Flux and concentration footprint modelling: State of the art

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Environmental Pollution 152 (2008) 653e666 www.elsevier.com/locate/envpol

Flux and concentration footprint modelling: State of the art ¨ . Rannik a, J. Rinne a, A. Sogachev a, T. Markkanen a,c, T. Vesala a,*, N. Kljun b, U K. Sabelfeld d,e, Th. Foken c, M.Y. Leclerc f a

Department of Physical Sciences, University of Helsinki, PO Box 64, FIN-00014 Helsinki, Finland Institute for Atmospheric and Climate Science, ETH Zurich, Universita¨tsstrasse 16, CH-8092 Zurich, Switzerland c Department of Micrometeorology, University of Bayreuth, D-95440 Bayreuth, Germany d Institute of Computational Mathematics and Mathematical Geophysics, Russian Academy of Sciences, Novosibirsk, Russia e c/o Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D-10117 Berlin, Germany f Laboratory for Environmental Physics, The University of Georgia, 1109 Experiment Street, Griffin, GA 30223, USA b

Received 29 May 2006; received in revised form 25 May 2007; accepted 29 June 2007

The estimation of footprints for the real world with a multitude of interesting gaseous and particulate substances remains a complex problem. Abstract Since early 1990s, the development of footprint models has been rapid with presently four different approaches being available: (i) analytical models, (ii) Lagrangian stochastic particle dispersion models, (iii) large-eddy simulations, and (iv) closure models. Parameterizations of some of these approaches have been developed, simplifying the original algorithms for use in practical applications. The paper provides a review of the footprint modelling. It also discusses our present understanding of the theoretical background, the most successful modelling approaches, as well as the usage and benefits of the footprint concept as it relates to flux measurements. There has recently been a trend emerging in modelling the behavior of the footprint functions using a less idealized, more realistic description of inhomogeneities, vegetation structure and topography, ultimately for reactive compounds. The estimation of footprints for application in the real world, complete with a multitude of interesting gaseous and particulate substances, remains a complex problem. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Footprint; Source area; Turbulent flux; Atmospheric turbulence; Atmospheric pollutants

1. Introduction There has been an explosion of direct flux measurements’ sites since the 1990s, with the advent of robust, affordable high-quality instrumentation and the implementation and widespread use of the eddy-covariance (EC) technique. The most direct and most common technique to measure trace gas fluxes is based on the eddy-covariance technique. The commercial availability of high frequency-response gas analyzers for various atmospheric compounds in particular has

* Corresponding author. Tel.: þ358 9 191 50862; fax: þ358 9 191 50717. E-mail address: [email protected] (T. Vesala). 0269-7491/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.envpol.2007.06.070

significantly improved. The EC technique facilitates direct turbulent flux measurements affecting marginally the natural gas transfer between the surface and air. It also provides a tool to estimate exchange over larger areas than, for example, a single measuring chamber. However, whereas a chamber confines a known source area for its measurement, the area represented by EC measurements is a complex function of the observation level, surface roughness length and canopy structure together with meteorological conditions (wind speed and direction, turbulence intensity and atmospheric stability). Most often, fluxes are calculated using 0.5e1 h time averages and different flux values represent different source areas, although the difference between the areas corresponding two successive values may be small. In the case of an inhomogeneous surface, knowledge of

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both the source area and strength is needed to interpret the measured signal. Only over an extended homogeneous surface, the position of the sensor is not an issue as long as it is located within the constant flux layer. The footprint defines the field of view of the flux/concentration sensor and reflects the influence of the surface on the measured turbulent flux (or concentration). Strictly speaking, a source area is the fraction of the surface (mostly upwind) containing effective sources and sinks contributing to a measurement point (see Kljun et al., 2002). The footprint is then defined as the relative contribution from each element of the surface area source/sink to the measured vertical flux or concentration (see Schuepp et al., 1990; Leclerc and Thurtell, 1990). Functions describing the relationship between the spatial distribution of surface source/sink and a signal are called the footprint function or the source weight function (see Schmid, 1994). The fundamental definition of the footprint function f is given by the integral equation of diffusion (Wilson and Swaters, 1991; see also Pasquill and Smith, 1983). h¼

Z

    f x;b x Q b x db x;

ð1Þ

x

where h is the quantity being measured at location x (note that x is a vector) and Qðb xÞ is the source emission rate/sink strength in the surface-vegetation volume c. h can be the concentration or the vertical eddy flux and f is then concentration or flux footprint function, respectively. The determination of the footprint function f is not a straightforward task and several theoretical approaches have been derived over the previous decades. They can be classified into four categories: (i) analytical models, (ii) Lagrangian stochastic particle dispersion models, (iii) largeeddy simulations, and (iv) ensemble-averaged closure models. Additionally, parameterizations of some of these approaches have been developed, simplifying the original algorithms for use in practical applications (e.g., Horst and Weil, 1992, 1994; Schmid, 1994; Hsieh et al., 2000; Kljun et al., 2004a). The criterion of a 100:1 fetch to measurement height ratio was long held as the golden rule guiding internal boundary layer (IBL) estimation and nowadays is still used as a ruleof-thumb to crudely approximate the source area of flux measurements over short canopies in daytime conditions. However, the unsatisfactory nature of the 100:1 ratio and the related footprint predictions were explicitly discussed 15 years ago by Leclerc and Thurtell (1990). This study was also the first application of the Lagrangian stochastic approach, with stability effects included in footprint estimation. Accordingly, simple analytical models, such as the one by Schuepp et al. (1990), have become widely used and integrated into the eddy-covariance software. With the addition of realistic velocity profiles and stability dependence, the advent and level of sophistication of Horst and Weil’s (1992, 1994) analytical solution further expanded the scope of this approach. Whereas their 1992 model can be applied only numerically, the 1994 model provides an approximate analytical solution. The latter,

later made two-dimensional by Schmid (1997), has been widely used, thanks to the additional insight provided to experiments over patchy surfaces. For the first time in micrometeorology, determining an area of influence on measurements using physically based criteria, instead of using empirical criteria, became feasible. Further development to footprint analysis was given by Leclerc et al. (1997) who applied more complicated numerical LES model to the footprint problem. Somewhat surprisingly, the older numerical modelling approach for turbulent BL, the closure model, was first discussed related to footprint estimation by Sogachev et al. (2002). A thorough overview over the development of the footprint concept is given in (Schmid, 2002) with Foken and Leclerc (2004) providing more recent information on the subject. Table A1 lists the most important studies on footprint modelling. The objective of this paper is to provide a review of the status of the footprint concept. We mainly focus on most recent approaches in modelling flux footprints (approaches (ii)e(iv)) and discuss most of the recent developments since 2002, when a thorough review article by Schmid (2002) was published. Rather than providing an exhaustive list of the most recent articles, this paper highlights the present understanding of the theoretical background, the most successful modelling methods and the usage of footprint concepts in quality assessment of surface flux measurements (see also Vesala et al., 2004). Lagrangian simulations are discussed in more detail since they are most commonly used. The development of footprint calculations for reactive gases and aerosol particles is still in its infancy, which is addressed. Finally, possible future research directions are discussed. 2. Background The footprint problem essentially deals with the calculation of the relative contribution to the mean concentration or flux at a fixed point in the presence of an arbitrary given source of a compound. To treat this problem numerically, two fundamentally different approaches are used: (1) conventional deterministic methods based on semi-empirical turbulent diffusion equation and closure assumptions (e.g. Schmid, 1994), or other filtration models, like large-eddy simulations, and (2) stochastic models based either on simulation of the Eulerian random velocity field, or stochastic Lagrangian models based on Lagrangian trajectory simulations. The deterministic approach deals directly with the equation governing the mean concentration but is restricted by the use of the Boussinesq hypothesis whose limitations should be additionally studied (e.g., Bysova et al., 1991). For instance, this hypothesis cannot be true if the concentration is calculated close to sources (Monin and Yaglom, 1975). More generally, high-order closure methods are developed, but different closure hypotheses also should be made (Monin and Yaglom, 1975). Stochastic models do not require any closure hypotheses, but the difficult problem is to construct an adequate parameterization of the Eulerian velocity field, or, as in the case of the Lagrangian approach, the stochastic Langevin equation whose trajectories have the desired turbulence characteristics.

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The source area naturally depends on measurement height and wind direction. The footprint is also sensitive to both atmospheric stability and surface roughness, as first pointed out by Leclerc and Thurtell (1990). This stability dependence has been further investigated by Kljun et al. (2002), comparing crosswind-integrated footprints predicted by a three-dimensional Lagrangian simulation (Fig. 1a). Here, the measurement height and roughness length were fixed to 50 and 0.05 m, respectively, whereas the friction velocity, vertical velocity scale, Obukhov length and boundary layer height were varied to represent strongly and forced convective, neutral and stable conditions. It can be seen that the peak location is closer to the receptor and less skewed in the upstream direction with increasingly convective conditions. In unstable conditions, the turbulence intensity is high, resulting in the upward transport of any compound and a shorter travel distance/time. Typically, the location of the footprint peak ranges from a few times the measurement height (unstable) to a few dozen times (stable). Note also the small contribution of the downwind turbulent diffusion in convective cases. Concentration footprints tend to be longer (Fig. 1b). In terms of the Lagrangian framework, this can be explained as follows. The flux footprint value over a horizontal area element is proportional to the difference of the numbers or particles (passive tracers) crossing the

655

measurement level in the upward and downward directions. Far from the measurement point, the number of upward and downward crossings of imaginary particles or fluid elements across an imaginary xey plane typically tends to be about the same and thus the up- and downward movements are counterbalanced decreasing the respective fractional flux contribution of those source elements to the flux system. In contrast to the flux footprint, each crossing contributes positively to the concentration footprint not depending on the direction of the trajectory. This increases the footprint value at distances further apart from the receptor location. In the lateral direction, the stability influences footprints in a similar fashion (Fig. 2). Mathematically, the surface area of influence on the entire flux goes to infinity and thus one must always define the %-level for the source area (see Schmid, 1994). Often 50, 75 or 90%-source areas contributing to a point flux measurement are considered. Corresponding cases for the 50%level source areas presented in Figs. 1 and 2 are shown in Fig. 3. The spatial extent encompassed by a concentration

(a)

(a)

(b) (b)

Fig. 1. (a) Crosswind-integrated footprint for flux and (b) concentration measurements for four different cases of stabilities (adapted from Kljun et al., 2002 with kind permission of Springer Science and Business Media). The locations of the respective peaks are indicated by vertical lines.

Fig. 2. (a) Three-dimensional flux footprint for the strongly convective and (b) the stable case as in Fig. 1 (adapted from Kljun et al., 2002 with kind permission of Springer Science and Business Media).

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measurement extends much further upwind than the flux footprint, especially in neutral and stable cases. Finally, one should mention the fundamental mathematical difference between concentration and flux footprint functions. From the definition (Eq. (1)), the footprint function for the concentration is a Green function of the turbulent diffusion problem, i.e., the concentration can be obtained via a superposition of solutions generated by a unit source of a passive scalar. f in Eq. (1) is the Green’s function of an advectione diffusion equation (Eulerian approach) or transition probability (Lagrangian approach), and in both cases, it is between 0 and 1 and integrates to unity over the volume (see Finnigan, 2004). In principle, it is possible to derive an analogous expression for the flux footprint function. But then it turns

(a)

out that the flux footprint is not the Green’s function of the flux transport equation and the expression with the relevant unit source involves some unclosed terms leading to additional closure assumptions (Finnigan, 2004). This may result in a complex behavior of the flux footprint function; in particular, it may be even negative for a complex, convergent flow over a hill (Finnigan, 2004). In a horizontally homogeneous shear flow, the flux footprint ff does satisfy 1 > ff > 0. The vertical distribution of the source/sink can also lead to an anomalous behavior (e.g., Markkanen et al., 2003). Then, the flux footprint represents in fact a combined footprint function that is a source strength-weighted average of the footprints of individual layers. Because of the principle of superposition, the combined function may become negative if one or more of the layers have a source strength that is opposite in sign to the net flux between vegetation and atmosphere (Lee, 2003). The combined function is not anymore a footprint function in the sense of Eq. (1) and we suggest that it would be called (normalized ) flux contribution function (see also Markkanen et al., 2003). 3. Recent footprint model development 3.1. Lagrangian stochastic trajectory approach The stochastic Lagrangian models describe the diffusion of a scalar by means of a stochastic differential equation, a generalized Langevin equation, dXðtÞ ¼ VðtÞdt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dVðtÞ ¼ aðt; XðtÞ; VðtÞÞdt þ C0 3ðXðtÞ; tÞdWðtÞ;

(b)

Fig. 3. (a) 50%-level source area for flux and (b) concentration measurements for four different cases of stabilities as in Fig. 1 (adapted from Kljun et al., 2002 with kind permission of Springer Science and Business Media). The square indicates the receptor location.

where X(t) and V(t) denote trajectory coordinates and velocity as a function of time t, C0 is the Kolmogorov constant, 3 is the mean dissipation rate of turbulent kinetic energy and W(t) describes the three-dimensional Wiener process. This equation determines the evolution of a Lagrangian particle in space and time by combining the evolution of trajectory as a sum of deterministic drift a and random terms. The drift term is to be specified for each specific model (Thomson, 1987). The Lagrangian stochastic approach can be applied to any turbulence regime, thus allowing footprint calculations for various atmospheric boundary layer flow regimes. For example, in the convective boundary layer, turbulence statistics are strongly non-Gaussian and for realistic dispersion simulation a non-Gaussian trajectory model has to be applied. However, most Lagrangian trajectory models fulfill the main criterion for construction of Lagrangian stochastic models, the wellmixed condition (Thomson, 1987), for only one given turbulence regime. As one of a few, Kljun et al. (2002) presented a footprint model based on a trajectory model for a wide range of atmospheric boundary layer stratification conditions. For overview of Lagrangian trajectory models for different flow types, including non-Gaussian turbulence, see Wilson and Sawford (1996). Lagrangian trajectory simulation can be performed forward as well as backward in time (more details below).

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It should be noted, however, that the Lagrangian stochastic approach is rigorously justified only in the case of stationary isotropic turbulent flow. Even in the case of homogeneous but non-isotropic turbulence the justification problem remains unsolved; in particular there are several different stochastic models, which satisfy the well-mixed condition (Thomson, 1987; Sabelfeld and Kurbanmuradov, 1998). This is often called as the uniqueness problem (for details, see the discussion in Kurbanmuradov et al., 1999, 2001; Kurbanmuradov and Sabelfeld, 2000). The stochastic Lagrangian method is nevertheless very convenient in footprint application: once the form of the parameterization is chosen, the stochastic Langevin type equation is solved by a very simple scheme (e.g. Sawford, 1985; Thomson, 1987; Sabelfeld and Kurbanmuradov, 1990). The approach needs only a one-point probability density function (pdf) of the Eulerian velocity field, and is much more efficient in numerical calculations than the stochastic Eulerian approach. The Lagrangian stochastic trajectory model together with appropriate simulation methods and corresponding estimators for concentration or flux footprints are usually merged into a Lagrangian footprint model. For a detailed overview of the estimation of concentrations and fluxes by the Lagrangian stochastic method, the concentration and flux footprints in particular, see Kurbanmuradov et al. (2001). 3.1.1. Forward and backward models The conventional approach of using a Lagrangian model for footprint calculation is to release particles at the surface point source and track their trajectories downwind of this source towards the measurement location (e.g., Leclerc and Thurtell, 1990; Horst and Weil, 1992; Rannik et al., 2000). Particle trajectories and particle vertical velocities are sampled at the measurement height. In case of horizontally homogeneous and stationary turbulence, the mean concentration at the measurement location (x, y, z) due to a sustained surface source Q located at height z0 can be described as: C ¼ hcðx; y; zÞi ¼

ni N X  1X 1   Q x  Xij ; y  Yij ; z0 ; N i¼1 j¼1 wij 

where N is number of released particles and ni is the number of intersections of particle trajectory i with the measurement height z; wij, Xij and Yij denote the vertical velocity and the coordinates of particle i at the intersection moment, respectively. Similarly, the mean flux is given by: F ¼ hwðx; y; zÞcðx; y; zÞi ¼

ni N X  1X w   ij Q x  Xij ; y  Yij ; z0 : N i¼1 j¼1 wij 

The concentration footprint and the flux footprint can be determined as follows: fC ¼

1 v2 C ; Q vxvy

ð2Þ

fF ¼

657

1 v2 F : Q vxvy

ð3Þ

Alternatively, it is possible to calculate the trajectories of a Lagrangian model in a backward time frame (cf. Thomson, 1987; Flesch et al., 1995; Flesch, 1996; Kljun et al., 2002). In this case, the trajectories are initiated at the measurement point itself and tracked backward in time, with a negative time step, from the measurement point to any potential surface source. The particle touchdown locations and touchdown velocities are sampled and mean concentration and mean flux at the measurement location can be described as: C ¼ hcðx; y; zÞi ¼

ni N X  2X 1   Q Xij ; Yij ; z0   N i¼1 j¼1 wij

and F ¼ hwðx; y; zÞcðx; y; zÞi ¼

ni N X   2X w  i0Q Xij ; Yij ; z0 ;   N i¼1 j¼1 wij

where wi0 is the initial (release) vertical velocity of the particle i and wij is the particle touchdown velocity. Again, the concentration footprint and the flux footprint are determined using Eqs. (2) and (3). The above backward estimators for concentration and flux do not assume homogeneity and stationarity of the turbulence field. The calculated trajectories can be used directly without a coordinate transformation. Therefore, if inhomogeneous probability density functions of the particle velocities are applied, backward Lagrangian footprint models hold the potential to be applied efficiently over inhomogeneous terrain. 3.1.2. Advantages and disadvantages of Lagrangian models The benefits of Lagrangian models include their capability to consider both Gaussian and non-Gaussian turbulence. While flow within the surface layer is nearly Gaussian, non-Gaussianity characterizes flow fields of both canopy layer and convective mixed layer. Another benefit of the Lagrangian stochastic models over analytical ones is their applicability in near-field conditions, i.e. in conditions when fluxes of constituents are disconnected from their local gradients, providing thus proper description for within canopy dispersion. This makes it possible to locate trace gas sources/sinks within a canopy. Baldocchi (1997), Rannik et al. (2000, 2003), Finnigan (2004) and Mo¨lder et al. (2004) studied the qualitative effects of canopy turbulence on the footprint function. In the case of tall vegetation, the footprint prediction depends primarily on two factors: canopy turbulence and the source/sink levels inside the canopy. These factors become of particular relevance for observation levels close to the treetops (Rannik et al., 2000; Lee, 2003; Markkanen et al., 2003). The LS models are not uniquely defined in three-dimensional flow fields. In one dimension the solution is, however, unique. Also, the Lagrangian stochastic models require a pre-defined wind field. Thus, as a weakness of these models,

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the performance of Lagrangian stochastic models depends on how well they are constructed for a particular flow type and how good the description of wind statistics is. However, the description of wind statistics inside a canopy becomes uncertain due to poor understanding of stability dependence of the canopy flow as well as of Lagrangian correlation time. The wind statistics necessary for LS footprint simulations originate from similarity theory, experimental data or from an output from a flow model capable to produce wind statistics. Combined with closure model results, the LS approach has been applied to study the influence of transition in surface properties on the footprint function. The first attempt was done by Luhar and Rao (1994) and by Kurbanmuradov et al. (2003). They derived the turbulence field of the two-dimensional flow over a change in surface roughness using a closure model and performed Lagrangian simulations to evaluate the footprint functions. The LS approach was combined also with LES simulation of convective boundary layer turbulence to infer concentration footprints (Cai and Leclerc, 2007). In this case, the LS simulations were performed for subgrid-scale turbulent dispersion. Finally the long computing-times due to a large number of trajectories required for producing statistically reliable results is an unavoidable weakness of Lagrangian stochastic footprint models. To overcome this, Hsieh et al. (2000) proposed an analytical model derived from Lagrangian model results. More recently, a simple parameterization based on a Lagrangian footprint model was proposed by Kljun et al. (2004a). This parameterization allows the determination of the footprint from atmospheric variables that are usually measured during flux observation programs (see Appendix 1). There are features that characterize specifically either forward or backward models. The central benefit of backward models over their forward counterpart is their ability to estimate footprints for both inhomogeneous and non-stationary flow field, whereas applicability of forward models is restricted by the so called inverse plume assumption. Application of backward model, however, includes numerical details to be properly solved. The method can be sensitive under some conditions to trajectory initial velocities, which have to be drawn according to Eulerian joint pdf. This can be solved by a numerical spin-up procedure, as did Kljun et al. (2002). In addition, the backward estimator for surface flux footprint involves a numerically unstable sum of the inverse of interception velocities, producing unrealistic spikes at the areas where statistics are poor. This problem does not occur with forward estimators as only the information on the vertical transfer direction of each particle is used to resolve the flux footprint. 3.2. Large-eddy simulation approach The large-eddy simulation (LES) approach is free of the drawback of a pre-defined turbulence field and, using NaviereStokes equations, resolves the large eddies while parameterizing subgrid-scale processes; this approach pre-supposes that most of the flux is contained in the large eddies: since these are directly resolved, this method provides a high-level

of realism to the flow despite complex boundary conditions (e.g. Hadfield, 1994). This technique is considered the technique of choice for many cases not ordinarily studied using simpler models and can include the effect of pressure gradient. However, it is computationally even more expensive than the Lagrangian approach. Furthermore it is limited to relatively simple flow conditions by the number of grid points in flow simulations. Nevertheless, the method has been applied to simulate footprints in the convective boundary layer (Leclerc et al., 1997) and to model the turbulence structure inside forest canopies (Shen and Leclerc, 1997). The LES approach shows potential for application in future studies, and is ideally suited to tackle footprint descriptions in inhomogeneous conditions. The LES approach provides a valuable ‘data set’, against which simpler footprint models can be verified. In principle, the LES approach can also cope with horizontal inhomogeneity as shown in Shen and Leclerc (1994, 1995), but is limited in practice by computational demands. The first LES arose from the studies of Moeng and Wyngaard (1988). The large-eddy simulation is a sophisticated model, which directly computes the three-dimensional, time-dependent turbulence motions with scales equal to or greater than twice the grid size, and only models the subgrid-scale motions (SGS). Typically, it predicts the three-dimensional velocity fields, pressure and turbulent kinetic energy. Depending on the purpose, it can also simulate the turbulent transport of moisture, carbon dioxide, and pollutants. There are several parameterizations available in treating the subgrid scales. One of the most widely used simulations is that originally developed by Moeng (1984) and Moeng and Wyngaard (1988) and modified by Leclerc et al. (1997), Su et al. (1998), and by Patton et al. (2001) and Vila`-Guerau de Arellano et al. (2005) for adaptation to include canopy and boundary layer scalar transport. Often, the SGS is parameterized using the 1.5 order of closure scheme. Depending on the large-eddy simulation used, the LES can contain a set of cloud microphysical equations and thermodynamic equations, and can predict the temperature, mixing ratios and pressure. Some LES also include terrain-following coordinate system. A spatial cross-average and temporal average is applied to the simulated data once the simulation has reached quasi steady-state equilibrium. Typical boundary conditions are periodic with a rigid lid applied to the top of the domain so that gravity waves are absorbed and reflection of the domain from the upper portion of the domain decreased. Recently, LES studies have been applied to canopy turbulence and have been shown to reproduce many observed characteristics of airflow within and immediately above a plant canopy, including skewness, coherent structures and two-point statistics. Concentrations and flux footprints have been studied using the LES, by examining the behavior of tracers released from multiple sources inside a forest canopy. 3.3. Closure model approach Another way to solve the NaviereStokes equations (NS), beside LES, is to apply ensemble-averaged NS and to use

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some empirical information to close the set of equations. This closure model approach is potentially an effective tool for estimating footprints over heterogeneous and hilly surfaces. It thus is surprising that its first practical applications appeared just recently. Presenting the atmospheric boundary layer SCADIS (Scalar Distribution) model based on 1.5 closure, Sogachev et al. (2002) demonstrated flux footprint estimation as one of the several possible model applications. Sogachev and Lloyd (2004) described in detail the footprint modelling technique based on the calculation of the individual contribution from each surface cell to a vertical flux at a receptor point. This is carried out by means of a comparative analysis of vertical flux fields formed by source cells consecutively activated. The fields are then normalized, yielding a contribution from each cell to the flux (flux contribution function). The actual footprint function can be constructed from the contribution footprint by weighting it according to known source strengths in each cell. Footprint functions modelled by SCADIS were compared with footprints derived from both analytical and Lagrangian stochastic approaches for condition of uniform surface (e.g. Schuepp et al., 1990; Leclerc and Thurtell, 1990; Kormann and Meixner, 2001). The best agreement was obtained in neutral conditions. A comparison of the results suggested that the closure method

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is suitable for practical applications and it has been effectively used also for inhomogeneous surfaces (Sogachev et al., 2004). Fig. 4a illustrates a complex case where the forest (canopy height h) is disrupted by clearings. The sensor was located at the h-normalized distance (20) from the clearing edge and footprints were calculated for three measurement heights using the SCADIS model. The wind was from the left. Both the enhanced turbulence exchange over the clearing due to advection of the turbulent kinetic energy from the forest canopy and the upward airflow on the downwind side of the clearing enhanced the contribution of the open surface. If the forcings of the clearing and the stand have same signs, a cumulative flux can exceed a unity behind disturbance (Fig. 4b). Klaassen et al. (2002) reported a similar situation for heat fluxes behind a forest edge. Fig. 4c, d shows the impact of the horizontal distance from the edge to the footprint. Here, the receptor was fixed at the h-normalized height of 1.5. The inputs required for the model calculations are of two different types. The first group includes the characteristics of the area over which the calculations are performed. These include spatial coordinates and the vertical and horizontal characteristics of the vegetation (structure, photosynthetic characteristics, etc.). The second group consists of meteorological

Fig. 4. Examples of footprints (a, c) and cumulative fluxes (b, d) derived by the SCADIS model (Sogachev et al., 2002) for sensor located at different distances (x/h) from a clearing and at different heights (z/h) above a forest. Vegetation height (h) is 15 m and Leaf Area Index is 3 m2 m2. Dotted line (a) encloses the forest boundary.

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parameters at any given point in time (radiation conditions and vertical profiles of wind speed, temperature, moisture and passive scalars). The model is adjusted to use information regarding wind speed and scalars obtained from synoptic levels and after that predict their distribution within the atmospheric boundary layer. Relative simple input information (usually collected at the site of interest or provided by large scale models) and low computing cost with limited amount of constants in closure equations make SCADIS attractive for practical applications. The approach has also been applied for estimating footprints for existing flux measurement sites in Tver region in European Russia (Sogachev and Lloyd, 2004) and Hyytia¨la¨ in Finland (Sogachev et al., 2004). Besides surface heterogeneities, the latter study also included considerations of local topography, which can be implemented in the SCADIS model. Vertical fluxes and footprint behavior over a few simplified landscape types were investigated by Sogachev et al. (2005a). The behavior of both scalar fluxes and flux footprints near a forest edge were investigated in detail for the Florida AmeriFlux site (Sogachev et al., 2005b) and Bankenbosch forest in the Netherlands (Klaassen and Sogachev, 2006). In Sogachev et al. (2005b) additional proofs of creditability of the closure approach were given by comparison of footprints predicted by SCADIS and two different LS models (Thomson, 1987; Kurbanmuradov and Sabelfeld, 2000). In summary, Sogachev and Lloyd’s (2004) approach on the usage of the closure model for footprint calculations is basically simple in its realization (activation of surface elements) and it could be implemented in other ensemble-averaged boundary layer models, regardless of their order of the closure. Although computing costs may be high in three-dimensional calculations and some uncertainties may remain for very low measurement heights, closure models can precisely link sources and observed fluxes under condition of vegetation heterogeneities and hilly terrain. 4. Validation of footprint models Validations of footprint models are often only a comparison of different footprint models. According to Foken and Leclerc (2004) only a few experimental data sets of tracer experiments are available for validation purposes. While analytical footprint predictions were often evaluated using results of Lagrangian footprint models, there is no such simple possibility for the evaluation of Lagrangian footprint models. LS footprint models consist of a dispersion model and an estimation scheme for the footprint function. LS dispersion models have been tested against dispersion experiments in numerous cases for different turbulence regimes (e.g. Thomson, 1987; Reynolds, 1998; Kurbanmuradov and Sabelfeld, 2000; Kljun et al., 2002) and therefore the ability of LS models to reproduce dispersion statistics is well established. Only a few footprint results from Lagrangian models were compared with experimental data: Mo¨lder et al. (2004) dealt with the validation of footprint models as a way to compare both analytical and Lagrangian models against experimental data. Before

this study, Leclerc et al. (1997) compared LES and Lagrangian simulations against tracer flux data and Finn et al. (1996) considered analytical and Lagrangian models over a canopy of short roughness. Kljun et al. (2004b) compared forward and backward Lagrangian models against data from tracer release experiments in a wind tunnel. Such independent comparisons between models and experimental validation help the assessment of model sensitivity. In general, the investigated footprint models agree well with the tracer experiment. Even though these comparisons gave promising results, there is still a need for further experimental data allowing for validation of footprint models, the LS models in particular. Foken and Leclerc (2004) pointed out that complex validation experiments are expensive, and hence prohibitive for the vast majority of university researchers. Nevertheless, the authors show that ongoing experiments can also be used to validate footprint models, when two or more well defined and neighboring surfaces with significantly different fluxes can be studied. These issues are important if footprint models are to be used as a tool to define experimental requirements and validate experimental data. An application of this method was successfully made by Go¨ckede et al. (2005) with two flux stations over bare soil and a meadow. A third flux station with a footprint area covering both surfaces was used to validate the footprint model, because the contributions of both surfaces changed with the stability and the wind velocity. Earlier investigations used a similar approach: Soegaard et al. (2003) operated five ground-level EC systems over five different crop fields together with a sixth set-up on top of a higher mast to enable landscape-wide flux measurements. The agreement between high-level values and those integrated from groundlevel using a re-formulated version of the models of Gash (1986) and Schuepp et al. (1990) was good. Hsieh et al. (2000) developed an analytical model based on Lagrangian dispersion model and dimensional analysis. They found a good agreement with model predictions and measured fluxes over a transect from a desert to an irrigated potato field. More recently, Marcolla and Cescatti (2005) compared three analytical footprint models over a meadow with different surface characteristics and one of the models (Schuepp et al., 1990) overestimated the footprint. 5. Application of footprint models for quality assessment of flux measurements Since experiments are mostly performed in a heterogeneous landscape and analytical footprint models use parameters designed largely for use over homogeneous surfaces, the input parameters (like roughness length or Obukhov length) must be averaged over a heterogeneous terrain. However, averaging of input parameters remains, at best, only a first-order approximation, since flux aggregation is a better averaging method than parameter aggregation (Avissar and Pielke, 1989). One way of using footprint models is to limit their application to cases where all underlying assumptions are fulfilled, but this is an unrealistic scenario for most experimental research and thus the models are often utilized in conditions not fully in

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agreement with the basic premises of such models. Fortunately, related uncertainties may be dampened down because analytical models are quite robust and provide realistic results even under real, non-ideal conditions working well within a limited range of possible roughness length and stabilities (Go¨ckede et al., 2004). To overcome these deficits Go¨ckede et al. (2006) revised their approach using the Lagrangian model by Rannik et al. (2003) and included a flux aggregation technique by means of averaged roughness lengths (Hasager and Jensen, 1999). The benefit of this approach was to provide a better estimation of the footprint from areas near the tower and in larger distances (Leclerc et al., 2003a,b). The necessary land use information can be taken from topographical maps or better remote sensing products (Reithmaier et al., 2006; Kim et al., 2006). Footprint climatology (Amiro, 1998) can also be combined with a data quality protocol of flux measurements (Foken and Wichura, 1996; Foken et al., 2004). This combination was applied by Rebmann et al. (2005) using the approach by Go¨ckede et al. (2004) on the evaluation of the carbon dioxide flux towers of the CarboEurope network. Footprint climatology can be also a valuable tool in comparison of different flux measurement techniques. Reth et al. (2005) compared chamber and eddy-covariance measurements. They found a better comparison result (50% error) as usually reported in the literature, if only the chambers within the exact footprint area were compared with the flux measurements. 6. Footprints of reactive gases and aerosol particles Recently, the effect of the chemistry on the fluxes measured above vegetation canopies of non-inert trace gases has gained attention (Strong et al., 2004; Meixner et al., 2005; Rinne et al., 2007). The chemical degradation of any compound emitted from the surface is likely to reduce the upward flux measured some distance above the ground and alter the footprint function as well. The matter is further complicated by the fact that measurements of such chemically active compounds as isoprene or monoterpenes are commonly conducted above deep vegetation canopies. Two approaches have been applied to study the effect of chemistry on measured fluxes. There exist several large-eddy simulations’ studies (e.g. Patton et al., 2001; Vinuesa and Vila`-Guerau de Arellano, 2003; Vila`-Guerau de Arellano et al., 2004), and few Lagrangian model studies (Strong et al., 2004; Rinne et al., 2007). Strong et al. (2004) have examined the effect of chemistry on isoprene fluxes measured above a deciduous forest. The chemistry was calculated using a first-order differential equation, resulting in an exponential decay function. This kind of function can be written as: cm ðtÞ ¼ cm ð0Þexp

X i

! km;i ci t

 t ¼ cm ð0Þexp  ; t

ð4Þ

where cm(t) is the concentration of the measured compound at the time t after the release of the air parcel, cm(0) the

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concentration at the time of the release of the parcel, km,i is the reaction constant between the measured compound and the compound i, ci is the concentration of the reactant compound, and t is the chemical lifetime of the compound, defined as: !1 X t¼ km;i ci : ð5Þ i

We combine a simplified chemistry (constant chemical lifetime) scheme with a stochastic Lagrangian model by Markkanen et al. (2003) to study the effect of the chemical degradation, using the cumulative one-dimensional footprint functions such as shown in Fig. 5. We can see that, while the cumulative footprint of an inert trace gas approaches unity with increasing distance from the measurement point, the footprint of the chemically active species remains lower. The more reactive the compound, the more sensitive is the flux to the compound’s degradation. Also, the level of the emission affects the total flux measured as the air parcels originating below the canopy have on the average longer transport times and thus more time for chemical degradation. In the model used for the calculation of Fig. 5, as well as in the models by Strong et al. (2004) and Rinne et al. (2007), a constant chemical lifetime was assumed. However, during the transport below, within and above the canopy, air parcels are exposed to different levels of solar short-wave radiation and to different ambient concentrations of reactive gases. This leads to a varying lifetime according to the instantaneous location of air parcels. This further calls for model development with a more sophisticated embedded transportechemistry interaction. In addition to chemical reactions, new physical factors enter into transport processes of aerosol particles. Compared to reactive gases, particles can be called multi-reactive (see Suni et al., 2003) due to coagulation and phase transition processes. Gas molecules are perfectly carried by air movements, whereas aerosol particles have inertia and they do not necessarily follow the trajectories of the smallest eddies. In addition, their gravitational settling velocity may be significant. Particles may also undergo coagulation, which changes the number concentration. New particles may be formed by nucleation and subsequent condensational growth can make them large enough to be detected by sensor, or evaporation of preexisting particles can lower their size below a detection limit of the sensor. The observed concentration is thus changed by particle inertia, gravitational settling velocity, coagulation or phase transition processes. The effect of the particle inertia on its aerodynamic properties can be considered by means of the Stokes number (see e.g. Seinfeld and Pandis, 1998). St ¼

tD u0 D2p rp Cc u0 ¼ ; LD 18mLD

ð6Þ

where tD is the relaxation time of the particle, u0 is the characteristic fluid velocity and LD is the characteristic length associated with the flow of interest. The relaxation time depends

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Fig. 5. Cumulative footprints of inert (solid line) and chemically active trace gases with atmospheric lifetime of 1 h (dashed line) and 5 min (dash-dotted line). Two source heights are considered. The forest parameterization was derived for the Hyytia¨la¨ Scots pine flux site (Finland) and described by Markkanen et al. (2003). Measurement height is 22 m and canopy height is 14 m. Friction velocity is 1 m/s.

on the particle aerodynamic diameter Dp, particle density rp, the Cunningham correction factor Cc and the dynamic viscosity of the carrier gas (air) m. The Stokes number can be interpreted as the ratio of the particle stop distance to the length scale of the flow. A small number implies that the particle is able to adopt the fluid very quickly. We can make a simple order-of-magnitude estimate how large particles must be so that they would still follow the curved flow patterns associated to small eddies. For that we conservatively assume that u0 equals 1 m/s and LD is set at 1 mm (length scale of small eddy). For simplicity rp is set at 1000 kg/m3 and Cc can be assumed to be 1 (see Seinfeld and Pandis, 1998). When 20  106 N s m2 is used for m, the particle diameter Dp must be about 19 mm so that the Stokes number would be 1. A particle having the diameter of 19 mm is very large by atmospheric standards and the result means that practically the effect of particle inertia on transport by eddies can be probably ignored in most cases. If 19-mm particles follow paths of small eddies they follow even better the larger eddies. If we modified the values of other input parameters, the numerical result would naturally change but in any case, within reasonable range of parameter values, the qualitative conclusion would remain the same. The more detailed analysis of relative turbulent motion between the particle and the turbulent surroundings is provided by Walklate (1986). It was revealed that turbulent slip could become significant for water droplets greater than 40 mm in crop canopies, so that they would not follow the turbulent airflow of the surroundings.

The gravitational settling velocity of 10-mm-sized (spherical) particle is only 3.5 mm/s (Hinds, 1999). Thus, it can be ignored for small particles but should be considered for transport of large particles, like plant diaspores (e.g. Tackenberg, 2003). In Lagrangian dispersion models, the settling is included by modification of the Lagrangian time scale (Jarosz et al., 2004) TL tp ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    ; 1þ

bVs sw

2

ð7Þ

where b is the empirical parameter, Vs is the gravitational settling velocity and sw is the standard deviation of the vertical wind component. For 40-mm-sized particles, the correction is the order of 1%, when 3 is used for b and 1 m/s for sw. However, the authors are not informed on any studies related to footprints of large particles, although there are numerous articles on particle dispersion. One should be capable of implementing coagulation, formation and phase transition processes to at least Lagrangian and closure models. Challenges thus remain before we can have confidence in treating the motion of heavy particles or buoyant gases (Wilson and Sawford, 1996). 7. Discussion: the next generation of footprint models A researcher focusing more on the practical work of flux and concentration measurements and not willing to conduct complicated footprint simulations may hesitate between

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analytical models and more sophisticated approaches. It is known that there exists a satisfying correspondence between Lagrangian and analytical models (e.g., Kljun et al., 2003) when stability is moderate and a measurement height is less than about 0.1 times the boundary layer height. When atmospheric stability approaches free convection, in very stable cases or for measurement heights outside the surface layer, there are significant discrepancies between both approaches, i.e. large differences in the peak location and the shape of the footprint (e.g., Kljun et al., 2003). In stable conditions, footprints using analytical models tend to be larger (further upstream). In convective conditions, on the other hand, the analytical models are not capable to predict downwind contribution of sources/sinks to the flux. Furthermore, analytical models are often invalid for many of the observed conditions and the general statement is that analytical models should be taken with caution and are in most cases incorrect. Subsequently, another question to be posed could be: are there LES, Lagrangian or closure models freely available and how can one access and apply them? As always, Lagrangian simulations codes and executable closure models are available through contacting most investigators who distribute them freely. Furthermore, there exists also an analytical model based on a Lagrangian stochastic dispersion model (Hsieh et al., 2000), and Kljun et al. (2004a,b) proposed a simple parameterization based on a dimensional-analytic scaling approach applied for a thoroughly tested Lagrangian footprint model. These two approaches are much more straightforward to apply than full-scale models. Appendix 1 provides information on easy-to-use footprint estimation tools. That said, one must remember that the development of footprint models is ongoing and many of their uncertainties need to be considered in the near future. Footprint models assume turbulent conditions and their underlying assumptions are limited by a stability range from 1 < z/L < 1 (z: height; L: Obukhov length) and a friction velocity u*  0.1 m/s. It is assumed that there are no gravity, stationary or standing waves, drainage flow, coherent structures, low-level jets aloft, etc. In general, the estimation of nocturnal footprints is prone to errors and most of the models, especially analytical ones, do not work in very stable conditions. The large scale advection component e most often undetected and therefore not measured e to the measured turbulent flux originating well outside the footprint area can be important and bias fluxes by several hundreds of % (Leclerc et al., 2003a,b; Sogachev et al., 2005b). Thus, in general, the total footprint is the sum of the turbulent exchange between the immediate surrounding surface and the atmosphere with an advective component, which in the current footprint definitions is assumed to be negligible. Lee (2004) has presented a model for advection applicable to footprint investigations. Based on the current status of footprint studies, the main future research directions are:  As regard the dependence of canopy turbulence on forest structure and influence of stability on footprint functions over forest canopies, Lagrangian simulation studies have

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raised up the question of uncertainties related to forest structure (Rannik et al., 2003). Both measurements and modelling can improve our understanding of canopy turbulence and dispersion processes. Note that flux measurements are often carried out in the roughness sublayer (layer between the canopy top and inertial sublayer) where turbulence is enhanced and the footprint distribution is in general more contracted than that based on the inertial sublayer similarity functions (Lee, 2003).  Restrictions of both applicability and deviations from footprint estimates based on the assumption of horizontal homogeneity should be clarified and changes in surfaces’ properties should be determined for actual conditions. Furthermore, the interaction between the footprint and internal boundary layers (Horst, 2000) should prompt more discussion along with the blending height concept and the extension of footprint models above the surface layer.  Numerical studies of flow and scalar dispersion in complex measurement conditions should be performed to evaluate the influence of topography and canopy heterogeneity on flux measurements and ecosystem exchange estimation. Tools, like the LES, capable of dealing with complex flows should be developed and applied to actual measurement conditions. Here we see also the potential in the use of ensemble-averaged closure models, which have been largely neglected in footprint studies. Footprint models should allow for active (such as chemically reactive gases or aerosol particles) scalars and large (order of 10 mm or larger) particles. The matter was studied by Strong et al. (2004) and Rinne et al. (2007).  Many of the above-mentioned points (inhomogeneities with high roughness, topography and interest in reactive compounds) are more pronounced in an urban environment. As the number of urban flux sites continues to grow, there exists also a demand for development of models capable of tackling most essential features of urban environment. Upscaling fluxes at the landscape/urban area level is an area of high relevance for large urban areas. Kim et al. (2006) proposed a new, potentially useful upscaling scheme. Finally, one may ask: is footprint modelling more than just a theoretical exercise and has it improved our understanding of surfaceeatmosphere exchange processes? Many field campaigns have been redesigned in the light of footprint considerations as, for example, the campaign of Kaharabata et al. (1997) or the NASA FIFE campaign in the late eighties. The NASA ABLE 3-B experiment also was planned using footprint predictions and flux site inter-comparisons were made using footprint calculations in order to interpret spatial flux differences. Aircraft flux measurements revealed the potential for extrapolating surface exchange models developed for a given location to another location (Ogunjemiyo et al., 2003). Here, footprint analysis played a key role as in the study by Kustas et al. (2006), where the

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spatial variability of sensible and latent heat fluxes was investigated by remote sensing experiments and tower and aircraft-based observations. It was revealed that larger length scales contribute to moisture than to the sensible heat. Sarkar and Hobbs (2003) developed a footprint method for estimating odor fluxes from a landfill site. The micrometeorological suitability of a flux site could be evaluated a priori, using footprint calculations, rather than a posteriori identification, as suggested by Leclerc et al. (2003a). Footprint methodologies have been successfully used by the CarboEurope community for obtaining information about the footprint climatology, the percentage of the investigated area in the footprint and footprint dependent data quality (Rebmann et al., 2005). However, when detailed footprint climatology is used for analysis of experimental results, one must remember that the present footprint models should be used thoughtfully because of the underlying assumptions inherent to modelling. Acknowledgments Support from ACCENT-BIAFLUX, CarboEurope, Nordic Centre of Excellence NECC, REBECCA project by Helsinki Environment Research Centre HERC, Nessling Foundation and Academy of Finland is acknowledged. Hermanni Riikonen is thanked for technical help. Appendix 1. The most important studies on footprint modelling are listed and commented in Table A1. Besides the detailed studies listed in Table A1, easy-to-use footprint estimates for measurements over forest canopies are needed, e.g. similar to the parameterized model for ABL as Table A1 Overview about the most important footprint models (if no remark: analytical model) (adopted from Foken, 2006) Author

Remarks

Pasquill (1972) Gash (1986) Schuepp et al. (1990)

First model description, concept of effective fetch Neutral stratification, concept of cumulative fetch Use of source areas, but neutral stratification and averaged wind velocity Lagrangian footprint model

Leclerc and Thurtell (1990) Horst and Weil (1992) Schmid (1994, 1997) Leclerc et al. (1997) Baldocchi (1997) Rannik et al. (2003, 2000) Kormann and Meixner (2001) Kljun et al. (2002)

Sogachev and Lloyd (2004) Strong et al. (2004)

One-dimensional footprint model Separation of footprints for scalars and fluxes LES model for footprints Footprint model within forests Lagrangian model for forests Analytical model with exponential wind profile Three-dimensional Lagrangian model for various turbulence stratifications with backward trajectories Boundary layer model with 1.5 order closure Footprint model with reactive chemical compounds

presented by Hsieh et al. (2000). Markkanen et al. (2003) presented footprint statistics as a function of the structure parameter and density of the forest. The parameterization by Kljun et al. (2004a) is available at http://footprint.kljun.net. The SCADIS closure model was also simplified (two-dimensional domain, neutral stratification, flat topography, etc.) and provided with a user-friendly menu. The operating manual for the set of basic and new created programs, called ‘‘Footprint calculator’’, was presented by Sogachev and Sedletski (2006) and is available freely by request to the authors or from Nordic Centre for Studies of Ecosystem Carbon Exchange (NECC) site (http://www.necc.nu/NECC/home.asp). References Avissar, R., Pielke, R.A., 1989. A parameterization of heterogeneous land surface for atmospheric numerical models and its impact on regional meteorology. Monthly Weather Review 117, 2113e2136. Amiro, B.D., 1998. Footprint climatologies for evapotranspiration in a boreal catchment. Agricultural and Forest Meteorology 90, 195e201. Baldocchi, D., 1997. Flux footprints within and over forest canopies. Boundary-Layer Meteorology 85, 273e292. Bysova, N.L., Garger, E.K., Ivanov, V.N., 1991. Experimental Studies of Atmospheric Diffusion and Calculation of Pollutant Dispersion. Gidrometeoizdat, Leningrad (in Russian). Cai, X., Leclerc, M.Y., 2007. Forward-in-time and backward-in-time dispersion in the convective boundary layer: asymmetry and equivalence. Boundary-Layer Meteorology 123, 201e218. Finn, D., Lamb, B., Leclerc, M.Y., Horst, T.W., 1996. Experimental evaluation of analytical and Lagrangian surface layer flux footprint models. Boundary-Layer Meteorology 80, 283e308. Finnigan, J.J., 2004. The footprint concept in complex terrain. Agricultural and Forest Meteorology 127, 117e129. Flesch, T.K., Wilson, J.D., Yee, E., 1995. Backward-time Lagrangian stochastic dispersion models and their application to estimate gaseous emissions. Journal of Applied Meteorology 34, 1320e1332. Flesch, T.K., 1996. The footprint for flux measurements, from backward Lagrangian stochastic models. Boundary-Layer Meteorology 78, 399e404. Foken, T., 2006. Angewandte Meteorologie, Mikrometeorologische Methoden, second ed. Springer, Berlin. Foken, T., Leclerc, M.Y., 2004. Methods and limitations in validation of footprint models. Agricultural and Forest Meteorology 127, 223e234. Foken, T., Wichura, B., 1996. Tools for quality assessment of surface-based flux measurements. Agricultural and Forest Meteorology 78, 83e105. Foken, T., Go¨ckede, M., Mauder, M., Mahrt, L., Amiro, B.D., Munger, J.W., 2004. Post-field data quality control. In: Lee, X., Massman, W.J., Law, B. (Eds.), Handbook of Micrometeorology: A Guide for Surface Flux Measurement and Analysis. Kluwer, Dordrecht, pp. 181e208. Gash, J.H.C., 1986. A note on estimating the effect of a limited fetch on micrometeorological evaporation measurements. Boundary-Layer Meteorology 35, 409e413. Go¨ckede, M., Rebmann, C., Foken, T., 2004. Use of footprint models for data quality control of eddy covariance measurements. Agricultural and Forest Meteorology 127, 175e188. Go¨ckede, M., Markkanen, T., Hasager, C.B., Foken, T., 2005. Use of footprint modelling for the characterisation of complex measuring sites. BoundaryLayer Meteorology 118, 635e655. Go¨ckede, M., Markkanen, T., Hasager, C.B., Foken, T., 2006. Update of footprint-based approach for the characterisation of complex measurement sites. Boundary-Layer Meteorology 118, 635e655. Hadfield, M.G., 1994. Passive scalar diffusion from surface sources in the convective boundary layer. Boundary-Layer Meteorology 69, 417e448. Hasager, C.B., Jensen, N.O., 1999. Surface-flux aggregation in heterogeneous terrain. Quarterly Journal of Royal Meteorological Society 125, 2075e 2102.

T. Vesala et al. / Environmental Pollution 152 (2008) 653e666 Hinds, W.C., 1999. Aerosol Technology. Properties, Behavior, and Measurement of Airborne Particles, second ed. Wiley, New York. Horst, T.W., Weil, J.C., 1992. Footprint estimation for scalar flux measurements in the atmospheric surface layer. Boundary-Layer Meteorology 59, 279e296. Horst, T.W., 2000. An intercomparison of measures of special inhomogeneity for surface fluxes of passive scalars. In: Proceedings of the 14th Symposium on Boundary Layer and Turbulence, Aspen, CO. American Meteorological Society, Boston, MA, pp. 11e14. Horst, T.W., Weil, J.C., 1994. How far is far enough? The fetch requirements for micrometeorological measurement of surface fluxes. Journal of Atmospheric and Oceanic Technology 11, 1018e1025. Hsieh, C.I., Katul, G., Chi, T., 2000. An approximate analytical model for footprint estimation of scalar fluxes in thermally stratified atmospheric flows. Advances in Water Resources 23, 765e772. Jarosz, N., Loubet, B., Huber, L., 2004. Modelling airborne concentration and deposition rate of maize pollen. Atmospheric Environment 38, 5555e 5566. Kaharabata, S., Schuepp, P.H., Ogunjemiyo, S., Shen, S., Leclerc, M.Y., Desjardins, R.L., MacPherson, J.I., 1997. Footprint considerations in boreas. Journal of Geophysical Research 102, 29113e29124. Kim, J., Guo, Q., Baldocchi, D.D., Leclerc, M.Y., Xu, L., Schmid, H.P., 2006. Upscaling fluxes from tower to landscape: overlaying flux footprints over high resolution (IKONOS) images of vegetation cover. Agricultural and Forest Meteorology 136, 132e146. Klaassen, W., van Breugel, P.B., Moors, E.J., Nieveen, J.P., 2002. Increased heat fluxes near a forest edge. Theoretical and Applied Climatology 72, 231e243. Klaassen, W., Sogachev, A., 2006. Flux footprint simulation downwind of a forest edge. Boundary-Layer Meteorology 121, 459e473. Kljun, N., Rotach, M.W., Schmid, H.P., 2002. A 3-D backward Lagrangian footprint model for a wide range of boundary layer stratifications. Boundary-Layer Meteorology 103, 205e226. Kljun, N., Kormann, R., Rotach, M.W., Meixner, F.X., 2003. Comparison of the Lagrangian footprint model LPDM-B with an analytical footprint model. Boundary-Layer Meteorology 106, 349e355. Kljun, N., Calanca, P., Rotach, M.W., Schmid, H.P., 2004a. A simple parameterisation for flux footprint predictions. Boundary-Layer Meteorology 112, 503e523. Kljun, N., Kastner-Klein, P., Fedorovich, E., Rotach, M.W., 2004b. Evaluation of a Lagrangian footprint model using data from a wind tunnel convective boundary layer. Agricultural and Forest Meteorology 127, 189e201. Kormann, R., Meixner, F.X., 2001. An analytic footprint model for neutral stratification. Boundary-Layer Meteorology 99, 207e224. ¨ ., Sabelfeld, K., Vesala, T., 1999. Direct and adKurbanmuradov, O., Rannik, U joint Monte Carlo algorithms for the footprint problem. Monte Carlo Methods and Applications 5, 85e112. Kurbanmuradov, O.A., Sabelfeld, K.K., 2000. Lagrangian stochastic models for turbulent dispersion in the atmospheric boundary layer. BoundaryLayer Meteorology 97, 191e218. ¨ ., Sabelfeld, K.K., Vesala, T., 2001. Kurbanmuradov, O., Rannik, U Evaluation of mean concentration and fluxes in turbulent flows by Lagrangian stochastic models. Mathematics and Computers in Simulation 54, 459e476. ¨ ., Sabelfeld, K., Vesala, T., 2003. Kurbanmuradov, O., Levykin, A.I., Rannik, U Stochastic Lagrangian footprint calculations over a surface with an abrupt change of roughness height. Monte Carlo Methods and Applications 9, 167e188. Kustas, W.P., Anderson, M.C., French, A.N., Vickers, D., 2006. Using a remote sensing field experiment to investigate flux-footprint relations and flux sampling distributions for tower and aircraft-based observations. Advances in Water Resources 29, 355e368. Leclerc, M.Y., Shen, S.H., Lamb, B., 1997. Observations and large-eddy simulation modeling of footprints in the lower convective boundary layer. Journal of Geophysical Research 102, 9323e9334. Leclerc, M.Y., Meskhidze, N., Finn, D., 2003a. Footprint predictions comparison with a tracer flux experiment over a homogeneous canopy of intermediate roughness. Agricultural and Forest Meteorology 117, 17e34.

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Leclerc, M.Y., Karipot, A., Prabha, T., Allwine, G., Lamb, B., Gholz, H.L., 2003b. Impact of non-local advection on flux footprints over a tall forest canopy: a tracer flux experiment. Agricultural and Forest Meteorology 115, 17e34. Leclerc, M.Y., Thurtell, G.W., 1990. Footprint prediction of scalar fluxes using a Markovian analysis. Boundary-Layer Meteorology 52, 247e258. Lee, X., 2003. Fetch and footprint of turbulent fluxes over vegetative stands with elevated sources. Boundary-Layer Meteorology 107, 561e579. Lee, X., 2004. A theory for scalar advection inside canopies and applications to footprint modelling. Agricultural and Forest Meteorology 127, 131e 141. Luhar, A.K., Rao, K.S., 1994. Source footprint analysis for scalar fluxes measured over an inhomogeneous surface. In: Gryning, S.E., Milan, M.M. (Eds.), Air Pollution Modeling and its Applications. Plenum Press, pp. 315e323. Marcolla, B., Cescatti, A., 2005. Experimental analysis of flux footprint for varying stability conditions in an alpine meadow. Agricultural and Forest Meteorology 135, 291e301. ¨ ., Marcolla, B., Cescatti, A., Vesala, T., 2003. FootMarkkanen, T., Rannik, U prints and fetches for fluxes over forest canopies with varying structure and density. Boundary-Layer Meteorology 106, 437e459. Meixner, F.X., Ammann, C., Rummel, U., Simon, E., Trebs, I., Dlugi, R., Falge, E., Foken, T., Zetzsch, C., 2005. Biosphere-atmosphere exchange of reactive trace gases of tall vegetation canopies: the roˆle of characteristic chemical, plant physiological, and turbulent time scales. Geophysical Research Abstracts. EGU05-A-09507, AS2.04-1TH2O-003. Moeng, C.-H., 1984. A large-eddy simulation model for the study of planetary boundary-layer turbulence. Journal of Atmospheric Sciences 41, 2052e 2062. Moeng, C.-H., Wyngaard, J., 1988. Spectral analysis of large-eddy simulation of the convective boundary layer. Journal of Atmospheric Sciences 45, 3573e3587. Mo¨lder, M., Klemedtsson, L., Lindroth, A., 2004. Turbulence characteristics and dispersion in a forest e verification of Thomson’s random-flight model. Agricultural and Forest Meteorology 127, 203e222. Monin, A.S., Yaglom, A.M., 1975. Statistical Fluid Mechanics. Vol. 2. M.I.T. Press, Cambridge, MA. Ogunjemiyo, S.O., Kaharabata, S.K., Schuepp, P.H., MacPherson, I.J., Desjardins, R.L., Roberts, D.A., 2003. Methods of estimating CO2, latent heat and sensible heat fluxes from estimates of land cover fractions in the flux footprint. Agricultural and Forest Meteorology 117, 125e144. Pasquill, F., 1972. Some aspects of boundary layer description. Quarterly Journal of Royal Meteorological Society 98, 469e494. Pasquill, F., Smith, F.B., 1983. Atmospheric Diffusion, third ed. Wiley, New York. Patton, E.G., Davis, K.J., Barth, M.C., Sullivan, P.P., 2001. Decaying scalars emitted by a forest canopy: a numerical study. Boundary-Layer Meteorology 100, 91e129. ¨ ., Aubinet, M., Kurbanmuradov, O., Sabelfeld, K.K., Markkanen, T., Rannik, U Vesala, T., 2000. Footprint analysis for the measurements over a heterogeneous forest. Boundary-Layer Meteorology 97, 137e166. ¨ ., Markkanen, T., Raittila, J., Hari, P., Vesala, T., 2003. Turbulence Rannik, U statistics inside and over forest: influence on footprint prediction. Boundary-Layer Meteorology 109, 163e189. Rebmann, C., Go¨ckede, M., Foken, Th., Aubinet, M., Aurela, M., Berbigier, P., Bernhofer, C., Buchmann, N., Carrara, A., Cescatti, A., Ceulemans, R., Clement, R., Elbers, J.A., Granier, A., Gru¨nwald, Th., Guyon, D., Havrankova, K., Heinesch, B., Knohl, A., Laurila, T., Longdoz, B., Marcolla, B., Markkanen, T., Miglietta, F., Moncrieff, J., Montagnani, L., Moors, E., Nardino, M., Ourcival, J.-M., Rambal, S., ¨ ., Rotenberg, E., Sedlak, P., Unterhuber, G., Vesala, T., Rannik, U Yakir, D., 2005. Quality analysis applied on eddy covariance measurements at complex forest sites using footprint modelling. Theoretical and Applied Climatology 80, 121e141. Reithmaier, L., Go¨ckede, M., Markkanen, T., Knohl, A., Churkina, A., Rebmann, C., Buchmann, N., Foken, T., 2006. Use of remotely sensed land use classification for a better evaluation of micrometeorological

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