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Soil Physics

Trace Gas Emission in Chambers

A Non-Steady-State Diffusion Model

^a Rubenstein School of the Environment and Natural Resources
^b Dep. of Physics, Univ. of Vermont, Burlington, VT 05602
^c USDA-ARS, Natural Resources Research Center, Fort Collins, CO
^d current address: Altos Imaging, Hinesburg, VT 05461

^* Corresponding author (glivings{at}madriver.com)

	ABSTRACT

TOP ABSTRACT INTRODUCTION MODEL FORMULATION MODEL EVALUATION MODEL PERFORMANCE CONCLUSIONS REFERENCES

 
Non-steady-state (NSS) chambers are widely used to measure trace gas emissions from the Earth's surface to the atmosphere. Unfortunately, traditional interpretations of time-dependent chamber concentrations often systematically underestimate predeployment exchange rates because they do not accurately represent the fundamental physics of diffusive soil gas transport that follows chamber deployment. To address this issue, we formally derived a time-dependent diffusion model applicable to NSS chamber observations and evaluated its performance using simulated chamber headspace CO₂ concentration data generated by an independent, three-dimensional, numerical diffusion model. Using nonlinear regression to estimate the model parameters, we compared the performance of the non-steady-state diffusive flux estimator (NDFE) to that of the linear, quadratic, and steady-state diffusion models that are widely cited in the literature, determined its sensitivity to violation of the primary assumptions on which it is based, and addressed some of the practicalities of its application. In sharp contrast to the other models, NDFE proved an accurate and robust estimator of trace gas emissions across a wide range of soil, chamber design, and deployment scenarios.

Abbreviations: H–M–P, Hutchinson–Mosier–Pedersen • NDFE, non-steady-state diffusive flux estimator • NSS, non-steady state

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MODEL FORMULATION MODEL EVALUATION MODEL PERFORMANCE CONCLUSIONS REFERENCES

 
NON-STEADY-STATE CHAMBERS (Livingston and Hutchinson, 1995; Hutchinson and Livingston, 2002) are widely used to measure rates of trace gas exchange between the Earth's surface and the atmosphere. Indeed, no other method has contributed more to current understanding of the magnitude and spatiotemporal variability of trace gas exchange rates or their process-level controls. Data from such studies have promoted understanding of C and nutrient dynamics, facilitated development of land use management strategies, and helped establish the relative importance of various greenhouse gas sources and sinks.

Despite these important contributions, theoretical and empirical evidence indicates that flux densities derived from NSS chambers systematically and often significantly underestimate the rate of emissions that prevailed before chamber deployment (Matthias et al., 1978; Hutchinson and Mosier, 1981; Jury et al., 1982; Samuelsson, 1990; Anthony et al., 1995; Wagner et al., 1997; Pedersen et al., 2001; Davidson et al., 2002; Pumpanen et al., 2003; Hibbard et al., 2004; Livingston et al., 2005). This bias results not from inherent limitations of chambers themselves, but from long-held misconceptions regarding the interpretation of chamber headspace concentration data and, in particular, from the use of flux estimation models that fail to accurately represent the fundamental physics of diffusive soil gas transport in the presence of a NSS chamber.

In practice, the predeployment flux density is estimated by first fitting an assumed model of chamber headspace concentration with time to the observed data and then projecting the instantaneous rate of change at the moment of chamber deployment. The most widely used model by far is the linear model, which assumes that emissions into the chamber headspace are constant throughout the deployment period. In fact, however, the rate of transport of a diffusing trace gas into the chamber headspace necessarily declines throughout deployment because any increase in the headspace concentration results in an immediate decline in the subsurface vertical concentration gradient driving that transport (Matthias et al., 1978; Hutchinson and Mosier, 1981; Jury et al., 1982; Samuelsson and Pettersson, 1984; Rolston, 1986; Hutchinson et al., 2000). The error in applying a linear model to inherently nonlinear concentration data has long been assumed negligible if recommended guidelines regarding chamber design, deployment, and sampling are followed to foster the appearance of linearity in the observed concentration data (Livingston and Hutchinson, 1995; Davidson et al., 2002; Hibbard et al., 2004); however, the resultant error is not negligible and thus the use of linear models has ensured that predeployment emission rates have been systematically and often substantially underestimated in nearly all NSS chamber applications (Matthias et al., 1978; Jury et al., 1982; Anthony et al., 1995; Hutchinson et al., 2000; Livingston et al., 2005).

In recognition of this issue, nonlinear models, such as the physically based model proposed by Hutchinson and Mosier (1981) and the quadratic model explored by Wagner et al. (1997), were applied to NSS chamber observations, although both approaches are limited in their applicability or interpretation. For example, the physical significances of the fitted polynomial coefficients of the quadratic model are not necessarily either apparent or meaningful. In turn, the diffusion model advanced by Hutchinson and Mosier (1981) is compromised by its assumption of steady-state conditions at every point in time (Hutchinson and Mosier, 1981; Hutchinson and Livingston, 2002) and, as originally implemented, limited in its applicability (Anthony et al., 1995; Pedersen, 2000), although Pedersen (2000) and Pedersen et al. (2001) mathematically extended this model to permit its application to any number and spacing of observations with time and to reduce its sensitivity to measurement error. We refer hereafter to this approach as the H–M–P model.

The non-steady-state diffusive flux estimator (NDFE) recently introduced by Livingston et al. (2005) is, to date, the only trace gas emissions model derived from time-dependent diffusion theory applicable to NSS chamber concentration data. We present here the formal derivation of that model and an evaluation of its performance relative to that of the linear, quadratic, and H–M–P models commonly cited in the literature, its sensitivity to violation of the primary assumptions on which it is based, and an assessment of some of the practicalities of its application.

	MODEL FORMULATION

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Assumptions and Derivation
Molecular diffusion is the principal mechanism driving trace gas exchange between soil and the atmosphere in terrestrial ecosystems (Penman, 1940; Jury et al., 1982; Ghildyal and Tripathi, 1987). Assuming that Fick's laws apply (Rolston and Moldrup, 2002) and that the soil has uniform properties across space and time except for a vertically distributed zero-order trace gas source, the rate of this exchange is described by the standard diffusion equation in one dimension:

[1]

where c is the trace gas concentration [M L^–3 air] at time t [T] and depth z [L soil] below the soil surface, is its depth-dependent zero-order source strength [M L^–3 soil T^–1], is the soil's air-filled porosity [L³ air L^–3 soil], and D_p is the soil gas diffusion coefficient [L³ air L^–1 soil T^–1] as defined by Rolston (1986) and Rolston and Moldrup (2002).

We assume that at t = 0, a NSS chamber enclosing an air volume V [L³ air] is deployed across soil area A [L² soil] with its vertical side walls inserted into the soil to sufficient depth that subsurface gas transport is limited to the vertical domain during the deployment period. Equation [1] can then be solved using Laplace transforms following the procedures in Carslaw and Jaeger (1959, Ch. 12, Sections 12.1–12.4). We define the Laplace transform of the concentration

[2]

where the variable p is inherent to the transform and obtain the subsidiary equation

[2]

where the steady-state concentration profile in the soil at t = 0 is defined as c₀(z). The latter satisfies the equation

[3]

Substitution of the source strength (z) from Eq. [3] into Eq. [2] yields

[4]

where we have defined q² p/D_p. The solution that converges as z approaches infinity is

[5]

Integration constant K is obtained using mass continuity at z = 0. We start from

[6]

where m_s [M] is the mass of the trace gas in the soil, and f is its flux density [M L^–2 soil T^–1]. At any time, the mass of the trace gas m_c [M] in the enclosed air volume above the soil surface is related to its concentration C_t [M L^–3 air] in that air volume by m_c = C_tV. Assuming that air in the chamber is uniformly mixed throughout, the concentration of the trace gas within the chamber headspace must be equal to its concentration in the air-filled soil pore spaces at the surface, z = 0, at all times, i.e., C_t = c(0,t). Mass continuity requires that

[8]

so that Eq. [6] becomes

[7]

The subsidiary equation of Eq. [7] can be written as

[8]

Substitution of Eq. [5] into Eq. [8] yields the value of K. With this value, Eq. [5] becomes

[9]

where c₀'(0) is the spatial derivative of the initial concentration evaluated at z = 0. The Laplace transform of Eq. [9] is given in Carslaw and Jaeger (1959, Appendix 5 [1 and 15]).

Trace gas concentration as a function of depth and time is thus

[10]

where erfc is the complementary error function. The flux density f₀ across the soil-atmosphere interface at t = 0 is

[13]

Substituting the latter into Eq. [10], defining the time constant (V/A)²(D_p)^–1 with units [T], and recognizing that C₀ = c₀(0), we evaluate the resulting equation at z = 0 to obtain the final form for the headspace trace gas concentration as a function of time:

[11]

Note that C_t is independent of the subsurface source distribution, as previously indicated by the numerical studies of Hutchinson et al. (2000), and that Eq. [11] applies only to trace gas emission from soil to the atmosphere; a follow-up study will address estimating the downward flux of trace gases having a soil sink. Also note that, for brevity and convenience, we refer hereafter to the ratio V/A as the effective chamber height, h, emphasizing that V represents the total aboveground enclosed air volume, including not only that of the chamber headspace but also, in recirculating systems, that of the pump, external gas concentration sensor, sample conditioning modules, and connecting tubing.

Parameter Estimation and Interpretation
Model parameters C₀, f₀, and are readily estimated by iteratively fitting Eq. [11] to the observed C_t using nonlinear regression to minimize the sum of the squared residuals between the observed and modeled values. Alternatively, f₀ and alone may be estimated if C₀ can be measured with negligible error. The latter may prove challenging in field applications, however, due to measurement variability, sample timing error, and spatial or temporal variability in the trace gas concentration distribution and surface flux density. We generally recommend fitting all three model parameters despite the attendant loss of one degree of freedom for error.

The precision with which parameters are estimated with nonlinear regression is less well defined than for linear regression. Guidelines for evaluating goodness-of-fit and for computing asymptotic approximations of the confidence intervals about the parameters are outlined in Ratkowsky (1990) and Rawlings et al. (1998), but are most applicable when the number of observations is large relative to the number of fitting parameters.

Parameters C₀ and f₀ represent the ambient trace gas concentration and surface flux density at the moment of chamber deployment. The experimental time constant may be thought of as a measure of chamber feedback. For example, for any given combination of trace gas, soil, chamber, and deployment period, represents a characteristic time during which the concentration gradient driving gas transport into the chamber headspace will tend to vanish in response to rising headspace concentration. When is small, the soil concentration gradient, and thus diffusive gas transport into the chamber, will decline more quickly than when is large.

Figure 1 examines chamber headspace concentration increase with time for various values of . Note that the larger the ratio of the deployment period to , the greater the decline in the concentration increase during that deployment period, and thus the more pronounced the curvature in the C_t vs. t observations. Correct data interpretation, therefore, clearly requires fitting a theoretical model with more than two parameters to the observed data. To understand why, we examine the case when the deployment time is short relative to , i.e., the regime where the C_t vs. t curve appears to be highly linear and linear regression coefficients of determination (R²) are characteristically high. A series expansion of Eq. [11] when t/ is small yields

[15]

Unlike most expansions that have integer exponents such that each term leads the next by an order of magnitude, this expansion has the atypical form of integer powers of the expansion variable t/ intertwined with half-integer powers. As a consequence, the (t/)^3/2 term is not negligible relative to the linear term and is thus the reason why a linear regression fit to the headspace concentration data underestimates f₀ even for small values of t/. For example, at t/ = 0.01, the (t/)^3/2 term is 7.5% of the linear term. Thus, if has a value of 600 min, a linear model applied to observed concentration data during a deployment period as short as 6 min (0.01) would underestimate f₀ by 7.5% even though the C_t vs. t response would resemble a straight line because of the weak curvature of the (t/)^3/2 term. We also note that the quadratic model advanced by Wagner et al. (1997) recognizes the need to account for the curvature in the data, but does so by introducing a quadratic term t² instead of (t/)^3/2 and assumes that the coefficient of the quadratic term is unrelated to f₀.

View larger version (16K): [in this window] [in a new window]   Fig. 1. Chamber headspace CO2 concentration increase (Ct – C0) as a function of deployment time (t) and effective chamber height (h) on soil with air-filled porosity = 0.3. Closed circles represent simulated concentrations computed using the numeric diffusion model. Solid lines represent the results of least-squares fits to the simulated concentration data using the non-steady-state diffusive flux estimator, NDFE.

	MODEL EVALUATION

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The numeric diffusion model was parameterized assuming the soil was unvegetated and uniform with respect to total porosity (0.5), air-filled porosity (0.3 unless otherwise noted), tortuosity (as defined by Sallam et al., 1984), and pH (6.5). The zero-order CO₂ source (50 µg CO₂–C m^–2 s^–1) was assumed to be exponentially distributed with a maximum at the soil surface and a 10-cm relaxation depth. Properties for CO₂ were chosen at 20°C and 100 kPa air pressure, and we assumed equilibrium between gaseous and dissolved CO₂ and among dissolved carbonate species in soil water. Chambers were assumed to be cylindrical with a 30-cm diameter and effective height of 5 to 100 cm as noted. In most scenarios, modeled chamber sidewalls were inserted into the soil to a depth sufficient to limit diffusive gas transport beneath the chamber to the vertical domain; otherwise, the insertion depth was specified and lateral transport beyond the basal area of the chamber was allowed and accounted for.

Except as noted, CO₂ flux density estimates were computed from chamber headspace concentrations recorded at 0, 5, 10, 20, and 30 min (n = 5) during 30-min simulated chamber deployments. Although this is a modest number of observations relative to the number of fitted parameters, it was chosen to represent the limited number of samples that often characterize protocols that combine field sampling and laboratory analyses, as opposed to in situ trace gas measurement methods. The limited number of observations in this study is also justifiable in that the simulated data are exact in the sense that they do not contain random experimental error. The impact of the number of samples on the accuracy and precision of replicate flux estimates is formally addressed below.

We compared NDFE performance with that of alternative flux estimation models commonly applied to NSS chamber observations, namely the (i) linear, (ii) quadratic (Wagner et al., 1997), and (iii) steady-state diffusion model (H–M–P) advanced by Hutchinson and Mosier (1981), Pedersen (2000), and Pedersen et al. (2001). Linear and quadratic model estimates were computed using Microsoft Excel's LINEST function. Estimates for the H–M–P model were computed using software provided by Pedersen et al. (2001, N2O, v1.0). The NDFE model parameters were estimated using Excel's Solver "add-in" tool. The only constraint imposed on NDFE solutions was that all fitted parameters had to be greater than zero. The central method of differencing was chosen to estimate partial derivatives of the objective and constraint functions and the convergence criterion was set at 10^–6.

Our experience suggests that convergence is rapid, particularly when the C_t vs. t data are visually nonlinear, and that parameter estimates are repeatable across a wide range of starting values. Although the latter need only be approximate, we chose starting values for C₀ and f₀ based on their linear-regression estimates. Starting values for were assigned by adding a random number between 1 and 1000 to the value = h²/D_air where D_air is the diffusivity of CO₂ in free air. As is generally recommended in nonlinear regression applications, multiple starting values for each flux estimate were explored to ensure that resultant estimates were based on global, not local, minima in the residual sum of squares. Exploratory analyses also indicated that parameter scaling was important to model performance, but required only that units of the three parameters be chosen such that their numerical values were within a few orders of magnitude of one another.

An annotated script for computing NDFE flux densities from NSS chamber observations using Microsoft Excel is available via the Internet at http://arsagsoftware.ars.usda.gov (verified 15 Apr. 2006). Also available to aid verifying implementation of the NDFE are simulated chamber headspace CO₂ concentration data for which the predeployment parameters are known.

	MODEL PERFORMANCE

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Goodness of Fit
In sharp contrast to other flux estimation approaches applied to NSS chamber observations, NDFE proved to be a robust and unbiased estimator of trace gas flux density across a wide range of soil, chamber design, and deployment situations. As evidence, we examined the goodness-of-fit of selected flux estimation models to the simulated C_t vs. t "experimental" data and evaluated the accuracy of the model parameters. Goodness-of-fit (²) between modeled and observed headspace concentrations was measured as the sum of the squared residuals:

[16]

where y_i and _i are the ith chamber headspace concentration values obtained from the numeric simulation and any one of the theoretical models, respectively, and N represents the number of observations from which the model parameters were estimated. This definition of ² is relative in that when the same data are fitted to the different models, the fit that yields a significantly smaller ² indicates the better model. Flux estimation accuracy was measured as relative error, i.e., the percentage deviation of the "estimated" flux density from the "known" predeployment flux density.

Figure 2 examines ² as a function of chamber feedback parameter for each of the flux estimation models examined given a fixed deployment time of 30 min. We chose as the abscissa because, for a fixed deployment time, determines the extent of the departure from linearity in the experimental data. The data demonstrate that NDFE models the C_t vs. t NSS chamber response far more accurately than the linear, quadratic, or H–M–P flux estimation models across the entire range of examined. Not surprisingly, the linear model, which ignores chamber feedback altogether, most poorly fit the experimental data. The quadratic and H–M–P models similarly performed better than the linear model but not nearly as well as the NDFE. We note for all models that ² decreased systematically with increasing and thus with increasing linearity in the C_t vs. t data. This is to be expected in that the C_t vs. t response approaches a straight line when the chamber feedback time is large relative to the deployment time. Any model that produces a straight line as will thus better fit the experimental data. Nonetheless, even for the maximum examined (h = 100 cm, = 0.3), the NDFE model ² was 2.5 to four orders of magnitude smaller than that of the competing flux estimation models because of its improved characterization of chamber feedback. When was small relative to the deployment time, i.e., when curvature in the C_t vs. t data was pronounced, however, NDFE goodness-of-fit was nearly five orders of magnitude better than that of the quadratic or H–M–P models and six orders of magnitude better than that of linear regression.

View larger version (16K): [in this window] [in a new window]   Fig. 2. Goodness-of-fit (2) of selected trace gas flux estimation models to simulated chamber headspace concentrations as a function of the experimental time constant . Data points represent observations for effective chamber heights h = 5, 10, 20, 50, and 100 cm on soil with air-filled porosity = 0.3 and 30-min deployment periods. The solid lines are not fits, but connect the data points to aid the eye.

View larger version (15K): [in this window] [in a new window]   Fig. 3. Flux (f0) estimation accuracy of selected models as a function of scaled deployment time (t/). Data points represent seven soil air-filled porosities () ranging from 0 to 0.5 and five effective chamber heights (h) ranging from 5 to 100 cm. Deployment period was 30 min. Non-steady-state diffusive flux estimator relative errors vary from –0.1 to –0.2%. The solid lines are not fits, but connect the data points to aid the eye.

The first assumption we evaluated is that trace gas transport into the NSS chamber headspace occurs largely through molecular diffusion. Although this assumption may apply in most measurement situations (Jury et al., 1982; Ghildyal and Tripathi, 1987; Rolston and Moldrup, 2002), it is well recognized that surface–atmosphere trace gas exchange rates are sometimes influenced by pressure-driven flow induced by natural processes (Scotter and Raats, 1969; Kimball, 1983; Thorstenson and Pollock, 1989; Massman and Farrier, 1992) or by inappropriate chamber design or deployment protocol (Rolston, 1986; Lund et al., 1999; Hutchinson and Livingston, 2001). To explore the latter issue, we modified the numeric diffusion model of Ishii et al. (1989) to simulate C_t when subsurface gas transport was driven by both diffusion and pressure flow processes. Note that our analysis applies only for an exponential trace gas source distribution with a maximum at the soil surface.

We considered first the instantaneous pressure perturbation that can occur when the chamber is not vented, or when the vent is blocked or poorly designed. In these situations, the air enclosed by the chamber is forced to flow downward or upward across the soil surface in response to its compression on chamber deployment or its expansion on air sample withdrawal, thus vertically displacing the trace gas concentration profile beneath the chamber (Hutchinson and Livingston, 2001). For example, a 1-mm vertical compression of the headspace enclosed by a chamber of 30-cm diameter and 20-cm effective height is equivalent to a 0.5 kPa positive pressure perturbation and would force 71 cm³ of headspace air across the soil surface.

Figure 4 demonstrates that pressure flow induced by even a small instantaneous pressure perturbation in an unvented chamber can seriously impact NDFE performance. For example, a perturbation of as little as 0.15 kPa below or above ambient resulted in an increase in ² by more than three orders of magnitude. Additionally, positive perturbations induced at deployment lead to significant underestimation of f₀ (–12.4% for P = 0.5 kPa), whereas negative perturbations simulating sample withdrawal at t = 10 min spawned small but increasing positive bias (2.1% for P = –0.5 kPa). Most importantly, however, f₀ relative error proved negligible under all situations examined when the chamber was properly vented (P = 0 kPa). Chambers, therefore, should incorporate properly designed vents and seals to minimize any such pressure perturbations (Hutchinson and Livingston, 2001, 2002).

View larger version (12K): [in this window] [in a new window]   Fig. 4. Non-steady-state diffusive flux estimator goodness-of-fit (2) to simulated headspace concentrations (closed symbols) and flux (f0) estimation accuracy (open symbols) in response to an instantaneous pressure perturbation (P) induced at t = 0 min (positive perturbation) or at t = 10 min (negative perturbation). Data represent an experimental time constant = 610 min. The solid lines are not fits, but connect the data points to aid the eye.

Third, we examined the NDFE model assumption that gas transport into the chamber headspace is a one-dimensional process, i.e., that chamber wall insertion depth is sufficient to effectively preclude lateral diffusion driven by increasing gas concentration in soil beneath the chamber. If the insertion depth is insufficient, however, the observed headspace concentration change with time will be less than if the chamber walls were inserted to an infinite depth, thus prohibiting lateral diffusion altogether. Figure 5 examines NDFE ² and flux estimation accuracy as a function of chamber wall insertion depth and soil air-filled porosity ( = 0.3 or 0.5). These data show that, for a fixed effective chamber height and deployment time, flux estimates are highly unreliable unless wall insertion depths are sufficient that lateral diffusion presumably no longer significantly impacts observed chamber headspace concentrations. These data also show that greater insertion depths are required the more rapid the chamber feedback (smaller ) to achieve comparable flux estimation accuracy. In close agreement with guidelines forwarded by Hutchinson and Livingston (2001, 2002) these data indicate that maximum flux estimation accuracy (f₀ relative error –0.1%) requires chamber wall insertion depths of at least 5 to 10 cm when the experimental time constant = 610 min, but a depth of at least 20 cm when = 180 min. Because failure to follow these guidelines can lead to substantial flux estimation error, deployment and quality control protocols should consider the chamber wall insertion depth for each measurement situation.

View larger version (13K): [in this window] [in a new window]   Fig. 5. Non-steady-state diffusive flux estimator goodness-of-fit (2) to simulated headspace concentrations (closed symbols) and flux (f0) estimation accuracy (open symbols) as a function of chamber wall insertion depth. Data are for a fixed effective chamber height h = 20 cm and 30-min deployment periods on soil with air-filled porosity = 0.3 or 0.5. Lines are not fits, but connect the data points to aid the eye.

The results demonstrate that NDFE ² and flux estimation accuracy were little influenced by the presence of an interfacial layer 0 to 10 mm in depth or by mixing rates expected in the chamber headspace above the interfacial layer. For example, f₀ relative errors 0.2% were observed for mixing rates of at least 100 x D_air. When mixing was 30 x D_air, f₀ was underestimated by <1%, and by only 3% when the mixing rate was as low as 10 x D_air. These data could also be interpreted to suggest that mechanical mixing may improve flux estimation accuracy by ensuring near-uniform gas concentrations throughout the chamber headspace. Indeed, under neutrally buoyant and calm conditions and particularly for taller chambers, headspace mixing by internal temperature-related and external turbulence-related processes may prove inadequate. As noted above, however, care must be taken to ensure that such systems induce minimal pressure flow.

Sensitivity to Measurement Error
Measurement error, i.e., the random variability attributable to sampling, sample handling, and sample analysis, varies greatly between applications, ranging from <0.1% for in situ measurement techniques to as much as 1.5 to 2% for methods that require extensive sample handling. To evaluate the effect of this error on NDFE performance, we used a series of Monte Carlo studies that added random error to the numerically simulated C_t data from which trace gas flux densities were then estimated using the NDFE. The error added to each observation was drawn from a normal distribution with a zero mean and assumed a fixed nominal measurement precision. Each analysis was based on 1000 simulated flux estimates, each of which, in turn, were based on five concentration observations as described above.

Figure 6 examines the effect of random measurement error on flux estimation accuracy for fixed 30-min deployments. Not surprisingly given the limited number of observations on which each NDFE estimate was based (n = 5), both the accuracy and precision of ₀ decreased with increasing measurement error, although greater accuracy was retained the larger the value of t/. Notably, the mean relative error in ₀ remained far less than that for any of the alternative flux estimation models operating on data without measurement error (see Fig. 3). Nevertheless, the range across which individual flux estimates varied as measurement error was raised to ±2% increased approximately threefold and the frequency of errors shifted from that of a normal to that of a strongly positively skewed distribution as is evidenced by the positive bias in the mean estimates of f₀. This shift in the distribution of errors arises because of the variable sensitivity of ₀ to error in estimating . Overall, ₀ is only weakly underestimated when is overestimated due to measurement error; that is, when the observed C_t vs. t response appears to be more linear than would be observed without measurement error. Conversely, ₀ is overestimated often by as much as 10% when is underestimated because the C_t vs. t response is more curvilinear than would otherwise be expected.

View larger version (11K): [in this window] [in a new window]   Fig. 6. Flux (f0) estimation accuracy of the non-steady-state diffusive flux estimator as a function of headspace concentration (Ct) measurement precision and scaled deployment time (t/). Data are for fixed effective chamber heights h = 10 and 20 cm, and 30-min deployment periods on soil with air-filled porosity = 0.3. Data plotted represent the mean ±95% CI derived from 1000 simulated flux estimates. The dashed line represents flux estimation accuracy when Ct was measured without error.

View larger version (12K): [in this window] [in a new window]   Fig. 7. Flux (f0) estimation accuracy of the non-steady-state diffusive flux estimator as a function of the number of observations used in the estimate and scaled deployment time (t/). Data are for fixed effective chamber heights h = 10 and 20 cm, and 30-min deployment periods on soil with air-filled porosity = 0.3. The simulated measurement precision of the headspace gas concentration (Ct) was ±1.5%. Data plotted represent the mean ±95% CI derived from 1000 simulated flux estimates. The dashed line represents flux estimation accuracy when Ct was measured without error.

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION MODEL FORMULATION MODEL EVALUATION MODEL PERFORMANCE CONCLUSIONS REFERENCES

	ACKNOWLEDGMENTS

 
This research was supported in part by U.S. Department of Agriculture–Agricultural Research Service Specific Cooperative Agreement no. 58-5402-3-301 with the University of Vermont. We are also indebted to D. Rolston, who commented on an early draft of this manuscript.

Received for publication September 27, 2005.

	REFERENCES

TOP ABSTRACT INTRODUCTION MODEL FORMULATION MODEL EVALUATION MODEL PERFORMANCE CONCLUSIONS REFERENCES

 

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