Tristan Senga Kiesse
email: tristan.senga-kiesse@inra.fr

Michael S Corson

Gwenola Le Galludec

Aurélie Wilfart

Sensitivity of greenhouse gas emissions to extreme differences in forage production of dairy farms

Keywords: animal feeding, environmental impact, extreme value, forage production

Dairy farms follow a wide variety of management practices depending on the economic and environmental objectives that farmers target. Management practices are reflected, among other things, by a wide variety of forage-production and animal-feeding strategies, which influence productivity and greenhouse gas (GHG) emissions of dairy farms. Many studies focus on agricultural systems with "average" characteristics to generalize results to a larger percentage of existing systems. This approach, however, ignores systems with uncommon characteristics, whose environmental impacts may be unusually large (e.g. due to inefficiencies) or small (e.g. due to innovations). To address this issue, we used Extreme-Value Theory (EVT) as a statistical tool to identify dairy farms with forage-related characteristics that could be considered "extreme". Application of EVT to a dataset of 78 dairy farms in Normandy (France) identified 15-20% of dairy farms with extreme minimum amounts of dry matter (DM) of grass from pastures or maize silage and 10-15% of farms with extreme maximum amounts of one or the other. Mean amounts of DM between the minimum sample and maximum sample differed by a factor of ca. 3 for grass and a factor of 1.5 for maize silage. Consequently, the maximum sample for grass or maize silage had mean estimated gross GHG emissions (kg CO 2 eq) 13% lower or 25% higher, respectively, than those of the minimum sample. Four strategies were identified for setting DM of grass and maize silage at extreme levels, which influenced milk production and GHG emissions.

Extreme changes in the DM of one forage were generally compensated by changes in those of the other forage and concentrated feed fed at non-extreme levels, which influenced enteric methane emissions, manure management or the amount of feeds purchased.

Introduction

Management practices of dairy farms are widely studied due to their influence on farm inputs, which in turn influence farm productivity and environmental impacts. Studying effects of management changes on greenhouse gas (GHG) emissions and farm profitability is useful for developing farm-level GHG mitigation strategies that could be financially viable [START_REF] Lovett | Greenhouse gas emissions from pastoral based dairying systems: The effect of uncertainty and management change under two contrasting production systems[END_REF]. For instance, animal-feeding strategies directly influence enteric methane (CH 4 ) emissions of dairy cattle, which contribute ca. 67% of GHG emissions of the global livestock sector [START_REF] Dollé | Les gaz à effet de serre en élevage bovin : évaluation et leviers d'action[END_REF][START_REF] Gerber | Productivity gains and greenhouse gas emissions intensity in dairy systems[END_REF][START_REF] Gerber | Tackling climate change though livestock -A global assessment of Emissions and mitigation opportunities. Food and Agriculture Organization of the United Nations (FAO)[END_REF]. Thus, studies that predict CH 4 emissions from dairy cattle often consider feed and forage intake as input variables [START_REF] Bannink | A model of enteric fermentation in dairy cows to estimate methane emission for the Dutch National Inventory Report using the IPCC Tier 3 approach[END_REF]. The quantities of inputs used to feed animals (either directly, such as concentrated feed, or indirectly, such as fertilizers) vary greatly among farms, which may cause these farms' environmental impacts (e.g. GHG emissions) to vary greatly as well. Many studies of agricultural systems focus on "average" characteristics to generalize results to a larger percentage of existing systems. This approach, however, ignores systems with uncommon characteristics, whose environmental impacts may be unusually large (e.g. due to inefficiencies) or small (e.g. due to innovations).

This study was concerned with modeling and predicting consequences of "extreme events" (e.g., uncommon farm characteristics), which can help understand how variability in dairy farm characteristics influences their environmental impacts. Dairy farms are confronted with extreme climate events, financial instability and environmental concerns such as climate change. In this context, farmers are developing various agricultural strategies to ensure food production, and these strategies are reflected in a wide variety of farm characteristics. This raises the issue of identifying dairy farm inputs with uncommon values, which affect their productivity and GHG emissions. Extreme-Value Theory (EVT) is the most popular approach to model extreme events. To analyze agricultural systems accurately, an increasing number of studies combine statistical analysis with quantitative models of environmental processes. For instance, statistics-based sensitivity analysis can be performed to assess the relative influence of input variables on model outputs [START_REF] Confalonieri | Comparison of sensitivity analysis techniques: A case study with the rice model WARM[END_REF][START_REF] Drouet | Sensitivity analysis for models of greenhouse gas emissions at farm level. Case study of N(2)O emissions simulated by the CERES-EGC model[END_REF]. Likewise, uncertainty analysis is used to assess the robustness of predicted environmental impacts of agricultural systems [START_REF] Basset-Mens | Uncertainty of global warming potential for milk production on a New Zealand farm and implications for decision making[END_REF][START_REF] Chen | Influence of emission-factor uncertainty and farmcharacteristic variability in LCA estimates of environmental impacts of French dairy farms[END_REF][START_REF] Cucurachi | A Protocol for the Global Sensitivity Analysis of Impact Assessment Models in Life Cycle Assessment[END_REF]. However, statistical analyses have generally failed to study variables with accuracy when these variables deviate from average trends. In contrast, EVT considers the stochastic nature of maximum and minimum values of a process [START_REF] Embrechts | Modelling Extremal Events for Insurance and Finance[END_REF][START_REF] Charras-Garrido | Extreme Value Analysis : an Introduction[END_REF]. Initially developed to study flood levels, EVT is now applied to finance, insurance, meteorological events and environmental sciences. For instance, EVT was applied to examine trends in hydrological extremes due to the anticipated intensification in the hydrological cycle caused by climate change [START_REF] Katz | Statistics of extremes in hydrology[END_REF]. Likewise, EVT was used to model hospitalization and death peaks due to cardiovascular disease in Canada [START_REF] Chiu | Mortality and morbidity peaks modeling: An extreme value theory approach[END_REF] and to assess water quality by analyzing marine biomarkers [START_REF] Durrieu | Dynamic extreme values modeling and monitoring by means of sea shores water quality biomarkers and valvometry[END_REF]. EVT increases the rigor of studies of extreme events by enabling one to statistically test and identify threshold values and suitable probability distribution functions for modeling the tails of distributions. To our knowledge, EVT has never been used to analyze dairy farms.

Based on a sample of dairy farms in Normandy (France), this study aimed to (1) identify those with "extreme" forage-related characteristics and (2) compare their milk production and GHG emissions. Groups of farms identified with extreme forage-related characteristics were also compared according to other characteristics, such as their size (e.g. number of livestock units (LU), utilized agricultural area (UAA)), areas of forage crops and sources of GHG emissions.

Material and methods

Data

The dataset consisted of 78 dairy farms in Normandy surveyed in 2014 by the French Livestock Institute (IDELE). Annual data for a variety of annual farm inputs, characteristics and outputs were collected for each farm: areas of pastures and silage maize in the forage area, grass yield, number of LU, milk produced per cow, amount of concentrated feed fed, estimated CH 4 emissions from dairy cows and heifers (Table 1). The main dairy cow breeds were Holstein (minimum, mean and maximum of 45%, 89% and 100%, respectively) on 54 farms, Normande (61%, 89% and 100%, respectively) on 18 farms, and cross-breeds (50%, 63% and 100%, respectively) on 6 farms. GHG emissions (CH 4 , nitrous oxide and carbon dioxide) due to production and transport of purchased feeds, purchased fertilizers and on-farm energy consumption were estimated. Milk production per cow was calculated as total milk production per year divided by the number of dairy cows. to French national contexts that considers factors such as feeding levels and proportion of concentrated feed in the diet [START_REF] Sauvant | Quantification of the main digestive processes in ruminants: the equations involved in the renewed energy and protein feed evaluation systems[END_REF]. The main sources of GHG emissions were enteric fermentation from cattle (50%), manure storage and management (20%), organic and inorganic fertilization (9%), production and transport of inputs (e.g. feeds (12.5%), fertilizers (3%), and animals (<1%)), and on-farm energy consumption (5%) (Supplementary Table A1). Emissions were estimated from information provided by farmers.

For instance, the amount of manure produced per year per farm was estimated from the type of animal housing, milk production (for dairy cows) and calving period (for suckler cows).

Farmers also supplied additional information, such as the average duration of manure storage and amounts of off-farm effluents imported. Likewise, farmers provided amounts of mineral fertilizers and effluents applied per ha, feed purchased and total energy (electricity and diesel) consumed based on information in their bills and accounting books. GHG emissions from these sources were estimated using emission factors (e.g. kg CO 2 eq per kg of feed, kWh of electricity or L of diesel purchased). Carbon sequestration (e.g. by pastures) was not considered. These contributions to GHG emissions led to a strong correlation between GHG (and enteric CH 4 ) emissions and milk produced per cow (Figure 1). We calculated each farm's production of each forage per LU (kg dry matter (DM)/LU/year) by dividing the total DM of grass from pastures (DM Grass ) or maize silage (DM Maize ) (equal to area (ha) × yield (t DM/ha)) by the number of LU. Since the yield of silage maize was not available for dairy farms in our dataset, we used the mean regional yield of 19.3 t DM/ha in 2014 (ARVALIS-Institut du Végétal). DM Grass and DM Maize varied greatly among farms (coefficients of variation (CV) = 47% and 38%, respectively), but the variability in DM Maize was due only to that of the area of silage maize, since we used the mean yield. Distributions of the amount of concentrated feed fed had similar characteristics as those of DM Grass and DM Maize (CV = 35%) (Table 1). Milk production per cow and estimated GHG and CH 4 emissions, however, varied less (CV = 10-18%) (Table 1). Annual GHG and CH 4 emissions were analyzed according to two functional units: farm (kg CO 2 eq/farm/year) and l of milk produced (kg CO 2 eq/l/year).

Extreme-Value Theory

Consider a sequence of independent and identically distributed (iid) random variables (rv)

with realizations , with its maximum sequence being denoted by Within the EVT framework, limit laws for the (normalized and centered) of the iid inputs belong to asymptotic distributions of type Fréchet, Weibull or Gumbel [START_REF] Embrechts | Modelling Extremal Events for Insurance and Finance[END_REF]. EVT thus enables identification of probability distribution functions of inputs affected by extreme variations. Applying EVT to a dataset consists of extracting a maximum sequence, , which contains values at maximum values of inputs, then fitting it using appropriate probability distribution functions such as the generalized Pareto distribution (GPD) [START_REF] Coles | An introduction to statistical modeling of extreme values[END_REF]. EVT also enables one to extract a minimum sequence and then fit it using extreme value distributions.

Generalized Pareto distribution

For a fixed threshold , consider the distribution of the excesses above , extracted from observations of the rv The excess distribution function ( ) above can be approximated by a GPD function, with parameter θ=( ), defined by the empirical distribution function

( ) [ ( ( ) )] ( ) Equation 1
for and ( ) , with and the scale and shape parameters, respectively [START_REF] Beirlant | Statistics of extremes : theory and applications[END_REF]. GPD also enables one to approximate distribution function of values below another fixed threshold extracted from observations of the rv

Threshold selection

Selecting the threshold requires a compromise between selecting too-small and too-large values of , which affects the bias and variance of the estimated GPD parameters. Setting too small increases the sample size of extremes and the bias of estimated parameters to the point that the fitted GPD may not approximate the tail well. Conversely, setting too large decreases the sample size of extremes, increasing the variance in estimated parameters.

Finally, must be sufficiently large to ensure that the GPD is a reliable model [START_REF] Scarrott | A review of extreme value threshold estimation and uncertainty quantification[END_REF]. Several methods are available to select , such as automatic selection of a threshold based on the Hill estimator [START_REF] Grama | Statistics of extremes by oracle estimation[END_REF][START_REF] Durrieu | Nonparametric adaptive estimation of conditional probabilities of rare events and extreme quantiles[END_REF] and graphical assessment using a mean residual life function and threshold stability properties. We chose to test four minimum and maximum extreme quantiles each and use them to calculate the corresponding values of [START_REF] Chiu | Mortality and morbidity peaks modeling: An extreme value theory approach[END_REF].

Statistical tests and root mean squared error

Once the minimum or maximum sample was extracted, the adequacy of its distribution to the GPD was tested using the Kolmogorov-Smirnov (K-S) test. For the K-S test, the null hypothesis H 0 was that the extracted sample followed a GPD, with results considered significant at α=5%. For each quantile tested, the extracted sample was also compared to the fitted GPD using the root mean squared error (RMSE). The Mann-Whitney (M-W) U test was also performed to test differences in mean farm characteristics between the minimum and maximum samples, with the differences considered significant at α=5%. For the M-W test, the null hypothesis H 0 was that characteristics of farms in the minimum and maximum samples followed the same distribution. We used the "evd" [START_REF] Stephenson | evd: Extreme Value Distributions[END_REF] and "extremefit" [START_REF] Durrieu | extremefit: Estimation of Extreme Conditional Quantiles and Probabilities[END_REF] packages of R software (R Core Team, 2018) to apply EVT.

Results

Extreme samples for dry matter of grass from pastures

Quantiles of the entire dataset were tested as potential thresholds for the minimum (10%, 15%, 20% and 25%) and maximum (75%, 80%, 85% and 90%) samples (Table 2).

Distributions of extracted minimum and maximum samples changed as the quantile increased (Figure 2 and Figure 3). The K-S test did not reject the null hypothesis of the adequacy of the GPD for any of the quantiles considered. For the minimum sample (GrassMIN), the 10% and 15% quantiles had the smallest RMSE but did not have the largest p-values of the K-S test (Table 2). Because the sample size of the 10% quantile was 7, we selected the 15% quantile as the minimum threshold, corresponding to DM Grass of 1708 kg/LU/year, to obtain a larger sample size of 11.

For the maximum sample (GrassMAX), we selected the 90% quantile, which had the largest p-value, the smallest RMSE and sample size of 10, as the maximum threshold, corresponding to DM Grass of 4946 kg/LU/year (Table 2). Other choices of thresholds were obviously possible.

Extreme samples for dry matter of maize silage

EVT was also applied to observations of DM Maize . Like for DM Grass , the K-S test did not reject the null hypothesis (Table 3). Thus, to obtain sufficiently large sample sizes, minimum and maximum quantiles of 20% and 85%, respectively, were selected for minimum (MaizeMIN) and maximum samples (MaizeMAX) of DM of maize silage, respectively, corresponding to threshold values of 3647 and 6443 kg/LU/year, respectively. According to these thresholds, three farms were in both GrassMIN and MaizeMAX, five farms were in both GrassMAX and MaizeMIN, and one farm was in both GrassMAX and MaizeMAX (Figure 4).

Milk production and emissions by sample

On average, GrassMAX and GrassMIN produced less and more milk per cow (6269 and 7954 l/cow/year, respectively) than the mean of the entire dataset (7210 l/cow/year) (Table 4).

Similar differences in means were observed for annual GHG and enteric CH 4 emissions (kg CO 2 eq) per farm and per l of milk produced (Table 5). Amounts of concentrated feed fed for GrassMAX and GrassMIN were less than the dataset mean.

Conversely, MaizeMIN and MaizeMAX produced less and more milk per cow (5785 and 7351 l/cow/year, respectively) than the dataset mean (7210 l/cow/year) (Table 4). Similar differences in means were observed for annual GHG and enteric CH 4 emissions per farm (Table 5). Per l of milk produced, annual GHG and enteric CH 4 emissions for MaizeMIN and MaizeMAX were generally slightly greater than or equal to, respectively, the dataset mean.

Amounts of concentrated feed fed for MaizeMIN and MaizeMAX were less or slightly greater, respectively, than the dataset mean (Table 5). 3.4. Sensitivity to extreme differences in DM of grass from pastures and maize silage

Milk production and farm sizes

GrassMAX had mean DM Grass 3 times as high as GrassMIN, and its farms produced 21% less milk per cow, had 33% less DM Maize and fed 13% less concentrated feed per l of milk (Table 4, Figure 5). In contrast, MaizeMAX had mean DM Maize 150% higher than that MaizeMIN, and its farms produced 27% more milk per cow, had 36% less DM Grass and fed 7% more concentrated feed per l of milk (Table 4, Figure 5). The difference in the amount of milk produced per cow between the minimum and maximum samples of DM Grass or DM Maize was significant according to the M-W test (i.e. the null hypothesis was rejected) (Table 4).

Although MaizeMAX or GrassMIN produced more milk per cow than MaizeMIN or GrassMAX, respectively, their farms did not differ significantly in mean size (e.g. number of LU, UAA) (Table 4) or replacement rate (29-31% among the four extreme samples) (Supplementary Table A2).

Cross-comparison of the same end of the distribution of each forage showed that GrassMIN produced 27% more milk per cow than MaizeMIN (Table 4). Conversely, MaizeMAX produced 17% more milk per cow than GrassMAX. Meanwhile, MaizeMAX fed 21% more concentrated feed per l of milk than GrassMAX, while MaizeMIN fed 2% less concentrated feed per l of milk than GrassMIN (Table 4).

Figure 5. Boxplots of milk production per cow (l/cow/year), concentrated feed fed (g/l of milk produced), grass from pastures and maize silage (kg dry matter (DM)/livestock unit/year) for farms in minimum ("MIN") and maximum ("MAX") samples of DM of grass from pastures and maize silage

GHG and enteric methane emissions per farm

Per farm, GrassMAX emitted 13% less gross GHGs than GrassMIN (non-significant), while

MaizeMAX emitted 25% more than MaizeMIN (significant) (Table 5, Figure 6). The same trend was observed for enteric CH 4 emissions, which were the main source (48-54%) of GHG emissions among the four extreme samples. The other sources  manure storage and management, purchased feeds, organic and inorganic fertilization, energy consumption and purchased fertilizers  contributed a mean of 18-21%, 9-14%, 8-10%, 5-6% and 2-3%, respectively, among the four extreme samples. Compared to MaizeMIN, percentage contributions to gross GHG emissions of enteric CH 4 emissions, manure storage and management, and organic and inorganic fertilization for MaizeMAX differed significantly:

11% lower, 12% lower and 32% higher, respectively (Supplementary Table A2). The percentage contribution of purchased feeds was also higher (by 37%), but not significantly so.

Compared to GrassMIN, percentage contributions of these sources for GrassMAX did not differ significantly, except for that of purchased feeds, which was significantly lower. Onfarm energy consumption and purchased fertilizers did not differ significantly between GrassMAX and GrassMIN or between MaizeMAX and MaizeMIN.

Figure 6. Distributions of annual gross GHG and enteric methane (CH 4 ) emissions (top) per farm per year (kg CO 2 eq/farm/year) and (bottom) per l of milk produced (kg CO 2 eq/l/year) for farms in minimum ("MIN") and maximum ("MAX") samples of dry matter (DM) of grass from pastures and maize silage

GHG and enteric methane emissions per l of milk produced

Per l of milk produced, GrassMAX emitted 17% more gross GHGs than GrassMIN (significant), while MaizeMAX emitted 1% less than MaizeMIN (non-significant) (Table 5, Figure 6). Enteric CH 4 emissions followed the same trend: 20% larger for GrassMAX than GrassMIN and 11% smaller for MaizeMAX than MaizeMIN (both significant). Crosscomparison of the same end of the distribution of each forage showed that GrassMIN emitted 17% less GHGs than MaizeMIN, but GrassMAX emitted nearly the same amount of GHGs as MaizeMAX (1% difference). In comparison, GrassMIN emitted 21% less enteric CH 4 than

MaizeMIN, but GrassMAX emitted 10% more enteric CH 4 than MaizeMAX.

Discussion

Differences in forages and concentrated feed, milk production and cattle management

Extreme changes in the amount of DM of a given type of forage (grass or maize silage) were generally compensated by changes in the other forage and concentrated feed fed at nonextreme levels. To feed their animals, MaizeMAX maintained grass from pastures and concentrated feed at levels similar to those of MaizeMIN. In contrast, GrassMAX fed less maize silage than GrassMIN but maintained concentrated feed at a level similar to that of GrassMIN. Meanwhile, an extreme increase in DM Maize was associated with increased milk production, since maize silage yields a large amount of DM and increases animal performance when included in animal diets [START_REF] Beauchemin | Nutritional management for enteric methane abatement: a review[END_REF]. Although most farms had Holstein cows (45-100% of cows on 69% of the farms), the prevalence of Normande cows (61-100% of cows on 23% of the farms) may have increased variability around the relation between DM Maize and milk production. Management practices for cattle, in terms of replacement rate, did not differ significantly between farms in minimum and maximum samples of each forage.

Meanwhile, the stocking rate (i.e. LU/ha Main Forage Area) decreased significantly from GrassMIN to GrassMAX but not from MaizeMIN to MaizeMAX.

Land management strategies

As expected, significant differences in the percentage of UAA in pastures and maize silage were found between farms in minimum and maximum samples of each forage (Supplementary Table A2), since the area of each forage was used to calculate DM. On a 70 ha commercial farm in the Netherlands, [START_REF] Vellinga | Maize silage for dairy cows: mitigation of methane emissions can be offset by land use change[END_REF] studied effects of increasing the proportion of maize silage in feed rations due to increasing the area of maize silage grown. They examined several land-use scenarios, such as converting permanent grasslands to silage maize, either in monoculture or in rotations with temporary grasslands.

For instance, increasing the silage maize area by 4 ha decreased GHG emission intensity by 11 g CO 2 eq/kg milk [START_REF] Vellinga | Maize silage for dairy cows: mitigation of methane emissions can be offset by land use change[END_REF]. However, the practicality of changing feeding strategies by changing land use depends on the size of the farm. Farms with small total area or grassland area may have difficulty making certain changes. Maximizing DM Grass also requires future studies that investigate how land use and management influence other services that grasslands provide besides forage production, such as modifying the quality (e.g. fatty-acid contents) of milk and meat and mitigating GHG emissions by sequestering soil organic carbon. Models have thus been developed to simulate farm systems, as well as agronomic and environmental services provided by grasslands [START_REF] Rotz | The Integrated Farm System Model: A Tool for Developing more Economically and Environmentally Sustainable Farming Systems for the Northeast[END_REF][START_REF] Graux | Ensemble modelling of climate change risks and opportunities for managed grasslands in France[END_REF].

Differences in GHG emissions by production strategy

The results highlighted production strategies, reflected through the management of DM Grass and DM Maize , that influenced both economic (milk production) and environmental (GHG and enteric CH 4 emissions) outputs from dairy farms. Thus, the strategies for setting DM Grass and DM Maize at extreme levels can be extrapolated, resulting in a change in annual GHG emissions whose direction (decrease or increase) tends to depend on the FU chosen: gross emissions per farm (kg CO 2 eq/farm/year) or emission intensity of milk (kg CO 2 eq/l milk). One strategy is to minimize DM Maize , which implies minimizing gross emissions but producing less milk than the dataset mean, meaning that emission intensity increases. A second strategy is to minimize DM Grass , which keeps gross emissions nearly constant and maximizes milk production, meaning that emission intensity decreases. A third strategy is to maximize DM Grass, which implies decreasing gross emissions and milk production (the former more than the latter), meaning that emission intensity decreases. Finally, a fourth strategy is to maximize DM Maize , which implies increasing gross emissions and milk production (the former more than the latter), meaning that emission intensity increases. In these four strategies, CH 4 emissions tended to vary in the same way as total GHG emissions, contributing approximately half of the farms' GHG emissions.

These four strategies also reflected farmers' management practices. For instance, feed purchases depend on a farm's degree of feed self-sufficiency [START_REF] Coquil | Transition to self-sufficient mixed crop-dairy farming systems[END_REF]. The contribution of purchased feed to gross GHG emissions was the smallest for GrassMAX and largest for GrassMIN. Likewise, from MaizeMIN to MaizeMAX, the contribution of organic and inorganic fertilization to gross GHG emissions increased but that of manure storage and management decreased. Meanwhile, no strategies resulted in a significant change in on-farm energy (diesel and electricity) consumption or amount of fertilizers purchased.

From a product-oriented viewpoint, decreasing GHG emissions per l of milk is a primary objective, though debate continues over the total and per-ha impacts of the intensive dairy farms required to reach such high levels of milk production and which functional unit(s) best represent environmental impacts of dairy farms [START_REF] Salou | Environmental impacts of dairy system intensification: the functional unit matters![END_REF]. In practice, management changes to address both GHG emissions and farm profitability should be a compromise between the four strategies highlighted to develop farm-level GHG mitigation strategies that could be financially viable.

EVT

Applying EVT identified dairy farms with extremely low DM Grass and DM Maize . Hazards that would further decrease DM Grass and DM Maize would have greater consequences on these extreme minimum samples than on the other samples. Thus, one research perspective is to assess the risks that dairy farms face if they already produce small amounts of forage, as well as potential consequences of economic or climate perturbations on their productivity and environmental impacts. One must consider, however, the sensitivity of threshold values to various factors, such as a dairy farm's type (extensive, intensive) and geographic location.

Doing so would provide useful information about variability in threshold values and the factors that influence it. Furthermore, sensitivity of a sample's GHG emissions to the choice of threshold values should be studied further. For instance, a threshold that is too small (or large) may lead to including (or excluding) values that are not (or are) extreme, affecting the relevance of results. In our study, one main criterion for choosing threshold values was to have a sufficiently large sample of dairy farms with extreme values of the inputs considered, to ensure the representativeness of the minimum and maximum samples extracted.

Nonetheless, each extreme sample contained only 10-13 farms, which limited the representativeness of farms with extreme differences in forages and influenced the significance of statistical tests performed to assess differences between the samples. In a sufficiently large dataset, testing the adequacy of fit of the extreme value distribution to the GPD and measuring the error may be the only criteria needed to choose a threshold value.

Figure 1 .

 1 Figure 1. Pairwise scatter plots of areas of pastures and silage maize in UAA (%), number of livestock units (LU), concentrated feed fed (g/l of milk produced), milk production per cow (l), and estimated greenhouse gas (GHG) and enteric methane (CH 4 ) emissions (kg CO 2 eq/farm/year) of 78 dairy farms in Normandy (France) in 2014, with Spearman's rho rank correlation coefficients ( = 78).

Figure 2 .

 2 Figure 2. Distributions of the maximum sample of dry matter (DM) of grass from pastures (kg/livestock unit (LU)/year) extracted as a function of dataset quantile (75%, 80%, 85% and 90%)

Figure 4 .

 4 Figure 4. Bivariate plot of annual gross greenhouse gas (GHG) emissions (kg CO 2 eq) of farms (a) per farm and (b) per l of milk produced, as a function of annual dry matter (kg per livestock unit (LU) per year) of grass (with solid lines for maximum and minimum thresholds) and maize silage (with dashed lines for maximum and minimum thresholds)

Table 1 .

 1 Descriptive statistics of annual characteristics of 78 dairy farms in Normandy (France) in 2014. 1Q = 1 st quartile, 3Q = 3 rd quartile, CV = coefficient of variation, GHG = greenhouse gas

	Characteristic	Min 1Q		Median Mean 3Q	Max CV (%)
	Utilized Agricultural Area (UAA) (ha)	30.6 82.5	104.9 118.7 149.4 258.1	44
	Main Forage Area (ha)	21.5 56.7	76.7 83.5 98.1 204.4	45
	Area of pastures (ha)	13.8 35.4	45.4 55.8 73.2 161.5	59
	Area of pastures in UAA (%)	15.2 34.6	44.9 48.1 59.0 100.0	40
	Area of silage maize (ha)	0.0 18.2	24.7 27.1 36.1 59.6	48
	Area of silage maize in UAA (%)	0.0 17.9	24.3 24.1 31.2 44.6	38
	Number of livestock units	32	81	97	108	130	201	36
	Replacement rate (%)	0.0 26.0	31.5 30.5 36.0 55.0	32
	Grass yield (t DM/ha/year)	1.9	4.7	5.9	6.1	7.1 12.6	32
	Concentrated feed fed (g/l milk)	0.0 144.5	184.0 184.8 16.8 319.0	35
	Milk production per cow (l/year)	4271 6278	7387 7210 8283 9490	18
	Gross GHG emissions (kg CO 2 eq/farm/year)	3830 6173	6725 6618 7160 8118	12
	Enteric CH 4 emissions (kg CO 2 eq/farm/year)	2537 3421	3576 3566 3729 4408	10
	IPCC Tier 3 methodology (IPCC, 2006) was used to predict CH 4 emissions (g CH 4 /kg of
	digestible organic matter) from dairy cows and heifers, based on an empirical model adapted

Table 2 .

 2 Results of applying Extreme-Value Theory to extract minimum ("MIN") and maximum ("MAX") samples of dry matter (DM) of grass from pastures (kg/livestock unit (LU)/year) as a function of multiple quantiles of the entire dataset of 78 farms in Normandy (France). Bold values indicate the quantiles retained as thresholds. RMSE = root mean squared error, K-S = Kolmogorov-Smirnov.

			DM of grass	Sample	Estimated	Estimated		K-S adequacy
	Sample Quantile	(kg/LU/year)	size	shape ̂	scale ̂	RMSE	test (p-value)
	MIN	10%	1586	7	-0.50	451.02	489.23	0.10
		15%	1708	11	-0.26	400.02	482.11	0.33
		20%	1868	15	-0.30	471.12	531.47	0.55
		25%	2088	19	-0.29	547.12	646.07	0.11
	MAX	75%	3614	19	-0.28	1528.51	1790.08	0.23
		80%	4176	16	-0.20	1185.48	1458.05	0.62
		85%	4480	13	-0.22	1115.73	1356.73	0.66
		90%	4946	10	-0.18	919.31	1142.26	0.85

Table 3 .

 3 Results of applying Extreme-Value Theory to extract minimum ("MIN") and maximum ("MAX") samples of dry matter (DM) of maize silage (kg/livestock unit (LU)/year) as a function of multiple quantiles of the entire dataset of 78 farms in Normandy (France). Bold values indicate the quantiles retained as thresholds. RMSE = root mean squared error, K-S = Kolmogorov-Smirnov.

			DM of maize silage	Sample	Estimated	Estimated		K-S adequacy
	Sample Quantile	(kg/LU/year)	size	shape ̂	scale ̂	RMSE	test (p-value)
	MIN	10%	2860	5	-0. 34	1517.92	1024.74	0.57
		15%	3439	9	-0. 19	1457.39	1205.25	0.78
		20%	3647	13	-0. 03	1237.25	1152.08	0.49
		25%	3811	17	0.03	1124.43	1120.97	0.34
	MAX	75%	5872	19	0.01	1306.87	1863.52	0.90
		80%	6115	16	0.02	1282.16	1853.90	0.84
		85%	6443	13	0.06	1225.65	1820.91	0.67
		90%	6935	8	0.07	1318.36	1945.92	0.61

Table 4 .

 4 Mean and coefficient of variation (CV) of annual farm characteristics for the entire dataset and for farms in minimum ("MIN") and maximum ("MAX") samples of dry matter (DM) of grass and maize silage. % chg. = percentage increase from minimum to maximum sample. M-W p = p-value of Mann-Whitney test for the difference between minimum and

	maximum samples. UAA = Utilized Agricultural Area, MFA = Main Forage Area, LU =
	livestock unit
				Grass DM sample			Maize silage DM sample
			Entire	MIN	MAX			MIN	MAX
	Variable	Statistic	dataset	sample	sample	% chg. M-W p	sample	sample	% chg. M-W p
	UAA (ha)	Mean	118.7	121.7	124.8	+3%	0.56	112.7	127.8	+13%	0.27
		CV	44%	26%	56%			51%	36%
	MFA (ha)	Mean	83.5	76.2	103.8	+36%	0.39	94.2	90	-5%	0.75
		CV	45%	39%	52%			45%	41%
	Grass DM	Mean	3031	1325	5862	+342%	<0.001	3987	2531	-36%	0.07
	(kg/LU/year)	CV	47%	23%	12%			37%	56%
	Maize silage	Mean	4836	5939	3931	-33%	0.02	2409	7669 +150%	<0.001
	DM (kg/LU/	CV	38%	10%	76%			51%	19%
	year)
	Number of LU	Mean	108	119	93	-21%	0.10	111	102	-8%	0.55
		CV	36%	24%	44%			39%	30%
	Concentrated	Mean	185	177	153	-13%	0.39	173	186	+7%	0.85
	feed fed (g/l	CV	35%	25%	47%			55%	34%
	milk prod./year)
	Milk prod. per	Mean	7210	7954	6269	-21%	0.04	5785	7351	+27%	<0.001
	cow (l/year)	CV	18%	12%	24%			19%	15%

Table 5 .

 5 Mean and coefficient of variation (CV) of annual gross greenhouse gas (GHG) and enteric methane (CH 4 ) emissions (kg CO 2 eq) by functional unit (FU) (per farm or per l of milk produced) for the entire dataset and for farms in minimum ("MIN") and maximum ("MAX") samples of dry matter (DM) of grass from pastures and maize silage. % chg. =

	percentage change from minimum to maximum sample. M-W p = p-value of Mann-Whitney
	test for the difference between emissions for farms in minimum and maximum extracted
	samples of DM of each forage.
					Grass DM sample			Maize silage DM sample
				Entire	MIN	MAX	%	M-W	MIN	MAX	%
	FU	Variable	Statistic	dataset	sample	sample	chg.	p	sample	sample	chg.	M-W p
	Farm	GHG	Mean	6618	7044	6113 -13%	0.13	5549	6962 +25%	<0.001
		emissions	CV	12%	7%	20%			16%	9%
		Enteric CH 4	Mean	3566	3430	3667	-6%	0.05	3208	3616 +12%	0.001
		emissions	CV	10%	13%	4%			9%	8%
	l of milk	GHG	Mean	0.93	0.89	0.99 +11%	0.03	0.98	0.97	-1%	0.93
	produced	emissions	CV	14%	11%	13%			13%	17%
		Enteric CH 4	Mean	0.50	0.46	0.56 +21%	0.003	0.56	0.50	-11%	0.02
		emissions	CV	13%	8%	13%			11%	12%

Table A2 .

 A2 Mean and coefficient of variation (CV) of additional farm characteristics for the entire dataset and for farms in minimum ("MIN") and maximum ("MAX") samples of dry matter (DM) of grass and maize silage. % chg. = percentage increase from minimum to maximum sample. M-W p = p-value of Mann-Whitney test for the difference between minimum and maximum samples. % chg. = percentage increase from minimum to maximum sample. UAA = Utilized Agricultural Area, LU=Livestock Unit, MFA=Main Forage Area, GHG = greenhouse gas

				Grass DM sample			Maize silage DM sample
			Entire	MIN	MAX	%	M-W	MIN	MAX	%	M-W
	Variable	Statistic	dataset	sample	sample	chg.	p	sample	sample	chg.	p
	Area of pastures in	Mean	48.1	31.3	70.3 +124% <0.001	72.3	38.5	-47% <0.001
	UAA (%)	CV	40%	36%	28%			26%	35%
	Area of silage maize in	Mean	24.1	30.7	14.5	-53% <0.001	13.7	31.7 +131% <0.001
	UAA (%)	CV	38%	22%	60%			62%	14%
	Replacement rate (%)	Mean	30.5	29.5	31.1	+5%	0.36	29.2	29.9	+2%	0.91
		CV	33%	17%	29%			34.2.%	25.6%
	Stocking rate (LU per	Mean	1.4	1.6	0.9	-42% <0.001	1.2	1.1	-8%	0.89
	ha of MFA)	CV	26%	19%	24%			31%	21%
	Enteric methane	Mean	49.6	48.0	52.6	+10%	0.09	53.9	47.7	-11% <0.001
	emissions in total GHG	CV	9%	6%	14%			9%	7%
	(%)
	Manure storage and	Mean	19.5	19.2	19.8	+3%	0.54	20.7	18.2	-12%	0.02
	management in total	CV	10%	10%	9%			7%	17%
	GHG (%)
	Purchased feeds in total	Mean	12.5	15.4	8.8	-43% <0.001	9.3	12.8	+37%	0.09
	GHG (%)	CV	33%	16%	50%			48%	29%
	Organic and inorganic	Mean	9	8.9	8.9	-0.1%	1	7.7	10.1	+32%	0.02
	fertilization in total	CV	25%	29%	25%			29%	25%
	GHG (%)
	Energy consumption in	Mean	5.3	5.4	5.3	-3%	0.87	5.7	5.8	+3%	0.43
	total GHG (%)	CV	27%	12%	22%			39%	20%
	Fertilizers in total GHG	Mean	3.4	3.5	3.4	-4%	1	2.5	3.0	+18%	0.54
	(%)	CV	52%	52%	69%5			66%	67%

Acknowledgments

We thank the Animal Physiology and Livestock Systems Division (PHASE) of INRA for its financial support. We also thank the French Livestock Institute (IDELE) for providing the dataset and two anonymous referees for their careful review and helpful comments, which led to considerable improvement in the article.

Supplementary Materials

489

Table A1. Descriptive statistics of the percentage of total greenhouse gas (GHG) emissions 490 due to methane emissions, manure storage and management, purchased feeds organic and