or multi-paddock grazing (MP), representing the best case scenario for rotational grazing [6,8,10]. GHG emissions were evaluated for the cow-calf farms under the three different grazing strategies using LCA approach. In addition, soil organic carbon (SOC) stock under the same grazing strategies were calculated using the soil carbon parameters measured by Teague et al. [15]. Based on both GHG emission and SOC stock values, we developed net C emission budgets for different farm transition scenarios, namely transiting from HC to MP, from HC to LC and from LC to MP.

The Study System
The goal of our study is to assess the GHG emissions and carbon sequestration from different grazing management options for representative cow-calf enterprises in the Southern Great Plain (SGP) region. A detailed description of these three management practices can be found in Teague et al. [15]. The life cycle we consider includes the entire production period from the start of the breeding season in April, to the point when the weaned calves are sold, in November of the following year as depicted in Figure 1. Cattle transportation from the site of cow-calf production to the next phase of production is not considered. The functional unit, to which all the environmental loads in the LCA are related, is generally defined as 1 kg live or carcass weight if the entire beef production systems are studied, which includes cow-calf, backgrounding and finishing systems [11]. As the boundary of our LCA study is limited to the calf-cow production system, we define the functional unit as one marketed beef calf as in Ogino et al. [16], so that our result can be easily compared with literature value from other regions. In addition, to capture the stocking density differences of different grazing strategies, we have also used one hectare of rangeland as the alternative functional unit.
Beef producers and regional extension experts have described the current cow calf production conditions in SGP area as follows. Most ranchers in the SGP area follow a breeding season from April to August, resulting in calf births from January to May. Weaning occurs from September to November. The production cycle of the year-1 cohort overlaps with the production cycles of the year-2 cohort. Specifically, during the breeding period and part of the gestation period, the cows still feed the previous cohort of calves. Similarly, during the lactation period of the current cohort, the cows will breed and become pregnant with the next cohort. Based on the contemporary production data, Figure 1a-c describe the timeline of the calf production cycle for the mature cows, 1st-year heifers and 2nd-year heifers respectively. Note that in Figure 1b, the female cattle start as 1st-year heifers, but turn into 2nd-year heifers at their second breeding season and eventually become mature cows at the end of our defined production cycle. Similarly, in Figure 1c, the female cattle start as 2nd-year heifers, then become mature cows after weaning their firstborn calves. For all female cattle for reproduction purpose, for simplicity Sustainability 2015, 7 13503 we assume the same average pregnancy rate, lactation rate and weaning rate as 90%, 86% and 82%, respectively.  For demonstration purpose a uniform gestation period for all cows from July to the following April is assumed. Take Figure 1a, the production timeline for mature cows for example, based on a 283-day pregnancy this gives 133 days of overlap between pregnancy and lactation and 150 days of pregnancy without lactation. Prior to each gestation period, we assume a 98-day period of lactating only for the previous cohort. On average, pregnancy rate, lactation rate and weaning rate are 90%, 86% and 82%, respectively. Thus, in this production cycle described in Figure 1a, 82% of the cows spend 196 days lactating but not pregnant, 150 days pregnant but not lactating, and 266 days both lactating and pregnant, totaling 612 days. Among the rest, 14% of cows didn't calve, based on the 86% lactating rate. It was assumed that all cows that didn't calve were neither lactating nor pregnant during the entire 612-day period. About 4% of cows give birth but do not raise a calf to the weaning stage. We assume these cows did not lactate during the entire 612-day period. Based on Figure 1 we assume these 4% of cows are pregnant for a period of 416 days in the production cycle and are neither lactating nor pregnant for the remaining 196 days.
Of the 332 weaned calves, except for the 53 that are retained as replacement heifers, the rest of the calves are sold immediately after weaning in November. The replacement heifers get bred for the first time during the next year's breeding season, and will be pregnancy checked at around 19 months old. They typically reach their mature weight when they wean their firstborn calves. After that they will stop growing and will remain on the ranch until ten to eleven years old. Figure 1b,c resemble Figure 1a in the breeding season, gestation period and lactation period. However, as 1st-year heifers are growing through the production season, and 2nd-year heifers also grow part of the production season, activities that attributes to enteric methane emission are different for 1st, 2nd-year heifers and mature cows even at the same production stage, as marked in Figure 1a-c.
In SGP region, the cattle are grazed on native prairie pasture 100% of the time. Supplemental hay is rarely used in the SGP region, except for the years of severe drought, therefore we will not take it into account in our study. In this region cotton seed meal is commonly used as a protein supplement during the winter when grass protein is low and we calculate the GHG emissions associated with this source of feed. The cows, 1st and 2nd-year heifers and bulls are fed 0.908 kilograms (2 pounds) of supplemental protein per head per day for 120 days.
Descriptions of the breakdown of cattle numbers for LC, HC and MP grazing can be found in Table 1.The total area of the representative farm in SGP area is defined as 4000 hectare. Given that 1 Animal Unit (AU) equals 450 kg, the stocking rates for LC grazing herd in SGP area is 14 AUs 100 ha −1 , while those for HC and MP are both 27 AUs 100 ha −1 , as described in Teague et al. [15]. In SGP region, the Typical Animal Mass (TAM) for cows and bulls are 500 and 900 kg·head −1 respectively. The 1st-year heifers weight 408 kg on average during the production cycle, as they start to breed in April at 12 months of age weighting 315 kg, and reach the mature cow weight of 500 kg by the time they wean their firstborn calves (Figure 1b). The 2nd-year heifers start at 431 kg at beginning of the 2nd breeding season and their weights stabilize at 500 kg after weaning their firstborn calves (Figure 1c). Therefore the weighted average weight for 2nd-year heifer is 487 kg during the production cycle. The TAM for calves is 40 kg at birth and 220 kg at weaning age, so we used the average value of 130 kg per head. Based on expert's opinion, the herd under LC grazing is comprised of 299 mature cows, 53 1st-year heifers, 53 2nd-year heifers, and 13 bulls. Based on the 82% weaning rate, 332 calves are weaned under LC grazing strategy. The proportion of cows, heifers, bulls and calves under HC and MP grazing are the same as the LC system. Pregnancy rates as well as the lactating and weaning rates across three different grazing strategies are assumed the same as the average corresponding rates described previously in this section. This assumption is based on previous literature findings, which reported no differences in pregnancy rate for cows associated with different stocking-rate treatments [17][18][19][20], cows maintained on continuous grazing and variable rotational grazing treatments [21,22], and cows under continuous and rotational stocking at identical stocking rates [23,24].

GHG Emissions
In this section we describe the methodologies used in GHG emission calculations. The environmental loads associated with beef cow-calf production are animal body, supplemental feed production and animal management. Specifically, five components of GHG emission on a typical SGP cow-calf farm were included: enteric methane emission, manure methane emission, manure nitrous oxide (N2O) emission, supplemental protein CO2 emission and GHG emission from farm energy use and fertilizer use. For the first three components, GHG emissions were first calculated using IPCC [25] on a per production cycle basis, then converted to the annual basis to be compatible in time frame with GHG emissions the last two components and carbon sequestration.
In addition, all gases were converted to CO2 equivalents (CO2e) to account for the global warming potential, where CO2 = 1, CH4 = 25 and N2O = 298 [26]. To compare GHG emission with carbon sequestration, CO2e is also converted to carbon equivalents (CE), where 2  with the unit of measurement being kg·CH4·head −1 ·year −1 . Here 1 i = denotes cows that are both lactating and pregnant; 2 i = denotes cows that are lactating but not pregnant; 3 i = denotes cows that pregnant but not lactating; 4 i = denotes cows that are neither pregnant nor lactating; 5 i = denotes 1st-year heifers that are growing only; 6 i = denotes 1st-year heifers that are growing and pregnant; 7 i = denotes 2nd-year heifers that are growing and lactating; 8 i = denotes 2nd-year heifers that are growing, pregnant and lactating; 9 i = denotes bulls and 10 i = denotes calves. According to IPCC [25], "the Tier 2 method should be used if enteric fermentation is a key source category for the animal category that represents a large portion of the country's total emissions." Clearly, Tier 2 approach should be adopted for beef cattle. Using Tier 2 approach is calculated as: where Ym is the methane conversion factor, which is the percent of gross energy in feed converted to methane. Based on IPCC [25], 6.5 1.0 m Y = ± . Lower bound is more appropriate for feed with high digestibility and high energy value, and vice versa. Without better information, the mean value is chosen for our baseline analysis. For calves fed entirely on milk, IPCC [25] specified that 0 m Y = . However, as their rumens develop, the calves also starts to emit methane, IPCC [25] did not provide any information on m Y for this category. Unaware of any enteric methane emission data for calves in SGP region, we will use the methane emission rate measured by Westberg et al. [28] for four calves on pasture owned by Washington State University Department of Animal Sciences. At the time of measurement, these four suckling calves were at 4 months in age weighing 206 kg on average, and were temporarily separated from their mothers. Since the average calf weight is 130 kg in our production cycle, we will proportionally adjust the measurement of Westberg et al. (2001), which was 2. Milk is the amount of milk produced (kg·day −1 ) and fat is the fat content of milk (%). According to EPA [29], the monthly lactation estimates for beef cows from January to December are respectively, 1.
REG is the ratio of net energy available in diet for growth to digestible energy consumed:

Manure CH4
On a cow-calf farm, cattle directly deposit dung and urine on the native prairie pasture where they graze all year long. Storage and treatment of manure, which occur very often when large animals are managed in a confined area, such as the feedlot, is not applicable to the cow-calf production in SGP region. Based on IPCC [25], compared to manure stored or treated as a liquid, manure deposited on pastures and rangelands tend to produce less manure CH4.
To calculate manure methane emission, the method provided by ICF Consulting [ICF] [30] is used, which is very similar to the Tier 2 method in IPCC [25], but is more informative in that instead of treating the North American region as a whole, it treats each state in U.S. separately. Overall, the manure methane emission can be calculated as: where i VS stands for the volatile solid produced by all the animals in subcategory i per year and it can be computed from: . Note that to maintain the consistency of unit we have changed the value of B0 accordingly, which, according to ICF [30], took the value of 2.72 (Unit: ft 3 per lb VS). Also note that 1 ft 3 = 0.0413 lbs [30]. ij WS is the percent of animal i's manure managed in manure system j. For all animal categories we assume a 100% pasture/range/paddock manure system, for which the methane conversion factor (MCF) for manure system is 1.4% in both Texas and Oklahoma.

Manure N2O
According to IPCC [25], the N2O emissions generated by manure in the system "pasture, range and paddock" occur directly and indirectly from the soil. Besides urine and dung, no other forms of manure such as organic N addition and synthetic fertilizer are applied on the native pasture. Therefore, direct N2O emissions from urine and dung inputs to grazed soils can be calculated as (Unit: kg N2O/production period): where direct F (Unit: kg N/production period ) is the amount of non-volatilized nitrogen excreted by grazing animals on pasture, range and paddock in a production period. The value of direct F can be estimated by ICF [31] method, that is: Kjeldahl Nitrogen per day per 1000 kg mass, or N K (Unit: kg N/day), takes the value of 0.34. Figure 1 shows that calves stay on the pasture for a total of 462 days (231 days for each cohort) during the production period, while the other subcategories of animals are assumed to stay on the farm throughout the production period.
3 EF (Unit: kg N2O/kg N) is the emission factor for N2O emissions from urine and dung deposited by grazing animals on pasture, range and paddock. According to IPCC [25], 3 EF takes a default value of 0.02 with uncertainty range between 0.007 and 0.06. Without better information, we will choose the default value of 3 EF for all three grazing strategies.
In addition to direct emissions of N2O, emissions of N2O also take place in two indirect channels, (1) through the volatilization of nitrogen as NH3 and oxides of nitrogen and deposition of these gases and their products back onto soils, and (2) through leaching and runoff of nitrogen. Based on IPCC [25], leaching and runoff are unlikely to occur for dryland regions, where precipitation is lower than evapotranspiration most time of the year. Therefore we assume that under the semi-arid climate of SGP region, leaching and runoff do not occur on cow-calf farms, as they are typically non-irrigated and unfertilized. Thus, indirect N2O emissions can be calculated as: where indirect F is the amount of nitrogen deposited by grazing animals on pasture, range and paddock (Unit: kg N/ production period), which can be calculated as: where φ is the fraction of volatilized nitrogen as NH3 and NOx, which takes a default value of 0.2 with uncertainty range between 0.05 and 0.5 [27]. 4 EF is the emission factor for N2O emissions atmospheric deposition of nitrogen on soils, which takes a default value of 0.01 with uncertainty range between 0.002 and 0.05 [27]. Without further information, we will assume default values for both φ and 4 EF .

Protein Supplement
In the SGP region cottonseed meal is used as the main source of supplemental protein. The majority of cotton seed available as supplement for beef cattle in Texas and Oklahoma is from central pivot irrigation (pers. comm. with Dr. Paul DeLaune, Environmental Soil Scientist, Texas A&M AgriLife Research, Vernon, Texas.). According to van Zeist et al. [32], the yield of cotton is divided into fibers (38%) and seed (62%), of which 5% of seed is reserved for replanting. The yields of this cotton seed from central pivot irrigation for industrial production purpose is 3030 kg·ha −1 and field production GHG emissions is 884 CE·ha −1 [33]. We assume that 57% of the GHG emission is attributable to the cottonseed for non-reproduction purpose and calculated the emissions at 503.88 kg·CE·ha −1 . Thus, the field level GHG emission for cottonseed production is estimated as 0.17 kg CE per kg of cottonseed in the field.
Energy required for crushing cottonseed is 1.25 mm BTU per ton of cottonseed crushed, when natural gas is used as the main source of thermal energy [34]. According to EPA [27], the carbon content coefficient is 14.47 kg·C·per·mm Btu for natural gas; therefore, the GHG emission is 18.09 kg CE per ton of cottonseed crushed, or 0.02 kg CE per kg of cottonseed crushed. Together, GHG emission is 0.19 kg CE per kg of cottonseed. For 1000 kg of cottonseed yield, industry-wide yields are 160 kg cottonseed oil, 455 kg cottonseed meal, 270 kg husks, 83.5 kg linters and 31.5 kg being lost. If we assume that 45.5% of GHG emission on cottonseed is attributable to cotton seed meal, then GHG emission is 0.19 kg CE per kg of cottonseed meal produced.
According to U.S. Department of Agriculture (USDA)-National Agricultural Statistics Service (NASS) Crop Production report released in August 12, 2015, the 2014 harvested area for upland cotton production in Texas was 1,861,519 hectare (4,600,000 acres). In addition, based on the statistics provided by National Cattlemen's Beef Association [35], for the year 2014 there were 4350,000 cows that calved in Texas. Given that each cow is fed 0.908 kg of supplemental protein per head per day in winter for 120 days, and that the yields of this cotton seed from central pivot irrigation as 3030 kg·ha −1 as we assumed above, together the calved cows consume 8.4% of the annual cotton harvested if they were planted under central pivot irrigation. Therefore the land currently used for cotton production is sufficient to meet the cow-calf farm's protein supplementation requirement. As no land use conversion to cotton production is necessary, GHG emission associated with land use change effect will not be considered in this paper.

Energy Use CO2
Ryan and Tiffany [36] reported fuel related energy expenses of $10.24 per head for cow-calf operators in 1995, with the energy use breakdown data as 6.07, 0.74, 1.62 gallons for diesel, gasoline and LP gas respectively, and 59.24 kWh for electric. Based on 124,884 Btu per gallon and 3413 Btu per kWh, the energy use were 758,046, 92,414, 202,312 and 202,186 Btu for diesel, gasoline, LP gas and electric respectively. According to the conversion unit provided in Del Grosso, Walsh and Duffield [37], the GHG emission from diesel, gasoline, LP gas and electric are 54.89, 6.48, 12.62, 34.87 kg CO2 equivalents respectively on a per cow basis.

GHG Emissions-Sensitivity Analysis
GHG emission calculation methods for the baseline scenario were provided in the previous sections. For continuous grazing strategies (LC and HC) only the baseline scenarios were considered. To gain a better insight into rotational grazing, both the baseline scenario and several alternative scenarios were considered. Compared to continuous grazing strategies, rotational grazing improved grass composition [15] and forage quality [38]. Therefore two factors related to forage quality are allowed to alter: the methane conversion factor m Y in Equation (1) and the feed digestibility factor ( DE ) in Equations (3) and (4). In addition, under rotational grazing strategy the animals are confined for a short period to a grazing paddock that is much smaller compared to that in continuous grazing. Thus, the lower value of Ca is also chosen, which is the coefficient to calculate

Soil Organic Carbon (SOC) Stock
With the soil organic matter (%) available in Teague et al. [15], we calculated the soil organic matter as: Here 1 k = refers to soil of depth 0-15 cm; 2 k = refers to soil of depth 15-30 cm and 3 k = refers to soil of depth 30-60 cm. The values of 1 density for LC, HC and MP were obtained from Teague et al. [15] Sustainability 2015, 7 13512 as 0.98, 1.06 and 0.91 Mg·m −3 . Bulk density at other soil depth levels was not measured in Teague et al. [15]. Therefore the values of 2 1 / density density and 3 1 / density density were first obtained from the average soil density values of the 6-year, 26-year and 60-year old restored grassland as measured by Potter, Torbert, Johnson, and Tischler [39]. Assuming the bulk density in soil depth of 0-15 cm (average bulk density among bulk density of soil depth of 0-5 cm, 5-10 cm and 10-15 cm) as 100%, then the bulk density in the soil depth of 15-30 cm (weighted average bulk density of 15-20 cm and 20-30 cm) and 30-60 cm (weighted average of 30-40 cm and 40-60 cm) are 115.1% and 120.8% respectively, taking the average of the three grassland sites. Percentages of soil organic matter (SOM) at various soil depths are denoted as % k SOM , which can be obtained from Teague et al. [15] for LC, HC and MP grazing strategies. The corresponding SOC can be computed given that SOM contains 58% carbon [40].

Carbon Sequestration
Teague et al. [15] studied three ranches practicing LC, HC and MP strategies in each of three adjacent counties. The same grazing strategy has been practiced for at least 9 years before the measurements were taken. Of the three ranches currently practicing MP grazing, two were converted from HC to MP, and one was originally under LC. The conversion occurred 10 years previously for the first two ranches and 20 years previously for the third ranch before the year of measurement. One of the limitations of this study is that the carbon stock measurement from previous years is unavailable. The farms with LC, HC and MP practices for each county, however, are located right across the fences from one another or nearby. Therefore, it is reasonable to assume that the MP practicing farms initially had the same SOC stock as the neighboring HC or LC farms if the conversion had not occurred. Similar to Stephenson et al. [41], we first establish the carbon stock for LC and HC was established as the benchmark, and consider the relative changes in carbon stock were considered when the conversion to different grazing practices occurred.
We considered three transitions: HC to MP, HC to LC and LC to MP. For sensitivity analysis purposes, three timing scenarios are also considered, namely transition occurring 10, 15 and 20 years prior to transition. To calculate carbon sequestration for each scenario, we adopted the methods used in Gascoigne et al. [42]  noting that when applying this method to the data in Potter et al. [39], it generated 412 kg·ha −1 sequestration rate per year, very close to their regression result of 447 kg·ha −1 sequestration rate. When there is no management improvement, for example, if HC or LC is always in practice without conversion to MP, then we assume the soil reaches an equilibrium and the C sequestration is zero [43]. Table 2 demonstrated GHG emissions for LC, HC and MP grazing strategies. Note that for the MP baseline scenario, we assumed the same parameters as continuous grazing in LCA approach. Such assumption is later on relieved in MP alternative scenarios to account for the potential improvement in grass quality and the less required energy as animals are confined in a much smaller paddock under MP scenario.

GHG Emissions
On a per calf basis, the total emission is 8034.90 kg·CO2e·calf −1 ·year −1 for LC, HC and MP baseline scenario, due to the same pregnancy and weaning rates assumed for the three grazing strategies based on previous literature observations. Though no difference in pregnancy rates were found under different stocking rate and management strategy, Arthington, Bohlen and Roka [18] showed pounds of calf weaned per acre of dedicated land was greater for high compared to medium and low stocking rates. Therefore the three grazing systems will differ in carbon emissions per pounds of calf weaned. Due to lack of field measured data in SGP area and no comparable literature values, pounds of calf weaned will not be used as the functional unit in our paper. To make a comparison with the carbon sequestration, total carbon emission is also computed on a per hectare basis, for which the value for LC is proportionally lower than those for HC and MP, due to the different stocking rates. For a cow-calf farm in Upper Midwestern U.S. that provides 75 calves for beef production, Pelletier et al. [44] reported a total GHG emission of 599 ton CO2e, which is equivalent to 7986.67 kg·CO2e·calf −1 . This overall emission is only 0.6% lower than the value calculated in this paper. However, on a cow-calf system in Japan, Ogino et al. [16] reported a total GHG emission of 4550 kg·CO2e·calf −1 which is 43.3% lower than the GHG emissions found in this study.
Among the sources of GHG emission presented in Table 2, we can calculate the emission distribution as 79.6% for enteric CH4, 1.9% for manure CH4, 15.9% for manure N2O, 1.2% for protein supplement and 1.4% for energy use. Clearly enteric methane accounts for the majority of all emissions, with emission from manure (CH4 and N2O) ranking next. GHG emissions from the two other sources only account for less than 3% of the total emission, which is almost negligible. Enteric methane accounts for 43.4% of the total GHG production for the cow-calf farm in Upper Midwestern U.S. [44] and 61.2% of the total emission in Japan [16], both of which are much lower than our estimation.
The vast gaps in percentage of GHG emissions from different subcategories of cow-calf production can be explained largely by the different production practices resulting from regional variation in climate and grazing conditions [4]. For example, the representative farm studied in Pelletier et al. [44] likely used supplemental forage to sustain cows during winter. However, in the SGP region we studied, cows graze year round on native pasture and generally no harvested forage is necessary for the winter months. As a result, feed production, though not needed for SGP region, is the second largest emission category in Pelletier et al. [44], accounting for 32.9% of the total emission. Similar differences can be found in the cow-calf system in Japan [16], which reported 18.4% and 8.3% of the total emission attributable to feed production and feed transport. Feed transport, which is not applicable in our study, is necessary in the study of Ogino et al. [16], due to the 25% of imported feed from U.S. and China.
The major reason that SGP region has much higher GHG emissions for cow-calf production is due to the much higher enteric CH4 emissions, which is caused by relatively lower feed quality on the unfertilized rangeland compared to that on the fertilized pasture. As our sensitivity analyses on MP grazing show, if we take account of the potential of grass quality improvement by MP grazing [38], as well as the reduced grazing energy due to much smaller paddock sizes of MP, then MP grazing reduces the total GHG emission by 33% to 5413.69 kg·CO2e·calf −1 year −1 , as reflected by the MP Alternative 1 scenario in Table 2. Even if the reduced grazing energy potentially caused by MP grazing is not accounted for, the total GHG will still be lowered to 5645.22 kg·CO2e·calf −1 ·year −1 (MP Alternative 2). Both figures are lower than that reported by Pelletier et al. [44]. However, if the benefit for MP grazing in LCA analysis is only limited to reduced grazing energy while the grass quality remains the same, then GHG emission will be lowered by less than 5% from the baseline scenario (MP Alternative 3).
These results underline the importance of grass quality improvement in reducing methane CH4 emission, and in return the total GHG emission. With the potential to increase grass quality and digestibility [38], MP grazing could lead to a big reduction in GHG emission. Note that our results on methane reduction by MP grazing is consistent with the field study conducted by DeRamus et al. [45], who measured methane emissions of cattle on different grazing management practices using methane collection equipment, and found a 22% reduction in annual methane emission from MP grazing when compared with continuous grazing. Table 3 shows that the SOC stock up to 60 cm depth for LC, HC and MP management practices are 122.6, 93.9 and 129.2 Mg·ha −1 respectively. Therefore, MP grazing led to the highest SOC stock, with LC slightly lagging behind by only 5%. HC has the lowest SOC stock, which is 27% lower than that for MP. Note that our results on SOC stock were comparable to the findings of Potter et al. [39], also in SGP region, where the SOC stock for the 6-year, 26-year and 60-year restored grassland, which were original agricultural land, were 110.0, 103.1 and 132.6 Mg·ha −1 respectively. We can see that our SOC stock value for the MP scenario is very close to that of the 60-year restored grassland; even SOC stock for LC is much higher than the 6-year and 26-year restored grassland. However, SOC stock for HC scenario fell far behind, which suggests HC grazing generated very little or no carbon sequestration over the time. This observation coincides with that of Follett and Reed [46], which showed improved grazing management, with its introduction of legumes, control of undesirable species and enhanced grass productivity, generally increases carbon sequestration. Meanwhile, moderate continuous grazing leads to a higher margin of C sequestered over GHG emission than heavy continuous grazing [12]. From Table 4 we can see that carbon sequestration rates varied greatly for different scenarios, with highest C sequestration rate for the 10-year transition scenario and the lowest C sequestration for the 20-year scenario. This is because if the same accrual of SOC stock occurs under a shorter period of time, then SOC accumulation rate for each year is in return higher, which means higher SOC sequestration. Note that SOC stock may not increase with the same accrual rate each year. As pointed out by Follett and Reed [46], relatively high C sequestration occurs on recently restored rangelands (up to 2.75 Mg·C·ha −1 ·year −1 ) while lower C sequestration occurs (0 to 1.6 Mg·C·ha −1 ·year −1 ) on rangelands managed the same way over the long term.

Carbon Sequestration
Given any transition period, the highest sequestration always occurred during the transition from HC to MP, while the sequestration rate under transition from HC to LC ranks second, generating slightly lower C sequestration, and the sequestration rate from LC to MP is much lower, generating only 20% of the highest C sequestration rate. Therefore C sequestration rate is closely tied to the initial land use practice. Transitioning from LC to MP generates a C sequestration rate of 330 kg·C·ha −1 ·year −1 , which is consistent with most carbon sequestration rates reported by the literature. For example, Stephenson et al. [41] reports a C sequestration rate of MP grazing lies within the range of 120 to 400 kg·C·ha −1 ·year −1 , and in the NGP region, Liebig et al. [12] report C sequestration rates of 390 to 460 kg·C·ha −1 ·year −1 for the unfertilized native prairie and fertilized crested wheatgrass. Soussana et al. [47] also obtained annual C sequestration rates between 191 and 491 C·ha −1 ·year −1 when modeling the European grassland management systems. After converting from HC to MP, the C sequestration rate was estimated as a much higher value of 1765 kg·C·ha −1 ·year −1 , which is consistent with the finding of Soussana et al. [48], who reported a C sequestration rate of 2400 ± 700 kg·C·ha −1 ·year −1 . This is because when poor management lowered SOC stock over the time, a transition to an improved practice such as MP will increase SOC stock at a higher rate [43]. To be conservative on the GHG mitigation potential of MP grazing, GHG emission from only MP baseline scenario in Table 2 is used to estimate carbon balance. Similar to Liebig et al. [12], who observed that unfertilized native prairie was a net sink for GHGs, our analysis also indicated that cow-calf farms converting from continuous to rotational grazing in SGP region could be either net carbon sinks or low carbon sources for decades. For example, under the intermediate 15-year scenario, cow-calf farms converting from HC to LC, from HC to MP and from LC to MP are likely net sinks for GHG with a net C sequestration rate of 2002.8, 1731.6 and 89.5 kg·C·ha −1 ·year −1 respectively. Even the most conservative 20-year scenario (Table 4), cow-calf farms converting from HC to LC and from HC to MP are net sinks for GHG with a net C sequestration rate of 1414.5 and 1253.3 kg·C·ha −1 ·year −1 . A transition from LC to MP will generate a low net carbon emission rate of 20.5 kg·C·ha −1 ·year −1 for the 20-year scenario. Therefore, it is worth noting that even though the GHG emissions in SGP region are higher on a per calf basis compared to the values reported in other regions of the world [16,44], net GHG emissions are likely negative when we take the carbon sequestration into account (Table 4). This is consistent with results from NGP region reported by Liebig et al. [12] where, when using a modest annual SOC sequestration rate of 0.17 tons·C·ha −1 with the continuously grazed forage base, both heavy and moderately stocked grazing strategies produced substantial carbon sinks of −0.618 and −0.783 tons·CO2e·ha −1 ·year −1 respectively. Overall these systems yielded −0.026 and −0.145 tons·CO2e·kg −1 animal gain while the enteric methane was reported to be 0.484 and 0.176 tons·CO2e·ha −1 ·year −1 .
If we assume a soil C sequestration rate of 330 kg·C·ha −1 ·year −1 as from LC to MP as reported in the 20-year scenario (Table 4), it will take an additional 116 years for the current SOC stock under MP practice to reach that of the native prairie reported by Potter et al. [39], which averaged 160.78 Mg·C·ha −1 in central Texas. Likewise if the sequestration rate of 440 kg·C·ha −1 ·year −1 from LC to MP as reported in the 15-year scenario is assumed, then it will take 87 years to reach the SOC level of the native prairie. Thus, the upward trend in C sequestration will likely continue for a number of decades. When assuming the C sequestration rate of 1765 kg·C·ha −1 ·year −1 under 20-year scenario, which occurred during the transition of HC to MP, it only takes an additional 38 years for the SOC stock under HC grazing to reach the SOC stock value of an average native prairie, or 23 years under the 15-year scenario. This coincides with the conclusion of Smith [43] and suggests that this high carbon sequestration rate is unlikely to last over the long term.
However, for decades after converting from the continuous grazing to MP grazing practice, the contribution of GHG emission by cow-calf farm in the beef production link is non-significant, non-existent, or negative relative to carbon sequestration rates for ruminants feeding solely on grazed perennial pastures. This is contrary to many commonly reported LCA analyses indicating that cow-calf enterprises account for the highest GHG emission in the beef production [11], which is misleading as most of these analyses do not consider the GHG sequestration in the ecosystem being studied. In addition, these analyses do not consider the GHG emissions generated by cropping practices [45,49] and soil erosion [50] associated with grains fed during the non-grazing portions of the production cycle. Net GHG emissions differ when the C sequestration under the different grazing practices is considered which underlines the importance of taking into account both GHG emission and C sequestration simultaneously in such analyses.

Conclusions
Using the LCA approach, this paper calculated GHG emissions of the cow-calf farms in the SGP region. Results show that the overall GHG emissions and main GHG emission sources of the SGP region differ from those of the rest of the U.S. such as the NGP region, and other countries in the world. This indicates the importance of the LCA analysis on a regional basis. In SGP, where according to our findings, overall GHG emissions are higher than the other regions, almost 80% of GHG emissions are from enteric methane. There is great potential of reducing GHG emission by increasing grass quality and digestibility, which could reduce total GHG emissions by as much as 30%. Compared to continuous grazing, MP grazing can improve grass quality [38] as well as grass production [15], thus, MP grazing strategy is potentially a good option to reduce GHG emission on a cow-calf farm.
Unlike most published work that isolates the analyses of GHG emission and C sequestration, our paper used field observed SOC data to estimate the C sequestrations for different grazing management systems. Contrary to the publications claiming that cow-calf farms are the most significant GHG emission source in the beef production link, our results show that cow-calf farms converting from HC to LC or MP practices in SGP region are likely net carbon sinks. In our study, the highest SOC stock occurred upon converting to MP grazing indicates that among the three different grazing practices we analyzed, MP has the highest carbon sequestration rate. Combined with its potential to significantly lower GHG emissions, we conclude that MP serves as the best carbon mitigation option.