Vertical source/sink strengths distributions and net flux of carbon dioxide (CO2) within and above a broadleaved Koreanpine forest canopy were modeled by inverse Lagrangian dispersion analysis method using only mean concentration profiles measured in Changbai mountain research station of China. The model results compared well with eddy covariance CO2 flux measurements above the canopy, and the average correlation coefficient is 0.89; however, the model overestimated nocturnal CO2 flux about two to three times larger than those measured results. The time-depth evolutions of CO2 source/sink strength and the relationship between CO2 flux and photosynthetically active radiation (PAR) were also discussed.
Raupach [1988, 1989a, 1989b] proposed the “Localized Near Field” (LNF) theory of dispersion in plant canopies, which showed that the scalar concentration profile within the canopy is the result of contributions from both local and distant sources. The LNF theory coupled with the distribution of vertical profiles of the standard deviation of wind speed (σw(z)) and the Lagrangian timescales (TL(z)) within the canopy, so that expressions between source strength and mean concentration profile can be derived and solved. Up to the present, a number of authors [Raupach et al., 1992; Denmead and Raupach, 1993; Denmead, 1995; Katul et al., 1997; Massman and Weil, 1999; Leuning et al., 2000; Leuning, 2000] used this approach, which with good agreement reported between modeled turbulent fluxes and those measured above the canopy. However, all the studies depend on the eddy covariance measurement to determine the friction velocity and the atmospheric stability parameter, which limited the method to be used widely. Wang et al.  used the method to simulate the water vapor source/sink strength and evapotranspiration only using the gradient measurement of microclimate parameters, which produced good agreement between modeled evapotranspiration and those measured by the open-path eddy covariance measurement system.
 The object of this study is to investigate the inverse Lagrangian dispersion analysis method developed by Raupach [1988, 1989a, 1989b] for broadleaved Koreanpine forest in Changbai Mountain, China, for estimating both CO2 source/sink distribution and CO2 flux from 17 to 22 June 2003, only using the gradient measurement of microclimate parameters. The calculated CO2 flux results are compared with the results measured by the open-path eddy covariance measurement system mounted on the tower above the canopy. The daily variation of CO2 flux source/sink distribution within the canopy and the relationship between CO2 flux and photosynthetically active radiation are also discussed.
2.1. Relationship Between the CO2 Concentration Profiles and Source/Sink Density
 The CO2 concentration field within a canopy can be viewed as the linear superposition of contributions from all sources. Raupach [1987, 1989a, 1989b] divided the concentration field into a “near-field” part, Cn(z), dominated by the contribution of nearby sources, and a “far-field” part, Cf(z), resulting from the contribution of distant sources. The near-field is defined as the distance from the source that fluid particles move in a time less than TL, the characteristic Lagrangian timescale of the turbulence, whereas the far-field is the distance from the source a fluid particle travels in times more than TL. Atmospheric turbulence causes transport of the CO2 from the source to the observation point. In the near-field, transport is dominated by coherent eddies, while in the far-field transport is essentially diffusive [Leuning et al., 2000]. Thus the CO2 concentration at height z can be written as
where C(z) is the CO2 concentration at height z. Suppose zR is a reference height. Then the difference of the CO2 concentration between height z and zR is given by
 The LNF theory assumed that the canopy is horizontally homogeneous at each height so that net transport is only in the vertical direction; that near-field transport can be described as if it occurred in a Gaussian homogeneous turbulent flow characterized by a standard deviation in vertical velocity σw(z) and a Lagrangian timescale TL(z); and that the contribution to the concentration field from far-field source is strictly diffusive [Katul et al., 2001].
 The near-field concentration profile is determined by the vertical distribution and strength of the sources, and the vertical profiles of σw(z) and TL(z). Designating the height of a given source as z0, the near-field concentration profile is given by
where S(z) is CO2 concentration source/sink strength at height z; kn is a “near-field kernel” whose analytical approximation is given by Raupach [1989a] as
On the basis of the gradient-diffusion relationship, Cf(z) − Cf(zR) is found to be
where Fg is the CO2 flux density from the underlying ground. So we get the relationship between C(z) and S(z) from equations (2), (3), (4), and (5).
 When the canopy is divided into m horizontally homogeneous layers with thickness Δzj and source strength Sj, and Fg is known, the discrete form of the relationship between C(z) and S(z) can be written as
where Dij is the coefficients of the dispersion matrix, Ci measured CO2 concentration at n heights distributed within and above the canopy, and CR is the concentration at a reference height. When Fg is unknown, the discrete form can be written as
where D0j is the coefficients for Fg. Both Dij and D0j can be solved from equations (3) and (5).
2.2. Profile for σw(z) and TL(z) in Neutral Atmospheric Stability
 To solve C(z) from S(z), or the inverse question, we need the profile for σw(z) and TL(z) within and above the canopy. On the basis of the study of Leuning et al. , the nondimensional profile of TLu*/hc and σw/u* in neutral atmospheric stability can be shown in Figure 1, where u* is the friction velocity, hc is the average canopy height. The lines shown in Figure 1 can mostly be written as [Raupach, 1987, 1989b; Leuning, 2000]
where a, b, d and θ are parameters given in Table 1. To ensure a smooth profile for σw/u*, the following function [Raupach, 1987, 1989b; Leuning, 2000[is used to describe the profile for 0 < z/hc < 0.8,
where x = z/hc; y = σw/u*.
Table 1. Variables and Parameters Used to Describe Normalized Profile of σw/u* and TLu*/hc, Given by Leuning 
z/hc − 0.8
2.3. Correction for the Effects of Stability on σw and TL
 The above profiles of σw/u* and TLu*/hc are appropriate for neutral atmospheric stability, but they can lead errors to estimating CO2 source/sink distribution and evapotranspiration using the inverse Lagrangian analysis, particularly when the atmosphere is stably stratified [Leuning et al., 2000; Leuning, 2000]. Leuning  used the stability function for heat and velocity to correct for the effects of stability on σw and TL using ζ as the stability parameter, which is the function of the Obukhov length. It is well known that the calculation of the Obukhov length requires eddy covariance measurement of sensible heat fluxes, which are always very small at night and errors in the measurements often cause the Obukhov length to have the wrong sign according to above-canopy temperature gradients. To overcome this problem, the gradient Richardson number Ri was used to take the place of ζ as the stability parameter. The correction for the effects of stability on σw and TL can be respectively written as
where σw (0) and TL(0) are given by equations (7) and (8) for neutral conditions, σw (Ri) and TL(Ri) are vertical velocity standard deviation and Lagrangian timescale as functions of the stability parameter Ri, and Ri is calculated from potential temperatures and wind speeds measured at two heights above the canopy using
in which zg = is the geometric mean height, with z2 > z1. Pruitt  state that the relationship between the stability function and Ri are
The friction velocity u* is also calculated by wind speeds measured at two heights above the canopy using
where k is von Karman constant, which is equal to 0.4. d is the zero-plane displacement for momentum, and approximated to 19.5 m [Liu et al., 1997]. ∂u/∂z is calculated by
 When the source/sink profile and Fg are known, the CO2 flux Fc can be calculated using
3. Materials and Methods
3.1. Study Site
 Measurements were carried on in a broadleaved Koreanpine forest near Changbai Mountain Research Station of Forest Ecosystem of Chinese Ecosystem Research Network (CERN), Chinese Academy of Sciences (41°31′49″∼42°25′18″N, 127°42′55″∼128°05′45″E), from 17 to 22 June 2003. The dominated species are Pinus koraiensis, Tilia amurensis, Quercus mongolica, Fraxinus mandshurica and Acermono etc. The mean forest canopy height hc is 26 m, the soil type of this area is upland dark brown forest soil, and the average slope of the study site is about 2–4%. The main wind direction during the study period is SW, with a fetch more than 1.5 km. According to the relationship of a sensor height to upwind fetch (1:100) [Dellwik and Jensen, 2005], this site is considered to be an ideal place to carry out studies of mass and energy exchange between forest and atmosphere through micrometeorological methods. All measurement equipment used in the study is mounted on an observation tower (42°24′9″N, 128°05′45″E, 761 m altitude), which is 61.8 m high.
3.2. Measurement of Wind Speed, Temperature, PAR and CO2 Concentration Profiles
 Wind speeds were measured using A100R sensors (Vector Instruments) located at two heights above the canopy, 49.8 and 61.8 m. Temperature was observed using HMP45C sensors (Vaisala), which were housed in 41002 radiations shield (Vaisala), at seven heights within and above the canopy: 2, 8.0, 22.0, 25.5, 32.0, 49.8 and 61.8 m. PAR was observed using LI190SB (LI-COR, Inc.) at 32 m. The measurements of mean CO2 profiles were carried out using a LI820 profiling system (LI-COR, Inc.) at seven levels (2, 8.0, 22.0, 25.5, 32.0, 49.8 and 61.8 m) above the ground surface, sequentially. All sensors were calibrated by Campbell scientific Inc. in time, and data were collected using data logger (CR23X, Campbell scientific Inc.). Each sensor measured every 2 s, and the average values were calculated by the data logger half hour.
3.3. Measurement of CO2 Flux
 An open-path eddy covariance measurement system, which consists of a data logger (CR5000, Campbell Scientific, Inc.) and three dimensional sonic anemometer (CSAT3, Campbell Scientific, Inc.), and an open path infrared gas analyzer (LI-7500, LI-COR, Inc.), is mounted on the observation tower at 40.0 m. While the analog signals from CSAT3 and LI-7500 and were sampled at 10 Hz using CR5000, the CO2 flux was calculated by eddy covariance technique with the 30 min averaging time. Density correction was done according to Webb et al. . The coordination rotation was done by Wu et al. .
3.4. Calculation Methods
 The data of wind speed, air temperature on two height (49.8 and 61.8 m) were used to calculate Ri and u* for each time on the basis of equations (11), (13) and (14). Leuning  pointed out that Ri should be in range [−2, 0.175], so an upper limit of Ri = 0.175 was used to take place of the calculated value when it is lager than 0.175, and a lower limit of Ri = −2 was used when the calculated value was lower than −2.
 Seven levels CO2 concentration data (2, 8.0, 22.0, 25.5, 32.0, 49.8 and 61.8 m) were used to calculate the CO2 source/sink. In order to gain a more smooth profile of the CO2 concentration, 1-D spline interpolation methods was used to calculate the concentration from 3.0 m to 62 m. On the basis of the calculated profile, 60 values were picked up for the calculation. So combining equations (3) and (5) with equations (7), (8), (9), (10), and (12), D0ij and Dij can be calculated. The CO2 source/sinks and Fg were solved from the nonlinear equations, presented by equation (6b), by a least squares method. The CO2 flux is also calculated by equation (15).
4. Results and Discussion
4.1. Comparison With Eddy Covariance Measurements
 Since it was not possible to test the simulated production of the analysis layer by layer, it was possible to compare its cumulative CO2 flux above the canopy with the independent eddy correlation measurements of height at 42 m. Figure 2 shows both CO2 fluxes in 6 days. Both the calculated and measured results have marked daily variation. The time series of the calculated fluxes closely followed those of the eddy correlation measurements. Cumulative fluxes of CO2 at above the canopy derived from the inverse Lagrangian analysis were in close agreement with eddy covariance measurements during daylight hours on all days shown in Figure 2. However, nocturnal fluxes from the inverse Lagrangian analysis were two to three times larger than those measured directly, especially on 22 June, which is same as the results reported by Leuning . The reasons for the apparent overestimation of the CO2 flux by the inverse Lagrangian analysis are still not clear. On the basis of the statistical calculations, as shown in Figure 3, the result of CO2 estimated by model is correlated to that of Eddy Covariance with the slope equaling to 1.0685, R2 equaling to 0.89, F test (P < 0.001).
4.2. CO2 Concentration Profiles and Source/Sink Strength
4.2.1. CO2 Concentration Profiles
Figure 4 shows the representative CO2 concentration profiles for one day in the study, 18 June 2003. The daily variation is large. The minimum values occur after midday, when the surface layer is well coupled to the atmospheric boundary layer. Such values are representative of the tropospheric CO2 concentration. At night, as the depth of the atmospheric boundary layer descends and the direction of CO2 exchange changes from biospheric uptake to emission, concentrations become quite elevated. So the typical feature in the profiles, as Figure 4 showed, is high concentration exists near ground at the early morning and night, which indicates a region of strong CO2 production there, and appears strong gradients; while low concentration exists at 95% canopy height during the midday, which indicates a region of CO2 sink there.
4.2.2. Source/Sink Distributions Within and Above the Canopy
Figure 5 shows the distribution of CO2 source/sink within and above the canopy, calculated by the inverse Lagrangian dispersion analysis, for the CO2 concentration profiles in Figure 4. It is evident that both the soil and canopy contributed to the total respiratory flux until 0500 local time (LT), and the soil was a net CO2 source all day long, with strong CO2 production due to the soil respiration. During most of the day, there is a second CO2 source belt at near 13 m, which is due to the stem and the lowest canopy layer respiration, but the strength is weak. Foliage in the top layer (about 90% canopy height) changed from a CO2 source in the morning (about 0400 LT or earlier) to a CO2 sink in the evening (2200 LT or later.). During the nighttime and early morning hours the simulated CO2 source/sink from 13 m to 22 m is positive, suggesting that stem and plant respiration are still the dominating source contributors vis à vis foliage CO2 simulation. The main sink belt exists at just at the average canopy height. According to the measured photosynthetically active radiation (PAR), the sink strength varied with the photosynthesis of the foliage (0600∼1800 LT), and reached to the maximum between 1000∼1300 LT.
 The Lagrangian model, with concentration measurements at only four levels inside the canopy (2, 8.0, 22.0, 25.5 m), correctly predicted the respiration dominance from 0000 to 0600 LT and from 1800 to 2400 LT. In fact, on the basis of the simulated source-sink profiles in Figure 5, much of the soil-plant respiration component is concentrated in the lower 3 m of the canopy; however, the influence of this respiration on the net vertical atmospheric CO2 fluxes persisted up to 25 m in the early morning hours (Figure 6).
Figure 7 shows the variation of modeled CO2 flux and PAR, with a higher respiration during the night and a higher photosynthetic uptake during the day. There is a relative low PAR and Fc near 1200 LT compared with the value at 1100 and 1400 LT, it is due to the energy balance of sunlit leaves often results in surface temperatures exceeding air temperatures, this situation promotes dark respiration [Amthor, 1994], decreases the solubility of CO2 relative to O2, thereby restraining the maximal rate of photosynthesis attained by sunlit leaves [Farquhar et al., 1980]. The relationship of Fc and PAR (Figure 8) reveals a markedly nonlinear response between them. So, there also present an obviously character that maximum Fc can also occur for some range approaching the maximum PAR.
 The inverse Lagrangian dispersion analysis for scalar transport that infers the source/sink and flux profiles from measured scalar concentration profiles was applied for CO2 within and above a broadleaved Koreanpine forest. After the comparison of the estimated results with measured ones by the open-path eddy covariance measurement system, we demonstrated the following:
 1. The inverse Lagrangian dispersion analysis method, using the gradient measurement of microclimate parameters as the input of the model, produced very well agreement between modeled and measured CO2 flux from 17 to 22 June 2003. The average precision is 89%.
 2. There is an obvious CO2 source belt at about 13 m within the canopy which is due to the stem and the lowest canopy layer respiration, but the strength is weak.
 3. The model over estimate nocturnal CO2 flux about two to three times larger than those measured results.
 4. The relationship of Fc and PAR reveals a marked nonlinear response between them.
 This work was funded by the National Natural Science Foundation of China project 30270280, the Foundation of Knowledge Innovation Program of Chinese Academy of Sciences YCXZY0203, the Foundation of Knowledge Innovation Program of Chinese Academy of Sciences KZCX1-SW-01-01A, and the National Natural Science Foundation of China project 30370293.