Abstract
Longterm monitoring stations for atmospheric pollutants are often equipped with lowresolution concentration samplers. In this study, we analyse the errors associated with using monthly average ammonia concentrations as input variables for bidirectional biosphereatmosphere exchange models, which are commonly used to estimate dry deposition fluxes. Previous studies often failed to account for a potential correlation between ammonia exchange velocities and ambient concentrations. We formally derive the exact magnitude of these errors from statistical considerations and propose a correction scheme based on parallel measurements using highfrequency analysers. In case studies using both modelled and measured ammonia concentrations and micrometeorological drivers from sites with varying pollution levels, we were able to substantially reduce bias in the predicted ammonia fluxes. Neglecting to account for these errors can, in some cases, lead to significantly biased deposition estimates compared to using highfrequency instrumentation or corrected averaging strategies. Our study presents a first step towards a unified correction scheme for data from nationwide air pollutant monitoring networks to be used in chemical transport and air quality models.
Introduction
Gaseous ammonia (NH_{3}) plays an important role in the atmosphere as part of the natural and anthropogenic N cycle and contributes to a number of adverse effects on the environment and public health^{1}. Recent developments allow the direct quantification of NH_{3} dry deposition and emission fluxes via the eddycovariance method^{2,3,4}; however, the necessary instrumentation is costly, longterm continuous studies are yet to be published, and the method is not trivially applicable in every environment. Alternative methods, such as the aerodynamic gradient technique, are even more labourintensive, usually require expensive wetchemical analyses, and are prone to errors in nonideal conditions^{5}.
A cost and labourefficient alternative to flux measurements is the use of socalled dry deposition inferential models. If they are properly validated against flux measurements in different ecosystems, they can be applied for regional estimates of NH_{3} dry deposition using only concentration measurements and a small number of (micro)meteorological variables as input data^{6,7,8,9,10}. These models are usually ran on a 30 minute basis, in accordance with the typical temporal resolution of flux measurements, or on an hourly basis within some largescale chemistry transport models (CTM), such as LOTOSEUROS^{11,12}. However, in national monitoring networks, such as the Measuring Ammonia in Nature (MAN) network in the Netherlands^{13}, often passive samplers or denuders (e.g. DELTA^{14}, or KAPS^{15,16}) are used to measure ambient NH_{3} concentrations, which typically only yield a temporal resolution of monthly averages. The impact of using such lowresolution concentration measurements as input data for bidirectional NH_{3} dry deposition inferential models has, to our knowledge, not been thoroughly investigated in the published literature, although they have regularly been used from local studies^{17,18} to integrated projects^{19}. In order to systematically assess potential bias introduced by using lowresolution concentration data, we exemplarily analysed a 1 year gapfree record of ambient NH_{3} concentrations predicted by the CTM LOTOSEUROS in conjunction with ECMWF (European Centre for MediumRange Weather Forecasts) meteorology as input data for an independent dry deposition inferential model by Massad et al.^{7}, as well as preliminary NH_{3} concentration measurements using quantum cascade laser (QCL) spectroscopy at a remote site in Germany. We investigated the potential magnitude of errors introduced by using lowresolution concentration measurements and formally derived the fundamental equations necessary for the development of correction schemes. Our study lays the groundwork for the characterisation of errors and estimation of sitespecific correction functions when using NH_{3} dry deposition models with lowresolution input data.
Methods
Dry deposition inferential modelling
Dry deposition of NH_{3} is most commonly modelled using parameterisations of a bigleaf canopy compensation point model, or a twolayer variant thereof when exchange with the soil or litterlayer is expected to be significant and can be parameterised within reasonable margins of uncertainty^{6,7,20}. We here use the parameterisation of Massad et al.^{7} in a onelayer configuration to ensure independence from the dry deposition module (DEPAC within LOTOSEUROS) involved in the generation of the synthetic data. In this model, the flux density of NH_{3} is predicted from the difference of the measured air NH_{3} concentration χ_{a} (μg NH_{3} m^{−3}) at the aerodynamic reference height z − d (m) and the (modelled) canopy compensation point concentration, χ_{c} (μg NH_{3} m^{−3}) (Fig. 1). The sign of this difference governs the direction of the flux (χ_{a} > χ_{c} leads to a deposition flux, with a negative sign by convention, and χ_{a} < χ_{c} leads to an emission flux). Furthermore, the magnitude of the predicted flux density is controlled by the magnitude (i) of χ_{a} − χ_{c}, and (ii) of a number of resistances towards deposition. Within this framework, the total net biosphereatmosphere exchange flux of NH_{3}, F (μg NH_{3} m^{−2} s^{−1}), is typically given as
where R_{a} (s m^{−1}) and R_{b} (s m^{−1}) are the aerodynamic and quasilaminar boundary layer resistance, respectively, and are here modelled as described in detail by Massad et al.^{7}. Instead of calculating the canopy compensation point (which is a function of both stomatal and cuticular resistance and, if applicable, their respective compensation points, and the air NH_{3} concentration), we can simplify the model scheme to strictly consist of serial resistances only (Fig. 1b). The effective ‘foliar compensation point’, χ_{f} (μg NH_{3} m^{−3}), is then given as a weighted average of both leaflayer pathways via
where χ_{w} (μg NH_{3} m^{−3}) and χ_{s} (μg NH_{3} m^{−3}) are the cuticular and stomatal compensation point, respectively, R_{w} (s m^{−1}) is the cuticular resistance, parameterised after Massad et al.^{7}, and R_{s} (s m^{−1}) is the stomatal resistance after Emberson et al.^{21}. In the Massad et al.^{7} parameterisation, χ_{w} is zero (i.e., only deposition to the cuticula is possible). R_{f} (s m^{−1}) is the ‘foliar resistance’, similar to the notation of Wichink Kruit et al.^{8}, and is given as
To further simplify the calculations, we define an exchange velocity, v_{ex} (m s^{−1}), as the inverse of the total resistance to NH_{3} exchange:
Note that R_{f} is not necessarily equal to the socalled canopy resistance, which is usually only used in unidirectional (depositiononly) models (i.e., they are only equal when χ_{f} is zero). Similarly, v_{ex} is not equal to the common concept of a deposition velocity, in which R_{f} is replaced by the canopy resistance and which is not used in conjunction with a compensation point.
Given these definitions, the net NH_{3} exchange flux can also be written as
where a positive flux indicates emission and a negative flux indicates deposition.
Flux prediction strategies for lowresolution input concentrations
Highfrequency concentration measurements are often cost and labourintensive, and usually not available within nationwide longterm monitoring networks. A number of different variants to predict longterm average or cumulative flux densities from lowfrequency concentration measurements can be found in the literature^{18,19,22,23,24,25,26,27,28}. We here discuss the case of monthly averages, which are a common outcome of passive sampler or denuder measurements, but the calculations remain the same for any other kind of averaging period. A sensitivity study for other averaging periods is given in the Supplementary Material.
Consider the case of (i) (micro)meteorological input data available at a sampling frequency the model is usually run at, e.g. 1 hour in our sample datasets, and (ii) ambient NH_{3} concentrations available at a lower sampling frequency, e.g. 1 month. We further assume that, from these data, reasonable flux predictions can only be made at the lowest available time scale, i.e. 1 month in this example. However, the model should still be run at a higher resolution in order to incorporate the effects of diurnal variations and daytoday variability in meteorological conditions. There are generally two straightforward strategies to predict monthly averaged NH_{3} fluxes under these conditions, the first being:
Here, an overbar \(\overline{x}\) denotes the arithmetic mean of some random variable x, and a prime x′ denotes the instantaneous deviation from \(\overline{x}\), i.e. \(x=\overline{x}+x^{\prime} \), similar to the notation commonly employed by the micrometeorological community. Consequently, equation (6) means that the model is run on an hourly basis with hourly meteorological input data, and the measured monthly average NH_{3} concentration is used as a substitute for hourly concentration values. In other words, it is assumed that the monthly average NH_{3} concentration is representative for hourly values and that internal mechanics of the model (such as the exponentially temperature dependent conversion of emission potentials to compensation points^{20}) effectively compensate the effect of the lowered input data resolution.
An alternative strategy would be to first calculate the exchange velocity and the compensation points at a high resolution (given that they are independent of χ_{a}), average them, and then calculate the monthly average flux from the monthly average of all other variables:
We will outline in the following section why both of these variants (equations (6) and (7)) will inevitably lead to biased results.
Derivation of the error term
A wellunderstood, but still often ignored fallacy is the assumption that the product of averages yields similar results to the average of products^{29}. However, this is generally only the case when all variables involved are completely independent and uncorrelated. Even if not all of these variables are formally linked within the governing equations of a dry deposition model, they may be correlated through their inherent dependence on external environmental factors (e.g. temperature, radiation, or turbulence). Meyers & Yuen^{30} were among the first to observe the impacts of ignoring this fallacy with regards to (unidirectional) inferential modelling of SO_{2} and O_{3} fluxes. For a bidirectional NH_{3} exchange scheme, the true mean flux over a certain period of time can be written as:
Recall our definition of \(\overline{x}\) and \(x^{\prime} \), from which it follows that \({x^{\prime} }^{2}\) is equal to the (non Besselcorrected) variance of x, and \(\overline{x^{\prime} \cdot y^{\prime} }\) to the covariance of two random variables x and y. With these additional definitions, we can calculate the true average flux from longterm average NH_{3} concentrations using the linearity of expected values and the definition of the covariance, as follows:
The difference between equation (7) and the last line of equation (9), i.e., the two covariance terms \(\overline{{v}_{{\rm{e}}{\rm{x}}}^{\prime} \cdot {\chi }_{{\rm{f}}}^{\prime} }\overline{{v}_{{\rm{e}}{\rm{x}}}^{\prime} \cdot {\chi }_{{\rm{a}}}^{\prime} }\) (μg NH_{3} m^{−2} s^{−1}), is equal to the exact error introduced when calculating average NH_{3} fluxes from average exchange velocities and measured longterm average concentration measurements. When directly calculating hourly fluxes with the longterm average NH_{3} concentrations used as a substitute for hourly values and averaging afterwards (i.e., using equation (6)), the error is equal to \(\overline{{v}_{{\rm{e}}{\rm{x}}}^{\prime} \cdot {\chi }_{{\rm{a}}}^{\prime} }\) (μg NH_{3} m^{−2} s^{−1}).
A first step towards bias elimination
If we run a dry deposition inferential model with only the ambient NH_{3} concentration as a longterm average and all other driving variables measured at a higher temporal resolution, as is usually the case for monitoring stations, where the measurement of meteorological variables at a high temporal resolution is not very difficult, only the last term of equation (9), i.e. the covariance of v_{ex} and χ_{a}, is unknown. We can expand it to take the form
Note that here \(\sqrt{\overline{{{v}_{{\rm{ex}}}^{\prime} }^{2}}}={\sigma }_{{v}_{{\rm{ex}}}}\) (m s^{−1}) and \(\sqrt{\overline{{{\chi }_{{\rm{a}}}^{\prime} }^{2}}}={\sigma }_{{\chi }_{{\rm{a}}}}\) (μg NH_{3} m^{−3}) are identical to the empirical standard deviation of v_{ex} and χ_{a}, respectively, and \(\overline{{v}_{{\rm{e}}{\rm{x}}}^{\prime} \cdot {\chi }_{{\rm{a}}}^{\prime} }\cdot {(\sqrt{\bar{{{v}_{{\rm{e}}{\rm{x}}}^{\prime} }^{2}}}\cdot \sqrt{\bar{{{\chi }_{{\rm{a}}}^{\prime} }^{2}}})}^{1}={r}_{{v}_{{\rm{e}}{\rm{x}}},{\chi }_{{\rm{a}}}}\) (−) is equal to the Pearson productmoment correlation of the two. Again, \({\sigma }_{{v}_{{\rm{ex}}}}\) is known and can trivially be calculated from higherresolution modelled estimates of v_{ex}. However, \({\sigma }_{{\chi }_{{\rm{a}}}}\) and \({r}_{{v}_{{\rm{ex}}},{\chi }_{{\rm{a}}}}\) remain unknown at this point.
A simple approach to calculate lessbiased fluxes from lowresolution concentration measurements could be based on accompanying highresolution concentration measurements at the same site for a limited amount of time. E.g., one would use a single ‘highfrequency’ (0.5 to 1 hour sampling rate) NH_{3} monitor to take parallel measurements at a monitoring site for a few months to gather the necessary data to derive correction factors, and then move the instrument to the next site. We can assume an increase of the variation in air NH_{3} concentrations with rising concentration levels, i.e. increasing \({\sigma }_{{\chi }_{{\rm{a}}}}\), with increasing mean \(\overline{{\chi }_{{\rm{a}}}}\), since (i) chemical measurement instruments often exhibit relative errors, and (ii) it is reasonable to suspect that, for instance, emissions from nearby sources would not only lead to a steady increase of the mean NH_{3} concentration, but also to a higher variability, depending on turbulent mixing, wind direction, and other factors. The most simple approach is to model this with a linear relationship:
where m_{0} (−) and b_{0} (μg NH_{3} m^{−3}) are the slope and intercept of the resulting regression line, respectively.
Modelling the correlation between the exchange velocity and the air NH_{3} concentration, \({r}_{{v}_{{\rm{ex}}},{\chi }_{{\rm{a}}}}\), is substantially less straightforward. A zerothorder approach would consist of simply taking the mean correlation over the measurement period used for deriving correction functions. However, this would eliminate the possibility of registering potential seasonality in \({r}_{{v}_{{\rm{ex}}},{\chi }_{{\rm{a}}}}\), and the next most simple alternative, a linear regression of \({r}_{{v}_{{\rm{ex}}},{\chi }_{{\rm{a}}}}\) against some environmental variable, would yield practically the same results if the slope of the regression is close to zero, leaving little reason not to favour at least a simple linear regression over the mean. Unfortunately, the choice of a suitable explanatory variable is far from trivial, as we essentially look for a correlation of a correlation, which is a somewhat illdefined and difficult to understand concept. We will here exemplarily perform a linear regression against temperature \({T}_{\mathrm{air}}(^\circ {\rm{C}})\), assuming that with rising temperature (as a measure for the energy content of the system), both volatilisation of NH_{3} and buoyancy will increase and the corr\elation between the two might become stronger. However, this is merely an educated guess and not bound to be the most suitable model, nor is temperature guaranteed the most suitable regressor. In fact, we suspect that especially at remote sites with little to no diurnal variation in air NH_{3} concentrations, but pronounced variation in v_{ex} (which is strongly linked to atmospheric turbulence), most variables with a strong diurnal cycle would work similarly well as a predictor for \({r}_{{v}_{{\rm{ex}}},{\chi }_{{\rm{a}}}}\). The model is given as:
with the slope m_{1} (−) and intercept b_{1} (−). Equation (9) then becomes:
Comparison of flux prediction strategies
A 12month gapfree set of ‘synthetic’ input data was generated by running the Eulerian grid model LOTOSEUROS^{31} in conjunction with ECMWF meteorology for the year 2016. Through a oneway nesting procedure a simulation over Germany was performed on a resolution of 0.125° longitude by 0.0625° latitude, approximately 7 by 7 km^{2}. The high resolution domain is nested in a European domain with a resolution of 0.5° longitude by 0.25° latitude, approximately 28 by 28 km^{2}. Emissions include the TNO MACCIII European emission inventory for the year 2014. For Germany, the national emission inventory of the German Environmental Protection Agency (UBA) was used to prescribe the gridded emissions. LOTOSEUROS is one of the few CTMs that include SO_{2}NH_{3} codeposition and bidirectional surfaceatmosphere exchange of NH_{3}^{12,32}.
We here used data from one grid cell in the Allgäu region in southern Germany (47°41′34.80′′ N, 10°2′6.00′′ E) (Fig. 2a). Average temperature during the year of 2016 was 8.1 °C, total precipitation 1690 mm, and the average NH_{3} concentration was 5.6 μg NH_{3} m^{−3} (highest hourly means up to 60.6 μg NH_{3} m^{−3}) at an (aerodynamic) reference height of 2.5 m above zeroplane displacement. The annual course of the leaf area index was modelled as implemented in the DEPAC deposition module within LOTOSEUROS^{12,21}. We here exemplarily used landuse parameters for grassy seminatural vegetation; results for other landuse classes can be found in Fig. S1 of the Supplementary Material.
Additionally, we tested the correction scheme for measured data from a flux tower in the Bavarian Forest in Germany (Fig. 2b) at 807 m a.s.l. (base of the tower), 48°56′50.27′′ N, 13°25′12.22′′ E^{33}. NH_{3} concentrations were measured using a QCL absorption spectrometer from Aerodyne Research Inc., Billerica, MA, USA (cf. Zöll et al.^{2}. for a detailed instrument description) at 31 m above ground level, and with an original sampling rate of 10 Hz averaged to 1 concentration value per hour. Turbulence measurements were taken with a sonic anemometer (model R3, Gill Instruments Ltd., Lymington, UK) at the same height, as well as temperature and relative humidity using HC2S3 probes (Campbell Scientific, Inc., Logan, UT, USA). Leaf area index and canopy height were not measured at the site and parameterised as proposed in Massad et al.^{7}. Annual average temperature at the site was 7.4 °C, total precipitation was 1047 mm, and the average of NH_{3} concentrations used in this study (approximately 56% data coverage of the year) was 1.1 μg NH_{3} m^{−3} (maximum 14.5 μg NH_{3} m^{−3}). Measured ambient NH_{3} concentrations at this site are preliminary, but have undergone common quality procedures, such as despiking, and system performance tests with regard to flow rate, temperature, and pressure stability. These data will be published in an ecological context in the near future. The purpose of using this dataset is solely meant for assessing the correction scheme, thus absolute numbers should not be cited for verifying ecosystemspecific thresholds. We also note that we here used the Massad et al.^{7} parameterisation in its original form, despite the findings of Schrader et al.^{34} regarding a likely too large nonstomatal (cuticular) resistance in this parameterisation. While this leads to relatively low predicted fluxes, the derivation of the error term is unaffected. An additional case study for a moorland site in southern Scotland can be found in Fig. S2 in the Supplementary Material.
The dry deposition inferential model was run for four different scenarios for each site:

1.
‘control’: all variables at hourly resolution; flux calculation on hourly basis and subsequent averaging to monthly average fluxes (equation (8)).

2.
‘direct’: monthly average NH_{3} concentrations and all other variables at hourly resolution; flux calculation on hourly basis with hourly NH_{3} concentrations substituted by their monthly averages; subsequent averaging to monthly average fluxes (equation (6)).

3.
‘monthly’: monthly average NH_{3} concentrations and all other variables at hourly resolution; calculation of exchange velocities and foliar compensation points on hourly basis; subsequent averaging to monthly average exchange velocities and foliar compensation points, and calculation of monthly fluxes via equation (7).

4.
‘corrected’: same as ‘monthly’, but with added correction terms from equations (11)–(13).
Note that ‘corrected’ can also be written as equal to ‘direct’ plus only the correction term for the covariance of v_{ex} and χ_{a}. Monthly deposition fluxes for months with gaps in the measured dataset were calculated by multiplying the arithmetic mean flux density of a given month with the number of data points at 100% coverage (assuming no bias of the gaps towards a certain time of the day).
Code and data availability
Synthetic data are available at reasonable request from M. Schaap. Measured data will be published separately after final analysis. A Python 2.7 implementation of the Massad et al.^{7} parameterisation can be requested from the lead author. An open source version of LOTOSEUROS is publicly available.
Results and Discussion
Figure 3a,b exemplarily shows the results of the comparison between the different averaging strategies for the synthetic dataset, using the parameterisation of the dry deposition model for seminatural ecosystems. During some months, the relative error reaches over 100% higher predicted deposition compared to ‘control’ in a given month (e.g., January and April). The lowest error introduced by using uncorrected averaging strategies is in August (54% for the ‘direct’ variant, and 58% for the ‘monthly’ variant). Overall, the uncorrected variants overestimate total NH_{3} dry deposition for the year 2016 roughly by a factor of two (Table 1). There is no clear dependency of the magnitude of the relative error on environmental drivers apparent from our observations; however, the magnitude of the error is naturally strongly linked to \({r}_{{{\rm{v}}}_{{\rm{ex}}}{,{\rm{\chi }}}_{{\rm{a}}}}\). Consequently, the performance of a correction scheme is directly proportional to the certainty with which the correlation of the exchange velocity and the air NH_{3} concentration can be estimated. It also directly follows from a special case of equation (9), when χ_{f} is assumed to be zero, that the use of average deposition velocities instead of effective deposition velocities in a unidirectional framework is affected by the exact same type of error. In fact, due to the implicit integration of the compensation point in the deposition velocity, the error can be expected to be larger. Similar observations have been made by Matt & Meyers^{35} and Meyers & Yuen^{30} for SO_{2} and O_{3}, in which they attempted to reduce the error by employing day and nightsampling strategies. The proposed correction approach leads to a strong improvement during all months (Table 1, Fig. 3a,b), especially considering its relative simplicity. These findings are also confirmed by running the model for different synthetic datasets with different landuse types (Table S1 and Fig. S1 in the Supplementary Material).
For the measured data (Fig. 3c,d), the picture is somewhat less clear, due to alternating over and underestimations of averaged predicted deposition with respect to the control. While in some months, uncorrected ‘direct’ or ‘monthly’ flux prediction strategies give the best approximation to flux calculation using highfrequency data (e.g., May or October), the sum of all deviations from ‘control’ is still lowest for the ‘corrected’ variant (Table 1). However, the mean absolute deviation from ‘control’ is lowest for the ‘direct’ variant, albeit by a very small margin. Reasons for this less clear performance can be found in the very uncertain regression of \({r}_{{v}_{{\rm{ex}}},{\chi }_{{\rm{a}}}}\) against ambient temperature (Fig. 4d), leading to all estimates of \({r}_{{v}_{{\rm{ex}}},{\chi }_{{\rm{a}}}}\) close to its arithmetic mean, although it clearly changes throughout the measurement period. Furthermore, the error introduced by the different averaging strategies is already much lower (<25%) than for the synthetic dataset, which indicates a strong sitespecificity of \({r}_{{v}_{{\rm{ex}}},{\chi }_{{\rm{a}}}}\). This is supported by the observation that other measured datasets exhibit much larger errors (Fig. S2 in the Supplement).
The higher than observed anticorrelation between NH_{3} concentrations and the exchange velocities may be due to the large difference in NH_{3} levels between the synthetic and measured dataset (Fig. 2). Firstly, in source areas primary emissions cause nighttime concentration maxima to occur, whereas exchange rates are highest during late morning hours when PBL growth has diluted the NH_{3} concentrations. CTM modelled data from the gridcell that includes the Bavarian Forest site (grey lines in Fig. 4d) exhibit a weaker anticorrelation \({r}_{{v}_{{\rm{ex}}},{\chi }_{{\rm{a}}}}\) than the synthetic dataset from the Allgäu region. LOTOSEUROS’ resolution may explain why it is still somewhat more negative than observed: In each grid cell emissions of NH_{3} take place, causing a slight nighttime maximum. In reality, the stagnant conditions do not allow these emissions to reach a hill site such as the Bavarian Forest measurement tower. Hence, the implicit spatial mixing may explain the stronger anticorrelation found in the measurements.
It is evident from Fig. 5, that, unfortunately for the purposes of correcting biased monthly flux estimates, the known part of the error term (\(\overline{{v}_{{\rm{ex}}}\text{'}\cdot {\chi }_{{\rm{f}}}\text{'}}\)) contributes much less to the total error than the unknown part (\(\overline{{v}_{{\rm{ex}}}\text{'}\cdot {\chi }_{{\rm{a}}}\text{'}}\)). Consequently, the choice between ‘direct’ and ‘monthly’ flux calculation strategies does not substantially change the magnitude of the error. The assumption of a relative error in measured air NH_{3} concentrations appears to be justified from our observations with both modelled and measured concentrations (Fig. 4a,c). However, modelling the correlation of the exchange velocity and air NH_{3} concentrations remains a challenge, as difficulties in the interpretation lead to difficulties in the conceptualisation of an adequate model for \({r}_{{v}_{{\rm{ex}}},{\chi }_{{\rm{a}}}}\) (Fig. 4b,d). Also note that, for NH_{3}, both deposition, and emission can occur. We make no distinction between the two in our analysis, as all sites show net deposition on the monthly scale and no artificial management events were modelled. Contrary to deposition velocity models, information about the direction of the flux is removed from the exchange velocity by explicitly separating it from the compensation point in the derivations. Equation (12) appears to work acceptably well for modelling \({r}_{{v}_{{\rm{ex}}},{\chi }_{{\rm{a}}}}\) in the synthetic dataset, but not very well for the measured one. A better course of action than the one presented here might, for example, be based on a multivariate regression using more than one environmental driving factor. However, many potential candidate variables are highly correlated, and the number of parameters of such a multivariate model may quickly approach the number of data points, leading to an increased risk of overfitting and questionable predictive value. We have investigated the potential of fitting the correction factors on a smaller timescale than the averaging period, thereby increasing the number of data points for the regression, but this has been rather unsuccessful in terms of reducing uncertainty. With simple regression approaches, an adequate correction function will certainly be sitespecific, and it will not be universally valid for different parameterisations of biosphereatmosphere exchange schemes. With the increasing availability of optical highfrequency NH_{3} measurement instruments, fitting ecosystemtype and environmental condition specific multivariate correction functions, thereby potentially eliminating the need for sitespecific parallel measurements, is a promising outlook, but we assume that the number of NH_{3} concentration measurements currently available is simply too low for this task. However, truly siteindependent correction functions that can be readily applied in existing modelling schemes may not even be possible to derive, as they likely depend on a multitude of factors which are not routinely measured. The relationship between v_{ex} and χ_{a} may be vastly different depending on, for example, the N status of the ecosystem of interest, atmospheric composition, and even the measurement period. In agricultural ecosystems, for instance, there are times when the concentration is largely driven by emission fluxes from the surface, and times when the ambient concentration will drive the flux. The same can be the case for forests before and after leaffall^{36}. Further research is necessary to develop an optimal strategy to handle these challenges.
Readers should be aware that the observations and derivations made in this study are strictly only valid for a model parameterisation where both v_{ex} and χ_{f} are not directly dependent on highfrequency observations of the air NH_{3} concentration, χ_{a}, such as the parameterisation of Massad et al.^{7}. For instance, Wichink Kruit et al.^{8} used an air NH_{3} concentration dependent formulation for the cuticular compensation point to approximate saturation effects within leaf surface water layers. In this case, it would be advisable to use a corrected formulation based on the ‘direct’ variant, so that one χ_{a}dependent covariance in the correction term can be eliminated. Other than that, all derivations demonstrated here remain the same, should be adaptable to other parameterisations and model structures in a straightforward manner, and they are valid for any arbitrary averaging period.
Conclusions
We have demonstrated and formally shown that commonly used averaging strategies for the prediction of longterm average fluxes from longterm average measurements of NH_{3} concentrations (e.g., from denuder or passive sampler records) and highfrequency micrometeorology are biased. The magnitude and variation of this bias is dependent on the biosphereatmosphereexchange scheme used, and measurement site characteristics, such as surface, parameters, pollution level and the distance to NH_{3} sources. The magnitude of errors in predicted fluxes introduced by using uncorrected averaging schemes is directly proportional to the (anti)correlation of NH_{3} exchange velocities and ambient concentrations, which is expected to be significant due to saturation effects on wet leaf surfaces^{37,38,39}, deposition historydependent compensation points^{7,8}, and their inherent dependence on the same environmental variables. Relative errors of up to 100% deviation from ‘control’ and higher were observed in the synthetic dataset, whereas measured data showed both over and underestimations of less than 25% that compensated each other over the course of the measurement period. The proposed correction scheme consists of

1.
Measuring timeseries of average NH_{3} concentrations with lowfrequency, lowcost monitoring equipment,

2.
Measuring meteorological drivers at a highfrequency with standard instrumentation,

3.
Taking parallel measurements with a highfrequency NH_{3} monitor for a limited time to parameterise functions to estimate the standard deviation of NH_{3} concentrations (equation (11)), and the correlation of air NH_{3} concentrations with the exchange velocity (equation (12)),

4.
Calculating corrected monthly average fluxes using equation (13).
The results of our first tests appear promising, but uncertainties in estimating aforementioned correlation have to be overcome in the future. In its current state, lowfrequency concentration measurements need to be accompanied by highfrequency measurements for a certain (yet to be determined) amount of time to derive valid sitespecific correction functions. Indepth model structure analyses and multisite studies, especially at those with higher NH_{3} concentrations and possibly emission fluxes, may give further valuable insight into the exact mechanics behind the dominant source of the error: the correlation of the NH_{3} exchange velocities and air NH_{3} concentrations.
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Acknowledgements
F.S., M.S., U.Z., and C.B. greatly acknowledge financial support by the German Environmental Protection Agency (UBA) under grant no. FKZ 3715512110. F.S., U.Z., and C.B. are additionally funded by the German Federal Ministry of Education and Research (BMBF) within the Junior Research Group NITROSPHERE under support code FKZ 01LN1308A. We highly appreciate the contribution of additional data by C.R. Flechard. We would like to thank J. Rüffer, J.P. Delorme, and M. Lewandowski from Thünen Institute for technical assistance in the field, and we highly acknowledge local support with the flux tower measurements by B. Beudert, W. Breit, and L. Höcker from the Nationalparkverwaltung Bayerischer Wald. Many thanks to J.W. Erisman for helpful comments on the manuscript.
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F.S. and C.B. had the idea and designed the study. M.S. and R.K. contributed the synthetic data, U.Z. carried out the measurements. F.S. carried out all analyses and wrote the manuscript with equal contributions from all coauthors.
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Schrader, F., Schaap, M., Zöll, U. et al. The hidden cost of using lowresolution concentration data in the estimation of NH_{3} dry deposition fluxes. Sci Rep 8, 969 (2018). https://doi.org/10.1038/s41598017180216
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DOI: https://doi.org/10.1038/s41598017180216
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