We first consider the case of a single area source with a single concentration sampler (target). The source varies with time. The method is based upon the general superimposition principle (Thomson et al., 2007), which relates the concentration at a given location C(x,t) to the source strength S(t) and the background concentration Cbgd(t) using a transfer function D(x,t), which has the dimensions of a transfer resistance (s m-1). C(x,t)=D(x,t)×S(t)+Cbgd(t) Here x denotes the location of the sensor and t the time. The concentration and source units are in µgN-NH3m-3 and µgN-NH3m-2s-1, respectively. The superimposition principle implies that the studied tracer must be conservative, which is a reasonable hypothesis for NH3 whose reaction time with acids in the atmosphere is below the transport time for spatial scales below 1000 m (Nemitz et al., 2009). Moreover, in Eq. (1), we assume a spatially homogeneous area source with strength S(t). The spatial homogeneity of the source is less trivial for NH3 than other gas released in agriculture as the source itself depends on the concentration at the surface. However, Loubet et al. (2010) have shown that the heterogeneity of the source can be neglected as long as the dimension of the source is larger than 20 m. Hence, this study is limited to source areas with fetch larger than 20 m and a spread of the concentration samplers over a domain smaller than 1000 m. Moreover, it is interesting to note that for infinitely spread fields, the transfer resistance is linearly linked to the transfer matrix (see Supplement Sect. S1)

Since we consider time-averaging concentration samplers, we develop the time-averaged equation of Eq. (1) over an integration time period τ: C(x)‾=Dx×S‾+Cbgd‾, where the overbars denote a time average over the period τ. Similar to turbulent flux calculations, the first part of the right-hand side of Eq. (2) is decomposed using the Reynolds decomposition of a random variable (Kaimal and Finnigan, 1994), giving C(x)‾=D(x)‾×S‾+Cbgd‾+D′xS′‾, where D(x)′S′‾ is the time covariance between D(x,t) and S(t). If the averaged background concentration Cbgd‾ is a known quantity, Eq. (3) can be easily manipulated to give an estimation of the averaged source strength S‾, the quantity we want to infer: S‾=Cx‾-Cbgd‾Dx‾(I)-D′x×S′‾Dx‾(II).

In the right-hand side of Eq. (4), (I) can be calculated from measured Cbgd‾ and C(x)‾ and D(x)‾, which is itself calculated with dispersion models. Conversely, (II) is a priori unknown and depends on the correlation between the source strength and the transfer function D(x)′S′‾. Hence, if (II) is neglected, the inferred source S‾ is biased. The relative bias of the method is then δS‾S‾=D′xS′‾Dx‾×S‾. Hence we show in Eq. (5) that time averaging leads to a relative bias which can be quantified by the time covariance between the transfer function and the source strength. However, this quantity is by nature unknown since the dynamics of S(t) is unknown. Determining D(x)′S′‾ requires knowledge of the source dynamics, which can be obtained from measurements with a micrometeorological method. It can alternatively be approached by modelling using state-of-the-art ammonia exchange processes as we do here.

In addition to the bias, which is term (II) in Eq. (4), evaluating term (I) is encompassed with errors related to the uncertainties in Cbgd‾, C(x)‾ and D(x)‾. In particular, cases in which D(x)‾ is small may lead to large errors in inferring the source term S. This is linked to the conditioning of the inverse problem and is discussed in Supplement Sect. S2.

If we generalise the approach to multiple sources and multiple receptors, then the transfer function becomes a matrix D(xi,Sj,t), which is the contribution of source Sj to concentration at a target located at xi. For reading purposes we simplify the matrix notation to Di,j. Equation (3) then becomes C1⋮CM‾=D1,1⋯D1,M⋮⋱⋮DN,1⋯DN,M‾×S1⋮SM‾+Cbgd‾+D′1,1⋯D′1,M⋮⋱⋮D′N,1⋯D′N,M×S′1⋮S′M‾, which in condensed notation gives C(xi)‾=Di,j‾×Sj‾+Cbgd‾+Di,j′×Sj′‾. If the number of targets is equal to the number of sources, the problem can be solved by inversion of a linear system. If the number of targets is larger than the number of sources, the problem is a multiple linear regression type with unknowns Sj‾ and Cbgd‾. The third term on the right-hand side of Eq. (6b) is a bias which is a priori unknown and which we will evaluate in this study.

The inferred sources, Siinferred‾, were derived from Eqs. (3) or (6) assuming the covariance term (last term on right-hand side) was null. The method used to infer the source was either a simple division (Eq. 3) or an optimisation of the linear system using the linear model function lm in R (package stats, R version 3.2.3), with either M = 1 (single source) or M = 9 (multiple sources): D1,1⋯D1,M⋮⋱⋮DN,1⋯DN,M‾×S1inferred⋮SMinferred‾=C1⋮CN‾-Cbgd‾. The bias δSi was then evaluated as the difference between the inferred sources Siinferred‾ and the modelled sources Siobs‾ averaged over each period: δSi=Siinferred‾-Siobs‾. As shown in Eqs. (3) and (6) the overall mean bias δSi contains (i) a bias term due to the inference method which is dependent mainly on the conditioning of the matrix Di,j (see Supplement Sect. S2) and (ii) a bias term which is intrinsically linked to the covariance between Di,j and Sj (Eqs. 3 and 6). Thus, with Eq. (8) we evaluate the sum of the two biases without distinction. In order to infer the sources, the elements of the dispersion matrix Di,j need to be determined. The next part details how these were estimated with a dispersion model.

A range of meteorological conditions were simulated based on the half-hourly meteorological data of the FR-Gri ICOS site in 2008. In total 13 periods of 28 days were considered, which spanned the whole year except the last 2 days of the year. Each period consisted of 1344 half-hourly data.

In order to evaluate the influence of the concentration averaging period on the source inference, several integration periods τ were tested: 0.5 (no integration), 3, 6, 12, 24, 48 and 168 h (7 days). In practice the concentrations were computed at each sensor location using Eq. (6) over 0.5 h: at that timescale, which corresponds to the spectral gap, the covariance term is assumed to be negligible (Van der Hoven, 1957). Then the averaged concentrations were computed for all integration periods.

Several scenarios were considered and summarised in Table 1.

The background concentration Cbgd‾ was either supposed known and fixed to the prescribed values (C1–C4) or was inferred (C5–C7).

Figure 5 reports an example source inference, which shows the progressive smoothing of the source with integration period. We first see that the source strength corresponding to Γ=105 leads to ammonia emissions ranging from 0 to ∼ 1 µgNH3m-2s-1 in the winter, which corresponds to 0.71 kg N ha-1 day-1. Over the entire year, the maximum emissions occur during the hottest days and reach up to 7.1 kg N ha-1 day-1. Regarding the inference method, it can be seen in that example that, up to 24 h, the variability in emissions over the period is captured quite well.

In this simpler case shown in Fig. 6, the fractional bias of the inferred emissions is mostly negative for the combination in which the ratio of sensor height to plot dimension is small and integration times are larger than 6 h. According to Eq. (5), this means that the covariance term D′S′‾ is negative for these conditions, meaning that any increase in source strength S at a time t is correlated with a decrease in the transfer function D(t)and vice versa. This is expected as S(t) increases with the surface temperature (Eq. 10) and is proportional to Razref+RbNH3-1 (Eq. 9), while D(t) is proportional to the aerodynamic resistance Ra(zref), as shown in Supplement Sect. S1. Hence, over daily periods, S and D are negatively correlated: S increases during the day and decreases at night (due to temperature and wind speed daily patterns), while D decreases during the day and increases at night (mainly due to wind speed patterns). This is expected to be a general feature for NH3 surface fluxes as the daily variability reproduced by the model used in this study is representative of most situations from mineral and organic fertilisation to urine patches or seabird colonies (Ferrara et al., 2014; Flechard et al., 2013; Milford et al., 2001; Móring et al., 2016; Personne et al., 2015; Riddick et al., 2014; Sutton et al., 2013).

The median bias δSi tends to increase in magnitude with the sensor height for large fields (xplot=100 and 200 m), while it decreases for smaller fields (xplot=25 and 50) when sensor height gets close to the field boundary layer height. Furthermore, δSi becomes positive and very large when sensors are above the field boundary layer height (Fig. 6). For large fields, the increase in the magnitude of the bias with lower sensor height is expected as D decreases with height in absolute value. For small fields, the decrease in the bias corresponds to a loss of information as D gets close to zero when the sensor gets closer to the field boundary layer height. For heights above this limit, we observe a change in sign of the bias, which can be explained by the fact that the sensor concentration footprint is not in the source during stable conditions (at night), while it is in the source under unstable conditions during the day. The inference method will hence not work if at least one sensor is not below the plot boundary layer height.

We also note that for integration periods equal to or below 3 h, the fractional bias is slightly positive, which can be explained by the positive correlation between S and D at small timescales. This is because of the influence of u∗ on T(z0): for a given solar radiation and air temperature over small timescales (< 3 h), an increase in u∗ leads to a decrease in T(z0), which leads to an exponential increase in the surface compensation point according to Eq. (10). However, at the same time, Ra(z)-1 decreases, but linearly with u∗. The resulting ammonia emissions calculated with Eq. (9) nevertheless increases because the exponential effect of temperature overcomes the linear effect of the exchange velocity (data not shown). This effect is more visible for large fields than small fields because over small fields an additional effect is that when u∗ decreases, the footprint increases and the source “seen” by the targets hence decreases because it incorporates a fraction of zero emissions sources.

Overall, the median fractional bias for weekly integrated emissions over a 25 m field and sensor heights below 0.5 m was overall -8 % with an interquartile (-14 to -2 %). We can conclude that the bias of the NH3 emissions is reproducible within ±6 %. We can also conclude that it would be better to place the concentration sensor at a low height to minimise the bias of the method.

The inference method depends on the turbulence at the site and especially on the main drivers of the dispersion, which are the friction velocity and the stability regime. Indeed, Fig. 7 shows that the relative root-mean-square residual of the inferred source (RRMSR) decreases with increasing u∗ at long integration periods and is larger in slightly stable than near-neutral or slightly unstable conditions. Figure 7 also shows that under stable conditions or low u∗ the RRMSR increases by more than an order of magnitude (up to 50 %) when integration periods increase from 6 to 12 h, which catches most of the source variance. We also see that under near-neutral or high u∗ conditions, the third quartile of the RRMSR remains below 10 % for all integration periods. Finally, we also see that the larger third quartiles at short integration periods are obtained with intermediate u∗ values or slightly unstable conditions. A similar response of the bias to u∗ and 1/L was reported by Fig. 6 in Flesch et al. (2004) and Fig. 3 in Gao et al. (2009) in controlled source experiments. While Gao et al. (2009) attributed the bias of the inference method to parameterisation of the stability dependence of the turbulent parameters (z/L), in this study this cannot happen since we use the same parameterisation for prescribing the concentration and inferring it. In our case, the interpretation is to be linked with Eq. (5): the smaller u∗ or the most stable conditions also correspond to the larger time derivatives of source strength (driven by surface temperature and surface exchange resistances) as well as the larger time derivatives of transfer function D. We hence expect that under such conditions, the covariance between the transfer function and the source strength will be larger than under near-neutral conditions. In a more heuristic view, under low turbulence, large time derivatives of concentrations are expected above a source due to low mixing (small changes in mixing lead to large variations in concentrations).

We conclude that the inference method with a long integration period will lead to very moderate biases for locations with near-neutral conditions and high wind speed, but may lead to much larger bias under stable conditions and low wind speed as soon as the integration period reaches 12 h.

We first consider strategy C1, which is the simplest configuration, in which plots are independent, background concentration is known and one target is used above each plot. Figure 9 shows that for the given treatment range (∼ 1–100 µg NH3 m-2 s-1), the fractional mean bias is lower than 0.2 in magnitude for the treatment emitting the most (treatment 3, Γ=106), lower than 0.4 for the intermediate treatment (treatment 2, Γ=105) and up to 8 for the treatment emitting the least (treatment 1, Γ=105); here we considered the 0.25–0.75 quantiles. The bias of the highest treatment (treatment 3) actually behaves similarly to a single-source case (Fig. 6), with a median bias around 10 % for 48 h integration periods. This is expected because treatment 1 and treatment 2 have a much smaller emissions strength and hence little influence on the concentration above the treatment 3 plots, which therefore behave in a similar manner to a single source. As a consequence, this bias in treatment 3 is mainly due to the anti-correlation between D and S, which increases with integration periods. The fractional mean bias is very large for treatment 1 even for small integration periods. The bias can either be positive or negative, showing that this method does not allow for a correct estimation of the smallest sources.

Figure 10 shows that the bias remains low as long as sensor heights are low enough to catch a sufficient part of the field footprint. When only a single height is used (strategy C1) this means that the sensor should be placed at 0.5 m or below for the field size we have tested here (25 m). The result is similar for a pair of sensors (strategy C3). For the lowest treatment though, the bias (and its variability) remain high whatever the height. It is interesting to notice that the heights which were found to provide an optimal inference of NH3 sources (below 0.5 m) are smaller than ZINST (the height at which the vertical flux can be approximated by the horizontal flux) reported by Wilson et al. (1982) (which were 0.9 m for 40 m diameter circular sources, and which we estimate as 0.65 m based on a power law extrapolation as in Laubach et al., 2012). It is also important to note that this height should vary with both the roughness length z0 and displacement height as was shown by Wilson et al. (1982) for ZINST.

Increasing the plot size from 25 to 200 m in width reduces the bias of the two highest source treatments for which the median bias reaches values around 10 %, while the interquartiles remain stable (Fig. 11). Conversely, in treatment 1 (Γ=104), the bias increases. It is expected that the bias in a multiple-source configuration never becomes smaller than the bias in a single-source problem, which is a limit linked to the time integration (covariance between the source and the concentration; see Eqs. 3 and 6). It is also expected that the biases remain higher than the single-source case until the source size increases sufficiently so that the concentration generated by a block on the neighbour fields becomes negligible compared to the concentration generated by the source below. This is what we observe in treatment 2 (Γ=105) and treatment 3 (Γ=06), with treatment 2 showing a median bias of -13 % (larger than in the single-source case) for the 200 m wide field, while the bias of the largest source tends to be -10 % [-17 %, -1 %], which is the range observed for a single source.

A central question is the capability of the inference method to resolve small or large differences in emissions from the nearby blocks. Indeed, we can speculate that small differences will be hard to resolve while large differences will lead to large bias. In order to determine the resolution power of the method, we compared the performance of the inference method with a set of three treatments: the first treatment had Γ=0 to mimic a reference field receiving no nitrogen; the second treatment had a constant Γ=1000 corresponding to a small emissions (0.7 kg N ha-1 day-1), and the third treatment Γ was successively set to increasing values from 1500 to 105 (70 kg N ha-1 day-1). In this section we consider the background to be known (sensitivity to the background concentration will be evaluated in the next section).

Figure 12 shows the median and interquartile biases of the cumulated emissions for the longest integration period of 168 h over the ratio of the high-to-low source treatments. The bias of the largest source always remained around 14 %, which is larger than the single-source case. The bias of the lowest source increased with increasing inter-treatment source ratios from 13 to 40 %. In fact we find that the fractional bias increased approximately as a power function of the ratio of the two predicted sources (dotted lines, 0.11 x0.256).

As pointed out by Flesch et al. (2004), the knowledge of the background concentration is essential in a source inference problem. Retrieving the background necessitates having at least Nsources+1 sensors. Hence only strategies with two heights per plot or which assume identical emissions in treatment repetitions can be evaluated in their capacity of retrieving the background (strategy C2 to C7). In order to evaluate the sensitivity of the method when the background concentration varies with time, we set a realistic background concentration as a linear combination of u∗ and air temperature (Ta) with a mean of 6 µgNH3m-3 and a standard deviation of 0.1 µgNH3m-3. This test was performed with a range of treatments in order to elucidate the correlations between varying background and varying treatments. We see in Fig. 13 that the concentration, which follows a realistic pattern, is well retrieved even over the longest integration period of 168 h. However, we see that for the treatments with the largest source contrast (Γ=1000 and 105), the background concentration can be overestimated even for small integration periods (6 h). The median residual of the background concentration was smaller in magnitude than 0.05 µgNH3m-3, except for the case with very large differences among treatments (0, 1000, 10 000), for which the residual reached 0.1 and 0.5 µgNH3m-3 for the 6 h and 24 h or 168 h integration periods. Furthermore, the background concentrations were overestimated for the largest source ratios and underestimated for the lowest source ratios and longer integration periods (24 and 168 h).

Finally, to identify which strategy is the most suitable for retrieving the emissions from the multi-plot configuration, we compared all strategies for a simulation with a variable background (set as in the previous section) and two source ratios of 2 and 20 between treatments 2 and 3 (treatment 1 being a zero-source reference). We found, as expected, that strategies with known backgrounds have low biases compared to strategies that calculate the background, except for strategy C7, which provided biases similar to strategy C3, which is the strategy equivalent to C7 but with a known background (Fig. 14). We also see that incorporating some knowledge of the sources by assuming plots from the same treatment have the same emissions gave slightly better estimates when the background is known (strategies C2 and C4 compared to C3). This is however not true when the background is unknown, in which case the magnitude of the bias increases up to a median of 0.7 (strategies C5 and C6 compared to C7). It is due to compensation between background concentration and source strength as we have seen in Fig. 14 that the background concentration was overestimated in such cases. We also see, as expected, that the strategies with two sensors placed at different heights above each plot lead to better evaluations of the emissions. Overall, the strategy based on two sensors above each plot, which also assumes that sources are independent, seems to be the most robust (strategy C7). This strategy does not assume the background is known, nor does it assume the plots have similar emissions, which is more adapted to reality. Indeed, even though the same amount of nitrogen is applied in each repetition plot, the emissions may vary due to soil heterogeneity and advection. We finally obtain a median bias for strategy C7 which is -16 % with an interquartile [-8–22 %]. It is important to stress though that the minimums and maximums are further away, which indicates that under some rarer circumstances, the method may overestimate the sources by 12 % or underestimate them by 40 %. These cases correspond to integration periods with very low wind speeds and stable conditions.

Crenna et al. (2008) have clearly shown that the optimal sensor positioning should be so that each sensor preferentially sees a single source, and reversely, each source should preferentially influence a single sensor. In this study the source–sensor geometry was especially designed in a way that minimises the condition number by placing the sensors in the middle of each plot. For the smallest source (xplot=25 m), the conditioning number ranged from 1.97 to 3.01 (median 2.42) for sensors located at 0.25 m, and increased to 2.6–6.9 (median 3.2) for sensors at 0.5 m, 4.7–150 (median 21) for sensors at 1.0 m, and 40–165 000 (median 640) for sensors at 2 m. This shows that including at least one sensor per block at heights lower than the field width divided by 20 would ensure that the conditioning number remains lower than in most trials reported by Crenna et al. (2008).

By comparing different strategies we have found that the strategies using two sensors over each source systematically led to improved performances (C3 versus C1 and C6 versus C5, Fig. 14). This is also in line with the results of Crenna et al. (2008), who showed that using more sensors separated spatially improves the performance of the inference method. Hence we can conclude that the inference method we used is based on a well-conditioned system which leads to robust results of the least-square optimisation. This is further illustrated by the real-case example (Figs. 15–17), which shows a good reproducibility among block repetitions. Indeed, good reproducibility among repetitions is a check for evaluating the quality of the inference method in real test cases. The use of the Bayesian inference method would however also be valuable in the set-up we propose here.

The use of time-averaging sensors for estimating ammonia sources was already reported by Sanz et al. (2010), Theobald et al. (2013), Carozzi et al. (2013a, b), Ferrara et al. (2014) and Riddick et al. (2016a, 2014). All these studies have shown the feasibility of these measurements; however only a few of them allow the estimation of the impact of averaging: Riddick et al. (2014) measured emissions from a bird colony on Ascension Island with WindTrax using both several ALPHA samplers in a transect across the colony and a continuous analyser for ammonia (AiRRmonia, Mechatronics, NL) downwind. They also averaged the continuous sampler concentrations to evaluate the effect of averaging on the emissions estimates. They found as we do here that averaging over monthly periods would lead to systematic underestimations from -9 to -66 %. They also found that estimations from diffusive samplers would lead to average underestimations of -12 %. This is very close to what we find here for a single source over 1 week (Fig. 6). In a similar comparison Riddick et al. (2016b) found that time integration led to slight overestimations with the integration approach, which is within the range of statistics of the bias we have found for the larger area sources (third quartile in Fig. 6).

We should bear in mind that the use of time-averaging sensors in the inference method is also highly dependent on the surface layer turbulent structure as shown by Fig. 7. We find, as expected, that stable conditions or low wind speed conditions are those that lead to the highest potential bias (as shown by the third quartile under stable conditions at the bottom of Fig. 7). This is a well-known limitation of inverse dispersion modelling which was reported by Flesch et al. (2009, 2004) and which suggested that inverse dispersion would be inaccurate for u∗<0.15 m s-1 and |z/L|<1. However, both our study and the studies of Riddick et al. (2014, 2016b) show that this is not as much of an issue for ammonia emissions. Indeed, this is due to the fact that ammonia emissions follow a daily cycle with low emissions at night and high emissions during the day. This is firstly because the ground surface compensation point concentration (Cpground) has an exponential dependency on surface temperature as assumed in Eq. (10) based on known thermodynamical equilibrium constants (Flechard et al., 2013). This is secondly due to the fact that ammonia emission is a diffusion-based process which is limited by the surface resistances, as modelled in Eq. (9), which leads to small fluxes when Razref and RbNH3 become large, which happens during low wind speeds (they are both roughly inversely proportional to wind speed) and stable conditions, which also happens at night (Flechard et al., 2013). In real situations, the combination of small turbulence and high surface concentration leads to a further decrease in the flux, which is dependent on the difference between Cpground and the concentration in the atmosphere above (a feature which was not accounted for in this study as this would imply a higher degree of complexity in the modelling approach). This means that the results we found in this study would not apply for species with an emissions pattern with different temporal dynamics (either constant or anti-correlated with surface temperature or wind speed).