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Abstract. Chambers are widely used to measure surface fluxes of nitrous oxide (N2O). Usually linear regression is used to calculate the fluxes from the chamber data. Non-linearity in the chamber data can result in an underestimation of the flux. Non-linear regression models are available for these data, but are not commonly used. In this study we compared the fit of linear and non-linear regression models to determine significant non-linearity in the chamber data. We assessed the influence of this significant non-linearity on the annual fluxes. For a two year dataset from an automatic chamber we calculated the fluxes with linear and non-linear regression methods. Based on the fit of the methods 32% of the data was defined significant non-linear. Significant non-linearity was not recognized by the goodness of fit of the linear regression alone. Using non-linear regression for these data and linear regression for the rest, increases the annual flux with 21% to 53% compared to the flux determined from linear regression alone. We suggest that differences this large are due to leakage through the soil. Macropores or a coarse textured soil can add to fast leakage from the chamber. Yet, also for chambers without leakage non-linearity in the chamber data is unavoidable, due to feedback from the increasing concentration in the chamber. To prevent a possibly small, but systematic underestimation of the flux, we recommend comparing the fit of a linear regression model with a non-linear regression model. The non-linear regression model should be used if the fit is significantly better. Open questions are how macropores affect chamber measurements and how optimization of chamber design can prevent this.


Introduction
Nitrous oxide (N 2 O) is one of the main contributors to the greenhouse effect causing global warming (Denman et al., 2007) ronmental concern (IPCC, 2007), moreover because they affect atmospheric chemistry (Crutzen, 1981).Globally, soils are the major source of N 2 O emissions and arable land is the largest anthropogenic source (Denman et al., 2007).In Europe annual emissions are reported for grassland up to 28 kg N 2 O−N ha −1 (Dobbie and Smith, 2003), for arable land up to 17 kg N 2 O−N ha −1 (Jungkunst et al., 2006).Emissions will continue to rise with increasing world population and agricultural production (IPCC, 2000).Chamber methods are widely used to measure N 2 O fluxes, especially non-flowthrough, non-steady-state (NFT NSS) chambers (Rochette and Eriksen-Hamel, 2008).The fluxes derived from the chamber data have been used for comparison studies, e.g. to compare the impact of various agricultural management practices on the fluxes.Furthermore, these data have been used to calculate annual emissions for national inventories (IPCC, 2006), and to calibrate detailed process models (Del Grosso et al., 2000;Li et al., 1992;Riedo et al., 1998).Chamber data consist of concentration measurements in the headspace of the chamber over time; the flux is derived from the concentration change over time.Usually, this change is assumed to be linear in time and the flux is determined with linear regression.However, already from the introduction of NFT NSS chamber methodology non-linear, decreasing concentration changes in the chamber headspace have been reported (Denmead, 1979).Use of linear regression will in these cases lead to an underestimation of the flux.This might not be a problem for comparison studies, which use the fluxes relative to each other.For national inventories and modeling though, the absolute values are of interest and an underestimation in the absolute value of the flux would result in biased errors in these studies (Rochette and Eriksen-Hamel, 2008).
Exponential (Matthias et al., 1978) or quadratic (Wagner et al., 1997) regression models have a better fit on non-linear chamber data than the linear regression model (Kroon et al., 2008;Hutchinson and Mosier, 1981).The non-linear regression models give higher flux estimates for these data: differences are reported up to 127% (Pedersen, 1999(Pedersen, , 2000;;Hutchinson et al., 1993;Kroon et al., 2008;Hutchinson and Mosier, 1981).To obtain a reliable flux, a linear regression model can only be used when Introduction

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Interactive Discussion the non-linearity in the chamber data is not significant (Rochette and Eriksen-Hamel, 2008).However, there is no common practice to define this significance.The best way would be to compare the fit of linear and non-linear regression models.Examples in literature include comparison of the coefficient of determination and the adjusted coefficient of determination (Kutzbach et al., 2007), comparison of chi-square (Kroon et al., 2008), calculation of the significance of the non-linear term in the regression model (Wagner et al., 1997), or simply looking at the shape of the concentration-time curve (Anthony et al., 1995).More commonly only the fit of the linear regression model is verified, using the coefficient of determination r 2 (e.g.Yamulki and Jarvis, 1999;Velthof and Oenema, 1995;Breuer et al., 2000;Matthias et al., 1980).There are two problems associated with this method.Firstly, not only non-linear behaviour but also random measurement inaccuracies lower r 2 .The influence of measurement inaccuracies is larger for small fluxes than for large fluxes.Therefore a lower r 2 is accepted for smaller fluxes than for larger fluxes and no overall lower limit for r 2 can be defined as criterion for acceptance of the linear regression model (e.g.Yamulki and Jarvis, 1999;Velthof and Oenema, 1995).Secondly, even for concentration data with a high r 2 for the linear regression, the underestimation of the flux determined with linear regression can be considerable.Modeling studies have demonstrated that there can still be a difference up to 16%, even for data with r 2 >0.99 (Conen and Smith, 2000;Pedersen et al., 2001).Thus, the question is how to determine if non-linearity in the concentration change in the chamber over time is significant and if a non-linear regression method should be used.
In this study we use two years of automatic chamber data for nitrous oxide from intensively managed grassland on clay.For this dataset, our aim is to: quantify the differences between the N 2 O fluxes derived from chamber data with linear and non-linear regression; evaluate how the goodness of fit of the regressions can be used as indicator for significant non-linearity in the chamber data; assess the impact of significant non-linearity on the daily and annual N 2 O fluxes.
Throughout this paper, fluxes derived with linear, quadratic and exponential regression are referred to as linear, quadratic and exponential fluxes, respectively.Chamber data refers to the concentrations measured in the chamber during each closure period.Instantaneous fluxes refer to fluxes derived from one closure period of the chamber; daily and annual fluxes are calculated from these fluxes.

Site and data
The N 2 O flux chamber data used in this study were collected in the context of the EU GREENGRASS project (Soussana et al., 2007;Flechard et al., 2007) from 14 July 2002 to 13 July 2004 on grassland on clay soil near Lelystad in the Netherlands.Nitrogen was added as fertilizer and slurry and through grazing; average annual nitrogen addition was 286 kg N ha −1 .The grass was harvested two to four times a year.The automatic chamber system had one non-insulated chamber with a size of 0.7 m×0.7 m and a height of 0.3 m.The air inside the chamber was mixed continuously by a fan.This chamber alternated between two positions where an aluminium base was inserted 0.05 m in the soil.Gas concentration measurements inside the chamber were performed with a gas chromatograph (GC; Interscience Compact GC, The Netherlands) located in a cabin approximately 15 m North-East of the automatic chamber (with prevailing West-Southwest wind directions).N 2 O was measured using a 1/8 Molsieve 5A column with a length of 2 m.The temperature of the column oven was 50 • C and the flow rate of the carrier gas (N 2 ) was 20 ml min −1 .The GC was fitted with an electron capture detector (ECD) for N 2 O. Measurement accuracy of the GC-ECD was ∼0.5 ppb, corresponding with a flux accuracy of 0.2 g N 2 O−N ha −1 d −1 .
Gas concentrations were determined at 0, 5, 15 and 25 min after closing the box.
High and low calibration standards were applied to the GC at 10 and 20 min after Introduction

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Full closing the chamber.The system measured for 30 min with the chamber in the first position (A), then another 30 min with the chamber in the second position (B).A third interval of 30 min was used to measure the ambient air concentration at different measurement heights, during which interval the chamber remained in position B. In this way chamber data for each position were obtained every 1.5 h.Instrument failure resulted in some longer periods with missing data.Chamber data (n=14 167) was accepted if all measured concentrations were above 250 ppb, if the begin concentration was below 600 ppb, if the measured concentrations of the reference gases were within an acceptable range and if the concentration changes over time were consistently positive or negative, taking into account the measurement accuracy.For chamber data with a standard deviation smaller than 2 ppb, the flux was assumed to be zero.Of the chamber data 25% was accepted (n=3549), of which 6% was set to zero; 23% was discarded because of errors in the measured concentrations, and 52% was discarded because of inconsistent concentration changes.Mainly data with small concentration changes (i.e.small fluxes) were discarded.In

Regression methods
From the accepted chamber data the flux ) was calculated by using the function lsqcurvefit and quadratic regression with constraints using the function fmincon.
For linear regression, the model fitted to the chamber data is given by in which C t (ppb) is the measured concentration at time t, a 0 (ppb) and a 1 (ppb min −1 ) are regression parameters.In this case, d C/d t t=0 is equal to a 1 .
For quadratic regression, the model fitted to the chamber data is given by ) and b 2 (ppb min −2 ) are regression parameters (Wagner et al., 1997).Now, d C/d t t=0 is equal to parameter b 1 .Parameter b 0 represents the concentration at t=0 and b 2 •t 2 can be regarded an extra loss term as compared to the linear regression (Wagner et al., 1997).The graph of a quadratic equation can take a parabolic shape, with a minimum or maximum and a change in sign of the slope.
One of the assumptions we made is that d C/d t is consistently positive or negative.To make certain that d C/d t does not change sign during the closure period of 25 min, the quadratic regression was applied with the following constraint: For exponential regression, a model based on Fick's law is fitted to the chamber data: in which c max (ppb), c 0 (ppb) and k (min −1 ) are regression parameters (de Mello and Hines, 1994;Matthias et al., 1978).Regression parameter c 0 represents the concentration at time t=0, regression parameter c max the maximum concentration that can be Introduction

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Full Screen / Esc Printer-friendly Version Interactive Discussion reached in the chamber and k a rate constant.In this case, the concentration change at time t=0 is given by

Goodness of fit
The goodness of fit for each regression was determined by the sum of squared errors SSE, the coefficient of determination r 2 and the adjusted coefficient of determination r 2 a (Neter et al., 1996): where n is the number of observed concentrations (n=4), p is the number of regression parameters, C t is the observed and Ĉt the modeled concentration.For the fluxes and each goodness-of-fit measure the median values were determined.
The goodness-of-fit of the regression methods has to be compared to decide whether linear or non-linear regression should be used.In this comparison the different number of regression parameters of the linear and non-linear regression methods has to be taken into account, as is the case in r 2 a .Therefore data is defined as significantly nonlinear if r 2 a of the non-linear regression methods is larger than r 2 a of the linear regression method.Introduction

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Full For this purpose missing values in the time series of daily fluxes had to be estimated.Simple linear interpolation was not suited, because of larger gaps in the data (up to 33 days).The uncertainty in cumulative fluxes increases sharply with linear interpolation of gaps larger than seven days (Smith and Dobbie, 2001;Weitz et al., 1999;Parkin, 2008).Therefore, in this study gaps shorter than 7 days were filled by linear interpolation, all other gaps were filled with the median of the background fluxes (e.g.Flechard et al., 2007).Background fluxes were defined per regression method as the daily fluxes smaller than the mean daily flux.The median was chosen as a measure for the central tendency, to reduce the influence of extreme negative fluxes.

Results and discussion
3.1 Fluxes  According to Fig. 3 for the instantaneous fluxes the difference between the linear fluxes and the quadratic and exponential fluxes is 59% and 162%, respectively.Similar studies in literature found average differences ranging from 30% to 127% for instantaneous fluxes (Anthony et al., 1995;Pedersen, 2000;Kroon et al., 2008).The average difference found for the quadratic regression method is within this range, the average difference found for the exponential regression method is above this range.

Goodness of fit
In Table 1 the goodness of fit of the three regression methods is presented as the median value of SSE, r 2 and r 2 a .In general, for the exponential regression method the median for SSE is smallest and the median values for r 2 and r 2 a are largest; this indicates that in general the exponential regression method has the best fit.The median for SSE and r 2 for the linear and quadratic regression method suggest that in general the quadratic regression has a better fit than the linear regression method.However, the median of r 2 a is larger for the linear than for the quadratic regression method.This shows that the general better fit of the quadratic regression method is no more than has to be expected by the addition of an extra regression parameter.
Although the general fit of the exponential regression method is very good, the reliability of the result of the exponential regression is questionable.As the method is based on physical principles, the regression parameters should also stay within a physical acceptable range.The parameter k has a physical interpretation (Hutchison and Mosier, 1981;Matthias et al., 1978): physical maximum for k is given: For a significant part of the data (8%) the regression result gives a k above this value.In Fig. 4 the regression parameter k is plotted against the relative difference between the linear and exponential flux.It shows that the difference between the linear and exponential flux is directly related to the value of k.The maximum difference (k=0.286min −1 ) is around 700%; this is plotted in Fig. 3b as a dashed line.The data on the left side of the line gives an overestimation in the exponential flux due to a too high value of k.In literature overestimation of flux by exponential regression methods has been shown before (Kroon et al., 2008;Pedersen et al., 2001).
Exponential regression has to be firmly constrained with measurements at the start of the closure period to prevent overestimation of the flux.For this dataset we decided not to use the exponential fluxes.In the remainder of this study we only use the results of the linear and quadratic regression methods.

Significant non-linearity
Non-linearity in the chamber data is defined to be significant if r 2 a is smaller for the linear regression method than for the non-linear regression methods, in this study the quadratic regression method.Overall the linear regression method has a higher r 2 a than the quadratic regression method, but for 32% of the data r 2 a is higher for the quadratic than linear regression.These data are regarded significant non-linear.Apparently, the linearity of chamber data can vary throughout the measurement period.
In Fig. 5 the percentage of data with significant non-linearity is depicted for bins of linear r 2 (bin width 0.05).A distinction has been made between fluxes smaller and larger than the median flux.In this dataset significant non-linearity is found more often for the larger fluxes (48%) than for the smaller fluxes (15%).This is also evident from Introduction

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Fig. 3a, where part of the smaller quadratic fluxes is almost equal to the linear fluxes, whereas the larger fluxes show a larger difference.
Comparing this measure for non-linearity with the linear coefficient of determination r 2 , we find no clear relation.Significant non-linearity is found for chamber data with linear r 2 >0.5.As expected, with increasing r 2 (0.85<r 2 <1.00) the percentage data with significant non-linearity decreases, but even 35% of data with r 2 >0.95 is significantly non-linear.We conclude that the linear r 2 is not a good measure to check whether linear regression or non-linear regression is better for calculation of the flux from chamber data.

Impact of significant non-linearity on daily annual fluxes
To determine the impact of significant non-linearity on annual fluxes, the linear fluxes were replaced by quadratic fluxes for data with significant non-linearity.Figure 6a and b shows how this affects the instantaneous and daily fluxes, respectively.The larger fluxes (>70 g N 2 O−N ha −1 d −1 ) mainly show non-linear behaviour, resulting in a figure almost equal to that for quadratic fluxes (Fig. 3a).Because of the log-normal distribution of instantaneous fluxes over a day, the larger fluxes are expected to affect the daily means more than the smaller ones.Indeed, the difference between the linear and non-linear daily fluxes increases as compared to the instantaneous fluxes.In Fig. 7 the annual fluxes, calculated with the linear regression method, the quadratic regression method and the mixture of both is given.The annual fluxes range from 0.6 to 2.9 kg N 2 O−N ha −1 .These are low compared to other studies on this site (Flechard et al., 2007;Velthof et al., 1996), but this is due to the long periods with missing data that are filled with the median background flux.The relative difference between the linear and mixed annual flux ranges from 21% to 53%, which is equal to an underestimation in the annual linear flux of 17% to 35%.It appears that in years with high peak emissions Introduction

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Full Screen / Esc Printer-friendly Version Interactive Discussion linear and mixed flux of 34%.

Causes of non-linearity
In this study 32% of the chamber data shows significant non-linearity, meaning that 68% of the chamber data is linear.The interesting question is why the concentration changes linearly in some cases and non-linearly in other.
The first main cause for non-linearity is a decreasing diffusion as a result of the increasing concentration in the chamber headspace (Gao and Yates, 1999;Conen and Smith, 2000;Healy et al., 1996;Matthias et al., 1978;Pedersen et al., 2001;Gao and Yates, 1998).This feedback between the concentration in the headspace of the chamber and the soil concentration profile cannot be avoided, but its influence can be minimized by proper chamber height and closure period.Rochette and Eriksen-Hamel (2008) reviewed the improvements in NFT NSS chamber methodology and proposed for chamber height in combination with closure period a value ≥40 cm h −1 (e.g.chamber height=20 cm; closure period=30 min).However, even then the flux is still slightly affected.Model studies show that for chamber data meeting this criteria linear regression underestimates the real flux by 8% (Conen and Smith, 2000) or 16% (Pedersen et al., 2001).
The second main cause for non-linearity in chamber data is leakage, when the N 2 O concentration in the headspace of the chamber is higher than the N 2 O concentration in the free air.There can be leaks in the system itself, but this can be avoided with sufficient care (Conen and Smith, 2000).Lateral diffusion of N 2 O beneath the base of the chamber can also cause leakage.The amount of leakage is related to the depth of the chamber base (Hutchinson and Livingston, 2001).Rochette and Eriksen-Hamel (2008) proposed for the base insertion in combination with the closure period a value ≥12 cm h −1 (e.g.insertion depth=6 cm; closure period=30 min).For soils with an air filled porosity of 0.3 m m −1 this will give an underestimation of the linear flux <1%.However, for soils with larger air filled porosities these insertion depths will not prevent leakage through the soil, resulting in larger underestimations ( Livingston, 2001), even more in windy conditions.Macropores, such as root and worm holes or shrinkage cracks, provide another way for leakage from the chamber through the soil.Shrinkage cracks are mainly found in clay and peat soils, where the frame of the chamber can act as a natural starting point for cracking.The abundance of macropores is related to the soil conditions and therefore changes over time.For longer measurement periods in particular there is a risk for leakage through macropores.We are not aware of chambers designed to prevent leakage through macropores, or of literature on the influence of macropores on chamber data.We expect fast leakage through macropores and therefore a large underestimation of the linear flux.
The hypothesis we derive from these studies is that even with proper chamber design and deployment (following Rochette and Eriksen-Hamel, 2008) non-linearity in chamber data due to a decreasing diffusion can cause an underestimation in the linear flux up to 16%.Larger underestimations due to more significant non-linearity should be attributed to leakage through the soil, either in dry conditions or through macropores.This hypothesis is hard to verify, because the causes of non-linearity in the chamber data are most often not known.However, a couple of examples support this hypothesis: (Kroon et al., 2008) (underestimation 44% to 145%) proved leakage of the chamber by the use of a trace gas; in the study of (Anthony et al., 1995) (underestimation 34%) leakage through the coarse sandy soil is suggested (Conen and Smith, 2000).In the present study (underestimation 17% to 35%) leakage through cracks might be expected in dry periods, because clayey soils are expected to develop cracks in such periods.

Conclusion
It is clear from the present and other studies that the differences between fluxes calculated with the linear and non-linear regression methods can be large.Yet not all chamber data is significantly non-linear, so the impact of this non-linearity on the an-Introduction

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Full We suggest that underestimations this large (>16%) are due to leakage through the soil.More research is needed to unravel the exact causes of non-linearity in chamber data, to find out how macropores influence chamber measurements and how chamber design can be optimized to prevent this.As long as non-linearity cannot be prevented, non-linear regression models should be used for significantly non-linear data.The only constraint we pose on using significant non-linear chamber data is that the concentration change is consistently positive or negative.
Smaller underestimations (<16%) in the linear flux can even occur in non-leaking chambers meeting all criteria for proper chamber design and deployment.Significant non-linearity cannot be recognized from the goodness of fit of solely linear regression.Therefore we join with Rochette and Eriksen-Hamel (2008) to recommend more than 3 concentration measurements per closure period and a comparison of the linear and a non-linear regression model on the chamber data.The underestimation might appear small compared to temporal and spatial uncertainties in the N 2 O flux, but it gives a systematic error (bias) in the linear flux, not a random one.This is an undesirable characteristic if national emissions are calculated or models are calibrated using chamber data.Besides, the absolute error is larger for larger fluxes, which have most impact on the annual totals.
We recommend the quadratic regression method as non-linear regression method rather than the exponential regression method.Exponential regression has to be firmly constrained with measurements at the start of the closure period to prevent overestimation of the flux.Although quadratic regression has no theoretical basis other than leakage in general, it is more robust having only few concentration measurements.Moreover, it can never give a worse result than linear regression, because the quadratic term tends to go to zero with increasing linearity.Full
is de height of the chamber, d C/d t t=0 (ppb N 2 O min −1 ) is the concentration change at time t=0, M m (g mol −1 ) molar weight of nitrogen and V m (m 3 mol −1 ) molar volume, calculated based on air temperature and pressure.The factor f converts the fluxes from ng N 2 O m 2 min −1 to g N 2 O−N ha −1 d −1 .The term d C/d t t=0 was derived with linear, quadratic or exponential regression.All regressions were performed using MATLAB (MathWorks Inc., 2005, version 7.1.0.246).Linear regression on the chamber data was performed using the function polyfit, exponential regression with constraints of daily and annual fluxes Daily fluxes were defined as the daily mean of the fluxes from positions A and B together; 416 daily fluxes could be calculated.Annual fluxes were calculated as the cumulative flux over 365 days and were determined for four partly overlapping periods starting at 25July 2002, 7 November 2002, 20 February 2003 and 5 June 2003.

Figure 2
Figure2shows the calculated linear fluxes and the complete time series of daily linear fluxes after gap-filling.The graph illustrates the specific temporal behavior of N 2 O fluxes with background levels around zero and a few strong emission peaks.These peaks occured during the three summer periods, following fertilizer application and precipitation (not shown).It also shows the large gaps in the dataset that are mainly due to instrument failure.In Fig.3the quadratic and exponential fluxes are plotted against their linear counterparts.In general, the linear fluxes are smallest and the exponential fluxes are largest, which is also evident from the median flux of the three regression methods (Table1).
in which D p (m 2 min −1 ) is the actual diffusion coefficient of N 2 O in soil, d (m) is the depth of a plane with a constant concentration, and h (m) is the chamber height.The minimum value for k is zero.Setting the soil diffusion coefficient equal to the diffusion coefficient in free air(Pritchard and Currie, 1982) a conservative estimate of the not as large.For this dataset taking significant non-linearity into account, gives a difference in the annual flux up to 53%, what is equal to an underestimation in the linear flux of 35%.

Fig. 1 .Fig. 2 .
Fig. 1.Typical examples of a rejected, non-consistent flux, a zero flux, a positive flux and a negative flux.

Fig. 3 .
Fig. 3. Comparison of instantaneous linear fluxes with (a) quadratic fluxes and (b) exponential fluxes; fluxes in g N 2 O−N ha −1 d −1 .The solid line is the fitted linear regression line, forced through zero, of which the equation and r 2 are given.The dotted line indicates the line 1:1.The exponential fluxes left of the dashed line are non-reliable.

Fig. 4 .
Fig. 4. The results of exponential regression: the rate constant k versus the relative difference between the exponential and linear flux; 2% of the data lies outside the range of the figure.The dotted lines indicate the physical maximum value for k.

Fig. 6 .
Fig. 6.Comparison of (a) instantaneous and (b) daily linear fluxes with mixed fluxes; fluxes in g N 2 O−N ha −1 d −1 .The solid line is the fitted linear regression line, forced through zero, of which the equation and r 2 are given.The dotted line indicates the line 1:1.
. Increasing N 2 O emissions therefore are of envi-

Table 1 .
Results and the goodness of fit of the regressions on the chamber data a .