Soil Science Society of America Journal 65:49-58 (2001)
© 2001 Soil Science Society of America
DIVISION S-1-SOIL PHYSICS
Stochastic Diffusion Model for Estimating Trace Gas Emissions with Static Chambers
Asger R. Pedersena,
Søren O. Petersenb and
Finn P. Vintherb
a Department of Biostatistics, University of Aarhus, Vennelyst Boulevard 6, DK-8000 Aarhus C, Denmark
b Department of Crop Physiology and Soil Science, Danish Institute of Agricultural Sciences, Research Centre Foulum, P. O. Box 50, DK-8830 Tjele, Denmark
Corresponding author (asger{at}biostat.au.dk)
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ABSTRACT
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Trace gas emission measurements are frequently based on static chamber methods, where the trace gas accumulation within an enclosed headspace is followed over time. This study addressed the statistical part of trace gas measurements by comparing the typical approach, linear regression analysis, with a new method proposed by A.R. Pedersen, which is based on a stochastic extension of the diffusion model described by G.L. Hutchinson and A.R. Mosier. The new method provides an estimate of the emission rate, the standard error, P values, confidence intervals, estimates of model parameters, and a set of methods for validation of the assumed model. It was applied to data of N2O emissions from a peat meadow with the groundwater level at 20- and 40-cm depths, respectively. Furthermore, the three models mentioned above were compared in a simulation study using parameter values representative for the observed data. The simulations demonstrated that the assumptions underlying linear regression were violated, that the standard t test for significance did not have the expected properties, and that R2 was a poor diagnostic for detecting deviations from these assumptions. The Hutchinson and Mosier estimator was not as biased as the linear regression estimator, but the method often failed because a necessary condition was not satisfied by the data, a large standard error was indicated, and the method did not provide a test of significance for the estimated emission rate. The new method provided a good description of the data and useful diagnostics for testing it, and due to its ability to use more observations (longer time series), it had a negligible failure rate and bias.
Abbreviations: High GWL, high groundwater level (15 cm below the surface) treatment • Low GWL, low groundwater level (40 cm below the soil surface) treatment • WFPS, water-filled pore space
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INTRODUCTION
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IN THE PAST, most studies of trace gas exchange between soils and the atmosphere have employed static chambers, and even though micrometeorological techniques become increasingly accessible in terms of economy and operation, chamber methods will probably continue to play an important role in trace gas studies, for example when measuring emissions at high spatial resolution (Ambus and Christensen, 1985; Clayton et al., 1994; Oenema et al., 1997), or when relationships between gas fluxes and soil environmental characteristics are investigated (Dunfield et al., 1995; Petersen, 1999). Also, the accumulation of gases beneath a soil cover can lower the detection limit considerably (Hutchinson and Livingston, 1993).
The assessment of trace gas emissions from soil by static chamber methods consists of a measurement part, where concentration changes within a confined headspace are monitored, and a statistical part, where the emission rate is estimated from the observed concentrations. It has been demonstrated, both theoretically and in practice, that the concentration of a gas beneath a soil cover will not increase linearly with time because of a declining concentration gradient (Matthias et al., 1978; Hutchinson and Mosier, 1981; Anthony et al., 1995; Healy et al., 1996), although, in general, a sampling strategy is chosen where this problem is assumed to be negligible.
In this study, we considered the statistical part by comparing a new statistical method with existing ones. The new method, recently developed by Pedersen (2000) and successfully applied to emissions of N2O from an arable soil, is based on a stochastic extension of the deterministic nonlinear model described by Hutchinson and Mosier (1981). It provides an estimate of the emission rate, the standard error, a test of significance of the estimate, and methods for validation of the assumed model for the concentration measurements.
The stochastic diffusion model presented below links measured concentrations with the deterministic diffusion model proposed by Hutchinson and Mosier (1981) by means of a noise specification driven by a Wiener process (Karatzas and Shreve, 1988). Deviations of measured concentrations from the deterministic model have, of course, physical explanations (e.g., nonideal experimental conditions, small violations of the assumptions underlying the deterministic model, and analytical errors). Such deviations are unexplained by the deterministic model, but are captured by the stochastic model, as demonstrated by Pedersen (2000). Accordingly, the stochastic model can also be used for simulating observed trace gas emissions (see Fig. 1)
and for performing simulation studies.
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Fig. 1. (a) An example of the deterministic diffusion model. (b) An arbitrary simulated trajectory from the stochastic extension of the model in (a). (c) Simulated trace gas concentrations extracted from the trajectory in (b)
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In this paper, we apply the new statistical method and existing ones to data of N2O emissions recently obtained from a field study, and the assumptions and requirements of the different methods are discussed. In addition, a simulation study is carried out to further investigate the methods.
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THEORY
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Linear regression and the expression proposed by Hutchinson and Mosier (1981) are probably the two most widely used methods for estimating trace gas emission rates with chamber methods; however, both are problematic. Typically the concentration is measured initially and at two time points after placement of the soil cover.
As will be clear from the results of this study, essentially all assumptions underlying the linear regression method are violated in situations with nonlinear trace gas accumulation in the chamber headspace (see Fig. 2)
. The Hutchinson and Mosier (1981) method solves the nonlinearity problem by assuming the nonlinear deterministic diffusion model defined by
 | (1) |
where Ct denotes the trace gas concentration beneath the soil cover at time t after deployment, 
denotes the concentration in some plane of (assumed) constant concentration at depth d near the upper limit of the production zone, 
, in which Dp denotes the effective gaseous diffusion coefficient in the soil, and A and V denote, respectively, the cross-sectional area and the volume of the soil cover. The emission rate corresponding with Eq. [1] is given by
 | (2) |
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Fig. 2. Linear regression systematically underestimates the true emission rate when the concentration curve exhibits declining gradients. Unfortunately, the nonlinearity cannot be detected by means of R2, when only three measurement time points are used. For the fictive data in the plot, 
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Hence, if 
and can be estimated from the measured trace gas concentrations, then r0 can be estimated by inserting the estimates of and into Eq. [2]. Let C0, C1, and C2 denote, respectively, the initial measured concentration and two measurements made after successive time periods of length 
. If
 | (3) |
then the following values of and
 | (4) |
 | (5) |
makes the first model (Eq. [1]) fit perfectly to the observed concentrations. Notice that we use a caret to distinguish a model parameter (unknown), for example, from its estimated value, 
, calculated from the data. The condition in Eq. [3] does, however, not ensure that 
is positive, since this requires the stronger condition
 | (6) |
In any case, the emission rate can be estimated by
 | (7) |
Being a perfect-fit estimator obviously makes Eq. [7] highly sensitive to random variations in the trace gas concentrations. In particular, this means that the condition in Eq. [3] is often not satisfied in practice (Anthony et al., 1995), implying that 
cannot be calculated. The condition of Eq. [3], for instance, is not satisifed for the simulated data in Fig. 1c. Anthony et al. (1995) applied the Hutchinson and Mosier (1981) method 2224 times and reported failure in 
45% of these cases because the condition in Eq. [3] was not satisfied by the data. Moreover, since the estimator (Eq. [7]) is based on a perfect fit to Eq. [1], the standard error cannot be estimated from the data, and the method does not provide a statistical test for significance of the estimated emission rate. Finally, the ability of Eq. [1] to describe the data cannot be tested when only three measurements are used. On the other hand, the method only applies to exactly three time-equidistant measurements.
To perform a proper statistical analysis, that is, to estimate the standard error of the emission rate estimator, to test the significance of the estimated emission rate, and to test the ability of the applied model to describe the data, there is a need for more than just two additional measurement time points after placement of the soil cover. A rather straightforward extension of the Hutchinson and Mosier (1981) method that is able to utilize more than three observations is least squares estimation. However, least squares estimation ignores serial correlation in the data, which is likely to introduce bias, and does not provide a fully specified statistical model that can be validated and which enables statistical testing of the significance of the estimated emission rate. Moreover, the least squares estimates must be found iteratively by means of some numerical procedure.
In an attempt to overcome the problems of the Hutchinson and Mosier (1981) method outlined above, Pedersen (2000) proposed to interpret data as originating from a noisy version of Eq. [1], that is, by modeling the measured trace gas concentrations by the following stochastic extension of Eq. [1]
 | (8) |
where W is a standard Wiener process, and 
2 is an unknown positive parameter determining the level of noise in the concentrations. A standard Wiener process is a continuous-time stochastic process with continuous trajectories and independent increments Wt - WS 
N(0, t - s), and plays a key role in the definition of stochastic differential equations (Karatzas and Shreve, 1988) of which Eq. [8] is an example. Stochastic differential equations are natural stochastic extentions of ordinary differential equations, and in the present example the application of Eq. [8] implies that the trace gas concentration is modeled by a stochastic diffusion process with mathematical expectation given by Eq. [1] and a variance that increases with time, reflecting the fact that the ability of Eq. [1] to predict the concentration decreases with time. Notice that we assume Eq. [8] as a model for measured rather than actual concentrations; that is, we do not separate analytical errors (defined as the sum of errors that make measured concentrations different from actual concentrations) from random deviations of the actual concentrations from the deterministic diffusion model. As is the case for most statistical models, all sources of randomness are pooled. More details about the stochastic diffusion model given by Eq. [8] can be found in Pedersen (2000) or Cox et al. (1985). Let C0, C1, ..., Cm, (m = n–1) denote a series of concentration measurements, and assume for a moment that the observation time points are equidistant with time step . Finally, let
Then the estimates of and derived from Eq. [8] by the methods proposed by Pedersen (2000) are given by
 | (9) |
and
 | (10) |
These estimates exist (can be calculated), and is positive, if and only if
 | (11) |
To ensure that is positive we need the stronger condition
 | (12) |
As in Eq. [7], the corresponding estimator of the emission rate is given by inserting Eq. [9] and [10] into Eq. [2]. For 
the estimators Eq. [9] and [10] are exactly the estimators Eq. [4] and [5], and the conditions in Eq. [11] and [12] are exactly the conditions in Eq. [3] and [6]. More importantly, however, by the method above we are able to include more observations in the statistical analysis. For , the standard error of 0 and the corresponding 95% confidence interval for r0 can be estimated. Moreover, the hypothesis 
can be tested by calculating the P value: 
, where SE(0) denotes the estimated standard error of 0, and F2
denotes the distribution function of the 2 distribution with one degree of freedom. The exact formula for the standard error can be found in Pedersen (2000). The importance of including more than three observations in the statistical analysis is further emphasized by the statistical properties of 0. As n increases, the failure probability tends to zero and 0 approaches the true emission rate given by Eq. [2]. These facts are theoretical properties of the statistical model defined by Eq. [8] and can be proven mathematically (Pedersen, 2000). Also, for longer time series, say n > 5, it is possible to test the ability of the Eq. [8] to describe the data by means of graphical techniques and statistical goodness-of-fit tests. Finally, everything above based on Eq. [8] can be performed with nonequidistant measurement time points, which may be important in practice if data are lost at some time point, or if a less rigid sampling scheme is preferred.
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MATERIALS AND METHODS
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Field Site and Sampling Conditions
Data for evaluation of the statistical models were obtained from a field study carried out in a peat soil with permanent grass for cattle grazing. The meadow (2 ha) was ditched and drained and the groundwater level 40 cm below the soil surface (Low GWL). A part of the meadow (0.3 ha) was encircled by a 2-m-deep sheet piling and a pumping system that kept the groundwater level 15 cm below the surface (High GWL). The peat soil (5-10 cm depth) contained 37.7% organic C, 2.8% total N, 1.2 kg-1 total P, pHCaC12 4.1, a bulk density of 0.31 g cm-3, and a total porosity of 76.5%.
Gas sampling for determination of N2O emission was conducted twice, on 10 Aug. and 7 Sept. 1999, respectively, using static chambers without vents (Conen and Smith, 1998). Five cylindrical white metal containers per treatment (16-cm diam., 15-cm height) with sharpened edges were inserted into the ground to a depth of 5 cm with as little disturbance as possible. The containers were then closed with a septum for gas sampling. Atmospheric gas samples were taken to represent time zero, although it is acknowledged that sampling of individual chambers would be required to account for any disturbance effects of chamber deployment. Subsequently, samples of 5 mL were taken from each chamber headspace at intervals of 30 min. On 10 August, 10 time series 
were sampled, five at plots with low groundwater level, and five at plots with high groundwater level. On 7 September, similar time series 
were sampled.
After gas sampling, the containers were removed, and soil was sampled (0-5 and 5-10 cm depth) for determination of soil moisture after drying overnight at 105°C. Soil temperature at 10-cm depth averaged 16°C on 10 August and 15°C on 7 September, and changed 
1°C during chamber deployment.
Gas samples were stored in 3-mL Venoject tubes (Terumo Corp., Tokyo, Japan), transported to the laboratory, and analyzed for N2O concentration within 48 h of sampling. Analysis of N2O concentration was done using a Varian 3300 gas chromatograph (Varian Associates, Sunnyvale, CA) equipped with a 63Ni-electron capture detector. Details of the gas chromatographic analysis are given in Maag and Vinther (1996).
Choice of Data for Analysis
To minimize nonlinearity, linear regression was applied to only the first three observations in each time series. Due to minor variations in the actual time steps used between samplings, the requirement of the Hutchinson and Mosier (1981) method that the three measurement time points be equidistant could not be fulfilled. Hence, a direct comparison of the Hutchinson and Mosier (1981) method with linear regression was not possible. However, the estimation method based on Eq. [8] is identical to the Hutchinson and Mosier (1981) method for and time-equidistant observations, but also applicable to and nonequidistant observation time points. As a substitute for the Hutchinson and Mosier (1981) method, we could therefore apply the method based on Eq. [8] to exactly the same data that linear regression was applied to. A major advantage of Eq. [8] is, however, that it is able to utilize more observations, so we also applied the stochastic diffusion method to the complete time series. The ability of Eq. [8] to describe the data was tested by graphical techniques and goodness-of-fit tests based on estimated uniform residuals (Pedersen, 2000).
Simulation Study
Assuming Eq. [8] and realistic parameter values derived from the measured data, we performed a simulation study of the properties of the different methods, for example, the bias of the linear regression estimator, the behavior of the R2 diagnostic and the t statistic associated with linear regression, the failure rate, bias and standard error of the Hutchinson and Mosier (1981) estimator, and similar properties of the emission rate estimator based on Eq. [8]. Specifically, we studied the choice of time step between consecutive concentration measurements with the three models.
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RESULTS AND DISCUSSION
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In Fig. 3
the measured N2O concentrations are plotted against time since deployment. The declining concentration gradients were apparent at both High GWL and Low GWL on 10 August and at Low GWL on 7 September, but less clear at High GWL on the last date. The extreme N2O concentration 22 min after deployment for the fourth chamber in the High GWL August data set was considered to be an outlier and excluded from the analyses. The sampling interval, 30 min, and therefore total deployment time was relatively long due to practical limitations, which increased the potential for disturbance of N2O emission rates. However, curvilinear headspace accumulation of N2O has also been observed in studies with much shorter deployment times and only 10-min sampling intervals (Anthony et al., 1995; Chang et al., 1998; Pedersen, 2000).
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Fig. 3. Measured N2O concentrations (µL L-1 N2O) against time (minutes) since deployment. The number (1–5) at each curve is for identification
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There were vertical gradients in soil moisture at both groundwater levels and on both sampling days (Table 1), although the difference between Low GWL and High GWL was only significant at the 5- to 10-cm depth.
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Table 1. Gravimetric soil moisture at the two ground water levels (GWL), Low GWL and High GWL, measured at two depth intervals.
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Table 2 presents emission rates and the associated statistics for the measurements shown in Fig. 3, as determined by three different estimation methods. The first two, linear regression and the so-called HM, were based on the first three sampling time points (0, 30, and 60 min); as discussed above HM corresponds exactly with the model of Hutchinson and Mosier (1981), although the stochastic diffusion model was applied due to nonequidistant sampling times. The last calculation method was the stochastic diffusion model applied to all sampling times.
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Table 2. Results for the two data sets (results for the August data are in the upper half) after applying linear regression, as well as the method based on Eq. [8] to the first three observations in each time series, and Eq. [8] also to the complete time series. The estimated emission rates and standard errors are in µg N2O-N m-2 h-1. The rightmost column contains P values for goodness of fit tests for Eq. [8] (Pedersen, 2000)
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Graphical tests and goodness-of-fit tests (exemplified in Fig. 4) convincingly supported the ability of Eq. [8] to describe the data. Except for the chambers Low GWL-3 and High GWL-2 in the August data set, the estimated emission rates were consistently lower for the linear regression method than for the method based on Eq. [8]. A similar conclusion was reached by Anthony et al. (1995), as well as by Healy et al. (1996) (in a simulation study) and Pedersen (2000), while other studies have not found evidence for curvilinearity in N2O accumulation (Clayton et al., 1994; Ambus and Christensen, 1995; Petersen, 1999). Notice also that, although the data clearly exhibited declining concentration gradients with respect to time, the R2 values for the linear regression method were always 
0.88 (Table 2), thus confirming the conclusion of Hutchinson and Livingston (1993) that is not a good diagnostic for testing the applicability of linear regression in this context. In fact, for all but the chambers Low GWL-3 and High GWL-2 in the August data set the three observations used by linear regression were distributed around the estimated straight line exactly as in Fig. 2 (i.e., the second observation above the line and the two other observations below the line). This conflicts with the assumption of linear regression that the deviations from the straight line are random and uncorrelated, and clearly indicates that the concentration curve is concave. In contrast, Eq. [8] estimates considerable serial correlation in these data (results not shown).
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Fig. 4. Model evaluation for the chamber Low GWL-3 in the September data set. (a) The solid curve is the estimated expected values for Eq. [8], and the slopes of the lines are the emission rates estimated by linear regression (dashed) and the method based on Eq. [8] (dotted). (b) Quantile plot of the estimated uniform residuals. If Eq. [8] is a valid model for the data, the points should be scattered unsystematically around the identity line
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Although we believe that the linear regression results are questionable, the relatively large P values compared with those of the method based on Eq. [8] are in part due to the relatively small sample sizes 
. However, extending the number of observations merely increases the nonlinearity and hence the bias of the linear regression estimator (underestimates even more). The P values for the method based on Eq. [8] were very small, which is probably related to a low noise level in these data. This may also be the reason for the relatively low failure rate of (the substitute for) the Hutchinson and Mosier (1981) method. Anthony et al. (1995) and Pedersen (2000) found failure rates close to 50%.
The analysis described above, as well as similar analyses described by Pedersen (2000), quite convincingly show that Eq. [8] provides a good description of this kind of data, so it seemed reasonable to apply it to simulation studies. The purpose of the simulations described below is merely to illustrate some types of information that can be derived from the model, and therefore the presentation is based on a set of selected, but realistic parameter values, namely averages of the values estimated for the 10 data sets: 
. Moreover, we use the average initial concentration for initialization of the model (i.e., 
corresponding with 
). An arbitrary simulated trajectory with these parameter values is shown in Fig. 5
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Fig. 5. An arbitrary simulated trajectory from Eq. [8] with parameter values given by the average of the values for the 10 plots in the two data sets. The solid line is the corresponding deterministic model curve given by Eq. [1]. Time is measured in minutes since deployment, and the concentration is measured in µL L-1 N2O
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In practice there will be a limit to how frequently measurements can be taken, realistically every 10 min with manual sampling, although with automated systems can be much lower (e.g., Ambus and Robertson, 1998). The simulated deployment time was set to 4 h in accordance with the time span covered in this study. Assuming Eq. [8] with the parameter values given above, we were able to assess the properties of the different estimators of the emission rate. For the given parameter values and given values of and n this was done by simulating 10000 time series of N2O concentrations according to Eq. [8] by means of the Milstein scheme (Kloeden and Platen, 1992). Various theoretical properties of emission rate estimators could then be estimated by their sample equivalents; for example, the failure probability of a given estimator could be estimated by the percentage of the 10000 time series for which the estimator could not be calculated. For linear regression we considered the bias (B = mathematical expectation of the estimator less the actual value), the relative bias 
, the standard error (SE), the coverage of the usual 95% confidence interval, and the R2 diagnostic. If the t test is valid in this context, the coverage should be exactly 95%, because the t test is an exact test under the assumptions of linear regression. For the method of Hutchinson and Mosier (1981) we considered the failure probability, the conditional bias (Bc, i.e., the bias given the method does not fail), the relative conditional bias 
, and the conditional standard error (SEc, the standard error given the method does not fail). Table 3 shows these quantities calculated by means of simulation for and different values of .
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Table 3. Various characteristics of linear regression and the Hutchinson and Mosier (1981) method calculated by means of simulation for n = 3 and parameter values given by the average of the values for the 10 plots in the two data sets. The quantities B, SE, Bc, and SEc are in 10-4 µL L-1 N2O min-1, and the numbers in brackets in the R2 column are the standard deviations of the simulated R2 values
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The last 7 rows in Table 3 
clearly show the linear regression estimator to be negatively biased, and even rather concentrated (small SE) around the biased value. As might be expected, B increases and SE decreases with . The coverages also show that the assumptions of linear regression are not fulfilled, implying that the t test claims significance too often for small values of , and too seldom for high values of . Finally, the results show that R2 is essentially unable to detect the nonlinearity from only three observation time points.
The results in Table 3 for the Hutchinson and Mosier (1981) method show that it tends to be positively biased (although in absolute values much less than the linear regression estimator), that it fails quite often, and that it has a relatively large standard error. It may be noticed that the bias of the linear regression estimator decreases with , whereas the failure probability and the relative conditional bias of the Hutchinson and Mosier (1981) estimator, RBc, seem to be minimal somewhere in the range 
. This illuminates a fundamental difference between the two methods. Linear regression is a local procedure in this context, because it approximates the tangent of the concentration curve at time zero, which requires small values of . The Hutchinson and Mosier (1981) method is a global procedure that attempts to model the whole curve in order to estimate the derivative at time zero by the mathematical derivative of the estimated curve. Hence, the Hutchinson and Mosier (1981) method favors values of for which the complete observation period captures the curvature.
A large simulation study (data not shown) suggested that the optimal value of for the Hutchinson and Mosier (1981) method depends primarily on the value of . This might be expected since, for fixed values of the remaining parameters, essentially determines how much of the curvature of the concentration curve can be observed within a given time span. For a range of values, and keeping all other parameter values constant, we have determined the optimal value of using failure probability as the primary criterion and relative conditional bias as the secondary criterion. The results are shown in Table 4.
Notice from Table 4 that the Hutchinson and Mosier (1981) estimator can have remarkably good properties in terms of failure probability and relative conditional bias with the proper choice of . However, it is not easy to transform these findings into practical recommendations regarding the choice of because 
reflects not only the measurement situation, but also site-specific soil characteristics. In any case, it is possible to overcome many problems with linear regression and the Hutchinson and Mosier (1981) method by measuring the trace gas concentration more than three times and using the method based on Eq. [8]. In Table 5 we have chosen equal to 1 and 10 min and calculated various characteristics of the emission rate estimator defined by Eq. [8] as a function of sampling times. Notice from Table 5 that even three or four observation time points in addition to time zero dramatically improved the properties of the estimator. However, to obtain a coverage of approximately 95%, it would be necessary to increase the number of observation time points further. This is in accordance with the theoretical statistical properties of the test, which is asymptotic, implying that the coverage approaches 95% as n increases.
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Table 5. Various characteristics of the emission rate estimator proposed by Pedersen (2000) calculated by means of simulation for equal to 1 and 10 min, and parameter values given by the average of the values for the 10 plots in the two data sets. The quantities Bc and SEc are in 10-4 µL L-1 N2O min-1
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By means of Eq. [8] it is possible to extract information about several parameters besides the emission rate, including , , and , and the corresponding 95% confidence intervals (Pedersen, 2000). Particularly, the parameters and may be of interest, although in a somewhat indirect sense for . The estimate and 95% confidence interval for immediately provides an estimate, (V/A), and a 95% confidence interval, (V/A)1 Dp/d (V/A)h, for the ratio Dp/d. Here l and h denote, respectively, the lower and upper limit of the 95% confidence interval for . If either Dp or d is estimated from other soil characteristics, then a (conditional) estimate of the other quantity can be calculated by, respectively, 
, with corresponding (conditional) 95% confidence intervals. The estimates and 95% confidence intervals are conditional in the sense that they assume that one of the quantities is known with certainty, although in practice this will usually be estimated by some other method. For the data presented in this study, Dp for the 0- to 10-cm depth interval was estimated with the model described by Freijer (1994) using the parameters given for a model soil with bulk physical characteristics quite similar to the peat soil studied here, as well as the actual air-filled porosities calculated for each sample. These Dp values were in turn used for estimating the depth d assumed by Hutchinson and Mosier (1981) to represent a depth below which the N2O concentration is constant (see Table 6). In practice there may not be such a well-defined boundary condition, but rather a vertical distribution of N2O production and consumption that reflects the specific combination of environmental variables at each depth.
There was no relationship between groundwater level and d on either sampling day. Most values ranged from 5- to 20-cm depth, and greater values were characterized by high uncertainty. The range of d values thus indicates that the N2O emitted was, at least partly, produced above the groundwater table. Taking the water-filled pore space (WFPS) at the 5- to 10-cm depth as representative for the unsaturated zone, N2O emissions are plotted against WFPS in Fig. 6
. There was a positive relationship between N2O flux and soil moisture above 40% WFPS, most strongly for High GWL, the relationship was different on the two sampling days despite very similar soil temperatures, and may be related to a decline in N mineralization rate between August and September (data not available). The relationship between N2O emissions and WFPS was probably caused by an effect of soil moisture on biological N2O production. Autotrophic nitrification has been shown to produce high proportions of N2O in acid soils (Martikainen, 1985), and in the present system there could have been an interaction with low O2 availability, which is also known to stimulate N2O production from nitrifiers (Goreau et al., 1980). However, N2O would also be a major product of denitrification at the WFPS values observed (Davidson, 1991). The mechanisms behind N2O released from this system cannot be elucidated without more detailed information about soil conditions at the site of N2O production.
The organic soil type of the experimental site was characterized by a high diffusibility (Freijer, 1994) and groundwater levels close to the soil surface (i.e., conditions where the nonlinearity of trace gas accumulation can be strong; Livingston and Hutchinson, 1995). However, pronounced nonlinearity and, hence, the need for analytical tools that can handle this type of data, may be encountered in a variety of situations, depending on the particular combination of soil environmental conditions and sampling strategy.
The proposed stochastic diffusion model is very robust against data variability, and it must be emphasized that the method cannot, and should not, replace careful data examination to identify excessive noise, as well as systematic errors and spurious results, so the experience of the analyst remains essential. Nevertheless, adoption of this method may reduce the need for disregarding data due to nonlinearity, and thereby help preserve statistical information.
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SUMMARY AND CONCLUSIONS
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The methods proposed provide a solution to the calculation of trace gas emission rates that can replace linear regression or the method of Hutchinson and Mosier (1981), and it offers new possibilities for data analysis. For only three equidistant observation time points the estimator is identical to the Hutchinson and Mosier (1981) estimator. The power of the new approach is the ability to use longer time series of trace gas concentration measurements, thus making a complete set of statistical methods available (e.g., estimation of the standard error of the emission rate estimator, test of significance of the estimated emission rate, estimation of a 95% confidence interval for the emission rate, model validation tools).
An essential property of the emission rate estimator proposed is that it approaches the true emission rate as the number of observations increases. The statistical properties of the estimator are also improved as the number of observations increase, a feature that is not shared by linear regression or the Hutchinson and Mosier (1981) method (see Table 3). The new method is extremely flexible with respect to the distribution of observation time points. Particularly, the curvature of the concentration curve need not be avoided, as is the case with linear regression, since the method proposed can use observations from any part of the concentration curve. In a sense, the method actually needs observations from the nonlinear part of the concentration curve. Moreover, the stochastic diffusion model, in contrast to the Hutchinson and Mosier (1981) method, does not require time-equidistant observations.
Nonlinear concentration curves could potentially be modeled purely empirically by polynomial regression, for example, quadratic or cubic regression. Although this would probably lead to less biased estimators than linear regression, polynomial regression (including linear regression) assumes that repeated measurements within a chamber are uncorrelated, which is highly questionable. Moreover, quadratic and cubic regression models estimate, respectively, four and five parameters from the data as opposed to the present method, which estimates only three parameters. This is an important difference, since, generally, more parameters require more data to obtain the same precision. Finally, being purely empirical, polynomial regression models do not utilize the scientific knowledge that is available in terms of, for example, the Hutchinson and Mosier diffusion model, which in general makes them much less robust to random fluctuations in data. The problem increases with the polynomial order.
The statistical methods described here and in Pedersen (2000) have been implemented in a freeware stand-alone computer program that can be obtained from the second author upon request. Besides improving the statistical analysis of data, the program also facilitates simulation studies, as exemplified in this study.
Received for publication March 1, 2000.
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REFERENCES
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